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2,869,038,154,728 | arxiv | \section{\label{sec:level1}Introduction}
The discovery of superconductivity in doped LaFeAsO triggered intensive research on layered FeAs systems\cite{LaFeAsOF_YKamihara_01}.
So far, the superconducting (SC) transition temperature {$T_{\rm sc}$} of $R$FeAsO has rapidly been raised up to 54\,K\cite{PrFeAsOF_ZARen_01,NdFeAsOF_ZARen_01, CeFeAsOF_GFChen_01,SmOFFeAs_XHChen_01,GdFeAsO_JYang_01}.
Furthermore, a similar high {$T_{\rm sc}$} exceeding 35\,K was also discovered upon doping the related compounds $A$Fe$_2$As$_2$ ($A$=Ca, Sr, Ba).\cite{AFe2As2_CKrellner_01,SrFe2As2_GFChen_01}
$A$Fe$_2$As$_2$ has the well-known ThCr$_2$Si$_2$-type structure, which can be regarded as replacing the (R$_2$O$_2$)$^{2+}$ layer in $R$FeAsO by a single divalent ion ($A^{2+}$) layer keeping the same electron count.\cite{AFe2As2_CKrellner_01}
$R$FeAsO and $A$Fe$_2$As$_2$ were found to present very similar physical properties.
Undoped $R$FeAsO compounds ($R$=La-Gd) present a structural transition at $T_0{\sim}$150\,K followed by the formation of a spin-density wave (SDW) at a slightly lower temperature {$T_{\rm N}$}${\sim}$140\,K.
$A$Fe$_2$As$_2$ exhibits a similar structural distortion, whereas the SDW forms at the same or a slightly lower temperature\cite{SrFe2As2_MTegel_01}.
Electron or hole doping leads to the suppression of the SDW and to the onset of superconductivity.
This connection between a vanishing magnetic transition and the simultaneous formation of a SC state is reminiscent of the behavior in the cuprates and in the heavy-fermion systems, suggesting the SC state in these doped FeAs systems to be of unconventional nature.
In addition, the SDW state is strongly coupled to the lattice distortion in this system and therefore it is of high interest to reveal the relationship between the lattice, the magnetism and superconductivity.
The magnetic structures of the FeAs system have been investigated by neutron diffraction study on $R$FeAs(O,F) for $R$=La, Ce, and Nd, and {BaFe$_2$As$_{2}$}\cite{LaFeAsOF_CdlCruz_01,CeFeAsOF_JZhao_01,BaFe2As2_QHuang_01}.
The antiferromagnetic (AFM) reflection was observed around the (1,0) or (0,1) position with respect to the Fe-As layer.
This corresponds to Fe moments with AFM coupling along the $a$ or $b$ direction, called the columnar structure.
In contrast, interlayer couplings is not unique among these compounds, suggesting a relatively weak coupling between the layers.
The size of the estimated Fe moment is less than 1\,{${\mu}_{\rm B}$}, which is much smaller than theoretical predictions.
So far the magnetic structure of these iron arsenides have not been determined uniquely, especially the relation of the direction of the AFM ordering with respect to the short or long Fe-Fe distances is not settled.
This is one of the fundamental questions on the magnetic properties of these materials and is indispensable for understanding the interplay between magnetism and superconductivity.
Among the reported FeAs systems, {SrFe$_2$As$_{2}$} should be a suitable compound to study the detailed magnetic structure.
By the substitution of Sr with K or Cs, the superconductivity appears with the maximum {$T_{\rm sc}$} of 38\,K.\cite{SrFe2As2_GFChen_01,SrFe2As2_KSasmal_01}
The parent compound {SrFe$_2$As$_{2}$} undergoes a first order transition at $T_0$=205\,K, where both, the SDW and the structural transition, occur simultaneously.\cite{SrFe2As2_AJesche_01}
A detailed x-ray diffraction study clarified that the structural transition at {$T_0$} is from tetragonal($I4/mmm$) to orthorhombic($Fmmm$) and therefore similar to BaFe$_2$As$_2$\cite{SrFe2As2_MTegel_01,SrFe2As2_AJesche_01}.
A stronger magnetism in {SrFe$_2$As$_{2}$} is inferred from the higher ordering temperature, the larger value of the Pauli like susceptibility above $T_0$ as well as the larger Fe hyperfine field observed in M\"ossbauer experiments.\cite{SrFe2As2_MTegel_01,SrFe2As2_AJesche_01}
Therefore, a larger ordered moment is expected for this compound, which should allow a detailed analysis.
In this paper, we report a neutron powder diffraction study on {SrFe$_2$As$_{2}$}.
The precise analysis of the observed magnetic Bragg peaks on {SrFe$_2$As$_{2}$}\,\,allowed us to uniquely determine the magnetic structure.
The magnetic propagation vector of {SrFe$_2$As$_{2}$} is {\textit{\textbf{q}}}=(1\,0\,1), thus the AFM coupling is realized in the longer Fe-Fe direction within the Fe-As layer.
The stacking along the $c$-direction is also AFM.
The direction of the magnetic moment is parallel to the $a$-axis as well.
A remarkable agreement was obtained in the temperature evolution of the magnetic moment and structural distortion obtained from independent measurements.
These facts clearly demonstrate the close relationship between magnetism and lattice distortion in {SrFe$_2$As$_{2}$}.
The details on the sample preparation of {SrFe$_2$As$_{2}$} have been described in Ref.~\onlinecite{AFe2As2_CKrellner_01}.
Neutron powder diffraction experiments were carried out on the two-axis diffractometer E6, installed at Helmholtz Center Berlin, Germany.
A double focusing pyrolytic graphite (PG) monochromator results in a high neutron flux at the sample position.
Data were recorded at scattering angle up to 110$^{\circ}$ using a two dimensional position sensitive detector (2-D PSD) with the size of 300${\times}$300\,mm$^2$.
Simultaneous use of the double focusing monochromator and 2-D PSD in combination with the radial oscillating collimator gives a high efficiency for taking diffraction patterns.
The neutron wavelength was chosen to be 2.4\,\AA\,\,in connection with a PG filter in order to avoid higher-order contaminations.
As a compensation for the high efficiency, error in the absolute values in the lattice constants become relatively large.
Since the detailed absolute values were already obtained from the x-ray diffraction\cite{SrFe2As2_AJesche_01}, we relied on these data and focus here on the details of the magnetic structure.
Fine powder of {SrFe$_2$As$_{2}$} with a total mass of ${\sim}$2\,g was sealed in a vanadium cylinder as sample container.
The standard $^4$He cryostat was used to cool the sample down to 1.5\,K well below {$T_0$}.
Neutron diffraction patterns were taken at different temperatures between 1.5\,K and 220\,K and the obtained diffraction patterns were analyzed by the Rietveld method using the software RIETAN-FP\cite{RIETAN_FIzumi_01}.
The software VESTA\cite{VESTA_KMomma_01} was used to draw both crystal and magnetic structures.
Figure~\ref{f1} shows neutron diffraction patterns of {SrFe$_2$As$_{2}$} taken at (a)\,$T$=220\,K ($T>T_0$) and (b)\,$T$=1.5\,K ($T<T_0$).
Results of the Rietveld analysis, the residual intensity curve, and tick marks indicating the expected reflection angle, are also plotted in the figure.
For the high temperature $T>${$T_0$}, the observed pattern is well reproduced by assuming the crystal structure with the space group $I4/mmm$ as shown in Fig.~\ref{f1}(a).
These results are consistent with that reported from x-ray diffraction study, including the positional parameter of As, $z_{\rm As}$=0.3602(2), where the number in the parenthesis indicate the uncertainty at the last decimal point.
The conventional reliability factors $R_{\rm wp}$=3.73\,\% $R_{\rm I}$=3.18\,\% and $R_{\rm F}$=1.93\,\% of the present analysis indicate the high quality of the present analysis.
Small impurity peaks observed at around 63$^{\circ}$ and 84$^{\circ}$ which originate from Cu and Al were excluded from the analysis.
Below {$T_0$}=205\,K, some nuclear Bragg reflections became broad.
The inset of Fig.~\ref{f1} shows the Bragg peak profile around 57.5$^{\circ}$ where the 1\,1\,2 Bragg peak of the high temperature tetragonal phase was observed.
The peak intensity drops and broadens on passing through {$T_0$}, as expected for the orthorhombic distortion where the lattice constant $a$ becomes longer than $b$.
The reflection profile could be well reproduced by the lattice distortion from the tetragonal to orthorhombic lattice reported by the x-ray diffraction study\cite{SrFe2As2_AJesche_01}.
Within the errors, the positional parameter of As in the orthorhombic phase $z_{\rm As}$=0.3604(2) is the same as that above $T_0$.
In addition to the lattice distortion, additional reflections were also observed at $T$=1.5\,K as indicated by arrows in Fig.~\ref{f1}(b).
These superlattice peaks are most prominent at low scattering angles, and the corresponding peaks were not observed in the x-ray diffraction\cite{SrFe2As2_MTegel_01,SrFe2As2_AJesche_01}.
Therefore, the origin of these superlattice peaks should be magnetic.
Figure~\ref{f2} shows the temperature dependence of the integrated intensity of the superlattice reflection around 43$^{\circ}$ which can be indexed as 1\,0\,3 as described later.
The left axis is set to be proportional to the size of the magnetic moment, ${\sqrt{I/I_0}}$, where $I_0$ is the intensity at the lowest temperature.
The 1\,0\,3 reflection appears below $T_0$= 205\,K and shows a sharp increase in its intensity, which is clearly seen in the profile shown in the inset.
The magnetic moment at 201\,K, just 4\,K below {$T_0$}, already attains 70\,\% of that at 1.5\,K.
These findings strongly support the first order transition at {$T_0$} in {SrFe$_2$As$_{2}$}.
In the same figure, the relative lattice distortion and the muon precession frequency of {SrFe$_2$As$_{2}$} taken from Ref.\onlinecite{SrFe2As2_AJesche_01} both normalized to the saturated values are plotted.
A remarkable agreement of the temperature dependences is seen for these quantities obtained from independent measurements.
This clearly demonstrates the close relationship between magnetism and lattice distortion in {SrFe$_2$As$_{2}$}.
Hereafter, we analyze the magnetic structure of {SrFe$_2$As$_{2}$}.
For simplicity, in the magnetic structure analysis the structural parameters were fixed to the best value determined from the nuclear Bragg peaks.
In the following, three representative models shown in the right panel of Fig.~\ref{f3} are considered.
At first, the direction of the AFM coupling with respect to the orthogonal axis is examined.
This corresponds to the difference between Model I with the propagation vector {\textbf{\textit{q}}}=(1\,0\,1) and Model II with {\textbf{\textit{q}}}=(0\,1\,1).
The difference between these models appears in the scattering angle arising from the subtle difference between $a$ and $b$.
The difference can be clearly seen in the comparison for 1\,0\,1(upper panel) and 1\,2\,1(bottom) reflections.
When Model I is used as a reference, the Model II gives a higher scattering angle for 1\,0\,1 which then corresponds to 0\,1\,1, and a lower angle for 1\,2\,1 (2\,1\,1).
The comparison in Fig.~\ref{f3} clearly show that only Model I gives the correct positions of the 1\,0\,1 and 1\,2\,1 reflections.
Therefore, the magnetic propagation vectors of {SrFe$_2$As$_{2}$} is determined to be {\textbf{\textit{q}}}=(1\,0\,1).
In a second step, we try to analyze the direction of AFM moment, which corresponds to Model I and III.
In Model I, the magnetic moment is set to be parallel to the AFM coupling in the Fe-As plane, ${\mu}{\parallel}a$, whereas it is perpendicular in Model III.
The magnetic diffraction intensity is depending on the angle between the magnetic moment and scattering vector.
The powder average of this angle factor for the orthorhombic symmetry is given as follows;
\begin{equation}
{\langle}q^2{\rangle}=1-(h^2a^{*2}\cos^2{\psi}_a+k^2b^{*2}\cos^2{\psi}_b+l^2c^{*2}\cos^2{\psi}_c)d^2
\end{equation}
where $h, k, l$ are the reflection indexes, $a^*, b^*, c^*$ are the primitive lattice vectors in reciprocal space, and $d$ is the spacing of the ($hkl$) in real space.
${\psi}_a$, ${\psi}_b$, ${\psi}_c$ are the angles between the magnetic moment and crystallographic axes $a, b$ and $c$.
This factor affects the relative magnetic intensities of 1\,0\,1, 1\,0\,3, and 1\,2\,1 diffraction peaks.
Model I gives the comparable intensity for all three peaks, in good agreement with experimental results.
In contrary, Model III results in a strong 1\,0\,1 and a vanishing 1\,2\,1 peak, which does not at all fit with the measured intensities.
As a result, the calculation based on the Model I gives an excellent agreement for the position and intensity of all magnetic reflections.
Therefore, we can conclusively determine the direction of the magnetic moment to be $a$.
By using the structure Model I, the size of the Fe magnetic moment is deduced to be 1.01(3)\,{${\mu}_{\rm B}$} with good reliability factors $R_{\rm wp}$=3.54\,\% $R_{\rm I}$=2.49\,\% and $R_{\rm F}$=1.63\,\%.
The determined magnetic structure of {SrFe$_2$As$_{2}$} is shown in Fig.~\ref{f4}; the AFM coupling occurs along the longer $a$ direction and the moment orients in the same direction.
The magnetic ordering in {SrFe$_2$As$_{2}$} within the FeAs layer seems identical to that in BaFe$_2$As$_{2}$\cite{BaFe2As2_QHuang_01}, in LaFeAs(O,F)\cite{LaFeAsOF_CdlCruz_01} and in CeFeAs(O,F)\cite{CeFeAsOF_JZhao_01}, although the stacking order is not common.
This supports the importance of the FeAs intralayer coupling as well as the weak interlayer coupling.
We now turn to the discussion of the relation between the structural distortion and the magnetic structure.
As mentioned before, the magnetic order in the FeAs system occurs either after the structural distortion or simultaneously.
A simple approach would be to consider a superexchange path between the nearest Fe neighbors.
Since the positional parameter of As and the lattice constant $c$ does not exhibit significant change, both the packing of the Fe-As layer and the Fe-As distance stay constant on passing through $T_0$.
Thus, the distortion within the Fe-As layer at {$T_0$} mainly results in a slight change in the Fe-Fe distance and the Fe-As-Fe bond angle.
The longer $a$ lattice constant leads to a slightly smaller bond angle along $a$, $\angle$(Fe-As-Fe)$_a$=71.2$^{\circ}$ as compared to that along $b$,$\angle$(Fe-As-Fe)$_b$=72.1$^{\circ}$, in other words, the difference is less than 1$^{\circ}$.
It is unlikely that such minor distortions lead to significant differences in exchange parameters.
In contrast, it should be noted that the observed orthorhombic distortion and the columnar magnetic structure in {SrFe$_2$As$_{2}$} are consistent with the results of a band structure calculation\cite{SrFe2As2_AJesche_01}.
The calculation gives both a comparable distortion and the correct magnetic structure in which the AFM arrangement is along the long $a$-axis.
These calculation predict the distortion to be stable only for the columnar state, not for other magnetic structures or the non-magnetic state, indicating the strong coupling of magnetic order and orthorhombic distortion.
Similar results were also obtained for LaFeAsO.\cite{LaFeAsO_yildirim_01,LaFeAsOF_PVSushko_01}
For this compound, T. Yildirim discusses the lattice distortion to be related to a lifting of the double degeneracy of the columnar state within a localized spin model for a frustrated square lattice\cite{LaFeAsO_yildirim_01}.
However, I. I. Mazin \textit{et al.} questioned the applicability of such a model for these layered FeAs systems\cite{LaFeAsO_IIMazin_01}.
A detailed neutron scattering study on single crystals is expected to give further insights into this fascinating coupling.
The size of the magnetic moment of 1.01\,{${\mu}_{\rm B}$} determined by the present neutron diffraction study is the largest among the $R$FeAsO and the ternary arsenide $A$Fe$_2$As$_2$ reported so far: 0.36\,{${\mu}_{\rm B}$} for LaFeAs(O,F)\cite{LaFeAsOF_CdlCruz_01}, 0.8\,{${\mu}_{\rm B}$} for CeFeAs(O,F)\cite{CeFeAsOF_JZhao_01}, 0.87\,{${\mu}_{\rm B}$} for BaFe$_2$As$_2$\cite{BaFe2As2_QHuang_01}.
This corresponds to the stronger magnetism deduced for {SrFe$_2$As$_{2}$} from bulk measurements, i.e. from the higher {$T_0$} and the larger hyperfine field in M\"ossbauer experiments.
On the other hand, it should be noted that the size of the ordered moment obtained in our neutron diffraction study is almost twice as large as that deduced in other microscopic measurements, ${\mu}$SR and M\"ossbauer spectroscopy.
Since the neutron diffraction intensity is proportional to the square of the size of the moment, this large difference can hardly be attributed to an experimental error.
Similar differences were obtained for {BaFe$_2$As$_{2}$} and $R$FeAsO as well\cite{LaFeAsOF_HHKauss_01,BaFe2As2_MRotter_01}.
This points to a general problem.
While the neutrons directly probe the density of the magnetic moment, M\"ossbauer and ${\mu}$SR rely on a scaling of the observed quantity, i.e. precession frequency and the hyperfine field.
It might be that these scaling procedures, which are well established for stable magnetic Fe systems with large moment, are not appropriate for the unusual magnetism in these layered FeAs systems.
\begin{acknowledgments}
We thank H.-H. Klaus and H. Rosner for stimulating discussions.
\end{acknowledgments}
|
2,869,038,154,729 | arxiv |
\section{\label{sec:introduction} Introduction}
Similar to the majority of the stable isotopes beyond iron,
$^{206,207,208}$Pb and $^{209}$Bi are synthesized by the rapid
\mbox{($r$-)} and slow \mbox{($s$-)} neutron capture processes.
However, this mass region is particularly interesting because the
$r$-process abundances are dominated by the decay of
the short lived $\alpha$-unstable transbismuth isotopes
\cite{cow99}.
This feature provides an important consistency check for the
$r$-process abundance calculations in the actinide region,
since the integrated $r$ residuals are constrained by the
difference between the solar abundance values and the
respective $s$-process components. Reliable $r$-process
calculations are required for the interpretation of the
observed Th and U abundances in the ultra metal-poor (UMP)
stars of the Galactic halo. Since these stars are considered
to be as old as the Galaxy, the observed Th and U abundances
can be used as cosmo-chronometers, provided the original
Th and U abundances are inferred from $r$-process models.
This dating mechanism has the advantage of being independent
of the yet uncertain $r$-process site~\cite{cow99,sch02,kra04}.
Apart from its relevance for establishing the basic constraints
for the $r$-process chronometry in general, $^{206}$Pb contains
also dating information in itself. The $^{206}$Pb/$^{238}$U
cosmochronometer was first introduced by Clayton in 1964~\cite{cla64}.
The $^{238}$U produced by the $r$ process decays with a half life
of $t_{1/2}=4.5\times 10^9$ yr over a chain of $\alpha$ and
$\beta$ decays ending at $^{206}$Pb. Therefore, its radiogenic abundance
component, $N^{206}_{c}$, can be used to constrain
the age of the parent isotope $^{238}$U, and hence the age ($\Delta_r$) of
the $r$-process. Unlike the more direct $r$-process abundance
predictions derived from the Th and U abundances in UMP stars,
this procedure requires a Galactic evolution model, which
describes the supernova rate or the frequency of the $r$-process
events~\cite{fow60}. The drawback of this clock
arises from the difficulty
to isolate the cosmo-radiogenic component of $^{206}$Pb accurately
enough from the additional abundance components.
Apart from these astrophysical aspects, the neutron capture cross
section of $^{206}$Pb is also of importance for the design of fast
reactor systems based on a Pb/Bi spallation source. Because
24.1\% of natural lead consists of $^{206}$Pb, its ($n, \gamma$)
cross section influences the neutron balance of the reactor~\cite{ado03}.
There have been several measurements of the $^{206}$Pb($n, \gamma$) cross section, which show discrepancies that are
difficult to understand (see Sec.~\ref{sec:comp}). The aim of this work is to
perform a new independent measurement with higher accuracy and in this way
to determine the $s$-process contribution to the $^{206}$Pb
abundance, $N^{206}_s$, more reliably.
In fact, the $s$-process abundance of this isotope is almost
completely determined by the stellar ($n, \gamma$) cross section,
nearly independent of the stellar model used \cite{rat04}. Therefore,
the uncertainty of $N^{206}_s$ arises mostly from the cross
section uncertainty.
Potential sources of systematic error have been substantially reduced in
the present measurement, which was performed at the CERN n\_TOF
installation. The new setup, and in particular the detectors themselves, were
optimized for very low neutron sensitivity. Furthermore, the detectors were
mounted at $\sim$125$^\circ$ with respect to the incident neutron
beam in order to minimize the correction for angular distribution
effects. The experimental details are presented in
Sec.~\ref{sec:measurement}, followed by the adopted data analysis
procedures and an evaluation of the various systematic uncertainties
in Sec.~\ref{sec:analysis}. The deduced resonance parameters and the
corresponding Maxwellian averaged capture cross sections in the
stellar temperature regime are presented in Sec.~\ref{sec:results}.
Based on these new data, first astrophysical implications for the
$s$-process abundance of $^{206}$Pb are discussed in
Sec.~\ref{sec:implications}.
\section{\label{sec:measurement} Measurement}
The time-of-flight (TOF) measurement was performed at the CERN n\_TOF
installation~\cite{preport} using a set of two C$_6$D$_6$ detectors.
Neutrons were produced by a 20~GeV proton beam on a lead spallation
target. The spallation source was surrounded by a 6~cm thick water
layer, which served as a coolant and as a moderator for the initially
fast neutron spectrum. The beam was characterized by intense bunches
of (3 to 7)$\times$10$^{12}$ protons, a width of 6~ns (rms), and
a repetition rate of only 0.4 Hz. This extremely low duty-cycle allows
one to perform ($n, \gamma$) measurements over a broad neutron energy
interval from 1~eV up to 1~MeV and to achieve favorable background
conditions. Data were recorded by means of an advanced acquisition
system with zero dead time, based on 8~bit Flash-Analog-to-Digital
Converters (FADC), with 500~MHz sampling rate and 8~MB buffer
memory~\cite{abb05}.
The measurement was performed with an enriched metal sample 8.123~g
in mass and 20~mm in diameter. The sample was enriched to 99.76\% in
$^{206}$Pb with small impurities of $^{207}$Pb (0.21\%) and
$^{208}$Pb (0.03\%).
Capture events were registered with two C$_6$D$_6$ $\gamma$-ray detectors
optimized for very low neutron sensitivity~\cite{pla03}. A sketch
of the experimental setup is shown in Fig.~2 of Ref.~\cite{dom06}.
The absolute value of the neutron fluence was determined by regular
calibration measurements with an 0.5~mm thick gold sample and by
using the saturated resonance technique~\cite{mac79} for the first
gold resonance at $E_n=4.9$~eV. The energy differential neutron flux
was determined with a relative uncertainty of $\pm$2\% from the
flux measurement with a $^{235,238}$U fission chamber calibrated by
Physikalisch-Technische Bundesanstalt (PTB)~\cite{ptb}. The
neutron intensity at the sample position was also monitored by means
of a \mbox{200-$\mu$g/cm$^2$} thick $^6$Li foil in the neutron beam
about 2.5~m upstream of the capture sample. The $^6$Li foil was
surrounded by four silicon detectors outside of the beam for recording
the $^3$H and $\alpha$ particles from the ($n, \alpha$) reactions.
Compared to previous measurements~\cite{miz79,all73}, the present setup
had the advantage that the detectors were placed at $\sim$125$^{\circ}$
with respect to the incident neutron beam. In this way, the corrections
for angular distribution effects of the prompt capture $\gamma$-rays
were strongly reduced. This configuration led also to a substantial
reduction of the background from in-beam $\gamma$-rays scattered
in the sample~\cite{abb03}.
\section{\label{sec:analysis} Capture data analysis}
The response function of the C$_6$D$_6$ detectors needs to be modified
such that the detection probability for capture cascades becomes independent
of the cascade multipolarity. This was accomplished by application
of the Pulse Height Weighting Technique (PHWT)~\cite{mac67}. Based on
previous experience~\cite{abb04,dom06,dom06b}, the weighting functions
(WFs) for the gold and lead samples were obtained by means of Monte Carlo
calculations. The accuracy of the WFs was verified with the method
described in Ref.~\cite{abb04}, by which the calculated WFs were applied
to Monte Carlo simulated capture $\gamma$-ray spectra. Using this
procedure, the uncertainty of the WFs was estimated to be smaller than
0.5\% for the samples used in the present experiment.
The weighted count rate $N^w$ is then transformed into an experimental
yield,
\begin{equation}\label{eq:yexp}
Y^{exp} = f^{sat} \frac{N^w}{N_n E_c},
\end{equation}
\noindent
where the yield-normalization factor $f^{sat}$ is determined by
calibration measurements using the saturated 4.9~eV resonance in gold.
$N_n$ denotes the neutron flux and $E_c$ the effective binding energy.
The yield in Eq.~(\ref{eq:yexp}) is still subject to several corrections.
The common effects of the background and of the low energy cutoff in
the pulse height spectra of the $\gamma$ detectors are described in
Secs.~\ref{sec:background} and \ref{sec:threshold}, respectively.
The measurement on $^{206}$Pb is particularly sensitive to the
angular distribution of the prompt capture $\gamma$-rays. The
impact of this effect is described in Sec.~\ref{sec:angular}.
\subsection{\label{sec:background}Backgrounds}
A major source of background is due to in-beam $\gamma$-rays,
predominantly from neutron captures in the water moderator, which
travel along the neutron flight tube and are scattered in the
$^{206}$Pb sample. This background exhibits
a smooth dependence on neutron energy, with a broad maximum around
$E_n \approx 10$~keV. The shape of this background was determined
from the spectrum measured with an isotopically pure $^{208}$Pb sample,
which contains practically no resonances in the investigated neutron
energy range. This spectrum was properly scaled and used as a point-wise
numerical function in the R-matrix analysis of the $^{206}$Pb capture
yield (see Sec.~\ref{sec:results}).
Another type of background arises in the analysis of resonances with a
dominant neutron scattering channel, $\Gamma_n >> \Gamma_\gamma$. In
such cases, there are about $\Gamma_n/\Gamma_\gamma$ scattered neutrons
per capture event. These scattered neutrons can be captured in the
detectors or in surrounding materials, thus mimicking true capture
events. This effect was estimated to be negligible for all the resonances
listed in Table~\ref{tab:RP}.
\subsection{\label{sec:threshold}Digital threshold}
As mentioned in Sec.~\ref{sec:measurement}, FADCs were used for
recording directly the analog output signals of the C$_6$D$_6$
detectors. Without any further discrimination, 8~MB of data
would have been acquired per proton pulse in each detector.
Depending on the sample, this enormous amount of data could be
reduced by factors of 20 to 100 by using a zero suppression
algorithm (see Ref.~\cite{abb05} for details). By this method
events below a certain pulse-height are discriminated by a
constant digital threshold analogous to conventional data
acquisition systems, where an electronic threshold is used to
reduce backgrounds and dead time effects.
Due to this threshold, the pulse height spectra of the C$_6$D$_6$
detectors exhibit a low energy cutoff at a certain value of the
signal amplitude (see Fig.~\ref{fig:PHS}).
In this experiment the threshold was set at a $\gamma$-ray energy
of 320~keV. If the pulse height spectra of the $^{206}$Pb
sample and of the gold sample used for normalization would have
the same shape, the fraction of weighted counts below this threshold
would nearly cancel out in the expression for the yield,
\begin{eqnarray}
Y^{exp} \propto \frac{\sum_{0keV}^{320~keV} W^{\rm Pb}_i R^{\rm Pb}_i +
\sum_{320~keV}^{E_c} W^{\rm Pb}_i R^{\rm Pb}_i}{\sum_{0keV}^{320~keV} W^{\rm Au}_i R^{\rm Au}_i +
\sum_{320~keV}^{E_c} W^{\rm Au}_i R^{\rm Au}_i} \nonumber \\
\approx \frac{\sum_{320~keV}^{E_c} W^{\rm Pb}_i R^{\rm Pb}_i}{\sum_{320~keV}^{E_c} W^{\rm Au}_i R^{\rm Au}_i}.
\end{eqnarray}
\noindent
Here, the $W_i$ and $R_i$ are the corresponding weighting factors
and response functions for a certain time of flight channel,
respectively. However, this approximation is only valid within 4 to
5\%, because the pulse height spectra of captures
on $^{206}$Pb and $^{197}$Au differ significantly near threshold
(Fig.~\ref{fig:PHS}).
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{./figure1.eps}
\caption{\label{fig:PHS} Pulse height spectra for the 4.9~eV resonance in
gold (grey) and for the 3.3~keV resonance in $^{206}$Pb (black), arbitrarily
scaled. The dashed lines are the MC-calculated $\gamma$-ray spectra for the
two resonances. The linear scale used in the inset illustrates the large
difference between the simulated spectra below a threshold of 300~keV.}
\end{figure}
This effect has been taken into account in the determination of the
experimental capture yield by simulating the capture cascades of each
isotope as described in detail in Refs.~\cite{abb04,dom06,dom06b}.
Fig.~\ref{fig:PHS} shows that the experimental spectra above the
digital threshold are well reproduced by the simulations. With this
correction the experimental yield becomes
\begin{eqnarray}
Y^{exp} \propto \frac{f^t_{\rm Pb}}{f^t_{\rm Au}} \frac{\sum_{320~keV}^{E_c}
W^{\rm Pb}_i R^{\rm Pb}_i}{\sum_{320~keV}^{E_c} W^{\rm Au}_i R^{\rm Au}_i}.
\end{eqnarray}
For the adopted digital threshold the yield of the 4.9~eV
resonance in $^{197}$Au needs to be scaled by a factor $f^t_{\rm Au} =
1.071(3)$, whereas the yield of the resonances in $^{206}$Pb required
a correction of $f^t_{\rm Pb} = 1.021(5)$ due to their harder spectrum.
Hence, the correction factor of the final experimental yield was $f^t =
f^t_{\rm Pb}/f^t_{\rm Au} = 0.952(4)$.
\subsection{\label{sec:angular}Angular distribution effects}
Neutron capture with orbital angular momentum $l>0$ leads to an
aligned state in the compound nucleus, perpendicular to the
direction of the incident neutron. Given the small multiplicity
($m=1$ to 2) of the capture cascades in $^{206}$Pb, most of the
prompt $\gamma$-rays registered with the C$_6$D$_6$ detectors
still carry this anisotropy, which affects the measured yield.
The angular distribution is in general given by,
\begin{eqnarray}\label{eq:W}
W(\theta) = \sum_k A_k P_k(cos\theta) = 1 + A_2
P_2(cos\theta) + \nonumber\\
+ A_4 P_4(cos\theta) + A_6 P_6(cos\theta),
\end{eqnarray}
\noindent
where $P_{k}(cos\theta)$ are the Legendre polynomials of order $k$
and $A_{k}$ are coefficients, which depend on the initial ($J$)
and final ($J'$) spin values, on the multipolarity ($l$) of the
transition, and on the degree of alignment. The angular distribution
effects in the capture yield are minimized (although not avoided)
by setting the detectors at 125$^{\circ}$. Since each C$_6$D$_6$
detector covers a substantial solid angle, capture $\gamma$-rays
are registered around 125$^{\circ} \pm \Delta\theta$. For the
actual setup of the present measurement one finds
$\Delta\theta \approx 28^{\circ}$.
\subsubsection{Resonances with spin $J=1/2$}
For resonances with $J = 1/2$ it can be assumed that they
decay directly to the ground state ($J^{\pi} = 1/2^-$) or to the first or
second excited states with $J^{\pi} = 5/2^-$ and $J^{\pi} = 3/2^-$,
respectively (see also Fig.~\ref{fig:levels}). In these cases, one
finds that $A_2 = A_4 = A_6 = 0$. Therefore, only resonances with
spin $J > 1/2$ may be affected by angular distribution effects.
\subsubsection{Resonances with spin $J=3/2$}
In order to quantify the uncertainty due to the angular distribution
of the prompt $\gamma$-rays emitted from excited states with $J^{\pi}
= 3/2^-$ the de-excitation patterns reported in Ref.~\cite{miz79} have
been used (Table~\ref{tab:ftheta}).
\begin{table}
\caption{\label{tab:ftheta} Measured decay patterns from resonances
with spin $J=3/2$ \cite{miz79}. The systematic uncertainty in the yield
of each resonance due to the angular distribution of the involved
transitions are given in the last column.}
\begin{ruledtabular}
\begin{tabular}{lccccc}
$E_{\circ}$ (keV) & \multicolumn{4}{c}{Intensity $I_{\gamma}$ (\%)}
& $\sigma^{3/2^-}_\theta$\\
\hline
& \multicolumn{4}{c}{$E_{\gamma}$ (keV)} \\
& 6737.9 & 6168.6 & 5840.8 & 4114.5 & \\
\hline
3.36 & 76.0(27) & 2.5(8) & 8.58(11) & 13.0(8) & $\pm$10\% \\
3.36$^a$ & 60 & 2.5 & 24.5 & 13 & $\pm$8\% \\
10.86 & & 100 & & & $\pm$2\% \\
21.87 & & 100 & & & $\pm$2\% \\
42.07 & & & 100 & & $\pm$10\% \\
\end{tabular}
\end{ruledtabular}
$^a$ This work.
\end{table}
\begin{figure}[h]
\includegraphics[width=0.25\textwidth]{./figure2.eps}
\caption{\label{fig:levels} Level scheme and decay patterns for $^{207}$Pb
\cite{miz79}. All energies are in keV.}
\end{figure}
For the first resonance at 3.36~keV, fair agreement has been found
between the relative intensities of Ref.~\cite{miz79} and the rather
coarse values deduced from the experimental pulse height spectrum
(Table~\ref{tab:ftheta} and Fig.~\ref{fig:PHS}), which suffer from
uncertainties due to background subtraction, limited counting
statistics and poor energy resolution of the C$_6$D$_6$ detectors.
Therefore, an uncertainty of about 20\% has to be ascribed to the
quoted $\gamma$-ray intensities.
The estimated effect of the angular distribution on the capture
yield ($\sigma^{3/2^-}_\theta$) is given in the last column of
Table~\ref{tab:ftheta}. These values were obtained via Monte Carlo
simulations of the experimental setup, using the energies and
intensities listed in Table~\ref{tab:ftheta} and the prescription
of Ref.~\cite{fer65}. The main uncertainty in the calculation of
the angular distribution effects arises from the unknown admixtures
of different multipolarities (M1+E2) for the transitions connecting
the original excited state $J^{\pi} =3/2^-$ with any of the three
lowest states (paths (a), (b) and (c) in Fig.~\ref{fig:levels}).
As shown in Table~\ref{tab:ftheta}, the decay pattern and the
corresponding effect on the capture yield $\sigma^{3/2^-}_\theta$
vary abruptly from one resonance to another. It is therefore
difficult to assess a common systematic uncertainty for the
remaining ${3/2^-}$ resonances. Assuming that the four resonances
listed in Table~\ref{tab:ftheta} constitute a representative
sample, one may consider their standard deviation of $\sigma =
4$\% as a realistic estimate of the systematic uncertainty due to
angular distribution effects.
Resonances with $J^{\pi} =3/2^+$ can be assumed to decay directly to the
ground state through an E1 transition. In this case we have estimated an
effect of 10\% in the capture yield with respect to the isotropic case.
However, since $3/2^+$ resonances appear at a relatively high neutron energy,
the final effect in the MACS is practically negligible (see below Sec.~\ref{sec:macs}).
\subsubsection{Resonances with spin $J=5/2$}
For resonances in $^{207}$Pb with $J^{\pi}=5/2^+$ the most probable decay
would be through an electric dipole transition to the first excited state
with $J^{\pi}=5/2^-$ and/or to the second excited state with $J^{\pi}=3/2^-$
(paths (b) and (c) in Fig.~\ref{fig:levels}). Under these assumptions,
the effect on the capture yield would be -12\% for path (b) and \mbox{9\%}
for path (c). However, mixtures of both decay paths would partly compensate
the correction for angular distribution effects. Adopting one standard
deviation of the two extreme cases $\sigma_\theta^{5/2^+} \simeq
10$\% would, therefore, represent a rather conservative estimate of the
corresponding uncertainty. Nevertheless, even such a relatively large
uncertainty for the cross section of $J^{\pi}=5/2^+$ resonances would
have negligible consequences for the Maxwellian averaged cross
section because these resonances contribute very little to the total
capture cross section (see Sec.~\ref{sec:macs}).
\subsection{\label{sec:summary uncertainties} Summary of uncertainties}
With the WFs calculated via the Monte Carlo technique, the accuracy
of the PHWT has been investigated in detail by the n\_TOF collaboration
\cite{abb04}. It has been shown that the capture yield can
be determined from the measured raw data with an accuracy better than
2\%.
Other sources of systematic uncertainty pertaining to this measurement
are due to the energy dependence of the neutron flux ($\pm$2\%) and to
the background due to in-beam $\gamma$-rays ($\pm$1\%). In the particular
case of the ($n, \gamma$) cross section of $^{206}$Pb, the uncertainty
introduced by the angular distribution of the capture $\gamma$-rays
has to be considered as well. This effect has been estimated to
contribute an uncertainty of $\pm 4\%$ for resonances with $J^\pi = 3/2^-$
and less than $\pm 10\%$ for resonances with $J^\pi = 3/2^+, 5/2^+$.
\section{\label{sec:results}Results}
A total of 61 capture levels were analyzed in the neutron energy range from 3~keV up to 570~keV using the R-matrix code SAMMY
\cite{lar06}. In the analysis, the orbital angular momenta $l$ and the
resonance spins $J$ were adopted from Ref.~\cite{mug06}. Some of the $l$ and $J$
parameters listed in Table~\ref{tab:RP} are tentative or arbitrary if
missing in Ref.~\cite{mug06}. We list all the parameters used in our analysis
so that the final values can be recalculated if necessary.
The capture yield $Y(E_\circ,\Gamma_n,\Gamma_\gamma)$
was parameterized with the Reich-Moore formalism, and a channel radius
of 9.5~fm was used for all partial waves. This parameterized yield was
fitted to the corrected experimental yield by variation of the capture
width $\Gamma_\gamma$ and/or neutron width $\Gamma_n$,
\begin{equation}\label{eq:Y}
f^t \times Y^{exp} = B + Y(E_\circ,\Gamma_n,\Gamma_\gamma),
\end{equation}
\noindent
where $f^t$ is the global yield correction factor given in
Sec.~\ref{sec:threshold}.
The term $B$ describing the background was parameterized as an analytical
function of the neutron energy in the range between 1~eV and
30~keV. Beyond 30~keV, $B$ was best described by means of a numerical
function (pointwise) determined from the measurement of the $^{208}$Pb sample (see
Ref.~\cite{dom06c} for details). The uncertainties quoted for the energy of
each resonance are only the statistical errors obtained from the fits of the
capture data performed with SAMMY.
\begin{longtable*}{ccccccccc}
\caption{Resonance parameters derived from the R-matrix analysis of the
$^{206}$Pb($n, \gamma$) data.}\label{tab:RP}\\
\hline
\hline
$E_{\circ}$ & $l$ & $J$ & $\Gamma_{\gamma}$ & $\Delta{\Gamma_{\gamma}}$ & $\Gamma_{n}$ & $\Delta{\Gamma_{n}}$ & $K_r ^a$ & $\Delta{K_r}$ \\
(eV) & & & (meV) & (\%) & (meV) & (\%) & (meV) & (\%)\\
\hline
3357.93(0.04) & 1 & 3/2 & 78.1 & 3 & 235 & &117 & 2 \\
10865.0(0.4) & 1 & 3/2 & 64.9 & 9 & 44.1 & 8 &52.5 & 6 \\
11296.0(0.5) & (1) & (1/2) & 455 & & 44.6 & 7 &40.6 & 7 \\
14220.0(0.6) & 1 & (1/2) & 152 & 6 & 1560 & &139 & 5 \\
16428.0(0.4) & 0 & 1/2 & 2268 & 9 & 936 & 5 &662 & 5 \\
19744.0(1.3) & 1 & (1/2) & 156 & 7 & 2581 & &147 & 6 \\
19809.0(0.9) & 1 & (3/2) & 295 & & 71.6 & 8 &115 & 6 \\
21885.0(0.9) & 1 & 3/2 & 121 & 6 & 875 & &212 & 5 \\
25112.0(0.9) & 1 & 3/2 & 438 & 9 & 326 & 8 &374 & 6 \\
25428(5) & 1 & 1/2 & 254 & 7 & 48901 & &253 & 7 \\
36200(6) & 1 & 1/2 & 312 & 14 & 35700 & &309 & 13 \\
37480.0(1.9) & 1 & (3/2) & 151 & 15 & 890 & &258 & 13 \\
39028(2) & 1 & (1/2) & 346 & & 93.0 & 36 &73.3 & 28 \\
40647(2) & 1 & (1/2) & 163 & 23 & 884 & &138 & 19 \\
42083.0(1.7) & 1 & (3/2) & 419 & 21 & 1419 & 91 &647 & 26 \\
47534(2) & (1) & (1/2) & 184 & 34 & 1000 & &155 & 29 \\
59233.0(0.2) & (2) & (3/2) & 322 & 16 & 1000 & &487 & 12 \\
63976(3) & (2) & 5/2 & 151 & 17 & 1110 & &400 & 15 \\
65990(10) & 0 & 1/2 & 1186 & 9 & 82200 & &1169 & 9 \\
66590(6) & 1 & 3/2 & 198 & 19 & 9530 & &387 & 19 \\
70352(7) & 1 & 1/2 & 163 & 34 & 10780 & &161 & 34 \\
80388(4) & 2 & 3/2 & 1490 & 8 & 7005 & &2457 & 6 \\
83699(6) & (2) & (3/2) & 351 & 16 & 8000 & &673 & 15 \\
88509(6) & 2 & 5/2 & 375 & 13 & 7996 & &1076 & 12 \\
91740(4) & (1) & (3/2) & 298 & 25 & 1000 & &460 & 19 \\
92620(13) & 0 & 1/2 & 991 & 15 & 32000 & &961 & 15 \\
93561(6) & 2 & 3/2 & 125 & 37 & 7001 & &246 & 37 \\
94743(7) & 2 & (3/2) & 241 & 20 & 7000 & &465 & 20 \\
101220(7) & 2 & (5/2) & 119 & 26 & 8000 & &351 & 25 \\
114380(5) & 1 & (3/2) & 655 & 24 & 2500 & &1037 & 19 \\
114602(6) & 2 & (5/2) & 366 & 19 & 5600 & &1030 & 18 \\
118100(6) & 2 & (5/2) & 390 & 16 & 5100 & &1087 & 15 \\
124753(47) & 1 & 3/2 & 2972 & 9 & 300000 & &5886 & 9 \\
125312(7) & 2 & (3/2) & 2783 & 10 & 21005 & &4915 & 9 \\
126138(38) & (1) & (3/2) & 319 & 32 & 100000 & &635 & 32 \\
140570(23) & 2 & 3/2 & 1387 & 11 & 103000 & &2736 & 11 \\
145201(6) & (2) & (3/2) & 518 & 30 & 3100 & &888 & 26 \\
146419(24) & 0 & 1/2 & 6092 & 8 & 176000 & &5888 & 8 \\
150880(7) & (1) & (1/2) & 554 & 48 & 4400 & &492 & 43 \\
151290(13) & 2 & 5/2 & 457 & 23 & 19000 & &1340 & 22 \\
191217(48) & (1) & (1/2) & 767 & 28 & 96977 & &761 & 27 \\
196990(37) & 1 & 1/2 & 584 & 45 & 64000 & &579 & 44 \\
198618(34) & 2 & 3/2 & 2730 & 10 & 132108 & &5350 & 10 \\
274630(22) & 1 & (1/2) & 514 & 65 & 32000 & &506 & 64 \\
276984(49) & 2 & 3/2 & 2481 & 13 & 112000 & &4854 & 13 \\
313400(18) & 2 & (3/2) & 1020 & 32 & 22000 & &1950 & 31 \\
314340(84) & 2 & 5/2 & 964 & 24 & 179000 & &2875 & 24 \\
356098(22) & 2 & (5/2) & 676 & 35 & 31000 & &1985 & 35 \\
357465(87) & 2 & 3/2 & 1998 & 24 & 455000 & &3979 & 24 \\
406200(55) & 2 & 5/2 & 656 & 51 & 102000 & &1955 & 51 \\
407200(41) & 2 & 3/2 & 2906 & 24 & 71000 & &5583 & 23 \\
416370(127) & 2 & 5/2 & 2722 & 16 & 307000 & &8096 & 16 \\
433340(32) & 2 & (5/2) & 4122 & 17 & 47000 & &11368 & 16 \\
434604(37) & (2) & (3/2) & 4695 & 23 & 58000 & &8687 & 21 \\
443412(13) & (2) & (5/2) & 2375 & 21 & 14000 & &6092 & 18 \\
466320(49) & (1) & (3/2) & 5413 & 15 & 90000 & &10211 & 14 \\
469080(76) & 2 & 3/2 & 3222 & 19 & 161000 & &6317 & 19 \\
471789(28) & (3) & (5/2) & 792 & 36 & 41000 & &2330 & 35 \\
476310(172) & 0 & 1/2 & 5252 & 18 & 374000 & &5180 & 17 \\
510690(51) & (2) & (3/2) & 3123 & 18 & 86000 & &6026 & 18 \\
572245(181) & 2 & 5/2 & 3838 & 13 & 793194 & &11460 & 13 \\
\hline
\end{longtable*}
$^a$ Capture kernel $K_r = g \Gamma_{\gamma} \Gamma_n/\Gamma$, with
$g=J+1/2$.\\
\subsection{\label{sec:comp} Comparison to previous work}
The radiative neutron capture cross section of $^{206}$Pb has been measured at
ORNL~\cite{mac64,all73,miz79}, at RPI~\cite{bar69} and at IRMM~\cite{bor07}.
As representative examples of these measurements we consider in this section
two measurements made at ORELA~\cite{all73,miz79}, a more complete analysis~\cite{mus80} of the ORELA
capture data~\cite{all73} made in combination with transmission data~\cite{hor79} and the recent experiment
made at IRMM~\cite{bor07}.
In order to compare these four data sets with the present
results (Table~\ref{tab:RP}), the ratio of the capture kernels are shown in
Fig.~\ref{fig:ratios}.
\begin{figure}[h]
\includegraphics[width=0.48\textwidth]{./figure3.eps}
\caption{\label{fig:ratios}(Color online) Ratio between the capture kernels reported in
Ref.~\cite{all73} (top), Ref.~\cite{miz79} (2nd), Ref.~\cite{mus80} (3rd) and Ref.~\cite{bor07}
(bottom) and the kernels determined here.}
\end{figure}
The values reported in Ref.~\cite{all73} show a relatively good agreement
with our results, except for the first two resonances at 3.3~keV and
14.25~keV, which are lower by $\sim$50\% (see
Fig.~\ref{fig:ratios}). However, these two resonances and the resonance at
16.428~keV are important because of their dominant contribution to the MACS
in the energy range between 5~keV and 20~keV.
It is difficult to determine the source of discrepancy, thus no correlation has been found between the
discrepancies and the spins of the resonances. The latter could probably help to
determine if there is any effect related to the angular distribution of the
prompt capture $\gamma$-rays or to the WF used in the previous measurement.
In the second measurement at ORELA~\cite{miz79} the discrepancies versus our present
results are smaller (see Fig.~\ref{fig:ratios}), but the capture kernels are
systematically larger, on average 20$\pm$5\% higher. This could probably reflect
that the WF used in Ref.~\cite{miz79} is overweighting the relatively hard
pulse height spectrum of $^{207}$Pb. Indeed, similar discrepancies have been
found in the past for $^{56}$Fe~\cite{mac87}, where the pulse height spectrum
is also considerably harder than that of the $^{197}$Au sample used for yield
normalization.
The posterior analysis~\cite{mus80} of the ORELA capture data~\cite{all73} in
combination with transmission~\cite{hor79} shows, on average, better agreement with the
capture areas reported here (see Fig.~\ref{fig:ratios}).
Finally, the results reported in the measurement at IRMM~\cite{bor07} show
the best agreement with the capture kernels of n\_TOF (see
Fig.~\ref{fig:ratios}). At E$_n \leq 40$~keV both
measurements agree within a few percent. At higher energy the fluctuations
are larger, but the agreement is still good within the quoted error bars.
As an illustrative example, the capture yield measured
at n\_TOF for the first resonance at 3.3~keV is compared in the top panel of Fig.~\ref{fig:resos} versus the yield
calculated from the resonance parameters reported in
Refs.~\cite{bor07,miz79,mug06}. Obviously, the
IRMM and n\_TOF results show good agreement in both the capture area and
the resonance energy.
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{./figure4.eps}
\includegraphics[width=0.45\textwidth]{./figure5.eps}
\caption{\label{fig:resos}(Color online) (Top figure) The bold red line represents an R-matrix fit to our
experimental capture yield starting from the initial parameters (solid green
line) in Ref.~\cite{mug06}. The dashed and dott-dashed curves correspond to
the capture yields determined in Ref.~\cite{bor07} and Ref.~\cite{miz79},
respectively. (Bottom figure) The fitted capture yield in
the 10-30~keV energy range (thin red line).}
\end{figure}
\subsection{\label{sec:macs} Maxwellian averaged capture cross section}
The Maxwellian averaged cross section (MACS) was determined using the SAMMY
code in the range of thermal energies relevant for stellar nucleosynthesis, i.e. from $kT$=5~keV up to
$kT$=50~keV. As discussed in the previous section, our results agree best with the values reported in
Ref.~\cite{bor07}. The latter data set seems also to be the most complete in
terms of number of analyzed resonances, with about 283 levels. Therefore our results were
complemented with resonances from Ref.~\cite{bor07} in order to avoid any
discrepancy due to resonances missing in Table~\ref{tab:RP}. The contribution
of these supplementary resonances to the MACS is $<0.1$\% at $kT=5$~keV and 6\% at
$kT=25$~keV. The fact that this correction starts to be significant towards
$kT \gtrsim 25$~keV is not relevant for the study of the nucleosynthesis of
$^{206}$Pb. Indeed, as it is discussed below in Sec.~\ref{sec:implications},
$^{206}$Pb is mostly synthesized between the He-shell flashes of the
asymptotic giant branch stars. These intervals between pulses provide about
95\% of the neutron exposure via the $^{13}$C($\alpha, n$)$^{16}$O reaction, which operates
at a thermal energy of $kT = 8$~keV. At this stellar temperature less than
0.5\% of the MACS is due to the supplemented resonances.
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{./figure6.eps}
\caption{\label{fig:macs} (Color online) Maxwellian averaged ($n, \gamma$) cross sections
for $^{206}$Pb from the resonance parameters of this work (bold red) compared
to the IRMM measurement~\cite{bor07} (dashed), to the recommended data of
Ref.~\cite{bao00} (grey) and to the compiled data of Ref.~\cite{mug06}
(solid green).}
\end{figure}
The uncertainties shown in Fig.~\ref{fig:macs} are only statistical.
The systematic uncertainties of the MACS quoted in Table~\ref{tab:macs}
include all contributions discussed in Sec.~\ref{sec:summary
uncertainties}.
\begin{table}[htbp]
\caption{\label{tab:macs} Maxwellian averaged cross section for
$^{206}$Pb.}
\begin{ruledtabular}
\begin{tabular}{cccc}
Thermal energy $kT$ & MACS & $\sigma_{stat}$ & $\sigma_{sys}$\\
(keV) & (mbarn) & (\%) &(\%)\\
\hline
5 & 21.3 & 1.8 & 5 \\
8 & 20.4 & 1.8 & 3 \\
10 & 19.4 & 1.9 & 3 \\
12 & 18.4 & 2.0 & 3 \\
15 & 17.1 & 2.1 & 3 \\
20 & 15.6 & 2.2 & 3 \\
25 & 14.7 & 2.3 & 3 \\
30 & 14.2 & 2.3 & 4 \\
40 & 13.5 & 2.2 & 4 \\
50 & 12.8 & 2.1 & 4 \\
\end{tabular}
\end{ruledtabular}
\end{table}
Assuming systematic uncertainties of 4\% and 10\%
for $3/2^-$ and $5/2^-$ resonances, respectively, the final
uncertainties are completely dominated by the 4\% uncertainty of
the $3/2^-$ resonances. A change of 10\%
in the cross section of the fewer $5/2^+$ resonances has a negligible
influence on the MACS at $kT = 5$~keV, it contributes only 0.5\% at
$kT = 25$~keV and increases linearly up to 1\% at $kT = 50$~keV.
An effect of 10\% in the capture yield of the $3/2^+$ resonances makes only a
1\% difference in the MACS at $kT = 25$~keV and it becomes also negligible
towards lower stellar temperatures.
The 3\% systematic uncertainty of the experimental method
itself originates from the PHWT, the neutron flux shape,
and the use of the saturated resonance technique.
In summary, the MACS of $^{206}$Pb can now be given with total
uncertainties of 5\% and 4\% at the stellar temperatures corresponding to 5~keV
and 25~keV thermal energies, respectively. This improvement with respect to the previously
recommended values of Ref.~\cite{bao00} becomes particularly important for
determining the $s$-process contribution to the production of lead and bismuth in the Galaxy.
\section{\label{sec:implications}The $s$-process abundance of $^{206}{\rm \bf Pb}$ as a
constraint for the U/Th clock}
The $s$-process production of $^{206}$Pb takes place in low mass
asymptotic giant branch (\textsc{agb}) stars of low metallicity
\cite{tra01}, where about 95\% of the neutron exposure is provided by
the $^{13}$C($\alpha, n$)$^{16}$O reaction at a thermal energy
of $kT \approx 8$~keV. At this stellar temperature the present
MACS is about 20\% lower and two times more accurate (see Fig.~\ref{fig:macs}) than the values from
Ref.~\cite{bao00}, which have been commonly used so far for stellar nucleosynthesis
calculations. The additional neutron irradiation provided by the
$^{22}$Ne($\alpha, n$)$^{25}$Mg reaction at the higher thermal energy of
$kT=23$~keV during the He shell flash is rather weak.
With the new MACS the $s$-process abundance of $^{206}$Pb has been
re-determined more accurately. A model calculation was carried out for
thermally pulsing \textsc{agb} stars of 1.5 and 3 $M_{\odot}$ and
a metallicity of [Fe/H] = $-$0.3. The abundance of $^{206}$Pb is
well described by the average of the two stellar models, which
represent the so-called main component~\cite{arl99}. Since the
contribution of $^{206}$Pb by the strong component is only 2\%,
the main component can be used to approximate the effective production
of $^{206}$Pb during Galactic chemical evolution (GCE)~\cite{GAB98,tra99,tra01}.
This approach yields an $s$-process abundance of $^{206}$Pb, which
represents 70(6)\% of the solar abundance value $N_\odot^{206} = 0.601(47)/10^6$Si~\cite{lod03}. The same
calculation made with the older MACS recommended by Bao et al.~\cite{bao00}
yields 64\%. The uncertainty on the calculated $s$-process abundance is
mostly due to the uncertainty on the solar abundance of lead
(7.8\%)~\cite{and89b}. The contribution from the uncertainty on the MACS at
8~keV is less than 2\%. Finally, the contribution from the $s$-process model
is $\pm$3\%. The latter corresponds to the mean root square deviation
between observed and calculated abundances for $s$-process only
isotopes~\cite{arl99}. This uncertainty is justified for $^{206}$Pb because
its nucleosynthesis is dominated by the main component and it is only
marginally affected ($\sim$2\%) by the strong component~\cite{GAB98,arl99,tra99,tra01}.
Furthermore, because of the much lower cross sections
of $^{208}$Pb and $^{209}$Bi, the synthesis of $^{206}$Pb remains practically
unaffected by the $\alpha$-recycling after $^{209}$Bi~\cite{rat04}. This
lends further confindence that the production of $^{206}$Pb, and hence its
uncertainty, follows the same trend as the main $s$-process component.
\begin{figure}[h]
\includegraphics[width=0.45\textwidth]{./figure7.eps}
\caption{\label{fig:Nc206} Estimate of the radiogenic component of $^{206}$Pb
using the Fowler's model with different nucleosynthetic assumptions (see
labels in curves) and the $r$-process age $\Delta_r = t_U - 4.6$~Gyr (vertical dashed line) derived from the age of the
Universe $t_U$~\cite{ben03}.}
\end{figure}
In order to estimate a constraint for the $r$-process abundance of $^{206}$Pb
one needs to take into account its radiogenic contribution, $N_c^{206}$, due to
the decay of $^{238}$U. As it is shown in the following, this component is
relatively small but can not be neglected. Based on the schematic model of
Fowler, which assumes an exponential decrease of the $r$-process yield during
GCE~\cite{cla64} (supernova rate $\Lambda = (0.43t)^{-1}$~Gyr$^{-1}$) and using the current best estimates for the age of the
Universe ($t_U = 13.7 \pm 0.2$~Gyr)~\cite{ben03}, one obtains $N_c^{206} =
0.027(2)/10^6$Si (see Fig.~\ref{fig:Nc206} and Table~\ref{tab:N}). This number, combined with our result for $N^{206}_s$,
yields an $r$-process residual,
\begin{equation}\label{eq:n206}
N^{206}_{r} = N^{206}_{\odot} - N^{206}_s - N^{206}_c = 0.153\pm0.063.
\end{equation}
The uncertainty in this result includes contributions of 8.4\% from $N^{206}_c$
(corresponding to the uncertainty on the solar abundance of $^{238}$U~\cite{and89b}), 7.8\% from the total solar
abundance of $^{206}$Pb, $N^{206}_{\odot}$~\cite{lod03,and89b}, and 8.6\% from the
determination of $N^{206}_{s}$ as discussed above. This means that, apart from
the uncertainties related with the simplified assumptions in the GCE model of Fowler, the
$r$-process abundance can be reliably constrained between 16\% and 36\% of
the solar $^{206}$Pb.
\begin{table}
\caption{\label{tab:N} Radiogenic abundance of $^{206}$Pb,
$N_c^{206}$ (Si=10$^6$), derived from the model of Fowler and the age of the Universe
(see Fig.~\ref{fig:Nc206}). $r$-Process residuals obtained via eq.~\ref{eq:n206}.}
\begin{ruledtabular}
\begin{tabular}{lccc}
GCE & $N^{206}_c = R N^{238}_\odot$ &\multicolumn{2}{c}{$ N^{206}_r = N^{206}_{\odot} - N^{206}_s - N^{206}_c$}\\
(Fig.~\ref{fig:Nc206}) & $10^6$Si & $10^6$Si & $N^{206}_r/N^{206}_\odot$(\%)\\
\hline
43\% SN rate & 0.027(2) & 0.15(6) & 26(10) \\
Sudden & 0.058(5) & 0.12(6) & 20(10) \\
Uniform & 0.0161(14) & 0.16(6) & 27(10) \\
\end{tabular}
\end{ruledtabular}
\end{table}
The $r$-process residuals derived here are consistent with $r$-process
model calculations available in the literature, i.e., $N^{206}_r =
26.6$\%~\cite{cow99}. More recent calculations yield $N^{206}_r$ values between 27\%
and 35\%~\cite{kra04}.
One can also derive hard limits for the $r$-process abundance, considering the two
extreme cases of sudden nucleosynthesis ($\Lambda \rightarrow \infty$) and
uniform nucleosynthesis ($\Lambda \rightarrow 0$). This
yields constraints between 10\% and 37\% of solar $^{206}$Pb (see
Table~\ref{tab:N}).
The situation is rather different for the corresponding $^{207}$Pb/$^{235}$U
ratio, which has been investigated as a potential clock in the
past~\cite{bee85}.
In this case, the $s$-process abundance of $^{207}$Pb was recently
determined to be $N^{207}_s = 77(8)$\%~\cite{dom06b}. A similar calculation
to that shown in Fig.~\ref{fig:Nc206} gives $N_c^{207} = 0.150(13)$ (see Table~\ref{tab:N207}). The
latter value reflects the large relative radiogenic abundance of $^{207}$Pb, $N_c^{207}/N_\odot^{207} =
22$\%, due to the much shorter half-life of $^{235}$U.
From the total solar abundance of $^{207}$Pb~\cite{lod03} and the $N_s^{207}$
and $N_c^{207}$ values quoted above, the $r$-process residual becomes $N^{207}_r = 0.003 \pm 0.073$,
which means that $N^{207}_r$ can not be larger than 11\% of the $^{207}$Pb
abundance in the solar system, $N_\odot^{207} = 0.665(52)$~\cite{lod03} (Table~\ref{tab:N207}).
\begin{table}
\caption{\label{tab:N207} Radiogenic abundance of $^{207}$Pb,
$N_c^{207}$ (Si=10$^6$), derived from the model of Fowler and the age of
the Universe. $r$-Process residuals obtained via eq.~\ref{eq:n206}.}
\begin{ruledtabular}
\begin{tabular}{lccc}
GCE & $N^{207}_c = R N^{235}_\odot$ &\multicolumn{2}{c}{$ N^{207}_r = N^{207}_{\odot} - N^{207}_s - N^{207}_c$}\\
& $10^6$Si & $10^6$Si & $N^{207}_r/N^{207}_\odot$(\%)\\
\hline
43\% SN rate & 0.150(13) & 0.003(73) & 0(11) \\
90\% SN rate & 0.08(7) & 0.073(72) & 11(11) \\
Uniform & 0.047(4) & 0.106(72) & 16(11) \\
\end{tabular}
\end{ruledtabular}
\end{table}
This result is in contrast with $r$-process
model calculations, which yield values between 22.7\% and
25.3\%, with a relative uncertainty of 15-20\%~\cite{cow99,kra04}.
The $s$-process abundances of $^{206,207}$Pb are rather reliable
and not very sensitive to details of the stellar models~\cite{rat04,dom05}.
Therefore, this discrepancy indicates that $r$-process abundances
might have been overestimated, possibly because the odd-even
effect is not properly reproduced by the ETFSI-Q mass model implemented
in the $r$-process calculations~\cite{cow99,kra04}. Indeed, one needs to
increase the supernova rate in the standard Fowler model from 43\% up to 90\% ($\Lambda =
(0.90t)^{-1}$~Gyr$^{-1}$) in order to achieve
agreement between these $r$-process constraints and the latter $r$-process
calculations~\cite{cow99,kra04}. Obviously the less realistic uniform
scenario would also provide agreement with the abundances from these
r-process models (see Table~\ref{tab:N207}).
However the situation has been improved recently after more detailed $r$-process model
calculations~\cite{kra07} predicted a new $N_r^{207}$ value, which is 35\% lower
than the previous one of Ref.~\cite{cow99}. This yields $N^{207}_r/N^{207}_\odot = 16.8$\%,
which is substantially closer (considering an uncertainty of 20\%) to the upper limit of 11\% derived here. In
this case a good agreement would be found for a more reasonable increase of
the supernova rate to 55\% in the Fowler model.
These constraints for the $r$-process abundances of $^{206,207}$Pb
become relevant for the validation of $r$-process model calculations and
hence, for the reliable interpretation of actinide abundances
observed in UMP stars and their use as cosmochronometers.
The $s$-process aspects will be more rigorously investigated
in a comprehensive study of the Pb/Bi region~\cite{bis06}, where the role of
stellar modeling and GCE will be discussed with a complete set of new cross
sections for the involved isotopes, including the present data for
$^{206}$Pb, and recent results for $^{204}$Pb~\cite{dom06c},
$^{207}$Pb~\cite{dom06b} and $^{209}$Bi~\cite{dom06}.
\section{\label{sec:summary} Summary}
The neutron capture cross section of $^{206}$Pb as a function of the neutron
energy has been measured with high resolution at the CERN n\_TOF installation using
two C$_6$D$_6$ detectors. Capture widths and/or radiative kernels could
be determined for 131 resonances in the neutron energy interval from
3~keV up to 620~keV. Systematic uncertainties of 3\%, 5\%, and
$\lesssim$10\% were obtained for resonances with spin-parities of
1/2$^{\pm}$, 3/2$^-$ and 5/2$^+$, respectively. The Maxwellian averaged
cross sections were found to be significantly smaller by 10\% to 20\%
compared to values reported earlier~\cite{bao00}, resulting in a
correspondingly enhanced $s$-process production of $^{206}$Pb. First
calculations with a standard \textsc{agb} model yield an $s$-process
component of 70(6)\% for the $^{206}$Pb abundance.
Combined with an estimate of the radiogenic production of $^{206}$Pb, the
$r$-process abundance is constrained between 16\% and 36\% of the solar
$^{206}$Pb abundance, well in agreement with $r$-process model calculations
reported in the literature~\cite{cow99,kra04}. A similar analysis for $^{207}$Pb shows
agreement only with most recent $r$-process model calculations~\cite{kra07}.
|
2,869,038,154,730 | arxiv | \section{Introduction}
Boron presents a wealth of possible two-dimensional (2D) allotropes, and boron sheets exhibit various structural polymorphs containing mostly hexagonal and triangular lattices \cite{Tang2007,Penev2012,Zhou2014a,Zhang2015c,Mannix2015,Feng2016a}. Low-buckled borophene sheets in the form of triangular lattices have been predicted years ago \cite{Tang2007, Kunstmann2006}, and have recently been synthesized on Ag substrates \cite{Mannix2015}. Electronic, magnetic, mechanical and optical properties of triangular boron sheets have recently been investigated by first-principles calculations \cite{Mortazavi2016b,Lherbier2016}. These 2D boron sheets, so-called borophene, were even predicted to exhibit superconductive behavior \cite{Penev2016}. Recent first-principles calculations confirmed that borophene sheets can serve as an ideal anode electrode material with high electrochemical performance for Mg, Na, and Li ion batteries, which outperform other 2D materials \cite{Mortazavi2016c}. These outstanding physical properties of borophene place it as a direct rival for graphene in a serie of applications \cite{Novoselov2004,Geim2007,Wehling2008}. Nevertheless, in spite of the promising applications of borophene, studies related to its thermal and mechanical properties at finite temperatures are still very limited. In particular, the thermal conductivity of borophene films remained almost unexplored. The mechanical properties of flat boron sheets with different vacancy ratios have recently been investigated via classical molecular dynamics (MD) simulations \cite{Le2016}, where it was shown that their mechanical properties depend significantly on atomic structures, loading direction, and temperature.
Motivated by the most recent experimental advances in the fabrication of corrugated triangular borophene sheets and their wide potential applications \cite{Mannix2015}, we believe it is fundamental to provide a comprehensive understanding of the thermal and mechanical properties of borophene sheets. Therefore, in the present work, we study the lattice thermal conductivity and mechanical properties of corrugated borophene at room temperature by performing extensive reactive molecular dynamics simulations. In general, molecular dynamics simulations are a powerful tool to predict and understand the physical properties of novel materials \cite{Mortazavi2015a,Abadi2016,Mortazavi2016a,Pereira2016}, at a fraction of the computational cost of first-principles calculations. In the present investigation we report on the direction dependent elastic modulus, stress-strain response and phonon thermal conductivity of corrugated borophene at room temperature. Our reactive MD simulations show that corrugated borophene sheets present highly anisotropic thermal and mechanical properties, and can guide future studies on the thermal and mechanical properties of borophene films.
\section{Methods: Molecular dynamics modelling}
All molecular dynamics simulations were carried out with LAMMPS \cite{Plimpton1995}, where the atomic equations of motion were time-integrated with the velocity Verlet algorithm \cite{Verlet1967}. The ReaxFF \cite{Weismiller2010} potential was used to model the atomic interactions in borophene sheets. Periodic boundary conditions were applied in the planar directions to remove the effects of free atoms at the edges. In this way, we studied infinite borophene sheets and not 2D boron nanoribbons. In the out of plane direction (z-direction), we defined a $20$ \AA~ vacuum by fixing the simulation box size along that direction. All MD simulations in this work were carried out at room temperature ($300$ K).
First we investigated the mechanical properties of single-layer boron sheets. In this case, the equations of motion were integrated with a small time step of $0.25$ fs. Before applying the loading conditions, the structures were relaxed to zero stress using a Nos\'e-Hoover barostat and thermostat (NPT) for $1.25$ ps. Uniaxial tension in the armchair and zigzag directions (notations are indicated in figure \ref{fig1}) were imposed by applying a constant engineering strain rate of $2.5 \times 10^8$ $s^{-1}$, one direction at a time. Because we are dealing with single-layer boron sheets, the stress perpendicular to the sheet (in the z-direction) is zero. Therefore, in order to guarantee uniaxial stress conditions, zero stress condition in the edge parallel to the tensile direction was achieved by relaxing the size of the simulation box in the direction perpendicular to the tensile direction using a barostat set to zero pressure. The macroscopic stress tensor is given by the virial theorem \cite{Marc1985,Zimmerman2004}:
\begin{equation}
\mathbf{\sigma} = \frac{1}{V} \sum_{a \in V} \left[ -m^{a} \mathbf{v}^a \otimes \mathbf{v}^a + \frac{1}{2} \sum_{a \ne b} \mathbf{r}^{ab} \otimes \mathbf{f}^{ab} \right]
\end{equation}
here, $m^a$ and $\mathbf{v}^a$ are the mass and the velocity vector of atom $a$, respectively. The symbol $\otimes$ denotes the tensor product of two vectors. $\mathbf{r}^{a}$ denotes the position of atom $a$. $\mathbf{r}^{ab} = \mathbf{r}^{b} - \mathbf{r}^{a}$ is the distance vector between atoms $a$ and $b$. $\mathbf{f}^{ab}$ is the force on atom $a$ due to atom $b$. $V$ is the volume of the structure. For boron sheets we defined $V=A \times t$, where $A$ is the surface area of the sheet, and $t$ is the sheet nominal thickness. In all simulations we assumed a nominal thickness of $2.9$ \AA~ for single-layer borophene sheets \cite{Mannix2015}.
The non-equilibrium molecular dynamics (NEMD) method was employed to study the phonon thermal conductivity of borophene. In this method, simulations were performed for borophene samples of increasing length with a fixed nominal width of $6.6$ nm. The phononic thermal conductivity of an infinite borophene sheet, as well as an effective phonon mean free path were extracted from the size dependence of the thermal conductivity. Due to the non-equilibrium conditions in these simulations, it was necessary to use a smaller integration time step, namely a time increment of $0.2$ fs. After obtaining the equilibrated structures at $300$ K, we fixed several rows of boron atoms at the two opposing ends of the simulation sample. Next, the simulation box (excluding the fixed atoms) was divided into $22$ slabs along the sample length, and a $20$ K temperature difference between the first and 22nd slabs was imposed in the system. To this aim, the temperature in these two slabs was controlled at the desired values ($310$ K and $290$ K) by independent Nos\'e-Hoover thermostats, while the remaining slabs were not connected to any thermostats or heat baths, effectively evolving in a microcanonical ensemble. Therefore, a constant heat flux was imposed in the system by continuously adding or removing energy in the thermostated slabs at a rate $dq/dt$. The heat flux along a cartesian direction can then be obtained from:
\begin{equation}
J_x=\frac{1}{A} \frac{dq}{dt}.
\end{equation}
Here, $A$ is the cross sectional area of the borophene sheets, once again assuming a nominal thickness of $2.9$ \AA~ \cite{Mannix2015}. The temperature of each slab is taken and the average kinetic energy of the atoms in the slab, and converted via the equipartition theorem. After a transient period, the system reaches a steady-state heat transfer condition, and a constant temperature gradient, $dT/dx$, is established along the sample length. The NEMD simulations were performed for at least $6$ ns in the steady state regime, during which the heat current and the temperature gradient were time-averaged. Finally, the thermal conductivity was obtained from Fourier's law:
\begin{equation}
\kappa = \frac{J_x}{dT/dx}.
\end{equation}
In order to calculate the phonon density of states, we simulated a borophene sheet at room temperature under microcanonical conditions for $100$ ps, during which we record the trajectories and velocities. We then post-process the trajectories to calculate the DOS from the Fourier transform of the normalized velocity autocorrelation function, such that:
\begin{equation}
DOS(\omega) = \int_0^{\infty} \frac{ \langle \mathbf{v}(t) \cdot \mathbf{v}(0) \rangle}{\langle \mathbf{v}(0) \cdot \mathbf{v}(0) \rangle} \exp(-i \omega t) dt
\end{equation}
where $\omega$ is the angular frequency and $\mathbf{v}$ is the atomic velocity.
\section{Results}
The atomic structure of corrugated borophene is shown in figure \ref{fig1}. The structure of borophene can be defined by introducing the $\alpha$ and $\beta$ lattice constants and the bucking height, $\Delta$ indicated in figure \ref{fig1}. Based on the ReaxFF results for the relaxed structure, the borophene $\alpha$, $\beta$ and $\Delta$ lattice constants were $2.1$ \AA, $3.18$ \AA~ and $0.76$ \AA, respectively. We note that according to first-principles DFT calculations \cite{Mannix2015} the $\alpha$ and $\beta$ lattice constants of borophene were predicted to be $1.7$ \AA~ and $2.9$ \AA, respectively, while $\Delta$ was calculated at $0.80$ \AA. This indicates that ReaxFF does not accurately predict the in-plane lattice parameters of borophene, at least if the first principles calculations are to be taken as accurate. Nonetheless, we will show in what follows that our simulation results for the elastic modulus and the lattice thermal conductivity are in excellent agreement with first-principles calculations.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{fig1.png}
\caption{Top and side views of the atomic structure of borophene. Two colors, green and purple, are used to indicate the height of the atoms. Lattice constants $\alpha$ and $\beta$, as well as the bucking height $\Delta$ are indicated. We studied mechanical and thermal transport properties along the in-plane directions, indicated as armchair and zigzag in analogy with graphene.}
\label{fig1}
\end{figure}
Figure \ref{fig2} plots the uniaxial stress-strain curves of the borophene sheet at $300$ K along armchair and zigzag directions. Stress values were calculated assuming a nominal thickness of $2.9$ \AA~ for single-layer borophene films.
The stress-strain curves include an initial linear region which is followed by a nonlinear trend up to a peak value, the sample tensile strength.
As the strain is increased further, the stress drops suddenly, which is a typical indication of a brittle fracture mechanism.
Anisotropic tensile strengths of $147$ GPa and $182$ GPa are obtained when the borophene membrane is stretched along zigzag and armchair directions, respectively, as shown in figure \ref{fig2}.
Both values are larger than a recent first principles prediction \cite{Mortazavi2016b}.
Meanwhile, the strain at the tensile strength, know as the fracture strain is estimated at $25$\% for the zigzag direction and $8.8$\% for the armchair direction, while the corresponding first principles values are $14.5$\% and $10.5$\% \cite{Mortazavi2016b}.
The calculated 2D Young's modulus of borophene are $188$ N m$^{-1}$ and $403$ N m$^{-1}$ in the zigzag and armchair directions, respectively. Remarkably, these values are within $10$\% of the results predicted by first-principles DFT calculations, namely $170$ N m$^{-1}$ \cite{Mannix2015} and $163$ N m$^{-1}$ \cite{Mortazavi2016b} for elongation along the zigzag direction and $398$ N m$^{-1}$ \cite{Mannix2015} and $382$ N m$^{-1}$ \cite{Mortazavi2016b} when stretched along the armchair direction.
The lower Young's modulus and higher fracture strain in the zigzag direction in comparison to the armchair one are due to the buckling along the zigzag direction of borophene, shown in figure \ref{fig1}.
The agreement of our results for the 2D Young's modulus with first principles calculations reveals that the utilized ReaxFF potential can accurately describe the atomic interaction at low strain levels within the elastic region, even if it does not yield the same lattice parameters as first principles calculations. This observation will also be valid for our thermal conductivity evaluation in the next few paragraphs.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{fig2.png}
\caption{Uniaxial stress-strain curves of the borophene sheet at $300$ K. Stress values were calculated assuming a nominal thickness of $2.9$ \AA~ for single-layer borophene films.}
\label{fig2}
\end{figure}
The behavior of the strees-strain curves shown in figure \ref{fig2} are quite different. Notably there seem to be two distinct linear regimes when the sample is stretched along the zigzag direction. We atribute this behavior to the buckling of the structure along the zigzag direction. In the first linear region the buckling of borophene is decreased due to the strain, and the sample becomes flatter. In the second regime the now in-plane B-B bonds are stretched up to their rupture point.
In figure \ref{fig3}, the deformation of a single-layer corrugated borophene sheet stretched along its zigzag direction is depicted. For borophene sheets under uniaxial tensile loading along the zigzag and armchair directions, we found that the structures extend uniformly and remain defect-free up to strain levels close to rupture. For a borophene membrane stretched along the zigzag direction, shortly before rupture the first B-B bond breakages occur (figure \ref{fig3}(a)), resulting in the formation of pentagons (figure \ref{fig3}(a) insets). Instantly after the formation of these initial pentagon defects the ultimate tensile strength is reached, and the coalescence of defects happen, resulting in the formation of a crack that grows rapidly perpendicular to the loading direction and leads to the sample rupture (figure \ref{fig3}(b)). Based on our reactive atomistic modeling, corrugated borophene presents a brittle failure mechanism at room temperature. This conclusion is corroborated by the fact that the initial defect formation and sample rupture occur at very close strain levels.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{fig3.png}
\caption{Uniaxial deformation process of single-layer borophene stretched along the zigzag direction (a) shortly before the rupture and (b) at failure point. The stress values are the uniaxial stress distribution along the loading direction. The OVITO package was used for the illustration of this figure \cite{Stukowski2010}.}
\label{fig3}
\end{figure}
Non-equilibrium molecular dynamics simulations for borophene sheets were performed along armchair and zigzag directions, for samples of increasing length in order to assess possible size effects on the thermal conductivity. In NEMD simulations the calculated lattice thermal conductivity presents a strong size dependency when the sample length is smaller than the phonon mean-free path of the structure \cite{Schelling2002,Xu2014,Fugallo2014,Zhu2014,Zhang2015,Barbarino2015,Neogi2015,Majee2016,Mortazavi2016a,Pereira2016}. Accordingly, the thermal conductivity of borophene sheets obtained in our simulations increased with the sample length, $L$. As a common approach, we can express the length dependence of the thermal conductivity with the ballistic to diffusive transition equation
\begin{equation}
\kappa(L) = \frac{\kappa}{1+\Lambda_{eff}/L},
\label{eq:fit}
\end{equation}
where $\kappa$ is the intrinsic thermal conductivity of an infinite-sized sample, and $\Lambda_{eff}$ is an effective phonon mean free path for the material \cite{Neogi2015,Mortazavi2016a,Pereira2016}. Notice that from this expression we have $\kappa(L)=\kappa/2$ when $L=\Lambda_{eff}$. Therefore, adjusting the above equation to the data points obtained from NEMD simulations with different lengths we can determine both free-parameters, $\kappa$ and $\Lambda_{eff}$.
Figure \ref{fig4} shows the data points calculated from NEMD simulations at $300$ K, and the solid lines indicate the corresponding fit to the data, from which we calculate conductivities of $75.9 \pm 5.0$ W m$^{-1}$K$^{-1}$ and $147 \pm 7.3$ W m$^{-1}$K$^{-1}$ along zigzag and armchair directions respectively. Meanwhile, the effective phonon mean free path take values of $16.7 \pm 1.7$ nm and $21.4 \pm 1.0$ nm for the zigzag and armchair directions. The data shows a clear anisotropy in the conduction of heat along the in-plane directions of borophene, with the conductivity along the armchair direction being about twice as large as in the zigzag direction. Such feature could be explored in the construction of future phononic devices \cite{Wagner2016}.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{fig4.png}
\caption{Thermal conductivity of borophene as a function of length along armchair and zigzag directions. Data points from NEMD simulations and lines from equation \ref{eq:fit}. Thermal conductivities are $75.9 \pm 5.0$ W m$^{-1}$K$^{-1}$ and $147 \pm 7.3$ W m$^{-1}$K$^{-1}$ along zigzag and armchair directions, respectively.}
\label{fig4}
\end{figure}
{Finally, we investigated the effect of mechanical strain on the the thermal conductivity of borophene along each direction independently, as shown in figure \ref{fig5}. When the sample is strained along the armchair direction there is a significant increase in thermal conductivity along that direction. Meanwhile, when the sample is strained along the zigzag direction there is a much smaller increase in thermal conductivity along that direction. For a strain of 8\% along the armchair direction the thermal conductivity increases by a factor of 3.5 (~250\%), whereas for the same amount of strain along the zigzag direction the increase is only by a factor of 1.2 (~20\%).
This feature reinforces the prospective application of borophene in the construction of phononic devices \cite{Wagner2016}. }
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{fig5.png}
\caption{Normalized thermal conductivity of borophene under strain. Notice the significant increase when the sample is strained along the armchair direction.}
\label{fig5}
\end{figure}
\section{Discussion}
In the development of this study, several interatomic potentials have been tested to describe the structure of borophene, including the Tersoff potential \cite{Tersoff1988,Tersoff1988a,Matsunaga2000,Mortazavi2015a}, and different versions of ReaxFF.
We have found that the ReaxFF parameter set developed by Weismiller et al. \cite{Weismiller2010} yields the closest predictions to first-principles results.
Even though the 2D Young's modulus predicted by reative MD are within $10$\% of first principles results, and the observed trends being in accordance with previous DFT results, the tensile strength and fracture strain are both much larger than predicted by DFT calculations \cite{Mortazavi2016b}.
In fact, along the zigzag direction, the failure strain of borophene ($\sim 25$\%) is close to that of graphene ($20$\%-$27$\%) \cite{Lee2008,Le2016a,Mortazavi2016}.
We believe this overestimation in tensile strength and failure strain could probably be decreased by adjusting the cutoff length for B-B bonds in the ReaxFF parameter set.
One should also consider that according to DFT calculations the equilibrium B-B bond lengths are not equal for the two different bonds in corrugated borophene \cite{Mannix2015,Mortazavi2016b}, whereas in ReaxFF, all B-B bonds are treated using a single set of parameters.
Nonetheless, we atribute the anisotropy in 2D Young's modulus and fracture strain to the buckling along the zigzag direction of borophene. In our interpretation, this buckling is also responsible for the appearance of two linear regimes in the strees-strain curve along the zigzag direction. Where the first part is due to the flattening of the borophene sheet and the second is due to the stretching of in-plane bonds.
The observed anisotropy in the lattice thermal conductivity is consistent with the results for the elastic moduli discussed above, where the elastic constant along the armchair direction was found to be larger than along the zigzag direction by a factor of $2$.
In order to complement our understanding of the underlying mechanism resulting in the anisotropic thermal conductivity of borophene, we calculated the phonon density of states (DOS) as the Fourier transform of the velocity autocorrelation function.
Figure \ref{fig6} shows the vibrational density of states projected onto the in-plane directions, as well as the out-of-plane direction.
Out-of-plane phonons have the largest DOS in the low-frequency regime (below $20$ THz), being most likely the main heat carriers in borophene.
Nonetheless, in the low-frequency region up to $16$ THz we observe a larger DOS for vibrations projected onto the zigzag direction.
We also observe a peak around $14$ THz for vibrations along the zigzag direction, which is caused by flat bands in the phonon dispersion and yield lower phonon group velocities along that direction.
Finally, since there are more phonon modes for scattering along the zigzag direction, it could also shorten the mean-free-path and phonon lifetimes along that direction.
\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{fig6.png}
\caption{Calculated vibrational density of states (DOS) for pristine borophene along different directions.}
\label{fig6}
\end{figure}
Our simulation results are remarkably close to the prediction presented in a very recent first-principles based study of the thermal properties of borophene \cite{Sun2016}. In their work, Sun et al. did not assume a thickness for borophene sheets, but if we scale their results with the nominal thickness used in our simulations we obtain $\kappa_{zigzag} \approx 72$ W m$^{-1}$K$^{-1}$ and $\kappa_{armchair} \approx 145$ W m$^{-1}$K$^{-1}$ . Thus a factor of $2$ anisotropy is observed in the in-plane thermal conductivities, which is attributed by them to the larger phonon group velocities along the armchair direction.
This observation is consistent with the smaller variation we found in the effective phonon mean free path along armchair and zigzag directions.
The difference in group velocities reported by Sun et al., along with our DOS analysis provides a consistent explanation for the predicted anisotropy in the in-plane thermal conductivity of borophene \cite{Mortazavi2016,Pereira2016}.
However, since their calculations are based on the quasi-harmonic approximation, which considers only three-phonon scattering processes, one could expect their conductivity to be larger than ours, which accounts for all possible anharmonic scattering processes. In fact, a comparison between their first-principles calculations and our MD results leads to the conclusion that scattering events including more than three phonons do not have a significant impact in the thermal conductivity of borophene.
\section{Summary}
We performed reactive molecular dynamics simulations to explore the mechanical response and the lattice thermal conductivity of free-standing corrugated borophene at room temperature.
The ReaxFF potential developed by Weismiller et al. was used to model atomic interactions in borophene sheets.
Although the ReaxFF potential overestimate the in-plane lattice parameters of borophene, we find that it predicts the elastic modulus and thermal conductivity of this novel material in excellent agreement with first-principles calculations, with a much lower computational cost.
According to our reactive molecular dynamics simulations, the 2D Young's modulus of borophene was estimated to be $188$ N m$^{-1}$ and $403$ N m$^{-1}$ along zigzag and armchair directions, respectively, which are within $10$\% of first-principles predictions.
This anisotropy in elastic moduli is attributed to the buckling of the borophene crystal structure along the zigzag direction.
We also performed non-equilibrium molecular dynamics simulations to calculate the lattice thermal conductivity of borophene sheets of increasing length
The intrinsic thermal conductivity of an infinite borophene sheet, as well as an effective phonon mean free path were obtained from the ballistic to diffusive transition equation. At room temperature, the thermal conductivity of borophene along its zigzag and armchair directions were predicted to be $75.9 \pm 5.0$ W m$^{-1}$K$^{-1}$ and $147 \pm 7.3$ W m$^{-1}$K$^{-1}$ , respectively.
Meanwhile, the effective phonon mean free path take values of $16.7 \pm 1.7$ nm and $21.4 \pm 1.0$ nm for the zigzag and armchair directions.
In this case, the anisotropy is attributed to differences in the density of states of low-frequency phonons, which correlate to lower group velocities and possibly shortens phonon lifetimes along the zigzag direction.
{We also observe that when borophene is strained along the armchair direction there is a significant increase in thermal conductivity along that direction. Meanwhile, when the sample is strained along the zigzag direction there is a much smaller increase in thermal conductivity along that direction. For a strain of 8\% along the armchair direction the thermal conductivity increases by a factor of 3.5 (~250\%), whereas for the same amount of strain along the zigzag direction the increase is only by a factor of 1.2 (~20\%).}
Interestingly, the anisotropy ratio for the thermal conductivity and elastic modulus were found to be almost the same, and also consistent with recent first-principles calculations.
Furthermore, comparing our phonon thermal conductivity predictions with first-principles calculations based on the quasi-harmonic approximation, which considers only three-phonon scattering processes, indicates that scattering events including more than three phonons do not have a significant impact in the thermal conductivity of borophene.
Our predictions for the elastic moduli and lattice thermal conductivity are in agreement with recent first principles results, at a fraction of the computational cost.
We expect our simulations to serve as a guide for future experiments concerning the mechanical and thermal properties of borophene and related novel 2D materials.
\section{Acknowledgements}
The authors would like to thank L.D. Machado for a critical reading of the manuscript. BM and TR greatly acknowledge the financial support by European Research Council for COMBAT project (Grant No. 615132). MQL was supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under the grant number: 107-02-2017-02. LFCP acknowledges financial support from the Brazilian government agency CAPES for the project ``Physical properties of nanostructured materials'' (Grant No. 3195/2014) via its Science Without Borders program and provision of computational resources by the High Performance Computing Center (NPAD) at UFRN.
|
2,869,038,154,731 | arxiv | \section{Introduction}
The noncommutative Schwartz space~${\mathcal{S}}$ is a weakly amenable m-convex
Fr\'echet algebra whose properties have been investigated in several
recent papers, see e.g.~\cite{TC,PD,KP,KP1}. It is not difficult to see that as a Fr\'echet space,~${\mathcal{S}}$ is nuclear. From this, we can easily deduce the following
analogue of Grothendieck's inequality, which we call
\emph{Grothendieck's inequality in~${\mathcal{S}}$}: there exists a
constant $K>0$ %
so that for any continuous bilinear
form $u\c{\mathcal{S}}\times{\mathcal{S}}\to{\mathbb{C}}$ and any $n\in{\mathbb{N}}$, there exists $k\in{\mathbb{N}}$ such
that for every $m\in{\mathbb{N}}$ and any $x_1,\ldots,x_m,y_1,\ldots,y_m\in{\mathcal{S}}$,
we have
\begin{equation}
\Big|\sum_{j=1}^mu(x_j,y_j)\Big|\leqslant K\|u\|_n^* \,\|(x_j)\|_k^{\rc}\,\|(y_j)\|_k^{\rc}
\label{target-inequality}
\end{equation}
The norms on the right hand side %
arise naturally from the
definition of~${\mathcal{S}}$, as explained in Section~\ref{section:2} below. Our
goal in this note is to show that in fact $k=2n+1$ always suffices,
and that this is best possible.
This appears to be the first result concerning Grothendieck's inequality in the category of Fr\'echet algebras; to the best of our knowledge, all previous results along these lines concern Banach spaces (including C${}^*$-algebras, general Banach algebras and operator spaces). For Fr\'echet algebras, Grothendieck's inequality seems to have a specific flavour. Every Fr\'echet space (and a fortiori, every Fr\'echet algebra) which appears naturally in analysis is nuclear, meaning that all tensor product topologies are equal. Since Grothendieck's inequality can be understood as the equivalence of two tensor products, it seems that we can take inequality~\eqref{target-inequality} for granted. The interesting question that remains is then optimality.
This paper is divided into four sections. In the remainder of this
section, we recall a C${}^*$-algebraic version of Grothendieck's
inequality due to Haagerup, and then review the definition and the basic
properties of~${\mathcal{S}}$ which we require. In Section~2 we explain how
nuclearity gives Grothendieck's inequality in~${\mathcal{S}}$, and we estimate the constants $K$ and $k$. Section~3 then settles the optimality question for~$k$ via a matricial construction. %
We conclude with a short section containing several reformulations of
the
inequality.%
\subsection{Grothendieck's inequality}
Pisier's survey article~\cite{GP} is a comprehensive reference for Grothendieck's inequality. This presents many equivalent formulations and applications of this famous result, and recounts its evolution from `commutative'~\cite{AG} to `noncommutative'. %
Of these reformulations and extensions, Haagerup's noncommutative version most closely resembles~(\ref{target-inequality}), and we state it here for the convenience of the reader.
\begin{theorem}[{\cite{UH}, \cite[Theorem~7.1]{GP}}]
Let~$A$ and~$B$ be C${}^*$-algebras. For any bounded bilinear
form~$u\colon A\times B\to \mathbb C$ and any finite sequence $(x_j,y_j)$
in $A\times B$, we have
\begin{equation*} \left|\sum u(x_j,y_j)\right| \leqslant 2\|u\|\,\|(x_j)\|^{\rc}\,\|(y_j)\|^{\rc}
\end{equation*}
where $\|(x_j)\|^{\rc} := \max\big\{
\big\|\sum x_j^*x_j\big\|^{\frac12}
,
\big\|\sum x_jx_j^*\big\|^{\frac12}
\big\}$.
\end{theorem}
\subsection{The noncommutative Schwartz space}
Let
\[s=\Big\{\xi=(\xi_j)_{j\in{\mathbb{N}}}\in{\mathbb{C}}^{{\mathbb{N}}}\colon |\xi|_n:=\Big(\sum_{j=1}^{+\infty}|\xi_j|^2j^{2n}\Big)^{\frac12}<+\infty\,\,\text{for all}\,\,n\in{\mathbb{N}}\Big\}\]
denote the so-called space of rapidly decreasing sequences. This space becomes Fr\'echet when endowed with the sequence $(|\cdot|_n)_{n\in{\mathbb{N}}}$ of norms defined above.
The basis $(U_n)_{n\in{\mathbb{N}}}$ of zero neighbourhoods of $s$ is defined by $U_n:=\{\xi\in s\colon |\xi|_n\ls1\}$. The topological dual~$s'$ of $s$ is the so-called space of slowly increasing sequences, namely
\[\Big\{\eta=(\eta_j)_{j\in{\mathbb{N}}}\in{\mathbb{C}}^{{\mathbb{N}}}\colon |\eta|_n':=\Big(\sum_{j=1}^{+\infty}|\eta_j|^2j^{-2n}\Big)^{\frac12}<+\infty\,\,\text{for some}\,\,n\in{\mathbb{N}}\Big\}\]
where the duality pairing is given by
$
\langle\xi,\eta\rangle:=\sum_{j\in{\mathbb{N}}}\xi_j\overline{\eta_j}$ for $\xi\in s$, $\eta\in s'$.
The \textit{noncommutative Schwartz space} ${\mathcal{S}}$ is the Fr\'echet space $L(s',s)$ of all continuous linear operators from $s'$ into $s$, endowed with the topology of uniform convergence on bounded sets. %
The formal identity map $\iota\c s\hookrightarrow s'$ is a continuous embedding and defines a product on ${\mathcal{S}}$ by
$xy:=x\circ\iota\circ y$ for $x,y\in{\mathcal{S}}$.
There is also a natural involution on ${\mathcal{S}}$ given by
$\langle x^*\xi,\eta\rangle:=\langle\xi,x\eta\rangle$ for $x\in{\mathcal{S}}$, $\xi,\eta\in s'$.
With these operations, ${\mathcal{S}}$ becomes an m-convex Fr\'echet $*$-algebra. The inclusion map ${\mathcal{S}}\hookrightarrow{\mathcal{K}}(\ell_2)$ is continuous, and in fact it is a spectrum-preserving $*$-homomorphism~\cite{PD}. %
Moreover~\cite[Proposition~3]{KP}, an element $x\in {\mathcal{S}}$ is positive (i.e., $x=y^*y$ for some $y\in {\mathcal{S}}$), if and only if the spectrum of $x$ is contained in~$[0,+\infty)$, or equivalently $\langle x\xi,\xi\rangle\geq 0$ for all $\xi\in s'$.
On the other hand, by~\cite[Cor. 2.4]{PD} %
and~\cite[Theorems~8.2, 8.3]{MF}, the topology of %
${\mathcal{S}}$ %
cannot be given by a sequence of C$^*$-norms. This %
causes some technical inconvenience (e.g.~there is no bounded
approximate identity in~${\mathcal{S}}$) meaning we cannot apply C$^*$-algebraic
techniques directly. %
\section{The inequality}\label{section:2}
Let $(\|\cdot\|_n)_{n\in{\mathbb{N}}}$ be a non-decreasing sequence of norms which gives the topology of~${\mathcal{S}}$. For $u\c{\mathcal{S}}\times{\mathcal{S}}\to{\mathbb{C}}$ a continuous bilinear form, we write
\[\|u\|^*_n:=\sup\{|u(x,y)|\colon x,y\in\mathcal{U}_n\}\]
where $\mathcal{U}_n=\{x\in{\mathcal{S}}\c\,\|x\|_n\ls1\}$; similarly, for a functional $\phi\in {\mathcal{S}}'$, we write
\[ \|\phi\|^*_n:=\sup\{ |\phi(x)|\c\, x\in\mathcal{U}_n\}.\] %
Following
Pisier~\cite[p.~316]{GP2}, for $k\in{\mathbb{N}}$ and $x_1,x_2,\dots,x_m\in {\mathcal{S}}$, we write
\[ \|(x_j)\|_k^{\rc}=\max\Big\{\Big\|\sum_{j=1}^mx_j^*x_j\Big\|_{k}^{\frac12},\Big\|\sum_{j=1}^mx_jx_j^*\Big\|_{k}^{\frac12}\Big\}.\]
Relative to our choice of norms $\|\cdot\|_n$, we
have now defined each term in our hoped-for
inequality~(\ref{target-inequality}). We will now reformulate it using
tensor products.
For C$^*$-algebras, such a reformulation is standard. Indeed, by~\cite[Theorem~1.1]{UH} (formulated along the lines of~\cite[Theorem~2.1]{KS}), Haagerup's noncommutative Grothendieck inequality entails the existence of a $K>0$ such that for any C$^*$-algebras $A,B$ and $z$ in the algebraic tensor product $A\otimes B$, we have $\|z\|_{\pi}\leqslant K\|z\|_{ah}$
where $\|\cdot\|_{\pi}$ is the projective tensor norm and
$\|\cdot\|_{ah}$ is the absolute Haagerup tensor norm~\cite[p.~164]{KS} on~$A\otimes B$, given by
\[ \|z\|_{ah}=\inf\Big\|\sum_{j=1}^m|x_j|^2\Big\|^{\frac12}\Big\|\sum_{j=1}^m|y_j|^2\Big\|^{\frac12}.\]
Here $|x|=\big(\tfrac12(x^*x+xx^*)\big)^{\frac12}$ for $x$ an element of a C$^*$-algebra, and the infimum is taken over all representations $z=\sum_{j=1}^{m} x_j\otimes y_j$ where $(x_j,y_j)\in A\times B$.
We proceed similarly for ${\mathcal{S}}$. For $x\in{\mathcal{S}}$, let
$|x|^2=\tfrac12(x^*x+xx^*)\in {\mathcal{S}}$
and consider the sequence
of \textit{absolute Haagerup tensor norms} %
$(\|\cdot\|_{ah,n})_{n\in{\mathbb{N}}}$ on the algebraic tensor product ${\mathcal{S}}\otimes{\mathcal{S}}$ given by
\[\|z\|_{ah,n}:=\inf\Big\|\sum_{j=1}^m|x_j|^2\Big\|_n^{\frac12}\Big\|\sum_{j=1}^m|y_j|^2\Big\|_n^{\frac12}\]
where the infimum runs over all ways to represent $z=\sum_{j=1}^mx_j\otimes y_j$ in ${\mathcal{S}}\otimes{\mathcal{S}}$. As usual, we write $(\|\cdot\|_{\pi,n})_{n\in{\mathbb{N}}}$ for the sequence of projective tensor norms on ${\mathcal{S}}\otimes {\mathcal{S}}$.
Just as in the C$^*$-algebra case, inequality \eqref{target-inequality} will follow once we show that the sequences of projective and absolute Haagerup tensor norms are equivalent on ${\mathcal{S}}\otimes{\mathcal{S}}$. In fact, the equivalence of these norms follows immediately from the nuclearity of~${\mathcal{S}}$ (see~\cite[Theorem~28.15]{MV} and~\cite[Ch. 21, \S2, Theorem~1]{J} for details). On the other hand, the optimal values of $k$ and~$K$ (depending on~$n$ and our choice of norms~$(\|\cdot\|_n)_{n\in{\mathbb{N}}}$) for which~\eqref{target-inequality} hold are not given by such general considerations. These optimal parameters will be denoted by $\kappa(n):=k_{best}$ and $K_n:=K_{best}$.
Henceforth, we focus only on the sequence of norms $(\|\cdot\|_n)_{n\in{\mathbb{N}}}$ where
\[\|x\|_n:=\sup\{|x\xi|_n\colon \xi\in U^{\circ}_n\},\qquad n\in{\mathbb{N}},\ x\in{\mathcal{S}}\]
and $U_n^{\circ}=\{\xi\in s'\c\,\,|\xi|_n'\ls1\}$. In other words, $\|x\|_n$ is the norm of~$x\in {\mathcal{S}}$, considered as a Hilbert space operator from $H_n':=\ell_2((j^{-n})_j)$ to $H_n:=\ell_2((j^n)_j)$.
This sequence does indeed induce the topology of~${\mathcal{S}}$.
In this context, we will estimate~$K_n$ and compute the exact values of~$\kappa(n)$.
We start with the following result, which can be compared with~\cite[Lemma~8]{KP}. To fix some useful notation, for $n\in\mathbb N$ we define an infinite diagonal matrix $d_n:=\operatorname{diag}(1^n,2^n,3^n,4^n,\dots)$\label{dn-def} which we consider as an isometry $d_n\c\ell_2\to H_n' $ and simultaneously as an isometry $d_n\c H_n\to\ell_2$.
\goodbreak
\begin{prop}
\label{inequalities}
Let~$n\in\mathbb N$. We have
\begin{enumerate}
\item[(i)]
$\|x\|_n=\sup\{\langle x\xi,\xi\rangle\colon \xi\in U^\circ_n\}$
for every positive $x\in{\mathcal{S}}$;
\item[(ii)] $\|x\|_n^2\leqslant\|x^2\|_{2n}$ for every self-adjoint $x\in{\mathcal{S}}$; and
\item[(iii)] $\|x\|_n^2\leqslant\|x^*x\|_{2n}^{\frac12}\|xx^*\|_{2n}^{\frac12}$ for every $x\in{\mathcal{S}}$.
\end{enumerate}
Moreover, inequalities~(ii) and~(iii) are sharp.
\end{prop}
\begin{proof} (i) Observe that $\|x\|_n=\|d_nxd_n\|_{{\mathcal{B}}(\ell_2)}$. Furthermore, since~$x$ is positive, $d_nxd_n$ is positive %
and we have
\begin{align*}
\|x\|_n & =\|d_nxd_n\|_{{\mathcal{B}}(\ell_2)}
=\sup\{\langle xd_n\xi,d_n\xi\rangle\colon |\xi|_{\ell_2}\ls1\} \\
&=\sup\{\langle x\xi,\xi\rangle\colon |\xi|_n'\ls1\}.
\end{align*}
(ii) For $x$ self-adjoint, we have \[\|x^2\|_{2n}=\|d_{2n}x^2d_{2n}\|_{{\mathcal{B}}(\ell_2)}=\|d_{2n}x\|^2_{{\mathcal{B}}(\ell_2)},\]
and by~\cite[Proposition~II.1.4.2]{BB},
\[\|x\|_n=\|d_nxd_n\|_{{\mathcal{B}}(\ell_2)}=\nu(d_nxd_n)=\nu(d_{2n}x)\leqslant\|d_{2n}x\|_{{\mathcal{B}}(\ell_2)},\]
where $\nu(\cdot)$ denotes the spectral radius. This gives the desired inequality.
(iii) Since ${\mathcal{S}}\hookrightarrow{\mathcal{B}}(\ell_2)$, any $x\in {\mathcal{S}}$ is also a Hilbert space operator, and the block-matrix operator
$\left[\begin{smallmatrix}
(xx^*)^{1/2} & x \\
x^* & (x^*x)^{1/2}
\end{smallmatrix}\right]$
is positive in ${\mathcal{B}}(\ell_2\oplus\ell_2)$ (see e.g.%
~\cite[p.~117]{VP}). Equivalently,
\begin{equation}
|\langle x\xi,\eta\rangle|^2\leqslant\langle(xx^*)^{1/2}\eta,\eta\rangle\langle(x^*x)^{1/2}\xi,\xi\rangle\qquad\forall\,\xi,\eta\in\ell_2.
\label{positive-ineq1}
\end{equation}
For $m\in{\mathbb{N}}$, let us write $p_m:=\left[\begin{smallmatrix}
I_m & 0 \\0 & 0
\end{smallmatrix}\right]
$
where $I_m\in{\mathcal{M}}_m$ is the identity matrix. Now fix $n\in{\mathbb{N}}$ and choose $\xi,\eta\in H_n'$. Then $p_m\xi,p_m\eta\in\ell_2$ for all $m\in{\mathbb{N}}$ and \eqref{positive-ineq1} gives
\[|\langle p_mxp_m\xi,\eta\rangle|^2\leqslant\langle p_m(xx^*)^{1/2}p_m\eta,\eta\rangle\langle p_m(x^*x)^{1/2}p_m\xi,\xi\rangle.\]
Since $(p_m)_{m\in{\mathbb{N}}}$ is an approximate identity in ${\mathcal{S}}$ (see~\cite[Proposition~2]{KP}), we obtain
\[|\langle x\xi,\eta\rangle|^2\leqslant\langle(xx^*)^{1/2}\eta,\eta\rangle\langle(x^*x)^{1/2}\xi,\xi\rangle.\]
Taking the supremum over all $\xi,\eta$ in the unit ball of $H_n'$ we get
\[\|x\|_n^2\leqslant\|(xx^*)^{1/2}\|_n\|(x^*x)^{1/2}\|_n.\]
Applying (ii) to the positive operators $(xx^*)^{1/2}$ and $(x^*x)^{1/2}$ we conclude that
$\|x\|_n^2
\leqslant\|x^*x\|_{2n}^{\frac12}\|xx^*\|_{2n}^{\frac12}$.
For sharpness, observe that if~$x$ is a diagonal rank one matrix unit then we have equality in both~(ii) and~(iii).
\end{proof}
\begin{prop}
\label{prop-fourth-root}
For any $n,m\in{\mathbb{N}}$ and $x_1,\ldots,x_m,y_1,\ldots,y_m\in{\mathcal{S}}$, we have
\begin{multline*}
\sum_{j=1}^m\|x_j\|_n\|y_j\|_n \\
\leqslant\frac{\pi^2}{6}\Big\|\sum_{j=1}^mx_j^*x_j\Big\|_{2n+1}^{\frac14}
\Big\|\sum_{j=1}^mx_jx_j^*\Big\|_{2n+1}^{\frac14}
\Big\|\sum_{j=1}^my_j^*y_j\Big\|_{2n+1}^{\frac14}
\Big\|\sum_{j=1}^my_jy_j^*\Big\|_{2n+1}^{\frac14}.
\end{multline*}
\end{prop}
\begin{proof}
Let $C:=\frac{\pi^2}{6}$ and let~$p\in {\mathbb{N}}$. We claim that
\[
\sum_{k=1}^m\|x_k^*x_k\|_p\leqslant C\Big\|\sum_{k=1}^mx_k^*x_k\Big\|_{p+1}.
\]
By the Cauchy--Schwarz inequality and Proposition~\ref{inequalities}(iii) this will then imply the desired inequality. To establish the claim, %
let $\xi^1,\ldots,\xi^m\in U_p^{\circ}$ and let us write $(e_{j})_{j\in{\mathbb{N}}}$ for the standard basis vectors in~$\ell^2$. We have
\[\sum_{k=1}^m\langle x_k^*x_k\xi^k,\xi^k\rangle=\sum_{k=1}^m\sum_{i,j=1}^{+\infty}\langle x_k^*x_ke_j,e_i\rangle (ij)^p\overline{\xi_i^k}i^{-p}\xi_j^kj^{-p}.\]
Applying the Cauchy--Schwarz inequality to summation over $i,j\in{\mathbb{N}}$ gives
\[\sum_{k=1}^m\langle x_k^*x_k\xi^k,\xi^k\rangle\leqslant\sum_{k=1}^m\Bigl(\sum_{i,j=1}^{+\infty}|\langle x_k^*x_ke_j,e_i\rangle|^2(ij)^{2p}\Bigr)^{\frac12}.\]
Since $x^*x$ is positive for any $x\in{\mathcal{S}}$, and for positive operators $y\in{\mathcal{S}}$ we have $|y_{ij}|^2\leqslant y_{ii}y_{jj}$ (where $y_{ij}:=\langle ye_j,e_i\rangle$), this implies that
\begin{align*}
\sum_{k=1}^m\langle x_k^*x_k\xi^k,\xi^k\rangle & \leqslant\sum_{j=1}^{+\infty}\Big\langle\Bigl(\sum_{k=1}^mx_k^*x_k\Bigr)j^pe_j,j^pe_j\Big\rangle \\
& \leqslant\sum_{j=1}^{+\infty}j^{-2}\sup_{i\in{\mathbb{N}}}\Big\langle\Bigl(\sum_{k=1}^mx_k^*x_k\Bigr)i^{p+1}e_i,i^{p+1}e_i\Big\rangle \\
& \leqslant C\Big\|\sum_{k=1}^mx_k^*x_k\Big\|_{p+1}.
\end{align*}
By Proposition~\ref{inequalities}(i), for any ${\varepsilon}>0$ there are $\xi^1,\ldots,\xi^m\in U_p^{\circ}$ with
\[
\sum_{k=1}^m\|x_k^*x_k\|_p<\sum_{k=1}^m\langle x_k^*x_k\xi^k,\xi^k\rangle+{\varepsilon}<C\Big\|\sum_{k=1}^mx_k^*x_k\Big\|_{p+1}+{\varepsilon}.\]
Taking the infimum over ${\varepsilon}>0$ yields the claim.
\end{proof}
\goodbreak
As a straightforward consequence of Proposition~\ref{prop-fourth-root}, we obtain:
\begin{theorem}[Grothendieck's inequality in~${\mathcal{S}}$]
\label{gt-s}
There is a constant $K\leqslant\frac{\pi^2}{6}$ such that $\|z\|_{\pi,n}\ls2K\|z\|_{ah,2n+1}$ for any $n\in{\mathbb{N}}$ and $z\in{\mathcal{S}}\otimes{\mathcal{S}}$. Moreover,
every continuous bilinear form $u\c{\mathcal{S}}\times{\mathcal{S}}\to{\mathbb{C}}$ satisfies inequality~(\ref{target-inequality}) with $k=2n+1$, for any $n,m\in {\mathbb{N}}$ and any $x_1,\ldots,x_m,y_1,\ldots,y_m\in{\mathcal{S}}$. %
In particular, taking $u(x,y):=\phi(x)\phi(y)$ where $\phi\in{\mathcal{S}}'$, we obtain
\begin{equation}
\sum_{j=1}^m|\phi(x_j)|^2\leqslant K(\|\phi\|_n^*\,\|(x_j)\|^{\rc}_{2n+1})^2%
\label{gt-functional}
\end{equation}
\end{theorem}
\begin{remark}
This shows that
$\kappa(n)\ls2n+1$. On the other hand, it is easy to show that
$\kappa(n)>2n-1$. Indeed, if not, then~\eqref{gt-functional} would hold with
$2n+1$ replaced by some ${\ell}\leqslant 2n-1$. Take $m\in{\mathbb{N}}$, define
$\xi_m:=\sum_{j=1}^mj^ne_j$ and $\phi_m\in{\mathcal{S}}'$ by $\phi_m(x):=\langle
x\xi_m,\xi_m\rangle$. Then for $x_j:=e_{jj},\,j=1,\ldots,m$ we get $\|(x_j)\|^{\rc}_\ell=m^\ell$
and
$\|\phi_m\|_n^*=m$. On the other hand, $\sum_{j=1}^m|\phi_m(x_j)|^2$
is equivalent (up to a constant) to $m^{4n+1}$. Therefore
\eqref{gt-functional} takes the form $m^{4n+1}\leqslant Cm^{2\ell+2}$ for some constant $C$ (independent of
$m$). Letting $m$ tend to infinity, we obtain ${\ell}\gs2n-\frac12$,
a contradiction. Hence $\kappa(n)\in \{2n,2n+1\}$.
\end{remark}
\section{Optimality}
We will now show that $\kappa(n)=2n+1$. For this, we will use the tensor product formulation, noting that
\[\kappa(n)=\min\Big\{k\in{\mathbb{N}}\colon \sup\Big\{\frac{\|z\|_{\pi,n}}{\|z\|_{ah,k}}\colon z\in{\mathcal{S}}\otimes{\mathcal{S}},\,z\neq0\Big\}<\infty\Big\}.\]
Recall the diagonal operator $d_n$ defined on page~\pageref{dn-def}
above. Since every $x\in{\mathcal{S}}$ is an operator on $\ell_2$ via the
canonical inclusions $\ell_2\stackrel{d_n}\hookrightarrow H_n'\xrightarrow{x}H_n\stackrel{d_n}\hookrightarrow\ell_2$,
it is clear that if $x\in{\mathcal{S}}$, then $d_nx$ and $xd_n$ are both operators on $\ell_2$. This leads to the following observation.
\begin{prop}
\label{elemntary-equalities}
If $z=\sum_{j=1}^kx_j\otimes y_j\in{\mathcal{S}}\otimes{\mathcal{S}}$, then
\[\|z\|_{\pi,n}=\Big\|\displaystyle\sum_{j=1}^kd_nx_jd_n\otimes d_ny_jd_n\Big\|_{\pi}.\]
\end{prop}
\begin{proof}
Write \[\Delta_nz=\sum_{j=1}^kd_nx_jd_n\otimes d_ny_jd_n\in{\mathcal{B}}(\ell_2)\otimes{\mathcal{B}}(\ell_2).\] If $\Delta_nz=\sum_{l=1}^ma_l\otimes b_l\in{\mathcal{B}}(\ell_2)\otimes{\mathcal{B}}(\ell_2)$ and $\sum_{l=1}^m\|a_l\|\|b_l\|<\|\Delta_nz\|_{\pi}+{\varepsilon}$ for some ${\varepsilon}>0$, then $z=\sum_{l=1}^md_n^{-1}a_ld_n^{-1}\otimes d_n^{-1}b_ld_n^{-1}$ and
\[
\|z\|_{\pi,n} \leqslant\sum_{l=1}^m\|d_n^{-1}a_ld_n^{-1}\|_n\|d_n^{-1}b_ld_n^{-1}\|_n
=\sum_{l=1}^m\|a_l\|\|b_l\|<\|\Delta_n z\|_{\pi}+{\varepsilon}.
\]
This gives $\|z\|_{\pi,n}\leqslant\|\Delta_nz\|_{\pi}$. The reverse inequality is proved similarly.
\end{proof}
We also need the following well-known fact.
\begin{prop}
\label{self-adjoint-haagerup}
If $\H$ is a Hilbert space and $x_1,\ldots,x_m\in{\mathcal{B}}(\H)$, then
\[\Big\|\sum_{j=1}^mx_j\otimes x_j^*\Big\|_h=\Big\|\sum_{j=1}^mx_jx_j^*\Big\|.\]
\end{prop}
\begin{proof}
By~\cite[Theorem~4.3]{RRS}, the Haagerup norm on the left hand side is equal to the completely bounded norm of the map on~${\mathcal{B}}(\H)$ given by~$a\mapsto \sum_{j=1}^m x_jax_j^*$, which is completely positive, so attains its completely bounded norm at the identity operator.
\end{proof}
\begin{theorem}%
For every $n\in{\mathbb{N}}$, we have $\kappa(n)=2n+1$.
\end{theorem}
\begin{proof}
By Theorem~\ref{gt-s}, it only remains to show that $\kappa(n)>2n$. %
Choose $k_n\in{\mathbb{N}}$ sufficiently large that
$k+1\ls2^k(k^{\frac{1}{4n}}-1)$ for all $k\geqslant k_n$.
This inequality ensures that for every $k\geqslant k_n$, if we define
\[i_1=2^k,\quad i_{k+1}=\lfloor k^{\frac{1}{4n}}2^k\rfloor,\quad
i_j=i_{k+1}+j-(k+1),\ 2\leqslant j\leqslant k,\] then
$i_1<i_2<\dots<i_{k+1}$. %
Denote by $(e_{ij})_{i,j\in{\mathbb{N}}}$ the standard matrix units, and for
$j=2,\ldots,k+1$, consider the self-adjoint operators
\[x_j:=e_{i_1,i_j}+e_{i_j,i_1}\in\mathcal{M}_{i_{k+1}}\subset{\mathcal{S}}.\]
Let
\[z_k:=\sum_{j=2}^{k+1}x_j\otimes x_j.\] Since
$d_nx_jd_n=i_1^ni_j^n(e_{i_1,i_j}+e_{i_j,i_1})$ and
$(d_nx_jd_n)^2=i_1^{2n}i_j^{2n}(e_{i_1,i_1}+e_{i_j,i_j})$,
by Propositions~\ref{elemntary-equalities} and~\ref{self-adjoint-haagerup} we obtain
\begin{align*}
\|z_k\|_{\pi,n} & =\Big\|\sum_{j=2}^{k+1}d_nx_jd_n\otimes d_nx_jd_n\Big\|_{\pi} \\
& \geqslant\Big\|\sum_{j=2}^{k+1}d_nx_jd_n\otimes d_nx_jd_n\Big\|_h
=\Big\|\sum_{j=2}^{k+1}(d_nx_jd_n)^2\Big\|
=i_1^{2n}\sum_{j=2}^{k+1}i_j^{2n}.
\end{align*}
On the other
hand, \[|x_j|^2=x_j^2=e_{i_1,i_1}+e_{i_j,i_j}\quad\text{and}\quad
\sum_{j=2}^{k+1}d_{2n}x_j^2d_{2n}=i_1^{4n}k
e_{i_1,i_1}+\sum_{j=2}^{k+1}i_j^{4n}e_{i_j,i_j}.\]
Therefore %
\begin{align*}
\|z_k\|_{ah,2n} &\leqslant\Big\|\sum_{j=2}^{k+1}|x_j|^2\Big\|_{2n}
=\Big\|\sum_{j=2}^{k+1}d_{2n}x_j^2d_{2n}\Big\|
=\max\{i_1^{4n}k,i_{k+1}^{4n}\} \\
& \leqslant i_1^{4n}k+i_{k+1}^{4n}.
\end{align*}
Hence
\begin{align*}
\frac{\|z_k\|_{\pi,n}}{\|z_k\|_{ah,2n}}>\frac{i_1^{2n}\sum_{j=2}^{k+1}
{i_j^{2n}}}{i_1^{4n}k+i_{k+1}^{4n}}
>
\frac{i_1^{-2n}i_2^{2n}}{1+k^{-1}i_1^{-4n}i_{k+1}^{4n}}\to \infty \text{ as $k\to \infty$},
\end{align*}
by our choice of $i_1,\dots,i_{k+1}$. So $\kappa(n)>2n$ as required.
\end{proof}
\section{Reformulations of the inequality}
Here we give several different ways of stating our inequality; in each case, an analogous result may be found in~\cite{GP}. The methods here are fairly standard, so full proofs are often omitted. Throughout, we write $K=\sup_{n\in{\mathbb{N}}}K_n\leqslant \pi^2/6$.
\subsection{Grothendieck's inequality with states}
Given $\xi\in U_n^{\circ}$, let
$\phi_{\xi}\in {\mathcal{S}}'$ be given by $\phi_\xi(x)=\langle
x\xi,\xi\rangle$, $x\in {\mathcal{S}}$. We call an element of the closed convex hull
of $\{ \phi_\xi\colon \xi\in U_n^\circ\}$ an %
\textit{$n$-state} on~${\mathcal{S}}$. %
Note that by Proposition~\ref{inequalities}(i), for any positive element $x\in {\mathcal{S}}$ we have
$\|x\|_n=%
\sup\{\phi(x)\colon \phi\in V_n\}$,
where $V_n\subseteq {\mathcal{S}}'$ is the set of all $n$-states on~${\mathcal{S}}$.
The next result may be deduced from Theorem~\ref{gt-s} by closely following the Hahn--Banach Separation argument of~\cite[\S23]{GP}.
\begin{theorem}
\label{gt-s-states}
For any continuous bilinear form $u\c{\mathcal{S}}\times{\mathcal{S}}\to{\mathbb{C}}$ and $n\in{\mathbb{N}}$, there are $(2n+1)$-states $\phi_1,\phi_2,\psi_1,\psi_2$ on ${\mathcal{S}}$ with
\[|u(x,y)|\leqslant K\|u\|_n^*\bigl(\phi_1(x^*x)+\phi_2(xx^*)\bigr)^{\frac12}\bigl(\psi_1(y^*y)+\psi_2(yy^*)\bigr)^{\frac12}\] for all $x,y\in{\mathcal{S}}$.
\end{theorem}
\subsection{`Little' Grothendieck inequality}
As a consequence we obtain the following `little' Grothendieck inequality in~${\mathcal{S}}$. Recall that if $T\c X\to Y$ is a linear %
map between Fr\'echet spaces, then $\|T\|_{n,k}:=\sup\{\|Tx\|_k\colon \|x\|_n\ls1\}$.
\begin{theorem}
For any Fr\'echet-Hilbert space~$H$, if $u_1,u_2\c{\mathcal{S}}\to H$ are continuous linear maps, $k,m,n\in{\mathbb{N}}$ and $x_1,\ldots,x_m,y_1,\ldots,y_m\in{\mathcal{S}}$, then
\begin{equation*}
\Big|\sum_{j=1}^m\langle u_1(x_j),u_2(y_j)\rangle_k\Big|\leqslant K\|u_1\|_{n,k}\,\|u_2\|_{n,k}\,\|(x_j)\|_{2n+1}^{\rc}\,\|(y_j)\|_{2n+1}^{\rc}.%
\end{equation*}
Equivalently, for any $k,n\in{\mathbb{N}}$ there are $(2n+1)$-states $\phi_1,\phi_2,\psi_1,\psi_2$ such that for all $x,y\in{\mathcal{S}}$ we have
\begin{multline*}|\langle u_1(x),u_2(y)\rangle_k|\leqslant K\|u_1\|_{n,k}\|u_2\|_{n,k}\\
\times \bigl(\phi_1(x^*x)+\phi_2(xx^*)\bigr)^{\frac12}\bigl(\psi_1(y^*y)+\psi_2(yy^*)\bigr)^{\frac12}.
\end{multline*}
\end{theorem}
\begin{proof}
Apply Theorems~\ref{gt-s} and \ref{gt-s-states} to $u_k(x,y):=\langle u_1(x),u_2(y)\rangle_k$ for $k\in{\mathbb{N}}$.
\end{proof}
Using the same argument as in the proof of Theorem~\ref{gt-s-states} we can obtain an equivalent version of the `little' Grothendieck inequality.
\begin{theorem}
For any Fr\'echet-Hilbert space $H$, if $u\c{\mathcal{S}}\to H$ is a continuous linear map and $k,n\in{\mathbb{N}}$, then there exist $(2n+1)$-states $\phi_1,\phi_2$ on ${\mathcal{S}}$ such that for all $x\in{\mathcal{S}}$ we have
\[\|ux\|_k\leqslant\sqrt{K}\|u\|_{n,k}\big(\phi_1(x^*x)+\phi_2(xx^*)\big)^{\frac12}.\]
\end{theorem}
\renewcommand{\bibliofont}{\normalsize\baselineskip=17pt}
\bibliographystyle{plain}
|
2,869,038,154,732 | arxiv | \section{Introduction}
The primordial inflation~\cite{Guth:1980zm} is perhaps one of the most important
paradigms of the early Universe, for a recent review,
see~\cite{Mazumdar:2010sa}. Inflation
is responsible for stretching the initial perturbations to the
observable scales in
the cosmic microwave background radiation (CMBR), and seeding the
initial perturbations for the large scale structure
formation~\cite{WMAP}. Since the future observational constraints will
provide a better understanding of the inflationary dynamics and its
potential, it is then important to reach
the desired level of accuracy by studying higher order quantum
corrections to the
cosmological perturbations. For a review on cosmological perturbation
theory, see~\cite{Mukhanov:1990me}.
The correlators at loop level in an inflationary setup appear to be
plagued by infrared (IR) divergences for {\it soft} quantum
modes whose wavelengths are extremely large, known as the super Hubble
fluctuations. The divergences appear when these fluctuations are
summed over in the loops \cite{IRLiterature, Weinberg:2005vy, Seery:2010kh, Riotto:2008mv,
Dimastrogiovanni:2008af, Burgess:2009bs, Giddings:2010nc,
Burgess:2010dd, Koivisto:2010pj}. Regularizing the
integrals via IR cutoffs does not solve the problem, but turns
the divergences into large {\it Logarithmic} corrections depending on
the cutoff (``box size''). There has been a debate about the question
if such correction are physical or not~~\footnote{ There are attempts to address the IR issue
by studying the pre inflationary phase in the early Universe, which
modifies the long wavelength
behavior of the solutions of the field equations, thus ameliorating or
even canceling the IR divergences
\cite{Koivisto:2010pj,Marozzi:2011da}. However,
this approach has a certain degree of arbitrariness, depending on the
choice of a specific pre-de Sitter scenario. For example, a possible
solution is to
evolve the perturbations from a contracting phase to the expanding
phase in a singularity free bouncing cosmology, where gravity becomes
asymptotically free in the ultra violet regime~\cite{Biswas:2005qr}.
Modification of the perturbation equations capable of ameliorating
the IR issue (as they affect the spectral index) can arise also
in models such as chain inflation, due to the interactions among the different
components of the system
\cite{Burgess:2005sb,Chialva:2008zw}.}.
Typically, the IR divergences are a signal of an ill-posed physical
question, and therefore their
resolution depends on our understanding of the physical system and the
approach we adopt to tackle the problem. For instance, an unphysical
element in the approach which leads to the divergence could be due to
an erroneous definition of an initial and final vacuum in a scattering
process, as it happens in the case of soft photons or
gluons emitted as a result of any quantum
process~\cite{Weinberg:1965nx}. The IR divergences
may also arise if the perturbation theory has been wrongly organized
without taking into account the relevant scales that make some contributions
unsuppressed, as for the case of field theory at finite temperatures~\cite{LeBellac}.
Another point of view on this problem has instead considered
the dependence on the box size (IR cutoff) of the
correlators as a physical input. The question then arises -- whether the
correlators, depending on the box size, had then to be averaged over
a distribution of boxes partitioning the Universe on super Hubble scales,
or if the correct physical interpretation would be to fix the box size to the desired
scale of observation and keep the
Log-enhancements~\footnote{\label{criticdecompcutoff} This also
seems to be the point in Ref.~\cite{Gerstenlauer:2011ti, Byrnes:2010yc}, as the
authors distinguish between IR and non-IR perturbations with respect to some
typical observer's scales $q_0$ and $L^{-1}$, and therefore the spectrum which they
claim to be IR-safe would then depend on this cut-off procedure. We
comment more on this in section \ref{amendingprocedure}.}.
Both these approaches have negative aspects -- the first one does not take
account of the fact that the observer is not capable of averaging
over boxes larger than its Hubble patch, while the second approach still
suffers from the ambiguity on how to define the box size (for example,
via comoving coordinates or physical ones), which therefore makes the
result for the correlators ambiguous.
In this paper, we will point out that the issue of IR divergences in
gravitational loops in the
cosmological correlators can be solved
by defining a {\it local observer} who is responsible for measuring
the observable quantities. We
will argue that once the observer and
observables are well defined, the IR divergences will turn out to be
an artifact of some unphysical assumptions,
usually taken in the definition of the perturbation theory.
Our implementation of the principle of locality will be different from
what discussed in other studies in the literature~\footnote{The issue
of locality has been discussed, with a different approach, also in
\cite{Urakawa:2009my, Urakawa:2010kr}. Their proposal, however,
entails considering non-standard gauge transformations that are
singular at large scales and results in
new gauge conditions \cite{Urakawa:2009my} or in the necessity of
using perturbation fields {\em different} than the usual curvature
and tensor ones, with a prescribed set of boundary conditions,
in order to avoid the IR divergences \cite{Urakawa:2010kr}.}.
In fact, our resolution is to redefine the standard perturbation theory in
the in-in formalism based on one physical principle:
\begin{itemize}
\item {\it any observable quantity should be defined in terms of a local observer
who is measuring those observables}.
\end{itemize}
As a result, the observable quantities are shown not to have IR divergences,
nor to depend on the IR cutoffs.
The paper is organized as follows; in section \ref{Ininformalism},
we start with a brief introduction of the standard in-in formalism and the
quantization of the perturbations. We then provide a general discussion of the
IR behavior of loop correlators in section \ref{GeneralBehaviour},
using the example of self-energy diagrams, and
discuss the ambiguities in the cutoff procedure for regularizing the integrals.
We present our resolution of the IR issue in section
\ref{physicalproposal}. First, in section \ref{conceptualpoint}, we
discuss the nature of the true observables and how the in-in
formalism, as it stands, fails to account for them. Then, in
\ref{IRallorders} we show
that all IR divergences at all orders arise only from certain specific
contributions to the correlators, and can be traced back to the
definition of the correlators using the background unperturbed metric;
finally we discuss how to eliminate the IR divergences by
amending the standard in-in computations in section
\ref{amendingprocedure}, and provide a detailed
example at one-loop for a specific case in section
\ref{oneloopexample}. Finally, we will briefly discuss
the non-Gaussianities in section \ref{Npointfunctions} and conclude in
section \ref{conclusion}.
\section{Basic formalism}\label{Ininformalism}
The appearance of large IR corrections in the
perturbation theory in an inflationary background is directly linked to
the presence of super-Hubble fluctuations of light
fields~\cite{Linde:1983gd,Linde:1986fd,Vilenkin:1982wt,Vilenkin:1983xq,Linde:1993xx,Linde:2005ht},
it is therefore not related to any specific property of the field,
i.e. graviton or scalar field, running
inside the loops. Here, we will briefly introduce the basics of
perturbation theory as it
is usually formulated using the in-in formalism and the path integral
quantization, for a review, see~\cite{Weinberg:2005vy}; at the same time, we
list our conventions.
The cosmological perturbation theory begins with the
definition of a threading and a slicing of a spacetime (foliation) in
the unperturbed background metric,
which is fully homogeneous and isotropic. Then
all the fields (including the metric) are written distinguishing the background
and the perturbations according to the chosen threading and slicing,
for a review, see~\cite{Weinberg:2005vy}.
The background is considered to be classical, while the perturbation
fields are quantized.
Let us consider a scalar field $\Phi$ (similar procedure will follow for tensor fields,
gravitons, once a polarization tensor is specified), and write it in
terms of the background $\phi_0(t)$ and the
perturbations $\phi(\vec x, t)$ as:
\begin{equation} \label{Backgroundandperturbations}
\Phi(\vec x, t) = \phi_0(t) + \phi(\vec x, t) \, .
\end{equation}
The field $\phi(\vec x, t)$ is expanded on a basis of eigenfunctions
$Y_{\vec k}(\vec x)$ of the
Laplace-Beltrami operator $\nabla^2$ with eigenvalues $-k^2$. Our conventions
are: $k$ is the comoving wavenumber related to the physical
momentum $p$ by
\begin{equation}\label{co-phys}
k=ap\,,~~
\end{equation}
and the conformal time is defined via the background metric
\begin{equation}
ds^2= -dt^2+a^2\delta_{ij}dx^i dx^j.
\end{equation}
as
\begin{eqnarray}
d\eta =a^{-1} \, dt\,,
\end{eqnarray}
where $a$ is the scale factor. We set the reduced Planck mass,
$M_\text{P}^2=1$. The field can then be promoted to a quantum
field, written as~\footnote{We will denote quantum fields, both fundamental and
composite, with $\widehat{~~}$\,.}:
\begin{equation} \label{fieldexpansion}
\widehat \phi(\vec{x}, \eta) = \int {d^3k \ov (2\pi)^{{3 \ov 2}}} \left[Y_{\vec{k}}(\vec{x}) u_k(\eta)
\widehat a_{\vec{k}} +Y_{\vec{k}}^*(\vec{x}) u_k^*(\eta) \widehat a_{\vec{k}}^\dagger \right]\,,
\end{equation}
and quantized in the usual way,
\begin{equation}
[\widehat a_{\vec{k}}^\dagger,~ \widehat a_{\vec{k}'}] = \delta(\vec k-\vec k') \, .
\end{equation}
The Whightman function is defined as:
\begin{equation} \label{WhightmanfunctionModeexpansionState}
W(\vec{x}, \eta, \vec{x}', \eta') = \langle \Omega| \widehat\phi(\vec{x}, \eta)\widehat\phi(\vec{x}', \eta')|\Omega\rangle
= \int {d^3k \ov (2\pi)^{3}} Y_{\vec{k}}(\vec{x})Y_{\vec{k}}^*(\vec{x})u_k(\eta)u_k^*(\eta') \, ,
\end{equation}
where $|\Omega\rangle $ is the vacuum state.
For convenience, we adopt planar coordinates, i.e.
$Y_{\vec{k}}(\vec{x}) = e^{i \vec{k} \cdot \vec{x}}$, and, as a shorthand
notation, we call Whightman function also
\begin{equation} \label{GenWhightman}
W_k(\eta, \eta') = u_k(\eta)u_k^*(\eta') \, .
\end{equation}
The power spectrum, $P_{k}$, is related to the Whightman function. In
particular, at late times
\begin{equation} \label{spectrumtree}
P_{k} = {k^3 \ov 2\pi^2} \lim_{\eta \to 0} W_k(\eta, \eta) \, .
\end{equation}
The mode functions which are entering the expansion of the field are such that $u_k(\eta) = {\mu_k / z}$,
where $\mu_k$ is the solution of the field equation:
\begin{equation} \label{modeequation}
\mu_k'' +\left(k^2- {z'' \ov z}\right)\mu_k=0.
\end{equation}
The quantity, ${z'' / z}$, depends on the background, it is often
called the gravitational source term for
the seed perturbations~\cite{Mukhanov:1990me}.
In a (quasi) de Sitter case, at leading order in the slow-roll parameters, the
general solution of the mode equation is given by:
\begin{equation} \label{modesolution}
u_k(\eta) = {\mu_k \ov z}=
c_1{\sqrt{-\pi \eta} \ov 2z}H^{(1)}_\nu(-k\eta)+c_2{\sqrt{-\pi \eta} \ov 2z}H^{(2)}_\nu(-k\eta) \, .
\end{equation}
where $H^{(1,~2)}$ are the Hankel functions. The flat space solution
(Bunch-Davies vacuum) is recovered for $\eta \to -\infty$, if $c_1=
e^{i\left(\nu + {1 \ov 2}\right){\pi \ov 2}}, \, c_2=0$.\\
In the in-in formalism, the expectation value of any operator
$\widehat{\mathcal{O}}$ at a given time $\eta_0$ is given by:
\begin{equation} \label{ininamplitudes}
\langle\Omega| \widehat{\mathcal{O}}(\eta_0) |\Omega\rangle =
\langle\Omega|\bar{T}\left(e^{i\int_{-\infty}^{\eta_0}d\eta H_I}\right) \widehat{\mathcal{O}}(\eta_0) T\left(e^{-i\int_{-\infty}^{\eta_0}d\eta H_I}\right)|
\Omega\rangle\,.
\end{equation}
In the case of the two-point function, expanding in the interaction Hamiltonian leads to:
\begin{equation} \label{twopointwithcorrections}
\mathcal{A}(k_1, k_2, \eta) =
\mathcal{A}_{\text{tree}}(k_1, k_2, \eta)+
\mathcal{A}_{\text{loops}}(k_1, k_2, \eta)\,,
\end{equation}
where the correlators, ${\cal A}$, depend on $k_i^2$ and indices are
contracted using the background metric. The time $\eta$ is also
defined by the unperturbed metric.
\section{Infrared divergences}
\label{GeneralBehaviour}
We discuss here the general features of IR divergences using the example of
the lowest-order gravitational loop diagrams that we are typically
interested in: the
self energy diagrams, such as those given
by the cubic interaction vertices, and the ``bubble'' diagram. We present
these two diagrams in Figs.~(\ref{fig1}) and (\ref{fig2}). The fields running in
the loop can be different, however a generic theory with gravitation
will involve a graviton loop.
\begin{figure
\begin{center}
\includegraphics[width=4cm]{bubblesssg.eps}
\end{center}
\caption{Correction to the scalar two point function from an
intermediate scalar and graviton.}
\label{fig1}
\begin{center}
\includegraphics[width=4cm]{seagullssgg.eps}
\end{center}
\caption{Correction to the scalar two point function from a graviton
bubble.}
\label{fig2}
\end{figure}
The computation of these diagrams has always been
performed in a pure de Sitter space~\cite{IRLiterature,
Weinberg:2005vy, Seery:2010kh, Riotto:2008mv,
Dimastrogiovanni:2008af, Burgess:2009bs,
Giddings:2010nc,Burgess:2010dd, Koivisto:2010pj}. However, some authors
have also discussed the case of a quasi de Sitter background
\cite{Giddings:2010nc,Koivisto:2010pj}. In order to discuss the IR
divergences, we will not need to know the
whole expression for these diagrams. Those can be found, for example
in Refs.~\cite{Riotto:2008mv,Dimastrogiovanni:2008af,Burgess:2009bs,Giddings:2010nc}.
Furthermore, our observations are largely independent of de Sitter or
quasi de Sitter. Let us briefly list here the main
differences. In a pure de Sitter case the scalar metric
perturbations are pure gauge, and the only physical perturbations are
the tensor ones, i.e. that of the gravitons. In
Eqs.~(\ref{modeequation},~\ref{modesolution}), $\nu$ becomes equal to
${3 / 2}$ in a pure de Sitter case.
In a quasi de Sitter case the de Sitter {\em scale invariance} is
broken, which results in a modification of the term ${z'' / z}$ in
Eq.~(\ref{modeequation}). It is evident from the experiments that the
breaking is very small, within the observable ranges of scales relevant
for the CMBR measurements, which corresponds to roughly $7$ e-foldings
of inflation.
It can be parametrized with the slow-roll parameter:
$
\epsilon = -{\dot H / H^2} \, .
$
The solution to the field Eqs.~(\ref{modeequation},~\ref{modesolution}) now has
$
\nu = {3 / 2}+{(1-n)/ 2},
$
where $n$ is the spectral index~\footnote{Which is a function of
$\epsilon$ and other small scale breaking parameters. The precise
values of the spectral index for gravitons and scalars are
different, see ~\cite{Mazumdar:2010sa}.}. For a red-tilded spectrum,
as the one observed for scalar perturbations, i.e. $n-1<0$, the IR
divergences of the
loop integrals become worse than that of the pure de Sitter case.
The computation of the diagrams is quite involved. Their structure is
not immediately transparent for what concerns the physical
interpretation of the IR divergences. In particular, the
diagrams have two types of integrations - one over the
(conformal) times for each of the vertices, and the other
over the loop momentum. There are two possible sources of
IR ``divergences''. One comes from the time integrals and it is
present only for some kind of interactions \cite{Seery:2010kh}. In our
scenario it will not appear, while, in
theories where it does, it is believed to be cured in realistic models
by using the dynamical
renormalization group techniques~\cite{Burgess:2009bs} (see also
\cite{Seery:2010kh}). The other IR divergence comes from the
momentum integral and it is always present. We will concentrate on
this latter one.
We briefly discuss the issues regarding the regularization (``box
approach'') of the IR
divergent integrals. This analysis is actually important to understand
that when the dependence on the cutoffs is not eliminated, the IR
issue is not fully understood, although
the corrections can be made
sufficiently small. The important point here is that there
is no unique choice for the IR cutoffs, rendering the results for
the observable quantities ambiguous.
For instance, the main difference for the IR divergence in the
momentum integration
arises from the choice of imposing a cutoff on the
{\em physical} or on the {\em comoving} momentum. To understand
its consequences, we consider the
example of a pure de Sitter case and the simplest IR divergent
integral, that for example comes from Fig. \ref{fig2}:
\begin{equation} \label{divergentpart}
\Lambda^{(\phi)}(\eta) = {1 \ov (2\pi)^3}\int d^3k W_k(\eta, \eta)\,.
\end{equation}
By choosing the cutoffs on the {\em physical}
momentum, $p_{UV/IR} = M_{UV/IR}$, where $M_{UV/IR}$ does
not depend on time, as in \cite{Burgess:2009bs}, one finds that for small $\eta$
\begin{equation} \label{deSitterDivergencePhysicalCutoff}
\Lambda(\eta) \sim {1 \ov 2\pi^2}\int^{a M_{UV}}_{a M_{IR}} {dk \ov k}
H^2 = {H^2 \ov 2\pi^2} \log\left({M_{UV} \ov M_{IR}}\right) \, .
\end{equation}
The rationale behind this choice is that
this kind of cutoffs does not break de Sitter
scale invariance.
Instead, choosing the cutoffs on the
comoving wavenumber as, $k_{UV/IR} = K_{UV/IR}$, where $K_{IR}$ is
independent of time, as in Ref.~ \cite{Giddings:2010nc}, and $K_{UV} =
a(\eta)\mu$, where $\mu$ is
the renormalization scale, one finds
\begin{eqnarray} \label{deSitterDivergenceScaleBreakingCutoff}
\Lambda(\eta) \sim {1 \ov 2\pi^2}\int^{K_{UV}}_{K_{IR}} {dk \ov k}
H^2 = {H^2 \ov 2\pi^2} \log\left({K_{UV} \ov K_{IR}}\right) \approx {H^2 \ov 2\pi^2} N_\eta ,
\end{eqnarray}
where $N_\eta$ is the number of efoldings from the beginning of
inflation up to time $\eta$ and we have neglected the UV contribution
proportional to $\log{\mu \ov H}$. This kind of cutoffs does break de Sitter scale
invariance. The rationale behind this choice is that we integrate over
modes that
were sub-Hubble at the beginning of inflation, $K_{IR}=a_iH_i$.
We see that Eqs.~(\ref{deSitterDivergencePhysicalCutoff},
\ref{deSitterDivergenceScaleBreakingCutoff}) give very different
results. First principles do not instruct us to prefer a cutoff
either on the
physical or the comoving momenta\footnote{There are different
valid physical motivations suggesting different kind of
cutoffs, for example finiteness of inflation for the comoving one, or causality and/or
superhorizon scale invariance for the physical one, but there is not
an utterly univocal
principle that truly stands above the others.}, and thus the scale
dependence of the ``observable''
changes depending on the cutoff, which clearly makes it
unphysical. The fact that the correlators grow with the
IR cutoff, that is the ``size of the box'' $L = K_{IR}^{-1}$ also creates
difficulty in approximating the
ensemble averages via the spatial averages, which are performed in
practical observations. In fact, the RMS deviation between the spatial
and the ensemble averages goes like
$\Delta P_k \sim {P_k / (kL)^{3/2}}$ \cite{WeinbergNew}, in the case
of the spectrum, and, since
$P_k$ grows with $L$ due to the IR
divergences of the quantum loops, it could become non-negligible and
should be taken into account for precise measurements and predictions,
for not too
large boxes and for certain scales (in particular for scales not too
different from the ``box size''). The solution to this problem is to
apply the ergodic theorem only to the true IR-safe correlators.
To resolve the issue of IR divergences, we must then
propose a recipe for the definition of a sensible perturbation theory
that does not have any of the ambiguities we have presented.
\section{Eliminating IR divergences}
\label{physicalproposal}
Our solution of the infrared issue will be centered on the concept of
{\em local observer}, as we said in the introduction. In particular,
our implementation of the principle of locality, will be
different from the one dealt with, for example, in
\cite{Urakawa:2009my, Urakawa:2010kr}, which consists in using gauge
transformations that are singular at large scales. It will also
differ, as we will see, from the approach in
\cite{Gerstenlauer:2011ti, Byrnes:2010yc}, which, in particular, uses
an explicit cutoff $q_0$, of the size of the typical scale of observations,
when proposing how to resum the contributions from scales larger than
$q_0^{-1}$~\footnote{As we
will discuss at the end of section \ref{amendingprocedure}, this
proposal is therefore an improved version of the ``physical box
size'' cutoff technique that we discussed in the introduction:
effectively the result is equivalent to just using from the start an
infrared cutoff equal to $q_0$, with
all the conceptual problems we outlined.}.
Our presentation is divided in three parts: in section
\ref{conceptualpoint} we investigate the physical point at the origin of
the infrared divergences, by discussing what the true observables are
and why
the in-in formalism, as it stands now, does not properly account for
them. In section \ref{IRallorders} we demonstrate that {\em all}
infrared divergences from momentum integrals in gravitational loops
have a common origin, which is precisely the unphysical
element of the formalism presented in
section \ref{conceptualpoint}.
Finally, in section \ref{amendingprocedure} we outline our solution of
the infrared issue proposing a more physical redefinition of the in-in
formalism. A detailed example of how our proposal is implemented is given in
section \ref{oneloopexample}.
Our notation will be as follows:
we will write the background FRW metric as $g_{\mu\nu}$:
\begin{equation} \label{backgroundmetric}
ds^2= g_{\mu\nu}dx^\mu dx^\nu = -dt^2+a^2\delta_{ij}dx^i dx^j.
\end{equation}
and, in our conventions, the background quantities, which are
contracted using the background metric, have no labels: for example
$k^2 = \delta^{ij}k_i k_j$.
The perturbed quantities are instead indicated by ''over-bar''.
Thus, the local classical metric is given by (neglecting the vector
part):
\begin{equation} \label{truelocalmetric}
\overline{ds}^2=
\overline g_{\mu\nu}dx^\mu dx^\nu =
-\overline N^2dt^2+a(t)^2 \overline h_{ij}(dx^i + \overline N^i dt)(dx^j + \overline N^j dt)
\end{equation}
where $N^i, N$ are the shift and lapse functions, and~\footnote{The
shift and lapse functions are
Lagrangian multipliers, determined by the constraints as functions of
the perturbations $\zeta, \gamma$ defined in
Eq.~(\ref{threemetricperturb}). In Eqs.~(\ref{truelocalmetric}),
(\ref{threemetricperturb}), we made a
gauge choice, for what concerns the parametrization of the metric and
the choice of coordinates $(\vec x, t)$. In particular,
we distinguish scalar and true tensor part, so that
$\gamma_i^i=\partial^i \gamma_{ij} = 0$.
The choice of a gauge does not introduce any element of
non-physicality or arbitrariness in the
description of the observable.}
\begin{equation} \label{threemetricperturb}
\overline h_{ij} dx^idx^j=
e^{2\zeta(\vec x, t)}(e^{\gamma(\vec x, t)})_{ij} dx^idx^j
\, .
\end{equation}
The quantities contracted with this metric will have an ''over-bar'',
for example $\overline k^2 = \overline h^{ij}k_i k_j$.
\subsection{True observables and in-in
formalism} \label{conceptualpoint}
We will here argue that the usual formalism for the
perturbation theory does not define correctly the observables from the
point of view of a local observer, and this is the origin of the IR
divergences.
Let us look closely at the definition of observables in the in-in
formalism. For definiteness, let us consider the two-point function as a working example
(our considerations apply also to $N$-point functions, as we will
discuss in section \ref{Npointfunctions}). The definition of the
observables in the perturbative in-in formalism is given by
the right-hand side of Eq.~(\ref{twopointwithcorrections}). The important
element in this formula is that the observables are defined in terms
of the background quantities, in particular the
background metric (entering for example in $k^2$). However,
the background metric has no physical meaning and we claim that
this is the reason of an appearance of the IR divergences.
In fact, any local observer uses true local clocks and ruler. The observer has
no notion of a ``background metric'' and ``perturbations'' on it, but
instead he/she uses the local metric for his/her measurements. We will
show that when writing the perturbative correlators from the point of view
of the local observer, that is using the local metric, the IR
divergences {\it are} cured.
In more details, the IR divergences arise for the following reason. In
the in-in formalism, the perturbations ($\zeta, \gamma$) are promoted
to quantum fields ($\widehat\zeta, \widehat\gamma$), and
path-integrated using the background metric of Eq.~(\ref{backgroundmetric})
to contract the indices. We will show that in the correlators at loop
level, {\em all} IR divergences arise when at least one of the momenta
running in the loops lines goes to zero. In that limit the correlator becomes effectively {\it
disconnected} and leads to the IR divergences.
We represent this, schematically, in Fig.~\ref{disconnectedbubble}.
\begin{figure
\begin{center}
\includegraphics[width=8cm]{Disconnected.eps}
\end{center}
\caption{A schematic representation of the loop
corrections in an IR limit as shown in the text. A generic correlator can be written,
taking care of the time-ordering, as a
product of Whigtman functions integrated over the momenta running in
the internal lines, with delta functions at vertices. In the IR
limit of a diagram line (for example, here in the picture, a
graviton's one), the total correlator factorizes
into the product of a disconnected two-point function at small
momentum times the rest of the correlator with a certain
``vertex''. The disconnected two-point function
(when $q\rightarrow 0$) leads to the IR divergences.
We will show that when redefining the correlators using the local
metric, the disconnected contributions disappear.}
\label{disconnectedbubble}
\end{figure}
We claim that it is precisely these operations of expansion over the
background metric and quantum averaging of the perturbations in effectively
disconnected contributions that make no sense from the
point of view of an observer.
These are the unphysical elements of the standard in-in perturbation
theory, because a local observer will not be sensitive to the
background metric Eq.~(\ref{backgroundmetric}), but to the local
metric Eq.~(\ref{truelocalmetric}), whose
classical value in the quantum theory is
\begin{equation} \label{classicallocalmetric}
\overline g_{\mu\nu} = \langle\, \widehat{\overline g}_{\mu\nu} \,\rangle \, ,
\end{equation}
where, in particular,
\begin{equation} \label{classicallocalthreemetric}
\overline h_{ij} =
\langle e^{2\widehat\zeta(\vec x, t)}(e^{\widehat\gamma(\vec x, t)})_{ij} \rangle
\, .
\end{equation}
We will show in section \ref{amendingprocedure} that by correctly
defining the correlators in terms of
the local classical metric Eqs.~(\ref{truelocalmetric}),
(\ref{classicallocalthreemetric}) these {\it disconnected} pieces
will be removed and the infrared issue cured. Our IR-safe correlators
will be different from those defined in other proposals, as for
example in \cite{Gerstenlauer:2011ti, Byrnes:2010yc} or
\cite{Urakawa:2009my, Urakawa:2010kr}.
\subsection{IR divergences at all order}\label{IRallorders}
We will now prove that indeed {\em all} IR divergences from momenta
integrals originate in the
unphysical expansion of the local metric
in perturbations and their path
integration in the IR limit
where higher order correlators become disconnected, as we have claimed
in the previous section.
Here, we will focus on the two-point function for scalar perturbations, but our analysis
can be extended to the general case of $N$-point functions and to
gravitons.
Let us first review in some detail the path-integral formulation of the in-in
perturbation theory, for a general reference see \cite{Weinberg:2005vy}. The
Lagrangian we are keen to discuss is the one of a
nearly massless scalar field, $\sigma$, which could be the inflaton,
minimally coupled to gravity,
\begin{equation}\label{actionsigma}
S= {1 \ov 2}\int\sqrt{-\text{det}(\overline g)}
\left[\overline R-\overline g_{\mu\nu}\partial_{\mu}\sigma\partial_{\nu}\sigma-2V(\sigma)\right] \, ,
\end{equation}
where $V(\sigma)$ is very flat, otherwise arbitrary. Using the ADM
formalism~\cite{Arnowitt:1962hi}, the action becomes
\begin{multline}
S = {1 \ov 2}\int\sqrt{-\text{det}(\overline h)}~
{a^3 \ov 2}~e^{3\zeta}\Bigg[\overline N\overline
R^{(3)}-2\overline NV(\sigma)+\overline N^{-1}\Big(\overline E^j{}_i\overline E^i{}_j-(\overline E^i{}_i)^2\Big)+
\nonumber\\
+ \overline N^{-1}\Big(\dot\sigma-\overline h^{ij} \overline N_i\partial_j\sigma\Big)^2-
\overline Na^{-2}e^{-2\zeta}[\exp{(-\gamma)}]^{ij}\partial_i\sigma\partial_j\sigma\Bigg]
\;,
\end{multline}
where
\begin{equation} \label{tensorEij}
\overline E_{ij}\equiv \frac{1}{2}\Big(\dot{\overline h}_{ij}-\nabla_i \overline N_j-\nabla_j\overline N_i\Big)\;;
\end{equation}
$\nabla_i$ is the
three-dimensional covariant derivative calculated with the
three-metric $\overline h_{ij}$; and $\overline R^{(3)}$ is the curvature scalar calculated with
this three-metric:
$$
\overline R^{(3)}=a^{-2}e^{-2\zeta}\Big[e^{-\gamma}\Big]^{ij}~\overline R^{(3)}_{ij}.
$$
All spatial indices $i$, $j$, etc. are lowered and raised with the
metric $\overline h_{ij}$ and its reciprocal. In particular,
by computing the derivatives, such as
$\dot{\overline h}_{ij}$ in Eq.~(\ref{tensorEij}), one can find a kinetic
term for $\zeta$ and $\gamma$, where indices are contracted by
$\overline h_{ij}$.
We choose the gauge
where the scalar field is homogeneous, and $\zeta \neq 0$.
The two-point function in the standard in-in
perturbation theory (using the
path-integral formulation) is obtained by expanding the metric
$\overline{h}_{ij}$ in terms of the perturbations $\zeta, \gamma$. In
this section we will not indicate quantum fields with
$\;\widehat{\mbox{}}\;$ in order not to clutter the equations. The
formalism also requires doubling the field degrees of freedom in order
to account for the time ordering and anti-ordering in
Eq.~(\ref{ininamplitudes}):
$\zeta \to \{\zeta_+, \zeta_-\}$, and similarly for $\gamma$. It will
appear to be
convenient to change basis to
$\zeta_C = {1 \ov 2}(\zeta_+ + \zeta_-)$,
$\zeta_\Delta = \zeta_+ - \zeta_-$, and the analogous for the $\gamma$.
The
perturbative expansion in terms of the perturbations $\zeta, \gamma$
is obtained by functionally Taylor-expand the exponential
$e^{i\int_{\eta_{\text{in}}}^{\eta}d\tau \int d^3x \sqrt{-det\,\overline h} \, \mathcal{L}(\zeta_C,\gamma_C,\zeta_\Delta,\gamma_\Delta)}$
in the path integral in powers of the perturbations
$\zeta_C,\gamma_C,\zeta_\Delta,\gamma_\Delta$:
\begin{eqnarray} \label{twopointininlagrangian}
\int {d^3 k_1 \ov (2\pi)^3} & \int {d^3 k_1 \ov (2\pi)^3}
e^{i\vec k_1\cdot \vec x_1 + i\vec k_2\cdot \vec x_2}\langle\zeta(x_1, \eta) \zeta(x_2, \eta)\rangle = \nonumber \\
& \int {d^3 k_1 \ov (2\pi)^3}\int {d^3 k_1 \ov (2\pi)^3}
e^{i\vec k_1\cdot \vec x_1 + i\vec k_2\cdot \vec x_2}
\int_{\mathcal{C}} \mathcal{D}\zeta_C\mathcal{D}\zeta_\Delta\mathcal{D}\gamma_C\mathcal{D}\gamma_\Delta
\, \zeta_C(x_1, \eta) \zeta_C(x_2, \eta) \, \nonumber \\
& \sum_{\substack{n, m, \\ u, v \geq 0}} {1 \ov n! m! u! v!}
\prod_{a=0}^n\biggl(\int d^3x_{(a)}d\eta_{(a)}\zeta_C\text{{\small$(\vec x_{(a)}, \eta_{(a)})$}}{\delta \ov \delta \zeta_C\text{{\small$(\vec x_{(a)}, \eta_{(a)})$}}}\biggr) \nonumber \\
& \qquad\prod_{b=0}^m\biggl(\int d^3x_{(b)}d\eta_{(b)} \gamma_{\text{{\tiny$Ci_{(b)}j_{(b)}$}}}\text{{\small$(\vec x_{(b)}, \eta_{(b)})$}}{\delta \ov \delta \gamma_{\text{{\tiny$Ci_{(b)}j_{(b)}$}}}\text{{\small$(\vec x_{(b)}, \eta_{(b)})$}}}\biggr) \nonumber \\
& \qquad\prod_{c=0}^u\biggl(\int d^3x_{(c)}d\eta_{(c)} \zeta_\Delta\text{{\small$(\vec x_{(c)}, \eta_{(c)})$}}{\delta \ov \delta \zeta_\Delta\text{{\small$(\vec x_{(c)}, \eta_{(c)})$}}}\biggr) \nonumber \\
& \qquad\prod_{d=0}^v\biggl(\int d^3x_{(d)}d\eta_{(d)} \gamma_{\text{{\tiny$\Delta\,i_{(d)}j_{(d)}$}}}\text{{\small$(\vec x_{(d)}, \eta_{(d)})$}}{\delta \ov \delta \gamma_{\text{{\tiny$\Delta\,i_{(d)}j_{(d)}$}}}\text{{\small$(\vec x_{(d)}, \eta_{(d)})$}}}\biggr) \nonumber \\
& \quad\qquad \;\;
e^{i\int_{\eta_{\text{in}}}^{\eta}d\tau \int d^3x
\sqrt{-det\,\overline h} \,
\mathcal{L}(\zeta_C,\gamma_C,\zeta_\Delta,\gamma_\Delta, \overline h_C, \overline h_\Delta)}
\Psi_{\text{vac}, C}(\eta_{\text{in}})\Psi_{\text{vac}, \Delta}(\eta_{\text{in}})
\, ,
\end{eqnarray}
where $\mathcal{C}$ is the closed time path, defined by
$\zeta_+(\eta_{\text{in}})= \zeta_-(\eta_{\text{in}})$ (similarly for other fields), and the vacuum
function is $\Psi_{\text{vac}, C/\Delta}$. Finally,
$
\mathcal{L}(\zeta_C,\gamma_C,\zeta_\Delta,\gamma_\Delta, \overline h_C, \overline h_\Delta) =
\mathcal{L}(\zeta_+,\gamma_+, \overline h_+)-\mathcal{L}(\zeta_-,\gamma_-, \overline h_-),
$ and, {\em after} the functional derivations,
$\zeta_C,\gamma_C,\zeta_\Delta,\gamma_\Delta \to 0$,
$\overline h_{ij} \to \delta_{ij}$, in
$\mathcal{L}$ for the usual Taylor expansion around the
unperturbed background.
We are now ready to investigate what kind of IR divergences from
momentum integrals are present in the
cosmological perturbation theory in a (quasi) de Sitter background.
We find that {\em all} these IR divergences, at all
orders, appear when a momenta running in a loop line goes to zero, as we
claimed. The proof of this is as follows: in the perturbation theory over (quasi) de
Sitter, the Whightman functions go as~\footnote{More precisely, in
the (quasi) de Sitter case, they would go as
$\sim |\vec p|^{-(4-n)}$, where $n$ is the spectral index.}:
\begin{equation}
\sim {1 \ov |\vec p|^3} = {1 \ov |\vec k-\vec q|^3}
= {1 \ov (|\vec k|^2+|\vec q|^2-2|\vec k||\vec q|\cos{\theta})^{3 \ov 2}}\,,
\end{equation}
and therefore this would diverge only when $\vec p = 0$. This happens when
the momentum running in the line goes to zero.
In particular, there are no {\it collinear} divergences arising when two
momenta are parallel, because the denominator of the Whightman
functions (from which all other propagators can be obtained) never
goes to zero in that case. This is very different from what happens
in a scattering amplitude on Minkowski background, where such
collinear divergences are present, for example when a soft gluon (or
graviton) line originates from a massless line yielding a
contribution to the diagram of the form:
$\sim {1 \ov p^\mu p_\mu} = {1 \ov (k-q)^\mu (k-q)_\mu} = {1 \ov -2k^\mu q_\mu}$, which
diverges when $k^\mu$ is parallel to $q^\mu$.
The IR divergences in momentum loops in perturbation theory
can therefore all be
recovered by taking the small momentum limit of the propagators in the
relevant loop lines. Now we want to prove the other claim of ours:
that all the IR divergences appear from
disconnected contributions to the correlators. The relevant
propagators in the usual $\pm$ basis are:
\begin{eqnarray}
G^{-+}(\eta, \eta') & = W(\vec{x}, \eta, \vec{x}', \eta') \,,
\qquad
G^{+-}(\eta, \eta') = W(\vec{x}, \eta', \vec{x}', \eta) \,, \nonumber \\
G^{++}(x,y) & = \theta(\eta-\eta') G^{-+}(\eta, \eta')
+ \theta(\eta'-\eta) G^{+-}(\eta, \eta') \,, \\
G^{--}(x,y) & = \theta(\eta-\eta') G^{+-}(\eta, \eta')
+ \theta(\eta'-\eta) G^{-+}(\eta, \eta') \,,\nonumber
\end{eqnarray}
where $W(\vec{x}, \eta, \vec{x}', \eta')$ is the Whightman function (recall
Eq.~(\ref{WhightmanfunctionModeexpansionState})). Note that
$G^{++} + G^{--} = G^{+-} + G^{-+}$.
In the new, more convenient, ($C, \Delta$) basis the correlation functions are
\begin{equation}
\left(\begin{array}{cc}
G_C & G_R \\
G_A & 0 \\
\end{array}\right)
= Q \left( \begin{array}{cc}
G^{++} & G^{+-} \\
G^{-+} & G^{--} \\
\end{array} \right) Q^T \,,
\qquad
Q = \left( \begin{array}{rrr}
{1 \ov 2} && {1 \ov 2} \\
1 && -1 \\
\end{array} \right) \,.
\label{Wmatrix}
\end{equation}
The advanced and retarded propagators are related by $G_A(x,y)
= G_R(y,x)$ and vanish in the coincidence limit. The convenience in
using this basis descends from the fact that in the
IR limit (small momentum), the propagators behave as follows
\begin{equation}
G_C(q \equiv |\vec q|, \eta, \eta') \underset{q \to 0}{\simeq}
W(q, \eta, \eta')|_{q \to 0} \sim { H^2 \ov q^{(4-n)}}\,, \qquad
G_R(q, \eta, \eta') \underset{q \to 0}{\sim} \theta(\eta-\eta')
{(\eta^{2\nu}-\eta^{'\,2\nu}) \ov (\eta \eta)^{'\,{{3 \ov 2}-\nu}}}
\end{equation}
and therefore the IR divergences coming from the vanishing of
the momentum in loop lines are accounted for by the $G_C$ propagator
only. We concentrate therefore on the expansion in $\zeta_C, \gamma_C$.
Let us then see how all these IR divergences arise from disconnected
contributions to the correlators. It is easy to do it now
that we have shown that they all arise when one (or more)
momenta in a diagram line
goes to zero. Indeed, the small momentum limit of the propagators can be
found using the expansion of the perturbation fields
given by Eq.~(\ref{fieldexpansion}), where - we recall - we have chosen
planar coordinates: $Y_{\vec{k}}(\vec{x}) = e^{i \vec{k} \cdot \vec{x}}$.
In this limit, $Y_{\vec{k}}(\vec{x}) \sim 1$, and thus the momenta of
the lines sent to this IR limit drop out of the
delta functions at the interaction vertices in
Eq.~(\ref{twopointininlagrangian}). Therefore, in the IR limit
the contributions of these lines to the two-point
function become disconnected, and
Eq.~(\ref{twopointininlagrangian}) reads, once written in momentum space, as:
\begin{multline} \label{twopointininlagrangiandecoupled}
\!\!\!\!\langle \zeta(k_1, \eta) \zeta(k_2, \eta)\rangle_{\text{IR, all orders}} \!\!= \!\!
\sum_{n, m \neq (0, 0)} {1 \ov n! m!} \!\!
\prod_{\substack{0 \leq a \leq n \\ 0 \leq b \leq m}}\biggl(\!\int d\eta_{(a)} d\eta_{(b)}\!\biggr)
\langle \prod_{a=0}^n \zeta_C(\eta_{(a)})\rangle_{_\text{(IR)}}
\langle\prod_{b=0}^m \gamma_{C\,i_{(b)}j_{(b)}}(\eta_{(b)})\rangle_{_\text{(IR)}} \\
\int_{\mathcal{C}}\mathcal{D}\zeta_C\mathcal{D}\zeta_\Delta\mathcal{D}\gamma_C\mathcal{D}\gamma_\Delta
\, \zeta_C(k_1, \eta) \zeta_C(k_2, \eta) \,
\prod_{a=0}^n\biggl(\int d^3x_{(a)}{\delta \ov \delta \zeta_C(\vec x_{(a)}, \eta_{(a)})}\biggr) \\
\prod_{b=0}^m\biggl(\int d^3x_{(b)}{\delta \ov \delta \gamma_{Ci_{(b)}j_{(b)}}(\vec x_{(b)}, \eta_{(b)})}\biggr)
\,
\, e^{i\int_{\eta_{\text{in}}}^{\eta}d\tau \int d^3x \sqrt{-\text{det}\overline h} \, \mathcal{L}(\zeta_C,\gamma_C,\zeta_\Delta,\gamma_\Delta, \overline h_C, \overline h_\Delta)}
\Psi_{\text{vac}, C}(\eta_{\text{in}})\Psi_{\text{vac}, \Delta}(\eta_{\text{in}}) \, ,
\end{multline}
where again, having Taylor expanded,
$\zeta_C,\gamma_C,\zeta_\Delta,\gamma_\Delta \to 0$, $\overline h_{ij} \to \delta_{ij}$, in
$\mathcal{L}$ after the functional derivations. The label
$_{_\text{(IR)}}$ indicates the small momentum limit of the
propagators (infrared divergent).
We have thus proven that the equation (\ref{twopointininlagrangiandecoupled})
comprises all the IR-divergent parts of the two-point correlator from
loop momenta integrations, and
these appear as disconnected contributions.
With a more schematic and shortcut notation, we can write
Eq.~(\ref{twopointininlagrangiandecoupled}) as
\begin{multline} \label{compacttwopointIR}
\langle \zeta(k_1, \eta) \zeta(k_2, \eta)\rangle_{\text{IR, all orders}} = \\
\sum_{n, m \neq (0, 0)} {1 \ov n! m!} \,
\langle \prod_{u=0}^n\zeta\rangle_{\text{{\tiny IR}}} \,
\langle\prod_{v=0}^m\gamma_{i_{(v)}j_{(v)}}\rangle_{\text{{\tiny IR}}} \,
{\partial^n \ov \partial \zeta_{_{\bar h}}^n} \,
{\partial^m \ov \partial \gamma_{_{\bar h} i_{(v)}j_{(v)}}^m}
\,\, \langle \zeta(k_1, \eta) \zeta(k_2, \eta)\rangle \, .
\end{multline}
With
the suffix $\mbox{}_{_{\bar h}}$ we indicate that the derivatives
do not act on the external fields, $\zeta(k_1, \eta), \zeta(k_2, \eta)$,
of the correlator, but on the
fields entering the correlator via its dependence on
$\overline{h}_{ij} = e^{2 \zeta}(e^{\widehat \gamma})_{ij}$.
The form of Eq.~(\ref{compacttwopointIR}) makes it even more
evident that the IR divergences are a
result of the expansion of the local perturbed metric over the
long-wavelength perturbations and the path integration in
the limit where higher order correlators become disconnected.
This result completes and extends that in Ref.~
\cite{Giddings:2010nc}, where it was shown
that some IR divergences in certain one-loop in-in computations could be
recovered from semiclassical expansions around the perturbed
metric~\footnote{In Ref.~\cite{Byrnes:2010yc}, it was also
argued that some loop IR-divergence could be obtained in this way,
but the $\delta N$ formalism was used instead, which was questioned
in \cite{Giddings:2010nc}.}.
We have here proven, using a rigorous field theory
formalism, that all the IR divergences in momentum integrals in
gravitational loops at all orders are accounted for by the
equation (\ref{compacttwopointIR}).
The demonstration applies to $N$-point function for all $N$'s as well.
Since now we see that all IR
divergences in gravitational loops have the same origin, whose nature
we can understand, we will propose a procedure to fully resolve them on
the basis of the physical concepts discussed in section
\ref{conceptualpoint}. In the next section we turn to this point.
\subsection{Defining a different in-in perturbation theory}
\label{amendingprocedure}
We set out now to define a new perturbative expansion in the in-in
formalism, capable of curing the infrared issue. We will outline our recipe, and
comment on how it differs from other proposals concerning the IR
issue. For definiteness, we will be illustrating
our proposal on the two-point correlator $ \langle \zeta(k_1, \eta)
\zeta(k_2, \eta)\rangle$, but, just as before, our considerations also apply
to $\langle \gamma \gamma\rangle$, or any other correlator of $N$
fields, for all $N$'s.
In the usual perturbation theory around the background metric, the
two-point correlator is given by a series
\begin{equation} \label{notationtwopoint}
\langle \zeta(k_1, \eta) \zeta(k_2, \eta)\rangle =
\mathcal{A}(k, \eta)\delta(\vec k_1+\vec k_2) =
\bigl(\mathcal{A}(k, \eta)_{_\text{tree}} +
\mathcal{A}(k, \eta)_{_\text{loops}}\bigr) \delta(\vec k_1+\vec k_2)
\end{equation}
We stress that we do not take any infrared limit of sorts here, but
instead we are considering the full perturbative series
($\mathcal{A}(k, \eta)_{_\text{loops}} =\mathcal{A}(k,
\eta)_{_\text{1-loop}}+\mathcal{A}(k, \eta)_{_\text{2-loop}}+
\ldots$) with the loops contributions including the ultraviolet, finite and
infrared parts altogether (as we are not concerned here with the
ultraviolet divergences, we will assume that all correlators here
and in the following are suitably renormalized).
The background metric enters in this formula because it is used to
contract the comoving momenta as (recall that we use the ADM formalism)
\begin{equation}
k^2 = h_{_\text{bckgr}}^{ij} \,k_j k_j = \delta^{ij} \,k_j k_j
\end{equation}
where we have used $h_{_\text{bckgr}}^{ij} = \delta^{ij}$.
Motivated by our results and understanding in the previous sections,
we set out to define a new perturbation theory, where scales are not
defined by contracting with the
background metric, but with the field
\begin{equation} \label{classicaleffectivethreemetric}
\overline h_{ij} =
\langle e^{2\widehat\zeta(\vec x, t)}(e^{\widehat\gamma(\vec x, t)})_{ij} \rangle
\, .
\end{equation}
The idea, let us repeat it once again,
is that in this way we should be able to take into account the
actual local metric that defines scales for the observer.
We want to see if in
this way the infrared issue will be cured or not.
Please, observe that in this definition, the notation $\langle ~~
\rangle$ indicates a quantum expectation value in the in-vacuum and
that $\widehat\zeta, \,\widehat\gamma$ are the full quantum
fields. Therefore the definition of $\overline h$ is the standard
definition of the classical field as the in-vacuum quantum average of
a quantum operator. In
particular, it is not based on any infrared limit or large scale
averaging or any infrared/large scale
definition/quantity/procedure.
As the background metric was entering the correlators by contracting
the wavenumbers $k$, the new field $\overline h$ (the local metric)
will have to be used
to contract the wavenumbers in place of the background metric, as
\begin{equation} \label{replacement}
\overline k^2 =(\overline{h})^{ij} \,k_j k_j
= \langle e^{-2\widehat \zeta(\vec x, t)}(e^{-\widehat \gamma(\vec x, t)})^{ij}\rangle\; k_i k_j .
\end{equation}
Note that this quantity is very different from the one defined in
Refs~\cite{Gerstenlauer:2011ti, Byrnes:2010yc}, which is
\begin{equation} \label{kappacut}
\kappa(q_0, L)^2 = e^{-2\zeta_{ir}(\vec x, t)}(e^{-\gamma_{ir}(\vec x, t)})_{ij}k^i k^j \, ;
\end{equation}
the latter quantity is in fact defined by using the large scale behavior
\begin{equation} \label{infraredfieldstasetal}
\zeta_{ir}/\gamma_{ir} = (2\pi)^{-{3 \ov 2}}\underset{L^{-1}< q \ll q_0}{\int} d^3q
e^{i \vec k \cdot \vec x} \zeta_{\vec k}/\gamma_{\vec k} \; ,
\end{equation}
where $q_0$ is a suitable cutoff defining an infrared/large scale
limit where the fields become classical\footnote{In
Refs~\cite{Gerstenlauer:2011ti, Byrnes:2010yc}, $q_0$ is
the scale of observation. There is a similar quantity -defined however
in position space- in \cite{Urakawa:2010kr} where $q_0$ is given
by the Hubble rate.}. As we said, we use instead the full quantum
fields and the usual quantum vacuum expectation value, see equation
(\ref{classicaleffectivethreemetric}). The
logic at the basis of the proposal in
Refs~\cite{Gerstenlauer:2011ti, Byrnes:2010yc} and of equation
(\ref{kappacut}) is the definition of a different background,
characterized by the scale $q_0$, taking into account superhorizon
modes, on which to
do a new perturbation theory. Their power spectrum is defined as
the Fourier transform of a spatial average at two points
$\vec x_1$ and $\vec x_2 =\vec x_1 + \vec y$ with $q_0^{-1} \sim y$ and
then distances are rescaled with the new background metric, which is
not fully local as well (it depends on $q_0^{-1}\sim |\vec x_2 -\vec x_1|$).
We, instead, put the
physical concept of locality at the basis of our redefinition of the
perturbation theory, asking what a local observer would actually
use: (\ref{classicaleffectivethreemetric}) is not a background
metric, but the local metric that the observer would use to define
scales and a local quantities. Further differences among the
other proposals and ours will soon appear.
At first sight, at this point we can construct two kind of correlators
using $\overline h$ that we might think to use to define a new
perturbation theory:
\begin{equation} \label{perturbcorrtrasf}
\mathcal{A}'(\overline k, \eta) =
\mathcal{A}'(\overline k, \eta)_{_\text{tree-level}} +
\mathcal{A}'(\overline k, \eta)_{_\text{loops}} \, ,
\end{equation}
or
\begin{equation} \label{perturbcorrnontrasf}
\mathcal{A}(\overline k, \eta) =
\mathcal{A}(\overline k, \eta)_{_\text{tree-level}} +
\mathcal{A}(\overline k, \eta)_{_\text{loops}} \, .
\end{equation}
Here, the first object $\mathcal{A}'(\overline k, \eta)$ is the
transformed two-point correlator under the transformation $k \to
\overline k$ acting as a change of coordinates. The second object
$\mathcal{A}(\overline k, \eta)$ is instead the {\em non-transformed}
correlator evaluated on the newly contracted $\overline k$. We claim
that the latter is the one we should use for the new perturbation
theory.
In fact, it is straightforward to see that (\ref{perturbcorrtrasf})
does not give any new perturbation theory, because
$\mathcal{A}'(\overline k, \eta) = \mathcal{A}(k, \eta)$, since the
correlators transform as scalar under the transformation $k \to
\overline k$, and thus we would be just re-writing in new coordinates the
same old perturbation theory, with all the same problems.
Instead $\mathcal{A}(\overline k, \eta)$ is obviously different from
$\mathcal{A}(k, \eta)$ and thus defines a new perturbation theory.
It is straightforward to obtain the explicit expression of
$\mathcal{A}(\overline k, \eta)$ in terms of $\mathcal{A}(k, \eta)$,
by using standard techniques in calculus and the action
of diffeomorphisms. We briefly illustrate the passages:
\begin{itemize}
\item we take the
correlator $\mathcal{A}(k, \eta)$ (which comprises the tree-level and
all loop contributions, see equation (\ref{notationtwopoint})) and
transform its $k$-dependence into a
$k(\overline k)$-dependence using
\begin{equation} \label{replacementinverse}
k^2 =(\overline{h}^{-1})^{ij} \,\overline k_j \overline k_j
= \langle e^{2\widehat \zeta(\vec x, t)}(e^{\widehat \gamma(\vec x, t)})^{ij}\rangle\; \overline k_i \overline k_j ,
\end{equation}
\item we then expand in Taylor series:
\begin{equation} \label{expandamplitudeoverlinek}
\mathcal{A}(k(\overline k), \eta) =
\mathcal{A}(\overline k, \eta)\big|_{\zeta_{_{\bar h}}=\gamma_{_{\bar h}} = 0}
+ \!\!\!\!
\sum_{\substack{n, m \geq 0 \\ \{n, m\} \neq \{0, 0\}}} {1 \ov n! m!} \langle(\widehat\zeta)^n\rangle \langle(\widehat\gamma^{ij})^m\rangle
\partial_{\zeta_{_{\bar h}}}^n \partial_{\gamma^{ij}_{_{\bar h}}}^m
\mathcal{A}(k, \eta)\big|_{\substack{\zeta_{_{\bar h}}=\gamma_{_{\bar h}} = 0 \\ (k^2 = \overline k^2)}}\, ,
\end{equation}
\item we finally note that the first term
on the right hand side of Eq.~(\ref{expandamplitudeoverlinek}), i.e.
\begin{eqnarray} \label{IRsafefirstway}
\mathcal{A}(\overline k, \eta)\big|_{\zeta_{_{\bar h}}=\gamma_{_{\bar h}} = 0} & =
\mathcal{A}(k(\overline k), \eta) - \!\!\!\!
\sum_{\substack{n, m \geq 0 \\ \{n, m\} \neq \{0, 0\}}} {1 \ov n! m!} \langle(\widehat\zeta)^n\rangle \langle(\widehat\gamma^{ij})^m\rangle
\partial_{\zeta_{_{\bar h}}}^n \partial_{\gamma^{ij}_{_{\bar h}}}^m
\mathcal{A}(k, \eta)\big|_{\substack{\zeta_{_{\bar h}}=\gamma_{_{\bar h}} = 0 \\ (k^2 = \overline k^2)}}\,
\end{eqnarray}
is precisely the non-transformed
correlator evaluated on $\overline k$, which we were looking for. Once
again, we stress that we have not taken any infrared limit or similar,
and that all higher order contribution include all ultraviolet, finite
and infrared parts.
\end{itemize}
In fact, at this point it is not yet clear that the infrared issue has been
cured in this way. In order to understand that, we need to look more
carefully at the loops contribution in
$\mathcal{A}(\overline k, \eta)$ to assess better how the new
formalism, centered around $\overline h$, differs from the old one
centered around the background metric.
Thus, by substituting (\ref{notationtwopoint}),
(\ref{perturbcorrtrasf}) in equation (\ref{IRsafefirstway}), we find
that~\footnote{Here we have also used
$\mathcal{A}'(\overline k, \eta) = \mathcal{A}(k, \eta)$ to write
everything more neatly in terms of $\overline k$.}
\begin{equation} \label{IRsafefirstwayloops}
\mathcal{A}(\overline k, \eta)_{_\text{loops}} =
\mathcal{A}'(\overline k, \eta)_{_\text{loops}} -
\sum_{\substack{n, m \geq 0 \\ \{n, m\} \neq \{0, 0\}}} {1 \ov n! m!} \langle(\widehat\zeta)^n\rangle \langle(\widehat\gamma^{ij})^m\rangle
\partial_{\zeta_{_{\bar h}}}^n \partial_{\gamma^{ij}_{_{\bar h}}}^m
\mathcal{A}(k, \eta)\big|_{\substack{\zeta_{_{\bar h}}=\gamma_{_{\bar h}} = 0 \\ (k^2 = \overline k^2)}}\, .
\end{equation}
Clearly, $\mathcal{A}(\overline k, \eta)_{_\text{loops}}$ is different
from zero, because the first
term
on the right hand side
is not equal to the the second term (the sum over disconnected
contributions), and thus their difference is not
null. In fact the first term is the standard loops contribution, not
given by disconnected diagrams. Recall that all terms at each order include
their ultraviolet, finite and infrared parts altogether.
However, because of our results in section \ref{IRallorders}, we see
that {\em the infrared
divergent parts} do cancel between the first and the second term on
the right hand side of equation
(\ref{IRsafefirstwayloops})\footnote{Note the crucial
minus sign in front of all the higher order terms in our equation
(\ref{IRsafefirstway}). Such minus has nothing
to do with an expansion in $k$ of
$\overline k^2 =
\langle e^{-2\widehat \zeta(\vec x, t)}(e^{-\widehat \gamma(\vec x, t)})^{ij}\rangle\; k_i k_j$,
which would instead give
\begin{equation} \label{expandamplitudek}
\mathcal{A}'(\overline k(k), \eta) =
\mathcal{A}(k, \eta)\big|_{\substack{\zeta_{_{\bar h}}=\gamma_{_{\bar h}} = 0 \\ (k^2 = \overline k^2)}}
+
\sum_{n, m \neq \{0, 0\}} {1 \ov n! m!} \langle(\widehat\zeta)^n\rangle \langle(\widehat\gamma^{ij})^m\rangle
\partial_{\zeta_{_{\bar h}}}^n \partial_{\gamma^{ij}_{_{\bar h}}}^m
\mathcal{A}(k, \eta)\big|_{\substack{\zeta_{_{\bar h}}=\gamma_{_{\bar h}} = 0 \\ (k^2 = \overline k^2)}}\,.
\end{equation}
because Wick's
theorem always forces $n$ and $m$ to be even in order to have non-zero
correlators. This also shows that (\ref{IRsafefirstway}) is not
equivalent to a gauge/coordinate transformation.} and thus
$\mathcal{A}(\overline k, \eta)$ is free of infrared
divergences: looking at equation (\ref{compacttwopointIR}) it is
straightforward to see that infrared divergences/Log-enhancements,
{\em and only those}, do cancel out exactly order by order.
We stress that, instead, the ultraviolet (renormalized) and
finite terms do not cancel between the first and second term and so
the loop contributions
$\mathcal{A}(\overline k, \eta)_{_\text{loops}}$ in the new
perturbation theory around $\overline h$ are 1) different
from those in the standard perturbation theory using the background
metric (thus the new perturbation theory is a non-trivial modification
and in principle testable),
and 2) have the bonus that the IR divergences are absent (without
using any IR cutoff procedure to define the new theory).
As a comment, let us observe that, being fully expressed as a
combination of usual standard coordinate-transformed correlators, as
visible in equation (\ref{IRsafefirstway}), also the new
correlators will be sharing the same properties of the old ones (such
as conservation and alike) at
large scales (large $\overline k$). Furthemore, since the field $\zeta$
is fully gauge invariant
and, at the level of field equations,
(\ref{replacement}) acts as a coordinate transformation,
$\zeta$ will be conserved on superhorizon $\overline k$-scales.
Note also that we could have obtained our result using
different gauge-fixed quantities (such as the inflaton
perturbation) or even gauge-invariant variables such as the Bardeen
potentials --
in all cases the physical interpretation of the IR-divergences linked
to the metric/clocks of the local observer is the
same (a local dressing and rescaling of the metric or Hubble rate, see
section \ref{physicalproposal}) and
the prescription for the IR-safe correlator,
Eq.~(\ref{IRsafefirstway}), changes only in the
fact that the chosen perturbation variables must be used in place of
$\zeta, \gamma$. Note that the action is indeed gauge
independent, and therefore the vertices are well-defined.
\vspace{0.4cm}
As a final comment, it can be useful to compare our proposal with the one in
Refs~\cite{Gerstenlauer:2011ti, Byrnes:2010yc}. As we have already
shown that their proposal is based on a quantity
$\kappa(q_0, L)^2 = e^{-2\zeta_{ir}}(e^{-\gamma_{ir}})_{ij}k^i k^j$,
see Eq.~(\ref{kappacut}), defined using $q_0, L$ to
define the classical fields $\zeta_{ir}, \gamma_{ir}$ via
the IR or large scale limit given by Eq.~(\ref{infraredfieldstasetal}).
We do not use any large scale limit in our case, but employ the
quantum expectation
value of the full quantum fields in our
Eqs.~(\ref{classicaleffectivethreemetric}), (\ref{replacement}).
Moreover, the IR-safe two-point correlator~\footnote{Actually,
\cite{Gerstenlauer:2011ti, Byrnes:2010yc} discuss the spectrum,
which is the two-point function rescaled by $k^3$, but this is not
relevant here.} is also defined in a different way than in our case.
Refs.~\cite{Gerstenlauer:2011ti, Byrnes:2010yc} define it via a
transformation $k \to \kappa(q_à, L)$ of the Fourier Transform of a
{\em spatial} average at {\em pair of points}:
\begin{equation} \label{Akappacut}
\mathcal{A}'(\kappa(q_0, L), \eta)_{\text{tree}}.
\end{equation}
In other words, their proposal is to fix a certain physical cutoff
$q_0$ corresponding to the typical scale of the observation
( $q_0^{-1}\sim |\vec x_2 -\vec x_1|$, with $\vec x_2, \vec x_1$ the
pair of points of the average),
and define as ``infrared'' all wave numbers $k < q_0$, essentially
decomposing the fields as $\zeta = \zeta_{ir}^{k < q_0} + \zeta^{k > q_0}$,
promoting only $\zeta^{k > q_0}$ to quantum fields (analogously is done
for $\gamma$) and declaring the infrared safe correlators to be just
the tree-level result written in the new variables, see Eq.~(\ref{kappacut}).
The first question that arises here is: since the correlator in
Eq.~(\ref{Akappacut}) is
only the tree-level contribution, supplemented with the IR effects in
Eq.~(\ref{kappacut}), what would be the IR-safe loop
corrections to it in the proposal of \cite{Gerstenlauer:2011ti,
Byrnes:2010yc}? The paper does not explicitly discuss this issue. It is however clear
that because of the decomposition
$\zeta = \zeta_{ir}^{k < q_0} + \zeta^{k > q_0}$, the loop corrections would
have the form
\begin{equation}
\mathcal{A}'(\kappa, \eta)_{\text{loop, with IR cutoff $q_0$}}.
\end{equation}
and thus the total correlator is:
\begin{equation} \label{totcorrkappacut}
\mathcal{A}'(\kappa, \eta)_{\text{tree}} +
\mathcal{A}'(\kappa, \eta)_{\text{loop, with IR cutoff $q_0$}}.
\end{equation}
It is straightforward to see then, that, although the ``IR-fields'' are
resummed in $\kappa$, equation (\ref{kappacut}) is indeed just given
by a coordinate transformation~\footnote{In fact, equation (\ref{Akappacut})
is similar to (\ref{expandamplitudek}), that is different from our
(\ref{IRsafefirstway}).}, and the final outcome amounts in effect
just to the use of an
infrared cutoff $q_0$ in the loops. That is, there is in effect no
difference with the
usual ``box cutoff'' approach, with all the issues that
we have discussed before.
It follows that there are big differences with our current
proposal: we {\em do not} introduce any infrared cutoff whatsoever,
not even an ``observational'' one like $q_0$, nor we decompose the
fields as $\zeta = \zeta_{ir}^{k < q_0} + \zeta^{k > q_0}$. What we do is really
to define a new perturbation theory on the basis of the quantum expectation
value $\overline h =\langle e^{2\widehat\zeta}e^{\widehat\gamma}
\rangle$ of the full quantum operator to define scales, instead than
the standard perturbation theory around the background metric. The new
perturbative correlators are given by the series
Eq.~(\ref{IRsafefirstway}), and the higher order corrections to
correlators are non-trivial and do not reduce to the use of a cutoff
in the old perturbation theory. It is straightforward to generalize to all
$N$-point functions for both $\zeta$ and $\gamma$.
\subsection{Example at one-loop}\label{oneloopexample}
We now give a detailed example at one-loop that our procedure do
cancel all IR divergences
(Log-cutoff enhancements) while redefining the perturbative series, in
the theory with the Lagrangian in
(\ref{actionsigma}).
To keep the example simple, let us consider a pure de Sitter case, so that
$\zeta = 0, \sigma = \sigma(\vec x, t)$. In this case, the only
physical metric perturbations are gravitons. We wish to compute the
IR-safe two-point correlator
$\langle \sigma(\overline k, \eta) \sigma(\overline k', \eta)\rangle$,
at one loop, following our recipe culminating in equations
(\ref{IRsafefirstway}).
We start by computing the one-loop correlator using the standard in-in
perturbation theory. We recall here the known results from Ref.~\cite{Giddings:2010nc}.
The one-loop in-in loop corrections are given by the diagrams
in Figs.~\ref{fig1} and \ref{fig2}. The relevant interaction
Lagrangians are obtained by expanding the full Lagrangian
in perturbation up to second order, and they read
\begin{equation}\label{lthree}
{\cal L}_3 = {a \ov 2}\gamma^{ij}\partial_i\sigma\partial_j\sigma\ ,
\end{equation}
\begin{equation}\label{lfour}
{\cal L}_4 = -{a \ov 4}\gamma^{il}\gamma_{l}^{\;j}\partial_i\sigma\partial_j\sigma \, ,
\end{equation}
where $a$ is scale factor of the de Sitter background. The computation
is performed by using Eq.~(\ref{ininamplitudes}),
and the details are quite complicated and of little interest here.
We will give some of the intermediate steps, details can be found in
Refs.~\cite{Dimastrogiovanni:2008af, Giddings:2010nc}.
From the diagram in Fig.~\ref{fig1}, we obtain two contributions
\begin{eqnarray}
A_k(\eta) & = -8{\rm Re}\int {d^3q \ov (2\pi)^3}
\int_{-\infty}^\eta{d\eta_1 \ov (H\eta_1)^2}
\int_{-\infty}^{\eta_1}{d\eta_2 \ov (H\eta_2)^2}\omega_{ij,kl}(\vec q) k^ik^jk^kk^l
W_q(\eta_1,\eta_2) W_{k-q}(\eta_1,\eta_2) \nonumber \\
& \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \times W_k(\eta,\eta_1)W_k(\eta,\eta_2) \\
B_k(\eta)& = 4{\rm Re}\int {d^3q \ov (2\pi)^3} \int_{-\infty}^\eta\prod_{i=1}^2
{d\eta_i \ov (H\eta_i)^2}\omega_{ij,kl}(\vec q) k^ik^jk^kk^l
W_q(\eta_1,\eta_2) W_{k-q}(\eta_1,\eta_2)
W_k(\eta_1,\eta)W_k(\eta,\eta_2) ,
\end{eqnarray}
where $W_a(\eta, \eta')$ is given by Eq.~(\ref{GenWhightman}). From Fig.~\ref{fig2}, we obtain
\begin{equation}
C_k(\eta)= {\rm Re}\left[ (-2i) \int {d^3q \ov (2\pi)^3} \int
\frac{d\eta'}{(H\eta')^2} \omega^{\phantom{i}l}_{i\phantom{l},lj} k^i k^j W_q(\eta',\eta'),
W_k(\eta,\eta')^2\right]\ .
\end{equation}
The definition of the sum over graviton polarizations $\omega_{ij,kl}$ is
\begin{eqnarray} \label{polarsum}
\omega_{ij,kl}(\vec q) & =\sum_s \epsilon^s_{ij}(\vec q)\epsilon^{s*}_{kl}(\vec q)&& =
\delta_{ik}\delta_{jl}+ \delta_{il}\delta_{jk}-\delta_{ij}\delta_{kl} \\
&&& ~~ + \delta_{ij}\vec {\hat q}_k\vec {\hat q}_l + \delta_{kl}\vec {\hat q}_i\vec {\hat q}_j- \delta_{ik}\vec {\hat q}_j\vec {\hat q}_l - \delta_{il}\vec {\hat q}_j\vec {\hat q}_k-\delta_{jk}\vec {\hat q}_i\vec {\hat q}_l-\delta_{jl}\vec {\hat q}_i\vec {\hat q}_k + \vec {\hat q}_i\vec {\hat q}_j\vec {\hat q}_k\vec {\hat q}_l\nonumber
\end{eqnarray}
where $\vec {\hat q}$ is the unit vector in the direction $\vec{q}$,
and $\epsilon_{ij}$ is the graviton polarization tensor. After evaluating the integrals and regularizing it with IR and
ultraviolet cutoffs, we get
\begin{eqnarray}
\langle \sigma(k, \eta) \sigma(k', \eta)\rangle_{\text{one-loop}} &
= A_k(\eta)+B_k(\eta)+C_k(\eta) \nonumber \\
& = {2 \ov 15}\log(\Lambda_{IR}){H^4 \ov M_{\text{P}}^4}
{1\ov (2\pi)^2 k^3}k^2\eta^2 \nonumber \\
& ~~ + \left\{\text{finite terms and UV
corrections}\right\}+{\cal O}\left({H^4 \ov M_{\text{P}}^4}\right)
\end{eqnarray}
Therefore, adding the three-level result
\begin{equation}
\langle \sigma(k, \eta) \sigma(k', \eta)\rangle_{\text{tree}} =
(2 \pi^3)\delta(k+k') \; {H^2 \ov M_{\text{P}}^2}{1 \ov 2k^3}(1+ k^2\eta^2) \, ,
\end{equation}
the total result for the two-point correlator is~\footnote{In \cite{Giddings:2010nc}, only the result for
$k\eta \to 0$ were presented.}:
\begin{eqnarray} \label{ininoneloopsigmatwopoint}
\mathcal{A}(k, \eta) & = \langle \sigma(k, \eta) \sigma(k, \eta)\rangle_{\text{tree}}
+ \langle \sigma(k, \eta) \sigma(k, \eta)\rangle_{\text{one-loop}} & \nonumber \\
& = \left[{H^2 \ov M_{\text{P}}^2}{1 \ov 2k^3}(1+ k^2\eta^2)+{2 \ov 15}\log(\Lambda_{IR}){H^4 \ov M_{\text{P}}^4}
{1\ov (2\pi)^2 k^3}k^2\eta^2 \right. \nonumber \\
& ~~ \left. + \left\{\text{finite terms and UV
corrections}\right\}\right]+ {\cal O}\left({H^4 \ov M_{\text{P}}^4}\right)
\end{eqnarray}
We have reinstated the Planck mass to show explicitly the suppression
factors ${H^2 / M_{\text{P}}^2}$. Note that $|\gamma|^2 \sim {H^2 / M_{\text{P}}^2}$.
We now wish to apply our procedure to obtain the
IR-safe correlator up to one-loop order, starting from this
result. We follow the steps in
section \ref{amendingprocedure}. The first one is to
compute the terms of the series in Eq.~(\ref{IRsafefirstway}) up to the order
of interest. Since we are working up
to one-loop, this means up to the order $\sim {H^4 \ov
M_{\text{P}}^4}$, which is quadratic in $\gamma$. Looking at
Eq.~(\ref{ininoneloopsigmatwopoint}), we see that we need to compute up to
the second order derivative of $\langle \sigma(k, \eta)
\sigma(k, \eta)\rangle_{\text{tree}}$, and no derivatives of
$\langle \sigma(k, \eta) \sigma(k, \eta)\rangle_{\text{one-loop}}$.
The second order expansion in $\gamma$ gives
\begin{eqnarray}
{1 \ov 2}\langle\widehat\gamma^{ij}\widehat\gamma^{lf}\rangle
\partial_{\gamma^{ij}_{_{\bar h}}}\partial_{\gamma^{lf}_{_{\bar h}}}
\langle \sigma(k, \eta)\sigma(k', \eta)\rangle_{\text{tree}}\big|_{k^2 = \overline k^2}
& = {1 \ov 2}\sum_{i, j} \langle\widehat\gamma^{il}\widehat\gamma_l^{\;j}\rangle \overline k_i \overline k_j
\partial_{k^2}\langle \sigma(k, \eta)\sigma(k',\eta)\rangle_{\text{tree}}\big|_{k^2 = \overline k^2}
\nonumber \\
& + {1 \ov 2}\sum_{i, j}\langle \widehat\gamma^{ij}\widehat\gamma^{kl}\rangle
\overline k_i \overline k_j \overline k_k \overline k_l \partial_{k^2}^2
\langle \sigma(k, \eta)\sigma(k', \eta)\rangle_{\text{tree}}\big|_{k^2 = \overline k^2}.
\end{eqnarray}
After some manipulation, we obtain
the contributions
\begin{multline}
\label{twokeystwopointderivative}
{1 \ov 2}\sum_{i, j} \langle\gamma^{il}\gamma_l^{\;j}\rangle \overline k_i \overline k_j
\partial_{k^2}\langle \sigma(k, \eta) \sigma(k', \eta)\rangle_{\text{tree}}\big|_{k^2 = \overline k^2} =
-{2 \ov 3}\log(\Lambda_{IR}){H^4 \ov M_{\text{P}}^4}
{1\ov (2\pi)^2 \overline k^3}\overline k^2\eta^2 + \\
~~ + \left\{\text{disconnected finite terms and UV
corrections}\right\}
\end{multline}
\begin{multline}
\label{fourkeystwopointderivative}
{1 \ov 2}\sum_{i, j}\langle \gamma^{ij}\gamma^{kl}\rangle
\overline k_i \overline k_j \overline k_k \overline k_l \partial_{k^2}^2
\langle \sigma(k, \eta)\sigma(k', \eta)\rangle_{\text{tree}}\big|_{k^2 = \overline k^2} =
{4 \ov 5}\log(\Lambda_{IR}){H^4 \ov M_{\text{P}}^4}
{1\ov (2\pi)^2 \overline k^3}\overline k^2\eta^2 + \\
~~ + \left\{\text{other disconnected finite terms and UV
corrections}\right\}.
\end{multline}
We now apply Eqs.~(\ref{IRsafefirstway}), subtracting
Eqs.~(\ref{twokeystwopointderivative},~\ref{fourkeystwopointderivative})
form Eq.~(\ref{ininoneloopsigmatwopoint}), to obtain
\begin{eqnarray}
\mathcal{A}_{\text{IR-safe}}(\overline k, \overline k', \eta) & = (2 \pi^3)\delta(\overline k+\overline k') &
& \left[{H^2 \ov M_{\text{P}}^2}{1 \ov 2\overline k^3}(1+ \overline k^2\eta^2)+
\right. \nonumber \\
& & & \left. + \left\{\text{new finite terms and UV
corrections}\right\}\right]+ {\cal O}\left({H^4 \ov M_{\text{P}}^4}\right).
\end{eqnarray}
As we can see, the dependence of
$\log(\Lambda_{IR})$ has disappeared and there is no trace of any
other IR regulator. The new finite terms and ultraviolet corrections
are given by the difference between those in equation
(\ref{ininoneloopsigmatwopoint}) and the disconnected contributions in
(\ref{twokeystwopointderivative}), (\ref{fourkeystwopointderivative});
they are non-zero as (\ref{ininoneloopsigmatwopoint}) is not given by
disconnected diagrams.
\subsection{$N$-point functions and non-Gaussianities}\label{Npointfunctions}
What we have said about the two-point function can be extended to the
most general case of $N$-point functions.
It is interesting then to discuss the loop correction to the
three-point function and the bispectrum, which is the first indication of
non-Gaussianities in the primordial perturbations.
The latter ones are usually
quantified with a set of parameters. The one which is related to the
three-point function is called $f_{NL}$, having defined
\begin{equation}\label{shape-func}
\langle
\zeta_{\vec{k}_1}(\eta)\zeta_{\vec{k}_2}(\eta)\zeta_{\vec{k}_3}(\eta)\rangle
\equiv \delta(\sum_i \vec{k}_i) (2\pi)^3 F(\vec{k}_1,\vec{k}_2,\vec{k}_3, \eta),
\end{equation}
and written the bispectrum as
\begin{equation}
F(\vec{k}_1,\vec{k}_2,\vec{k}_3, \eta) \equiv -{6 \ov 5}
f_{NL}[P^{(\zeta)}P^{(\zeta)} + \text{2 permutations}]
\end{equation}
The parameter $f_{NL}$ will receive corrections at one-loop.
In particular, in the standard approach, it suffers from IR
Logarithmic-enhancements (although mitigated by some dependence on the
spectral index $1-n$) \cite{Giddings:2010nc}. Such corrections can
make non-Gaussianities much more pronounced theoretically than what is
actually measured, see~\cite{WMAP}.
Instead, if we use our IR-safe proposal to calculate the
three-point function corrected up to one loop, one does not find such
Logarithmic-enhancements, but only the ultraviolet corrections (which are
renormalized). We will not compute them here as it goes beyond
the scope of this paper.
\section{Conclusion}\label{conclusion}
In this work we have proposed a method to solve the IR issues of
cosmological correlators. The solution we propose here is based on the
physical principle -- every observable (correlator) should be defined in terms of a
local measurement. Of course, when one in practice
approximates the ensemble averaging of the correlators with the
spatial averaging, one introduces by definition a certain degree of
non-locality. We have discussed
the connection between the IR issue and this approximation
at the end of section \ref{GeneralBehaviour}.
Our proposal consists in a modification of the formalism enabling us
to ask physical questions.
In particular, we emphasize that quantities must be defined locally by
local observers and we implement this principle by defining a new
perturbation theory in the in-in formalism.
We do not see any breaking of perturbation theory due to IR
corrections~\cite{ArkaniHamed:2007ky}, as
our definition of the perturbation theory is IR-safe. Furthermore,
there is no ambiguity related to the choice of regularization of the
IR divergent integrals, as the final result does not depend on it.
\section*{Acknowledgments}
The authors would like to thank David Lyth, Mischa Gerstenlauer, Arthur
Hebecker and Gianmassimo Tasinato for helpful discussions.
D.C. is supported by a Postdoctoral F.R.S.-F.N.R.S. research
fellowship via the Ulysses Incentive Grant for the Mobility in Science
(promoter at the Universit\'e de Mons: Per Sundell).
|
2,869,038,154,733 | arxiv | \section{Introduction}
Exceptional objects and exceptional sequences have been studied in various contexts. The terminology
{``exceptional"} was first used by Rudakov and his school \cite{Ru} when dealing
with vector bundles on the complex projective plane $\bbP^2$, or more generally for del Pezzo surfaces.
They proposed a useful way to construct exceptional objects from given ones by means of left and right mutations. These mutation operators induce an action of the braid group $B_n$ on the set of exceptional sequences of length $n$.
Bondal and Polishchuk conjectured
in \cite{BP} that the semi-product of the braid group $B_n$ with $\bbZ^n$ acts transitively on the set of complete exceptional sequences for any triangulated category which is generated by an exceptional
sequence of length $n$.
For a finitary category $\A$ over an algebraically closed field, which is a hereditary length category,
Schofield \cite{Schofield} exhibited an algorithm to construct all exceptional objects in $\A$. This algorithm is in fact effective for any field by Ringel \cite{Ringel96}.
By using Schofield's algorithm, Crawley-Boevey \cite{CB} showed that the
braid group acts transitively on the set of complete exceptional sequences in the module category
of a finite dimensional hereditary algebra over an algebraically closed field.
A simplification of the proof and a generalization to hereditary Artin algebras were
given by Ringel \cite{Rex}. Later on,
Meltzer \cite{Mel} proved the transitivity for the category of coherent sheaves over a weighted projective line. More generally, the transitivity also holds for any exceptional curve (cf. \cite{KM}).
Given a finitary hereditary abelian category $\A$, we have the so-called Ringel--Hall algebra $\cH(\A)$ {\cite{Ringel0,Ringel1}}.
For a finite dimensional hereditary algebra $A$ over a finite field, which is of finite representation type, Ringel proved that $\cH(A):=\cH(\modcat A)$ is isomorphic to the positive part
of the quantized enveloping algebra associated to $A$. For any hereditary algebra $A$, Green \cite{Green} introduced a bialgebra structure on $\cH(A)$
and showed that the composition subalgebra $C(A)$ of $\cH(A)$ generated by simple $A$-modules
gives a realization of the positive part of the corresponding quantized enveloping algebra.
Based on works of Ringel and Green,
Xiao \cite{X97} defined the antipode for the extended Ringel--Hall algebra $\tilde{\cH}(A)$ and proved that $\tilde{\cH}(A)$ has a Hopf algebra structure.
Moreover, he showed that the reduced Drinfeld double Hall algebra of $A$ provideds a realization of the whole quantized enveloping algebra associated to $A$.
The exceptional objects play central roles in the connections of $\cH(A)$ with Lie theory. Indeed, as mentioned above, the composition subalgebra $C(A)$ is defined via simple $A$-modules, which form a complete exceptional sequence when suitably ordered. Moreover, each exceptional module belongs to $C(A)$ for any hereditary algebra $A$ by \cite{ZP,CX}. But in general, an exceptional sequence may not provide building blocks for $C(A)$. The reason is that the simple generators can not be built from a general exceptional sequence by using the Ringel--Hall multiplication.
This problem has been resolved in \cite{CX} by introducing new operators in $\cH(A)$, the so-called left and right derivations, to serve as a kind of {``division"}.
However, the situation is different at the level of categories. Namely,
the subcategories generated by two mutation-equivalent exceptional sequences (in the same orbit under the action of the braid group) always coincide, see for instance \cite{CB}.
The purpose of this paper is to establish a framework rather than $\cH(\A)$ in which mutation-equivalent exceptional sequences play the same role. We intend to deal with the reduced Drinfeld double Hall algebra $D(\A)$, which can be roughly thought as two copies of $\cH(\A)$. In fact, we will show that the subalgebras of $D(\A)$ generated by mutation-equivalent exceptional sequences coincide. This follows from explicit expressions of mutation formulas in $D(\A)$, see Propositions \ref{formula for left mutation} and \ref{formula for right mutation}. These formulas give an accurate recursive algorithm in $D(\A)$ to express each exceptional object in terms of any given complete exceptional sequence. In particular, for a finite dimensional hereditary algebra $A$, we obtain the explicit recursion formulas in $D(A)$ to express each exceptional module via simple modules. We remark that the results in \cite{Sheng} imply that the derived Hall algebra of $\A$, which can be thought as infinite copies of $\cH(\A)$, also fits our desired framework.
The paper is organized as follows. Section 2 gives a brief review on Ringel--Hall algebras and their Drinfeld doubles, as well as the braid group action on the set of exceptional sequences. The main results of this paper are stated in Section 3, but two alternative proofs of Proposition \ref{formula for left mutation} are given in Sections 5 and 6 respectively.
In Section 4 we treat the projective line case as a typical example, and in Section 7, we give the paralleled results in Lie algebra case. The last section includes some applications.
Throughout the paper, let $k$ be a finite field with $q$ elements, and $v=\sqrt{q}$.
Let $\A$ be a finitary (essentially small, Hom-finite) hereditary abelian $k$-category, and we always assume that the endomorphism ring $\End_{\A} X$ for each indecomposable object $X\in\A$ is local. {For each object $X\in\A$ and positive integer $n$, we denote by $nX$ the direct sum of $n$ copies of $X$.} Let $\P$ be the set of isoclasses (isomorphism classes) $[M]$ of objects $M$ in $\A$, and ${\rm ind}\A$ be the complete set of indecomposable objects in $\A$. We choose a representative $V_\alpha\in\alpha$ for each $\alpha\in\P$. Let $\lz_1,\lz_2\in\P$, we denote by $V_{\lz_1+\lz_2}$ the direct sum $V_{\lz_1}\oplus V_{\lz_2}$. Let $K(\A)$ denote the Grothendieck group of $\A$, and we denote by $\mathfrak{r}=\mathfrak{r}_{\A}$ the rank of $K(\A)$. We write $\hat{M}\in K(\A)$ for the class of an object $M\in\A$. For a finite set $X$, we denote by $|X|$ its cardinality. For each $\lz\in \P$ we denote by $a_\lambda$ the cardinality of the automorphism group of $V_\lambda$. Let $A$ be always a finite dimensional hereditary $k$-algebra, and denote by $\modcat A$ the category of finite dimensional (left) $A$-modules. $\bbX$ is always a weighted projective line over $k$, and we denote by $\coh\bbX$ the category of coherent sheaves on $\bbX$.
\section{Preliminaries}
In this section, we review the definitions of Hall algebras, exceptional sequences and quantum binomial coefficients, and collect some necessary known results. Let $\A$ be a finitary hereditary abelian $k$-category.
\subsection{Drinfeld double Hall algebras}
Given objects $A, B\in\A$, let $$\lr{A,B}=\dim_k\Hom_{\A}(A,B)-\dim_k\Ext_{\A}^1(A,B).$$
This descends to a bilinear form $$\lr{-,-}:K(\A)\times K(\A)\longrightarrow\mathbb{Z},$$ known as the \emph{Euler form}. The \emph{symmetric Euler form} $(-,-)$ is given by $(\az,\bz)=\lr{\az,\bz}+\lr{\bz,\az}$ for any $\az,\bz\in K(\A)$.
The \emph{Ringel--Hall algebra} $\mathcal {H}(\A)$ is by definition the $\mathbb{C}$-vector space with the basis $\{u_{\az}|\az\in\P\}$ and with the multiplication given by $u_{\az}u_{\bz}=v^{\lr{\az,\bz}}\sum_{\lambda\in\P}g_{\az\bz}^{\lambda}u_{\lambda}$ for all $\az,\bz\in\P$, where $g_{\az\bz}^{\lambda}$ is the number of subobjects $X$ of $V_{\lambda}$ such that $V_{\lambda}/X$ and $X$ lie in the isoclasses $\az$ and $\bz$, respectively. By Riedtmann-Peng formula \cite{Riedtmann,Peng},
$$g_{\az\bz}^{\lambda}=\frac{|\Ext^1_{\A}(V_{\az},V_{\bz})_{V_\lambda}|}{|\Hom_{\A}(V_{\az},V_{\bz})|}\cdot \frac{a_{\lambda}}{a_{\az}a_{\bz}},$$
where $\Ext^1_{\A}(V_{\az},V_{\bz})_{V_\lambda}$ denotes the subset of $\Ext^1_{\A}(V_{\az},V_{\bz})$ consisting of equivalence classes
of exact sequences in $\A$ of the form $0\ra V_{\bz}\ra V_{\lambda}\ra {V_{\az}}\ra 0.$
Let $\bfK=\bbC[K(\A)]$ be the group algebra of the Grothendieck group $K(\A)$. We denote by $K_{\az}$ (resp. $K_{M}$) the element of $\mathbf{K}$ corresponding to the class $\az\in K(\A)$ (resp. the object $M\in\A$). The \emph{extended Ringel--Hall algebra} is the vector space $\tilde{\mathcal {H}}(\A):=\mathcal {H}(\A)\otimes_{\mathbb{C}}\mathbf{K}$ with the structure of an algebra (containing $\mathcal {H}(\A)$ and $\mathbf{K}$ as subalgebras) by imposing the relations
$$K_{\az}u_M=v^{(\az,\hat{M})}u_MK_{\az}, \text{\ for\ any\ } \az\in K(\A), M\in\A, $$ where and elsewhere we write $u_M$ for $u_{[M]}$ if $M$ is an object in $\A$. Green \cite{Green} introduced a topological bialgebra structure on $\tilde{\mathcal {H}}(\A)$ by defining the comultiplication as follows:
$$\Delta(u_\lambda K_{\gz})
=\sum\limits_{\az,\bz\in\P}v^{\lr{\az,\bz}}\frac{a_{\az}a_{\bz}}{a_{\lambda}}g_{\az\bz}^{\lambda}u_{\az}K_{\bz+\gz}\otimes u_{\bz}K_{\gz}, \text{\ for\ any\ } \lambda\in\P, \gz\in K(\A).$$
Moreover, there exists the so-called Green's pairing
$(-,-): \tilde{\mathcal {H}}(\A)\times\tilde{\mathcal {H}}(\A)\to \mathbb{C}$
given by the formula $$(u_\lambda K_{\az},u_{\lambda'}K_{\bz})=\frac{\delta_{\lambda,\lambda'}}{a_{\lambda}}v^{-(\az,\bz)}.$$
This pairing is non-degenerate and satisfies that $(ab,c)=(a\otimes b, \Delta(c))$ for any $a,b,c\in\tilde{\mathcal {H}}(\A)$. That is, it is a Hopf pairing.
Now we introduce the reduced Drinfeld double of the topological bialgebra $\tilde{\mathcal {H}}(\A)$.
For this let $\tilde{\mathcal {H}}^{\pm}(\A)$ be two copies of $\tilde{\mathcal {H}}(\A)$ viewed as algebras, and write their basis
as $\{u_\lambda^{\pm}K_{\gz}|\lambda\in\P, \gz\in K(\A)\}$, and the comultiplications are given by
$$\Delta(u_\lambda^{\pm} K_{\gz})
=\sum\limits_{\az,\bz\in\P}v^{\lr{\az,\bz}}\frac{a_{\az}a_{\bz}}{a_{\lambda}}g_{\az\bz}^{\lambda}u_{\az}^{\pm}K_{^{\pm}\bz+\gz}\otimes u_{\bz}^{\pm}K_{\gz}. $$
The \emph{Drinfeld double} of the topological bialgebra $\tilde{\mathcal {H}}(\A)$ with respect
to the Green's pairing $(-,-)$ is the associative algebra $\tilde{D}(\A)$, defined as the free product of
algebras $\tilde{\mathcal {H}}^+(\A)$ and $\tilde{\mathcal {H}}^-(\A)$ subject to the following relations $D(a,b)$ for all $a,b\in\tilde{\mathcal {H}}(\A)$:
\begin{equation}\label{Drinfeld relation}\sum(a_1,b_2)b_1^+\ast a_2^-=\sum(a_2,b_1)a_1^-\ast b_2^+,\end{equation}
where we write $\Delta(c^{\pm})=\sum c_1^{\pm}\otimes c_2^{\pm}$ in the Sweedler's notation for $c\in\tilde{\mathcal {H}}(\A)$ (see for example \cite{BurSch}).
The \emph{reduced Drinfeld double} $D(\A)$ is the quotient of $\tilde{D}(\A)$ by the two-sided ideal
$$I=\langle {K_{\az}\otimes1-1\otimes K_{-\az} } , \az\in K(\A)\rangle.$$
For each object $M\in\A$ and $t\in\mathbb{Z}$, we sometimes also write $K_M^{\pm t}$ for $K_{\pm t\hat{M}}$.
We have the following fundamental result of the reduced Drinfeld double Hall algebras for finitary hereditary abelian categories due to Cramer.
\begin{Thm}{\rm(\cite{cram})}\label{cramer}
Let $\A$ and $\B$ be two finitary hereditary abelian $k$-categories such that there
exists an equivalence of triangulated categories $F: D^b(\A)\cong D^b(\B)$.
Then there is an algebra isomorphism $\Phi: D(\A)\cong D(\B)$ uniquely determined by the following property.
For any $\az\in K(\A)$, we have $\Phi(K_{\az})=K_{F(\az)}$. Next, for any object $M\in\mathcal{A}$ such that $F(M)=N[-n]$ with $N\in\mathcal{B}$ and $n\in\bbZ$, we have
$$\Phi(u_{M}^{\pm})=v^{-n\langle N,N\rangle}u_{N}^{\pm \bar{n}}K_{N}^{\pm n},$$
where $\bar{n}=+$ (resp. $-$) if $n$ is even (resp. odd).
\end{Thm}
\subsection{Exceptional sequences}
An object $V_{\az}\in\A$ is called \emph{exceptional} if $\Ext_{\A}^1(V_{\az},V_{\az})$ vanishes and $\End_{\A}V_{\az}$ is a skew field. In our case, it is a field. A pair $(V_{\az},V_{\bz})$ of exceptional objects in $\A$ is called \emph{exceptional} provided $$\Hom_{\A}(V_{\bz},V_{\az})=\Ext_{\A}^1(V_{\bz},V_{\az})=0.$$ A sequence $(V_{\az_1},V_{\az_2},\cdots,V_{\az_r})$ of exceptional objects in $\A$ is called \emph{exceptional} provided any pair $(V_{\az_i},V_{\az_j})$ with $1\leq i<j\leq r$ is exceptional.
It is said to be \emph{complete} if $r=\mathfrak{r}_{\A}$, the rank of the Grothendieck group $K(\A)$.
For any exceptional pair $(V_{\az},V_{\bz})$, let $\mathscr{C}$$(V_{\az},V_{\bz})$ be
the smallest full subcategory of $\A$ which contains the objects $V_{\az}, V_{\bz}$, and
is closed under kernels, cokernels and extensions. As an exact extension-closed full subcategory of $\A$, $\sC(V_{\az},V_{\bz})$ is itself abelian and hereditary, which is
derived equivalent to the module category $\modcat \Lz$, where and elsewhere $\Lz$ is a finite dimensional
hereditary $k$-algebra with two isoclasses of
simple modules {(\cite{HR1,HR2,LM}).}
According to {\cite{Bondal,CB,CX,Rex}}, for any exceptional pair $(V_{\az},V_{\bz})$, there exist unique objects $L(\az,\bz)$ and $R(\az,\bz)$ with the property that $(L(\az,\bz),V_{\az})$ and $(V_{\bz},R(\az,\bz))$ are both exceptional pairs in $\A$. The objects $L(\az,\bz)$ and $R(\az,\bz)$ are called the \emph{left mutation} and \emph{right mutation} of $(V_{\az},V_{\bz})$, respectively.
Moreover, if $\cE=(V_{\az_1},V_{\az_2},\cdots,V_{\az_r})$ is an exceptional sequence, for $1\leq i<r$, we define
$$L_i\cE:=(V_{\az_1},\cdots,V_{\az_{i-1}},L(\az_i,\az_{i+1}),V_{\az_i},V_{\az_{i+2}},\cdots,V_{\az_r}),$$
$$R_i\cE:=(V_{\az_1},\cdots,V_{\az_{i-1}},V_{\az_{i+1}},R(\az_i,\az_{i+1}),V_{\az_{i+2}},\cdots,V_{\az_r}).$$
Then $L_i\cE$ and $R_i\cE$ are both exceptional sequences.
Recall that the braid group $\mathcal {B}$$_r$ in $r-1$ generators $\sigma_1,\sigma_2,\cdots,\sigma_{r-1}$ is the group with these generators and the relations $\sigma_{i}\sigma_{i+1}\sigma_{i}=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$ for all $1\leq i<r-1$ and $\sigma_{i}\sigma_{j}=\sigma_{j}\sigma_{i}$ for $j\geq i+2$. The braid group $\mathcal {B}$$_r$ acts on the set of all exceptional sequences of length $r$ via $\sigma_i\cE:=L_i\cE$ and $\sigma_i^{-1}\cE:=R_i\cE$.
Two exceptional sequences of length $r$ are said to be \emph{mutation equivalent} if they belong to the same $\mathcal {B}$$_r$-orbit.
\subsection{Quantum binomial coefficients}
In Hall algebras, the following notations and relations are frequently-used. Let $i,l$ be two nonnegative integers and $0\leq i\leq l$, set
\begin{equation*}
[l]=\frac{v^l-v^{-l}}{v-v^{-1}},~[l]!=\prod_{t=1}^l[t],~\left[\begin{smallmatrix}
l\\i
\end{smallmatrix}\right]=\frac{[l]!}{[i]![l-i]!},
\end{equation*}
and
\begin{equation*}
|l]=\frac{{q}^l-1}{{q}-1},~|l]!=\prod_{t=1}^l|t],~\left|\begin{smallmatrix}
l\\i
\end{smallmatrix}\right]=\frac{|l]!}{|i]!|l-i]!}.
\end{equation*}
Then $|l]=v^{l-1}[l],~|l]!=v^{\left(\begin{smallmatrix}
l\\2
\end{smallmatrix}\right)}[l]!,~\left|\begin{smallmatrix}
l\\i
\end{smallmatrix}\right]=v^{i(l-i)}\left[\begin{smallmatrix}
l\\i
\end{smallmatrix}\right].$ For a polynomial $f\in$$\mathbb{Z}$$[v,v^{-1}]$ and a positive integer $d$, we denote by $f(v)_d$ the polynomial obtained from $f$ by replacing $v$ by $v^d$.
The following lemma is well-known (see for example \cite{R96}).
\begin{Lem}\label{zuhe}
For any integer $l>0$,
\begin{equation}
\sum_{i=0}^l(-1)^iv^{i(i-1)}\left|\begin{smallmatrix}
l\\i
\end{smallmatrix}\right]=0.
\end{equation}
\end{Lem}
For any exceptional object $V_{\lz}\in\A$, set $u_{\lz}^{\pm(t)}=(1/{[t]^!_{\epsilon(\lz)}})u_{\lz}^{\pm t}$ in the reduced Drinfeld double Hall algebra $D(\A)$, where $\epsilon(\lz)=\dim_k\End_{\A}V_{\lz}$.
We have the identity $u_{\lz}^{\pm(t)}=(v^{\epsilon(\lz)})^{t(t-1)}u_{t\lz}^{\pm}$.
\section{Main results}
In this section we state our main results of this paper. For any exceptional sequence $\cE=(V_{\az_1},V_{\az_2},\cdots,V_{\az_{r}})$ in $\A$, we denote by $D_{\cE}(\A)$ the subalgebra of $D(\A)$ generated by the elements $\{u_{\az_i}^\pm, K_{\az_i}|1\leq i\leq r\}$, and call $D_{\cE}(\A)$ the subalgebra generated by $\cE$.
\begin{Thm}\label{Main theorem}
Let $\A$ be a finitary hereditary abelian $k$-category. If two exceptional sequences $\cE_1$ and $\cE_2$ are mutation equivalent, then $D_{\cE_1}(\A)=D_{\cE_2}(\A)$.
\end{Thm}
\begin{pf}
Let $r$ be the length of $\epsilon_1$ and $\epsilon_2$. By induction, we assume that $\cE_2$ is obtained from $\cE_1$ by one step left mutation (resp. one step right mutation), i.e., $\cE_2=L_i(\cE_1)$ (resp. $\cE_2=R_i(\cE_1)$) for some $1\leq i\leq r-1$. Moreover, by the definitions of $L_i$ and $R_i$, we know that $\cE_1$ and $\cE_2$ only differ in two components. Thus we assume that $\cE_1$ and $\cE_2$ have length $r=2$ without loss of generality. In this case, by \cite{HR1,HR2,LM} we know that the category $\mathscr{C}$$(\cE_i)$ is derived equivalent to $\modcat \Lz$ for some hereditary $k$-algebra $\Lz$ with two isoclasses of simple modules for $i=1,2$. By Cramer's Theorem \ref{cramer}, which states that the Drinfeld double Hall algebra is derived invariant, we further assume that $\mathscr{C}$$(\cE_1)$ is just $\modcat \Lz$. Now the result follows from Proposition \ref{formula for left mutation} (\emph{resp}. Proposition \ref{formula for right mutation}), which gives an explicit expression of $u_{\gz}^{\pm}$ for the left mutation $L(\az,\bz)=V_{\gz}$ (\emph{resp}. of $u_{\lz}^{\pm}$ for the right mutation $R(\az,\bz)=V_{\lz}$) in $u_{\az}^{\pm}, u_{\bz}^{\pm}, K_{\pm\az}, K_{\pm\bz}$ for any exceptional pair $(V_{\az},V_{\bz})$ in $\modcat \Lambda$.
\end{pf}
Let $(V_{\az},V_{\bz})$ be an exceptional pair in $\modcat \Lz$. Denote by $\epsilon(\az)=\lr{\az,\az}, \epsilon(\bz)=\lr{\bz,\bz},$ and $$n(\az,\bz)=\frac{\lr{\az,\bz}}{\lr{\az,\az}},\ m(\az,\bz)=\frac{\lr{\az,\bz}}{\lr{\bz,\bz}},\ n=|n(\az,\bz)|, \ m=|m(\az,\bz)|.$$
The following two propositions provide explicit formulas in the reduced Drinfeld double Hall algebra for the left mutation and right mutation respectively.
\begin{Prop}\label{formula for left mutation}
Let $(V_{\az},V_{\bz})$ be an exceptional pair in $\modcat \Lambda$ with the left mutation $L(\az,\bz)=V_{\gz}$.
\begin{itemize}
\item[(i)] If $\lr{\az,\bz}\leq 0$, then
$$u_{\gz}^{\pm}=\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{n-j}u_{\az}^{\pm(n-j)}u_{\bz}^{\pm}
u_{\az}^{\pm(j)};$$
\item[(ii)] If $n\Dim V_{\az}>\Dim V_{\bz}$, then $$u_{\gz}^{\pm}=v^{\epsilon(\bz)}K_{\pm\bz}\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)}u_{\az}^{\pm(n-j)}u_{\bz}^{\mp}
u_{\az}^{\pm(j)};$$
\item[(iii)] If $0<n\Dim V_{\az}<\Dim V_{\bz}$, then
$$u_{\gz}^{\pm}=K_{\mp n\az}\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)}u_{\az}^{\mp(n-j)}u_{\bz}^{\pm}
u_{\az}^{\mp(j)}.$$
\end{itemize}
\end{Prop}
\begin{Prop}\label{formula for right mutation}
Let $(V_{\az},V_{\bz})$ be an exceptional pair in $\modcat \Lambda$ with the right mutation $R(\az,\bz)=V_{\lz}$.
\begin{itemize}
\item[(i)] If $\lr{\az,\bz}\leq 0$, then
$$u_{\lz}^{\pm}=\sum_{j=0}^m(-1)^j(v^{\epsilon(\bz)})^{m-j}u_{\bz}^{\pm(j)}u_{\az}^{\pm}
u_{\bz}^{\pm(m-j)};$$
\item[(ii)] If $m\Dim V_{\bz}>\Dim V_{\az}$, then $$u_{\lz}^{\pm}=v^{\epsilon(\az)}\sum_{j=0}^m(-1)^j(v^{\epsilon(\bz)})^{-(m-1)(m-j)}u_{\bz}^{\pm(j)}u_{\az}^{\mp}
u_{\bz}^{\pm(m-j)}K_{\mp\az};$$
\item[(iii)] If $0<m\Dim V_{\bz}<\Dim V_{\az}$, then
$$u_{\lz}^{\pm}=\sum_{j=0}^m(-1)^j(v^{\epsilon(\bz)})^{-(m-1)(m-j)}u_{\bz}^{\mp(j)}u_{\az}^{\pm}
u_{\bz}^{\mp(m-j)}K_{\pm m\bz}.$$
\end{itemize}
\end{Prop}
Let $D_{\mathscr{E}}(\A)$ be the subalgebra of $D(\A)$ generated by all $K_{\az},\az\in K(\A)$, and all $u_{\lz}^{\pm}$ corresponding to exceptional objects $V_{\lz}$ in $\A$. {We call $D_{\mathscr{E}}(\A)$ the \emph{double composition algebra} of $\A$.}
\begin{Prop}\label{main cor}
Let $\mathcal{A}$ and $\mathcal{B}$ be two finitary hereditary abelian $k$-categories. If $\mathcal{A}$ and $\mathcal{B}$ are derived
equivalent, then $D_{\mathscr{E}}(\A)\cong D_{\mathscr{E}}(\mathcal{B})$.
\end{Prop}
\begin{pf}
By Cramer's Theorem \ref{cramer}, the derived equivalence between $\mathcal{A}$ and $\mathcal{B}$ implies an isomorphism between their Drinfeld double Hall algebras. Moreover, note that the derived equivalence preserves exceptional objects in both categories. Thus $D_{\mathscr{E}}(\A)\cong D_{\mathscr{E}}(\mathcal{B})$. This finishes the proof.
\end{pf}
\begin{Thm}\label{add conditions for A}
Let $\A$ be a finitary hereditary abelian $k$-category. Assume that $\A$ satisfies the following conditions:
\begin{itemize}
\item[(i)] there exists a complete exceptional sequence in $\A$;
\item[(ii)] the action of the braid group $\mathcal {B}$$_{\mathfrak{r}}$ on the set of complete exceptional sequences in $\A$ is transitive;
\item[(iii)] for any exceptional object $E\in\A$, there exists a complete exceptional sequence $\cE'$ in $\A$ such that $E\in\cE'$.
\end{itemize}
Then for any complete exceptional sequence $\cE$ in $\A$, $D_{\mathscr{E}}(\A)=D_{\cE}(\A)$.
\end{Thm}
\begin{pf}
Obviously, by definition we have an inclusion $D_{\cE}(\A)\subseteq D_{\mathscr{E}}(\A)$. To finish the proof we only need to show that for each exceptional object $E$ in $\A$, the generator $u_{E}^{\pm}$ of $D_{\mathscr{E}}(\A)$ belongs to $D_{\cE}(\A)$. For this, choose a complete exceptional sequence $\cE'$ in $\A$ such that $E\in\cE'$. By transitivity we obtain that $D_{\cE}(\A)=D_{\cE'}(\A)$ according to Theorem \ref{Main theorem}. This finishes the proof.
\end{pf}
\begin{Cor}\label{main examples}
Let $\A=\modcat A$ for some finite dimensional hereditary algebra $A$ or $\A=\coh\bbX$ for some weighted projective line $\bbX$. Then for any complete exceptional sequence $\cE$ in $\A$, $D_{\mathscr{E}}(\A)=D_{\cE}(\A)$.
\end{Cor}
\begin{pf}
By \cite{CB,Rex} and \cite{KM} we know that $\A$ satisfies the conditions in Theorem \ref{add conditions for A} in both cases. This finishes the proof.
\end{pf}
{\begin{Rem}(1) In the case $\A=\modcat A$, the simple $A$-modules form a complete exceptional sequence when suitably ordered. Hence the double composition algebra of $\A$ can be defined via simple modules, which in fact has been adopted as its original definition, see for example \cite{Ringel1, X97}.
Moreover, in this case, the above result is closely related to \cite[Corollary 5.3]{CX}. For this we only need to observe that, under the natural embedding $$\cH(\A)\hookrightarrow D(\A); u_{\az}\mapsto u^+_{\az}, \az\in\P,$$ the left and right derivations $_{i}\delta, \delta_{i}$ of $\cH(\A)$ associated to each simple $A$-module $S_i$ are related to the adjoint operator $[u_{i}^-, -]$ (see for example \cite[Proposition 3.1.6]{Lus}).
(2) In the case $\A=\coh\bbX$, the double composition algebra $D_{\mathscr{E}}(\A)$ appears in different contexts, which will be discussed in detail in Subsection \ref{composition algebra of coh subsection}.
\end{Rem}}
\section{Typical example}
In this section, we consider $\A$ as the category $\coh\bbP^1$ of coherent sheaves on the projective line $\mathbb{P}$$^1$ over $k$. The Ringel--Hall of $\coh\bbP^1$ has been widely studied, which are closely related to the quantum affine algebra $U_{v}(\hat{sl_2})$, see for example \cite{Ka, BK, Sch2006, BurSch}. This is our original example for Theorem \ref{Main theorem}, and the content in this section itself is of interest.
For the category $\coh\bbP^1$, it is well-known that the exceptional objects coincide with the indecomposable vector bundles, which are all line bundles by \cite{Grothendieck} and hence have the form $\co(i), i\in\mathbb{Z}$. Moreover, each exceptional pair in $\coh\bbP^1$ has the form $(\co(i),\co(i+1)), i\in\mathbb{Z}$. In this case $n=n(\co(i),\co(i+1))=2$ and $m=m(\co(i),\co(i+1))=2$.
The left (resp. right) mutation of $(\co(i),\co(i+1))$ is $\co(i-1)$ (resp. $\co(i+2)$), which comes from the following Auslander--Reiten sequence for $j=i$ (resp. $j=i+1$):
$$0\longrightarrow \co(j-1)\longrightarrow \co(j)\oplus\co(j)\longrightarrow \co(j+1)\longrightarrow 0.$$
We will give explicit expressions in the reduced Drinfeld double Hall algebra of $\coh\bbP^1$ for the elements $u_{\co(i-1)}^{\pm}$ and $u_{\co(i+2)}^{\pm}$ corresponding to the left and right mutations of the exceptional pair $(\co(i),\co(i+1))$ respectively.
Recall from \cite{Sch2006} that for any $j\in\mathbb{Z}$,
\begin{equation}\label{yu1}\Delta(u_{\co(j)}^\pm)=u_{\co(j)}^{\pm}\otimes 1+K_{{\co}(j)}^{\pm}\otimes u_{\co(j)}^{\pm}+\sum_{k>0}\Thz_k^{\pm}K_{{\co}(j-k)}^{\pm}\otimes u_{\co(j-k)}^{\pm}.\end{equation}
The elements $\Thz_k^{\pm}, k\geq 0$ play important roles in the study of $D(\coh\bbP^1)$, for its concrete definition we refer to \cite[Example 4.12]{Sch2006}. We emphasize that $\Thz_1^{\pm}=(v-v^{-1})\sum\limits_{S\in\cS_1}u_{S}^{\pm}$, where $\cS_1$ denotes the set of torsion sheaves of degree one, and $|\cS_1|=q+1$. Let {$\delta= \widehat{{\co}(1)}-\widehat{{\co}}$}. Then
\begin{equation}\label{yu2}\Delta(\Thz_1^\pm)=\Thz_1^\pm\otimes1+K_{\pm\delta}\otimes\Thz_1^\pm~~\mbox{and}~~ (\Thz_1^+,\Thz_1^-)=q-q^{-1}.\end{equation
\begin{Prop}\label{left formula for P1} For any $i\in\mathbb{Z}$,
$$u_{\co(i-1)}^{\pm}=K_{{\co}(i+1)}^{\pm}\sum_{j=0}^2(-1)^jv^{j-1}u_{\co(i)}^{\pm(2-j)}
u_{\co(i+1)}^{\mp}u_{\co(i)}^{\pm(j)}.$$\end{Prop}
\begin{pf} By duality, we only show the formula for $u_{\co(i-1)}^{+}$.
Using $(\ref{yu1})$ and $(\ref{yu2})$, we obtain that
\begin{equation*}\begin{split}
u_{\co(i)}^+u_{\co(i+1)}^-&=u_{\co(i+1)}^-u_{\co(i)}^++\Thz_1^-K_{{\co}(i)}^{-}(u_{\co(i)}^+,u_{\co(i)}^-)\\&=
u_{\co(i+1)}^-u_{\co(i)}^++(q-1)^{-1}\Thz_1^-K_{{\co}(i)}^{-},
\end{split}\end{equation*} and
\begin{equation*}\begin{split}
u_{\co(i)}^+\Thz_1^-&=\Thz_1^-u_{\co(i)}^++K_{-\delta}u_{\co(i-1)}^+(\Thz_1^+K_{{\co}(i-1)},\Thz_1^-)\\&=
\Thz_1^-u_{\co(i)}^++(q-q^{-1})K_{-\delta}u_{\co(i-1)}^+.
\end{split}\end{equation*}
Hence, $\Thz_1^-=(q-1)[u_{\co(i)}^+, u_{\co(i+1)}^-]K_{{\co}(i)}$ and
$u_{\co(i-1)}^+=(q-q^{-1})^{-1}K_{\delta}[u_{\co(i)}^+, \Thz_1^-].$
Observing that $K_{\delta}$ is central and $K_{{\co}(i)}K_{\delta}=K_{{\co}(i+1)}$, we obtain that
$$u_{\co(i-1)}^+=K_{{\co}(i+1)}
(v^{-1}u_{\co(i)}^{+(2)}u_{\co(i+1)}^--u_{\co(i)}^+
u_{\co(i+1)}^-u_{\co(i)}^++vu_{\co(i+1)}^-u_{\co(i)}^{+(2)}).$$
\end{pf}
\begin{Prop} For any $i\in\mathbb{Z}$,
$$u_{\co(i+2)}^{\pm}=\sum_{j=0}^2(-1)^jv^{j-1}u_{\co(i+1)}^{\pm(j)}
u_{\co(i)}^{\mp}u_{\co(i+1)}^{\pm(2-j)}K_{{\co}(i)}^{\mp}.$$\end{Prop}
\begin{pf}
By duality, we only show the formula for $u_{\co(i+2)}^{+}$. Using similar arguments as those in the proof of Proposition \ref{left formula for P1}, the element $\Thz_1^+$ can be expressed by the elements in $D(\coh\bbP^1)$ corresponding to the exceptional pair $(\co(i), \co(i+1))$ as follows:
$$\Thz_1^+=(q-1)[u_{\co(i)}^-,u_{\co(i+1)}^+]K_{{\co}(i)}^-.$$
On the other hand, by definition $\Thz_1^+=(v-v^{-1})\sum\limits_{S\in\cS_1}u_{S}^+$, and for any $S\in\cS_1$,
$$g_{\co(i),S}^{\co(i)\oplus S}=g_{S,\co(i+1)}^{\co(i+2)}=1,\ \ g_{S,\co(i+1)}^{S\oplus \co(i+1)}=q.$$
Hence,
\begin{equation*}\begin{split}
\Thz_1^+u_{\co(i+1)}^+&=(v-v^{-1})\sum\limits_{S\in\cS_1}u_{S}^+u_{\co(i+1)}^+\\
&=(v-v^{-1})\sum\limits_{S\in\cS_1}v^{\lr{S,\co(i+1)}}(qu_{S\oplus \co(i+1)}^++u_{\co(i+2)}^+)\\
&=(v-v^{-1})\sum\limits_{S\in\cS_1}(vu_{S\oplus \co(i+1)}^++v^{-1}u_{\co(i+2)}^+)\\
&=(v-v^{-1})\sum\limits_{S\in\cS_1}u_{\co(i+1)}^+u_{S}^++(1-q^{-1})\sum\limits_{S\in\cS_1}
u_{\co(i+2)}^+\\
&=u_{\co(i+1)}^+\Thz_1^++(1-q^{-1})(1+q)u_{\co(i+2)}^+.
\end{split}\end{equation*}
Therefore, \begin{equation*}\begin{split}
u_{\co(i+2)}^+&=(1-q^{-1})^{-1}(1+q)^{-1}[\Thz_1^+, u_{\co(i+1)}^+]\\
&=q(1+q)^{-1}[[u_{\co(i)}^-,u_{\co(i+1)}^+]K_{{\co}(i)}^-,u_{\co(i+1)}^+]\\
&=(vu_{\co(i+1)}^{+(2)}u_{\co(i)}^--u_{\co(i+1)}^+u_{\co(i)}^-u_{\co(i+1)}^++v^{-1}u_{\co(i)}^-u_{\co(i+1)}^{+(2)})K_{{\co}(i)}^-.
\end{split}\end{equation*}
\end{pf}
\section{The first proof for Proposition \ref{formula for left mutation}}
In this section, we present a fundamental proof for Proposition \ref{formula for left mutation} by using explicit comultiplicatin formulas in the reduced Drinfeld double Hall algebra $D(\Lz):=D(\modcat \Lambda)$ of the module category $\modcat \Lambda$.
For the exceptional pair $(V_{\az},V_{\bz})$ in $\modcat \Lambda$, if $\lr{\az,\bz}\leq 0$, then $(V_{\az},V_{\bz})$ are two simple modules in $\modcat \Lambda$, where $V_{\bz}$ is the simple projective module and $V_{\az}$ is the simple injective module. In this case, the formula in Proposition \ref{formula for left mutation} (i) is well-known, see for example \cite[Proposition 4.3.3]{CX}. Hence we only need to prove the statements of Proposition \ref{formula for left mutation} (ii) and (iii) in the following, in which case $n=n(\az,\bz)$.
\subsection{The case of $n\Dim V_{\az}>\Dim V_{\bz}$}
That is, $(V_{\az},V_{\bz})$ are the slice modules in the preprojective or preinjective component of $\modcat \Lambda$.
The left mutation $V_{\gamma}$ of $(V_{\az},V_{\bz})$ is given by the following Auslander--Reiten sequence by \cite{CX}
$$0\to V_{\gamma}\to V_{n\az}\to V_{\bz}\to 0.$$
We have the following result:
\begin{Lem}{\rm (\cite{CX})}\label{lem4.2}
If $f: V_{n\az}\to V_{\bz}$ is an epimorphism, then $\Ker f\cong V_{\lambda_i}\oplus V_{i\az}$ with $0\leq i\leq n-1$, where $V_{\lambda_0}\cong V_{\gz}$. Moreover, each $V_{\lambda_i}$ is a submodule of $V_{\gz}$ and has no direct summands which are isomorphic to $V_{\az}$.
\end{Lem}
We will give the proof of Proposition \ref{formula for left mutation} (ii) by considering whether $(V_{\az},V_{\bz})$ lies in the preinjective component or in the preprojective. We only give the proof for the result of $u_{\gz}^+$, since the formula for $u_{\gz}^-$ can be obtained dually.
\subsubsection{$(V_{\az},V_{\bz})$ lies in the preinjective component}
In this case, for any $1\leq j\leq n$, each non-zero morphism from $V_{j\az}$ to $V_{\bz}$ is an epimorphism, whose kernel has similar description as above:
\begin{Lem}\label{ker}
Let $1\leq j\leq n$ and $f:V_{j\az}\to V_{\bz}$ be an epimorphism, then $\Ker(f)\cong V_{\lambda_{j,i}}\oplus V_{i\az}$ for some $0\leq i\leq j-1$, where $V_{\lambda_{j,i}}$ contains no direct summands which are isomorphic to $V_{\az}$, and it fits into the following exact sequence $$\xymatrix{0\ar[r]&V_{\lz_{j,i}}\ar[r]&V_{(j-i)\az}\ar[r]&V_{\bz}\ar[r]&0.}$$
Moreover, $[\Ext_{\Lz}^1(V_{\bz},V_{\lz_{j,i}}):\End_{\Lz}V_{\bz}]=1$.
\end{Lem}
\begin{pf}
For the epimorphism $f:V_{j\az}\to V_{\bz}$, we have the exact sequence
$$\xymatrix{0\ar[r]& \Ker f\oplus V_{(n-j)\az}\ar[r]& V_{j\az}\oplus V_{(n-j)\az}\ar[r]^-{f\choose 0}& V_{\bz}\ar[r]& 0.}$$ Then in a similar way to \cite[Lemma 4.1.2]{CX}, we can prove the first statement.
Since $\Hom_{\Lz}(V_{\bz},V_{\lz_{j,i}})=0$, we obtain that
$\lr{V_{\bz},V_{\lz_{j,i}}}=-\dim_k\Ext_{\Lz}^1(V_{\bz},V_{\lz_{j,i}}).$ On the other hand, $\lr{V_{\bz},V_{\lz_{j,i}}}=\lr{\bz,(j-i)\az-\bz}=-\epsilon(\bz)=-\dim_k\End_{\Lz}(V_{\bz})$.
Hence, $[\Ext_{\Lz}^1(V_{\bz},V_{\lz_{j,i}}):\End_{\Lz}V_{\bz}]=1$.
\end{pf}
\begin{Lem}\label{yin1}
Let $0\leq j\leq n-1$. Then
$$u_{(n-j)\az}^+u_{\bz}^-=u_{\bz}^-u_{(n-j)\az}^++\sum_{i=0}^{n-j-1}\sum_{\lz_{n-j,i}:g_{\bz,\lz_{n-j,i}}^{(n-j-i)\az}\neq0}v^{-\epsilon(\bz)}K_{-\bz}
u_{\lambda_{n-j,i}+i\az}^+.$$
\end{Lem}
\begin{pf}
Note that $$\Delta(u_{\bz}^-)=u_{\bz}^-\otimes 1+K_{-\bz}\otimes u_{\bz}^-+\mbox{else}$$ and
$$\Delta(u_{(n-j)\az}^+)=u_{(n-j)\az}^+\otimes 1+K_{(n-j)\az}\otimes u_{(n-j)\az}^++\sum v^{-\epsilon(\bz)}a_{n-j}u_{\bz}^+K_{(n-j)\az-\bz}\otimes u_{k_{n-j}}^++\mbox{else},$$
where the sum on the right-hand side is taken over all the isoclasses $k_{n-j}$'s satisfying $g_{\bz, k_{n-j}}^{(n-j)\az}\neq 0$, and by Lemma \ref{ker}, $\lr{{\bz},k_{n-j}}=\lr{\bz,(n-j)\az-\bz}=-\epsilon(\bz)$ and
\begin{equation*}\begin{split}a_{n-j}&=\frac{|\Ext_{\Lz}^1(V_{\bz},V_{\lz_{n-j,i}}\oplus V_{i\az})_{V_{(n-j)\az}}|}{|\Hom_{\Lz}(V_{\bz},V_{\lz_{n-j,i}}\oplus V_{i\az})|}
\\&=\frac{|\Ext_{\Lz}^1(V_{\bz},V_{\lz_{n-j,i}})_{V_{(n-j-i)\az}}|}{|\Hom_{\Lz}(V_{\bz},V_{\lz_{n-j,i}})|}\\&=q^{\epsilon(\bz)}-1.\end{split}\end{equation*}
Thus, by $(\ref{Drinfeld relation})$ we obtain the desired identity.
\end{pf}
\begin{Lem}\label{yin2}
Let $0\leq j\leq n-1$. Then
\begin{equation*}\begin{split}u_{(n-j)\az}^+u_{\bz}^-u_{j\az}^+&=
(v^{\epsilon(\az)})^{j(n-j)}\left|\begin{smallmatrix}
n\\j
\end{smallmatrix}\right]u_{\bz}^-u_{n\az}^++\\&(v^{\epsilon(\az)})^{j(n-j)}
\sum_{i=0}^{n-j-1}\sum_{\lz_{n-j,i}:g_{\bz,\lz_{n-j,i}}^{(n-j-i)\az}\neq0}v^{-\epsilon(\bz)}K_{-\bz}
\left|\begin{smallmatrix}
i+j\\j
\end{smallmatrix}\right]u_{\lz_{n-j,i}+(i+j)\az}^+.\end{split}\end{equation*}
\end{Lem}
\begin{pf}
Observing that $$u_{(n-j)\az}^+u_{j\az}^+=(v^{\epsilon(\az)})^{j(n-j)}g_{(n-j)\az,j\az}^{n\az}u_{n\az}^+=(v^{\epsilon(\az)})^{j(n-j)}
\left|\begin{smallmatrix}
n\\j
\end{smallmatrix}\right]u_{n\az}^+$$
and \begin{equation*}\begin{split}u_{\lz_{n-j,i}+i\az}^+u_{j\az}^+&=v^{\lr{(n-j)\az-\bz,j\az}}g_{\lz_{n-j,i}+i\az,j\az}^{\lz_{n-j,i}+(i+j)\az}
u_{\lz_{n-j,i}+(i+j)\az}^+\\&
=(v^{\epsilon(\az)})^{j(n-j)}g_{i\az,j\az}^{(i+j)\az}u_{\lz_{n-j,i}+(i+j)\az}^+\\&
=(v^{\epsilon(\az)})^{j(n-j)}\left|\begin{smallmatrix}
i+j\\j
\end{smallmatrix}\right]u_{\lz_{n-j,i}+(i+j)\az}^+,\end{split}\end{equation*}
we finish the proof by Lemma $\ref{yin1}$.
\end{pf}
$\mathbf{Proof~~of~~Proposition}~~\ref{formula for left mutation}~~(ii)~~\mathbf{for~~the~~preinjective~~case}$:
\begin{equation*}\begin{split}
&\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)}u_{\az}^{+(n-j)}u_{\bz}^-u_{\az}^{+(j)}\\
=&\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)+(n-j)(n-j-1)+j(j-1)}u_{(n-j)\az}^+u_{\bz}^-u_{j\az}^+\\
=&\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{j(j-1)-j(n-j)}u_{(n-j)\az}^+u_{\bz}^-u_{j\az}^+\\
=&(\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{j(j-1)}\left|\begin{smallmatrix}
n\\j
\end{smallmatrix}\right])u_{\bz}^-u_{n\az}^++\\
&v^{-\epsilon(\bz)}K_{-\bz}\sum_{j=0}^{n-1}\sum_{i=0}^{n-1-j}(-1)^j(v^{\epsilon(\az)})^{j(j-1)}
\left|\begin{smallmatrix}
i+j\\j
\end{smallmatrix}\right]\sum_{\lz_{n-j,i}:g_{\bz,\lz_{n-j,i}}^{(n-j-i)\az}\neq0}u_{\lambda_{n-j,i}+(i+j)\az}^+.
\end{split}\end{equation*}
By Lemma \ref{zuhe} the first term in the final equation vanishes. Moreover, let $i+j=t$, then using $\sum_{j=0}^{n-1}\sum_{i=0}^{n-1-j}=\sum_{t=0}^{n-1}\sum_{j=0}^t$ we obtain that
\begin{equation*}\begin{split}&\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)}u_{\az}^{+(n-j)}u_{\bz}^-u_{\az}^{+(j)}\\
=&v^{-\epsilon(\bz)}K_{-\bz}\sum_{t=0}^{n-1}\sum_{j=0}^{t}(-1)^j(v^{\epsilon(\az)})^{j(j-1)}
\left|\begin{smallmatrix}
t\\j
\end{smallmatrix}\right]\sum_{\lz_{n-j,t-j}:g_{\bz,\lz_{n-j,t-j}}^{(n-t)\az}\neq0}u_{\lambda_{n-j,t-j}+t\az}^+\\
\overset{t=0}{=\!=}& v^{-\epsilon(\bz)}K_{-\bz}u_{\gamma}^+,\end{split}\end{equation*}
where we have used Lemma \ref{zuhe} again for any $t\neq 0$ in the last equation above. This finishes the proof.
\subsubsection{$(V_{\az},V_{\bz})$ lies in the preprojective component}
\begin{Lem}\label{shang}
Let $1\leq j\leq n-1$, and \begin{equation}\label{cok}\xymatrix{0\ar[r]&V_{j\az}\ar[r]^-{\psi}&V_{\bz}\ar[r]^-{\varphi}&V_{c_j}\ar[r]&0}\end{equation} be a short exact sequence in $\modcat \Lambda$. Then
\begin{itemize}
\item[(1)] $\Hom_{\Lz}(V_{\bz},V_{c_j})\cong \End_{\Lz}V_{\bz}\cong\End_{\Lz}V_{c_j};$
\item[(2)] $V_{c_j}$ is indecomposable, $a_{c_j}=a_{\bz}$, and $g_{c_j,j\az}^{\bz}=1.$
\end{itemize}
\end{Lem}
\begin{pf}
Applying the functor $\Hom_{\Lz}(V_{\bz},-)$ to the sequence $(\ref{cok})$, we obtain that
$$\Hom_{\Lz}(V_{\bz},V_{c_j})\cong \End_{\Lz}V_{\bz}.$$
Since $V_{\bz}$ is exceptional, $\End_{\Lz}V_{\bz}$ is a field. So $\Hom_{\Lz}(V_{\bz},V_{c_j})$ is an one-dimensional vector space over $\End_{\Lz}V_{\bz}$. Hence, $V_{c_j}$ is indecomposable, and thus $\End_{\Lz}V_{c_j}$ is a local $k$-algebra. We claim that $\End_{\Lz}V_{c_j}$ is a field. Indeed, for any $0\neq b\in\End_{\Lz}V_{c_j}$, there exists $g\in\End_{\Lz}V_{\bz}$ such that $b\varphi=\varphi g$. Clearly, $g\neq0$, so $g$ is an isomorphism. Suppose that $b$ is nilpotent, say $b^m=0$ for some positive integer $m$. Then $\varphi g^m=b^m \varphi=0$, and thus $\varphi=0$. This is a contradiction.
Applying the functor $\Hom_{\Lz}(-,V_{c_j})$ to the sequence $(\ref{cok})$, we have the following exact sequence:
$$\xymatrix{0\ar[r]&\End_{\Lz}V_{c_j}\ar[r]^-{\varphi^\ast}&\Hom_{\Lz}(V_{\bz},V_{c_j})\ar[r]^-{\psi^\ast}&\Hom_{\Lz}(V_{j\az},V_{c_j})}.$$
We claim that $\psi^\ast=0$. Indeed, for any $f\in\Hom_{\Lz}(V_{\bz},V_{c_j})$, we can write $f=a\varphi$ for some $a\in\End_{\Lz}V_{\bz}$, since $\Hom_{\Lz}(V_{\bz},V_{c_j})$ is one-dimensional over $\End_{\Lz}V_{\bz}$. So $\psi^\ast f=f\psi=a\varphi\psi=0$. Hence, $\End_{\Lz}V_{c_j}\cong\Hom_{\Lz}(V_{\bz},V_{c_j})$. Thus, $\End_{\Lz}V_{c_j}\cong\End_{\Lz}V_{\bz}$ as fields, and then $a_{c_j}=a_{\bz}$.
Since $\Hom_{\Lz}(V_{\bz},V_{c_j})$ as an $\End_{\Lz}V_{c_j}$-vector space is one-dimensional, it is easy to see that $g_{c_j,j\az}^{\bz}=1.$
\end{pf}
\begin{Lem}\label{yin3}
Let $1\leq j\leq n-1$. Then
$$u_{(n-j)\az}^+u_{\bz}^-=u_{\bz}^-u_{(n-j)\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(v^{\epsilon(\az)})^{-i(n-j)}u_{c_i}^-K_{-i\az}u_{(n-j-i)\az}^+.$$
\end{Lem}
\begin{pf}
Note that $$\Delta(u_{\bz}^-)=u_{\bz}^-\otimes 1+K_{-\bz}\otimes u_{\bz}^-+\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}b_iu_{c_i}^-K_{-i\az}\otimes u_{i\az}^-+\mbox{else},$$
where $$b_i=v^{\lr{\bz-i\az,i\az}}g_{c_i,i\az}^{\bz}\frac{a_{c_i}a_{i\az}}{a_{\bz}}=(v^{\epsilon(\az)})^{-i^2}a_{i\az}.$$
On the other hand, we have $$\Delta(u_{(n-j)\az}^+)=u_{(n-j)\az}^+\otimes 1+\sum_{i=1}^{n-j-1}a_iu_{(n-j-i)\az}^+K_{i\az}\otimes u_{i\az}^++K_{(n-j)\az}\otimes u_{(n-j)\az}^++\mbox{else},$$
where \begin{equation*}\begin{split}a_i&=(v^{\epsilon(\az)})^{(n-j-i)i}\frac{|\Ext_{\Lz}^1(V_{(n-j-i)\az},V_{i\az})_{V_{(n-j)\az}}|}{
|\Hom_{\Lz}(V_{(n-j-i)\az},V_{i\az})|}\\
&=(v^{\epsilon(\az)})^{(n-j-i)i}/(q^{\epsilon(\az)})^{(n-j-i)i}\\
&=(v^{\epsilon(\az)})^{-(n-j-i)i}.\end{split}\end{equation*}
Hence, by $(\ref{Drinfeld relation})$, we obtain that
\begin{equation*}\begin{split}
u_{(n-j)\az}^+u_{\bz}^-&=u_{\bz}^-u_{(n-j)\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}b_ia_{n-j-i}a_{i\az}^{-1}u_{c_i}^-K_{-i\az}
u_{(n-j-i)\az}^+\\
&=u_{\bz}^-u_{(n-j)\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(v^{\epsilon(\az)})^{-i^2-(n-j-i)i}u_{c_i}^-K_{-i\az}
u_{(n-j-i)\az}^+\\
&=u_{\bz}^-u_{(n-j)\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(v^{\epsilon(\az)})^{-i(n-j)}u_{c_i}^-K_{-i\az}
u_{(n-j-i)\az}^+.
\end{split}\end{equation*}
\end{pf}
\begin{Lem}\label{leading term}
$$u_{n\az}^+u_{\bz}^-=u_{\bz}^-u_{n\az}^++\sum_{i=1}^{n-1}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(v^{\epsilon(\az)})^{-ni}u_{c_i}^{-}K_{-i\az}u_{(n-i)\az}^+
+v^{-\epsilon(\bz)}K_{-\bz}u_{\gamma}^+.$$
\end{Lem}
\begin{pf}
Note that $$\Delta(u_{\bz}^-)=u_{\bz}^-\otimes 1+K_{-\bz}\otimes u_{\bz}^-+\sum_{i=1}^{n-1}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}b_iu_{c_i}^-K_{-i\az}\otimes u_{i\az}^-+\mbox{else},$$ where $b_i=(v^{\epsilon(\az)})^{-i^2}a_{i\az}$. Moreover,
$$\Delta(u_{n\az}^+)=u_{n\az}^+\otimes 1+\sum_{i=1}^{n-1}a_iu_{(n-i)\az}^+K_{i\az}\otimes u_{i\az}^++K_{n\az}\otimes u_{n\az}^++\tilde{a}_0u_{\bz}^+K_{\gz}\otimes u_{\gz}^++\mbox{else},$$
where $a_i=(v^{\epsilon(\az)})^{-(n-i)i}$, and $\tilde{a}_0=v^{\lr{\bz,n\az-\bz}}\frac{|\Ext_{\Lz}^1(V_{\bz},V_{\gz})_{V_{n\az}}|}{|\Hom_{\Lz}(V_{\bz},V_{\gz})|}
=v^{-\epsilon(\bz)}(q^{\epsilon(\bz)}-1)$.
Hence, by $(\ref{Drinfeld relation})$, we obtain the desired identity.
\end{pf}
\begin{Lem}\label{gexiang}
Let $1\leq j\leq n-1$. Then $u_{(n-j)\az}^+u_{\bz}^-u_{j\az}^+=$
$$(v^{\epsilon(\az)})^{j(n-j)}\left|\begin{smallmatrix}
n\\j
\end{smallmatrix}\right]u_{\bz}^-u_{n\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}
(v^{\epsilon(\az)})^{j(n-j)-ni}\left|\begin{smallmatrix}
n-i\\j
\end{smallmatrix}\right]u_{c_i}^-K_{-i\az}u_{(n-i)\az}^+.$$
\end{Lem}
\begin{pf}
For any positive integers $i$ and $j$, $$u_{i\az}^+u_{j\az}^+=(v^{\epsilon(\az)})^{ij}g_{i\az,j\az}^{(i+j)\az}u_{(i+j)\az}^+
=(v^{\epsilon(\az)})^{ij}\left|\begin{smallmatrix}
i+j\\j
\end{smallmatrix}\right]u_{(i+j)\az}^+.$$
Hence, by Lemma $\ref{yin3}$, we obtain that
\begin{equation*}\begin{split}&u_{(n-j)\az}^+u_{\bz}^-u_{j\az}^+=u_{\bz}^-u_{(n-j)\az}^+u_{j\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(v^{\epsilon(\az)})^{-i(n-j)}u_{c_i}^-K_{-i\az}u_{(n-j-i)\az}^+u_{j\az}^+\\
=&(v^{\epsilon(\az)})^{j(n-j)}\left|\begin{smallmatrix}
n\\j
\end{smallmatrix}\right]u_{\bz}^-u_{n\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}
(v^{\epsilon(\az)})^{-i(n-j)+(n-j-i)j}\left|\begin{smallmatrix}
n-i\\j
\end{smallmatrix}\right]u_{c_i}^-K_{-i\az}u_{(n-i)\az}^+\\
=&(v^{\epsilon(\az)})^{j(n-j)}\left|\begin{smallmatrix}
n\\j
\end{smallmatrix}\right]u_{\bz}^-u_{n\az}^++\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}
(v^{\epsilon(\az)})^{j(n-j)-ni}\left|\begin{smallmatrix}
n-i\\j
\end{smallmatrix}\right]u_{c_i}^-K_{-i\az}u_{(n-i)\az}^+.\end{split}\end{equation*}
\end{pf}
$\mathbf{Proof~~of~~Proposition}~~\ref{formula for left mutation}~~(ii)~~\mathbf{for~~the~~preprojective~~case}$:
By Lemmas \ref{leading term} and \ref{gexiang} we get
\begin{equation*}\begin{split}
&\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)}u_{\az}^{+(n-j)}u_{\bz}^-u_{\az}^{+(j)}
\\=&\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{j(j-1)-j(n-j)}u_{(n-j)\az}^+u_{\bz}^-u_{j\az}^+\\
=&(\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{j(j-1)}\left|\begin{smallmatrix}
n\\j
\end{smallmatrix}\right])u_{\bz}^-u_{n\az}^++\sum_{i=1}^{n-1}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(v^{\ez(\az)})^{-ni}u_{c_i}^-K_{-i\az}u_{(n-i)\az}^+\\
&+\sum_{j=1}^{n-1}\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(-1)^j(v^{\ez(\az)})^{j(j-1)-ni}\left|\begin{smallmatrix}
n-i\\j
\end{smallmatrix}\right]u_{c_i}^-K_{-i\az}u_{(n-i)\az}^++
v^{-\epsilon(\bz)}K_{-\bz}u_{\gz}^+.
\end{split}\end{equation*}
For simplicity we denote the right-hand side of the last equality by $R_1+R_2+R_3+R_4$. By Lemma \ref{zuhe} we have $R_1=0$. In order to finish the proof it remains to show that $R_2+R_3=0$. In fact, by convention, we set $u_{c_n}^-=0$. Then
\begin{equation*}\begin{split}
&R_2+R_3\\
=&\sum_{j=0}^{n-1}\sum_{i=1}^{n-j}\sum_{c_i:g_{c_i,i\az}^{\bz}\neq0}(-1)^j(v^{\ez(\az)})^{j(j-1)-ni}\left|\begin{smallmatrix}
n-i\\j
\end{smallmatrix}\right]u_{c_i}^-K_{-i\az}u_{(n-i)\az}^+ \quad(\text{set\ }t=n-i)\\
=&\sum_{j=0}^{n-1}\sum_{t=j}^{n-1}\sum_{c_{n-t}:g_{c_{n-t},(n-t)\az}^{\bz}\neq0}
(-1)^j(v^{\ez(\az)})^{j(j-1)-n(n-t)}\left|\begin{smallmatrix}
t\\j
\end{smallmatrix}\right]u_{c_{n-t}}^-K_{-(n-t)\az}u_{t\az}^+\\
=&\sum_{t=0}^{n-1}\sum_{j=0}^{t}(-1)^j(v^{\ez(\az)})^{j(j-1)}\left|\begin{smallmatrix}
t\\j\end{smallmatrix}\right]\sum_{c_{n-t}:g_{c_{n-t},(n-t)\az}^{\bz}\neq0}(v^{\ez(\az)})^{-n(n-t)}u_{c_{n-t}}^-K_{-(n-t)\az}u_{t\az}^+\\
=&(v^{\ez(\az)})^{-n^2}u_{c_n}^-K_{-n\az} \quad(\text{here\ we\ have\ used\ Lemma\ \ref{zuhe}}) \\
=&0.
\end{split}\end{equation*}
\subsection{The case of $0<n(\az,\bz)\Dim V_{\az}<\Dim V_{\bz}$}
That is, $(V_{\az},V_{\bz})$ are the two projective modules in $\modcat \Lambda$ with $V_{\az}$ simple projective. Moreover, $V_{\gz}$ is the simple injective $\Lz$-module which can be determined by an arbitrary short exact sequence of the form
$$0\longrightarrow V_{n\az}\longrightarrow V_{\bz}\longrightarrow V_{\gz}\longrightarrow 0.$$
In this case, Lemma $\ref{shang}$ holds for $1\leq j\leq n$, and Lemmas $\ref{yin3}$ and $\ref{gexiang}$ hold for $0\leq j\leq n-1$. In particular, $V_{c_n}\cong V_{\gz}$. Then the proof of Proposition \ref{formula for left mutation} (iii) is analogous to
the proof of Proposition \ref{formula for left mutation} (ii) for the preprojective case, which will be omitted here.
\begin{Rem} Dually, Proposition \ref{formula for right mutation} can be proved in a quite similar way.
\end{Rem}
\section{The second proof for Proposition \ref{formula for left mutation}}
In this section, we produce a short proof for Proposition \ref{formula for left mutation}, which benefits from the explicit expression of Cramer's isomorphism in Theorem \ref{cramer}.
For the same reason as that in Section 5, we only prove the statements (ii) and (iii) in Proposition \ref{formula for left mutation}.
Thus for the exceptional pair $(V_{\az},V_{\bz})$ in $\modcat\Lz$, we assume $\lr{\az,\bz}>0$ and then $n=n(\az,\bz)$. Denote by $D^b(\sC(V_{\az},V_{\bz}))$ the bounded derived category of $\sC(V_{\az},V_{\bz})$ with the translation functor $[1]$.
\subsection{The case of $n\Dim V_{\az}>\Dim V_{\bz}$}
Set $V_{\bz'}=V_{\bz}[-1]\in D^b(\sC(V_{\az},V_{\bz}))$.
Then $\sC(V_{\az},V_{\bz'})$ is derived equivalent to $\sC(V_{\az},V_{\bz})$.
By Cramer's Theorem \ref{cramer}, we have the isomorphism
$$\Phi:D(\sC(V_{\az},V_{\bz'}))\cong D(\sC(V_{\az},V_{\bz})),$$
which preserves $u_{\az}^{\pm}$ and $u_{\gz}^{\pm}$, and sends $u_{\bz'}^{\pm}$ to $v^{-\epsilon(\bz)}u_{\bz}^{\mp}K_{\pm\bz}.$
Note that $V_{\az}$ and $V_{\bz'}$ are simple in $\sC(V_{\az},V_{\bz'})$, and the left mutation $L(V_{\az},V_{\bz'})=V_{\gz}$, which is given by the standard exact sequence
$$0\longrightarrow V_{\bz'}\longrightarrow V_{\gz}\longrightarrow V_{n\az}\longrightarrow 0.$$
By Proposition \ref{formula for left mutation} (i), the following equality holds in $D(\sC(V_{\az},V_{\bz'}))$:
$$u_{\gz}^{\pm}=\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{(n-j)}u_{\az}^{\pm(n-j)}u_{\bz'}^{\pm}
u_{\az}^{\pm(j)}.$$
Using Cramer's isomorphism $\Phi$, we obtain the following equality in $D(\sC(V_{\az},V_{\bz}))$:
$$u_{\gz}^{\pm}=\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{(n-j)}u_{\az}^{\pm(n-j)}(v^{-\ez(\bz)}u_{\bz}^{\mp}K_{\pm\bz})
u_{\az}^{\pm(j)}.$$
Notice that \begin{equation*}\begin{split}
u_{\az}^{\pm(n-j)}u_{\bz}^{\mp}K_{\pm\bz}
u_{\az}^{\pm(j)}&=v^{(-\bz,(n-j)\az-\bz)}K_{\pm\bz}u_{\az}^{\pm(n-j)}u_{\bz}^{\mp}u_{\az}^{\pm(j)}\\
&=v^{2\ez(\bz)-n(n-j)\ez(\az)}K_{\pm\bz}u_{\az}^{\pm(n-j)}u_{\bz}^{\mp}u_{\az}^{\pm(j)}.
\end{split}\end{equation*}
Hence, $$u_{\gz}^{\pm}=v^{\epsilon(\bz)}K_{\pm\bz}\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)}u_{\az}^{\pm(n-j)}u_{\bz}^{\mp}
u_{\az}^{\pm(j)}.$$
\subsection{The case of $0<n\Dim V_{\az}<\Dim V_{\bz}$}
Set $V_{\az'}=V_{\az}[1]$. Then $\sC(V_{\az'},V_{\bz})$ is derived equivalent to $\sC(V_{\az},V_{\bz})$.
By Cramer's Theorem \ref{cramer}, we have the isomorphism
$$\Phi:D(\sC(V_{\az'},V_{\bz}))\cong D(\sC(V_{\az},V_{\bz})),$$
which preserves $u_{\bz}^{\pm}$ and $u_{\gz}^{\pm}$, and sends $u_{\az'}^{\pm}$ to $v^{\epsilon(\az)}u_{\az}^{\mp}K_{\mp\az}.$
Note that $V_{\az'}$ and $V_{\bz}$ are simple in $\sC(V_{\az'},V_{\bz})$, and the left mutation $L(V_{\az'},V_{\bz})=V_{\gz}$, which is given by the standard exact sequence
$$0\longrightarrow V_{\bz}\longrightarrow V_{\gz}\longrightarrow nV_{\az'}\longrightarrow 0.$$
By Proposition \ref{formula for left mutation} (i), the following equality holds in $D(\sC(V_{\az},V_{\bz'}))$:
$$u_{\gz}^{\pm}=\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{(n-j)}u_{\az'}^{\pm(n-j)}u_{\bz}^{\pm}
u_{\az'}^{\pm(j)},$$ since $\epsilon(\az')=\epsilon(\az).$
Using Cramer's isomorphism $\Phi$, we obtain the following equality in $D(\sC(V_{\az},V_{\bz}))$:
$$u_{\gz}^{\pm}=\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{(n-j)}{(v^{\epsilon(\az)}u_{\az}^{\mp}K_{\mp\az})}^{(n-j)}u_{\bz}^{\pm}
{(v^{\epsilon(\az)}u_{\az}^{\mp}K_{\mp\az})}^{(j)}.$$
Notice that for any positive integer $t$, $$(v^{\ez(\az)}u_{\az}^{\mp}K_{\mp\az})^{(t)}=(v^{\ez(\az)})^{-t^2}K_{\mp t\az}u_{\az}^{\mp(t)}.$$
Thus, $$u_{\gz}^{\pm}=\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{(n-j)-(n-j)^2-j^2} K_{\mp (n-j)\az}u_{\az}^{\mp(n-j)} u_{\bz}^{\pm} K_{\mp j\az}u_{\az}^{\mp(j)}.$$
Moreover, observe that
\begin{equation*}\begin{split}
u_{\az}^{\mp(n-j)} u_{\bz}^{\pm} K_{\mp j\az}=&v^{-(j\az, (n-j)\az-\bz)} K_{\mp j\az} u_{\az}^{\mp(n-j)} u_{\bz}^{\pm}\\
=&(v^{\ez(\az)})^{-2j(n-j)+nj} K_{\mp j\az} u_{\az}^{\mp(n-j)} u_{\bz}^{\pm},
\end{split}\end{equation*}
and $$(n-j)-(n-j)^2-j^2 -2j(n-j)+nj=-(n-1)(n-j).$$
Hence, $$u_{\gz}^{\pm}=K_{\mp n\az}\sum_{j=0}^n(-1)^j(v^{\epsilon(\az)})^{-(n-1)(n-j)}u_{\az}^{\mp(n-j)}u_{\bz}^{\pm}
u_{\az}^{\mp(j)}.$$
\begin{Rem} Analogously, we can prove Proposition \ref{formula for right mutation} by using Cramer's Theorem \ref{cramer}.
\end{Rem}
\section{The versions of Lie algebras}
Let $\A$ be a finitary hereditary abelian $k$-category as before. Let $D^b(\A)$ be the bounded derived category of $\A$ with the suspension functor $T$. Denote by $\mathcal {R}(\A)$ the root category $D^b(\A)/T^2$. The automorphism $T$ of $D^b(\A)$ induces an automorphism of $\mathcal {R}(\A)$, denoted still by $T$. Then $T^2=1$ in $\mathcal {R}(\A)$.
By using the Ringel--Hall algebra approach, Peng and Xiao \cite{PengXiao2} defined a Lie algebra $\cal{L}(\cR):=\cal{L}(\cR(\A))$ via the root category $\mathcal {R}(\A)$. The nilpotent part of $\cal{L}(\cR)$ has a basis $\{u_{[M]}, u_{[TM]}|M \in\rm{ind}\A\}$.
To simplify the notation, in what follows we write $V_{T\az}:=T(V_{\az})$ and $u_{T\az}:=u_{V_{T\az}}$ for each ${\az}\in\P$.
In this section we will show that the main results in Section 3 also hold under the framework of $\cal{L}(\cR)$.
Let $(V_{\az},V_{\bz})$ be an exceptional pair in $\A$.
Recall that $$n(\az,\bz)=\frac{\lr{\az,\bz}}{\lr{\az,\az}},\ m(\az,\bz)=\frac{\lr{\az,\bz}}{\lr{\bz,\bz}},\ n=|n(\az,\bz)|, \ m=|m(\az,\bz)|.$$
Now, let us recall the left mutation $L_{\az}(\bz)$ and right mutation $R_{\bz}(\az)$ of $(V_{\az},V_{\bz})$ in the root category $\mathcal {R}(\A)$ (see for example \cite{Bondal}).
If $(V_{\az},V_{\bz})$ is orthogonal, i.e., $\Hom(V_{\az},V_{\bz})=0=\Hom(V_{\az},V_{T\bz})$, then $L_{\az}(\bz)=V_{T\bz}$ and $R_{\bz}(\az)=V_{T\az}$. Otherwise, there exists $i=0$ or $1$, such that $\Hom(V_{\az},V_{T^i\bz})\neq 0$. In this case, the left and right mutations are determined respectively by the following canonical triangles
$$\xymatrix{V_{T\bz}\ar[r]& L_{\az}(\bz)\ar[r]& nV_{T^i\az}
\ar[r]& V_{\bz}}$$ and
$$\xymatrix{V_{\az} \ar[r]& mV_{T^i\bz}
\ar[r]& R_{\bz}(\az)\ar[r]& V_{T\az}.}$$
The following result provides explicit formulas in the Lie algebra $\cal{L}(\cR)$ for the left mutation and right mutation respectively.
For the proof we refer to \cite[Proposition 7.3]{LinPeng} {(see also \cite{R4, PengXiao2})}, where the proof is stated for tubular algebra cases, but it is in fact effective more generally.
\begin{Prop}\label{formula for left mutation--Lie}
Let $\A$ be a finitary hereditary abelian $k$-category. Let $(V_{\az},V_{\bz})$ be an exceptional pair in $\A$ with $\Hom(V_{\az},V_{T^i\bz})\neq 0$ for $i=0$ or 1. Then
$$(-1)^n(n!)u_{[L_{\az}(\bz)]}=({\rm{ad}} u_{T^i\az})^n u_{T\bz}$$ and
$$(m!)u_{[R_{\bz}(\az)]}=({\rm{ad}} u_{T^i\bz})^m u_{T\az}.$$
\end{Prop}
For any exceptional sequence $\cE=(V_{\az_1},V_{\az_2},\cdots,V_{\az_r})$ in $\A$, the subalgebra generated by $\{u_{\az_i}, u_{{T\az_i}}|1\leq i\leq r\}$ is denoted by $\cal{L}_{\cE}(\cR)$.
As an immediate consequence of Proposition \ref{formula for left mutation--Lie}, we have the following
\begin{Prop}\label{Main Prop for Lie algbera}
Let $\A$ be a finitary hereditary abelian $k$-category. If two exceptional sequences $\cE_1$ and $\cE_2$ are mutation equivalent, then $\cal{L}$$_{\cE_1}(\cR)=\cal{L}$$_{\cE_2}(\cR)$.
\end{Prop}
Let $\cal{L}_{\mathscr{E}}(\cR)$ be the subalgebra of $\cal{L}(\cR)$ generated by all $u_{[M]}$'s and
$u_{[TM]}$'s with $M$ exceptional in $\A$, which is called the {\em composition Lie algebra} of $\mathcal {R}(\A)$. By definition, it is easy to see that the composition Lie algebra is invariant under derived equivalences.
Moreover, in a similar way to the proof in Theorem \ref{add conditions for A}, we obtain the following
\begin{Prop}\label{Lie algebra gen by exc seq}
Assume the conditions in Theorem \ref{add conditions for A} hold. Then for each complete exceptional sequence $\cE$ in $\A$, $L_{\cE}(\cR)=\cal{L}_{\mathscr{E}}(\cR)$.
\end{Prop}
\begin{Rem}
(1) In the case $\A=\modcat A$, the simple $A$-modules form a complete exceptional sequence when suitably ordered. Hence the composition Lie algebra can be defined via simple modules, {which in fact has been adopted as its original definition, see for example \cite{Ringel1990, PengXiao2}.}
(2) In the case $\A=\coh\bbX$, the composition Lie algebra is a nice model to realize the loop algebra of the Kac--Moody algebra associated to $\bbX$, for details we refer to Subsection \ref{composition lie algebra subsection}.
\end{Rem}
\section{Applications}
\subsection{Analogues of quantum Serre relations }
We provide some analogues of quantum Serre relations for any exceptional pair $(V_{\az},V_{\bz})$ in $\A$.
\begin{Prop}\label{Serre's relation}
Let $(V_{\az},V_{\bz})$ be an exceptional pair in $\A$,
\begin{itemize}
\item[(i)] if $\lr{\az,\bz}\leq0$, then
$$\sum_{j=0}^{n+1}(-1)^ju_{\az}^{\pm(n+1-j)}u_{\bz}^{\pm}u_{\az}^{\pm(j)}=0
=\sum_{j=0}^{m+1}(-1)^ju_{\bz}^{\pm(m+1-j)}u_{\az}^{\pm}u_{\bz}^{\pm(j)};$$
\item[(ii)] if $\lr{\az,\bz}>0$, then
$$\sum_{j=0}^{n+1}(-1)^j(v^{\ez({\az})})^{nj}u_{\az}^{\pm(n+1-j)}u_{\bz}^{\mp}u_{\az}^{\pm(j)}=0
=\sum_{j=0}^{m+1}(-1)^j(v^{\ez({\bz})})^{-mj}u_{\bz}^{\pm(m+1-j)}u_{\az}^{\mp}u_{\bz}^{\pm(j)}.$$
\end{itemize}
\end{Prop}
\begin{pf}
If $\lr{\az,\bz}\leq0$, then $(V_{\az},V_{\bz})$ are two relative simple objects in $\sC(V_{\az},V_{\bz})$, thus the first statement is well-known (see \cite{Ringel1,Ringel2}). If $\lr{\az,\bz}>0$, then one can use a similar proof as that for Proposition \ref{formula for left mutation} in Section 5 or Section 6 to obtain the result.
\end{pf}
\subsection{Lusztig's symmetries}
First of all, let us recall the isomorphism between the double composition algebra $D_{\mathscr{E}}(A):=D_{\mathscr{E}}(\modcat A)$ and the corresponding quantum group $U_q(\mathfrak{g})$ (cf. \cite{X97}), which is defined on generators by
$$\theta:D_{\mathscr{E}}(A)\longrightarrow U_q({\fg}),~~u_i^+\mapsto E_i,~~u_i^-\mapsto -v^{-\epsilon(i)}F_i,~~K_i\mapsto \tilde{K}_i.$$
The notations used here for elements of $U_q(\mathfrak{g})$ are the same as those in \cite[Chapter 3]{Lus}. Lusztig \cite[Chapter 37]{Lus} defined four families of symmetries as automorphisms of $U_q(\mathfrak{g})$, which are denoted by $T_{i,e}^{'}$ and $T_{i,e}^{''}$, where $e=\pm1$ and $i\in I$.
In the left mutation formula in Proposition $\ref{formula for left mutation}$(i), $V_{\az}$ and $V_{\bz}$ are two simple $\Lambda$-modules, say $\az=i$ and $\bz=j$. By the standard exact sequence
$$0\longrightarrow V_j\longrightarrow V_{\gz}\longrightarrow nV_i\longrightarrow 0,$$ where
$n=-\frac{2(i,j)}{(i,i)}$, we obtain that
$\dim_kV_{\gz}=\dim_kV_j+n\dim_kV_i=\epsilon(j)+n\epsilon(i)$. Besides,
$\epsilon(\gz)=\lr{\gz,\gz}=\lr{j+ni,j+ni}=\epsilon(j)+n\cdot(i,j)+n^2\epsilon(i)=\epsilon(j)
$. So $-\dim_kV_{\gz}+\epsilon(\gz)=-n\epsilon(i)$. Hence,
\begin{equation*}\begin{split}\theta(u_{\gz}^-)&=\sum_{r=0}^n(-1)^r(v^{\epsilon(i)})^{n-r}(-v^{-\epsilon(i)}F_i)^{(n-r)}(-v^{-\epsilon(j)}F_j)
(-v^{-\epsilon(i)}F_i)^{(r)}\\&=
(-1)^{n+1}v^{-\epsilon(j)}\sum_{r=0}^n(-1)^rv^{-r\epsilon(i)}F_i^{(n-r)}F_jF_i^{(r)}\\&=
(-1)^{n+1}v^{-\epsilon(j)}T_{i,1}^{'}(F_j).\end{split}\end{equation*}
Since $\epsilon(j)=\epsilon(\gz)$, we obtain that
\begin{equation}\label{L1}T_{i,1}^{'}(F_j)=(-1)^{n+1}v^{\epsilon(\gz)}\theta(u_{\gz}^-).\end{equation}
Similarly, we obtain that (see also \cite{CX,R4})
\begin{equation}T_{i,1}^{''}(E_j)=v^{-\dim_kV_{\gz}+\epsilon(\gz)}\theta(u_{\gz}^+).\end{equation}
In the right mutation formula in Proposition $\ref{formula for right mutation}$(i), $V_{\az}$ and $V_{\bz}$ are two simple $\Lambda$-modules, say $\az=i$ and $\bz=j$. By the standard exact sequence
$$0\longrightarrow mV_j\longrightarrow V_{\lz}\longrightarrow V_i\longrightarrow 0,$$ where
$m=-\frac{2(i,j)}{(j,j)}$, we obtain that $\dim_kV_{\lz}=\epsilon(i)+m\epsilon(j)$ and $\epsilon(\lz)=\epsilon(i)$, thus $-\dim_kV_{\lz}+\epsilon(\lz)=-m\epsilon(j)$. Hence,
\begin{equation*}\begin{split}\theta(u_{\lz}^-)&=\sum_{r=0}^m(-1)^r(v^{\epsilon(j)})^{m-r}(-v^{-\epsilon(j)}F_j)^{(r)}(-v^{-\epsilon(i)}F_i)
(-v^{-\epsilon(j)}F_j)^{(m-r)}\\
&=(-1)^{m+1}v^{-\epsilon(i)}\sum_{r=0}^m(-1)^rv^{-r\epsilon(j)}F_j^{(r)}F_iF_j^{(m-r)}\\&=
(-1)^{m+1}v^{-\epsilon(i)}T_{i,-1}^{''}(F_j).\end{split}\end{equation*}
Since $\epsilon(i)=\epsilon(\lz)$, we obtain that \begin{equation}T_{i,-1}^{''}(F_j)=(-1)^{m+1}v^{\epsilon(\lz)}\theta(u_{\lz}^-).\end{equation}
Similarly, we obtain that (see also \cite{CX,R4})
\begin{equation}\label{L4}T_{i,-1}^{'}(E_j)=v^{-\dim_kV_{\lz}+\epsilon(\lz)}\theta(u_{\lz}^+).\end{equation}
We remark that all four kinds of Lusztig's symmetries have appeared in Equations $(\ref{L1}-\ref{L4})$.
\subsection{Double composition algebras for weighted projective lines}\label{composition algebra of coh subsection}
The \emph{composition algebra} $C(\bbX)$ of $\coh\bbX$ is defined by Schiffmann \cite{Sch2004} as the subalgebra of the Ringel--Hall algebra $\cH(\coh\bbX)$ generated by $u_{\co(l\vc)}, l\in \bbZ$; $T_r, r\in\bbZ_{>0}$ and $u_{S_{ij}}, 1\leq i\leq t, 1\leq j\leq p_i-1$, which has been used to realize a certain ``positive part" of the quantum loop algebra of the Kac-Moody algebra associated to $\bbX$.
Later on, Burban and Schiffmann \cite{BurSch} defined an extended version of the composition algebra, denoted by {$U(\bbX):=C(\coh\bbX)\otimes \bf{K}$}, where $\bfK$ is the group algebra $\bbC[K(\coh\bbX)]$. They showed that $U(\bbX)$ is a topological bialgebra and then defined $DU(\bbX)$ as its \emph{reduced Drinfeld double}. Meanwhile, Dou, Jiang and Xiao \cite{DJX} defined the \emph{``double composition algebra"} $DC(\bbX)$ of $\coh\bbX$ as the subalgebra of $D(\coh\bbX)$ generated by two copies of $C(\bbX)$, say $C^{\pm}(\bbX)$, together with the group algebra $\bfK$.
\begin{Prop}\label{identity for composition algebra for wpl}
Keep notations as above. The following subalgebras of $D(\coh\bbX)$ coincide: $DC(\bbX)=DU(\bbX)=D_{\mathscr{E}}(\coh\bbX)$.
\end{Prop}
\begin{pf} It has been mentioned in \cite{DJX} that $DC(\bbX)=DU(\bbX)$. By \cite[Corollary 5.23]{BurSch}, the subalgebra $DU(\bbX)$ is generated by $u^{\pm}_{\co(\vx)}, \vx\in\bbL$ together with the torus {$\bf{K}$}. Note that each line bundle $\co(\vx)$ is exceptional in $\coh\bbX$. Hence by the definition of $D_{\mathscr{E}}(\coh\bbX)$, it contains $DU(\bbX)$ as a subalgebra. On the other hand, since $\bigoplus_{0\leq\vx\leq \vc}\co(\vx)$ forms a tilting object in $\coh\bbX$,
they can be arranged as a complete exceptional sequence of $\coh\bbX$. Then by the Main Theorem we obtain that $D_{\mathscr{E}}(\coh\bbX)\subseteq DU(\bbX)$, which completes the proof.
\end{pf}
\begin{Rem}
{The above proposition shows that for the category $\coh\bbX$, the reduced Drinfeld double composition algebra $DU(\bbX)$ defined by Burban and Schiffmann, and the ``double composition algebra" $DC(\bbX)$ defined by Dou, Jiang and Xiao, both coincide with the double composition algebra $D_{\mathscr{E}}(\coh\bbX)$ defined via exceptional objects. Moreover, by Corollary \ref{main examples}, they can be generated by any complete exceptional sequence in $\coh\bbX$.}
\end{Rem}
\subsection{Composition Lie algebras for weighted projective lines}\label{composition lie algebra subsection}
Now we consider the composition Lie algebra $\cal{L}(\cR):=\cal{L}(\cR(\coh\bbX))$ of $\coh\bbX$. In Theorem 2 of \cite{CB}, Crawley-Boevey found a series of elements in $\cal{L}(\cR)$ to satisfy the generating relations of the loop algebra of the Kac--Moody algebra associated to $\bbX$.
We emphasize that the subalgebra of $\cal{L}(\cR)$ generated by these elements, which we denote by $\cal{L}$$_c(\cR)$, is nothing else but the composition Lie algebra $\cal{L}_{\mathscr{E}}(\cR)$.
\begin{Prop}\label{composition Lie algebra for wpl}
Keep notations as above. Then $\cal{L}$$_c(\cR)=\cal{L}_{\mathscr{E}}(\cR)$.
\end{Prop}
\begin{pf}
We first claim that $\cal{L}_{\mathscr{E}}(\cR)\subseteq\cal{L}$$_c(\cR)$. In fact, note that $\cal{L}$$_c(\cR)$ contains the elements $u_{[M]}$ and $u_{[TM]}$ for $M=\co, \co(\vc)$ or $S_{ij}, 1\leq i\leq t, 1\leq j\leq p_i-1$. Moreover, these elements form a complete exceptional sequence in the following way
$$(\co,\co(\vc);S_{1,p_1-1},\cdots, S_{11};\cdots; S_{t,p_t-1},\cdots, S_{t1}).$$ Hence the claim follows from Proposition \ref{Lie algebra gen by exc seq}.
On the other hand, we need to show that $\cal{L}$$_c(\cR)\subseteq \cal{L}_{\mathscr{E}}(\cR).$ By definition we know that $c, h_{\star 0}, e_{\star r}, f_{\star r}$ and $e_{v 0}, f_{v0}$ belong to the $\cal{L}_{\mathscr{E}}(\cR)$ for $r\in\bbZ, v=ij, 1\leq i\leq t, 1\leq j\leq p_i-1$. It suffices to show that the other generators of $\cal{L}$$_c(\cR)$ can be generated by these elements. In fact, for any $r\neq 0$, $h_{\star r}=[e_{\star r}, f_{\star 0}]\in\cal{L}_{\mathscr{E}}(\cR)$. Moreover, by induction on the index $j$ (we write $\star=i0$ for convenience), we can obtain $e_{v r}, f_{v r}, h_{v r}$ belong to $\cal{L}_{\mathscr{E}}(\cR)$ for $v=ij$ and $r\neq 0$ by using the relations $e_{vr}=[e_{v0}, h_{v'r}], f_{vr}=[h_{v'r}, f_{v0}]\ (v'=i,j-1)$ and $h_{vr}=[e_{vr}, f_{v0}]$.
\end{pf}
\subsection{Bridgeland's Hall algebras and Modified Ringel--Hall algebras}
Let $\A$ be a finitary hereditary abelian $k$-category with enough projectives. In order to give an intrinsic realization of the entire quantized enveloping algebra via Hall algebras, Bridgeland \cite{Br} considered the Hall algebra of 2-cyclic complexes of projective objects in $\A$, and achieved an algebra $\mathcal {D}\mathcal {H}_{{\rm red}}(\mathcal{A})$, called the {\em reduced Bridgeland's Hall algebra} of $\A$, by taking some localization and reduction. By \cite{Yan,ZHC}, $\mathcal {D}\mathcal {H}_{{\rm red}}(\mathcal{A})$ is isomorphic to the reduced Drinfeld double Hall algebra $D(\A)$. In order to generalize Bridgeland's construction to any hereditary abelian category $\A$, Lu and Peng \cite{LP} defined an algebra $\cM\cH_{\bbZ/2,{\rm tw,red}}(\A)$, called the {\em reduced modified Ringel--Hall algebra} of $\A$, and proved that it is also isomorphic to $D(\A)$.
Hence, our results can also be applied to $\mathcal {D}\mathcal {H}_{{\rm red}}(\mathcal{A})$ and $\cM\cH_{\bbZ/2,{\rm tw,red}}(\A)$.
\section*{Acknowledgments}
The authors are grateful to Bangming Deng, Jie Xiao and Jie Sheng for their helpful comments. In particular, Section 6 comes from Deng's suggestion.
|
2,869,038,154,734 | arxiv | \section{Introduction}
Support vector machine (SVM)~\cite{cristianini2000} is a popular methodology applicable to extensive domains requiring prediction or statistical analysis including text categorization, medical and biological study, economics, psychology and social science. The methodology learns the coefficient vector ${\bm{w}}:=\left[w_{1},\dots,w_{d}\right]^\top$ of a linear classifier $\left<{\bm{w}},\cdot\right>$ from a set of examples, each of which consists in $d$ features and a binary class label.
Usually, domain knowledge is exploited when designing features so that the selected features are possibly correlated to the binary category. In many cases, for each selected feature, the sign of the correlation to the binary category is known in advance. For example, when predicting the pathogen concentration in a river, we may use water quality indices. In the field of the water quality engineering~\cite{KatKobOis-jwh19,Varela18a,KatKobIto15a}, no one doubts the fact that EC, SS, BOD, TN, TP and WT are positively correlated to the concentration of \textit{Escherichia coli} (\textit{E. coli}) in a river, and DO and FW have a negative correlation~\cite{KatKobOis-jwh19}. The hydrogen exponent is also informative for \textit{E. coli} count prediction because the organism cannot survive in acidic or basic water.
It would be ideal if the signs of coefficients ${\bm{w}}$ coincided with the signs of true correlations to the binary class labels. In case that $h$-th feature variable $x_{h}$ is positively correlated to the class label, then a positive weight coefficient $w_{h}$ is expected to achieve a better generalization performance than a negative coefficient. However, the standard SVM learning cannot prevent the learned weight from being negative if the correlation is weak, and vice versa.
In this paper, we discuss constraining the signs of the coefficients explicitly in SVM learning.
A simple approach to the constrained optimization is the \emph{projected gradient} method~\cite{Bertsekas99} in which a step for projection onto the feasible region is inserted at each iteration of the gradient method. We focus on \emph{Pegasos} method~\cite{Shalev-Shwartz11a-pegasos}, a popular gradient method for SVM learning with the \emph{sublinear convergence} guaranteed.
We consider inserting the projection step into each of Pegasos iteration, and analyzed the number of iterations to ensure a solution ${\bm{w}}$ to be $\epsilon$-\emph{accurate} (this term is defined in Section III). We have found that the required number of iterations to ensure $\epsilon$-accuracy is bounded by twice the iteration numbers taken by the original Pegasos method.
The time complexity for each iteration of the Pegasos-based method is $O(nd)$, which is equal to that for the original Pegasos.
The projected gradient method suffers from a drawback which is the incapability of the judgement for attaining $\epsilon$-accurate solution.
We present an alternative algorithm that possesses the clear termination criterion guaranteeing the $\epsilon$-accuracy for the solution.
The alternative learning algorithm does not solve the primal problem directly, but solves the dual problem of the sign constrained SVM learning problem. To maximize the dual objective function, the \emph{Frank-Wolfe} framework~\cite{Levitin66,jaggi13icml} is adopted, which allows the proposed algorithm to inherit the sublinear convergence property from the Frank-Wolfe framework.
Each iteration of the Frank-Wolfe framework consists of two steps: the \emph{direction finding} step and the \emph{line search} step. In this study, we have obtained the following findings: In the Frank-Wolfe algorithm for solving the dual problem to SVM learning under sign constraints,
both the direction finding and line search steps can be performed within $O(nd)$ computational cost.
Hence, we have reached an analytical result that each iteration of the Frank-Wolfe algorithm takes $O(nd)$ computational cost which is same as the time complexity of each iteration in the projected gradient algorithm. This suggests that the proposed Frank-Wolfe algorithm totally shares all the advantages of the projected gradient algorithm, and furthermore overcomes a drawback: lack of the termination criterion.
Exploitation of domain knowledge is a straightforward application of sign constraints, although sign constraints is effective when utilizing exploitation of a task structure.
We found that the sign constraints are advantageous to the task structure of the biological sequence classification~\cite{LiaNob03-jcb,KatTsuAsa05a}.
Predicting some properties such as structural similarity and cellular functionality of a protein is a central issue in bioinformatics to understand the mechanism in the cell~\cite{KinKatTsu04a,KatNag10a}. A protein is a sequence of amino acids. More similar amino-acid sequences share a common ancestor with higher probability, tending to have common properties such as structural similarity and cellular function. Hence, sequence alignment is a typical methodology to infer the properties of unknown proteins. In this study, we found that combining sequence similarities with sign-constrained SVM provides an effective methodology for protein function inference. We shall demonstrate the efficiency on the new application of sign-constrained SVM to protein function inference.
This paper is organized as follows: After discussing related work in next section, the learning problem to be minimized is formulated in Section~\ref{s:scsvm}. The projected gradient algorithm and the analysis are analyzed in Section~\ref{s:pega}. In Section~\ref{s:dual}, the dual problem is presented and, in Section~\ref{s:stdfw}, the Frank-Wolfe algorithm for solving the dual problem is analyzed. Experimental results for convergence behaviors and the application to biological sequence classification are presented in Section~\ref{s:exp}. In the last section, this paper is concluded with future work for exploring the potential of sign constraints.
\section{Related Work}
\label{s:related}
For discriminative learning, use of sign constraints has not been discussed well so far, although non-negative least square regression for other applications has been explored extensively.
The applications of non-negative least square regression include non-negative image restoration~\cite{Henrot2013-icassp},
face representation~\cite{YangfengJi2009-icmla,HeZheHu13}, microbial analysis~\cite{CaiGuKen17}, image super-resolution~\cite{DonFuShi16}, spectral analysis~\cite{QiangZhang07-asrc}, tomographic imaging~\cite{JunMa2013-algo}, and sound source localization~\cite{YuanqingLin2004-icassp}.
Some of them used non-negative least square regression as a key ingredient for the non-negative matrix factorization~\cite{lee2001algorithms}.
Non-negative least squares estimation can be computed efficiently. A basic algorithm for the non-negative least regression is the active set method~\cite{Lawson1995solving} which is accelerated by integrating projected gradient approach~\cite{DongminKim2010-siamjsc}.
These algorithms are computationally stable and fast. However, these approaches help to minimize the empirical risk only for the case that the loss function is the square error.
We are aware of an existing report in which prior knowledge for coefficients of linear classifiers is exploited~\cite{Fernandes2017}, as with our approach. They add a term penalizing the violation to the prior knowledge. The penalizing strength is controlled by a constant coefficient.
The learning problem discussed in our study is an extreme case of their formulation~\cite{Fernandes2017}.
They employ a gradient method for minimizing the penalized empirical risk, although the details of the algorithm were not provided.
This paper focuses on how to minimize the empirical risk under sign constraints in well-appointed framework, to rigorously analyze the convergence rate and the computational cost.
Without sign constraints, many optimization algorithms are available for generic empirical risk minimization~\cite{Roux12a-sag,Schmidt2016-sag,Johnson13a-svrg,LinXiao2014-siamjo,Defazio2014-nips}.
Many of them require a step size which is in need to be chosen manually in advance, although theoretically guaranteed step sizes are too small to converge the solution to the minimum for practical use.
The Frank-Wolfe algorithm developed in this paper works without any step-size as well as any hyper-parameter for optimization.
Proposal for use of sign constraints in biological sequence analysis is another contribution of this paper. To find the evidence for homology between two biological sequences, sequence similarity has been used frequently~\cite{SmiWat81,AltGisMil90,Pea90}. Liao et al~\cite{LiaNob03-jcb} devised a method, named \emph{SVM-pairwise}, that combines the sequence comparison approach with SVM. In SVM-pairwise, the sequence similarities to all known sequences are used to train SVM as a feature vector. Supported with the remarkable success of SVM-pairwise, a large number of similar techniques have been developed for different tasks in the computational biology~\cite{LanDenCri04,LanBieCri04,LiuZhaXu14,OguMum06,KatTsuAsa05a}. In this paper, it is demonstrated that the generalization power of SVM-pairwise can be improved significantly by exploiting the effect of sign constraints.
\section{Sign-Constrained SVM}
\label{s:scsvm}
In this section, we consider imposing sign constraints for SVM learning. The sign constraints are posed for the coefficients for which the signs of the correlations between the corresponding features and the class labels are available in advance. Let us denote by ${\mathcal{I}}_{+}$ and ${\mathcal{I}}_{-}$ the index set of features known to be correlated to the class label positively and negatively in advance, where ${\mathcal{I}}_{+}\cap{\mathcal{I}}_{-}=\emptyset$ and ${\mathcal{I}}_{+}\cup{\mathcal{I}}_{-}\subseteq [d]$. The sign constraints are defined as
\begin{tsaligned}
& \forall h\in{\mathcal{I}}_{+}, \,\, w_{h}\ge 0
\quad
\text{and}
\quad
\forall h'\in{\mathcal{I}}_{-}, \,\, w_{h'}\le 0.
\end{tsaligned}
We may eliminate the non-positive constraints because these constraints can be transformed to the non-negative constraints by negating features for ${\mathcal{I}}_{+}$ in advance. Hereinafter, we assume that this preprocess is performed in advance. Then, ${\mathcal{I}}_{-}=\emptyset$. We introduce a constant vector ${\bm{\sigma}}:=\left[\sigma_{1},\dots,\sigma_{d}\right]^\top\in\{0,1\}^{d}$ such that
\begin{tsaligned}
\sigma_{h}:=\mathds{1}[h\in{\mathcal{I}}_{+}]
\end{tsaligned}
where $\mathds{1}[x]=1$ if the logical argument $x$ is true; otherwise, $\mathds{1}[x]=0$. Then, the sign constraints can be rewritten simply as ${\bm{\sigma}}\odot{\bm{w}}\ge{\bm{0}}$ where the operator $\odot$ denotes the Hadamard product.
The feasible region is expressed as
\begin{tsaligned}
{\mathcal{S}} :=
\left\{
{\bm{w}}\in{\mathbb{R}}^{d}\,\middle|\,
{\bm{\sigma}}\odot{\bm{w}}\ge{\bm{0}}_{d}
\right\}.
\end{tsaligned}
The sign-constrained SVM learning problem is described as
\begin{tsaligned}\label{eq:scsvm-primal}
\text{min}\quad
&
P({\bm{w}})
\quad
\text{wrt}
\quad
{\bm{w}}\in{\mathcal{S}},
\end{tsaligned}
where
\begin{tsaligned}\label{eq:scsvm-primalobj-def}
&
P({\bm{w}})
:=
\frac{\lambda}{2}\lVert{\bm{w}}\rVert^{2}
+
\frac{1}{n}
\sum_{i=1}^{n}
\max(0,1-y_{i}\left<{\bm{x}}_{i},{\bm{w}}\right>).
\end{tsaligned}
It can be seen that the optimization problem~\eqref{eq:scsvm-primal}
is the standard SVM learning problem if ${\bm{\sigma}}={\bm{0}}_{d}$.
In this study, two algorithms, a projected gradient algorithm and a Frank-Wolfe algorithm, were developed for solving the learning problem~\eqref{eq:scsvm-primal}. The two algorithms are presented in Section \ref{s:pega} and \ref{s:stdfw}, respectively.
We shall use ${\bm{w}}_{\star}$ to denote the solution optimal to our learning problem~\eqref{eq:scsvm-primal}.
A solution ${\bm{w}}$ is said to be $\epsilon$-\emph{accurate} when $P({\bm{w}})-P({\bm{w}}_{\star})\le\epsilon$.
\section{Projected Gradient Algorithm}
\label{s:pega}
Our projected gradient algorithm for the sign constrained SVM was developed by inserting the projection step into each step of Pegasos method~\cite{Shalev-Shwartz11a-pegasos}.
The convergence theory of the original Pegasos method is based on the property that the optimal solution to the standard SVM learning problem is in the ball with radius $\sqrt{1/\lambda}$. Imposing the sign constraints breaks down this property, although we found that, if the sign constraints are imposed, the norm of the optimal solution is still bounded. The optimal solution lies in a slightly larger ball with radius $\sqrt{2/\lambda}$, denoted by ${\mathcal{B}}:=\left\{ {\bm{w}}\in{\mathbb{R}}^{d}\,\middle|\,\lVert{\bm{w}}\rVert\le\sqrt{2/\lambda}\right\}$.
The projected gradient algorithm developed in this study starts with ${\bm{w}}^{(1)}:={\bm{0}}$. At the $t$-th iteration, the solution is updated as
\begin{tsaligned}\label{eq:scpega-update}
{\bm{w}}^{(t+1)}:=
\Pi_{{\mathcal{B}}}\left(\Pi_{{\mathcal{S}}}\left({\bm{w}}^{(t)}-\frac{1}{\lambda t}\nabla P({\bm{w}}^{(t)})
\right)\right).
\end{tsaligned}
Therein, $\Pi_{{\mathcal{B}}}({\bm{x}})$ and $\Pi_{{\mathcal{S}}}({\bm{x}})$ denote the Euclidean projection from a point ${\bm{x}}\in{\mathbb{R}}^{d}$ onto the ball ${\mathcal{B}}$ and the feasible region ${\mathcal{S}}\subseteq{\mathbb{R}}^{d}$, respectively.
The projection onto ${\mathcal{S}}$ can be expressed as
\begin{tsaligned}
\Pi_{{\mathcal{S}}}({\bm{v}}) := \mathop{\textrm{argmin}}\limits_{{\bm{w}}\in{\mathcal{S}}}\lVert{\bm{v}}-{\bm{w}}\rVert
= {\bm{v}} + \max({\bm{0}},-{\bm{\sigma}}\odot{\bm{v}}).
\end{tsaligned}
Assume that $\forall i\in[n]$, $\lVert{\bm{x}}_{i}\rVert\le R$.
To analyze the convergence of the algorithm \eqref{eq:scpega-update},
we observe the following properties:
\begin{itemize}
%
\item ${\mathcal{B}}\cap{\mathcal{S}}$ is a closed convex set;
%
\item ${\bm{w}}_{\star}\in{\mathcal{B}}\cap{\mathcal{S}}$;
%
\item
$\forall{\bm{w}}\in{\mathbb{R}}^{d}$;
$\Pi_{{\mathcal{B}}}\left(\Pi_{{\mathcal{S}}}\left({\bm{w}}\right)\right)=\Pi_{{\mathcal{B}}\cap{\mathcal{S}}}\left({\bm{w}}\right)$;
%
\item
$\forall t\in{\mathbb{N}}$,
$\lVert\nabla P({\bm{w}}_{t})\rVert \le \sqrt{2\lambda} + R$.
%
\end{itemize}
Combining these properties with the proof techniques
used by Shalev-Shwartz et al~\cite{Shalev-Shwartz11a-pegasos},
we obtain the following theorem.
\begin{theorem-waku}
It then holds that, $\forall T\in{\mathbb{N}}$, $\exists T'\in[T]$,
the primal objective error is bounded as
\begin{tsaligned}
P({\bm{w}}^{(T')})-P({\bm{w}}_{\star}) \le
\frac{(\sqrt{2\lambda}+R)^{2}\log(T)}{\lambda T}.
\end{tsaligned}
\end{theorem-waku}
A weak point of the projected gradient method is the lack of a termination criterion. There is no way to check the $\epsilon$-accuracy, because the minimal value $P({\bm{w}}_{\star})$ is usually unknown and thereby the objective error $P({\bm{w}}^{(t)})-P({\bm{w}}_{\star})$ cannot be assessed.
\begin{figure}[t!]
\centering
\begin{tabular}{ll}
&
\includegraphics[width=0.45\linewidth]{001-demo894-06.try01.ax2.iter002.030.eps}
\end{tabular}
\caption{%
Iteration of Frank-Wolfe algorithm. In $t$-th iteration, the dual problem~\eqref{eq:prob-scsvm-dual} in which the dual objective $D$ is replaced to its linear approximation~\eqref{eq:lmo-scsvm} is solved. The solution is moved to the point at which the dual objective $D$ is maximized on the line segment between the current solution ${\bm{\alpha}}^{(t-1)}$ and the optimal solution to the sub-problem, say ${\bm{u}}_{t}$.
\label{fig:demo894-06-fwiter}%
}
\end{figure}
\begin{figure}[t!]
\centering
\begin{tabular}{lll}
(a) &
\\
\multicolumn{2}{l}{%
\includegraphics[width=0.73\linewidth]{002-demo890-01.try06-w-symbols.eps}%
}%
\\
(b) &
(c)
\\
\includegraphics[width=0.49\linewidth]{003-demo890-01.try09.eps}
&
\includegraphics[width=0.41\linewidth]{004-demo890-01.try10.eps}
\end{tabular}
\caption{
Objective function for line search problem, say $\zeta(\eta)$, is a
piecewise quadratic function.
In the $h$-th interval $[\theta_{h},\theta_{h+1}]$,
the function is expressed in the form of
$\zeta(\eta)=a_{h}\eta^{2}+b_{h}\eta+c_{h}$.
There are three cases in the line search problem.
(a) One of three cases
in which $\nabla\zeta(0)>0>\nabla\zeta(1)$.
The point $\eta$ with the gradient vanishing
is the maximizer of the function $\zeta(\eta)$
in this case.
(b) In case of $0\ge\nabla\zeta(0)$,
the function $\zeta(\eta)$ attains the maximum
at $\eta=1$.
(c) In case of $\nabla\zeta(1)\ge 0$,
$\zeta(\eta)$ is maximized at $\eta=0$.
\label{fig:demo890-01-zeta-3cases}
}
\end{figure}
\section{Dual Problem}
\label{s:dual}
The projected gradient algorithm described in the previous section suffers from the lack of a clear criterion for when iterations should be terminated.
To obtain a termination criterion, we consider solving a dual problem instead of the primal problem. The following problem is dual to the primal problem~\eqref{eq:scsvm-primal}:
\begin{tsaligned}\label{eq:prob-scsvm-dual}
\text{max}\quad&
D({\bm{\alpha}})
\quad
\text{wrt}\quad
{\bm{\alpha}}\in[0,1]^{n},
\\
\text{where}\quad&
D({\bm{\alpha}}):=
-\frac{\lambda}{2}\lVert{\bm{w}}({\bm{\alpha}})\rVert^{2} + \frac{1}{n}\left<{\bm{1}},{\bm{\alpha}}\right>,
\\
&
{\bm{w}}({\bm{\alpha}}):=\Pi_{{\mathcal{S}}}\left(\frac{1}{\lambda n}{\bm{X}}{\bm{\alpha}}\right).
\end{tsaligned}
where a $d$-dimensional vector $y_{i}{\bm{x}}_{i}$ is stored in the $i$-th column of the matrix ${\bm{X}}\in{\mathbb{R}}^{d\times n}$.
The dual problem for the sign-constrained SVM is reduced to that of the standard SVM when no sign constraints are imposed, which can be seen as follows: in the case of no sign constraints, ${\mathcal{S}}={\mathbb{R}}^{d}$, leading to $\Pi_{{\mathcal{S}}}({\bm{v}})={\bm{v}}$ and ${\bm{w}}({\bm{\alpha}})={\bm{X}}{\bm{\alpha}}/(\lambda n)$; substituting them into \eqref{eq:prob-scsvm-dual}, the dual problem for the standard SVM is derived.
This fact suggests that the difference in the dual objective functions between the sign-constrained and standard SVMs is the presence or absence of the projection onto the feasible region. The projection operation in the dual objective for the sign-constrained SVM precludes the direct use of the standard optimization approach to learning the standard SVM, and thereby makes the optimization problem challenging.
The primal variable optimal to the primal problem~\eqref{eq:scsvm-primal} can be recovered by ${\bm{w}}_{\star}={\bm{w}}({\bm{\alpha}}_{\star})$ where ${\bm{\alpha}}_{\star}$ is a solution optimal to the dual problem~\eqref{eq:prob-scsvm-dual}, which suggests that iterations in some dual optimization algorithm can be stopped when the duality gap is sufficiently small (i.e. $P({\bm{w}}({\bm{\alpha}}))-D({\bm{\alpha}})\le\epsilon$). From the nature of the duality:
\begin{tsaligned}
\forall{\bm{\alpha}}\in[0,1]^{n},\qquad P({\bm{w}}({\bm{\alpha}}))\ge P({\bm{w}}_{\star}) \ge D({\bm{\alpha}}),
\end{tsaligned}
the duality gap lower than a pre-defined threshold $\epsilon$ ensures the primal objective error below the threshold $\epsilon$.
\section{Frank-Wolfe Algorithm}
\label{s:stdfw}
In this section, we present a dual optimization algorithm based on the Frank-Wolfe framework~\cite{Levitin66,jaggi13icml} for learning the sign constrained SVM. The Frank-Wolfe-based algorithm presented in this section is alternative to the projected gradient algorithm described in the previous section.
In general, the Frank-Wolfe framework performs optimization within a convex polyhedron. The feasible region of the dual problem for the sign-constrained SVM is a hyper-cube which is in a class of convex polyhedrons, suggesting that the dual problem can be a target of the Frank-Wolfe framework.
\begin{algorithm}[t!]
\caption{
Frank-Wolfe algorithm for solving the dual problem \eqref{eq:prob-scsvm-dual}.
\label{algo:stdfw-svm}}
\Begin{
Let ${\bm{\alpha}}^{(0)}\in[0,1]^{n}$\;
\For{$t:=1$ \KwTo $T$}{
${\bm{u}}_{t}\in\mathop{\textrm{argmax}}\limits_{{\bm{u}}\in[0,1]^{n}}\left<\nabla D({\bm{\alpha}}^{(t-1)}),{\bm{u}}\right>$\;
${\bm{q}}_{t}:={\bm{u}}_{t}-{\bm{\alpha}}^{(t-1)}$\;
%
$\eta_{t}\in\arg\max_{\eta\in[0,1]}D({\bm{\alpha}}^{(t-1)}+\eta{\bm{q}}_{t})$\;
%
${\bm{\alpha}}^{(t)} := {\bm{\alpha}}^{(t-1)}+\eta_{t}{\bm{q}}_{t}$\;
}
}
\end{algorithm}
As described in Algorithm~\ref{algo:stdfw-svm}, each iteration of the Frank-Wolfe framework consists of two steps, respectively, referred to as the \emph{direction finding step} and the \emph{line search step}.
In the direction finding step, a linear programming problem is solved.
In the sub-problem, the dual objective is replaced with its linear approximation
around a previous solution:
\begin{tsaligned}
{\bm{u}}\mapsto
\left<\nabla D({\bm{\alpha}}^{(t-1)}),{\bm{u}}\right>
+
D({\bm{\alpha}}^{(t-1)}).
\end{tsaligned}
The linear approximation is maximized within the convex polyhedron, yielding
a linear programming problem. We denote by ${\bm{u}}_{t}$
the optimal solution to the sub-problem at $t$-th iteration.
In the line search step, the dual objective is maximized on the line segment between the previous solution and the solution optimal to the linear programming problem, say ${\bm{\alpha}}^{(t-1)}$ and ${\bm{u}}_{t}$, respectively. The updated solution can be expressed as ${\bm{\alpha}}^{(t)}:={\bm{\alpha}}^{(t-1)}+\eta_{t}{\bm{q}}_{t}$ where
\begin{tsaligned}
\eta_{t}\in
\mathop{\textrm{argmax}}\limits_{\eta\in[0,1]}
D({\bm{\alpha}}^{(t-1)}+\eta{\bm{q}}_{t})
\text{ and }
{\bm{q}}_{t}:={\bm{u}}_{t}-{\bm{\alpha}}^{(t-1)}.
\end{tsaligned}
As long as both the sub-problems involved in the two steps are exactly solved at each iteration,
the sublinear convergence is ensured.
In addition, the accuracy of the solution is guaranteed by use of the duality gap $P({\bm{w}}({\bm{\alpha}}))-D({\bm{\alpha}})$ for a stopping criterion. If the computational cost of each iteration in the Frank-Wolfe algorithm is within $O(nd)$, then the new approach share all the favorable properties of the projected gradient method and simultaneously overcomes the shortcoming.
These discussions suggest that
both the direction finding step and the line search step need to be computed within $O(nd)$, in order to keep the computational cost of each iteration from exceeding $O(nd)$.
\subsection{Direction Finding Step}
\label{ss:lmo}
It is shown here that $O(nd)$ computation accomplishes the direction finding step in the Frank-Wolfe algorithm (Algorithm~\ref{algo:stdfw-svm}). The direction finding step requires to solve the following sub-problem
\begin{tsaligned}\label{eq:lmo-scsvm}
\text{min}\quad&
\left<\nabla D({\bm{\alpha}}^{(t-1)}), {\bm{u}}\right>
\quad
\text{wrt}\quad
{\bm{u}}\in[0,1]^{n}.
\end{tsaligned}
The sub-problem~\eqref{eq:lmo-scsvm} is in a class of linear programming problems.
The direction finding step consumes $O(n^{3})$ computation if we resort to a general-purpose linear programming solver. The computational cost $O(n^{3})$ is computationally taxing when training with a large set of training examples.
In this study, we have found a closed-form solution optimal to the linear programming problem~\eqref{eq:lmo-scsvm}. The $i$-th entry in the optimal solution ${\bm{u}}_{t}\in[0,1]^{n}$ is expressed as
\begin{tsaligned}\label{eq:lmosol-scsvm}
u_{i,t}
=
\begin{cases}
1 \qquad \text{if }y_{i}\left<{\bm{w}}({\bm{\alpha}}^{(t-1)}),{\bm{x}}_{i}\right> < 1,
\\
0 \qquad \text{if }y_{i}\left<{\bm{w}}({\bm{\alpha}}^{(t-1)}),{\bm{x}}_{i}\right> \ge 1.
\end{cases}
\end{tsaligned}
\tslong{See Subsection~\ref{ss:deriv-eq:lmosol-scsvm}
for the derivation. }%
\tsshort{See the supplement
for the derivation. }%
The optimal solutions to the sub-problem~\eqref{eq:lmo-scsvm} is not unique.
The sub-linear convergence is ensured even if taking any point among the set of the optimal solutions.
\textbf{Time Complexity of Direction Finding Step: }
The procedure and the time complexities for computation of the closed-form solution \eqref{eq:lmosol-scsvm}
are summarized as follows.
\begin{center}
\begin{tabular}{ll}
Compute ${\bm{v}}^{(t-1)} := {\bm{X}}{\bm{\alpha}}^{(t-1)}/(\lambda n)$;
& $O(nd)$.
\\
Compute ${\bm{w}}^{(t-1)} := \Pi_{{\mathcal{S}}}({\bm{v}}^{(t-1)})$;
& $O(d)$.
\\
Compute
${\bm{z}}^{(t-1)} := {\bm{X}}^\top{\bm{w}}^{(t-1)}$;
& $O(nd)$.
\\
$\forall i\in[n]$,
$ u_{i,t} := \mathds{1}(1\le z_{i}^{(t-1)})$;
& $O(n)$.
\end{tabular}
\end{center}
Therein, $z_{i}^{(t-1)}$ is the $i$-th entry in the vector ${\bm{z}}^{(t-1)}$,
and the function $\mathds{1}(\cdot)$ takes the value of one if the logical argument is
true; otherwise zero.
The line search problem can, thus, be solved in $O(nd)$
when $d$ is within $O(n)$.
\subsection{Line Search Step}
\label{ss:lnsrch}
It has been seen that the direction finding step can be performed within $O(nd)$ computational cost. It shall be shown here that the line search step requires only $O(nd)$ cost, too.
Line search is an operation that finds an optimal solution, denoted by $\eta_{\star}\in[0,1]$,
to the sub-problem:
\begin{tsaligned}\label{eq:prob-lnsrch}
\text{max}\quad
&
\zeta(\eta)
\quad
\text{wrt}\quad
\eta\in[0,1],
\\
\text{where}\quad
&
\zeta(\eta):=D({\bm{\alpha}}+\eta{\bm{q}}),
\quad
{\bm{\alpha}} \in [0,1]^{n},
\\
& {\bm{q}}\in [0,1]^{n}-{\bm{\alpha}},\,\, {\bm{q}}\ne{\bm{0}}_{n}.
\end{tsaligned}
In case of ${\bm{q}}={\bm{0}}_{n}$, it is obvious that any $\eta\in[0,1]$ is optimal. Here, our discussion is limited to the case ${\bm{q}}\ne {\bm{0}}_{n}$ as given in \eqref{eq:prob-lnsrch}.
The following is the key lemma for solving the line search problem.
\begin{lemma-waku}\label{lem:lnsrch-is-piesequadfun}
The objective function of the sub-problem given in \eqref{eq:prob-lnsrch},
say $\zeta:[0,1]\to{\mathbb{R}}$, is a differentiable and concave piecewise quadratic function
and its derivative is continuous and monotonically decreasing.
Namely, there exist an integer $d_{t}\in[d+1]$,
coefficients $(a_{k},b_{k},c_{k})$ for $k\in[d_{t}]$
with $a_{k}\le 0$ and
$\theta_{0},\dots,\theta_{d_{t}+1}$ such that
$0 = \theta_{1} < \theta_{2} < \dots < \theta_{d_{t}+1} = 1$
and the function can be expressed as
$\forall k\in[d_{t}]$,
$\forall \eta\in[\theta_{k},\theta_{k+1}]$,
%
\begin{tsaligned}\label{eq:01-lnsrch-is-piesequadfun}
\zeta(\eta) = a_{k}\eta^{2} + b_{k}\eta + c_{k},
\end{tsaligned}
and $\forall k\in[d_{t}-1]$,
\begin{tsaligned}\label{eq:02-lnsrch-is-piesequadfun}
2 a_{k}\theta_{k+1} + b_{k} = 2 a_{k+1}\theta_{k+1} + b_{k+1}.
\end{tsaligned}
\end{lemma-waku}
\tslong{See Subsection~\ref{ss:proof-lem:lnsrch-is-piesequadfun}
for the proof of Lemma~\ref{lem:lnsrch-is-piesequadfun}. }%
\tsshort{See the supplement
for the proof of Lemma~\ref{lem:lnsrch-is-piesequadfun}. }%
\textbf{Solution to Line Search: }
Lemma~\ref{lem:lnsrch-is-piesequadfun} suggests that
three cases described in Figure~\ref{fig:demo890-01-zeta-3cases}.
From this observation, it turns out that
an optimal solution to the line search problem \eqref{eq:prob-lnsrch} can be
given in a closed-form as
\begin{tsaligned}\label{eq:sol-to-lnsrch}
\eta_{\star}
=
\begin{cases}
0 \qquad& \text{if }b_{1} \le 0,
\\
1 \qquad& \text{if }2 a_{d_{t}+1} + b_{d_{t}+1} \ge 0,
\\
-
\frac{b_{k_{\star}}}{2a_{k_{\star}}}
\qquad&
\text{if $b_{1} > 0$, $2 a_{d_{t}+1} + b_{d_{t}+1} < 0$, $a_{k_{\star}}< 0$,}
\\
\theta_{k_{\star}}
\qquad&
\text{if $b_{1} > 0$, $2 a_{d_{t}+1} + b_{d_{t}+1} < 0$, $a_{k_{\star}}= 0$}
\end{cases}
\end{tsaligned}
where $k_{\star}\in[d_{t}]$ here is the index of one of
$d_{t}$ intervals,
in which the derivative vanishes at some point. Namely,
\begin{tsaligned}
2 a_{k_{\star}}\theta_{k_{\star}} + b_{k_{\star}} \ge 0 \quad\text{ and }\quad
2 a_{k_{\star}}\theta_{k_{\star}+1} + b_{k_{\star}} \le 0.
\end{tsaligned}
Note that, from the definition of $k_{\star}$, it must hold that $b_{k_{\star}}=0$
in the case
that $b_{1} > 0$, $2 a_{d_{t}+1} + b_{d_{t}+1} < 0$ and $a_{k_{\star}}= 0$.
\textbf{Endpoints and Coefficients of Piecewise Quadratic Function $\zeta$: }
Here we describe
how to determine the endpoints
$\theta_{1},\theta_{2},\dots,\theta_{d_{t}+1}\in[0,1]$
and the coefficients of the piecewise quadratic function $\zeta$,
say $(a_{k},b_{k},c_{k})$ for $k=1,\dots,d_{t}$.
Let ${\bm{v}}_{0}:=\left[v_{1,0},\dots,v_{d,0}\right]^{\top}$
and ${\bm{v}}_{q}:=\left[v_{1,q},\dots,v_{d,q}\right]^{\top}$
have the following entries, $\forall h\in[d]$,
\begin{tsaligned}
v_{h,0} := \frac{1}{\lambda n}\left<{\bm{f}}_{h},{\bm{\alpha}}\right>, \quad
v_{h,q} := \frac{1}{\lambda n}\left<{\bm{f}}_{h},{\bm{u}}-{\bm{\alpha}}\right>.
\end{tsaligned}
where ${\bm{f}}_{h}\in{\mathbb{R}}^{n}$ is the $h$-th column vector in ${\bm{X}}^{\top}$.
Let
\begin{multline}
\Theta
:=
\Bigg\{
\theta\in[0,1]
\,\Bigg|\,
\exists h\in{\mathcal{I}}_{+}
\\
\text{ s.t. }
v_{h,q} \ne 0, \,
\theta = - \frac{v_{h,0}}{v_{h,q}}
\Bigg\}
\cup \{ 0, 1\}.
\end{multline}
The number of intervals $d_{t}$ can be set to
the cardinality of the set $\Theta$ minus one
(i.e. $d_{t} = \text{card}(\Theta)-1$)
and the endpoints $\theta_{1},\dots,\theta_{d_{t}+1}$ of $d_{t}$
intervals are determined by sorting the elements in $\Theta$
so that $\theta_{1} < \theta_{2} < \dots < \theta_{d_{t}+1}$.
In this setting, it can be observed that
equation~\eqref{eq:01-lnsrch-is-piesequadfun} is satisifed
if coefficients are given as
$\forall k\in[d_{t}]$,
\begin{tsaligned}\label{eq:coef-piesequadfun}
& a_{k}:=- \frac{\lambda}{2}
\sum_{h\in{\mathcal{H}}_{k}} v_{h,q}^{2},
\\
&
b_{k}:=
\frac{1}{n}\left<{\bm{1}},{\bm{u}}-{\bm{\alpha}}\right>
-
\lambda \sum_{h\in{\mathcal{H}}_{k}} v_{h,q}v_{h,0},
\\
&
c_{k}
:=
\frac{1}{n}\left<{\bm{1}},{\bm{\alpha}}\right>
-
\frac{\lambda}{2}
\sum_{h\in{\mathcal{H}}_{k}} v_{h,0}^{2}
\end{tsaligned}
where
\begin{tsaligned}
{\mathcal{H}}_{k}
:=
{\mathcal{I}}_{0}
\cup
\left\{
h\in{\mathcal{I}}_{+}
\,\middle|\,
v_{h,0} + \frac{\theta_{k}+\theta_{k+1}}{2} v_{h,q} > 0
\right\}.
\end{tsaligned}
\textbf{Time Complexity of Line Search Step: }
The procedure required to compute the solution \eqref{eq:sol-to-lnsrch}
and the time complexity of each step is described as follows.
\begin{center}
%
\begin{tabular}{ll}
Compute ${\bm{v}}_{0}$ and ${\bm{v}}_{q}$; & $O(nd)$.
\\
Determine $\Theta$; & $O(d)$.
\\
Sort the elements in $\Theta$; & $O(d\log d)$.
\\
Compute ${\mathcal{H}}_{k}$ for $k\in[d_{t}]$; & $O(d^{2})$.
\\
Compute coefficients $(a_{k},b_{k},c_{k})$
for $k\in[d_{t}]$; & $O(d^{2})$.
\\
Find $k_{\star}$; & $O(d)$.
\\
Compute a solution \eqref{eq:sol-to-lnsrch};
& $O(1)$.
\end{tabular}
\end{center}
Hence the line search problem can be solved in $O(nd)$
when $d$ is within $O(n)$.
\subsection{Convergence Analysis}
\label{ss:fwconv}
We have seen that the direction finding step and
the line search step can be performed exactly,
meaning that our algorithm inherits the theoretical guarantee
for the convergence property of the Frank-Wolfe algorithm.
In general, the objective error of the Frank-Wolfe algorithm
for a minimization problem is bounded as
\begin{tsaligned}
F({\bm{x}}^{(T)}) - F({\bm{x}}_{\star})
\le
\frac{2C_{F}}{T+2}
\end{tsaligned}
where $C_{F}$ is the curvature of the
continuously differentiable and convex objective function $F$
to be minimized~\cite{jaggi13icml}.
The curvature~\cite{jaggi13icml} is defined as
\begin{tsaligned}\label{eq:curv-def}
C_{F} :=
\sup
\Big\{
\frac{2}{\gamma^{2}}{\mathcal{D}}_{F}({\bm{y}}_{\gamma}\,;\,{\bm{x}})
\,\Big| & \,
{\bm{u}}, {\bm{x}}\in{\mathcal{M}},\, \gamma\in(0,1],\,
\\
&
{\bm{y}}_{\gamma}:=(1-\gamma){\bm{x}} + \gamma{\bm{u}}
\Big\},
\end{tsaligned}
where ${\mathcal{M}}$ is the domain of the minimization problem
and ${\mathcal{D}}_{F}({\bm{y}}\,;\,{\bm{x}})$ is the Bregman divergence.
We obtained the following bound for the curvature of
the objective in the dual problem~\eqref{eq:prob-scsvm-dual}.
\begin{lemma-waku}\label{lem:curv-scsvm}
Assume that the norm of every feature vector is
$R$, at most (i.e. $\forall i\in[n]$, $\lVert{\bm{x}}_{i}\rVert\le R$).
The curvature for $F:=-D$ with ${\mathcal{M}}:=[0,1]^{n}$
is then bounded as $C_{F}\le R^{2}/\lambda$.
\end{lemma-waku}
\tslong{See Subsection~\ref{ss:proof-lem:curv-scsvm}
for the proof of Lemma~\ref{lem:curv-scsvm}. }%
\tsshort{See the supplement
for the proof of Lemma~\ref{lem:curv-scsvm}. }%
This lemma leads to our main result:
\begin{theorem-waku}
%
Suppose that $\forall i\in[n]$, $\lVert{\bm{x}}_{i}\rVert\le R$.
Each iteration of Algorithm~\ref{algo:stdfw-svm} can be
performed within $O(nd)$ computational cost.
The objective error is less than or equal to a positive scalar $\epsilon>0$
(i.e. $D({\bm{\alpha}}_{\star}) - D({\bm{\alpha}}^{(T)}) \le \epsilon$) for any $T\in{\mathbb{N}}$
such that
%
\begin{tsaligned}
T \ge \frac{2R^{2}}{\lambda \epsilon} - 2.
\end{tsaligned}
\end{theorem-waku}
\begin{table}[t!]
\centering
\caption{ROC scores for amino acid sequence classification.
\label{tab:demo387-06}}
{
\begin{tabular}{|c|cc|}
\hline
Class
& \shortstack{Conventional \\ SVM-pairwise}
& \shortstack{Sign Constrained \\ SVM-pairwise}
\\
\hline
1 & 0.731 (0.009) & \textbf{\underbar{0.749}} (0.010) \\
2 & 0.681 (0.013) & \textbf{\underbar{0.756}} (0.012) \\
3 & 0.735 (0.013) & \textbf{\underbar{0.758}} (0.012) \\
4 & 0.745 (0.010) & \textbf{\underbar{0.768}} (0.009) \\
5 & 0.709 (0.017) & \textbf{\underbar{0.784}} (0.008) \\
6 & 0.625 (0.007) & \textbf{\underbar{0.692}} (0.012) \\
7 & 0.628 (0.023) & \textbf{\underbar{0.702}} (0.030) \\
8 & 0.664 (0.018) & \textbf{\underbar{0.733}} (0.018) \\
9 & 0.603 (0.019) & \textbf{\underbar{0.681}} (0.022) \\
10 & 0.712 (0.008) & \textbf{\underbar{0.739}} (0.010) \\
11 & 0.523 (0.026) & \textbf{\underbar{0.561}} (0.029) \\
12 & 0.894 (0.013) & \textbf{\underbar{0.905}} (0.012) \\
\hline
\end{tabular}
}
\end{table}
\section{Experiments}
\label{s:exp}
In this section, we illustrate how fast the optimization algorithm converge in real-world data, and empirically show how well the sign constraints work for the task of biological sequence classification.
\subsection{Convergence}
We examine the convergence of the projected gradient and the Frank-Wolfe algorithms on the Cora and MNIST datasets.
Cora dataset is a collection of 15,396 academic papers related to computer science~\cite{SenNamBil08}. Each paper is annotated manually to provide a hierarchical classification. The dataset has ten categories at the top hierarchy. We divided the top categories into two groups to pose a binary classification problem. The positive class consists of IR, AI, OS, HA, and Pr, and the negative class is from Da, EC, Ne, DA, and HCI, resulting 10,778 and 4,618 papers belong to the positive and negative class, respectively. The task here is to classify each paper from the normalized word frequencies in its title. After removing stop words, 12,644 words used in the title of 15,396 papers were found. Each 12,644-dimensional feature vector is normalized with Euclidean norm.
The MNIST dataset is the most popular dataset for benchmarking machine learning methodologies. The dataset has 60,000 handwritten digit images. Each image has 784 pixels. For this simulation, images of odd digits (i.e. `1', `3', `5', `7', `9') and even digits (i.e. `0', `2', `4', `6', `8') were labeled as positive and negative, respectively. Each feature vector is normalized to transform each of examples to a unit vector.
A hundred iterations of the projected gradient and the Frank-Wolfe algorithms were run and the primal objective values $P({\bm{w}})$ were monitored 55 times at different iterations. The iteration numbers at which the objective function was evaluated were spaced evenly on a logarithmic scale (i.e. $t=1,2,3,\dots,50,53,55,\dots,87,92,96,100$). The minima among these objective errors monitored so far were recorded. For the Frank-Wolfe Algorithm, the duality gaps $P({\bm{w}})-D({\bm{\alpha}})$ were also evaluated. The regularization parameter $\lambda$ was varied with $\lambda=10^{-6}/n, 10^{-4}/n, 10^{-2}/n$ where $n=15,396$ for Cora and $n=60,000$ for MNIST.
In order to examine how fast each algorithm converges, the true objective error should be assesed, although it is impossible because $P({\bm{w}}_{\star})$ is usually unknown. To approximate the objective error, we ran the Frank-Wolfe algorithm to compute the dual objective value at $T':=1,000$th iteration. Although $D({\bm{\alpha}}^{(T')})$ might be slightly smaller than the true minimal primal objective value $\min_{{\bm{w}}\in{\mathcal{S}}}P({\bm{w}})$, the value $P({\bm{w}})-D({\bm{\alpha}}^{(T')})$ was regarded as the objective error.
\tsshort{%
The upper row in Figure~3~of the supplement plots the objective errors against iteration numbers on Cora, respectively.
}%
\tslong{%
The upper row in Figure~3~plots the objective errors against iteration numbers on Cora, respectively.
}
For all $\lambda$ on Cora, the projected gradient algorithm achieved smaller objective errors at around ten iterations, whereas the objective errors of the Frank-Wolfe were much smaller than those of the projected gradient after 15 iterations.
\tsshort{%
For MNIST (See Figure~4~in the supplement),
}%
\tslong{%
For MNIST (See Figure~4),
}%
the Frank-Wolfe converged to the minimum much faster than the projected gradient. The lower row depicts the duality gaps produced from the Frank-Wolfe. Recall that the duality gap can be a clear stopping criterion ensuring the quality of the solution.
The duality gaps were indeed converged to zero, implying that the duality gap can be utilized as a stopping criterion.
\subsection{Biological Sequence Classification}
\label{ss:svmpairwise}
Here we demonstrate that sign constraints are a powerful methodology for boosting the prediction performance for SVM-pairwise~\cite{LiaNob03-jcb} which is a binary classifier of biological sequences. SVM-pairwise predicts the existence/absence of a property of a query protein from its amino-acid sequence.
Conventional analysis of biological sequences is based on pairwise comparison between two sequences using a sequence similarity measure such as the Smith Waterman score~\cite{SmiWat81}.
SVM-pairwise was developed to predict the remote homology by combining the sequence comparison with supervised machine learning. The feature vector taken by SVM-pairwise is the sequence similarities to each of sequences for training. If $n$ proteins are in a training dataset, the feature vector has $n$ entries, $x_{1},\dots,x_{n}$. If we assume that the first $n_{+}$ proteins in the training set are annotated as positive and the rest of $n_{-}:=n-n_{+}$ proteins negative, then the first $n_{+}$ entries $x_{1},\dots,x_{n_+}$ are the sequence similarities to positive proteins and $x_{n_{+}+1},\dots,x_{n}$ are the similarities to negative proteins. Since each feature vector for training is $n$-dimensional, the data matrix ${\bm{X}}$ is square
\tsshort{%
(See Figure~5~in the supplement).
}%
\tslong{%
(See Figure~5).
}%
The prediction score for a query protein ${\bm{x}}$ is the difference from
\begin{tsaligned}
w_{1}x_{1} + \dots + w_{n_{+}}x_{n_{+}}
\end{tsaligned}
to
\begin{tsaligned}
- w_{n_{+}+1}x_{n_{+}+1} - \dots - w_{n}x_{n}.
\end{tsaligned}
From this view, it is preferable that the first $n_{+}$ weight coefficients $w_{1},\dots,w_{n_{+}}$ are non-negative and that the remaining $n_{-}$ coefficients $w_{ n_{+}+1},\dots,w_{n}$ are non-positive. Nevertheless, the pairwise SVM does not ensure these requirements.
These observations motivated us to impose these constraints explicitly as
$w_{1} \ge 0, \dots, w_{n_{+}}\ge 0$
and
$w_{n_{+}+1}\le 0, \dots, w_{n}\le 0$.
We examined the effectiveness of sign constraints on 3,583 yeast proteins with the annotations for these cellular functions and the Smith-Waterman similarities available from https://noble.gs.washington.edu/proj/sdp-svm/. The annotation of cellular function of a protein was existence or absence for each of 12 functional categories. We posed 12 independent binary classification task. A half of 3,583 proteins were selected randomly and used for training and the rest were used for testing.
We performed five-fold cross-validation within a training subset to determine the value of the regularization constant.
Weight coefficients ${\bm{w}}\in{\mathbb{R}}^{n}$ were determined by conventional SVM-pairwise and sign-constrained SVM-pairwise, respectively. For each of 12 binary classification tasks, ten ROC scores for conventional SVM-pairwise and ten ROC scores for sign-constrained SVM-pairwise were obtained by repeating this procedure ten times.
The one-sample t-test was performed to detect the significant difference between two ROC scores.
The average of ROC scores and the standard deviations over ten repetitions were shown in Table~\ref{tab:demo387-06}.
The best ROC scores are bold-faced and the underlined scores do not have a statistical significant difference from the best performance.
\tsshort{%
Additional results are given in the supplement.
}%
\tslong{%
Additional results are given in Subsection~\ref{ss:protein}.
}%
For all of 12 tasks, sign constrained modification consistently improved the ROC scores, suggesting that the sign constraints are a promising technique for SVM-pairwise framework.
\section{Conclusion}
\label{s:concl}
In this paper, we discussed the projected gradient algorithm and the Frank-Wolfe algorithm for learning the linear SVM under sign constraints. We provided solid analyses for the computational cost of each iteration and the convergence rate both for the two optimization algorithms. We found that both algorithms were ensured to have the sublinear convergence rate and the $O(nd)$ computational cost for each iteration. Finally, we proposed to introduce sign constraints for the SVM-pairwise approach and empirically demonstrated the promising prediction performances on 12 binary classification tasks for cellular functional prediction. SVM-pairwise is an approach specialized for computational biology, although similar approaches have also been discussed in more general settings~\cite{Tsuda02esann,Ng01nips}. Benchmarking the effects of sign constraints on SVM-pairwise in other applications will be the subject of future work.
\section*{Acknowledgment}
This research was performed by the Environment Research and Technology Development Fund JPMEERF20205006 of the Environmental Restoration and Conservation Agency of Japan and supported partially by JSPS KAKENHI Grant Number 19K04661.
\bibliographystyle{plain}
|
2,869,038,154,735 | arxiv | \section{$Q^2$ dependence of $(\sigma_A - \sigma_P)$}
The $Q^2$ and $\nu$ dependence of
$(\sigma_A - \sigma_P)$ is contained in two spin dependent nucleon
form factors
$G_1(\nu, Q^2)$ and $G_2(\nu, Q^2)$, viz.
\begin{equation}
\sigma_A - \sigma_P =
{16 m \pi^2 \alpha \over 2 m \nu - Q^2}
\Biggl(
m \nu G_1 (\nu, Q^2) - Q^2 G_2 (\nu, Q^2)
\Biggr).
\end{equation}
Following Anselmino, Ioffe and Leader \cite{ansel}, the $Q^2$
dependent quantity
\begin{equation}
I (Q^2) = m^3 \int_{ {Q^2 \over 2m} }^{\infty} {d \nu \over \nu}
G_1 (\nu, Q^2)
\end{equation}
can be used to interpolate between polarised photoproduction and
deep inelastic scattering. The Drell-Hearn-Gerasimov sum-rule Eq.(1)
is the statement:
\begin{equation}
I(0) = - {1 \over 4} \kappa^2_N.
\end{equation}
In high $Q^2$ deep inelastic scattering
\begin{equation}
m^2 \nu G_1 (\nu, Q^2) \rightarrow g_1 (x, Q^2), \ \ \
m \nu^2 G_2 (\nu, Q^2) \rightarrow g_2 (x, Q^2)
\end{equation}
where $x= {Q^2 \over 2m \nu}$ is the Bjorken variable.
In the deep inelastic limit
\begin{equation}
I (Q^2) =
{2 m^2 \over Q^2}
\int_0^1 dx g_1 (x, Q^2).
\end{equation}
As $Q^2$ tends to $\infty$, the first moment of $g_1$
is given by
\cite{mink,kod,larin,altr}
\begin{equation}
\int^1_0 dx g_1(x,Q^2)
= {1 \over 2} \sum_{\rm q} {\rm e_q^2} \Delta q_{\rm inv}
\Bigl\{1 + \sum_{\ell\geq 1} c_\ell\,\bar{g}^{2\ell}(Q)\Bigr\}
+ {\cal O}\Bigl({1 \over Q^2}\Bigr)
\end{equation}
Here
$\Delta q_{\rm inv}$ is the q flavoured,
gauge and scale
invariant axial charge \cite{bcft}.
The coefficient $c_\ell$ is calculable in
$\ell$-loop perturbation theory \cite{larin}.
The current experimental value of the singlet axial
charge
\begin{equation}
g_A^0|_{\rm inv} = \Delta u_{\rm inv} + \Delta d_{\rm inv} +
\Delta s_{\rm inv}
\end{equation}
is \cite{expt, badelek}
\begin{equation}
g_A^0|_{\rm inv} = 0.28 \pm 0.07.
\end{equation}
This number is four standard deviations below the value 0.58
that it would have taken if
$\Delta s_{\rm inv}$ were zero \cite{ej}.
The large violation of OZI in $g_A^0|_{\rm inv}$
is commonly associated
with the $U_A(1)$ axial anomaly in QCD [18-23].
The axial anomaly does not contribute to the anomalous magnetic
moment
for the reasons outlined in Section 4.
This source of OZI violation gives a vanishing contribution
to the Drell-Hearn-Gerasimov integral
$I(0)$ at photoproduction.
Strange resonance excitations
$\gamma N \rightarrow K \Lambda^*$,
and
$\gamma N \rightarrow K \Sigma^*$
have the potential to make an important contribution to $I(0)$
at photoproduction.
These resonance excitations contribute to polarised deep inelastic
scattering
only at non-leading twist.
They are not present in $g_A^0|_{\rm inv}$.
Several models [57,79-86] have been proposed for the non-leading
twist $Q^2$ variation of $I(Q^2)$,
which changes sign between polarised photoproduction,
Eq.(28), and deep inelastic scattering, Eqs.(31-33).
Bag model \cite{ji} and QCD sum-rule \cite{bbk,stein}
calculations are available for the twist four
${\cal O}({1 \over Q^2})$ corrections to $g_1$
in Eq.(31).
These calculations do not yet include the physics of the axial
anomaly, which is
potentially very important in the flavour singlet channel \cite{ioffe}.
Ioffe and collaborators \cite{ansel, iof92} have proposed a
simple phenomenological model for the $Q^2$ dependence of
the inelastic part of
$(\sigma_A - \sigma_P)$.
They argue that the contribution from resonance production,
denoted $I^{\rm res}(Q^2)$,
has a strong $Q^2$ dependence for small $Q^2$ and then drops
rapidly with $Q^2$.
The non-resonant part of $I(Q^2)$ is then parametrised by a
smooth function which they took as the sum of a monopole and
a dipole term, viz.
\begin{equation}
I(Q^2) =
I^{\rm res} (Q^2) + 2 m^2 \Gamma^{\rm as}
\Biggl( {1 \over Q^2 + \mu^2}
- {C \mu^2 \over (Q^2 + \mu^2)^2} \Biggr).
\end{equation}
Here
\begin{equation}
\Gamma^{\rm as} = \int_0^1 dx g_1 (x, \infty)
\end{equation}
and
\begin{equation}
C = 1 + {1 \over 2}
\ {\mu^2 \over m^2} \ {1 \over \Gamma^{\rm as}} \
\Bigl( {1 \over 4} \kappa^2 + I^{\rm res}(0) \Bigr).
\end{equation}
Using vector meson dominance arguments, the mass parameter
$\mu$ was identified with rho meson mass, $\mu^2 \simeq m_{\rho}^2$.
Karliner's analysis \cite{ikar} suggests that
$I^{\rm res}_p(0) = -1.01$.
(There is no elastic contribution to (DHG).)
In this approach,
the ``discrepancy'' in the Drell-Hearn-Gerasimov sum-rule
$(I - I^{\rm res})_p(0) = +0.21$
is associated with
the non-resonant vector meson dominance term.
The E143 Collaboration at SLAC have recently announced the first
low-$Q^2$ measurements of $I_p(Q^2)$ and $I_d(Q^2)$ at
$Q^2 = 0.5$GeV$^2$ and $Q^2 = 1.2$GeV$^2$ \cite{e143}.
Both the proton and deuteron data points are consistent
with the predictions
of \cite{iof92, soff}.
The $Q^2 = 1.2$GeV$^2$ data is consistent with the QCD
sum-rule calculations \cite{bbk, stein} of the twist
four contribution to Eq.(31) with coefficients
$c_\ell$ which are evaluated
to ${\cal O}(\alpha_s^3)$ \cite{larin, alta}.
(Theoretical uncertainties due to higher order coefficients
may be difficult to estimate \cite{mart}.)
\section {Conclusions and Opportunities}
Photoproduction spin sum-rules offer a new window on the
spin structure of the nucleon that complements the
information that we can learn from polarised deep inelastic
scattering experiments.
The first direct measurements of the spin photoproduction
cross-sections
$\sigma_A$ and $\sigma_P$ will soon be available from the
ELSA, GRAAL, LEGS and MAMI facilities up to $\sqrt{s} \leq 2.5$GeV.
These experiments will make a precise check of $\gamma N \rightarrow
N^{*}$ resonance contributions to the Drell-Hearn-Gerasimov and Spin
Polarisability sum-rules.
Further important information on the photoproduction spin structure
of the nucleon will come from measuring:
\begin{itemize}
\item
Non-resonant contributions associated, in part, with vector
meson dominance (Section 6) and
\item
Strange resonances
($\gamma N \rightarrow K \Lambda^{*}$)
and
($\gamma N \rightarrow K \Sigma^{*}$)
as possible sources
of the Karliner ``discrepancy'' (Sections 4 and 5);
\item
Tests of spin dependent Regge theory to distinguish
the Lorentz structure of pomeron-like exchanges Eqs.(12-14).
This requires varying $\sqrt{s} \geq 10$GeV
to ensure that we are in the region
where the pomeron dominates over other Regge contributions
\cite{pvl1} (Section 2).
Regge behaviour provides a good description of
$(\sigma_A + \sigma_P)$ starting at $\sqrt{s} \simeq 2.5$GeV.
\end{itemize}
When combined with measurements of the $Q^2$
dependence of
$(\sigma_A - \sigma_P)$ from SLAC and TJNAF, we will soon have a
much more complete picture of the nucleon's
internal
spin structure.
\pagebreak
\vspace{1.0cm}
{\large \bf Acknowledgements: \\}
\vspace{3ex}
It is a pleasure to thank S.J. Brodsky, R.J. Crewther, N. d'Hose,
B. Metsch, W. Melnitchouk, H. Petry, B. Schoch, D. Sch\"utte and
A.W. Thomas for helpful discussions on various aspects of DHG
physics. This work was supported by a Research Fellowship of the
Alexander von Humboldt Foundation.
|
2,869,038,154,736 | arxiv | \section{Introduction}
\label{intro}
The last few years have seen
the emergence of a Standard $\Lambda$CDM Model of cosmology
motivated by and consistent with
a wide range of observations, including
the cosmic microwave background,
distant supernovae,
big-bang nucleosynthesis,
large-scale structure,
the abundance of rich galaxy clusters,
and
local measurements of the Hubble constant
(e.g.\ \citealt{Tegmark04b}).
The power spectrum of fluctuations
(of temperature, density, flux, shear, etc.)
is the primary statistic used to constrain cosmological parameters
from observations of the cosmic microwave background
(\citealt{Spergel03}),
of galaxies
(\citealt{Tegmark04a};
\citealt{Cole05};
\citealt{Sanchez05};
\citealt{Eisenstein05}),
of the Lyman alpha forest
(\citealt{Seljak05};
\citealt{LHHHRS05};
\citealt{VH05}),
and of weak gravitational lensing
(\citealt{HYG02};
\citealt{PLWM03};
\citealt{TJ04};
\citealt{Sheldon04}).
From a cosmological standpoint,
the most precious data lie at large, linear scales,
where fluctuations preserve the imprint
of their primordial generation.
A generic, albeit not universal, prediction of inflation
is that primordial fluctuations should be Gaussian.
At large, linear scales,
observations are consistent with fluctuations being Gaussian
\citep{Komatsu03}.
However,
much of the observational data,
especially those involving galaxies,
lies in the translinear or nonlinear regime.
It remains a matter of ongoing research to elucidate
the extent to which nonlinear data can be used to constrain cosmology.
We recently began
\citep{RH05}
a program to measure quantitatively,
from cosmological simulations,
the Fisher information
content of the nonlinear matter power spectrum
(specifically, in the first instance,
the information
about the initial amplitude of the linear power spectrum).
For Gaussian fluctuations,
the power spectrum contains all possible information
about cosmological parameters.
At nonlinear scales,
where fluctuations are non-Gaussian,
it is natural to start by measuring information
in the power spectrum,
although it seems likely
that additional information
resides in the 3-point and higher order correlation functions
(\citealt{TJ04};
\citealt{SS05}).
Measuring the Fisher information in the power spectrum
involves measuring the covariance matrix of power.
For Gaussian fluctuations,
the expected covariance of estimates of power
is known analytically,
but at nonlinear scales
the covariance of power must be estimated from simulations.
A common way to estimate the covariance matrix of a quantity
is to measure its covariance over an ensemble of computer simulations
(\citealt{MW99};
\citealt{SZH99};
\citealt{ZE05}; \citealt{ZDEK05};
\citealt{RH05}).
However,
a reliable estimate of covariance can be computationally expensive,
requiring many, perhaps hundreds
(\citealt{MW99}; \citealt{RH05})
of realizations.
On the other hand it is physically obvious that
the fluctuations in the values of quantities over
the different parts of a single simulation
must somehow encode the covariance of the quantities.
If the covariance could be measured from single simulations,
then it would be possible to measure covariance
from fewer, and from higher quality, simulations.
In any case,
the ability to measure covariance from a single simulation
can be useful in identifying simulations whose statistical
properties are atypical.
A fundamental difficulty with estimating
covariances from single simulations in cosmology
is that the data are correlated over all scales, from small to large.
As described by
\citet{Kunsch89},
such correlations invalidate some of the
``jackknife''
and
``bootstrap''
schemes suggested in the literature.
In jackknife,
variance is inferred
from how much a quantity varies
when some segments of the data are kept, and some deleted.
Bootstrap is like jackknife,
except that deleted segments are replaced with other segments.
As part of the work leading to the present paper,
we investigated a form of the bootstrap procedure,
in which we filled each octant of a simulation cube
with a block of data selected randomly from the cube.
Unfortunately,
the sharp edges of the blocks introduced undesirable small scale power,
which seemed to compromise the effort to measure covariance of power reliably.
Such effects can be mitigated by tapering
\citep{Kunsch89}.
However,
it seemed to us that bootstrapping, like jackknifing,
is a form of re-weighting data,
and that surely the best way to re-weight data
would be to apply the most slowly possible varying weightings.
For a periodic box,
such weightings would be comprised of
the largest scale modes, the fundamentals.
In the present paper, \S\ref{estimate},
we consider applying an arbitrary weighting to
the density of a periodic cosmological simulation,
and we show how the power spectrum
(and its covariance, and the covariance of its covariance)
of the weighted density are related to the true power spectrum
(and its covariance, and the covariance of its covariance).
We confirm mathematically the intuitive idea
that weighting with fundamentals yields the most reliable estimate
of covariance of power.
Multiplying the density in real space by some weighting
is equivalent to convolving the density in Fourier space
with the Fourier transform of the weighting.
This causes the power spectrum
(and its covariance, and the covariance of its covariance)
to be convolved with the Fourier transform of the square
(and fourth, and eighth powers)
of the weighting.
The convolution does least damage when the weighting window is
as narrow as possible in Fourier space,
which means composed of fundamentals.
In \S\ref{weightings}
we show how to design a best set of weightings,
by minimizing the expected variance of
the resulting estimate of covariance of power.
These considerations lead us to recommend a specific
set of $52$ weightings,
each consisting of a combination of fundamental modes.
This paper should have stopped neatly at this point.
Unfortunately, numerical simulations, described in a companion paper
\citep{RH06},
revealed an unexpected (one might say insidious),
substantial discrepancy at nonlinear scales between
the variance of power estimated by the weightings method
and the variance of power estimated by the ensemble method.
In \S\ref{beatcoupling}
we argue that this discrepancy arises from beat-coupling,
a nonlinear gravitational coupling to the large-scale
beat mode between closely spaced nonlinear wavenumbers,
when the power spectrum is measured from Fourier modes
at anything other than infinitely sharp sets of wavenumbers.
Surprisingly,
in cosmologically realistic simulations,
the covariance of power
is dominated at nonlinear scales
by this beat-coupling to large scales.
We discuss the beat-coupling problem in \S\ref{discussion}.
Beat-coupling is relevant to observations
because real galaxy surveys
yield Fourier modes in
finite bands of wavenumber $k$,
of width $\Delta k \sim 1/R$
where $R$ is a chararacteristic linear size of the survey.
Section~\ref{summary} summarizes the results.
\section{Estimating the covariance of power from an ensemble of weighted density fields}
\label{estimate}
The fundamental idea of this paper is
to apply an ensemble of weightings to a
(non-Gaussian, in general) density field,
and to estimate the covariance of the power spectrum
from the scatter in power between different weightings.
This section derives the relation
between the power spectrum of a weighted density field
and the true power spectrum,
along with its expected covariance, and the covariance of its covariance.
It is shown,
equations~(\ref{piapprox}), (\ref{DphatiDphatjf}), and (\ref{Dphati2Dphatj2g}),
that the expected ((covariance of) covariance of)
shell-averaged power of weighted density fields
is simply proportional to the true
((covariance of) covariance of) shell-averaged power,
provided that two approximations are made.
The two approximations are, firstly,
that the power spectrum and trispectrum
are sufficiently slowly varying functions of their arguments,
equations~(\ref{Papprox}) and (\ref{Tapprox}),
and, secondly,
that power is estimated in sufficiently broad shells in $k$-space,
equation~(\ref{broadshellapprox}).
The required approximations are most accurate if the
weightings contain only the largest scale Fourier modes,
such as the weightings containing only fundamental modes
proposed in \S\ref{weightings}.
As will be discussed in \S\ref{beatcoupling},
the apparently innocent assumption, equation~(\ref{Tapprox}),
that the trispectrum is a slowly varying function of its arguments,
is incorrect,
because it sets to zero some important beat-coupling contributions.
However, it is convenient to pretend
in this section and the next, \S\ref{estimate} and \S\ref{weightings},
that the assumption~(\ref{Tapprox}) is true,
and then to consider
in \S\ref{beatcoupling}
how the results are modified
when the beat-coupling contributions to the trispectrum are included.
Ultimately we find,
\S\ref{largescale},
that the weightings method remains valid
when beat-couplings are included,
and,
\S\ref{notquiteweightings},
that the minimum variance weightings derived in \S\ref{weightings},
while no longer exactly minimum variance,
should be close enough to remain good for practical application.
This section is necessarily rather technical,
because it is necessary to distinguish carefully
between various flavours of power spectrum:
estimated versus expected;
unweighted versus weighted;
non-shell-averaged versus shell-averaged.
Subsections~\ref{P} to \ref{ddP}
present expressions for the various power spectra,
their covariances, and the covariances of their covariances.
Subsections~\ref{subtractmeanP}
and \ref{subtractmeanddP}
show how the expressions are modified when,
as is usually the case,
deviations in power must be measured relative to an estimated
rather than an expected value of power.
\subsection{The power spectrum}
\label{P}
Let $\rho({\bmath r})$ denote the density of a statistically homogeneous
random field at position ${\bmath r}$ in a periodic box.
Choose the unit of length so that the box has unit side.
The density $\rho({\bmath r})$ might represent, perhaps, a realization
of the nonlinearly evolved distribution of dark matter, or of galaxies.
The density could be either continuous or discrete (particles).
Expanded in Fourier modes $\rho({\bmath k})$,
the density $\rho({\bmath r})$ is\footnote{
The same symbol $\rho$ is used in both real and Fourier space.
The justification for this notation is that $\rho$ is the
same vector in Hilbert space
irrespective of the basis with respect to which it is expanded.
See for example \cite{H05} for a pedagogical exposition.
}
\begin{equation}
\label{rhok}
\rho({\bmath r}) =
\sum_{{\bmath k}} \rho({\bmath k}) \e^{- 2\upi \im {\bmath k} . {\bmath r}}
\ .
\end{equation}
Thanks to periodicity,
the sum is over an integral lattice of wavenumbers,
${\bmath k} = \{k_x, k_y, k_z\}$
with integer $k_x$, $k_y$, $k_z$.
The expectation value $\langle \rho({\bmath r}) \rangle$ of the density
defines the true mean density $\overline{\rho}$,
which without loss of generality we take to equal unity
\begin{equation}
\label{rhobar}
\overline{\rho} \equiv \langle \rho({\bmath r}) \rangle = 1
\ .
\end{equation}
The deviation $\Delta\rho({\bmath r})$ of the density from the mean is
\begin{equation}
\label{Drhobar}
\Delta\rho({\bmath r}) \equiv \rho({\bmath r}) - \overline{\rho}
\ .
\end{equation}
The expectation values of the Fourier amplitudes vanish,
$\langle \rho({\bmath k}) \rangle = 0$,
except for the zero'th mode,
whose expectation value equals the mean density,
$\langle \rho({\bmath 0}) \rangle = \overline{\rho}$.
The Fourier amplitude $\rho({\bmath 0})$ of the zero'th mode is the actual
density of the realization,
which could be equal to, or differ slightly from,
the true mean density $\overline{\rho}$,
depending on whether the mean density of the realization
was constrained to equal the true density, or not.
Because the density field is by assumption statistically homogeneous,
the expected covariance of Fourier amplitudes
$\rho({\bmath k})$
is a diagonal matrix
\begin{equation}
\label{DrhokDrhok}
\left\langle \Delta\rho({\bmath k}_1) \, \Delta\rho({\bmath k}_2) \right\rangle
=
1_{{\bmath k}_1 + {\bmath k}_2}
P({\bmath k}_1)
\ .
\end{equation}
Here $1_{\bmath k}$ denotes the discrete delta-function,
\begin{equation}
1_{\bmath k} =
\left\{
\begin{array}{ll}
1 & \mbox{if ${\bmath k} = {\bmath 0}$} \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
and $P({\bmath k})$ is the power spectrum.
Note that there would normally be an extra factor of $\overline{\rho}^{-2}$
on the left hand side of equation~(\ref{DrhokDrhok}),
but it is fine to omit the factor here
because the mean density is normalized to unity,
equation~(\ref{rhobar}).
The reason for dropping the factor of $\overline{\rho}^{-2}$
is to maintain notational consistency with
equation~(\ref{Pi}) below
for the power spectrum of weighted density
(where the deviation in density is necessarily
{\em not\/} divided by the mean).
The symmetry $P(-{\bmath k}) = P({\bmath k})$
in equation~(\ref{DrhokDrhok})
expresses pair exchange symmetry.
Below, \S\ref{shell},
we will assume that the density field is statistical isotropic,
in which case the power is a function $P(k)$
only of the scalar wavenumber $k \equiv |{\bmath k}|$,
but for now we stick to the more general case
where power is a function $P({\bmath k})$ of vector wavenumber ${\bmath k}$.
\subsection{The power spectrum of weighted density}
\label{Pw}
Let
$w_i({\bmath r})$
denote the $i$'th member of a set of real-valued weighting functions,
and let $\rho_i({\bmath r})$ denote the density weighted by the
$i$'th weighting
\begin{equation}
\rho_i({\bmath r}) \equiv w_i({\bmath r}) \rho({\bmath r})
\ .
\end{equation}
The Fourier amplitudes $\rho_i({\bmath k})$
of the weighted density are convolutions
of the Fourier amplitudes of the weighting and the density:
\begin{equation}
\label{rhoik}
\rho_i({\bmath k}) =
\sum_{{\bmath k}^\prime} w_i({\bmath k}^\prime) \rho({\bmath k} - {\bmath k}^\prime)
\ .
\end{equation}
Reality of the weighting functions implies
\begin{equation}
w_i(-{\bmath k}) = w_i^\ast({\bmath k})
\ .
\end{equation}
The expected mean $\overline{\rho}_i({\bmath r})$ of the weighted density
is proportional to the weighting,
\begin{equation}
\label{rhobari}
\overline{\rho}_i({\bmath r})
\equiv
\langle \rho_i({\bmath r}) \rangle
= w_i({\bmath r})
\end{equation}
in which a factor of $\overline{\rho}$ on the right hand side has been omitted
because the mean density has been normalized to unity, equation~(\ref{rhobar}).
The deviation $\Delta\rho_i({\bmath r})$ of the weighted density from the mean is
\begin{equation}
\label{Drhobari}
\Delta\rho_i({\bmath r}) \equiv \rho_i({\bmath r}) - \overline{\rho}_i({\bmath r})
\ .
\end{equation}
In Fourier space the expected mean $\overline{\rho}_i({\bmath k})$ of the weighted density is
\begin{equation}
\label{rhobarik}
\overline{\rho}_i({\bmath k})
\equiv
\langle \rho_i({\bmath k}) \rangle
= w_i({\bmath k})
\end{equation}
and the deviation $\Delta\rho_i({\bmath k})$ of the weighted density from the mean is
\begin{equation}
\label{Drhobarik}
\Delta\rho_i({\bmath k}) \equiv \rho_i({\bmath k}) - \overline{\rho}_i({\bmath k})
\ .
\end{equation}
The deviations $\Delta\rho_i({\bmath k})$
in the Fourier amplitudes of the weighted density are convolutions
of the weighting and the deviation in the density
\begin{equation}
\label{Drhoik}
\Delta\rho_i({\bmath k}) =
\sum_{{\bmath k}^\prime} w_i({\bmath k}^\prime) \Delta\rho({\bmath k} - {\bmath k}^\prime)
\end{equation}
similarly to equation~(\ref{rhoik}).
The expected covariance between
two weighted densities $\rho_i({\bmath k}_1)$ and $\rho_j({\bmath k}_2)$
at wavenumbers ${\bmath k}_1$ and ${\bmath k}_2$ is,
from equations~(\ref{DrhokDrhok}) and (\ref{Drhoik}),
\begin{equation}
\label{rhoikrhojk}
\left\langle \Delta\rho_i({\bmath k}_1) \, \Delta\rho_j({\bmath k}_2) \right\rangle
=
\sum_{{\bmath k}^\prime} w_i({\bmath k}^\prime) w_j({\bmath k}_1 + {\bmath k}_2 - {\bmath k}^\prime)
P({\bmath k}_1 - {\bmath k}^\prime)
\ .
\end{equation}
The weighting breaks statistical homogeneity,
so the expected covariance matrix of Fourier amplitudes
$\rho_i({\bmath k})$,
equation~(\ref{rhoikrhojk}),
is not diagonal.
Nevertheless we {\em define} the power spectrum $P_i({\bmath k})$
of the $i$'th weighted density
by the diagonal elements of the covariance matrix,
the variance
\begin{equation}
\label{Pi}
P_i({\bmath k})
\equiv
\left\langle \Delta\rho_i({\bmath k}) \, \Delta\rho_i(- {\bmath k}) \right\rangle
\ .
\end{equation}
Note that this definition~(\ref{Pi}) of the power spectrum $P_i({\bmath k})$
differs from the usual definition of power in that
the deviations $\Delta\rho_i({\bmath k})$ on the right
are Fourier transforms of the deviations $\Delta\rho_i({\bmath r})$
{\em not\/} divided by the mean density $\overline{\rho}_i({\bmath r}) = w_i({\bmath r})$
(dividing by the mean density would simply unweight the weighting,
defeating the whole point of the procedure).
The power spectrum $P_i({\bmath k})$ defined by equation~(\ref{Pi})
is related to the true power spectrum $P({\bmath k})$ by,
equation~(\ref{rhoikrhojk}),
\begin{equation}
P_i({\bmath k})
=
\sum_{{\bmath k}^\prime} \left| w_i({\bmath k}^\prime) \right|^2
P({\bmath k} - {\bmath k}^\prime)
\ .
\end{equation}
Now make the approximation that the power spectrum
$P({\bmath k} - {\bmath k}^\prime)$
at the wavenumber ${\bmath k} - {\bmath k}^\prime$ displaced by ${\bmath k}^\prime$ from ${\bmath k}$
is approximately equal to the power spectrum $P({\bmath k})$
at the undisplaced wavenumber ${\bmath k}$
\begin{equation}
\label{Papprox}
P({\bmath k} - {\bmath k}^\prime)
\approx
P({\bmath k})
\ .
\end{equation}
This approximation is good
provided that the power spectrum $P({\bmath k})$
is slowly varying as a function of wavenumber ${\bmath k}$,
and that the displacement ${\bmath k}^\prime$ is small compared to ${\bmath k}$.
In \S\ref{weightings} we constrain the weightings $w_i({\bmath k}^\prime)$
to contain only fundamental modes,
${\bmath k}^\prime = \{k^\prime_x, k^\prime_y, k^\prime_z\}$
with $k^\prime_x$, $k^\prime_y$, $k^\prime_z$ = $0, \pm 1$,
so that the displacement ${\bmath k}^\prime$ is as small as it can be
without being zero,
and the approximation~(\ref{Papprox})
is therefore as good as it can be.
The approximation~(\ref{Papprox})
becomes exact in the case of a constant, or shot noise,
power spectrum $P({\bmath k})$,
except at ${\bmath k} - {\bmath k}^\prime = {\bmath 0}$.
Under approximation~(\ref{Papprox}),
the power spectrum of the $i$'th weighted density is
\begin{equation}
\label{Piapproxw}
P_i({\bmath k})
\approx
P({\bmath k})
\sum_{{\bmath k}^\prime} \left| w_i({\bmath k}^\prime) \right|^2
\end{equation}
which is just proportional to the true power spectrum $P({\bmath k})$.
Without loss of generality,
let each weighting $w_i({\bmath k}^\prime)$
be normalized so that the factor
on the right hand side of equation~(\ref{Piapproxw}) is unity
\begin{equation}
\label{wnorm}
\sum_{{\bmath k}^\prime} \left| w_i({\bmath k}^\prime) \right|^2
= 1
\ .
\end{equation}
Then the power spectrum $P_i({\bmath k})$
of the weighted density
is approximately equal to the true power spectrum $P({\bmath k})$
\begin{equation}
\label{Piapprox}
P_i({\bmath k})
\approx
P({\bmath k})
\ .
\end{equation}
Thus, in the approximation~(\ref{Papprox})
and with the normalization~(\ref{wnorm}),
measurements of the power spectrum $P_i({\bmath k})$ of weighted densities
provide estimates of the true power spectrum $P({\bmath k})$.
The plan is to use the scatter in the estimates of power
over a set of weightings to estimate the covariance matrix of power.
\subsection{The covariance of power spectra}
\label{cov}
Let $\widehat{P}({\bmath k})$
denote the power spectrum of unweighted density at wavevector ${\bmath k}$
measured from a simulation,
the hat distinguishing it from the true power spectrum $P({\bmath k})$:
\begin{equation}
\label{Phat}
\widehat{P}({\bmath k}) \equiv \Delta\rho({\bmath k}) \Delta\rho(-{\bmath k})
\ .
\end{equation}
Below, \S\ref{shell}, we will invoke statistical isotropy,
and we will average over a shell in $k$-space,
but in equation~(\ref{Phat})
there is no averaging
because there is just one simulation,
and just one specific wavenumber ${\bmath k}$.
Because of statistical fluctuations,
the estimate $\widehat{P}({\bmath k})$ will in general differ from the true power $P({\bmath k})$,
but by definition
the expectation value of the estimate equals the true value,
$\langle \widehat{P}({\bmath k}) \rangle = P({\bmath k})$.
The deviation $\Delta\widehat{P}({\bmath k})$ in the power
is the difference between the measured and expected value:
\begin{equation}
\label{DPhat}
\Delta\widehat{P}({\bmath k})
\equiv
\widehat{P}({\bmath k}) - P({\bmath k})
\ .
\end{equation}
The expected covariance of power
involves the covariance of the covariance of unweighted densities
\begin{eqnarray}
\label{DrhokDrhokDrhokDrhok}
\lefteqn{
\Bigl\langle
\bigl[ \Delta\rho({\bmath k}_1) \Delta\rho({\bmath k}_2) - 1_{{\bmath k}_1 + {\bmath k}_2} P({\bmath k}_1) \bigr]
}
&&
\nonumber
\\
&\times&
\bigl[ \Delta\rho({\bmath k}_3) \Delta\rho({\bmath k}_4) - 1_{{\bmath k}_3 + {\bmath k}_4} P({\bmath k}_3) \bigr]
\Bigr\rangle
\nonumber
\\
&=&
\left( 1_{{\bmath k}_1 + {\bmath k}_3} 1_{{\bmath k}_2 + {\bmath k}_4} + 1_{{\bmath k}_1 + {\bmath k}_4} 1_{{\bmath k}_2 + {\bmath k}_3} \right)
P({\bmath k}_1) P({\bmath k}_2)
\nonumber
\\
&&
\mbox{}
+
1_{{\bmath k}_1 + {\bmath k}_2 + {\bmath k}_3 + {\bmath k}_4}
T({\bmath k}_1, {\bmath k}_2, {\bmath k}_3, {\bmath k}_4)
\label{eta}
\end{eqnarray}
which is a sum of a reducible, Gaussian part,
the terms proportional to $P({\bmath k}_1) P({\bmath k}_2)$,
and an irreducible, non-Gaussian part,
the term involving the trispectrum
$T({\bmath k}_1, {\bmath k}_2, {\bmath k}_3, {\bmath k}_4)$.
Equation~(\ref{eta}) essentially defines what is meant by the trispectrum
$T$.
Exchange symmetry implies that the trispectrum function
is invariant under permutations of its 4 arguments.
The momentum-conserving delta-function $1_{{\bmath k}_1 + {\bmath k}_2 + {\bmath k}_3 + {\bmath k}_4}$
in front of the trispectrum $T$ expresses translation invariance.
It follows from equation~(\ref{DrhokDrhokDrhokDrhok}) that
the expected covariance of estimates of power is
\begin{eqnarray}
\label{DPhatDPhat}
\lefteqn{
\left\langle \Delta\widehat{P}({\bmath k}_1) \Delta\widehat{P}({\bmath k}_2) \right\rangle
}
&&
\\
\nonumber
&=&
\left( 1_{{\bmath k}_1 + {\bmath k}_2} + 1_{{\bmath k}_1 - {\bmath k}_2}\right)
P({\bmath k}_1)^2
+
T({\bmath k}_1, -{\bmath k}_1, {\bmath k}_2, -{\bmath k}_2)
\ .
\end{eqnarray}
\subsection{The covariance of power spectra of weighted density}
\label{covw}
Similarly to equations~(\ref{Phat}) and (\ref{DPhat}),
let $\widehat{P}_i({\bmath k})$
denote the power spectrum of the $i$'th weighted density at wavevector ${\bmath k}$
measured from a simulation
\begin{equation}
\label{Phati}
\widehat{P}_i({\bmath k}) \equiv \Delta\rho_i({\bmath k}) \Delta\rho_i(-{\bmath k})
\end{equation}
and let $\Delta\widehat{P}_i({\bmath k})$ denote the deviation
between the measured and expected value
\begin{equation}
\label{DPhati}
\Delta\widehat{P}_i({\bmath k})
\equiv
\widehat{P}_i({\bmath k}) - P_i({\bmath k})
\ .
\end{equation}
The expected covariance between the power spectra
of the $i$'th and $j$'th weighted densities is,
from equations~(\ref{Drhoik}) and (\ref{DrhokDrhokDrhokDrhok}),
\begin{eqnarray}
\label{DPhatiDPhatj}
\lefteqn{
\left\langle \Delta\widehat{P}_i({\bmath k}_1) \Delta\widehat{P}_j({\bmath k}_2) \right\rangle
=
}
&&
\nonumber
\\
\nonumber
\lefteqn{
\sum_{{\bmath k}_1^\prime + {\bmath k}_1^{\prime\prime} + {\bmath k}_2^\prime + {\bmath k}_2^{\prime\prime} = {\bmath 0}}
w_i({\bmath k}_1^\prime) w_i({\bmath k}_1^{\prime\prime})
w_j({\bmath k}_2^\prime) w_j({\bmath k}_2^{\prime\prime})
}
&&
\\
\nonumber
\lefteqn{
\Bigl[
\bigl(
1_{{\bmath k}_1 - {\bmath k}_1^\prime + {\bmath k}_2 - {\bmath k}_2^\prime}
+
1_{{\bmath k}_1 - {\bmath k}_1^\prime - {\bmath k}_2 - {\bmath k}_2^{\prime\prime}}
\bigr)
P({\bmath k}_1 {-} {\bmath k}_1^\prime) P(- {\bmath k}_1 {-} {\bmath k}_1^{\prime\prime} )
}
&&
\\
\lefteqn{
\mbox{}
+
T({\bmath k}_1 {-} {\bmath k}_1^\prime, -{\bmath k}_1 {-} {\bmath k}_1^{\prime\prime}, {\bmath k}_2 {-} {\bmath k}_2^\prime, -{\bmath k}_2 {-} {\bmath k}_2^{\prime\prime})
\Bigr]
\ .
}
&&
\end{eqnarray}
\subsection{The covariance of shell-averaged power spectra}
\label{shell}
Assume now that the unweighted density field $\overline{\rho}({\bmath r})$
is statistically isotropic,
so that the true power spectrum $P(k)$ is a function only of
the absolute value $k \equiv |{\bmath k}|$ of its argument.
In estimating the power $P(k)$ from a simulation,
one would typically average the measured power
over a spherical shell $V_k$ of wavenumbers in $k$-space.
Actually the arguments below generalize immediately
to the case where the power is not isotropic,
in which case $V_k$ might be chosen to be some localized patch in $k$-space.
However,
we shall assume isotropy,
and refer to $V_k$ as a shell.
Let $\hat{p}(k)$
denote the measured power averaged over a shell $V_k$
about scalar wavenumber $k$
(the estimated shell-averaged power $\hat{p}(k)$ is written in lower case
to distinguish it from the estimate $\widehat{P}({\bmath k})$ of power
at a single specific wavevector ${\bmath k}$):
\begin{equation}
\label{phat}
\hat{p}(k) \equiv \frac{1}{N_{\bmath k}} \sum_{{\bmath k} \in V_k} \widehat{P}({\bmath k})
\ .
\end{equation}
Here $N_{\bmath k}$ is the number of modes $\rho({\bmath k})$ in the shell $V_k$.
We count $\rho({\bmath k})$ and its complex conjugate $\rho(-{\bmath k})$
as contributing two distinct modes,
the real and imaginary parts of $\rho({\bmath k})$.
The expectation value of the estimates $\hat{p}(k)$
of shell-averaged power equals the true shell-averaged power $p(k)$
\begin{equation}
\label{p}
\langle \hat{p}(k) \rangle
= p(k)
\equiv \frac{1}{N_{\bmath k}} \sum_{{\bmath k} \in V_k} P({\bmath k})
\ .
\end{equation}
The deviation $\Delta\hat{p}(k)$
between the measured and expected value of shell-averaged power is
\begin{equation}
\label{Dphat}
\Delta\hat{p}(k) \equiv \hat{p}(k) - p(k)
= \frac{1}{N_{\bmath k}} \sum_{{\bmath k} \in V_k} \Delta\widehat{P}({\bmath k})
\ .
\end{equation}
The expected covariance of shell-averaged estimates of power is,
from equations~(\ref{Dphat}) and (\ref{DPhatDPhat}),
\begin{eqnarray}
\label{DphatDphat}
\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle
&=&
\frac{1}{N_{{\bmath k}_1} N_{{\bmath k}_2}}
\Bigl[
2
\sum_{{\bmath k}_1 \in V_{k_1} \cap V_{k_2}}
P({\bmath k}_1)^2
\\
\nonumber
&+&
\sum_{{\bmath k}_1 \in V_{k_1} , \ {\bmath k}_2 \in V_{k_2}}
T({\bmath k}_1, -{\bmath k}_1, {\bmath k}_2, -{\bmath k}_2)
\Bigr]
\ .
\end{eqnarray}
In the usual case,
the shells $V_{k}$ would be taken to be non-overlapping,
in which case the intersection
$V_{k_1} \cap V_{k_2}$
in equation~(\ref{DphatDphat})
is equal either to $V_{k_1}$
if $V_{k_1}$ and $V_{k_2}$ are the same shell,
or to the empty set
if $V_{k_1}$ and $V_{k_2}$ are different shells.
\subsection{The covariance of shell-averaged power spectra of weighted density}
\label{shellw}
Similarly to equation~(\ref{phat}),
let $\hat{p}_i(k)$ denote
the measured shell-averaged power spectrum
of the $i$'th weighted density at wavenumber $k$
\begin{equation}
\label{phati}
\hat{p}_i(k) \equiv \frac{1}{N_{\bmath k}} \sum_{{\bmath k} \in V_k} \widehat{P}_i({\bmath k})
\ .
\end{equation}
The expectation value of the estimates $\hat{p}_i(k)$ is
(compare eq.~(\ref{p}))
\begin{equation}
\label{pi}
\langle \hat{p}_i(k) \rangle = p_i(k)
\equiv \frac{1}{N_{\bmath k}} \sum_{{\bmath k} \in V_k} P_i({\bmath k})
\ .
\end{equation}
In the approximation~(\ref{Papprox}) of a slowly varying power spectrum,
and with the normalization~(\ref{wnorm}),
the expected shell-averaged power spectrum $p_i(k)$ of the weighted density
is approximately equal to the
shell-averaged power spectrum $p(k)$ of the unweighted density
(compare eq.~(\ref{Piapprox}))
\begin{equation}
\label{piapprox}
p_i(k) \approx p(k)
\ .
\end{equation}
The deviation $\Delta\hat{p}_i(k)$
between the measured and expected values is
(compare eq.~(\ref{Dphat}))
\begin{equation}
\label{Dphati}
\Delta\hat{p}_i(k)
\equiv
\hat{p}_i(k) - p_i(k)
=
\frac{1}{N_{\bmath k}} \sum_{{\bmath k} \in V_k} \Delta\widehat{P}_i({\bmath k})
\ .
\end{equation}
The expected covariance of shell-averaged power spectra of weighted densities is,
from equations~(\ref{Dphati}) and (\ref{DPhatiDPhatj}),
\begin{eqnarray}
\label{DphatiDphatj}
\lefteqn{
\left\langle \Delta\hat{p}_i(k_1) \Delta\hat{p}_j(k_2) \right\rangle
=
\frac{1}{N_{{\bmath k}_1} N_{{\bmath k}_2}}
}
&&
\nonumber
\\
\nonumber
\lefteqn{
\sum_{{\bmath k}_1^\prime + {\bmath k}_1^{\prime\prime} + {\bmath k}_2^\prime + {\bmath k}_2^{\prime\prime} = {\bmath 0}}
w_i({\bmath k}_1^\prime) w_i({\bmath k}_1^{\prime\prime})
w_j({\bmath k}_2^\prime) w_j({\bmath k}_2^{\prime\prime})
\sum_{{\bmath k}_1 \in V_{k_1} , \ {\bmath k}_2 \in V_{k_2}}
}
&&
\\
\nonumber
\lefteqn{
\Bigl[
\bigl(
1_{{\bmath k}_1 - {\bmath k}_1^\prime + {\bmath k}_2 - {\bmath k}_2^\prime}
+
1_{{\bmath k}_1 - {\bmath k}_1^\prime - {\bmath k}_2 - {\bmath k}_2^{\prime\prime}}
\bigr)
P({\bmath k}_1 {-} {\bmath k}_1^\prime) P(- {\bmath k}_1 {-} {\bmath k}_1^{\prime\prime} )
}
&&
\\
\lefteqn{
\quad
\mbox{}
+
T({\bmath k}_1 {-} {\bmath k}_1^\prime, -{\bmath k}_1 {-} {\bmath k}_1^{\prime\prime}, {\bmath k}_2 {-} {\bmath k}_2^\prime, -{\bmath k}_2 {-} {\bmath k}_2^{\prime\prime})
\Bigr]
\ .
}
&&
\end{eqnarray}
Assume, analogously to approximation~(\ref{Papprox}) for the power spectrum,
that the trispectrum function
$T({\bmath k}_1 {-} {\bmath k}_1^\prime, -{\bmath k}_1 {-} {\bmath k}_1^{\prime\prime}, {\bmath k}_2 {-} {\bmath k}_2^\prime, -{\bmath k}_2 {-} {\bmath k}_2^{\prime\prime})$
in equation~(\ref{DphatiDphatj})
is sufficiently slowly varying,
and the displacements ${\bmath k}_1^\prime$, ${\bmath k}_1^{\prime\prime}$, ${\bmath k}_2^\prime$, ${\bmath k}_2^{\prime\prime}$
sufficiently small,
that
\begin{eqnarray}
\label{Tapprox}
\lefteqn{
T({\bmath k}_1 {-} {\bmath k}_1^\prime, -{\bmath k}_1 {-} {\bmath k}_1^{\prime\prime}, {\bmath k}_2 {-} {\bmath k}_2^\prime, -{\bmath k}_2 {-} {\bmath k}_2^{\prime\prime})
}
\nonumber
\\
&\approx&
T({\bmath k}_1, - {\bmath k}_1, {\bmath k}_2, - {\bmath k}_2)
\ .
\end{eqnarray}
In \S\ref{beatcoupling}
we will revisit the approximation~(\ref{Tapprox}),
and show that in fact it is not true,
in a way that proves to be interesting and observationally relevant.
In this section and the next, \S\ref{weightings},
however, we will continue to assume that the approximation~(\ref{Tapprox}) is valid.
In the approximations~(\ref{Papprox}) and (\ref{Tapprox})
that the power spectrum and trispectrum
are both approximately constant for small displacements of their arguments,
the covariance of shell-averaged power spectra,
equation~(\ref{DphatiDphatj}),
becomes
\begin{eqnarray}
\label{DphatiDphatjapprox}
\lefteqn{
\left\langle \Delta\hat{p}_i(k_1) \Delta\hat{p}_j(k_2) \right\rangle
\approx
\frac{1}{N_{{\bmath k}_1} N_{{\bmath k}_2}}
}
&&
\nonumber
\\
\nonumber
\lefteqn{
\sum_{{\bmath k}_1^\prime + {\bmath k}_1^{\prime\prime} + {\bmath k}_2^\prime + {\bmath k}_2^{\prime\prime} = {\bmath 0}}
w_i({\bmath k}_1^\prime) w_i({\bmath k}_1^{\prime\prime})
w_j({\bmath k}_2^\prime) w_j({\bmath k}_2^{\prime\prime})
}
&&
\\
\nonumber
\lefteqn{
\sum_{{\bmath k}_1 \in V_{k_1} , \ {\bmath k}_2 \in V_{k_2}}
\Bigl[
\bigl(
1_{{\bmath k}_1 - {\bmath k}_1^\prime + {\bmath k}_2 - {\bmath k}_2^\prime}
+
1_{{\bmath k}_1 - {\bmath k}_1^\prime - {\bmath k}_2 - {\bmath k}_2^{\prime\prime}}
\bigr)
P({\bmath k}_1)^2
}
&&
\\
\lefteqn{
\qquad\qquad\qquad
\mbox{}
+
T({\bmath k}_1, -{\bmath k}_1, {\bmath k}_2, -{\bmath k}_2)
\Bigr]
\ .
}
&&
\end{eqnarray}
Consider the Gaussian ($P^2$) part of this expression~(\ref{DphatiDphatjapprox}).
In the true covariance of shell-averaged power, equation~(\ref{DphatDphat}),
the Gaussian part of the covariance is a diagonal matrix,
with zero covariance between non-overlapping shells.
By contrast,
the Gaussian part of the covariance of power of weighted densities,
equation~(\ref{DphatiDphatjapprox}),
is not quite diagonal.
In effect,
the Gaussian variance in each shell
is smeared by convolution with the weighting function,
causing some of the Gaussian variance
near the boundaries of adjacent shells
to leak into covariance between the shells.
In \S\ref{weightings},
we advocate restricting
the weightings $w_i({\bmath k})$ to contain only fundamental modes,
which keeps smearing to a minimum.
Whatever the case,
if each shell $V_k$
is broad compared the extent of the weightings $w_i({\bmath k})$ in $k$-space,
then the smearing is relatively small, and can be approximated as zero.
Mathematically, this broad-shell approximation
amounts to approximating
\begin{eqnarray}
\label{broadshellapprox}
\lefteqn{
\sum_{{\bmath k}_1 \in V_{k_1} , \ {\bmath k}_2 \in V_{k_2}}
1_{{\bmath k}_1 - {\bmath k}_1^\prime + {\bmath k}_2 - {\bmath k}_2^\prime}
+
1_{{\bmath k}_1 - {\bmath k}_1^\prime - {\bmath k}_2 - {\bmath k}_2^{\prime\prime}}
}
&&
\\
\nonumber
&\approx&
\sum_{{\bmath k}_1 \in V_{k_1} , \ {\bmath k}_2 \in V_{k_2}}
1_{{\bmath k}_1 + {\bmath k}_2}
+
1_{{\bmath k}_1 - {\bmath k}_2}
=
\sum_{{\bmath k}_1 \in V_{k_1} \cap V_{k_2}}
2
\ .
\end{eqnarray}
In the broad-shell approximation~(\ref{broadshellapprox}),
the expected covariance of shell-averaged power
spectra of weighted densities,
equation~(\ref{DphatiDphatjapprox}),
simplifies to
\begin{equation}
\label{DphatiDphatjf}
\left\langle \Delta\hat{p}_i(k_1) \Delta\hat{p}_j(k_2) \right\rangle
\approx
f_{ij}
\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle
\end{equation}
where the factor $f_{ij}$ is
\begin{equation}
\label{fij}
f_{ij} \equiv
\sum_{{\bmath k}_1^\prime + {\bmath k}_1^{\prime\prime} + {\bmath k}_2^\prime + {\bmath k}_2^{\prime\prime} = {\bmath 0}}
w_i({\bmath k}_1^\prime) w_i({\bmath k}_1^{\prime\prime})
w_j({\bmath k}_2^\prime) w_j({\bmath k}_2^{\prime\prime})
\ .
\end{equation}
In real (as opposed to Fourier) space, the factor $f_{ij}$ is
\begin{equation}
f_{ij}
= \int w_i({\bmath r})^2 w_j({\bmath r})^2 \, \dd^3 r
\ .
\end{equation}
Equation~(\ref{DphatiDphatjf}) is the most basic result of the present paper.
It states that the expected covariance between estimates of
power from various weightings is proportional to the
true covariance matrix of power.
The nice thing about the result~(\ref{DphatiDphatjf})
is that the constant of proportionality $f_{ij}$
depends only on the weightings $w_i({\bmath k})$ and $w_j({\bmath k})$,
and is independent both of the power spectrum $P({\bmath k})$
and of the wavenumbers $k_1$ and $k_2$ in the covariance
$\left\langle \Delta\hat{p}_i(k_1) \Delta\hat{p}_j(k_2) \right\rangle$.
\subsection{The covariance of the covariance of shell-averaged power spectra of weighted density}
\label{ddP}
Equation~(\ref{DphatiDphatjf})
provides the formal mathematical justification
for estimating the covariance of power
from the scatter in estimates of power
over an ensemble of weightings of density.
In \S\ref{weightings} we will craft the
weightings $w_i({\bmath k})$ so as to minimize the expected variance
of the estimated covariance of power.
The resulting weightings are ``best possible'',
within the framework of the technique.
To determine the minimum variance estimator,
it is necessary to have an expression for
the (co)variance of the covariance of power,
which we now derive.
The expected covariance between estimates
$\Delta\hat{p}_i(k_1) \Delta\hat{p}_i(k_2)$
of covariance of power
is a covariance of covariance of covariance of densities,
an 8-point object.
This object involves, in addition to the 8-point function,
a linear combination of products of lower-order functions adding to 8 points.
The types of terms are
(cf.\ \citealt{VH01})
\begin{equation}
\label{eightpt}
2^4, \quad 2 \cdot 3^2, \quad 2^2 \cdot 4, \quad 2 \cdot 6, \quad 3 \cdot 5, \quad 4^2, \quad 8
\end{equation}
in which $2^4$ signifies a product of four 2-point functions,
$2 \cdot 3^2$ signifies a product of a 2-point function with two 3-point functions,
and so on, up to $8$, which signifies the 8-point function.
We do not pause to write out all the terms explicitly,
because in the same slowly-varying and broad-shell approximations
that led to equation~(\ref{DphatiDphatjf}),
the covariance of covariance of power spectra of weighted densities simplifies to
\begin{eqnarray}
\label{Dphati2Dphatj2g}
\lefteqn{
\Bigl\langle
\bigl[ \Delta\hat{p}_i(k_1) \Delta\hat{p}_i(k_2) - \left\langle \Delta\hat{p}_i(k_1) \Delta\hat{p}_i(k_2) \right\rangle \bigr]
}
&&
\nonumber
\\
&&
\times
\bigl[ \Delta\hat{p}_j(k_3) \Delta\hat{p}_j(k_4) - \left\langle \Delta\hat{p}_j(k_3) \Delta\hat{p}_j(k_4) \right\rangle \bigr]
\Bigr\rangle
\nonumber
\\
&\approx&
g_{ij}
\Bigl\langle
\bigl[ \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) - \left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle \bigr]
\nonumber
\\
&&
\times
\bigl[ \Delta\hat{p}(k_3) \Delta\hat{p}(k_4) - \left\langle \Delta\hat{p}(k_3) \Delta\hat{p}(k_4) \right\rangle \bigr]
\Bigr\rangle
\end{eqnarray}
where $g_{ij}$ is, analogously to equation~(\ref{fij}),
\begin{eqnarray}
\label{gij}
\lefteqn{
g_{ij}
\equiv
\sum_{{\bmath k}_1^\prime + {\bmath k}_1^{\prime\prime} + {\bmath k}_2^\prime + {\bmath k}_2^{\prime\prime} + {\bmath k}_3^\prime + {\bmath k}_3^{\prime\prime} + {\bmath k}_4^\prime + {\bmath k}_4^{\prime\prime} = {\bmath 0}}
}
&&
\\
\nonumber
\lefteqn{
\quad
w_i({\bmath k}_1^\prime) w_i({\bmath k}_1^{\prime\prime}) w_i({\bmath k}_2^\prime) w_i({\bmath k}_2^{\prime\prime})
w_j({\bmath k}_3^\prime) w_j({\bmath k}_3^{\prime\prime}) w_j({\bmath k}_4^\prime) w_j({\bmath k}_4^{\prime\prime})
\ .
}
&&
\end{eqnarray}
In real (as opposed to Fourier) space, the factors $g_{ij}$ are
\begin{equation}
g_{ij}
= \int w_i({\bmath r})^4 w_j({\bmath r})^4 \, \dd^3 r
\ .
\end{equation}
Equation~(\ref{Dphati2Dphatj2g}) states,
analogously to equation~(\ref{DphatiDphatjf}),
that the expected covariance of covariance of power spectra
of weighted densities is proportional to the
true covariance of covariance of power.
As with the factors $f_{ij}$, equation~(\ref{fij}),
the constants of proportionality $g_{ij}$, equation~(\ref{gij}),
depend only on the weightings $w_i({\bmath k})$ and $w_j({\bmath k})$,
and are independent of the power spectrum $P({\bmath k})$
or of any of the higher order functions,
and are also independent of the wavenumbers $k_1$, ..., $k_4$
in the covariance,
a gratifyingly simple result.
\subsection{Subtracting the mean power}
\label{subtractmeanP}
The deviation $\Delta\hat{p}_i(k)$
of the shell-averaged power spectrum of the $i$'th weighted density
was defined above, equation~(\ref{Dphati}),
to be the difference between the measured value
$\hat{p}_i(k)$
and the expected value
$p_i(k)$
of shell-averaged power.
However, the expected power spectrum
$p_i(k)$
(the true power spectrum)
is probably unknown.
Even if the true power spectrum is known in the linear regime
(because the simulation was set up with a known linear power spectrum),
the true power spectrum in the non-linear regime is not known precisely,
but must be estimated from the simulation.
In practice, therefore, it is necessary to measure the deviation in power
not from the true value,
but rather from some estimated mean value.
Two strategies naturally present themselves.
The first strategy is to take the mean power spectrum
to be the measured power spectrum
$\hat{p}(k)$
of the unweighted density of the simulation.
In this case the deviation
$\Delta\hat{p}^\prime_i(k)$
between the measured shell-averaged power spectra
of the weighted and unweighted densities is
(the deviation $\Delta\hat{p}^\prime_i(k)$
is primed to distinguish it
from the deviation $\Delta\hat{p}_i(k)$,
eq.~(\ref{Dphati}))
\begin{equation}
\label{Dphati1}
\Delta\hat{p}^\prime_i(k)
\equiv
\hat{p}_i(k) - \hat{p}(k)
\ .
\end{equation}
The second strategy is to take the mean power spectrum
to be the average over weightings of the measured power spectra
of weighted densities,
$N^{-1} \sum_i \hat{p}_i(k)$.
In this case the deviation
$\Delta\hat{p}^\prime_i(k)$
between the measured shell-averaged power spectra
and their average is
(with the same primed notation for the deviation
$\Delta\hat{p}^\prime_i(k)$
as in eq.~(\ref{Dphati1});
it is up to the user to decide which strategy to adopt)
\begin{equation}
\label{Dphati2}
\Delta\hat{p}^\prime_i(k)
\equiv
\hat{p}_i(k) - \frac{1}{N} \sum_i \hat{p}_i(k)
\ .
\end{equation}
The advantage of the first strategy,
equation~(\ref{Dphati1}),
is that the power spectrum $\hat{p}(k)$ of
the unweighted density is the most accurate
(by symmetry)
estimate of the power spectrum
that can be measured from a single simulation.
Its disadvantage is that measurements of power spectra of weighted densities
yield (slightly) biassed estimates of the power spectrum of unweighted density,
because the approximation~(\ref{Papprox}) can lead to a slight bias
if, as is typical, the power spectrum $P({\bmath k})$ is not constant.
In other words, the approximation $p_i(k) \approx p(k)$,
equation~(\ref{piapprox}), is not an exact equality.
Although the bias is likely to be small, it contributes systematically
to estimates of deviations of power,
causing the covariance of power to be systematically over-estimated.
The second strategy,
equation~(\ref{Dphati2}),
is unaffected by this bias,
but the statistical uncertainty is slightly larger.
Probably the sensible thing to do is to apply both strategies,
and to check that they yield consistent results.
To allow a concise expression for the covariance of power to be written down,
it is convenient to introduce $v_i({\bmath k})$,
defined to be the Fourier transform of the squared real-space weighting,
$v_i({\bmath r}) \equiv w_i({\bmath r})^2$,
\begin{equation}
\label{vi}
v_i({\bmath k}) \equiv \sum_{{\bmath k}^\prime + {\bmath k}^{\prime\prime} = {\bmath k}}
w_i({\bmath k}^\prime) w_i({\bmath k}^{\prime\prime})
\ .
\end{equation}
The normalization condition~(\ref{wnorm}) on the weightings $w_i({\bmath k})$
is equivalent to requiring
\begin{equation}
v_i({\bmath 0}) = 1
\ .
\end{equation}
In terms of $v_i({\bmath k})$,
the factors $f_{ij}$,
equation~(\ref{fij}),
relating the expected covariance matrix of power spectra of weighted densities
to the true covariance matrix of power are
\begin{equation}
\label{fijv}
f_{ij}
=
\sum_{\bmath k} v_i({\bmath k}) v_j(-{\bmath k})
\ .
\end{equation}
An expression is desired for the covariance of power
in terms of the deviations $\Delta\hat{p}^\prime_i(k)$,
equations~(\ref{Dphati1}) or (\ref{Dphati2}),
instead of $\Delta\hat{p}_i(k)$.
For this, a modified version of $v_i({\bmath k})$ is required.
For strategy one,
equation~(\ref{Dphati1}),
\begin{equation}
\label{vpi1}
v^\prime_i({\bmath k})
=
\left\{
\begin{array}{ll}
0 & ({\bmath k} = {\bmath 0}) \\
v_i({\bmath k}) & ({\bmath k} \neq {\bmath 0})
\end{array}
\right.
\end{equation}
whereas for strategy two,
equation~(\ref{Dphati2}),
\begin{equation}
\label{vpi2}
v^\prime_i({\bmath k})
=
v_i({\bmath k}) - \frac{1}{N} \sum_i v_i({\bmath k})
\ .
\end{equation}
In either case,
the expected covariance
$\left\langle \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_j(k_2) \right\rangle$
of estimates of shell-averaged power spectra is
related to the true covariance
$\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle$
of shell-averaged power by
(compare eq.~(\ref{DphatiDphatjf}))
\begin{equation}
\label{DphatiDphatjfp}
\left\langle \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_j(k_2) \right\rangle
\approx
f^\prime_{ij}
\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle
\end{equation}
where the factors $f^\prime_{ij}$ are
(compare eq.~(\ref{fijv}))
\begin{equation}
\label{fpijv}
f^\prime_{ij}
=
\sum_{\bmath k} v^\prime_i({\bmath k}) v^\prime_j(-{\bmath k})
\ .
\end{equation}
The approximation~(\ref{DphatiDphatjfp})
is valid under the same assumptions made
in deriving the approximation~(\ref{DphatiDphatjf}),
namely the slowly-varying approximations~(\ref{Papprox}) and (\ref{Tapprox}),
and the broad-shell approximation~(\ref{broadshellapprox}).
\subsection{Subtracting the mean covariance of power}
\label{subtractmeanddP}
The expression~(\ref{gij})
for the covariance of covariance of power
must likewise be modified
to allow for the fact that the deviations in power must be measured
as deviations not from the true power spectrum but from
either (strategy~1) the power spectrum of the unweighted density,
or (strategy~2) the averaged power spectrum of the weighted densities.
For this purpose it is convenient to define $u_i({\bmath k})$
to be the Fourier transform of the fourth power of the real-space weighting,
$u_i({\bmath r}) \equiv v_i({\bmath r})^2 = w_i({\bmath r})^4$,
\begin{equation}
\label{ui}
u_i({\bmath k}) \equiv \sum_{{\bmath k}^\prime + {\bmath k}^{\prime\prime} = {\bmath k}}
v_i({\bmath k}^\prime) v_i({\bmath k}^{\prime\prime})
\ .
\end{equation}
In terms of $u_i({\bmath k})$,
the factors $g_{ij}$,
equation~(\ref{gij}),
relating the expected covariance of covariance of power spectra of weighted densities
to the true covariance of covariance of power are
\begin{equation}
\label{giju}
g_{ij}
=
\sum_{\bmath k} u_i({\bmath k}) u_j(-{\bmath k})
\ .
\end{equation}
To write down an expression for
the covariance of the covariance of the deviations
$\Delta\hat{p}^\prime_i(k)$
instead of
$\Delta\hat{p}_i(k)$,
define a modified version $u^\prime_i({\bmath k})$
of $u_i({\bmath k})$
by
\begin{equation}
\label{upi}
u^\prime_i({\bmath k}) \equiv \sum_{{\bmath k}^\prime + {\bmath k}^{\prime\prime} = {\bmath k}}
v^\prime_i({\bmath k}^\prime) v^\prime_i({\bmath k}^{\prime\prime})
\end{equation}
which is the same as equation~(\ref{ui})
but with primed $v^\prime_i({\bmath k})$,
equations~(\ref{vpi1}) or (\ref{vpi2}),
in place of $v_i({\bmath k})$.
Then the covariance of the covariance of the deviations
$\Delta\hat{p}^\prime_i(k)$
is related to the true covariance of covariance of shell-averaged power by
(compare eq.~(\ref{Dphati2Dphatj2g}))
\begin{eqnarray}
\label{Dphati2Dphatj2gp}
\lefteqn{
\Bigl\langle
\bigl[ \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2) - \left\langle \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2) \right\rangle \bigr]
}
&&
\nonumber
\\
&&
\times
\bigl[ \Delta\hat{p}^\prime_j(k_3) \Delta\hat{p}^\prime_j(k_4) - \left\langle \Delta\hat{p}^\prime_j(k_3) \Delta\hat{p}^\prime_j(k_4) \right\rangle \bigr]
\Bigr\rangle
\nonumber
\\
&\approx&
g^\prime_{ij}
\Bigl\langle
\bigl[ \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) - \left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle \bigr]
\nonumber
\\
&&
\times
\bigl[ \Delta\hat{p}(k_3) \Delta\hat{p}(k_4) - \left\langle \Delta\hat{p}(k_3) \Delta\hat{p}(k_4) \right\rangle \bigr]
\Bigr\rangle
\end{eqnarray}
where the factors $g^\prime_{ij}$ are
(compare eq.~(\ref{giju}))
\begin{equation}
\label{gpiju}
g^\prime_{ij}
=
\sum_{\bmath k} u^\prime_i({\bmath k}) u^\prime_j(-{\bmath k})
\ .
\end{equation}
Equation~(\ref{Dphati2Dphatj2gp})
gives the expected covariance of the difference between
the estimate of covariance
$\Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2)$
and its expectation value
$\left\langle \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2) \right\rangle$,
but this latter expectation value is again an unknown quantity.
What can actually be measured is the difference between
the estimate
$\Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2)$
and its average over weightings
$N^{-1} \sum_i \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2)$.
To write down an expression for the covariance of the covariance
relative to the weightings-averaged covariance
rather than the expected covariance,
define a modified version $u^{\prime\prime}_i({\bmath k})$ of $u^\prime_i({\bmath k})$,
equation~(\ref{upi}), by
\begin{equation}
\label{uppi}
u^{\prime\prime}_i({\bmath k})
\equiv
u^\prime_i({\bmath k}) - \frac{1}{N} \sum_i u^\prime_i({\bmath k})
\ .
\end{equation}
Then the covariance of the covariance of the deviations
$\Delta\hat{p}^\prime_i(k)$
is related to the true covariance of covariance of shell-averaged power by
(compare eqs.~(\ref{Dphati2Dphatj2g}) and (\ref{Dphati2Dphatj2gp}))
\begin{eqnarray}
\label{Dphati2Dphatj2gpp}
\lefteqn{
\Bigl\langle
\bigl[ \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2) - \frac{1}{N} \sum_k \Delta\hat{p}^\prime_k(k_1) \Delta\hat{p}^\prime_k(k_2) \bigr]
}
&&
\nonumber
\\
&&
\times
\bigl[ \Delta\hat{p}^\prime_j(k_3) \Delta\hat{p}^\prime_j(k_4) - \frac{1}{N} \sum_l \Delta\hat{p}^\prime_l(k_3) \Delta\hat{p}^\prime_l(k_4) \bigr]
\Bigr\rangle
\nonumber
\\
&\approx&
g^{\prime\prime}_{ij}
\Bigl\langle
\bigl[ \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) - \left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle \bigr]
\nonumber
\\
&&
\times
\bigl[ \Delta\hat{p}(k_3) \Delta\hat{p}(k_4) - \left\langle \Delta\hat{p}(k_3) \Delta\hat{p}(k_4) \right\rangle \bigr]
\Bigr\rangle
\end{eqnarray}
where the factors $g^{\prime\prime}_{ij}$ are
(compare eqs.~(\ref{giju}) and (\ref{gpiju}))
\begin{equation}
\label{gppiju}
g^{\prime\prime}_{ij}
=
\sum_{\bmath k} u^{\prime\prime}_i({\bmath k}) u^{\prime\prime}_j(-{\bmath k})
\ .
\end{equation}
Approximations~(\ref{Dphati2Dphatj2gp}) and (\ref{Dphati2Dphatj2gpp})
are valid under the same approximations as
approximations~(\ref{DphatiDphatjf}) and (\ref{Dphati2Dphatj2g}),
namely
the slowly-varying approximations~(\ref{Papprox}) and (\ref{Tapprox}),
and the broad-shell approximation~(\ref{broadshellapprox}).
\section{Minimum variance weightings}
\label{weightings}
It was shown in \S\ref{estimate}
that the expected covariance between shell-averaged power spectra
of weighted densities is proportional to the true covariance of
shell-average power, equation~(\ref{DphatiDphatjfp}).
It follows that the scatter in estimates of power from different weightings
can be used to estimate the true covariance of power.
In this section we use minimum variance arguments
to derive a set of $52$ weightings,
equation~(\ref{wkminvar}),
which we recommend, \S\ref{recommend}, for practical application.
In this section as in the previous one, \S\ref{estimate},
we continue to ignore the beat-coupling contributions
to the (covariance of) covariance of power.
These beat-couplings are discussed in \S\ref{beatcoupling},
which in \S\ref{notquiteweightings} concludes
that the minimum variance weightings derived in the present section,
although no longer precisely minimum variance,
should be satisfactory for practical use.
\subsection{Fundamentals and symmetries}
\label{fund}
In the first place,
we choose to use weightings $w_i({\bmath k})$
that contain only combinations of fundamental modes,
that is,
${\bmath k} = \{k_x, k_y, k_z\}$
with $k_x$, $k_y$, $k_z$ running over $0, \pm 1$.
By restricting the weightings to fundamental modes only,
we ensure that the two approximations
required for equation~(\ref{DphatiDphatjfp}) to be valid
are as good as can be.
The first approximation was
the slowly-varying approximation,
that both the power spectrum $P({\bmath k})$
and the trispectrum $T({\bmath k}_1, -{\bmath k}_1, {\bmath k}_2, -{\bmath k}_2)$
remain approximately constant, equations~(\ref{Papprox}) and (\ref{Tapprox}),
when their arguments are displaced by the extent of the weightings $w_i({\bmath k}^\prime)$,
that is,
by amounts ${\bmath k}^\prime$ for which $w_i({\bmath k}^\prime)$ is non-zero.
The second approximation was
the broad-shell approximation,
that the shells $V_k$ over which the estimated power $\hat{p}_i(k)$ is averaged
are broad compared to the extent of the weightings $w_i({\bmath k}^\prime)$,
which reduces the relative importance of smearing of Gaussian variance
from the edges of adjacent shells into covariance between the shells.
In the second place,
we choose to use weightings that are symmetrically related to each other,
which seems a natural thing to do given the cubic symmetry of a periodic box.
Choosing a symmetrically related set of weightings
not only simplifies practical application of the procedure,
but also simplifies the mathematics of determining a best set
of Fourier coefficients $w_i({\bmath k})$,
as will be seen in \S\ref{minvar} below.
There are $48$ rotational and reflectional transformations of a cube,
corresponding to choosing the $x$-axis in any of 6 directions,
then the $y$-axis in any of 4 directions perpendicular to the $x$-axis,
and finally the $z$-axis in either of the 2 directions
perpendicular to the $x$- and $y$-axes.
To the rotational and reflectional transformations
we adjoin the possibility of translations
by a fraction (half, quarter, eighth) of a box along any of the 3 axes,
for a net total of
$48 \times 8^3 = 24{,}576$
possible transformations.
In practice, however,
the minimum variance weightings $w_i({\bmath k})$
presented in \S\ref{theweightings}
prove to possess a high degree of symmetry,
greatly reducing the number of
distinct weightings.
\subsection{How to derive minimum variance weightings}
\label{minvar}
For brevity,
let $\widehat{X}_i$ denote an estimate of the covariance of shell-averaged power
from the $i$'th weighted density
(the arguments $k_1$ and $k_2$ on $\widehat{X}_i$ are suppressed,
since they play no role in the arguments that follow)
\begin{equation}
\label{Xhati}
\widehat{X}_i
\equiv
\frac{1}{f^\prime} \,
\Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_i(k_2)
\ .
\end{equation}
The quantity $f^\prime$ here is any diagonal element
\begin{equation}
f^\prime \equiv f^\prime_{ii}
\end{equation}
of the matrix of factors $f^\prime_{ij}$
defined by equation~(\ref{fpijv});
the diagonal elements $f^\prime_{ii}$ are identically equal for all $i$
because the weightings $w_i({\bmath k})$ are by assumption symmetrically related.
The factor $1/f^\prime$ in equation~(\ref{Xhati})
ensures that
$\widehat{X}_i$
is,
in accordance with equation~(\ref{DphatiDphatjfp}),
an
estimate
of the true covariance of shell-averaged power,
which we abbreviate $X$,
\begin{equation}
\label{Xi}
\langle \widehat{X}_i \rangle
\approx X
\equiv
\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle
\ .
\end{equation}
The approximation~(\ref{Xi}) is valid
under the assumptions made in deriving equation~(\ref{DphatiDphatjfp}),
namely the slowly-varying approximations~(\ref{Papprox}) and (\ref{Tapprox}),
and the broad-shell approximation~(\ref{broadshellapprox}).
Let $N$ denote the number of weightings.
Because the weightings are by assumption symmetrically related,
it follows immediately that the best estimate
of the true covariance of shell-averaged power
$\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle$
will be a straight average over the ensemble of weightings
\begin{equation}
\label{Xhat}
\widehat{X}
=
\frac{1}{N} \sum_{i} \widehat{X}_i
\ .
\end{equation}
It remains to determine the best Fourier coefficients $w_i({\bmath k})$
for a representative weighting $i$.
The best set is that which minimizes the expected variance
$\langle \Delta\widehat{X}^2 \rangle \equiv \langle (\widehat{X}{-}X)^2 \rangle$
of the estimate~(\ref{Xhat}).
According to equation~(\ref{Dphati2Dphatj2gp}),
this expected variance
$\langle \Delta\widehat{X}^2 \rangle$
is approximately proportional to a factor that depends on the weightings
\begin{equation}
\label{DXhat2}
\langle \Delta \widehat{X}^2 \rangle
=
\frac{1}{N^2} \sum_{ij}
\langle \Delta \widehat{X}_i \Delta \widehat{X}_j \rangle
\simpropto
\frac{1}{(f^\prime N)^2} \sum_{ij} g^\prime_{ij}
\end{equation}
multiplied by another factor that is independent of weightings,
namely the true covariance of covariance of power,
the expression to the right of the coefficient $g^\prime_{ij}$
in equation~(\ref{Dphati2Dphatj2gp}).
Note that the variance $\langle \Delta \widehat{X}^2 \rangle$
is the expected variance
$\langle (\widehat{X}{-}X)^2 \rangle$
about the true value $X$,
so it is $g^\prime_{ij}$, equation~(\ref{gpiju}),
not $g^{\prime\prime}_{ij}$, equation~(\ref{gppiju}),
that appears in equation~(\ref{DXhat2}).
Equation~(\ref{DXhat2})
shows that minimizing the variance
$\langle \Delta\widehat{X}^2 \rangle$
with respect to the coefficients $w_i({\bmath k})$ of the weightings
is equivalent to minimizing the quantity on the right hand side
of the proportionality~(\ref{DXhat2}).
From equations~(\ref{fpijv}), (\ref{upi}), and (\ref{gpiju})
it follows that this factor can be written
\begin{equation}
\label{DXhat2u}
\frac{1}{(f^\prime N)^2} \sum_{ij} g^\prime_{ij}
=
\frac{1}{u^\prime({\bmath 0})^2}
\sum_{{\bmath k}}
\left| u^\prime({\bmath k}) \right|^2
\end{equation}
where $u^\prime({\bmath k})$ denotes the average of $u^\prime_i({\bmath k})$,
equation~(\ref{upi}),
over weightings
\begin{equation}
\label{up}
u^\prime({\bmath k})
\equiv
\frac{1}{N}\sum_{i} u^\prime_i({\bmath k})
\ .
\end{equation}
Note that $f^\prime = u^\prime({\bmath 0})$.
Equation~(\ref{DXhat2u}) shows that minimizing the variance
$\langle \Delta\widehat{X}^2 \rangle$
involves computing $u^\prime({\bmath k})$, equation~(\ref{up}).
We evaluate
$u^\prime({\bmath k})$
using an algebraic manipulation program (Mathematica) as follows.
A representative weighting $w_i({\bmath k})$
contains $27$ non-zero Fourier coefficients,
since by assumption it contains only combinations of fundamental modes.
The coefficients $w_i({\bmath k})$ and $w_i(-{\bmath k})$,
which are complex conjugates of each other,
effectively contribute two coefficients, the real and imaginary parts of $w_i({\bmath k})$.
First,
evaluate $v_i({\bmath k})$, equation~(\ref{vi}), in terms of the coefficients $w_i({\bmath k})$
of the representative weighting.
The $v_i({\bmath k})$ are non-zero for 125 values of ${\bmath k}$,
those whose components
$k_x$, $k_y$, $k_z$ run over $0, \pm 1, \pm 2$.
Each $v_i({\bmath k})$ is a quadratic polynomial in the $27$ Fourier coefficients.
Next, modify $v_i({\bmath k})$ to get $v^\prime_i({\bmath k})$, equation~(\ref{vpi1}),
by setting the coefficient for ${\bmath k} = {\bmath 0}$ to zero.
Again,
each $v^\prime_i({\bmath k})$ is a quadratic polynomial in the $27$ Fourier coefficients.
For definiteness, we adopt strategy one, equation~(\ref{vpi1}),
rather than strategy two, equation~(\ref{vpi2}).
That is, we assume that the deviation $\Delta\hat{p}^\prime_i(k)$
in the power spectrum of the $i$'th weighting of density
is being measured relative to the power spectrum of the unweighted density,
rather than relative to the average of the power spectra of the weighted densities.
In the end it turns out, \S\ref{theweightings2},
that the minimum variance solution is the same for both strategies,
so there is no loss in restricting to strategy one.
Next, evaluate $u^\prime_i({\bmath k})$, equation~(\ref{upi}).
The $u^\prime_i({\bmath k})$ are non-zero for 729 values of ${\bmath k}$,
those whose components
$k_x$, $k_y$, $k_z$ run over $0, \pm 1, \mbox{...}, \pm 4$.
Each $u^\prime_i({\bmath k})$ is a quartic polynomial in the $27$ Fourier coefficients.
Next, evaluate $u^\prime({\bmath k})$, equation~(\ref{up}),
the average of $u^\prime_i({\bmath k})$ over weightings $i$.
Consider first averaging $u^\prime_i({\bmath k})$ over the
48 different rotational and reflectional transformations
of the weighting.
The averaged result $u^\prime({\bmath k})$
possesses rotational and reflectional symmetry,
so that $u^\prime({\bmath k})$
is equal to its value at ${\bmath k}$
with components permuted and reflected
in such a way that
$0 \leq k_x \leq k_y \leq k_z \leq 4$,
of which there are $35$ distinct cases.
The rotationally and translationally symmetrized function $u^\prime({\bmath k})$
can be computed by averaging the values of $u^\prime_i({\bmath k})$
at $729$ values of ${\bmath k}$
into $35$ distinct bins.
The symmetrized function satisfies $u^\prime({\bmath k}) = u^\prime(-{\bmath k})$,
so is necessarily real.
Thus the absolute value sign around $u^\prime({\bmath k})^2$
in equation~(\ref{DXhat2u})
can be omitted.
Now consider averaging the $u^\prime_i({\bmath k})$ over
translations by half a box in each dimension.
There are $2^3 = 8$ such translations,
and each translation is characterized by a triple
$s_x$, $s_y$, $s_z$
giving the number of half boxes translated in each dimension,
either zero or one for each component.
The effect of the translation is to multiply each
coefficient $w_i({\bmath k})$
by $(-)^{s_x k_x + s_y k_y + s_z k_z}$,
that is,
by $\pm 1$ according to whether
$s_x k_x + s_y k_y + s_z k_z$ is even or odd.
The sign change carries through the definitions~(\ref{vi}) of $v_i({\bmath k})$
and (\ref{vpi1}) of $v^\prime_i({\bmath k})$
to the definition~(\ref{upi}) of $u^\prime_i({\bmath k})$,
and thence to the definition~(\ref{up}) of $u^\prime({\bmath k})$.
That is,
the effect of a translation by half a box is to multiply
$u^\prime({\bmath k})$
by $(-)^{s_x k_x + s_y k_y + s_z k_z}$.
It follows that, after averaging over translations,
$u^\prime({\bmath k})$ vanishes if any component of ${\bmath k}$ is odd,
leaving only cases where all components of ${\bmath k}$ are even.
Consequently,
$u^\prime_i({\bmath k})$ need be evaluated only at the $125$ wavevectors ${\bmath k}$
all of whose components are even.
The symmetrized function $u^\prime({\bmath k})$
can be computed by averaging the values of $u^\prime_i({\bmath k})$
at the $125$ values of ${\bmath k}$
into the $10$ distinct bins with even
$0 \leq k_x \leq k_y \leq k_z \leq 4$.
It is amusing that increasing the number of weightings
(by a factor $8$, if all translations yield distinct weightings)
actually decreases the computational work
required to find the best Fourier coefficients $w_i({\bmath k})$.
Adjoining translations by a quarter of a box
simplifies the problem of finding the minimum variance solution
for the coefficients $w_i({\bmath k})$ even further.
There are $4^3 = 64$ such translations,
and each translation is characterized by a triple
$s_x$, $s_y$, $s_z$, each component running over $0$ to $3$,
giving the number of quarter boxes translated in each dimension.
The effect of the translation is to multiply each
coefficient $w_i({\bmath k})$
by $\im^{s_x k_x + s_y k_y + s_z k_z}$.
The effect propagates through to the symmetrized function
$u^\prime({\bmath k})$,
which is therefore non-zero
only for the $27$ wavevectors ${\bmath k}$ all of whose components are multiples of $4$.
The symmetrized function $u^\prime({\bmath k})$
can be computed by averaging the values of $u^\prime_i({\bmath k})$
at the $27$ values of ${\bmath k}$
into the $4$ distinct bins with
$0 \leq k_x \leq k_y \leq k_z \leq 4$
and
each component a multiple of $4$.
One more step,
adjoining translations by an eighth of a box,
reduces the problem of finding the minimum variance solution
to a triviality.
After adjoining translations by an eighth of a box,
the symmetrized function $u^\prime({\bmath k})$
vanishes except at ${\bmath k} = {\bmath 0}$.
The function to be minimized,
the right hand side of equation~(\ref{DXhat2u}),
is therefore identically equal to $1$,
and any arbitrary weighting therefore yields a minimum variance solution.
Though amusing,
the result is not terribly useful,
because it involves a vast number, $48 \times 8^3 = 24{,}576$,
of weightings.
Physically,
if there are enough weightings,
then together they exhaust the information
about the covariance of power,
however badly crafted the weightings may be.
As will be seen in \S\ref{theweightings},
there are much simpler solutions that achieve the absolute
minimum possible variance, for which
the right hand side of equation~(\ref{DXhat2u})
equals $1$,
with far fewer weightings.
The argument above
has shown that the problem of finding the minimum variance solution
for $w_i({\bmath k})$
attains its simplest non-trivial form
if the weightings are generated from a representative weighting
by rotations, reflections, and translations by quarter of a box,
a total of
$48 \times 4^3 = 3{,}072$
symmetries.
In this case,
the weighting-dependent factor in
the variance of covariance of power,
the right hand side of equation~(\ref{DXhat2u}),
becomes a rational function,
a ratio of two $8$th order polynomials
in the $27$ Fourier coefficients $w_i({\bmath k})$,
the numerator being a sum
$\sum u^\prime({\bmath k})^2$ of squares of $4$ quartics,
and the denominator $u^\prime({\bmath 0})^2$ the square of a quartic.
It is this function that we minimize in \S\ref{theweightings}
to find a best set of weightings.
The minimum variance solution is
independent of the overall normalization of the coefficients $w_i({\bmath k})$,
since the quantity being minimized,
the ratio on the right hand side of equation~(\ref{DXhat2u}),
is independent of the normalization of $w_i({\bmath k})$.
Once the minimum variance solution for the coefficients $w_i({\bmath k})$
has been found,
the coefficients can be renormalized to satisfy the
normalization condition~(\ref{wnorm})
that ensures that the estimates $\hat{p}_i(k)$
of the shell-averaged power spectra of weighted densities
are estimates of the true shell-averaged power $p(k)$,
equations~(\ref{pi}) and (\ref{piapprox}).
\subsection{Minimum variance weightings}
\label{theweightings}
The previous subsection, \S\ref{minvar},
described how to obtain the coefficients $w_i({\bmath k})$
that minimize the expected variance
of the estimate of covariance of shell-averaged power
that comes from averaging over an ensemble of weightings
that contain only combinations of fundamental modes,
and that are symmetrically related to each other by
rotations, reflections, and translations by quarter of a box.
\wfig
Numerically,
we find not one but three separate sets of minimum variance weightings
(with hindsight, the sets are simple enough that they might
perhaps have been found without resort to numerics).
Each set consists of symmetrical transformations of
a weighting generated by a single mode,
namely
$\{1,0,0\}$, $\{1,1,0\}$, and $\{1,1,1\}$
respectively for each of the three sets.
Because each individual weighting has a rather high degree of symmetry,
each set has far fewer than the
$48 \times 4^3 = 3{,}072$ weightings
expected if all symmetrical transformations yielded distinct weightings.
Each of the three sets is generated by the weighting
\begin{equation}
\label{wkminvar}
w_i({\bmath k})
=
\left\{
\begin{array}{ll}
\e^{\pm \im \upi / 8} / \sqrt{2} & \mbox{if } {\bmath k} = \pm {\bmath k}_i \\
0 & \mbox{otherwise}
\end{array}
\right.
\end{equation}
where ${\bmath k}_i$ is one of the three possibilities
\begin{equation}
\label{kminvar}
{\bmath k}_i
=
\left\{
\begin{array}{ll}
\{1,0,0\} & \mbox{set one: $12$ weightings} \\
\{1,1,0\} & \mbox{set two: $24$ weightings} \\
\{1,1,1\} & \mbox{set three: $16$ weightings.}
\end{array}
\right.
\end{equation}
In real space, the weighting $w_i({\bmath r})$
corresponding to $w_i({\bmath k})$ of equation~(\ref{wkminvar}) is
\begin{equation}
\label{wrminvar}
w_i({\bmath r})
=
\sqrt{2} \cos \Bigl[ 2\upi \Bigl( {\bmath k}_i . {\bmath r} + \frac{1}{16} \Bigr) \Bigr]
\ .
\end{equation}
The complete set of
$12$ ($24$, $16$)
weightings for each set is obtained
as follows.
In set one (two, three),
a factor of $6$ ($12$, $8$)
comes from the cubic (dodecahedral, octohedral)
symmetry of permuting and reflecting the components
$k_x$, $k_y$, $k_z$ of ${\bmath k}$,
or equivalently the components
$x$, $y$, $z$ of ${\bmath r}$.
A further factor of $2$ comes from
multiplying $w_i(\pm{\bmath k})$ by $\pm\im$,
equivalent to translating by quarter of a box,
or $1/16 \rightarrow 5/16$
in equation~(\ref{wrminvar}).
The three minimum variance solutions are absolute minimum variance,
in the sense that each set not only minimizes
the expression on the right hand side of equation~(\ref{DXhat2u}),
but it solves
$u^\prime({\bmath k}) = 0$ for ${\bmath k} \neq {\bmath 0}$.
This means that
it is impossible to find better solutions
in which all the weightings are symmetrically related to each other,
which is the condition
under which equation~(\ref{DXhat2u}) was derived.
With the minimum variance solutions in hand,
it is possible to go back and examine
the covariance
$\langle \Delta\hat{p}^\prime_i \Delta\hat{p}^\prime_j \rangle$,
equation~(\ref{DphatiDphatjfp}),
between estimates of power from different weightings $i$ and $j$,
either within the same set, or across two different sets.
Estimates of power between two different sets are uncorrelated:
the covariance $\langle \Delta\hat{p}^\prime_i \Delta\hat{p}^\prime_j \rangle$
is zero if $i$ and $j$ are drawn from two different sets.
If on the other hand the weightings $i$ and $j$ are drawn from the same set,
then it turns out that only half of the weightings,
the $6$ ($12$, $8$) weightings related by the
cubic (dodecahedral, octohedral) symmetry of
permuting and reflecting $k_x$, $k_y$, $k_z$,
yield distinct estimates of deviation in power.
The covariance matrix
$\langle \Delta\hat{p}^\prime_i \Delta\hat{p}^\prime_j \rangle$
of estimates of power
between the $6$ ($12$, $8$)
cubically (dodecahedrally, octohedrally) related weightings
is proportional to the unit matrix.
However, translating a weighting by quarter of a box,
$w_j(\pm{\bmath k}) = \pm\im \, w_i(\pm{\bmath k})$,
yields an estimate of deviation of power
that is minus that of the original weighting,
$\Delta\hat{p}^\prime_j = - \Delta\hat{p}^\prime_i$.
Actually, this is exactly true
only if the slowly-varying and thick-shell approximations
are exactly true
(of course,
the thick-shell approximation is never exactly true).
Thus
translating a weighting by quarter of a box
should yield an estimate of deviation in power
that is highly anti-correlated with the original;
which should provide a useful check of the procedure.
Translating a weighting by half a box simply changes its sign,
$w_j(\pm{\bmath k}) = - w_i(\pm{\bmath k})$.
This yields an estimate of deviation of power
that equals exactly (irrespective of approximations)
that of the original weighting,
so yields no distinct estimate of deviation in power.
These redundant translations by half a box
have already been omitted from the set of $12$ ($24$, $16$) weightings.
The value of $f^\prime$,
the factor that converts, equation~(\ref{Xhati}),
estimates $\widehat{X}_i$ of the covariance of power from a weighted density field
to an
estimate of the true covariance of power is
\begin{equation}
\label{fpminvar}
f^\prime = 1/2
\end{equation}
the same factor for each of the three sets.
The expected covariance matrix
$\langle \Delta\widehat{X}_i \Delta\widehat{X}_j \rangle$
of estimates of covariance of power
equals $g^\prime_{ij}$ times the true covariance of covariance of power,
according to equation~(\ref{Dphati2Dphatj2g}).
The factors $g^\prime_{ij}$,
equation~(\ref{gpiju}),
are
\begin{equation}
\label{gpijminvar}
g^\prime_{ij}
=
\left\{
\begin{array}{ll}
3/8 & \mbox{if ${\bmath k}_i = {\bmath k}_j$} \\
1/8 & \mbox{if ${\bmath k}_i = - {\bmath k}_j$} \\
1/4 & \mbox{otherwise.}
\end{array}
\right.
\end{equation}
Equation~(\ref{gpijminvar})
is valid for weightings $i$, $j$
both within the same set and across different sets.
The case ${\bmath k}_i = {\bmath k}_j$ in equation~(\ref{gpijminvar})
occurs not only when $i = j$,
but also when the weightings $i$ and $j$
are related by translation by quarter of a box.
The case ${\bmath k}_i = - {\bmath k}_j$ in equation~(\ref{gpijminvar})
occurs not only when the weightings $i$ and $j$
are parity conjugates of each other,
but also when they are parity confugates
translated by quarter of a box.
The factors $g^{\prime\prime}_{ij}$,
equation~(\ref{gppiju}),
which relate the covariance
$\langle (\widehat{X}_i{-}\widehat{X}) (\widehat{X}_j{-}\widehat{X}) \rangle$
of estimates $\widehat{X}_i$
relative to their measured mean $\widehat{X}$, equation~(\ref{Xhat}),
as opposed to their expected mean $X$, equation~(\ref{Xi}),
are
\begin{equation}
\label{gppijminvar}
g^{\prime\prime}_{ij}
=
\left\{
\begin{array}{ll}
1/8 & \mbox{if ${\bmath k}_i = {\bmath k}_j$} \\
- 1/8 & \mbox{if ${\bmath k}_i = - {\bmath k}_j$} \\
0 & \mbox{otherwise.}
\end{array}
\right.
\end{equation}
An estimate of the uncertainty in the estimate $\widehat{X}$
can be deduced by measuring the variance
$N^{-1} \sum_i (\widehat{X}_i{-}\widehat{X})^2$
in the fluctuations
about the measured mean $\widehat{X}$.
There is of course no point in attempting to estimate the uncertainty from
$N^{-2} \sum_{ij} \Delta\widehat{X}_i \Delta\widehat{X}_j$,
which is identically zero.
The true variance
$\langle \Delta\widehat{X}^2 \rangle$
can be estimated
from the measured variance
$N^{-1} \sum_i (\widehat{X}_i{-}\widehat{X})^2$
by
\begin{equation}
\label{DXhat2est}
\langle \Delta\widehat{X}^2 \rangle
\approx
{2 \over N}
\sum_i (\widehat{X}_i - \widehat{X})^2
\end{equation}
in which the factor of $2$ comes from
(but note the caveat at the end of \S\ref{notquiteweightings})
\begin{equation}
\label{ggminvar}
{N^{-2} \sum_{ij} g^\prime_{ij} \over N^{-1} \sum_{i} g^{\prime\prime}_{ii}}
=
{1/4 \over 1/8}
=
2
\end{equation}
which corrects for the neglected covariance in the measured variance.
\subsection{Minimum variance weightings for strategy two}
\label{theweightings2}
The minimum variance weightings derived above assumed,
for definiteness,
strategy one,
in which the deviation
$\Delta\hat{p}^\prime_i({\bmath k})$
in power is taken
to be relative to the power spectrum of the unweighted density,
equation~(\ref{Dphati1}),
An alternative strategy, strategy two,
is to take the deviation
$\Delta\hat{p}^\prime_i({\bmath k})$
in power
to be relative to the average of the power spectra of the weighted densities,
equation~(\ref{Dphati2}).
Strategy two yields an estimate of covariance of power that
has potentially less systematic bias,
but potentially greater statistical uncertainty.
As it happens,
the minimum variance solution for strategy one,
\S\ref{theweightings},
proves also to solve the minimum variance problem for strategy two.
Thus the minimum variance solution weightings are the same for both strategies.
Mathematically,
expectation values of covariances for the two methods
differ in that $v^\prime_i({\bmath k})$
is given for strategy one by equation~(\ref{vpi1}),
and for strategy two by equation~(\ref{vpi2}).
However,
for the minimum variance weightings $w_i({\bmath k})$ of strategy one,
equation~(\ref{wkminvar}) and its symmetrical transformations,
it turns out that
$N^{-1} \sum_i v_i({\bmath k})$,
the term subtracted from $v_i({\bmath k})$ in strategy two,
equation~(\ref{vpi2}),
is equal to $v_i({\bmath 0})$ if ${\bmath k} = {\bmath 0}$,
and zero otherwise.
This is exactly the same as
the term subtracted from $v_i({\bmath k})$ in strategy one,
equation~(\ref{vpi1}).
It follows that
$v^\prime_i({\bmath k})$ is the same for the two strategies.
Although the minimum variance set of weightings is the
same for both strategies,
the two strategies will in general yield
different estimates of the covariance of power.
\subsection{More minimum variance weightings}
\label{minvarmore}
The three minimum variance sets of weightings found (numerically)
in \S\ref{theweightings}
all take the same form, equation~(\ref{wkminvar}),
differing only in that they are generated by a different single mode,
with wavevectors
$\{1,0,0\}$, $\{1,1,0\}$, and $\{1,1,1\}$
respectively.
One can check that
the result generalizes to higher order weightings,
in which the wavevector ${\bmath k}_i$ in equation~(\ref{wkminvar})
is any wavevector with integral components
(such as $\{2,0,0\}$, $\{2,1,0\}$, and so on).
That is, for any wavevector
${\bmath k}_i$ with integral components,
the weightings generated
from the weighting of equation~(\ref{wkminvar})
by rotations, relections, and translations by quarter of a box,
form a minimum variance set.
All the results of \S\ref{theweightings} (and \S\ref{theweightings2})
carry through essentially unchanged.
In particular, all equations~(\ref{wrminvar})--(\ref{ggminvar})
remain the same.
The disadvantage of including higher order weightings
is that the estimates $\widehat{X}_i$
of the covariance of power
become increasingly inaccurate as the wavenumber
$|{\bmath k}_i|$ of the weighting increases,
because
the slowly-varying approximations~(\ref{Papprox}) and (\ref{Tapprox}),
and the broad-shell approximation~(\ref{broadshellapprox}),
become increasingly poor
as $|{\bmath k}_i|$ increases.
The advantage of including higher order weightings is that
the more weightings, the better the statistical estimate,
at least in principle.
However,
the gain from more weightings is not as great as one might hope.
The Cram\'{e}r-Rao inequality
(\citealt{KS67};
see e.g.\ \citealt{H05} for a pedagogical derivation)
states that the inverse variance of the best possible unbiassed estimate
$\widehat{X}$ of the parameter $X$
must be less than or equal to the Fisher information $F$
(see \citealt{TTH97})
in the parameter $X$
\begin{equation}
\label{CramerRao}
\langle \Delta\widehat{X}^2 \rangle^{-1}
\leq
F
\equiv
- \Bigl\langle {\partial^2 \ln {\cal L} \over \partial X^2} \Bigr\rangle
\end{equation}
where ${\cal L}$ is the likelihood function.
To the extent that
the estimates $\widehat{X}_i$ are Gaussianly distributed
(that is,
the likelihood function is a Gaussian in the estimates $\widehat{X}_i$
\begin{equation}
{\cal L}
\simpropto
\exp \Bigl[ - \frac{1}{2}
\sum_{ij} \langle \Delta\widehat{X}_i \Delta\widehat{X}_j \rangle^{-1}
(\widehat{X}_i - X ) ( \widehat{X}_j - X)
\Bigr]
\end{equation}
with covariance
$\langle \Delta\widehat{X}_i \Delta\widehat{X}_j \rangle$
independent of $X$),
which could be a rather poor approximation,
the Fisher information $F$ in the parameter $X$
approximates the sum of the elements of the inverse covariance matrix,
\begin{equation}
\label{Fisher}
F
\approx
\sum_{ij} \langle \Delta\widehat{X}_i \Delta\widehat{X}_j \rangle^{-1}
\ .
\end{equation}
In the present case,
the covariance matrix
$\langle \Delta\widehat{X}_i \Delta\widehat{X}_j \rangle$
is proportional to $g^\prime_{ij}$,
so in approximation that $\widehat{X}_i$ are Gaussianly distributed,
the Fisher information $F$ is proportional to
\begin{equation}
\label{Fisherg}
F
\simpropto
\sum_{ij} {g^\prime_{ij}}^{-1}
\ .
\end{equation}
With the coefficients $g^\prime_{ij}$
given by equation~(\ref{gpijminvar}),
the quantity on the right hand side of equation~(\ref{Fisherg})
proves to be a constant,
independent of the number of estimates $\widehat{X}_i$
\begin{equation}
\label{Fisherminvar}
\sum_{ij} {g^\prime_{ij}}^{-1}
=
4
\ .
\end{equation}
This constancy of the Fisher information $F$
with respect to the number of estimates
suggests that
there is no gain at all in adjoining more and more estimates.
However, this conclusion
is true only to the extent, firstly, that the slowly-varying
and broad-shell approximations are good,
and, secondly, that
the estimates $\widehat{X}_i$ are Gaussianly distributed,
neither of which assumptions necessarily holds.
All one can really conclude
is that the gain in statistical accuracy
from including more estimates is likely to be limited.
There is however another important consideration
besides the accuracy of the estimate of the covariance matrix of power:
it is desirable
that the estimated covariance matrix be,
like the true covariance matrix,
strictly positive definite,
that is, it should have no zero (or negative) eigenvalues.
As noted by
\cite{PS05},
if a matrix is estimated as an average over $N$ estimates,
then its rank can be no greater than $N$.
Thus,
to obtain a positive definite
covariance matrix of power for $N$ shells of wavevector,
at least $N$ distinct estimates $\widehat{X}_i$
are required.
In \S\ref{recommend} below
we recommend estimating the covariance of power from
an ensemble of $12 + 24 + 16 = 52$ weightings.
This will yield a positive definite covariance matrix
only if the covariance of power is estimated
over no more than $52$ shells of wavenumber.
Since, as noted in \S\ref{theweightings},
weightings related by translation by quarter of a box
yield highly anti-correlated estimates of power,
hence highly correlated estimates of covariance of power,
a more conservative approach would be to consider
that the $52$ weightings yield
only $26$ effectively distinct estimates of covariance of power,
so that the covariance of power can be estimated
over no more than $26$ shells of wavenumber.
If
(strategy two)
the deviation of power
is measured relative to
the measured mean over symmetrically related weightings,
a (slightly) different mean for each of the $3$
sets of weightings,
then $3$ degrees of freedom are lost,
and the covariance of power can be estimated
over no more than $52 - 3 = 49$ shells of wavenumber,
or more conservatively
over no more than $26 - 3 = 23$ shells of power.
\subsection{Recommended strategy}
\label{recommend}
Here is a step-by-step recipe for applying the weightings method
to estimate the covariance of power from a periodic simulation.
\begin{enumerate}
\item
Select the weightings $w_i$.
We recommend the minimum variance sets of weightings
given by equation~(\ref{wkminvar})
and its symmetrical transformations.
If the weightings are restricted to contain
only combinations of fundamental modes,
then there are three such sets of weightings,
equation~(\ref{kminvar}),
and the three sets together provide
$N = 12 + 24 + 16 = 52$ distinct weightings.
\item
For each weighting,
measure the shell-averaged power spectrum $\hat{p}_i(k)$
of the weighted density field,
equations~(\ref{phati}) and (\ref{Phati}).
\item
For each weighting,
evaluate the deviation $\Delta\hat{p}_i(k)$
in the shell-averaged power
as the difference between $\hat{p}_i(k)$
and,
either (strategy one)
the shell-averaged power $\hat{p}(k)$ of the unweighted density,
or (strategy two)
the mean
$N^{-1} \sum_i \hat{p}_i(k)$
over symmetrically related weightings.
The advantage of strategy one is that
the statistical error is potentially smaller,
whereas the advantage of strategy two is that
the systematic bias is potentially smaller.
In strategy two,
it makes sense to subtract the mean separately
for each symmetrically related set of weightings,
because the systematic bias is (slightly) different
for each set.
We recommend trying both strategies one and two,
and checking that they yield consistent results.
\item
Estimate the covariance matrix of shell-averaged power
from the average over all $N$ ($52$) weightings
\begin{equation}
\label{Xest}
\bigl\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \bigr\rangle_{\rm est}
=
{2 \over N}
\sum_{i} \Delta\hat{p}_i(k_1) \Delta\hat{p}_i(k_2)
\ .
\end{equation}
The factor of $2$ in equation~(\ref{Xest})
is $1/f^\prime = 2$, equation~(\ref{fpminvar}),
necessary to convert the average over weightings
to an estimate of the true covariance of power,
equation~(\ref{Xhati}).
\end{enumerate}
\vartwofig
\section{Beat-coupling}
\label{beatcoupling}
This paper should have ended at this point.
Unfortunately,
numerical tests,
described in detail in the companion paper
\citep{RH06}
revealed a serious problem.
Figure~\ref{vartwofig}
shows the problem.
It shows the median and quartiles
of variance of power measured by the weightings method
in each of 25 ART $\Lambda$CDM simulations of $128 \, h^{-1} {\rmn{Mpc}}$ box size,
compared to the variance of power measured over the ensemble of
the same 25 simulations.
Although the two methods agree at linear scales,
the weightings method gives a systematically larger variance at nonlinear scales.
The discrepancy reaches almost an order of magnitude
at the smallest scales measured,
$k \sim 5 \, h^{-1} {\rmn{Mpc}}$.
The reader is referred to \citet{RH06} for details
of the simulations and their results.
This section diagnoses and addresses the problem.
The next section, \S\ref{discussion},
discusses the problem
and its relevance to observations.
\subsection{The cause of the problem: beat-coupling}
\label{cause}
The physical cause of the problem
illustrated in Figure~\ref{vartwofig}
traces to a nonlinear coupling of products of Fourier modes
closely spaced in wavenumber
to the large-scale beat mode between them.
This beat-coupling, as we refer to it,
occurs only when power is measured
from Fourier modes with a finite spread in wavevector,
and therefore appears in the weightings method
(and in observations -- see \S\ref{observation} below)
but not in the ensemble method.
The beat-coupling is surprisingly large,
to the point that,
as seen in Figure~\ref{vartwofig},
it actually dominates the variance of power at nonlinear scales.
More specifically,
in the ensemble method,
the power spectrum of a periodic simulation
is measured from the variance
$\Delta\rho({\bmath k}) \Delta\rho(-{\bmath k})$
of Fourier modes.
In the weightings method
on the other hand,
the power spectrum
receives contributions not only from the variance,
but also from the covariance
$\Delta\rho({\bmath k}) \Delta\rho(-{\bmath k}{-}\bvarepsilon)$
between modes a small wavevector $\bvarepsilon$ apart.
This covariance vanishes in the mean,
but it couples to large-scale modes
$\Delta\rho(\bvarepsilon)$
through quadratic nonlinearities.
That is,
the correlation between the product
$\Delta\rho({\bmath k}) \Delta\rho(-{\bmath k}{-}\bvarepsilon)$
and the large-scale mode
$\Delta\rho(\bvarepsilon)$
is the bispectrum
\begin{equation}
\langle \Delta\rho({\bmath k}) \Delta\rho(-{\bmath k}{-}\bvarepsilon) \Delta\rho(\bvarepsilon) \rangle
=
B({\bmath k}, -{\bmath k}{-}\bvarepsilon, \bvarepsilon)
\ .
\end{equation}
The bispectrum is zero for Gaussian fluctuations,
but is driven away from zero by nonlinear gravitational growth.
\subsection{Tetrahedron}
\label{tetrahedron}
The place where, prior to this section, we inadvertently
discarded the large-scale beat-coupling, is equation~(\ref{Tapprox}),
where we made the seemingly innocent approximation
that the trispectrum
$T({\bmath k}_1, {\bmath k}_2, {\bmath k}_3, {\bmath k}_4)$
is a slowly varying function of what appears to be its arguments,
${\bmath k}_1$ to ${\bmath k}_4$.
This assumption is false,
as we now show.
For a statistically isotropic field (as considered in this paper),
the trispectrum depends on six scalar arguments.
This follows from the fact that
a spatial configuration of four points is determined by
the six lengths of
the sides of the tetrahedron whose vertices are the four points.
In Fourier space,
the configuration
is an object four of whose sides are
equal to the wavevectors ${\bmath k}_1$ to ${\bmath k}_4$.
The object forms a closed tetrahedron
(because $\sum_i {\bmath k}_i = {\bmath 0}$),
whose shape
is determined by the six lengths of the sides of the tetrahedron.
\tetrahedronfig
Figure~\ref{tetrahedronfig}
illustrates the configuration of interest in the present paper,
that for the trispectrum in equation~(\ref{DphatiDphatj}).
Rewritten as a function of six scalar arguments,
the trispectrum
of equation~(\ref{DphatiDphatj})
is
\begin{eqnarray}
\label{Tsix}
\lefteqn{
T({\bmath k}_1 {-} {\bmath k}_1^\prime, -{\bmath k}_1 {-} {\bmath k}_1^{\prime\prime}, {\bmath k}_2 {-} {\bmath k}_2^\prime, -{\bmath k}_2 {-} {\bmath k}_2^{\prime\prime})
=
}
&&
\\
\nonumber
\lefteqn{
T\bigl( | {\bmath k}_1 {-} {\bmath k}_1^\prime | , | {\bmath k}_1 {+} {\bmath k}_1^{\prime\prime} | , | {\bmath k}_2 {-} {\bmath k}_2^\prime | , | {\bmath k}_2 {+} {\bmath k}_2^{\prime\prime} | ,
| {\bmath k}_1 {-} {\bmath k}_2 {-} {\bmath k}_1^\prime {-} {\bmath k}_2^{\prime\prime} | , \varepsilon \bigr)
}
&&
\end{eqnarray}
where the wavevector $\bvarepsilon$ is defined by
\begin{equation}
\label{epsilon}
\bvarepsilon
\equiv
- ( {\bmath k}^\prime_1 + {\bmath k}^{\prime\prime}_1 )
=
{\bmath k}^\prime_2 + {\bmath k}^{\prime\prime}_2
\end{equation}
which is small but not necessarily zero.
The invalid approximation~(\ref{Tapprox}) is equivalent to approximating
\begin{eqnarray}
\label{Tapprox1}
\lefteqn{
T\bigl( | {\bmath k}_1 {-} {\bmath k}_1^\prime | , | {\bmath k}_1 {+} {\bmath k}_1^{\prime\prime} | , | {\bmath k}_2 {-} {\bmath k}_2^\prime | , | {\bmath k}_2 {+} {\bmath k}_2^{\prime\prime} | ,
| {\bmath k}_1 {-} {\bmath k}_2 {-} {\bmath k}_1^\prime {-} {\bmath k}_2^{\prime\prime} | , \varepsilon \bigr)
}
\nonumber
\\
&\approx&
T( k_1 , k_1 , k_2 , k_2 , | {\bmath k}_1 {-} {\bmath k}_2 | , 0 )
\ .
\end{eqnarray}
The problem with this approximation is apparent.
Although primed wavenumbers are small compared to unprimed ones,
so that the approximation in the first five arguments is reasonable,
in the last argument it is not valid
to approximate a finite wavenumber $\varepsilon$,
however small, by zero.
A valid approximation is, rather,
\begin{eqnarray}
\label{Tapprox2}
\lefteqn{
T\bigl( | {\bmath k}_1 {-} {\bmath k}_1^\prime | , | {\bmath k}_1 {+} {\bmath k}_1^{\prime\prime} | , | {\bmath k}_2 {-} {\bmath k}_2^\prime | , | {\bmath k}_2 {+} {\bmath k}_2^{\prime\prime} | ,
| {\bmath k}_1 {-} {\bmath k}_2 {-} {\bmath k}_1^\prime {-} {\bmath k}_2^{\prime\prime} | , \varepsilon \bigr)
}
\nonumber
\\
&\approx&
T( k_1 , k_1 , k_2 , k_2 , | {\bmath k}_1 {-} {\bmath k}_2 | , \varepsilon )
\ .
\end{eqnarray}
As an example of the large-scale beat-coupling contributions
to the trispectrum
that arise from the beat wavevector $\bvarepsilon$,
consider perturbation theory.
\subsection{Perturbation theory}
\label{PT}
In perturbation theory (PT),
the trispectrum can be split into snake and star
contributions
(\citealt{SZH99}; \citealt{SS05})
\begin{eqnarray}
\label{Tab}
\lefteqn{
T({\bmath k}_1, {\bmath k}_2, {\bmath k}_3, {\bmath k}_4)
=
}
&&\!\!\!\!\!\!
\nonumber
\\
&&\!\!\!\!\!\!
4 \, P(k_1) P(k_2)
P(k_{13}) F_2({\bmath k}_1,-{\bmath k}_{13}) F_2({\bmath k}_2,{\bmath k}_{13})
\nonumber
\\
&&\!\!\!\!\!\!
\
\mbox{}
+
\mbox{cyclic (12 snake terms)}
\nonumber
\\
&&\!\!\!\!\!\!
\mbox{}
+
P(k_1) P(k_2) P(k_3)
\bigl[
F_3({\bmath k}_1,{\bmath k}_2,{\bmath k}_3) + \mbox{perm.~(6 terms)}
\bigr]
\nonumber
\\
&&\!\!\!\!\!\!
\
\mbox{}
+
\mbox{cyclic (4 star terms)}
\end{eqnarray}
where
${\bmath k}_{ij} \equiv {\bmath k}_i + {\bmath k}_j$,
and the second-order PT kernel $F_2$ is given by
\begin{equation}
F_2({\bmath k}_1,{\bmath k}_{2})
=
\frac{5}{7}
+ \frac{x}{2} \left( \frac{k_1}{k_2} + \frac{k_2}{k_1} \right)
+ \frac{2}{7}\ x^2
\label{F2}
\end{equation}
with $x \equiv \hat{{\bmath k}}_1 \cdot \hat{{\bmath k}}_2$.
In the case of interest,
where the trispectrum is that of equation~(\ref{Tsix}),
4 of the 12 snake terms produce a coupling to large scales,
those where the beat wavenumber
$k_{13}$
in equation~(\ref{Tab})
is small.
In the (valid) approximation~(\ref{Tapprox2}),
the pertinent PT trispectrum is
\begin{eqnarray}
\label{TapproxPT}
\lefteqn{
T\bigl( | {\bmath k}_1 {-} {\bmath k}_1^\prime | , | {\bmath k}_1 {+} {\bmath k}_1^{\prime\prime} | , | {\bmath k}_2 {-} {\bmath k}_2^\prime | , | {\bmath k}_2 {+} {\bmath k}_2^{\prime\prime} | ,
| {\bmath k}_1 {-} {\bmath k}_2 {-} {\bmath k}_1^\prime {-} {\bmath k}_2^{\prime\prime} | , \varepsilon \bigr)
}
\nonumber
\\
&\approx&
T( k_1 , k_1 , k_2 , k_2 , | {\bmath k}_1 {-} {\bmath k}_2 | , 0 )
\nonumber
\\
&&
\mbox{}
+
16 \, P(k_1) P(k_2)
P(\varepsilon) F_2({\bmath k}_1,-\bvarepsilon) F_2({\bmath k}_2,\bvarepsilon)
\end{eqnarray}
in which the term on the last line represents the large-scale beat-coupling contribution
incorrectly ignored by the approximation~(\ref{Tapprox1}).
In equation~(\ref{DphatiDphatj})
for the covariance of shell-averaged power,
this trispectrum, equation~(\ref{TapproxPT}),
is angle-averaged over the directions
of ${\bmath k}_1$ and ${\bmath k}_2$.
The angle-averaged second-order PT kernel is
\begin{equation}
\int F_2({\bmath k},\bvarepsilon) \, {do_{\bmath k} \over 4\upi}
=
\frac{17}{21}
\end{equation}
and it follows that the last line of equation~(\ref{TapproxPT}),
when angle-averaged, is
$16 (17/21)^2 P(k_1) P(k_2) P(\varepsilon)$.
Following the same arguments that led from equation~(\ref{DphatiDphatj})
to equation~(\ref{DphatiDphatjf}),
and then to equation~(\ref{DphatiDphatjfp}),
but with the beat-coupling term now correctly retained in the trispectrum,
one finds that
equation~(\ref{DphatiDphatjfp})
for the expected covariance of shell-averaged power
spectra of weighted densities
is modified to
\begin{eqnarray}
\label{DphatiDphatjfLPT}
\lefteqn{
\left\langle \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_j(k_2) \right\rangle
\approx
f^\prime_{ij}
\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle
}
&&
\nonumber
\\
&&
\qquad
\mbox{}
+
4 \, R_a P(k_1) P(k_2)
\sum_{\bmath k} v^\prime_i({\bmath k}) v^\prime_j(-{\bmath k}) P(k)
\end{eqnarray}
where
$v^\prime_i({\bmath k})$ is defined by equations~(\ref{vi})
and (\ref{vpi1}) or (\ref{vpi2}),
and the constant $R_a$ is
\begin{equation}
\label{RaPT}
R_a = 4 \left( \frac{17}{21} \right)^2 \approx 2.62
\ .
\end{equation}
The reason for writing equation~(\ref{DphatiDphatjfLPT})
in this form, with the constant $R_a$ separated out,
is that,
as will be seen in \S\ref{hierarchical},
the same expression remains valid in the hierarchical model,
but with $R_a$ the 4-point hierarchical snake amplitude.
Figure~\ref{vartwofig}
includes lines showing the predicted PT result
for the variance of shell-averaged power of weighted density,
equation~(\ref{DphatiDphatj}),
both with (solid lines)
and without (dashed lines)
beat-coupling.
The PT variance with beat-coupling was obtained
by numerically integrating the PT expression~(\ref{Tab})
for the trispectrum~(\ref{Tsix}) in
equation~(\ref{DphatiDphatj})
(that is, without making the approximations~(\ref{Tapprox2})
or (\ref{DphatiDphatjfLPT})),
with the minimum variance weightings~(\ref{wkminvar}),
and then multiplying by the factor $1/f^\prime = 2$,
equation~(\ref{fpminvar}).
From this
the PT variance without beat-coupling was obtained
by setting $P(\varepsilon) = 0$.
The variance without beat-coupling agreed well with a direct
PT evaluation of equation~(\ref{DphatDphat}).
Figure~\ref{vartwofig}
shows that
the beat-coupling contribution
predicted by perturbation theory
seems to account reasonably well for
the extra variance that appears at nonlinear scales
in the weightings versus the ensemble method.
We will return to equation~(\ref{DphatiDphatjfLPT})
in \S\ref{largescale} below,
but first consider the hierarchical model
as a prototype of the trispectrum beyond perturbation theory.
\subsection{Hierarchical model}
\label{hierarchical}
Perturbation theory is valid only in the translinear regime.
The behaviour of the trispectrum in the fully nonlinear regime
is less well understood.
Available observational and $N$-body evidence
(\citealt{CBH96}; \citealt{HG99}; \citealt{SF99}; \citealt{Baugh04}; \citealt{Croton04})
is consistent
with a hierarchical model of higher order correlations.
In the hierarchical model
\citep{Peebles80},
the trispectrum is a sum of snake and star terms
\begin{eqnarray}
\label{Thier}
\lefteqn{
T({\bmath k}_1, {\bmath k}_2, {\bmath k}_3, {\bmath k}_4) =
}
&&\!\!\!\!\!\!
\nonumber
\\
&&\!\!\!\!\!\!
R_a \bigl[ P(k_1) P(k_2) P(k_{13}) + \mbox{cyclic~(12 snake terms)} \bigr]
\nonumber
\\
&&\!\!\!\!\!\!
\mbox{}
+
R_b \bigl[ P(k_1) P(k_2) P(k_3) + \mbox{cyclic~(4 star terms)} \bigr]
\ .
\end{eqnarray}
The PT trispectrum,
equation~(\ref{Tab}),
shows a hierarchical structure
with hierarchical amplitudes $R_a$ and $R_b$
that are not constant,
but rather depend on the shape of the trispectrum tetrahedron.
At highly nonlinear scales,
Scoccimarro \& Frieman (1999) suggested an ansatz,
dubbed hyperextended perturbation theory (HEPT),
that the hierarchical amplitudes
go over to the values predicted by perturbation theory
for configurations collinear in Fourier space.
For power law power spectra $P(k) \propto k^n$,
HEPT predicts 4-point amplitudes
\begin{equation}
\label{Q4}
R_a = R_b = {54 - 27\ 2^n + 2\ 3^n + 6^n \over 2 \, (1 + 6\ 2^n + 3\ 3^n + 6\ 6^n)}
\ .
\end{equation}
As pointed out by \citet{SF99}
and
\cite{H00},
HEPT is not entirely consistent because it predicts
a covariance of power
$\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \rangle$
that violates the Schwarz inequality when $k_1 \gg k_2$.
In the hierarchical model with constant hierarchical amplitudes,
4 of the 12 snake terms produce a coupling to large
scales in the trispectrum of interest,
equation~(\ref{Tsix}).
In the (valid) approximation~(\ref{Tapprox2}),
the hierarchical trispectrum is
\begin{eqnarray}
\label{Tapproxhier}
\lefteqn{
T\bigl( | {\bmath k}_1 {-} {\bmath k}_1^\prime | , | {\bmath k}_1 {+} {\bmath k}_1^{\prime\prime} | , | {\bmath k}_2 {-} {\bmath k}_2^\prime | , | {\bmath k}_2 {+} {\bmath k}_2^{\prime\prime} | ,
| {\bmath k}_1 {-} {\bmath k}_2 {-} {\bmath k}_1^\prime {-} {\bmath k}_2^{\prime\prime} | , \varepsilon \bigr)
}
\nonumber
\\
&\approx&
T( k_1 , k_1 , k_2 , k_2 , | {\bmath k}_1 {-} {\bmath k}_2 | , 0 )
\nonumber
\\
&&
\mbox{}
+
4 \, R_a \, P(k_1) P(k_2)
P(\varepsilon)
\end{eqnarray}
in which the term on the last line represents the large-scale beat-coupling contribution.
The hierarchical trispectrum~(\ref{Tapproxhier})
looks similar to (slightly simpler than)
the PT trispectrum~(\ref{TapproxPT}).
Following the same arguments as before,
one recovers the same expression~(\ref{DphatiDphatjfLPT})
for the expected covariance of shell-averaged power
spectra of weighted densities.
\subsection{Covariance of shell-averaged power spectra including large-scale coupling}
\label{largescale}
Suppose that either perturbation theory, \S\ref{PT},
or the hierarchical model, \S\ref{hierarchical},
offers a reliable guide to
the coupling of the nonlinear trispectrum to large scales,
so that equation~(\ref{DphatiDphatjfLPT}) is a good approximation
to the expected covariance of shell-averaged power spectra
of weighted densities.
Make the further assumption that the power spectrum
is approximately constant
over the large-scale wavevectors
represented in $v^\prime_i({\bmath k})$
\begin{equation}
\label{PapproxL}
P(k) \approx P(2 k_b) = \mbox{constant}
\quad
\mbox{for $v_i^\prime({\bmath k}) \neq 0$}
\end{equation}
where $k_b$ is the wavenumber at the box scale.
The factor $2$ in $2 k_b$ in equation~(\ref{PapproxL})
appears as a reminder that
the wavevectors ${\bmath k}$ in $v^\prime_i({\bmath k})$ are,
equations~(\ref{vi}) and (\ref{vpi1}) or (\ref{vpi2}),
sums of pairs of wavenumbers ${\bmath k}^\prime$
represented in the weighting $w_i({\bmath k}^\prime)$.
For example, if the weightings are taken
to be the minimum variance weightings
given by equation~(\ref{wkminvar}),
then $k_b = k_i$
where $k_i$ is the wavenumber of the weighting.
Approximation~(\ref{PapproxL})
is in the same spirit as, but distinct from,
the earlier approximation~(\ref{Papprox})
that the power spectrum is a slowly varying function.
Note that equation~(\ref{PapproxL}) does {\em not\/}
require that $P(0) \approx P(2 k_b)$
(which would certainly not be correct,
because $P(0) = 0$),
because $v^\prime_i({\bmath 0})$ is zero,
which is true a priori in strategy one,
equation~(\ref{vpi1}),
and ends up being true a posteriori in strategy two,
equation~(\ref{vpi2}),
by the argument in \S\ref{theweightings2}.
In the approximation~(\ref{PapproxL}),
the summed expression on the right hand side of
equation~(\ref{DphatiDphatjfLPT}) is
\begin{equation}
\label{vPapprox}
\sum_{\bmath k} v^\prime_i({\bmath k}) v^\prime_j(-{\bmath k}) P(k)
\approx
f^\prime_{ij}
P(2 k_b)
\end{equation}
and equation~(\ref{DphatiDphatjfLPT}) reduces to
\begin{eqnarray}
\label{DphatiDphatjfL}
\left\langle \Delta\hat{p}^\prime_i(k_1) \Delta\hat{p}^\prime_j(k_2) \right\rangle
&\approx&
f^\prime_{ij}
\bigl[
\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle
\nonumber
\\
&&
\mbox{}
+
4 \, R_a P(k_1) P(k_2) P(2 k_b)
\bigr]
\end{eqnarray}
with the term on the last line
being the large-scale beat-coupling contribution.
Equation~(\ref{DphatiDphatjfL})
provides the fundamental justfication
for the weightings method
when beat-coupling is taken into account.
It states that
the covariance of shell-averaged power spectra of weighted densities
is proportional to the sum of
the true covariance
$\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle$
of shell-averaged power,
and a beat-coupling term
$4 \, R_a P(k_1) P(k_2) P(2 k_b)$
proportional to power at (twice) the box wavenumber $k_b$.
The crucial feature of equation~(\ref{DphatiDphatjfL})
is that the constant of proportionality $f^\prime_{ij}$,
equation~(\ref{fpijv}),
depends only on the weightings $w_i({\bmath k})$,
and is independent either of
the power spectrum $P(k)$
or of the wavenumbers $k_1$ and $k_2$.
In the limit of infinite box size,
the beat-coupling contribution
to the covariance of power spectra of weighted densities
in equation~(\ref{DphatiDphatjfL})
goes to zero,
$P(2 k_b) \rightarrow P (0) = 0$
as $k_b \rightarrow 0$,
and the covariance becomes proportional
to the true covariance
$\left\langle \Delta\hat{p}(k_1) \Delta\hat{p}(k_2) \right\rangle$
of power.
However, in cosmologically realistic simulations,
such as illustrated in Figure~\ref{vartwofig}
and discussed further in \S\ref{discussion},
the beat-coupling contribution,
far from being small,
is liable to dominate at nonlinear scales.
Beyond perturbation theory or the hierarchical model,
the weightings method remains applicable
just so long as the hierarchical amplitude $R_a$
in equation~(\ref{DphatiDphatjfL})
is independent of the weightings $ij$.
In general,
$R_a$ could be any arbitrary function of $k_1$, $k_2$,
and the box wavenumber $k_b$.
\subsection{Not quite minimum variance weightings}
\label{notquiteweightings}
Section~\ref{weightings}
derived sets of minimum variance weightings
valid when the covariance, and the covariance of covariance,
of power spectra of weighted densities
took the separable forms given by
equations~(\ref{DphatiDphatjfp}) and (\ref{Dphati2Dphatj2gp}).
When beat-scale coupling is included,
the covariance of power,
equation~(\ref{DphatiDphatjfL}),
still takes the desired separable form
(as long as the hierarchical amplitude $R_a$
is independent of the weightings $ij$),
but the covariance of covariance of power
(eq.~(\ref{Dphati2Dphatj2gpL}) of Appendix~\ref{minvarL})
does not.
In Appendix~\ref{minvarL},
we discuss what happens to the minimum variance derivation
of \S\ref{weightings}
when beat-coupling is included.
We argue that
the minimum variance weightings
of \S\ref{theweightings}
are no longer exactly minimum variance,
but probably remain near minimum variance,
and therefore fine to use in practice.
The factor $2$ on the right hand side of equation~(\ref{DXhat2est})
is no longer correct when beat-coupling is included,
but may remain a reasonable approximation.
\section{Discussion}
\label{discussion}
As shown in \S\ref{beatcoupling},
the covariance of nonlinear power
receives beat-coupling contributions from large scales
whenever power is measured from Fourier modes $\rho({\bmath k})$
that have a finite spread in wavevector ${\bmath k}$,
as opposed to being delta-functions at single discrete wavevectors.
Physically,
the large-scale beat-coupling arises
because a product
$\Delta\rho({\bmath k}) \Delta\rho(-{\bmath k}{-}\bvarepsilon)$
of Fourier amplitudes of closely spaced wavevectors
couples by nonlinear gravitational growth
to the beat mode $\Delta\rho(\bvarepsilon)$ between them.
The beat-coupling contribution does not appear
when covariance of power is measured from ensembles of
periodic box simulations,
because in that case power is measured
from products of Fourier amplitudes
$\Delta\rho({\bmath k}) \Delta\rho(-{\bmath k})$
at single discrete wavevectors.
Here the ``beat'' mode is the mean mode,
${\bmath k} {-} {\bmath k} = {\bmath 0}$,
whose fluctuation is by definition always zero,
$\Delta\rho({\bmath 0}) = 0$.
There is
on the other hand
a beat-coupling contribution
when covariance of power is measured by
the weightings method,
because the Fourier modes of weighted density
are spread over more than one wavevector.
For weightings constructed from combinations of fundamental modes,
as recommended in
\S\ref{weightings},
the covariance of power spectra of weighted densities
receives beat-coupling contributions from power near the box fundamental $k_b$.
The beat-coupling and normal contributions
to the variance
$\left\langle \Delta\hat{p}^\prime_i(k)^2 \right\rangle$
of nonlinear power
are in roughly the ratio
$P(2 k_b) / P(k)$
of power at the box scale
to power at the nonlinear scale,
according to equation~(\ref{DphatiDphatjfL}).
In cosmologically realistic simulations,
box sizes are typically around the range
$10^2$--$10^3 h^{-1} {\rmn{Mpc}}$.
This is just the scale at which the
power spectrum goes through a broad maximum.
For example,
in observationally concordant $\Lambda$CDM models,
power goes through a broad maximum
at $k_{\rm peak} \approx 0.016 \, h \, {\rmn{Mpc}}^{-1}$
(e.g.\ \citealt{Tegmark04a}),
corresponding to a box size
$4\upi/k_{\rm peak} \sim 800 \, h^{-1} {\rmn{Mpc}}$.
Power at the maximum
is about $25$ times
greater than power at the onset
$k_{\rm trans} \approx 0.3 \, h \, {\rmn{Mpc}}^{-1}$
of the translinear regime,
$P(k_{\rm peak}) / P(k_{\rm trans}) \approx 25$,
and the ratio
$P(k_{\rm peak}) / P(k)$
of power increases at more nonlinear wavenumbers $k$.
It follows that in cosmologically realistic simulations
the beat-coupling contribution to the covariance of power
is liable to dominate the normal contribution.
This is consistent with the numerical results
illustrated in Figure~\ref{vartwofig}
and discussed by \citet{RH06},
which show that the variance of power
measured by the weightings method
(which includes beat-coupling contributions)
substantially exceeds, at nonlinear scales,
the variance of power measured by the ensemble method
(which does not include beat-coupling contributions).
\subsection{Relevance to real galaxy surveys}
\label{observation}
In real galaxy surveys,
measured Fourier modes inevitably have finite width $|\Delta{\bmath k}| \sim 1/R$,
where $R$ is a characteristic linear size of the survey.
The characteristic size $R$
varies from $100 \, h^{-1} {\rmn{Mpc}}$ to a few $1000 \, h^{-1} {\rmn{Mpc}}$
(an upper limit is set by the comoving horizon distance,
which is about $10^4 \, h^{-1}{\rmn{Mpc}}$
in the concordant $\Lambda$CDM model).
It follows that the covariance of nonlinear power
measured in real galaxy surveys
is liable to be dominated not by the ``true'' covariance of power
(the covariance of power in a perfect, infinite survey),
but rather by the contribution from
beat-coupling to power at the scale of the survey.
This means that one must take great care
in using numerical simulations
to estimate or to predict the covariance
of nonlinear power expected in a galaxy survey.
The scatter in power over an ensemble of periodic box simulations
will certainly underestimate the covariance of power
by a substantial factor at nonlinear scales,
because of the neglect of beat-coupling contributions.
A common and in principle reliable procedure
is to estimate the covariance of power of a galaxy survey
from mock surveys ``observed''
with the same selection rules as the real survey
from numerical simulations large enough to encompass the entire survey
(e.g.\ \citealt{CDS01};
\citealt{PB03};
\citealt{Tegmark04a};
\citealt{VMC04};
\citealt{Blaizot05};
\citealt{FSO05};
\citealt{Cole05};
\citealt{Eisenstein05};
\citealt{Park05}).
It is important that numerical simulations
be genuinely large enough to contain a mock survey.
One should be wary about estimating covariance of power
from mock surveys
extracted from small periodic boxes replicated many times
(e.g.\ \citealt{Yang04}),
since such boxes are liable to be missing power at precisely those wavenumbers,
the inverse scale size of the mock survey,
where beat-coupling should in reality be strongest.
Beat-coupling arises from a real gravitational
coupling to large scale modes,
and the simulation from which a mock survey is extracted
must be large enough to contain such modes.
Further,
it would be wrong
to take, say,
a volume-limited subsample of a galaxy survey,
and then to estimate the covariance of power
from an ensemble of periodic numerical simulations
whose size is that of the volume-limited subsample.
A volume-limited subsample of observational data
retains beat-coupling contributions
to the covariance of power,
whereas periodic box simulations do not.
\section{Summary}
\label{summary}
This paper falls into two parts.
In the first part,
\S\ref{estimate} and \S\ref{weightings},
we proposed a new method, the weightings method,
that yields an estimate of the covariance of the power spectrum
of a statistically homogeneous and isotropic density field
from a single periodic box simulation.
The procedure is to
apply a set of weightings to the density field,
and to measure the covariance of power
from the scatter in power over the ensemble of weightings.
In \S\ref{estimate}
we developed the formal mathematical apparatus
that justifies the weightings method,
and in \S\ref{weightings}
we derived sets of weightings that
achieve minimum variances estimates of covariance of power.
Section~\ref{recommend}
gives a step-by-step recipe for applying the weightings method.
We recommend a specific set of 52 minimum variance weightings
containing only combinations of fundamental modes.
In the second part of this paper,
\S\ref{beatcoupling} and \S\ref{discussion},
we discuss an unexpected glitch in the procedure,
that emerged from the periodic box numerical simulations
described in the companion paper \citep{RH06}.
The numerical simulations showed that,
at nonlinear scales,
the covariance of power measured by the weightings method
substantially exceeded that measured
over an ensemble of independent simulations.
In \S\ref{beatcoupling}
we argue from perturbation theory
that the discrepancy between the weightings and ensemble methods
arises from ``beat-coupling'',
in which products of closely spaced Fourier modes
couple by nonlinear gravitational growth
to the large-scale beat mode between them.
Beat-coupling is present whenever nonlinear power is measured from Fourier modes
that have a finite spread in wavevector,
as opposed to being delta-functions at single discrete wavevectors.
Beat-coupling affects the weightings method,
because Fourier modes of weighted densities have a finite width,
but not the ensemble method,
because the Fourier modes of a periodic box are delta-functions of wavevector.
As discussed in \S\ref{discussion},
beat-coupling inevitably affects real galaxy surveys,
whose Fourier modes necessarily have a finite width
of the order of the inverse scale size of the survey.
Surprisingly,
at nonlinear scales,
beat-coupling is liable to dominate
the covariance of power of a real survey.
One would have thought that the covariance of power at nonlinear
scales would be dominated by structure at small scales,
but this is not true.
Rather, the covariance of nonlinear power
is liable to be dominated by beat-coupling
to power at the largest scales of the survey.
A common and valid procedure
for estimating the covariance of power from a real survey
is the mock survey method,
in which artificial surveys are ``observed''
from large numerical simulations,
with the same selection rules as the real survey.
It is important that
mock surveys be extracted from genuinely large simulations,
not from many small periodic simulations stacked together,
since stacked simulations miss the large-scale power
essential to beat-coupling.
Finally, it should be remarked that,
although this paper has considered only the covariance
of the power spectrum,
it is likely that,
in real galaxy surveys and cosmologically realistic simulations,
beat-coupling contributions dominate
the nonlinear variance and covariance
of most other statistical measures,
including higher order $n$-point spectra
such as the bispectrum and trispectrum,
and $n$-point correlation functions in real space,
including the $2$-point correlation function.
\section*{Acknowledgements}
We thank Nick Gnedin, Matias Zaldarriaga and Max Tegmark for helpful
conversations, and Anatoly Klypin and Andrey Kravtsov for making the MPI
implementation of ART available to us and for help with its application.
{\sc grafic} is part of the {\sc cosmics}
package, which was developed by Edmund Bertschinger under NSF grant
AST-9318185. The simulations used in this work were performed at the
San Diego Supercomputer Center using resources provided by the National
Partnership for Advanced Computational Infrastructure under NSF
cooperative agreement ACI-9619020.
This work was supported by NSF grant AST-0205981 and by
NASA ATP award NAG5-10763.
|
2,869,038,154,737 | arxiv | \section{ Introduction}
Symbiotic stars (SS) are a class of variable stars consisting of a
cool giant, a hotter object, either a hot subdwarf or a compact object,
and an emission nebula.
The optical variability of symbiotics may take different forms and
time scales. One form is of cyclic variations, due to the varying aspects
of the revolving binary system, with or without an apparent eclipse in the
light curve (LC). Binary periods of SS are of the order of 1 to a few
years. Another type of variability has an explosive character, in the form
of a single outburst, as for symbiotic novae, or multiple events. The time
scales of these variations are quite long: the decay times of the
outburst of symbiotic novae range between a few months to more than a
century. The cool giant in some symbiotic systems shows also intrinsic
variability such as radial pulsations of Mira-type.
Variability of SS light on short time scale of minutes and hours has not
been reported much in the literature. Recently, however, such variations
on this time scale, reminiscing the flickering phenomenon in cataclysmic
variables, have been discovered in symbiotics (Sokoloski et al. 2001).
We have analysed anew the historical light curve of BF Cyg. This analysis
resulted in the discovery of new elements in the long-term light curve of
the star which may give us new clues on the nature of this system.
In this paper we present this analysis of the long-term light
curve of BF Cyg, covering 114 years of observations. The characteristics of
the symbiotic system BF Cyg are described in Section 2, and the data
sets used on our analysis are reviewed in Section 3. In Section 4 we describe
the time series analysis and the periodicities detected. In Section 5
we discuss the physical interpretation of these periodicities.
\section {Brief description of BF Cyg}
BF Cyg is a very bright object (V $\simeq 9 $). This makes it a popular
target for observations on a wide wavelength range. This is also probably
the reason why the record of measurements of its magnitude goes back in
history for over 115 years (see below). Similar to many SS stars, and in
particular to the prototype Z And, the long-term light curve of BF Cyg
shows two kinds of optical variability. One is a regular periodic
modulation, with large changes in its amplitude. The other type of the
long range variability of this star is of explosive character, taking the
form of repeating outbursts (see Figure 1).
The cool component is a fairly normal M5 giant and the system is
classified as an S-type symbiotic with near-IR colors consistent with those
of normal cool giants (Kenyon \& Fernandez-Castro, 1987; Munari et al. 1992).
The IUE ultraviolet continuum suggests the presence of a hot
subdwarf with a temperature $>60000 K$ (Gonz\'{a}les-Riestra, Cassatella \& Fernandez-Castro 1990).
Photometric variability with an apparent modulation period of 754 days
was first noted by Jacchia (1941).
The system has been extensively studied in the optical and the ultraviolet
wavelength ranges (Mikolajewska et al. 1989; Fernandez-Castro et al. 1990;
Gonz\'{a}les-Riestra et al. 1990; Skopal et al. 1997).
Its optical and
ultraviolet emission lines vary in phase or/and in anti phase with the
photometric minima. The orbital nature of this variation is confirmed by
infrared radial velocities data and the spectroscopic orbital period
is 757.2, nearly identical to the photometric one (Fekel et al. 2001).
\section {The long-term light curve of BF Cyg}
We collected data from three large photometric measurement sets
retrievable for this system, in order to reconstruct its historical light
curve.
A regular photometric monitoring of BF Cyg for the years 1890-1940 is
available from the Harvard plates (Jacchia, 1941). While few plates were
collected in the first ten years, the data are more frequent after the
year 1900.
A second data set is composed of the photographic measurements
of Skopal et al. (1995).
A third large data set is the visual magnitude estimates collected by the
American Variable Stars Association (AAVSO). These are often daily
estimates of the magnitude of the system. For homogeneity we averaged the
AAVSO data over a time interval of 24 days, similar to the sampling
interval
of the Jacchia (1941) data set. The AAVSO at our hands is updated to Dec
2004.
\begin{figure*}
\includegraphics[width=130mm]{figure1.eps}
\caption{A 114 years light curve of the symbiotic star BF Cyg, from the year
1890 up until Dec 2004. Dots refer to m$_{pg}$ and crosses indicate visual
data transformed to m$_{pg}$. The solid line is a 3d degree polynomial, and a 9
term harmonic wave based on 3 inependent periods, fitted to the data by least squares.
See text for further explanations.}
\end{figure*}
In Fig. 1 we present the long-term light curve (LC) of BF Cyg from 1890 up
to the present. In this figure, the AAVSO data have been scaled to the
photographic scale, by adding a factor .61 that takes into account the (B-V)
color of the system at quiescence (Munari et al. 1992) and the transformation
B=m$_{pg}$ +0.11 (Allen, 1973). This curve covers the photometric behavior
of BF Cyg from 1890 to Dec 2004. There is however a considerable gap in the
distribution of the data points between JD 2429986 and JD 2434486 which
affects significantly the spectral window function of this time series
(see Section 4.1) The solid line in the figure will be explained in
Section 4.2. A sample of the data is shown in Table 1. Table 1 in
full will be accessible only electronically.
\begin{tabular}{@{}llrcc@{}} \\
\multicolumn {2} {|c|}{\bf Table 1. The m $_{pg}$ or scaled m$_{v}$ data of Fig. 1}\\
\\
\\
Julian day & magnitude \\
\\
2411547.9& 12.22 \\
2411570.8& 12.70 \\
2411616.4& 12.30 \\
2411684.9& 11.87 \\
\\
\end{tabular}
The system underwent a dramatic 3.5 magnitudes brightening event around
the year 1894. This outburst is followed by slow fading of the star that
continues until the very present time. This behavior is reminiscent of
that of the small class of symbiotic novae, that have one single major
outburst recorded in their historical light curve. Actually, there is
hardly a typical light curve
of symbiotic novae, and the fading from the outburst shows a large variety
of behavior. It can be gradual, such as in RR Tel or AG Peg, or
characterized by large light oscillations as in V1329 Cyg or by a deep
minimum as in PU Vul (Viotti, 1993).
In BF Cyg, a few major events of sudden brightening of the system by 1-2
magnitudes are superposed on the decline from
the major 1894 outburst. The duration of such an explosive event may last
a few years, and may include episodes of short term flares of
brightening by a few tenths of magnitude lasting a few days. In
between outbursts, the system exhibits periods of relative quiescence that
last for a few years. In the last eight years BF Cyg is in one of these
quiescence states, while the system is now back to the
brightness level measured by Jacchia (1941) on the few Harvard patrol
plates obtained in the year 1890 before the large 1894 outburst.
A periodic oscillation of about 754 days was already recognized by
Jacchia (1941), with amplitude $\sim $1 magnitude. Throughout the years
other values of the photometric orbital periodicity have been suggested by
various investigators. For example P=757.3 days (Pucinskas, 1970),
P=756.8 days (Mikolajewska et al. 1989), P=757.2 days (Fekel et al. 2001).
As already noted by Jacchia (1941), however, there were epochs in the history
of the star, during which the binary modulation was very small, sometime
nearly disappearing entirely from the LC.
\section{Time Series Analysis}
The time series that we analyze in this work is the fading branch of the
light curve of BF Cyg that follows the outburst of the star in 1894. This
means that the few old measurements of the star magnitude prior to this
event have been discarded. We have detrended the series from the long-term
fading effect by subtracting from it a polynomial of 3d degree that was
fitted to the data by the least squares method. The dots (photographic
magnitudes) and crosses (scaled visual magnitudes) in Figure 2 are
the detrended data points and they constitute the LC that we shall refer
to in this work. The solid line is a running mean of the data points over
the time interval of 757 days. The use of this line will be explained in
Section 4.2.
\begin{figure}
\includegraphics[width=87mm]{figure2.eps}
\caption{Dots (mpg magnitudes) and crosses (visual transformed to mpg) represent
the detrended LC of BF Cyg that is analysed in this work. Solid
line is a running means curve over 757 days of the data points.}
\end{figure}
As a first step we computed the power spectrum (PS) of this LC (Scargle 1982).
Figure 3 displays this PS in the frequency range corresponding to
the period interval between 40000 and 100 days. The upper bound is the
entire length of the LC. The lower bound is the period corresponding to
one half of the Nyquist frequency of the series. The insert shows the window
function created by the non uniform sampling of the LC by the available measurements.
Two groups of high peaks are clearly identified in the PS. One is at the red
end of the spectrum, around the frequency .00015 days$^{-1}$, and the other
around 0.0013 days$^{-1}$. We shall discuss them in turns in the next two
sections.
\begin{figure}
\includegraphics[width=87mm]{figure3.eps}
\caption{Power spectrum of the light curve of BF Cyg shown in Figure 2. Insert is the window function.}
\end{figure}
\subsection{Periods between 1000 and 40000 days}
Figure 4 is a blowup of the PS shown in Figure 3 in the period range
between 1000 and 40000 days. We also considered a time series of 46 points
obtained from the observed LC by binning it into 46 bins of 757 days
width. A third LC that we also considered is the running mean shown in
Figure 2 as a solid line. These two LCs have identical PS to that shown in
Figure 4. Table 2 lists the frequencies, in days$^{-1}$, of the highest
peaks seen in Figure 4, as well as their corresponding periods, in days,
and the corresponding peak power.
\begin{figure}
\includegraphics[width=3 inch]{figure4.eps}
\caption{Blowup of the red end of the power spectrum shown in Figure 3.}
\end{figure}
The highest peak (a2) in Figure 4, corresponding to the period of $\simeq$6400
days represents a periodicity in the outburst events of the star that is
also apparent in Figure 1. This periodicity is well established at a high level
of statistical confidence as demonstrated in Appendix A (available electronically).
\begin{tabular}{@{}llrcc@{}} \\
\multicolumn {5} {|c|}{\bf Table 2. Peaks in the power spectrum }\\
\\
Frequency & Period & Power & &\\
days$^{-1}$ & days & sigma unit & \\
\\
4.4954 10$^{-5}$ & 22245 & 5.28 & & \\
8.0871 10$^{-5}$ & 12365 & 6.39 & a1 & P1 \\
1.1679 10$^{-4}$& 8562.5 & 8.55 & a2$^{'}$ & alias\\
1.5670 10$^{-4}$ & 6381.8 & 11.48 & a2 & P1/2 \\
1.9461 10$^{-4}$ & 5138.5 & 4.80 & & \\
2.2454 10$^{-4}$ & 4453.6 & 5.08 & b & P4 \\
2.7043 10$^{-4}$ & 3697.8 & 6.13 & b'/a4$^{'}$&alias \\
3.1832 10$^{-4}$ & 3141.5 & 5.88 & a4 & P1/4\\
4.7596 10$^{-4}$ & 2101.0 & 3.96 & a6 & P1/6 \\
6.1963 10$^{-4}$ & 1613.9 & 3.60 & a8 & P1/8 \\
7.7327 10$^{-4}$ & 1293.2 & 2.94 & a10 & P1/10\\
12.522 10$^{-4}$ & 798.61 & 3.79 & p3 &P3 \\
12.861 10$^{-4}$ & 777.55 & 5.44 & &alias \\
13.200 10$^{-4}$ & 757.57 & 5.42 & p2 &P2 \\
13.539 10$^{-4}$ & 738.59 & 4.61 & & alias \\
14.517 10$^{-4}$ & 688.84 & 3.10 & & \\
\\
\\
\end{tabular}
Nearly all other peaks in Figure 4 are related to this frequency in the
following way. The peak denoted a1 corresponds to a period that is
very nearly twice the period of peak
a2: P1 $\sim$12400 days. Peaks a4, a6, a8 and a10 are the 4th, 6th, 8th and
10th harmonics of the P1 periodicity. The peak marked a2' between a1 and
a2 is an alias of the dominant peak a2, due to the gap in the distribution
of the data points of the LC along the time axis seen in Figure 2 (Section 3).
Its frequency is the sum of the frequency of a2 and of the frequency of the
highest peak in the spectral window function, shown as insert in Figure 3
(the high peak at the right hand side of the spectral window function corresponds
to the yearly cycle of 365 days).
Indeed, the PS of each of the two subgroups of the data points that constitute
the LC, when calculated separately, shows no trace of the a2' peak seen in Figure 4.
Peak b is quite prominent in the PS. It corresponds to the period P4$\sim$4450
days. By itself it is not very significant statistically. We believe that
it is nonetheless a significant feature of the LC of this star. The reason
is that within the statistical uncertainties in the frequency values, its
frequency satisfies the equation 1/P4-2/P1=1/P5, where 2/P1 is the frequency of
the highest peak a2, and P5$\sim$14700 days. The significance of the P5 periodicity
will become apparent in Section 4.2. We show there that a similar relation
1/P2-1/P3=1/P5 exists between P5 and two other prominent periodicities in the LC of the
star, P2 and P3. The peak marked b'/a4' is an alias of its two neighbours
on both sides, due to the gap in the data, as discussed above.
Further evidence to the validity of our interpretation of the PS seen in Figure 3 is
given in Appendix B (available electronically).
By least squares fitting, which will be described in the next section, we obtain the
following best estimates of the values of the 3 periods P1, P4 and P5:
P1=12750$\pm$400 days, P4= 4436$\pm$40 days, P5=14580$\pm$400 days.
\begin{figure}
\includegraphics[width=75mm]{figure5.eps}
\caption{Detrended light curve of BF Cyg folded onto the period P1=12750 days.
The cycle is exhibited twice.}
\end{figure}
Figure 5 presents the detrended LC of BF Cyg (Figure 2), folded onto the
period P1. The figure presents the cycle twice. The presence of the P1
peak in the PS is due to the apparent alternating height of the 6 recorded
successive outbursts, as presented by the two unequal maxima in the P1
cycle.
The last recorded outburst reached its maximum light around JD
2448040. Based on our best fit values of the 3 periodicities P1, P2 and
P3 (see Section 4.2), we predict that BF Cyg will reach the formal maximum point
of its next outburst around JD 2454236. In practice it means that the system will
be found at the height of an outburst in mid-year ($\sim$May) 2007.
\subsection{Periods between 200 and 1000 days}
Figure 6 is a blowup of the PS shown in Figure 3 in the period range 500 to
1000 days. Among the four peaks that dominate this figure, the one marked
p2 corresponds to the known binary period of the system P2=757 days. The peak
marked p3 corresponds to the period P3=798 days. The two unmarked neighbors of
the p2 peak correspond to the periods 777 and 738 days. They are aliases of
the p2 peak, created by the gap in the distribution of the observed points, discussed
in Sections 3. Detailed explanation of this claim is given in appendix C (available electronically).
\begin{figure}
\includegraphics[width=75mm]{figure6.eps}
\caption{Blowup of the section of the PS shown in Figure 3, containing the
peaks around the binary orbital periodicity of the system.}
\end{figure}
Best estimates for the value of the periods P2 and P3, as well as of P1, P4 and P5,
were obtained in the following way. We present the observed LC by a 9 term Fourier series,
consisting of the P1 period with its first 5 higher even harmonics and the P4 periodicity,
along with the P3 and the P2 periods. This presentation has only 3 free parameters, the values
of P1, P2 and P3. The values of the other 6 periodicities are fixed by these 3. We find
the value of these 3 parameters that give the 9 term series that fits best the observed LC in
the least squares sense. The values of P1, P4 and P5 so obtained were given in Section 4.1.
For P2 and P3 and their ephemeris we obtain:
Min(P2)=JD 2451443 $\pm$20 + (757.3 $\pm$0.9) $\times $E
Min(P3)=JD 2451531 $\pm$23 + (798.8 $\pm$1.4) $\times $E
Here the JD numbers are times of minimum light and the error figures
indicate half of a 95\% confidence intervals around the corresponding
values. They are computed by bootstrap (Efron and Tibshirani 1993) as detailed in
Appendix D (available electronically).
Our P2 period is identical to the period already found by Pucinskas (1970),
Min= JD 2415065 +757.3 $\times$E and confirmed as the orbital one by
Fekel et al. (2001).
\subsubsection {Further discussion of periods P2 and P3 }
In order to better analyze the photometric variability of the star in the
frequency range of the binary cycle 1/P2, we removed from the observed LC
all the variations on time scale shorter than the P2 period. This was done
by computing the running mean curve of the observed LC, averaging all
points within a window of width 757 days. This curve is shown as a solid
line in Figure 2. The dots in Figure 7 are the residuals of the observed
LC after removing the running mean curve from it. The solid line in Figure
7a represents a 2 term harmonic series of the P2 and P3 periods.
\begin{figure*}
\includegraphics[width=97mm]{figure7.eps}
\caption{Dots in all frames are the residuals of the observed light curve
after removing the running mean curve obtained with a 757 days wide
window, shown as solid line in Figure 2. (a) Solid line is the best
fitted 3 terms series with the periods P2, P3 and P2/2. (b) Zoom on
the first section of the plot in frame a. (c) Zoom on the second
section of plot a. (d) Blowup of the last 5000 days of the light
curve. The second thinner solid line is the best fitted harmonic wave
with the Fekel's (2001) period of 757.2 days derived from radial
velocity data.}
\end{figure*}
In order to enable detailed examination, Figure 7b and 7c present,
respectively, zooms on the first and on the second subsections of the data
set of Figure 7a. Figure 7d is a blowup of the last 5000 days of the light
curve. The second, thin solid curve is the best fitted harmonic wave to the
data, with Fekel's et al. (2001) binary period derived from radial velocity
measurements.
Figure 7 demonstrates that the P2 orbital modulation, as well as the P3
oscillations, are present in the LC at quiescence states, as well as
during the outburst events. Except for the modulation due to the beat phenomenon,
the structure and amplitude are independent of the luminosity of the system.
In particular, the apparent binary variation has the same amplitude
at outburst maximum as during quiescence. It has also at the present epoch
the same average magnitude as it had nearly a century ago, when the star
was 3 magnitudes brighter.
The oscillations of the system with the P3 periodicity, simultaneously
with P2, explain in particular the epoch in the history of the star,
around JD 2427400, at which the 757 days oscillations have all but disappeared
from the light curve (see Figure 7b). This phenomenon was already noted by
Jacchia (1941) more than 60 years ago. A similar episode of a nearly
disappearance of the orbital modulation is apparent again around JD
2442000. At this time the system was at the middle of one of its outburst
events (Figure 2) and the intense activity of the star is masking the
phenomenon to some extent, but it is still noticeable in Figure 7c. These
two epochs of near disappearance of the 757 days variability are nodes of the
beat between the P2 and P3 oscillations. The beat period is P5=14577,
which is equal to the cycle of the beat between the 6376 and the 4436 periods
discussed in Section 4.1.
Based on our ephemeris of the two neighboring periods P2 and P3 we predict that
the 757 variability of the system will be damped again around the year 2013.
The presence of the second period P3 in the LC in addition to P2 also provides
an alternative explanation to the apparent shift in some of the minima of the
binary cycle. Jacchia (1941) already noted that an O-C diagram indicates an
apparent change in position of minima, which he thought to be correlated with
the brightness of the star. Skopal (1998) reports on a similar effect in more
recent parts of the LC. Skopal et al. (1997) report also on the presence of a
possible secondary minimum in some of the 757 days cycles of the star.
Figure 7 a,b and c show that most of these variations in the profile structure
of the $\sim$757 days oscillations find a natural explanation in the ever changing
phase difference between the P2 and P3 frequencies. The figures show that not
only the two phases of damped oscillations are well explained by the contribution
of the P3 periodicity. At epochs of quiescence of the star, when the $\sim$757 days
oscillations are not perturbed by an outburst activity, e.g. at
JD 2415000 $<$ t $<$ JD 2418000 or JD 2448000 $<$ t $<$JD 2453000, the 2 periods
LC fits the detailed structure of the observed oscillations quite well. It traces
faithfully the minima of the oscillations and it also gives an apparent second
minimum to some of the cycles. The interference of the two periods may
also explain the rather different values that investigators of this star
have derived from photometric time series at different epochs in the
history of the star (Jacchia 1941; Skopal 1998).
In particular we point out, that due to this interference, the minimum
brightness times of individual cycles of the combined periodicity do not
fall necessarily at the minimum times of the binary period. Therefore, the
times of least brightness do not overlap exactly the times of inferior
conjunction of the giant star. One example is near the conjunction
JD 2451395.2, determined by Fekel et al. (2001) from radial velocity
curves. From our synthetic LC we find that minimum light of the system was
reached only 48 days later, on JD 2451443. From the presently available
observed data, it is difficult to determine the time of minimum light with
the accuracy required to distinguish between these 2 dates.
Figure 7d
shows the last 5000 days of the observed LC, along with our best fit 2
periods curve. The second, thin solid line is the best fit to the data of a
harmonic wave with Fekel's period of 757.2 days. Figure 7d seems to
indicate that the data support the notion that the later of the two dates,
resulting from our 2 periods presentation of the LC is closer to the true
minimum of the system.
We may summarize this section with the claim that all the gross features
of the 1894-2004 LC of BF Cyg, following its outburst of 1894, are well
accounted for by a slow continuous decay, and 3 distinct periods P1=12750
days (+5 even harmonics), P2=757.3 days and P3=798.8 days. A
P4=14580 days period is also of significance although it is not presented
explicitly in the LC. It is the beat of the P2 and P3 periods and it is
manifested in the LC through the appearance of the P5=4436 d
periodicity. The solid line in Figure 1 is a plot of the 9 term series of
the 4 periods P1, P2, P3 and P4, with their corresponding harmonics,
superposed on a 3d degree polynomial representing the slow decay from the
1894 outburst.
\section{ Discussion}
\subsection{The outbursts cycle}
The 6 recorded outbursts of BF Cyg in the last 104 years occurred with a
constant time interval of $\sim$6376 days between them. Periodic or
quasi-periodic variations in the light of stars, with periods of thousands
of days and amplitudes of 2 or 3 magnitudes are known to exist in the class
of semi-regular giants (Mattei et al. 1988; Kiss et al. 1999).
It seems, however, that the 6376 days periodic
variability of BF Cyg is of a different nature. The structure of the LC
shows the characteristics of outbursts, rather than of pulsations and so it
is indeed interpreted by most researchers in the field
(e.g. Gonz\'{a}les-Riestra et al. 1990; Skopal et al. 1997).
In an attempt to understand the origin of the outbursts phenomenon and the
nature of the clock that regulates their appearances, the clock of the
solar cycle, with its similar time constant of $\sim$8000 days, comes to
mind. We hypothesize that the origin of the cyclic outbursts of BF Cyg
lies in some magnetic activity, driven by a magnetic dynamo process in the
outer layers of the cool giant component of this stellar system. It is
known that the 11/22 year solar cycle modulates the mass flux of the solar
wind, among other measured parameters of the Sun. The basic process
originates in an interaction between differential rotation and convection
motions (Babcock 1961; Ulrich \& Boyden 2005).
Solar-like cycle in Asymptotic Giant Branch stars
has been proposed by Soker (2000) as an explanation to the morphology of a
few planetary nebulae. Soker (2002) has also proposed that giant stars
that are members of symbiotic systems may harbor a magnetic dynamo
process.
The giant in BF Cyg may indeed possess one important ingredient of such a
process, namely, a relatively fast rotation. It is classified as a M5 star
(Kenyon \& Fernandez-Castro 1987; M\H{u}rset \& Schmid 1999).
No data are available for the rotation velocity of such late type stars in
the De Medeiros et al. (2000) tables, but from the general trend of the velocity
distribution of cool stars, very small mean velocity (2-3 km/sec) is expected.
Fekel et al (2001), on the other hand, derive a projected rotational velocity
of the giant in BF Cyg of $\sim$4.5 $\rm km\, s^{-1}$.
This makes the giant of BF Cyg a fast rotator relative to field giants
of similar spectral type. We note also that differential rotation, another
important ingredient of the magnetic dynamo mechanism, has been recently
measured in some active K-type giants (Weber, Strassmeier \& Washuettl 2005).
In the magnetic dynamo scenario, the repeating intensifications in the optical
luminosity of the system that take the form of the outbursts, are due to
periodic enhancement of the stellar wind from the cool giant of the
system, regulated by this dynamo process.
This results in an enhancement of the mass accretion rate onto the compact
star of the system. The intense optical luminosity originates in the vicinity
of the hot component, probably in a bloated gaseous shell around the WD star
(Munari 1989).
We proposed in the past a similar solar-like cycle as an explanation for
the $\sim $ 8400 days ($\sim $23 years) cycle that we discovered in the LC
of another symbiotic star - Z And (Formiggini \& Leibowitz 1994).
A comparison between BF Cyg and Z And reveals similarities between these
two systems and in their behavior both in quiescence and at outbursts.
Their cool component is a M5 star (M\H{u}rset \& Schmid 1999) and
the orbital period is almost the same: 758.8 days for Z And (Formiggini \&
Leibowitz 1994) and 757.3 for BF Cyg (section 4.2).
Both systems belong to the classical symbiotics family, and during
outbursts their hot component seems to maintain a constant bolometric
luminosity while expanding in radius (Mikolajewska \& Kenyon 1992).
One difference between the 2 systems is that in addition to its sequence
of 6 "small" outbursts, BF Cyg underwent the dramatic 1894 event of large
outburst, quite distinct from the 6 that followed. No such event has been
recorded in the history of Z And. In view of the 1894 event, BF Cyg should
perhaps be classified as symbiotic nova, as already suggested by Skopal et al.
(1997). That event seems to be of the scale and nature that are different from
those of the small outbursts. The abrupt increase by 4 magnitudes in its
luminosity, and the slow decay lasting over 100 years afterwards, are
probably a signature of a thermonuclear runaway process, similar to the
events that characterize systems such as AG Peg, V1016 Cyg as symbiotic
novae (Mikolajeswka \& Kenyon 1992).
\subsection {The brightness oscillations}
Near the P2=757.3 days binary period of BF Cyg we discovered the period
P3=798.8 days. This could be the periodicity of oscillations of the M giant
star of this system. However, we consider this interpretation unlikely for
the following reason. The giant of BF Cyg is not a Mira type star. It
does not show modulations in the near infrared as for Mira. In the
(J-H) vs. (H-K) color diagram (Whitelock 1994) it does not lie in the
region occupied by Miras.
We suggest that the P3 periodicity is the rotation period of the giant of
the system. The rotation of the giant component of SS's was already discussed
in the literature (e.g. Munari 1988). In general, the giant stars in symbiotic
systems are assumed to rotate synchronously with their binary revolution. This assumption
is based on theoretical considerations (Zahn 1977), taking into account the synchronization
time scale expected from the values of the radii, masses and binary separations that are
typical of symbiotics on one hand, and the estimated ages of these binary systems
on the other. The period P3 is close enough to P2 and therefore would not be an unusual
exception to this commonly believed rule.
Modulation of the star light at the rotation period of the giant is indeed expected in
models explaining the binary photometric variations on the basis of the reflection
effect (Kenyon 1986, Formiggini \& Leibowitz 1990). If the giant star is not strictly
isotropic in its photometric characteristics, e.g. if there are dark spots on its
photosphere or if there are areas of different reflectivity on its
surface, rotation of this star will modulate the luminosity of the system
in the rotation frequency.
Modulations at the rotation frequency are expected also in models whereby the
photometric binary variations are due to variation in the optical depth towards the
main light source of the system that are coupled to the orbital revolution. In such models
the WD and its close vicinity are seen by the observer through different regions of the
stellar wind of the giant star (Gonz\'{a}les-Riestra et al. 1990). The
stellar wind of the giant is not necessarily isotropic with respect to the
giant center. The non isotropic structure of the wind may also be coupled to the giant
rotation through the effect of the stellar magnetic field. In such a case, the opacity
toward the main light source of the system will vary at the giant rotation frequency,
in addition to its variation at the binary frequency.
The P5=14580 days periodicity is the beat period of the binary orbital
period P2 and the P3 period. If the latter is indeed the giant rotation period, P5
is the period of the tidal wave that propagates in the outer layers of the giant, in the
coordinate system of the rotating star. We suggested above that a magnetic
dynamo process may be the driving mechanism responsible for the outbursts
cycle. It is not unreasonable to speculate that a tidal wave in
differentially rotating convecting layers may also modulate the magnetic
field of the star at the tidal wave frequency, in addition to the modulation by the
main cycle of the dynamo. The apparent P4 periodicity, if found to be real, may be
the beat period of these two magnetic cycles in the outer layers of the giant.
Finally we note that as mentioned in Section 4.2.1 the structure and the amplitude of
the P2 and P3 oscillations are independent of the average luminosity of
the system. In particular it appears that the amplitudes of these modulations at maximum light
of outbursts are not significantly different from those during quiescence
states of the system.
This fact is consistent with the reflection interpretation of the binary modulation
(Kenyon 1986, Formiggini \& Leibowitz 1990). To first approximation the reflection and
the reprocessing of the hot component's radiation in the giant atmospheric layers, give
rise to optical luminosity that is a given fraction of the hot component
luminosity. An increase in the later will result in the same relative
increase in the luminosity of the heated hemisphere of the giant, hence
the independence of the amplitude of the P2 variations, expressed in
magnitude units, on the luminosity of the system.
This independence is also consistent with the modulating optical depth
interpretation. A given optical depth of an absorbing layer reduces the
flux of the emerging radiation by a given percentage of the flux of the
radiation impinging on it, i.e. it has the same amplitude in magnitude
units.
\section{Summary}
We identify 3 independent periodicities in the LC of the last 104 years of
the symbiotic star BF Cyg. One is a 6376 days (or twice this
value) periodicity in the occurrence of outbursts on the decaying branch
of the luminosity of the system, following its major outburst in 1894. The
second is the well known binary period of the system of 757.3 days. The
third one is 798.8 days, which we suggest is the rotation period of the
giant star of this symbiotic system. A 4th period which is a beat of the
6370 days cycle with the beat period of the binary revolution and the giant
rotation period may also be present in the light curve.
\section*{Acknowledgments}
We acknowledge with thanks the variable star observations from the AAVSO
International Database contributed by observers worldwide and used in this
research.
This research is supported by ISF - Israel Science Foundation of the
Israeli Academy of Sciences.
|
2,869,038,154,738 | arxiv | \section{Introduction}
In these short notes, we study a very simple
problem with a solution that is universally well-known:
spontaneous decay of a laser-driven two-level atom
which is coupled to the electromagnetic field in the vacuum state.
The techniques that we use to discretize the quantum
stochastic differential equation \cite{HuP84} that
describes the interaction between the two-level atom and
the laser field are very well-known
\cite{Kum85, Par88, LiP88, AtP06, BvH08, BHJ09}.
The discretized model consists of a repeated unitary
interaction of the two-level atom with subsequent field
qubits parametrized by a discretization parameter $\lambda$. The repeated interaction model easily leads
to a quantum stochastic difference equation that has
the QSDE we wish to simulate as its limit as $\lambda$ goes to zero.
Note that unitarity of the interaction is preserved in the
discretized model, which is a very desirable feature: e.g.\ after time evolution probabilities
will still always take values between $0$ and $1$. Furthermore,
the unitarity allows us to easily map the interaction on unitary gates in
a quantum computer.
The motivation for the work we present here is twofold and
aimed at a future with computers with more and more reliable
qubits:
\begin{enumerate}
\item We wish to emphasize that
quantum optics might prove a very fruitful field
of application for early quantum computers.
It is well-known how to discretize the type of
problems that arise in quantum optics
and the resulting quantum stochastic difference equations
are easy to implement on a quantum computer.
Moreover, the field of quantum optics historically
contains many interesting problems and techniques that
can serve as interesting benchmark problems for
early quantum computers.
\item On a quantum computer we can do a fully
coherent simulation of a system in interaction with
the electromagnetic field. That is, on a large enough
quantum computer, we can simulate
the complete unitary that describes the interaction between system and field, putting us
past standard analyses using master equations or
quantum filtering equations \cite{Bel92b, Car93, Dav69, BHJ07} because we also have a complete
description of the field to our availability. This could be
very useful if we wish to simulate non-Markovian networks
of systems interacting at different points with the same field,
possibly containing fully coherent feedback loops \cite{GoJ09, GoJ09b}.
\end{enumerate}
The remainder of these notes is organized as follows.
Section \ref{sec difference eq} introduces the QSDE
that we wish to simulate on a quantum computer: a laser driven
two-level atom in interaction with the vacuum EM-field. We discuss
repeated unitary interaction models, the resulting
quantum stochastic difference equations and discuss how
to take the limit to obtain quantum stochastic differential equations
\cite{Par88, LiP88, AtP06, BvH08, BHJ09}. Next, we
introduce the repeated interaction that leads to the
QSDE corresponding to the problem at hand: a laser
driven two-level atom in interaction with the vacuum
EM-field. Section \ref{sec quantum circuit} describes
how we have implemented the repeated interaction
model of Section \ref{sec difference eq} on the
IBMqx4 Tenerife quantum computer \cite{IBMQ}.
Section \ref{sec results} presents the results of
our simulations. We compute the reduced dynamics
of the two-level atom from the simulation results and
compare it to the dynamics given by the theoretical master
equation, derived from the underlying discrete model. We
also compute the conditional dynamics of the two-level atom
conditioned on both counting photons in the field and observing a field quadrature
in the field and compare the results to the quantum filtering
equations derived from the underlying discrete
model \cite{Brun02, GS04, BHJ09, GCMC18}. Reproducing
the correct quantum filtering equations would give an indication that
the simulation also reproduces the correlation between the field
and the two-level atom correctly.
In Section \ref{sec conclusion}, we formulate some conclusions from
our results. It should be noted that currently only very limited
simulations can be done due to the small number of reliable qubits in
the quantum computers that are currently available. In a 5 qubit
machine such as the IBMqx4, we only have 4 field qubits to our availability,
severely limiting our simulation capabilities. The work in these notes
should be seen through the lens of a hoped-for strong increase in the
computing capacity in the near future.
\section{Quantum stochastic difference equations}\label{sec difference eq}
We will now first introduce the problem that we will be studying
in this paper. We consider a two-level atom with a
coupling to the electromagnetic field. We will let the
two-level atom be driven by a laser. We will not model
the laser by an additional channel in the field that is
in an coherent state, but will directly introduce the Rabi oscillations
induced by the laser field as a Hamiltonian term in the
QSDE. We will use the following
notation throughout the paper: $\sigma_x, \sigma_y$ and $\sigma_z$
are the standard Pauli matrices. Furthermore, $\sigma_+$ and $\sigma_-$
are the \emph{raising} and \emph{lowering operator}
matrix of the two-level atom, respectively.
The interaction of the laser driven two-level atom with
the vacuum electromagnetic field is given by the
following quantum stochastic differential
equation (QSDE) in the sense of \cite{HuP84}
\begin{equation}\begin{split}\label{eq QSDE}
dU_t = \Big\{\sqrt{\kappa}\sigma_- dA_t^* - \sqrt{\kappa}\sigma_+ dA_t -\frac{1}{2}\kappa \sigma_+\sigma_- dt
- i\omega \sigma_+ \sigma_- dt - i\frac{\Omega}{2} \sigma_y dt \Big\}U_t, \ \ \ U_0 = I,
\end{split}\end{equation}
where $\kappa$ is the \emph{decay rate}, $\omega$ is the \emph{transition frequency} of the two-level
atom and $\Omega$ is the frequency of the \emph{Rabi oscillations} induced by a laser field.
In order to simulate Eqn \eqref{eq QSDE} on a quantum
computer, we first need to introduce the discrete
models (see e.g.\ \cite{BHJ09} for a detailed introduction) that
in a suitable limit will converge to a quantum stochastic differential
equation \cite{Par88, LiP88, AtP06, BvH08} such as Eqn \eqref{eq QSDE}.
To this end we first define
a time interval $[0,T]$. We divide this time interval into
$N$ equal sub intervals of length $T/N$.
We define $\lambda := \sqrt{T/N}$, i.e.\
we have $N$ sub intervals of length $\lambda^2$.
With each sub interval we associate a two-level quantum
system representing the slice of the (truncated) field that interacts
with the two-level atom at that moment.
In this way we obtain a repeated interaction
\begin{equation}\label{eq repeated interaction}
U(l) = \overleftarrow{\prod}_{i = 1}^l M_i = M_l M_{l-1}\ldots M_2M_1, \ \ \ \ 1 \le l \le N.
\end{equation}
Here $M_i$ is a unitary operator that couples
the two-level atom and the $i$th field slice which is here also
represented by a two-level system. Note that we take all $M_i$'s to be
identical apart from the fact that they all act on their own
slice of the field. Furthermore, we will let the $M_i$'s
be a function of $\lambda$, such that if $\lambda$
goes to $0$ (i.e.\ $N$ goes to infinity) the $M_i$'s converge
to the identity map $I$. That is, we will get more and
more interactions, with smaller and smaller effect.
We now introduce linear operators
$M^\pm, M^+, M^-$ and $M^0$ \cite{BHJ09}, acting on the
two-level atom Hilbert space, in such a way that we have the following
decomposition
\begin{equation}\label{eq decomposition}
M_i - I = M^\pm \otimes \Delta \Lambda(i) + M^+\otimes \Delta A^*(i) + M^- \otimes \Delta A(i) + M^0 \otimes \Delta t(i),
\end{equation}
where the \emph{discrete quantum noises} (see e.g. \cite{BHJ09}) are given by
\begin{equation*}\begin{split}
&\Delta \Lambda(i) :=(\sigma_+ \sigma_-)_i, \ \ \ \
\Delta A^*(i) := \lambda(\sigma_+)_i, \\
&\Delta A(i) := \lambda(\sigma_-)_i,\ \ \ \
\Delta t(i) := \lambda^2 I_i.
\end{split}\end{equation*}
Note that this decomposition uniquely defines
the coefficients $M^\pm, M^+, M^-$ and $M^0$
and note that these coefficients are all a function of $\lambda$,
which we leave implicit to keep our notation light. Furthermore, we usually
omit the tensor products in Eqn \eqref{eq decomposition}
to keep the notation light.
We can now write Eqn \eqref{eq repeated interaction} as the
following \emph{quantum stochastic difference equation} (see e.g. \cite{BHJ09})
\begin{equation}\begin{split}\label{eq difference equation}
\Delta U(l) :={ }&U(l)-U(l-1) = \\
={ }&\Big\{M^\pm \Delta \Lambda(l) +
M^+\Delta A^*(l) + M^- \Delta A(l) + M^0 \Delta t(l)\Big\}U(l-1),
\end{split}\end{equation}
where $1 \le l \le N$ and $U(0) = I$. We now have the following theorem
due to Parthasarathy \cite{Par88} (weak convergence), Parthasarathy and Lindsay \cite{LiP88} (weak convergence of the quantum flow) and
Attal and Pautrat \cite{AtP06} (strong convergence uniform on compact time intervals).
We state the theorem without giving the precise meaning of the mode of convergence
of the repeated interaction model to the unitary solution of the QSDE
because we would need to introduce further mathematical
details that would make us stray too far from the main narrative of
this article (see however \cite{AtP06}, or \cite{BvH08}).
\begin{theorem}\label{thm main thm} {\bf (Parthasarathy, Lindsay, Attal and Pautrat \cite{Par88, LiP88, AtP06})}
Suppose the
following limits exist:
\begin{equation}\begin{split}\label{eq definition}
&S := \lim_{\lambda \to 0} M^\pm + I, \ \ \ \ L := \lim_{\lambda \to 0} M^+ \\
&L^\dag := \lim_{\lambda \to 0} - M^-S^*, \ \ \ \ H := \lim_{\lambda \to 0} i M^0 + \frac{i}{2} L^* L,
\end{split}\end{equation}
then it follows that $S$ is unitary, $L^\dag$ is the
adjoint of $L$, i.e. $L^* = L^\dag$, $H$ is self-adjoint and $U([t/\lambda^2])$
(where the brackets [ ] stand for rounding down to an integer)
converges to a unitary $U_t$ given by the following QSDE
\begin{equation}\label{eq general QSDE}
dU_t = \Big\{(S-I)d\Lambda_t + LdA_t^* - L^*SdA_t - \frac{1}{2}L^*Ldt - iHdt\Big\}U_t, \ \ \ \ U_0 = I.
\end{equation}
\end{theorem}
We proceed by guessing an interaction unitary $M_i$ in the
repeated interaction Eqn \eqref{eq repeated interaction} and check
via Thm \ref{thm main thm} that it leads to the correct limit coefficients
to reproduce in the limit our system of interest
which is given Eqn \eqref{eq QSDE}. Note that there are several $M_i$s that
will lead to the correct limit system. We will take the following $M_i$ and will show
that it indeed leads to Eqn \eqref{eq QSDE} in the limit:
\begin{equation}\begin{split}\label{eq Mi}
M_i{ } &= \exp\Big(\sqrt{\kappa}\sigma_-\otimes \lambda\sigma_+ - \sqrt{\kappa}\sigma_+\otimes \lambda\sigma_-\Big)
\exp\Big( -i \omega \sigma_+\sigma_-\otimes \lambda^2 I_i\Big)
\exp\Big( -i \frac{\Omega}{2} \sigma_y \otimes \lambda^2 I_i\Big) \\
&= \tiny\begin{pmatrix}
e^{ -i \omega \lambda^2}\cos(\frac{\Omega\lambda^2}{2}) & -e^{ -i \omega \lambda^2}\sin(\frac{\Omega\lambda^2}{2}) & 0 & 0 \\
\cos(\sqrt{\kappa}\lambda)\sin(\frac{\Omega\lambda^2}{2}) & \cos(\sqrt{\kappa}\lambda)\cos(\frac{\Omega\lambda^2}{2}) & e^{ -i \omega \lambda^2} \sin(\sqrt{\kappa}\lambda)\cos(\frac{\Omega\lambda^2}{2}) & -e^{ -i \omega \lambda^2} \sin(\sqrt{\kappa}\lambda)\sin(\frac{\Omega\lambda^2}{2}) \\
-\sin(\sqrt{\kappa}\lambda)\sin(\frac{\Omega\lambda^2}{2}) & -\sin(\sqrt{\kappa}\lambda)\cos(\frac{\Omega\lambda^2}{2}) & e^{ -i \omega \lambda^2}\cos(\sqrt{\kappa}\lambda)\cos(\frac{\Omega\lambda^2}{2}) & -e^{ -i \omega \lambda^2}\cos(\sqrt{\kappa}\lambda)\sin(\frac{\Omega\lambda^2}{2}) \\
0 & 0 & \sin(\frac{\Omega\lambda^2}{2}) & \cos(\frac{\Omega\lambda^2}{2}) \\
\end{pmatrix}.
\end{split}\end{equation}
A short calculation then reveals
\begin{equation}\begin{split}\label{eq coefficients}
&M^0 = \begin{pmatrix}
\frac{e^{ -i \omega \lambda^2}\cos(\sqrt{\kappa}\lambda)\cos(\frac{\Omega\lambda^2}{2})-1}{\lambda^2} & \frac{-e^{ -i \omega \lambda^2}\cos(\sqrt{\kappa}\lambda)\sin(\frac{\Omega\lambda^2}{2})}{\lambda^2} \\
\frac{ \sin(\frac{\Omega\lambda^2}{2})}{\lambda^2} & \frac{\cos(\frac{\Omega \lambda^2}{2})-1}{\lambda^2} \\
\end{pmatrix}, \\
&M^- = \begin{pmatrix}
\frac{-\sin(\sqrt{\kappa}\lambda)\sin(\frac{\Omega\lambda^2}{2})}{\lambda} & \frac{-\sin(\sqrt{\kappa}\lambda)\cos(\frac{\Omega\lambda^2}{2})}{\lambda}\\
0 & 0
\end{pmatrix}, \\
& M^+ = \begin{pmatrix}
0 & 0 \\
\frac{e^{ -i \omega \lambda^2} \sin(\sqrt{\kappa}\lambda)\cos(\frac{\Omega\lambda^2}{2})}{\lambda} & \frac{-e^{ -i \omega \lambda^2} \sin(\sqrt{\kappa}\lambda)\sin(\frac{\Omega\lambda^2}{2})}{\lambda}
\end{pmatrix}, \\
&M^\pm = \begin{pmatrix}
e^{ -i \omega \lambda^2}\big(1-\cos(\sqrt{\kappa}\lambda)\big)\cos(\frac{\Omega\lambda^2}{2}) & e^{ -i \omega \lambda^2}(\cos(\sqrt{\kappa}\lambda)-1)\sin(\frac{\Omega\lambda^2}{2})\\
(\cos(\sqrt{\kappa}\lambda)-1)\sin(\frac{\Omega\lambda^2}{2}) & (\cos(\sqrt{\kappa}\lambda)-1)\cos(\frac{\Omega\lambda^2}{2}) \\
\end{pmatrix}.
\end{split}\end{equation}
Using the definition of $S, L$ and $H$ in Eqn \eqref{eq definition}, we find
\begin{equation*}\begin{split}
&S = I, \ \ \ \ L = \sqrt{\kappa} \sigma_- , \ \ \ \ H = \omega \sigma_+ \sigma_- + \frac{\Omega}{2}\sigma_y.
\end{split}\end{equation*}
That is, we have found a repeated interaction model
that converges to the QSDE of
Eqn \eqref{eq QSDE}.
\section{The quantum circuit}\label{sec quantum circuit}
The various contributions to the interaction between the atom and the field,
given in Eqn \eqref{eq Mi}, can easily be mapped to elementary quantum
gates.
The following quantum circuit \cite{NiC00} implements this interaction for a single time
slice:
\begin{center}
\makebox{\Qcircuit @C=1em @R=.7em {
\lstick{atom} & \gate{R_y(\Omega\lambda^2)} &
\gate{R_z(\omega\lambda^2)}
& \ctrl{1} & \gate{R_y(2\sqrt{\kappa}\lambda)} & \ctrl{1} & \qw \\
\lstick{field} & \qw & \qw
& \targ\ & \ctrl{-1} & \targ & \qw
}}
\end{center}
For modeling multiple time slices, different qubits are used to
describe the field at different times, and coupled to
the atom qubit. Currently, this limits the simulation to
a maximum of four time slices on the IBMqx4 computer.
If operations based on classical bits are enabled, we
could reuse the same qubit for the field by measuring this qubit, and rotating it back to its $| 0\rangle$
state when the outcome is $1$. Unfortunately, this is currently only
implemented in simulators and not in real hardware.
At the end of the simulation, all qubits (both field and atom) were measured.
The atom was measured in the $x$, $y$, and $z$ basis, whereas the field qubits
were measured in the $x$ and $z$ basis. Statistics were collected over 10,240 runs
for each combination of measurement directions of atom and field.
\section{Results}\label{sec results}
\subsection{Master equation}
The repeated interaction model given by Eqn \eqref{eq Mi}
leads to the following discrete time master equation \cite[Eqn 4.3, page 265]{BHJ09}
for the state $\rho_l$ of the two-level atom:
\begin{equation}\label{eq discrete master}
\Delta \rho_l := \rho_l - \rho_{l-1} =\mathcal{L}(\rho_{l-1})\lambda^2,\ \ \ \
\rho_0 = \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix},
\end{equation}
where the discretized Lindblad operator is
given by \cite[Eqn 4.4, page 265]{BHJ09}
\begin{equation}\label{eq discrete Lindblad}
\mathcal{L}(\rho) := M^+ \rho {M^+}^* + \lambda^2 M^0 \rho {M^0}^* + M^0\rho + \rho {M^0}^*.
\end{equation}
Here $M^+$ and $M^0$ are given by Eqn \eqref{eq coefficients}.
Figure \ref{fig master equation} compares the
theoretical results given by the master
equation Eqn \eqref{eq discrete master} and the
results obtained with the IBMqx4 Tenerife quantum computer.
\begin{figure}
\centering
\includegraphics[width=0.6\textwidth]{master.png}
\caption{Calculated expectation values for the Pauli spin operators for the two-level atom,
using parameters $\kappa=1$, $\omega=0$ and $\Omega=12$.
Results are plotted for the master equation Eqn \eqref{eq discrete master},
and calculations on the actual
IBMqx4 Tenerife quantum computer. The results of the master equation
are obtained using a time step $\lambda^2=0.01$. The IBMqx4 curve consists of
collected results of experiments with time steps $\lambda^2=0.16$, $0.14$, and $0.10$,
each averaged over 10240 runs.}
\label{fig master equation}
\end{figure}
\subsection{Homodyne quantum filter}
We now turn to the situation where we are not simply
tracing over the field, but condition on observations
made in the field. Suppose that for $l = 1,2,3,4$,
we have observed $\lambda*\sigma_x$ in the field:
\begin{equation*}
\Delta Y(l) := \lambda * \mbox{"the outcome of the\ } \sigma_x \mbox{-measurement of the\ }l\mbox{th field qubit"}.
\end{equation*}
Physically, this corresponds to the case where we observe
the field with a homodyne detection setup after the interaction
with the two-level atom. We can condition the time evolution of the density
matrix on an observed homodyne photo current detection record.
The conditioned density matrix obeys the following \emph{discrete
quantum filtering equation for homodyne detection} \cite[Section 5.2, page 274]{BHJ09}
\begin{equation}\begin{split}\label{eq discrete quantum filter homodyne}
\Delta \rho_l = \mathcal{L}(\rho_{l-1})\lambda^2 + \frac{\mathcal{J}(\rho_{l-1})- \mbox{tr}[\mathcal{J}(\rho_{l-1})] \Big( \rho_{l-1}+\lambda^2\mathcal{L}(\rho_{l-1})\Big)}
{1-\lambda^2 \mbox{tr}[\mathcal{J}(\rho_{l-1})]^2}
\Big(\Delta Y(l) - \mbox{tr}[\mathcal{J}(\rho_{l-1})]\lambda^2\Big),
\end{split}\end{equation}
where $\mathcal{L}$ is given by Eqn \eqref{eq discrete Lindblad} and $\mathcal{J}$ and the initial state $\rho_0$
are given by
\begin{equation*}
\mathcal{J}(\rho) := M^+\rho + \rho{M^+}^* + \lambda^2M^+\rho{M^0}^* + \lambda^2M^0 \rho{M^+}^*, \ \ \ \
\rho_0= \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix},
\end{equation*}
Here $M^+$ and $M^0$ are given by Eqn \eqref{eq coefficients}.
Figures \ref{fig homodyne z}, \ref{fig homodyne x} and \ref{fig homodyne y} compare the
theoretical results given by the quantum filter
equation Eqn \eqref{eq discrete quantum filter homodyne} and the
results obtained with the IBM Qiskit \cite{Qiskit} simulator and
the IBMqx4 Tenerife quantum computer.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{homodyne_z.png}
\caption{Time evolution of the expectation value of the Pauli $\sigma_z$ operator
of the two-level atom,
in the homodyne detection scheme, for the four trajectories that have accumulated
the most statistics. The trajectory is given by the title above the plots, which shows
$\Delta Y$. The filter equation results are from
Eqn \eqref{eq discrete quantum filter homodyne}.
The IBMqx4 results are averaged over 10,240 runs. The simulator results are obtained
from the IBM Qiskit simulator, using the same quantum assembly code that was used
on the IBMqx4 Tenerife computer, and are averaged over 102,400 runs.}
\label{fig homodyne z}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{homodyne_x.png}
\caption{As Figure \ref{fig homodyne z}, for the $\sigma_x$ operator.}
\label{fig homodyne x}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{homodyne_y.png}
\caption{As Figure \ref{fig homodyne z}, for the $\sigma_y$ operator.}
\label{fig homodyne y}
\end{figure}
\subsection{Counting quantum filter}
Finally, we turn to the situation where we condition on having
observed $\sigma_+\sigma_-$ in the field for $l=1,2,3,4$.
That is, we have the following observations:
\begin{equation*}
\Delta Y(l) = \frac{1 + \mbox{"the outcome of the\ } \sigma_z \mbox{-measurement
of the\ }l\mbox{th field qubit"}}{2}.
\end{equation*}
Physically, this corresponds to the case where we
observe the field with a photo detector after the interaction with the two-level
atom. We can condition the time evolution of the density matrix on an observed
photo detection record. The conditioned density matrix obeys the following
\emph{discrete quantum filtering equation for photon counting} \cite[Section 5.3, page 276]{BHJ09}
\begin{equation}\begin{split}\label{eq discrete quantum filter photon counting}
\Delta \rho_l{ }&= \mathcal{L}(\rho_{l-1})\lambda^2 + \frac{\frac{M^+\rho_{l-1}{M^+}^*}{\mbox{tr}[\rho_{l-1} {M^+}^* M^+]} - \rho_{l-1}-\lambda^2\mathcal{L}(\rho_{l-1})}
{1-\lambda^2 \mbox{tr}[\rho_{l-1} {M^+}^* M^+]}
\Big(\Delta Y(l) - \mbox{tr}[\rho_{l-1} {M^+}^* M^+]\lambda^2\Big), \\
\rho_0{ }&= \begin{pmatrix} 0 & 0 \\ 0 & 1\end{pmatrix}.
\end{split}\end{equation}
Here $M^+$ is given by Eqn \eqref{eq coefficients}.
Figures \ref{fig counting z}, \ref{fig counting x} and \ref{fig counting y} compare the
theoretical results given by the quantum filter
equation Eqn \eqref{eq discrete quantum filter photon counting} and the
results obtained with the IBM Qiskit simulator and
the IBMqx4 Tenerife quantum computer.
\begin{figure}
\centering
\includegraphics[width=\textwidth]{counting_z.png}
\caption{Time evolution of the expectation value of the Pauli $\sigma_z$ operator
of the two-level atom,
in the photon counting scheme, for the case where no photons were detected (0000),
and where a single photon was detected after the fourth (0001), third (0010), second (0100)
and first (1000) time step. The filter equation results are from Eqn \eqref{eq discrete quantum filter photon counting}.
The IBMqx4 results are averaged over 10,240 runs. The simulator results are obtained
from the IBM Qiskit simulator, using the same quantum assembly code that was used
on the IBMqx4 Tenerife computer, and are averaged over 102,400 runs.}
\label{fig counting z}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{counting_x.png}
\caption{As Figure \ref{fig counting z}, for the $\sigma_x$ operator.}
\label{fig counting x}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\textwidth]{counting_y.png}
\caption{As Figure \ref{fig counting z}, for the $\sigma_y$ operator.}
\label{fig counting y}
\end{figure}
\section{Conclusion}\label{sec conclusion}
In these notes we have shown that it is fairly
straightforward to implement quantum stochastic
differential equations on a quantum computer.
The mathematical theory \cite{Kum85, Par88, LiP88, AtP06, BvH08, BHJ09}
behind the necessary discretization of the equations
is well worked out and easily translated into a
quantum circuit. It is possible with the very limited
capacity of the currently available quantum computers
to simulate some simple quantum optical
features described by a QSDE (e.g. a Rabi oscillation).
We have also seen that the filter equations are
to a large extent correctly reproduced on the
IBMqx4 Tenerife quantum computer. This provides
confidence that the (quantum) correlations between
the atom and the field are accounted for correctly.
This opens the door to fully coherent simulations of
systems that interact with the field at different points,
even including fully coherent feedback loops \cite{GoJ09, GoJ09b}.
Naturally, this is only possible when quantum computers
are available with more and more reliable qubits.
As can be seen in Figures \ref{fig counting z} and \ref{fig counting x},
the time evolution between counts in the photon counting scheme seems to be accounted for correctly,
however, the jump operation does not seem to be accurate to the theory.
This can be understood though: there are relatively few jumps and there
is already quite a bit of noise on the results when the field is in the vacuum
state. This means there is an identity component in the jump operator that
leads to a deviation from the theory (in which this noise is not present, but could
be modeled).
With respect to the discrete quantum filtering equations \cite{BHJ09}, we
note that
they completely coincide with the IBM Qiskit simulator results. However,
they are much less computationally intensive and could still be used if the
number of qubits is larger than the simulator can deal with. Note, however,
that in the simulator it is also possible to recycle the field qubits after they
have been measured. This is currently not yet possible on
the real IBMQ hardware.
It will be interesting to see future quantum computers simulate more
complex benchmark problems originating from quantum optics.
|
2,869,038,154,739 | arxiv | \section{Introduction}
Entanglement entropy quantifies the amount of information about a quantum state that is lost upon restriction to a subsystem. Typically, by ``subsystem'' one means a spatial region, but in general it can be any subalgebra of observables that belong to the theory in question. This algebraic point of view has received increased interest in the context of defining entanglement entropy in gauge theories \cite{Casini:2013rba, Casini:2014aia, Radicevic:2014kqa, Radicevic:2015sza, Soni:2015yga, Ghosh:2015iwa, Aoki:2015bsa}, but the tools unearthed in this body of work can also be used to understand another deep question: the relation between entanglement entropy and field-theoretic dualities.
The crux of the algebraic approach to entanglement lies in constructing the density matrix $\rho_V$ associated to any subalgebra $\A_V$ of observables \cite{Ohya:2004, Balachandran:2013}. (We will often use ``subsystem'' to refer either to the rule $V$ that picks out the subalgebra, or to the subalgebra $\A_V$ itself.) This density operator is the unique element of $\A_V$ that is positive semi-definite, has unit trace in some natural representation, and reproduces the expectation values of all operators in the subsystem via $\avg \O = \Tr(\rho_V \O)$. The entanglement entropy of $\A_V$ is then defined as the von Neumann entropy $S = - \Tr(\rho_V \log \rho_V)$. When $V$ is a spatial region and $\A_V$ is the maximal algebra of observables in $V$, the density matrix defined in this algebraic way coincides with the one obtained by tracing out the degrees of freedom outside $V$ in the density matrix for the whole system. However, even if these conditions are not fulfilled, $\rho_V$ is still a legitimate density operator, and its entropy reflects the fact that certain measurements on the system are not accessible to us.
A general approach like this is indispensable when studying dualities. Typically, when two theories map to each other, there is a small mismatch between a theory that respects duality and a theory that we are used to dealing with, and the subalgebras dual to each other are not necessarily maximal algebras on any spatial region. For example, when a photon is dualized to a scalar in $d = 2$ spatial dimensions, the scalar zero mode obeys a certain compactness condition. At weak gauge theory coupling, the zero mode operator is absent altogether. The lack of this operator in the strongly coupled (``ordered'') regime of the scalar theory leads to both an area law and a term that corresponds to the topological entanglement entropy in the dual weakly coupled gauge theory \cite{Agon:2013iva}.
One upshot of this discussion is that there is no unique notion of entanglement entropy associated to a spatial region. Instead, for each region one can define a multitude of algebras associated to it, and each algebra choice comes with its own entanglement entropy. The maximal algebra is a natural choice, and indeed most of the current intuition comes from the entropy associated to this algebra, via the tracing out procedure. Nevertheless, this is still just a choice, and other algebras associated to a region --- for instance, one that differs from the maximal one by only one generator --- lead to (in principle) different measures of entanglement. We emphasize that this is not a mere UV ambiguity in the definition of the entropy (see also \cite{Casini:2013rba}), in the same way that a choice of boundary conditions in a path integral is not merely a UV effect. This analogy is not accidental; we will demonstrate below that a particular non-maximal choice of subalgebra can be represented as a tracing out of a full density matrix while summing over boundary conditions at the entangling edge.
The purpose of this paper is to study simple Ising systems on a lattice and to very explicitly show how the same entanglement entropy is exhibited on both sides of various dualities. In particular, we will focus on Kramers-Wannier (KW) dualities of the Ising model in $d = 1$ and $d = 2$ \cite{Savit:1979ny}, and on Jordan-Wigner/bosonization dualities of the Ising model in $d = 1$ dimensions. Working with the Ising model affords us a great degree of transparency, but our conclusions generalize to KW dualities of other Abelian theories in different dimensions.
\section{Entropy in a single spin}
Before sinking our teeth into pairs of dual theories, let us first warm up using a rather trivial example. We will use notation that immediately generalizes to more complicated cases. Consider a system consisting of a single spin with a two-dimensional Hilbert space $\H$. The algebra of Hermitian operators acting on this spin is $\A = \{\sigma^\mu\}$, where $\sigma^0 = \1$ and the remaining three operators, $\sigma^x$, $\sigma^y$, and $\sigma^z$, satisfy the commutation relations of the usual Pauli matrices. This algebra is generated by two operators, say $\sigma^x$ and $\sigma^z$; other operators are obtained as products of these two. There exist four subalgebras of $\A$:
\bel{
\{\1\},\ \{\1,\,\sigma^x\},\ \{\1,\,\sigma^y\},\ \trm{and}\ \{\1,\,\sigma^z\}.
}
Other subsets of $\A$ are not algebras because they are not closed under multiplication.
Each of the above algebras has an associated reduced density matrix. For a subalgebra $\A_V = \{\1,\O\}$, the general density operator can be written as $\rho_V = \rho_\1 \1 + \rho_\O \O$, where $\rho_\1$ and $\rho_\O$ are numbers. These coefficients are uniquely determined by solving a system of two linear equations coming from the requirement
\bel{\label{def rho}
\Tr(\rho_V \O) = \avg \O.
}
This uniqueness persists in more complicated examples with arbitrarily large algebras. For the trivial subalgebra $\A_V = \{\1\}$, the reduced density matrix is $\rho_V = \rho_\1 \1$, where $\rho_\1$ is chosen to ensure the operator has unit trace, in agreement with \eqref{def rho}.
The coefficients in the expansion
\bel{\label{expansion}
\rho_V = \sum_{\O \in \A_V} \rho_\O \O
}
will all depend on the representation of the trace on the l.h.s.~of \eqref{def rho}. In principle, we can choose \emph{any} representation, and we would get a legal density matrix. The von Neumann entropy associated to $\rho_V$ does depend on this choice. For instance, the trivial algebra $\A_V = \{1\}$ can be represented as the identity operation on a space of arbitrary dimension $D$, and the entropy would then be $\log D$. There is no reason to believe that one representation is more fundamental than another; each is a different yardstick for measuring the entropy. In this paper we will always employ the natural choice that comes from the original Hilbert space on which the full algebra $\A$ was defined, and when comparing entropies of different algebras we will make sure to only compare the entropies associated to representations of the same dimensionality.
With this comment in mind, we choose to represent the operators $\1$ and $\O$ as $2\times 2$ matrices acting on vectors in $\H$. If $\A_V = \{\1\}$, the reduced density matrix is $\rho_V = \frac12\1$ and the entropy is $S_V = \log 2$ regardless of the original state of the system. This is natural, as having access only to the identity operator means that we have no way of measuring anything about the system, so we can do no better than to express it as a completely mixed state.
If $\A_V = \{\1, \sigma^z\}$, say, things are more interesting. If $\avg{\sigma^z} = 0$, the reduced density matrix is again $\rho_V = \frac12 \1$, describing a mixed state since we have no information whether the system is in the $+1$ or $-1$ eigenstate of one of the other two operators. The entropy is again $\log 2$. If $\avg{\sigma^z} = 1$, however, the reduced density matrix describes a pure state, $\rho_V = \qvec{\!\!\uparrow}\qvecconj{\uparrow\!\!}$, and the entropy is zero; this time the observable algebra is enough to determine all information about the state of the system.
The setup described so far has a very nice property that generalizes to all $\Z_2$ models we study in this paper: all operators except for the identity have zero trace, and all operators square to the identity. This allows us to multiply both sides of \eqref{expansion} with an operator $\O$ and take the trace, getting
\bel{\label{coefficients}
\rho_\O = \frac{\avg\O}{\Tr\, \1}.
}
If we know the coefficients $\rho_\O$ of the full density matrix, we automatically know all the reduced density matrices: we just project the sum $\rho = \sum_{\O \in \A} \rho_\O \O$ to $\rho_V = \sum_{\O \in \A_V} \rho_\O \O$. In other examples, we may want to express the operators in $\A_V$ as acting on a smaller Hilbert space $\H_V$, in which case we need to restrict the sum to $\O \in \A_V$ \emph{and} to rescale all the surviving coefficients $\rho_\O$ by $\frac{\dim \H}{\dim \H_V}$. Doing this for a maximal algebra on a spatial subset $V$ gives the reduced matrix $\rho_V = \Tr_{\bar V} \rho$.
\section{Ising-Ising duality ($d = 1$)}
Consider the quantum Ising model defined on a chain with $L$ sites. Any conceivable operator in this model can be written as $\sigma^{\mu_1}_1 \ldots \sigma^{\mu_L}_L$, where $\sigma^\mu_i$ is a Pauli matrix acting on site $i$.\footnote{For clarity, we omit the $\otimes$ symbols and factors of $\1$. In proper notation, $\sigma_i^\mu$ would be written as $\1_1 \otimes \ldots \otimes \1_{i - 1} \otimes \sigma^\mu_i \otimes \1_{i + 1} \otimes \ldots \otimes \1_L$.} The version of the Ising model that possesses a Kramers-Wannier (KW) dual does not contain all of these operators; the needed algebra is generated by operators $\sigma^z_i$ and $\sigma^x_i \sigma^x_{i + 1}$ for $i = 1, \ldots, L - 1$. This choice reflects the adoption of open boundary conditions ($\sigma^z$ cannot be measured at the edges of the system).\footnote{There exists a version with a $\sigma^z$ at only one edge, but it would give us the same results as this one.} The Hamiltonian is
\bel{
H = -\sum_{i = 1}^{L-1} \sigma^x_i \sigma^x_{i + 1} + h \sum_{i = 2}^{L - 1} \sigma^z_i,
}
and the Hilbert space $\H$ is taken to be $2^{L - 1}$-dimensional, with the spin on site 1 always being in state $\qvec +$ that satisfies $\sigma^x_1 \qvec + = \qvec +$.
At strong coupling ($h \gg 1$), there are two orthogonal ground states corresponding to two different boundary conditions,
\bel{
\qvec{\Omega_\downarrow^{h \gg 1}} = \qvec{+\!\!\downarrow\downarrow\ldots\downarrow\downarrow}, \quad \qvec{\Omega_\uparrow^{h \gg 1}} = \qvec{+\!\!\downarrow\downarrow \ldots \downarrow\uparrow},
}
with $\sigma^z \qvec{\!\!\uparrow} = \qvec{\!\!\uparrow}$ and $\sigma^z \qvec{\!\!\downarrow} = -\qvec{\!\!\downarrow}$ as usual. At weak coupling ($h \ll 1$) the unique ground state is
\bel{
\qvec{\Omega^{h \ll 1}} = \qvec{++\ldots+}.
}
The KW dual of this system is an Ising model defined on the links of the above chain. We define the dual algebra via generators $\tau^z_{i,\, i+1} = \sigma^x_i \sigma^x_{i + 1}$ and $\tau^x_{i - 1,\, i} \tau^x_{i,\, i + 1} = \sigma^z_i$, so the Hamiltonian of the dual space is
\bel{\label{1d dual H}
H = - \~h \sum_{i = 1}^{L-1} \tau^z_{i,\, i+1} + \sum_{i = 2}^{L - 1} \tau^x_{i-1,\,i} \tau^x_{i,\,i+1}, \quad \~h = \frac1h.
}
The $2^{L - 1}$-dimensional Hilbert space of Ising spins on links is isomorphic to the original Hilbert space, and we will denote its elements with $|\cdot\}$. The natural mappings between the two spaces map ground states to each other, and in terms of basis elements they are
\algnl{
\begin{split}
\qvec{\pm}_i \qvec{\pm}_{i + 1} \mapsto |\!\uparrow\}_{i,\,i+1},&\quad
\qvec{\pm}_i \qvec{\mp}_{i + 1} \mapsto |\!\downarrow\}_{i,\,i+1}. \\
|\pm\}_{i-1,\,i}|\pm\}_{i,\,i+1} \mapsto \qvec{\!\downarrow}_i,&\quad
|\pm\}_{i-1,\,i}|\mp\}_{i,\,i+1} \mapsto \qvec{\!\uparrow}_i.
\end{split}
}
This implements the standard picture of spin flips being dualized to kinks/domain walls by a KW transformation.
The lack of individual $\tau^x$ operators in the dual picture means that there is no measurement that would distinguish between states related by a global spin flip in the $\tau^x$ eigenbasis. This is not so in the original picture, where the first spin is fixed to be in the $\qvec +$ state, so all the other individual $\sigma^x_i$ eigenvalues can be measured, and there is no lack of information on the overall spin flip in the $\sigma^x$ basis. Rather, the global spin-flip symmetry of the dual model corresponds to the $\qvec{\!\uparrow}_L \mapsto \qvec{\!\downarrow}_L$ symmetry in the original model (note that $\sigma_L^z$ is also not an observable).
Let us now study entanglement entropy on both sides of the duality. Consider first a set of neighboring sites $V$ in the original picture. Assuming that $V$ is away from the edges of the system, the maximal algebra $\A_V$ supported on $V$ is generated by $|V|$ operators $\sigma^z_i$ and $|V| - 1$ operators $\sigma^x_i \sigma^x_{i + 1}$. The entanglement entropy that we wish to compute is the von Neumann entropy of the matrix $\rho_V = \sum_{\O \in \A_V} \rho_\O \O$ represented as an operator on the Hilbert space $\H_V$ of spins in $V$. The coefficients in this expansion are, according to \eqref{coefficients},
\bel{
\rho_\O = \frac{\avg \O}{\dim \H_V}.
}
At strong coupling, the reduced density operator $\rho_V^{h \gg 1}$ is built out of all the operators with nonzero expectation values in the state $\qvec{\Omega_\downarrow^{h\gg 1}}$. (The result will be the same in the other ground state, of course.) The operators with nonzero vevs are all possible products of $\sigma^z$'s, and the density matrix is
\bel{\label{1d rho strong}
\rho_V^{h \gg 1} = \frac1{2^{|V|}}\left(\1 - \sum_{i \in V} \sigma_i^z + \sum_{i < j} \sigma^z_i \sigma^z_j - \ldots + (-1)^{|V|} \prod_{i \in V} \sigma^z_i \right).
}
These matrices are all diagonal in the $\sigma^z$ eigenbasis, and it takes a simple counting exercise to determine that all the diagonal entries except for $\qvec{\!\downarrow\ldots\downarrow}\qvecconj{\downarrow\ldots\downarrow\!}$ are zero. Thus, $\rho_V^{h \gg 1}$ describes a pure state and the entanglement entropy at strong coupling is
\bel{
S^{h \gg 1}_V = 0.
}
At weak coupling, the situation is inverted. The only operators with nonzero vevs are products of $\sigma^x_i \sigma^x_{i + 1}$, and all the vevs are equal to one. The reduced density operator is
\bel{
\rho^{h \ll 1}_V = \frac1{2^{|V|}}\left(\1 + \sum_{i < j} \sigma^x_i \sigma^x_j + \sum_{i < j < k < l} \sigma^x_i \sigma^x_j \sigma^x_k \sigma^x_l + \ldots \right).
}
Like before, it is sufficient to work in the $\sigma^x$ basis and count the $\pm1$ terms on the diagonal. The result is that matrix elements at positions $\qvec{+\ldots+}\qvecconj{+\ldots+}$ and $\qvec{-\ldots-}\qvecconj{-\ldots-}$ will each equal $1/2$, while all others will be zero. The weak-coupling reduced density matrix thus represents a mixed state with entropy
\bel{
S^{h \ll 1}_V = \log 2.
}
\begin{figure}[tb!]
\begin{center}
\begin{tikzpicture}[scale = 2]
\draw[step = 0.5, dotted] (-1.9, 0) -- (1.9, 0);
\draw[step = 0.5, dotted] (-1.9, -1) -- (1.9, -1);
\foreach \x in {-1.75, -1.25, ..., 1.75}
\draw (\x, 0) node[pink] {$\bullet$};
\foreach \x in {-1.5, -1, ..., 1.5}
\draw (\x, -1) node[lightgray] {$\bullet$};
\draw[thick] (-0.98, -1) -- (0.98, -1);
\draw (-1, -1) node {$\circ$};
\draw (1, -1) node {$\circ$};
\draw[thick, red] (-.73, 0) -- (.73, 0);
\foreach \x in {-0.5, 0, 0.5} {
\draw (\x, -1) node {$\bullet$};
\draw (\x, -1.1) node[anchor = north] {$\tau^z$};
\draw (\x, 0.1) node[anchor = south, red] {$\sigma^x \sigma^x$};
};
\foreach \x in {-0.75, -0.25, 0.25, 0.75} {
\draw (\x, 0) node[red] {$\bullet$};
\draw (\x, -0.9) node[anchor = south] {$\tau^x \tau^x$};
\draw (\x, -0.1) node[anchor = north, red] {$\sigma^z$};
\draw (\x, -0.5) node {$\updownarrow$};
};
\end{tikzpicture}
\end{center}
\caption{\small \textsc{(color online)} KW duality in $d = 1$. Above: the original picture. Thick red dots are the set $V$, and operators generating its maximal algebra are explicitly labeled. Below: dual picture. Black circles denote edge sites without $\tau^z$ operators; all operators in the dual algebra are also labeled.}
\label{fig 1d}
\end{figure}
In the dual picture, the algebra $\A_V$ maps to the non-maximal algebra $\~\A_{\~V}$ on the region $\~V$ with $|V| + 1$ sites (see Fig.~\ref{fig 1d}). This dual algebra is generated by operators $\tau^z_i$ for $i \in \~V - \del \~V$ and $\tau^x_i \tau^x_{i + 1}$ for $i \in \~V$; in other words, $\~\A_{\~V}$ is obtained by removing the edge operators $\tau^z_i$ from the maximal algebra on $\~V$. Note that the dimension of the Hilbert space on $\~V$ is different from the dimension of the corresponding Hilbert space on $V$. As discussed in the Introduction, this means that the entanglement entropy that is naturally calculated in the dual picture can be greater than the entropy in the original picture, as the algebras of the observables are the same but their representations differ. In order to meaningfully compare entropies on both sides of the duality, we will take dual operators to act on the $2^{|\~V|}$-dimensional Hilbert space $\H_{\~V}$ that has the first spin (at one edge of $\~V$) fixed to $|+\}$. This parallels the need to choose the spin on the end of the chain to be in the $\qvec+$ state.
Direct calculation can verify that the entanglement entropies are the same, as they should be since the dual subalgebras are isomorphic and the representations have the same dimension. It is instructive to see how this works out. The ground states at weak dual coupling are
\bel{
|\Omega_1^{\~h \ll 1}\} = |+-+\ldots\},\quad |\Omega_2^{\~h \ll 1}\} = |-+-\ldots\},
}
and operators with nonzero vevs in these states are $\tau_i^x \tau_{i+1}^x$ and their products. The reduced density matrix $\rho_{\~V}^{\~h \ll 1}$ is found to be the pure state matrix $|+-+\ldots\}\{+-+\ldots|$ in both ground states, and the entanglement entropy is
\bel{
S_{\~V}^{\~h \ll 1} = 0.
}
At strong dual coupling, the ground state is
\bel{
|\Omega^{\~h \gg 1}\} = |\!\uparrow\ldots \uparrow\},
}
and the operators with nonzero vevs are $\tau^z_i$ and their products. With the inclusion of the edge operators, the reduced density matrix on $\H_{\~V}$ takes the form
\bel{\label{1d rho dual strong}
\rho_{\~V}^{\~h \gg 1} = \frac1{2^{|\~V|}} \left(\1 + \sum_{i \in \~V - \del \~V} \tau^z_i + \ldots \right) \otimes \1.
}
The matrix in the parentheses is a pure state density matrix, $|\!\!\uparrow\ldots\uparrow\}\{\uparrow\ldots\uparrow\!\!|$. The entire von Neumann entropy of $\rho_{\~V}^{\~h \gg 1}$ comes from the identity operator at the edge where no boundary condition has been imposed:
\bel{
S_{\~V}^{\~h \gg 1} = \log 2.
}
These calculations show that the naturally defined entanglement entropies of dual systems are
\bel{
S_{\~V}^{\~h} = S_V^{h}.
}
While this may seem like a foregone conclusion given that the density matrices are evidently mapped to each other by duality, we point out that the origin of the entropy (when present) is different on the two sides of duality. In the original picture, the entropy came from mixing of the two states related by a global spin flip; in the dual picture, the same entropy came from the edge mode alone. This is a very simple example of UV/IR correspondence engendered by duality.
\section{Ising-gauge duality ($d = 2$)}
\subsection{Setup}
Let us now define a $\Z_2$ gauge theory on a square $L\times L$ lattice. Operators and states are defined on links $\ell$, and the most general operator has the form $\prod_\ell \sigma^{\mu_\ell}_\ell$. Gauge-invariant states are those elements of the full Hilbert space $\H_0$ that are invariant under Gauss operators $G_i = \prod_\mu \sigma^x_{(i,\, \mu)}$ at any site $i$, with the product over all directions $\mu$ of links emanating from $i$. These states form the gauge-invariant Hilbert space $\H$. Operators that map $\H$ to $\H$ form the gauge-invariant algebra $\A$, which is generated by all the operators $\sigma^x_\ell$ and by products $W_p = \prod_{\ell \in p} \sigma^z_\ell$ around each plaquette $p$. The $W_p$ are magnetic operators (Wilson loops) and they create closed loops of electric flux. The $\sigma^x_\ell$ are electric operators and they create pairs of vortices (``magnetic flux insertions'').
As done for the Ising model, in order to define a theory with a KW dual, we choose that $\A$ has no generators on the edge of the lattice. In other words, we regard an electric operator $\sigma^x_\ell$ as unphysical/unobservable if $\ell$ does not belong to exactly two plaquettes. Now all the remaining gauge-invariant operators can be mapped to operators in the Ising model on the dual lattice via
\bel{
W_p = \tau_p^z, \quad \sigma^x_\ell = \tau^x_p \tau^x_q,
}
with $p$ and $q$ being two plaquettes that both contain the link $\ell$. As before, we see that the system with no operators at its edges gets dualized to a system with no individual $\tau^x$ operator on any site.
\begin{figure}[tb!]
\begin{center}
\begin{tikzpicture}[scale = 2]
\foreach \x in {-1, -0.5,..., 1} \draw (\x, -1) node[lightgray] {$\bullet$};
\foreach \x in {-1, -0.5,..., 1} \draw (\x, 1) node[lightgray] {$\bullet$};
\foreach \x in {-1, -0.5,..., 1} \draw (1, \x) node[lightgray] {$\bullet$};
\foreach \x in {-1, -0.5,..., 1} \draw (-1, \x) node[lightgray] {$\bullet$};
\foreach \x in {-0.75, -0.25, ..., 0.75} {
\draw (-1.25, \x) node[red] {$\circ$};
\draw (1.25, \x) node[red] {$\circ$};
\draw (\x, -1.25) node[red] {$\circ$};
\draw (\x, 1.25) node[red] {$\circ$};
};
\foreach \x in {-0.75, -0.25,..., 0.75}
\foreach \y in {-0.75, -0.25,..., 0.75}
\draw (\y, \x) node[red] {$\bullet$};
\draw[step = 0.5, red, yshift = -0.25cm] (-1.23, -0.75) grid (1.23, 1);
\draw[step = 0.5, red, xshift = -0.25cm] (-0.75, -1.23) grid (1, 1.23);
\draw[step = 0.5, dotted] (-1.9, -1.9) grid (1.9, 1.9);
\draw[step = 0.5, thick] (-1, -1) grid (1, 1);
\end{tikzpicture}
\end{center}
\caption{\small \textsc{(color online)} Duality in $d = 2$: Thick black lines are the set $V$, and grey circles denote edge sites in $\del V$. Electric operators $\sigma^x$ are defined on all thick black lines, and magnetic operators $W$ are defined on all thick black plaquettes. All red circles together form the dual set $\~V$. Operators $\tau^z$ on filled-in circles belong to the dual algebra $\~\A_{\~V}$, as do all $\tau^x\tau^x$ pairs on sites connected by red lines. }
\label{fig 2d}
\end{figure}
The mapping of Hilbert spaces presents more subtleties than in $d = 1$. The Hilbert space $\~\H$ of the Ising model has $2^{(L-1)^2}$ dimensions, as it is a product of two-dimensional Hilbert spaces on each plaquette. The gauge-invariant Hilbert space $\H$ has the same dimension, but the full space $\H_0$ has a much greater dimension, $2^{2L(L - 1)}$, being a product of two-dimensional spaces on each link. The full Hilbert space cannot be mapped to the Ising model; only gauge-invariant states map. Since the entanglement entropy in gauge theories is typically (if tacitly) calculated using the full Hilbert space \cite{Gromov:2014kia, Kitaev:2005dm, Hamma:2005zz, Levin:2006zz, Radicevic:2014kqa, Radicevic:2015sza, Soni:2015yga, Pretko:2015zva, Aoki:2015bsa, Buividovich:2008gq, Buividovich:2008yv, Casini:2013rba, Casini:2014aia, Donnelly:2011hn, Donnelly:2012st, Donnelly:2014gva, Ghosh:2015iwa}, we might expect large differences between entanglement entropies that are naturally calculated on the two sides of KW duality. This does not happen: the inclusion of Gauss law operators $G_i$ in the algebra of observables effectively projects the full Hilbert space down to the physical one. We will later explain how this works in detail.
We take the gauge theory Hamiltonian to be
\bel{
H = g \sum_\ell \sigma^x_\ell - \sum_p W_p.
}
The strong coupling ground state is degenerate, just like it was for the Ising chain. There are two physical ground states. Both contain a product of states $\qvec -$ at each interior link, and they differ in how the Gauss law is realized at the edge of the system. (Each of these states is obtained from the other by adding an (unobservable) electric flux loop along the system edge.) These states confine electric fields, and excitations are loops of electric flux. At weak coupling, the ground state $\qvec{\Omega^{g \ll 1}}$ has $W_p = 1$ on each plaquette. This is satisfied by $\prod_\ell \qvec{\!\!\downarrow}_\ell$ and by all other states obtained by acting on this one with products of $G_i$. The only gauge-invariant ground state is the sum of all of these states, and this is $\qvec{\Omega^{g \ll 1}}$. This ground state can also be expressed as the unweighted sum over all possible electric flux loop excitations of either strong coupling state $\qvec{\Omega^{g \gg 1}_{1/2}}$.
The dual Hamiltonian is
\bel{
H = \sum_{\avg{p,\, q}} \tau^x_p \tau^x_q - \~g\sum_p \tau^z_p, \quad \~g = \frac1g.
}
which is just the higher-dimensional analogue of \eqref{1d dual H}. As before, we denote the dual quantum states by $|\cdot\}$. At strong dual coupling, the ground state is
\bel{
|\Omega^{\~g\gg 1}\} = \prod_p |\!\uparrow\}_p,
}
and at weak dual coupling the two ground states are the two possible ``checkerboard'' tilings of the lattice by $|+\}$ and $|-\}$ states,
\bel{
|\Omega_1^{\~g \ll 1}\} = \prod_{i,\, j = 1}^{L - 1} |(-1)^{i + j}\}_{(i,\, j)},\quad |\Omega_2^{\~g \ll 1}\} = \prod_{i,\, j = 1}^{L - 1} |(-1)^{i + j + 1}\}_{(i,\, j)}.
}
\subsection{Entanglement in gauge theory}
The entanglement entropy on the gauge theory side has been calculated for both strong and weak coupling \cite{Soni:2015yga, Ghosh:2015iwa, Donnelly:2011hn, Radicevic:2015sza, Casini:2013rba}. In order to emphasize and further illustrate our operator approach to density matrices, we present a telegraphic derivation these well-known results. We will focus on the entropy associated to the maximal algebra that can be placed upon a set of links $V$.
At weak coupling, all products of Wilson loops satisfy $\avg{\prod_p W_p} = 1$, and all electric operators and their products have vanishing vevs --- with the exception of Gauss operators, $G_i$, whose products all satisfy $\avg{\prod_i G_i} = 1$. All these operators with nonzero vevs mutually commute. Thus, the reduced density operator for the algebra $\A_V$ can be written as
\bel{
\rho_V^{g \ll 1} = \frac{1}{2^{\#\trm{links}(V)}} \left(\1 + \sum_p W_p + \sum_i G_i + \sum_{p \neq q} W_p W_q + \sum_{i \neq j} G_i G_j + \sum_{i,\, p} G_i W_p + \ldots \right),
}
where the denominator is the dimension of the Hilbert space of spins on all links in $V$ (without any regard for gauge invariance), and the sums include all Gauss operators and elementary Wilson loops that generate $\A_V$. This density operator can also be written as
\bel{
\rho_V^{g \ll 1} = \frac{2^{\#\trm{stars}(V)}}{2^{\#\trm{links}(V)}} \left(\1 + \sum_p W_p + \sum_{p \neq q} W_p W_q + \ldots\right) \prod_i \frac{\1 + G_i}2.
}
(Note that a Gauss operator $G_i$ is in $\A_V$ if and only if all links emanating from site $i$ are in $V$; such configurations of links are called ``stars.'') Each operator $\frac12(\1 + G_i)$ projects onto the space of states that obey the Gauss law at site $i$. This extremely convenient fact allows us to forget about the original set of degrees of freedom inside $V$ and to work just with gauge-invariant basis vectors. However, we must still work with gauge-variant degrees of freedom at edge sites of $V$, as the associated Gauss operators will not be in $\A_V$. This shows how the general operator prescription reduces to the one studied by extended Hilbert space and/or superselection sector techniques \cite{Soni:2015yga, Ghosh:2015iwa}.\footnote{This conclusion holds generally, not just in the ground state at weak coupling. Gauss operators belong to the center of $\A_V$, they always have unit expectation values for gauge-invariant states, and hence any density matrix can be written as a projector to the gauge-invariant subspace.}
The von Neumann entropy of $\rho_V^{g \ll 1}$ is easy to compute when the matrix is stripped of the projection operators and expressed in the basis that diagonalizes the Wilson loops. The density matrix is then diagonal and uniformly mixes the $2^{|\del V| - 1}$ different basis vectors that correspond to states with $W_p = 1$ and $G_i = 1$ at all plaquettes and stars. (Here we assume that $V$ does not contain disconnected components.) Its entropy takes the familiar form
\bel{
S^{g \ll 1}_V = (|\del V| - 1) \log 2.
}
At strong coupling, the situation is somewhat simpler: operators with nonzero vevs are all products of $\sigma^x$'s with $\avg{\sigma^x_{\ell_1} \ldots \sigma^x_{\ell_n}} = (-1)^n$. Gauss operators are a special case of these, and do not need to be treated separately. The reduced density matrix is
\bel{
\rho^{g \gg 1}_V = \frac1{2^{\#\trm{links}(V)}} \left(\1 - \sum_\ell \sigma^x_\ell + \sum_{\ell \neq \ell'} \sigma^x_\ell \sigma^x_{\ell'} - \ldots\right).
}
In the electric basis this matrix is diagonal and by simple inspection we see that the entry corresponding to the basis vector $\prod_{\ell \in V} \qvec{-}_\ell$ is equal to unity; therefore other entries must be zero, and the matrix is pure. This again reproduces the well-known result
\bel{
S^{g \gg 1}_V = 0.
}
\subsection{Entanglement in the dual picture}
Just like in the previous Section, the maximal algebra on a set of links does not map to a maximal algebra on a set of sites on the dual lattice. Instead, the dual algebra $\~\A_{\~V}$ lacks individual $\tau^x$ generators on any site but contains an extra set of $\tau^x_p \tau^x_q$ generators that wrap the edge of the original region $V$ (see Fig.~\ref{fig 2d}).
At strong dual coupling, the operators with nonzero vev are $\tau_p^z$ and their products. Note that duals to Gauss operators, i.e.~products of $\tau^x_p \tau^x_q$ along closed contours, have vevs equal to $1$ not by virtue of the state being special, but rather purely algebraically, because each $\tau^x_p$ in that product is repeated an even number of times and the full product just gives the identity. These are not independent observables the way $G_i$'s were in the gauge theory.
The reduced density operator is thus
\bel{
\rho_{\~V}^{\~g \gg 1} = \frac 1{2^{|\~V|}} \left(\1 + \sum_p \tau^z_p + \sum_{p \neq q} \tau^z_p \tau^z_q + \ldots + \prod_p \tau^z_p \right) \otimes \sideset{}{'}\prod_{p \in \del \~V} \1_p,
}
where the indices $p$ and $q$ in the parentheses run over the interior of the dual region, $\~V - \del\~V$. The product over edge degrees of freedom is primed to denote that it runs over $|\del \~V| - 1$ sites; this is because we need the representation of the dual subalgebra to have the same dimension as in the original picture, and hence we fix one boundary site to always be in the $|+\}$ state. This is a straightforward generalization of the $d = 1$ case \eqref{1d rho dual strong}, and the entropy is simply
\bel{
S_{\~V}^{\~g \gg 1} = (|\del \~V| - 1) \log 2.
}
Since $|\del V| = |\del\~ V|$, the dual entropies $S_{\~V}^{\~g \gg 1}$ and $S_{V}^{g \ll 1}$ are equal, as they should be.
At weak dual coupling, the operators with nonzero expectations are $\tau^x_p \tau^x_q$ pairs and their products, \emph{except} for the products along closed loops of links. The reduced density matrix is
\bel{
\rho_{\~V}^{\~g \ll 1} = \frac1{2^{|\~V|}} \left(\1 + \sum_{p\neq q} (-1)^{|p - q|} \tau^x_p \tau^x_q + \ldots \right).
}
By the same trick used in its dual strongly coupled gauge theory case, we notice that, in the $\tau^x$ eigenbasis, the vector with alternating $+$ and $-$ states is an element of $\H_{\~V}$ that gives a unit entry on the diagonal of $\rho_{\~V}^{\~g \ll 1}$. Since this is a density matrix, all other entries must be zero, and the entanglement entropy is
\bel{
S_{\~V}^{\~g \ll 1} = 0.
}
We see that $S_{\~V}^{\~g} = S_V^{g}$ holds in $d = 2$, just like it did in $d = 1$. This time the interesting effect is the topological piece of the weak-coupling entropy, $-\log 2$, and the corresponding term in the strongly coupled scalar entropy. In the gauge theory this term is well-understood: due to the Gauss law, the total electric flux passing through the edge $\del V$ must be zero, this leads to a constraint on the types of states that the interior can be in, and therefore the entropy is smaller than the area law term that one may na\"ively expect. In the dual picture, the reduced density matrix uniformly mixes all edge modes (modulo an overall spin flip) even though the interior is in an ordered state. When computing an expectation value, this mixing can be implemented as a sum over all possible domain walls on a spin chain located on the entangling edge $\del \~V$. In the case at hand, there are $2^{|\del \~V| - 1}$ different domain walls.
This phenomenon is already known for the dual of $U(1)$ gauge theory \cite{Agon:2013iva}. The entanglement entropy of a compact scalar at small radius was found to come from the sum over configurations with different windings along the entanglement edge in the replica path integral. In the language of the present paper, the sum over winding sectors follows from the fact that the edge $\del\~V$ does not admit any position operators as observables. Adding these edge operators would project us to a sector with zero winding; equivalently stated, the reduced density matrix of this enlarged algebra would have to be that of a pure state in order to reproduce expectation values of the newly added operators. There exist related discussions in the contexts of $d = 3$ gauge theories \cite{Pretko:2015zva}, self-dual higher form gauge theories \cite{Ma:2015xes}, and the $d = 1$ Ising model \cite{Ohmori:2014eia}.
\section{Bosonization ($d = 1$)}
As our final example, we study the Jordan-Wigner transformation between the Ising chain and a system of Majorana fermions. This is a rather simple setup, but it will provide us with an example of a system where a nonlocal set of generators is needed to form the subalgebra of one side of the duality.
This time, we work with an Ising chain on $L$ sites with all operators present, and with Hamiltonian
\bel{
H = -\sum_{i = 1}^{L-1} \sigma^x_i \sigma^x_{i + 1} + h \sum_{i = 1}^{L} \sigma^z_i.
}
The dual Majorana operators are
\bel{
c_i = \sigma^x_i \prod_{j < i} \sigma^z_j ,\quad d_i = \sigma^y_i \prod_{j < i} \sigma^z_j .
}
These operators are Hermitian and any two nonidentical ones anticommute. Often, $d_i$ is written as $c_{i + 1/2}$. The dual Hamiltonian is
\bel{
H = i \sum_{i = 1}^{L - 1} d_i c_{i + 1} - i h \sum_{i = 1}^L c_i d_i.
}
(Note that due to the anticommutation between all $c$'s and $d$'s, their products must be multiplied by $i$ to give Hermitian operators.) The Majoranas $c_i$ and $d_i$ act on the $2^i$-dimensional Hilbert space of a complex fermion at sites $1$ through $i$, which is in turn isomorphic to the Hilbert space of $i$ spins.
Let us consider the entropy associated to the algebra generated by a set of adjacent $c$'s and $d$'s on sites $V$. At strong coupling ($h \gg 1$), the ground state is a ``Majorana superconductor,'' a condensate of pairs of fermions that are coupled by the $h$ term in $H$. Individual Majorana fermions have vanishing vevs, and the only operators with nonvanishing expectations are the identity and products of pairs $i c_i d_i$. In the spin language, these are the $\sigma^z_i$ operators that detect that the ground state is ordered.
When choosing the representation of the reduced algebra $\A_V$, it is important to keep it the same dimension as in the original spin picture. Thus, even though all operators have trailing $\sigma^z$'s or $\1$'s going all the way to the beginning of the chain, we choose to represent all operators as matrices acting on the $2^{|V|}$-dimensional space of spins/fermions living in $V$. If the algebra $\A_V$ contains both Majoranas on each site in $V$, the strong coupling entropy is zero, in complete analogy with \eqref{1d rho strong}. If not, i.e.~if the system's right edge $V$ cuts between sites $i$ and $i + \frac12$, this is equivalent to removing $\sigma_i^z$ but leaving $\sigma^x_i$ in the subalgebra. If the system's left edge cuts between $i - \frac12$ and $i$, the situation is a bit more complicated because then both $\sigma^y_{i - 1}$ and $\sigma^x_{i - 1}$ will appear in the algebra, but the latter operator will only appear in products with other operators on sites $i$ and onwards.
At weak coupling, the spin system has two ground states (this time we have not imposed boundary conditions to lift this degeneracy). They are related by a global spin flip. In the Majorana picture, however, the degeneracy comes from the edge Majoranas $c_1$ and $d_L$ which are free (while all the others are paired up by the $d_i c_{i + 1}$ terms in $H$). If we pick the algebra $\A_V$ such that it doesn't split any of these pairs of Majoranas, we will get zero entropy. If, however, we pick the algebra such that it is a maximal algebra on a set of spin sites, then it will necessarily cut through two pairs of coupled Majoranas. The final result in this case is an entropy of $\log 2$. In the spin language this entropy came from the mixing of two states on $V$ related by a global spin flip, and in the fermion language it came from the dangling Majoranas at the edge of $V$. Once again, this is an example of a UV/IR connection.
\section{Outlook}
The main purpose of this paper was to provide a point of view from which entanglement entropy becomes an object that is naturally preserved by various dualities. The price we paid was the need to generalize entanglement entropy away from the usual ``tracing out'' procedure. From the point of view of this usual procedure, the generalized notion of entanglement entropy amounts to introducing summations over sectors labeled by eigenvalues of operators removed from a maximal algebra on a spatial region. In gauge theories, this summation over superselection sectors has been the subject of a lot of attention \cite{Casini:2013rba, Casini:2014aia, Radicevic:2014kqa, Radicevic:2015sza, Soni:2015yga, Ghosh:2015iwa, Aoki:2015bsa}, and in this paper we have shown that it can be understood to follow from excluding Gauss operators at the edges of the system from the observable subalgebra. In scalar theories, we have seen how excluding edge operators leads to a sum over winding sectors around the edge (or solitonic configurations on the edge), a variant of which was already considered in \cite{Agon:2013iva} in order to study entanglement in a scalar dual to a $d = 2$ Maxwell theory.
The entropy of a non-maximal algebra on a spatial region $V$ may seem undeserving of the name ``entanglement'' entropy. In particular, this entropy can be defined even in $d = 0$, as we did for the case of a single spin, and here there are no spatial regions to entangle. However, it seems that distinguishing between purely spatial entanglement entropy and these other entropies is not productive, as duality mixes up these notions. Moreover, at least in the case of Majorana fermions, there is even a way to map each generator of the Ising chain algebra to a Majorana operator on a separate spatial site, giving a direct geometric interpretation to each spin operator.
Requiring additional symmetries is a useful way to tame the multitude of subalgebras that can be placed on a spatial region. For instance, if we work on a lattice with spherical symmetry and we pick $V$ to be a ball, then requiring that $\A_V$ be spherically symmetric significantly restricts the set of allowed subalgebras, as we can now only remove a generator from all points at the edge. In fact, demanding enough symmetry may pick out a unique algebra (up to differences leading to nonuniversal terms only), as evidenced by the fact that supersymmetric Renyi entropies of certain superconformal theories on spherical regions (computed via localization of the replica trick path integral) agree across dualities without any manual summation over superselection sectors \cite{Nishioka:2013haa}.
Our results easily generalize to other Abelian theories. In particular, $\Z_k$ and $U(1)$ theories with known duals all follow the pattern of mapping a maximal algebra to a non-maximal one. In $d = 3$, where a gauge theory maps to another gauge theory via electric-magnetic duality, our results rather reassuringly imply that the maximal algebra on one side (the ``electric center'' choice) will map to the ``magnetic center'' choice on the other side of the duality. We have not touched upon dualities of nonabelian theories, as these are much more complex, but we expect that a similar story will hold.
A duality that we have so far not mentioned at all is holography. The algebraic approach to entanglement entropy in gravitational theories is still nascent \cite{Donnelly:2015hta, Donnelly:2016auv, Giddings:2015lla}, and even bulk operator reconstruction based on boundary data is very much an active field of research (see, for instance, \cite{Morrison:2014jha, Jafferis:2014lza, Hamilton:2006az, Kabat:2011rz, Dong:2016eik, Almheiri:2014lwa, Headrick:2014cta, Wall:2012uf, Czech:2012bh}). Nevertheless, holographic dualities qualitatively behave like the dualities studied in this paper: they are strong-weak coupling dualities, there exists a UV/IR connection \cite{Susskind:1998dq}, and entanglement entropy in the bulk appears to be equal to the one on the boundary, inasmuch as we know how to define entanglement in quantum gravity (see e.g.~\cite{Jafferis:2015del, Faulkner:2013ana}). Based on this and on the intuition developed in this paper, a reasonable speculation at this stage would be that a maximal algebra in a subregion does not holographically map to a maximal algebra in a dual subregion, so any statement about the duality of subregions must be supplemented with rules about how to exclude certain operators (or how to sum over corresponding sectors in the path integral) on at least one side of the duality. It would be interesting to understand what boundary operators are missed by the construction of \cite{Dong:2016eik}, which proves that local bulk operators in the entanglement wedge of a boundary region $V$ are dual to boundary operators supported only in $V$. For example, local boundary operators in $V$ are dual to classical gravity backgrounds, i.e.~to solitons that live in the entire bulk, so we should expect that the algebra dual to local bulk operators in the entanglement wedge can contain local operators in the boundary only in some approximate sense. Making these speculations precise is a difficult but extremely rewarding task left for future work.
\section*{Acknowledgments}
It is a pleasure to thank Chao-Ming Jian, Jen Lin, Steve Shenker, and Itamar Yaakov for discussions. The author is supported by a Stanford Graduate Fellowship.
|
2,869,038,154,740 | arxiv | \section*{References}
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\end{document}
|
2,869,038,154,741 | arxiv | \section{Introduction}
The study of how the structural parameters of galaxies are linked to their dynamics and stellar properties via scaling relations and how these change as a function of redshift, environment and wavelength constitute some of the most used probes to understand the formation and evolution of galaxies. The Fundamental Plane (FP, \citealt{Dressler1987,Djorgovski1987,Bender1992,Saglia1993,Donofrio2022,Donofrio2023}) and its projections are among the most important galaxy scaling relations. The FP links the structural properties of early-type galaxies (ETGs), such as their surface brightnesses and effective radii, with their dynamics. One of its projection is the Kormendy relation (KR, \citealt{Kormendy1977}). The KR links the effective radius $R_\mathrm{e}$ of an ETG to its surface brightness within that radius $\left \langle \mu \right \rangle_{\mathrm{e}}$, $\left \langle \mu \right \rangle_{\mathrm{e}} = \alpha + \beta \log{R_{\mathrm{e}}}$. The KR is a physical correlation, presumably reflecting the difference in the origin of bright and faint ellipticals and bulges. It provides information on how the size and light distribution in ETGs evolve as a function of redshift. This implies that the KR can be used to understand whether ETGs have completed their mass and size growth at their redshift or if there is still significant growth up to $z=0$. Furthermore, the KR can also be used to study the bulge formation of disk galaxies \citep{Gadotti2009}: bulges can be disentangled from pseudo-bulges, being the latter $3\sigma$ outliers with respect to the best-fitting KR \citep{Gao2020}.
The KR has been extensively studied in the literature \citep{Ziegler1999,LaBarbera2003,Longhetti2007,Rettura2010,Saracco2014}. However, there are still contrasting conclusions drawn from it. In particular, they depend on how the sample of galaxies is selected, whether it is based on colours, S\'ersic indices or stellar populations (e.g. \citealt{Saracco2010,Andreon2016,Fagioli2016}). In \citealt{Tortorelli2018}, we investigated the effect of the sample selection and we found that a selection purely based on colours may bias the KR parameters estimate, while galaxies selected through S\'ersic indices (ETGs), visual inspection (ellipticals) and spectra (passives) constitute a more coeval population.
Additionally, the morphologies and, therefore, the sizes of galaxies may appear different at different wavelengths. Since different wavelengths probe different regions and physical processes inside the galaxies, the KR does not necessarily hold or have the same parameters at all wavelengths. The surface brightness in the near-infrared is less affected by gas and dust extinction. It is dominated by the older stellar population, which constitutes galaxies' main stellar mass component, especially ETGs. Hence, the contrast between the near-infrared and the optical observations translates into the contrast between the underlying mass component (older stellar population) and the younger stellar population component. Therefore, the wavelength range used to study the KR may impact the conclusions that we can draw from it. Studies at low-redshift of the wavelength dependence of the KR have been conducted by, e.g. \citealt{LaBarbera2010}. These studies show an evolution of the KR slope as a function of wavelength for Sload Digital Sky Survey (SDSS) ETGs selected via colours in the form of a steepening of the relation from the $\textit{g}$ to the $\textit{K}$ observed band. However, studies at higher redshifts ($z > 0.3$) with samples of ETGs consistently selected at different redshifts are still missing.
In this letter, we use data from the Hubble Space Telescope (HST) Frontier Fields (FF) survey \citep{Lotz2017} to study the KR of ETGs as a function of wavelength in three clusters at intermediate redshift, namely Abell\,S1063, at $z = 0.348$, MACS\,J0416.1-2403, at $z = 0.396$, and MACS\,J1149.5+2223, at $z = 0.542$. We build the KR for ETGs selected via S\'ersic indices only, following the conclusions in \citealt{Tortorelli2018} about consistently selecting ETGs at different redshifts.
The structural parameters are measured using a new \textsc{python} package we develop called \textsc{morphofit} \citep{Tortorelli2023}. We use the data analysis pipeline first introduced in \citealt{Tortorelli2018} and refined in \citealt{Tortorelli2023} that involves galaxies' surface brightness profile fit in images of increasing sizes.
The letter is structured as follows. In Sect. \ref{section:dataset}, we describe the photometric and spectroscopic data used for this analysis. In Sect. \ref{section:struct_param_estimate} we summarise the structural parameters estimate process using \textsc{morphofit}. Sect. \ref{section:kormendy_wave} presents the results on the Kormendy relation behaviour as a function of wavelength. We provide the main conclusions in Sect. \ref{section:conclusions}.
Unless otherwise stated, we give errors at the $68$ per cent confidence level (hereafter c.l.), and we report the circularised effective radii. Throughout this paper, we use $H_0 = 70 \mathrm{km\ s^{-1}\ Mpc^{-1}}$ in a flat cosmology with $\Omega_{\mathrm{M}}$ = 0.3 and $\Omega_{\Lambda}$ = 0.7. In the adopted cosmology, $1\arcsec$ corresponds to $4.921\ \mathrm{kpc}$ at $z = 0.348$, to $5.340\ \mathrm{kpc}$ at $z = 0.396$ and to $6.364\ \mathrm{kpc}$ at $z = 0.542$.
\section{Dataset}
\label{section:dataset}
We analyse the three clusters Abell S1063 (AS1063) at $z = 0.348$, MACS J0416.1-2403 (M0416) at $z = 0.396$ and MACS J1149.5+2223 (M1149) at $z = 0.542$. AS1063, M0416 and M1149 have a wealth of multi-wavelength and wide-field data from photometry and spectroscopy \citep{Rosati2014,Karman2015,Grillo2016,Mercurio2021}. The photometric data we use in this study are available at the STScI MAST Archive\footnote{https://archive.stsci.edu/prepds/frontier/}. They belong to the FF programme\footnote{http://www.stsci.edu/hst/campaigns/frontier-fields/HST-Survey} (PI: J. Lotz, \citealt{Lotz2017}), aiming at combining the power of \textit{HST} and \textit{Spitzer} with the natural gravitational telescope effect of massive high-magnification clusters of galaxies. These datasets allow us, for instance, to test predictions of the $\Lambda \mathrm{CDM}$ model \citep{Annunziatella2017,Sartoris2020}, to measure the Hubble constant value \citep{Grillo2018,Grillo2020}, to refine weak and strong lensing models to map the total mass distribution in clusters \citep{Gruen2013,Caminha2016,Bergamini2019,Granata2022} and to serendipitously discover very distant lensed galaxies up to $z \sim 6$ \citep{Vanzella2016,Balestra2018}.
In order to measure the structural parameters (i.e. effective radii and surface brightnesses), we use the $0.060\ \mathrm{arcsec/pixel}$ images in all seven optical/NIR bands of the FF programme. The three optical bands \textit{F435W, F606W, F814W} belong to the Advanced Camera for Surveys (ACS), while the four NIR bands \textit{F105W, F125W, F140W, F160W} to the Wide Field Camera 3 (WFC3) IR imager. They span a wavelength range from $\sim 3500$ \AA , corresponding to the observed Johnson \textit{B}-band filter, to $\sim 17400$ \AA, corresponding to the observed \textit{H} filter. This roughly corresponds to the rest-frame ranges: \textit{u} to \textit{J} band for AS1063 and M0416, and near-UV to \textit{Y} band for M1149. We use the \textit{drz} science images, the \textit{rms} images and the \textit{exp} exposure time map images.
\begin{table*}[ht!]
\small
\centering
\begin{tabular}{l c c c c c c c c c c}
\hline
\hline
&&\textbf{AS1063}&&&\textbf{M0416}&&&\textbf{M1149}&\\
\hline
& $\alpha$ & $\beta$ & $\sigma$ & $\alpha$ & $\beta$ & $\sigma$ & $\alpha$ & $\beta$ & $\sigma$\\
\textit{F435W} & 20.97 $\pm$ 0.11 & 3.11 $\pm$ 1.08 & 0.65 $\pm$ 0.01 & 21.04 $\pm$ 0.13 & 2.81 $\pm$ 0.69 & 0.70 $\pm$ 0.01 & 20.21 $\pm$ 0.66 & 4.69 $\pm$ 2.14 & 0.94 $\pm$ 0.01\\
\textit{F606W} & 19.12 $\pm$ 0.11 & 3.86 $\pm$ 0.61 & 0.72 $\pm$ 0.01 & 19.06 $\pm$ 0.11 & 2.85 $\pm$ 0.34 & 0.65 $\pm$ 0.01 & 19.00 $\pm$ 0.29 & 2.70 $\pm$ 1.02 & 0.50 $\pm$ 0.01\\
\textit{F814W} & 18.19 $\pm$ 0.11 & 3.72 $\pm$ 0.54 & 0.74 $\pm$ 0.01 & 18.04 $\pm$ 0.11 & 3.29 $\pm$ 0.38 & 0.74 $\pm$ 0.01 & 17.56 $\pm$ 0.22 & 3.27 $\pm$ 0.83 & 0.63 $\pm$ 0.01\\
\textit{F105W} & 17.72 $\pm$ 0.12 & 4.30 $\pm$ 0.64 & 0.81 $\pm$ 0.01 & 17.56 $\pm$ 0.10 & 3.38 $\pm$ 0.48 & 0.81 $\pm$ 0.01 & 16.85 $\pm$ 0.35 & 3.74 $\pm$ 1.80 & 0.72 $\pm$ 0.01\\
\textit{F125W} & 17.47 $\pm$ 0.13 & 4.31 $\pm$ 0.72 & 0.82 $\pm$ 0.01 & 17.18 $\pm$ 0.12 & 3.60 $\pm$ 0.67 & 0.85 $\pm$ 0.01 & 16.39 $\pm$ 0.72 & 4.81 $\pm$ 3.46 & 0.84 $\pm$ 0.01\\
\textit{F140W} & 17.32 $\pm$ 0.13 & 4.22 $\pm$ 0.75 & 0.83 $\pm$ 0.01 & 17.07 $\pm$ 0.13 & 3.76 $\pm$ 0.68 & 0.89 $\pm$ 0.01 & 16.06 $\pm$ 0.27 & 5.45 $\pm$ 1.39 & 0.86 $\pm$ 0.01\\
\textit{F160W} & 17.21 $\pm$ 0.14 & 5.28 $\pm$ 0.87 & 0.91 $\pm$ 0.01 & 16.85 $\pm$ 0.11 & 3.62 $\pm$ 0.54 & 0.81 $\pm$ 0.01 & 16.00 $\pm$ 0.29 & 5.08 $\pm$ 1.58 & 0.86 $\pm$ 0.01\\
\hline
\hline
\end{tabular}
\caption{The table reports the best-fitting slopes, intercepts, observed scatters and their $1\sigma$ errors obtained by fitting the KR to the ETG samples. }
\label{table:KR_parameters}
\end{table*}
The AS1063 and M1149 spectroscopically confirmed cluster members we use in our analysis are the same as those selected in \citealt{Tortorelli2018} using \textit{VLT/MUSE} IFU spectra. They are $95$ for AS1063, having redshifts in the range $0.335 \le z \le 0.361$ \citep{Karman2015}, and $68$ with redshifts $0.52 \le z \le 0.57$ \citep{Grillo2016} for M1149. \textit{HST} imaging and \textit{MUSE} IFU spectra allow us to be fully complete down to a magnitude value of $22.5$ in the \textit{F814W} waveband \citep{Caminha2016}. This limit corresponds roughly to a stellar mass value of $M_{*} \sim 10^{9.8} \mathrm{M_{\odot}}$ and $M_{*} \sim 10^{10.0} \mathrm{M_{\odot}}$ for AS1063 and M1149, respectively, for the typical spectral energy distribution of the sources we are interested in and considering a Salpeter initial mass function (IMF) \citep{Salpeter1955}. The cluster members for M0416 are also those for which MUSE spectra are available. They are $119$ with redshifts $0.38 \le z \le 0.41$. The description of the member selection is detailed in \citealt{Annunziatella2017} and \citealt{Caminha2017}.
In \citealt{Tortorelli2018}, we defined and compared four different galaxy samples according to (a) S\'ersic indices: early-type (`ETG'), (b) visual inspection: `ellipticals', (c) colours: `red', (d) spectral properties: `passive'. We showed that the KR built with the `ETG' sample is fully consistent with the ones obtained with the `elliptical' and `passive' samples. On the other hand, the KR slope built with the `red' sample is only marginally consistent with those obtained with the other samples. Therefore, in this work, we analyse the results solely using the sample of ETGs selected via S\'ersic indices.
\section{Structural parameters estimate with \textsc{morphofit}}
\label{section:struct_param_estimate}
In order to build the KR, we need to estimate the structural parameters of spectroscopically confirmed cluster members. We do this using the \textsc{python} package \textsc{morphofit}\footnote{https://pypi.org/project/morphofit/} \citep{Tortorelli2023}. The package uses \textsc{SExtractor} \citep{Bertin1996} and \textsc{GALFIT} \citep{Peng2011} to automatically fit the surface brightness profile of a set of user-defined galaxies. The code is highly parallelisable, making it suitable for modern wide-field photometric surveys. A complete description of the software features can be found in \citealt{Tortorelli2023}.
The methodology for the structural parameters estimate is a refined version of the one highlighted in \citealt{Tortorelli2018}. To deal with the cluster-crowded environment and the intracluster light contribution, we adopt an iterative approach that analyses images of increasing size (from stamps to the full images) using different point-spread-function (PSF) images, background estimation methods and sigma image creation.
The analysis with \textsc{morphofit} starts by running \textsc{SExtractor} in forced photometry mode on the drizzled images of the three clusters in all seven wavebands. The structural parameters estimated with \textsc{SExtractor} constitute the initial values of the surface brightness profile fits with \textsc{GALFIT}. Since \textsc{GALFIT} requires a PSF image for the light profile convolution, and it has been proven that different PSFs may lead to different structural parameters estimate \citep{Vanzella2019}, we build four different PSF images with four different methods for each cluster and each waveband (see Fig. 2 in \citealt{Tortorelli2023}) to average out systematic effects arising from a specific model PSF. We use the \textsc{SExtractor} catalogue to select stars on the images based on their loci on the magnitude-size (MAG\_AUTO vs FLUX\_RADIUS \textsc{SExtractor} parameters) and magnitude-maximum surface brightness (MAG\_AUTO vs MU\_MAX \textsc{SExtractor} parameters) planes and then cut stamps of $50$ pixels size around them. The sample of stars is further refined based on a signal-to-noise ($100 \le S/N \le 800$) and isolability criterion (no detected sources around).
\begin{figure*}[ht!]
\centering
\includegraphics[width=9cm]{./figures/tortorelli_figure1a.pdf}
\includegraphics[width=9cm]{./figures/tortorelli_figure1b.pdf}
\caption{This figure shows the best-fitting KR slope $\beta$ (left panel) and intercept $\alpha$ (right panel) evolution as a function of wavelength for AS1063, M0416 and M1149 (blue, green and red points, respectively). The scatter points refer to the best-fitting KR slopes and intercepts obtained from the linear regression on the galaxy structural parameters measured with \textsc{morphofit}. The dashed lines in the left panel represent the best-fitting linear relations that describe the trend of increasing slope values as a function of wavelengths for the three clusters. The coloured bands group the scatter points belonging to the same waveband, which are displaced for plot clarity. The left plot legend shows the slope values and their $1\sigma$ uncertainties.}
\label{KR_slopes_intercepts_wavelength}
\end{figure*}
\textsc{morphofit} then cuts stamps around the spectroscopically confirmed cluster members and fits their light profiles and those of the neighbouring objects in the stamps with \textsc{GALFIT}. We fit single S\'ersic profiles with initial values of the positions, magnitudes, half-light radii, axis ratios, and position angles given by the \textsc{SExtractor} parameters XWIN\_IMAGE, YWIN\_IMAGE, MAG\_AUTO, FLUX\_RADIUS, BWIN\_IMAGE/AWIN\_IMAGE, and THETAWIN\_IMAGE, respectively. The initial values of the S\'ersic indices are kept fixed at $2.5$. The fit is performed on each member for all the seven wavebands with different combinations of four kinds of PSF images, two background estimate methods and two different ways of providing \textsc{GALFIT} with sigma images (see Sect. 2 and 4 in \citealt{Tortorelli2023} for more details). The final galaxy structural parameters in the seven wavebands are an error-weighted mean of the parameters estimated with the different combinations.
These structural parameters are, in turn, used as initial values for the light profile fit of galaxies in increasing-size images, first on image regions and then on the full images. We use the same combination of PSF images, background and sigma images as the fit on stamps. The final structural parameters we use to build the KR at different wavelengths are a weighted mean of those obtained with the different combinations in the seven wavebands on the full images. The structural parameters for the three clusters are provided as supplementary material to this letter (Appendix \ref{appendix:cat_description}).
\section{The Kormendy relation as a function of wavelength}
\label{section:kormendy_wave}
We use the structural parameters estimated in section \ref{section:struct_param_estimate} to select the sample of ETGs we use to build the KR. Following \citealt{Tortorelli2018}, we fit galaxies with single S\'ersic profiles and we select ETGs as those having S\'ersic indices in the \textit{F814W} waveband, $\mathrm{n_{\textit{F814W}}} \ge 2.5$. Additionally, we limit our analysis to galaxies brighter than the completeness limit, $m_{\mathrm{\textit{F814w}}} \le 22.5\ \mathrm{ABmag}$, and we do not consider BCGs in the fit. The number of galaxies satisfying this criterion is 42, 50, and 35 for AS1063, M0416 and M1149, respectively.
We perform the linear regression analysis of the KR, $\left \langle \mu \right \rangle_{\mathrm{e}} = \alpha + \beta \log{\mathrm{R}_{\mathrm{e}}}$, as in \citealt{Tortorelli2018}, using the Bivariate Correlated Errors and intrinsic Scatter estimator (BCES, \citealt{Akritas1996}) method. $\left \langle \mu \right \rangle_{\mathrm{e}}$ and $R_{\mathrm{e}}$ are in units of $\mathrm{mag\ arcsec^{-2}}$ and $\mathrm{kpc}$, respectively, and the surface brightness is corrected for the cosmological dimming effect. We report in Table \ref{table:KR_parameters} the slopes, intercepts, scatters, and their $1\sigma$ uncertainties of the best-fitting KRs built with the ETG samples for the three clusters in the seven wavebands.
Figure \ref{KR_slopes_intercepts_wavelength} shows the slope and intercept evolution with wavelength for AS1063, M0416 and M1149. We fit a linear relation to the slopes as a function of wavelength. We find the three clusters' slopes evolve with wavelength, smoothly increasing their values from the observed \textit{B}-band to the observed \textit{H}-band. This result extends the trend of KR slope increase with wavelength already found at low redshift by \citealt{LaBarbera2010} to intermediate redshift. In our work, the NIR slopes have higher values and are only marginally consistent within the errors with the values quoted by \citealt{LaBarbera2010}, despite our errors being larger due to the much smaller number of objects.
A steeper KR implies that the difference in the surface brightness between small and large ETGs is larger than that obtained with a shallower KR. This physically implies that smaller ETGs are more centrally concentrated than larger ETGs in the NIR regime with respect to the optical one. Since different wavebands probe different stellar populations in the galaxy, the smooth increase in slope from optical to NIR also implies that smaller-size ETGs have stronger internal stellar population gradients than galaxies with larger effective radii. Additionally, this result points to the fact that the analysis of the KR at different redshifts should be conducted in the same rest-frame bands. Otherwise, the wavelength evolution may impact our conclusions from the KR analysis.
The trend of the intercepts shows an expected behaviour as a function of wavelength. The intercepts of the three clusters get brighter at larger wavelengths since the sample of ETGs is expected to be constituted by a population of galaxies with passive spectral energy distributions. The trend as a function of redshift is also consistent with what we already found in \citealt{Tortorelli2018}, meaning that the intercepts get fainter at lower redshift due to the passivisation of the stellar population. The scatter points of AS1063 and M0416 are almost consistent at all wavelengths, given their smaller difference in redshifts with respect to M1149.
\section{Conclusions}
\label{section:conclusions}
In this letter, we investigate how the KR parameters change as a function of the wavelength range probed. We perform this analysis using spectroscopically confirmed cluster members of the three FF clusters, Abell S1063 ($z=0.348$), MACS J0416.1-240($z=0.396$) and MACS J1149.5+2223 ($z=0.542$) in seven photometric wavebands from the observed \textit{B}-band to the observed \textit{H}-band.
The galaxy structural parameters for the KR are measured using the \textsc{python} package \textsc{morphofit}, following a refined version of the methodology of increasing image size already adopted in \citealt{Tortorelli2018}.
We measure the galaxy structural parameters for the KR using the \textsc{python} package \textsc{morphofit} \citep{Tortorelli2023}, following the methodology of increasing image size already adopted in \citealt{Tortorelli2018}. We use the structural parameters to select ETGs as those having S\'ersic indices, $\mathrm{n_{\textit{F814W}}} \ge 2.5$, and magnitude brighter than the completeness limit, $m_{\mathrm{\textit{F814w}}} \le 22.5\ \mathrm{ABmag}$.
We build the KR across the whole range of available wavelengths, and we find that the KR intercepts follow an expected trend in becoming fainter at lower redshift due to the passivisation of the ETG stellar populations. We also find that the slopes of the KRs smoothly increase with wavelength for all three clusters.
This result extends the conclusions already found by \citealt{LaBarbera2010} at low-redshift with SDSS to intermediate redshifts. The slope increase with wavelength implies that smaller-size ETGs are more centrally concentrated in the near-infrared than larger-size ETGs with respect to the optical regime. Since different wavelengths probe different stellar populations, this also implies that smaller-size ETGs have stronger internal stellar population gradients with respect to larger-size ETGs. The investigation of the slope change with wavelength suggests that studies addressing the KR evolution should be conducted at similar rest-frame wavebands at different redshifts for a robust comparison.
\begin{acknowledgements}
We acknowledge financial contributions by PRIN-MIUR 2017WSCC32 "Zooming into dark matter and proto-galaxies with massive lensing clusters" (P.I.: P.Rosati), INAF ``main-stream'' 1.05.01.86.20: "Deep and wide view of galaxy clusters (P.I.: M. Nonino)" and INAF ``main-stream'' 1.05.01.86.31 "The deepest view of high-redshift galaxies and globular cluster precursors in the early Universe" (P.I.: E. Vanzella). The CLASH Multi-Cycle Treasury Program is based on observations made with the NASA/ESA Hubble Space Telescope. The Space Telescope Science Institute is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. Based on observations made with the European Southern Observatory Very Large Telescope (ESO/VLT) at Cerro Paranal, under programme IDs 60.A-9345(A), 095.A-0653(A), 294.A-5032(A) and 186.A-0798(A).
\end{acknowledgements}
\bibliographystyle{aa}
|
2,869,038,154,742 | arxiv | \section*{Introduction}
\vspace{-2mm}
The systematic study of $A$-hypergeometric $D$-modules, also known as GKZ-systems, was initiated by Gelfand, Graev, Kapranov, and Zelevinski \cite{GGZ}, \cite{GKZ}. These are systems of linear partial differential equations in several complex variables that generalize classical hypergeometric equations.
They are determined by a matrix $A=(a_1 \cdots a_n)=(a_{i,j})$ with columns $a_k\in{\mathbb Z}^d$ and a parameter vector $\beta\in{\mathbb C}^d$.
Let $x_1,x_2,\dots,x_n$ be coordinates on ${\mathbb C}^n$, with corresponding partial derivatives $\partial_1,\partial_2,\dots,\partial_n$, so that the Weyl algebra $D$ on ${\mathbb C}^n$ is generated by $x_1,\dots,x_n,\partial_1,\dots,\partial_n$.
Let
\[
I_A \defeq \<\partial^u-\partial^v\mid u,v\in{\mathbb N}^n, Au=Av\> \subseteq {\mathbb C}[\partial_1,\ldots, \partial_n]
\]
denote the \emph{toric ideal} of $A$.
Denote by $E_i\defeq\sum_{j=1}^n a_{i,j} x_j \partial_j$ the $i$th \emph{Euler operator} of $A$. The \emph{$A$-hypergeometric $D$-module} with parameter $\beta\in{\mathbb C}^d$ is the left $D$-module
\[
M_A(\beta)\defeq D/D\cdot\< I_A, E_1-\beta_1,\ldots, E_d-\beta_d\>.
\]
For any choice of $A$ and $\beta$, the module $M_A(\beta)$ is \emph{holonomic} \cite{GGZ,adolphson}. When $\beta\in{\mathbb C}^d$ is generic, the dimension of the space of germs of holomorphic solutions of $M_A(\beta)$ at a nonsingular point, also known as its \emph{(holonomic) rank}, is equal to the \emph{normalized volume} ${\rm \operatorname{vol}}(A)$ of the matrix $A$, see \eqref{eqn:normalized-volume} \cite{GKZ, adolphson}. In general, this is only a lower bound; see \cite{SST} for the case when $I_A$ is homogeneous and \cite{MMW} for the general case.
The set
\[
{\mathcal E}(A)\defeq\{\beta\in{\mathbb C}^d \mid \; {\operatorname{rank}} (M_A (\beta))>{\rm \operatorname{vol}}(A)\}
\]
is called the \emph{exceptional arrangement} of $A$, which is an affine subspace arrangement of codimension at least two that is closely related to the local cohomology modules of the toric ring ${\mathbb C}[\partial]/I_A$ \cite{MMW}.
A parameter $\beta \in {\mathcal E}(A)$ is called a \emph{rank jumping} parameter.
There is a combinatorial formula to compute the rank of $M_A(\beta)$ in terms of the \emph{ranking lattices} ${\mathbb E}^\beta$ of $A$ at $\beta$ \cite{berkesch}, with previous results in
\cite{CDD} with $d=2$ and in
\cite{okuyama} when $d=3$ or $\beta$ is \emph{simple} (see also Section \ref{sec:Simple-rank-jumping}).
Unfortunately, the presence of alternating signs in this formula do not yield a strong upper bound for the rank of $M_A(\beta)$; however, if $\beta$ is simple, it quickly follows that the rank of $M_A(\beta)$ is at most $(d-1){\rm \operatorname{vol}}(A)$, see Corollary \ref{cor:upper-bound-F-simple-rank}.
We show that this bound is tight by constructing a sequence of examples for which the ratio ${\operatorname{rank}}(M_A(\beta))/{\rm \operatorname{vol}}(A)$ tends to $d-1$, see Theorem \ref{thm:family-examples}.
In addition, we prove that the equality cannot hold for any example with simple parameter $\beta$ and that our examples are minimal in certain sense, see Remark \ref{rem:remark-minimal-volume}. Another interesting feature of these examples is that ${\mathcal E}(A)$ contains all the lattice points in the convex hull of the columns of $A$ and the origin.
On the other hand, there are other known upper bounds for the holonomic rank of $M_A(\beta)$. In particular,
\[
{\operatorname{rank}} (M_A (\beta))\leq \begin{cases}
4^d \cdot {\rm \operatorname{vol}}(A) & \text{if $I_A$ is homogeneous~\cite{SST},}\\
4^{d+1}\cdot {\rm \operatorname{vol}}(A) & \text{otherwise~\cite{BFM-parametric}.}
\end{cases}
\]
It was shown in \cite{Fer-exp-growth} that these upper bounds are qualitatively effective, i.e., there is some $a>1$ such that for any $d\geq 3$, there is a $(d\times n$)-matrix $A_d$ and a parameter $\beta_d\in {\mathbb C}^d$ such that
\[
{\operatorname{rank}}(M_{A_{d}}(\beta_d))\geq a^d {\rm \operatorname{vol}}(A_d).
\]
However, the maximum possible value of $\sqrt[d]{{\operatorname{rank}}(M_A(\beta))/{\rm \operatorname{vol}}(A)}$ that has, up until now, appeared in the literature is $\sqrt[3]{7/5}\approx 1.1187$, see \cite[Example 2.6]{Fer-exp-growth}, which was first considered in \cite{MW}. The supremum of the value of $\sqrt[d]{{\operatorname{rank}}(M_A(\beta))/{\rm \operatorname{vol}}(A)}$ over the examples in the current note is $\sqrt[5]{4}\approx 1.3195$, i.e., $\sqrt[d]{d-1}$ for $d=5$. It is still an open problem to find the supremum of the set of values of $\sqrt[d]{{\operatorname{rank}}(M_A(\beta))/{\rm \operatorname{vol}}(A)}$ for variation among the set of full rank $(d\times n)$-matrices $A$ and $\beta\in {\mathbb C}^d$, for $d\geq 3$ and $n\geq d+2$.
\subsection*{Acknowledgements}
We are grateful to Laura Felicia Matusevich and Uli Walther for helpful discussions over the years on bounding the rank of an $A$-hypergeometric system.
\section{Lower bounds for the normalized volume}
\label{sec:lowerBounds}
Fix a $(d\times n)$-integer matrix $A=(a_1 \cdots a_n)$, where $a_i\in{\mathbb Z}^d$ denotes the $i$th column of $A$.
With the convention that $0\in{\mathbb N}$,
assume that ${\mathbb Z} A\defeq \sum_{j=1}^n {\mathbb Z} a_j ={\mathbb Z}^d$ and that the affine semigroup ${\mathbb N} A\defeq\sum_{j=1}^n {\mathbb N} a_j$ is positive, meaning that ${\mathbb N} A\cap(-{\mathbb N} A)=\{\bf{0}\}$.
We also assume for simplicity that all the columns of $A$ are distinct from each other and the origin.
Identify $A$ with its set of columns, and for any subset $F$ of $A$, denote by $\Delta_{F}$ the convex hull in ${\mathbb R}^d$ of the origin and $F$.
We also identity $F$ with its index set $\{j\mid a_j \in F\}$.
Given a lattice $\Lambda$ such that $F\subseteq \Lambda\subseteq {\mathbb Q} F\cap{\mathbb Z}^d$,
the \emph{normalized volume} of $F$ in $\Lambda$ is the integer
\begin{equation}\label{eqn:normalized-volume}
{\rm \operatorname{vol}}_{\Lambda}(F)
= \dim({\mathbb R} F)! \cdot
\dfrac{{\rm \operatorname{vol}}_{{\mathbb R} F}(\Delta_F)}{[ {\mathbb Z}^d\cap {\mathbb Q} F : \Lambda]},
\end{equation}
where ${\rm \operatorname{vol}}_{{\mathbb R} F}(\cdot)$ denotes Euclidean volume in ${\mathbb R} F$. We write ${\rm \operatorname{vol}}(A)$ for ${\rm \operatorname{vol}}_{{\mathbb Z} A}(A) = {\rm \operatorname{vol}}_{{\mathbb Z}^d}(A)$.
A subset $F$ of the columns of the matrix $A$ is a \emph{face} of $A$, denoted $F\preceq A$, if ${\mathbb R}_{\geq0}F$ is a face of the cone ${\mathbb R}_{\geq0}A\defeq\sum_{j=1}^n{\mathbb R}_{\geq0} a_j$ and $F= A\cap{\mathbb R} F$.
The codimension of a nonempty face $F$ of $A$ is ${\rm codim}(F)\defeq d-\dim ({\mathbb R} F)$, with the convention that ${\rm codim} (\varnothing)=d$.
\begin{lemma}\label{lem:adding-a-point}
If $\tau$ is a proper subset of $A$ with $\Delta_\tau \cap A=\tau$, then there exists a column $a$ of $A\setminus \tau$ such that $\Delta_{\tau\cup\{a\}}\cap A=\tau\cup\{a\}$. Moreover, if $\Delta_{\tau}$ is not full dimensional, then $a$ may be chosen so that $\dim (\Delta_{\tau\cup\{a\}})=\dim (\Delta_\tau)+1$.
\end{lemma}
\begin{proof}
If $\Delta_\tau\subseteq {\mathbb R}^d$ is full dimemsional, then choose any column $a\in A\setminus \tau$.
Since $a\notin \Delta_\tau$, the vector $a$ is a vertex of $\Delta_{\tau \cup\{a\}}$, and the rest of the vertices of $\Delta_{\tau \cup\{a\}}$ are vertices of $\Delta_\tau$.
In particular, if there exists a vector $a'\in (\Delta_{\tau\cup\{a\}}\cap A)\setminus(\tau\cup\{a\})$, then $a'$ is not a vertex of $\Delta_{\tau\cup\{a\}}$ and $\Delta_\tau\subsetneq\Delta_{\tau\cup\{a'\}} \subsetneq \Delta_{\tau\cup\{a\}}$.
Thus, $a$ can be replaced by $a'$. Also, notice that
\[
(\Delta_{\tau\cup\{a'\}}\cap A)\setminus(\tau\cup\{a'\})\subsetneq(\Delta_{\tau\cup\{a\}}\cap A)\setminus(\tau\cup\{a\}).
\]
We can thus repeat this process of replacement of $a$ until the equality $\Delta_{\tau\cup\{a''\}}\cap A=\tau\cup\{a''\}$ holds for some $a''$ in $A\setminus\tau$.
On the other hand, if $\Delta_\tau\subseteq {\mathbb R}^d$ is not full dimensional,
let $a\in A\setminus \tau$ be such that $\dim (\Delta_{\tau\cup\{a\}})=\dim (\Delta_\tau)+1$. Such a choice of $a$ exists because the rank of $A$ is $d$.
Since $\Delta_\tau$ is a facet of $\Delta_{\tau \cup\{a\}}$ and $\Delta_\tau \cap A=\tau$, no point in $(\Delta_{\tau \cup\{a\}}\cap A)\setminus \tau$ is in ${\mathbb R} \tau$, and the result follows.
\end{proof}
\begin{lemma}\label{lem:volume-lower-bound}
If $F\preceq A$ is a face of $A$, then
\begin{equation}\label{eqn:volume-F-ineq}
{\rm \operatorname{vol}}(A)\geq {\rm \operatorname{vol}}_{{\mathbb Z}^d\cap {\mathbb Q} F}(F)+n-|F| -{\rm codim} (F),
\end{equation} where $|F|$ is the cardinality of $F$. In particular, ${\rm \operatorname{vol}}(A)\geq n-d +1$.
\end{lemma}
\begin{proof}
Since $F\preceq A$ is a face of $A$, $\Delta_F\cap A=F$.
By Lemma~\ref{lem:adding-a-point}, there is a set $\sigma$ of ${\rm codim}(F)$ linearly independent columns of $A\setminus F$ such that $\Delta_{F\cup\sigma}$ is full dimensional and $\Delta_{F\cup\sigma}\cap A=F\cup\sigma$.
The normalized volume in the lattice ${\mathbb Z}^d$ of $F\cup\sigma$ is at least ${\rm \operatorname{vol}}_{{\mathbb Z}^d\cap {\mathbb Q} F}(F)$.
Again by Lemma~\ref{lem:adding-a-point}, there is a column $a$ of $A\setminus (F\cup \sigma)$ such that no other column of $A\setminus (F\cup \sigma)$ lies in
$\Delta_{F\cup\sigma\cup\{a\}}$, the convex hull of the ${\rm codim}(F)+|F|+1$ points of $F\cup \sigma\cup\{a\}$ and the origin.
In fact, $n-({\rm codim}(F)+|F|+1)$ more columns of $A\setminus(F\cup\sigma)$ can be iteratively found in this way. Notice that each time a new point is added to $F\cup\sigma$ using Lemma~\ref{lem:adding-a-point}, the normalized volume of the convex hull of the new set is increased at least by one. This proves the first statement. The second statement follows from the first one by taking $F=\varnothing$.
\end{proof}
Notice that for any face $F\preceq A$, $n-|F| -{\rm codim} (F)\geq 0$, and equality holds if and only if $A$ is a \emph{pyramid} over $F$, so ${\mathbb Z}^d= {\mathbb Z} F \oplus \left(\bigoplus_{j\notin F} {\mathbb Z} a_j\right)$.
Further, if $A$ is a pyramid over $F$, then equality holds in \eqref{eqn:volume-F-ineq} because ${\mathbb Z}^d\cap {\mathbb Q} F={\mathbb Z} F$ and ${\rm \operatorname{vol}}(A)={\rm \operatorname{vol}}_{{\mathbb Z} F}(F)$, see \cite[Lemma 3.5]{reducibility}.
The converse is not true; a counterexample is provided in Remark \ref{rem:remark-minimal-volume}.
On the other hand, if equality holds in \ref{eqn:volume-F-ineq}, then all the lattice points in $\Delta_A\setminus \Delta_F$ are columns of $A$.
Denote the toric ring associated to $A$ by $S_A\defeq{\mathbb C}[\partial]/I_A\cong {\mathbb C} [{\mathbb N} A]$.
\begin{prop}\label{prop:volume-normal}
If ${\rm \operatorname{vol}}(A)= n-d+1$, then $S_A$ is normal.
\end{prop}
\begin{proof}
Let $H$ be an affine hyperplane such that all the columns of $A$ not in $\tau\defeq H\cap A$ belong to the open half space determined by $H$ not containing the origin; such a hyperplane exists because ${\mathbb N} A$ is positive and $\bf{0}$ is not a column of $A$.
For any simplex $\sigma\subseteq \tau$ such that $\Delta_{\sigma}\cap A=\sigma$, ${\rm \operatorname{vol}}_{{\mathbb Z}^d}(\sigma)=1$ because otherwise, by adding a point of $A\setminus\sigma$ using Lemma~\ref{lem:adding-a-point} and taking the convex hull iteratively, the volume would increase by at least one in each step and the normalized volume of $A$ would be larger that $n-d+1$.
Now, since ${\rm \operatorname{vol}}_{{\mathbb Z}^d}(\sigma)=1$, $\sigma$ forms a basis in the lattice ${\mathbb Z}^d$ and ${\mathbb N} \sigma ={\mathbb Z}^d \cap {\mathbb R}_{\geq 0} \sigma$. Since ${\mathbb R}_{\geq 0}A$ equals the union of the cones ${\mathbb R}_{\geq 0}\sigma$ for simplices $\sigma\subseteq \tau$ with $\tau$ as above, it follows that ${\mathbb R}_{\geq 0}A\cap {\mathbb Z}^d ={\mathbb N} A$, and hence, $S_A$ is normal.
\end{proof}
\begin{cor}\label{cor:nonCM-inequality}
If $S_A$ is not Cohen--Macaulay, then $d\geq 2$, $n\geq d+2$, and ${\rm \operatorname{vol}}(A)\geq n-d+2$.
\end{cor}
\begin{proof}
If either $d=1$ or $n-d=1$, then under our hypotheses on $A$, $S_A$ is Cohen--Macaulay. On the other hand, if ${\rm \operatorname{vol}}(A)< n-d+2$, then $S_A$ is normal by Lemma~\ref{lem:volume-lower-bound} and Proposition~\ref{prop:volume-normal}, which implies that $S_A$ is Cohen--Macaulay by~\cite[Theorem 1]{Hochster}.
\end{proof}
The inequality in Corollary~\ref{cor:nonCM-inequality} is sharp; for any $d\geq 2$ and $n\geq d+2$, there is a pointed matrix $A$ as above with ${\rm \operatorname{vol}}(A)=n-d+2$ such that $S_A$ is not Cohen--Macaulay.
To see this, notice first that for $d=2$ and $n=d+2=4$, the matrix
\[
A=\left(\begin{array}{cccc}
1 & 1 & 0 & 0\\
0 & 1 & 2 & 3\end{array}\right)
\]
satisfies that ${\rm \operatorname{vol}}(A)=4$ and $S_A$ is not Cohen--Macaulay. On the other hand, in order to produce examples with $n\geq 5$, it is enough to modify this example by adding the columns $(0,k)^t$ for $k=4, \ldots, n-1$, and this operation keeps $S_A$ invariant up to isomorphism.
To construct more examples with the same value of $n-d$ but larger $d$, it is enough to consider a pyramid over the previous example. This alters $S_A$ by tensoring over ${\mathbb C}$ with a polynomial ring in a number of variables equal to the increment of $d$.
\section{Rank versus volume in the simple case}
\label{sec:Simple-rank-jumping}
In this section, we recall some notations and results from \cite{berkesch}.
For a face $F\preceq A$, consider the union of the lattice translates
\begin{align*}
{\mathbb E}_F^\beta\defeq
\big[{\mathbb Z}^d\cap(\beta+{\mathbb C} F) \big]\smallsetminus({\mathbb N} A+{\mathbb Z} F) = \bigsqcup_{b\in B_F^\beta} (b+{\mathbb Z} F),
\end{align*}
where $B_F^\beta \subseteq {\mathbb Z}^d$ is a set of lattice translate representatives.
As such, $|B^\beta_{F}|$ is the number of translates of ${\mathbb Z} F$ appearing in ${\mathbb E}_F^\beta$, which is by definition equal to the difference between $[{\mathbb Z}^d\cap {\mathbb Q} F:{\mathbb Z} F]$ and the number of translates of ${\mathbb Z} F$ along $\beta+{\mathbb C} F$ that are contained in ${\mathbb N} A + {\mathbb Z} F$.
Given the set
$\mathcal{J}(\beta)\defeq\{(F,b)\mid F\preceq A,\, b\in B_F^\beta,\, {\mathbb E}_F^\beta\neq\varnothing\}$,
the \emph{ranking lattices} of $A$ at $\beta$ are defined to be
\begin{align*}
{\mathbb E}^\beta
\defeq \bigcup_{(F,b)\in {\mathcal J}(\beta)} (b+{\mathbb Z} F).
\end{align*}
Note that the ranking lattices of $A$ at $\beta$ is precisely the union of those sets $(b+{\mathbb Z} F)$ contained in ${\mathbb Z}^d \setminus {\mathbb N} A$ such that $\beta\in (b+{\mathbb C} F)$. This is closely related to the set of holes of the affine semigroup ${\mathbb N} A$, namely the set $({\mathbb Z}^d \cap {\mathbb R}_{\geq 0}A)\setminus {\mathbb N} A$.
The main result in \cite{berkesch} states that the rank of $M_A(\beta)$ can be computed from the combinatorics of ${\mathbb E}^\beta$ and $\Delta_A$. An explicit formula for the rank is given when the rank jumping parameter $\beta$ is \emph{simple} (for a face $G\preceq A$), meaning that the set of maximal pairs $(F,b)$ in ${\mathcal J}(\beta)$ with respect to inclusion on $b+{\mathbb Z} F$ all correspond to a unique face $G\preceq A$ (see also \cite{okuyama} for this particular case); in this case,
\begin{equation}\label{eqn:formula-simple-rank-jump}
{\operatorname{rank}} (M_A (\beta))={\rm \operatorname{vol}}(A)+ |B_G^\beta|\cdot ({\rm codim}(G)-1)\cdot {\rm \operatorname{vol}}_{{\mathbb Z} G}(G).
\end{equation}
We now provide some consequences of this result.
\begin{cor}\label{cor:upper-bound-F-simple-rank}
If $d\geq3$ and $\beta\in{\mathbb C}^d$ is simple for the face $F\preceq A$, then
\[
{\operatorname{rank}} (M_A (\beta))\leq {\rm codim}(F)\cdot {\rm \operatorname{vol}}(A).
\]
In particular, if $\beta\in{\mathcal E}(A)$ is simple, then
\[
{\operatorname{rank}} (M_A (\beta))\leq (d-1)\cdot {\rm \operatorname{vol}}(A).
\]
\end{cor}
\begin{proof}
The first statement follows from~\eqref{eqn:formula-simple-rank-jump} and the definition of normalized volume in \eqref{eqn:normalized-volume}. Indeed,
\begin{equation}\label{eqn:inequalities}
|B_F^\beta|\cdot {\rm \operatorname{vol}}_{{\mathbb Z} F}(F)\leq [{\mathbb Z}^d \cap {\mathbb Q} F: {\mathbb Z} F]\cdot {\rm \operatorname{vol}}_{{\mathbb Z} F}(F)
={\rm \operatorname{vol}}_{{\mathbb Q} F\cap {\mathbb Z}^d}(F)\leq {\rm \operatorname{vol}}(A).
\end{equation}
We can assume without loss of generality that ${\rm \operatorname{vol}}(A)\geq 2$, since otherwise $A$ is a simplex and ${\mathcal E}(A)=\varnothing$.
For the second statement, notice first that if ${\rm codim}(F)=d$, then
${\rm \operatorname{vol}}_{{\mathbb Z} F}(F)=1=|B_F^\beta|$ and
\begin{equation}
{\operatorname{rank}} (M_A(\beta))={\rm \operatorname{vol}}(A)+d-1 \leq (d-1)\cdot {\rm \operatorname{vol}}(A),\label{eqn:ineq2}
\end{equation}
since $d\geq 3$ and ${\rm \operatorname{vol}}(A)\geq 2$.
Thus, we can assume that ${\rm codim}(F)\leq (d-1)$ and the second upper bound follows from the first one.
\end{proof}
We can improve the bound in Corollary~\ref{cor:upper-bound-F-simple-rank} as follows.
\begin{cor}\label{cor:sharper-upper-bound-F-simple-rank}
If $d\geq3$ and $\beta\in{\mathbb C}^d$ is simple for the face $F\preceq A$, then
\begin{equation}\label{eqn:sharper-inequality}
{\operatorname{rank}} (M_A (\beta))\leq {\rm codim}(F)\cdot {\rm \operatorname{vol}}(A)-({\rm codim}(F)-1)(n-|F|-{\rm codim}(F)).
\end{equation}
In particular, if $\beta\in{\mathcal E}(A)$ is simple, then
\begin{equation}\label{eqn:strict-inequality-for-simple}
\frac{{\operatorname{rank}} (M_A (\beta))}{{\rm \operatorname{vol}}(A)} < (d-1).
\end{equation}
\end{cor}
\begin{proof}
The proof of~\eqref{eqn:sharper-inequality} follows from the first inequality in \eqref{eqn:inequalities} and \eqref{eqn:volume-F-ineq}. For~\eqref{eqn:strict-inequality-for-simple}, notice first that if ${\rm codim}(F)=1$, then ${\operatorname{rank}} (M_A(\beta))={\rm \operatorname{vol}}(A)$ by \eqref{eqn:formula-simple-rank-jump}; otherwise, $({\rm codim}(F)-1)\geq 1$.
By \cite[Corollary~9.2]{MMW}, ${\mathcal E}(A)= \varnothing$ is equivalent to $S_A$ being Cohen--Macaulay.
Thus, for the case when ${\rm codim}(F)=d$, it is enough to use that the inequality in \eqref{eqn:ineq2} is in fact strict, because ${\rm \operatorname{vol}}(A)\geq 4$ by Corollary \ref{cor:nonCM-inequality}.
For the remaining cases, it is now enough to see that $n-|F|-{\rm codim}(F)\geq 1$.
By way of contradiction, assume that $A$ is a pyramid over $F$, so that any $\beta\in{\mathbb C}^d$ can be written uniquely as $\beta=\beta_F +\beta_{\overline{F}}$ with $\beta_F \in {\mathbb C} F$, $\beta_{\overline{F}}\in {\mathbb C} \overline{F}$ and ${\operatorname{rank}} (M_A(\beta))= {\operatorname{rank}}( M_F(\beta_F))$, see \cite[Lemma 3.7]{reducibility}.
Since $\beta\in{\mathcal E}(A)$, it follows that $\beta_F\in{\mathcal E}(F)$.
Notice also that if $F\preceq G\preceq A$, then ${\mathbb E}_G^\beta={\mathbb E}_G^{\beta'}$ for any $\beta'\in \beta+{\mathbb C} F$.
Since $\beta$ is simple for $F$, the generic vectors $\beta' \in \beta + {\mathbb C} F$ are also simple for $F$ and ${\operatorname{rank}}(M_A(\beta'))={\operatorname{rank}} (M_A(\beta))$.
Thus,
\[
{\operatorname{rank}} (M_F(\beta'_F)) ={\operatorname{rank}} (M_F(\beta_F))>{\rm \operatorname{vol}}_{{\mathbb Z} F}(F).
\]
It follows that for generic $\gamma \in {\mathbb C} F$, ${\operatorname{rank}} (M_F(\gamma))>{\rm \operatorname{vol}}_{{\mathbb Z} F}(F)$, which is a contradiction, as this should be equality by~\cite{adolphson}.
\end{proof}
\begin{theorem}
The set
\[
{\mathcal E}_2(A)\defeq\{\beta \in {\mathbb C}^d \mid \; {\operatorname{rank}} (M_A (\beta))\geq 2 \cdot {\rm \operatorname{vol}}(A)\}
\]
is an affine subspace arrangement of codimension at least three.
\end{theorem}
\begin{proof}
The exceptional arrangement ${\mathcal E}(A)$ is known to be a finite union of translates of linear subspaces ${\mathbb C} G$ for faces $G\preceq A$ of codimension at least two \cite[Corollary 9.4 and Porism 9.5]{MMW}. Moreover, it is shown in \cite[Theorem~2.6]{MMW} that rank of $M_A(\beta)$ is upper-semicontinuous as a function of $\beta$ with respect to the Zariski topology. Thus, on each irreducible component $C$ of ${\mathcal E}(A)$ the rank of $M_A(\beta)$ is constant outside a Zariski closed subset of $C$ of codimension at least three. Moreover, this codimension three set is also an affine subspace arrangement; see the argument after Definition 4.7 in \cite{berkesch}. It is thus enough to find, for any codimension two component $C$, a set of parameters $\beta\in C$ such that ${\operatorname{rank}} (M_A(\beta))<2 \cdot {\rm \operatorname{vol}}(A)$ and whose Zariski closure is $C$. Indeed, if $C$ has codimension two, we have that $C=b+ {\mathbb C} G$ for some face $G\preceq F$ of codimension two and some $b\in {\mathbb C}^d$.
Notice that for any proper face $G'\preceq A$ not containing $G$, the intersection $C\cap ({\mathbb Z}^d +{\mathbb C} G')$ is at most a countably and locally finite union of translates of the linear space ${\mathbb C} G\cap {\mathbb C} G'$ of codimension at least three. Since there are only finitely many such faces $G'$, the set of parameters $\beta \in C$ such that $\beta\notin ({\mathbb Z}^d +{\mathbb C} G')$ for any such $G'$ is not contained in any Zariski closed set of codimension three. For such $\beta$ and $G'$, it follows that ${\mathbb E}^\beta_{G'}=\varnothing$. Then, for $C$ as above and $\beta\in C$, the only possible faces involved in $\mathcal{J}(\beta)$ are $G$ and the two facets containing $G$. Thus, \cite[Section 5.3 and Example 6.21]{berkesch} yield the inequality
\begin{equation*}
{\operatorname{rank}} (M_A (\beta))\leq{\rm \operatorname{vol}}(A)+ |B_G^\beta| ({\rm codim}(G)-1) {\rm \operatorname{vol}}_{{\mathbb Z} G}(G),\end{equation*}
where equality holds if $\beta$ is $G$-simple. By the proof of Corollary~\ref{cor:sharper-upper-bound-F-simple-rank} applied to the codimension two face $G$, it follows that ${\operatorname{rank}} (M_A(\beta))<2\cdot {\rm \operatorname{vol}}(A)$ for a set of parameters $\beta\in C$ that is not contained in any codimension three Zariski closed set.
\end{proof}
\section{A sequence of examples in the simple case}
\label{sec:examples}
In this section, we prove that for any $d\geq 3$, the strict inequality~\eqref{eqn:strict-inequality-for-simple} from Corollary \ref{cor:sharper-upper-bound-F-simple-rank} is sharp for simple parameters $\beta$.
\begin{theorem}\label{thm:family-examples}
There is a sequence of full rank $(d\times (2d-1))$ integer matrices $\{A_{d,b}\}_{b=2}^\infty$ for which there is a simple parameter $\beta\in{\mathbb C}^d$ for which
\[
\lim_{b\to\infty}
\frac{{\operatorname{rank}} (M_{A_{d,b}}(\beta))}{{\rm \operatorname{vol}}(A_{d,b})}
= d-1.
\]
In fact, the set of simple parameters $\beta\in {\mathbb C}^d$ that maximize the ratio ${\operatorname{rank}} (M_{A_{d,b}}(\beta))/{\rm \operatorname{vol}}(A_{d,b})$ is a line through the origin.
\end{theorem}
Consider the following $(d\times (2d-1))$-matrix with $d\geq 3$:
\begin{equation}\label{eqn:Adb}
A_{d,b}
= (a_1 \; a_2\; \cdots a_{2d-1})
\defeq \left(\begin{array}{ccc}
I_{d-1} & I_{d-1} & \bf{0}_{d-1}\\
\bf{0}_{d-1}^t & \bf{1}_{d-1}^t & b
\end{array}\right),
\end{equation}
where $b\geq 2$ is an integer, $I_{d-1}$ denotes the identity matrix of rank $d-1$, $\bf{1}_{d-1}$ is the column vector consisting of $d-1$ entries of $1$, and $\bf{0}_{d-1}$ is the zero column vector of length $d-1$.
Note that ${\mathbb Z} A_{d,b}={\mathbb Z}^d$. We now compute the normalized volume of $A_{d,b}$ in this lattice.
To do this, for $j\in {\mathbb Z}$, set $h^{(j)}\defeq (0,\ldots,0,j)^t$.
\begin{lemma}\label{lem:lemma-volume}
The normalized volume of $A_{d,b}$ in \eqref{eqn:Adb} is $b+d-1$.
\end{lemma}
\begin{proof}
The polytope $\Delta_A$ can be decomposed as the union of two polytopes in ${\mathbb R}^d$ that intersect in a common facet.
One of these polytopes is the convex hull of the origin, the first $2(d-1)$ columns of $A_{d,b}$, and the lattice point $h^{(1)}$. This is a prism with height $1$ and base equal to a unit $(d-1)$-simplex, so its normalized volume in ${\mathbb Z}^d$ is $d$. The second polytope is the convex hull of $h^{(1)}$ and the last $d$ columns of $A_{d,b}$, which is a $d$-simplex. This $d$-simplex is the lattice translation by $h^{(1)}$ of the $d$-simplex that is the convex hull of the origin, the first $(d-1)$-columns of $A_{d,b}$, and $h^{(b-1)}$; therefore, its normalized volume in ${\mathbb Z}^d$ is $b-1$.
\end{proof}
\begin{remark}\label{rem:b-copies-of-F}
The last column of $A_{d,b}$ is $b\cdot e_d$, where $e_d$ is the $d$th standard basis vector in ${\mathbb C}^d$. The face $F_b\defeq \{a_{2d-1}\}\preceq A$ has normalized volume $1$ in the lattice ${\mathbb Z} F_b$ and
\[
{\mathbb Z}^d \cap {\mathbb C} F_b=\bigcup_{k=0}^{b-1} \left(h^{(k)}+{\mathbb Z} F_b\right)
\]
consists of $b$ translated copies of ${\mathbb Z} F_b$.
\end{remark}
\begin{remark}\label{rem:remark-minimal-volume}
The normalized volume of $F_b\defeq \{a_{2d-1}\}$ in the lattice ${\mathbb Z}^d\cap {\mathbb Q} F_b$ is $b$. In particular, equality holds in \eqref{eqn:volume-F-ineq} for $A=A_{d,b}$ and $F=F_b$.
\end{remark}
\begin{prop}\label{prop:exceptional}
The exceptional arrangement of $A_{d,b}$ is a finite union of lines parallel to ${\mathbb C} e_d$.
\end{prop}
\begin{proof}
The first $d-1$ columns of $A_{d,b}$ and its last column are linearly independent, and their nonnegative hull is precisely the first orthant ${\mathbb R}_{\geq 0}^d$. Thus, ${\mathbb R}_{\geq 0}A_{d,b}={\mathbb R}_{\geq 0}^d$ and ${\mathbb R}_{\geq 0}A_{d,b}\cap {\mathbb Z}^d={\mathbb N}^d$.
To determine the set of holes of ${\mathbb N} A$, given by ${\mathbb N}^d\setminus {\mathbb N} A$, can be written as a finite union of lattice translates of ${\mathbb N} F_b={\mathbb N} b e_d$,
notice first that the affine semigroup $S\subseteq {\mathbb N}^d$ generated by the first $2(d-1)$ columns of $A$ is normal, and their lattice span is ${\mathbb Z}^d$. Note also that, for $F_b\defeq \{a_{2d-1}\}$,
\[
{\mathbb N} A_{d,b}\cap {\mathbb C} F_b={\mathbb N} F_b={\mathbb N} b e_d,
\]
so ${\mathbb C} F_b \cap {\mathbb N} A ={\mathbb N} F_b={\mathbb N} b e_d$.
In order to complete the description of ${\mathbb N}^d\setminus {\mathbb N} A$, denote by $\Delta_b$ the simplex given by the convex hull of the following points in ${\mathbb N} A$:
\[
{\bf 0}, \,
b a_d=b (e_1+e_d), \,
b a_{d+1}=b (e_2+e_d),
\, \ldots, \,
b a_{2d-2}=b(e_{d-1}+e_d), \,
a_n=b e_d.
\]
Since ${\mathbb N} A= S+{\mathbb N} b e_d$, the set of holes of ${\mathbb N} A$ is the union of the sets $c+{\mathbb N} b e_d$, where $c$ runs through the lattice points:
\[
{\mathbb Z}^d\cap \Delta_b\setminus
\left(
{\mathbb R}_{\geq 0} (e_1 + e_d )
\,\cup\,
{\mathbb R}_{\geq 0} (e_2 + e_d )
\,\cup\, \cdots \,\cup\,
{\mathbb R}_{\geq 0} (e_{d-1}+e_d)
\,\cup\,
\left(b e_d + \sum_{k=1}^{d-1} {\mathbb R}_{\geq 0} e_k\right)
\right).
\]
It now follows from \cite{MMW,berkesch} that the exceptional arrangement of $A$ is
\[
{\mathcal E}(A)=\bigcup_{k=1}^{d-1} \bigcup_{m=0}^{b-2} (m e_k+{\mathbb C} F_b).
\qedhere
\]
\end{proof}
By the proof of Proposition~\ref{prop:exceptional},
if $b\geq 3$, then all the lattice points in $\Delta_{A_{d,b}}$ belong to ${\mathcal E} (A_{d,b})$.
\begin{lemma}\label{lem:rank-max}
If $F_b\defeq \{a_{2d-1}\}$, then the function $\beta \in {\mathbb C}^d \mapsto {\operatorname{rank}} (M_{A_{d,b}}(\beta))$ reaches its maximum exactly when $\beta \in {\mathbb C} F_b$, and this maximum value is $(d-1)b+1$.
\end{lemma}
\begin{proof}
We first show that if $\beta \in {\mathbb C} F_b$, then the rank of $M_{A_{d,b}}(\beta)$ is $(d-1)b+1$. In this case, the ranking lattices at $\beta$ are
\[
{\mathbb E}^{\beta} =
\bigcup_{j=1}^{b-1} \left(h^{(j)}+{\mathbb Z} F_b\right).
\]
Thus, by \cite{berkesch}, the rank jump at $\beta$ is equal to
\[
{\operatorname{rank}}(M_A(\beta) - {\rm \operatorname{vol}}(A) =
|B_{F_b}^{\beta}|\cdot {\rm \operatorname{vol}}_{{\mathbb Z} F_b}(F_b)\cdot ({\rm codim} (F_b)-1),
\]
where ${\rm codim} (F_b)=d-1$, ${\rm \operatorname{vol}}_{{\mathbb Z} F_b}(F_b)=1$, and $|B_{F_b}^{\beta}|=b-1$ by Remark \ref{rem:b-copies-of-F}, since
\[
{\mathbb Z} F_b\subseteq ({\mathbb N} A_{d,b} +{\mathbb Z} F_b)\cap (\beta+ {\mathbb C} F)\cap {\mathbb Z}^d.
\]
Thus,
${\operatorname{rank}} (M_{A_{d,b}} (\beta))={\rm \operatorname{vol}}_{{\mathbb Z}^d}(A_{d,b})+ (b-1)(d-1)$,
which gives the desired equality by Lemma \ref{lem:lemma-volume}.
In order to prove that this is the maximum value of ${\operatorname{rank}} (M_{A_{d,b}} (\beta))$, it is enough to observe that when $\beta$ lies in a component of the form $(m e_k+{\mathbb C} F_b)\subseteq {\mathcal E}(A)$ with $m\neq 0$, then the computation of the rank jump is analogous to the previous case, but the number $|B_{F_b}^{\beta}|$ will be smaller.
This is the case because
\[
me_k, \,
m e_k +e_d, \,
m e_k +2 e_d, \, \ldots, \,
m e_k+ m e_d\in {\mathbb N} A,
\]
and hence there are $(m+1)$-translated copies of ${\mathbb Z} F_b$ in ${\mathbb N} A\cap (m e_k+{\mathbb C} F_b)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:family-examples}]
The result now follows immediately from Lemmas \ref{lem:rank-max} and \ref{lem:lemma-volume}.
\end{proof}
\raggedbottom
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2,869,038,154,743 | arxiv | \section{Introduction}
\label{intro}
Kohn-Sham (KS) density functional theory (DFT) (see, e.g., \cite{Koh-RMP-99}) is a successful method for electronic structure calculations, thanks to its unique combination of low computational cost and reasonable accuracy. In the Kohn-Sham formalism, the total energy of a many-electron system in the external potential $\hat{V}_{ne}=\sum_i v_{ne}({\bf r}_i)$ is rewritten as a functional of the one-electron density $\rho({\bf r})$,
\begin{equation}
E[\rho]=T_s[\rho]+\int d{\bf r}\,v_{ne}({\bf r})\,\rho({\bf r})+U[\rho]+E_{\rm xc}[\rho].
\label{eq_Erho}
\end{equation}
In Eq.~(\ref{eq_Erho}), $T_s[\rho]$ is the kinetic energy of a non-interacting system of fermions (usually called KS system) having the same one-electron density $\rho$ of the physical, interacting, system. The Hartree energy $U[\rho]$ is the classical repulsion energy, $U[\rho]=\frac{1}{2}\int d{\bf r}\int d{\bf r}'\rho({\bf r})\rho({\bf r}')|{\bf r}-{\bf r}'|^{-1}$, and the exchange-correlation functional $E_{\rm xc}[\rho]$ must be approximated. Minimization of Eq.~(\ref{eq_Erho}) with respect to the spin-orbitals forming the KS determinant lead to the KS equations. Thus, instead of the physical problem, in KS DFT we solve the hamiltonian of a model system of non-interacting fermions, and we recover the energy of the physical system via an approximate functional.
Despite its success in scientific areas ranging from material science to biology, approximate KS DFT is far from being perfect, and many fundamental issues still need to be addressed. In particular, KS DFT encounters difficulties in handling near-degeneracy correlation effects (rearrangement of electrons within partially filled shells), and in taking into account long-range van der Waals interaction energies (crucial, e.g., for layered materials and biomolecules). In principle, all the shortcomings of KS DFT come from our lack of knowledge of the exchange-correlation functional, and a huge effort is put nowadays in trying to improve the approximations for $E_{\rm xc}[\rho]$ (for recent reviews see, e.g., \cite{Mat-SCI-02,PerRuzTaoStaScuCso-JCP-05}).
An alternative strategy to overcome the problems of DFT is range separation: the electron-electron interaction is split into a long-range and a short range part, and the two are treated at different levels of approximation
\cite{StoSav-INC-85,Sav-INC-96,LeiStoWerSav-CPL-97,IikTsuYanHir-JCP-01,KamTsuHir-JCP-02,TawTsuYanYanHir-JCP-04,PolSavLeiSto-JCP-02,TouColSav-PRA-04,AngGerSavTou-PRA-05,TouGorSav-TCA-05,GolWerSto-PCCP-05,GolWerStoLeiGorSav-CP-06,GerAngMarKre-JCP-07,FroTouJen-JCP-07,FroJen-PRA-08,HeyScuErn-JCP-03,HeyScu-JCP-04,HeyScu-JCP-04b,VydHeyKruScu-JCP-06,VydScu-JCP-06,HenIzmScuSav-JCP-07,BaeNeu-PRL-05,BaeLivNeu-CP-06,LivBae-PCCP-07}.
Prof. Hirao has been a pioneer in this field, investigating the effect of range separation on the exchange energy with remarkable success (see, e.g., \cite{IikTsuYanHir-JCP-01,KamTsuHir-JCP-02,TawTsuYanYanHir-JCP-04}).
The variant of range separation that we consider here \cite{StoSav-INC-85,Sav-INC-96,LeiStoWerSav-CPL-97,PolSavLeiSto-JCP-02,TouColSav-PRA-04,AngGerSavTou-PRA-05,TouGorSav-TCA-05,GolWerSto-PCCP-05,GolWerStoLeiGorSav-CP-06,GerAngMarKre-JCP-07,FroTouJen-JCP-07,FroJen-PRA-08} can be viewed as a way to
remove the constraint that the model system be non-interacting: instead of the KS system, one can define a long-range-only-interacting system (whose wavefunction is thus multideterminantal) having the same density of the physical system. The remaining part of
the energy is then approximated with a short-range exchange-correlation functional.
The resulting long-range-only hamiltonian, being weakly interacting (and without the electron-electron cusp), can be treated at a reasonable computational cost with standard wavefunction methods: in general, the needed configuration space to achieve good accuracy is small, and often second-order perturbation theory suffices. At the same time, this long-range interaction, albeit small, can make the corresponding wavefunction capture near-degeneracy effects and long-range van der Waals energies. Provided that the energy functionals are correctly redefined, there is no double counting of the energy, and the method is in principle exact, as it is KS DFT.
As mentioned, this range-separated multideterminant DFT needs an approximation for the short-range exchange-correlation functional. One can follow the same path as for KS DFT: start with the local-spin-density approximation (LSDA), consistently constructed as the difference between the standard LSD functional and the exchange-correlation energy of an electron gas with long-range-only interaction \cite{PazMorGorBac-PRB-06}, and then add gradient corrections (GGA) \cite{GolWerSto-PCCP-05,TouColSav-JCP-05,GolWerStoLeiGorSav-CP-06,FroTouJen-JCP-07}, and eventually
meta-gradient corrections (mGGA). However, this path, which proved highly successful for KS DFT, may not be the best for a scheme in which long-range correlations are explicitly taken into account by wavefunction methods. Indeed, in most cases there is no improvement when passing from LSDA to GGA \cite{GolWerSto-PCCP-05,GolWerStoLeiGorSav-CP-06,GolStoThiSch-PRA-07}, with the exception of hydrogen-bonded complexes \cite{GolLeiManMitWerSto-PCCP-08}.
In recent years we have extended the ``Overhauser model'', an approximate method to calculate the short-range part of the pair density in the uniform electron gas, to systems of nonuniform density \cite{GorSav-PRA-05,GorSav-PM-06,GorSav-IJMPB-07,GorSav-JCTC-07}, finding that it yields an accurate description of the short-range part of the spherically- and system-averaged pair
density (intracule density) of small atoms. In Ref.~\cite{GorSav-IJMPB-07} we have combined the Overhauser equations with the Kohn-Sham equations in a self-consistent way, recovering full CI total energies within 1~mH for the He isoelectronic series. In Ref.~\cite{GorSav-JCTC-07} we have shown that, unlike all the available correlation functionals \cite{KatRoySpr-JCP-06}, the model works equally well for the high-density limit of the He and the Hooke's atom series.
Thus, on one hand, the Overhauser model seems to be a very good candidate to construct short-range correlation energy functionals. On the other hand, we have tested it only on systems dominated by dynamical correlation: in the He atom, the Overhauser model yields essentially the exact KS correlation energy. However, when we move to systems with strong static correlation we expect the Overhauser model to be unable to yield good results. The combination of the Overhauser model with range-separated multideterminant DFT seems then natural: it can be viewed as a way to produce an adapted short-range correlation functional for the range-separated multideterminant DFT, or as a way to add the description of static correlation to the Overhauser model.
In this work we combine the Overhauser model with range-separated multideterminant DFT, applying it to the case of the H$_2$ molecule along the dissociation curve, thus analyzing also the case of strong static correlation, as the dissociation limit is approached. The paper is organized as follows. After briefly reviewing in Secs.~\ref{sec_multDFT} and \ref{sec_Overh} the basic equations of range-separated multideterminant DFT and of the extended Overhauser model, we first analyze in Sec.~\ref{sec_Ovh2exact}, using very accurate variational wavefunctions \cite{RycCenKom-CPL-94,CenKomRyc-CPL-95,CenKut-JCP-96}, how the ``exact'' electron-electron interaction which appears in the Overhauser model (and that it is usually approximated with a physically-motivated interaction) should be as the H$_2$ molecule is stretched. This analysis shows the difficulty of modeling static correlation within the Overhauser model. Since the model is only able to describe correlation, we combine it with a generalized OEP scheme for multideterminant DFT, which is described in Sec.~\ref{sec_mdOEP}. The combination of the two methods is then presented in Sec.~\ref{sec_mdOverh}, with results for the H$_2$ molecule. The last Sec.~\ref{sec_conc} is devoted to conclusions and perspectives.
\section{Multideterminant DFT via range separation}
\label{sec_multDFT}
Hohenberg and Kohn \cite{HohKoh-PR-64} introduced a universal functional of the density $F[\rho]$, which can be written as a constrained minimum search \cite{Lev-PNAS-79},
\begin{equation}
F[\rho]=\min_{\Psi\to\rho}\langle\Psi|\hat{T}+\hat{V}_{ee}|\Psi\rangle.
\label{eq_F}
\end{equation}
In Eq.~(\ref{eq_F}) the expectation of the kinetic energy operator $\hat{T}=-\frac{1}{2}\sum_i\nabla_i^2$ plus the Coulomb electron-electron repulsion operator $\hat{V}_{ee}=\sum_{i>j}|{\bf r}_i-{\bf r}_j|^{-1}$ is minimized over all wavefunctions yielding the density $\rho$. The universality of the functional $F[\rho]$ stems from the fact that $\hat{T}$ and $\hat{V}_{ee}$ are the same for every electronic system of given particle number $N=\int \rho({\bf r})d{\bf r}$. Kohn and Sham \cite{KohSha-PR-65} introduced another functional, $T_s[\rho]$ of Eq.~(\ref{eq_Erho}), by replacing $\hat{V}_{ee}$ in Eq.~(\ref{eq_F}) with zero,
\begin{equation}
T_s[\rho]=\min_{\Phi\to\rho}\langle\Phi|\hat{T}|\Phi\rangle,
\label{eq_Ts}
\end{equation}
and used $T_s[\rho]$ for approximating an important part of $F[\rho]$. In Eq.~(\ref{eq_Ts}), and in the rest of this paper, $\Phi$ denotes a non-interacting wavefunction (thus in the majority of cases a single Slater determinant). Similarly, we can introduce a functional
$F_{\rm LR}^\mu[\rho]$ for a long-range-only interaction $\hat{W}_{\rm LR}^\mu$ (here chosen using the error function, with the real parameter $\mu$ governing the cutoff of the short-range part),
\begin{equation}
\hat{W}_{\rm LR}^\mu=\frac{1}{2}\sum_{i\neq j} \frac{{\rm erf}(\mu|{\bf r}_i-{\bf r}_j|)}{|{\bf r}_i-{\bf r}_j|},
\end{equation}
by defining
\begin{equation}
F_{\rm LR}^\mu[\rho]=\min_{\Psi^\mu\to\rho}\langle\Psi^\mu|\hat{T}+\hat{W}_{\rm LR}^\mu|\Psi^\mu\rangle.
\end{equation}
In this way we have
\begin{eqnarray}
\lim_{\mu\to\infty} F_{\rm LR}^\mu[\rho] & = & F[\rho] \\
\lim_{\mu\to 0} F_{\rm LR}^\mu[\rho] & = & T_s[\rho].
\end{eqnarray}
We can then write the total energy of a given many-electron system as
\begin{equation}
E[\rho]=F_{\rm LR}^\mu[\rho]+\int d{\bf r}\,v_{ne}({\bf r})\,\rho({\bf r})+\int d{\bf r}\int d{\bf r}'\rho({\bf r})\rho({\bf r}') \frac{{\rm erfc}(\mu|{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|}+E_{\rm xc}^\mu[\rho],
\label{eq_Etotmu}
\end{equation}
where ${\rm erfc}(x)=1-{\rm erf}(x)$ is the complementary error function. As in KS DFT then, minimization is performed over the wavefunction $\Psi^\mu$,
\begin{eqnarray}
E_0 & = & \min_{\Psi^\mu}\Biggl\{\langle\Psi^\mu|\hat{T}+\hat{W}_{\rm LR}^\mu|\Psi^\mu\rangle+
\int d{\bf r}\,v_{ne}({\bf r})\,\rho_{\Psi^\mu}({\bf r})+ \nonumber \\
& + & \int d{\bf r}\int d{\bf r}'\rho_{\Psi^\mu}({\bf r})\rho_{\Psi^\mu}({\bf r}') \frac{{\rm erfc}(\mu|{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|}+E_{\rm xc}^\mu[\rho_{\Psi^\mu}]\Biggr\},
\label{eq_E0DFTmd}
\end{eqnarray}
where $\rho_{\Psi^\mu}$ is the density corresponding to $\Psi^\mu$. Eq.~(\ref{eq_E0DFTmd}) yields an effective,
long-range-only-interacting hamiltonian to be solved with a chosen wavefunction method. The short-range
exchange-correlation functional $E_{\rm xc}^\mu[\rho]$ is then defined as the energy needed to make Eq.~(\ref{eq_Etotmu})
exact,
\begin{equation}
E_{\rm xc}^\mu[\rho]=F[\rho]-F_{\rm LR}^\mu[\rho]-\int d{\bf r}\int d{\bf r}'\rho({\bf r})\rho({\bf r}') \frac{{\rm erfc}(\mu|{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|}.
\end{equation}
For instance, the correct LSD approximation to $E_{\rm xc}^\mu[\rho]$ is
\begin{equation}
E_{\rm xc}^{\mu, {\rm LSD}}[\rho]=\int\rho({\bf r})\left\{\epsilon_{xc}(\rho_\uparrow({\bf r}),\rho_\downarrow({\bf r}))-\epsilon^\mu_{xc}(\rho_\uparrow({\bf r}),\rho_\downarrow({\bf r}))\right\},
\end{equation}
where $\epsilon_{xc}(\rho_\uparrow({\bf r}),\rho_\downarrow({\bf r}))$ is the exchange-correlation energy per electron of the standard uniform electron gas (with Coulomb electron-electron interaction) and $\epsilon^\mu_{xc}(\rho_\uparrow({\bf r}),\rho_\downarrow({\bf r}))$ is the exchange-correlation energy per electron of a uniform electron gas with interaction ${\rm erf}(\mu r_{12})/r_{12}$ \cite{PazMorGorBac-PRB-06}.
An exact expression for $E_{\rm xc}^\mu[\rho]$ is found from the adiabatic connection formula \cite{Sav-INC-96,Yan-JCP-98}:
\begin{equation}
E_{\rm xc}^\mu[\rho]=\int_\mu^\infty d \mu' \int_0^\infty 4\pi r_{12}^2 f^{\mu'}(r_{12}) \frac{2}{\sqrt{\pi}}e^{-\mu'^2 r_{12}^2}d r_{12}-\int d{\bf r}\int d{\bf r}'\rho({\bf r})\rho({\bf r}') \frac{{\rm erfc}(\mu|{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|},
\label{eq_adia}
\end{equation}
where $f^{\mu}(r_{12})$ is the spherically and system-averaged pair density (intracule density)
obtained by integrating $|\Psi^\mu|^2$
over all variables but $r_{12}=|{\bf r}_2-{\bf r}_1|$,
\begin{equation}
f^\mu(r_{12}) = \frac{N(N-1)}{2}\sum_{\sigma_1...\sigma_N}
\int |\Psi^\mu({\bf r}_{12},{\bf R},{\bf r}_3,...,{\bf r}_N)|^2
\frac{d\Omega_{{\bf r}_{12}}}{4\pi} d{\bf R} d{\bf r}_3...d{\bf r}_N,
\label{eq_intra}
\end{equation}
with ${\bf R}=({\bf r}_1+{\bf r}_2)/2$. The gaussian damping appearing in Eq.~(\ref{eq_adia}) comes from the derivative of the long-range interaction ${\rm erf}(\mu r_{12})/r_{12}$ with respect to $\mu$, and shows that the exchange-correlation energy is determined by the short-range part of the intracule density. Notice that when $\mu=0$ Eqs.~(\ref{eq_adia}) yields the KS exchange-correlation energy functional from a nonlinear adiabatic connection \cite{Sav-INC-96,Yan-JCP-98}.
\section{The extended Overhauser model}
\label{sec_Overh}
The extended Overhuaser model consists in writing an effective Schr\"odinger-like equation for the intracule density $f(r_{12})$ of a given system. The basic idea is the following \cite{GorSav-PRA-05,GorSav-PM-06,GorSav-IJMPB-07}. We start with the observation that the intracule density $f(r_{12})$ couples to any electron-electron interaction operator depending only on the interelectronic distance, $\hat{W}=\sum_{i> j}w(|{\bf r}_i-{\bf r}_j|)$, in the same way as the density $\rho({\bf r})$ couples to any local one-body potential operator $\hat{V}=\sum_i v({\bf r}_i)$, i.e.,
\begin{eqnarray}
\langle\Psi|\hat{W}|\Psi\rangle & = & \int d{\bf r}_{12} f(r_{12}) w(r_{12}), \\
\langle\Psi|\hat{V}|\Psi\rangle & = & \int d{\bf r} \rho({\bf r}) v({\bf r}).
\end{eqnarray}
We can then follow the Hohenberg and Kohn philosophy but with the roles of $\rho({\bf r})$ and $f(r_{12})$, and of $\hat{V}_{ne}$ and $\hat{V}_{ee}$, interchanged. That is, in analogy with Eq.~(\ref{eq_F}) we can define a system-dependent functional $G[f]$,
\begin{equation}
G[f]=\min_{\Psi\to f}\langle \Psi|\hat{T}+\hat{V}_{ne}|\Psi\rangle,
\label{eq_G}
\end{equation}
so that the total energy of a given physical system is equal to
\begin{equation}
E[f]=G[f]+\int d{\bf r}_{12} \frac{f(r_{12})}{r_{12}}.
\label{eq_Ef}
\end{equation}
Like Kohn and Sham, we can define another functional by setting $\hat{V}_{ne}$ equal to zero in Eq.~(\ref{eq_G}),
\begin{equation}
T_{\rm f}[f]=\min_{\Psi\to f}\langle \Psi|\hat{T}|\Psi\rangle.
\end{equation}
The functional $T_{\rm f}[f]$ corresponds to the internal kinetic energy of a free (zero external potential) cluster of fermions having the same intracule density of the physical system. The fermions of this cluster interact with an effective interaction $w_{\rm eff}(r_{12})$ which has the same role of the KS potential for the KS system. In practice, this effective interaction must be approximated. Moreover, for $N>2$ electrons the cluster equation become a complicated many-body problem, so that other approximations are needed. As in the original Overhauser model for the uniform electron gas \cite{Ove-CJP-95,GorPer-PRB-01}, we can approximate the cluster equation with a set of radial geminals $g_i(r_{12})$,
\begin{eqnarray}
& & \left[-\frac{1}{r_{12}}\frac{d^2}{dr_{12}^2}r_{12}
+\frac{\ell (\ell+1)}{r_{12}^2}+w_{\rm eff}(r_{12})\right] g_i(r_{12}) = \epsilon_i\,
g_i(r_{12})
\nonumber \\
& & \sum_i \vartheta_i|g_i(r_{12})|^2 = f(r_{12}),
\label{eq_eff}
\end{eqnarray}
whose occupancies $\vartheta_i$ must be defined (e.g., in a determinantal-like way as in the original Overhauser model \cite{GorPer-PRB-01}).
In practice, trying to solve the whole many-electron Schr\"odinger equation by means of Eqs.~(\ref{eq_Ef})-(\ref{eq_eff}) is a daunting task. The idea is rather \cite{GorSav-PRA-05,GorSav-PM-06,GorSav-IJMPB-07} to couple this ``average-pair-density-functional theory'' with a density functional scheme: Eqs.~(\ref{eq_eff}) can be generalized to any $f^\mu(r_{12})$ along the adiabatic connection of DFT. In Refs.~\cite{GorSav-PRA-05,GorSav-IJMPB-07} we started from the effective interaction
$w_{\rm eff}^{\rm KS}(r_{12})$ which, when inserted in Eqs.~(\ref{eq_eff}), gives the intracule density corresponding to the Kohn-Sham system, $f_{\rm KS}(r_{12})$ (that can be obtained from the KS determinant). We then wrote
an approximation for $w_{\rm eff}^\mu(r_{12})$ along the long-range adiabatic connection of DFT as
\begin{equation}
w_{\rm eff}^\mu(r_{12})=w_{\rm eff}^{{\rm KS}}(r_{12})+w_{\rm eff}^{c,\mu}(r_{12}).
\label{eq_weff}
\end{equation}
The only term that needs to be approximated is then $w_{\rm eff}^{c,\mu}(r_{12})$, an effective interaction that should essentially ``tell'' to the intracule density that, while the electron-electron interaction is turned on (i.e. as $\mu$ increases), the one-electron density $\rho({\bf r})$ does not change. As the information on $\rho({\bf r})$ has been ``washed away'' in the integration over the center of mass ${\bf R}$ of Eq.~(\ref{eq_intra}), this constraint can be imposed only in an approximate way. For two-electron atoms, for which Eq.~(\ref{eq_eff}) is exact with one geminal \cite{GorSav-PM-06}, $g=\sqrt{f}$, a simple approximation for $w_{\rm eff}^{c,\mu}(r_{12})$ is \cite{GorSav-PRA-05,GorSav-IJMPB-07,GorSav-JCTC-07}
\begin{equation}
w_{\rm eff}^{c,\mu}(r_{12})=\frac{{\rm erf}(\mu\, r_{12})}{r_{12}}-\left(\frac{4\pi}{3}\overline{r}_s^3\right)^{-1}\int_{|{\bf x}|\leq \overline{r}_s}\frac{{\rm erf}(\mu |{\bf r}_{12}-{\bf x}|)}{|{\bf r}_{12}-{\bf x}|}\,d{\bf x},
\label{eq_wc}
\end{equation}
where $\overline{r}_s$ is a screening length associated to the radius of a sphere containing on average one electron \cite{Ove-CJP-95,GorPer-PRB-01,GorSeiSav-PCCP-08}. The physical idea behind Eq.~(\ref{eq_wc}) is to mimic the constraint of fixed one-electron density by screening the electron-electron interaction over a length associated to the ``space'' available to each electron (which is determined by the density).
Indeed, for the He isoelectronic series Eqs.~(\ref{eq_adia}), (\ref{eq_eff}) and (\ref{eq_wc}), combined self-consistently with the Kohn-Sham equations, recover the full CI total energy within 1 mH \cite{GorSav-PRA-05,GorSav-IJMPB-07,GorSav-JCTC-07}.
\begin{figure}
\includegraphics[width=8cm]{f_1p4.pdf}
\includegraphics[width=8cm]{f_3.pdf}
\includegraphics[width=8cm]{f_4p5.pdf}
\includegraphics[width=8cm]{f_6.pdf}
\caption{Intracule densities $f(r_{12})$ for the H$_2$ molecule at different internuclear distances $R$ for the physical system (from the accurate variational wavefunction described in Subsec.~\ref{subsec_cencek}) and for the KS system (from the density corresponding to the same accurate variational wavefunctions).}
\label{fig_f}
\end{figure}
\section{The Overhauser model for the H$_2$ molecule: how things should be}
\label{sec_Ovh2exact}
For a closed-shell physical electronic system (atom, molecule) with $N=2$ particles, the Schr\"odinger equation describing the internal degrees of freedom of a cluster of fermions having the same intracule density $f(r_{12})$ is exactly given by \cite{GorSav-PM-06,GorSav-IJMPB-07,GorSav-JCTC-07}
\begin{equation}
\left[-\frac{1}{r_{12}}\frac{d^2}{d r_{12}^2} r_{12}+w_{\rm eff}(r_{12})\right]\sqrt{f(r_{12})}=\epsilon \sqrt{f(r_{12})}.
\label{eq_eff2ele}
\end{equation}
As a first study, we calculate and analyze the ``exact'' Overhauser interaction $w_{\rm eff}(r_{12})$ at full coupling strength (i.e., for electron-electron interaction $1/r_{12}$, corresponding to $\mu=\infty$) for the H$_2$ molecule at different values of the internuclear distance $R$, and we compare it with the approximation of Eq.~(\ref{eq_wc}). To this purpose, we need extremely accurate
intracule densities $f(r_{12})$, which are described in the next Subsec.~\ref{subsec_cencek}.
\subsection{Intracule densities from accurate variational wavefunctions}
\label{subsec_cencek}
We use the accurate variational wavefunctions of Refs.~\cite{RycCenKom-CPL-94,CenKomRyc-CPL-95,CenKut-JCP-96}, which are expanded in explicitly correlated gaussian geminals,
\begin{eqnarray}
\Psi({\bf r}_1,{\bf r}_2) & = & (1+\hat{P}_{12})(1+\hat{i}_e)\sum_{k=1}^K c_k\psi_k({\bf r}_1,{\bf r}_2)
\label{eq_expa}\\
\psi_k({\bf r}_1,{\bf r}_2) & = & e^{-\alpha_k|{\bf r}_1-{\bf r}_{Ak}|^2}e^{-\beta_k|{\bf r}_2-{\bf r}_{Bk}|^2}e^{-\gamma_k r_{12}^2},
\label{eq_gaussgem}
\end{eqnarray}
where ${\bf r}_{Ak}$ and ${\bf r}_{Bk}$ are centers that lie on the internuclear axis, $\hat{P}_{12}$ means permutation of ${\bf r}_1$ and ${\bf r}_2$, and $\hat{i}_e$ is the inversion operator with respect to the center of the molecule. The parameters appearing in Eqs.~(\ref{eq_expa})-(\ref{eq_gaussgem}) are determined variationally by minimizing the energy with
the conjugate gradient method (for more details on the wavefunction and the algorithms employed, see Refs.~\cite{RycCenKom-CPL-94,CenKomRyc-CPL-95,CenKut-JCP-96}). The expansion length $K=1200$ in Eq.~(\ref{eq_expa})
is used, resulting in energies with the extraordinary accuracy of $10^{-10}$ Hartree.
The intracule densities $f(r_{12})$ from these extremely accurate wavefunctions can be easily calculated, since all the needed integrals are analytic. We also calculated the one electron densities $\rho({\bf r})$, and the intracule densities $f_{\rm KS}(r_{12})$ corresponding to the KS system, which can be obtained by inserting in Eq.~(\ref{eq_intra}) the KS wavefunction $\frac{1}{2}\sqrt{\rho({\bf r}_1)}\sqrt{\rho({\bf r}_2)}$. In Fig.~\ref{fig_f} we
show the intracule densities $f(r_{12})$ and $f_{\rm KS}(r_{12})$ for the internuclear distances $R=1.4$, $3.0$, $4.5$ and $6.0$ a.u. Although mathematically the wave
function of Eqs.~(\ref{eq_expa})-(\ref{eq_gaussgem}) is cuspless, we see that the very elaborate ansatz
permits to describe
the exact linear behaviour of the intracule density for $r_{12}\to 0$, up to extremely short
distances. Fig.~\ref{fig_fu2} shows the same quantities multiplied by the volume element $4\pi r_{12}^2$. This figure better visualizes the transition from dynamical to static correlation. In Fig.~\ref{fig_HL} we also report the same quantities in the extreme stretched case, $R=20$, obtained from the simple Heitler-London wavefunction.
\begin{figure}
\includegraphics[width=8cm]{fu2_1p4.pdf}
\includegraphics[width=8cm]{fu2_3.pdf}
\includegraphics[width=8cm]{fu2_4p5.pdf}
\includegraphics[width=8cm]{fu2_6.pdf}
\caption{The same intracule densities of Fig.~\ref{fig_f} multiplied by the volume element $4\pi r_{12}^2$.}
\label{fig_fu2}
\end{figure}
\begin{figure}
\includegraphics[width=8.5cm]{fu2_20.pdf}
\caption{The intracule density $f(r_{12})$ multiplied by the volume element $4\pi r_{12}^2$ for the H$_2$ molecule in the extreme stretched case $R=20$. The physical $f(r_{12})$ has been calculated from the simple Heitler-London wavefunction, and $f_{\rm KS}(r_{12})$ from its corresponding density.}
\label{fig_HL}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{weff_1p4.pdf}
\includegraphics[width=8cm]{weff_3.pdf}
\includegraphics[width=8cm]{weff_4p5.pdf}
\includegraphics[width=8cm]{weff_6.pdf}
\caption{The ``exact'' effective Overhauser interaction $w_{\rm eff}(r_{12})$ for the intracule densities of the physical and of the KS systems of Fig.~\ref{fig_f}.}
\label{fig_weff}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{wc_1p4.pdf}
\includegraphics[width=8cm]{wc_3.pdf}
\includegraphics[width=8cm]{wc_4p5.pdf}
\includegraphics[width=8cm]{wc_6.pdf}
\caption{The difference $w_{\rm eff}^c(r_{12})$ between the ``exact'' effective Overhauser interaction for the intracule density of the physical and of the KS systems of Fig.~\ref{fig_weff}. The Coulomb repulsion $1/r_{12}$ is also reported.}
\label{fig_wc}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{weff_20.pdf}
\includegraphics[width=8cm]{wc_20.pdf}
\caption{Same as Figs.~\ref{fig_weff}-\ref{fig_wc} for the extreme stretched molecule, using the simple Heitler-London wavefunction.}
\label{fig_wHL}
\end{figure}
\subsection{Accurate Overhauser potentials}
From the accurate intracule densities of the previous subsection we can calculate, by inversion, the corresponding ``exact'' Overhauser interaction $w_{\rm eff}(r_{12})$,
\begin{equation}
w_{\rm eff}(r_{12})=\frac{1}{\sqrt{f(r_{12})}}\frac{1}{r_{12}}\frac{d^2}{d r_{12}^2}\left(r_{12}\sqrt{f(r_{12})}\right)+{\rm const.}
\label{eq_inv}
\end{equation}
The inversion of Eq.~(\ref{eq_inv}) is done numerically, by finite differences. In Fig.~\ref{fig_weff} we report the effective Overhauser interactions that, when inserted in Eq.~(\ref{eq_eff2ele}), give the physical and the KS intracule, corresponding, respectively, to $\mu=\infty$ and $\mu=0$ along the long-range adiabatic connection of Sec.~\ref{sec_multDFT} (or to $\lambda=1$ and $\lambda=0$ along the usual linear adiabatic connection in which $\hat{V}_{ee}$ is simply multiplied by $\lambda$). We see that $w_{\rm eff}(r_{12})$ for large $r_{12}$ goes to the same constant for both the KS and the physical system, as it should be \cite{GorSav-IJMPB-07} (of course if we go to too large $r_{12}$ we start to observe the wrong harmonic wall due to the gaussian asymptotic decay of our wavefunction). The difference between the effective Overhauser interaction for the physical and the KS system gives $w_{\rm eff}^{c,\mu\to\infty}\equiv w_{\rm eff}^c$ of Eq.~(\ref{eq_weff}), and is reported in Fig.~\ref{fig_wc}, where also the Coulomb repulsion $1/r_{12}$ is shown. From this figure, we see that, when the system is still dominated by dynamical correlation, as in the case $R=1.4$ and $R=3$, $w_{\rm eff}^c(r_{12})$ is essentially a screened Coulomb interaction. That is, for short-range it behaves as $1/r_{12}$, and then for large $r_{12}$ goes to zero much faster than $1/r_{12}$. In such cases, the approximation of Eq.~(\ref{eq_wc}), which at $\mu=\infty$ reads
\begin{eqnarray}
w_{\rm eff}^c(r_{12}) = & \frac{1}{r_{12}}
+\frac{r_{12}^2}{2\overline{r}_s^3}-\frac{3}{2 \overline{r}_s}
\qquad & r_{12}\le \overline{r}_s \nonumber \\
w_{\rm eff}^c(r_{12}) = & 0 & r_{12}>\overline{r}_s. \label{eq_wcCoulomb}
\end{eqnarray}
can work reasonably well, with a screening length $\overline{r}_s\sim R$. However, as $R$ grows and the system starts to be dominated by static correlation, we see that the approximation of Eq.~(\ref{eq_wcCoulomb}) cannot work: the ``exact'' $w_{\rm eff}^c(r_{12})$ still decays much faster than $1/r_{12}$ for large $r_{12}$, but at short range is more repulsive than the Coulomb interaction! I.e., we need an ``overscreened'' interaction. This is completely evident in the extreme stretched case $R=20$ of Fig.~\ref{fig_wHL}, again obtained from the simple Heitler-London wavefunction.
\section{Generalized optimized effective potential method for multideterminant DFT}
\label{sec_mdOEP}
In recent years,
the focus of a large part of the scientific community working on improving the approximations for $E_{xc}[\rho]$ has shifted from seeking explicit functionals of the density (like the generalized gradient approximations), to implicit functionals, typically using the exact exchange $E_{\rm x}[\rho]$, which is only explicitly known in terms of the Kohn-Sham orbitals $\phi_i({\bf r})$. The corresponding Kohn-Sham potential must then be computed with the optimized effective potential (OEP) method (for a recent review, see \cite{KumKro-RMP-08}). The OEP scheme can be generalized to the multideterminant range-separated DFT by first noticing that we
can divide $E_{xc}^\mu[\rho]$ into exchange and correlation in two different ways \cite{TouGorSav-TCA-05}: we can define the exchange energy with respect to the KS determinant $\Phi$,
\begin{equation}
E_x^\mu[n] =
\langle \Phi |\hat{V}_{ee}-\hat{W}_{\rm LR}^\mu|\Phi\rangle
-\int d{\bf r}\int d{\bf r}'\rho({\bf r})\rho({\bf r}') \frac{{\rm erfc}(\mu|{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|},
\end{equation}
and then define the usual correlation energy functional $E^\mu_c[\rho]$ as the energy missed by the KS wavefunction,
\begin{equation}
E^\mu_c[\rho]=E^\mu_{\rm xc}[\rho]-E_{\rm x}^\mu[\rho],
\label{eq_Eccomp}
\end{equation}
but we can also define a multideterminantal (md) exchange
functional \cite{TouGorSav-TCA-05} by using the wavefunction $\Psi^\mu$,
\begin{equation}
E_{\rm x, md}^\mu[\rho] =
\langle \Psi^\mu |\hat{V}_{ee}-\hat{W}_{\rm LR}^\mu|\Psi^\mu\rangle-
\int d{\bf r}\int d{\bf r}'\rho({\bf r})\rho({\bf r}') \frac{{\rm erfc}(\mu|{\bf r}-{\bf r}'|)}{|{\bf r}-{\bf r}'|},
\label{eq_Exmd}
\end{equation}
and then a corresponding correlation energy that recovers the energy missed by $\Psi^\mu$ (which is smaller than the energy missed by the KS determinant $\Phi$),
\begin{equation}
E^\mu_{\rm c, md}[\rho]=E^\mu_{\rm xc}[\rho]-E_{\rm x, md}^\mu[\rho].
\label{eq_Ecmd}
\end{equation}
Then, with this latter definition of the correlation energy, the generalized OEP-like scheme for multideterminant DFT becomes \cite{TouGorSav-TCA-05}
\begin{equation}
E_0=\inf_{v^\mu}\left\{\langle\Psi^\mu_{v^\mu}| \hat{T}+\hat{V}_{ee}+\hat{V}_{ne}
|\Psi^\mu_{v^\mu}\rangle+E^\mu_{\rm c, md}[\rho_{\Psi^\mu_{v^\mu}}]\right\},
\label{eq_OEPmd}
\end{equation}
where $\Psi^\mu_{v^\mu}$ is obtained by solving the Schr\"odinger equation
corresponding to the hamiltonian
\begin{equation}
\hat{H}^\mu= \hat{T}+\hat{W}_{\rm LR}^\mu+\hat{V}^\mu, \qquad \hat{V}^\mu=\sum_i v^\mu({\bf r}_i).
\label{eq_Hmu}
\end{equation}
Notice that this multideterminant OEP scheme is different from the one recently
proposed in Ref.~\cite{WeiDelGor-JCP-08}. In Eq.~(\ref{eq_OEPmd}) the weak long-range interaction $\hat{W}_{\rm LR}^\mu$ automatically selects the configuration space needed to yield an accurate solution for the hamiltonian $\hat{H}^\mu$ of Eq.~(\ref{eq_Hmu}), while in Ref.~\cite{WeiDelGor-JCP-08} the configuration space is chosen essentially by hand, using physical and chemical intuition.
In the next Sec.~\ref{sec_mdOverh} we use the Overhauser model to approximate $E^\mu_{\rm c, md}[\rho]$, and we apply our combined formalism to the case of the H$_2$ molecule.
\section{Multideterminant DFT combined with the Overhauser model}
\label{sec_mdOverh}
From the adiabatic connection formalism we can easily write an exact formula for $E^\mu_{\rm c, md}[\rho]$,
\begin{equation}
E^\mu_{\rm c, md}[\rho]=\int_\mu^\infty d\mu'\int_0^\infty 4 \pi \,r_{12}^2\, \left[f^{\mu'}(r_{12})-f^{\mu}(r_{12})\right]\frac{2}{\sqrt{\pi}}e^{-\mu'^2 r_{12}^2}\,dr_{12},
\label{eq_ecmd1}
\end{equation}
which shows that $E^\mu_{\rm c, md}[\rho]$ is determined by the change in the short-range part of the
intracule density when the electron-electron interaction increases from ${\rm erf}(\mu r_{12})/r_{12}$ to
the full Coulomb repulsion $1/r_{12}$. By adding and subtracting $f_{\rm KS}(r_{12})$, Eq.~(\ref{eq_ecmd1}) can also be written as
\begin{equation}
E^\mu_{\rm c, md}[\rho]=E_c^\mu[\rho]-\int_0^\infty 4\pi\,r_{12}^2\,\left[f^{\mu}(r_{12})-f_{\rm KS}(r_{12})\right]\,\frac{{\rm erfc}(\mu r_{12})}{r_{12}}\,dr_{12},
\label{eq_ecmd2}
\end{equation}
where $E_c^\mu[\rho]$ is the correlation energy of Eq.~(\ref{eq_Eccomp}), defined with respect to the KS determinant.
We computed $f^\mu(r_{12})$ within the Overhauser model, Eq.~(\ref{eq_eff}) with one geminal $g=\sqrt{f}$, using the simple screened potential of Eqs.~(\ref{eq_weff})-(\ref{eq_wc}) with the screening length $\overline{r}_s=R$. For each internuclear distance $R$, the intracules $f^\mu(r_{12})$ have been calculated for 33 values of $\mu$ between $\mu=0.01$ and $\mu=20$.
By numerical integration we then computed
\begin{equation}
\frac{\partial E_c^\mu[\rho]}{\partial \mu}=\int_0^\infty 4 \pi \,r_{12}^2\, \left[f^{\mu}(r_{12})-f_{\rm KS}(r_{12})\right]\frac{2}{\sqrt{\pi}}e^{-\mu'^2 r_{12}^2}\,dr_{12},
\end{equation}
and we fitted the values of $\frac{\partial E_c^\mu[\rho]}{\partial \mu}$ with the derivative of the function
\begin{equation}
E_c^\mu[\rho]=\frac{a_4}{b^{10}}-\frac{a_1 \mu^6+a_2\mu^7+a_3\mu^8+a_4\mu^{10}}{(1+b^2\mu^2)^5},
\end{equation}
which has the correct asymptotic behaviors \cite{GorSav-PRA-06}. We also computed, again by numerical integration, the second term on the right-hand-side of Eq.~(\ref{eq_ecmd2}) in order to obtain $E^\mu_{\rm c, md}[\rho]$.
We then implemented the generalized OEP scheme of Eq.~(\ref{eq_OEPmd}) by first minimizing the effective potential $v^\mu({\bf r})$ at the ``generalized-exchange''-only level, and by adding $E^\mu_{\rm c, md}[\rho]$ only as a final correction. Since $E^\mu_{\rm c, md}[\rho]$ is very small, we do not expect substantial changes by implementing a full self-consistent scheme. Our procedure can be summarized with the equation
\begin{equation}
E_0=\left(\inf_{v^\mu}\langle\Psi^\mu_{v^\mu}| \hat{T}+\hat{V}_{ee}+\hat{V}_{ne}
|\Psi^\mu_{v^\mu}\rangle\right)+E^\mu_{\rm c, md}[\rho_{\Psi^\mu_{v^\mu}}],
\label{eq_E0md}
\end{equation}
where $E^\mu_{\rm c, md}[\rho_{\Psi^\mu_{v^\mu}}]$ is calculated with the final density resulting from the minimization in the first term on the right-hand-side of Eq.~(\ref{eq_E0md}).
To carry out the minimization with respect to the potential $v^\mu({\bf r})$ in Eq.~(\ref{eq_E0md}) we proceeded as follows.
We parametrized the potential $v^\mu({\bf r})$ with a simple two-parameter form, by adding to the physical external potential a gaussian centered on each atom, $c\, e^{-\gamma r^2}$. The minimization of the expectation $\langle\Psi^\mu_{v^\mu}| \hat{T}+\hat{V}_{ee}+\hat{V}_{ne} |\Psi^\mu_{v^\mu}\rangle$ with respect to the two parameters $c$ and $\gamma$ is done by calculating at each step full-CI wavefunctions $\Psi^\mu_{v^\mu}$ for the hamiltonian with electron-electron interaction ${\rm erf}(\mu r_{12})/r_{12}$ and external potential $v^\mu({\bf r})$. All calculations were done at the cc-V5Z basis-set level. We also produced with MOLPRO \cite{Molpro-PROG-02} full CI reference results for the physical hamiltonian, for comparison. Our simple parametrization of the potential $v^\mu$, containing only two parameters, is enough to yield at $\mu=0$ the HF energy within 0.5 mH, which is the accuracy we sought in this study. This way, we avoid all the well-known problems of the OEP method in finite basis set \cite{StaScuDav-JCP-06} at the price of obtaining only an upper bound for our minimization problem (yet, with the reasonable accuracy of 0.5 mH).
\begin{figure}
\includegraphics[width=8.5cm]{Ecmd_1p4.pdf}
\includegraphics[width=8.5cm]{Ecmd_2.pdf}
\includegraphics[width=8.5cm]{Ecmd_4.pdf}
\caption{The short-range correlation energy for range-separated multideterminant DFT as a function of the cutoff parameter $\mu$ for the H$_2$ molecule at three different values of the internuclear distance $R$. Dots ($\bullet$) are ``exact'' values (see text in Sec.~\ref{sec_mdOverh}), solid lines are the results from the Overhuaser model, and the dashed lines are the LDA values.}
\label{fig_Ecmd}
\end{figure}
In Fig.~\ref{fig_Ecmd} we report the results for $E^\mu_{\rm c, md}[\rho]$ for three different values of the internuclear distance $R$. The dots ($\bullet$) are the ``exact'' values of $E^\mu_{\rm c, md}[\rho]$, i.e., the full-CI total energies obtained with MOLPRO minus the energies corresponding to the first term on the right-hand-side of Eq.~(\ref{eq_E0md}), $\inf_{v^\mu}\langle\Psi^\mu_{v^\mu}| \hat{T}+\hat{V}_{ee}+\hat{V}_{ne}
|\Psi^\mu_{v^\mu}\rangle$. The solid line is $E^\mu_{\rm c, md}[\rho]$ from the Overhauser model, and the dashed line is the LDA result, obtained from the parametrization of Ref.~\cite{PazMorGorBac-PRB-06}, in which $E^\mu_{\rm c, md}[\rho]$ for the uniform electron gas has been calculated with Quantum Monte Carlo methods. We see from this figure that when the system is still dominated by dynamical correlation, as in the $R=1.4$ and the $R=2$ cases, the Overhauser model yields, even at $\mu=0$ (i.e. for pure KS DFT), correlation energies with errors of $\sim 5$~mH (while LDA is off by $\sim 60$~mH), which reduce to 1~mH at $\mu=0.5$ (where the LDA error is still $\sim 10$~mH).
We focuse here on the value $\mu\sim 0.5$ since it is the one commonly used in practical applications \cite{AngGerSavTou-PRA-05,TouGorSav-TCA-05,GolWerSto-PCCP-05,GolWerStoLeiGorSav-CP-06,GerAngMarKre-JCP-07,FroTouJen-JCP-07,FroJen-PRA-08}.
When the system starts to be dominated by static correlation, as in the $R=4$ case, the Overhauser model with the simple screened potential of Eq.~(\ref{eq_wc}) gives, at $\mu=0$, errors very close to those of LDA ($\sim 20$~mH), which are still of the order of $\sim 10$~mH at $\mu=0.5$. As the molecule approaches the dissociation limit, $R\to\infty$, the exact $E^\mu_{\rm c, md}[\rho]$ tends to the limiting behavior in which $E^{\mu=0}_{\rm c, md}[\rho]= E_c^{\rm KS}[\rho]$, and $E^{\mu}_{\rm c, md}[\rho]= 0$ for any $\mu>0$. This is due to the fact that, as $R\to\infty$, the long-range only wavefunction $\Psi^\mu$, even at very small $\mu$ (i.e., with an infinitesimal interaction), becomes essentially exact and equal to the Heitler-London wavefunction, so that the functional should be just equal to zero. In this limit, the Overhauser model is wrong for small $\mu$ (because, as explained in Sec.~\ref{sec_Ovh2exact}, it misses the ``overscreening'' at short range), but, for $\mu \gg 0$, yields $E^\mu_{\rm c, md}[\rho]$ that go to zero much faster than LDA, as it can be already grasped from the third panel of Fig.~\ref{fig_Ecmd}. It is thus still more suitable than LDA to be combined with the range-separated multideterminant DFT, but it definitely needs some improvement.
Notice that the Overhauser model would yield much more accurate results if we were able to compute $E^{\mu}_{\rm c, md}[\rho]$ by using in Eq.~(\ref{eq_weff}) instead of $w_{\rm eff}^{\rm KS}(r_{12})$ the interaction $w_{\rm eff}^\mu(r_{12})$ which yields the intracule $f^\mu(r_{12})$ associated to the wavefunction $\Psi^\mu$. This way, we would use the information available in $\Psi^\mu$ to the maximum extent, and we would not have the problems associated to the ``overscreening'' discussed in Sec.~\ref{sec_Ovh2exact}. This possibility will be investigated in future work.
\section{Conclusions and perspectives}
\label{sec_conc}
We have presented a preliminary study of the combination of range-separated multideterminant DFT with the Overhauser model, with an application to the paradigmatic case of the H$_2$ molecule. We have first analyzed, by means of very accurate variational wavefunctions, the failure of the Overhauser model in describing static correlation and we have then used it to produce an adapted short-range correlation functional for range-separated multideterminant DFT. The results are very good for internuclear distances close to equilibrium, and are still encouraging as the molecule is stretched. Indeed, in the dissociation limit the exact short-range correlation functional should go to zero for any $\mu>0$, and the Overhauser model yields short-range correlation energies that go to zero faster than LDA as $\mu$ increases.
Future work will address the study of better approximations for the unknown Overhauser electron-electron interaction, and the development of a more efficient scheme to combine it with range-separated multideterminant DFT.
\section*{Acknowledgements}
We thank W. Cencek for providing us with the accurate geminal wave functions of the H$_2$ molecule.
This work was supported by the ANR (National French Research Agency) under Grant n.~ANR-07-BLAN-0271.
|
2,869,038,154,744 | arxiv | \section{Introduction}
\label{intro}
We call {\it{pluricomplex Green function}} $G_A$ of a compact set $A\subset\mathbb{C}^n$, $n\geq 1$, the plurisubharmonic function defined as$$\displaystyle G_{A}:= {\sup}^* \bigg\{ v \in PSH(\mathbb{C}^n):\text{ } v|_{A}\leq 0,\text{ } v(z)\leq \frac{1}{2}\log(1+\|z\|^2)+O(1)\bigg\},$$where ${\sup}^*$ denotes the upper semi-continuous regularization of the upper envelope, and $PSH(\mathbb{C}^n)$ denotes the set of plurisubharmonic functions in $\mathbb{C}^n$. The set $A$ is called {\it{L-regular}} if $G_A$ is continuous. In this case, the set $\{G_A = 0\}$ is the polynomially convex envelope $\hat{A}$ of $A$. We also consider, for an open bounded set $U\subset \mathbb{C}^n$, the {\it{Green function of $A\subset U$ relative to $U$}} defined by $$\displaystyle G_{A,U}:= {\sup}^* \big\{ v \in PSH(U):\text{ } v\leq 0,\text{ } v|_{A}\leq -1\big\}.$$
Let $U_a:=\{G_A<a\}$ for $a\in\mathbb{R}^{+}\setminus \{0\}$. If $A$ is not pluripolar and $\hat{A}\subset U_a$, then a relation between $G_A$ and $G_{A,U_a}$ holding in $U_a$ is given by Proposition 5.3.3 in \cite{K}: \begin{equation}\label{relation}
\displaystyle G_A=a(G_{A,U_a}+1).
\end{equation}
A compact $A\subset \mathbb{C}^n$ is said to {\it{satisfy the {\L}S} condition} if there exists an open set $U$ containing it and two constants $c,c'>0$ such that its pluricomplex Green function $G_A$ verifies the following regularity condition : $$\forall z\in U, G_A(z)\geq c\cdot dist(z, A)^{c'},$$where $dist$ denotes the euclidean distance (see for instance \cite{BG} or \cite{BG2}).\\
\noindent For technical reasons, we will take in this work $c' = \frac{1}{c}$, which does not change the definition, as we are not interested here in finding the optimal constants.\\
On a compact set $A\subset \mathbb{C}^n$ verifying the {\L}S condition, as well as the HCP condition (i.e. the H\"olderian continuity of $G_A$), for example a semi-algebraic compact set, we have the rapid approximation property of continuous functions by polynomials. Relatively few examples of compacts satisfying the {\L}S condition are known. Some examples are given in \cite{PiI}. Let us also note that Pierzcha{\l}a showed in \cite{PiII} that a compact verifying the {\L}S condition is polynomially convex. Bia{\l}as and Kosek \cite{BK} construct such sets using holomorphic dynamics.
In the same vein, we show that the so-called filled Julia sets in $\mathbb{C}$ satisfy the {\L}S condition. More precisely, our main goal is to show the following result concerning the filled Julia set of a polynomial $f:\mathbb{C}\rightarrow \mathbb{C}$, i.e. the set of points $z\in\mathbb{C}$ whose orbit $(f^n(z))_n$ is bounded : \\
\noindent {\bf{Theorem. }}{\it{The filled Julia set of a polynomial $f:\mathbb{C}\rightarrow \mathbb{C}$ of degree $\geq 2$, if its interior is non empty, satisfies the {\L}S condition.}}\\
The differentials operators operators $\partial$ and $\overline{\partial}$ will be understood in the sense of currents. Recall that a continuous function $u$ from an open set of $\mathbb{C}^n$ into $\mathbb{R}$ is pluriharmonic (harmonic if $n=1$) if and only if $\partial \overline{\partial} u =0$ (see for example Theorem 2.28 in \cite{L}). \\
In Section 2, we recall some definitions and elementary facts about holomorphic dynamics in one dimension, and we prove a useful lemma concerning the regularity of filled Julia sets. More precisely, we prove that the filled Julia set $K$ of a polynomial of degree $d\geq 2$ with non-empty interior satisfies $\overline{\mathring{K}}=K$. In Section 3, we define in $\mathbb{C}^n$, $n\geq 1$, the set of obstruction points to the {\L}S condition, and we prove that the complementary of this set is big, in a sense which will be specified. Section 4 is devoted to the proof of the main theorem previously stated.
\section{Dynamics in $\mathbb{C}$}
We start by recalling some definitions related to one-dimensional holomorphic dynamics. Let us consider a polynomial $f:\mathbb{C}\rightarrow \mathbb{C}$ of degree $d\geq 2$.
We call \textit{Fatou set} of $f$, denoted $\mathcal{F}$, the largest open subset in which the family of iterations $f^n$ is equicontinuous.
\textit{The Julia set of} $f$, denoted $J$, is the complement of $\mathcal{F}$ in $\mathbb{C}$. Let us note for what follows that $J$ is not pluripolar.
We call \textit{filled Julia set of} $f$ the set $K$ of points $z\in \mathbb{C}$ whose orbit $(f^n(z))_n$ is bounded.
Note that $K$ is compact, as $\infty$ is a superattractive fixed point of $f$, hence belonging to $\mathcal{F}$. The complement of $K$ is the basin of attraction of infinity. We have $\partial K = J$ and $G_K = G_J$. \\
Under very little restrictive conditions, the set $K$ is of non-empty interior. It is the case, for instance, for $f(z) = z^2+a$, with $a$ in the interior of the Mandelbrot set.\\
We construct the subharmonic function $G:\mathbb{C}\rightarrow\mathbb{R}^+$, limit in $L^1_{loc}$ of the sequence $ (\log(1+|f^n|)/d^n)_n$. It is known that $G$ is continuous (and even H\"olderian), harmonic in $\mathcal{F}$, and that it verifies $G(z)=0$ if and only if $z\in K$, and also that $G(z)-\log|z| = O(1)$ at infinity. By uniqueness, $G$ is therefore the pluricomplex Green function of $K$ (and of $J$). It satisfies by construction the invariance property \begin{equation}\label{invariance}G\circ f = d\cdot G.\end{equation}The measure $\frac{i}{\pi}\partial \overline{\partial} G$ is a probability measure of support exactly $J$ (see e.g. \cite{G}). We first show a preliminary lemma about the filled Julia set.
\begin{lem}\label{julia}
The filled Julia set $K$ of a polynomial of degree $d\geq 2$ with non-empty interior satisfies $\overline{\mathring{K}}=K$.
\end{lem}
\begin{proof}
First recall the following equidistribution result (see e.g. Theorem 1.10 in \cite{G} or Theorem 6.1 in \cite{FS2}). Let $G$ the Green function associated to a polynomial $f$ of degree $d\geq 2$. Then, for all $x\in \mathbb{C}$ (except possibly for a totally invariant set consisting of at most two points in the Fatou set, see e.g. \cite{Be}), we have the following weak convergence of measures :\begin{equation}\label{equidistribution}\displaystyle\lim_{n\rightarrow +\infty}\frac{1}{d^n}f^{n*}\delta_x:=\lim_{n\rightarrow +\infty}\frac{1}{d^n}\sum_{y:f^n(y)=x}\delta_y = \frac{i}{\pi}\partial \overline{\partial} G,\end{equation}
\noindent where $\delta_x$ is a Dirac measure with support in $\mathring{K}$, except for the exceptional points previously mentioned. Note that $\partial K$ is invariant by $f$ and by $f^{-1}$, and hence so is $\mathring{K}$. By Equation (\ref{equidistribution}), since the support of $f^{n*}\delta_x$ is included in $\mathring{K}$ for all $n$, every open subset of $\mathbb{C}$ intersecting $J=\partial K$ also intersects $\mathring{K}$, thus $\overline{\mathring{K}}=K$.
\qed
\end{proof}
\section{Study of the obstruction to the {\L}S condition}
For $n\geq 1$, let \begin{equation}\label{Oc}\displaystyle O_c:=\{z\in\mathbb{C}^n: dist(z,A)< 1, G_A(z)< c \cdot dist(z,A)^{1/c}\}.\end{equation}
Note that the sequence of open sets $O_c$ is increasing with $c$ for $c<1$. The {\L}S condition is satisfied by a compact nonpolar set $A\subset\mathbb{C}^n$, L-regular and polynomially convex, if and only if the set $$\displaystyle I:=\bigcap_{c>0} \overline{O_c}\subset \partial A$$ is empty. We call $I$ {\it{the set of obstruction points to the {\L}S condition}}.
\begin{exemple}[\cite{BK}] Counter-example : If $A$ is the union of two disks of radius $1$, tangent to each other at the origin, then it does not satisfy the {\L}S condition; the set of obstruction points is $I=\{0\}\neq \emptyset$.
\end{exemple}
\noindent The following result provides more insight into the structure of the complementary of $O_c$. We prove it for $n\geq 1$.
\begin{pro}\label{lemme1}
Let $A\subset\mathbb{C}^n$, $n\geq 1$, be a nonpolar, L-regular and polynomially convex compact set. Suppose that the pluricomplex Green function $G_A$ is pluriharmonic outside of A (harmonic if $n=1$).\\
Then, there exists $c_0>0$ such that $\forall c\in \ ]0,c_0]$, $\partial A$ is included in the boundary of the open set $\{z \in \mathbb{C}^n: G_A(z)>c \cdot dist(z,A)^{1/c}\}$.
\end{pro}
\begin{proof}
Let $\mu$ denote the positive measure $\displaystyle \frac{i}{\pi}\partial \overline{\partial} G_A\wedge \omega^{n-1}$ on $\mathbb{C}^n$, where $$\displaystyle \omega := \frac{i}{2\pi}\partial \overline{\partial} \log (1+\|z\|^2)$$ is the Fubini-Study form. Note that the support of the measure $\mu$ is exactly $\partial A$. Indeed, $ supp(\mu)\subset\partial A$ since $\frac{i}{\pi}\partial \overline{\partial} G_A=0$ in $\mathbb{C}^n\setminus \partial A$ by hypothesis. On the other hand, if there existed $x\in \partial A\setminus supp(\mu)$, then $G_A$ would be (pluri)harmonic in a neighborhood of $x$, hence null in this neighborhood, which can not happen because $A$ is polynomially convex.
\noindent Let us suppose by contradiction that $\forall c_0>0,\text{ }\exists c\in ]0,c_0],\text{ }\exists x \in \partial A,\text{ }\exists r>0,\text{ }$
$B(x,r)\cap \{z \in \mathbb{C}^n, G_A(z)>c \cdot dist(z,A)^{1/c}\}=\emptyset.$
\noindent Thus we can take $\displaystyle c'\in \left]0,\frac{1}{4n}\right[$, $x'\in\partial A$, and $r'>0$, such that $$G_A(z)\leq c'\cdot dist(z,A)^{\frac{1}{c'}}, \ \forall z \in B(x',r').$$
\noindent Denote $r_0:=\frac{r'}{2}$. Then, $\forall r<r_0$, $\forall x\in B(x',r_0)\cap \partial A$, the Chern-Levine-Nirenberg inequality implies : $$\displaystyle \mu\big(B(x,r)\big) \leq k\cdot r^{-2n}\sup_{B(x,2r)} G_A \leq c'\cdot k\cdot (2r)^{\frac{1}{c'}-2n},$$
for some constant $k>0$ independent of $r$, $r_0$, $x'$ and $c'$.
With the notation $\displaystyle \nu:=\frac{\mu}{\mu\big(B(x',r_0)\big)}\mathbf{1}_{B(x',r_0)}$, where $\displaystyle\mathbf{1}_{B(x',r_0)}$ is the characteristic function of $\displaystyle B(x',r_0)$, the measure $\nu$ is a probability measure, and we can rewrite the previous inequality : $\forall r>0$, $\forall x\in B(x',r_0)\cap \partial A$, $$\displaystyle \nu\big(B(x,r)\big)\leq \frac{ c'\cdot k}{{\mu\big(B(x',r_0)\big)}}\cdot (2r)^{\frac{1}{c'}-2n}.$$
Then, by Frostman Lemma (see for example Lemma 10.2.1 in \cite{Be}), the Hausdorff dimension of $\partial A\cap B(x_0,r_0)$ is strictly greater than $2n$ for our choice $c'<\frac{1}{4n}$, which gives a contradiction. (Recall that Frostman Lemma ensures that, if $m$ is a probability measure on a metric space $E$ verifying $m\big(B(x,r)\big)<q \cdot r^{\alpha}$ for all $x\in E$, $r>0$, with fixed $q>0$, $\alpha >0$, then the Hausdorff dimension of $E$ is greater than $\alpha$).
We thus conclude that $\exists c_0>0$, $\forall c\in ]0,c_0]$, $\forall x\in\partial A$, $\forall r>0$: $$\displaystyle \displaystyle B(x,r)\cap \{z \in \mathbb{C}^n, G_A(z)>c \cdot dist(z,A)^{1/c}\}\neq \emptyset,$$
which proves the statement.
\qed
\end{proof}
\section{Proof of the main theorem}
We will need the following result of Poletsky (Corollary p. 170 in \cite{Po}, see also \cite{Po2}), generalized by Rosay (\cite{Ro}). Let $U$ a connected complex manifold of dimension $n\geq 1$. We denote by $\mathcal{H}_{z,U}$ the set of holomorphic functions $h:V_h\rightarrow U$ from a neighbourhood $V_h$ of $\overline{\Delta}=\{|z|\leq 1\}\subset\mathbb{C}$ (possibly depending on $h$) into $U$ such that $h(0)=z$. We also denote by $PSH(U)$ the set of plurisubharmonic functions defined on $U$.
\begin{pro}\label{poletsky} Let $u:U\rightarrow \mathbb{R}$ be an upper semi-continuous function. With the previous notations, the function defined by $$\displaystyle \tilde{u}(z):=\frac{1}{2\pi}\inf_{f\in \mathcal{H}_{z,U}}\int_0^{2\pi} u(f(e^{i\theta}))d\theta ,$$ if it is not everywhere equal to $-\infty$, belongs to $PSH(U)$ and verifies $\tilde{u}\leq {u}$. Moreover, this function $\tilde{u}$ is maximal among all the functions in $PSH(U)$ verifying this inequality.
\end{pro}
\begin{rmq}We deduce from Proposition \ref{poletsky} the following property of antisubharmonic functions, i.e. functions with subharmonic opposite. Let $B:=B(a,r)\subset\mathbb{C}$ be an open ball, $\displaystyle u:\overline{B}\rightarrow\mathbb{R}$ a continuous function, antisubharmonic in $B$. Then $\hat{u}:B\rightarrow\mathbb{R}$ is an harmonic function, with the same boundary values as $u$, in the sense that $\displaystyle \lim_{z\rightarrow z_0}\hat{u}=u(z_0)$ for $z_0\in \partial B$. \\
Indeed, given a continuous function $g:\overline{B}\rightarrow \mathbb{R}$, denote by $\tilde{g}:B\rightarrow \mathbb{R}$ the solution of the Dirichlet problem in $B$ with boundary condition $g_{|_{\partial B}}$, that is to say, the unique continuous function defined on $\overline{B}$ which is harmonic in $B$ and equal to $g$ on $\partial{B}$. Then $v:=\max(\tilde{u},\hat{u})$ is a subharmonic function with the same values as $u$ on $\partial B$. Since $u$ is antisubharmonic, we have $\tilde{u}\leq u$. Thus $$\hat{u}\leq v\leq u.$$ Since $\hat{u}$ is maximal among the subharmonic functions which are $\leq u$ in $B$ and equal to $u$ on $\partial B$, we conclude that $\hat{u}=v$, and hence $\tilde{u}=\hat{u}$. \\
Thanks to Theorem 3.1.4 in \cite{K}, the conclusion is the same if $B$ is a ball in $\mathbb{C}^n$, when substituting the expression "harmonic function" by "maximal plurisubharmonic function", and the expression "antisubharmonic function" by "antiplurisubharmonic function".\end{rmq}
Let $U\subset \mathbb{C}^n$, $n\geq 1$, be a bounded open set. Denote by $\lambda$ the normalized Lebesgue measure on the unit circle $\partial\mathbb{U}\subset\mathbb{C}$. Denote also by $\Lambda_{z,U}$ the set of measures of the form $h_*\lambda(\cdot):=\lambda(h^{-1}(\cdot))$, where $h:V_h\rightarrow U$ is an holomorphic function defined in a neigborhood $V_h$ (possibly depending on $h$) of the closed unit disk $\overline{\mathbb{U}}$, such that $h(0)=z$. Note that that the Dirac measure $\delta_z$ belongs to $\Lambda_{z,U}$. An immediate consequence of Proposition \ref{poletsky} is the following corollary, where $\mathbf{1}_G$ denotes the characteristic function of $G\subset\mathbb{C}^n$ :
\begin{cor}\label{klimek}Let $U\subset \mathbb{C}^n$ be a bounded open set, and $A\subset U$ a $L$-regular nonpolar compact set satisfying $\overline{\mathring{A}}=A$. Then $$\displaystyle \frac{1}{2\pi}\inf_{f\in \mathcal{H}_{z,U}}\int_0^{2\pi} -\mathbf{1}_{A}\circ f(e^{i\theta})d\theta=-\sup_{\mu_z\in\Lambda_{z,U}}\mu_z(A) =G_{A,U}(z).$$\end{cor}
\noindent Recall that we denote by $K$ the filled Julia set of a polynomial application $f:\mathbb{C}\rightarrow\mathbb{C}$ of degree $\geq 2$, and $dist(\cdot,\cdot)$ the euclidean distance on $\mathbb{C}^n$. Let us prove the main result stated in the introduction :
\begin{thm}\label{thm} Let $K\subset \mathbb{C}$ be the filled Julia set of a polynomial $f:\mathbb{C}\rightarrow \mathbb{C}$ of degree $d\geq 2$, of non-empty interior. Then $K$ satisfies the {\L}S condition.\end{thm}
\begin{proof}
For $b\in \mathbb{R}^{+}\setminus \{0\}$, denote $U_b:=\{G_K<b\}\subset \mathbb{C}$. For $l\in \mathbb{R}^{+}\setminus \{0\}$, denote also $K_l:=\{z\in\mathbb{C}\text{ }|\text{ }dist(z,K)\leq l\}$. Then choose $a>0$ such that $\displaystyle K_{2}\subset f^{-1}(U_a)$. Note that $f^{-1}(U_a)=U_{\frac{a}{d}}\subset\subset U_a$ by (\ref{invariance}). Denote by $\mathcal{C}_a$ the corona $U_a\setminus f^{-1}(U_a)$. There exists $\delta\in ]0,1[$ such that \begin{equation}\label{delta}\displaystyle G_{K_{2, U_a}}+1\geq\delta (G_{K,U_a}+1)\text{ on }\mathcal{C}_a.\end{equation} Take $\displaystyle c\in \left]0,\frac{\delta}{2a}\right[$, sufficiently small to have $\overline{O_c}\subset f^{-1}(U_a)$ and $\left(\frac{1}{c^2}\right)^c<2$.
We have $\forall \epsilon\in ]0,2]$, $\forall y\in U_a$,
\begin{align*}
\displaystyle c\cdot dist(y,K)^{\frac{1}{c}}&\geq \inf_{\mu_y\in \Lambda_{y,U_a}}\int_{U_a} c\cdot dist(\cdot,K)^{\frac{1}{c}}d\mu_y\\
&\geq \inf_{\mu_y\in \Lambda_{y,U_a}}\int_{U_a\setminus K_{\epsilon}} c\cdot dist(\cdot,K)^{\frac{1}{c}}d\mu_y \\
&\geq \left(\min_{\overline{U_a}\setminus \mathring{K_{\epsilon}}}c\cdot dist(\cdot, K)^{\frac{1}{c}}\right)\inf_{\mu_y\in \Lambda_{y,U_a}}\int_{U_a\setminus K_{\epsilon}}d\mu_y \\
&= c\epsilon^{\frac{1}{c}}(G_{K_\epsilon,U_a}+1)(y).
\end{align*}
The first inequality comes from the fact that the Dirac measure $\delta_y$ belongs to $\Lambda_{y,U_a}$. The last inequality comes from Corollary \ref{klimek}, whose application is allowed by Lemma \ref{julia}. Then taking $\epsilon=\left(\frac{1}{c^2}\right)^c<2$, we obtain in $U_a$: \begin{equation}\label{estimation}\displaystyle c\cdot dist(\cdot,K)^{\frac{1}{c}}\geq \frac{1}{c}(G_{K_\epsilon,U_a}+1).\end{equation}
Now suppose, by contradiction, that $O_c\neq \emptyset$ (see Equation (\ref{Oc}) for definition). Thanks to the fact that $c<\frac{\delta}{2a}$, we can choose $x\in O_c\setminus \{G_{K}<\frac{2ac^2}{\delta}dist(\cdot,K)^{\frac{1}{c}}\}$.
Note that $x\in O_c$ implies a "slow growth" of $(f^n(x))_n$, in the sense that $\forall n\geq 1$ such that $f^n(x)\notin O_c$, we have
$$\displaystyle\frac{1}{d^n}c\cdot dist(f^n(x),K)^{\frac{1}{c}}\leq G_K(x)< c\cdot dist(x,K)^{\frac{1}{c}},$$ and hence
\begin{equation}\label{slow}\displaystyle dist(f^n(x),K)< d^{nc} dist(x,K).\end{equation}
\noindent Since $\displaystyle U_a\setminus K = \bigcup_{i\geq 0}f^{-i}(\mathcal{C}_a)$ by (\ref{invariance}), there exists $N>0$ such that $f^N(x)\in \mathcal{C}_a$. Equations (\ref{slow}), (\ref{estimation}), (\ref{delta}), (\ref{relation}), then (\ref{invariance}), give
\begin{align*}
\displaystyle c\cdot dist(x,K)^{\frac{1}{c}} &\geq \frac{c}{d^N} dist\left(f^N(x),K\right)^{\frac{1}{c}}\\
&\geq \frac{1}{cd^N}(G_{K_{\epsilon},U_a}+1) \circ f^N(x) & \\
\\ &\geq \frac{\delta}{cd^N}(G_{K,U_a}+1)\circ f^N(x)\\
&= \frac{\delta}{ca}G_K(x).
\end{align*}
But this contradicts our assumption $x\notin \{G_{K}<\frac{2ac^2}{\delta}dist(x,K)^{\frac{1}{c}}\}$. We conclude that $O_c=\emptyset$. In other words, $K$ satisfies the {\L}S condition.
\qed
\end{proof}
\begin{rmq}We note that if $f$ is assumed to be {\bf{hyperbolic}}, that is to say if $f$ do not have critical points in $J$, there exist a constant $b>0$ and a neighborhood of $K$ in which \begin{equation}\label{control}\displaystyle dist\left(f(\cdot), K\right)\geq b\cdot dist(\cdot, K).\end{equation}Indeed, it is sufficient to etablish this inequality outside $K$. Let then $V$ be a neigborhood of $K$ in which $|f'|\geq a$ for some $a>0$, let $z\in V\setminus K$, and $z_0\in J$ such that $f(z_0)\in J$ achieves the distance $dist(f(z),J)$. Then Theorem 1 of \cite{D} shows the existence of a constant $k>0$ (depending only on the degree of $f$) and of a point $z_1\in J=\partial K$, such that $$\displaystyle dist(f(z),K)=dist\left(f(z),f(z_0)\right)\geq a\cdot k \cdot dist(z,z_1)\geq a\cdot k \cdot dist(z,K).$$
\noindent In the particular case where $b\geq 1$ in (\ref{control}), we obtain a simpler proof of Theorem \ref{thm}, and a more quantitative estimation for $c$ in Equation (\ref{Oc}). Indeed, suppose $O_c\neq \emptyset$ with $O_c\subset \subset V$. We can choose $x\in O_c$ such that $f(x)\notin O_c$. Then, (\ref{slow}) together with (\ref{control}) give $$\displaystyle c>\frac{\log{b}}{\log d}.$$
\end{rmq}
\begin{acknowledgements}
We thank Laurent Gendre and Ta\"ib Belghiti for their reading and suggestions, as well as Marta Kosek and the whole organization committee of the conference "On Constructive Theory of Functions" for their invitation to expose a preliminary version of this article in Poland.
\end{acknowledgements}
|
2,869,038,154,745 | arxiv | \section{Introduction}
Complex organic molecules have been detected toward a range of astrophysical environments, including low-mass protostars \citep{vanDishoeck95, Cazaux03, Bottinelli07}; however, the origins of these complex molecules as well as their fates are uncertain. Commonly-suggested formation routes for the detected molecules include various gas-phase reactions starting with thermally evaporated CH$_3$OH ice, atom-addition reactions on dust grains, and UV- and cosmic ray-induced chemistry in the icy grain mantles that form during the pre-stellar stages \citep[][for a review]{Charnley92, Nomura04, Herbst09}. The focus is currently on an ice formation pathway \citep[e.g.][]{Garrod06, Garrod08} because of the failures of gas phase chemistry to explain the observed abundances of some of the most common complex molecules toward low-mass protostars, especially HCOOCH$_3$.
Recent experiments on the photochemistry of CH$_3$OH-rich ices have shown that 1) UV irradiation of CH$_3$OH ices at 20--70~K result in the production of large amounts of the complex molecules observed around protostars and 2) the chemistry has a product branching ratio which is temperature and ice composition dependent \citep{Oberg09d}. A key result is that HCOOCH$_3$ and other HCO-X ices are only abundantly produced in CO:CH$_3$OH mixtures. A similar result has been reported for proton-bombarded CH$_3$OH and CH$_3$OH:CO ices \citep{Bennett07b}. CO:CH$_3$OH ice mixtures are probably common since CH$_3$OH forms from hydrogenation of CO \citep[e.g.][]{Watanabe03,Cuppen09}. CO evaporates at 17--25~K on astrophysical time scales \citep{Bisschop06}, and thus HCO-X ices will mainly form in cold, UV-exposed ices. At higher temperatures, closer to the protostar, UV irradiation of the remaining pure CH$_3$OH ice will instead favor the production of C$_2$H$_5$OH, CH$_3$OCH$_3$ and (CH$_2$OH)$_2$.
In light of this proposed formation scenario, protostars rich in CH$_3$OH ice are natural targets when searching for complex molecule sources. The low-mass protostar B1-b is such a source. From previous SCUBA and 3 mm continuum maps, B1-b consists of two cores B1-bN and B1-bS, separated by 20'' \citep{Hirano99}. The cores are similar with $T_{\rm dust}\sim18$~K, $M=1.6-1.8\:M_{\odot}$ and $L_{\rm bol}=2.6-3.1\:L_{\odot}$. The {\it Spitzer Space Telescope} only observed one protostar, to the south-west of the B1-bS dust core at 03:33:20.34, +31:07:21.4 (J2000), but it was still named B1-b -- to avoid confusion it will be referred to as the `protostar' position. From the {\it Spitzer} $c2d$ (From Molecular Cores to Planet Forming Disks) ice survey the B1-b protostar has a CH$_3$OH ice abundance of 11\% with respect to H$_2$O ice \citep{Boogert08}, corresponding to $\sim5\times10^{-6}$ with respect to H$_2$. The fractional abundance of H$^{13}$CO$^+$ decreases by a factor of 5 toward the SCUBA core, indicative of CO freeze-out \citep{Hirano99}. A HCO-dominated complex chemistry is thus expected where the ice is exposed to UV radiation.
The B1-b region is complicated by a number of outflows, which may enhance the UV field through shocks and open cavities through which stellar UV photons can escape \citep{Jorgensen06, Walawender09, Hiramatsu10}. The outflow of most interest to this study runs in the south-west direction from B1-b protostar, though it is not known whether B1-b is its source or just happens to lie in its path. The outflow is not observed north-east of the B1-b protostar, where the B1-b SCUBA core is situated. It is therefore unclear whether the outflow terminates at the protostar or continues into the core, hidden from view. If it does penetrate into the SCUBA core, this may increase the UV flux in the region by orders of magnitude, enhancing both the UV photochemistry and photodesorption of ices.
The latter is important, since once formed, the complex molecules must be (partly) released into the gas phase to be observable at millimeter wavelengths. The second reason for targeting B1-b is the detection of a large CH$_3$OH gas phase column density of $\sim$2.3--2.5$\times10^{14}$ cm$^{-2}$ toward the protostar \citep{Oberg09a,Hiramatsu10} and thus ice evaporation. The narrow line width and low excitation temperature of the observed CH$_3$OH suggest non-thermal ice evaporation -- the abundances are consistent with UV photodesorption. Assuming that complex ice chemistry products desorb non-thermally at an efficiency similar to CH$_3$OH ($\sim$10$^{-3}$ molecules per incident UV photon \citep{Oberg09d}), the protostellar environment should contain a cold gas-phase fingerprint of the complex ice composition; B1-b may offer the first possibility to observe the earliest stages of complex molecule formation, untainted by either warm ice chemistry or gas-phase processing.
Addressing these predictions, we present IRAM 30-m telescope observations of a $80''\times80''$ CH$_3$OH map of the B1-b protostellar envelope and nearby dust cores, followed by a search for HCOOCH$_3$, CH$_3$CHO, CH$_3$OCH$_3$, C$_2$H$_5$OH and HOCH$_2$CHO toward two of the identified CH$_3$OH peaks. The CH$_3$OH gas abundances and the complex abundances and upper limits with respect to CH$_3$OH are then discussed in terms of different ice desorption mechanisms and complex molecule formation scenarios. The results are finally compared with complex molecule observations toward other low-mass protostars and outflows.
\section{Observations and data reduction}
The B1-b protostellar envelope and the nearby SCUBA cores were mapped in three CH$_3$OH transitions in February 2009 using the HEterodyne Receiver Array HERA \citep{Schuster04} on the Institut de RadioAstronomie Millim\'etrique (IRAM) 30-m telescope together with the VESPA autocorrelator backend (Tables \ref{tab:obs} and \ref{tab:ch3oh}). The $80''\times80''$ map is centered on R.A 03:33:21.3 and Dec 31:07:36.7 (J2000), the estimated midpoint between B1b-S and B1b-N \citep{Hirano99}. VESPA was used with a 320 kHz channel spacing resulting in a velocity resolution of 0.78 km s$^{-1}$ (channel spacing $\delta$V of $\sim$0.39 km s$^{-1}$) at 242 GHz. Observations were carried out through two separate undersampled (8'' spacing) raster maps using beam-switching with a throw of 200'' in azimuth. The two maps were off-set by +4'', + 4'' and the combination of the two maps thus results in a single over-Nyquist-sampled map -- the half power beamwidth of the 30-m telescope at 242 GHz is $\sim$10''. The rms is $\sim$50~mK in the first map and $\sim$130~mK in the second map in a 0.39 km s$^{-1}$ channel. The average rms over the entire map was estimated to $\sim$90~mK. The system temperature varied between 600 and 1200~K.
The second set of observations aimed at detecting complex molecules were carried out in June 2009 with the IRAM 30-m telescope, using the new EMIR receiver (Table \ref{tab:obs}). The positions used for pointing correspond to the CH$_3$OH maximum toward the center of the SCUBA core at 03:33:20.80, 31:07:40.0 (J2000), termed the 'core' position, and a local maximum at 03:33:21.90, 31:07:22.0 (J2000) termed the 'outflow' position. The targeted lines are listed in Table \ref{tab:comp} and the frequencies are taken from the JPL molecular database and the Cologne Database for Molecular Spectroscopy \citep{Muller01}. The focus is on lines with low excitation energies (15--100~K) because of the previously discovered low excitation temperature of CH$_3$OH toward the same source. The chosen settings contain at least two lines for HCOOCH$_3$, CH$_3$CHO, CH$_3$OCH$_3$ and C$_2$H$_5$OH (and additional HOCH$_2$CHO and C$_2$H$_5$CN lines) to allow us to constrain the excitation temperatures of any detected species. The observations were carried out using three different receiver settings, each including a combination of the EMIR 90 and 150 GHz receivers (Table \ref{tab:obs}). At these wavelengths, the beam sizes are $\sim$27 and 19'', respectively. All three settings were used toward the core position, and two toward the outflow position. Each receiver was connected to a unit of the autocorrelator, with a spectral resolution of 40~MHz and a bandwidth of 120~MHz, equivalent to an unsmoothed velocity resolution of $\sim$0.1 km s$^{-1}$. Typical system temperatures were 100-150 K. All observations were carried out using wobbler switching with a 140'' throw (100'' in azimuth and 100'' in elevation).
For both sets of observations, pointing was checked every $\sim$2 hr on J0316+413 with a typical pointing accuracy of $\sim$2--3''. All intensities reported in this paper are expressed in units of main-beam brightness temperature, which were converted from antenna temperatures in paKo using reported main beam and forward efficiencies (B$_{\rm eff}$ and F$_{\rm eff}$) of $\sim$75 and 95\% with the E90 receiver, $\sim$69 and 93\% with E150 receiver and a main beam efficiency of 52\% at 2 mm for the HERA receiver. The rms in mK are reported in Table \ref{tab:obs}.
The data were reduced with the CLASS program, part of the GILDAS software package (see http://www.iram.fr/IRAMFR/GILDAS). Linear (first-order) baselines were determined from velocity ranges without emission features, and then subtracted from the spectra. Some velocity channels showed spikes, which were replaced by interpolating the closest two good channels in the map, while no correction was made to the complex molecule spectra -- there were no spikes close to any expected line positions and thus no correction was necessary. The complex organics observation were smoothed to velocity resolutions of 0.4-0.8 km s$^{-1}$ toward the core position and 2-4 km km s$^{-1}$ toward the outflow position respectively to achieve a maximum signal-to-noise while still barely resolving the expected lines.
\section{Results}
\subsection{CH$_3$OH}
Figure \ref{fig:int} shows the integrated CH$_3$OH map superimposed on a SCUBA dust map of the B1-b region. The dust map contains one elongated core, rather than the two sources observed by \citet{Hirano99}. The SCUBA core in the remainder of the paper refers to the dust core in Fig. \ref{fig:int}, while the `core' position refers to the nearby CH$_3$OH maximum probed for complex molecules. The dust and molecular maps share some features, such as the N-S elongation and the 'tail' toward the south-west, but the centers of the two maps are offset -- the CH$_3$OH column density peaks in between the SCUBA core and the `protostar' suggestive of ice desorption on the west side of the dust core due to irradiation from the protostar traveling through outflow cavities, as discussed in more detail below.
A velocity channel map of the same CH$_3$OH line shows that in addition to the quiescent, elongated CH$_3$OH emission that overlaps with the western half of the dust emission, there are at least two high velocity components, suggestive of two perpendicular outflows which are either directed along the line of sight or which are too young to have any resolved component in the plane of the sky (Fig. \ref{fig:map}). The two outflow positions are almost coincidental with B1-bS as reported by \citet{Hirano99} and may thus be evidence for a very young protostellar object embedded in the B1-bS core.
Figure \ref{fig:ch3oh_sp} shows the extracted spectra at two of the CH$_3$OH intensity peaks in the channel map -- the `core' and southeast `outflow' positions. The CH$_3$OH emission toward the `core' consists of narrow, symmetric lines with small wings. Overall these spectra seem to be dominated by quiescent gas. In contrast the CH$_3$OH lines toward the 'outflow' position are more than four times broader and asymmetric, consistent with the assignment to an outflow. The spectra toward the `core' are similar to those observed by \citet{Hiramatsu10} toward the same line of sight. They did however not find any evidence for outflow activity toward B1-b. The lack of channel maps in \citet{Hiramatsu10} prevents direct comparison, but the detected outflows here are on a much smaller scale compared to the outflows they do observe and may thus have been overlooked.
Table \ref{tab:ch3oh} lists the observed CH$_3$OH transitions as well as the line FWHM and integrated intensities toward the core and outflow positions. To facilitate comparison with observations of complex molecules and previous CH$_3$OH observations at lower frequencies, the CH$_3$OH intensities were extracted both directly from the HERA maps and from maps convolved with 19'' and 27'' beams, the IRAM 30-m beam sizes at 131 and 89 GHz, respectively. Because of the asymmetry in the lines, the integrated intensities were calculated by integrating from -10 to +5 km s$^{-1}$ around each line center, rather than integrating the fitted Gaussians used to estimate the FWHM.
The excitation temperature of CH$_3$OH toward each position is estimated using rotational diagrams \citep{Goldsmith99} in Fig. \ref{fig:ch3oh_rot} based on the CH$_3$OH map intensities convolved with a 27'' beam. The `core' and `outflow' positions only contain three CH$_3$OH lines each and the derived temperatures are therefore uncertain by at least a factor of two. The best fits results in 10~K for the `core' and 8~K for the `outflow'. For the `protostar' position the new data are combined with previous IRAM 30-m data observed at $\sim$97 GHz \citep{Oberg09a}. The resulting excitation temperature is $11\pm1$~K, confirming the estimated rotational temperatures toward the `core' and `outflow'. At the densities expected in molecular cloud cores, CH$_3$OH is easily sub-thermally excited -- rotational temperatures below 25~K are expected at densities up to 10$^6$ cm$^{-3}$ even if the kinetic temperature is above 100~K \citep{Bachiller98}. NH$_3$ maps of the same region however confirm that in this case the kinetic temperature is close to the CH$_3$OH rotational temperature, $\sim$10--12~K \citep{Bachiller90}.
\subsection{Search for complex molecules}
Building on the CH$_3$OH map, we searched for complex molecules toward the `core' and `outflow' positions. All acquired spectra are shown in the Appendices. Figure \ref{fig:comp_sp} shows six blow-ups focusing on the targeted complex molecules. The two lowest lying transitions of HCOOCH$_3$ ($E_{\rm up}$ = 18~K) and of CH$_3$CHO ($E_{\rm up}$ = 28~K) are detected toward the core region (Fig. \ref{fig:comp_sp}a,d) at the 5-$\sigma$ level. Higher lying transitions of the two molecules are not detected at the 3-$\sigma$ level, nor are any transitions of C$_2$H$_5$OH, HOCH$_2$CHO or C$_2$H$_5$CN. One 2-$\sigma$ transition of CH$_3$OCH$_3$ is tentatively seen and its intensity is consistent with non-detections of the other lines. No complex molecule lines are detected toward the outflow (Fig. \ref{fig:comp_sp}). The two securely detected species, HCOOCH$_3$ and CH$_3$CHO, both display narrow line profiles between 0.5--0.9 km s$^{-1}$. The HCOOCH$_3$ lines show no evidence of asymmetry, while the CH$_3$CHO lines have high-velocity wings suggestive of an emission contribution from the 'O2' outflow.
All data for the targeted lines are listed in Table \ref{tab:comp}, including molecule transitions, catalogue frequencies, $E_{\rm low}$, FWHM of detected lines, integrated intensities for detected lines and integrated intensity upper limits for non-detections. The upper limits on higher lying HCOOCH$_3$ and CH$_3$CHO lines are consistent with the $\sim$10~K excitation temperature of CH$_3$OH toward the same line of sight. Thus a 10~K excitation temperature is assumed when calculating column densities.
Using these excitation temperatures, the total HCOOCH$_3$ column density (A + E) toward the B1-b core is $8.3\pm2.8\times10^{12}$ cm$^{-2}$, averaged over the telescope beam, and the total (A + E) CH$_3$CHO column density is $5.4\pm1.4\times10^{12}$ cm$^{-2}$, which corresponds to abundances with respect to CH$_3$OH of 2.3 and 1.1\% respectively. The different beam sizes at different frequencies are accounted for by comparing the HCOOCH$_3$ to the CH$_3$OH column density derived with a 27'' beam and the CH$_3$CHO to the CH$_3$OH column density derived with the 19'' beam. The upper limit abundances for the other molecules are $\lesssim$0.8, $<$1.0\% and $<$1.1\% for CH$_3$OCH$_3$ , C$_2$H$_5$OH and HOCH$_2$CHO toward the core. The most significant upper limit toward the outflow is the total HCOOCH$_3$ abundance of $<$1.4\% with respect to CH$_3$OH. The calculated abundances and upper limits of complex molecules are summarized in Table \ref{tab:abund}. The lack of detections toward the outflow does not prove that it contains significantly less complex molecules than the `core', but the outflow position is definitely not richer in complex organics than the core region. The outflow upper limits are comparable to the detected abundances toward another low-mass outflow L1157 \citep{Arce08}
\section{Discussion}
\subsection{The origin of the CH$_3$OH gas}
The CH$_3$OH average abundance toward the quiescent CH$_3$OH `core' is $\sim2\times10^{-9}$ with respect to N$_{\rm H_2}$, assuming a SCUBA dust intensity conversion factor of N$_{\rm H_2}/S_{\rm \nu}^{\rm beam}$ of 1.3$\times10^{20}$ cm$^{-2}$ (mJy beam$^{-1}$)$^{-1}$ \citep{Kauffmann08}. This abundance is too high to explain with gas-phase formation of CH$_3$OH through any mechanism investigated so far, i.e. gas phase reactions in quiescent cores result in CH$_3$OH abundances below 10$^{-13}$ \citep{Garrod07}. The observed CH$_3$OH gas is thus a product of ice desorption.
The protostar B1-b belongs to a small set of young low-mass stellar objects with CH$_3$OH ice abundances of $>$10\% with respect to H$_2$O \citep[][Bottinelli et al. submitted to A\&A]{Boogert08}. CH$_3$OH ice has been previously observed to vary on small scales, however, and the ice abundance towards the `core' and `outflow' may be as low as 1\% with respect to H$_2$O ice. Still, $<$0.5\% of the CH$_3$OH ice must evaporate to account for the observed CH$_3$OH abundance of $\sim$2$\times10^{-9}$ toward the `core', assuming a H$_2$O abundance of $\sim5\times10^{-5}$ with respect to H$_2$.
The extent of the CH$_3$OH gas ($\sim$10000$\times$15000 AU) together with the low excitation temperature of CH$_3$OH and the offset between the CH$_3$OH `core' and the `protostar' exclude thermal ice evaporation as a major contributor to the observed abundances -- a low-mass protostar cannot heat more than a few 100 AU to the sublimation temperature of CH$_3$OH of $\sim$80~K. This leaves non-thermal evaporation of ices, which can be broadly divided into two categories: ice evaporation due to grain sputtering in shocks and less violent non-thermal ice evaporation due to UV irradiation, cosmic rays or release of chemical energy \citep{Jones96, Shen04, Garrod07}. Grain sputtering is often invoked to explain excess CH$_3$OH in protostellar outflows \citep[e.g.][]{Bachiller01} and is probably responsible for the factor of a few higher abundances of CH$_3$OH gas in the two outflow positions around the dust core, using the same conversion factor as above to derive the dust column densities from the SCUBA map. The narrow CH$_3$OH lines in the remainder of the map suggest a different origin for the gas toward the protostar and `core' regions.
The asymmetric distribution of CH$_3$OH compared to the dust core further suggest that ice is desorbed by irradiation from the direction of the protostar, illuminating the south-west side of the core. The exact nature of the irradiation is difficult to constrain without detailed modeling, but UV photodesorption is known to be efficient \citep{Oberg09d}. The dust core has a visual extinction of $>$100 mag. For radiation to affect the `core' region thus requires extensive cavities in the interstellar medium. As discussed in the Introduction, there are several large-scale outflows crossing the B1 region, which could have carved out such cavities \citep{Jorgensen06, Walawender09}. In particular Fig. 1 shows that there is a visible outflow in the 'right' direction originating either from the B1-b protostar or from the nearby B1-d protostar to the southwest. In either case, the outflow may continue on towards the 'core', hidden from observations by the large column of dust, resulting in UV irradiation of the icy grains on the side of the dust core closest to the protostar.
UV irradiation can be generated in a number or ways within such an outflow cavity. If an outflow is still active in the direction of the `core' position, the UV radiation may originate in the jet shock itself \citep{Reipurth01}. Some outflows are also known to become hot enough to emit x-rays \citep{Pravdo01}, which may directly desorb the ice or desorb it through secondary UV photons. The narrow line width of the CH$_3$OH emission suggest however that the CH$_3$OH gas does not originate close to shocked gas. It seems more likely that a past outflow has opened a cavity between the protostar and the `core' position and that UV radiation from this neighboring low-mass protostar protostar is desorbing the ices. Low-mass protostars are known to have excess UV fluxes compared to the interstellar radiation field, originating in the boundary layer between the protostar and the accretion disk \citep{Spaans95}.
UV photodesorption is thus a probable source of the CH$_3$OH gas distribution towards the B1-b dust core. Observations of UV tracers are however needed to confirm this scenario, since we cannot completely rule out that the CH$_3$OH are the left-overs from grain-sputtering by the same shock that opened up the cavity, if it passed through the area 10$^4$-10$^5$ years ago, the typical depletion time-scale at molecular cloud densities.
\subsection{The origin of complex molecules toward B1-b}
The similar line widths and excitation temperature upper limits of HCOOCH$_3$ and CH$_3$CHO compared to CH$_3$OH suggest that the complex molecules toward the B1-b core originate from photodesorbed ice as well. This is a new potential source of complex organic molecules around low-mass protostars compared to what has been previously suggested in the literature, where complex molecules have been observed in a small warm core or disk where ices have evaporated thermally \citep[e.g.][]{Bottinelli07,Jorgensen05} or in shocks following grain sputtering \citep[e.g.][]{Arce08}. In difference to sputtering and thermal desorption, which quickly destroys the entire ice mantle in a small region, UV photodesorption can release a small fraction of the ice over large quiescent regions. This offers the opportunity to study ice chemistry {\it in situ} as it evolves around protostars, as previously suggested in \citet{Oberg09a, Oberg09d}.
Non-thermal desorption of HCOOCH$_3$ and CH$_3$CHO is only a probable source of the observed gas if the molecules can be produced efficiently on grains, however. As discussed in the Introduction, such a production channel exists. UV processing of CO:CH$_3$OH-rich ices produces complex molecules at abundances that are consistent with the observed abundances of a few \% with respect to CH$_3$OH.
Moreover this formation path explains the relative abundances of HCOOCH$_3$ and CH$_3$OCH$_3$ or C$_2$H$_5$OH. Figure \ref{fig:reaction} illustrates how the addition of CO affects the complex CH$_3$OH chemistry to produce the observed molecules. Quantitatively, UV irradiation of a CH$_3$OH:CO 1:10 ice mixture at 20~K under laboratory conditions results in the production of a CH$_3$OCH$_3$/HCOOCH$_3$ ratio of $<$0.1 and irradiation of a pure CH$_3$OH ice in CH$_3$OCH$_3$/HCOOCH$_3$ ratios of $>$1.3. While the laboratory results cannot be directly extrapolated to B1-b, a CO-rich ice is certainly needed to produce the observed low CH$_3$OCH$_3$(C$_2$H$_5$OH)/HCOOCH$_3$ ratios; without CO in the ice, the most recent model predict ratios of $>$10, rather than the observed $<$0.5 \citep{Garrod08}.
\subsection{Comparison with previous complex chemistry observations}
With energetic CO:CH$_3$OH ice processing followed by non-thermal desorption established as the most probable explanation for the observed complex molecules in B1-b, it is interesting to see how the chemistry toward B1-b compares with complex chemistry observations toward other low-mass sources: NGC1333-IRAS 2A, 4A and 4B, IRAS 16293-2422 and L1157. All sources have been observed with single-dish telescopes where the emission region is assumed to be unresolved. These observations thus contain emission from both hot and cold material -- dependent on the density profile one or the other component may dominate. In addition IRAS 16293-2422 has been studied extensively using interferometry. In these observations two cores, A and B, are resolved and emission from the cold envelope can be excluded.
Because of assumptions on the emitting region for complex molecules in most previous studies, comparisons of absolute abundances are difficult between different sources using literature values. Assuming the same source sizes of CH$_3$OH and complex molecules toward protostars observed with single-dish telescopes result in abundances of a few \% with respect to CH$_3$OH, comparable to those observed toward B1-b \citep{Maret05, Bottinelli04a, Bottinelli07}. Where beam averaged or resolved values of complex molecule abundances are actually reported, e.g. toward the low-mass outflow L1157 and the hot cores (or accretion disks) of the two IRAS 16293-2422 cores, the abundances are also of the order of a few \% \citep{Bisschop08, Arce08}. There is thus no evidence that the complex molecule abundances with respect to CH$_3$OH toward B1-b are special compared to other low-mass sources, ranging from outflows to hot cores.
Table \ref{tab:compar} lists the abundance ratios of different complex molecules toward the different low-mass sources. The B1-b CH$_3$CHO/HCOOCH$_3$ ratio falls in between what has been previously measured toward the A and B cores in IRAS 16293. The upper limits on the CH$_3$OCH$_3$/ HCOOCH$_3$ and C$_2$H$_5$OH/HCOOCH$_3$ ratios are comparable to the single-dish observations toward NGC1333 2A, 4A and 4B, and possibly toward the IRAS 16293 envelope as well, but significantly lower compared to the resolved observations toward the IRAS 16293 A and B cores. In resolved observations toward NGC 1333 2A there is also a CH$_3$OCH$_3$ detection pointing to a higher CH$_3$OCH$_3$ to HCOOCH$_3$ ratio in the warm region close to the protostar \citep{Jorgensen05}. This is entirely consistent with a scenario where complex molecules form in cold CO:CH$_3$OH ice mixtures in the outer envelope, followed by further formation of CH$_3$OHCH$_3$ and C$_2$H$_5$OH in the warmer CO-depleted CH$_3$OH ice close to the protostar (Fig. \ref{fig:cartoon}). In the cold parts complex ices are only released non-thermally through e.g. UV photodesorption, while closer to the protostar thermal desorption is likely to dominate. This agrees with the low excitation temperatures in the single dish observations compared to the observations that resolve the innermost part of the envelope.
\section{Conclusions}
\begin{enumerate}
\item Cold CH$_3$OH gas (excitation temperature of $\sim$10~K) is abundant and widespread toward B1-b. Quiescent CH$_3$OH is most abundant in between the B1-b SCUBA dust core and the B1-b protostar, indicative of UV photodesorption of ice on the side of the quiescent dust core because of radiation from the protostar escaping through outflow cavities.
\item Cold HCOOCH$_3$ and CH$_3$CHO are both detected and CH$_3$OCH$_3$ is tentatively detected toward the quiescent CH$_3$OH peak. No complex molecules are observed toward an outflow associated with B1-b, but 3-$\sigma$ upper limits with respect to CH$_3$OH are only slightly lower compared to the quiescent core. In addition an asymmetry in the CH$_3$CHO emission lines is suggestive of some contribution from a small outflow included in the beam.
\item Assuming a 10~K excitation temperature the calculated beam-averaged abundances are 2.3\% for HCOOCH$_3$, 1.1\% for CH$_3$CHO, $\lesssim$0.8\% for CH$_3$OCH$_3$ and $<$1.0-1.1\% for C$_2$H$_5$OH and HOCH$_2$CHO with respect to CH$_3$OH.
\item Building on recent experiments, the observations of large abundances of HCO-containing molecules are explained by UV/cosmic ray processing of cold CH$_3$OH:CO ice followed by non-thermal desorption of a fraction of the produced organic ice.
\item The beam-averaged abundances with respect to CH$_3$OH and the ratios between different complex molecules are similar to other single-dish observations toward low-mass protostars. In contrast resolved observations of the cores around protostars are more abundandant in complex molecules without a HCO-group. This is consistent with a complex ice chemistry that evolves from HCO-rich to HCO-poor as the ice warms up and the CO-ice evaporates when the icy dust falls toward the protostar.
\end{enumerate}
Including B1-b, there are still only a handful of observations of complex molecules toward low-mass protostars. In addition few complex molecules are detected toward each source. This makes it difficult both to determine the prevalence of a complex organic chemistry around low-mass protostars and to firmly establish the main formation path of such molecules. The detections of complex molecules in the vicinity of the CH$_3$OH-ice-rich protostar suggests that targeting other CH$_3$OH-ice rich protostars may increase our sample of complex molecule detections significantly. Maybe more important from a formation pathway point of view is to increase our understanding on how the complex chemistry varies between warm, luke-warm and cold regions i.e. how it varies with distance from the protostar. This should be pursued both on larger scales with single dish observations and on smaller scales with interferometers -- similarly to what has been done toward IRAS 16293 and NGC1333 2a \citep[e.g.][]{Jorgensen05,Bisschop08}. In the meantime the complex molecules observed toward B1-b, and their exceptionally low excitation temperatures and line widths, provides clear evidence, for the efficient formation of complex ices during star- and planet-formation.
\acknowledgments
We are grateful to the IRAM staff for help with the observations and reduction of the resulting data. This work has benefitted from discussions with Herma Cuppen, Robin Garrod and Lars Kristensen and from comments by an anonymous referee. Umut Yildiz carried out the second round of observations at the IRAM 30-m with the financial support of RadioNet. Support for K.~I.~\"O is provided by NASA through Hubble Fellowship grant awarded by the Space Telescope Science Institute, which is operated by the Association
of Universities for Research in Astronomy, Inc., for NASA, under contract NAS 5-26555. Astrochemistry in Leiden is supported by a SPINOZA grant of the Netherlands Organization for Scientific Research (NWO). The Centre for Star and Planet Formation is funded by the Danish National Research Foundation and the University of Copenhagen's programme of excellence.
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2,869,038,154,746 | arxiv | \section{Introduction}\label{int}
Denote by $\zeta_N$ an $N$-th primitive root of unity. Let $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N])$ denote the category of mixed Tate motives unramified over $\mathbb{Z}[\zeta_N][1/N]$. By the main result of \cite{dg}, the motivic fundamental groupoid of $\mathbb{P}^1-\{0,\mu_N,\infty\}$ can be realized in the category of $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N])$.
We call the Lie algebra of the maximal pro-unipotent subgroup of motivic fundamental group of $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N])$ the motivic Lie algebra of $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N])$.
From Proposition 2.3 in \cite{dg}, the motivic Lie algebra of $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N])$ is graded free Lie algebra.
Since the sub-Tannakian category generated by the function ring of the motivic fundamental groupoid of $\mathbb{P}^1-\{0,\mu_N,\infty\}$ is $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N]$ for $N=1$ by \cite{brown} and for $N=2,3,4,6,8$ by \cite{del}, from \cite{dg} we know that the motivic Lie algebra of $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N])$ has an induced depth filtration for $N=1,2,3,4,6,8$.
In \cite{del}, P. Deligne proves that the depth-graded motivic Lie algebra of $\mathcal{MT}(\mathbb{Z}[\zeta_N][1/N])$ is a free Lie algebra bi-graded by weight and depth for $N=2,3,4,6,8$. While the structure of depth-graded motivic Lie algebra of $\mathcal{MT}(\mathbb{Z})$ is not fully understood up to now.
L. Schneps gives the structure of depth-graded motivic Lie algebra of $\mathcal{MT}(\mathbb{Z})$ in depth two \cite{sch}. And A. B. Goncharov's work \cite{gon} give the structure of depth-graded motivic Lie algebra of $\mathcal{MT}(\mathbb{Z})$ in depth three.
F. Brown gives some conjectural description of the structure of the depth-graded motivic Lie algebra of of $\mathcal{MT}(\mathbb{Z})$ in all depth in \cite{depth}.
It's widely believed that the Lie subalgebra of depth-graded motivic Lie algebra generated by the depth one part only has the period polynomial relations in depth two among the generators (in \cite{li} we call this statement the non-degenerated conjecture). In this paper, we will show that from an isomorphism conjecture of K.Tasaka \cite{tasaka} we can deduce Brown's matrix conjecture and the non-degenerated conjecture. Thus we reduce the well-konwn non-degenerated conjecture to a purely linear algebra problem which probably are more easy to handle.
And from the analysis in \cite{en}, our results give partial evidence to Brown's homological conjecture about depth-graded motivic Lie algebra in \cite{depth}.
\section{Mixed Tate motives}\label{mtm}
Denote $\mathcal{MT}(\mathbb{Z})$ the category of mixed Tate motives over $\mathbb{Z}$. The references about mixed Tate motives are \cite{bf}, \cite{dg}. $\mathcal{MT}(\mathbb{Z})$ is a neutral Tannakian category over $\mathbb{Q}$. Denote $\pi_1(\mathcal{MT}(\mathbb{Z}))$ the fundamental group $\mathcal{MT}(\mathbb{Z})$, then we have
\[
\pi_1(\mathcal{MT}(\mathbb{Z}))=\mathbb{G}_m\ltimes U.
\]
Where $U$ is pro-unipotent algebraic group with free Lie algebra generated by the formal symbol $\sigma_{2n+1}$ in weight $2n+1$ for $n\geq 1$.
By \cite{dg}, the motivic fundamental groupoid of $\mathbb{P}^1-\{0,1,\infty\}$ can be realized in the category $\mathcal{MT}(\mathbb{Z})$.
Denote ${}_0\Pi_1$ the motivic fundamental groupoid of $\mathbb{P}^1-\{0,1,\infty\}$ from the tangential base point $\overrightarrow{1}_0$ at $0$ to the tangential base point $\overrightarrow{-1}_1$ at $1$. Its function ring over $\mathbb{Q}$ is
\[\mathcal{O}({}_0\Pi_1)=\mathbb{Q}\langle e^0,e^1\rangle,\]
where $\mathbb{Q}\langle e^0,e^1\rangle$ is equipped with the shuffle product.
Denote by ${}_x\Pi_y$ the de-Rham realization of motivic fundamental groupoid of $\mathbb{P}^1\backslash\{0,1,\infty\}$ from $x$ to $y$ where $x,y\in \{\overrightarrow{1}_0,\overrightarrow{-1}_1\}$. We write $\overrightarrow{1}_0$, $\overrightarrow{-1}_1$ as $0$, $1$ respectively for short. Denote by $G$
the group of automorphisms of the groupoid ${}_x\Pi_y$ for $x,y\in{0,1}$ which respect to the following structures:
(1) (Groupoid structure) The composition maps
\[
{}_x\Pi_y \times {}_y\Pi_z\rightarrow {}_x\Pi_z
\]
for all $x,y,z\in\{0,1\}$.
(2) (Inertia) The automorphism fixes the elements
\[\mathrm{exp}(e_0)\in {}_0\Pi_0(\mathbb{Q}),\, \mathrm{exp}(e_1)\in {}_1\Pi_1(\mathbb{Q}),\]
where $e_0, e_1$ respectively denotes the differential $\frac{dz}{z},\frac{dz}{1-z}$.
From Proposition 5.11 in \cite{dg}, it follows that ${}_x\Pi_y$ is an $G$-torsor.
We have a natural morphism
\[
\varphi:\mathrm{U}^{dR}\rightarrow G\simeq{}_0\Pi_1.
\]
From \cite{brown} $\varphi$ is injective. Denote by $\mathfrak{g}$ the corresponding Lie algebra of $\mathrm{U}^{dR}$, we have an injective map
\[
i:\mathfrak{g}\rightarrow \mathrm{Lie}\, G \simeq(\mathbb{L} (e_0,e_1),\{ \; , \;\}).
\]
Where ($\mathbb{L}(e_0,e_1),\{\;,\;\}$) is the free Lie algebra generated by $e_0,e_1$ with the following Ihara Lie bracket
\[
\{f,g\}=[f,g]+D_f(g)-D_g(f)
\]
and $D_f$ is a derivation on $\mathbb{L}(e_0,e_1)$ which satisfies $D_f(e_0)=0,\;D_f(e_1)=[e_1,f]$ for $f\in \mathbb{L}(e_0,e_1)$.
We denote by $\mathfrak{h}$ the Lie algebra ($\mathbb{L}(e_0,e_1),\{\;,\;\}$) for short. There is a natural decreasing depth filtration on $\mathfrak{h}$ defined by
\[
\mathfrak{D}^r\mathfrak{h}=\{\xi\in \mathfrak{h}\mid \mathrm{deg}_{e_1}\;\xi\geq r\}.
\]
And define the weight grading by the total degree of $e_0,e_1$ for the elements of $\mathfrak{h}$.
From the injective map $i$, there is an induced depth filtration on $\mathfrak{g}$, define
\[
\mathfrak{dg}=\oplus_{r\geq 1}\mathfrak{D}^r\mathfrak{g}/\mathfrak{D}^{r+1}\mathfrak{g}
\]
with induced Lie bracket as depth graded motivic Lie algebra of $\mathcal{MT}(\mathbb{Z})$.
By Th$\acute{e}$or$\grave{e}$me 6.8(i) in\cite{dg}, we have
$i(\sigma_{2n+1})=(\mathrm{ad}\,e_0)^{2n}(e_1)+$ terms of degree $\geq 2 $ in $e_1$. So $\mathfrak{dg}_1$ is essentially the $\mathbb{Q}$-linear combination of $\overline{\sigma}_{2n+1}=(\mathrm{ad}\,e_0)^{2n}(e_1)$, $n\geq 1$ in $\mathfrak{h}$.
Here we give the definition of restricted even period polynomial:
\begin{Def}\label{polynomial}
For $N\geq 3$, the restricted even period polynomial of weight $N$ is the polynomial $p(x_1,x_2)$ of degree $N-2$ which satisfies\\
(i) $p(x_1,0)=0$, i.e. p is restricted;\\
(ii) $p(\pm x_1,\pm x_2)=p(x_1,x_2)$, i.e. p is even;\\
(iii) $p(x_1,x_2)+p(x_1-x_2,x_1)-p(x_1-x_2,x_2)=0$.\\
Denote $\mathbb{P}_N$ the set of even restricted period polynomials of weight $N$.
\end{Def}
For $\mathbb{Q}$-vector space, denote by $\mathrm{Lie}(V)$ the free Lie algebra generated by the vector space V. Denote by $\mathrm{Lie}_n(V)$ elements of $\mathrm{Lie}(V)$ with exactly $n$ occurrences of the formal Lie bracket $[\;,\;]$.
For $n\geq2$, define \[\alpha: \mathbb{P} \otimes\underbrace{\mathfrak{d}\mathfrak{g}_1\otimes\cdots\otimes\mathfrak{d}\mathfrak{g}_1}_{n-2}\rightarrow\mathrm{Lie}_n(\mathfrak{d}\mathfrak{g}_1)\]
by
\[\alpha:\sum p_{r,s}x_1^{r-1}x_2^{s-1}\otimes \overline{\sigma}_{i_1}\otimes\cdots\otimes\overline{\sigma}_{i_{n-2}}\mapsto \sum p_{r,s}[\cdots[[\overline{\sigma}_r,\overline{\sigma}_s],\overline{\sigma}_{i_1}],\cdots,\overline{\sigma}_{i_{n-2}}]\]
where $[\,,\,]$ is the formal lie bracket. Denote by $\beta:\mathrm{Lie}_n(\mathfrak{d}\mathfrak{g}_1)\rightarrow \mathfrak{d}\mathfrak{g}_n$ the map that replacing the formal Lie bracket by the induced Ihara bracket.
The following conjecture is well-known.
\begin{conj}\label{non}{(non-degenerated conjecture)}
For $n\geq2$, the following sequence
\[\mathbb{P} \otimes\underbrace{\mathfrak{d}\mathfrak{g}_1\otimes\cdots\otimes\mathfrak{d}\mathfrak{g}_1}_{n-2}\xrightarrow{\alpha}\mathrm{Lie}_n(\mathfrak{d}\mathfrak{g}_1)\xrightarrow{\beta} \mathfrak{d}\mathfrak{g}_n\]
is exact.
\end{conj}
\section{Universal enveloping algebra}\label{uea}
Denote by $\mathcal{U}\mathfrak{h}$ the universal enveloping algebra of $\mathfrak{h}$ and denote by $\mathbb{Q}\langle e_0,e_1\rangle$ the non-commutative polynomial ring in symbol $e_0,e_1$. From Proposition $5.9$ in \cite{dg} we know that $\mathcal{U}\mathfrak{h}$ is isomorphic to $\mathbb{Q}\langle e_0,e_1\rangle$ as a vector space. But the new multiplication structure on $\mathbb{Q}\langle e_0,e_1\rangle$ which are transformed from $\mathcal{U}\mathfrak{h}$ are rather subtle. It's not the usual concatenation product .
Denote $gr_{\mathfrak{D}}^r\mathbb{Q}\langle e_0,e_1\rangle$ the elements of $\mathbb{Q}\langle e_0, e_1\rangle$ with exactly $r$ occurrences of $e_1$. We have the following map:
\[
\begin{split}
\rho:&gr_{\mathfrak{D}}^r\mathbb{Q}\langle e_0,e_1\rangle\rightarrow \mathbb{Q}[y_0,y_1,\cdots,y_r]\\
&e_0^{a_0}e_1e_0^{a_1}e_1\cdots e_1e_0^{a_r}\mapsto y_0^{a_0}y_1^{a_1}\cdots y_r^{a_r}
\end{split}
\]
The map $\rho$ is the polynomial representation of $\mathbb{Q}\langle e_0, e_1\rangle$ defined by F. Brown.
In \cite{depth}, F. Brown introduced a $\mathbb{Q}$-bilinear map $\underline{\circ}:\mathbb{Q}\langle e_0, e_1\rangle\otimes_{\mathbb{Q}}\mathbb{Q}\langle e_0, e_1\rangle\rightarrow \mathbb{Q}\langle e_0, e_1\rangle $ which in the polynomial representation can be written as
\[
\begin{split}
f\underline{\circ}g(y_0,\cdots,&y_{r+s})=\sum_{i=0}^sf(y_i,y_{i+1},\cdots,y_{i+r})g(y_0,\cdots,y_i,y_{i+r+1},\cdots, y_{r+s})+\\
&(-1)^{deg f+r}\sum_{i=1}^s f(y_{i+r},\cdots, y_{i+1}, y_i)g(y_0,\cdots,y_{i-1},y_{i+r},\cdots, y_{r+s})
\end{split}
\]
for $f\in \mathbb{Q}[y_0,\cdots,y_r]=\rho(gr_{\mathfrak{D}}^r\mathbb{Q}\langle e_0, e_1\rangle)$, $g\in \mathbb{Q}[y_0,\cdots,y_s]=\rho(gr_{\mathfrak{D}}^s\mathbb{Q}\langle e_0, e_1\rangle)$.
Since by the general theory of Lie algebra, the natural action of $\mathfrak{h}$ on $\mathcal{U}\mathfrak{h}$ is the form $(a, b_1\otimes b_2\otimes\cdots b_r)\mapsto a\otimes b_1\otimes \cdots \otimes b_r$ in $\mathcal{U}\mathfrak{h}$ for $a, b_1,\cdots, b_r\in \mathfrak{h}$. By Proposition 2.2 in \cite{depth}, we have
\[
a_1\circ a_2\circ\cdots\circ a_r=a_1\underline{\circ}(a_2\underline{\circ}(\cdots(a_{r-1}\underline{\circ} a_r)\cdots))
\]
for $a_i\in \mathfrak{h}\in \mathbb{Q} \langle e_0, e_1\rangle,1\leq i\leq r-1$, $a_r\in \mathbb{Q}\langle e_0,e_1\rangle$.
The above formula is still not enough to give a very clear picture of the new multiplication $\circ$ on $\mathbb{Q}\langle e_0, e_1\rangle$. But it's enough for our purpose.
We first introduce some notation from K.Tasaka \cite{tasaka}.
Denote by
\[
S_{N,r}=\{(n_1,...,n_r)\in\mathbb{Z}^r\mid n_1+...+n_r=N,n_1,...,n_r\geq3:odd\}.
\]
We write $\overrightarrow{m}=(m_1,...,m_r)$ for short, while \[\mathbf{Vect}_{N,r}=\{(a_{n_1,...,n_r})_{\overrightarrow{n}\in S_{N,r}}\mid a_{n_1,...,n_r}\in \mathbb{Q}\}.\]
For a matrix $P=\left(p\dbinom{m_1,...,m_r}{n_1,...,n_r}\right)_{\substack {\overrightarrow{m}\in S_{N,r}\\\overrightarrow{n}\in S_{N,r}}}$, the action of $P$ on $a=(a_{m_1,...,m_r})_{\overrightarrow{m}\in S_{N,r}}$ means
\[
aP=\left(\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,...,m_r}p\dbinom{m_1,...,m_r}{n_1,...,n_r}\right)_{\overrightarrow{n}\in S_{N,r}}
\]
Denote by $\mathbb{P}_{N,r}$ the $\mathbb{Q}$-vector space spanned by the set
\[
\{x_1^{n_1-1}\cdots x_r^{n_r-1}\mid(n_1,...,n_r)\in S_{N,r}\}.
\]
Obviously there is an isomorphism
\[\pi:\mathbb{P}_{N,r}\longrightarrow\mathbf{Vect}_{N,r}\]
\[\sum_{\overrightarrow{n}\in S_{N,r}}a_{n_1,...,n_r}x^{n_1-1}_1\cdots x^{n_r-1}_r\longmapsto(a_{n_1,...,n_r})_{\overrightarrow{n}\in S_{N,r}}.\]
Denote by
\[
\mathbf{W}_{N,r}=\{p\in\mathbb{P}_{N,r}\mid p(x_1,...,x_r)=p(x_2-x_1,x_2,x_3,...,x_r)-p(x_2-x_1,x_1,x_3,...,x_r)\}.
\]
Denote by
\[
e\dbinom{m_1,...,m_r}{n_1,...,n_r}=\delta\dbinom{m_1,...,m_r}{n_1,...,n_r}+\sum_{i=1}^{r-1}\delta\dbinom{m_2,...,m_i,m_{i+2},...,m_r}{n_1,...,n_{i-1},n_{i+2},...,n_r}b^{m_1}_{n_i,n_{i+1}}.
\]
Which the $b^m_{n,n'}$ are defined by
\[
b^m_{n,n'}=(-1)^n\dbinom{m-1}{n-1}+(-1)^{n'-m}\dbinom{m-1}{n'-1}.
\]
and $\delta\dbinom{m_1,...,m_r}{n_1,...,n_r}=1$ if $\overrightarrow{m}=\overrightarrow{n}$, $\delta\dbinom{m_1,...,m_r}{n_1,...,n_r}=0$ if $\overrightarrow{m}\neq\overrightarrow{n}$.
And the matrix $E^{(r-i)}_{N,r},\,i=0,1,...,r-2$ are defined by
\[
E^{(r-i)}_{N,r}=\left(\delta\dbinom{m_1,...,m_i}{n_1,...,n_i}e\dbinom{m_{i+1},...,m_r}{n_{i+1},...,n_r}\right)_{\substack {\overrightarrow{m}\in S_{N,r}\\\overrightarrow{n}\in S_{N,r}}}.
\]
We write $E^{(r)}_{N,r}$ as $E_{N,r}$.
And denote by
\[
C_{N,r}=E^{(2)}_{N,r}\cdot E^{(3)}_{N,r}\cdots E^{(r-1)}_{N,r}\cdot E_{N,r}
\]
for $r=2$. And denote by $C_{N,r}$ the one row, one column matrix $1$ for $N>1,\mathrm{odd}$, $r=1$.
By \cite{schneps}, we know $\pi(\mathbf{W}_{N,2})=\mathrm{Ker}\;E_{N,2}$.
For $r\geq 3$, K.Tasaka proved that
\[
(\pi(\mathbf{W}_{N,r})(E_{N,r}-I_{N,r})\subseteq \mathrm{Ker}\; E_{N,r},
\]
which $I_{N,r}$ denotes the identity matrix $$\left(\delta\dbinom{m_1,...,m_r}{n_1,...,n_r}\right)_{\substack {\overrightarrow{m}\in S_{N,r}\\\overrightarrow{n}\in S_{N,r}}}.$$
Furthermore, K. Tasaka proposed the following conjecture.
\begin{conj}\label{taisha} {(Tasaka conjecture)}
The linear map $$\eta:\pi(\mathbf{W}_{N,r})\rightarrow \mathrm{Ker}\;E_{N,r}$$
$$a_{\overrightarrow{m}}\mapsto (a_{\overrightarrow{m}})(E_{N,r}-I_{N,r})$$
is an isomorphism.
\end{conj}
In \cite{tasaka}, K.Tasaka suggests a way to prove the injectivity in the above conjecture. But there is a gap in his proof. We prove the injectivity for $r=3$ in \cite{li}.
In \cite{depth}, F. Brown proposed the following conjecture
\begin{conj}\label{fbro}{(Brown's matrix conjecture)} The rank of the matrices $C_{N,r}$ satisfy
\[
1+\sum_{N,r>0}\mathrm{rank}\;C_{N,r} x^{N}y^r=\frac{1}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2}
\]
\end{conj}
Now we can state our main result.
\begin{Thm}\label{im}
Tasaka conjecture $\Rightarrow$ Brown's matrix conjecture $\Rightarrow$ non-degenerated conjecture.
\end{Thm}
\section{Calculation}
In this section we will prove Theorem \ref{im}. In fact we will prove a little bit more.
First we will need the following result about Lie algebra.
\begin{prop}\label{ideal}
Let $\mathfrak{L}$ be a Lie algebra over $\mathbb{Q}$, denote by $\mathcal{U}\mathfrak{L}$ its universal envelope algebra. $\mathfrak{M}$ is a Lie ideal in $\mathfrak{L}$, denote by $\mathcal{U}\mathfrak{L}(\mathfrak{M})$ the two-sided ideal generated by $\mathfrak{M}$ in $\mathcal{U}\mathfrak{L}$. Then we have
\[
\mathfrak{L}\cap(\mathcal{U}\mathfrak{L}(\mathfrak{M}) )=\mathfrak{M}.
\]
\end{prop}
\noindent{\bf Proof}:
We have the following commutative diagram
\[
\xymatrix{
0 \ar[r] & \mathfrak{M} \ar[d] \ar[r] & \mathfrak{L} \ar[d] \ar[r] & \mathfrak{L}/\mathfrak{M} \ar[d] \ar[r] & 0 \\
0 \ar[r] & \mathcal{U}\mathfrak{L}(\mathfrak{M}) \ar[r] & \mathcal{U}\mathfrak{L} \ar[r] & \mathcal{U}(\mathfrak{L}/\mathfrak{M}) \ar[r] & 0 }
\]
And the first row and second row are short exact sequences. By the Poincar\'{e}-Birkhoff-Witt theorem in Lie algebra, we know that the three vertical maps are all injective. $\mathfrak{L}\cap(\mathcal{U}\mathfrak{L}(\mathfrak{M}) )=\mathfrak{M}$ follows by diagram chasing. $\hfill\Box$\\
Recall the injective Lie algebra homomorphism in Section \ref{mtm}
\[
i:\mathfrak{g}\rightarrow \mathfrak{h}.
\]
Since the Lie algebra $\mathfrak{h}$ is bigraded by weight and depth, the map $i$ induces a natural injective Lie algebra homomorphism
\[
\overline{i}:\mathfrak{dg}\rightarrow \mathfrak{h}.
\]
The maps $i$ and $\overline{i}$ induce the natural injective algebra homomorphisms on enveloping algebra
\[
\mathcal{U}i:\mathcal{U}\mathfrak{g}\rightarrow \mathcal{U}\mathfrak{h}=(\mathbb{Q}\langle e_0,e_1\rangle,\circ)
\]
and
\[
\mathcal{U}\overline{i}:\mathcal{U}\mathfrak{dg}\rightarrow \mathcal{U}\mathfrak{h}=(\mathbb{Q}\langle e_0,e_1\rangle,\circ).
\]
As $\mathfrak{g}$ is a free Lie algebra generated by elements $\sigma_{2n+1}$ for $n\geq 1$ in weight $2n+1$. We have $\mathcal{U}\mathfrak{g}=\mathbb{Q}\langle \sigma_3, \sigma_5,\cdots, \sigma_{2n+1},\cdots\rangle$ (the non-commutative polynomial ring generated by the symbol $\sigma_{2n+1}$ for $n\geq 1$) with the usual concatenation product.
For $r=0$, denote by $L_r$ the rational field $\mathbb{Q}$. For $r\geq 1$, denote by $L_r$ the $\mathbb{Q}$-linear space generated by elements
\[
\overline{\sigma}_{n_1}\circ \overline{\sigma}_{n_2}\circ\cdots\circ\overline{\sigma}_{n_r}=(\mathrm{ad}\;e_0 )^{n_1-1}e_1\circ (\mathrm{ad}\;e_0 )^{n_2-1}e_1\circ\cdots \circ (\mathrm{ad}\;e_0 )^{n_r-1}e_1
\]
in $\mathcal{U}\mathfrak{h}$ for $n_i\geq 3$, odd, $1\leq i\leq r$.
Define the map $B_r:L_1\otimes_{\mathbb{Q}} L_{r-1}\rightarrow L_r$ by
\[
B_r(\overline{\sigma}_{n_1}\otimes (\overline{\sigma}_{n_2}\circ \cdots\circ\overline{\sigma}_{n_r}))=\overline{\sigma}_{n_1}\circ \overline{\sigma}_{n_2}\circ \cdots\circ\overline{\sigma}_{n_r}.
\]
and define the map $A_r:\mathbb{P}\otimes_{\mathbb{Q}} L_{r-2}\rightarrow L_1\otimes_{\mathbb{Q}} L_{r-1}$ by
\[
A_r:((\sum_{n_1,n_2\geq 3,\mathrm{odd}} p_{n_1,n_2}x_1^{n_1-1}x_2^{n_2-1})\otimes (\overline{\sigma}_{n_3}\circ\cdots\cdots \circ\overline{\sigma}_{n_r}))=\sum_{n_1,n_2\geq 3,\mathrm{odd}}p_{n_1,n_2}\overline{\sigma}_{n_1}\otimes (\overline{\sigma}_{n_2}\circ \cdots\circ\overline{\sigma}_{n_r} ).
\]
We have the following lemma
\begin{lem}\label{equi}
The non-degenerated conjecture for all $r\geq 2$ is equivalent to that the following sequence is exact
\[
0\rightarrow \mathbb{P}\otimes_{\mathbb{Q}} L_{r-2}\xrightarrow{A_r} L_1\otimes_{\mathbb{Q}} L_{r-1}\xrightarrow{B_r} L_r\rightarrow 0.
\]
for all $r\geq 2$.
\end{lem}
\noindent{\bf Proof}: If
\[\tag{1}
\overline{x}=\sum_{\overrightarrow{n}\in S_{N,r}}a_{n_1,\cdots, n_r}\overline{\sigma}_{n_1}\otimes (\overline{\sigma}_{n_2}\circ\cdots\circ \overline{\sigma}_{n_r})\in \mathrm{Ker}\;B_r,
\]
then by definition we will have
\[\tag{2}
\sum_{\overrightarrow{n}\in S_{N,r}}a_{n_1,\cdots,n_r}\sigma_{n_1}\sigma_{n_2}\cdots \sigma_{n_r}\in \mathfrak{D}^{r+1}\mathcal{U}\mathfrak{g}=\mathfrak{D}^{r+1}\mathbb{Q}\langle \sigma_3,\cdots, \sigma_{2n+1},\cdots\rangle.
\]
Since $\mathcal{U}\mathfrak{g}$ is a non-commutative polynomial ring, from formula $(1)$ and $(2)$ we have
\[
\begin{split}
\sum_{\overrightarrow{n}\in S_{N,r}}&a_{n_1,\cdots,n_r}\sigma_{n_1}\sigma_{n_2}\cdots \sigma_{n_r}\\
&=\sum_{i=2}^r\sum_{\overrightarrow{m}\in S_{N,r}}b^i_{m_1,\cdots, m_i, m_{i+1},\cdots,m_r}[[\cdots[\sigma_{m_1},\sigma_{m_2}],\cdots], \sigma_{m_i}]\sigma_{m_{i+1}}\cdots \sigma_{m_r}
\end{split}
\]
for some $b^i_{\overrightarrow{m}},\overrightarrow{m}\in S_{N,r}$ and
\[
\sum_{(m_1,\cdots,m_i)\in S_{N-m_{i+1}-\cdots-m_r}}b^i_{m_1,\cdots,m_i,m_{i+1},\cdots, m_r}[[\cdots[\sigma_{m_1},\sigma_{m_2}],\cdots],\sigma_{m_i}]\subseteq \mathfrak{D}^{i+1}\mathfrak{g}
\]
On one hand, if the non-degenerated conjecture is true for all depth, we will have $\overline{x}\subseteq \mathrm{Im}\;A_r$, i.e. $\mathrm{Im}\; A_r=\mathrm{Ker}\;B_r$.
And since it's obvious that $B_r$ is surjective. While $A$ is injective follows from $\mathrm{Im}\;A_r=\mathrm{Ker}\;B_r$ in depth $r-1$ and the fact that $\mathbb{P}\otimes L_1 \cap L_1\otimes \mathbb{P}=\{0\}$. We deduce that from the non-degenerated conjecture for all depth we will have the short exact sequence for all $r\geq 2$.
On the other hand, if the sequence is exact for all $r\geq 2$, let
\[\tag{3}
\overline{x}=\sum_{\overrightarrow{m}\in S_{N,r}}b_{m_1,\cdots, m_r}\{\{\cdots\{\overline{\sigma}_{m_1},\overline{\sigma}_{m_2}\},\cdots\},\overline{\sigma}_{m_r}\}=0
\]
in $\mathfrak{dg}_r$, which $\{\,,\,\}$ denotes the induced Ihara Lie bracket on $\mathfrak{dg}$.
Then
\[\tag{4}
x=\sum_{\overrightarrow{m}\in S_{N,r}}b_{m_1,m_2,\cdots,m_r}[[\cdots[\sigma_{m_1},\sigma_{m_2}],\cdots],\sigma_{m_r}]\in gr_{\mathfrak{D}}^{r+1}\mathcal{U}\mathfrak{g},
\]
which $[\,,\,]$ denotes the formal Lie bracket on the non-commutative polynomial ring $\mathcal{U}\mathfrak{g}$. Rewrite $x$ as
\[\tag{5}
x=\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}\sigma_{m_1}\sigma_{m_2}\cdots \sigma_{m_r}.
\]
Denote
\[
x_{\mathfrak{d}}=\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}\overline{\sigma}_{m_1}\otimes (\overline{\sigma}_{m_2}\circ \overline{\sigma}_{m_3}\circ\cdots \circ \overline{\sigma}_{m_r} )\in L_1\otimes_{\mathbb{Q}} L_{r-1},
\]
then from $(3),(4)$ and $(5)$, we have $x_{\mathfrak{d}}\in \mathrm{Ker}\; B_r$.
Since $\mathrm{Im}\;A_r =\mathrm{Ker}\; B_r$, we have
\[
x_{\mathfrak{d}}=\sum_{\overrightarrow{m}\in S_{N,r}}c_{m_1,m_2,m_3,\cdots,m_r}\overline{\sigma}_{m_1}\otimes (\overline{\sigma}_{m_2}\circ \overline{\sigma}_{m_3}\circ\cdots \circ \overline{\sigma}_{m_r})
\]
and $p=\sum\limits_{m_1,m_2\geq 3,\mathrm{odd}}c_{m_1,m_2,m_3,\cdots,m_r}x_1^{m_1-1}x_2^{m_2-1}\in \mathbb{P}$. Let $\iota: \mathbb{P}\rightarrow \mathfrak{g}$ be the map
\[
\iota:\sum_{r,s\geq 3,\;\mathrm{odd}}p_{r,s}x_1^{r-1}x_2^{s-1}\mapsto \sum_{r,s\geq 3,\;\mathrm{odd}}p_{r,s}[\sigma_r,\sigma_s].
\]
From $\mathrm{Im}\;A_i=\mathrm{Ker}\;B_i$ for $i=2,\cdots,r$, we deduce inductively that
\[
x\in \mathcal{U}\mathfrak{g}(\iota(\mathbb{P}) ),
\]
which $\mathcal{U}\mathfrak{g}(\iota(\mathbb{P}) )$ means the two-sided ideal generated by $\iota({\mathbb{P}})$ in $\mathcal{U}\mathfrak{g}$. By Proposition \ref{ideal}, $x$ belongs to the Lie ideal generated by $\iota(\mathbb{P})$ in $\mathfrak{g}$. So from the short exact sequence for all depth we can deduce the non-degenerated conjecture in all depth.
$\hfill\Box$\\
The following lemma reduces the non-degenerated conjecture to a dimension conjecture of $L_r$ in each weight $N$ for all $r$.
\begin{lem}\label{r}
Denote by $L_{N,r}$ the weight $N$ part of $L_r$, then
the formula
\[
1+\sum_{N,r>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r}x^Ny^r=\frac{1}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2}
\]
is equivalent to that the following sequence is exact
\[
0\rightarrow \mathbb{P}\otimes_{\mathbb{Q}} L_{r-2}\xrightarrow{A_r} L_1\otimes_{\mathbb{Q}} L_{r-1}\xrightarrow{B_r} L_r\rightarrow 0.
\]
for all $r\geq 2$. Which $\mathbb{O}(x)=\frac{x^3}{1-x^2}$, $\mathbb{S}(x)=\frac{x^{12}}{(1-x^4)(1-x^6)}$.
\end{lem}
\noindent{\bf Proof}:
$'\Rightarrow'$
It's clear that $B_r$ is surjective and $\mathrm{Im}\;A_r\subseteq \mathrm{Ker}\; B_r$ for all $r\geq 2$.
Since
\[
\sum_{N>0}\mathrm{dim}_{\mathbb{Q}}L_{N,1}x^N=\mathbb{O}(x),
\]
from the dimension formula we have
\[
\tag{6}
\mathrm{dim}_{\mathbb{Q}}\;L_{N,r}x^N-\sum_{N>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r-1}\cdot\mathbb{O}(x)+\sum_{N>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r-2}\cdot\mathbb{S}(x)=0
\]
for all $r\geq 2$.
It's obvious that $A_2$ is injective, $B_2$ is surjective and $\mathrm{Im}\;A_2\subseteq\mathrm{Ker}\;B_2$. So from formula $(6)$ in $r=2$, we have $\mathrm{Im}\;A_2=\mathrm{Ker}\;B_2$.
Inductively, we can deduce that $A_r$ is injective from $\mathrm{Im}\;A_{r-1}=\mathrm{Ker}\;B_{r-1}$ and Goncharov's result $\mathbb{P}\otimes_{\mathbb{Q}}L_1\cap L_1\otimes_{\mathbb{Q}}\mathbb{P}=0$.
Then $\mathrm{Im}\;A_r=\mathrm{Ker}\;B_r$ follows from formula $(6)$ and the fact that $A_r$ is injective, $B_r$ is surjective and $\mathrm{Im}\;A_r\subseteq \mathrm{Ker}\; B_r$.
$'\Leftarrow'$
It's clear that
\[
\sum_{N>0}\mathrm{dim}_{\mathbb{Q}}L_{N,1}x^N=\mathbb{O}(x).
\]
Then from the short exact sequence we have
\[
\mathrm{dim}_{\mathbb{Q}}\;L_{N,r}x^N-\sum_{N>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r-1}\cdot\mathbb{O}(x)+\sum_{N>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r-2}\cdot\mathbb{S}(x)=0
\]
for all $r\geq 2$. So
\[
1+\sum_{N,r>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r}x^Ny^r=\frac{1}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2}.
\] $\hfill\Box$\\
\begin{rem}\label{i}
In fact, if we only know that
\[
1+\sum_{N,r>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r}x^Ny^r\geq \frac{1}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2},
\]
which inequality means the coefficient of the term $x^Ny^r$ in the left side is bigger than the corresponding coefficient in the right side for all $N,r>0$, then we can still deduce the short exact sequence exactly the same way as in the proof of Lemma \ref{r}.
\end{rem}
Now we investigate the polynomial representation of $L_{N,r}$. From the main result of Section \ref{uea}, we have
\[
\rho(\overline{\sigma}_{m_1}\circ\overline{\sigma}_{m_2}\circ\cdots\circ \overline{\sigma}_{m_r})=(y_1-y_0)^{m_1-1}\underline{\circ}((y_1-y_0)^{m_2-1}\underline{\circ}(\cdots((y_1-y_0)^{m_{r-1}-1}\underline{\circ}(y_1-y_0)^{m_r-1})))
\]
for $\overrightarrow{m}=(m_1,m_2,\cdots,m_r)\in S_{N,r}$.
For $\overrightarrow{n}=(n_1,n_2,\cdots,n_r)\in S_{N,r}$, the coefficient of $y_1^{n_1-1}y_2^{n_2-1}\cdots y_r^{n_r-1}$ in $\rho(\overline{\sigma}_{m_1}\circ\overline{\sigma}_{m_2}\circ\cdots\circ \overline{\sigma}_{m_r}) $ is
\[
c\binom{m_1,m_2,\cdots,m_r}{n_1,n_2,\cdots,n_r}.
\]
Which $c\binom{m_1,m_2,\cdots,m_r}{n_1,n_2,\cdots,n_r}$ is the $(m_1,m_2,\cdots,m_r)$-th row, $(n_1,n_2,\cdots,n_r)$-th column term of the matrix $C_{N,r}$.
Now we have
\begin{prop}\label{rep}
The following map
\[
\widetilde{\eta}:W_{N,r}\rightarrow \mathbf{Vect}_{N,r}
\]
\[
(a_{m_1,\cdots,m_r})_{\overrightarrow{m}\in S_{N,r}}\mapsto \left(\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,\cdots,m_r}\delta\binom{m_1}{n_1}e\binom{m_2,\cdots,m_r}{n_2,\cdots,n_r} \right)_{\overrightarrow{n}\in S_{N,r}}
\]
satisfy $\widetilde{\eta}(W_{N,r})\subseteq \mathrm{Ker}\;E_{N,r}$. And furthermore, $\widetilde{\eta}(a)+\eta(a)=0$ for $$a=(a_{m_1,\cdots,m_r})_{\overrightarrow{m}\in S_{N,r}}\in W_{N,r}.$$
\end{prop}
\noindent{\bf Proof}:
Consider the natural action of $\mathfrak{dg}$ on $\mathcal{U}\mathfrak{h}=(\mathbb{Q}\langle e_0,e_1\rangle,\circ )$. By the main results of Section \ref{uea}, we know that
\[
\sum_{({m_2,\cdots,m_r)}\in S_{N-n_1,r-1}}a_{n_1,m_2,\cdots,m_r}e\binom{m_2,\cdots,m_r}{n_2,\cdots,n_r}
\]
is the coefficient of $y_2^{n_2-1}\cdots y_r^{n_r-1}$ in the polynomial representation of
\[
\sum_{(m_2,\cdots,m_r)\in S_{N-n_1,r-1}}a_{n_1,m_2,\cdots,m_r}\overline{\sigma}_{m_2}\circ(e_1e_0^{m_3-1}e_1\cdots e_1e_0^{m_r-1} ).
\]
for $(n_2,\cdots,n_r)\in S_{N-n_1,r-1}$. Furthermore,
\[
\sum_{\overrightarrow{m},\overrightarrow{n}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}\delta\binom{m_1}{n_1}e\binom{m_2,\cdots,m_r}{n_2,\cdots,n_r}e\binom{n_1,n_2,\cdots,n_r}{k_1,k_2,\cdots,k_r}
\]
is the coefficient of $y_1^{k_1-1}y_2^{k_2-1}\cdots y_r^{k_r-1}$ in the polynomial representation of
\[
\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}\overline{\sigma}_{m_1}\circ \overline{\sigma}_{m_2}\circ(e_1e_0^{m_3-1}e_1\cdots e_1e_0^{m_r-1}).
\]
If $a=(a_{m_1,\cdots,m_r})_{\overrightarrow{m}\in S_{N,r}}\in W_{N,r}$, then
\[
\begin{split}
&\;\;\;\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}\overline{\sigma}_{m_1}\circ \overline{\sigma}_{m_2}\circ(e_1e_0^{m_3-1}e_1\cdots e_1e_0^{m_r-1})\\
&=\frac{1}{2}\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}(\overline{\sigma}_{m_1}\circ \overline{\sigma}_{m_2}-\overline{\sigma}_{m_2}\circ \overline{\sigma}_{m_1})\circ(e_1e_0^{m_3-1}e_1\cdots e_1e_0^{m_r-1})\\
&= \frac{1}{2}\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}\{\overline{\sigma}_{m_1}, \overline{\sigma}_{m_2}\}\circ(e_1e_0^{m_3-1}e_1\cdots e_1e_0^{m_r-1}) \\
&=0.
\end{split}
\]
So we have
\[
\sum_{\overrightarrow{m},\overrightarrow{n}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}\delta\binom{m_1}{n_1}e\binom{m_2,\cdots,m_r}{n_2,\cdots,n_r}e\binom{n_1,n_2,\cdots,n_r}{k_1,k_2,\cdots,k_r}=0,
\]
i.e. $\widetilde{\eta}(W_{N,r})\subseteq \mathrm{Ker}\;E_{N,r}$.
Similarly, in order to prove $\widetilde{\eta}(a)+\eta(a)=0$ for $a=(a_{m_1,\cdots,m_r})_{\overrightarrow{m}\in S_{N,r}}\in W_{N,r}$, it suffices to show that the coefficient of the term $y_1^{n_1-1}y_2^{n_2-1}\cdots y_r^{n_r-1}$ in the polynomial representation of
\[
-\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}[\overline{\sigma}_{m_1}\circ(e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1})-e_1e_0^{m_1-1}e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1}]
\]
is equal to the coefficient of the term $y_1^{n_2-1}y_2^{n_3-1}\cdots y_{r-1}^{n_r-1}$ in the polynomial representation of
\[\tag{7}
\sum_{(m_2,m_3,\cdots,m_r)\in S_{N-n_1,r-1}}a_{n_1,m_2,\cdots,m_r}\overline{\sigma}_{m_2}\circ(e_1e_0^{m_3-1}\cdots e_1e_0^{m_r-1})
\]
for all
$\overrightarrow{n}\in S_{N,r}$.
From Proposition $2.2$ in \cite{depth}, we have
\[
\begin{split}
&\;\;\;\;\;\overline{\sigma}_{m_1}\circ(e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1})\\
&=\overline{\sigma}_{m_1}e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1}-e_1\overline{\sigma}_{m_1}e_0^{m_2-1}\cdots e_1e_0^{m_r-1}+e_1e_0^{m_2-1}(\overline{\sigma}_{m_1}\circ(e_1e_0^{m_3-1}\cdots e_1e_0^{m_r-1}))
\end{split}
\]
and the $y_0$ part of the polynomial representation of $(7)$ is
\[
f_0(y_0,y_1,\cdots,y_{r-1})=\sum_{(m_2,\cdots,m_r)\in S_{N-n_1,r-1}}a_{n_1,m_2,\cdots,m_r}((y_1-y_0)^{m_2-1}-y_1^{m_2-1} )y_2^{m_3-1}\cdots y_{r-1}^{m_r-1}.
\]
So we have
\[
\begin{split}
&\;\;\;\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}[\overline{\sigma}_{m_1}\circ(e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1})
-e_1e_0^{m_1-1}e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1}]\\
&=\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}[\overline{\sigma}_{m_1}e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1}-e_1\overline{\sigma}_{m_1}e_0^{m_2-1}\cdots e_1e_0^{m_r-1}\\
&\;\;\;\;\;\;\;\;\;\;\;+e_1e_0^{m_2-1}(\overline{\sigma}_{m_1}\circ(e_1e_0^{m_3-1}\cdots e_1e_0^{m_r-1})) -e_1e_0^{m_1-1}e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1} ]
\end{split}
\]
Since the coefficient of $y_1^{n_1-1}y_2^{n_2-1}\cdots y_r^{n_r-1}$ in the polynomial representation of
\[
\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}e_1e_0^{m_1-1}(\overline{\sigma}_{m_2}\circ(e_1e_0^{m_3-1}\cdots e_1e_0^{m_r-1}))
\]
is equal to $(\eta(a))_{n_1,n_2,\cdots,n_r}$ minus the coefficient of $y_1^{n_1-1}y_2^{n_2-1}\cdots y_r^{n_r-1}$ in $$y_1^{m_1-1}f_0(y_1,y_2,\cdots,y_r).$$
And the polynomial representation of
\[
\begin{split}
&\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}[\overline{\sigma}_{m_1}e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1}-e_1\overline{\sigma}_{m_1}e_0^{m_2-1}\cdots e_1e_0^{m_r-1}\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\;\;\;\;\;\;\;-e_1e_0^{m_1-1}e_1e_0^{m_2-1}\cdots e_1e_0^{m_r-1} ]
\end{split}
\]
is
\[
\begin{split}
&\;\;\;\;f_1(y_0,y_1,\cdots,y_r)\\
&=\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}[(y_1-y_0)^{m_1-1}y_2^{m_2-1}-(y_2-y_1)^{m_1-1}y_2^{m_2-1}\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-y_1^{m_1-1}y_2^{m_2-1}]y_3^{m_3-1}\cdots y_r^{m_r-1}
\end{split}
\]
So to prove $\widetilde{\eta}(a)+\eta(a)=0$ it suffices to prove
\[\tag{8}
\begin{split}
\sum_{\overrightarrow{m}\in S_{N,r}}a_{m_1,m_2,\cdots,m_r}[&e_1e_0^{m_1-1}(\overline{\sigma}_{m_2}\circ(e_1e_0^{m_3-1}\cdots e_1e_0^{m_r-1}))\\
&+e_1e_0^{m_2-1}(\overline{\sigma}_{m_1}\circ(e_1e_0^{m_3-1}\cdots e_1e_0^{m_r-1}))]=0.
\end{split}
\]
and
\[\tag{9}
f_1(y_0,y_1,\cdots,y_r)-y_1^{m_1-1}f_0(y_1,y_2,\cdots,y_r)
\; \mathrm{has}\;\mathrm{trivial}\;y_1^{n_1-1}y_2^{n_2-1}\cdots y_r^{n_r-1}\;\mathrm{term}
\]
for $\overrightarrow{n}\in S_{N,r}$.
The the formula $(8)$ follows from $a_{m_1,m_2,m_3,\cdots,m_r}+a_{m_2,m_1,m_3,\cdots,m_r}=0$.
And the formula $(9)$ follows from that $a=(a_{m_1,\cdots,m_r})_{\overrightarrow{m}\in S_{N,r}}\in W_{N,r}$.$\hfill\Box$\\
\begin{rem}
From Proposition \ref{rep} we obtain Tasaka's result \cite{tasaka} \[\eta(W_{N,r})\subseteq \mathrm{Ker}\;(E_{N,r}-I_{N,r})\]
immediately. See the proof Proposition $5.5$ in \cite{li} for a proof of the fact $$\widetilde{\eta}(a)+\eta(a)=0$$ based on explicit matrix calculation. Also see the proof of Lemma $4.9$ in \cite{dmn} for a proof based on polynomial representation.
\end{rem}
Now we can prove our main results.\\
\noindent{\bf Proof of Theorem \ref{im}}: From linear algebra, we have
\[\tag{10}
\mathrm{Ker}\; C_{N,r}\cong\mathrm{Ker}\;(E^{(2)}_{N,r}E^{(3)}_{N,r}\cdots E^{(r-1)}_{N,r})\oplus \mathrm{Im}\;(E^{(2)}_{N,r}E^{(3)}_{N,r}\cdots E^{(r-1)}_{N,r})\cap \mathrm{Ker}\; E_{N,r}.
\]
By the definition of $C_{N,r}$, view $C_{N,r}$ as linear transformation on the vector space $\mathbf{Vect}_{N,r}$, then we have
\[\tag{11}
\mathrm{Ker}(E^{(2)}_{N,r}E^{(3)}_{N,r}\cdots E^{(r-1)}_{N,r})\cong \bigoplus_{m>1,\mathrm{odd}}\mathrm{Ker}\; C_{N-m,r-1}.
\]
If Conjecture \ref{taisha} (Tasaka conjecture) is true, then from Proposition \ref{rep} and the fact
\[
\mathrm{Im}\;(E_{N,r}^{(2)}E_{N,r}^{(3)}\cdots E_{N,r}^{(r-2)})\cap W_{N,r}\cong \bigoplus_{m>0,\mathrm{even}}\mathbb{P}_m\otimes _{\mathbb{Q}}\mathrm{Im}\;C_{N-m.r-2}
\]
we have
\[\tag{12}
\mathrm{Im}\;(E^{(2)}_{N,r}E^{(3)}_{N,r}\cdots E^{(r-1)}_{N,r})\cap \mathrm{Ker}\; E_{N,r}\cong \bigoplus_{m>0,\mathrm{even}}\mathbb{P}_{m}\otimes_{\mathbb{Q}}\mathrm{Im}\; C_{N-m,r-2}.
\]
From formula $(10), (11)$ and $(12)$, we have
\[\tag{13}
\begin{split}
\sum_{N,r>0}\mathrm{dim}_{\mathbb{Q}} \mathrm{Ker}\; C_{N,r} x^N y^r=&\sum_{N,r>0}\mathrm{dim}_{\mathbb{Q}} \mathrm{Ker}\; C_{N,r} x^N y^r\cdot {\mathbb{O}(x)y}\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\mathbb{S}(x)y^2\cdot (1+\sum_{N,r>0}\mathrm{dim}_{\mathbb{Q}} \mathrm{Im}\; C_{N,r} x^Ny^r).
\end{split}
\]
From formula $(13)$, we have
\[\tag{14}
1+\sum_{N,r>0}\mathrm{rank}\;C_{N,r}x^Ny^r=\frac{1}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2}.
\]
The polynomial representation of element $\overline{\sigma}_{m_1}\circ \overline{\sigma}_{m_2}\circ\cdots \circ \overline{\sigma}_{m_r}$ in $L_{N,r}$ for $\overrightarrow{m}\in S_{N,r}$ is
\[\tag{15}
(y_1-y_0)^{m_1-1}\underline{\circ}((y_1-y_0)^{m_2-1}\underline{\circ}(\cdots\underline{\circ}(y_1-y_0)^{m_r-1})\cdots))
\]
And the coefficient of the term $y_1^{n_1-1}y_2^{n_2-1}\cdots y_r^{n_r-1}$ in the formula $(15)$ is \\
the $(m_1,m_2,\cdots, m_r)$-th row, the $(n_1,n_2,\cdots,n_r)$-th column element of the matrix $C_{N,r}$.
So from formula $(14)$ we have
\[
1+\sum_{N,r>0}\mathrm{dim}_{\mathbb{Q}}\;L_{N,r}x^Ny^r\geq \frac{1}{1-\mathbb{O}(x)y+\mathbb{S}(x)y^2}.
\]
From Remark \ref{i} and Lemma \ref{equi}, we have the non-degenerated conjecture.$\hfill\Box$\\
\begin{rem}\label{dc}
Formula $(12)$ is essentially the Conjecture $4.12$ in \cite{dmn}, in the above proof we actually show that formula $(12)$ is a corollary of Tasaka's isomorphism conjecture. And see \cite{li} for the application of non-degenerated conjecture to motivic multiple zeta values.
\end{rem}
|
2,869,038,154,747 | arxiv | \section{Introduction}
IceCube Neutrino Observatory, the world's largest neutrino
detector, has detected 54 neutrino events within 1347 days with energy between 20 TeV to 2.3 PeV ~\cite{aartsen2014observation,kop}.
Shower events, most likely due to $\nu_e$ or
$\nu_\tau$ charge current $\nu N$ interactions and also due to neutral current $\nu N$ interactions of all flavors, dominate the event
list (39 including 3 events with 1--2 PeV energy) while track events,
most likely due to $\nu_\mu$ charge current $\nu N$ interactions,
constitute the rest. Among a total of 54 events about 21 could be due
to atmospheric neutrino ($9.0^{+8.0}_{-2.2}$) and muon ($12.6\pm 5.1$)
backgrounds. A background-only origin of all 54 events has been
rejected at 6.5-$\sigma$ level ~\cite{kop}. Therefore a
cosmic origin of a number of neutrino events is robust. The track
events have on average $\sim 1^\circ$ angular resolution, but the
dominant, shower events have much poorer angular resolution, $\sim
15^\circ$ on average ~\cite{kop}. Searching for sources of these events is now one of the major
challenges in astrophysics. Pinpointing the astrophysical sources
where these neutrinos are coming from is difficult due to large
uncertainty in their arrival directions.
High energy cosmic rays (CRs)
can interact with low energy photons and/or low energy protons to
produce neutrinos and high energy gamma rays inside the source or
while propagating to earth. So a multi-messenger study of neutrinos, Cosmic Rays (CRs)
and gamma-rays can identify the possible astrophysical sources. In our first attempt to search for sources we tried to see a correlation with Ultra-High Energy (UHE) CRs with the earlier 37 cosmic neutrino events ~\cite{Moharana:2015nxa}. A detail analysis of IceCube neutrino events with the Pierre Auger Observa (PAO) and Telescope Array (TA) has been done in collaboration ~\cite{Aartsen:2015dml}.
Here we study correlation of IceCube neutrino events with TeVCat, {\it Swift}-BAT 70 month X-ray source catalog ~\cite{2013ApJS..207...19B} and 3LAC source catalog ~\cite{Ackermann:2015yfk}. A similar study of correlation of IceCube neutrinos with the gamma ray sources has also been done ~\cite{Resconi:2015nva} to find a correlation of $Fermi$-LAT source with only track HESE events is also done ~\cite{anthony}. Recently a detail analysis of correlation showed at least 2 $\sigma$ result with extreme blazars ~\cite{Resconi2} and 3$\sigma$ with the starforming regions ~\cite{Emig:2015dma}. To do specific correlation study we use different cuts on the energy flux of these sources, and also different sets of source types, and showed the results of this study.
\begin{figure}[h]
\includegraphics[width=36pc]{allsource.png}
\caption{\label{skymap}Sky map of the 52 IceCube cosmic neutrino events with error circles and sources from different catalogs in Galactic coordinate system.}
\end{figure}
\section{IceCube neutrino events and Source catalogs}
For our analysis we consider all 52 IceCube detected neutrino events. Two track events (event numbers 28
and 32) are coincident hits in the IceTop surface array and are almost
certainly a pair of atmospheric muon background events
~\cite{aartsen2014observation}. Therefore we excluded them from our analysis.
Fig.~\ref{skymap} shows sky map of the 52 events in Galactic
coordinates with reported angular errors.
For the correlation analysis we have used 3 different source catalogs.
{\it Swift}-BAT 70 month X-ray source catalog ~\cite{2013ApJS..207...19B}, $Fermi$ Third Catalog of Active Galactic Nuclei (3LAC) ~\cite{Ackermann:2015yfk}, TeVCat ~\cite{2008ICRC....3.1341W}. The sky map in Fig.~\ref{skymap} shows the extragalactic sources from these catalogs.
{\it Swift}-BAT 70 month X-ray source catalog includes 1210 objects, and after excluding Galactic sources the number of sources become 785. In our previous study ~\cite{Moharana:2015nxa} we found 18 sources from this catalog that are correlated simultaneously with UHECRs and IceCube neutrino events. PAO collaboration has also found an anisotropy at $\sim 98.6\%$ CL in
UHECRs with energy $\ge 58$ EeV and within $\sim 18^\circ$ circles
around the AGNs in {\em Swift}-BAT catalog at
distance $\le 130$ Mpc ~\cite{PierreAuger:2014yba} . These 18 sources mostly have an X-ray energy flux $\ge10 ^{-11}$ ${\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$. So, in the present analysis we use all the sources from this catalog which have flux $\ge10 ^{-11}$ ${\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$. This condition decreased the number of sources to 687. In the sky map of Fig.~\ref{skymap} we have shown these 687 sources.
TeVCat contains sources that are detected with very high energy (VHE) gamma rays with energy $\ge 50$ GeV. It includes 161 sources, out of which 22 are unidentified sources. This is the highest energy source catalog, particularly interesting for $\nu$ production. Sky map in Fig~\ref{skymap} contains TeVCat sources that are not in the Galactic plane.
The {\it Third Catalog of Active Galactic Nuclei (AGNs)} detected by Fermi LAT (3LAC) ~\cite{Ackermann:2015yfk} is a subset of the {\it Fermi} LAT {\it Third Source Catalog (3FGL)} ~\cite{2015ApJS..218...23A}. The 3FGL catalog includes 3033 sources detected above a 4$\sigma$ significance (test statistic $>$ 25) on the whole sky, during the first 4 years of the Fermi mission (2008-2012). The original 3LAC sample includes 1591 AGNs from 3FGL, though 28 are duplicate associations. An additional cut had also been performed to exclude the Galactic plane region ($|b| \leq 10^\circ$) where the incompleteness of the counterpart catalogs significantly hinders AGN association. However, in this paper, we chose to study what we call the ``extended 3LAC" sample of 1773 sources, that includes sources of the Galactic plane, and that could be associated to several neutrino events. In the extended 3LAC sample, 491 sources are flat spectrum radio quasars (FSRQs), 662 are BL Lacs, 585 are blazars of unknown type (BCU), and 35 are non-blazar AGNs.
\section{Statistical method for Correlation study}
To study correlation between cosmic neutrinos and sources from different catalogs separately, we map the
Right Ascension and Declination $(RA, Dec)$ of the event directions and sources
into unit vectors on a sphere as
$$
{\hat x} = (\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta)^T,
$$
where $\phi = RA$ and $\theta = \pi/2 - Dec$. Scalar product of the
neutrino and source vectors $({\hat x}_{\rm neutrino}\cdot {\hat
x}_{\rm source})$ therefore is independent of the coordinate system.
The angle between the two vectors
\begin{equation}
\label{gamma}
\gamma = \cos^{-1} ({\hat x}_{\rm neutrino}\cdot
{\hat x}_{\rm source}),
\end{equation}
is an invariant measure of the angular correlation between the
neutrino event and source directions ~\cite{Virmani:2002xk,Moharana:2015nxa}. Following ref.~\cite{Virmani:2002xk} we use a
statistic made from invariant $\gamma$ for each neutrino direction
${\hat x}_i$ and source direction ${\hat x}_j$ pair as
\begin{equation}
\label{delta}
\delta\chi^2_i = {\rm min}_j (\gamma_{ij}^2/\delta\gamma_i^2),
\end{equation}
which is minimized for all $j$. Here $\delta\gamma_i$ is the
1-$\sigma$ angular resolution of the neutrino events. We use the exact
resolutions reported by the IceCube collaboration for each event
~\cite{aartsen2014observation}.
A value $\delta \chi^2_i \le 1$ is considered a ``good match'' between
the $i$-th neutrino and a source directions. We exploit
distributions of all $\delta\chi^2_i$ statistics to study angular
correlation between IceCube neutrino events and sources in catalog. The
distribution with observed data giving a number of ``hits'' or $N_{\rm
hits}$ with $\delta\chi^2 \le 1$ therefore forms a basis to claim
correlation. Note that in case more than one source direction from the catalog are
within the error circle of a neutrino event, the $\delta\chi^2$ value
for UHECR closest to the neutrino direction is chosen in this method.
We estimate the significance of any correlation in data by comparing
$N_{\rm hits}$ with corresponding number from null distributions.
We construct null distributions by randomizing only the $RA$ of the sources, keeping their $Dec$ the same as
their direction in the catalog. This {\it semi-isotropic null} is a quick-way to check
significance. We perform 100,000 realizations of drawing random numbers to
assign new $RA$ and $Dec$ values for each event to construct
$\delta\chi^2$ distributions in the same way as done with real data.
We calculate statistical significance of correlation
in real data or $p$-value (chance probability) using frequentists'
approach. We count the number of times we get a random data set that
gives equal or more hits than the $N_{\rm hits}$ in real data within
$\delta\chi^2 \le 1$ bin. Dividing this number with the total
number of random data sets generated (100,000) gives us the
$p$-value. We cross-check this $p$-value by calculating the Poisson
probability of obtaining $N_{\rm hits}$ within $\delta\chi^2 \le 1$
bin given the corresponding average hits expected from the null
distribution. We found the $N_{\rm hits}$ distribution in $\delta\chi^2 \le 1$ does not follow the Poisson distribution.
\section{Results and Discussions}
We used all 45 HBL (high-frequency peaked BL Lacs) type source listed in TeVCat for our first correlation study with neutrino events. A similar correlation study was carried out in ~\cite{Sahu:2014fua} using HBLs and neutrino data. Our study showed a $p$-value 0.58 with frequentists method while with Poisson distribution probability is 0.1, with 16 neutrinos correlating with different HBLs, almost the same as the null distribution. The distribution is shown in Fig.~\ref{hbl}.
{\it Swift}-BAT 70 month X-ray source included 657 sources with energy flux $10^{-11} {\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$. The study of correlation with neutrino events showed a $p$-value 0.825 with 39 $N_{\rm hits}$ for the real data and nearly 40 for null distribution, as shown in Fig.~\ref{swift}.
\begin{figure}[ht]
\begin{minipage}{16pc}
\includegraphics[width=16pc]{hbl}
\caption{\label{hbl}Correlation Study for all 45 HBL sources from TeVCat.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{18pc}
\includegraphics[width=16pc]{swift_11}
\caption{\label{swift}Correlation Study for {\it Swift} BAT X-ray catalog sources with energy flux more than $10^{-11} {\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$ also shown in ~\cite{rewin}.}
\end{minipage}
\end{figure}
The correlation study of all 1773 sources in the extended 3LAC catalog gives a $p$-value 0.806 with 41 $N_{\rm hits}$ in for real data, as shown in Fig.~\ref{3lac_all}. Most of the 3LAC sources are populated in the region of energy flux $10^{-11} {\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$, and the population decreases abruptly at higher flux. So, we took a set of sources with energy flux $\ge 10^{-11} {\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$. It decreased the number of sources in the set to 652, and the correlation study has a $p$-value 0.763 , with $N_{\rm hits}$ in $\delta\chi^2 \le 1$, 39, shown in Fig.~\ref{3lac_f}.
\begin{figure}[h]
\begin{minipage}{16pc}
\includegraphics[width=16pc]{3lac}
\caption{\label{3lac_all}Correlation Study for all 1773 sources of extended 3LAC catalog also shown in ~\cite{rewin}.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{16pc}
\includegraphics[width=16pc]{3lac_11}
\caption{\label{3lac_f}Correlation Study for sources from extended 3LAC catalog with energy flux $\ge$ $10^{-11} {\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$.}
\end{minipage}
\end{figure}
In order to do further study for different type of sources we used the 662 BL Lac source set from the extended 3LAC catalog. The correlation $p$-value for these sources is 0.764, shown in Fig. ~\ref{bllac}. Similarly for the 491 FSRQ sources from the extended 3LAC catalog the $p$-value is 0.784, shown in Fig.~\ref{fsrq}. For BL Lac and FSRQ sources we found 39 and 38 $N_{\rm hits}$ respectively.
\begin{figure}[h]
\begin{minipage}{16pc}
\includegraphics[width=16pc]{3lac_bll}
\caption{\label{bllac}Correlation Study of BL Lac sources from extended 3LAC catalog.}
\end{minipage}\hspace{2pc}%
\begin{minipage}{16pc}
\includegraphics[width=16pc]{3lac_fsrq}
\caption{\label{fsrq}Correlation Study of FSRQ sources from extended 3LAC catalog.}
\end{minipage}
\end{figure}
The correlation study of IceCube neutrino events with different type of sources as TeVCat HBL, 3LAC BL Lac and FSRQ is done but we have not found any statistically significant result for these sets. We have also put constraints on the energy flux of 3LAC catalog and the sources observed by {\it Swift} in 70 months of its observation, and the result is not significant. However with this type of study we can discard different type of extragalactic sources for IceCube neutrino events.
\begin{table*}\centering
\begin{tabular}{|c|c|c|c|} \hline
{Catalog Name} & {Source type} & {\# of sources} & {p-value}
\\ \hline
TeVCAT & HBL & 45 & 0.58 \\
$Swift$ Bat X-ray & energy flux > $10^{-11} \,{\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$ & 657 & 0.825 \\
3LAC (Extended) & All & 1773 & 0.806 \\
3LAC (Extended) & energy flux > $10 ^{-11} \,{\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$ & 652 & 0.763 \\
3LAC (Extended) & BL Lac & 662 & 0.764 \\
3LAC (Extended) & FSRQ & 491 & 0.786 \\
\hline
\end{tabular}
\caption{Results of correlation study. }
\label{tab:res}
\end{table*}
\section{Summary}
IceCube neutrino observatory has detected at least 54 neutrino events within energy 30 TeV-2 PeV. Sources for these events is still a puzzle for both particle physics and astrophysics. In our project we have tried to find correlation of the arrival direction of these events with direction of sources from TeVCat, {\it Swift} and 3LAC catalogs. In order to test correlation we have used invariant statistics, called the minimum $\delta \chi^2$, as in ~\cite{Virmani:2002xk,Moharana:2015nxa}. Out of 52 neutrino events, 16 were correlated with HBLs from TeVCat but the statistical significance of this correlation $p-$value is 0.58. Similarly we study correlation of neutrino events with sources from {\it Swift} and 3LAC having energy flux $\ge$ $10^{-11} {\rm{erg} \, \rm{cm^{-2}} \, \rm{sec}^ {-1}}$, for which we also found a poor statistical significance. The FSRQ and BL Lacs from 3LAC catalog also showed less significant statistics for the correlation study.
|
2,869,038,154,748 | arxiv | \section{Bootstrap-Based Test}
\label{sec:bootstrap}
\subsection{The Bootstrap Algorithm}
The permutation test bypasses the need to estimate the (unbiased) marginal distributions $f_{X}, f_{Y}$, which can be difficult and even impossible when $\wfun$ vanishes on part of the support of $f_{XY}$. Nevertheless, when we can estimate the univariate marginals consistently from the data, a bootstrap test is a viable alternative. Briefly, we generate independent samples from the estimated null distribution and compute the test statistic for each such sample. We then reject the null hypothesis if the observed test statistic is greater than the $1-\alpha$ quantile of the resulting bootstrap distribution. The test is summarized by Algorithm \ref{alg:bootstrap}.
\begin{algorithm}
\caption{{\small Bootstrap-Based Test of Quasi-Independence}} {\small
\begin{algorithmic}[1]
\Statex {\bf Input:} $\mathcal{D}$ - sample, $\wfun(x,y)$ - bias function, $T : \mathbb{R}^{2n} \to \mathbb{R}$ - a test statistic
\Statex {\bf Parameters:} $B$ - number of bootstrap samples
\State Estimate the marginals $\xcdfhat, \hat{F}_{Y}$.
\State Compute the test statistic $T_0 = T(\mathcal{D})$.
\For{$i=1$ to $B$}
\State Generate a bootstrap sample $\mathcal{D}_i$ by sampling with replacement $n$ i.i.d examples from $[\xcdfhat \hat{F}_{Y}]^{(w)}$.
\State Estimate the marginals $\hat{F}_{X,i}, \hat{F}_{Y,i}$ of the bootstrap sample $\mathcal{D}_i$.
\State Compute the test statistic $T_i = T(\mathcal{D}_i)$.
\EndFor
\State {\bf Output:} $P_{value} \equiv \frac{1}{B+1} \sum_{i=0}^{B} \indicator{ T_0 \ge T_i}$.
\end{algorithmic}
\label{alg:bootstrap}}
\end{algorithm}
As an alternative of estimating the marginal distributions, samples can be drawn under the null from the unbiased conditional distribution of $X$ given the observed $Y$ values $y_1,\ldots,y_n$; \cite{efron1999nonparametric} apply this approach to doubly truncated data.
\subsection{Estimating the Marginal Distributions}
\label{sec:bootstrap_marginal_estimation_null}
The next challenge is implementing step $1$ of Algorithm \ref{alg:bootstrap}, namely estimating the univariate marginals, given a known bias function $\wfun(x,y)$.
Naturally, under a bias-sampling regime, the underlying marginals may not be identifiable, unless additional modelling assumptions, either on $F_{XY}$ or $\wfun(x,y)$, are made.
\subsubsection{Case 1: Estimating the Marginal Distributions Under Quasi-independence}
\label{sec:marginal_estimation_under_null}
For a valid test, it is enough to estimate $F_{X}$ and $F_{Y}$ in the observable region under the null hypothesis of quasi-independence. A general algorithm for estimation of the marginal densities in \eqref{eq:quasi_independence} under quasi-independence is developed next. \cite{bickel1991large} provide a somewhat similar algorithm for the case where $X$ is discrete, which reduces to the selection bias model of \cite{vardi1985empirical}.
Let $\tilde{X}\sim \tilde{F}_X, \tilde{Y}\sim \tilde{F}_Y$, where $\tilde{F}_X,\tilde{F}_Y$ are the cumulative distribution functions of $\tilde{f}_X,\tilde{f}_Y$ in \eqref{eq:quasi_independence}. Under quasi-independence, by the law of total probability, the density of $X$ can be written as ${f}_{X}(x)=\mathbb{E}\{w(x,\tilde{Y})\}\tilde{f}(x)/\mathbb{E}\{w(\tilde{X},\tilde{Y})\}$. Thus, an estimate of $\tilde{F}_Y$ yields an estimate for $\mathbb{E}\{w(x,\tilde{Y})\}$, which can be used to build an inverse weighting estimate for $\tilde{F}_X$:
\begin{equation} \label{eq:estim_update}
\widehat{\tilde{F}}_X(x)= \frac{\sum_{i=1}^n \indicator{x_{i}\le x}
\hat{\mathbb{E}}\{w(x_i,\tilde{Y})\}^{-1}} {\sum_{i=1}^n \hat{\mathbb{E}}\{w(x_i,\tilde{Y})\}^{-1}}.
\end{equation}
This estimate can be used in turn to estimate $\tilde{F}_Y$, suggesting an iterative procedure as described in Algorithm \ref{alg:est_marg}.
For the important case $\wfun(x,y) = \indicator{x<y}$, the algorithm reduces to the standard product-limit (PL) estimator for left and right truncated data, implemented, for example, in the DTDA package of R \citep{moreira2010dtda}. For more details and an extension to more than two variables see Supp. Materials, Section \ref{sec:IterativeAlgorithm_appendix}.
\begin{algorithm}
\caption{{\small Estimation of Marginal Distributions Under Quasi-independence}} {\small
\begin{algorithmic}[1]
\Statex {\bf Input:} $\mathcal{D}$ - sample, $\wfun(x,y)$ - bias function, $d(F_1,F_2)$ - distance function.
\Statex {\bf Parameters:} $\epsilon$ - convergence criterion.
\State Generate initial estimates $\tilde F^{new}_X,\tilde F^{new}_Y$ and set $\tilde F^{old}_X=\tilde F^{old}_Y\equiv 0$.
\While{$d(\tilde F^{old}_X,\tilde F^{new}_X)+d(\tilde F^{old}_Y,\tilde F^{new}_Y)>\epsilon$}
\State Set $\tilde F^{old}_X=\tilde F^{new}_X$ and $\tilde F^{old}_Y=\tilde F^{new}_Y$.
\State Calculate $\mathbb{E}_{\tilde F^{new}_{Y}}\{w(x,{Y})\}$, and update $\tilde F^{new}_X$ using \eqref{eq:estim_update}.
\State Calculate $\mathbb{E}_{\tilde F^{new}_{X}}\{w({X},y)\}$, and update $\tilde F^{new}_Y$ using the equivalent for $\tilde{F}_Y$ of \eqref{eq:estim_update}.
\EndWhile
\State {\bf Output:} $\tilde F^{new}_X$ and $\tilde F^{new}_Y$.
\end{algorithmic}
\label{alg:est_marg}}
\end{algorithm}
While Algorithm \ref{alg:est_marg} provides a general procedure to estimate a distribution under independence, for testing purposes it may result in low power. Consider the truncation model $\wfun(x,y) = \indicator{x<y}$. The PL estimators are consistent under the null hypothesis, but using them in our test leads to low power, even in seemingly very extreme situations of a strong dependence. For a test to perform reasonably well for moderate sample sizes, $F_{X}$ and $F_{Y}$ should be estimated well not only under the null, but also under the alternative hypothesis (see Section \ref{sec:simulations} and Supp. Materials, Section B). The next example demonstrates this claim.
\noindent \textbf{Example: Difficulties in Detecting Quasi-Independence}
Consider the case where data is generated from a uniform
bivariate distribution over $\{0 < x, y < 1: \hspace{0.03in} |x − y| < 0.3\}$, and let $\wfun(x,y)=\indicator{x<y}$ be the standard truncation model.
We drew $n=500$ samples from this model and estimated the univariate marginals CDFs using the PL estimators. The left panel of Figure \ref{example} shows the sampled data points. The green and red curves in the right panel are the resulting PL estimates of $\tilde{F}_X$ and $\tilde{F}_Y$, respectively. Because the unbiased variables $\tilde{X}$ and $\tilde{Y}$ are exchangeable, they share the same underlying marginal distribution, depicted by the blue line. The product-limit estimates differ considerably from the true marginal distribution. When such CDFs generate the truncated data, the probability of a selection $(\tilde X < \tilde Y)$
is small, and when it happens, the values of $\tilde X$ and $\tilde Y$ tend to be close, yielding a scatter plot somewhat similar to the observed data. Indeed, the middle panel of Figure \ref{example} shows pairs obtained by generating independent
variables from the estimated product-limit curves $\xcdfhat$, $\hat{F}_{Y}$ and retaining only observations satisfying $\tilde X< \tilde Y$.
This example shows that independent variables under selection bias can produce data similar to that obtained by strongly dependent variables. Applying the bootstrap test using estimates of the marginal that are consistent only under the null independence assumption may result in a test with low power.
\begin{comment}
\begin{figure}[!h]
\begin{minipage}{1.\columnwidth}
\begin{center}
\begin{tabular}{cc}
\begin{tabular}{c}
\includegraphics[trim={0 2 0 0}, width=0.3\columnwidth]{../Figures/simulations/UniformStrip_rho_0.3_scatter.png}
\end{tabular}
&
\hspace{-0.1in}
\begin{tabular}{c}
\includegraphics[width=0.3\columnwidth]{../Figures/simulations/UniformStrip_rho_0.3_KM_marginal.png}
\end{tabular}\\
\begin{minipage}{0.374\columnwidth}
\vspace*{0.25in}
\hspace*{-0.45in}
\begin{tabular}{c}
\includegraphics[trim={0 0 0 1.5cm}, width=1.27\columnwidth]{../Figures/simulations/UniformStrip_rho_0.3_marginal_scatter.png}
\end{tabular}
\end{minipage}
&
\hspace{0.2in}
\begin{minipage}{0.3\columnwidth}
\vspace{-0.1in}
\caption{(top left): a scatterplot of samples drawn from the true underlying distribution. (bottom left): a scatterplot of samples drawn from the truncated independence distribution, using the PL estimates. (top right): the PL estimates of the univariate marginal CDFs $\xcdfhat$ and $\hat{F}_{Y}$ are biased and do not resemble the true underlying CDF $F_{X}=F_{Y}$.}
\label{example}
\end{minipage}
\end{tabular}
\end{center}
\end{minipage}
\end{figure}
\end{comment}
\begin{figure}[!h]
\begin{tabular}{ccc}
\includegraphics[trim={0 2 0 0}, width=0.3\columnwidth]{UniformStrip_rho_0.3_scatter.png}
&
\hspace{-0.1in}
\includegraphics[trim={0 0 0 1.5cm}, width=0.3\columnwidth]{UniformStrip_rho_0.3_marginal_scatter.png}
&
\hspace*{-0.1in}
\includegraphics[width=0.3\columnwidth]{UniformStrip_rho_0.3_KM_marginal.png}
\end{tabular}
\caption{(left): a scatterplot of samples drawn from the true underlying distribution. (middle): a scatterplot of samples drawn from the truncated independence distribution, using the PL estimates. (right): the PL estimates of the univariate marginal CDFs $\xcdfhat$ and $\hat{F}_{Y}$ are biased and do not resemble the true underlying CDF $F_{X}=F_{Y}$.}
\label{example}
\end{figure}
A possible solution is to find estimators for the marginal CDFs that are consistent also under the alternative hypothesis of quasi-dependence. However, as the model is not identifiable under the alternative \citep{cheng2007nonparametric}, such estimators can be calculated only under additional assumptions, either on $\wfun(x,y)$, or on the underlying joint distribution (or both). We next demonstrate this through two different settings.
\subsubsection{Case 2: Strictly Positive $\wfun(x,y)>0$}
The problem of estimating non-parametrically a general multivariate distribution $F$ using weighted data is well known (e.g., \cite{vardi1985empirical}) and for $w>0$ the non-parametric maximum likelihood estimator (NPMLE) is given by:
\be
\hat{F}_{XY}^n(t, s) = \frac{\sum_{i=1}^{n} \indicator{X_i\leq t, Y_i\leq s}w(X_i, Y_i)^{-1}}{\sum_{i=1}^{n}w(X_i, Y_i)^{-1}}.
\label{eq:non_parametric_MLE}
\ee
Estimators for $F_{X}$ and $F_{Y}$ can be then obtained by marginalization of Equation \eqref{eq:non_parametric_MLE},
\be
\xcdfempir(t) =\hat{F}_{XY}^n(x, \infty) = \frac{\sum_{i=1}^{n} \indicator{X_{i}\leq t}w(X_i,Y_i)^{-1}}{\sum_{i=1}^{n}w(X_i, Y_i)^{-1}}.
\label{eq:marginal_estimation_under_positive_weight_function}
\ee
The estimator above can be used whenever $w>0$ in the entire support of $F_{XY}$.
By the law of large numbers, $n^{-1}{\sum_{i=1}^{n} \indicator{X_{i}\leq t}w(X_i, Y_i)^{-1}}\to F_X(t)/\mathbb{E}_{f_{XY}} \{\wfun(X,Y)\}$ a.s. and $n^{-1}{\sum_{i=1}^{n}w(X_i, Y_i)^{-1}} \to 1/\mathbb{E}_{f_{XY}} \{\wfun(X,Y)\}$ a.s. so by the continuous mapping theorem $\xcdfempir(t) \to F_{X}(t)$ a.s. By similar arguments, $\hat{F}_{Y}^n(s) \to F_{Y}(s)$ a.s.
\subsubsection{Case 3: Left Truncation $\wfun(x,y) = \indicator{x<y}$}
In contrast to the former case, when $\wfun(x,y)$ is a truncation function, estimating the marginals under quasi-independence is more challenging due to the actual loss of data and identifiability issues. Nevertheless, for certain types of truncation mechanisms, additional assumptions on the joint distribution may allow to reconstruct the marginals. In particular, the most familiar type of truncation in the statistical literature is left (or right) truncation, described by $\wfun(x,y) = \indicator{x<y}$.
The next proposition shows that under exchangeability, the empirical distribution function of the joint sample $X_1,\ldots,X_n,Y_1,\ldots,Y_n$ is a consistent estimator for the marginals (the proof is in the Supp. Methods, Section \ref{sec:SI_proofs}).
\begin{proposition}
Let $F_{XY}(x,y)$ be an exchangeable joint distribution having a density $f_{XY}(x,y) \equiv f_{XY}(y,x)$ and let $\mathcal{D} \sim [F_{XY}^{(w)}]^n$ be a sample with the truncation weight function $\wfun(x,y) = \indicator{x<y}$. Let $\hat{F}^{(w),n}_X, \hat{F}^{(w),n}_Y$ be the empirical CDFs of $F_{X}^{(w)}, F_{Y}^{(w)}$, respectively. Define:
\be
\xcdfempir(x) = \hat{F}_{Y}^n(x) = \frac{\hat{F}^{(w),n}_X(x)+\hat{F}^{(w),n}_Y(x)}{2}=\frac{1}{2n}\Big[\sum_{i=1}^n \indicator{X_i\le x} + \sum_{i=1}^n \indicator{Y_i\le x} \Big].
\label{eq:exchange_estimator}
\ee
Then $\xcdfempir, \hat{F}_{Y}^n \to F_{X}=F_{Y}$ a.s.
\label{prop:consistent_estimator_under_exchangeability}
\end{proposition}
\section{Discussion}
\label{sec:discussion}
In this paper we address the problem of testing quasi-independence in the presence of a general bias function and in possibly
non-monotone settings.
As demonstrated using real-life data sets, testing independence naively, while ignoring the bias function, can either create spurious dependence or mask true dependence.
We introduce two general machineries to simulate samples under the biased null distribution, namely, the permutations and bootstrap approaches, and examine the challenges possessed by both.
Concretely, the former requires drawing permutations from a (general) non-uniform distribution, while the latter requires consistent estimation of the univariate marginals under the alternative hypothesis.
We tackle the first challenge by utilising an MCMC scheme and an importance sampling methodology.
Our simulation study indicates that for large sample size, the latter approach may suffer some degradation in performance, due to the difficulty in finding a proposal distribution suited for a general bias function.
On the bootstrap front, we identify two settings in which consistent estimators can be derived, both under the null and the alternative.
We also introduce a new algorithm for estimating the marginal CDFs under the null, for a general bias function.
This is of independent interest in cases where the null hypothesis is not rejected.
Importantly, both the permutations and bootstrap approaches can be combined with different statistics, thus result in a different test.
As shown in simulations and real-life data sets, the choice of statistic can affect the power of the resulting test.
An appealing feature of the methodologies developed here is that they can be easily adapted to cases where the bias function is not known, but can be estimated from the data.
An important instance of such cases is that of censoring. In particular, in observational studies, left truncation right censoring settings are frequently encountered.
Our tests can accommodate such settings and it would be interesting to explore this approach further to more general cases of censoring and biased sampling.
We demonstrate the merit of our proposed tests, using simulated and real-life data sets, and showed that even for truncated data they attain similar and often higher power in most settings considered here, compared to the recent minP2 test.
Lastly, some theoretical aspects of the proposed algorithms are yet to be explored and left for future work.
In particular we conjecture that, under the assumption of quasi-dependence, both the WP and bootstrap test are consistent.
\section{Simulation Studies}
\label{sec:simulations}
We investigated, using simulation, the performances of the weighted permutation (WP) and bootstrap tests, and compared them to that of \cite{tsai1990testing}'s and the minimum $P_{value}$ (minP2) test of \cite{chiou2018permutation}.
The latter are applicable only to truncated data of the form $\wfun(x,y)=\indicator{x<y}$.
We implemented the simulations in $R$, with time consuming parts implemented in {\it c++} using the $rcpp$ package (\cite{eddelbuettel2011rcpp}). Scripts reproducing all figures and tables are available online at \url{https://github.com/YanivTenzer/TIBS}. P-values for the minP2 test were calculated using the package permDep in R \citep{permDep}, version $1.0.3$ (Aug. 14th, 2019) from \url{https://github.com/stc04003/permDep}.
We calculated the rejection rate (power) at a significance level $\alpha=0.05$ by averaging results of $500$ replications. As the tests are computationally demanding, we used small sample sizes of $n=100$ and $n=200$ observations for uncensored and censored settings, respectively, in order to perform extensive simulations under different settings. We used $B=1000$ permuted or bootstrap null datasets for all tests except minP2 for which we used only $B=100$ null datasets and $100$ replications, as it was much slower.
The average running times of the tests on a standard laptop with an i7 2.8Ghz dual core Intel processor were $\approx\!0.02$ seconds for Tsai's test, $\approx\!0.11$ seconds for the new weighted permutation test, $\approx\!2.86$ seconds for the bootstrap test
and $\approx\!93.45$ seconds for the minP2 test.
\subsection{Truncation, $\wfun(x,y)=\indicator{x<y}$}
\label{sec:truncation}
We study the performances of the tests under truncation for various dependence models with and without censoring. The censoring variable, $C$, was sampled from Gamma distributions, with the shape and scale parameters set such that roughly $25 \%-30\%$ of the observations were censored for each model.
We first simulated data under monotone dependence models with an exchangeable joint distribution, where consistent estimators of the marginals exist (as shown in Section \ref{sec:bootstrap}) and we expect the bootstrap procedure to perform well. We generated $X$ and $Y$ from a standard bivariate Gaussian distribution with different correlations $\rho$ (Norm($\rho$)). In addition, we generated $X$ and $Y$ with standard Gaussian marginal distributions under two copula models: (i)
The Gumbel copula ({\bf GC}) with dependence parameter $\theta=1.6$ (Kendall's $\tau = 0.375$), and (ii) The Clayton copula ({\bf CC}) with dependence parameter $\theta=0.5$ ($\tau = 0.2$).
Although both the Gumbel and Clayton copulas produce monotone dependence structures, the two are different in nature - while the former provides upper tail dependence structure the latter produces lower tail dependence \citep{nelsen2007introduction}.
Figure \ref{fig:monotone_exchangeable} in the Supp. Materials presents scatterplots of simulated pairs from the three models.
The results are summarized in Table \ref{tab:simulation_study}, with the test having the highest power shown in boldface. As expected, in the Gaussian settings, under the null distribution (i.e., $\rho=0$), all tests achieve the correct $\alpha=0.05$ error rate. Under the alternative, the bootstrap procedure demonstrates favorable performance in all three settings. In the Gaussian settings, for $\rho<0$ the WP test consistently outperforms minP2. The minP2 has the lowest power in this setting. For $\rho>0$, the WP test has poorer performance, probably due to the difficulty of sampling permutations consistent with the truncation, while Tsai's test shows the second highest power, after the bootstrap.
\begin{table}
{\tiny
\begin{center}
\scalebox{1}{
\begin{tabular}{clccccccc}
\hline
& & \multicolumn{4}{c}{Uncensored ($n=100$)} & \multicolumn{3}{c}{Censored ($n=200$)} \\
\hline
Setting & Model & Tsai & minP2 & WP & Bootstrap & WP & MinP2 & Tsai \\
\hline
\multirow{10}{*}{\parbox{3.5cm}{\scriptsize Monotone \\ Exchangeable}}
& Norm($-0.9$) & {\bf 1} & {\bf 1} & {\bf 1} & {\bf 1} & \bf 1 & \bf 1 & \bf 1 \\
& Norm($-0.7$) & 0.998 & 0.960 & 0.998 & {\bf 1} & \bf 1 & 0.828 & \bf 1 \\
& Norm($-0.5$) & 0.764 & 0.510 & 0.742 & {\bf 0.998} & 0.813 & 0.284 & \bf 0.895 \\
& Norm($-0.3$) & 0.284 & 0.130 & 0.278 & {\bf 0.828} & 0.273 & 0.096 & \bf 0.366 \\
& Norm($0.0$) & 0.056 & 0.050 & 0.064 & 0.058 & 0.052 & 0.042 & 0.046 \\
& Norm($0.3$) & 0.178 & 0.110 & 0.118 & {\bf 0.780} & 0.097 & 0.083 &\bf 0.202 \\
& Norm($0.5$) & 0.352 & 0.140 & 0.194 & {\bf 1} & 0.211 & 0.103 & \bf 0.404\\
& Norm($0.7$) & 0.498 & 0.290 & 0.262 & {\bf 1} & 0.389 & 0.097 & \bf 0.729 \\
& Norm($0.9$) & 0.658 & 0.260 & 0.236 & {\bf 1} & 0.495 & 0.076 & \bf 0.907 \\
& GC ($\theta=1.6$) & 0.196 & 0.130 & 0.104 & {\bf 1} & 0.079 & 0.080 & \bf 0.192\\
& CC ($\theta=0.5$) & 0.110 & 0.130 & 0.074 & {\bf 0.782} & 0.126 & 0.127 & \bf 0.181 \\
\hline
\multirow{2}{*}{\parbox{3.5cm}{\scriptsize Monotone \\ Non-Exchangeable}}& LD ($\rho=0.0)$ & 0.042 & 0.010 & 0.046 & \textcolor{gray}{0.996} & 0.005 &0.048 &0.051 \\
& LD ($\rho=0.4)$ & {\bf 0.634} & 0.330 & 0.578 & \textcolor{gray}{0.704} & 0.060 & 0.056 & \bf 0.250 \\
\hline
\parbox{3.5cm}{\scriptsize Non-monotone \\ Exchangeable}
& CLmix($0.5$) & 0.278 & {0.140} & {\bf 0.412} & 0.338 & \bf 0.398& 0.120 & 0.308 \\
\hline
\multirow{12}{*}{\parbox{3.5cm}{\scriptsize Non-monotone, \\ Non-exchangeable}}
& CNorm($-0.9$)& 0.992 & {\bf 1} & {\bf 1} & \textcolor{gray}{1} & \bf 1 & 0.987 & 0.897 \\
& CNorm($-0.7$)& 0.844 & 0.950 & {\bf 0.992} & \textcolor{gray}{0.998} & \bf 0.972 & 0.609 & 0.635 \\
& CNorm($-0.5$)& 0.514 & 0.600 & {\bf 0.794} & \textcolor{gray}{0.988} & \bf 0.626 & 0.262 & 0.321 \\
& CNorm($-0.3$)& 0.176 & 0{\bf .290} & 0.272 & \textcolor{gray}{0.982} & \bf 0.212 & 0.138 & 0.157 \\
& CNorm($0.0$)& 0.042 & 0.020 & 0.046 & \textcolor{gray}{0.986} & 0.047 & 0.053 & 0.055 \\
& CNorm($0.3$)& 0.132 & 0.170 & {\bf 0.216} & \textcolor{gray}{0.998} & \bf 0.228 & 0.035 & 0.089 \\
& CNorm($0.5$)& 0.266 & 0.370 & {\bf 0.654} & \textcolor{gray}{1} & \bf 0.753 & 0.037 & 0.163 \\
& CNorm($0.7$)& 0.490 & 0.790 & {\bf 0.992} & \textcolor{gray}{1} & \bf 0.999 & 0.080 & 0.268 \\
& CNorm($0.9$)& 0.698 & {\bf 1} & {\bf 1} & \textcolor{gray}{1} & \bf 1 & 0.298 & 0.367 \\
\hline
\hline
\end{tabular}
}
\end{center}
\caption{Power at a significance level of $\alpha=0.05$ for left-truncated data ($\wfun(x,y)=\indicator{x<y}$) for uncensored (left 4 columns) and censored (right 3 columns) models. Norm($\rho$) - Bivariate normal distribution with correlation $\rho$. GC - The Gumbel Copula, CC - the Clayton Copula, with dependence parameter $\theta$. LD - Lifetime Distribution with correlation $\rho$, CLmix - an exchangeable mixture of two Clayton copulas, CNorm($\rho$) - non-exchangeable joint distribution with dependence parameter $\rho$. \label{tab:simulation_study}}
}
\end{table}
Following \cite{chiou2018permutation}, we next simulated data from a lifetime distribution (LD) where the joint distribution is non-exchangeable having marginal distributions $X \sim exp(5)$ and $Y \sim Weibull(3,8.5)$. We specified the dependence of $(X,Y)$ through a normal copula, where the strength of dependence is determined by the correlation parameter $\rho$. We simulated data under independent ($\rho=0$) and dependent ($\rho=0.4$) $F_{XY}$ before truncation. Figure \ref{monotone_non_exchangeable} in the Supp. Materials displays scatterplots of simulated pairs from both models.
The results are shown in the second part of Table \ref{tab:simulation_study}.
Although $X$ and $Y$ are not exchangeable, we applied the bootstrap procedure (shown in light gray) as well, using marginal estimates according to Equation \eqref{eq:exchange_estimator}, in order to investigate the impact of model miss-specification on its performance. The devastating impact of model miss-specification under the null distribution for the bootstrap procedure is now apparent: the test does not retain the desired rejection rate $\alpha=0.05$ under the null hypothesis ($\rho = 0$). For $\rho=0.4$, Tsai's test has the highest power, followed by the WP test and then minP2.
The third simulation study evaluates the performance of the tests under non-monotone dependence. Starting with a non-monotone exchangeable model, we simulated data from a mixture of two Clayton copulas with dependence parameters $\theta=0.5 \hspace{0.03in}(\tau = 0.2)$ and $\theta=-0.5 \hspace{0.03in}(\tau=-0.333)$, respectively, and equal population proportions. Figure \ref{non_monotone_exchangeable} in the Supp. Materials presents scatterplots of simulated pairs from the model. The third part of Table \ref{tab:simulation_study} presents the results. Our bootstrap approach has the highest power, followed by the WP test, Tsai's test and lastly minP2.
It is somewhat surprising that Tsai's test outperforms here minP2, as the former is tailored to monotone alternatives. %
Finally, we considered a non-monotone and non-exchangeable model.
We used a normal copula, CNorm($\rho$), with varied correlation coefficient $\rho$ to specify the joint distribution of $X$ and $Y$, where $X \sim Weibull(0.5,4)$ and $Y \sim U[0,16]$, set such that $\mathbb{E}(X)=\mathbb{E}(Y)=8$, and retained only pairs satisfying $Y \geq X$. Figure \ref{fig:non_monotone_non_exchangeanle} in the Supp. Materials displays scatterplots of simulated pairs from the models.
The last part of Table \ref{tab:simulation_study} presents the results. As expected, the bootstrap procedure (light gray) does not retain the desired rejection rate under the null hypothesis. Under the alternative, both the WP and minP2 tests outperform Tsai's procedure across the entire range of the dependence parameter $\rho$. This behaviour is expected because both tests were designed to detect non-monotone dependency. Our method has the highest power for all values of $\rho$. minP2 is more powerful than Tsai's test for strong (absolute) correlations and less powerful for weaker correlations.
\subsection{Strictly Positive Bias Functions}
\label{sec:strictly_positive}
Our new tests can be used to detect dependency in a general weighted model. To study the performance of our approach, we consider two cases of positive biased sampling.
In the first case we took $w(x,y)=x+y$, where $X$ and $Y$ were sampled from the log-normal bivariate distribution with zero mean, unit variance and correlation $\rho$. Recall that for a strictly positive $\wfun$, the inverse weighted Hoeffding statistic from Section \ref{sec:inverse_weight_stat} can be used as an alternative to the adjusted Hoeffiding statistic. We also compared here the bootstrap and MCMC approaches to the importance sampling approach from Section \ref{sec:importance_sampling}, with four different importance sampling distributions, described in the Supp. Methods, Section \ref{sec:IS_appendix}.
We applied six different sampling methods for p-value calculation for each of the two test statistics, resulting in twelve different tests.
When applying the Bootstrap, we estimated the univariate marginals using the weighted estimators in Equation \eqref{eq:marginal_estimation_under_positive_weight_function}.
Table \ref{tab:strictly_positive} shows the rejection rates of the various tests at a significance level $\alpha=0.05$, for sample size $n=100$ and for $\rho=0$ (independence) and $\rho=0.2$. All tests except the bootstrap seem to maintain the significance level at approximately $\alpha =0.05$ under the null. Under the alternative, the importance sampling approach with a uniform and 'grid' importance distribution is most powerful, but the MCMC approach is not far behind.
In the second example, we considered a bivariate Gaussian distribution $Norm(\rho)$ for $X$ and $Y$ as in Section \ref{sec:truncation}, and with $\wfun(x,y) \propto Norm(-\rho)$. The biased sampling function here masks the dependence so the observed pairs, $(X_i,Y_i)$, are independent Gaussian random variables. This example shows that biased sampling can not only create spurious dependencies, but can also mask true dependencies. Nevertheless, knowing $\wfun$ we can apply our tests and detect the dependence, as is shown in Table \ref{tab:strictly_positive}.
As expected, for all tests, as $\rho$ increases (we used only $\rho>0$ due to symmetry) the power increases quite rapidly. The importance sampling with a uniform distribution is usually the most powerful, with the MCMC approach very close or superior for strong correlation. The bootstrap approach shows poor performance for this case. As in the previous example, here too the inverse weighting statistic is inferior to the adjusted Hoeffding statistic. The relative success of the importance sampling schemes for both examples can be explained by the small sample size. As shown in the Supp. Materials, Section \ref{sec:IS_appendix}, when the sample size increases the importance sampling distributions become unrepresentative of the distribution $P_{\wmat}$, resulting in poor performance, whereas the MCMC approach is much more robust to changes in sample size.
\begin{table}[tb]
\centering
\begingroup\tiny
\begin{tabular}{rrrrrrr}
\hline
Model & WPIS & WPIS & WPIS & WPIS & WP & Bootstrap \\
IS-Dist. & Kou-McCullagh & Uniform & Monotone & Grid & & \\
\hline
LogNormal($\rho=0$) & 0.048 & 0.050 & 0.016 & 0.050 & 0.056 & 0.072 \\
IW & 0.050 & 0.038 & 0.016 & 0.036 & 0.060 & 0.080 \\
LogNormal($\rho=0.2$) & 0.606 & {\bf 0.676} & 0.004 & {\bf 0.676} & 0.602 & 0.632 \\
IW & 0.384 & 0.432 & 0.000 & 0.428 & 0.382 & 0.414 \\
\hline
Norm(0.0) & 0.046 & 0.048 & 0.032 & 0.040 & 0.046 & 0.004 \\
IW & 0.056 & 0.058 & 0.032 & 0.054 & 0.050 & 0.004 \\
Norm(0.1) & {\bf 0.084} & {\bf 0.084} & 0.052 & 0.080 & 0.082 & 0.006 \\
IW & 0.076 & 0.076 & 0.032 & 0.076 & 0.078 & 0.008 \\
Norm(0.3) & 0.296 & 0.322 & 0.134 & {\bf 0.350} & 0.302 & 0.052 \\
IW & 0.282 & 0.296 & 0.136 & 0.318 & 0.298 & 0.024 \\
Norm(0.5) & 0.582 & {\bf 0.640} & 0.298 & 0.630 & 0.582 & 0.166 \\
IW & 0.502 & 0.526 & 0.228 & 0.532 & 0.492 & 0.038 \\
Norm(0.7) & 0.856 & {\bf 0.860} & 0.516 & 0.850 & 0.852 & 0.334 \\
IW & 0.730 & 0.706 & 0.456 & 0.684 & 0.726 & 0.078 \\
Norm(0.9) & {\bf 0.964} & 0.914 & 0.712 & 0.918 & {\bf 0.964} & 0.672 \\
IW & 0.876 & 0.800 & 0.662 & 0.786 & 0.878 & 0.114 \\
\hline
\end{tabular}
\caption{Strictly positive bias functions; estimated power at a significance level of $\alpha=0.05$, and sample size $n=100$ of the permutations (WP), uniform importance sampling permutations (WPIS), and bootstrap tests. For each parameter settings and sampling method two statistics were applied, shown in separate lines: The adjusted Hoeffding statistic, with expectations estimated using permutations/bootstrap samples, and the inverse weighting statistic (IW). The top four rows represent a LogNormal distribution with $\wfun(x,y)=x+y$. The bottom rows represent a Gaussian distribution with correlation $\rho$, with $w(x,y)$ proportional to a Gaussian density with correlation $-\rho$. \label{tab:strictly_positive}}
\endgroup
\end{table}
\section{Introduction}
\label{sec:introduction
\vspace{-0.2cm} Testing independence of two random variables $X,Y$ is a fundamental statistical problem. Classical methods have focused on testing linear (Pearson’s correlation coefficient) or monotone (Spearman’s correlation, Kendall's tau) dependence, while other works focus on developing methods to capture complex dependencies (e.g. using Pearson's Chi-squared test). This classical problem keeps drawing attention from scholars with recent approaches focusing on omnibus tests employing computer-intensive methods \citep{gretton2008kernel,szekely2009brownian, heller2012consistent,heller2016consistent}.
A more challenging task is testing independence of two random variables when data is obtained through a general biased sampling mechanism. The most familiar example is that of truncation, where observations are restricted to a certain `observable region'. All cross-sectional samples are subject to some form of truncation, and the standard way of analysing such data is to assume independence between the truncation mechanism and the variables of interest (see \cite{chiou2018permutation} for more discussion and references). The results of the analysis can be highly biased if the independence assumption is violated. Yet, another important problem that exploits tests for truncated data is testing the Markov assumption in the Illness-Death model \citep{rodriguez2012nonparametric}.
The problem of quasi-independence was first dealt with in the framework of contingency tables (e.g., \cite{goodman1968analysis}) and was studied more recently in the framework of survival analysis where data are restricted by the condition $X\le Y$ (see, \cite{tsai1990testing} and the discussion below). \cite{bickel1991large} consider general biased regression models for which independence is equivalent to a zero regression coefficient. Another common framework where the problem naturally arises is in cross-sectional sampling designs; Section \ref{sec:real_data} provides several examples.
Biased sampling in general, and truncation in particular, may imply dependence in the sample that does not exist in the population. This fact was acknowledged more than a century ago by \cite{elderton1913correlation} who studied the correlation between an intellectually disabled child’s place in the family and the size of that family. They noticed that ``the size of the family must be as great or greater than the imbecile's place in it ... and there would certainly be correlation, if we proceeded to find it by the usual product moment method, but such correlation is, or clearly may be, wholly spurious".
In this paper, we study the problem of detecting dependency from general biased samples. We are given a sample of $n$ independent and identically distributed (i.i.d.) observations $\{(x_i, y_i)\}_{i=1}^n$ drawn from a joint distribution having a density $F^{(w)}_{X,Y}(dx,dy) \propto {\wfun(x,y)F_{XY}(dx,dy)}$, where $\wfun$ is a known non-negative function having a positive finite expectation with respect to $F_{XY}$. Here $F_{XY}$ is the joint distribution of the pair $(X,Y)$ in the population, and $F^{(w)}_{X,Y}$ is the distribution of observed pairs, tilted by $\wfun$, the sampling mechanism. The aim is to test the null hypothesis $H_0: F_{XY}(x,y)=F_{X}(x)F_{Y}(y)$ for all $(x,y)$. However, because $F_{XY}$ is not identifiable on $\{(x,y):\wfun(x,y)=0\}$, the goal is restricted to testing quasi-independence defined as $F_{XY}(dx,dy)=\tilde{F}_{X}(dx)\tilde{F}_{Y}(dy)$ for all $(x,y)\in \{(x',y'):\wfun(x',y')>0\}$, for some functions $\tilde{F}_{X},\tilde{F}_{Y}$ \citep{tsai1990testing}.
Testing quasi-independence under a biased-sampling regime is challenging and previous works mainly focused on simple truncation models. \cite{tsai1990testing} considered the problem of testing quasi-independence under left-truncation (and right censoring) based on the conditional Kendall's tau correlation coefficient. \cite{efron1999nonparametric} and \cite{martin2005testing} extended this method to the settings of double-truncation.
\cite{chen2007sequential} suggested an importance sampling algorithm to estimate the P-value under truncation models. \cite{emura2010testing} constructed a log-rank type statistic for the left-truncation setting.
\cite{chen1996product} proposed a conditional version of Pearson's product-moment correlation. These tests are powerful for monotone alternatives, but generally less powerful against non-monotone dependencies frequently encountered in real-life applications.
Some recent works accommodated non-monotone alternatives by utilizing local versions of Kendall's tau test \citep{rodriguez2012nonparametric, de2012markov}, and a weighted version of the local Kendall's tau \citep{rodriguez2016methods}. These tests are inefficient for small sample sizes, and their performance depends on the choice of pre-selected grids.
Testing quasi-independence by utilizing permutations was previously proposed by \cite{tsai1990testing}, \cite{efron1999nonparametric} and \cite{chiou2018permutation}. The test of \cite{chiou2018permutation} is based on a scan statistic that partitions the sample space into two groups according to a threshold value of $X$, and compares the distributions of $Y$ in the two groups. Although no formal results are established, the simulations indicate that the procedure has comparable power to the test of \cite{martin2005testing} for monotone alternatives, and performs much better for non-monotone alternatives. Yet, similarly to other aforementioned works, these works consider only the cases of one and two-sided truncation.
The current paper describes a new family of tests of independence for data from a general biased sampling design. The tests are not restricted to monotone alternatives and are based on a scan statistic similar to the one suggested by \cite{heller2016consistent}. As in \cite{heller2016consistent}, P-values are calculated using permutations, but here the permutation distribution under the null hypothesis is not uniform and generating permutations is a challenging task. We consider two approaches for calculation of P-values, the first employs a Markov-Chain Monte Carlo (MCMC) algorithm to sample from the resulting weighted permutation distribution, and the second uses importance sampling.
In addition, an alternative bootstrap test of independence is considered that requires consistent estimation of the univariate marginal distributions under the alternative hypothesis. We identify two settings under which consistent estimators of the marginals are attainable (under both the null and the alternative hypotheses).
The rest of the paper is organized as follows. Section \ref{sec:preliminaries} introduces the problem of testing quasi-independence and its relation to previous works. Section \ref{sec:permutations} derives the permutation tests and studies some of their theoretical properties. Section \ref{sec:bootstrap} presents an alternative bootstrap-based approach. Section \ref{sec:teststat} introduces the adjusted Hoeffding Statistic and describes its computation. An inverse weighting test for the case of strictly positive $w$, and implementation of the tests to left-truncated right-censored data are also discussed. The new methods are compared in simulations and applied to real-life data sets in Sections \ref{sec:simulations} and \ref{sec:real_data}, respectively. Section \ref{sec:discussion} completes the paper with a discussion.
\section{Permutation Test}
\label{sec:permutations}
\subsection{The Distribution of Permutations}
Under biased-sampling, different permutations are not equally likely under the null model, thus should not be uniformly sampled as in standard permutation tests. Therefore, the sampling mechanism should account for the discrepancy in weight of distinct permutations based on the data. Let $\pi(y)$ be the vector $y$ rearranged according to a permutation $\pi$, i.e. $\pi(y) = (y_{\pi(1)}, .., y_{\pi(n)})$, and
let $\pi(\mathcal{D})$ be the permuted sample: $\pi(\mathcal{D}) \equiv \big((x_1, y_{\pi(1)}), .., (x_n, y_{\pi(n)})\big)$.
For a sample $\mathcal{D}=\{(x_i, y_i)\}_{i=1}^n$, let $\wmat \in \mathbb{R}_{n\times n}$ be a weight-matrix defined by
$\wmat(i,j) \equiv \wfun(x_i, y_j).$
Let $S_n$ be the set of all permutations of $n$ elements, and for $\pi\in S_n$ consider the probability
\be
P_{\wmat}(\pi) \equiv \frac{1}{per(\wmat)} \prod_{i=1}^n \wmat(i,\pi(i)),
\label{eq:prob_perm}
\ee
where $per(\wmat) = \sum_{\pi \in S_n} \prod_{i=1}^n \wmat(i, \pi(i))$ is the normalizing constant, given by the permanent of the matrix $\wmat$.
\setcounter{claim}{0}
\begin{claim}
\label{claim:P_W_conditional}
Under $H_0$, the probability $P_{\wmat}(\pi)$ represents the probability of observing permuted datasets conditional on the marginal sets, that is, $P_{\wmat}(\pi(\mathcal{D})) = P_0(\pi \mid x, \mathcal{D}_y)$, where $P_0$ denotes the probability under quasi-independence.
\end{claim}
\begin{proof}
For a general weighted model, we have
%
\be
\label{eq:generalperm}
P(\pi(\mathcal{D}) \mid x, \mathcal{D}_y) = \frac{\prod_{i=1}^n f_{XY}^{(w)}(x_i, y_{\pi(i)}) }{\sum_{\pi'\in S_n} \prod_{i=1}^n f_{XY}^{(w)}(x_i, y_{\pi'(i)}) }.
\ee
%
Under the null, $f_{XY}^{(w)}(x,y)\propto w(x,y)\tilde{f_{X}}(x)\tilde{f_{Y}}(y)$, hence
\begin{eqnarray*}
P_0(\pi(\mathcal{D}) \mid x, \mathcal{D}_y) = \frac{\prod_{i=1}^n \tilde{f_{X}}(x_i) \tilde{f_{Y}}(y_{\pi(i)}) \wmat(i,\pi(i))}{\sum_{\pi' \in S_n} \prod_{i=1}^n \tilde{f_{X}}(x_i) \tilde{f_{Y}}(y_{\pi'(i)}) \wmat(i,\pi'(i))}
= P_{\wmat}(\pi).
\end{eqnarray*}
\end{proof}
When $\wfun(x,y)$ is a truncation function, $P_{\wmat}(\pi)$ is simply the uniform distribution over the set of valid permutations, i.e. permutations $\pi$ yielding permuted datasets $\pi(\mathcal{D})$ which are consistent with the truncation.
The next lemma shows that permuted data points drawn from Equation \eqref{eq:generalperm} follow the distribution of $n$ independent copies of $(X,Y) \sim F_{XY}^{(w)}$.
\begin{lemma}
\label{lemma:doulbe_randomization}
Let $\dataset \sim [F_{XY}^{(w)}]^n$ and conditionally on $\dataset$ let $\pi$ be a permutation having the conditional probability law given in Equation \eqref{eq:generalperm}. Then $\pi(\dataset) \sim [F_{XY}^{(w)}]^n$.
\begin{proof}
Using Equation \eqref{eq:generalperm} and the law of total probability, we have
\begin{eqnarray}
f_{\pi(\dataset)}\Big((x_1,y_1),\ldots,(x_n,y_n)\Big)& =&
\sum_{\pi \in S_n} \frac{\prod_{i=1}^n f_{XY}^{(w)}(x_i, y_i)}{\sum_{\pi'} \prod_{i=1}^n f_{XY}^{(w)}(x_i, y_{\pi'(i)}) } \prod_{i=1}^n f_{XY}^{(w)}(x_i, y_{\pi(i)}) \nonumber \\
& =& \prod_{i=1}^n f_{XY}^{(w)}(x_i, y_i).
\end{eqnarray}
\end{proof}
\end{lemma}
Recalling that under the null, Equation \eqref{eq:generalperm} reduces to \eqref{eq:prob_perm} (by Claim \ref{claim:P_W_conditional}), we have
\begin{corollary}
\label{cor:alpha}
Under the null, permuted data points drawn according to Equation \eqref{eq:prob_perm} follow the distribution of $n$ independent copies of $(X,Y) \sim [\tildeF_{X} \tildeF_{Y}]^{(w)}$.
\end{corollary}
\subsection{The Weighted-Permutation Test of Independence}
Let $T(\mathcal{D})$ be any test statistic. The permutation test consists of comparing $T(\mathcal{D})$ to its null distribution over all permuted samples $\pi(\mathcal{D})$, and calculating the P-value by the proportions of permutations $\pi_i$ with
test statistic $T(\pi_i(\mathcal{D}))$ exceeding $T(\mathcal{D})$. In practice, a large number of permutations, $B$, is sampled, and the P-value is approximated by
\be
{P}_{value} = \frac{1}{B+1} \sum_{i=0}^{B} \indicator{ T(\pi_i(\mathcal{D})) \geq T(\mathcal{D}) },
\label{eq:empirical_pval}
\ee
where $\pi_0$ is the identity permutation corresponding to $T(\mathcal{D})$. We formalize our weighted-permutation test of independence as shown in Algorithm \ref{alg:weighted_permutations}.
\begin{algorithm}
\caption{{\small Weighted Permutation Test of Quasi-Independence}} {\small
\begin{algorithmic}[1]
\Statex {\bf Input:} $\mathcal{D}$ - sample, $\wfun(x,y)$ - bias function, $T : \mathbb{R}^{2n} \to \mathbb{R}$ - a test statistic
\Statex {\bf Parameters:} $B$ - number of permutations
\State Generate $B$ permutations $\pi_1,..,\pi_{B} \sim P_{\wmat}$.
\State Compute the test statistic $T_0 \equiv T(\mathcal{D})$.
\For{$i=1$ to $B$}
\State Compute the test-statistic for the permuted dataset, $T_i = T(\pi_i(\mathcal{D}))$.
\EndFor
\State {\bf Output:} $P_{value} \equiv \frac{1}{B+1} \sum_{i=0}^{B} \indicator{T_i \geq T_0}$.
\end{algorithmic}
\label{alg:weighted_permutations}}
\end{algorithm}
%
Corollary \ref{cor:alpha} assures that under the null distribution $P_{value} \sim U[\frac{1}{B+1},...,\frac{B}{B+1},1]$ the type-1 error probability of the weighted permutation test is at most $\alpha$. (The addition of $1$ to the denominator and numerator in stage $6$, i.e. including the original sample, is necessary to ensure type-1 error below $\alpha$, but can be neglected in practice for large $B$.)
In order to implement Algorithm \ref{alg:weighted_permutations}, a method to sample weighted permutations $\pi_i \sim P_{\wmat}$ is required. An MCMC algorithm that generates such permutations is discussed next.
\subsection{Sampling Permutations using MCMC}
\label{sec:mcmc_permutations}
The case of sampling uniformly from a restricted set of permutations, i.e., permutations $\pi$ with $P_{\cal W}(\pi)>0$, was considered by \cite{diaconis2001statistical}. Their algorithm deals with the important special case of truncation with a 0/1 weight function. To enable sampling from a general distribution, we utilize the Metropolis-Hasting (MH) algorithm \citep{metropolis1953equation,hastings1970monte}. Let $\pi_t = (\pi_t(1),..,\pi_t(n))$ be the permutation at step $t$. Define the neighbours of $\pi_t$ to be all permutations obtained from $\pi_t$ by a single swap, that is,
\[
Neig(\pi_t) \equiv\{\pi_t^{i \leftrightarrow j} \equiv \big(\pi_t(1),..\pi_t(j),..,\pi_t(i), .., \pi_t(n)\big),\hspace{0.04in} \forall i < j\}.
\]
We then proceed according to the standard MH algorithm: at each iteration we sample a permutation uniformly from this set $\pi_t^{i \leftrightarrow j} \sim U [Neig(\pi_t)]$, as well as generate a uniform random number $u$ on $[0,1]$. Finally, we accept the new permutation only if
\[
u \leq \frac{P(\pi_t^{i \leftrightarrow j})}{P(\pi_t)} = \frac{\wmat(i,\pi_t(j)) \wmat(j,\pi_t(i))}{\wmat(i,\pi_t(i))\wmat(j,\pi_t(j))}.
\]
A similar algorithm was suggested by \cite{efron1999nonparametric} for doubly truncated data. However, for truncated data the weights are all 0 or 1, making the problem much simpler.
Algorithm \ref{alg:mcmc_permutations} describes our MCMC approach step-by-step.
\begin{algorithm}
\caption{{\small MCMC for Biased Sampling of Permutations}} {\small
\begin{algorithmic}[1]
\Statex {\bf Input:} $\mathcal{D}$ - sample, $\wfun(x,y)$ - bias function
\Statex {\bf Parameters:} $B$ - number of permutations, $M_0$ - 'burn-in' number of steps, $M$ - number of steps between two permutations.
\State Compute $\wmat(i,j) = \wfun(x_i,y_j), \quad \forall i,j=1,..,n$.
\State Set $\pi_0$ the identity permutation $\pi_0(i) = i$.
\For{$t=0$ to $M_0 + B M - 1$}
\State Sample $\pi_t^{i \leftrightarrow j} \sim U [Neig(\pi_t)]$, and $u \sim U[0,1]$.
\If {$u \leq \frac{\wmat(i,\pi_t(j)) \wmat(j,\pi_t(i))}{\wmat(i,\pi_t(i))\wmat(j,\pi_t(j))}$}
\State set $\pi_{t+1} \leftarrow \pi_t^{i \leftrightarrow j}$.
\Else
\State set $\pi_{t+1} \leftarrow \pi_t$.
\EndIf
\EndFor
\State {\bf Output:} The resulting $B$ permutations $\pi_{M_0},\pi_{M_0+M},..,\pi_{M_0+B M}$.
\end{algorithmic}
\label{alg:mcmc_permutations}}
\end{algorithm}
\subsection{Importance Sampling}
\label{sec:importance_sampling}
Due to the difficulty of sampling directly from the weighted permutations distribution $P_{\wmat}$, we propose here another approach: sample permutations $\pi_1,\ldots,\pi_B$ according to an importance probability law $P_{IS}$ such that $P_{IS}(\pi)>0$ whenever $P_{\wmat}(\pi)>0$, and calculate the P-value by
\begin{equation}
P_{value} \equiv \frac{\sum_{i=0}^{B} \frac{ P_{\wmat}(\pi_i)}{ P_{IS}(\pi_i)} \indicator{ T_i \geq T_0 } } {\sum_{i=0}^{B}\frac{P_{\wmat}(\pi_i)}{P_{IS}(\pi_i)}}.
\label{eq:IS_pval}
\end{equation}
The unknown term $per(\wmat)$ appearing in $P_{\wmat} $ (see Equation \eqref{eq:prob_perm}) is cancelled in the above equation, thus enabling us to compute the $P_{value}$ even when $P_{\wmat}$ is known only up to a normalizing constant.
\cite{chen2007sequential} suggest this importance sampling algorithm for statistical inference under truncated data. It is based on a simple sequential method to generate permutations under $P_{IS}$. \cite{kou2009approximating} have generalized one of the approaches proposed in \cite{chen2007sequential}
to estimate the permanent of general weight functions. We have derived several sequential importance sampling approaches, similar to those of \cite{chen2007sequential} and \cite{kou2009approximating}, that are applicable for general weight functions $\wmat$, and investigated their performances in testing.
\cite{harrison2012conservative} shows that for any test statistic satisfying mild invariance properties, the test that includes the identity permutation
in the P-value calculation in Equation \eqref{eq:IS_pval} controls the type-1 error at level $\alpha$ (see his Theorem 1).
This result applies directly to our case by considering our approach as testing conditionally on the data $\mathcal{D}$.
Although the correction above ensures validity, the importance sampling approach can perform very poorly if the importance distribution $P_{IS}$ is far from $P_{\wmat}$, for example when $P_{IS}$ is taken to be the uniform distribution over $S_n$, because in such cases the $B$ sampled permutations have very low probability under $P_{\wmat}$. It is thus challenging to suggest a distribution $P_{IS}$ that is both easy to calculate and sample from, as well as close enough to $P_{\wmat}$ for general $\wmat$ - see Supp. Materials, Section \ref{sec:IS_appendix} for more details.
\section{Preliminaries}
\label{sec:preliminaries}
Let $F_{XY}$ be a bivariate distribution function with a density $f_{XY}$ and univariate marginals $f_{X}, f_{Y}$. We consider $n$ independent pairs $(X_i,Y_i) \sim F_{XY}^{(w)}(x,y)$ of scalar continuous random variables, sampled from the joint density
\be
f^{(w)}_{X,Y}(x,y)=\wfun(x,y) f_{XY}(x,y)/\mathbb{E}_{f_{XY}} \{\wfun(X,Y)\},
\label{eq:weighted_density}
\ee
where $\wfun : \mathbb{R}^2 \to \mathbb{R}^+$ is a non-negative weight function such that $0 < \mathbb{E}_{f_{XY}}\{\wfun(X,Y)\} < \infty$. The marginals of the observed data are denoted by $f_{X}^{(w)}(x) = \int_{-\infty}^{\infty} f_{XY}^{(w)}(x,y) dy$ and $f_{Y}^{(w)}(y) = \int_{-\infty}^{\infty} f_{XY}^{(w)}(x,y) dx$.
The weighted independent density is defined as $[f_{X} f_{Y}]^{(w)}(x,y) = {\wfun(x,y) f_{X}(x) f_{Y}(y)}/{\mathbb{E}_{f_{X} f_{Y}} \{\wfun(X,Y)\}}$, with the corresponding weighted distribution $[F_{X} F_{Y}]^{(w)}$.
An important special case is that of a truncated sample in which
\begin{align}
\wfun(x,y) = \setindicator{\truncregion}(x,y) = \left\{ \begin{array}{lll}
1 & \quad (x,y) \in \truncregion \\
0 & \quad otherwise, \\
\end{array} \right.
\label{eq:truncation_function}
\end{align}
for some set $\truncregion \subset \mathbb{R}^2$. This special form of $\wfun(x,y)$ arises frequently in practice and was previously investigated by several authors, as discussed in Section \ref{sec:introduction}.
A strongly related concept to our problem is that of \emph{quasi-independence} in truncation models \citep{tsai1990testing}, which can be naturally extended to a general weight function $\wfun(x,y)$:
\begin{definition}(quasi-dependence)
We say that the joint distribution $F_{XY}^{(w)}$ is quasi-independent with respect to the weight function $\wfun$, if there exist density functions $\tilde{f_{X}}$ and $\tilde{f_{Y}}$, such that
\be \label{eq:quasi_independence}
F_{XY}^{(w)}(x,y) \propto \int\displaylimits_{-\infty}^{x}\int\displaylimits_{-\infty}^{y} \wfun(s,t) \tilde{f_{X}}(s) \tilde{f_{Y}}(t) dt ds, \quad \forall x,y \in \mathbb{R} \: .
\ee
Otherwise, we say that $F_{XY}^{(w)}$ is quasi-dependent with respect to the weight function $\wfun$.
\label{def:quasi_independence}
\end{definition}
Based on the sample $\mathcal{D} = \{(x_i,y_i)\}_{i=1}^n$, we aim at performing the following hypothesis testing for quasi-dependence:
\be
\begin{array}{rl}
H_0: & F_{XY}^{(w)} \textrm{\; is \; quasi-independent} \\
H_1: & \textrm{otherwise}
\end{array}
\label{eq:main_hypothesis_testing}
\ee
\begin{remark}
Quasi-dependence implies dependence. When $\wfun$ is strictly positive, quasi-dependence is simply dependence of $X$ and $Y$.
\end{remark}
\begin{remark}
Quasi-independence does not imply independence. If $\wfun(x,y) = 0$ for some $(x,y) \in \mathbb{R}^2$, it is possible to have quasi-independence without independence (and then we must have either $\tilde{F_{X}} \neq F_{X}$ or $\tilde{F_{Y}} \neq F_{Y}$). For the important case of $w(x,y)=\mathbbm{1}(x,y)$, \cite{cheng2007nonparametric} discuss the identifiability problem and its implications.
\end{remark}
We denote by $\mathcal{D}_x, \mathcal{D}_y$ the unordered samples comprising of $\{x_1,\ldots,x_n\}$ and $\{y_1,\ldots,y_n\}$, respectively. For convenience, we often keep the indices of the original data, but not the coupling between $x,y$, and to this end we use
the unordered sample $(x, \mathcal{D}_y)$ - i.e. we keep the original ordering of the $x_i$'s but only the marginal empirical distribution of the $y_i$'s.
\section{Real-Life Datasets}
\label{sec:real_data}
We applied the various tests to four data sets, shown in Figure \ref{fig:real_datasets}. P-values for the WP and bootstrap tests are based on $B=10^5$ samples, and for the minP2 test on $B=10^4$, due to computational restrictions.
\begin{enumerate}
\item \textbf{Time from Infection to AIDS in HIV Carriers} - A classical example of truncated data occur in AIDS retrospective studies, where the time from HIV infection to AIDS ($Y$) is restricted to be smaller than the time from HIV to sampling ($X$) \citep{lagakos1988nonparametric}. Here we analyze data on 295 AIDS cases, available in the DTDA package of R \citep{moreira2010dtda}. By design, the sample comprised only patients satisfying $0\leq X\leq Y$. The new WP, Tsai's and minP2 tests all obtained significant P-values of $0.001$, $0.005$, and $0.002$, respectively, suggesting that dependence exists between the two time variables.
To examine the effect of considering the truncation mechanism when testing for independence, we also performed a WP test with a constant $\wfun$, and the P-value remained significant and in fact was reduced to $10^{-5}$.
\begin{comment}
\item \textbf{Length of Hospitalization in Intensive-Care-Units} - Data on $137$ patients that were hospitalized in Intensive-Care-Units (ICUs) on a random day were collected in five Israeli hospitals (see \cite{mandel2010competing}). Let $Y$ and $X$ denote the length of stay in the ICU and the time from admission to sampling. Since only patients having $0 \leq X \leq Y$ are observable, the data are truncated with $\wfun(x,y)=\indicator{x \leq y}$. Previous works (e.g. \cite{mandel2010competing}) analyzed the data assuming that $X$ and $Y$ are quasi-independent, and it is therefore important to test the validity of this assumption. The new WP test, Tsai's test and minP2 test were all insignificant at the standard $\alpha=0.05$ significance level, with P-values of $0.298, 0.108$ and $0.346$, respectively.
\end{comment}
\item \textbf{Survival in the Channing House Community} - \cite{hyde1977testing} analyzed survival data for residents of the Channing House retirement community in Palo Alto, CA; the full dataset is available in the boot $R$ package \citep{R_boot}. The survival time, $Y$, of a resident is left truncated by the entering age to the community, $X$, and is right censored by the age at the end of followup. After removing five observations that were not consistent with the criterion $X<Y$, the data consisted of $n=457$ individuals, 282 ($61.7\%$) of which were censored. Quasi independence of survival time and entering time was tested applying the approach described in Section \ref{sec:censoring}.
Tsai's and minP2 tests were calculated for comparison.
The WP, Tsai's and minP2 tests all obtained non-significant P-values of
$0.854$, $0.099$ and $0.140$, respectively, showing no evidence for dependence between entering age and survival. As expected, when we ignored the truncation mechanism, a naive WP test of independence considering only the censoring and using $\wfun(x,y)=S(y-x)$ (see Equation \eqref{eq:censoring}) yielded a spurious signal of dependence, with a P-value of $10^{-5}$, demonstrating the need to account for biased sampling when testing.
\item \textbf{Time before and after infection in Intensive-Care-Units} -
Data on $137$ patients that were hospitalized in Intensive-Care-Units (ICUs) on a random day were collected in five Israeli hospitals as part of a national cross-sectional study \citep{mandel2010competing}. Infection data were collected from admission to the ICU until discharge or 30 days, whatever comes first. An important question was whether the time of infection is associated with the remaining time in the ICU.
To test this independence hypothesis, we use the sub-sample of patients who admitted to the ICU without delay and who acquired infections during their first 30 days of hospitalization in the ICU. Thus, we test independence of $X$, the time from admission to the ICU to infection and $Y$, the time from infection to discharge from the ICU. Due to the sampling mechanism, the data are length biased according to the total length of stay in the ICU, yielding the weight function $\wfun(x,y)=x+y$.
The new WP and bootstrap tests were both significant at the standard $\alpha=0.05$ level, with P-values $0.039$ and $0.031$, respectively, indicating that the time of acquiring infection shows a significant effect on prolonging the remaining time in the ICU.
When we used a WP test while ignoring the biased sampling function $\wfun$, the signal for dependence disappeared, and we got a P-value of $0.605$. Thus, for this dataset the biased sampling masks the true dependence between $X$ and $Y$, and our test that takes $\wfun$ into account was able to reveal it.
\item \textbf{Time to Promotion to the Rank of Full Professor} - The data consists of cross-sectional records on all faculty members of the Hebrew University of Jerusalem who were employed in $1998$. We tested whether the age at promotion to the associate professor rank depends on the service time in that rank. Let $A_{AP}$ and $A_{FP}$ denote the age at promotion to the ranks associate professor (AP) and full professor (FP) respectively; we test independence between $X = A_{AP}$ and $Y = A_{FP}−A_{AP}$ for associate professors promoted to full professor before the age of $65$, back then the retirement age in Israel. We used the sub-sample of $306$ faculty members who were promoted to the FP rank after 1980 and were younger than $65$ at sampling time (1998). As in \cite{mandel2012cross}, we assume that professors will stay in the university until age $65$. Assuming a stable entrance process to the AP rank, the cross-sectional study design leads to length biased sampling according to the length of service at the FP rank. The restriction of the data to professors who promoted after $1980$ resulted in the weight function $\wfun(x,y) = \min(65−x-y, 18) \indicator{x+y<65}
Thus, the weight is neither a truncation function nor strictly positive, and the only test applicable is the permutation test of Section \ref{sec:permutations}.
We applied the WP test with $B=10^5$ permutations,
and obtained a very small P-value of $0.00002$, meaning that the age of promotion to associate professor rank does depend on the service time in that rank.
Ignoring the biased sampling function $\wfun$ still yielded a significant P-value of $0.00060$.
\end{enumerate}
\begin{comment}
\vspace{-0.3in}
\begin{figure}[H]
\begin{center}
\begin{tabular}{cccc}
\hspace{-0.1in}$AIDS$ & \hspace{0.1in}$Channing$ $House$ & $Huji$ & $Infection$ \\
\hspace{-0.3in}
\includegraphics[width=0.26\columnwidth]{Figures/real_data/AIDS.png}
&
\hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{Figures/real_data/ChanningHouse.png}
&
\hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{Figures/real_data/huji.png}
&
\hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{Figures/real_data/Infection.png}
\end{tabular}
\end{center}
\vspace{-0.4in}
\caption{\footnotesize Scatterplots of four real datasets. Red points indicate the true data. Black '+' signs represent points $(x_i, y_{\pi(i)})$ for randomly drawn permutations $\pi$ under biased sampling: $\wfun(x,y)= \indicator{x<y}$ for the AIDS dataset, $\wfun(x,y)=\setindicator{\{x<y\}} S(y-x)$ for the Channing housing dataset, $\wfun(x,y)=min(65−x-y, 18) \indicator{x+y<65}$ for the Huji dataset and $\wfun(x,y)=x+y$ for the Infection dataset.}
\label{fig:real_datasets}
\end{figure}
\end{comment}
\vspace{-0.3in}
\begin{figure}[h]
\begin{center}
\begin{tabular}{cc}
\hspace{-0.4in}$AIDS$ & \hspace{-0.3in}$Channing$ $House$ \\
\hspace{-0.3in}
\includegraphics[width=0.45\columnwidth]{AIDS_with_w.png}
&
\hspace{-0.2in}
\includegraphics[width=0.45\columnwidth]{ChanningHouse_with_w.png} \\
\hspace{-0.4in} $Huji$ & \hspace{-0.5in} $Infection$ \\
\hspace{-0.2in}
\includegraphics[width=0.45\columnwidth]{huji_with_w.png}
&
\hspace{-0.2in}
\includegraphics[width=0.45\columnwidth]{Infection_with_w.png}
\end{tabular}
\end{center}
\vspace{-0.4in}
\caption{\footnotesize \footnotesize Scatterplots of four real datasets. Red points indicate the true data. Green '+' signs represent points $(x_i, y_{\pi(i)})$ for randomly drawn permutations $\pi$ under biased sampling. The biased sampling functions are shown as background in grayscale, with $\wfun(x,y)= \indicator{x<y}$ for the AIDS dataset, $\wfun(x,y)=\setindicator{\{x<y\}} S(y-x)$ for the Channing housing dataset, $\wfun(x,y)=min(65−x-y, 18) \indicator{x+y<65}$ for the Huji dataset and $\wfun(x,y)=x+y$ for the Infection dataset.}
\label{fig:real_datasets}
\end{figure}
\section*{Supplementary Materials}
\label{sec:supplement_material}
\renewcommand{\thesubsection}{\Alph{subsection}}
\subsection{Proofs}
\label{sec:SI_proofs}
The proof of Claim \ref{claim:P_W_conditional} is brought below:
\begin{proof}
For a general weighted model, we have
%
\be
\label{eq:generalperm}
P(\pi(\mathcal{D}) \mid x, \mathcal{D}_y) = \frac{\prod_{i=1}^n f_{XY}^{(w)}(x_i, y_{\pi(i)}) }{\sum_{\pi'\in S_n} \prod_{i=1}^n f_{XY}^{(w)}(x_i, y_{\pi'(i)}) }.
\ee
%
Under the null, $f_{XY}^{(w)}(x,y)\propto w(x,y)\tilde{f_{X}}(x)\tilde{f_{Y}}(y)$, hence
\begin{eqnarray*}
P_0(\pi(\mathcal{D}) \mid x, \mathcal{D}_y) = \frac{\prod_{i=1}^n \tilde{f_{X}}(x_i) \tilde{f_{Y}}(y_{\pi(i)}) \wmat(i,\pi(i))}{\sum_{\pi' \in S_n} \prod_{i=1}^n \tilde{f_{X}}(x_i) \tilde{f_{Y}}(y_{\pi'(i)}) \wmat(i,\pi'(i))}
= P_{\wmat}(\pi).
\end{eqnarray*}
\end{proof}
\noindent The proof of Proposition \ref{prop:consistent_estimator_under_exchangeability} is brought below:
\begin{proof}
Since $X$ and $Y$ are continuous exchangeable random variables, $P(X< Y)=1/2$ and therefore
$$
f_{XY}^{(w)}(x,y)=2\indicator{x < y} f_{XY}(x,y).
$$
Calculating the weighted marginal, we have
$$
f_{X}^{(w)}(t)=2\int_t^\infty f_{XY}(t,y)dy=2\int_t^\infty f_{XY}(y,t)dy,
$$
due to exchangeability. Similarly
$$
f_{Y}^{(w)}(t)=2\int_{-\infty}^t f_{XY}(x,t)dx .
$$
Thus (due to the continuity assumption),
$$
\frac{f_{X}^{(w)}(t)+f_{Y}^{(w)}(t)}{2}=\int_t^\infty f_{XY}(y,t)dy+\int_{-\infty}^t f_{XY}(x,t)dx = f_{Y}(t).
$$
The result follows by applying the Glivenko–Cantelli Theorem on $\hat{F}^{(w),n}_X$ and $\hat{F}^{(w),n}_Y$ .
\end{proof}
\subsection{Simulated Data}
This section of the supplementary materials contains the estimated power and scatterplots for simulated data of the various settings presented in Section \ref{sec:simulations}.
\subsubsection*{Monotone-Exchangeable Joint Distribution}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{cccc}
\hspace{0.01in}$Norm(\rho=0.9)$ & \hspace{0.07in} $Norm(\rho=-0.9)$ & $GC(\theta=1.6)$ & \hspace{0.15in}$CC(\theta=0.5)$ \\
\hspace{-0.3in}
\includegraphics[width=0.26\columnwidth]{Gaussian_truncation_0.9.jpg}
&
\hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{Gaussian_truncation_-0.9.jpg}
&
\hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{Gumbel_truncation_1.6.jpg}
&
\hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{Clayton_truncation_0.5.jpg} \\
\end{tabular}
\end{center}
\vspace{-0.2in}
\caption{\footnotesize Scatterplots of $400$ samples generated from the monotone-exchangeable distributions described in Section \ref{sec:truncation} of the main manuscript, under truncation $\indicator{X\leq Y}$. Observed points are shown in red. Points excluded due to the truncation are shown in gray. All plots show data with standard Normal margins. The left two plots are for a bi-variate Gaussian distribution. For the negative correlation $(\rho=-0.9)$ the dependence between $X$ and $Y$ is immediately apparent even when observing only data after truncation. For the positive correlation $(\rho=0.9)$ the dependence is harder to detect by observing only such data, in similar to Figure \ref{fig:example} in the main text. The right two figures show data for the Gumbel copula (GC) with dependence parameter $\theta=1.6$ ,and the Clayton copula (CC) with dependence parameter $\theta=0.5$.}
\label{fig:monotone_exchangeable}
\end{figure}
Figure ~\ref{fig:monotone_exchangeable} displays the scatterplots of samples generated from the monotone-exchangeable distributions described in Section \ref{sec:truncation} of the main manuscript. $X, Y$ were generated from a standard Gaussian distribution. We examined three possible dependence structures, as specified by three different copulas:
\begin{itemize}
\item Gaussian, with correlation parameter $\rho \in \{-0.9, -0.8,\ldots, 0.9\}$
\item Gumbel, with dependence parameter $\theta=1.6$
\item Clayton, with dependence parameter $\theta=0.5$
\end{itemize}
\subsubsection*{Monotone Non-Exchangeable Joint Distribution}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{cc}
\hspace{0.2in}$LD(\rho=0)$ & \hspace{0.2in}$LD(\rho=0.4)$\\
\includegraphics[width=0.302\columnwidth]{LD_truncation_0.jpg}
&
\hspace{-0.1in}
\includegraphics[width=0.3\columnwidth]{LD_truncation_0.4.jpg} \\
\end{tabular}
\end{center}
\vspace{-0.2in}
\caption{\footnotesize Scatterplots of samples generated from the monotone-non-exchangeable lifetime distribution (LD) described in Section \ref{sec:truncation} of the main manuscript, with otherwise the same settings as Figure \ref{fig:monotone_exchangeable}. We used a Gaussian copula with dependence parameter $\rho=0$ (left) and $\rho=0.4$ (right), where $X \sim Weibull(8.5, 3)$ and $Y \sim exp(0.2)$.}
\label{monotone_non_exchangeable}
\end{figure}
Figure ~\ref{monotone_non_exchangeable} displays the scatterplots of samples generated from the monotone non-exchangeable distribution appear in Section \ref{sec:truncation} of the main manuscript. We generated pairs ($X,Y$) with $X \sim exp(0.2)$ and $Y \sim Weibull(3,8.5)$. We specified the dependence of $(X,Y)$ through a normal copula, where the strength dependence is determined by the correlation parameter $\rho$. We considered two levels of dependence as measured by the pre-truncation $\rho= 0$ and $\rho=0.4$.
\subsubsection*{Non-Monotone Exchangeable Joint Distribution}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{c}
\hspace{0.15in}$CLmix(0.5)$\\
\includegraphics[width=0.302\columnwidth]{CLmix_truncation_0.5.jpg}
\end{tabular}
\end{center}
\vspace{-0.2in}
\caption{\footnotesize Scatterplots of samples generated from the non-monotone exchangeable distribution described in Section \ref{sec:truncation} of the main manuscript. $X, Y$ were generated from a mixture of two Clayton copulas with dependence parameters $\theta=0.5$ and $\theta=-0.5$.}
\label{non_monotone_exchangeable}
\end{figure}
Figure ~\ref{non_monotone_exchangeable} displays the scatterplot of samples generated from the non-monotone exchangeable distribution described in Section \ref{sec:truncation} of the main manuscript. $X, Y$ were generated from a mixture of two Clayton copulas with dependence parameter $\theta=0.5$ and $\theta=-0.5$.
\subsubsection*{Non-Monotone Non-Exchangeable Joint Distribution}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{cccc}
\hspace{0.01in}$CNorm(-0.9)$ & \hspace{0.07in} $CNorm(0)$ & \hspace{0.01in} $CNorm(0.5)$ & \hspace{0.15in}$CNorm(0.9)$ \\
\hspace{-0.3in}
\includegraphics[width=0.26\columnwidth]{nonmonotone_nonexchangeable_truncation_-0.9.jpg}
& \hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{nonmonotone_nonexchangeable_truncation_0.jpg}
& \hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{nonmonotone_nonexchangeable_truncation_0.5.jpg}
& \hspace{-0.2in}
\includegraphics[width=0.26\columnwidth]{nonmonotone_nonexchangeable_truncation_0.9.jpg}
\end{tabular}
\end{center}
\vspace{-0.4in}
\caption{\footnotesize Scatterplots of samples generated from the non-monotone non-exchangeable distributions described in Section \ref{sec:truncation}. We used a normal copula with varied correlation coefficient $\rho$ to specify the joint distribution of $X, Y$, with $X \sim Weibull(0.5,4)$ and $Y \sim U[0,16]$. Due to the long tail of the Weibull distribution, we cut the $X$-axis at $40$, and points with $X_i>40$ are not shown.}
\label{fig:non_monotone_non_exchangeanle}
\end{figure}
Figure ~\ref{fig:non_monotone_non_exchangeanle} displays the scatter plot of samples generated from the non-monotone non-exchangeable distributions described in Section \ref{sec:truncation}. We used a normal copula with varied correlation coefficient $\rho$ to specify the joint distribution of $X, Y$, with $X \sim Weibull(0.5,4)$ and $Y \sim U[0,1]$.
\subsubsection{Strictly Positive Biased Sampling}
\begin{figure}[!h]
\begin{center}
\begin{tabular}{ccc}
\hspace{0.2in}$LogNormal(\rho=0)$ & \hspace{0.2in}$LogNormal(\rho=0.5)$ & \hspace{0.2in}$LogNormal(\rho=0.9)$ \\
\includegraphics[width=0.3\columnwidth]{LogNormal_sum_0.jpg}
&
\hspace{-0.1in}
\includegraphics[width=0.3\columnwidth]{LogNormal_sum_0.5.jpg}
&
\hspace{-0.1in}
\includegraphics[width=0.3\columnwidth]{LogNormal_sum_0.9.jpg} \\
\hspace{0.2in}$Normal(\rho=0)$ & \hspace{0.2in}$Normal(\rho=0.5)$ & \hspace{0.2in}$Normal(\rho=0.9)$ \\
\includegraphics[width=0.3\columnwidth]{Gaussian_gaussian_0.jpg}
&
\hspace{-0.1in}
\includegraphics[width=0.3\columnwidth]{Gaussian_gaussian_0.5.jpg}
&
\hspace{-0.1in}
\includegraphics[width=0.3\columnwidth]{Gaussian_gaussian_0.9.jpg}
\end{tabular}
\end{center}
\vspace{-0.2in}
\caption{\footnotesize Scatterplots of samples generated from distributions with positive biased sampling. Points sampled from $f_{XY}^{(\wfun)}$ are shown in red. Points sampled from $f_{XY}$ are shown in gray.
The top panels show data for the LogNormal distribution described in Section \ref{sec:strictly_positive} of the main manuscript, sampled under the bias function $\wfun(x,y)=x+y$ for different values of $\rho$. The sampling mechanism increases the values of the observed $(x_i,y_i)$ and creates spurious dependence even under the null ($\rho=0$). The bottom panels show data for the normal distribution with correlation $\rho$, sampled under a bias function proportional to the normal density with correlation $-\rho$. The sampling mechanism cancels the dependence between $X$ and $Y$ and the observed data is independent for all values of $\rho$.}
\label{fig:log_normal}
\end{figure}
Figure ~\ref{fig:log_normal} displays the scatter plot of samples generated from the two distributions with positive biased sampling functions, described in Section \ref{sec:strictly_positive} in the main text: (1.) A LogNorml distribution with standard marginals and correlation $\rho$ with $\wfun(x,y)=x+y$, and (2.) a bi-variate Gaussian distribution with standard marginals and correlation $\rho$ with $\wfun$ proportional to a bi-variate Gaussian density with standard marginals and correlation $-\rho$.
\subsection{Computing the Expectations for the Statistic Under the Null}
\label{sec:fast_bootstrap}
Our two proposed tests are significantly faster than minP2, as shown in Section \ref{sec:simulations}. However, Tsai's test, which is specialized for truncation, is faster than our tests, especially compared to the bootstrap test which is about an order of magnitude slower than the weighted permutation test. It is thus natural to seek computational improvements for our tests, but this may have statistical, in addition to computational, consequences.
In particular, the computation of our test statistic $T$ in Equation \eqref{eq:modifed_hoeffding} requires estimation of the expected cell counts under the null $e_{i}^{jk}$. It is possible to use a different test statistic with different,
albeit wrong, expectations. The weighted permutation test with this statistic will still be valid, according to Corollary \ref{cor:alpha}, and a bootstrap test with this statistic can also be used. We examined two such modified statistics:
(i) A naive approach, where we ignore the biased sampling function $\wfun$ and
simply use the empirical marginal distributions $\xcdfhat, \hat{F}_{Y}$ as our estimators, resulting the $T$ statistic used by \cite{heller2012consistent}, and (ii) a fast bootstrap based approach, where
the marginals are estimated according to Equation \eqref{eq:expected_quardant}, but are not re-estimated
for each bootstrap sample. While these two approaches appear simpler and are faster,
they may suffer from significant loss of power. For example, for bivariate Gaussians with correlation coefficient $\rho=-0.4$,
our bootstrap approach achieves a power of $0.634$, while using the naive expectations yield only
power of $0.394$ and estimating the expectations only once via bootstrap achieves power of $0.406$. Similar trends of reduction in power were observed for other $\rho$ values.
\subsection{Algorithms for Importance Sampling}
\label{sec:IS_appendix}
We have investigated several algorithms for importance sampling, and in particular their accuracy and validity for testing.
In \cite{chen2007sequential}, a sequential importance sampling (SIS) algorithm for truncation was proposed.
\cite{kou2009approximating} generalized the algorithm for approximating the permanent and $\alpha$-permanent of general matrices. The validity of the importance sampling technique was established in \cite{harrison2012conservative}.
We describe in Algorithm \ref{alg:sis_permutations} the details of the monotonic SIS scheme we have implemented. The idea behind this scheme is to order the rows $i$ of $\wmat$ in a decreasing order of the variance of the log-weights $\log(\wmat(i,j))$. Then, starting with the row with highest variance, for each such row $i$ select $\pi(i)$ maximizing $\log(\wmat(i, \pi(i)))$. This procedure will increase the value of $\log(\wmat(i, \pi(i)))$ for the first rows significantly beyond their expectation, where as we reach the last rows and must settle for lower values of $\log(\wmat(i, \pi(i)))$, the variance between different choices is small and the overall contribution to the product $P_{\wmat}(\pi)$ is not significant. As a result, this scheme samples permutations $\pi$ with high values of $P_{\wmat}(\pi)$. Additional algorithms can be suggested by small modifications of Algorithm \ref{alg:sis_permutations}. In 'uniform' importance sampling, we simply sample permutations based on the uniform measure $P_{IS}(\pi)=\frac{1}{n!}$. The 'uniform grid' method interpolates between the uniform method, that produce permutations with $P_{\wmat}(\pi)$ too low, and the monotonic method that produce permutations with $P_{\wmat}(\pi)$ that may be too high. This is achieved by taking a grid of values $0 = \alpha_1 < \alpha_0 < .. < \alpha_G = 1$ and for each $\alpha_k$ modifying step $9$ of the algorithm to sample with probability proportional to $\wmat(\sigma(i),j)^{\alpha_k}$. In our simulations we used $G=10$ with equidistant $\alpha_i$ values.
Finally, the Kou-McCullagh algorithm skips the ordering steps 5,6 and computes the column-sums $C_i = \sum_j \wmat(i,j)$ over the remaining rows at each step. Step $9$ is replaced by sampling $\pi(i,j)$ with probability proportional to $\frac{\wmat(i,j)}{C_j-\wmat(i,j)}$ (see \cite{kou2009approximating} for more details).
We have studied the empirical performance of the SIS algorithms. To illustrate, we show their empirical rejection rate under the null for the LogNormal example with $\wfun(x,y)=x+y$. While for $n=100$ the different sampling schemes perform reasonably well, as shown in Table \ref{tab:strictly_positive}, as the sample size $n$ grows, approximating the $P-value$ becomes more challenging. We illustrate the limitations of the algorithms in an example with $n=1000$. Empirically, the MCMC approach is robust and maintains an approximately
uniform $U[0,1]$ P-values distribution under the null in all cases we have tested, while the importance sampling approaches are all overly conservative.
Figure \ref{fig:IS_distribution} shows the probabilities of the sampled permutations under different methods, and the corresponding test statistic. Figure \ref{fig:IS_log_prob_ratio} shows the variability in the importance sampling weights
$\frac{P_{\wmat}(\pi_i)}{P_{IS}(\pi_i)}$ of the different methods, causing a poor estimation of the P-value.
Figure \ref{fig:IS_pvals} shows the resulting performance. The Kou-McCullagh scheme is the least conservative scheme among all SIS methods, but is still far from a uniform $U[0,1]$ P-values distribution. The power of the SIS under the alternative will also be reduced due to their conservative behaviour.
\begin{algorithm}
\caption{{\small SIS Permutation Test of Quasi-Independence }} {\small
\begin{algorithmic}[1]
\Statex {\bf Input:} $\mathcal{D}$ - sample, $\wfun(x,y)$ - bias function.
\Statex {\bf Parameters:} $B$ - number of permutations.
\State Compute $\wmat(i,j) = \wfun(x_i,y_j), \quad \forall i,j=1,..,n$.
\State Set $\pi_0$ the identity permutation $\pi_0(i) = i$.
\For{$b=1$ to $B$}
\State Initialize the importance sampling and weighted (unnormalized) probabilities \hspace{1cm} $\quad \quad P_{IS}(\pi_b)=P_{\wmat}(\pi_b)=1$.
\State Compute $V_i = Var\Big(\log(\wmat(i,1)),..,\log(\wmat(i,n))\Big), \quad \forall i=1,..,n$.
\State Order the variables by decreasing order of $V_i$: $V_{\sigma(1)} \geq ... \geq V_{\sigma(n)}$.
\For {$i=1$ to $n$}
\State Sample $\pi_{b}(\sigma(i))$ with probabilities: \\ \hspace{3cm} $Pr(\pi_{b}(\sigma(i))=j) \propto \wmat(\sigma(i),j) \prod_{k=1}^{i-1} \indicator{\pi_b(\sigma(k)) \neq j}$.
\State Update the probability $P_{IS}(\pi_{b}) = P_{IS}(\pi_{b}) \times \frac{\wmat(\sigma(i),\pi_{b}(i))}{\sum_{j=1}^n \wmat(\sigma(i),j) \prod_{k=1}^{i-1} \indicator{\pi_b(\sigma(k)) \neq j}}$ .
\State Update the (unnormalized) probability $P_{\wmat}(\pi_{b}) = P_{\wmat}(\pi_{b}) \times \wmat(\sigma(i),\pi_{b}(i))$ .
\EndFor
\EndFor
\State {\bf Output:} $P_{value}$ computed from the $P_{IS}(\pi_b),P_{\wmat}(\pi_b)$'s according to Equation \eqref{eq:IS_pval}.
\end{algorithmic}
\label{alg:sis_permutations}}
\end{algorithm}
\newpage
\clearpage
\thispagestyle{empty}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.8\columnwidth, height=13cm]{Statistics_vs_log_P_W_inverse_weighting_n_1000.pdf}
\end{center}
\vspace{-0.2in}
\caption{\footnotesize Log probabilities under $P_{\wmat}$ for permutations sampled under different methods, vs. their computed statistic $T_i=T(\pi_i(\mathcal{D}))$ for the inverse weighting statistic from Section \ref{sec:inverse_weight_stat}.
For $n=1000$, a weighted sample was simulated from the independent LogNormal distribution, with $\wmat(x,y)=x+y$. Since $P_{\wmat}$ is known only up to a multiplicative constant, we show the log probabilities minus the log probability of the identity permutation.
Shown are the scaled log probabilities for $B=250$ random permutations for four importance sampling methods, and the MCMC method (green '+' signs), in addition to the ID permutation (red triangle). For each method, we also display the coefficient of variation diagnostic
of $\frac{P_{\wmat}(\pi)}{P_{IS}(\pi)}$, and the resulting p-value.
According to the MCMC Method, the permutations with log-probability close to the identity cover most of the probability space under $P_{\wmat}$.
The uniform (monotone) importance sampling methods sample permutations with too low (high) $P_{\wmat}$ values.
The monotone grid method interpolates between the two and covers a large range of the log-probabilities, including log-probabilities sampled under the MCMC method. The Kou-McCullagh method \cite{kou2009approximating} also samples permutations with log-probabilities close to the MCMC method. However, the importance sampling probabilities $P_{IS}(\pi_i)$ for the permutations sampled under these methods are vastly different from the true underlying probabilities $P_{\wmat}(\pi_i)$, as is evident by the large value of the coefficient of variation $CV(\frac{P_{\wmat}(\pi_i)}{P_{IS}(\pi_i)})$.
Although the true P-value is $\approx 0.24$ (as confirmed by extensive MCMC simulations with higher $B$), all importance sampling methods are too conservative here and give a P-value of $1$, in part due to the inclusion of the identity permutation in the P-value calculation.}
\label{fig:IS_distribution}
\end{figure}
\newpage
\clearpage
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.8\columnwidth, height=13cm]{IS_log_P_W_minus_log_P_IS_vs_Statistic2.pdf}
\end{center}
\vspace{-0.2in}
\caption{\footnotesize Log probabilities ratios $\log(\frac{P_{\wmat}}{P_{IS}})$ for permutations sampled under different methods, vs. their computed statistic $T_i=T(\pi_i(\mathcal{D}))$ for the inverse weighting statistic from Section \ref{sec:inverse_weight_stat}. Parameters are the same as in Figure \ref{fig:IS_distribution}.
Since $P_{\wmat}$ is known only up to a multiplicative constant, we show the log probabilities ratios minus the log probability ratio of the identity permutation.
For the MCMC Method, the ratio is $1$ and all permutations have equal weights, with $\approx 24\%$ of permutations $\pi_i$ having $T_i \geq T_0$. For the uniform and monotone methods we have $T_i\geq T_0$ for all permutations giving a too conservative P-value of $1$.
For the monotone-grid and Kou-McCullagh method, we do sample permutations with $T_i < T_0$, but their overall probabilities ratios are negligible, giving again a conservative $Pvalue \approx 1$.}
\label{fig:IS_log_prob_ratio}
\end{figure}
\newpage
\clearpage
\begin{figure}[!h]
\begin{center}
\includegraphics[width=0.8\columnwidth, height=13cm]{all_tests_validity_pvals_cumulative.pdf}
\end{center}
\vspace{-0.2in}
\caption{\footnotesize Cumulative p-value distribution for a the same independence LogNormal distribution from Figure \ref{fig:IS_distribution} and the same tests. The permutation test gives an approximate $U[0,1]$ distribution.
The bootstrap test is slightly non-calibrated, and gives low p-values under the null.
The different importance sampling tests are all too conservative, with the Kou-McCullagh technique performing the best.}
\label{fig:IS_pvals}
\end{figure}
\subsection{An Iterative Algorithm for Marginal Estimation}
\label{sec:IterativeAlgorithm_appendix}
This section deals with estimation of the marginal distributions under the null independence model presented in Section \ref{sec:marginal_estimation_under_null} in the main text. We consider the more general scenario of $k$ independent random variables ${X}_1\sim {F}_1,\ldots,{X}_k\sim {F}_k$ that become dependent due to selection bias. Specifically, we assume that observations are vectors $(X_1,\ldots,X_k)$ having the density
\begin{equation} \label{basic}
\frac{w(x_1,x_2,\ldots,x_k) F_1(dx_1) F_2(dx_2)\cdots
F_k(dx_k)}{\mathbb{E}\{w( X_1, X_2,\ldots , X_k)\}},
\end{equation}
where the non-negative weight function $w$ is known and has a finite
expectation with respect to $ F_1\times\cdots\times
F_k$. The aim is to estimate $ F_1,\ldots, F_k$ without any parametric assumptions. For identifiability, we assume that for
any measurable subset $\truncregion$ in the support of $ F_j$,
${E}\big[w( X_1,\ldots, X_k) \indicator{X_j\in \truncregion}\big]>0$ ($j=1,\ldots,k$); that is, observations from any subset of the support can be observed.
The problem of non-parametric estimation of a general multivariate
distribution $F$ using weighted data is well known (e.g., \cite{vardi1985empirical}) and the corresponding estimator
of $ F_1,\ldots, F_k$ can be obtained by marginalization. Specifically, generalizing Equation \eqref{eq:non_parametric_MLE} in the main paper, the estimator for $F_j$ is
\begin{equation} \label{naive}
\hat{F}^n_j(t)=\frac{\sum_{i=1}^n I\{{X_{ij}\le t}\}
\wfun(X_{i1},\ldots,X_{ik})^{-1}} {\sum_{i=1}^n \wfun(X_{i1},\ldots,X_{ik})^{-1}} \quad (j=1,\ldots,k),
\end{equation}
where $X_i$ is the vector of subject $i$, and $X_{ij}$ is its $j$'th coordinate.
Estimator \eqref{naive} does not exploit independence, and it is especially inappropriate when $\wfun$ vanishes in part of the support (i.e., truncation).
Let ${\tilde{E}}$ denote the conditional expectation operator,
${\tilde{E}}(w ; j,x)=\mathbb{E}\{w( X_1,\ldots, X_k)\mid X_j=x\}$.
For any $1\le j\le k$, the likelihood of the data
$(x_{i1},\ldots,x_{ik})$ ($i=1,\ldots,n$) can be factorized as follows (see Equation (\ref{basic})):
\begin{equation} \label{splitlike}
\prod_{i=1}^n \frac{w(x_{i1},\ldots,x_{ik})\prod_{\ell\ne j} F_i(dx_{i\ell})}
{{\tilde{E}}(w; j,x_{ij})} \times \prod_{i=1}^n \frac{{\tilde{E}}(w; j,x_{ij}) F_j(dx_{ij})}
{\mathbb{E}\{\tilde E(w; j, X_j)\}}.
\end{equation}
Due to independence, the first term in Equation \eqref{splitlike} does not depend on $F_j$. Thus, assuming all distributions except $F_j$ are known, the likelihood of
$n$ observations is proportional to the second term of Equation \eqref{splitlike},
and a natural non-parametric estimate of $F_j$ is $\hat{ F}_j(x)\propto \sum_i \indicator{x_{ij}\le
x} {\tilde{E}}(w;j,x_{ij})^{-1}$. This estimator assigns mass only to the observed points and gives rise to Algorithm \ref{alg:est_marg_app},
generalizing Algorithm \ref{alg:est_marg} in the main text.
\begin{algorithm}
\caption{{\small Estimation of Marginals Under Quasi-independence}} {\small
\begin{algorithmic}[1]
\Statex {\bf Input:} $(x_{i1},\ldots,x_{ik})$ ($i=1,\ldots,n$) - sample, $w$ - bias function, $d(F_1,F_2)$ - distance function.
\Statex {\bf Parameters:} $\epsilon$ - convergence criterion.
\State Generate initial estimates using Equation (\ref{naive}), and set $ F^{new}_j=\hat{F}^n_j$, $ F^{old}_j\equiv 0$.
\While{$\max_j d( F_j^{old}, F_j^{new})<\epsilon$}
\State Set $ F^{old}_j=F^{new}_j$ $(j=1,\ldots,k)$.
\For{$j=1,\ldots,k$}
\State Calculate $e^{new}_{ij}={\tilde{E}}(w; j,x_{ij})$ with
respect to $\prod_{\ell=1}^k F_\ell^{new}$, for all $x_{ij}$
$(i=1,\ldots,n)$.
\State Set
$$
F^{new}_j(x)= \frac{\sum_{i=1}^n \indicator{x_{ij}\le x}
[e^{new}_{ij}]^{-1}} {\sum_{i=1}^n [e^{new}_{ij}]^{-1}}.
$$
\EndFor
\EndWhile
\State {\bf Output:} $F^{new}_1,\ldots,F^{new}_k$.
\end{algorithmic}
\label{alg:est_marg_app}}
\end{algorithm}
Algorithm \ref{alg:est_marg_app} provides a non-parametric estimator for the model under the independence constraint. The most time consuming part of the algorithm is the calculation of $e^{new}_{ij}={\tilde{E}}(w; j,x_{ij})$ in step 5. In most cases, it seems
unavoidable to use full enumeration so the
complexity of each step 5 is $O(n^{k})$. For the important case of a linear weight function, the calculation of ${\tilde{E}}(w; j,x_{ij})$ is rather simple.
Let $w({\bf x})={\bf a}^{t}{\bf x}+b$, where the vector $\bf a$ and the
constant $b$ are known ($\bf a\equiv 1$ and $b=0$ is the sum-bias case $w(\textbf{x})=x_1+\cdots+x_k$). Then
\be
{\tilde{E}}(w ; j,x_{ij})={\bf a}^t\Big(
\mathbb{E} (X_1),\ldots,\mathbb{E}(X_{j-1}),x_{ij},\mathbb{E} (X_{j+1}),\ldots,\mathbb{E
} (X_k)\Big) + b
\ee
where the expectations are calculated for each coordinate separately with respect to
$ F_1^{new},\ldots, F_k^{new}$. Thus, the computational complexity of each step $5$
reduces to $O(n)$.
Several properties of Algorithm \ref{alg:est_marg_app} are readily established; their proofs are standard and hence are omitted.
\begin{proposition} \label{propincreas}
The likelihood increases in each step of the algorithm.
\end{proposition}
\begin{proposition}
The algorithm converges in the interior of the parameter space.
\end{proposition}
\begin{proposition}
The NPMLE is a fixed point of the algorithm.
\end{proposition}
\subsection{Implementation Details}
\label{sec:Implementation_appendix}
We list below several practical issues that were dealt with when implementing our TIBS algorithm.
\begin{itemize}
\item
{\bf Permutations MCMC parameters:} When sampling permutations, we seek permutations that are independent from each other. This is approximately achieved when enough MCMC steps are used between every two permutations chosen such that the Markov Chain is mixing and is close to the stationary distribution. The parameter $M$ of MCMC steps between two consecutive permutations was set to $M=2n$, where $n$ is the sample size. This choice was based on calculating the correlation between $\pi_t(i)$ and $\pi_{t+M}(i)$ as a function of $M$ and showing that this correlation decays.
In addition, a 'burn-in' parameter $M_0$ is often used in MCMC studies, which is the initial number of steps performed after starting from the identity permutation. Here, we chose to not include a 'burn-in' step, i.e. setting $M_0=0$ and taking the first permutation (the identity) as one of our permutations. This ensures that under the null we have one sample identical to the original sample, i.e. the empirical p-value cannot be lower than $\frac{1}{B}$. Some empirical arguments against the usage of 'burn-in' are given in \url{http://users.stat.umn.edu/~geyer/mcmc/burn.html}.
\item
{\bf Estimating expectations using MCMC:} The MCMC permutations algorithm was used to estimate the expected cell counts in the test statistic, as shown in Equation \eqref{eq:estimate_expected}, using the estimated transition probabilities $\hat{P}_{ij}$ in Equation \eqref{eq:P_ij_estimator}. For these estimators, we used {\it all} permutations generated during the MCMC algorithm, and not only the $B$ permutations selected later to produce permuted (null) samples. While consecutive permutations are strongly dependent and differ only in one pair ($\pi(i), \pi(j)$), adding
the estimator for $P_{ij}$ over all permutations reduces the variance of the expected cell estimators, compared to using only every $M$'th permutation, at a negligible additional computational cost.
\item
{\bf Removing cells with low counts:} The test statistic $T$ in Equation \eqref{eq:modifed_hoeffding} sums over all cells, including cells with small expected counts.
Such cells yield terms with high variance and may therefore reduce the power of the test statistic, especially recalling that we have variance not only in the observed counts $o_i^{jk}$ but we also use estimators $\hat{e}_i^{jk}$ for the expected counts.
To reduce the variance of these cells, we include in the test statistic the contribution of cells for a data point $i$ only if $\hat{e}_i^{jk}>1$ for all $j,k$. Otherwise, we simply skip these cells, for both the original sample and the permuted and bootstrap samples. Alternatively, one can replace the Pearson chi-square statistic by a likelihood-ratio test statistic that is less sensitive to cells with low counts.
\item
{\bf Dealing with ties:}
Even for a continuous distribution $F_{XY}$ and for our test statistic $T$ in Equation \eqref{eq:modifed_hoeffding}, we encounter ties data points $(x_i,y_i)$ determining the quadrants $Q_i^{jk}$ and data points $(x_j, y_{\ell})$ used as counts for the statistic. When $x_j=x_i$ and/or $y_{\ell}=y_i$ the data points lie on the boundaries of the quadrants $Q_i^{jk}$.
This can occur in the bootstrap test (as the same values may be sampled multiple times),
and also for permutation testing, where the permuted sample contains data points $(x_i, y_{\ell})$ and $(x_j, y_i)$ for some $j,{\ell}$. We can count these boundary points on one quadrant, another, or ignore them altogether.
While the effect of such ties is asymptotically negligible - they may affect the statistic and P-value calculation for small samples. For example, if we discard them, then for the original sample the sum of the four quadrants observed counts will be $n-1$, but for a permuted dataset this will be typically $n-2$.
This causes a small but systematic bias between the original and randomized (permuted or bootstrap) samples,
and may result in an invalid test with a non-uniform P-values distribution under the null.
To overcome this bias, our implementation uses a small perturbation of the quadrants $Q_i^{jk}$ (see Equation \eqref{eq:data_quartiles_def}), such that the center point used is $(x_i+\eps^{(x)}_i, y_i+\eps^{(y)}_i)$ with $\eps^{(x)}_i, \eps^{(y)}_i \iid N(0, 10^{-9})$. This randomly assigns any point (in the original or bootstrap/permuted sample) containing $x_i$ or $y_i$ (for example the original data point $(x_i,y_i)$) into one of the relevant quadrants, and does not affect the assignment of other points, thus ensuring that all $n$ points are counted for both the original and the randomized samples.
\end{itemize}
\begin{comment}
\item The following table presents all of the different tests and their applicability: \zuk{To remove?} \micha{I'm not sure we need that; if we leave it, Inverse Weighting should be in the rows}
\begin{table}
\scriptsize
\begin{center}
\begin{tabular}{|l||*{4}{c|}}\hline
\backslashbox{Pval Calc. Method}{ Test Statistic}
&\makebox[5em]{ \makecell{Adjusted \\ Hoeffding}}&\makebox[4em]{\makecell{Inverse \\ Weighted Hoeffding}}&\makebox[3em]{Tsai}
&\makebox[3em]{minP2}\\\hline\hline
Bootstrap & $\wfun^+$, $(\wfun^{01}, F_{XY}^{\leftrightarrow})$ & $\wfun^+$ & & \\ \hline
Permutations MCMC & V & $\wfun^+$ & $\wfun^{01}$ + $C$ & $\wfun^{01}$ + $C$ \\ \hline
Permutations IS & V & $\wfun^+$ & & \\ \hline
\end{tabular}
\caption{The list of applicable tests proposed by us and in the literature. $\wfun^+$ indicates positive $\wfun$.
$\wfun^{01}$ indicates truncation. $F_{XY}^{\leftrightarrow}$ indicates and exchangeable distribution. $C$ indicates dealing with censored data.}
\label{tab:different_tests}
\end{center}
\end{table}
\end{comment}
\section{Test Statistics}
\label{sec:teststat}
\subsection{The Adjusted Hoeffding Statistic}
While the permutation and bootstrap approaches can be applied with any test statistic, our goal is to modify an existing omnibus test to weighted models in general and to truncation in particular. For the latter, most tests used to date are tailored to specific alternatives, such as monotone dependence. A recent new approach studied by \cite{chiou2018permutation} can test against a general alternative, but uses either significant computational resources or permutations with different sample sizes, so its significance level is not guaranteed. We are inspired by some popular non-parametric tests of independence such as \cite{thas2004nonparametric}, \cite{heller2012consistent}, and \cite{heller2016consistent}, and for concreteness, we describe the approach of the latter, which generalizes a modified version of \cite{hoeffding1948non}. Our problem requires two major modifications. First, biased sampling should be taken into account when computing the null distribution of the test statistic using a bootstrap or permutations resampling approach; this was addressed in the previous sections. Second, the test statistic compares observed counts with their expectations under the null, and the computation of these expectations needs to be modified to accommodate biased sampling.
The test belongs to a family of tests which compare the observed counts $o_{\truncregion}$ to the expected counts $e_{\truncregion}$ for different sets $\truncregion \in \mathbb{R}^2$.
As in \cite{heller2016consistent}, our test statistic is based on Pearson's Chi-squared statistics, and we consider all partitions defined by the data $\mathcal{D}$. Specifically, each data point $(x_i,y_i) \in \mathcal{D}$ defines a partition of $\mathbb{R}^2$ into four quadrants:
\be
Q_{i}^{jk} \equiv \Big\{(x',y') \in \mathbb{R}^2 \: : \: \indicator{x'>x_i} = j, \indicator{y'>y_i} = k \Big\} \quad j,k \in \{0,1\}.
\label{eq:data_quartiles_def}
\ee
For example, $Q_{i}^{00} = (-\infty, x_i] \times (-\infty, y_i]$ and $P\big((X,Y)\in Q_{i}^{00}\big) = F_{XY}^{(w)}(x_i,y_i)$.
Let $(x_i, y_i)$ be a point in the sample $\mathcal{D}$. For a quadrant $Q_i^{jk}$, we denote the observed number of points by $o_{i}^{jk} \equiv o_{Q_i^{jk}}$ and the expected number of points under the null by $e_{i}^{jk} \equiv e_{Q_i^{jk}}$. We then compute, for each quadrant, the scaled squared difference between the observed and expected number of points under $H_0$. Finally, we sum over all the sample points to get our test statistic:
\be
T = \sum_{i=1}^n \sum_{j,k \in \{0,1\}} \frac{(o_{i}^{jk}-e_{i}^{jk})^2}{e_{i}^{jk}}.
\label{eq:modifed_hoeffding}
\ee
Estimating the expected values $e_{i}^{jk}$ requires the estimation of the null distribution, which may become highly non-trivial in the biased sampling setting. First, $F_{X}$ and $F_{Y}$ may be un-identifiable, and therefore using plug-in estimators of $F_{X}, F_{Y}$ may give a poor approximation of the distribution of the test statistic under the null. Second, evaluation of expectations or probabilities under the null may require computationally costly integration of the null distribution, $[f_{X} f_{Y}]^{(w)}$, as opposed to a simple multiplication of the empirical marginals in the standard setting. If the expectations $e_{i}^{jk}$ are not estimated correctly, the WP test is still valid according to Corollary \ref{cor:alpha} and the bootstrap approach can be also applied, but power can be severely reduced, as is shown in the Supp. Materials, Section \ref{sec:fast_bootstrap}.
\subsubsection{Computing Expectations under Biased Sampling}
A natural approach is to estimate the marginals $F_{X}$ and $F_{Y}$ under the null independence model and use them to calculate the expected count in a certain cell. However, as discussed in Section \ref{sec:bootstrap}, this approach works well only for special models. We therefore suggest here an alternative method that directly estimates the expected counts.
Let $P_{ij} \equiv P_{\wmat}(\pi(i)=j) =
\sum_{\pi \in S_n} P_{\wmat}(\pi) \indicator{ \pi(i)=j }$, and define the Bernoulli random variables $\xi^\pi_{ij} \equiv \indicator{\pi(i) = j }$,
so $P_{ij} = \mathbb{E}(\xi^\pi_{ij})$.
Let $\truncregion \subset \mathbb{R}^2$ be an arbitrary set. Given a sample $\mathcal{D}$, for any permutation $\pi$ of the data, denote the number of points in $\truncregion$ under $\pi$ (i.e., after permuting the data set $\mathcal{D}$) by
$\obsset_{\truncregion}(\pi)= \sum_{i=1}^n \indicator{ (x_i,y_{\pi(i)}) \in {\truncregion} }$,
and let $e_{\truncregion} \equiv \mathbb{E}_{P_{\wmat}} \{\obsset_{\truncregion}(\pi)\}$ be the expected number of data points in $\truncregion$, under the permutations distribution $P_{\wmat}$.
The $P_{ij}$ values determine the expected number $e_{\truncregion}$ for any set $\truncregion$ via the following claim:
\begin{claim}
For any $\truncregion \subset \mathbb{R}^2$, the expected number of points $e_{\truncregion}$ under the permutations distribution $P_{\wmat}$ is given by:
\be
e_{\truncregion} = \sum_{i,j=1}^n \indicator{(x_i,y_j) \in \truncregion} \mathbb{E}_{P_{\wmat}}(\xi^\pi_{ij}) = \sum_{i,j=1}^n \indicator{(x_i,y_j) \in \truncregion} P_{ij} .
\label{eq:expected_region}
\ee
\label{claim:expected_permutations}
\begin{proof}
By definition, we have:
\be
e_{\truncregion} = \mathbb{E}_{P_{\wmat}} \big[\sum_{i=1}^n \indicator{ (x_i,y_{\pi(i)}) \in \truncregion }\big] =
\sum_{i,j=1}^{n} \indicator{(x_i,y_j) \in \truncregion} P_{\wmat}\big( \pi(i)=j \big) =
\sum_{i,j=1}^{n} \indicator{(x_i,y_j) \in \truncregion} P_{ij}.
\ee
\end{proof}
\end{claim}
The probabilities $P_{ij}$ can be easily estimated using the MCMC scheme described in Algorithm \ref{alg:mcmc_permutations}: let $\pi_0$ be the identity permutation and $\pi_1, .., \pi_{B}$ be the sampled permutations, and define the following estimator:
\be
\hat{P}_{ij} \equiv \frac{1}{B+1} \sum_{b=0}^B \indicator{\pi_b(i)=j}.
\label{eq:P_ij_estimator}
\ee
When we sample permutations using an importance distribution $P_{IS}$ (see Section \ref{sec:importance_sampling}),
the above estimator for ${P}_{ij}$ is replaced by:
\be
\hat{P}_{ij}^{(IS)} \equiv \frac{\sum_{b=0}^B \indicator{\pi_b(i)=j} \frac{P_{\wmat}(\pi_b)}{P_{IS}(\pi_b)} }{\sum_{b=0}^B \frac{P_{\wmat}(\pi_b)}{P_{IS}(\pi_b)}}.
\label{eq:P_ij_estimator_IS}
\ee
Plugging Equation \eqref{eq:P_ij_estimator} (or Equation \eqref{eq:P_ij_estimator_IS}) into Equation \eqref{eq:expected_region} gives an estimator of $e_{\truncregion}$,
\be
{\hat{e}}_{\truncregion} = \sum_{i,j=1}^{n} \indicator{(x_i,y_j) \in \truncregion} \hat{P}_{ij},
\label{eq:estimate_expected}
\ee
which can be used in the Chi-squared statistic.
For the bootstrap approach, the estimate of the null distribution is used in a straightforward manner. Consider, for example, the bottom-left quadrant with respect to a point $(x_i,y_i)$, $Q_{i}^{00}$; given estimators $\xcdfhat, \hat{F}_{Y}$ of the univariate CDFs, a natural estimator for the mass (up to a normalizing constant) that the null puts on $Q_{i}^{00}$ is given by
\be
\widehat{e_{i}^{00}} = n[\xcdfhat \hat{F}_{Y}]^{(w)} (x_i,y_i) .
\label{eq:expected_quardant}
\ee
\subsection{An Inverse Weighting Statistic for Strictly Positive $\wfun$}
\label{sec:inverse_weight_stat}
When $\wfun$ is strictly positive, a test can utilize an inverse weighting approach.
For a set $\truncregion \in \mathbb{R}^2$, define the {\it inverse weighted} observed and expected counts
$o_{\truncregion}^{(\wfun)}$ and $e_{\truncregion}^{(\wfun)}$, respectively:
\begin{align}
o_{\truncregion}^{(\wfun)} &= \sum_{\ell=1}^n \setindicator{\{(x_{\ell}, y_{\ell}) \in \truncregion\}} \wfun(x_{\ell}, y_{\ell})^{-1}, \nonumber \\
e_{\truncregion}^{(\wfun)} &= n E_{[F_{X} F_{Y}]^{(w)}} \{ \setindicator{\truncregion} \wfun(X,Y)^{-1} \}.
\label{eq:weighted_obs_exp}
\end{align}
For the quadrants $Q_i^{jk}$, we can compute estimates for the expected weighted counts ${e_i^{jk}}^{(\wfun)} \equiv e_{Q_i^{jk}}^{(\wfun)}$ by multiplying the corresponding marginal inverse weighted counts. For example, for $Q_i^{00}$:
\be
{e_i^{00}}^{(\wfun)} = \frac{(\sum_{\ell=1}^n \wfun(x_{\ell}, y_{\ell})^{-1} \indicator{x_{\ell} \le x_i}) (\sum_{\ell=1}^n \wfun(x_{\ell}, y_{\ell})^{-1} \indicator{y_{\ell} \le y_i}) }{\sum_{\ell=1}^n \wfun(x_{\ell}, y_{\ell})^{-1}} .
\label{eq:weighted_counts}
\ee
Similarly to Equation \eqref{eq:modifed_hoeffding}, the weighted statistic is given by
\be
T^{(w)} = \sum_{i=1}^n \sum_{j,k \in \{0,1\}} \frac{({o_{i}^{jk}}^{(w)}-{e_{i}^{jk}}^{(w)})^2}{{e_{i}^{jk}}^{(w)}}.
\label{eq:weighted_statistic}
\ee
\subsection{Unknown $\wfun$: Left Truncated Right Censored Data}
\label{sec:censoring}
In some applications the biased sampling function $\wfun$ is unknown. However, our methodology is still applicable when $\wfun$ can be estimated consistently. In particular, we can tackle the important case of left truncation with censoring.
Consider the standard left-truncation right-censoring model where $(X_i,Y_i, C_i)$ are $n$ independent triplets, the joint density of $(X_i,Y_i)$ is proportional to $f_{XY}(x,y)\setindicator{\{x<y\}}$ for some density $f_{XY}$, and $(X_i,Y_i)$ are independent of the censoring variables $C_i$. We observe triplets $(X_i,\min(Y_i,X_i+C_i),\Delta_i$), where $\Delta_i=\setindicator{\{Y_i< X_i+C_i\}}$ are the censoring indicators with $\Delta_i=0$ and $1$ for censored and uncensored observations, respectively. We suggest testing independence based on the uncensored observations, where censored observations are used only for estimation of the weight function.
Specifically, the conditional density of an uncensored observation is simply
\be
\frac{S(y-x)f_{XY}(x,y)\setindicator{\{x<y\}}}{\mathbb{E}_{f_{XY}}[S( Y- X)\setindicator{\{ X< Y\}}]},
\label{eq:censoring}
\ee
where $S(t)=P(C>t)$ is the survival function of $C$. Thus, the density of uncensored observations has exactly the form of Equation \eqref{eq:weighted_density}, with $w(x,y)=\setindicator{\{x<y\}} S(y-x)$, a function involving a continuous and a truncated part. The methods developed in previous sections are flexible enough to accommodate such functions. The survival function $S$ can be estimated by the standard Kaplan-Meier estimator applied to the data $\{(\min(C_i,Y_i-X_i),1-\Delta_i), \: i=1,\ldots,n \}$.
|
2,869,038,154,749 | arxiv |
\section{Introduction}
Geometric optimization problems are central to geometry processing as most methods involve some form of optimization as a step in their pipeline.
Shape deformation problems are inherently nonlinear and therefore can be difficult to solve. In particular, problems involving near-isometric deformations are typically ill-conditioned due to the combination of high stretching and low bending resistance.
Moreover, physical objectives are often invariant with respect to rigid body motions and the alignment of a mesh in Euclidean space is considered to be a post-processing task.
However, this rigid body motion invariance can cause conceptual and numerical issues when working with nodal positions.
Thus it is beneficial to find degrees of freedom for mesh description and corresponding deformations which are rigid body motion invariant.
In this work, we study the Nonlinear Rotation-Invariant Coordinates (NRIC) that describe the immersion of a mesh using the edge lengths and dihedral angles of the mesh instead of the nodal positions.
Beyond their inherent invariance to rigid transformations, these coordinates offer additional benefits, such as their natural occurrence in discrete deformation energies and their representation of natural modes of deformation in a localized sparse fashion.
For example, when a human character (represented by a triangle mesh) lifts her straight arm, the induced variations in nodal positions comprise the entire arm.
However, the same variation encoded in the change of lengths and angles is limited to the shoulder region, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot the place where the actual physical work is done.
Prior work on shape interpolation by \citet{WiDrAl10} and \citet{FrBo11} showed that linear blending of the NRIC for a set of shapes already yields interesting nonlinear deformations.
However, since in general nodal positions that realize given edge lengths and dihedral angles may not exist, these methods rely on optimization in the space of nodal positions.
\paragraph*{Contribution}
The basis of our approach is the triangle inequalities and the integrability conditions derived by \citet{WaLiTo12}.
Our goal is to provide the machinery required to formulate and solve geometric optimization problems entirely in NRIC.
\begin{itemize}
\item We reformulate the integrability conditions using quaternions and use this to provide an implicit description of the NRIC manifold along with its tangent spaces.
\item We reformulate the nonlinear energy from \cite{HeRuSc14} in NRIC and provide its derivatives to equip the NRIC manifold with a Riemannian metric.
\item For solving (constrained) geometric optimization problems in the NRIC manifold, we describe an approach based on the augmented Lagrange method.
In this context, we illustrate how to efficiently handle the triangle inequality constraints using the natural barrier term in the nonlinear energy and a modified line search.
This also includes explicit formulas for the second derivatives of the integrability conditions, which are needed for evaluating the Hessian of objectives acting on NRIC.
\item Finally, we introduce a hybrid algorithm to construct nodal positions of a discrete surface from NRIC which do not necessarily fulfill the integrability conditions.
The algorithm uses an adaptive mesh traversal algorithm as initialization to a Gauß--Newton solver.
In our experiments, this proves to effectively reduce the number of required Gauß--Newton iterations.
Typically, a single iteration is sufficient or even no iteration is needed.
\end{itemize}
Experiments demonstrate the utility of our framework for various applications such as geodesics in shape space and paper folding.
Our approach is particularly well-suited to deal with near isometric deformations of discrete shell surfaces, which is underpinned by a variety of numerical examples.
\paragraph*{Organization}
The remainder of this paper is organized as follows.
After reviewing related work in Section \ref{sec:relatedWork}, we summarize the necessary background on the established discrete integrability conditions as introduced by \citet{WaLiTo12} in Section \ref{sec:background}.
We define our NRIC manifold and reformulate the integrability conditions using quaternions in Section \ref{sec:nric}.
In \autoref{sec:energy}, we discuss the nonlinear deformation energy.
Afterwards, we introduce a corresponding variational calculus in Section \ref{sec:varProblems}.
The robust reconstruction of nodal positions from lengths and angles is discussed in Section \ref{sec:recon}.
Finally, we show a series of applications in Section \ref{sec:results} and discuss limitations and challenges in Section \ref{sec:discussion}.
\section{Related Work}\label{sec:relatedWork}
In this section, we discuss relevant work on linear and nonlinear coordinates, rigidity of triangle meshes, shape interpolation, shape spaces and near-isometric deformation.
\paragraph*{Linear coordinates}
For solving problems in geometry processing, it can be useful to switch from the usual nodal coordinates to a different representation that is adapted to the given task. We distinguish between coordinates that depend linearly and nonlinearly on the nodal coordinates.
\emph{Differential coordinates} use discrete differential operators on a triangle mesh to define coordinates. Two examples are \emph{gradient-domain} approaches for meshes \cite{YuZhXuShBaGuSh04,SuPo04}, which operate on the gradients of functions, and the \emph{Laplace coordinates}~\cite{SoCoLiAlRoSe04,LiSoCoLeRoSe04}, which make use of the discrete Laplace--Beltrami operator.
Since the differential coordinates depend linearly on the nodal positions, the immersion that best matches given differential coordinates can be found by solving a linear least-squares problem.
While linearity of the coordinates facilitates computations, it also fundamentally limits their applicability.
For example, shape editing approaches that use linear coordinates often yield unnatural and distorted shapes when larger deformations are involved~\cite{BoSo08}.
\paragraph*{Nonlinear coordinates}
In classical differential geometry, the fundamental theorem of surfaces~\cite{Do76} states that two immersion of a surface to $\mathbb{R}^3$ differ by a rigid motion if and only if the first and second fundamental forms agree and provides integrability conditions that guarantee the existence of an immersion for a given first and second fundamental form.
This motivates using discrete analogs to the fundamental forms as coordinates for triangle mesh processing.
Explicitly, the list of all edges lengths and dihedral angles is used. Analogous to the classical theorem, the nodal positions of two meshes with the same combinatorics agree up to a global rigid motion if and only if all edge lengths and all dihedral angles agree.
Integrability conditions that guarantee the existence of nodal positions realizing a given vector of edge lengths and dihedral angles were derived by \citet{WaLiTo12}.
The integrability conditions are formulated using moving frames associated with the triangles of the mesh.
Already in earlier work, \citet{LiSoLe05,LiCoGaLe07} used moving frames to define coordinates on triangle meshes.
Our goal is to extend this line of work by providing the tools and structures needed for solving optimization problems that are formulated in the nonlinear coordinates.
\paragraph*{Rigidity}
While the existence and uniqueness results for the nonlinear coordinates require both, the edge lengths and the dihedral angles, rigidity results can already be obtained if only edge lengths are considered.
For convex polytopes, Cauchy's and Dehn's rigidity theorems \cite{De16} show rigidity and infinitesimal rigidity and \citet{Gl75} showed that \emph{almost all} simply-connected polyhedra are rigid.
An example of polyhedra that allow for isometric continuous deformations, which are non-rigid, is Cornelly's sphere \cite{Co77}.
In this paper, we will formulate infinitesimal rigidity in terms of NRIC.
In recent work, \citet{AmRo18} studied the dihedral rigidity of polyhedra and parametrized triangle meshes via dihedral angles.
Related to rigidity is the problem of computing an immersion from prescribed edge lengths.
Algorithms for this problem were proposed by \citet{BoEyKoBr15} and
\citet{ChKnPiSc18}.
Relaxing the concept of rigidity, conformal geometry identifies metrics that differ only by a conformal factor. \citet{CrPiSc11} study the numerical treatment of the integrability conditions for surfaces in this setting.
\paragraph*{Near-isometric deformations}
Isometric and near-isometric deformations are important for computational folding of piecewise flat or developable structures \cite{KiFlChMiShPo08,BoVoGoWa16,StGrCr18,RaHoSo18b}.
The computation of near-isometric deformation can be done by simulating elastic shells consisting of stiff material with low bending resistance \cite{BuZoGr06,SoVoWaGr12,NaPfBr13}.
These materials yield ill-conditioned problems that are difficult to solve numerically.
The NRIC perspective improves the numerical accessibility of such problems.
\paragraph*{Shape interpolation}
Shape interpolation, also called blending or morphing, is an important problem in geometry processing which is used for applications such as deformation transfer \cite{BaVlGrPo09,YaGaLaRoXi18}, motion processing \cite{PrKaChCoHo16}, example-based methods for shape editing \cite{FrBo11}, inverse kinematics \cite{SuPo04,Wa16}, and material design \cite{MaThGrGr11}.
Approaches to shape interpolation based on linear coordinates use non-linear operations for blending the coordinates.
For example, the gradient-domain approach of \citet{XuZhWaBa05} extracts the rotational components from deformation gradients via polar decomposition and applies nonlinear blending operations to these components.
While the nonlinear blending helps to compensate for linearization artifacts, it is a difficult task to estimate the local rotations that resolve large deformations.
\citet{KiGa08} and \citet{GaLaLiChShXi16} introduce improved nonlinear blending operations for the rotational components.
\citet{WiDrAl10} introduce a scheme for shape interpolation using nonlinear coordinates. Their method linearly blends edge lengths and dihedral angles
and uses a multi-scale shape matching algorithm for constructing interpolating shapes.
\citet{FrBo11} model the process of finding the shape that best matches the blended lengths and angles as a nonlinear least-squares optimization problem and solve it using a multi-resolution Gau\ss--Newton scheme.
A related approach by \citet{WuBoShRoBr10} blends edge lengths and the normal vectors of two example shapes and constructs the intermediate shapes using a mesh traversal algorithm based on a minimal spanning tree with dihedral angle differences as weights.
Model reduction approaches that enable real-time shape interpolation have been introduced by \citet{TyScSe15} and \citet{RaEiSe16}.
Related to shape interpolation are shape spaces, which are shape manifolds equipped with a Riemannian metric.
Shape spaces are used for various applications in computer vision, computational anatomy, and medical imaging. For a general introduction to shape space and their applications, we refer to the textbook of \citet{Yo10}.
\citet{KiMiPo07} introduced a Riemannian metric on spaces of triangle meshes and show that concepts from Riemannian geometry such as the exponential map and parallel transport can be used for geometry processing tasks like deformation transfer, shape interpolation, and extrapolation.
\citet{HeRuWa12,HeRuSc14} propose an alternative physically-based metric on the shape space of triangle meshes that reflects the viscous dissipation required to physically deform a thin shell.
\citet{BrTyHi16} derive a discrete curve shortening flow in shape space and use it for processing animations of deformable objects.
While the shape spaces study deformations of meshes with fixed connectivity, functional correspondences \cite{OvBeSoBuGu12} can be used to blend \cite{KoBrBrGlKi13} and analyze \cite{RuOvAzBeChGu13} pairs of meshes with different connectivity.
In recent work, the functional correspondences of intrinsic and extrinsic geometry \cite{CoSoBeGuOv17} and deformation fields \cite{CoOv19} have been studied.
\section{Background}\label{sec:background}
In this section, we briefly review the work by \citet{WaLiTo12} who introduced a discrete version of the fundamental theorem of surfaces.
We consider a simplicial surface, which is a simplicial complex $\mathcal{K} = (\mathcal{V}, \mathcal{E}, \mathcal{F})$ consisting of sets of vertices $\mathcal{V}$, edges $\mathcal{E} \subset \mathcal{V} \times \mathcal{V}$ and faces $\mathcal{F} \subset \mathcal{V} \times \mathcal{V} \times \mathcal{V}$ such that the topological space $\vert \simplicial \vert$ obtained by identifying each face with a standard two-simplex and gluing the faces along the common edges is a two-dimensional manifold.
A map $X \colon \mathcal{V} \to \mathbb{R}^3$ is called \emph{generic} if for every face, the three vertices are in general position, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot, there is no straight line in $\mathbb{R}^3$ containing the three vertices.
We define
\begin{align}\label{eq:vertexSpace}
\mathcal{N} := \{ X(\mathcal{V}) \mid X \colon \mathcal{V} \to \mathbb{R}^3 \text{ generic } \} \subset \mathbb{R}^{3|\vertices|} \, \, ,
\end{align}
which we denote the space of \emph{discrete surfaces}.
For any $X$, there is a unique map $\imm^\ast\colon \vert \simplicial \vert \to \mathbb{R}^3$ that is continuous, an affine map of each simplex, and interpolates $X$ at the vertices.
$\imm^\ast$ maps the faces of $\vert \simplicial \vert$ to triangles in \(\mathbb{R}^3\), and, if $\imm^\ast$ corresponds to a generic map $X$, none of the triangles degenerates.
We will need this property to ensure that the elastic energies we consider in Section~\ref{sec:energy} are well-defined.
\begin{wrapfigure}{r}{0.45\columnwidth}
\centering
\scalebox{0.8}{
\Large
\def100pt{90mm}
\import{figures/sketch_integrability_vertex/}{drawing.pdf_tex}
}
\caption{
Construction of integrability condition for the 6-loop of faces around a vertex.
Specifically, the transition rotation \(R_{50} = R_0(\theta_5) R_2(\gamma_{0})\) (orange) is constructed from the dihedral angle \(\theta_5\) and interior angle \(\gamma_0\) (both black).
It transforms the frame \(\frame_5\) into frame \(\frame_0\) (both blue, normal vector not shown), \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot \(\frame_0 = \frame_5 R_{50}\).
The other transition rotations are constructed in the same way and applying them sequentially yields the integrability condition \(R_{01}R_{12}R_{23}R_{34}R_{45}R_{50} \protect\overset{!}{=} \mathds{1}\).
}
\end{wrapfigure}
Since we assume that the underlying simplicial complex remains unchanged, by abuse of notation, we will often refer to the image $X(\mathcal{V})$ of the generic map simply as $X$.
For a discrete surface $X \in \mathcal{N}$, we denote by ${l}(X) = ({l}_e(X))_{e\in\mathcal{E}}$ its vector of edge lengths and by $ {\theta}(X) = ({\theta}_e(X))_{e\in\mathcal{E}}$ its vector of dihedral angles.
Wang \emph{et al}\onedot studied necessary and sufficient conditions that an arbitrary tuple $({l}, {\theta}) \in \mathbb{R}^{2|\edges|}$ is induced by a discrete surface.
The first necessary condition is the triangle inequality, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot
\begin{equation}
\label{eq:triangInequ}
\tag{T}
\mathcal{T}_f({l}) > 0 \quad \text{ for all } f \in \mathcal{F}\, ,
\end{equation}
where $\mathcal{T}_f({l}) = \begin{pmatrix} l_i + l_j - l_k & l_i - l_j + l_k & - l_i + l_j + l_k \end{pmatrix}$ for a face \(f \in \mathcal{F}\) with edge lengths \(l_i,l_j,l_k\) and the above inequality is to be understood componentwise.
Thus, by combining the maps for all faces and extending constantly to dihedral angles we obtain a linear map $\mathcal{T} \colon \mathbb{R}^{2|\edges|} \to \mathbb{R}^{3 |\faces|}$ and we see that \eqref{eq:triangInequ} defines an open convex polytope in \(\mathbb{R}^{2 |\edges|}\).
The next set of conditions is referred to as \emph{discrete integrability conditions}.
They ensure that we can integrate the local change of geometry induced by the lengths and angles to reconstruct the immersed discrete surface.
In other words, the immersion is invariant with respect to the start and order of the reconstruction.
For a face \(f \in \mathcal{F}\) with (immersed) edges \(E_1, E_2, \, E_3 \in \mathbb{R}^3\), one defines the \emph{standard discrete frame} $\frame_f$ as the orthogonal matrix with rows $\frac{E_1}{\lVert E_1 \rVert}$, $\frac{E_1 \times N_f}{\lVert E_1 \times N_f \rVert}$, and $N_f$, where $N_f \in S^2$ is the unit face normal.
The normal component requires our discrete surfaces to be globally orientable which we will assume in the following.
Then the transition between frames \(\frame_j\) and \(\frame_i\) of adjacent faces \(f_i, f_j \in \mathcal{F}\) and a common edge \(e \in \mathcal{E}\) can be described by a rotation matrix \(R_{ij}\) with \(\frame_j = \frame_i R_{ij}\).
This rotation decomposes into three elementary rotations, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot
\begin{equation}\label{eq:transRotation}
R_{ij} = R_2(\gamma_{e,i})R_0(-\theta_e)R_2(\gamma_{j,e}),
\end{equation}
where $R_k(\varphi) \in \mathit{SO}(3)$ denotes a rotation around the $k$th standard basis vector in $\mathbb{R}^3$ by $\varphi \in [0,2\pi]$ and \(\gamma_{e,j}\) and \(\gamma_{i,e}\) denote the angles between the common edge and the first vector of \(\frame_i\) resp.\ \(\frame_j\).
In particular, the transition rotations are completely determined by the lengths and angles using the law of cosines.
Now let $\mathcal{V}_0 \subset \mathcal{V}$ be the index set of interior vertices.
Then for each \(v\in\mathcal{V}_0\), which is the center of a \(n_v\)-loop of faces \(f_0,\ldots,f_{n_v\!-\!1}\) and edges $e_0, \ldots, e_{n_v\!-\!1}$ connected to \(v\), we obtain a \emph{closing condition}.
To this end, one chooses the frames \(\frame_0,\ldots,\frame_{n_v\!-\!1}\) such that $e_i$ always coincides with the first basis vector in $\frame_i$.
Consequently, the corresponding transition rotations simplify to \(R_{ij} = R_0(\theta_i) R_2(\gamma_{j})\), where \(\gamma_{j}\) is the interior angle at \(v\) in \(f_j\) with $j = i+1$ modulo $n_v$ and $\theta_i$ is the dihedral angle at $e_i$.
Applying the transition property $\frame_j = \frame_i R_{ij}$ sequentially along the loop, the identity \(\frame_0 = \frame_0 \prod_{i=0}^{n_v-1} R_{i,(i+1) \bmod n_v}\) must hold for immersed discrete surfaces.
This can be phrased as the integrability condition
\begin{equation}\tag{I}\label{eq:discreteIntegrMap}
\mathcal{I}_v({l},\theta) \coloneqq \prod\limits_{i=0}^{n_v-1} R_{i,(i+1) \bmod n_v} \overset{!}{=} \mathds{1}
\end{equation}
for the \(n_v\)-loop of faces around all \emph{interior} vertices $v\in\mathcal{V}_0$.
Note that the transition rotations in \eqref{eq:discreteIntegrMap} in fact depend on $({l},{\theta})$.
\citet{WaLiTo12} proved that the necessary conditions \eqref{eq:triangInequ} and \eqref{eq:discreteIntegrMap} are indeed sufficient (for simply connected surfaces).
In detail, their discrete fundamental theorem of surfaces reads:
\emph{If \(({l},{\theta}) \in \mathbb{R}^{2|\edges|}\) satisfies \eqref{eq:triangInequ} and \eqref{eq:discreteIntegrMap}, there exists $X \in \mathcal{N}$ (unique up to rigid body motions) such that ${l}(X) = {l}$ and ${\theta}(X) = {\theta}$.}
They also extended this to non-simply connected surfaces but to simplify the exposition, we restrict ourselves to the simply connected case.
\section{The NRIC manifold of edge lengths and dihedral angles}\label{sec:nric}
In this section, we consider the Nonlinear and Rotation-Invariant Coordinates (NRIC) given as a vector ${z}=(l_e,\theta_e)_{e \in \mathcal{E}}\in\mathbb{R}^{2|\edges|}$ that lists all the edge lengths and dihedral angles of a discrete surface. Using the integrability conditions, we describe the manifold of discrete surfaces as a submanifold of $\mathbb{R}^{2|\edges|}$ and derive a scheme for computing its tangent spaces.
\paragraph*{NRIC manifold}
We consider the map
\begin{equation}\label{eq:projection}
{Z} \colon \mathcal{N} \to \mathbb{R}^{|\edges|} \times \mathbb{R}^{|\edges|}\, , \quad X \mapsto \left( {l}(X), {\theta}(X) \right)
\end{equation}
that associates to any discrete surface the vector stacking its edge length and dihedral angles
The image of \eqref{eq:projection} describes the submanifold
\begin{equation}\label{eq:nricDefinition}
\mathcal{M} := {Z}(\mathcal{N}) = \{{z} \in \mathbb{R}^{2|\edges|} \mid \exists X \in \mathcal{N} \colon {Z}(X) = {z} \}.
\end{equation}
of $\mathbb{R}^{2|\edges|}$ that we call the NRIC manifold.
\paragraph*{Implicit description}
In the following, we will use the conditions \eqref{eq:discreteIntegrMap} and \eqref{eq:triangInequ} to derive an implicit description of \(\mathcal{M}\).
Directly using condition \eqref{eq:discreteIntegrMap} leads to nine scalar constraints per vertex, which is a redundant description since $\mathit{SO}(3)$ is a three-dimensional manifold.
Instead, we will introduce a reformulation using unit quaternions as an equivalent representation of spatial rotations.
To this end, let us first briefly recall the necessary basics of quaternions and their relation to spatial rotation such that this section is self-contained, for a detailed treatment we refer to standard textbooks such as \cite{Ha06}.
Quaternions can be understood as an extension of the complex numbers and are generally represented as \(q = a+b \boldsymbol{i} + c \boldsymbol{j} + d \boldsymbol{k} \), where \(a,b,c,d \in \mathbb{R}\) and \(\boldsymbol{i}, \boldsymbol{j}, \boldsymbol{k}\) are the so-called \emph{quaternion units}.
These units fulfill the fundamental identity \(\boldsymbol{i}^2 = \boldsymbol{j}^2 = \boldsymbol{k}^2 = \boldsymbol{ijk} = -1 \), from which the general multiplication of quaternions can be defined via distributive and associative law and thus quaternions form a noncommutative division ring \(\mathbb{H}\).
In this context, $a$ is called the real part of $q$ and $b,c,$ and $d$ the vector part, for which we also write $\mathrm{vec}(q) = (b,c,d) \in \mathbb{R}^3$.
Unit quaternions are those for which the product with their conjugate \(\bar q \coloneqq a-b \boldsymbol{i} - c \boldsymbol{j} - d \boldsymbol{k} \) is one, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot $q \bar q = a^2 + b^2 + c^2 + d^2 = 1$.
Points in three-dimensional space \(p \in \mathbb{R}^3\) can be identified with quaternions having vanishing real part, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot we write \(p = p_1 \boldsymbol{i} + p_2 \boldsymbol{j} + p_3 \boldsymbol{k}\).
Now, given a rotation \(Q\) around the unit vector \(u \in \mathbb{R}^3\) by angle \(\varphi \in [0, 2 \pi)\) we can define a corresponding \emph{unit} quaternion
\[q(u,\varphi) \coloneqq \cos \frac{\varphi}{2} + \left(u_1 \boldsymbol{i} + u_2 \boldsymbol{j} + u_3 \boldsymbol{k}\right) \sin \frac{\varphi}{2}.\]
Then one can verify that for any \(p \in \mathbb{R}^3\), the conjugation $qpq^{-1}$ with $q$ results in the rotated point \(Qp\).
The quaternion $-q(u, \varphi)$ would lead to the same rotation, thus the quaternions form a double covering of \(\mathit{SO}(3)\).
Furthermore, investigating this conjugation one realizes that the composition of two rotations given as unit quaternions \(q_1, q_2 \in \mathbb{H}\) is given by their product \(q_1q_2\) and hence this correspondence is a homomorphism between \(\mathit{SO}(3)\) and the unit quaternions.
Turning to the reformulation of the integrability conditions \eqref{eq:discreteIntegrMap}, recall that we needed rotations around the $0$th and $2$nd basis vector in $\mathbb{R}^3$ for which we now introduce the corresponding quaternions
\begin{equation}
q_0(\varphi) \coloneqq \cos \frac{\varphi}{2} + \boldsymbol{i} \sin \frac{\varphi}{2}, \quad q_2(\varphi) \coloneqq \cos \frac{\varphi}{2} + \boldsymbol{k} \sin \frac{\varphi}{2} \quad \text{ for } \varphi \in [0, 2 \pi).
\end{equation}
Then we identify the simplified transition rotation \(R_{ij} = R_0(\theta_i) R_2(\gamma_{j})\) from before with the quaternion
\begin{equation}
q_{ij} \coloneqq q_0(\theta_i)\, q_2(\gamma_{j}),
\end{equation}
where again \(f_0,\ldots,f_{n_v\!-\!1}\) and $e_0, \ldots, e_{n_v\!-\!1}$ are the \(n_v\)-loops of faces resp.\ edges connected to $v$, \(\gamma_{j}\) is the interior angle at \(v\) in \(f_j\) with $j = i+1$ modulo $n_v$, and $\theta_i$ is the dihedral angle at $e_i$.
To finally reformulate the condition \eqref{eq:discreteIntegrMap}, we need to deal with the ambiguity introduced by the double covering, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot that the identity rotation is represented by \(q = \pm 1\).
However, we see that in both cases the vector part \(\mathrm{vec}(q) \in \mathbb{R}^3\) is zero, which is indeed for unit quaternions already a sufficient condition to be plus or minus one.
Then we use this alternative characterization of the identity rotation to formulate the \emph{quaternion integrability conditions} as
\begin{equation}\tag{I\textsubscript{q}}\label{eq:quatIntegrMap}
\mathcal{Q}_v({l},\theta) \coloneqq \mathrm{vec}\left(\prod\limits_{i=0}^{n_v-1} q_{i,(i+1) \bmod n_v}\right) \overset{!}{=} 0
\end{equation}
for the \(n_v\)-loop of faces around all interior vertices $v\in\mathcal{V}_0$.
Now, we can rewrite the manifold defined in \eqref{eq:nricDefinition} as
\begin{align}\label{eq:nricDefinitionImplicit}
\mathcal{M} = \big\{{z} \in \mathbb{R}^{2|\edges|} \,\big|\, \mathcal{T}(z) > 0,\, \mathcal{Q}(z) =0 \big\}\, .
\end{align}
Here we have collected all constraints in a vector-valued functional $\mathcal{Q} \colon \mathbb{R}^{2|\edges|} \to \mathbb{R}^{3|\mathcal{V}_0|}$ with $\mathcal{Q} = (\mathcal{Q}_v)_{v \in \mathcal{V}_0}$.
Obviously, $\mathcal{Q}_v$ depends solely on the edge lengths of the adjacent faces of $v$ and the dihedral angles at edges centered at $v$.
Given ${z} \in \mathbb{R}^{2|\edges|}$ with $\mathcal{Q}({z}) = 0$ one can easily reconstruct vertex coordinates $X \in \mathcal{N}$ with ${Z}(X) ={z}$.
For a robust and stable reconstruction for $\mathcal{Q}({z}) \neq 0$, we refer to Section~\ref{sec:recon}.
\paragraph*{Tangent space}
The implicit formulation \eqref{eq:nricDefinitionImplicit} consists of the triangle inequalities defining an open convex polytope and of the nonlinear integrability conditions, which define a lower-dimensional, differential structure on $\mathcal{M}$.
Therefore, we can derive an implicit description of its tangent space solely based on $\mathcal{Q}$.
In detail, for ${z}\in\mathcal{M}$ the tangent space is given by
\begin{align} \nonumber
T_{z} \mathcal{M} &= \mathrm{ker}\, D\mathcal{Q}({z}) \coloneqq \{ w \in \mathbb{R}^{2|\edges|} \, | \, D\mathcal{Q}({z}) w = 0 \}\,,
\end{align}
where $D \mathcal{Q}({z})$ is a matrix in $\mathbb{R}^{3|\mathcal{V}_0|, 2|\edges|}$.
Partial derivatives of $\mathcal{Q}_v$ are given by the chain rule as
\begin{align}\label{eq:intFirstDeriv}
\partial_{{z}_k} \mathcal{Q}_v({z}) &= \mathrm{vec} \left(\sum_{i=0}^{n_v-1} q_{01}({z}) \ldots \partial_{{z}_k} q_{i,i+1}({z}) \ldots q_{n_v-1,0}({z})\right),
\end{align}
where the partial derivatives of a quaternion-valued map are to be understood componentwise as for vector-valued maps.
The gradient of $\mathcal{Q}_v$ can be computed with $O(n_v)$ cost and is sparse.
It has only $O(n_v)$ non vanishing entries, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot $\partial_{\theta_e} \mathcal{Q}_v \equiv 0$ if $v$ is not a vertex of the edge $e$ and $\partial_{{l}_e} \mathcal{Q}_v \equiv 0$ if the edge $e$ is not an edge of a triangle with vertex $v$.
We provide details on the gradient computation as well as an implementation in terms of a \textsc{Mathematica} notebook in the supplementary material.
\begin{figure}[ht]
\centering
\includegraphics[width=0.24\columnwidth]{steffens/SteffensPolyhedron.jpg}
\includegraphics[width=0.24\columnwidth]{steffens/steffens_neg.png}
\includegraphics[width=0.24\columnwidth]{steffens/steffens_arrows.png}
\includegraphics[width=0.24\columnwidth]{steffens/steffens_pos.png}
\caption{The tangent space reveals an infinitesimal isometric variation at the classical Steffen's polyhedron (middle).
Indeed, extrapolating in this positive (left) resp. negative (right) direction (solely in the $\theta$ component) allows for isometric deformations.
The extrapolation is implemented via an incremental addition of the infinitesimal isometric variation coupled with a back projection onto $\mathcal{M}$.
See also video in supplementary material for an animation. }
\label{fig:steffens}
\end{figure}
To illustrate the NRIC manifold and its tangent spaces, we will for the remainder of the section discuss an immediate application.
With the tangent space at hand, one can verify the \emph{infinitesimal rigidity} of a discrete surface with NRIC ${z}\in \mathcal{M}$.
In fact, a necessary condition for the existence of continuous one-parameter families of isometric deformations starting at ${z}$ is the existence of an infinitesimal isometric variation $w\in T_{z} \mathcal{M}$ with $P_{l} w = 0$ and $w\neq 0$, where $P_{l}$ is the projection onto the length component, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot $P_{l} ({l},{\theta}) = {l}$, see for example \autoref{fig:steffens}.
Note, however, that this is surely not a sufficient condition, which we can also observe in \autoref{fig:origami}.
Thus, we simply verify if the kernel of $D \mathcal{Q}({z})$ has a non-trivial intersection with the ${\theta}$ subspace of $\mathbb{R}^{2|\edges|}$, namely the kernel $\ker P_{l}$.
This intersection is given by $\ker \left( \begin{array}{c|c} B_{T_{z} \mathcal{M}} & B_{\theta} \end{array}\right)\,$,
where $B_{T_{z} \mathcal{M}}$ is a matrix whose columns form a orthonormal basis of $T_{z}\mathcal{M}$ and
$B_{\theta}$ is the canonical basis of $\ker P_{l}$.
We compute a singular value decomposition (SVD) of this matrix and evaluate the smallest singular value $\lambda_0$.
If $\lambda_0 = 0$, then there exists an infinitesimal isometric variation.
Otherwise, the singular value provides a quantitative measure for the lack of such an infinitesimal isometric variation.
\begin{figure}[ht]
\centering
\includegraphics[width=0.19\columnwidth]{origami_cylinder/GuestPellegrino_n12_arrow.png}
\includegraphics[width=0.19\columnwidth]{origami_cylinder/GuestPellegrino_n12_075.png}
\includegraphics[width=0.19\columnwidth]{origami_cylinder/GuestPellegrino_n12_100.png}
\includegraphics[width=0.19\columnwidth]{origami_cylinder/GuestPellegrino_n12_110.png}
\includegraphics[width=0.19\columnwidth]{origami_cylinder/GuestPellegrino_n12_120.png}\\
\parbox{0.19\linewidth}{\hfill}
\parbox{0.19\linewidth}{\centering ${\theta}^\ast\!= 0.75\cdot\bar {\theta}$}
\parbox{0.19\linewidth}{\centering reference}
\parbox{0.19\linewidth}{\centering ${\theta}^\ast\! = 1.1 \cdot \bar {\theta}$ }
\parbox{0.19\linewidth}{\centering ${\theta}^\ast \!= 1.2 \cdot \bar {\theta}$}
\caption{
Top: Almost isometric compression of an Origami cylinder as depicted in \citet{BoVoGoWa16} with the only infinitesimal isometric variation (left).
Optimizing \eqref{eq:quadEnergy} on NRIC manifold with $\delta\!=\!0$ and hard constraints on target angles ${\theta}^\ast$ along the upper horizontal edges (relative to the reference angle $\bar{\theta} = 2.257$) leads to non-isometric deformations with as small as possible edge length distortion (from left to right $4\%$, $0.3\%$, and $0.4\%$ average change of edge length).
See also supplementary video.
}
\label{fig:origami}
\end{figure}
In \autoref{fig:origami}, we show that the Origami cylinder considered by \citet{BoVoGoWa16} does not allow for an isometric deformation path which leads to a compression by folding.
Indeed, the only nontrivial infinitesimal isometric variation is indicated by arrows (top, right).
However, there is no nontrivial family of isometric deformation with this shape as the initial shape.
As discussed by \citet{BoVoGoWa16} the experimental paper deformation (top, left) is not isometric.
This is reflected by our criteria for infinitesimal isometric variations when we additionally enforce the dihedral angles on the upper and lower plate to remain constant which leads to $\lambda_0=0.015$ clearly indicating the nonexistence of such a variation.
\section{Nonlinear energy and geometry of the NRIC manifold}
\label{sec:energy}
So far, we have introduced a differential structure on $\mathcal{M}$.
Going forward, it will be essential to additionally consider an elastic deformation energy $\mathcal{W}$ between different NRIC as it provides a dissimilarity measure on $\mathcal{M}$.
Although different choices for $\mathcal{W}$ are possible, we will primarily focus on the hyperelastic deformation energy from \citet{HeRuSc14}.
To this end, we reformulate this energy in NRIC to define a physically-motivated Hessian structure on $\mathcal{M}$.
This will be a straightforward undertaking which underlines our claim that NRIC are a natural choice for computing deformations.
In particular, we will see that the local injectivity constraints inbuilt in this energy allow us to replace the triangle inequalities and thus reduce the number of constraints.
For comparison reasons, we will finally consider a simple quadratic deformation energy as it has been used in \cite{FrBo11}.
Based on models from mathematical physics, the hyperelastic energy used in \cite{HeRuSc14} consists of two separate contributions, i.e.
\begin{equation}
\label{eq:genericEnergy}
\mathcal{W} = \W_\mem + \delta^2\, \W_\bend.
\end{equation}
From a physical point of view, the first term $\W_\mem$ will measure the stretching of edges and triangles, i.e. local \emph{membrane distortions}.
Likewise, the second term $\W_\bend$ will measure the difference in bending between triangles, i.e. local \emph{bending distortions}.
In particular, the global weight $\delta$ represents the thickness of a thin elastic material represented by the discrete surface.
In the following, we will investigate separately how the membrane and bending energy introduced in \cite{HeRuSc14} can be reformulated in NRIC.
Note that the membrane energy has originally been proposed in \cite{HeRuWa12} whereas the bending energy has been taken from \cite{GrHiDeSc03}.
\paragraph{Membrane energy}
Let \(X\in\mathcal{N}\) be a discrete surface and \(\imm^\ast\colon \vert \simplicial \vert \to \mathbb{R}^3\) the corresponding continuous, piecewise linear map that interpolates the vertices.
The derivative of \(\imm^\ast\) is constant over each triangle, and, since \(X\) is a generic map, the derivative has full rank.
This implies that \(\imm^\ast\) induces a metric $G$ (also called first fundamental form) on $\vert \simplicial \vert$.
This metric is defined in the interior of the faces and along the edges. It enables measuring the length of arbitrary curves in $\vert \simplicial \vert$ and makes $\vert \simplicial \vert$ a metric space.
Two discrete surfaces, \(X\) and \(\tilde{\imm}\), induce two different metrics, $G$ and $\widetilde G$, on $\vert \simplicial \vert$.
The metric distortion tensor $\mathcal{G}[X , \tilde{\imm}]$ is defined as the symmetric tensor that at any point in the interior of a triangle satisfies \[G(\mathcal{G}[X ,\tilde{\imm}]v,w) = \widetildeG(v,w)\] for any pair $v,w$ of tangential vectors.
The membrane energy evaluates the trace and the determinant of $\mathcal{G}[X, \tilde{\imm}]$.
For our purpose, it is essential to be able to evaluate the distortion tensor for discrete surfaces
\begin{wrapfigure}{r}{0.2\columnwidth}
\vspace{-1.3em}
\hspace{-2em}
\small
\def100pt{100pt}
\subimport{figures/sketch_triangle/}{drawing.pdf_tex}
\end{wrapfigure}
given by their NRIC ${z}$ and $\tilde {z}$ directly without having to reconstruct vertex positions first.
In the following, we derive an explicit formula for $\mathcal{G}[{z} ,\tilde {z}]$.
For discrete surfaces, the metric and the distortion tensor are constant for every triangle.
Consider an arbitrary triangle \(f\) in $\mathbb{R}^3$. We parametrize \(f\) with an affine map $\phi\colon t \to f$, where $t$ is the right angled triangle in $\mathbb{R}^2$ shown in the inset figure.
The standard basis $b_1,b_2$ of $\mathbb{R}^2$ agrees with second edge and the negative of the first edge of $t$.
Then $\d\phi(b_1)=E_2(f)$ and $\d\phi(b_2)=-E_1(f)$, where $E_1(f),E_2(f)$ denote the edge vectors of $f$.
Thus, the metric on $t$ induced by $\phi$ is
\begin{equation} \label{eq:discFirstFundForm}
G\vert_f =
\begin{pmatrix}
\lVert \d\phi(b_1) \rVert^2 & \langle \d\phi(b_2), \d\phi(b_1)\rangle \\
\langle \d\phi(b_2), \d\phi(b_1)\rangle & \lVert \d\phi(b_2) \rVert^2
\end{pmatrix}
=
\begin{pmatrix}
\lVert E_2(f) \rVert^2 & - \langle E_1(f), E_2(f)\rangle \\
- \langle E_1(f), E_2(f)\rangle & \lVert E_1(f) \rVert^2
\end{pmatrix},
\end{equation}
The entries of the metric can be expressed in terms of the length of the edges of $f$.
The diagonal entries are the squared length of the second and the first edge.
The off-diagonal entries are given by scalar products of edge vectors and from linear algebra we recall that for two vectors \(v, w \in \mathbb{R}^3\) we have \(\langle v, w\rangle = \lVert v \rVert \, \lVert w \rVert \cos(\gamma) \), where \(\gamma\) is the angle between \(v\) and \(w\).
In our case, this is the interior angle of a triangle which can be computed from its edge lengths by the law of cosines.
For two NRIC ${z}$ and $\tilde {z}$, we can use the formula to compute the metrics $G\vert_f$ and $\widetildeG\vert_f$ for every $f$.
Then, the distortion tensor is given as \(\mathcal{G}[X, \tilde X]\vert_f := (G\vert_f)^{-1}\widetilde G\vert_f\).
We want to note that the resulting distortion tensor depends on the chosen domain and parametrization.
However, we consider isotropic materials for which the membrane energy depends only on the trace and determinant of the distortion tensor.
Since the determinant and the trace are invariant under coordinate transformations, we obtain the same results independently of the chosen domain and parametrization.
Similarly the roles of the edges could be exchanged, for example, one could consider the second and third edge.
This would alter the parametrization and therefore yield a different distortion tensor.
Still, the relevant quantities, the determinant and the trace of $\mathcal{G}$, would be the same.
Having established that the distortion tensor is completely given by the NRIC of discrete surfaces, we can now adapt the membrane energy from \cite{HeRuWa12} applying a nonlinear energy density to it, which has a global minimum at the identity.
\begin{definition}[Membrane energy]
\label{def:ltheta_membrane_energy}
For a simplicial surface \(\mathcal{K}\), we define the membrane energy on NRIC \({z}, \tilde {z} \in \mathbb{R}^{2|\edges|}\) as
\begin{equation}
\label{eq:ltheta_membrane_energy}
\mathcal{W}_{\mbox{{\tiny mem}}}[{z}, \tilde {z} ] = \sum_{f \in \mathcal{F}} a_f \cdot W_{\mbox{{\tiny mem}}}(\mathcal G[{z}, \tilde {z}]|_f),
\end{equation}
where
\begin{equation*}
W_{{\mbox{{\tiny mem}}}}(A) := \frac{\mu}{2}\mathrm{tr}\, A + \frac{\lambda}{4}\det A -\left(\mu+\frac{\lambda}{2}\right)\log \det A - \mu - \frac{\lambda}{4},
\end{equation*}
for positive material constants \(\mu\) and \(\lambda\) and $a_f$ is the area of $f$ computed from edge lengths by Heron's formula.
\end{definition}
For more explicit formulas of the energy in terms of edge lengths we refer to the appendix and for the energy's derivatives to the supplementary material.
\paragraph{Bending energy}
Next, we adapt the \emph{Discrete Shells} bending energy \cite{GrHiDeSc03} also used in \cite{HeRuSc14}.
One directly sees that expressing this energy in lengths and angles requires no further calculations, and as before we replace its primary variables by NRIC.
\begin{definition}[\emph{Discrete Shells} bending energy]
\label{def:ltheta_bending_energy}
For a simplicial surface \(\mathcal{K}\), we define the Discrete Shells bending energy on NRIC \({z}, \tilde {z} \in \mathbb{R}^{2|\edges|}\) as
\begin{equation}
\label{eq:ltheta_bending_energy}
\mathcal{W}_{\mbox{{\tiny bend}}}[{z}, \tilde {z}] = \sum_{e\in\mathcal{E}} \frac{(\theta_e - \tilde{\theta}_e)^2}{d_e}l_e^2,
\end{equation}
where \(d_e = \frac{1}{3}(a_f + a_{f'})\) for the two faces \(f\) and \(f'\) adjacent to \(e \in \mathcal{E}\), as before computed by Heron's formula.
\end{definition}
Finally, we combine the membrane and bending energy in a weighted sum.
\begin{definition}[Nonlinear deformation energy]
\label{def:ltheta_energy}
Let \(\mathcal{K}\) be a simplicial surface and let \({z}, \tilde {z} \in \mathbb{R}^{2|\edges|}\) be two NRIC.
The nonlinear deformation energy is defined by
\begin{equation} \label{eq:nonlinearEnergy}
\mathcal{W}_{nl}[{z}, \tilde {z}] = \mathcal{W}_{\mbox{{\tiny mem}}}[{z}, \tilde {z} ] + \delta^2\, \mathcal{W}_{\mbox{{\tiny bend}}} [{z}, \tilde {z} ],
\end{equation}
where $\mathcal{W}_{\mbox{{\tiny mem}}}$ is the membrane energy from \autoref{def:ltheta_membrane_energy}, $\mathcal{W}_{\mbox{{\tiny bend}}}$ is the bending energy from \autoref{def:ltheta_bending_energy}, and \(\delta\) represents the thickness of the material.
\end{definition}
\paragraph{Relationship with triangle inequalities}
One essential property of the membrane energy is that it allows us to control local injectivity via the built-in penalization of volume shrinkage, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot we have \(W_{\mbox{{\tiny mem}}}(\mathcal G[{z}, \tilde {z}]|_f) \to \infty\) for \(\tilde{a}_f \to 0\).
To see this, we recognize that $\det \mathcal{G}[{z}, \tilde {z}]|_f = (\det G\vert_f)^{-1} \det \widetilde G \vert_f = a_f^{-2}\,\tilde a_f^2$ and hence $-\log \det \mathcal{G}[{z}, \tilde {z}]|_f \to \infty$ when $\tilde{a}_f$ goes to zero.
This control over the local injectivity also has consequences for the consideration of the triangle inequalities.
Because of it, we also have that the energy diverges, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot \(W_{\mbox{{\tiny mem}}}(\mathcal G[{z}, \tilde {z}]|_f) \to \infty\) if one of the components of \(\mathcal{T}_f(l)\) approaches zero meaning that we get close to violating one of the triangle inequalities.
Especially, we set \(W_{\mbox{{\tiny mem}}}(\mathcal G[{z}, \tilde {z}]|_f) = \infty\) if \(\mathcal{T}_f(l) > 0\) does not hold.
This allows us to characterize the NRIC manifold \(\mathcal{M}\) by
\begin{equation}
\label{eq:manifoldViaEnergy}
\mathcal{M} = \big\{{z} \in \mathbb{R}^{2|\edges|} \,\big|\, \mathcal{W}_{nl}[{z}^\ast, z] < \infty \text{ for a fixed } {z}^\ast \in \mathcal{M} ,\, \mathcal{Q}(z) =0 \big\}\, ,
\end{equation}
avoiding the explicit dependence on the triangle inequalities \autoref{eq:triangInequ} we had before.
Note, however, that the integrability conditions \autoref{eq:quatIntegrMap} are still necessary as finite energy does not guarantee their attainment.
The characterization \autoref{eq:manifoldViaEnergy} will be helpful later on to devise efficient numerical schemes for solving variational problems on $\mathcal{M}$.
\paragraph{Quadratic model}
Previously, \citet{FrBo11} used a quadratic deformation model for NRIC, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot they considered the weighted quadratic energy
\begin{equation} \label{eq:quadEnergy}
\mathcal{W}_q[{z},{z}^\ast] = \sum_{\edge \in \edges} \alpha_e \lVert {l}_e - {l}_e^\ast \rVert^2 + \delta^2 \, \sum_{\edge \in \edges} \beta_e \lVert {\theta}_e - {\theta}_e^\ast \rVert^2\, .
\end{equation}
In fact, almost the same model has been used in \cite{GrHiDeSc03} to define the \emph{Discrete Shells} energy for physical simulations based on nodal positions.
The weights $\alpha = (\alpha_e)_e$ and $\beta = (\beta_e)_e$ can be chosen in different ways.
Typically, they are computed from edge lengths ${l}_e = {l}_e(\bar z)$
and areas $d_e = d_e(\bar {z})$ associated with edges and defined on some representative reference configuration $\bar{z}\in\mathbb{R}^{2|\edges|}$.
For example, the authors in \cite{GrHiDeSc03, FrBo11} set in a related context $\alpha_e = {l}_e^{-2}$ and $\beta_e = {l}_e^2 \, d_e^{-1}$, whereas \citet{HeRuSc16} have chosen $\alpha_e = d_e^{\vphantom{-2}} {l}_e^{-2}$, for $e \in \mathcal{E}$.
Here the (physical) parameter $\delta^2 > 0$ trades the impact on length variations off against angle variations and can be considered as the squared thickness of the material as before.
This quadratic energy has no inbuilt control over the local injectivity of the deformation and hence does not allow a characterization without explicit dependence on the triangle inequalities as in \autoref{eq:manifoldViaEnergy}.
We found that in many of our examples this decreased the numerical accessibility and increased the needed number of iterations and runtimes.
Nevertheless, as demonstrated by \citet{FrBo11}, it often leads to natural-looking deformations and we will consider it in some of our examples.
\paragraph{Riemannian metric}
For each ${z}\in \mathcal{M}$, a Riemannian metric $g_{z}$ is a symmetric, positive definite quadratic form on the tangent space $T_{z} \mathcal{M}$ measuring the cost of an infinitesimal variation in tangential direction.
In our context a tangential vector $w\in T_{z} \mathcal{M}$ splits into two components $w= (w_{l}, w_{\theta})$, where $w_{l}$ is the variation of edge lengths and $w_{\theta}$ the variation of dihedral angles.
Following Rayleigh's paradigm, $\frac12$ times the Hessian of an elastic deformation energy can be considered as a Riemannian metric on the space of discrete surfaces if it is positive definite.
Precisely, we obtain the metric for tangent vectors $v,w \in \mathbb{R}^{2|\edges|}$ via
\begin{align}\label{eq:metric}
g_{{z}}(v,w) = v^T\left(\frac12 \mathrm{Hess}\, \mathcal{W}[{z},{z}] \right)w\, .
\end{align}
As investigated in \cite{HeRuSc14}, this is true for the energy defined in \autoref{eq:nonlinearEnergy} with the choice of membrane and bending energies made above.
Furthermore, it holds for the quadratic energy \autoref{eq:quadEnergy} if we choose all weights to be positive.
With the metric at hand, one can define the Riemannian distance on $\mathcal{M}$ and compute for instance shortest geodesic curves, \emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot Section~\ref{sec:results}.
\section{Variational problems on the manifold}\label{sec:varProblems}
The quest for geometrically optimal, discrete surfaces often leads to variational problems.
However, in many applications, the corresponding objective functional can naturally be formulated in our coordinates, thus on the NRIC manifold \eqref{eq:nricDefinitionImplicit}, and its first and second variation can be computed easily.
To this end, one aims at solving a constrained optimization problem, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot given an objective functional $\mathcal{E} \colon \mathbb{R}^{2|\edges|} \to \mathbb{R}$ the task is to
\begin{equation}
\begin{aligned}
& \underset{{z}\, \in\, \mathbb{R}^{2|\edges|}}{\text{minimize}}
& & \mathcal{E}({z}) \\
& \text{subject to}
& & \mathcal{Q}_v({z}) = 0 \text{ for each } v \in {\mathcal{V}_0}, \\
&
& & \mathcal{T}_f({z}) > 0 \text{ for each } f \in {\mathcal{F}}.
\end{aligned}
\tag{OPT}
\label{eq:genericOpt}
\end{equation}
Due to non-convexity of the objective, in general, there is no guarantee for a unique, global minimizer for the optimization problem.
\begin{figure}[ht]
\begin{minipage}[c]{0.56\columnwidth}
\includegraphics[trim=70 25 75 25, clip,width=0.48\columnwidth]{strangulation/sphere.png}
\includegraphics[trim=75 25 75 25, clip,width=0.48\columnwidth]{strangulation/sphere_0_AL.png}\\
\includegraphics[trim=75 0 75 50, clip,width=0.48\columnwidth]{strangulation/sphere_2_AL.png}
\includegraphics[trim=75 25 75 25, clip,width=0.48\columnwidth]{strangulation/sphere_4_AL.png}
\end{minipage}
\begin{minipage}[c]{0.39\columnwidth}
\includegraphics[trim=100 0 80 0, clip,width=0.48\columnwidth]{strangulation/DYNA_not_brushed_hfh.png}
\includegraphics[trim=100 0 80 0, clip,width=0.48\columnwidth]{strangulation/DYNA_brushed_hfh_AL.png}
\end{minipage}
\caption{Left: Unit sphere (grey) with black constraint curves to be shortened by means of equality constraints on edge lengths along with different results for varying bending parameter $\delta^2= 10^{-\{0,2,4\}}$ in the nonlinear objective \autoref{eq:nonlinearEnergy}.
Right: Same experiment but with (orange) constraint areas where the target edge lengths were increased by $30\%$ to simulate a brushing tool.}
\label{fig:strangulation}
\end{figure}
A simple example of an objective functional $\mathcal{E}$ is given by the dissimilarity to some given ${z}^\ast$ on the linear space $\mathbb{R}^{2|\edges|}$ measured by the deformation energy, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot $\mathcal{E}({z}) = \mathcal{W}[{z}^\ast,z]$, where $\mathcal{W}$ is an elastic deformation energy as discussed in the previous section.
For example, in \autoref{fig:strangulation} we have used the nonlinear energy defined in \autoref{eq:nonlinearEnergy} along with coordinate constraints on a certain subset of edge lengths to simulate a ``constriction'' of a sphere along curves or creating cartoon-like characters by inflating for instance hands and feet (\emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \cite{KrNiPo14}).
\paragraph{Ensuring triangle inequalities}
One crucial problem we encounter when we try to solve \autoref{eq:genericOpt} are the triangle inequalities which lead to $3|\faces|$ inequality constraints causing the problem to be computationally expensive.
Therefore, we aim for an approach to deal with them efficiently rooted in our geometric setup from \autoref{sec:nric} and \autoref{sec:energy}.
We achieve this by a modified line search.
First, recall that the set \(\R^{2\numE}_\mathcal{T} = \left\{ {z} \in \mathbb{R}^{2|\edges|} \mid \mathcal{T}({z}) > 0 \right\} \) defines an open connected subset of $\mathbb{R}^{2|\edges|}$.
Therefore, if we start with an initial point ${z}^0$ fulfilling the triangle inequalities we only have to ensure that every iterate remains in the set.
Hence, in a line search method where we search for a new iterate ${z}^{k+1}$ along a direction $d^k$ we have to restrict this search to $\R^{2\numE}_\mathcal{T}$.
We accomplish this using backtracking, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot reducing the stepsize $\beta^k$ until ${z}^{k+1} = {z}^k + \beta^k d^k \in \R^{2\numE}_\mathcal{T}$ holds.
In implementations, this can easily achieved by setting $\mathcal{Q}_v({z}) = \infty$ if $\mathcal{T}_f({z}) \not > 0$ for any face $f$ adjacent to $v \in \mathcal{V}_0$.
We can obtain an even more natural approach when we work with the nonlinear membrane energy $\mathcal{W}_{\mbox{{\tiny mem}}}$.
Recall that in \autoref{sec:energy} we introduced the characterization \autoref{eq:manifoldViaEnergy} of $\mathcal{M}$ without explicit dependence on the triangle inequalities by exploiting the growth of $W_{\mbox{{\tiny mem}}}$ for triangles with vanishing area.
This now readily fits into our modified line search approach.
In fact, if we compare our nonlinear energy to interior point methods \cite[Chapter 19]{NoWr06} we see that the logarithmic penalty in the energy takes the role of a barrier term which ensures that we stay in the admissible set $\R^{2\numE}_\mathcal{T}$.
Overall, we see that in both cases we can treat the inequality constraints in the line search and hence apply algorithms for equality-constrained optimization with a considerably lower number of constraints.
Note, that this approach can be adapted for trust region methods by limiting the size of the trust region appropriately.
\paragraph{Augmented Lagrange}
Next, we describe our approach to solving these equality-constrained problems based on the augmented Lagrange method.
First, let us briefly recall the Lagrangian formulation of our problem.
In fact, this means we seek for a saddle point of the Lagrangian
\begin{equation} \tag{Lag} \label{eq:genericLagrangian}
L({z}, {\lambda}) = \mathcal{E}({z}) - \mathcal{Q}({z}) \cdot {\lambda}
\end{equation}
with ${z}\in \mathbb{R}^{2|\edges|}$ and Lagrange multiplier ${\lambda} \in \mathbb{R}^{3|\mathcal{V}_0|}$.
The necessary condition for a saddle point $({z},{\lambda}) \in \mathbb{R}^{2|\edges|} \times \mathbb{R}^{3|\mathcal{V}_0|}$ is
\begin{align} \label{eq:lagrangeNecessary}
D L({z},{\lambda}) = \left(D_{z} L({z},{\lambda}),\, D_{\lambda} L({z},{\lambda})\right)^T = \left(D_{z} \mathcal{E}({z}) - D_{z} \mathcal{Q}({z}) \cdot {\lambda},\, -\mathcal{Q}({z})\right)^T = 0\,,
\end{align}
where $D_{z}$ and $D_{\lambda}$ denote the Jacobian with respect to ${z}$ and ${\lambda}$, respectively.
Instead of directly applying Newton's method to this equation we consider the augmented Lagrange method \cite{He75, NoWr06}.
It is a combination of the Lagrangian approach with the quadratic penalty method where we construct a series of unconstrained optimization problems in ${z}$ to approximate the solution of \autoref{eq:genericOpt}.
For the sake of completeness, we briefly recall it here.
The augmented Lagrangian is defined by
\begin{equation} \label{eq:augmentedLagrangian}
L({z}, {\lambda}, {\mu}) = \mathcal{E}({z}) - \mathcal{Q}({z}) \cdot {\lambda} + \frac{{\mu}}{2}\, \lVert \mathcal{Q}(z) \rVert_2^2,
\end{equation}
and a sequence $({z}^k, {\lambda}^k, {\mu}^k)$ of approximate solutions, approximate Lagrangian multipliers, and penalty parameters is generated by alternating between minimizing $L(\,\cdot\, , {\lambda}^k, {\mu}^k)$ to obtain ${z}^{k+1}$ and computing updates to ${\lambda}^k$ and ${\mu}^k$.
Hereby, the penalty parameter ${\mu}$ is increased until we reach sufficient attainment of the equality constraints.
On the other hand, ${\lambda}$ is updated by an increasingly accurate estimation of the correct multipliers ${\lambda}^\ast$ solving \autoref{eq:lagrangeNecessary}.
This can be accomplished in various ways, one popular way which we choose to follow is to set ${\lambda}^{k+1} = {\lambda}^{k} - {\mu}^k\, \mathcal{Q}(z^{k+1})$.
Though we cannot expect the augmented Lagrange method to converge for arbitrary initial data,
under reasonable assumptions, one can prove that the sequence ${\lambda}^k$ obtained this way converges to ${\lambda}^\ast$, which significantly improves convergences compared to the quadratic penalty method, see for example \cite{NoWr06}.
We want to remark that though our problem \autoref{eq:genericOpt} involves strict inequality constraints, the local convergence theory for the augmented Lagrange method given in \cite[Chapter 17]{NoWr06} applies to our problem.
The triangle inequality constraints define an open set and thus \autoref{eq:genericOpt} can be seen as an equality-constrained problem over an open set.
As \cite[Theorem 17.5 \& 17.6]{NoWr06} are only concerned with local minimizers and provide local results, they still hold if the problem is only defined on an open set after possibly modifying constants describing local neighborhoods.
An explicit algorithmic description of the method with all involved parameters and derivatives will be provided in the appendix.
\paragraph{Unconstrained Optimization}
Using the augmented Lagrange method leads to a series of unconstrained optimization problems.
They are typically non-convex, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot we encounter indefinite Hessians $D^2_{z} L$ of the Lagrangian.
This means that a simple Newton's method with line search might not be an efficient and robust approach as we are not guaranteed to obtain a descent direction.
To rectify this, we choose a simple adaption suggested in \cite[Section 3.4]{NoWr06}.
First, we determine a shift ${\tau}^k$ such that the matrix $ D^2_{z} L({z}^k, {\lambda}^k, {\mu}^k) + {\tau}^k \mathds{1}$ is positive definite.
This achieved by starting with an initial estimate and then increasing ${\tau}^k$ until a Cholesky decomposition succeeds.
Then, a descent direction is obtained by solving the linear system
\begin{equation}
\label{eq:descentDir}
\left(D^2_{z} L({z}^k, {\lambda}^k, {\mu}^k) + {\tau}^k \mathds{1}\right)d^k = -D_{z} L({z}^k, {\lambda}^k, {\mu}^k).
\end{equation}
Along this direction we perform an Armijo-type backtracking line search.
Note again, that the local convergence theory for Newton-type methods is still valid even though we minimize over an open set defined by the strict triangle inequalities, \emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \cite[Chapter 1]{Be99}.
In some instances, we could speed-up the minimization by first performing a small number of iterations with a BFGS approximation of the Hessian.
To compute the descent direction as above, we need the gradient and the Hessian of our constraint functionals $\mathcal{Q}$.
We already evaluated $D_{z} \mathcal{Q} \in \mathbb{R}^{3|\mathcal{V}_0|, 2|\edges|}$ in \autoref{eq:intFirstDeriv} and compute for the Hessian of $\mathcal{Q}$
\[
D_{z}^2 \mathcal{Q} \cdot {\lambda} = \left(\sum_{v \in \mathcal{V}_0} \partial_{{z}_l} \partial_{{z}_k} \mathcal{Q}_v \cdot {\lambda}_v \right)_{l,k = 1,\ldots, 2|\edges|}
\]
the components as
\begin{align*}
\partial_{z_l} \partial_{z_k} \mathcal{Q}_v(z) &= \mathrm{vec} \left(\sum_{j=0}^{n-1} q_{01}({z}) \ldots \partial_{z_l} \partial_{z_k} q_{j,j+1}({z}) \ldots q_{n-1,0}({z})\right)\\
&\phantom{=} + \mathrm{vec} \left(\sum_{\substack{i,j=0\\i\neq j}}^{n-1} q_{01}({z}) \ldots \partial_{z_l} q_{i,i+1}({z}) \ldots \partial_{z_k} q_{j,j+1}({z}) \ldots q_{n-1,0}({z})\right),
\end{align*}
which can also be evaluated with $O(n_v)$ cost.
We provide further details on the Hessian computation in the supplementary material.
\section{Reconstruction of an immersion}
\label{sec:recon}
In the preceding sections, we discussed the geometry as well as constrained optimization problems on the NRIC manifold $\mathcal{M}$, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot in terms of edge lengths and dihedral angles.
The remaining task is to reconstruct for given ${z} \in \mathcal{M}$ an immersion $X \in \mathcal{N}$ of the simplicial surface in $\mathbb{R}^3$ with ${z}={Z}(X)$.
Beyond the computation of vertex coordinates for ${z}\in\mathcal{M}$, one frequently asks for an approximate immersion $X\in\mathcal{N}$ for ${z} \not\in \mathcal{M}$ such that ${Z}(X) \approx {z}$.
Indeed, the computation of just approximate immersions is required in case of
\begin{itemize}
\item modeling of deformations energies in terms of dihedral angles and edge lengths, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot using the linear embedding space $\mathbb{R}^{2|\edges|}$ instead of $\mathcal{M}$,
\item using a high tolerance for the fulfillment of the constraints in the augmented Lagrange or a penalty method,
\item coordinates ${z}$ which are only approximately computed numerically.
\end{itemize}
Thus, we ask for a reconstruction map \(\mathcal{R} \colon \mathbb{R}^{2|\edges|} \to \mathcal{N}\), such that \(\mathcal{R}\vert_\mathcal{M}\) is the right inverse of \({Z}\) with
\({Z} \circ \mathcal{R} = \mathds{1}_ \mathcal{M}\), where \(\mathds{1}_ \mathcal{M}\) is the identity on the NRIC manifold.
Let us emphasize that by the rigid body motion invariance of our NRIC approach, we obtain \(\mathcal{R} \circ {Z} (X) = Q X\), where \(X \in \mathcal{N}\) and \(Q \in \mathit{SE}(3)\) is some rigid body motion acting on the immersion.
\paragraph{Variational approach}
For some given ${z} \in \mathbb{R}^{2|\edges|}$, where not necessarily ${z} \in \mathcal{M}$, we are looking for the nodal positions $X \in \mathcal{N}$, such that the resulting ${Z}(X) \in \mathcal{M}$ is as close as possible to ${z}$.
Fr\"ohlich and Botsch \cite{FrBo11} have used a least squares functional to build a \emph{variational reconstruction}, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot they compute
\begin{equation}\label{eq:genericVarRecon}
\argmin_{X \in \mathbb{R}^{3|\vertices|}} \, \mathcal{W}\left[{z},\,{Z}(X)\right]
\end{equation}
with $\mathcal{W}\left[{z},\tilde {z}\right]$ describing the proximity of ${z}$ and $\tilde {z}$.
Note that the solution is only unique up to a rigid deformation.
A simple example of a quadratic functional $\mathcal{W}$ is given by \eqref{eq:quadEnergy} as it was used by Fr\"ohlich and Botsch.
They proposed a Gau\ss-Newton method \cite[Section 10.3]{NoWr06} to solve \eqref{eq:genericVarRecon}, however, for general ${z}\in \mathbb{R}^{2|\edges|}$, one still has to solve a high-dimensional and nonlinear optimization problem in $\mathbb{R}^{3|\vertices|}$.
If \({z} \in \mathcal{M}\) and the initialization of the Gau\ss-Newton method is close to the solution, it usually converges in only a few iterations.
However, if ${z}$ is far away from \(\mathcal{M}\) and the initialization is poor, artifacts may occur.
\paragraph{Constructive approach}
For ${z} \in \mathcal{M}$ a \emph{constructive reconstruction} of the immersion $X\in\mathcal{N}$ can be derived by means of frames and transition rotations, as they were used to define the integrability conditions (\emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot Section \ref{sec:background}).
This method was introduced by \citet{LiSoLe05} and further elaborated in \cite{WaLiTo12}.
Before we investigate a combination of the constructive and the variational approach for ${z}\notin\mathcal{M}$, let us briefly review the constructive reconstruction.
Assume we are given an admissible target \({z} = ({l},{\theta}) \in\mathcal{M}\).
Since the reconstruction from lengths and angles is only defined up to rigid body motions, we further assume that we are given the position of one vertex and the orientation of an adjacent triangle \(f_0\) in the form of a frame \(\frame_0\).
If $f\in\mathcal{F}$ is a neighboring triangle of \(f_0\), one can infer the induced transition rotations $R_{0f}$ from ${z}$ and thus determine $\frame_f = \frame_0 R_{0f}$. Repeating this iteratively, one can construct frames for all faces.
This algorithm is indeed well-defined on simply connected triangulations due to the integrability constraints, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot if there are two paths connecting a triangle \(f\) to \(f_0\), then the frames constructed along the two paths coincide.
Given frames for all faces and hence the orientation of all triangles, one can finally reconstruct the nodal positions.
\paragraph{Adaptive spanning trees}
Next, we take into account a violation of the integrability condition \eqref{eq:discreteIntegrMap} for ${z} \not\in \mathcal{M}$ and ask for a reconstruction of an approximate immersion.
Let us remark that \citet{WaLiTo12} handle non-admissible targets ${z} \notin \mathcal{M}$ when modeling surfaces via a modification of ${l}$ and $\theta$.
They study a least-square type functional and relax in a least square sense the identity $\frame_f = \frame_0 R_{0f}$ as well as \eqref{eq:edgeVectors}.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{reconstruction_dyna/Assembled.png}
$X_1$ \hspace*{12mm} $X_2$ \hspace*{5.5cm} (BFS) \hspace*{17mm} (MST) \hspace*{17mm} (SPT)
\caption{Left: Input shapes $X_1$ and $X_2$ (taken from \cite{PoRoMaBl15}) and reconstruction from linear average $({Z}(X_1) + {Z}(X_2))/2 \notin \mathcal{M}$ with the local violations of the integrability condition as color map. Note that violations are highly concentrated, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot in the armpits.
Rightmost shapes: reconstruction using breadth-first search (BFS), minimal spanning tree (MST) and shortest path tree (SPT). The triangulation is color-coded with respect to the order of traversal.
See video for an animation of the reconstruction order.}
\label{fig:dynaReconstruction}
\end{figure}
The direct frame-based reconstruction with a spanning tree of the dual graph built by breadth-first search is very sensitive to violations of the integrability.
In fact, the errors occurring when walking over such a violation propagate to all following frames and are even amplified, \emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \autoref{fig:dynaReconstruction}.
In addition, reconstructing nodal positions of a face along two different paths connecting it to the initial face \(f_0\), where at least one is passing a zone of violated integrability conditions, leads to substantially different results and thus visual artifacts.
However, the regions of violation appear frequently to be highly localized in practice, \emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \autoref{fig:dynaReconstruction}.
Thus, we build a spanning tree which traverses faces with violation of the integrability condition as late as possible in the mesh traversal for the reconstruction.
To this end, we consider the dual graph of $\mathcal{K}$ with weights based on the integrability condition.
Each dual edge corresponds to a primal edge \(e = (v,v ') \in \mathcal{E}\) and we can assign to this dual edge a scalar weight reflecting the lack of integrability \(w_e\) by averaging the violation of integrability at the two adjacent vertices \(v\) and \(v'\):
\begin{equation}\label{eq:edgeWeights}
w_e := \frac{\rvert\mathrm{tr}\,\, \mathcal{I}_{v}({z}) - 3\lvert + \rvert\mathrm{tr}\,\, \mathcal{I}_{v'}({z}) - 3\lvert}{2}\,,
\end{equation}
where $\mathcal{I}_{v}({z})$ is the matrix-valued map defined in \eqref{eq:discreteIntegrMap}.
Note that $\mathcal{I}_{v}({z}) \in \mathit{SO}(3)$ and that $\mathrm{tr}\, Q = 3 \Leftrightarrow Q= \mathds{1}$ for $Q \in \mathit{SO}(3)$.
Now, the weights \eqref{eq:edgeWeights} are used to build a spanning tree adapted to the problem.
The first variant is to construct a minimal spanning tree (MST) of the dual graph, which is built such that the sum of all edge weights in the tree is minimal.
Such a minimal spanning tree can be computed via Prim's algorithm and provides a way to traverse the dual graph while avoiding unnecessarily large violations of the integrability.
Another variant is to construct a shortest path tree (SPT), which is built such that the path distance from the root to any other vertex in the tree is the shortest in the whole dual graph.
This can be achieved by Dijkstra's algorithm and provides a way to traverse the dual graph such that for each face the sum of integrability violation along the dual path used for its reconstruction is minimal.
We compare both novel variants against the original breadth-first search (BFS) in \autoref{fig:dynaReconstruction}.
A pseudo code of the entire algorithm is given in the appendix.
Formally, the algorithm --- using either (MST) or (SPT) --- is only defined for ${z}\in\mathcal{M}$.
In particular, the triangle inequality is assumed to be defined.
However, the algorithm can easily be generalized for ${z} \in \mathbb{R}^{2|\edges|}$ with $\mathcal{T}_f({z}) \leq 0$ for some face $f$ by setting the interior angles of $f$ to zero.
By our definition of \eqref{eq:discreteIntegrMap} and edge weights \eqref{eq:edgeWeights} those triangles will be automatically considered as late as possible in the adaptive algorithm.
\paragraph{Preassembled tree}
The runtime of the tree-based reconstruction algorithm is dominated by the cost for the construction of the spanning tree.
Thus, if one aims at reconstructing numerous immersions of a discrete surface with the same connectivity and a very high resolution (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot many vertices) it would be desirable to use a preassembled spanning tree.
Of course, this preassembled tree has to be reasonable for a large set of lengths and angles.
If we are given samples $z_1, \ldots, {z}_n \in \mathcal{M}$ and corresponding edge weights $w^1, \ldots, w^n \in \mathbb{R}^{|\edges|}$, we simply set $ w_e = \max_i w^i_e$ for all $e \in \mathcal{E}$ and construct a spanning tree based on these weights.
\begin{figure}[t]
\includegraphics[width=\columnwidth]{reconstruction_cactus/Assembled.png}
\caption{Left: Input shapes $X_1$ and $X_2$ (small) and reconstruction from linear average $({Z}(X_1) + {Z}(X_2))/2 \notin \mathcal{M}$ with local violations of the integrability condition as color map.
Rightmost shapes: reconstruction order using a minimal spanning tree (order as colormap), visual artefacts along the body and final solution after one step of a Gau\ss-Newton smoothing.}
\label{fig:cactusReconstruction}
\end{figure}
\paragraph{Hybrid approach}
Just applying our constructive reconstruction algorithm works very well for ${z}\notin\mathcal{M}$ as long as the violations are localized as in \autoref{fig:dynaReconstruction}.
However, we observe imperfect results when the violations are distributed over larger areas, \emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \autoref{fig:cactusReconstruction}.
In this case, we suggest a hybrid method combining our robust constructive reconstruction and as a post processing the variational reconstruction.
In detail, we make use of the (still imperfect) output of our constructive reconstruction to initialize the variational reconstruction as in \eqref{eq:genericVarRecon}.
Typically, a single Gau\ss-Newton step is sufficient to smooth the result adequately (\emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \autoref{fig:cactusReconstruction}).
\section{Numerical experiments and comparisons}\label{sec:results}
In this section, we study qualitative and quantitative properties of the NRIC tools and demonstrate that in particular for modeling with near isometric deformations the NRIC manifold outperforms established methods that consider nodal positions as primal degrees of freedom.
To this end, we pick up the generic variational problem \autoref{eq:genericOpt} introduced in \autoref{sec:varProblems} together with the proposed augmented Lagrange method.
In the following, we discuss different objective functionals $\mathcal{E}$ in \autoref{eq:augmentedLagrangian} and depending on the application additional constraints.
Note, however, that the \emph{constraint functional} $\mathcal{Q}$ in \autoref{eq:augmentedLagrangian}, which describes the NRIC manifold implicitly via \autoref{eq:nricDefinitionImplicit}, remains unchanged.
\paragraph*{Elastic averages}
Let $X_1, \ldots, X_n \in \mathcal{N}$ be a set of example shapes (sharing the same connectivity).
Frequently, one is interested in a mean or average shape, \emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \cite{TyScSe15}.
Given an elastic deformation energy, a so-called weighted elastic average is defined to be the minimizer of a weighted sum of elastic energies for deformations from the input shapes to the free shape.
This can be translated directly to our NRIC manifold, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot for a given elastic deformation energy $\mathcal{W}$ on $\mathcal{M}$ and convex weights $\mu \in \mathbb{R}^n$ we define the weighted elastic NRIC average as a solution of \autoref{eq:genericOpt} with
\begin{equation} \label{eq:elasticAverageFunctional}
\mathcal{E}({z}) = \sum_{i = 1}^n \mu_i \, \mathcal{W}[ {Z}(X_i), {z} ]\, .
\end{equation}
In \autoref{fig:handsAverage}, we show (the reconstructions of) weighted elastic NRIC averages for a set of six hand shapes and different weights $\mu_1, \ldots, \mu_6$.
Here, we have used the nonlinear deformation energy \eqref{eq:nonlinearEnergy} in \eqref{eq:elasticAverageFunctional}.
\begin{figure}[ht]
\begin{minipage}[c]{0.3\columnwidth}
\setlength{\unitlength}{.3\linewidth}%
\begin{picture}(3,4)
\put(-0.2,2){ \includegraphics[width=1.2\unitlength]{hands_elastic_mean/hands_input_0.png} }
\put( 1.0,2){ \includegraphics[width=1.2\unitlength]{hands_elastic_mean/hands_input_1.png} }
\put( 2.2,2){ \includegraphics[width=1.2\unitlength]{hands_elastic_mean/hands_input_2.png} }
\put(-0.2,0){ \includegraphics[width=1.2\unitlength]{hands_elastic_mean/hands_input_3.png} }
\put( 1.0,0){ \includegraphics[width=1.2\unitlength]{hands_elastic_mean/hands_input_4.png} }
\put( 2.2,0){ \includegraphics[width=1.2\unitlength]{hands_elastic_mean/hands_input_5.png} }
\put(-0.1,1.8){\footnotesize $X_1$ }
\put( 1.0,1.8){\footnotesize $X_2$ }
\put( 2.2,1.8){\footnotesize $X_3$ }
\put(-0.1,-0.1){\footnotesize $X_4$ }
\put( 1.0,-0.1){\footnotesize $X_5$ }
\put( 2.2,-0.1){\footnotesize $X_6$ }
\end{picture}
\end{minipage}
%
\begin{minipage}[c]{0.69\columnwidth}
\setlength{\fboxsep}{0pt}%
\setlength{\fboxrule}{1pt}%
\hspace{0.5em}
\includegraphics[width=0.24\linewidth-0.5em,trim={0 1em 2cm 0},clip]{hands_elastic_mean/hands_NRIC_al1.png}
\includegraphics[width=0.24\linewidth,trim={0 1em 0 0},clip]{hands_elastic_mean/hands_NRIC_al2.png}
\includegraphics[width=0.24\linewidth,trim={0 1em 0 0},clip]{hands_elastic_mean/hands_NRIC_al3.png}
\includegraphics[width=0.24\linewidth,trim={0 1em 0 0},clip]{hands_elastic_mean/hands_NRIC_al4.png}\\[0.1em]
\parbox{0.24\linewidth}{\vspace{-1em}\begin{align*} \mu \equiv \tfrac16 \end{align*}}
\parbox{0.24\linewidth}{\vspace{-1em}\begin{align*} \mu_{1,2,6} &= 0.03 \\ \mu_3 &= 0.6 \\ \mu_{4,5} &= 0.15 \end{align*}}
\parbox{0.24\linewidth}{\vspace{-1em}\begin{align*} \mu_{2} &= 0.48 \\ \mu_{4,5} &= 0.07 \\ \mu_{6} &= 0.385 \end{align*}}
\parbox{0.24\linewidth}{\vspace{-1em}\begin{align*} \mu_{2,5} = 0.5 \end{align*}}
\end{minipage}
\caption{Reconstruction of nodal positions from elastic averages of six hand poses (grey) with different (convex) weights $\mu\in\mathbb{R}^6$ computed as minimizer of \eqref{eq:elasticAverageFunctional} on the NRIC manifold. }
\label{fig:handsAverage}
\end{figure}
\paragraph*{Isometric deformations via additional constraints}
Interesting applications can be described by considering \autoref{eq:genericOpt} along with the simple objective $\mathcal{E}({z}) = \mathcal{W}[{z}^\ast,z]$ but with additional, simple coordinate constraints.
For example, in \autoref{fig:strangulation} we have seen experiments where we posed lengths constraints ${l}^{\vphantom{ast}}_{i} = {l}^\ast_i$ for $i\in I$ on the coordinates ${z} = ({l},{\theta})$ for some index set $I \subset \mathcal{E}$ and prescribed target lengths ${l}^\ast$.
Similarly, we obtain an elegant way to simulate the \emph{isometric} folding of a (flat) sheet of paper given in NRIC as ${z}^\ast = ({l}^\ast, {\theta}^\ast)$ where ${\theta}^\ast = 0$.
To this end, we pose the length constraints ${l}^{\vphantom{ast}}_e = {l}^\ast_e$ for all $e \in \mathcal{E}$ along with ${\theta}_{i} = const \neq 0$ if $i\in I$ for some index set $I \subset \mathcal{E}$.
Note, that under these length constraints the nonlinear and quadratic energy approach agree if we compute the weights in \autoref{eq:quadEnergy} from the reference ${z}^\ast$.
For example, in \autoref{fig:paperFoldingCompareGauss} we impose the constraint ${\theta}_i = \pi/2$ for the edges on two short line segments on two neighboring sides of the sheet.
Since all edge lengths are fixed and all other dihedral angles are degrees of freedom for the minimization of \autoref{eq:nonlinearEnergy} on $\mathcal{M}$, we obtain a perfect isometric deformation as indicated by the vanishing discrete Gau\ss \, curvature (\autoref{fig:paperFoldingCompareGauss}, right).
In comparison, vertex-based methods as \cite{GrHiDeSc03} or \cite{HeRuWa12} do not achieve a perfect isometry---even when computed with a very high membrane stiffness (\autoref{fig:paperFoldingCompareGauss}, left).
For the optimization in nodal positions, we used the energy $X \mapsto \mathcal{W}_{nl}[{z}^\ast, {Z}(X) ]$ with a shell thickness parameter $\delta=10^{-3}$.
In fact, further reducing $\delta$ one observes numerical instabilities.
This is due to the fact that isometric deformations induce bending distortions only but optimizing bending energies in terms of nodal positions is a highly nonlinear singular perturbation problem that quickly triggers numerical issues.
Conversely, the corresponding bending energy in NRIC is quadratic.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\columnwidth]{paperfolds/two_side_back_nodal.png}
\includegraphics[width=0.48\columnwidth]{paperfolds/two_side_back_NRIC.png}
\caption{Paper folding with local constraints for dihedral angle: simulation in vertex space (left) leads to infinitesimal isometry violations whereas the result in NRIC is completely isometric (right).
The absolute value of discrete Gau\ss \, curvature (as angle defect) is shown using the color map $0\hspace{1mm}$\protect\resizebox{.1\linewidth}{!}{\protect\includegraphics{paperfolds/colorbar_red_log.png}}$\hspace{1mm}0.03$, which is zero everywhere on the right.
Furthermore, the corresponding histograms are plotted aside the surfaces.}
\label{fig:paperFoldingCompareGauss}
\end{figure}
Besides vanishing Gau\ss \, curvature, pure isometric deformations exhibit further characteristics, as illustrated in \autoref{fig:paperFoldingCompare}.
In this example, we have a very similar setup as in \autoref{fig:paperFoldingCompareGauss} but we pose the angle constraints on two opposite sides.
First, let us point out that we observed convergence of the augmented Lagrange method described above to different local minima when using different parameters for the increase of the penalty parameter ${\mu}$.
We show two different local minima in \autoref{fig:paperFoldingCompare} where we obtained the lowest energy value when increasing ${\mu}$ conservatively (shown on the right).
Now, since the NRIC results are perfectly isometric and rather smooth deformations of the flat sheet one can indeed observe effects predicted analytically by the Hartman-Nirenberg theorem \cite{HaNi59, Ho11}.
Loosely speaking, isometric deformations of a flat sheet can locally be described either as flat patches or segments of straight lines (rulings) going to the boundary.
In the middle and right columns of \autoref{fig:paperFoldingCompare} one can easily identify flat triangular regions as well as a cone-like bundle of straight lines propagating towards the boundary.
These structures are not reflected by the vertex-based numerical minimizer already discussed above (\autoref{fig:paperFoldingCompare}, far left).
\begin{figure}[ht]
\centering
\includegraphics[width=0.32\columnwidth]{paperfolds/opp_side_back_nodal_hr.png}
\includegraphics[width=0.32\columnwidth]{paperfolds/opp_side_back_NRIC_hr2.png}
\includegraphics[width=0.32\columnwidth]{paperfolds/opp_side_back_NRIC_hr1.png}\\
\includegraphics[width=0.32\columnwidth]{paperfolds/opp_side_side_nodal_hr.png}
\includegraphics[width=0.32\columnwidth]{paperfolds/opp_side_side_NRIC_hr2.png}
\includegraphics[width=0.32\columnwidth]{paperfolds/opp_side_side_NRIC_hr1.png}\\
\parbox{0.32\columnwidth}{\hfill}
\parbox{0.32\columnwidth}{\centering $\mathcal{E}({z}) \approx 159.6$}
\parbox{0.32\columnwidth}{\centering $\mathcal{E}({z}) \approx 150.3$}
\caption{Once more paper folding with local constraints for dihedral angle: simulation in vertex space (left) leads to infinitesimal isometry violations whereas the results in NRIC are completely isometric (middle, right).
The result in the middle shows a local minimum obtained by the augmented Lagrange method when increasing the penalty parameter ${\mu}$ (too) aggressively exhibiting higher deformation energy (shown below) than the right result where the penalty was increased more conservatively.
Triangle-averaged mean curvature is shown as color map $0\hspace{1mm}$\protect\resizebox{.1\linewidth}{!}{\protect\includegraphics{paperfolds/colorbar_pink.png}}$\hspace{1mm}\geq0.005$, flat triangular regions can only be seen in the NRIC simulations. }
\label{fig:paperFoldingCompare}
\end{figure}
\paragraph*{Time-discrete geodesics}
So far we have only considered static examples where a single shape was optimized subject to external forces or boundary conditions.
However, one can easily generalize \autoref{eq:genericOpt} to optimize for multiple shapes simultaneously, for instance, to simulate a kinematic behavior.
We focus on the computation of time discretized \emph{geodesics} in the NRIC manifold here.
On the manifold $\mathcal{M}$ with metric $g$ defined in \autoref{eq:metric} a geodesic connecting end points ${{z}}_A$ and ${{z}}_B$ in $\mathcal{M}$ is the curve ${{z}} \colon [0,1] \to \mathcal{M}$ minimizing the
\emph{path energy} $\int_0^1 g_{{{z}}(t)}(\dot{{z}}(t),\dot{{z}}(t))\d t$ subject to ${{z}}(0)={{z}}_A$ and ${{z}}(1) = {{z}}_B$.
In particular, the minimizer $({{z}}(t))_{0\leq t\leq 1}$ obeys the constant speed property $g_{{z}}(\dot{{z}}(t), \dot{{z}}(t)) = const$.
\citet{HeRuWa12} introduced the concept of \emph{time-discrete geodesics} (in a vertex-based approach) as a variational approximation of continuous geodesics.
For $K\in\mathbb{N}$, they consider a finite sequence ${{z}}_0, \ldots, {{z}}_K$ in $\mathcal{M}$ with ${{z}}_0={{z}}_A$ and ${{z}}_K={{z}}_B$ and define the \emph{time-discrete path energy}
\begin{align}\label{eq:discretePathEnergy}
E[{{z}}_0, \ldots, {{z}}_{K}] = K \, \sum_{k=1}^K \mathcal{W}[{{z}}_{k-1}, {{z}}_k]\, ,
\end{align}
where $\mathcal{W}$ is assumed to be a local approximation of the squared Riemannian distance and ${{z}}_k \approx {{z}}(k/K)$.
Minimizers $({{z}}_0, \ldots, {{z}}_K)$ of \eqref{eq:discretePathEnergy} for fixed end points ${{z}}_0$ and ${{z}}_K$ are said to be time-discrete geodesics.
In particular, they obey a discrete constant speed property, \emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot there is a uniform energy distribution $\mathcal{W}[{{z}}_{k-1}, {{z}}_k] \approx const$ along the curve.
\begin{figure}[ht]
\centering
\scalebox{1}{
\includegraphics[width=0.08\textwidth]{geodesic_plate/short_approx_geo_step_0.png}
\includegraphics[width=0.08\textwidth]{geodesic_plate/short_approx_geo_step_1.png}
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\includegraphics[width=0.08\textwidth]{geodesic_plate/short_approx_geo_step_4.png}
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\caption{Bottom: discrete geodesic in NRIC with input data from \cite{AmRo18} and membrane distortion as colormap ($0\hspace{1mm}$\protect\resizebox{.1\linewidth}{!}{\protect\includegraphics{paperfolds/colorbar_red_log.png}}$\hspace{1mm}\geq1$);
above: Linear interpolation in ambient space $\mathbb{R}^{2|\edges|}$ as in \cite{FrBo11} with energy distribution (green) vs. geodesic interpolation on $\mathcal{M}$ with constant energy distribution (orange).}
\label{fig:geodesicExample}
\end{figure}
The concept of discrete geodesics directly translates to the NRIC manifold and the path energy in \eqref{eq:discretePathEnergy} can be considered as an objective functional in \autoref{eq:genericOpt}.
Note, however, that this increases the number of free variables substantially.
In \autoref{fig:geodesicExample}, we show different time-discrete geodesics in NRIC where we use the quadratic deformation energy \autoref{eq:quadEnergy} in \autoref{eq:discretePathEnergy}.
In particular, we compare for end shapes being two oppositely bent plates our NRIC geodesic (orange) to the linear interpolation (green) in the embedding space $\mathbb{R}^{2|\edges|}$, which corresponds to a naive transfer of the projection approach by \citet{FrBo11}.
As indicated by the histogram plots, the discrete constant speed property can only be obtained for the NRIC formulation.
\begin{figure}[ht]
\centering
\includegraphics[width=\columnwidth]{geodesic_pill/Assembled.png}
\caption{Intermediate shapes at $t=1/3$ of a discrete geodesic between two perfectly isometric end shapes (grey) taken from \citet{DuVoTaMa16} obtained via NRIC optimization (orange) and vertex-based methods as in \cite{HeRuWa12} (blue) resp. \cite{FrBo11} (green).
Note that we preserve the isometry due to our hard length constraints. In contrast, the vertex-based methods get either stuck in local minima (blue) or reveal artifacts such as unnatural asymmetries (green).}
\label{fig:pillGeodesic}
\end{figure}
Furthermore, we can combine the computation of time-discrete geodesics with further constraints on the coordinates, \emph{e.g}\onedot} \def\Eg{\emph{E.g}\onedot to simulate isometric deformation paths.
For example, in \autoref{fig:pillGeodesic} we compare the computation of (almost) \emph{isometric} geodesic paths between perfectly isometric end shapes taken from \citet{DuVoTaMa16} to vertex-based methods.
A similar example is shown in \autoref{fig:monkeySaddleGeodesic}, where the first input shape ${z}_0 = ({l}^\ast, {\theta}^\ast)$ describes a hyperbolic monkey saddle and the second input shape is given by a reflection ${z}_K = ({l}^\ast, -{\theta}^\ast)$ of the saddle.
\autoref{fig:monkeySaddleGeodesic} demonstrates that our approach is able to realize a perfectly isometric deformation path (orange) by enforcing ${l}_k = {l}^\ast$ for all $0<k<K$, whereas vertex based optimization methods fail.
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\caption{Isometric geodesic paths. Left input shape $({l}^\ast, {\theta}^\ast)$ as hyperbolic monkey saddle, right input shape is the reflection $({l}^\ast, -{\theta}^\ast)$.
Comparison of discrete geodesics computed in NRIC (orange, perfectly isometric) and by methods based on nodal positions (\emph{i.e}\onedot} \def\Ie{\emph{I.e}\onedot \citet{HeRuWa12} (blue) and \citet{FrBo11}).
The latter approaches are not able to resolve pure isometric geodesics as indicated by a histogram of varying lengths on the right. See also supplementary video.}
\label{fig:monkeySaddleGeodesic}
\end{figure*}
\paragraph*{Timings}
Lastly, let us discuss the runtimes of the proposed method, where all computations were performed using a desktop computer with an Intel(R) Core(TM) i7-4790 CPU and 16 GB RAM.
In our framework, we use the Eigen library \cite{Eigen} for linear algebra tasks and CHOLMOD \cite{CHOLMOD} for the Cholesky decomposition.
At first, we list timings for the reconstruction performed without parallelization.
As a representative example, we report on timings measured on the Dyna dataset \cite{PoRoMaBl15} (\emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \autoref{fig:dynaReconstruction}) where $|\vertices| \approx 6.9k$.
The generation of (MST) or (SPT) takes about 12ms, the generation of a spanning tree via (BFS) takes about 1ms.
The traversal of a spanning tree takes about 5ms, and one Gau\ss-Newton iteration is done in 330ms (with 80\% spending in the linear solver).
Next, computing the entries of the Hessian of the constraint functional $\mathcal{Q}$ requires, again without parallelization, 16ms for the discrete surfaces considered in \autoref{fig:paperFoldingCompareGauss} and 112ms for the Dyna dataset considered in \autoref{fig:strangulation}.
Detailed timings for the optimization described in \autoref{sec:varProblems} are listed in \autoref{table:runtimes}.
Note, that the evaluation of the augmented Lagrangian $L$ and its derivatives requires substantially more time than computing the entries of $D^2_{z} \mathcal{Q}$.
This originates from computing the square of $D_{z} \mathcal{Q}$ and assembling the Hessian in CSR format because these operations do not benefit from parallelization in our current implementation.
Compared to computations in nodal positions, our method requires more memory due to the increased number of primal degrees of freedom.
However, because this number is approximately twice the number of nodal positions the total memory consumption only increases by a constant factor of approximately four.
\begin{table}[t]
\centering
\renewcommand{\arraystretch}{1.1}
\begin{tabular}{|lc||ccccc|}
\hline
\multicolumn{2}{|c||}{example} & \multicolumn{2}{c}{iterations} & \multicolumn{3}{c|}{avg.\ times per Newton iteration} \\
Figure & \(N\) & aug.\ Lagrange & Newton & evaluation & solve & line search \\
\hline \hline
\ref{fig:strangulation} (left, avg.) & 5220 & 18 & 83 & 30 ms & 23 ms & 3 ms \\
\ref{fig:strangulation} (right) & 41328 & 14 & 121 & 281 ms & 201 ms & 15 ms \\
\ref{fig:handsAverage} (avg.) & 36552 & 13 & 203 & 263 ms & 244 ms & 18 ms \\
\ref{fig:paperFoldingCompareGauss} & 6272 & 83 & 477 & 34 ms & 34 ms & 7 ms \\
\ref{fig:paperFoldingCompare} (right) & 24832 & 46 & 454 & 173 ms & 84 ms & 7 ms \\
\ref{fig:geodesicExample} (top) & 5760 & 10 & 15 & 16 ms & 69 ms & 2 ms \\
\ref{fig:geodesicExample} (bottom) & 36000 & 12 & 65 & 338 ms & 732 ms & 10 ms \\
\ref{fig:pillGeodesic} & 69936 & 11 & 44 & 626 ms & 301 ms & 22 ms \\
\ref{fig:monkeySaddleGeodesic} & 32240 & 11 & 173 & 218 ms & 193 ms & 5 ms \\
\hline
\end{tabular}
\caption{
Performance statistics of our approach on the different examples shown before.
From left to right: number of degrees of freedom, number of iterations of the augmented Lagrange method, total number of Newton iterations, average time for evaluation of function and derivatives per Newton iteration, average time for computing ${\tau}$ and solving the linear system per Newton iteration, and average time for line search per Newton iteration.
}
\label{table:runtimes}
\end{table}
\section{Conclusion}\label{sec:discussion}
We introduce a framework that allows us to pose and solve geometric optimization problems in terms of NRIC.
The framework is built on several novel concepts.
First, we introduce a Riemannian structure for the NRIC manifold stemming from an implicit description via integrability conditions and a physically-motivated nonlinear elastic energy.
In particular, we demonstrate how the notion of a tangent space can be used to identify infinitesimal isometric variations.
Second, we present an approach based on the augmented Lagrange method and a modified line search for solving generic optimization problems in NRIC.
Third, we develop a hybrid algorithm for the reconstruction of nodal positions from length and angle coordinates that uses a mesh traversal to initialize a Gau\ss--Newton solver.
We tested our framework on different problems including shape interpolation and paper folding.
A particular strength is the simulation of true isometric deformations---a task where well-established vertex-based methods often fail.
\paragraph*{Limitations and challenges}
We see great potential in using NRIC for geometric optimization problems and expect that the techniques we present will be further developed.
We plan to formulate an extended geodesic shape space calculus (\emph{cf}\onedot} \def\Cf{\emph{Cf}\onedot \cite{HeRuSc14}) including geodesic extrapolation and parallel transport in NRIC and expect to profit from the rigid motion invariance of the coordinates and their robustness for near-isometric deformation.
In the context of a statistical analysis of shapes, our NRIC formulation enables direct processing of input data without an a priori rigid co-registration.
To this end, our NRIC manifold is a natural starting point for the development of a corresponding Riemannian principal component analysis.
Though our experiments demonstrate the benefits of NRIC-based optimization, our current framework has several limitations and poses challenges in making the optimization more efficient.
First, the current implementation can only handle simply connected surfaces.
An extension to higher-genus surfaces would require to include integrability conditions along non-contractible paths that generate the fundamental group.
This would lead to more global constraints in our optimization problems.
Typical examples of surfaces in geometric modelling have only a small number of generators of the homology group.
However, this necessity of complicated constraints is a general limitation of our method compared to nodal positions.
Second, a fundamental challenge is to reduce the number of degrees of freedom and integrability conditions.
Our current framework works with $2|\edges|$ variables and $3|\vertices|$ integrability conditions per shape.
This implies a larger number of variables compared to optimization in nodal positions, which in turn means increased memory requirement and more costly iterations.
Here, it might be worthwhile to explore model reduction approaches.
Furthermore, the triangle inequality constraints are in general challenging to take care of in the implementation.
We found in all our experiments that the proposed adapted line search, especially in conjunction with the nonlinear deformation energy, was able to handle them robustly.
Finally, we aim to account for point constraints in our NRIC-based optimization.
These type of constraints frequently appear, for instance, in physical simulations as forces or boundary conditions.
This could be accomplished by performing a partial reconstruction of the points with attached constraints using an explicit formula that results from tracing the paths in \autoref{alg:frame_direct}.
Then the derivatives of the explicit formula need to be computed with respect to NRIC to enable their use in optimization problems which might be a feasible task for modern automatic differentiation frameworks.
Nonetheless, this would introduce highly nonlinear and nonlocal terms to the optimization potentially limiting the performance of our method.
This introduces the challenge of devising different ways to combine NRIC-based modeling with point constraints.
|
2,869,038,154,750 | arxiv | \section{Introduction}
\label{intro}
The radio population of a star-forming galaxy -- dominated by supernova
remnants (SNRs), {\sc Hii} regions, and radio pulsars -- contains much
information on the evolution and properties of such galaxies. To better
understand such sources in M31, we have surveyed M31 at 325 MHz with the Very
Large Array\footnote{The VLA is operated by The National Radio Astronomy
Observatory, which is a facility of the National Science Foundation operated
under cooperative agreement by Associated Universities, Inc.} (VLA) in order
to identify radio sources intrinsic to this galaxy. In the introduction of
\citeauthor{paper1} \citeyear{paper1} (hereafter referred to as Paper I), we
discussed some of the advantages of surveying M31 at low frequencies, mainly
that the large field of view of the VLA at this frequency allows one to image
the optical disk of M31 in one pointing. However, observing at low frequencies
also has advantages in classifying the detected sources. Low frequency
observations ($\nu \la 1$~GHz) have increased sensitivity to the
steep--spectrum ($\alpha \la -0.5$) population of a galaxy, mostly composed of
SNRs and radio pulsars, than higher frequency observations. These two
different class of objects provide separate pieces of information: the SNR
population traces recent massive star formation and individual SNRs both
highlight the local interstellar medium (ISM) and are crucial to understanding
the cosmic ray population of the galaxy, while radio pulsars are very useful
in determining the line-of-sight electron density and magnetic field and
contain information on the total core-collapse supernova rate of a galaxy.
In a low-frequency survey, these sources are noticeably brighter than the
flat--spectrum population of a galaxy, composed mainly of HII regions and
Pulsar Wind Nebulae (PWNe). By comparing our images to higher frequency
images of M31, we can separate these different components of the radio
population of M31 from each other.
In addition, low--frequency observations coupled with higher--frequency ones
allow the detection of spectral curvature, a powerful probe
of a source and its surroundings. Two processes which can cause
low--frequency spectral turnover are free--free absorption, indicative of
ionized material along the line of sight, and synchrotron self--absorption,
which is believed to occur in active galactic nuclei (AGN; \citeauthor{odea98}
\citeyear{odea98}). Simultaneous 74 and 333 MHz
observations have detected free-free absorption caused by ionized gas inside
SNR Cas A \citep{casa}, and 74 and 327 MHz observations of SNR W49B discovered
a foreground HII region obscured by a molecular cloud \citep{w49b}.
Low--frequency surveys of the center of star-burst galaxy M82 allowed the
authors to calculate the properties of its ionized ISM through the detection
of free-free absorption (e.g. \citeauthor{wills} \citeyear{wills};
\citeauthor{allen} \citeyear{allen}). In this case, unlike for the
the Milky Way cases discussed above, spectral turnover was detected
above 325~MHz. Low frequency
spectral turnover has also been observed in compact, distant, powerful radio
galaxies known as Gigahertz Peaked sources (GPS) believed to be examples of
AGN powered radio galaxies whose jets have yet to penetrate through the ISM of
the galaxy \citep{odea98}. In addition, a radio spectrum with low--frequency
values {\it above} the power--law extrapolation of higher frequency
measurements is the signature of nonlinear shock acceleration, an important
process in explaining the observed cosmic ray spectrum, and has been observed
in young SNRs such as Tycho's SNR and Kepler's SNR \citep{ellison}.
In Paper I, we described a 325 MHz (``GLG'') survey of M31 conducted
with the A-configuration of the VLA. In this survey, we imaged 405
radio sources between 6\arcsec~(the beam size)~and 170\arcsec~in
extent, and calculated for each source its angular distance from the
center of M31 ($\theta_{M31}$), the projected radius of the source
assuming it is in the optical disk of M31 ($R_{M31}$), its major axis
($\theta_M$), minor axis ($\theta_m$), position angle ($\theta_{PA}$), peak
intensity ($A$), integrated flux ($S$), and - through comparisons with
the 36W \citep{36w}, 37W \citep{37w}, Braun \citep{braun}, and NVSS
\citep{nvss} radio surveys - its spectral index $\alpha$ and spectral
curvature parameter $\varphi$, using the convention:
\begin{equation}
S_{\nu} \propto \nu^{\alpha} e^{-\varphi(\nu^{-2.1})}
\end{equation}
To determine if a ``GLG'' source had a counterpart in one of these other
catalogs, we used a limiting distance of:
\begin{equation}
\label{radctrp}
r=Max\left(\frac{\theta_M^{GLG}+\theta_m^{GLG}}{2},\frac{\theta_M+\theta_m}{2}\right)
\end{equation}
where $\theta_M^{GLG}$ is the major axis of the GLG source, $\theta_m^{GLG}$
is the minor axis of the GLG source (these are not the deconvolved values but
the size of the source in our images), $\theta_M$ is the deconvolved major
axis of the catalog source, and $\theta_m$ is the deconvolved minor axis of
the catalog source \citep{braun}. Only a ``good match'' -- one where the
GLG source had only one match in the other catalog and its counterpart only
had no other matches in the GLG source list -- was used to calculate
$\alpha$ and $\varphi$.
In addition, we classified each source into one of four morphological
categories:
\begin{itemize}
\item Unresolved (``U''): The 281 GLG sources with
$\theta_M < 2 \times \theta_m$ and $\theta_M \la 15$\arcsec.
\item Elongated (``El''): The 16 sources with $\theta_M > 2 \times \theta_m$
and $\theta_M \la 15$\arcsec.
\item Complex (``C''): The 51 sources with $\theta_M \ga 15$\arcsec~that have
structure on scales smaller than $\sim$10\arcsec.
\item Extended (``Ex''): The 57 sources with $\theta_M \ga 15$\arcsec~that do
not have structure on scales smaller than $\sim$10\arcsec. These sources were
detected by running the source-finding algorithm {\sc SFIND}, part of the
{\sc MIRIAD} software package, on images smoothed to
20\arcsec$\times$20\arcsec. A subset of these category are the Good Extended
Sources (``ExG''), the two sources which satisfy the above criteria and have a
counterpart in one of the aforementioned higher frequency radio surveys.
Extended sources without counterparts in one of these surveys are {\it most
likely spurious}, but were included in the final source list for completeness.
\end{itemize}
This source list is provided in Table 3 of Paper I.
As described in Paper I, a statistical analysis of the angular distribution
($\theta_{M31}$) of sources, the radial distribution of sources ($R_{M31}$),
flux density ($S_{325}$), and spectral index ($\alpha$) distribution of
sources revealed that the majority of detected sources are background radio
galaxies. The primary purpose of this paper is to identify sources intrinsic
to M31, determine their nature, and discuss their properties. In addition, we
have also identified interesting background and foreground sources. This
paper is structured as follows: Section \ref{classification} discusses the
information used to classify sources, Section \ref{interesting} presents the
final classification of sources, and Section \ref{conclusion} summarizes
our results.
In this paper, $S_{\nu}$ is the flux density of a source at a
frequency $\nu$ MHz, and the distance to M31 is assumed to be 780$\pm$13~kpc
\citep{stanek}.
\section[]{Classification of Observed sources}
\label{classification}
In Paper I, we provided statistical information which helped determine the
general properties of the GLG sources. However, we also want to determine the
nature of each observed source in order to sort out background radio galaxies
from sources in M31 and the Milky Way, as well as to differentiate between the
different kinds of sources that comprise these categories. In order to
accomplish this, we used three different pieces of information: a source's
radio spectral properties, its morphological characteristics, and its
multi-wavelength (from the far--IR to the X--ray) properties. While it was
not possible to determine the nature of all 405 GLG sources, we were able to
classify 125 of them based on the information and methodology described below.
\subsection{Radio Spectrum Classifications}
\label{specclass}
We defined three spectral categories in order to help separate different
classes of sources -- steep spectrum ($\alpha \leq -1.3$~and
$\mid\varphi\mid \leq 3\sigma_{\varphi}$, where $\sigma_{\varphi}$ is
the error in $\varphi$), flat spectrum ($-0.5 \leq \alpha \leq 0.5$~and~
$\mid\varphi\mid \leq 3\sigma_{\varphi}$), and sources with spectral curvature
( $\mid\varphi\mid > 3\sigma_{\varphi}$).\footnote{Only ``U'', ``El'', ``C'',
and ``ExG'' sources are included in these categories, since the 55 ``Ex'' are
unlikely to be real sources. See Section 2.3 of Paper I for more
details.} In the cases of sources with spectral curvature, the source
must have been detected at two other radio frequencies -- most
commonly 610 and 1400~MHz. 105 GLG sources fell into at least one of
these categories. Table \ref{steepspec} lists the steep spectrum
sources, Table \ref{flatspec} lists the flat spectrum sources, and
Table \ref{turnspec} lists the sources with spectral curvature.
Figure \ref{specsrcs} shows the positions of these sources with
respect to the optical disk of M31. The distribution of steep
spectrum and flat spectrum sources is uniform on the sky, while the
distribution of turnover sources is condensed around the optical disk
of M31. However, this most likely is a selection effect -- as
mentioned in Paper I, $\varphi$ was only calculated for those GLG
sources with counterparts in at least three other radio surveys. As
shown in Figure 1 of Paper I, these surveys only overlap around the
optical disk of M31.
This classification scheme was defined in such a way to separate different
physical classes of sources.
\paragraph{Steep Spectrum Sources}
With the advent of deep low--frequency surveys, Ultra-Steep Spectrum sources
(USS;~$\alpha \leq -1.3$) have been increasingly studied and were discovered
to be efficient tracers of High-z Radio Galaxies (\citeauthor{pedani03}
\citeyear{pedani03}; \citeauthor{hzrg} \citeyear{hzrg}) and relic radio
galaxies (\citeauthor{komis94} \citeyear{komis94}; \citeauthor{kaplan00}
\citeyear{kaplan00}). In addition, pulsars -- either in M31 or in
the Milky Way -- would fall in this category, since pulsars have an average
spectral index of $\alpha\sim-1.8$ \citep{lorimer95}. Thirty-six
non-Extended sources fall into the steep--spectrum category, $\sim$10\% of the
total non-''Ex'' population. This percentage of steep--spectrum sources is
much higher than in previous surveys, e.g. 0.5\% in \citet{hzrg} and
2.7\% in \citet{aaron74}. This difference is probably due to the increased
sensitivity of both the present survey and the higher frequency surveys used
for comparison than those used in these previous studies.
\paragraph{Flat Spectrum Sources}
Flat spectrum sources are most likely dominated by a combination of FRI/FRII
radio galaxies -- both are radio galaxies whose radio emission is powered by a
central AGN, but the radio emission of an FRI is core-dominated and in general
is less luminous that an FRII radio galaxy, whose radio emission is
lobe--dominated \citep{fr} -- and more exotic extragalactic objects
such as BL Lacs. HII regions, which often have a spectral index in this
frequency range of $\alpha \geq -0.1$, SNRs, whose spectral indices in general
fall between $\alpha \approx -1$ and $\alpha \approx -0.3$
(\citeauthor{filipovic98} \citeyear{filipovic98}; \citeauthor{greensnrcat}
\citeyear{greensnrcat}), and PWNe, which have a radio spectral index of
$\alpha \approx -0.1$ \citep{helfand87}, in M31 or the Milky Way, would also
fall in this category.
\paragraph{Sources with Spectral Curvature}
Sources with spectral curvature could be time-variable flat--spectrum radio
galaxies\footnote{If the brightness of a radio source substantially changed
between the higher frequency radio surveys and our 325~MHz observation, it
would manifest itself as curvature in the composite radio spectrum.}, or if
$\varphi>0$, normal galaxies (i.e galaxies whose emission is not dominated by
a central AGN; \citeauthor{im90} \citeyear{im90}), as well as AGN, where
spectral turnover is most likely due to synchrotron self--absorption
\citep{odea98}. In addition, if located in the optical disk of M31,
$\varphi>0$ could indicate a foreground HII region along the line of sight, or
the result of internal free-free absorption in a HII region or SNR. An
extragalactic source with $\varphi<0$, in addition to possibly being variable,
could indicate the presence of a compact steep spectrum core which dominates
the emission at low frequencies (\citeauthor{fanti90} \citeyear{fanti90};
\citeauthor{athreya97} \citeyear{athreya97}). If the source is located in
optical disk of M31, $\varphi<0$ may be the result of a HII/SNR complex --
if a SNR and HII region fall along the same line of sight, the HII region
could dominate at higher-frequencies and the SNR, due to its steeper radio
spectrum, would dominate at lower-frequencies, or a diffuse particle
acceleration, which has been observed in young SNRs \citep{ellison}.\\
\smallskip\\
The different kinds of sources in each of these spectral
classifications can be differentiated from each other by their morphological
and multi-wavelength properties discussed in Sections \ref{morphclass} and
\ref{multiwave}. The criteria used to differentiate them is discussed in
Section \ref{interesting}.
\subsection{Morphological Classifications}
\label{morphclass}
Classifying a source based on its morphology is the least ambiguous method
for determining the nature of the source used in this paper. Of the 405
sources detected, only the 51 ``Complex'' sources, shown in Figure
\ref{complex}, were sufficiently resolved to classify morphologically. All
but a few have the radio morphology of a FRI, of a FRII, or of a radio jet.
The classification of these sources is given in Section \ref{interesting}, and
sources with anomalous morphologies are discussed further in Section
\ref{weirdmorph}.
\subsection{Multi-wavelength Classifications}
\label{multiwave}
In order to classify sources using their properties at other wavelengths, the
GLG source list was compared to source lists derived from targeted near
infrared (IR), far infrared (FIR), optical, and X-ray observations of M31, as
well as all-sky surveys in these wavebands. We looked for counterparts across
the electromagnetic spectrum because the different wavebands are sensitive to
different kinds of objects, many of which could appear in the GLG source list.
The result of these comparisons is summarized in Table \ref{mwcomp}. In this
table, a counterpart is where a GLG source had at least one counterpart in
this catalog and a good counterpart is the where the GLG source had only one
counterpart in this catalog and no other GLG source had this catalog source as
a counterpart. For ``C'' sources, which were resolved in our survey,
a good counterpart could be spurious due to the large search radius
used. For these sources, we checked if the counterpart corresponded
to a region of high radio emission. The number of false counterparts
and false good counterparts were calculated by shifting the RA and
DEC of every source in the catalog and then determining the number of
counterparts and ``good'' counterparts. In Table \ref{mwcomp}, the
number of false counterparts and false ``good'' counterparts given is
the average number of counterparts/''good'' counterpart in the
different shifts, and the error is the standard deviation. The
amount of the shift was typically $\pm$1\arcmin in both RA and DEC,
but was larger for some catalogs (which we note when we discuss them).
\subsubsection{Near-Infrared}
\label{ir}
In order to determine if a non-extended GLG source is an IR source, we
searched for counterparts in the 2MASS Extended Source Catalog (ESC;
\citeauthor{2MASSesc} \citeyear{2MASSesc})\footnote{Only ESC sources within
1$^{\circ}$ of the center of M31 were included in this search.} and the 2MASS
Point Source Catalog (PSC; \citeauthor{2MASSpsc} \citeyear{2MASSpsc}). We did
not look for IR counterparts of ``Ex'' sources because their large size,
coupled with the high density of 2MASS sources in the direction of M31,
resulted in a high coincidence rate, and because we believe that most ``Ex''
sources are spurious. We did include ``ExG'' sources in this analysis. For
the ESC, the limiting distance for a counterpart between a PSC source and an
``U'' or ``El'' GLG source is defined to be:
\begin{eqnarray}
r=Max(3\times\sqrt{(\sigma_{\alpha}^{GLG})^2+(\sigma_{\delta}^{GLG})^2}~,
\nonumber\\
3\times\sqrt{(\theta_M^{ESC})^2+(\theta_m^{ESC})^2})
\end{eqnarray}
where $\sigma_{\alpha}^{GLG}$ is the error in RA of the GLG source,
$\sigma_{\delta}^{GLG}$ is the error in DEC of the GLG source,
$\theta_M^{ESC}$ is the major axis of the error ellipse of the ESC source,
and $\theta_m^{ESC}$ is the minor axis of the error ellipse of the ESC
source. The errors on position were used for the ``U'' and ``El'' GLG sources
instead of its extent because their extents are not well known. For a complex
GLG source, the limiting distance for a ESC counterpart was:
\begin{equation}
r=Max\left(\frac{\theta_M^{GLG}+\theta_m^{GLG}}{2},3 \times
\sqrt{(\theta_M^{ESC})^2+(\theta_m^{ESC})^2}\right)
\end{equation}
where $\theta_M^{GLG}$ is the major axis of the GLG source, $\theta_m^{GLG}$
is the minor axis of the GLG source, and $\theta_M^{ESC}$ and $\theta_m^{ESC}$
are the same as before. We used the measured size of the ``C'' GLG sources
instead of the error on their positions because they are resolved and the IR
emission need not be centered on the radio emission. Once this list
was compiled, we rejected associated we considered to be unlikely due
to the large offset between the radio and ESC source. The remaining
candidates are listed in Table \ref{jhkesc}, along with the IR
properties of the ESC counterparts. The number of likely associations
is consistent with that expected from the analysis presented in Table
\ref{mwcomp}.
For the 2MASS PSC, counterparts were detected using the same method described
above. Again, we rejected associations where there was a large offset
between the radio and IR source and, for resolved radio sources, no
correspondence between the IR and radio emission. As seen in Table
\ref{mwcomp}, the number of false good counterparts for ``U'' or
``El'' GLG sources is small. The results of this comparison, and the
IR properties of the PSC counterparts, are listed in Table \ref{jhkpsc}.
The presence of an IR counterpart in the PSC allowed us to determine if a GLG
source was either an FRI or FRII, background ``normal'' galaxy, a star in the
Milky Way, or none of the above. To determine if the observed IR magnitude
was consistent with that of a background FRI or FRII, we first calculated the
redshift of the GLG source if it is an FRI or an FRII by
simultaneously solving the following equations -- the luminosity-redshift
($L$--$z$) relation for FRI and FRIIs\footnote{This equation was
derived using galaxies with a redshift z$<$0.5. This analysis assumes that the
luminosity-redshift relationship of FRIs and FRIIs does not change at higher
redshift.}~\citep{zirbel}:
\begin{equation}
\log~L_{408}^{FRI}=(1.03\pm0.22)~\log~z + (1.92\pm0.13)
\end{equation}
\begin{equation}
\log~L_{408}^{FRII}=(26.97\pm0.32)~\log~z + (28.38\pm0.12)
\end{equation}
where $L_{408}^{FRI}$ is the luminosity in Watts
of an FR~I galaxy at $\nu=408$~MHz, $L_{408}^{FR~II}$ is the luminosity
in Watts of an FRII galaxy at $\nu=408$~MHz, and $z$ is the redshift of the
FR~I/FR~II; the luminosity--distance relationship:
\begin{equation}
L_{408}=S_{408}\times 4\pi d_L^2
\end{equation}
where $S_{408}$ is determined by using the spectral model defined in Equation
4 in Paper I; and $d_{L}$ the luminosity distance, is defined as:
\begin{equation}
d_L=\frac{c}{H_{0}}[z+\frac{1}{2}(1-q_0)z^2]
\end{equation}
where $c$ is the speed of light, $H_0$ is Hubble's Constant (assumed to be
71~km~sec$^{-1}$~Mpc$^{-1}$), and $q_0$, the deceleration parameter, is -0.56,
as determined using the latest cosmological results -- for the redshift of the
galaxy. Using this redshift, we calculated the expected K--magnitude
of such an FRI/FRII using the {\it K-z} relationship for radio galaxies in the
7C Redshift survey \citep{willott}:
\begin{equation}
K=17.37+4.53~\log~z-0.31(\log~z)^2
\end{equation}
As a result of this analysis, we concluded that none of these sources
satisfied the requirement for an FRI or FRII radio galaxy since the
required redshift were ridiculously high ($z\ga10$).
We used the IR colors of PSC counterparts to determine if a GLG source is a
background normal galaxy, a star in M31, or a star in the Milky Way. Since
these classes of objects have very different IR colors, once we determine the
IR colors of these objects in the field of M31 we can classify the nature of a
GLG source based on the IR colors of its PSC counterpart. The IR colors of
background normal galaxies was determined by looking at the IR colors of all
2MASS ESC sources within 1$^{\circ}$ of M31 (top-left panel of Figure
\ref{ircolors}), since that catalog is believed to be composed almost
entirely ($\sim$97\%) of galaxies\footnote{From ``Explanatory Supplement to
the 2MASS All Sky Data Release''
(http://www.ipac.caltech.edu/2mass/releases/allsky/doc/explsup.html)}. To
distinguish between IR sources in the Milky Way and IR sources in M31,
we looked at the IR color diagram of PSC sources within 5\arcmin~of the center
of M31 -- expected to be dominated by sources in M31 (bottom-left panel of
Figure \ref{ircolors}), and that of PSC sources $>$90\arcmin~from the
center of M31 -- which is dominated by foreground stars and background
galaxies (bottom-right panel of Figure \ref{ircolors}). As Figure
\ref{ircolors} shows, these two groups have a very different range of IR
colors. The band of sources in the IR color diagram of PSC sources
$>$90\arcmin~is the same as that seen in IR color diagrams of stars in the
Milky Way \citep{irstars}, and PSC counterparts which fall in this range are
believe to be stars in the Milky Way. The results of this analysis are
presented in Section \ref{interesting}.
\subsubsection{Far-Infrared}
\label{fir}
We looked for Far Infrared (FIR) counterpart of GLG sources in the
Schmidtobreick et. al., 2000 (``Knots'') catalog of FIR knots in M31
\citep{knots}, the Xu \& Helou catalog of FIR point sources in M31 \citep{xu},
and the IRAS Point Source Catalog (IRAS-PSC; \citeauthor{iraspsc}
\citeyear{iraspsc}), IRAS Serendipitous Source Catalog (IRAS-SSC;
\citeauthor{irasssc} \citeyear{irasssc}), and IRAS Faint Source Catalog
(IRAS-FSC; \citeauthor{irasfsc} \citeyear{irasfsc}). For these catalogs,
matches were determined using the same criteria used for radio catalogs,
given in Equation \ref{radctrp}. As seen in Table \ref{mwcomp}, only
for the IRAS-PSC and IRAS-FSC catalogs are the number of counterparts
statistically significant.\footnote{The number of false counterparts
and false good counterparts for the FIR surveys were calculated using
position shifts of 10\arcmin, instead of 1\arcmin~used for the other
surveys, because of the large reported beam size.} However, even in
these cases the offsets are rather large. Therefore, we are not able
to claim any overlap between the observed radio and FIR sources in
this field. However, {\it Spitzer} with its improved angular
resolution will improve this situation.
\subsubsection{Optical}
\label{optical}
The GLG source list was compared to several optical catalogs in order to
determine the nature of the source: the Winkler \& Williams SNR catalog
(WW; Ben Williams, private communications 2002), the Magnier et al. SNR
catalog (Mag; \citeauthor{magniersnr} \citeyear{magniersnr}), the Braun \&
Walterbos SNR catalog (BW~SNR; \citeauthor{bwsnr} \citeyear{bwsnr}), the
Walterbos \& Braun catalog of Ionized Nebulae (Ion; \citeauthor{wbion}
\citeyear{wbion}), the Ford \& Jacoby catalog of Planetary Nebulae (FJ~PN;
\citeauthor{fjpn} \citeyear{fjpn}), and the Pellet et. al. catalog of HII
regions (Pellet; \citeauthor{pellet} \citeyear{pellet}). For all of
these, we used the same criteria that we used for other radio catalogs
(Equation \ref{radctrp}). As seen in Table \ref{mwcomp}, the
counterparts to GLG sources in the WW SNR catalog are likely to be
real but the rest are most likely coincidental. Table \ref{optctrp}
lists the counterparts of GLG sources in this catalog.
In addition to these ``specialty'' optical catalogs, we compared the GLG
source list with the Magnier et. al. BVRI catalog of optical sources in the
field of M31 (MagOpt; \citeauthor{magstars} \citeyear{magstars}). While a
majority of the sources in this catalog are stars in M31, it also contains
background galaxies and foreground (Milky Way) stars. The criteria for a
counterpart in this catalog is the same as that used for the 2MASS PSC,
described in Section \ref{ir}. MagOpt counterparts of complex GLG sources
were also determined the same way they were for the 2MASS PSC, as described in
Section \ref{ir}. From the analysis present in Table \ref{mwcomp}, a
majority of the counterparts between MagOpt sources and ``U'' or
``El'' GLG sources are coincidental, but due to the small offsets
between our radio sources and these sources, as well as the lack of
optical color information for most of these counterparts, it is
difficult to separate the real from the false counterparts. As a
result, for completeness we include Table \ref{magstarctrp}, which
lists the GLG sources with a counterpart in this catalog, as well as
B, V, R, and I magnitude of the optical counterpart.
We also looked for optical counterparts of GLG sources in unpublished
H$\alpha$, [SII], and R--band images of M31 (generously provided by Ben
Williams).\footnote{Both the H$\alpha$ and [SII] images are
continuum subtracted.} These images have a larger FOV than the MagOpt catalog,
and the H$\alpha$ \& [SII] information helps determine if a source is a SNR.
Figure \ref{redgd} shows all GLG sources with R--band counterparts, Figure
\ref{halpgd} shows all sources with H$\alpha$ counterparts and Figure
\ref{siigd} shows all sources with [SII] counterparts. In the case of
GLG203, we believe the observed H$\alpha$ emission is an artifact of
the continuum subtraction process.
The final classification of GLG sources with optical counterparts is given in
Section \ref{interesting}.
\subsubsection{X-ray}
\label{xray}
In order to determine if a GLG source has an X-ray counterpart, we compared
the GLG source list to source lists derived from the {\it ROSAT} All--Sky
Survey (``RASS'') [both the faint \citep{RASSfaint} and bright
\citep{RASSbright}
source lists], the Second {\it ROSAT} PSPC survey of M31 \citep{rosat}, a
{\it Chandra} survey of bright X-ray sources in M31 Globular Clusters
\citep{m31glob}, a {\it Chandra} ACIS-I survey of the central region of M31
\citep{acis}, a {\it Chandra} HRC survey over most of the optical disk of M31
\citep{hrc}, a deep {\it Chandra} HRC survey of the central region of M31
\citep{kaaret}, and a {\it XMM--Newton} survey of the northern--half of M31
(Sergey Trudolyubov, private communication). The method of finding
counterparts in these X--ray catalogs was the same as that used to find
counterparts in the 2MASS PSC survey, described in Section
\ref{ir}.\footnote{For the ROSAT~PSPC survey, the
size of the X--ray source used was the $\sigma_{Pos}$ (Column~9) value of the
source given in Table 6 of \citet{rosat}. For the Chandra~ACIS survey, this
value is 1\arcsec~\citep{acis}, for the HRC~survey of the optical disk it is
10\arcsec~\citep{hrc}, for the HRC~survey of the center of M31 it is 1\arcsec~
\citep{kaaret}, for the list of {\it Chandra} sources in Globular cluster it is
5\arcsec~\citep{m31glob}, and for the {\it XMM--Newton} source list it is
3\arcsec.} From the analysis presented in Table
\ref{mwcomp}, a majority of RASS counterparts and all the counterparts from HRC
survey of the optical disk of M31 are probably coincidental. For the Second
{\it ROSAT} PSPC survey of M31 about one-third of the counterparts are
coincidental, but for the {\it Chandra} ACIS-I survey of the central region of
M31 and the {\it XMM--Newton} survey the counterparts are probably
real. Again, for these surveys we attempted to separate the real
counterparts from the false ones by using the offsets and -- for the
resolved radio sources -- the locations of the X-ray emission with
respect to the radio emission. Table \ref{xraygdctrp} lists the good
X-ray counterparts of GLG sources, as well as their Hardness Ratios
(HR1 and HR2) -- the X-ray equivalent of IR colors -- where
available. HR1 and HR2 are defined as:
\begin{equation}
HR1=\frac{H1+H2-S}{H1+H2+S}
\end{equation}
\begin{equation}
HR2=\frac{H2-H1}{H2+H1}
\end{equation}
with $S$ the count rate of photons with energy $E_{\gamma}$ between 0.1 and
0.4 KeV, $H1$ the count rate of photons with $E_{\gamma}=$0.4--0.9 KeV, and
$H2$ the count rate of photons with $E_{\gamma}=$0.9--2.0 KeV.
In addition to cross referencing the GLG source list with the aforementioned
X--ray source lists, we also looked for counterparts in images produced by the
{\it XMM--Newton} survey of the optical disk of M31 (image courtesy of Albert
Kong) and the {\it Chandra}--HRC survey of the southern disk of M31 (image
courtesy of Ben Williams). GLG025, GLG040, and GLG243 were discovered to have
faint uncatalogued counterparts in the {\it XMM--Newton} image of M31,
while no new counterparts were found in the {\it Chandra}--HRC image.
A diverse group of sources have substantial X--ray emission, from
extragalactic objects such as FRI/ FRII radio galaxies (e.g.
\citeauthor{brunetti97} \citeyear{brunetti97}, \citeauthor{harris02}
\citeyear{harris02}) and BL-Lacs \citep[e.g.][]{tagliaferri03} to galactic
objects such as SNRs, pulsars and X--ray binaries. In an attempt to
differentiate between these sources, we looked at the HR1 vs. HR2 diagrams of
these surveys/sources. Unfortunately, the boundaries between these groups are
ill--defined. However, the work of \citet{filipovic98} showed that sources
with HR2$<$0 are likely to be ``thermal'' SNRs, remnants whose X--ray emission
is dominated by the ejecta and swept-up ISM. Further classification of GLG
sources with X--ray counterparts is given in Section \ref{interesting}.
\section{Discussion}
\label{interesting}
Based on the information described above, we have attempted to determine
the nature of the radio sources detected in this observation. We placed
the GLG sources into five broad categories:
\begin{itemize}
\item Sources in M31,
\item Sources in the Milky Way,
\item ``Background'' sources (sources neither in M31 nor in the Milky Way),
\item Anomalous sources (source with spectral, morphological, and/or
multi-wavelength characteristics that defy classification), and
\item Sources for which there is insufficient information to classify them.
\end{itemize}
The first four categories will
be discussed in more detail in this Section. Based on the statistical
analysis discussed in Paper I and on population synthesis work done by
Carole Jackson (private communication), we expect that the fifth category is
dominated by background radio galaxies.
\subsection{M31 sources}
\label{m31srcs}
The discrete radio population of a normal galaxy is dominated by SNRs, HII
regions, radio pulsars, and Pulsar Wind Nebulae (PWNe). Each of these classes
of sources gives different and complementary pieces of information regarding
the properties of M31 -- SNRs, pulsars, and PWNe give information regarding
the death of past massive stars in M31 while HII regions provide information
on the current massive star population of M31. Detailed analysis of these
objects can also provide information on the ISM and cosmic ray population of
M31.
\subsubsection{Supernova Remnants, Pulsar Wind Nebulae, and Pulsars}
\label{snrs}
Type II SNRs and pulsars are both formed by the same phenomena -- the massive
explosion (supernova) that marks the death of a massive star. When the
star explodes, it produces a large shock wave that propagates through the ISM,
sweeping up material, that produces the SNR. Most of the time, a rapidly
spinning neutron star (pulsar) is produced, which in turn can generate a
highly--relativistic particle wind that creates a Pulsar Wind Nebula (PWN)
\citep{slane02}. Table \ref{snrcan} lists the 11 GLG sources that we believe
are SNR, PWN, or pulsar candidates. The columns of Table \ref{snrcan} are as
follows:
\begin{description}
\item[Column 1:] Name of source
\item[Column 2:] $R_{M31}$, as defined in Section 4.1 of Paper I, in kpc.
\item[Column 3:] Radio Spectral index $\alpha$
\item[Column 4:] Radio Spectral Curvature parameter $\varphi$
\item[Column 5:] Classification criteria, defined later in this section.
GLG011 also satisfies Criteria \#2 for a High--Frequency Variable (defined in
Section \ref{variables}), and GLG014 and GLG036 are also CESS sources
(defined in Section \ref{cess}).
\item[Column 6:] Size of source if at the distance of M31, in parsecs
\item[Column 7:] Isotropic 325~MHz Luminosity, $L_{325}$, of the source
assuming it is at the distance of M31, in units of mJy~kpc$^{2}$.
\item[Column 8:] Integrated radio luminosity of the source
($\nu=10$~MHz--$10^5$~MHz) assuming the source is at the distance of M31. This
number does not take into account any detected spectral curvature, since if
this source is a SNR or PWN any observed spectral turnover is most likely
extrinsic to the source.
\item[Column 9:] Emission measure, $EM$, of the source in pc~cm$^{-6}$, if the
spectral turnover is due to free-free emission. The EM is only given for
``spectral curvature'' sources with $\varphi > 0$ as defined in Section
\ref{specclass}. Only one SNR candidate is a ``spectral curvature'' source.
None of the PWN or pulsar candidates showed any spectral curvature.
\end{description}
A GLG source was classified as either a SNR, PWNe, or a pulsar if it was
inside the optical disk of M31 ($R_{M31} \la 27$~kpc), was either a ``U'' or
``El'' source, and had one of the following properties:
\begin{itemize}
\item A good counterpart in one of the optical SNR catalogs described in
Section \ref{optical}. (Criterion A in Table \ref{snrcan})
\item A good counterpart in the Pellet optical HII region catalog described
in Section \ref{optical} and has $\alpha<$-0.2. In the Milky Way, young SNRs
are often observed to be near HII regions. Since the progenitors of Type II
SNRs are short--lived, they often occur in regions with ongoing massive star
formation, which often have HII regions. (Criterion B in Table \ref{snrcan})
\item Is a ``spectral turnover'' source with $\varphi <$0. A SNR could have
$\varphi <$0 as a result of diffuse particle acceleration \citep{ellison}, or
if there is a flat--spectrum source, most likely a HII region or PWNe,
projected along the line of sight. In this case, the SNR would dominate the
emission observed at lower frequencies and the flat--spectrum source would
dominate the emission at higher frequencies since the SNR has a steeper
spectrum (in general, SNRs have a spectral index between
$\alpha \approx -1.0~\mbox{and}~\alpha \approx -0.3$) than these objects. A
similar effect can be seen if a radio pulsar -- which have a median spectral
index of $\alpha \approx -1.8$ \citep{lorimer95} -- lies along the sight.
However it is unlikely that a pulsar in M31 would be bright enough to outshine
a flat--spectrum source, even at 325~MHz. (Criterion C in Table \ref{snrcan})
\item A good X--ray counterpart with HR2$<$0, indicative of a thermal SNR
\citep{filipovic98}. (Criterion D in Table \ref{snrcan}) A pulsar or PWN
could also have a good X--ray counterpart with HR2$<$0, in the case the radio
emission would be dominated by the PWN or pulsar while the X--ray emission is
dominated by the thermal SNR.
\item A good counterpart in the ROSAT 2nd PSPC survey which has been classified
as a ``SNR'' on the basis of previous optical work. (Criterion E in Table
\ref{snrcan})
\end{itemize}
A source that met any of these criteria and has $-1.0<\alpha\leq-0.3$ was
classified as a SNR, while a source that met this criteria and has
$\alpha > -0.3$ was classified as a PWN, and a source that met this
criteria with $\alpha \leq -1.0$ was classified as a pulsar candidate.
While not every SNR, PWNe, or pulsar in the GLG catalog will satisfy
these requirements, this set of criteria should eliminate most, if not
all, background sources.
The distribution of these sources over the optical disk of M31, as shown in
Figure \ref{snrpos}, appears to be relatively uniform. Having identified the
SNR, PWNe, and pulsar candidates in the GLG source list, we compared their
properties to those of known objects of each type.
\paragraph{Supernova Remnants}
\label{snrcans}
The SNR candidates we identified in M31 are listed in Table
\ref{snrcan} and shown in Figure \ref{snrpics}. For comparison
purpose, we calculated the 325~MHz flux densities ($S_{325}$) and
major axis ($\theta_M$) of known SNRs in the Milky Way (MW), LMC, SMC
and M33 if at the distance of M31. For MW SNRs, we used radio
spectral information from the Green SNR catalog \citep{greensnrcat}
and distance information from Tables 1 and 2 of
\citet{case98}.\footnote{We did not use distances from Table 3 of
\citet{case98} because these distances were calculated using the
$\Sigma$--$D$ relationship, considered to be highly inaccurate.}
For LMC and SMC SNRs, we used data presented in Table 1a of
\citet{berkhuijsen86}, while the properties of M33 SNRs was obtained
from Table 4 of \citet{gordon99}. Figure \ref{snrbmajflux} shows
$S_{325}$ vs $\theta_M$ for these SNRs, as well as for the SNR
candidates in M31 identified above. The GLG SNR candidates fall
within the locus of points defined by SNRs in these other galaxies,
implying that they are most likely not background sources. However
the sample of known SNRs in the MW, LMC, SMC, and M33 are subject to a
myriad of selection effects, and there are GLG sources not classified
as SNR candidates that have similar $S_{325}$ and $\theta_M$. From
Figure \ref{snrbmajflux}, the SNR candidates detected in our survey --
if they indeed are SNRs -- appear to be the brightest, and therefore
probably youngest, SNRs in M31. These SNR candidates are brighter
than known radio SNRs in M31, e.g. \citet{lorant}, which were too
faint to be detected in our survey. Higher resolution radio imaging
as well as deeper X--ray imaging are needed to verify that they are
SNRs and to better determine their properties.
Of the SNR candidates identified, only GLG193 shows spectral turnover. If this
spectral turnover is due to free-free absorption, the implied EM towards this
source is $2.1 \times 10^5~$pc~cm$^{-6}$ -- more than order of magnitude
higher than the EM of ionized gas observed inside SNRs, e.g. Cas~A which has
an EM of $\sim 1.7 \times 10^4~$pc~cm$^{-6}$ \citep{casa}. This possibly
could be due to a foreground HII region, either in the Milky Way or in M31,
since the observed EM is similar to that of Galactic HII regions
\citep{shaver83}. However, the lack of H$\alpha$ emission in this region,
implies that any such HII region is obscured, possibly by a molecular cloud.
The amount of absorption needed to block an HII region would require so much
dust that M31 would not be visible if the obscuration was due to a Milky Way
source. Therefore, both the HII region and the obscuring molecular cloud need
to be in M31. However, this conclusion is extremely preliminary and more data
is needed in order to determine the nature of the spectral turnover observed
in this source.
\paragraph{Pulsar Wind Nebulae}
\label{pwnecans}
Pulsar Wind Nebulae (PWNe) are magnetic ``bubbles'' inflated by an
energetic wind produced by the central pulsar \citep{slane02}. The PWN
candidates we identified in M31 are listed in Table \ref{snrcan} and
shown in Figure \ref{snrpics}. The quintessential PWN
in the Milky Way is the Crab Nebula, with a radio luminosity $L_R$ of
$1.8 \times 10^{35}$~ergs~s$^{-1}$ \citep{helfand87} and is a bright emitter
at X--ray, $\gamma$--ray, and even TeV energies. Two of the three PWN
candidates detected in the GLG survey, GLG011 and GLG198, have higher radio
luminosities than the Crab Nebula -- GLG011 is $\sim70\times$~more luminous
and GLG198 is $\sim30\times$~more luminous. Both GLG011 and GLG068 have
X--ray counterparts; the X--ray luminosity\footnote{Measured between 0.3 and
7 keV.} ($L_X$) of GLG011 is $\sim2\times10^{36}$~erg~s$^{-1}$ (Albert Kong,
private communication), and the X--ray luminosity\footnote{Measured between
0.1 and 2 keV assuming a hydrogen column density
$N_{H}=9\times10^{20}~\mbox{cm}^{-2}$ and photon index $\Gamma=-2$.} of GLG068
is $\sim2.1 \times 10^{36}$~erg~s$^{-1}$ \citep{rosat}. The X--ray luminosity
of both of these sources is about an order of magnitude less than that of the
Crab Nebula, $L_{X}^{Crab} \sim 2.5\times10^{37}$~erg s$^{-1}$
\citep{helfand87}. Since the X--ray emission of a PWN reflects the current
energy injection from the pulsar while the radio emission reflects the
integrated energy input over the lifetime of the system, the ratio of $L_X$ to
$L_R$ is believed to be indicative of the system's age -- though the intrinsic
scatter in ${L_X}/{L_R}$ is large \citep{helfand87}. The ${L_X}/{L_R}$ ratio
of GLG011 and GLG068 is $\sim$1.6 and $\sim$21.6 respectively, well within the
${L_X}/{L_R}$ range of known PWNe \citep{helfand87}.
GLG011 also is a High--Frequency Variable (see Section \ref{variables} for
more information), and has a counterpart in the WENSS survey whose flux is
3.5$\sigma$ lower than the flux measured here (numbers are in Table
\ref{hfvcanlf}), but is not considered a Low--Frequency Variable using the
criteria defined in Section \ref{variables}. While PWNe are extremely dynamic
sources, they are not known to intrinsically vary by such a degree. This
variation might be an example of an ``Extreme Scattering Event'' (ESE), in
which extragalactic (outside the Milky Way) radio sources are observed to vary
greatly in amplitude at frequencies $\nu \la 2.7$~GHz. ESEs are possibly
related to large radio continuum loops in the Milky Way
(\citeauthor{fiedler94a} \citeyear{fiedler94a};~\citeauthor{fiedler94b}
\citeyear{fiedler94b}). While there are no Milky Way Galactic loops near
GLG011, the source does lie on the edge of a very large SNR/super-bubble in
M31. It is possible that the observed variability is caused by this
structure, although more information is needed. If this was the case,
than the intrinsic size of GLG011 must be very small and therefore it
could not be a PWN.
\paragraph {Pulsar Candidates}
\label{pulscans}
Of the three GLG sources identified as pulsar candidates, GLG014 and GLG036
fall under the definition of Compact Extremely Steep--Spectrum (CESS) sources,
which are discussed in detail in Section \ref{cess}. While these sources are
steep spectrum sources located inside the optical disk of M31, they
are most likely not radio pulsars in M31 because, if so, they would be
several orders of magnitude more luminous than any known galactic
pulsar. In addition, GLG036 and GLG205 are slightly extended, which
one would not expect from a pulsar. Additionally, GLG205 lies far off
the optical disk of M31, not expected for a radio pulsar in M31. We
conclude that we did not detect any pulsars in M31, not surprising
given the sensitivity of our survey.
\subsubsection{HII Regions}
\label{hiiregs}
HII regions, the regions of ionized material that surround new O or B stars,
are a major component of the radio emission of a normal galaxy. The radio
spectrum emission of an HII region is flat at higher radio frequencies
($\alpha \sim -0.1$) but turns over a lower frequencies as a result of
free--free absorption. For roughly half of all Milky Way HII regions this
turnover occurs at $\nu_t>325$~MHz \citep{shaver83}, so an HII region might
appear as a flat--spectrum source or a spectral turnover source in our survey.
To account for these two possibilities, a GLG source was classified as a HII
region if it was inside the optical disk of M31 ($R_{M31} \leq 27~\mbox{kpc}$),
not a Low--Frequency Variable or High--Frequency Variable (as defined in
Section \ref{variables}), and was either:
\begin{itemize}
\item The counterpart of an optical HII region and a flat spectrum source,
\item A spectral turnover source with a flat spectrum, as defined in Section
\ref{specclass}, that has not been identified as a SNR, or
\item A flat spectrum source, as defined in Section \ref{specclass}, with no
spectral turnover ($\mid$$\varphi$$\mid$$\leq\sigma_{\varphi}$) that has not
been identified as a SNR.
\end{itemize}
Two sources, GLG016 and GLG088 satisfy the second criteria, while none satisfy
the first or third. Unfortunately, a flat--spectrum radio galaxy (FSRG) would
also satisfy the second criteria. The presence of Far-IR emission,
while not seen in all HII regions, would indicate that a source is a
HII region since FSRGs are not strong FIR emitters. However, as
discussed in Section \ref{fir}, it is not currently possible to
reliably associate FIR objects with the radio sources detected in this
survey. As a result, we did not detect any strong HII region
candidates in the GLG source list. Since most Milky Way HII regions
when placed at the distance of M31 are much fainter than the detection
limit of this survey, this is not unexpected.
\subsection{Milky Way Sources}
\label{mwsrcs}
We believe that we have found two types of Milky Way radio sources in the
GLG catalog, radio stars and Planetary Nebulae. Radio emission has previously
been detected from a wide variety of stars: from pre-main sequence K--stars
\citep{kutner86} to O-B stars \citep{rauw02}. The cause of the radio emission
is equally varied -- the radio emission of O-B stars is expected to be
dominated by free-free emission from stellar winds \citep{rauw02}, while radio
emission from pre-main sequence stars is believed to be the result of
chromospheric emission \citep{kutner86}. In addition, radio emission has been
detected from binary systems \citep{richards93}, and possibly from stars
orbiting a microquasar \citep{marti98}.
A GLG source was categorized as a radio star if it had a bright optical/IR
counterpart or had IR colors consistent with that of a star. Three GLG sources
met this criteria: GLG023 -- which has an $I\approx13$ counterpart in
the MagOpt catalog while most sources in this catalog have $I=18-20$, GLG097
-- which has the IR colors of a star, and GLG116 -- which has a bright PSC
counterpart ($K\approx5$ as opposed to $K\sim13-15$). R--band images of these
sources is shown in Figure \ref{redgd}. All three have non-thermal radio
spectrum ($\alpha$=$-0.89$, $-0.92$, and $-0.79$,~respectively), consistent
with the processes mentioned above. GLG023 has an optical counterpart in the
\citet{berk88} catalog of bright stars in the field of M31, which has a B
magnitude of 15.04$\pm$0.19. This is similar to the B magnitude of its
counterpart in the MagOpt catalog ($14.9\pm0.04$), implying the source is not
variable and possibly ruling out a binary explanation for the radio emission.
The spectral class of this star is unknown. GLG097 is also in a catalog of
reference stars in the direction of M31 \citep{shokin98}, and the spectral
class of this star is also unknown. GLG116 is likely the radio
counterpart of SAO 36591, a K0 star \citep{roeser88}. Radio emission from K0
type stars have often been associated with Algol--Type Binaries, which often
are X--ray sources and typically have radio emission on the order of a few mJys
at $\nu \sim 5$~GHz \citep{richards93}. GLG116 has no X--ray counterpart,
and has $S_{5400}=22$~mJy, brighter than that of most observed Algol-type
binaries but not all, e.g Algol itself \citep{richards93}.
In addition to detecting radio stars, we believe that GLG347 may be a Planetary
Nebula in the Milky Way. GLG347 is located in the SE corner
of our FOV and has counterparts in the 5th Cambridge Survey (5C3.152), 3rd
Bologna Survey (B3 0043+398), and the Miyun 232 MHz survey
(MY 004309.7+394955.8). On the basis of these detections, it has previously
been identified as a radio galaxy \citep{simbad}. However, as shown in Figure
\ref{glg347pic}, located to the SE of this source is a region of increased
emission not classified as a source by $\mathcal{SFIND}$. Taken together,
these regions suggest a shell of diameter $\sim$45\arcsec~and a flux density
$S_{325}$=18.5~mJy. This is smaller and fainter than most galactic SNRs
\citep{greensnrcat}, and at $b \approx -23^{\circ}$, much further off
the galactic plane that most Milky Way SNRs, so we do not believe that is a
SNR. However, this source has a diameter, morphology and flux density similar
to that of planetary nebulae, and this is the basis of its classification
\citep{pneb}. There is an IR source in the center of the radio emission,
shown in Figure \ref{glg347pic}, with the IR colors of a star which could be
the source of the Planetary Nebula.
\subsection{Background Sources}
\label{extragal}
In this section, a ``background'' source is one neither in M31 or in the Milky
Way. The population of background sources is dominated by galaxies whose
radio emission is powered by a central AGN (primarily FRI and FRII radio
galaxies) -- particularly sources with $S_{325} > 10$~mJy, and galaxies
whose radio emission is dominated by synchrotron emission from relativistic
electrons and free--free emission from HII regions (normal galaxies,
\citeauthor{condon} \citeyear{condon}) -- expected to dominate at
$S_{325} < 10~\mbox{mJy}$ (Carole Jackson, private communication 2002).
Identifying these sources is crucial in determining whether or not a source is
in M31 or the Milky Way, since, as shown in Section 4 of Paper 1,
``background'' sources dominate the GLG source list.
Even though this class of sources is not the primary motivation nor the primary
focus of this paper, many of these are interesting objects: we detected a BL
Lac, a galaxy merger, several High--Frequency and Low--Frequency Variable
sources, and have identified a Giant Radio Galaxy (GRG) candidate and several
High-z Radio Galaxy (HzRG) candidates. This section will first briefly
describe the sources which we believe are typical background sources (FRI,
FRII, and normal galaxies) and then discuss the more exotic background sources
in detail.
\subsubsection{FRI \& FRII Radio Galaxies and Radio Jets}
\label{agn}
The most common source with $S_{325} \stackrel{>}{\sim} 5$~mJy are galaxies
whose radio emission is dominated by a central super-massive black hole
(Active Galactic Nuclei; AGN). These sources typically fall into three broad
morphological categories:
\begin{description}
\item{\bf{FRI:}} A galaxy whose radio emission appears core dominated
\citep{fr}. A GLG source was categorized as an FRI if it was either:
\begin{itemize}
\item resolved and the radio emission is dominated by a single, central
component (Criterion A in Table \ref{frIcan}). A resolved PWN may have
such a morphology, but we do not expect to be able to resolve a PWNe in M31.
\item an unresolved or elongated source with a counterpart in the ROSAT 2nd
PSPC survey categorized as a ``Galaxy" source which does not have a good IR
counterpart (Criterion B in Table \ref{frIcan}). As shown in Section
\ref{ir}, the detected IR emission from GLG sources is too bright for the
source to be an FRI/FRII explanation.
\item an unresolved or elongated source outside the optical disk of M31
with an X--ray counterpart and no IR counterparts (Criterion C in Table
\ref{frIcan}).
\end{itemize}
As mentioned in Section \ref{ir}, no GLG source with a counterpart in the
2MASS catalogs had a K--magnitude consistent with that of FRI. Table
\ref{frIcan} lists the nine GLG sources determined to be FRIs. One of these
is also a High--Frequency Variable, as defined in Section \ref{variables}, and
two are also High-z Radio Galaxy candidates, as defined in Section \ref{hzrgs}.
The IR counterpart of GLG122 is a star, and most likely unrelated to the
radio source. All but two of these sources have X--ray counterparts. This
Table is by no means complete since most likely a vast majority of unresolved
sources detected outside of the optical disk of M31 are FRIs, it only lists
those sources for which there is additional evidence that they are FRIs.
\item{\bf{FRII:}} A galaxy whose radio emission is dominated by radio jets
emitted from the central black hole, and often more luminous than FRIs
\citep{fr}. FRIIs may or may not have a ``core'' component that corresponds
to the central AGN. A GLG source was categorized as an FRII if it was
resolved and had the morphology described above. Table \ref{frIIcan} lists
the twenty-three GLG sources determined to be FRIIs. GLG015 and GLG078 are
also classified as anomalous sources because of H$\alpha$/IR emission detected
near these sources. GLG078, GLG129, GLG187, and GLG296 are also HzRG
candidates (Section \ref{hzrgs}), and GLG187, GLG296, and GLG358 are
Low--Frequency Variables (Section \ref{variables}). GLG220 and GLG212 form a
radio triple, shown in Figure \ref{radtrip}. Seven FRIIs have 2MASS
counterparts, and of these four have the IR colors of a normal galaxy. For
these four sources we believe that the radio emission is from the central AGN
but the IR emission is from the rest of the host galaxy. The IR counterparts
of the other three are most likely coincidental.
\item{{\bf Radio Jets:}} Radio jets are most commonly emitted from AGN, but
tend to be larger in angular size than FRIIs which is why they are in a
separate category. A GLG source was classified as having radio jets purely on
a morphological basis. Figure \ref{jetspic} shows the five radio jets
discovered in this survey: GLG031/GLG033, GLG045/GLG051, GLG054/GLG059, and
GLG266/GLG269, and GLG270/GLG271/GLG275. Between GLG031 and GLG033 is an
elongated 20 cm source in the Braun Catalog (Braun 3), not seen in our 325 MHz
image, oriented perpendicular to GLG031/GLG033, that possibly is the host
galaxy. The bright point sources
in GLG031 and GLG033 are most likely hot spots in the jet \citep{carilli98}.
GLG045/GLG051 is a one--sided jet, where GLG045 is the central core while
GLG051 is the jet itself. The bright point source
in GLG051 is most likely the hot spot produced at the termination point of the
jet \citep{carilli98}. In the case of GLG054/GLG059, GLG054 has the morphology
of a radio jet, and its extended emission is oriented towards GLG059. The
GLG266/GLG269 system also does not have the classical morphology of a radio
jet -- GLG269 is a steep spectrum ($\alpha \sim -2$) complex source with the
morphology of a one-sided jet, while GLG266 is a HzRG candidate (Section
\ref{hzrgs}) along the jet axis of GLG269. The GLG270/GLG271/GLG275 is a
classic radio jet, with GLG271 most likely the central engine that produces
GLG270 and GLG275.
\end{description}
\subsubsection{Normal Galaxies}
\label{galaxies}
While Section \ref{agn} identified background AGN, this section identifies
background normal galaxies -- those whose radio emission is not dominated by
a central AGN but from synchrotron emission emitted by relativistic electrons
(``cosmic rays'') in SNRs and in diffuse gas interacting with the galaxy's
magnetic field and free--free
emission produced by HII regions. Normal radio galaxies have optical
morphologies across the entire Hubble diagram \citep{condon}. A GLG source
was determined to be a normal galaxy if it satisfied at least one of the
following criteria:
\begin{description}
\item[Criterion I:] Has an 2MASS PSC counterpart with the IR colors of
a normal galaxy (as discussed in Section \ref{ir}).
\item[Criterion II:] Has a counterpart in the 2MASS ESC catalog.
\item[Criterion III:] Has a counterpart in the {\it ROSAT} 2nd PSPC
survey that was classified as a galaxy and in either the 2MASS ESC
or 2MASS PSC.
\end{description}
Table \ref{rgcan} lists the GLG sources identified as radio
counterparts to normal galaxies. As shown in this table, several of
these sources fall into other categories. In these cases, we believe that the
observed radio and IR emission come from different parts of the source
-- the radio emission from the central AGN and the IR emission from
the rest of the galaxy. GLG253 is located in the center of a z=0.293
galaxy cluster identified by the {\it XMM--Newton} satellite \citep{galclust}.
Two of these sources -- GLG050 and GLG203 -- are ``spectral curvature'' sources
as defined in Section \ref{specclass}. For GLG203 there is no evidence for
ionized material along the line of sight, so the observed spectral turnover is
most likely intrinsic. Intrinsic low--frequency spectral turnover has been
seen in many normal galaxies, and can be explained by a clumpy, non--thermal
plasma with electron temperatures $200~{\rm K} \leq T_e \leq1500~{\rm K}$ and
electron density $n_e=0.07-0.10$~cm$^{-3}$ \citep{im90}. For a typical galaxy,
the optical depth at 325~MHz of such a medium ($n_e=0.1$~cm$^{-3}$ and
$T_e=100$~K) is $\sim$0.002, an order of magnitude lower than that observed
in GLG203. However, the EM of these galaxies is similar to that of the
starburst galaxy M82 (\citeauthor{allen} \citeyear{allen};~\citeauthor{wills}
\citeyear{wills}). The cause of the negative spectral curvature observed in
GLG050 is unclear.
\subsubsection{High-z Radio Galaxies}
\label{hzrgs}
Three separate criteria, all sensitive to different kinds of High-z Radio
Galaxies (HzRGs), were used to identify HzRG candidates. The first criterion
used the fact that Ultra--Steep Spectrum (USS) sources, defined here to be
$\alpha \leq -1.3$, have been determined to be efficient tracers of HzRGs
(\citeauthor{hzrg} \citeyear{hzrg}; \citeauthor{pedani03} \citeyear{pedani03}).
This is believed to be an intrinsic property of HzRGs since the generic
spectrum of a radio galaxy lobe, assumed to behave like an optically thin
synchrotron source, steepens in spectral index by $\sim$0.5 around the
``bend'' frequency $\nu_{b}$ as one goes from low to high frequencies due to
synchrotron losses. The observed $\nu_{b}$ is proportional to $({1+z})^{-1}$,
so at higher redshifts the bend--frequency occurs at lower frequencies, causing
the object to have a steep spectral index when observed at frequencies above
the redshifted bend frequency. Inverse--Compton scattering off the CMB also
contributes to the steeping of the spectrum of HzRGs, since losses due to this
process increase as $(1+z)^4$ \citep{krolik91}. Using this information, a
source was identified as a HzRG if it was resolved with the morphology of an
FRI or an FRII and has $\alpha \leq -1.3$. The four sources that met this
``USS'' criterion are listed in Table \ref{hzrgcan}, implying a source density
of $\sim$1700~sr$^{-1}$. This is more than 10$\times$ higher than that
measured in a comparison between the WENSS and NVSS surveys \citep{hzrg}.
The second criterion is designed to detect two subclasses of potential HzRGs
which have a more compact morphology -- Gigahertz Peaked-Spectrum (GPS)
sources and Compact Steep Spectrum (CSS) sources \citep{kaplan00}. Both
classes are extremely powerful, compact\footnote{GPS sources are $\la$1~kpc in
size while CSS generally are $\sim$1--20~kpc in size \citep{odea98}.} sources
with radio spectra that turnover at low--frequencies. For GPS sources,
this turnover occurs at $\nu_t \sim$1~GHz, while for CSS sources this turnover
generally occurs at $\nu_t \la$500~MHz \citep{odea98}. At frequencies above
$\nu_t$, the spectral index of GPS sources is typically between
$\alpha \approx -0.5$ and $\alpha \approx -0.9$ \citep{odea98}, while for
CSS sources it is between $\alpha \approx -0.5$ and $\alpha \approx -1.5$
\citep{fanti90}. The spectral turnover is most likely due to synchrotron
self--absorption, but free--free absorption may also play a role
\citep{odea98}. A GLG source was determined to be a CSS or a GPS source
if it is a ``spectral curvature'' source with $\varphi >$~0 and spectral index
$\alpha <$-0.5 that has not been classified as a ``normal'' galaxy (see
Section \ref{galaxies} for details). Since only measurements at three
frequencies are available, we are unable to distinguish between CSS and GPS
sources, though a visual inspection of the spectra of these sources implies
that most of them are CSS sources. The sources that met this
``CSS/GPS'' criterion are listed in Table \ref{hzrgcan}.
High--resolution ($\theta_{res} < 1$\arcsec) VLA images of CSS sources have
shown that they are often doubles, triples, or even have jets associated with
them (\citeauthor{fanti90} \citeyear{fanti90}; \citeauthor{athreya97}
\citeyear{athreya97}). CSS sources with extended emission almost always have
cores with a flat or inverted spectrum ($-0.5<\alpha<1.0$) and extended
emission with a spectral index of $-1.0 < \alpha < -0.7$ \citep{athreya97}.
If these two components are of comparable strength than it is possible that at
higher frequencies ($\nu \sim$1~GHz) the ``flat--spectrum'' core dominates
while at lower frequencies ($\nu \sim$300~MHz) the steeper--spectrum extended
emission dominates, resulting in a spectrum with $\varphi <$0. CSS sources of
this nature were selected from the GLG sample by requiring that the source be
a ``spectral curvature'' source outside the optical disk of M31 with
$\varphi <$~0. The sources that met this ``CSS'' criterion are also
listed in Table \ref{hzrgcan}.
Table \ref{hzrgcan} lists the GLG sources that met at least one of these
criteria. GLG296 was previously identified as a HzRG candidate based on its
properties in the WENSS and NVSS catalogs \citep{hzrg}, and several
of these objects fall under other classifications as well. Follow up
radio and optical observations are needed to determine if these
objects are truly High-z Radio Galaxies.
\subsubsection{Galaxy Merger}
\label{glg247}
Galaxy mergers are powerful radio sources because the collision triggers star
formation, which in turn creates radio--bright sources such as HII regions and
SNRs. Radio observations of galaxy mergers, such as the ``Antennae'' galaxies
(NGC 4038/4039), have discovered that the radio emission from these sources is
concentrated in the region of brightest optical emission \citep{neff00}.
GLG247 is classified as a galaxy merger as a result of its radio morphology,
as well as its relationship with its IR, optical, and X--ray counterparts,
shown in Figure \ref{glg247pic}. GLG247 has a relatively steep spectrum,
$\alpha=-0.87$, so it is possible that the radio population is dominated by
SNRs like that of the ``Antennae'' galaxies \citep{neff00}, but higher
resolution radio images are needed to determine the source of the radio
emission. GLG247 could also be an ``X--shaped'' radio galaxy \citep{leahy84}.
\subsubsection{BL Lacs}
\label{bllac}
BL Lacs -- believed to be the radio emission from a jet aligned with our
line-of sight emitted by a super-massive black hole -- are some of the most
powerful radio sources known and have been detected throughout the EM spectrum,
from the radio up to TeV energies \citep{tagliaferri03}. The X--ray
counterpart of GLG105
was previously identified as a BL Lac candidate in the Einstein Slew Survey
\citep{perlman96}. In our survey, GLG105 is a a complex source with a
cometary morphology (shown in Figure \ref{glg105pic}) and a radio spectrum
with
spectral index $\alpha=-0.74\pm0.10$ and no curvature, but shows
variability at high--frequencies (see Section \ref{variables} for more
details). Similar variability has been observed in other BL~Lacs (e.g.
\citeauthor{padrielli87} \citeyear{padrielli87}). GLG105 has optical, IR, and
X--ray counterparts,
and its broadband SED (radio to optical) is shown in Figure \ref{glg105pic}.
This spectrum does not fit general spectral models of BL~Lacs, in which a
power--law extrapolation of the radio emission over-predicts
the observed optical/IR emission \citep{bllacsed}. This discrepancy can be
explained by variability between the different epochs of the observation, but
is interesting
nevertheless. Additionally, H$\alpha$ emission has been detected near the
``tail'' of GLG105 (shown in Figure \ref{glg105pic}), uncharacteristic
of BL~Lac objects. However, this source has a $log(f_x/f_V)=1.6$, consistent
with that of other known BL~Lacs \citep{maccacaro88}. Since GLG105 is located
within the optical disk of M31, it is possible that it is a very luminous
($L_X=2.4 \times 10^{38}$~ergs s$^{-1}$; \citeauthor{rosat} \citeyear{rosat})
X--ray source in M31, but is extremely unlikely due to the rarity of such
objects.
\subsubsection{Giant Radio Galaxies}
\label{grg}
As mentioned in Section \ref{agn}, some central AGN are known to produce
relativistic jets which -- if powerful enough -- inflate a ``cocoon'' that
expands first into the ISM of the host galaxy, and then into the surrounding
Intergalactic Galactic Medium (IGM). The magnetic fields in these cocoons
produce synchrotron emission, which is observed in the form of radio lobes. A
Giant Radio Galaxy (GRG) is a radio galaxy whose lobes span a projected
distance of $>1$~Mpc. Since radio lobes expand with time, GRGs must be very
old ($>10^8$ years), and located in under-dense regions relative to other,
more compact, FRIs and FRIIs. GRGs are, as a result of their large size, very
powerful probes of the IGM \citep{schoenmakers01}.
We believe that GLG242 and GLG260, shown in Figure \ref{grgpic}, located
4\arcmin~away from each other, together comprise a Giant Radio Galaxy
(GRG). (According to the search criteria of \citet{schoenmakers01}, the
radio lobes need to span 5\arcmin~in order for the source to be considered a
GRG candidate. However, since the redshift of this object is unknown, the
physical distance between these two objects could be $>1$~Mpc.) GLG242 is a
complex source which composed of a point source with an attached region of
extended emission oriented in the general direction of GLG260, an unresolved
source in our nomenclature but which, under close examination, has extended
emission in the direction of GLG242. GLG242 has a flux density of
$S_{325}=254$~mJy and a spectral index $\alpha=-1.07\pm0.42$, while GLG260 is
somewhat weaker ($S_{325}=176$~mJy) and has a flatter spectrum
($\alpha=-0.81\pm0.031$). These differences are similar to that seen in other
GRGs (Table 3 of \citeauthor{schoenmakers00} \citeyear{schoenmakers00}). When
combined, these sources have a total flux density of $S_{325}\sim$430~mJy --
well within the flux range of known GRGs (Figure 1a of
\citeauthor{schoenmakers01} \citeyear{schoenmakers01}), and a composite
spectral index $\alpha_{325}^{1400}=-0.94$ -- steeper than all but a handful
of known GRGs (Table A4 of \citeauthor{schoenmakers01}
\citeyear{schoenmakers01}).
A more detailed analysis of these sources is hampered by the lack of
information about their host galaxy. As seen in the 325~MHz and the 1.4~GHz
(NVSS) radio image of the GLG242/GLG260 (shown in Figure \ref{grgpic}) system,
there is no radio source between them. While most host galaxies of GRGs have
$S_{325}>3$~mJy (the 5$\sigma$ RMS limit in the area between these sources),
several host galaxies of GRGs have 325~MHz flux densities below this limit so
it is not inconceivable that this is the case for GLG242/GLG260 (Table
A3 of \citeauthor{schoenmakers01} \citeyear{schoenmakers01}). In addition,
there is no obvious optical/IR candidate for the host galaxy. There is an IR
point source located $\sim$20\arcsec~south of the center of GLG242, but
this source has the IR colors of a star (see Section \ref{ir} for details), so
it is probably unrelated. GLG260 has no counterparts in the Far-IR, IR,
optical, or X--ray catalogs/images searched. A J--band image of GLG242/GLG260
(also shown in Figure \ref{grgpic}) reveals the existence of a faint 2MASS
source (2MASS:00465464+4033383) on the line that connects GLG242 and GLG260,
but based on its IR-colors and its optical properties in the USNO B-1.0
catalog \citep{usno}, it is most likely a star.
\subsubsection{Variable Sources}
\label{variables}
Flux variability -- whether at low ($\nu<1$~GHz) or high ($\nu\geq1$~GHz)
frequencies, or on short ($t_{var}<$1~yr) or long ($t_{var}\geq$1~yr)
timescales -- provides a wealth of information about the source itself as well
as on the intervening ISM. Studies of variable radio sources have that they
typically vary on timescales longer than a year \citep{gregorini86}, and that
source which vary at low frequencies ($\nu \la 1~$GHz) have different
properties (spectral index, galactic latitude, variability magnitude) than
those which vary at high frequencies ($\nu \ga 1~$GHz)
(\citeauthor{cawthorne85} \citeyear{cawthorne85}; \citeauthor{gregorini86}
\citeyear{gregorini86}; \citeauthor{rys1990} \citeyear{rys1990}).
It is currently believed that variability at low and high frequencies are
caused by two separate processes:
\begin{itemize}
\item A process intrinsic to the source -- which is almost always an active
galactic nucleus (AGN) of some sort -- that dominates at higher frequencies
but can cause variability at lower frequencies.
\item A process extrinsic to the source, caused by material in the Milky Way,
that dominates variability at low frequencies -- Interstellar Scintillation
(ISS).
\end{itemize}
This intrinsic process is believed to occur in a majority, if not all, flat
spectrum sources -- most likely QSOs or BL Lacs \citep{rys1990} -- with
$S_{1400} \geq 10$~mJy \citep{carilli2003}. The extrinsic process is believed
to be Refractive Interstellar Scintillation (RISS), the result of radio waves
from the extragalactic source passing through a turbulent region of the ISM
\citep{rickett84}, is also used to explain slow flux variations observed in
Galactic pulsars \citep{cordes98}.
Using data available from the WENSS survey, we searched for low-frequency
variables (LFVs) in the GLG source list. Since the WENSS source list was
derived from observations conducted in 1991--1993 \citep{wenss} while the GLG
sources were observed in 2000, we are sensitive to LFVs that vary on a
$\sim$7--10 year timescale. In addition, since the WENSS FOV includes the
entire GLG FOV, we can detect LFVs across the entire FOV and see if the ISM of
M31 plays a role. A GLG source was identified as LFV if it satisfied one of
the following two criteria:
\begin{description}
\item [Criterion 1:] The GLG source had a good match in the WENSS catalog, and
had:
\begin{equation}
\Delta S_{325} \geq 5 \times \sqrt{(\sigma_S^{GLG})^2+(\sigma_S^{WENSS})^2}
\end{equation}
where $\Delta S_{325}$ is the absolute value of the difference in 325 MHz
flux densities between the two sources, $S_{325}^{GLG}$ is the measured flux
density of the source in the GLG catalog, $S_{325}^{WENSS}$ is the measured
flux density of the corresponding source in the WENSS catalog,
$\sigma_S^{GLG}$ is the error of $S_{325}^{GLG}$, and $\sigma_S^{WENSS}$ is
the error of $S_{325}^{WENSS}$
\item[Criterion 2:] The GLG source had no counterpart in the WENSS catalog,
and:
\begin{equation}
S_{325}^{GLG} \geq 30~\mbox{mJy} + 5\times \sigma_S^{GLG}
\end{equation}
where $S_{325}^{GLG}$ and $\sigma_S^{GLG}$ are defined as above. This criteria
looks for GLG sources that should have been, but were not, detected in the
WENSS survey, which has a completeness limit of 30~mJy \citep{wenss}.
\end{description}
Table \ref{lfvcan} lists the 10 GLG sources that were classified as LFV, and
for comparison purposes, a non-LFV GLG source (last line of Table). The
columns are as follows:
\begin{description}
\item[Column 1:] GLG name
\item[Column 2:] Morphology Type
\item[Column 3:] $R_{M31}$, as defined in Section 4.1 in Paper I, in kpc.
\item[Column 4:] Radio Spectral Index $\alpha$, calculated using Equation 7 in
Paper I.
\item[Column 5:] Spectral curvature parameter $\varphi$, calculated using
Equation 7 in Paper I.
\item[Column 6:] Name of WENSS counterpart.
\item[Column 7:] Distance between GLG source and WENSS counterpart in
arcseconds.
\item[Column 8:] $S_{325}^{GLG}$, as defined above, in mJy.
\item[Column 9:] $S_{325}^{WENSS}$, as defined above, in mJy
\item[Column 10:] Variation amplitude
$\frac{\Delta S_{325}}{\overline{S_{325}}}$, where $\Delta S_{325}$ is defined
above and $\overline{S_{325}}={S^{GLG}_{325}+S^{WENSS}_{325}}$/2
\end{description}
Table \ref{lfvcanhf} lists the 1.4 GHz properties of these LFVs. The columns
are as follows:
\begin{description}
\item[Column 1:] GLG name
\item[Column 2:] Name of 1.4 GHz counterparts
\item[Column 3:] Distance from GLG source to 1.4 GHz counterparts,
in arcseconds.
\item[Column 4:] Flux density of 1.4 GHz counterparts in mJy.
\item[Column 5:] $\overline{S_{1400}}$, the average flux density of the
1.4~GHz counterparts, in mJy.
\item[Column 6:] $\frac{\Delta S_{1400}}{\overline{S_{1400}}}$, where
$\Delta S_{1400}$ is the difference in flux density of the 1.4 GHz
counterparts, and $\overline{S_{1400}}$ is defined below.
\end{description}
A total of 95 GLG sources\footnote{Extended sources with no radio
counterparts were excluded from this sample since they are, most likely, false
detections.} either had a good counterpart
in the WENSS catalog, or had $S_{325}^{GLG} \geq$30~mJy with no counterparts
in the WENSS catalog. Of these, $\sim$11\% of them are LFVs, a significantly
higher percentage than that observed in previous LFVs searches. Additionally,
the observed variability is higher than in previous studies
\citep{stinebring00}, especially given the high Galactic latitude of M31
($|b|=22^{\circ}$). In addition, a large number of the LFVs in the GLG sample
are ``steep'' spectrum ($\alpha \leq -0.5$) and/or resolved sources, different
from previous LFV studies \citep{riley93}. For five of the steep--spectrum
LFVs it was possible to calculate $\varphi$, and two of them (GLG200 and
GLG340) show no spectral curvature -- unlike previously detected steep
spectrum LFVs \citep{riley93}. However, it is possible that lower frequency
observations of these two sources would show spectral flattening. Of the three
resolved LFVs -- two of which are FRIIs (GLG187 and GLG296) and one
is an Extended source (GLG009) -- only GLG009 and GLG296 have sufficient
information at 1400~MHz to determine if they are high-frequency variable (HFVs)
or have spectral curvature. GLG009 is a HFV, surprising because most HFVs
are compact \citep{rys1990} and there doesn't appear to be a compact source
inside GLG009. GLG296 is not a HFV, but does have spectral turnover similar
to that previously detected in LFVs. While the larger resolution of
the WENSS survey can explain a higher flux density in this survey because the
recorded flux is inflated by extended emission in the region resolved out in
the GLG survey, over half of the detected LFVs have higher fluxes in the GLG
survey than the WENSS survey. However, the fluxes of GLG009 and GLG296 is
lower in the GLG survey than in the WENSS survey so these source might not be
variables.
By using the fluxes of a GLG source's 1.4 GHz counterparts, we were able to
determine if a source is a HFV. The criteria were:
\begin{description}
\item[Criterion 1:] The GLG sources has two good 1400~MHz counterparts, and
the $S_{1400}$ of the two counterparts differed by more than
5$\sigma_{\rm diff}$, where $\sigma_{\rm diff}$ is the error in the difference.
\item[Criterion 2:] The GLG sources has more than two good 1400~MHz
counterparts, and for at least one of these counterparts:
\begin{equation}
|S_{1400}-\overline{S_{1400}}| \geq 5\times \sqrt{(\sigma_S)^2 +
(\sigma_{\bar{S}})^2}
\end{equation}
where $\sigma_S$ is the error in $S_{1400}$ of the 1400~MHz counterpart in
question and $\sigma_{\overline{S}}$ is the error of $\overline{S_{1400}}$, the
weighted average of all the 1400~MHz flux densities.
\end{description}
Table \ref{hfvcan} for comparison lists the six HFVs identified in the GLG
source list, and a
non-HFV GLG source (last line of Table). The columns are as follows:
\begin{description}
\item[Column 1:] Name of GLG source
\item[Column 2:] Radio spectral index, $\alpha$, as calculated in Section
4.3 in Paper I.
\item[Column 3:] Spectral curvature parameter, $\varphi$, as calculated in
Section 4.3 in Paper I.
\item[Column 4:] Names of 1.4 GHz counterparts.
\item[Column 5:] Distance from 1.4 GHz counterparts to GLG source, in
arcseconds.
\item[Column 6:] Flux density of 1.4 GHz counterparts, $S_{1400}$, in mJy.
\item[Column 7:] Variation amplitude
$\frac{\Delta S_{1400}}{\overline{S_{1400}}}$, where $\Delta S_{1400}$ is
the difference between the minimum and maximum $S_{1400}$ measurements and
$\overline{S_{1400}}$ is defined above.
\end{description}
Table \ref{hfvcanlf} lists the 325~MHz properties of these HFVs. The
columns are as follows:
\begin{description}
\item[Column 1:] Name of GLG source.
\item[Column 2:] Morphology type.
\item[Column 3:] $R_{M31}$, defined in Section 4.1 in Paper I, in kpc.
\item[Column 4:] Name of WENSS counterpart.
\item[Column 5:] Distance between GLG source and WENSS counterpart in
arcseconds.
\item[Column 6:] Flux density of source in GLG observation, $S_{325}^{GLG}$ in
mJy.
\item[Column 7:] Flux density of WENSS counterpart, $S_{325}^{WENSS}$ in mJy.
\item[Column 8:] Average 325~MHz flux density of source, $\overline{S_{325}}$,
in mJy.
\item[Column 9:] 325~MHz Variation amplitude,
$\frac{\Delta S_{325}}{\overline{S_{325}}}$, where $\Delta S_{325}$ and
$\overline{S_{325}}$ are defined above.
\end{description}
There are 120 sources in the GLG source list with two or more good 1400~MHz
counterparts, and only six (5\%) of them fit the above criteria for a HFV.
This percentage is slightly lower than previous searches (e.g.
\citeauthor{rys1990} \citeyear{rys1990};~\citeauthor{carilli2003}
\citeyear{carilli2003}), but the amount of variation observed is similar
\citep{rys1990}. As mentioned earlier, only GLG009 is both a LFV and HFV.
Two of the six HFVs were resolved -- GLG009, and the BL Lac candidate GLG105
which was described in Section \ref{bllac}. Of the four unresolved
HFVs, only one of them is a flat--spectrum source (GLG011). Only one of the
steep--spectrum HFVs (GLG004) showed any appreciable spectral turnover, and
only at the 2$\sigma$ level.
To determine if M31 has an effect on low--frequency and/or high--frequency
variability, we plotted $\frac{\Delta S}{\overline{S}}$ at both 325 and 1400
MHz as a function of $R_{M31}$, seen in Figure \ref{varpics}. There is no
observed correlation between M31 and flux variation at either 325 or 1400 MHz.
To summarize, in the GLG source list we detected 10 LFVs and seven HFVs. A
higher percentage of LFVs were found in the GLG survey than in other LFV
searches, most likely the result of this survey's sensitivity to fainter
sources. However, the variability is higher than seen in previous LFV surveys,
and much higher than expected for the Galactic latitude of M31. Many of these
have steep spectra, unlike in other surveys, and several were resolved.
However, the HFVs detected in the GLG survey have similar properties seen in
other HFV searches. Lastly, flux density variations at both 325 and 1400 MHz
appear to be uncorrelated with structures in M31.
\subsection{``Anomalous'' Sources}
\label{anomalous}
In this section, we discuss sources with spectral, morphological, and/or
multi-wavelength properties we do not understand.
\subsubsection{Compact, Extremely Steep Spectrum Sources}
\label{cess}
Table \ref{cesscan} lists the 22 GLG sources which we consider to be Compact,
Extremely Steep Spectrum (CESS) sources -- sources that are either ``U'' or
``El'' in our radio images, not classified as HFVs, LFVs (see Section
\ref{variables} for details), or as HzRGs (see Section \ref{hzrgs} for
details), and have a spectral index $\alpha \leq -1.6$. Images of these
sources are shown in Figure \ref{cesspics}. Only four of these sources have
been detected at other wavelengths -- GLG014 has an X--ray counterpart, GLG036
has an optical counterpart, and both GLG140 and GLG149 have IR counterparts,
though these may be coincidental. This survey is not the first time that such
objects have been detected -- a comparison of the NVSS survey and the 365~MHz
Texas survey catalogs found 74 sources with $\alpha \leq$-1.5 and four sources
with spectral index $\alpha <$-2.5 \citep{kaplan00}. This search covered 10
sr of sky, a much larger region than our observation, but only included sources
with $S_{365} \geq$200~mJy \citep{kaplan00}, a much higher cutoff than this
survey ($S_{325} \sim$3~mJy) -- so the higher density of CESS sources detected
in this survey could be a result of our sensitivity to fainter sources.
Two possible explanations for the observed steep--spectra observed are:
\begin{description}
\item[Option \#1:] These sources have flatter spectrum than those calculated
here, and their apparent spectrum is steep because of some intrinsic or
extrinsic effect.
\item[Option \#2:] These sources have intrinsically steep spectra.
\end{description}
These options are not mutually exclusive, and neither one necessarily explains
all the CESS sources.
Option \#1 essentially implies that the source is variable -- either
intrinsically
or extrinsically. As discussed in Section \ref{variables}, such variability
can either occur at low frequencies (in this case $\nu=325$~MHz) or at high
frequencies (in this case $\nu=1.4$~GHz). As mentioned earlier,
high--frequency variation, believed to be intrinsic to the source, occurs
predominantly in ``flat--spectrum'' sources ($\alpha \geq -0.5$;
\citeauthor{rys1990} \citeyear{rys1990}). To test this explanation, we
calculated the high--frequency
variation needed to explain the observed steep spectrum of CESS sources
assuming the source has an intrinsic spectral index of $\alpha=-0.5$. Table
\ref{cesshfv} shows the results of this analysis. The columns of this table
are:
\begin{description}
\item[Column 1:] Name of GLG source.
\item[Column 2:] 325~MHz flux density of GLG source in mJy.
\item[Column 3:] Extrapolated 1.4~GHz flux density of GLG source, in mJy,
assuming a spectral index of $\alpha=-0.5$, $S_{1400}^{exp}$.
\item[Column 4:] Measured (or upper limit) 1.4~GHz flux density, in mJy,
$S_{1400}^{meas}$. For sources with counterparts in multiple 1400~MHz
catalogs, this is the minimum measured value.
\item[Column 5:] Survey that this value for $S_{1400}$ comes from.
\item[Column 6:] $\frac{\Delta S_{1400}}{\overline{S_{1400}}}$ of this source,
defined in Section \ref{variables}.
\item[Column 7:] Average percent variation ($\overline{Var_{1400}}$) of this
source per year, equal to:
\begin{equation}
\overline{Var_{1400}}=100 \times \frac{\Delta S_{1400}}{t_{years}}
\end{equation}
where ${\Delta S_{1400}}$ is simply $|S_{1400}^{exp} - S_{1400}^{meas}|$ and
$t_{years}$ is the elapsed time between the GLG observation and 1.4~GHz
observation in Column~5.
\end{description}
The required values for $\frac{\Delta S_{1400}}{\overline{S_{1400}}}$ for all
but GLG046 and GLG152 are much higher than those observed
in the previous HFV surveys discussed in Section \ref{variables}. This does
not rule out intrinsic variability as the cause of CESS sources since the
1.4~GHz surveys used in the above calculation are much deeper than those used
in previous HFV surveys. However, at $S_{1400} \la 1-10$~mJy, normal
galaxies are an increasingly dominant fraction of the radio galaxy population
\citep{hopkins98}, and since these sources are not expected to be extremely
variable, it is not obvious that fainter 1.4~GHz sources will have higher flux
variability.
Variability at low--frequencies is also a possible explanation of the CESS
population.
As stated in Section \ref{variables}, low--frequency variation is caused by
two separate types of processes: intrinsic variation and interstellar
scintillation. If low--frequency variation is intrinsic, then the source
would be a Group 1 LFV (See Section \ref{variables}), and previous studies of
such LFVs have found that these sources have ``flat'' spectrum
($\alpha \geq -0.5$) \citep{gregorini86}. Therefore, we could analyze this
possibility in a similar fashion as we did for the high--frequency variation
possibility -- using the available 1.4 GHz data, calculate the 325~MHz of the
object at the time of 1.4~GHz observation, and then determine the variability
needed to explain the 325~MHz flux density measured in the GLG survey. Table
\ref{cesslfv} presents the results of this analysis. The columns of this
table are:
\begin{description}
\item[Column 1:] Name of GLG source.
\item[Column 2:] 325~MHz flux density of GLG source in mJy.
\item[Column 3:] Measured (or upper limit) 1.4~GHz flux density, in mJy,
$S_{1400}^{meas}$.
\item[Column 4:] Survey that this value for $S_{1400}$ comes from.
\item[Column 5:] 325~MHz flux density of GLG source, in mJy, assuming a
spectral index of $\alpha=-0.5$, $S_{325}^{exp}$.
\item[Column 6:] $\frac{\Delta S_{325}}{\overline{S_{325}}}$ of this source,
defined in Section \ref{variables}.
\item[Column 7:] Average percent variation ($\overline{Var_{325}}$) of this
source per year, equal to:
\begin{equation}
\overline{Var_{325}}=100 \times \frac{\Delta S_{325}}{t_{years}}
\end{equation}
where ${\Delta S_{325}}$ is simply $|S_{325}^{exp} - S_{325}^{meas}|$ and
$t_{years}$ is the elapsed time between the GLG observation and 1.4~GHz
observation in Column~5.
\end{description}
As was the case for High--Frequency variation, the variability required to
explain the observed steep--spectra of the CESS sources if they are flat
spectrum, intrinsically varying LFVs is much higher than that seen in previous
LFV studies (e.g. \citeauthor{rys1990} \citeyear{rys1990}; \citeauthor{riley93}
\citeyear{riley93}) for all CESS sources but
GLG046 and GLG152. Additionally, most Group 1 LFVs have optical counterparts,
not true for most of the CESS sources. Sensitivity to lower $S_{325}$ does not
necessarily help this explanation, since at lower $S_{325}$ normal galaxies
dominate the radio population (Carole Jackson, private communication 2002) and
these sources are not expect to vary with such magnitude.
If the CESS sources are LFVs, and the variation is not intrinsic, than the
probable explanation for the low--frequency variability is interstellar
scintillation (ISS). ISS, caused by variations in electron density
($\delta n_e$) along the line of sight, typically comes in two forms:
\begin{itemize}
\item Diffractive Interstellar Scintillation (DISS), which can cause flux
modulations of up to 100\% over timescales on the order of minutes and over a
bandwidth of the order of a MHz \citep{cordes98}.
\item RISS , the form of ISS believed
to be behind most low--frequency variation (see Section \ref{variables} for
more details), tends to cause a lower degree of flux modulation than DISS but
it occurs over a much longer timescale (on the order of hours) and a much
wider bandwidth (several MHz) than DISS \citep{cordes98}.
\end{itemize}
Neither form of ISS is a viable explanation for the CESS sources. For the
case of DISS, its limited bandwidth and short timescales make it an unlikely
cause. According to
the NE2001 model for the electron density of the Milky Way, the bandwidth
of DISS towards M31 is $\sim$0.4~kHz \citep{ne2001}. Since the observations
were done in spectral line mode with channel bandwidths of 10~kHz, the average
effect of DISS over the entire bandwidth is negligible. In addition, the
short timescales of DISS argue against it as the cause of the CESS sources.
Though extragalactic\footnote{In this context, ``extragalactic'' means outside
the Milky Way, not outside both the Milky Way and M31.} sources have a
DISS timescale roughly 3-30$\times$ larger than that of Galactic (Milky Way)
objects \citep{cordes98}, increasing the DISS timescale from minutes to
$\la$1~hour, since M31 was observed for a total of hour hours a 100\% increase
in 325~flux density over this timescale would be insufficient to create the
observed steep spectra of these sources.
RISS is unlikely to be the source of the CESS sources since the flux
modulation typically produced by RISS is too small. Flux density monitoring of
pulsars have found that flux variation caused by RISS in distant pulsars is
on the order of $\sim$5\%, insufficient to create the steep spectrum observed
here. Extragalactic LFVs that vary as a result of RISS typically have flux
variations on the order of a few percent \citep[e.g.][]{rys1990}, again
insufficient to create the steep spectra observed here.
While RISS and DISS are unlikely to be the cause of the CESS, this does not
rule out ISS playing a role. There have been a few instances, often called
Extreme Scattering Events \citep{fiedler94a}, where a source increases
in
flux by more than a magnitude over a bandwidth of several MHz and over a
timescale that ranges from minutes to hours. The most extreme example of this
is PSR B0655+64, whose 325~MHz flux density increased by a factor of $\sim$43
for roughly an hour over a bandwidth of $\sim$4~MHz \citep{galama97}. The
reason for this sudden amplification is not known, but is believed to be the
result of ISS rather than an intrinsic process because this pulsar has been
observed to be stable over timescale of minutes to weeks \citep{galama97}. If
something similar occurred to a source during our observation, than the
measured flux density would be on the order of $\sim$10$\times$ higher than the
actual flux density. To see if this is a possible explanation for the CESS
sources, we calculated the spectral index needed to explain the measured
higher--frequency flux densities assuming the observed 325~MHz flux densities
is 10$\times$ higher than the intrinsic value. The results are summarized in
Table \ref{cessess}. For ten of the CESS sources, the required spectral index
is reasonable ($\alpha \ga -0.7$). While it is a possibility, such extreme
events as that seen in PSR~B0655+64 are extremely rare -- the event described
above was the only one seen in a long--term pulsar monitoring project
\citep{galama97} -- so it is unlikely that something similar happened to many
objects during our observation. However, there is precedence for this sort of
behavior in the direction of M31 -- in the 36W survey, it was observed that
the flux density of 5C3.132 decreased by a factor of five in eight years
\citep{36w}.
As a result of the analysis above, we do not believe that variability --
whether intrinsic or the result of ISS -- does a good job of explaining the
observed steep spectrum for most of the CESS sources. Therefore, we are left
with Option \#2, that the CESS sources (or at least some of them) are
intrinsically steep-spectrum sources. The most common steep spectrum sources
are radio pulsars, High--z Radio Galaxies (HzRGs), and Relic Radio galaxies
(RRGs).
If a CESS source is a pulsar, then it must be unresolved and is in either M31
or the Milky Way. To test this possibility, we assume that a CESS source in
the optical disk of M31 ($R_{M31} \leq 27$~kpc) is a pulsar in M31 -- meaning
it has a distance of 780~kpc, while for a CESS source outside the optical disk
of M31 we assumed it is a pulsar in the Milky Way -- implying a distance of
$\sim$10 kpc. To determine if the pulsar possibility is reasonable, we
compared the luminosity of these sources with that of the brightest galactic
pulsars -- at $\nu=400~\mbox{MHz}$ it is PSR B1302-64 with a beamed
luminosity\footnote{Since radio emission from pulsars are not emitted
isotropically, we used $L=S~\times~d^{2}$ not $L=S~\times~4\pi d^2$ to account
for beaming.} $L_{400}$ of 26100 mJy\ kpc$^2$, while at $\nu=1.4~\mbox{GHz}$
the brightest pulsar is PSR B0736-40 with a luminosity $L_{1400}$ of 9700
mJy\ kpc$^2$ (numbers taken from the ATNF Pulsar Database). The results of
this analysis are shown in Table \ref{cesspsr}, whose columns are as follows:
\begin{description}
\item[Column 1:] GLG Name
\item[Column 2:] $R_{M31}$ in kpc, as defined in Section 4.1 in Paper I.
\item[Column 3:] $L_{400}$ in mJy kpc$^{2}$
\item[Column 4:] $L_{1400}$ in mJy kpc$^{2}$
\item[Column 5:] $\alpha_{L}^{400}$, the spectral index required for $L_{400}$
of the source to be equal to that of PSR B1302-64.
\item[Column 6:] $\alpha_{L}^{1400}$, the spectral index required for
$L_{1400}$ of the source to be equal to that of PSR B0736-40.
\end{description}
Both $L_{400}$ and $L_{1400}$ are the beamed luminosity. The luminosities of
CESS sources which might be pulsars in M31 are higher, by at least an order of
magnitude, than that of the brightest observed Galactic pulsars. The spectral
index required to make these luminosities consistent with that of the
brightest 400~MHz pulsar are much steeper than that of any known pulsar, those
for some of these sources the spectral index required to make the luminosity
consistent with that of the brightest 1.4~GHz pulsar is similar to that of the
steepest spectrum radio pulsars \citep{lorimer95}. For CESS sources outside
the optical disk, their radio luminosities at a distance of 10~kpc is
consistent with that of Milky Way pulsars in the Milky Way, but are unlikely
to be pulsars because of their high Galactic latitude. It is extremely
premature to suggest that any of these sources may be pulsars, since deep
optical imaging is needed to rule out the possibility that a source is a
HzRG/relic radio galaxy, objects expected to be faint optical sources
\citep{kaplan01}.
Another possibility is that the CESS sources are pulsars than emit giant
pulses, which have been detected from a wide variety of pulsars -- from young
pulsars like the Crab Pulsar \citep{lundgren95} to the millisecond pulsar
PSR~B1937+21 \citep{kinkhabwala00}. Even though giant pulses can be as much
as $4000\times$ brighter than the average pulse, they do not significantly
increase the integrated flux density \citep{lundgren95}. The most extreme
case of this is PSR B0540-69 in the LMC, which emits giant pulses but remains
undetected in continuum 1400~MHz observations. This pulsar has a spectral
index $\alpha_{640}^{1380} \leq -4.4$ \citep{johnston03}, a similar limit as
for many of the CESS sources in Table \ref{cesspsr}.
The second possibility for the intrinsic steep--spectrum explanation is that
they are High-z Radio Galaxies (HzRGs), which were discussed in Section
\ref{hzrgs}. In that section, the steep--spectrum HzRG candidates discussed
were all resolved radio sources with the morphology of an FR~II. However,
HzRG candidates often are much smaller than those sources -- the \citet{hzrg}
survey of Ultra Steep Spectrum ($\alpha\leq-1.3$; USS) sources found that USS
sources have a constant median angular size of $\sim$6\arcsec~between
S$_{1400}$=10~mJy--1~Jy \citep{hzrg}. Therefore, we expect to detect
unresolved USS sources in this survey. The density of CESS sources in this
survey ($\sim$9500 sr$^{-1}$) is much higher than in other surveys, e.g. the
density of USSs in the WENSS and NVSS survey is $\sim$151 sr$^{-1}$
\citep{hzrg}. A higher density of steep--spectrum sources in the GLG survey is
not completely unexpected since our survey is substantially deeper than the
WENSS survey (a flux limit of $S_{325} \approx$3~mJy as opposed to $S_{325}
\approx$18~mJy) and the 1.4 GHz surveys used for comparison are much
deeper than the NVSS survey. However, as mentioned earlier, starburst galaxies
dominate the population of sources with $S_{1400} \la 10$~mJy
-- which are not expected to have spectra as steep as these sources.
The third possibility for an intrinsically steep spectrum is relic radio
galaxies. These are objects which used to be FRII radio galaxies, but the
central AGN turned off causing the relativistic electrons insides the lobes of
these galaxies to lose energy through synchrotron emission and inverse Compton
Scattering off the Cosmic Microwave Background (CMB). The higher energy
electrons lose energy faster than the lower energy ones, steepening the
electron energy distribution and, consequently, the observed radio spectral
index \citep{komis94}. These sources can have spectral indices as steep as
$-6 \la \alpha \la -5$, but tend to be rather large with
$\theta_M \ga 10$\arcsec~\citep{kaplan00}. Only GLG006, GLG022, GLG036,
GLG059, and GLG305 have $\theta_M > 10$\arcsec, with GLG036 the largest with
$\theta_M=16.6$\arcsec. There is an R--band source on the edge of
GLG036, but it is unlikely the two are related. Also, GLG036 is one
of the weakest CESS sources detected.
For the reasons stated above, we do not believe that is possible at this time
to definitively state the nature of the observed CESS sources, and feel they
are most likely a mix of the possibilities described above. Deeper optical
imaging and another 325~MHz radio observation of M31 would be very helpful in
solving this mystery: optical imaging would determine whether or not these
sources are galaxies (HzRGs tend to have $R \la 20$~mag), and another radio
image would put a strong constraint on the variability of these sources.
These sources are much fainter than previous steep--spectrum samples, and
therefore are potentially extremely interesting.
\subsubsection{Anomalous Morphology and Multi--Wavelength Characteristics}
\label{weirdmorph}
In this section, we discuss sources with anomalous radio morphologies and/or
multi-wavelength properties.
\paragraph{GLG015}
\label{glg015}
GLG015 is a complex source located on the edge of the optical disk of M31
with a spectral index of $\alpha=-0.99\pm0.08$ and no spectral curvature, and
has the morphology of a compact FRII. However, GLG015
is bordered on two sides by two clumps of H$\alpha$ and [SII] emission
(see Figures \ref{halpgd} and \ref{siigd}) and is orientated such that it fits
between them without overlap. This emission has no counterparts in the R--band
image of this region, or in any of the Far-IR, IR, and X--ray catalogs, but
the bright spots in H$\alpha$/[SII] correspond to stars in the MagOpt catalog.
GLG015 most likely is a FRII coincidentally located in
this position and orientation, if it is connected to the H$\alpha$/[SII]
emission around it, the nature of this source is unknown.
\paragraph{GLG065}
\label{glg065}
GLG065 is a complex located just outside the optical disk of M31. GLG065 has
the most puzzling morphology of all sources in the GLG source list - as
shown in Figure \ref{complex} it appears to be a collection of four,
maybe five points sources arranged somewhat symmetrically. It is unlikely that
this source is a gravitational lens due to the the wide separation of
the sources (at $\sim$30\arcsec~this would be one the most widely separated
gravitational lenses known), and the lack of optical/IR emission expected from
the lensing object. It does have a passing resemblance to 3C315, a classic
``X--shaped'' radio source \citep{leahy84}. Another possible explanation is
that GLG065 is a collection of SNRs in M31, although the lack of H$\alpha$
emission in this region makes this explanation unlikely.
\paragraph{GLG078}
\label{glg078}
GLG078 is a complex, steep spectrum ($\alpha=-1.57$) source with significant
spectral turnover ($\varphi=0.10$) just beyond the optical disk of M31 and the
morphology of an asymmetric FRII. There is
a 2MASS PSC source located next to the center of the radio emission - but there
is no radio emission from the specific location of the IR source, and it has
the IR colors of a star in the Milky Way. The R--band image of GLG078 (shown
in Figure \ref{redgd}) shows the IR source mentioned above as well as a source
located at the SE edge of the radio emission. We believe that GLG078 is an
FR~II and that the R--band source is coincidental. Higher-resolution radio
images are required to better understand the nature of this source.
\section{Conclusion}
\label{conclusion}
In this paper we have presented the results of a four hour 325~MHz
survey of M31. A statistical analysis of the 405 radio sources
detected in this survey, done in Paper I \citep{paper1}, revealed that
most of these sources are background radio galaxies. In this paper we
present our attempt to determine the nature of these source using
their observed radio spectral properties, radio morphologies, and
their Far--IR/IR/Optical/X--ray properties of the detected sources.
With this information we classified 112 out of the 405 detected
sources, as summarized in Table \ref{srcsum}.
Sources in the GLG catalog could be placed into four broad categories: sources
in M31, sources in the Milky Way, extragalactic sources, and anomalous
sources. In M31, we have identified five supernova remnant candidates and
three pulsar wind nebula candidates, given in Section \ref{m31srcs}. The
supernova remnant candidates have properties (extent and radio luminosity)
similar to that of known supernova remnants in the Milky Way, M33, LMC, and
SMC as shown in Figure \ref{snrbmajflux}. Two of the pulsar wind nebula
candidates are more luminous than the Crab Nebula, but only by a factor of a
few. Further observations are needed to confirm the classification of these
sources and determine the morphology and basic physical parameters. The
objects are the brightest of their class in M31, and as such are individually
interesting.
In the Milky Way, we believed we have identified three radio stars based on
optical and near-IR information and a Planetary Nebula based on radio
morphology, as described in Section \ref{mwsrcs}. A wide variety of
extragalactic sources were identified, ranging from the ordinary (FRI and FRII
radio galaxies, star-forming galaxies, and radio jets) to the exotic (a BL Lac
candidate, a galaxy merger, a Giant Radio Galaxy candidate, and High-z Radio
Galaxy Candidates, including Compact Steep Spectrum and Gigahertz Peaked
Spectrum radio galaxies), using the criteria described in Section
\ref{extragal}. Last but not least, we have identified many anomalous sources
- ranging from highly variable sources (Section \ref{variables}) to compact
($\theta_M \la 10$\arcsec) extremely steep spectrum ($\alpha \leq -1.6$)
sources (Section \ref{cess}) to source with anomalous morphologies and/or
multi-wavelength properties (Section \ref{weirdmorph}).
To summarize, in just a four-hour 325~MHz observation of M31 we have
potentially identified the brightest supernova remnants and pulsar wind
nebulae in these galaxy, as well as identified a class of steep spectrum
sources whose nature is unknown. In this paper, we have a detailed a technique
for separating background radio sources from those intrinsic to M31, which
will be extremely useful in future radio surveys of this galaxy. While much
remains unknown about the radio population of M31, this survey is an important
step in understanding the properties of this galaxy.
\acknowledgements
TJW Lazio acknowledges that basic research in radio astronomy at the NRL
is supported by the Office of Naval Research. The authors wish to
thank the anonymous referee for his/her helpful comments,
Andrew Hopkins for providing us with the latest version of $\mathcal{SFIND}$;
Elias Brinks, Mike Garcia, Phil Kaaret, Albert Kong, Linda Schmidtobreick,
Sergey Trudolyubov, Rene Walterbos, Ben Williams, C. Kevin Xu, and the SIMBAD
help desk for providing us with source-lists and/or images; Carole Jackson for
providing her model of the flux distribution of background radio galaxies at
325~MHz; and Aaron Cohen, Elly Berkhuijsen, Jim Cordes, Rosanne DiStefano,
Mike Garcia, Dan Harris, John Huchra, Namir Kassim, Albert Kong, Pat Slane,
Krzysztof Stanek, Lorant Sjouwerman, Ben Williams, and Josh Winn for useful
discussions; and Harvard CDF for computer access.
The National Radio Astronomy Observatory is a facility of the National Science
Foundation operated under cooperative agreement by Associated Universities,
Inc.
This research has made use of NASA's Astrophysics Data System; of the SIMBAD
database, operated at CDS, Strasbourg, France; of the NASA/IPAC Extragalactic
Database (NED) which is operated by the Jet Propulsion Laboratory, California
Institute of Technology, under contract with the National Aeronautics and
Space Administration; of data products from the Two Micron All Sky Survey,
which is a joint project of the University of Massachusetts and the Infrared
Processing and Analysis Center/California Institute of Technology, funded by
the National Aeronautics and Space Administration and the National Science
Foundation; and of the NASA/ IPAC Infrared Science Archive, which is operated
by the Jet Propulsion Laboratory, California Institute of Technology, under
contract with the National Aeronautics and Space Administration.
\bibliographystyle{apj}
\clearpage
\begin{deluxetable}{ccc|cc}
\tablecolumns{5}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Steep Spectrum ($\alpha<-1.3$)~GLG Sources \label{steepspec}}
\tablehead{
\colhead{Name} & \colhead{Type} & \colhead{Flux [mJy]} &
\colhead{$\alpha$} & \colhead{$\varphi$}}
\startdata
GLG002 & U & 3.65$\pm$0.59 & -1.45$\pm$0.34 & 0.13$\pm$0.05 \\
GLG006 & U & 18.12$\pm$0.58 & $\leq$-3.18 & \nodata \\
GLG007 & U & 5.45$\pm$0.58 & $\leq$-2.37 & \nodata \\
GLG014 & U & 23.42$\pm$0.69 & $\leq$-3.35 & \nodata \\
GLG022 & El & 11.62$\pm$1.79 & $\leq$-2.81 & \nodata \\
GLG031 & C & 63.97$\pm$7.47 & -1.44$\pm$0.08 & \nodata \\
GLG036 & El & 38.38$\pm$2.59 & $\leq$-3.53 & \nodata \\
GLG047 & U & 8.84$\pm$0.75 & -1.64$\pm$0.10 & \nodata \\
GLG055 & U & 18.73$\pm$1.10 & $\leq$-2.01 & \nodata \\
GLG062 & U & 24.97$\pm$1.37 & $\leq$-2.94 & \nodata \\
GLG104 & U & 16.44$\pm$0.89 & $\leq$-2.07 & \nodata \\
GLG115 & C & 17.32$\pm$6.10 & $\leq$-2.19 & \nodata \\
GLG129 & C & 51.32$\pm$6.98 & -1.45$\pm$0.30 & -0.08$\pm$0.03 \\
GLG140 & U & 52.49$\pm$2.82 & $\leq$-3.00 & \nodata \\
GLG149 & U & 7.31$\pm$0.71 & $\leq$-2.22 & \nodata \\
GLG157 & U & 15.81$\pm$0.98 & -1.40$\pm$0.27 & 0.07$\pm$0.03 \\
GLG160 & U & 13.74$\pm$0.75 & $\leq$-2.99 & \nodata \\
GLG171 & U & 18.88$\pm$1.09 & $\leq$-2.90 & \nodata \\
GLG179 & U & 16.70$\pm$0.65 & -1.46$\pm$0.25 & 0.10$\pm$0.04 \\
GLG187 & C & 73.69$\pm$8.33 & -1.62$\pm$0.09 & \nodata \\
GLG212 & U & 18.49$\pm$1.14 & $\leq$-1.36 & \nodata \\
GLG216 & U & 28.95$\pm$1.59 & -1.30$\pm$0.07 & \nodata \\
GLG233 & C & 20.66$\pm$8.24 & -1.60$\pm$0.26 & 0.11$\pm$0.08 \\
GLG263 & U & 26.26$\pm$1.08 & $\leq$-1.60 & \nodata \\
GLG266 & U & 96.05$\pm$2.13 & -2.25$\pm$0.08 & \nodata \\
GLG269 & C & 73.17$\pm$11.99 & -2.06$\pm$0.14 & \nodata \\
GLG289 & U & 11.06$\pm$0.77 & $\leq$-1.94 & \nodata \\
GLG294 & U & 11.51$\pm$0.76 & $\leq$-2.01 & \nodata \\
GLG296 & C & 203.17$\pm$13.47 & -1.60$\pm$0.12 & 0.04$\pm$0.02 \\
GLG305 & U & 26.99$\pm$1.31 & $\leq$-3.11 & \nodata \\
GLG319 & U & 10.20$\pm$0.83 & $\leq$-2.45 & \nodata \\
GLG322 & U & 25.86$\pm$1.04 & $\leq$-1.59 & \nodata \\
GLG336 & U & 18.82$\pm$1.31 & $\leq$-1.38 & \nodata \\
GLG347 & U & 18.93$\pm$0.94 & $\leq$-1.38 & \nodata \\
GLG361 & U & 18.99$\pm$1.34 & $\leq$-1.38 & \nodata \\
GLG405 & U & 22.72$\pm$1.33 & $\leq$-2.75 & \nodata \\
\enddata
\tablecomments{The spectral curvature parameter $\varphi$ was calculated only
for those sources detected in three radio surveys, including this one.}
\end{deluxetable}
\begin{deluxetable}{ccc|cc}
\tablecolumns{5}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Flat Spectrum ($-0.5<\alpha<0.5$)~GLG Sources \label{flatspec}}
\tablehead{
\colhead{Name} & \colhead{Type} & \colhead{Flux [mJy]} &
\colhead{$\alpha$} & \colhead{$\varphi$}}
\startdata
GLG011 & U & 37.23$\pm$1.90 & -0.17$\pm$0.08 & -0.01$\pm$0.01 \\
GLG016 & U & 3.87$\pm$0.50 & -0.07$\pm$0.08 & \nodata \\
GLG021 & U & 8.57$\pm$0.76 & 0.16$\pm$0.48 & -0.07$\pm$0.07 \\
GLG027 & U & 36.57$\pm$1.62 & -0.45$\pm$0.23 & -0.04$\pm$0.03 \\
GLG034 & U & 12.95$\pm$0.46 & -0.29$\pm$0.04 & \nodata \\
GLG049 & U & 7.99$\pm$0.66 & -0.00$\pm$0.14 & \nodata \\
GLG066 & U & 4.25$\pm$0.49 & -0.49$\pm$0.17 & \nodata \\
GLG067 & U & 14.29$\pm$0.85 & -0.29$\pm$0.10 & \nodata \\
GLG068 & U & 4.19$\pm$0.42 & -0.25$\pm$0.09 & \nodata \\
GLG069 & U & 13.22$\pm$0.82 & 0.39$\pm$0.41 & -0.03$\pm$0.05 \\
GLG076 & U & 4.29$\pm$0.39 & $\leq$-0.37 & \nodata \\
GLG083 & El & 14.58$\pm$1.20 & -0.32$\pm$0.09 & \nodata \\
GLG088 & U & 13.90$\pm$0.82 & -0.04$\pm$0.05 & \nodata \\
GLG091 & U & 3.98$\pm$0.57 & $\leq$-0.31 & \nodata \\
GLG109 & U & 5.16$\pm$0.54 & $\leq$-0.49 & \nodata \\
GLG111 & U & 16.30$\pm$0.79 & $\leq$0.00 & \nodata \\
GLG112 & U & 10.11$\pm$0.46 & $\leq$0.00 & \nodata \\
GLG114 & U & 7.14$\pm$0.53 & -0.28$\pm$0.06 & \nodata \\
GLG131 & El & 2.96$\pm$2.16 & $\leq$-0.11 & \nodata \\
GLG134 & U & 6.42$\pm$0.55 & -0.39$\pm$0.11 & \nodata \\
GLG142 & U & 4.46$\pm$0.69 & $\leq$-0.39 & \nodata \\
GLG144 & U & 46.56$\pm$2.57 & -0.39$\pm$0.03 & \nodata \\
GLG159 & El & 4.51$\pm$1.92 & $\leq$-0.40 & \nodata \\
GLG163 & U & 8.30$\pm$0.72 & -0.34$\pm$0.08 & \nodata \\
GLG167 & U & 6.10$\pm$0.55 & -0.45$\pm$0.17 & 0.07$\pm$0.03 \\
GLG170 & U & 7.20$\pm$0.71 & -0.48$\pm$0.09 & \nodata \\
GLG172 & U & 9.01$\pm$0.59 & 0.18$\pm$0.31 & -0.08$\pm$0.04 \\
GLG198 & U & 11.79$\pm$0.79 & -0.08$\pm$0.20 & -0.07$\pm$0.03 \\
GLG215 & U & 4.20$\pm$0.45 & $\leq$-0.44 & \nodata \\
GLG222 & U & 19.11$\pm$0.83 & $\leq$-0.18 & \nodata \\
GLG224 & U & 6.38$\pm$0.76 & 0.23$\pm$0.21 & \nodata \\
GLG229 & U & 7.40$\pm$0.66 & -0.15$\pm$0.08 & \nodata \\
GLG240 & U & 14.37$\pm$0.84 & -0.32$\pm$0.05 & \nodata \\
GLG265 & U & 3.94$\pm$0.52 & $\leq$-0.31 & \nodata \\
GLG285 & U & 15.19$\pm$0.86 & -0.37$\pm$0.04 & \nodata \\
GLG286 & U & 6.84$\pm$0.87 & -0.38$\pm$0.10 & \nodata \\
GLG288 & U & 14.21$\pm$1.09 & -0.49$\pm$0.06 & \nodata \\
GLG302 & U & 31.42$\pm$1.77 & -0.20$\pm$0.10 & -0.00$\pm$0.01 \\
GLG323 & U & 287.12$\pm$9.88 & -0.45$\pm$0.01 & \nodata \\
GLG331 & U & 7.53$\pm$0.77 & 0.35$\pm$0.07 & \nodata \\
GLG381 & El & 11.96$\pm$2.41 & -0.43$\pm$0.14 & \nodata \\
GLG384 & U & 6.95$\pm$0.85 & -0.23$\pm$0.25 & \nodata \\
GLG391 & U & 23.22$\pm$1.49 & -0.48$\pm$0.04 & \nodata \\
GLG395 & U & 24.52$\pm$1.44 & -0.19$\pm$0.04 & \nodata \\
\enddata
\tablecomments{The spectral curvature parameter $\varphi$ was calculated only
for those sources detected in three radio surveys, including this one.}
\end{deluxetable}
\begin{deluxetable}{ccc|cc}
\tablecolumns{5}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{GLG Sources with Spectral Tunrnover ($\varphi \geq
3\sigma_{\varphi}$) \label{turnspec}}
\tablehead{
\colhead{Name} & \colhead{Type} & \colhead{Flux [mJy]} &
\colhead{$\alpha$} & \colhead{$\varphi$}}
\startdata
GLG009 & ExG & 45.87$\pm$4.46 & -1.28$\pm$0.06 & 0.10$\pm$0.013 \\
GLG039 & U & 25.98$\pm$1.12 & -1.21$\pm$0.10 & 0.11$\pm$0.016 \\
GLG046 & U & 5.76$\pm$0.64 & -2.01$\pm$0.16 & 0.18$\pm$0.028 \\
GLG048 & U & 16.00$\pm$0.72 & -1.18$\pm$0.14 & 0.08$\pm$0.019 \\
GLG050 & C & 140.35$\pm$9.10 & -0.57$\pm$0.00 & -0.03$\pm$0.000 \\
GLG059 & U & 30.69$\pm$0.94 & -1.77$\pm$0.08 & 0.17$\pm$0.011 \\
GLG077 & U & 530.96$\pm$26.03 & -0.58$\pm$0.00 & -0.03$\pm$0.003 \\
GLG078 & C & 109.30$\pm$8.91 & -1.57$\pm$0.00 & 0.10$\pm$0.000 \\
GLG084 & U & 8.52$\pm$0.58 & -0.92$\pm$0.16 & 0.13$\pm$0.024 \\
GLG087 & U & 14.75$\pm$0.99 & -0.78$\pm$0.14 & 0.08$\pm$0.023 \\
GLG090 & C & 49.89$\pm$6.24 & -0.68$\pm$0.00 & -0.03$\pm$0.000 \\
GLG101 & El & 12.83$\pm$1.21 & -1.26$\pm$0.16 & 0.16$\pm$0.026 \\
GLG133 & U & 7.57$\pm$0.70 & -1.36$\pm$0.23 & 0.14$\pm$0.033 \\
GLG135 & U & 189.63$\pm$9.61 & -0.55$\pm$0.01 & 0.02$\pm$0.005 \\
GLG152 & U & 7.68$\pm$0.56 & -1.63$\pm$0.23 & 0.19$\pm$0.034 \\
GLG154 & U & 60.07$\pm$3.05 & -1.04$\pm$0.03 & 0.03$\pm$0.006 \\
GLG158 & U & 32.59$\pm$1.85 & -1.24$\pm$0.12 & 0.09$\pm$0.019 \\
GLG186 & U & 710.23$\pm$35.47 & 0.98$\pm$0.03 & -0.14$\pm$0.006 \\
GLG193 & U & 8.63$\pm$0.68 & -0.93$\pm$0.14 & 0.07$\pm$0.023 \\
GLG203 & U & 36.97$\pm$1.71 & -0.94$\pm$0.08 & 0.05$\pm$0.012 \\
GLG208 & U & 8.48$\pm$0.65 & -1.07$\pm$0.16 & 0.09$\pm$0.025 \\
GLG210 & U & 11.16$\pm$0.67 & -0.70$\pm$0.00 & 0.08$\pm$0.001 \\
GLG232 & U & 30.36$\pm$2.00 & -0.24$\pm$0.00 & -0.05$\pm$0.001 \\
GLG250 & U & 13.75$\pm$0.71 & -1.32$\pm$0.14 & 0.14$\pm$0.021 \\
GLG274 & U & 13.73$\pm$0.75 & -1.30$\pm$0.17 & 0.11$\pm$0.026 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{c|cc|cc|c}
\tablecolumns{6}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Comparison with Near-IR, Far-IR, Optical and X--ray Surveys
\label{mwcomp}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{Survey} & \colhead{\# of GLG sources} &
\colhead{\# of GLG sources} & \colhead{\# of false} &
\colhead{\# of false} & \colhead{Reference} \\
\colhead{} & \colhead{with counterparts} &
\colhead{with good counterparts} & \colhead{counterparts} &
\colhead{good counterparts}
}
\startdata
2MASS ESC & 21 & 13 & 12.00$\pm$3.74 & 7.00$\pm$1.60 & \cite{2MASSesc} \\
\nodata & 20 & 12 & 10.87$\pm$3.35 & 6.37$\pm$2.06 & \nodata \\
2MASS PSC & 44 & 29 & 33.87$\pm$3.13 & 20.12$\pm$2.64 & \cite{2MASSpsc} \\
\nodata & 17 & 17 & 1.50$\pm$1.60 & 1.50$\pm$1.60 & \nodata \\
\hline
``Knots'' & 33 & 12 & 39.87$\pm$4.94 & 10.37$\pm$3.20 & \cite{knots} \\
Xu \& Helou & 12 & 10 & 14.75$\pm$4.89 & 8.12$\pm$2.90 & \cite{xu} \\
IRAS-PSC & 5 & 5 & 1.00$\pm$1.19 & 1.00$\pm$1.19 & \cite{iraspsc} \\
IRAS-SSC & 2 & 2 & 0.87$\pm$0.99 & 0.75$\pm$1.03 & \cite{irasssc} \\
IRAS-FSC & 5 & 5 & 0.62$\pm$0.74 & 0.62$\pm$0.74 & \cite{irasfsc} \\
\hline
WW SNR & 5 & 5 & 0.87$\pm$0.35 & 0.87$\pm$0.35 &
Ben Williams, private communication \\
Mag SNR & 1 & 1 & 0.62$\pm$0.74 & 0.62$\pm$0.74 & \cite{magniersnr} \\
BW SNR & 0 & 0 & 0.37$\pm$0.51 & 0.37$\pm$0.51 & \cite{bwsnr} \\
``Ion'' & 4 & 3 & 4.75$\pm$0.88 & 3.25$\pm$1.38 & \cite{wbion} \\
FJ PN & 0 & 0 & 0 & 0 & \cite{fjpn} \\
``Pellet'' & 19 & 4 & 20.62$\pm$2.87 & 7.25$\pm$2.37 & \cite{pellet} \\
MagOpt & \nodata & 20 & \nodata & 14.50$\pm$6.30 & \cite{magstars} \\
\hline
RASS & 12 & 8 & 7.25$\pm$1.83 & 5.50$\pm$1.06
& \cite{RASSbright}; \cite{RASSfaint} \\
\nodata & 10 & 8 & 6.12$\pm$1.64 & 4.37$\pm$0.91 & \nodata \\
{\it ROSAT} Survey & 37 & 32 & 13.75$\pm$4.80 & 10.50$\pm$4.10
& \cite{rosat} \\
\nodata & 29 & 26 & 11.75$\pm$3.95 & 9.25$\pm$3.80 & \nodata \\
{\it Chandra} Glob. Clus. & 0 & 0 & 0 & 0 & \cite{m31glob} \\
ACIS-I survey & 2 & 2 & 0 & 0 & \cite{acis} \\
HRC - Central Region & 0 & 0 & 0.25$\pm$0.46 & 0.25$\pm$0.46
& \cite{kaaret} \\
HRC - Optical Disk & 5 & 2 & 4.50$\pm$1.30 & 2.75$\pm$1.16 & \cite{hrc} \\
{\it XMM} North Fields & 5 & 5 & 0.25$\pm$0.46 & 0.25$\pm$0.46
& Sergey Trudolyubov, private communication
\enddata
\tablecomments{``RASS'' stands for {\it ROSAT} all sky survey, and {\it ROSAT}
survey represents the Second {\it ROSAT} PSPC Survey of M31. The other survey
abbreviations are defined in the text. The second line for the 2MASS ESC,
2MASS PSC, RASS and {\it ROSAT} surveys are the results of this analysis only
using ``U'' and ``El'' GLG sources. The horizontal lines delineate different
wavebands. The definition of a ``good'' counterpart is given in Section
\ref{intro}.}
\end{deluxetable}
\begin{deluxetable}{cc||cc|ccc|cc}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{IR Magnitudes and Colors of 2MASS ESC counterparts of GLG
sources \label{jhkesc}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{GLG} & \colhead{Source} & \colhead{} & \colhead{} &
\colhead{2MASS} & \colhead{ESC} & \colhead{Source} &
\colhead{} \\
\colhead{Name} & \colhead{Type} & \colhead{Name} & \colhead{Offset
[\arcsec]} & \colhead{J Mag.} & \colhead{H mag.} & \colhead{K Mag.}
& \colhead{J-H} & \colhead{H-K}}
\startdata
GLG018 & U & 2MASS-EXT00415185+4124424 & 0.45 & 14.9$\pm$0.19 & 14.1$\pm$0.22
& 13.3$\pm$0.15 & 0.83 & 0.82 \\
GLG163 & U & 2MASS-EXT00383316+4128509 & 0.23 & 13.1$\pm$0.09 &
12.2$\pm$0.11 & 11.7$\pm$0.10 & 0.85 & 0.55 \\
GLG184 & U & 2MASS-EXT00380553+4120166 & 0.84 & 13.7$\pm$0.07 &
12.8$\pm$0.08 & 12.4$\pm$0.09 & 0.97 & 0.33 \\
GLG203 & U & 2MASS-EXT00415339+4021174 & 0.42 & 12.9$\pm$0.04 &
12.1$\pm$0.05 & 11.6$\pm$0.05 & 0.79 & 0.52 \\
GLG204 & El & 2MASS-EXT00453674+4030033 & 0.30 & 15.4$\pm$0.11 &
14.6$\pm$0.12 & 14.3$\pm$0.15 & 0.82 & 0.30 \\
GLG209 & U & 2MASS-EXT00421611+4019360 & 0.48 & 14.3$\pm$0.08 &
13.7$\pm$0.14 & 13.0$\pm$0.11 & 0.51 & 0.75 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{cc||cc|ccccc|cc}
\tablecolumns{11}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{IR Magnitude and Colors of 2MASS PSC counterparts of GLG sources
\label{jhkpsc}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{GLG} & \colhead{Source} & colhead{} & \colhead{} &
\colhead{} & \colhead{2MASS} & \colhead{PSC} & \colhead{Sources}
& \colhead{} & \colhead{} & \colhead{} \\
\colhead{Name} & \colhead {Type} & \colhead{Name} & \colhead{Offset
[\arcsec]} & \colhead{J Mag.} & \colhead{H Mag.} &
\colhead{K Mag.} & \colhead{B Mag.} & \colhead{R Mag.} &
\colhead{J-H} & \colhead{H-K}}
\startdata
GLG002 & U & 00422667+4118043 & 0.63 & 15.6$\pm$0.10 & 14.8$\pm$0.12
& 14.2$\pm$0.10 & \nodata & \nodata & 0.81 & 0.60 \\
GLG003 & U & 00423301+4111423 & 1.76 & 16.7$\pm$0.19 & 15.7$\pm$0.19
& 15.2$\pm$0.20 & \nodata & \nodata & 1.00 & 0.43 \\
GLG017 & C & 00420221+4126468 & 14.1 & 15.7$\pm$0.06 & 15.0$\pm$0.09
& 14.7$\pm$0.10 & 18.9 & 18.5 & 0.74 & 0.24 \\
GLG018 & U & 00415182+4124420 & 0.27 & 15.5$\pm$0.08 & 14.5$\pm$0.08
& 13.8$\pm$0.06 & 17.6 & 19.1 & 1.00 & 0.68 \\
GLG030 & C & 00413926+4130352 & 1.49 & 15.9$\pm$0.09 & 15.2$\pm$0.11
& 14.5$\pm$0.08 & \nodata & \nodata & 0.68 & 0.68 \\
GLG050 & C & 00450182+4124532 & 1.10 & 17.2$\pm$0.22 & 16.1$\pm$0.18
& 15.8$\pm$0.21 & \nodata & \nodata & 1.03 & 0.38 \\
GLG078 & C & 00453965+4112170 & 4.13 & 13.9$\pm$0.02 & 13.6$\pm$0.02
& 13.6$\pm$0.03 & 15.6 & 14.5 & 0.26 & 0.04 \\
GLG079 & U & 00425297+4149210 & 0.60 & 16.4$\pm$0.12 & 15.4$\pm$0.11
& 14.7$\pm$0.10 & 19.5 & 18.5 & 0.93 & 0.69 \\
GLG097 & U & 00460201+4123183 & 0.59 & 12.4$\pm$0.02 & 12.2$\pm$0.01
& 12.1$\pm$0.02 & 13.9 & 13.1 & 0.24 & 0.06 \\
GLG105 & C & 00401380+4050047 & 8.30 & 16.1$\pm$0.12 & 15.1$\pm$0.10
& 14.7$\pm$0.11 & \nodata & \nodata & 0.96 & 0.47 \\
GLG116 & C & 00423543+4157468 & 2.61 & 6.2$\pm$0.02 & 5.3$\pm$0.01 &
5.1$\pm$0.02 & 10.8 & 9.2 & 0.84 & 0.18 \\
GLG122 & C & 00391105+4128491 & 9.28 & 14.1$\pm$0.02 & 13.6$\pm$0.03
& 13.5$\pm$0.03 & 16.3 & 15.5 & 0.50 & 0.12 \\
GLG127 & U & 00390735+4103449 & 0.72 & 16.6$\pm$0.16 & 15.9$\pm$0.21
& 15.2$\pm$0.17 & 19.0 & 18.7 & 0.69 & 0.67 \\
GLG134 & U & 00440838+4034463 & 0.75 & 16.7$\pm$0.17 & 15.7$\pm$0.14
& 14.9$\pm$0.11 & \nodata & \nodata & 0.99 & 0.74 \\
GLG163 & U & 00383316+4128506 & 0.59 & 14.4$\pm$0.06 & 13.9$\pm$0.10
& 13.3$\pm$0.06 & 13.4 & 13.2 & 0.52 & 0.57 \\
GLG166 & U & 00452433+4155366 & 1.35 & 16.2$\pm$0.30 & 15.5$\pm$0.13
& 14.7$\pm$0.11 & \nodata & \nodata & 0.72 & 0.77 \\
GLG184 & U & 00380555+4120166 & 0.60 & 14.7$\pm$0.08 & 14.0$\pm$0.10
& 13.3$\pm$0.06 & 15.1 & 14.8 & 0.68 & 0.66 \\
GLG203 & U & 00415342+4021176 & 0.31 & 14.0$\pm$0.07 & 13.4$\pm$0.09
& 12.5$\pm$0.05 & 13.3 & 11.5 & 0.61 & 0.82 \\
GLG204 & El & 00453673+4030035 & 0.36 & 16.1$\pm$0.10 & 15.5$\pm$0.12
& 14.6$\pm$0.09 & 18.0 & 18.4 & 0.66 & 0.84 \\
GLG209 & U & 00421612+4019359 & 0.43 & 15.0$\pm$0.05 & 14.2$\pm$0.06
& 13.6$\pm$0.06 & 15.1 & 15.5 & 0.81 & 0.57 \\
GLG225 & U & 00375290+4139165 & 0.90 & 16.5$\pm$0.13 & 15.7$\pm$0.16
& 15.0$\pm$0.12 & \nodata & \nodata & 0.86 & 0.69 \\
GLG247 & C & 00474961+4141504 & 0.50 & 15.7$\pm$0.11 & 14.9$\pm$0.12
& 14.3$\pm$0.11 & 14.6 & 16.6 & 0.77 & 0.55 \\
GLG253 & U & 00462509+4204273 & 0.23 & 16.2$\pm$0.13 & 15.3$\pm$0.14
& 14.7$\pm$0.12 & 18.0 & 18.8 & 0.90 & 0.61 \\
GLG287 & U & 00365438+4054591 & 0.17 & 16.2$\pm$0.30 & 15.8$\pm$0.17
& 15.2$\pm$0.13 & \nodata & \nodata & 0.41 & 0.61 \\
GLG335 & U & 00360794+4128444 & 0.91 & 15.1$\pm$0.06 & 14.2$\pm$0.06
& 13.6$\pm$0.05 & 15.4 & 14.2 & 0.88 & 0.56 \\
GLG387 & C & 00502389+4105115 & 1.54 & 17.3$\pm$0.25 & 16.4$\pm$0.24
& 15.5$\pm$0.15 & \nodata & \nodata & 0.93 & 0.87
\enddata
\end{deluxetable}
\begin{deluxetable}{c|cc}
\tablecolumns{5}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tablecaption{GLG sources with a counterpart in the WW~SNR catalog
\label{optctrp}}
\tablehead{
\colhead{GLG} & \colhead{Optical} & \colhead{Counterpart} \\
\colhead{Name} & \colhead {Name} & \colhead{Offset [\arcsec]}}
\startdata
GLG011 & WW~29 & 52.8 \\
GLG020 & WW~47 & 2.21 \\
GLG037 & WW~12 & 1.83 \\
GLG068 & WW~10 & 2.25 \\
GLG123 & WW~60 & 41.4 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{ccc||cc|cccc}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{GLG sources with a Good Match in the MagOpt Catalog
\label{magstarctrp}}
\tablehead{
\colhead{} & \colhead{GLG Source} & \colhead{} &
\colhead{} & \colhead{} & \colhead{MagOpt} & \colhead{Counterpart} &
\colhead{} & \colhead{} \\
\colhead{Name} & \colhead {Type} & \colhead{$S_{325}$ [mJy]} &
\colhead{Name} & \colhead{Offset [\arcsec]} &
\colhead{B Mag.} & \colhead{V Mag.} & \colhead{R Mag.} &
\colhead{I Mag.}}
\startdata
GLG008 & U & 17.4 & MagOpt20716 & 3.20 & \nodata & \nodata & \nodata &
19.6$\pm$0.12 \\
GLG014 & U & 23.4 & MagOpt26706 & 2.15 & \nodata & \nodata & \nodata &
20.1$\pm$0.19 \\
GLG019 & U & 1223.6 & MagOpt31940 & 0.34 & \nodata & \nodata & \nodata
& 20.2$\pm$0.13 \\
GLG023 & U & 105.9 & MagOpt31494 & 1.09 & 14.9$\pm$0.04 & 14.0$\pm$0.02
& 13.5$\pm$0.02 & 13.2$\pm$0.00 \\
GLG040 & U & 10.2 & MagOpt13395 & 2.54 & 21.2$\pm$0.07 & 20.5$\pm$0.05
& 20.0$\pm$0.11 & 19.5$\pm$0.10 \\
GLG061 & El & 23.2 & MagOpt12931 & 3.03 & \nodata & \nodata & \nodata &
20.5$\pm$0.24 \\
GLG083 & El & 14.5 & MagOpt38317 & 3.04 & \nodata & \nodata &
24.3$\pm$0.26 & 19.3$\pm$0.09 \\
GLG087 & U & 14.7 & MagOpt18329 & 3.08 & \nodata & \nodata & \nodata &
20.7$\pm$0.13 \\
GLG098 & U & 9.44 & MagOpt15970 & 1.97 & \nodata & \nodata & \nodata &
20.3$\pm$0.12 \\
GLG105 & C & 115.4 & MagOpt119137 & 8.44 & \nodata & 19.5$\pm$0.00 &
\nodata & 17.5$\pm$0.06 \\
GLG123 & U & 26.0 & MagOpt42565 & 4.15 & \nodata & \nodata & \nodata &
19.9$\pm$0.11 \\
GLG141 & U & 4.86 & MagOpt12449 & 0.71 & 20.0$\pm$0.10 & 19.5$\pm$0.01
& \nodata & 18.8$\pm$0.03 \\
GLG158 & U & 32.5 & MagOpt4317 & 2.71 & \nodata & \nodata & \nodata &
20.9$\pm$0.22 \\
GLG166 & U & 4.98 & MagOpt42862 & 0.64 & \nodata & \nodata &
22.9$\pm$0.00 & 17.8$\pm$0.00 \\
GLG189 & C & 12.5 & MagOpt61872 & 4.51 & \nodata & \nodata & \nodata
& 20.5$\pm$0.15 \\
GLG200 & U & 45.7 & MagOpt3337 & 1.94 & \nodata & \nodata & \nodata &
20.0$\pm$0.10 \\
GLG224 & U & 6.38 & MagOpt3946 & 1.47 & \nodata & \nodata & \nodata &
20.0$\pm$0.38 \\
GLG240 & U & 14.3 & MagOpt44371 & 0.85 & \nodata & \nodata & \nodata &
19.4$\pm$0.10 \\
GLG286 & U & 6.84 & MagOpt711 & 2.59 & \nodata & \nodata & \nodata &
20.5$\pm$0.17 \\
GLG293 & C & 53.0 & MagOpt10657 & 0.40 & \nodata & 20.0$\pm$0.04 &
\nodata & 18.3$\pm$0.06 \\
GLG337 & U & 13.7 & MagOpt47411 & 3.37 & \nodata & 20.5$\pm$0.11 &
\nodata & 18.2$\pm$0.00 \\
GLG350 & U & 9.26 & MagOpt1439 & 4.00 & \nodata & \nodata & \nodata &
20.5$\pm$0.18 \\
GLG388 & U & 146.4 & MagOpt48400 & 4.10 & \nodata & \nodata & \nodata
& 19.7$\pm$0.09 \\
\enddata
\tablecomments{The number assigned to each MagOpt source corresponds to its
row number in Table 6 of \citet{magstars}.}
\end{deluxetable}
\begin{deluxetable}{cc||cc|cc|c}
\tablecolumns{7}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Good X--ray counterparts of GLG sources \label{xraygdctrp}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{GLG} & \colhead{Source} & \colhead{} & \colhead{} &
\colhead{X--Ray} & \colhead{Sources} & \colhead{} \\
\colhead{Name} & \colhead{Type} & \colhead{Name} & \colhead{Offset
[\arcsec]} & \colhead{HR1} &
\colhead{HR2} & \colhead{$f_X$ [erg cm $^{-2}$ s$^{-1}$]} }
\startdata
GLG005 & U & r3-64 & 1.28 & 0.67$\pm$0.13 & 0.83$\pm$0.09 & \nodata \\
GLG011 & U & r3-70 & 1.09 & 0.34$\pm$0.18 & 0.47$\pm$0.17 & \nodata \\
GLG020 & U & XMM~North~40 & 1.13 & \nodata & \nodata & \nodata \\
GLG021 & U & RXJ0041.7+4126 & 1.47 & \nodata & \nodata & 7.11$\times
10^{-14}$ \\
GLG023 & U & XMM~North~46 & 0.49 & \nodata & \nodata & \nodata \\
GLG028 & U & RXJ0041.6+4103 & 3.71 & \nodata & \nodata & 5.52$\times
10^{-14}$ \\
GLG041 & U & RXJ0043.6+4054 & 10.9 & \nodata & \nodata & 1.22$\times
10^{-13}$ \\
GLG068 & U & RXJ0040.7+4055 & 6.17 & \nodata & \nodata & $2.85\times
10^{-14}$\\
GLG074 & C & RXJ0039.9+4111 & 3.11 & \nodata & \nodata & $<4.65\times
10^{-14}$ \\
GLG088 & U & RXJ0044.6+4145 & 6.99 & \nodata & \nodata & $<2.37\times
10^{-14}$ \\
GLG090 & C & RXJ0040.7+4048 & 6.11 & \nodata & \nodata & $1.8\times
10^{-14}$ \\
GLG105 & C & RXJ0040.2+4050 & 4.51 & 0.97$\pm$0.00 & 0.63$\pm$0.01 &
$3.54\times 10^{-12}$ \\
GLG128 & C & RXJ0043.9+4157 & 2.01 & \nodata & \nodata & $<4.05\times
10^{-14}$ \\
GLG163 & U & RASS-BRIGHT:1RXSJ003832.6+412858 & 9.31 & 0.93$\pm$0.12
& 0.21$\pm$0.20 & \nodata \\
GLG186 & U & RASS-FAINT:1RXSJ003825.4+413708 & 6.69 & 0.70$\pm$0.26
& 0.74$\pm$0.22 & \nodata \\
GLG193 & U & RXJ0046.1+4154 & 4.85 & -0.38$\pm$-0.34 & 0.82$\pm$0.84 &
$2.16\times 10^{-14}$ \\
GLG203 & U & RASS-BRIGHT:1RXSJ004154.0+402118 & 6.58 & 0.62$\pm$0.14
& 0.11$\pm$0.18 & \nodata \\
GLG209 & U & RASS-BRIGHT:1RXSJ004218.8+401942 & 0.85$\pm$0.11 &
0.29$\pm$0.16 & \nodata \\
GLG246 & U & RXJ0040.1+4021 & 6.94 & \nodata & \nodata & $1.38\times
10^{-14}$ \\
GLG250 & U & RXJ0047.5+4149 & 6.69 & \nodata & \nodata & $1.32\times
10^{-13}$ \\
GLG253 & U & XMM~North~244 & 5.21 & \nodata & \nodata & \nodata \\
GLG284 & U & XMM~North~268,RXJ0046.7+4208 & 4.80,4.61 &
\nodata,0.72$\pm$0.14 & \nodata,0.51$\pm$0.14 & \nodata,$5.82\times10^{-14}$ \\
GLG321 & U & XMM~North~222,RXJ0046.0+4220 & 2.84,3.18 &
\nodata,\nodata & \nodata,\nodata & \nodata,$2.58\times 10^{-14}$ \\
GLG342 & C & RXJ0039.5+4008 & 8.31 & \nodata & \nodata &
$2.13\times10^{-14}$ \\
\enddata
\tablecomments{``RASS'' sources are from the {\it ROSAT} All--Sky survey
(\cite{RASSfaint}; \cite{RASSbright}), ``RX'' sources are from the {\it ROSAT}
Second PSPC survey of M31 \citep{rosat}, ``r'' sources are from the
\citet{acis} X--ray survey of M31, ``HRC'' sources are from the \citet{hrc}
X--ray survey of M31, and ``XMM~North'' sources are from the {\it XMM} survey
of M31 (Sergey Trudolyubov, private communication). $f_X$ is for the 0.1--2.0
KeV band and assumes a power law with photon index $\Gamma=-2.0$ and
$N_H=9\times10^{20} \mbox{cm}^{-2}$, and is calculated using data from the
Second {\it ROSAT} PSPC Survey of M31 \citep{rosat}.}
\end{deluxetable}
\begin{deluxetable}{cc|cc||c|c|cc|c}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Radio Properties of M31 SNR and PWN Candidates \label{snrcan}}
\tablehead{
\colhead{Name} & \colhead{$R_{M31}$} & \colhead{$\alpha$} &
\colhead{$\varphi$} & \colhead{Criterion} & \colhead{Size} &
\colhead{$L_{325}$} & \colhead{$L_R$} & \colhead{EM} \\
\colhead{} & \colhead{[kpc]} & \colhead{} & \colhead{} &
\colhead{} & \colhead{[pc$\times$pc]} & \colhead{[mJy~kpc$^{2}$]} &
\colhead{[${\mbox{ergs}}~\mbox{s}^{-1}$]} & \colhead{[pc~cm$^{-6}$]}
}
\startdata
GLG020 & 3.5 & -0.83$\pm$0.05 & \nodata & A~ & $<$23~$\times$~$<$20 &
6.2$\times10^{7}$ & 2.3$\times10^{34}$ & \nodata \\
GLG037 & 9.4 & -0.95$\pm$0.09 & \nodata & A~ & $<$37~$\times$~$<$26 &
7.1$\times10^{7}$ & 2.1$\times10^{34}$ & \nodata \\
GLG056 & 12.2 & -0.60$\pm$0.14 & \nodata & B~ & $<$21~$\times$~$<$18 &
3.1$\times10^{7}$ & 2.3$\times10^{34}$ & \nodata \\
GLG123 & 13.0 & -0.76$\pm$0.17 & \nodata & A~ & $<$20~$\times$~$<$17 &
2.0$\times10^{8}$ & 9.0$\times10^{34}$ & \nodata \\
GLG193 & 13.6 & -0.93$\pm$0.14 & 0.07$\pm$0.02 & D~ & $<$20~$\times$~$<$19 &
6.6$\times10^{7}$ & 2.1$\times10^{34}$ & 2.1$\times10^{5}$ \\
\hline
GLG011 & 5.8 & -0.17$\pm$0.08 & \nodata & HFV(2)~A~ & $<$20~$\times$~$<$18 &
2.8$\times10^{8}$ & 1.2$\times10^{36}$ & \nodata \\
GLG068 & 8.4 & -0.25$\pm$0.09 & \nodata & A~E~ & $<$21~$\times$~$<$19 &
3.2$\times10^{7}$ & 9.7$\times10^{34}$ & \nodata \\
GLG198 & 15.4 & -0.08$\pm$0.20 & -0.07$\pm$0.02 & C~ & $<$23~$\times$~$<$17
& 9.0$\times10^{7}$ & 5.8$\times10^{35}$ & \nodata \\
\enddata
\tablecomments{The first five sources in this table are the SNR
candidates, and the second three are the PWN candidates. The
``Criterion'' column refers to the method of selecting the SNR/PWN
candidates described in Section \ref{snrs}. GLG011 is also a
High-Frequency Variable (HFV) sources, described in Section \ref{variables}.}
\end{deluxetable}
\clearpage
\begin{deluxetable}{cccc||c|c||ccc}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Properties of FRI Candidates \label{frIcan}}
\tablehead{
\colhead{Name} & \colhead{$S_{325}$~[mJy]} & \colhead{$\alpha$} &
\colhead{$\varphi$} & \colhead{Type} & \colhead{Criterion} &
\colhead{IR?} & \colhead{Optical?} & \colhead{X--Ray?}}
\startdata
GLG060 & 82.9$\pm$6.03 & -0.78$\pm$0.11 & \nodata & FRI~ & A~ & N & N & N
\\
GLG090 & 49.8$\pm$6.24 & -0.68$\pm$0.00 & -0.03$\pm$0.00 & FRI~ & A~ & N
& N & Y \\
GLG122 & 18.9$\pm$12.37 & -1.25$\pm$0.15 & \nodata & FRI~ & A~ & Y & N & N
\\
GLG186 & 710.2$\pm$35.47 & 0.98$\pm$0.03 & -0.14$\pm$0.00 & FRI~HzRG~ &
B~C~CSS & N & N & Y \\
GLG284 & 450.8$\pm$19.58 & -0.79$\pm$0.06 & \nodata & FRI~ & B~ & N &
N & Y
\enddata
\tablecomments{The criteria for determining if a source is a FRI radio
galaxy is described in Section \ref{agn}. GLG186 is also a High-z
Radio Galaxy candidate (HzRG), as described in Section \ref{hzrgs}.}
\end{deluxetable}
\begin{deluxetable}{cccc||c|c||ccc}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Properties of FRII Candidates \label{frIIcan}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{Name} & \colhead{$S_{325}$~[mJy]} & \colhead{$\alpha$} &
\colhead{$\varphi$} & \colhead{Type} & \colhead{Criterion} &
\colhead{IR?} & \colhead{Optical?} & \colhead{X--Ray?}
}
\startdata
GLG015 & 206.8$\pm$13.62 & -0.99$\pm$0.08 & \nodata & FRII~Anom.~Source~ &
FRII~Morph.~Anom.~Source~ & N & N & N \\
GLG017 & 27.0$\pm$7.05 & -1.27$\pm$0.09 & \nodata & FRII~NG~ &
FRII~Morph.~I~ & Y & N & N \\
GLG029 & 28.5$\pm$4.85 & -0.81$\pm$0.27 & \nodata & FRII~ & FRII~Morph.~
& N & N & N \\
GLG030 & 76.3$\pm$8.81 & -0.54$\pm$0.10 & \nodata & FRII~NG~ &
FRII~Morph.~I~ & Y & N & N \\
GLG050 & 140.3$\pm$9.10 & -0.57$\pm$0.00 & -0.03$\pm$0.00 & FRII~NG~ &
FRII~Morph.~I~ & Y & N & N \\
GLG074 & 191.5$\pm$12.89 & -0.85$\pm$0.09 & \nodata & FRII~ &
FRII~Morph.~ & N & N & Y \\
GLG078 & 109.3$\pm$8.91 & -1.57$\pm$0.00 & 0.10$\pm$0.00 &
FRII~HzRG~Anom.~Source~ & FRII~Morph.~HzRG~Anom.~Source~ & Y & N & N \\
GLG085 & 247.5$\pm$15.69 & -0.99$\pm$0.09 & \nodata & FRII~ &
FRII~Morph.~ & N & N & N \\
GLG128 & 438.2$\pm$26.92 & -1.07$\pm$0.09 & \nodata & FRII~ &
FRII~Morph.~ & N & N & Y \\
GLG129 & 51.3$\pm$6.98 & -1.45$\pm$0.30 & -0.08$\pm$0.03 & FRII~HzRG~ &
FRII~Morph.~HzRG~ & N & N & N \\
GLG156 & 29.5$\pm$8.61 & -1.16$\pm$0.11 & \nodata & FRII~ & FRII~Morph.~
& N & N & N \\
GLG161 & 30.7$\pm$7.40 & -0.58$\pm$0.17 & \nodata & FRII~ & FRII~Morph.~
& N & N & N \\
GLG162 & 91.2$\pm$12.45 & -0.79$\pm$0.05 & \nodata & FRII~ &
FRII~Morph.~ & N & N & N \\
GLG187 & 73.6$\pm$8.33 & -1.62$\pm$0.09 & \nodata & FRII~HzRG~LFV~ &
FRII~Morph.~HzRG~1~ & N & N & N \\
GLG189 & 12.5$\pm$11.71 & -1.29$\pm$0.45 & \nodata & FRII~ &
FRII~Morph.~ & N & N & N \\
GLG207 & 44.8$\pm$9.29 & -0.74$\pm$0.15 & \nodata & FRII~ & FRII~Morph.~
& N & N & N \\
GLG211 & 40.3$\pm$6.56 & -0.61$\pm$0.08 & \nodata & FRII~ & FRII~Morph.~
& N & N & N \\
GLG220 & 80.9$\pm$11.72 & -1.06$\pm$0.08 & \nodata & FRII~RT~ &
FRII~Morph.~Radio~Triple~Morph.~ & N & N & N \\
GLG296 & 203.1$\pm$13.47 & -1.60$\pm$0.12 & 0.04$\pm$0.02 & FRII~HzRG~LFV~ &
FRII~Morph.~USS~1~ & N & N & N \\
GLG358 & 120.6$\pm$10.68 & -1.17$\pm$0.06 & \nodata & FRII~LFV~ &
FRII~Morph.~2~ & N & N & N \\
GLG359 & 180.3$\pm$16.99 & -1.09$\pm$0.06 & \nodata & FRII~ &
FRII~Morph.~ & N & N & N \\
GLG369 & 79.1$\pm$9.29 & -1.15$\pm$0.08 & \nodata & FRII~ & FRII~Morph.~
& N & N & N \\
GLG370 & 213.3$\pm$15.00 & -0.99$\pm$0.04 & \nodata & FRII~NG~ &
FRII~Morph.~I~ & N & N & N \\
GLG377 & 342.6$\pm$21.72 & -0.88$\pm$0.04 & \nodata & FRII~ &
FRII~Morph.~ & N & N & N \\
\enddata
\tablecomments{The criterion for determine if a source is a FRII radio
galaxy is described in Section \ref{agn}. Several of these sources
fall in other categories as well, listed in the ``Type'' column. In
this column, ``Anom. Source'' means the source has anomalous
morphological and/or multi-wavelength properties and is discussed in
Section \ref{weirdmorph}, ``NG'' means the source also meets the
criteria of a ``normal galaxy'' as defined in Section \ref{galaxies},
``HzRG'' means it is a High-z Radio Galaxy candidate as defined in
Section \ref{hzrgs}, ``LFV'' means it is a Low-Frequency Variable as
defined in Section \ref{variables}, and ``RT'' means it is part of a
Radio Triple. The criteria for their inclusion in these categories
are also listed in the ``Criterion'' column, and are defined in their
respective sections.}
\end{deluxetable}
\begin{deluxetable}{cccc||c|c||cccc}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Properties of Normal Galaxies \label{rgcan}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{Name} & \colhead{$S_{325}$~[mJy]} & \colhead{$\alpha$} &
\colhead{$\varphi$} & \colhead{Type} & \colhead{Criterion} &
\colhead{IR?} & \colhead{Optical?} & \colhead{X--Ray?} &
\colhead{EM~[pc cm$^{-6}$]}
}
\startdata
GLG002 & 3.65$\pm$0.59 & -1.45$\pm$0.34 & 0.13$\pm$0.05 & NG~ & I~ & Y
& N & N & 4.0$\times10^{5}$ \\
GLG003 & 4.76$\pm$0.53 & -0.95$\pm$0.02 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG017 & 27.0$\pm$7.05 & -1.27$\pm$0.09 & \nodata & FRII~NG~ &
FRII~Morph.~I~ & Y & N & N & \nodata \\
GLG018 & 10.6$\pm$0.76 & -1.02$\pm$0.22 & \nodata & NG~NG~ & I~II~ & Y
& N & N & \nodata \\
GLG030 & 76.3$\pm$8.81 & -0.54$\pm$0.10 & \nodata & FRII~NG~ &
FRII~Morph.~I~ & Y & N & N & \nodata \\
GLG041 & 191.1$\pm$7.32 & -0.97$\pm$0.08 & 0.03$\pm$0.01 & NG~LFV~ &
III~1~ & N & N & Y & 1.0$\times10^{5}$ \\
GLG050 & 140.3$\pm$9.10 & -0.57$\pm$0.00 & -0.03$\pm$0.00 & FRII~NG~ &
FRII~Morph.~I~ & Y & N & N & 1.0$\times10^{5}$ \\
GLG079 & 9.08$\pm$0.73 & -1.04$\pm$0.05 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG105 & 115.4$\pm$8.11 & -0.74$\pm$0.10 & \nodata & NG~NG~BL-Lac~HFV~ &
I~III~Prev.~Ident.~1~ & Y & N & Y & \nodata \\
GLG127 & 13.5$\pm$0.78 & -0.62$\pm$0.05 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG134 & 6.42$\pm$0.55 & -0.39$\pm$0.11 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG163 & 8.30$\pm$0.72 & -0.34$\pm$0.08 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG166 & 4.98$\pm$0.62 & -0.76$\pm$0.08 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG184 & 8.78$\pm$0.65 & $\leq$-0.85 & \nodata & NG~NG~ & I~II~ & Y &
N & N & \nodata \\
GLG203 & 36.9$\pm$1.71 & -0.94$\pm$0.08 & 0.05$\pm$0.01 & NG~NG~NG~ &
I~II~III~ & Y & N & Y & 1.6$\times10^{5}$ \\
GLG204 & 70.0$\pm$4.29 & -0.65$\pm$0.04 & \nodata & NG~NG~ & I~II~ & Y
& N & N & \nodata \\
GLG209 & 10.4$\pm$0.89 & -0.75$\pm$0.09 & \nodata & NG~NG~ & I~II~ & Y
& N & Y & \nodata \\
GLG225 & 10.0$\pm$0.82 & -1.00$\pm$0.14 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG247 & 40.8$\pm$9.25 & -0.87$\pm$0.16 & \nodata & NG~Galaxy-Merger~ &
I~Morph.~Class. & Y & N & N & \nodata \\
GLG253 & 14.0$\pm$0.90 & -1.11$\pm$0.04 & \nodata & NG~ & I~ & Y & N &
Y & \nodata \\
GLG287 & 16.5$\pm$0.97 & -0.79$\pm$0.07 & \nodata & NG~ & I~ & Y & N &
N & \nodata \\
GLG335 & 8.89$\pm$0.77 & $\leq$-0.86 & \nodata & NG~ & I~ & Y & N
& N & \nodata \\
GLG387 & 88.3$\pm$10.98 & -0.78$\pm$0.08 & \nodata & NG~ & I~ & Y & N
& N & \nodata \\
\enddata
\tablecomments{The criteria used to determine if a source is a
``normal galaxy'' are defined in Section \ref{galaxies}. Several of
these sources fall into other categories, as listed in the ``Type''
column. FRII sources are discussed in Section \ref{agn}, ``LFV'' (Low
Frequency Variable) sources are discussed in Section \ref{variables},
as are ``HFV'' (High Frequency Variable) sources. GLG105 has
previously been identified as a BL~Lac, discussed in Section
\ref{bllac}m and GLG247 has the morphology of a galaxy merger, as
discussed in Section \ref{glg247}.}
\end{deluxetable}
\begin{deluxetable}{cccc||c|c||ccc}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Properties of HzRG Candidates \label{hzrgcan}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{Name} & \colhead{$S_{325}$~[mJy]} & \colhead{$\alpha$} &
\colhead{$\varphi$} & \colhead{Type} & \colhead{Criterion} &
\colhead{IR?} & \colhead{Optical?} & \colhead{X--Ray?}
}
\startdata
GLG078 & 109.3$\pm$8.91 & -1.57$\pm$0.00 & 0.10$\pm$0.00 &
FRII~HzRG~Anom.~Source~ & FRII~Morph.~USS~Anom.~Source~ & Y & N & N \\
GLG129 & 51.3$\pm$6.98 & -1.45$\pm$0.30 & -0.08$\pm$0.03 & FRII~HzRG~ &
FRII~Morph.~USS~ & N & N & N \\
GLG187 & 73.6$\pm$8.33 & -1.62$\pm$0.09 & \nodata & FRII~HzRG~LFV~ &
FRII~Morph.~USS~1~ & N & N & N \\
GLG296 & 203.1$\pm$13.47 & -1.60$\pm$0.12 & 0.04$\pm$0.02 & FRII~HzRG~LFV~ &
FRII~Morph.~USS~1~ & N & N & N \\
\hline
GLG084 & 8.52$\pm$0.58 & -0.92$\pm$0.16 & 0.13$\pm$0.02 & HzRG~ & CSS/GPS~
& N & N & N \\
GLG101 & 12.8$\pm$1.21 & -1.26$\pm$0.16 & 0.16$\pm$0.02 & HzRG~ & CSS/GPS~
& N & N & N \\
GLG133 & 7.57$\pm$0.70 & -1.36$\pm$0.23 & 0.14$\pm$0.03 & HzRG~ & CSS/GPS~
& N & N & N \\
GLG135 & 189.6$\pm$9.61 & -0.55$\pm$0.01 & 0.02$\pm$0.00 & HzRG~ & CSS/GPS~
& N & N & N \\
GLG152 & 7.68$\pm$0.56 & -1.63$\pm$0.23 & 0.19$\pm$0.03 & HzRG~CESS~ &
CSS/GPS~CESS~ & N & N & N \\
GLG154 & 60.0$\pm$3.05 & -1.04$\pm$0.03 & 0.03$\pm$0.00 & HzRG~ & CSS/GPS~
& N & N & N \\
GLG179 & 16.7$\pm$0.65 & -1.46$\pm$0.25 & 0.10$\pm$0.03 & HzRG~ & CSS/GPS~
& N & N & N \\
GLG183 & 23.1$\pm$1.28 & -0.84$\pm$0.14 & 0.05$\pm$0.02 & HzRG~ &
CSS/GPS~ & N & N & N \\
GLG210 & 11.1$\pm$0.67 & -0.70$\pm$0.00 & 0.08$\pm$0.00 & HzRG~ & CSS/GPS~
& N & N & N \\
\hline
GLG077 & 530.9$\pm$26.03 & -0.58$\pm$0.00 & -0.03$\pm$0.00 & HzRG~ & CSS
& N & N & N \\
GLG186 & 710.2$\pm$35.47 & 0.98$\pm$0.03 & -0.14$\pm$0.00 &
FRI~FRI~HzRG~ & B~C~CSS & N & N & Y \\
GLG232 & 30.3$\pm$2.00 & -0.24$\pm$0.00 & -0.05$\pm$0.00 & HzRG~ & CSS
& N & N & N \\
\enddata
\tablecomments{The criteria for categorizing sources as High-z Radio
Galaxy (HzRG) candidates are described in Section \ref{hzrgs}.
Several of these sources fall under other classifications as well,
as listed in the ``Type'' column. FRI and FRII sources are
described in Section \ref{agn}, Anomalous sources (``Anom. Source'')
are described in Section \ref{weirdmorph}, Low Frequency Variable
(``LFV'') sources are described in Section \ref{variables}, and
Compact Extremely Steep Spectrum (``CESS'') sources are described in
Section \ref{cess}. The criteria for cataloging these sources as such
are listed under the ``Criterion'' column and described in their
respective sections.}
\end{deluxetable}
\begin{deluxetable}{ccc|cc||cc|cc||c}
\tablecolumns{10}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Low-Frequency Variable sources \label{lfvcan}}
\setlength{\tabcolsep}{0.04in}
\tablehead{
\colhead{Name} & \colhead{Type} & \colhead{$R_{M31}$~[kpc]} &
\colhead{$\alpha$} & \colhead{$\varphi$} & \colhead{WENSS Name} &
\colhead{Offset~[\arcsec]} & \colhead{$S_{325}^{GLG}$~[mJy]}
& \colhead{$S_{325}^{WENSS}$~[mJy]} &
\colhead{$\frac{\Delta S_{325}}{\overline{S_{325}}}$}
}
\startdata
GLG009 & ExG & 5.4 & -1.28 & 0.10 & WNB0040.8+4103 & 1.70 & 45.87$\pm$4.46 &
84.00$\pm$4.00 & 0.58$\pm$0.09 \\
GLG041 & U & 21.9 & -0.97 & 0.03 & WNB0040.9+4038 & 2.14 & 191.13$\pm$7.32 &
254.00$\pm$3.90 & 0.28$\pm$0.03 \\
GLG187 & C & 52.8 & -1.62 & \nodata & WNB0043.1+4020 & 10.97 & 73.69$\pm$8.33
& 16.00$\pm$3.90 & 1.28$\pm$0.27 \\
GLG200 & U & 21.5 & -0.98 & 0.02 & \nodata & \nodata & 45.74$\pm$2.58 &
$\leq$30 & $\geq$0.41 \\
GLG259 & U & 61.1 & -0.96 & \nodata & \nodata & \nodata & 43.98$\pm$2.14 &
$\leq$30 & $\geq$0.37 \\
GLG280 & U & 66.6 & -0.70 & \nodata & \nodata & \nodata & 69.80$\pm$3.44 &
$\leq$30 & $\geq$0.79 \\
GLG296 & C & 42.2 & -1.60 & 0.04 & WNB0046.0+4120 & 0.73 & 203.17$\pm$13.47
& 277.00$\pm$3.60 & 0.30$\pm$0.05 \\
GLG340 & U & 22.1 & -1.15 & 0.02 & WNB0035.0+4008 & 2.80 & 263.31$\pm$11.66
& 330.00$\pm$3.50 & 0.22$\pm$0.04 \\
GLG354 & U & 66.5 & -1.06 & \nodata & WNB0047.0+4054 & 32.05 & 14.59$\pm$0.72
& 42.00$\pm$3.70 & 0.96$\pm$0.16 \\
GLG398 & U & 64.9 & -0.50 & \nodata & WNB0047.9+4111 & 3.29 & 55.34$\pm$3.02
& 26.00$\pm$3.50 & 0.72$\pm$0.12 \\
\hline
GLG359 & C & 81.2 & -1.09$\pm$0.06 & \nodata & WNB0045.8+4013 & 3.80 &
180.30$\pm$16.99 & 198.00$\pm$4.00 & 0.09$\pm$0.09
\enddata
\tablecomments{The ``Type'' column refers to the 325~MHz morphology of
the sources, as described in Section \ref{intro}. $R_{M31}$~is the
projected radius of the sources in M31, as defined in Section 4.1
Equation 3 of Paper I \citet{paper1}.}
\end{deluxetable}
\begin{deluxetable}{c||cc|ccc}
\tablecolumns{6}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{1.4 GHz Properties of Low-Frequency Variable sources
\label{lfvcanhf}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{Name} & \colhead{1.4~GHz Counterparts} &
\colhead{Offset~[\arcsec]} & \colhead{$S_{1400}$~[mJy]} &
\colhead{$\overline{S_{1400}}$~[mJy]} &
\colhead{$\frac{\Delta S_{1400}}{\overline{S_{1400}}}$}
}
\startdata
GLG009 & NVSS004336+412020,37W158C & 7.75,2.62 & 10.67$\pm$0.46,27.50$\pm$0.40
& 19.08$\pm$0.43 & 0.88$\pm$0.03 \\
GLG041 & NVSS004341+405428,Braun~162,37W168 & 1.80,0.55,1.27 &
66.11$\pm$0.46,54.00$\pm$2.70,66.50$\pm$3.40 & 65.77$\pm$0.45 & 0.19$\pm$0.06
\\
GLG187 & NVSS004552+403628 & 5.0 & 6.14$\pm$6.14 & \nodata & \nodata \\
GLG200 & NVSS004111+402410,37W075 & 0.55,1.12 & 13.47$\pm$0.46,15.50$\pm$0.90
& 14.48$\pm$0.71 & 0.13$\pm$0.07 \\
GLG259 & NVSS004550+402257 & 2.8 & 10.65$\pm$10.65 & \nodata & \nodata \\
GLG280 & NVSS004624+402215 & 0.3 & 24.18$\pm$24.18 & \nodata & \nodata \\
GLG296 & NVSS004846+413715,37W249 & 1.44,0.82 & 30.45$\pm$0.46,21.90$\pm$2.60
& 26.17$\pm$1.86 & 0.32$\pm$0.10 \\
GLG340 & NVSS003745+402515,37W008 & 1.45,2.25 & 61.11$\pm$0.46,56.40$\pm$11.30
& 58.75$\pm$7.99 & 0.08$\pm$0.19 \\
GLG354 & NVSS004948+41116 & 7.2
& 3.05$\pm$3.05 & \nodata & \nodata \\
GLG358 & NVSS003544+413015 & 2.7 & 21.67$\pm$21.67 & \nodata & \nodata \\
GLG374 & NVSS003522+41165 & 0.5 & 25.73$\pm$25.73 & \nodata & \nodata \\
GLG393 & NVSS003453+413131 & 0.8 & 99.77$\pm$99.77 & \nodata & \nodata \\
GLG398 & NVSS005043+41282 & 0.4 & 27.83$\pm$27.83 & \nodata & \nodata \\
\enddata
\end{deluxetable}
\begin{deluxetable}{c|cc||ccc||c}
\tablecolumns{7}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{High-Frequency Variable sources \label{hfvcan}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{Name} & \colhead{$\alpha$} & \colhead{$\varphi$} &
\colhead{1.4~GHz Counterparts} & \colhead{Offset~[\arcsec]} &
\colhead{$S_{1400}$~[mJy]} & \colhead{$\frac{\Delta
S_{1400}}{\overline{S_{1400}}}$}
}
\startdata
GLG004 & -0.84 & 0.04 & NVSS004318+411729,37W158A & 7.96,11.73 &
3.76$\pm$0.46,12.60$\pm$0.30 & 1.07$\pm$0.08 \\
GLG009 & -1.28 & 0.10 & NVSS004336+412020,37W158C & 7.75,2.62 &
10.67$\pm$0.46,27.50$\pm$0.40 & 0.88$\pm$0.03 \\
GLG011 & -0.17 & -0.01 & NVSS004251+412633,Braun~103,37W144 & 1.25,0.31,4.87
& 20.11$\pm$0.46,25.21$\pm$0.24,23.70$\pm$1.00 & 0.21$\pm$0.02 \\
GLG019 & -0.71 & -0.01 & NVSS004218+412926,Braun~61,37W115 & 0.22,0.18,0.49 &
363.30$\pm$0.46,351.46$\pm$1.45,373.40$\pm$15.00 & 0.06$\pm$0.04 \\
GLG105 & -0.74 & 0.00 & NVSS004013+40505,37W051 & 5.02,6.46 &
39.52$\pm$0.46,51.80$\pm$1.70 & 0.26$\pm$0.03 \\
GLG266 & -2.25 & \nodata & NVSS004126+401235,37W083B & 4.32,1.47 &
2.72$\pm$0.46,41.10$\pm$3.10 & 1.75$\pm$0.22 \\
\hline
GLG161 & -0.58 & -0.02 & NVSS003932+40441,37W036 & 2.45,3.44 &
9.71$\pm$0.46,9.70$\pm$0.60 & 0.00$\pm$0.07 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{ccc||cc|cc|cc}
\tablecolumns{9}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{325~MHz Properties of High-Frequency Variable sources
\label{hfvcanlf}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{GLG Name} & \colhead{Type} & \colhead{$R_{M31}$} &
\colhead{WENSS Name} & \colhead{Offset~[\arcsec]} &
\colhead{$S_{325}^{GLG}$~[mJy]} & \colhead{$S_{325}^{WENSS}$~[mJy]} &
\colhead{$\overline{S_{325}}$~[mJy]} &
\colhead{$\frac{\Delta S_{325}}{\overline{S_{325}}}$}
}
\startdata
GLG004 & U & 4.2 & \nodata & \nodata & 22.85$\pm$0.70 & \nodata & \nodata &
\nodata \\
GLG009 & Ex & 5.4 & WNB0040.8+4103 & 1.70 & 45.87$\pm$4.46 & 84.00$\pm$4.00
& 64.93$\pm$4.23 & 0.58$\pm$0.09 \\
GLG011 & U & 5.8 & WNB0040.1+4110 & 4.38 & 37.23$\pm$1.90 & 22.00$\pm$3.90 &
29.61$\pm$3.06 & 0.51$\pm$0.15 \\
GLG019 & U & 12.3 & WNB0039.5+4113 & 0.35 & 1223.64$\pm$61.20 &
1190.00$\pm$3.90 & 1206.82$\pm$43.36 & 0.02$\pm$0.05 \\
GLG105 & C & 11.0 & \nodata & \nodata & 115.49$\pm$8.11 & \nodata & \nodata &
\nodata \\
GLG266 & U & 30.5 & \nodata & \nodata & 96.05$\pm$2.13 & \nodata & \nodata &
\nodata \\
\enddata
\tablecomments{The ``Type'' column refers to the 325~MHz morphology of
the sources, as described in Section \ref{intro}. $R_{M31}$~is the
projected radius of the sources in M31, as defined in Section 4.1
Equation 3 of Paper I \citet{paper1}.}
\end{deluxetable}
\begin{deluxetable}{ccc|ccc||c|c||ccc}
\tablecolumns{11}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Properties of CESS Sources \label{cesscan}}
\setlength{\tabcolsep}{0.02in}
\tablehead{
\colhead{Name} & \colhead{Type} & \colhead{$R_{M31}$~[kpc]} &
\colhead{$S_{325}$~[mJy]} & \colhead{$\alpha$} &
\colhead{$\varphi$} & \colhead{Source Type} & \colhead{Criteria} &
\colhead{IR?} & \colhead{Optical?} & \colhead{X--Ray?}
}
\startdata
GLG006 & U & 8.5 & 18.1$\pm$0.58 & $\leq$-3.18 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG007 & U & 9.0 & 5.45$\pm$0.58 & $\leq$-2.37 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG014 & U & 7.3 & 23.4$\pm$0.69 & $\leq$-3.35 & \nodata & CESS~ &
CESS~ & N & N & N \\
GLG022 & El & 16.3 & 11.6$\pm$1.79 & $\leq$-2.81 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG036 & El & 20.8 & 38.3$\pm$2.59 & $\leq$-3.53 & \nodata & CESS~ &
CESS~ & N & Y & N \\
GLG046 & U & 18.8 & 5.76$\pm$0.64 & -2.01$\pm$0.16 & 0.18$\pm$0.02 & CESS~ &
CESS~ & N & N & N \\
GLG047 & U & 25.1 & 8.84$\pm$0.75 & -1.64$\pm$0.10 & \nodata & CESS~ &
CESS~ & N & N & N \\
GLG055 & U & 27.4 & 18.7$\pm$1.10 & $\leq$-2.01 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG059 & U & 26.0 & 30.6$\pm$0.94 & -1.77$\pm$0.08 & 0.17$\pm$0.01 &
RJ~CESS~ & Radio~Jet~Morph.~CESS~ & N & N & N \\
GLG062 & U & 26.4 & 24.9$\pm$1.37 & $\leq$-2.94 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG104 & U & 10.6 & 16.4$\pm$0.89 & $\leq$-2.07 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG140 & U & 38.0 & 52.4$\pm$2.82 & $\leq$-3.00 & \nodata & CESS~ &
CESS~ & N & N & N \\
GLG149 & U & 26.3 & 7.31$\pm$0.71 & $\leq$-2.22 & \nodata & CESS~ &
CESS~ & N & N & N \\
GLG152 & U & 39.3 & 7.68$\pm$0.56 & -1.63$\pm$0.23 & 0.19$\pm$0.03 &
HzRG~CESS~ & CSS/GPS~CESS~ & N & N & N \\
GLG160 & U & 19.1 & 13.7$\pm$0.75 & $\leq$-2.99 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG171 & U & 25.7 & 18.8$\pm$1.09 & $\leq$-2.90 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG263 & U & 65.6 & 26.2$\pm$1.08 & $\leq$-1.60 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG289 & U & 16.1 & 11.0$\pm$0.77 & $\leq$-1.94 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG294 & U & 16.1 & 11.5$\pm$0.76 & $\leq$-2.01 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG305 & U & 25.3 & 26.9$\pm$1.31 & $\leq$-3.11 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG319 & U & 22.2 & 10.2$\pm$0.83 & $\leq$-2.45 & \nodata & CESS~ & CESS~ &
N & N & N \\
GLG405 & U & 26.9 & 22.7$\pm$1.33 & $\leq$-2.75 & \nodata & CESS~ & CESS~ &
N & N & N \\
\enddata
\tablecomments{The ``Type'' column refers to the 325~MHz morphology of
the sources, as described in Section \ref{intro}. $R_{M31}$~is the
projected radius of the sources in M31, as defined in Section 4.1
Equation 3 of Paper I \citet{paper1}. The criteria for classifying a
sources as a CESS sources is described in Section \ref{cess}.
GLG059 is also part of a radio jet, as described in Section
\ref{agn}, while GLG152 is also a High-z Radio Galaxy (HzRG)
candidate, as described in Section \ref{hzrgs}. The criteria for
these identifications are listed in the ``Criteria'' column, and
discussed in the respective sections.}
\end{deluxetable}
\begin{deluxetable}{cc||ccc|cc}
\tablecolumns{7}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{High Frequency Variability explanation for CESS Sources
\label{cesshfv}}
\tablehead{
\colhead{Name} & \colhead{$S_{325}$~[mJy]} &
\colhead{$S_{1400}^{exp}$~[mJy]} &
\colhead{$S_{1400}^{meas}$~[mJy]} &
\colhead{Survey} &
\colhead{$\frac{\Delta S_{1400}}{\overline{S_{1400}}}$} &
\colhead{$\overline{\mbox{Var}_{1400}}$}
}
\startdata
GLG006 & 18.1 & 8.73 & $\leq$0.15 & Braun & $\geq$1.93 & $\geq$78 \\
GLG007 & 5.45 & 2.62 & $\leq$0.15 & Braun & $\geq$1.78 & $\geq$22 \\
GLG014 & 23.4 & 11.2 & $\leq$0.15 & Braun & $\geq$1.94 & $\geq$101 \\
GLG022 & 11.6 & 5.59 & $\leq$0.16 & Braun & $\geq$1.88 & $\geq$49 \\
GLG036 & 38.3 & 18.4 & $\leq$0.18 & Braun & $\geq$1.96 & $\geq$166 \\
GLG046 & 5.76 & 2.77 & 1.72 & Braun & 0.46 & 9 \\
GLG047 & 8.84 & 4.26 & 0.74 & Braun & 1.40 & 31 \\
GLG055 & 18.7 & 9.02 & $\leq$1.01 & Braun & $\geq$1.59 & $\geq$72 \\
GLG059 & 30.6 & 14.7 & 12.8 & NVSS & 0.14 & 24 \\
GLG062 & 24.9 & 12.0 & $\leq$0.29 & Braun & $\geq$1.90 & $\geq$106 \\
GLG104 & 16.4 & 7.92 & $\leq$1.34 & 37W & $\geq$1.42 & $\geq$27 \\
GLG140 & 52.4 & 25.2 & $\leq$2.50 & NVSS & $\geq$1.64 & $\geq$284 \\
GLG149 & 7.31 & 3.52 & $\leq$0.25 & Braun & $\geq$1.72 & $\geq$29 \\
GLG152 & 7.68 & 3.70 & 4.04 & NVSS & 0.08 & 4 \\
GLG160 & 13.7 & 6.62 & $\leq$0.15 & Braun & $\geq$1.91 & $\geq$58 \\
GLG171 & 18.8 & 9.09 & $\leq$0.23 & Braun & $\geq$1.89 & $\geq$80 \\
GLG263 & 26.2 & 12.6 & $\leq$2.50 & NVSS & $\geq$1.34 & $\geq$126 \\
GLG289 & 11.0 & 5.33 & $\leq$2.50 & NVSS & $\geq$0.72 & $\geq$35 \\
GLG294 & 11.5 & 5.55 & $\leq$2.50 & NVSS & $\geq$0.75 & $\geq$38 \\
GLG305 & 26.9 & 13.0 & $\leq$2.50 & NVSS & $\geq$1.35 & $\geq$131 \\
GLG319 & 10.2 & 4.91 & $\leq$0.25 & Braun & $\geq$1.80 & $\geq$42 \\
GLG405 & 22.7 & 10.9 & $\leq$2.50 & NVSS & $\geq$1.25 & $\geq$105 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{cc||ccc|cc}
\tablecolumns{7}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Low Frequency Variability explanation for CESS Sources
\label{cesslfv}}
\tablehead{
\colhead{Name} & \colhead{$S_{325}$~[mJy]} &
\colhead{$S_{1400}^{meas}$~[mJy]} & \colhead{Survey} &
\colhead{$S_{325}^{exp}$~[mJy]} &
\colhead{$\frac{\Delta S_{325}}{\overline{S_{325}}}$} &
\colhead{$\overline{\mbox{Var}_{325}}$}
}
\startdata
GLG006 & 18.1 & $\leq$0.15 & Braun & 0.31 & $\geq$1.93 & $\geq$161 \\
GLG007 & 5.45 & $\leq$0.15 & Braun & 0.31 & $\geq$1.78 & $\geq$46 \\
GLG014 & 23.4 & $\leq$0.15 & Braun & 0.31 & $\geq$1.94 & $\geq$210 \\
GLG022 & 11.6 & $\leq$0.16 & Braun & 0.34 & $\geq$1.88 & $\geq$102 \\
GLG036 & 38.3 & $\leq$0.18 & Braun & 0.38 & $\geq$1.96 & $\geq$345 \\
GLG046 & 5.76 & 1.72 & Braun & 3.58 & 0.46 & 19 \\
GLG047 & 8.84 & 0.74 & Braun & 1.54 & 1.40 & 66 \\
GLG055 & 18.7 & $\leq$1.01 & Braun & 2.10 & $\geq$1.59 & $\geq$151 \\
GLG059 & 30.6 & 12.8 & NVSS & 26.6 & 0.14 & 50 \\
GLG062 & 24.9 & $\leq$0.29 & Braun & 0.61 & $\geq$1.90 & $\geq$221 \\
GLG104 & 16.4 & $\leq$1.34 & 37W & 2.78 & $\geq$1.42 & $\geq$56 \\
GLG140 & 52.4 & $\leq$2.50 & NVSS & 5.18 & $\geq$1.64 & $\geq$591 \\
GLG149 & 7.31 & $\leq$0.25 & Braun & 0.53 & $\geq$1.72 & $\geq$61 \\
GLG152 & 7.68 & 4.04 & NVSS & 8.40 & 0.08 & 9 \\
GLG160 & 13.7 & $\leq$0.15 & Braun & 0.31 & $\geq$1.91 & $\geq$122 \\
GLG171 & 18.8 & $\leq$0.23 & Braun & 0.49 & $\geq$1.89 & $\geq$167 \\
GLG263 & 26.2 & $\leq$2.50 & NVSS & 5.18 & $\geq$1.34 & $\geq$263 \\
GLG289 & 11.0 & $\leq$2.50 & NVSS & 5.18 & $\geq$0.72 & $\geq$73 \\
GLG294 & 11.5 & $\leq$2.50 & NVSS & 5.18 & $\geq$0.75 & $\geq$79 \\
GLG305 & 26.9 & $\leq$2.50 & NVSS & 5.18 & $\geq$1.35 & $\geq$272 \\
GLG319 & 10.2 & $\leq$0.25 & Braun & 0.52 & $\geq$1.80 & $\geq$87 \\
GLG405 & 22.7 & $\leq$2.50 & NVSS & 5.18 & $\geq$1.25 & $\geq$219 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{cc||ccc}
\tablecolumns{5}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Extreme Scattering Event explanation for CESS Sources
\label{cessess}}
\tablehead{
\colhead{Name} & \colhead{$S_{325}$~[mJy]} &
\colhead{$S_{325}^{ESE}$~[mJy]} &
\colhead{$S_{1400}^{meas}$~[mJy]} & \colhead{$\alpha^{ESE}$}
}
\startdata
GLG006 & 18.1 & 1.81 & $\leq$0.15 & $\leq$-1.70 \\
GLG007 & 5.45 & 0.54 & $\leq$0.15 & $\leq$-0.87 \\
GLG014 & 23.4 & 2.34 & $\leq$0.15 & $\leq$-1.88 \\
GLG022 & 11.6 & 1.16 & $\leq$0.16 & $\leq$-1.32 \\
GLG036 & 38.3 & 3.83 & $\leq$0.18 & $\leq$-2.07 \\
GLG046 & 5.76 & 0.57 & 1.72 & 0.75 \\
GLG047 & 8.84 & 0.88 & 0.74 & -0.11 \\
GLG055 & 18.7 & 1.87 & $\leq$1.01 & $\leq$-0.41 \\
GLG059 & 30.6 & 3.06 & 12.8 & 0.98 \\
GLG062 & 24.9 & 2.49 & $\leq$0.29 & $\leq$-1.46 \\
GLG104 & 16.4 & 1.64 & $\leq$1.34 & $\leq$-0.13 \\
GLG140 & 52.4 & 5.24 & $\leq$2.50 & $\leq$-0.50 \\
GLG149 & 7.31 & 0.73 & $\leq$0.25 & $\leq$-0.72 \\
GLG152 & 7.68 & 0.76 & 4.04 & 1.13 \\
GLG160 & 13.7 & 1.37 & $\leq$0.15 & $\leq$-1.51 \\
GLG171 & 18.8 & 1.88 & $\leq$0.23 & $\leq$-1.42 \\
GLG263 & 26.2 & 2.62 & $\leq$2.50 & $\leq$-0.03 \\
GLG289 & 11.0 & 1.10 & $\leq$2.50 & $\leq$0.55 \\
GLG294 & 11.5 & 1.15 & $\leq$2.50 & $\leq$0.53 \\
GLG305 & 26.9 & 2.69 & $\leq$2.50 & $\leq$-0.05 \\
GLG319 & 10.2 & 1.02 & $\leq$0.25 & $\leq$-0.95 \\
GLG405 & 22.7 & 2.27 & $\leq$2.50 & $\leq$0.06 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{cc||cc|cc}
\tablecolumns{6}
\tablewidth{0pc}
\tableheadfrac{0.05}
\tabletypesize{\scriptsize}
\tablecaption{Pulsar explanation for CESS Sources \label{cesspsr}}
\tablehead{
\colhead{Name} & \colhead{$R_{M31}$~[kpc]} &
\colhead{$L_{400}$~[mJy~kpc$^2$]} &
\colhead{$L_{1400}$~[mJy~kpc$^2$]} & $\alpha_L^{400}$ &
\colhead{$\alpha_L^{1400}$}
}
\startdata
GLG006 & 8.5 & $\leq$5.6$\times10^{6}$ & $\leq$1.0$\times10^{5}$ &
$\leq$-29.1 & $\leq$-4.8 \\
GLG007 & 9.0 & $\leq$2.0$\times10^{6}$ & $\leq$1.0$\times10^{5}$ &
$\leq$-23.3 & $\leq$-3.9 \\
GLG014 & 7.3 & $\leq$7.1$\times10^{6}$ & $\leq$1.0$\times10^{5}$ &
$\leq$-30.3 & $\leq$-4.9 \\
GLG022 & 16.3 & $\leq$3.9$\times10^{6}$ & $\leq$1.1$\times10^{5}$ &
$\leq$-26.9 & $\leq$-4.5 \\
GLG036 & 20.8 & $\leq$1.1$\times10^{7}$ & $\leq$1.3$\times10^{5}$ &
$\leq$-32.7 & $\leq$-5.3 \\
GLG046 & 18.8 & 2.3$\times10^{6}$ & 1.8$\times10^{5}$ & -23.5 & -4.0 \\
GLG047 & 25.1 & 3.8$\times10^{6}$ & 4.9$\times10^{5}$ & -25.6 & -4.3 \\
GLG059 & 26.0 & 1.2$\times10^{7}$ & 1.3$\times10^{6}$ & -31.6 & -5.1 \\
GLG062 & 26.4 & $\leq$8.2$\times10^{6}$ & $\leq$2.0$\times10^{5}$ &
$\leq$-30.6 & $\leq$-5.0 \\
GLG104 & 10.6 & $\leq$6.5$\times10^{6}$ & $\leq$4.8$\times10^{5}$ &
$\leq$-28.6 & $\leq$-4.7 \\
GLG149 & 26.3 & $\leq$2.8$\times10^{6}$ & $\leq$1.7$\times10^{5}$ &
$\leq$-24.7 & $\leq$-4.1 \\
GLG160 & 19.1 & $\leq$4.4$\times10^{6}$ & $\leq$1.0$\times10^{5}$ &
$\leq$-27.7 & $\leq$-4.6 \\
GLG171 & 25.7 & $\leq$6.2$\times10^{6}$ & $\leq$1.6$\times10^{5}$ &
$\leq$-29.3 & $\leq$-4.8 \\
GLG289 & 16.1 & $\leq$4.4$\times10^{6}$ & $\leq$3.9$\times10^{5}$ &
$\leq$-26.7 & $\leq$-4.4 \\
GLG294 & 16.1 & $\leq$4.6$\times10^{6}$ & $\leq$3.7$\times10^{5}$ &
$\leq$-26.9 & $\leq$-4.5 \\
GLG305 & 25.3 & $\leq$8.6$\times10^{6}$ & $\leq$1.7$\times10^{5}$ &
$\leq$-31.0 & $\leq$-5.0 \\
GLG319 & 22.2 & $\leq$3.7$\times10^{6}$ & $\leq$1.7$\times10^{5}$ &
$\leq$-26.3 & $\leq$-4.4 \\
GLG405 & 26.9 & 7.7$\times10^{6}$ & 2.4$\times10^{5}$ & -30.2 & -4.9 \\
\hline
GLG055 & 27.4 & $\leq$1.2$\times10^{3}$ & $\leq$9.8$\times10^{1}$ & $\leq$12.6
& $\leq$1.1 \\
GLG140 & 38.0 & $\leq$2.8$\times10^{3}$ & $\leq$6.5$\times10^{1}$ & $\leq$7.7
& $\leq$0.4 \\
GLG152 & 39.3 & 5.4$\times10^{2}$ & 7.0$\times10^{1}$ & 16.9 & 1.7 \\
GLG263 & 65.6 & $\leq$1.8$\times10^{3}$ & $\leq$2.5$\times10^{2}$ & $\leq$11.0
& $\leq$0.8 \\
\enddata
\tablecomments{The sources above the horizontal line lie within the
optical disk of M31, while the sources below the line don't.}
\end{deluxetable}
\begin{table}
\caption{ Classification Summary \label{srcsum}}
\begin{center}
\begin{tabular}{rrcc}
\hline
\hline
{\bf Number of Classified Sources:} & & {\bf 112 } & \\
\hline
{\it M31 Sources}: & & {\it 8 } & \\
& SNR Candidates: & & 5 \\
& PWN Candidates: & & 3 \\
\hline
{\it Milky~Way Sources}: & & {\it 4 } & \\
& Radio Stars: & & 3 \\
& PN Candidates: & & 1 \\
\hline
{\it Extragalactic Sources}: & & {\it 66 } & \\
& FRIs: & & 6 \\
& FRIIs: & & 24 \\
& Radio Jets: & & 5 (11) \\
& Radio Triples: & & 1 (2) \\
& Normal Galaxies: & & 23 \\
& HzRG Candidates: & & 16 \\
& Galaxy Mergers: & & 1 \\
& BL Lacs: & & 1 \\
& GRG Candidates: & & 1 (2) \\
\hline
{\it Variable Sources:} & & {\it 15} & \\
& Low Frequency Variables: & & 10 \\
& High Frequency Variables: & & 6 \\
\hline
{\it Anomalous Sources:} & & {\it 25} & \\
& CESS sources: & & 22 \\
& Anom. Morph./Multiwave. Props.: & & 3 \\
\end{tabular}
\end{center}
{\scriptsize {\it Note:} Some GLG sources fall into multiple categories. SNR
stands for Supernova Remnant, PWN stands for Pulsar Wind Nebula, PN
stands for Planetary Nebula, HzRG stands for High--z Radio Galaxy, and
CESS stands for ``Compact Extremely Steep Spectrum.'' The number is
parenthesis indicate the number of GLG sources that fall into a
certain category, e.g. five radio jets were detected which comprise 11
GLG sources since, in some case, the components of the jets were
classified as separate GLG sources.} \\
\end{table}
\clearpage
\newpage
\figcaption{The location of steep--spectrum sources (upper left),
flat--spectrum sources (upper right), and sources with spectral turnover
(bottom), as defined in Section \ref{specclass}. The names are to the right
of the source location, and the optical image of M31 is from the Palomar All
Sky Survey. As mentioned in Section \ref{specclass}, the clustering of
sources with spectral turnover around the optical disk of M31 is a
selection effect, further described in the text. \label{specsrcs}}
\figcaption[]{Greyscale images of all Complex GLG sources overlaid with
contours. Ellipses denote the size and orientation of the source as
determined by the {\sc miriad} task {\sc sfind}. All the GLG sources are
roughly the same
size because {\sc sfind} determined the properties using the smoothed
maps described in Section 2.2.2 in Paper I. The intensity scale is in Jy
beam$^{-1}$. The values of the contours change for each source, and
correspond to $\frac{A}{i}$, where $A$ is the peak~flux of the source, and $i$
is a counter that begins at $i=1$ and increases by 1 for sources with
$\frac{I}{\sigma_{RMS}}\leq10$, by 2 for sources with
$10<\frac{I}{\sigma_{RMS}}\leq20$, and by 4 for sources with
$\frac{I}{\sigma_{RMS}}>20$, until $\frac{A}{i}<\sigma_{RMS}$.\label{complex}}
\figcaption{The IR colors of all ESC sources within 1$^\circ$ of M31
({\it top--left}), of good PSC counterparts of GLG sources ({\it top--right}),
of PSC sources within 5\arcmin~of M31 ({\it bottom--left}), and of PSC sources
$>$90\arcmin~away from M31 ({\it bottom--right}). In the top--left graph,
diamonds are good ESC counterparts of GLG sources and crosses are good PSC
counterparts of GLG sources, and in the bottom graphs crosses indicate good
counterparts of GLG sources. \label{ircolors}}
\figcaption[]{R--band (Greyscale) image of GLG sources with optical
counterparts, overlaid with radio contours (only for Complex sources) and
ellipses that show the extent and orientation of the GLG source. The colorbar
is in ADU counts. Since the R--band images were not calibrated, the
conversion between ADU counts and flux is unknown. (Ben Williams, private
communication) The contours change for each source, and correspond to
$\frac{A}{2i}$, where $A$ is the peak~flux of the source, and $i$ is a counter
that begins at $i=1$ and increases by 0.5 for sources with
$\frac{I}{\sigma_{RMS}}\leq10$, by 2 for sources with
$\frac{I}{\sigma_{RMS}}\leq20$, by 3 for sources with
$20<\frac{I}{\sigma_{RMS}}\leq50$, and by 10 for sources with
$50<\frac{I}{\sigma_{RMS}}$, until $\frac{A}{i}<\sigma_{RMS}$. \label{redgd}}
\figcaption[]{H$\alpha$ image of GLG sources with an H$\alpha$ counterpart.
The colorbar is in ADU counts, and the conversion to flux is
$3.28\times10^{-14} \frac{\rm erg}{\rm cm^2~ADU~s}$ (Ben Williams, private
communication). The contours change for each source, but follow the formula
given in Figure \ref{redgd}. \label{halpgd}}
\figcaption[]{[SII] image of GLG sources with a [SII] counterpart. The
colorbar is in ADU counts, and the conversion to flux is
$2.89\times10^{-14} \frac{\rm erg}{\rm cm^2~ADU~s}$ (Ben Williams, private
communication). The contours change for each source, but follow the formula
given in Figure \ref{redgd}. \label{siigd}}
\figcaption[]{Radio contour images of all the Complex sources with
good X--ray counterparts. The cross represents a {\it
Chandra}~HRC sources, while the ellipses represent {\it ROSAT}~PSPC
sources. The contour values change for each source, and correspond
to $\frac{A}{i}$, where $A$ is the peak~flux of the source, and $i$
is a counter that begins at $i=1$ and increases by 1 for sources
with $\frac{I}{\sigma_{RMS}}\leq10$ until
$\frac{A}{i}<\frac{\sigma_{RMS}}{2}$, and by 2 for sources with
$10<\frac{I}{\sigma_{RMS}}\leq30$, and by 4 for sources with
$\frac{I}{\sigma_{RMS}}>30$, until
$\frac{A}{i}<\sigma_{RMS}$.\label{xraycnt}}
\figcaption{325~MHz Greyscale images of SNR (top two rows) and PWN
(bottom row) candidates in the GLG source list. The intensity scale
is in Janskys. The ellipse corresponds to the orientation and
extent of the GLG source. \label{snrpics}}
\figcaption{The position of M31 SNR and PWN candidates with respect
to the optical disk of M31. Optical image courtesy of the Palomar All--Sky
Survey. \label{snrpos}}
\figcaption{$S_{325}$ vs $\theta_M$ of Milky Way (Xs), M33
(diamonds), LMC (triangles), and SMC (squares) supernova remnants if placed at
the distance of M31. SNR candidates in the GLG sources are denoted by the
thick ``+'' signs. Arrows indicate the upper limit on $\theta_M$ for GLG SNR
candidates. The line represents the observational limits of the GLG survey --
only SNRs with a flux density $S_{325} > 3~$mJy beam$^{-1}$ will be detected.
\label{snrbmajflux}}
\figcaption{325~MHz Radio image ({\it left}) and J--Band image ({\it right}) of
GLG347. The black contour indicates the location and extent of GLG347 and the
grey contours correspond to $S_{325}=1, 2, 3, 4~\mbox{and}~5$ mJy beam${}^{-1}$.
\label{glg347pic}}
\figcaption{325~MHz grey scale of Radio Triple GLG220 and GLG212. The black
ellipses indicate the location and extent of the GLG sources, and the white
contours correspond to $S_{325}=2, 3, 4, 5, 7$ mJy beam${}^{-1}$. \label{radtrip}}
\figcaption{Radio Jets GLG031/GLG033 ({\it upper--left}, previous page),
GLG045/GLG051({\it upper--right}, previous page), GLG054/GLG059
({\it bottom--left}, previous page), GLG266/GLG269 ({\it bottom--right},
previous page), GLG270/GLG271/GLG275 (this page). The black ellipses
are the GLG sources plus Braun 3 (middle ellipse in GLG031/GLG033), and the
white contours are the same as in Figure \ref{radtrip}. \label{jetspic}}
\figcaption{Radio ({\it left}) and Palomar All Sky Survey optical image
({\it right}) of GLG247 overlaid with radio contours (white) whose levels
are $S_{325}=2, 3, 4, 5, 7$ mJy beam${}^{-1}$. The black ellipse corresponds to the size
and orientation of the GLG source.\label{glg247pic}}
\figcaption{Radio image of BL~Lac candidate GLG105 ({\it upper--left}),
H$\alpha$ image of GLG105 ({\it upper--right}; courtesy of Ben Williams), and
broadband SED of GLG105 ({\it bottom}). The black contours in the top two
images are $S_{325}=2.5, 5, ..., 15$ mJy beam${}^{-1}$. The arrows in the bottom image
are upper--limits, while the diamond and triangle are the expected value for
an FRI and FRII, respectively, using the procedure described in Section
\ref{ir}. \label{glg105pic}}
\figcaption{\scriptsize NVSS image ({\it top--right}), Palomar All-Sky Survey
image
({\it top--left}), and 325~MHz radio image of Giant Radio Galaxy candidate
GLG242/GLG260 ({\it bottom}). White contours are $S_{325}=2.5, 5, ..., 15$
mJy beam${}^{-1}$.\label{grgpic}}
\figcaption{Raw ({\it top--row}) and Binned ({\it bottom--row}) Distribution of
$\frac{\Delta S}{\overline{S}}$ vs. $R_{M31}$ at 325~MHz ({\it left}) and
1400~MHz ({\it right}). \label{varpics}}
\figcaption{325~MHz images of CESS sources. The ellipses correspond to the
size and orientation of the GLG source. \label{cesspics}}
\end{document}
|
2,869,038,154,751 | arxiv | \section{Introduction}
\subsection{Motivations and General Framework}
In this paper, we shall be concerned with geometric functionals and
excursion probabilities for some nonlinear transforms evaluated on Fourier
components of spherical random fields. More precisely, let $\left\{ T(x)
\text{ }x\in S^{2}\right\} $ denote a Gaussian, zero-mean isotropic
spherical random field, i.e. for some probability space $(\Omega ,\Im ,P)$
the application $T(x,\omega )\rightarrow \mathbb{R}$ is $\left\{ \Im \times
\mathcal{B}(S^{2})\right\} $ measurable, $\mathcal{B}(S^{2})$ denoting the
Borel $\sigma $-algebra on the sphere. It is well-known that the following
representation holds, in the mean square sense{\ (see for instance \cit
{leonenko2}, \cite{marpecbook}, \cite{mal})}:
\begin{equation}
T(x)=\sum_{\mathbb{\ell }m}a_{\mathbb{\ell }m}Y_{\mathbb{\ell }m}(x)=\sum_
\mathbb{\ell }}T_{\mathbb{\ell }}(x)\text{ , }T_{\mathbb{\ell }}(x)=\sum_{m=
\mathbb{\ell }}^{\mathbb{\ell }}a_{\mathbb{\ell }m}Y_{\mathbb{\ell }m}(x
\text{ .} \label{norcia}
\end{equation}
where $\left\{ Y_{\mathbb{\ell }m}(.)\right\} $ denotes the family of
spherical harmonics, and $\left\{ a_{\mathbb{\ell }m}\right\} $ the array of
random spherical harmonic coefficients, which satisfy $\mathbb{E}a_{\mathbb
\ell }m}\overline{a}_{\mathbb{\ell }^{\prime }m^{\prime }}=C_{\mathbb{\ell
}\delta _{\mathbb{\ell }}^{\mathbb{\ell }^{\prime }}\delta _{m}^{m^{\prime
}};$ here, $\delta _{a}^{b}$ is the Kronecker delta function, and the
sequence $\left\{ C_{\mathbb{\ell }}\right\} $ represents the angular power
spectrum of the field. As pointed out in \cite{mp2012}, under isotropy the
sequence $C_{\mathbb{\ell }}$ necessarily satisfies $\sum_{\mathbb{\ell }
\frac{(2\mathbb{\ell }+1)}{4\pi }C_{\mathbb{\ell }}=\mathbb{E}T^{2}<\infty $
and the random field $T(x)$ is mean square continuous. Under the slightly
stronger assumption $\sum_{\mathbb{\ell }\geq L}(2\mathbb{\ell }+1)C_
\mathbb{\ell }}=O(\log ^{-2}L),$ the field can be shown to be a.s.
continuous, an assumption that we shall exploit heavily below.
Our attention will be focussed on the Fourier components $\left\{ T_{\mathbb
\ell }}(x)\right\} $, which represent random eigenfunctions of the spherical
Laplacian
\begin{equation*}
\Delta _{S^{2}}T_{\mathbb{\ell }}=-\mathbb{\ell }(\mathbb{\ell }+1)T_
\mathbb{\ell }}\text{ , }\mathbb{\ell }=1,2,...
\end{equation*
A lot of recent work has been focussed on the characterization of geometric
features for $\left\{ T_{\mathbb{\ell }}\right\} ,$ under Gaussianity
assumptions; for instance \cite{Wig1}, \cite{Wig2} studied the asymptotic
behaviour of the nodal domains, proving an earlier conjecture by Berry on
the variance of {(functionals of)} the zero sets of $T_{\mathbb{\ell }}.$ In
an earlier contribution, \cite{BGS} had focussed on the \textit{Defect} or
signed area, i.e. the difference between the positive and negative regions;
a Central Limit Theorem for these statistics and more general nonlinear
transforms of Fourier components was recently established by \cite{MaWi3}.
These studies have been motivated, for instance, by the analysis of
so-called Quantum Chaos {(see again \cite{BGS})}, where the behaviour of
random eigenfunctions is taken as an approximation for the asymptotics in
deterministic case, under complex boundary conditions. More often, spherical
eigenfunctions emerge naturally from the analysis of the Fourier components
of spherical random fields, as in (\ref{norcia}). In the latter
circumstances, several functionals of $T_{\mathbb{\ell }}$ assume a great
practical importance: to mention a couple, the squared norm of $T_{\mathbb
\ell }}$ provides an unbiased sample estimate for the angular power spectrum
$C_{\mathbb{\ell }},$
\begin{equation*}
\mathbb{E}\left\{ \int_{S^{2}}T_{\mathbb{\ell }}^{2}(x)dx\right\} =(2\mathbb
\ell }+1)C_{\mathbb{\ell }}\text{ ,}
\end{equation*
while higher-order power lead to estimates of the so-called polyspectra,
which have a great importance in the analysis of non-Gaussianity (see e.g.
\cite{marpecbook}).
The previous discussion shows that the analysis of nonlinear functionals of
\left\{ T_{\mathbb{\ell }}\right\} $ may have a great importance for
statistical applications, especially in the framework of cosmological data
analysis. In this area, a number of papers have searched for deviations of
geometric functionals from the expected behaviour under Gaussianity. For
instance, the so-called Minkowski functionals have been widely used as tools
to probe non-Gaussianity of the field $T(x)$, see \cite{matsubara} and the
references therein. On the sphere, Minkowski functionals correspond to the
area, the boundary length and the Euler-Poincar\'{e} characteristic of
excursion sets, and up to constants they correspond to the Lipschitz-Killing
Curvatures we shall consider in this paper, see \cite{RFG}, p.144. Many
other works have also focussed on local deviations from the Gaussianity
assumption, mainly exploiting the properties of integrated higher order
moments (polyspectra), see \cite{pietrobon1}, \cite{rudjord2}.
In general, the works aimed at the analysis of local phenomena are often
based upon wavelets-like constructions, rather than standard Fourier
analysis. The astrophysical literature on these issues is vast, see for
instance \cite{Wiaux}, \cite{Starck} and the references therein. Indeed, the
double localization properties of wavelets (in real and harmonic domain)
turn out usually to be extremely useful when handling real data.
In this paper, we shall focus on sequence of spherical random fields which
can be viewed as averaged forms of the spherical eigenfunctions, e.g.
\begin{equation*}
\beta _{j}(x)=\sum_{\mathbb{\ell }}b(\frac{\mathbb{\ell }}{B^{j}})T_{\mathbb
\ell }}(x)\text{ , }j=1,2,3...
\end{equation*
for $b(.)$ a weight function whose properties we shall discuss immediately.
The fields $\left\{ \beta _{j}(x)\right\} $ can indeed be viewed as a
representation of the coefficients from a continuous wavelet transform from
T(x),$ at scale $j$. More precisely, consider the kerne
\begin{eqnarray*}
\Psi _{j}(\left\langle x,y\right\rangle ) &:&=\sum_{\mathbb{\ell }}b(\frac
\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell
}(\left\langle x,y\right\rangle ) \\
&=&\sum_{\mathbb{\ell }}b(\frac{\mathbb{\ell }}{B^{j}})\sum_{m=-\mathbb{\ell
}}^{\mathbb{\ell }}Y_{\mathbb{\ell }m}(x)\overline{Y}_{\mathbb{\ell }m}(y
\text{.}
\end{eqnarray*
Assuming that $b(.)$ is smooth (e.g. $C^{\infty })$, compactly supported in
[B^{-1},B],$ and satisfying the partition of unity property $\sum_{j}b^{2}
\frac{\mathbb{\ell }}{B^{j}})=1,$ for all $\ell >B,$ where is a fixed
``bandwidth" parameter s.t. $B>1.$ Then $\Psi _{j}(\left\langle
x,y\right\rangle )$ can be viewed as a continuous version of the needlet
transform, which was introduced by Narcowich et al. in \cite{npw1}, and
considered from the point of view of statistics and cosmological data
analysis by many subsequent authors, starting from \cite{bkmpAoS}, \cit
{mpbb08}, \cite{pbm06}. In this framework, the following localization
property is now well-known: for all $M\in \mathbb{N}$, there exists a
constant $C_{M}$ such tha
\begin{equation*}
\left\vert \Psi _{j}(\left\langle x,y\right\rangle )\right\vert \leq \frac
C_{M}B^{2j}}{\left\{ 1+B^{j}d(x,y)\right\} ^{M}}\text{ ,}
\end{equation*
where $d(x,y)=\arccos (\left\langle x,y\right\rangle )$ is the usual
geodesic distance on the sphere. Heuristically, the fiel
\begin{equation*}
\beta _{j}(x)=\int_{S^{2}}\Psi _{j}(\left\langle x,y\right\rangle
)T(y)dy=\sum_{\mathbb{\ell }}b(\frac{\mathbb{\ell }}{B^{j}})T_{\mathbb{\ell
}(x)\text{ }
\end{equation*
is then only locally determined, i.e., for $B^{j}$ large enough its value
depends only from the behaviour of $T(y)$ in a\ neighbourhood of $x.$ This
is a very important property, for instance when dealing with spherical
random fields which can only be partially observed, the canonical example
being provided by the masking effect of the Milky Way on Cosmic Microwave
Background (CMB) radiation.
It is hence very natural to produce out of $\left\{ \beta _{j}(x)\right\} $
nonlinear statistics of great practical relevance. To provide a concrete
example, a widely disputed theme in CMB data analysis concerns the existence
of asymmetries in the angular power spectrum; it has been indeed often
suggested that the angular power $\left\{ C_{\mathbb{\ell }}\right\} $ may
exhibit different behaviour for different subsets of the sky, at least over
some multipole range, see for instance \cite{hansen2009}, \cite{pietrobon1}.
It is readily seen tha
\begin{equation*}
\mathbb{E}\left\{ \beta _{j}^{2}(x)\right\} =\sum_{\mathbb{\ell }}b(\frac
\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }
\text{ ,}
\end{equation*
which hence suggests a natural \textquotedblleft local\textquotedblright\
estimator for a binned form of the angular power spectrum. More precisely,
it is natural to consider some form of averaging and introduc
\begin{equation*}
g_{j;2}(z):=\int_{S^{2}}K(\left\langle z,x\right\rangle )\beta _{j}^{2}(x)d
\text{ .}
\end{equation*
For instance, should we consider the behaviour of the angular power spectrum
on the northern and southern hemisphere, we might focus o
\begin{equation*}
g_{j;2}(N):=\int_{S^{2}}K(\left\langle N,x\right\rangle )\beta _{j}^{2}(x)d
\text{ , }g_{j;2}(S):=\int_{S^{2}}K(\left\langle S,x\right\rangle )\beta
_{j}^{2}(x)dx\text{ ,}
\end{equation*
where $K(\left\langle a,.\right\rangle ):=\mathbb{I}_{[0,\frac{\pi }{2
]}(\left\langle a,.\right\rangle )$ is the indicator function of the
hemisphere centred on $a\in S^{2},$ and $N,S$ denote respectively the North
and South Poles (compare \cite{hansen2009},\cite{pietrobon1}, \cit
{bennett2012, planckIS} and the references therein). More generally, we
shall be concerned with statistics of the for
\begin{equation}
g_{j;q}(z):=\int_{S^{2}}K(\left\langle z,x\right\rangle )H_{q}(\beta
_{j}(x))dx\text{ ,} \label{mainproc}
\end{equation
where $H_{q}(.)$ is the Hermite polynomial of $q$-th order; for instance,
for $q=3$ these procedures can be exploited to investigate local variation
in Gaussian and non-Gaussian features (see \cite{rudjord2} and below for
more discussion and details).
\subsection{Main Result}
The purpose of this paper is to study the asymptotic behaviour for the
expected value of the Euler characteristic and other geometric functionals
for the excursion regions of sequences of fields such as $\left\{
g_{j;q}(.)\right\} $, and to exploit these results to obtain excursion
probabilities in nonGaussian circumstances. Indeed, on one hand these
geometric functionals are of interest by themselves, as they provide the
basis for implementing goodness-of-fit tests (compare \cite{matsubara}); on
the other hand, they provide the clue for approximations of the excursion
probabilities for $\left\{ g_{j;q}(.)\right\} ,$ by means of some weak
convergence results we shall establish, in combination with some now
classical arguments described in detail in the monograph \cite{RFG}.
It is important to stress that our results are obtained under a setting
which is quite different from usual. In particular, the asymptotic theory is
investigated in the high frequency sense, e.g. assuming that a single
realization of a spherical random field is observed at higher and higher
resolution as more and more refined experiments are implemented. This is the
setting adopted in \cite{marpecbook}, see also \cite{anderes},\cite{steinm}
for the related framework of fixed-domain asymptotics.
Because of the nature of high-frequency asymptotics, we cannot expect the
finite-dimensional distributions of the processes we focus on to converge.
This will require a more general notion of weak convergence, as developed
for instance by \cite{davydov}, \cite{dudley}. By means of this, we shall
indeed show how to evaluate asymptotically valid excursion probabilities,
which provide a natural solution for hypothesis testing problems. Indeed,
the main result of the paper, Theorem \ref{thm:exc:prob}, provides a very
explicit bound for the excursion probabilities of nonGaussian fields such as
(\ref{mainproc}), e.g.
\begin{equation}
\limsup_{j\rightarrow \infty }\left\vert \Pr \left\{ \sup_{x\in S^{2}}\tilde
g}_{j;q}(x)>u\right\} -\left\{ 2(1-\Phi (u))+2u\phi (u)\lambda
_{j;q}\right\} \right\vert \leq \exp \left( -\frac{\alpha u^{2}}{2}\right) ,
\label{mainresult}
\end{equation
where $\tilde{g}_{j;q}(x)$ has been normalized to have unit variance, $\phi
(.),\Phi (.)$ denote standard Gaussian density and distribution function,
\alpha >1$ is some constant and the parameters $\lambda _{j;q}$ have
analytic expressions in terms of generalized convolutions of angular power
spectra, see (\ref{porteaperte}), (\ref{porteaperte2}). See also \cit
{NardiSiegYakir} for some related results on the distribution of maxima of
approximate Gaussian random fields; note, however, that our approach is
quite different from theirs and the tools we use allow us to get much
stronger results in terms of the uniform estimates.
\subsection{Plan of the paper}
The plan of the paper is as follows: in Section 2 we review some background
results on random fields and geometry, mainly referring to the now classical
monograph \cite{RFG}. Section 3 specializes these results to spherical
random fields, for which some background theory is also provided, and
provides some simple evaluations for Lipschitz-Killing curvatures related to
excursion sets for harmonic components of such fields. More interesting
Gaussian subordinated fields are considered in Section 4, where some
detailed computations for covariances in general Gaussian subordinated
circumstances are also provided. Section 5 provides the main convergence
results, i.e. shows how the distribution of these random elements are
asymptotically proximal (in the sense of \cite{davydov}) to those of a
Gaussian sequence with the same covariances. This result is then exploited
in Section 6, to provide the proof of (\ref{mainresult}). A number of
possible applications on real cosmological data sets are discussed
throughout the paper.
\section{Background: random fields and geometry}
This section is devoted to recall basic integral geometric concepts, to
state the Gaussian kinematic fundamental formula, and to discuss its
application in evaluating the excursion probabilities. This theory has been
developed in a series of fundamental papers by R.J. Adler, J.E. Taylor and
coauthors (see \cite{adleraap, worsley}, \cite{tayloradler2003}, \cit
{tayloradler2009}, \cite{taylortakemuraadler}, \cite{adleradvapp}), and it
is summarized in the monographs \cite{RFG}, \cite{adlerstflour} which are
our main references in this Section (see also \cite{azaiswschebor}, \cit
{azaisbook} for a different approach, and \cite{taylorvadlamani}, \cit
{chengxiao}, \cite{adlersamo}, \cite{adlerblanchet} for some further
developments in this area; applications to the sphere have also been
considered very recently by \cite{chengschwar,chengxiao2}).
\subsection{Lipschitz Killing curvatures and Gaussian Minkowski functionals}
There are a number of ways to define Lipschitz-Killing curvatures, but
perhaps the easiest is via the so-called tube formulae, which, in its
original form is due to Hotelling \cite{Hotelling} and Weyl \cite{Wey39}. To
state the tube formula, let $M$ be an $m$-dimensional smooth subset of
\mathbb{R}^{n}$ such that $\partial M$ is a $C^{2}$ manifold endowed with
the canonical Riemannian structure on $\mathbb{R}^{n}.$ The tube of radius
\rho $ around $M$ is defined as
\begin{equation}
\text{Tube}(M,\rho )\ =\ \left\{ x\in \mathbb{R}^{n}:\;d(x,M)\leq \rho
\right\} ,
\end{equation
where,
\begin{equation}
d(x,M)\ =\ \inf_{y\in M}\Vert x-y\Vert .
\end{equation
Then according to Weyl's tube formula {(see \cite{RFG})}, the Lebesgue
volume of this constructed tube, for small enough $\rho ,$ is given by
\begin{equation}
\lambda _{n}(\text{Tube}(M,\rho ))\ =\ \sum_{j=0}^{m}\rho ^{n-j}\omega _{n-j
\mathcal{L}_{j}(M)\text{ }, \label{tube:formula}
\end{equation
where $\omega _{j}$ is the volume of the $j$-dimensional unit ball and
\mathcal{L}_{j}(M)$ is the $j^{th}$-Lipschitz-Killing curvature (LKC) of $M
. A little more analysis shows that $\mathcal{L}_{m}(M)=\mathcal{H}_{m}(M)$,
the $m$-dimensional Hausdorff measure of $M$, and that $\mathcal{L}_{0}(M)$
is the Euler-Poincar\'{e} characteristic of $M$. Although the remaining LKCs
have less transparent interpretations, it is easy to see that they satisfy
simple scaling relationships, in that $\mathcal{L}_{j}(\alpha M)=\alpha ^{j
\mathcal{L}_{j}(M)$ for all $1\leq j\leq m$, where $\alpha M=\{x\in \mathbb{
}^{n}:x=\alpha y\ $for some$\ y\in M\}$. Furthermore, despite the fact that
defining the $\mathcal{L}_{j}$ via (\ref{tube:formula}) involves the
embedding of $M$ in $\mathbb{R}^{n}$, the $\mathcal{L}_{j}(M)$ are actually
intrinsic, and so are independent of the ambient space.
Apart from their appearance in the tube formula (\ref{tube:formula}), there
are a number of other ways in which to define the LKCs. One such
(non-intrinsic) way which signifies the dependence of the LKCs on the
Riemannian metric is through the shape operator. Let $M$ be an $m
-dimensional $C^{2}$ manifold embedded in $\mathbb{R}^{n}$; then
\begin{equation}
\mathcal{L}_{k}(M)=K_{n,m,k}\int_{M}\int_{S(N_{x}M)}\text{Tr}(S_{\nu
}^{(m-k)})1_{N_{x}M}(-\nu )\,\mathcal{H}_{n-m-1}(d\nu )\mathcal{H}_{m-1}(dx),
\label{LK:equation}
\end{equation
where, $K_{n,m,k}= \frac{1}{(m-k)!}\frac{\Gamma\left( \frac{(n-k)}{2}\right
}{(2\pi)^{(n-k)/2}},$ and $S(N_{x}M)$ denotes a sphere in the normal space
N_{x}M$ of $M$ at the point $x\in M$.
Closely related to the LKCs are set functionals called the Gaussian
Minkowski functionals (GMFs), which are defined via a Gaussian tube formula.
Consider the Gaussian measure, $\gamma _{n}(dx)=(2\pi )^{-n/2}e^{-\Vert
x\Vert ^{2}/2}dx$, instead of the standard Lebesgue measure in (\re
{tube:formula}); the Gaussian tube formula is then given by
\begin{equation}
\gamma _{n}\left( (M,\rho )\right) =\sum_{k\geq 0}\frac{\rho ^{k}}{k!
\mathcal{M}_{k}^{\gamma _{n}}(M)\text{ }, \label{Gaussian:tube:formula}
\end{equation
where the coefficients $\mathcal{M}_{k}^{\gamma _{n}}(M)$'s are the GMFs
(for technical details, we refer the reader to \cite{RFG}). We note that
these set functionals, like their counterparts in (\ref{tube:formula}) can
be expressed as integrals over the manifold and its normal space (cf. \cit
{RFG}).
\subsection{Excursion probabilities and the Gaussian kinematic fundamental
formula}
A classical problem in stochastic processes is to compute the excursion
probability or the suprema probability
\begin{equation*}
P\left( \sup_{x\in M}f(x)\geq u\right) ,
\end{equation*
where, as before, $f$ is a random field defined on the parameter space $M$.
In the case when $f$ happens to be a centered Gaussian field with constant
variance $\sigma ^{2}$ defined on $M$, a piecewise smooth manifold, then by
the arguments set forth in Chapter 14 of \cite{RFG}, we have that
\begin{equation}
\Big|P\left\{ \sup_{x\in M}f(x)\leq u\right\} -\mathbb{E}\left\{ \mathcal{L
_{0}(A_{u}(f;M))\right\} \Big|<O\left( \exp\left(-\frac{\alpha u^{2}}
2\sigma ^{2}}\right)\right) , \label{eqn:sup:EC}
\end{equation
where $\mathcal{L}_{0}(A_{u}(f;M))$ is, as defined earlier, the Euler-Poinca
\'{e} characteristic of the excursion set $A_{u}(f;M)=\{x\in M:f(x)\geq u\}
, and $\alpha >1$ is a constant, which depends on the field $f$ and can be
determined (see Theorem 14.3.3 of \cite{RFG}).
At first sight, from (\ref{eqn:sup:EC}) it may appear that we may have to
deal with a hard task, e.g. that of evaluating $E\left\{ \mathcal{L
_{0}(A_{u}(f;M))\right\} $. This task, however, is greatly simplified due to
the \textit{Gaussian kinematic fundamental formula} (Gaussian-KFF) (see
Theorems 15.9.4-15.9.5 in \cite{RFG}), which states that, for a smooth
M\subset \mathbb{R}^{N}$
\begin{equation*}
\mathbb{E}(\mathcal{L}_{i}^{f}(A_{u}(f,M)))
\end{equation*
\begin{equation*}
=\sum_{\mathbb{\ell }=0}^{\dim (M)-i}\left(
\begin{array}{c}
i+\mathbb{\ell } \\
\mathbb{\ell
\end{array
\right) \frac{\Gamma \left( \frac{i}{2}+1\right) \Gamma \left( \frac{\mathbb
\ell }}{2}+1\right) }{\Gamma \left( \frac{i+\mathbb{\ell }}{2}+1\right)
(2\pi )^{-\mathbb{\ell }/2}\mathcal{L}_{i+\mathbb{\ell }}^{f}(M)\mathcal{M}_
\mathbb{\ell }}^{\gamma }([u,\infty )),
\end{equation*
e.g., in the special case of the Euler characteristic ($i=0$)
\begin{equation}
\mathbb{E}\left\{ \mathcal{L}_{0}^{f}(A_{u}(f;M))\right\} =\sum_{j=0}^
\mbox{dim}(M)}(2\pi )^{-j/2}\mathcal{L}_{j}^{f}(M)\mathcal{M}_{j}^{\gamma
}\left( [u,\infty )\right) , \label{eqn:GKF}
\end{equation
where $\mathcal{L}_{j}^{f}(M)$ is the $j$-th LKC of $M$ with respect to the
induced metric $g^{f}$ given by
\begin{equation*}
g_{x}^{f}(Y_{x},Z_{x})=\mathbb{E}\left\{ Yf(x)\cdot Zf(x)\right\} ,
\end{equation*
for $X_{x},Y_{x}\in T_{x}M$, the tangent space at $x\in M$. The Gaussian
kinematic fundamental formula will play a crucial role in all the
developments to follow in the subsequent sections.
\section{Spherical Gaussian fields}
In this Section we shall start from some simple results on the evaluation of
the expected values of Lipschitz-Killing curvatures for sequences of
spherical Gaussian processes. These results will be rather straightforward
applications of the Gaussian kinematic fundamental formula (\ref{eqn:GKF}),
and are collected here for completeness and as a bridge towards the more
complicated case of {nonlocal transforms of} Gaussian subordinated
processes, to be considered later.
Note first that for {a unit variance} Gaussian field on the sphere
f:S^{2}\rightarrow \mathbb{R}$, the{\ expected value} of the Euler-Poinca
\'{e} characteristic{\ of the excursion set $A_{u}(f;S^{2})=\{x\in
S^{2}:f(x)\geq u\}$ is given by}
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{0}(A_{u}(f,S^{2}))\right\}
\end{equation*
\begin{equation*}
=\mathcal{L}_{0}^{f}(S^{2})\mathcal{M}_{0}^{\gamma }([u,\infty ))+(2\pi
)^{-1/2}\mathcal{L}_{1}^{f}(S^{2})\mathcal{M}_{1}^{\gamma }([u,\infty
))+(2\pi )^{-1}\mathcal{L}_{2}^{f}(S^{2})\mathcal{M}_{2}^{\gamma }([u,\infty
))\text{ },
\end{equation*
fo
\begin{equation*}
\mathcal{M}_{0}^{\gamma }([u,\infty ))=\int_{u}^{\infty }\phi (x)dx\text{,
\,\,\,\,\mathcal{M}_{j}^{\gamma }([u,\infty ))=H_{j-1}(u)\phi (u),
\end{equation*
where $\phi (\cdot )$ denotes the density of a real valued standard normal
random variable, and $H_{j}(u)$ denotes the Hermite polynomials
\begin{equation*}
H_{j}(u)=(-1)^{j}\left( \phi (u)\right) ^{-1}\frac{d^{j}}{du^{j}}\phi (u
\text{ and }H_{-1}(u)=1-\Phi (u)\text{,}
\end{equation*
{while} $\mathcal{L}_{k}^{f}(S^{2})$ are the usual Lipschitz-Killing
curvatures, under the induced Gaussian metric, i.e
\begin{equation*}
\mathcal{L}_{k}^{f}(S^{2}):={\frac{(-2\pi )^{-(2-j)/2}}{2}
\int_{S^{2}}Tr(R^{(N-k)/2})Vol_{g^{f}}\text{ ;}
\end{equation*
here, $R$ is the Riemannian curvature tensor and $Vol_{g^{f}}$ is the volume
form, under the induced Gaussian metric, given by
\begin{equation*}
g^{f}(X,Y):=\mathbb{E}\left\{ Xf\cdot Yf\right\} =XY\mathbb{E}(f^{2})\text{
.
\end{equation*}
We recall that $\mathcal{L}_{0}(M)$ is a topological invariant and does not
depend on the metric; in particular, $\mathcal{L}_{0}(S^{2})\equiv 2.$
Moreover, because the sphere is an (even-) $2$-dimensional manifold,
\mathcal{L}_{1}^{f}(S^{2})$ is identically zero.
As mentioned before, we start from some very simple result on the Fourier
components and wavelets transforms of Gaussian fields, e.g. the expected
value of the Euler-Poincar\'{e} characteristic for two forms of harmonic
components, namel
\begin{equation*}
T_{\ell }(x)=\sum_{m=-\ell }^{\ell }a_{\ell m}Y_{\ell m}(x)\text{ and }\beta
_{j}(x)=\sum_{\mathbb{\ell }}b(\frac{\ell }{B^{j}})T_{\ell }(x)\text{ ,}
\end{equation*
the first representing a Fourier component at the multipole $\ell ,$ the
second a field of continuous needlet/wavelet coefficients at scale $j.$ We
normalize these processes to unit variance by takin
\begin{equation*}
\widetilde{T}_{\ell }(x)=\frac{T_{\ell }(x)}{\sqrt{\frac{2\ell +1}{4\pi
C_{\ell }}}\text{ , and }\widetilde{\beta }_{j}(x)=\frac{\beta _{j}(x)}
\sqrt{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})\frac{(2\ell +1)}{4\pi
C_{\ell }}}\text{ .}
\end{equation*
We start reporting some simple results on Lipschitz-Killing curvatures {of
excursion sets generated by} spherical Gaussian fields (see \cite{matsubara}
and the references therein for related expressions on $\mathbb{R}^{2}$ from
an astrophysical point of view). These results are straightforward
consequences of equation (\ref{eqn:GKF}).
\begin{lemma}
We have
\begin{eqnarray*}
\mathcal{L}_{2}^{\widetilde{\beta }_{j}}(S^{2}) &=&4\pi \frac{\sum_{\mathbb
\ell }}b^{2}(\frac{\ell }{B^{j}})(2\ell +1)C_{\mathbb{\ell }}P_{\ell
}^{\prime }(1)}{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})(2\ell
+1)C_{\ell }} \\
&=&4\pi \frac{\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})(2\ell +1)C_{\ell }\frac
\ell (\ell +1)}{2}}{\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})(2\ell +1)C_{\ell
}\mbox{ .}
\end{eqnarray*}
\end{lemma}
\begin{proof}
Recall first that, in standard spherical coordinate
\begin{equation*}
P_{\mathbb{\ell }}(\left\langle x,y\right\rangle )=P_{\mathbb{\ell }}(\sin
\vartheta _{x}\sin \vartheta _{y}\cos (\phi _{x}-\phi _{y})+\cos \vartheta
_{x}\cos \vartheta _{y})\text{ .}
\end{equation*
Some simple algebra then yield
\begin{equation*}
\left. \frac{\partial ^{2}}{\partial \vartheta _{x}\partial \vartheta _{y}
P_{\mathbb{\ell }}(\left\langle x,y\right\rangle )\right\vert _{x=y}=\left.
\frac{\partial ^{2}}{\sin \vartheta _{x}\sin \vartheta _{y}\partial \phi
_{x}\partial \phi _{y}}P_{\mathbb{\ell }}(\left\langle x,y\right\rangle
)\right\vert _{x=y}=P_{\mathbb{\ell }}^{\prime }(1)\text{ ,}
\end{equation*
an
\begin{equation*}
\left. \frac{\partial ^{2}}{\sin \vartheta _{x}\partial \vartheta
_{y}\partial \phi _{x}}P_{\mathbb{\ell }}(\left\langle x,y\right\rangle
)\right\vert _{x=y}=0\text{ .}
\end{equation*
The geometric meaning of the latter result is that the process is still
isotropic {under the new transformation,} whence the derivatives along the
two directions are still independent. We thus have tha
\begin{equation*}
\mathcal{L}_{2}^{\widetilde{\beta }_{j}}(S^{2})=\int_{S^{2}}\left\{ \det
\left[
\begin{array}{cc}
\left. \frac{\partial ^{2}}{\partial \vartheta _{x}\partial \vartheta _{y}
\Gamma (x,y)\right\vert _{x=y} & \left. \frac{\partial ^{2}}{\sin \vartheta
_{x}\partial \phi _{x}\partial \vartheta _{y}}\Gamma (x,y)\right\vert _{x=y}
\\
\left. \frac{\partial ^{2}}{\sin \vartheta _{y}\partial \phi _{y}\partial
\vartheta _{x}}\Gamma (x,y)\right\vert _{x=y} & \left. \frac{\partial ^{2}}
\sin \vartheta _{x}\sin \vartheta _{y}\partial \phi _{x}\partial \phi _{y}
\Gamma (x,y)\right\vert _{x=y
\end{array
\right] \right\} ^{1/2}\sin \vartheta d\vartheta d\phi
\end{equation*
\begin{equation*}
=4\pi \left\{ \det \left[
\begin{array}{cc}
\frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac
2\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell
}}b^{2}(\frac{\ell }{B^{j}})\frac{(2\mathbb{\ell }+1)}{4\pi }C_{\mathbb{\ell
}}} & 0 \\
0 & \frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }
\frac{2\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell }}^{\prime }(1)}{\sum_
\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})\frac{(2\mathbb{\ell }+1)}{4\pi }C_
\mathbb{\ell }}
\end{array
\right] \right\} ^{1/2}
\end{equation*
\begin{equation*}
=4\pi \frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})\frac{(2\mathbb
\ell }+1)}{4\pi }C_{\mathbb{\ell }}P_{\mathbb{\ell }}^{\prime }(1)}{\sum_
\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})\frac{(2\mathbb{\ell }+1)}{4\pi }C_
\mathbb{\ell }}}\text{ .}
\end{equation*
Now recall that $P_{\mathbb{\ell }}^{\prime }(1)=\frac{\mathbb{\ell }
\mathbb{\ell }+1)}{2},$ whence the claim is established.
\end{proof}
\begin{remark}
Note that since the random field $\beta _{j}$ is an isotropic Gaussian
random field, the Lipschitz-Killing curvatures of $S^{2}$ under the metric
induced by the field $\beta _{j}$ are given by
\begin{equation*}
\mathcal{L}_{i}^{\widetilde{\beta }_{j}}(S^{2})=\lambda _{j}^{i/2}\mathcal{L
_{i}(S^{2}),
\end{equation*
where $\mathcal{L}_{i}(S^{2})$ is the $i$-th LKC under the usual Euclidean
metric, and $\lambda _{j}$ is the second spectral moment of $\widetilde
\beta }_{j}$ (cf. \cite{RFG}). This result is true for all isotropic and
unit variance Gaussian random fields.
\end{remark}
The second auxiliary result that we shall need follows immediately from
Theorem 13.2.1 in \cite{RFG}, specialized to isotropic spherical random
fields with unit variance\emph{.}
\begin{lemma}
For the Gaussian isotropic field $\widetilde{\beta }_{j}:S^{2}\rightarrow
\mathbb{R}$ , such that $\mathbb{E}\widetilde{\beta }_{j}=0,$ $\mathbb{E
\widetilde{\beta }_{j}^{2}=1,$ $\widetilde{\beta }_{j}\in C^{2}(S^{2})$
almost surely, we have tha
\begin{equation}
\mathbb{E}\left\{ \mathcal{L}_{0}(A_{u}(\widetilde{\beta
_{j}(x),S^{2}))\right\} =2\left\{ 1-\Phi (u)\right\} +4\pi \left\{ \frac
\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{
\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell
}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }
\right\} \frac{ue^{-u^{2}/2}}{\sqrt{(2\pi )^{3}}}\mathcal{\ }, \label{feb1}
\end{equation
\begin{equation}
\mathbb{E}\left\{ \mathcal{L}_{1}(A_{u}(\widetilde{\beta
_{j}(x),S^{2}))\right\} =\pi \left\{ \frac{\sum_{\mathbb{\ell }}b^{2}(\frac
\ell }{B^{j}})C_{\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }P_{\mathbb
\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_
\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }}\right\} ^{1/2}e^{-u^{2}/2},
\label{feb2}
\end{equation
and finall
\begin{equation}
\mathbb{E}\left\{ \mathcal{L}_{2}(A_{u}(\widetilde{\beta
_{j}(x),S^{2}))\right\} =\left\{ 1-\Phi (u)\right\} 4\pi \text{ .}
\label{feb3}
\end{equation}
\end{lemma}
\begin{proof}
We start by recalling that, from Theorem 13.2.1 in \cite{RFG
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{i}(A_{u}(\widetilde{\beta
_{j}(x),S^{2}))\right\} =\sum_{\mathbb{\ell }=0}^{\dim (S^{2})-i}\left[
\begin{array}{c}
i+\mathbb{\ell } \\
\mathbb{\ell
\end{array
\right] \lambda ^{\mathbb{\ell }/2}\rho _{\mathbb{\ell }}(u)\mathcal{L}_{i
\mathbb{\ell }}(S^{2})\text{ ,}
\end{equation*
wher
\begin{equation*}
\left[
\begin{array}{c}
i+\mathbb{\ell } \\
\mathbb{\ell
\end{array
\right] :=\left(
\begin{array}{c}
i+\mathbb{\ell } \\
\mathbb{\ell
\end{array
\right) \frac{\omega _{i+\mathbb{\ell }}}{\omega _{i}\omega _{\mathbb{\ell }
}\text{ , }\omega _{i}=\frac{\pi ^{i/2}}{\Gamma (\frac{i}{2}+1)}\text{ ,}
\end{equation*
\begin{equation*}
\rho _{\mathbb{\ell }}(u)=(2\pi )^{-\mathbb{\ell }/2}\mathcal{M}_{\mathbb
\ell }}^{\gamma }([u,\infty ))=(2\pi )^{-(\mathbb{\ell }+1)/2}H_{\mathbb
\ell }-1}(u)e^{-u^{2}/2},
\end{equation*
so tha
\begin{equation*}
\rho _{0}(u)=(2\pi )^{-1/2}\sqrt{2\pi }(1-\Phi
(u))e^{u^{2}/2}e^{-u^{2}/2}=(1-\Phi (u))\text{ ,}
\end{equation*
\begin{equation*}
\text{ }\rho _{1}(u)=\frac{1}{2\pi }e^{-u^{2}/2}\text{ , }\rho _{2}(u)=\frac
1}{\sqrt{(2\pi )^{3}}}ue^{-u^{2}/2}.
\end{equation*
Here
\begin{equation*}
\lambda =\mathbb{E}\beta _{j;\vartheta }^{2}=\mathbb{E}\beta _{j;\phi }^{2
\text{ , }\beta _{j;\vartheta }=\frac{\partial }{\partial \vartheta }\beta
_{j}(\vartheta ,\phi )\text{ , }\beta _{j;\phi }=\frac{\partial }{\sin
\vartheta \partial \phi }\beta _{j}(\vartheta ,\phi )\text{ . }
\end{equation*
\begin{equation*}
\mathbb{E}\left\{ \widetilde{\beta }_{j;\vartheta }^{2}\right\} =\frac
\partial ^{2}}{\partial \vartheta ^{2}}\mathbb{E}\left\{ \widetilde{\beta
_{j}^{2}\right\} =\frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_
\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell }}^{\prime }(1
}{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{
\mathbb{\ell }+1}{4\pi }}\text{ ,}
\end{equation*
whenc
\begin{eqnarray*}
\mathbb{E}\left\{ \mathcal{L}_{0}(A_{u}(\widetilde{\beta
_{j}(x),S^{2}))\right\} &=&\left\{ 1-\Phi (u)\right\} \mathcal{L
_{0}(S^{2})+\lambda \frac{ue^{-u^{2}/2}}{\sqrt{(2\pi )^{3}}}\mathcal{L
_{2}(S^{2}) \\
&=&2\left\{ 1-\Phi (u)\right\} +\lambda \frac{ue^{-u^{2}/2}}{\sqrt{(2\pi
)^{3}}}4\pi \\
&=&2\left\{ 1-\Phi (u)\right\} +4\pi \left\{ \frac{\sum_{\mathbb{\ell
}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi
P_{\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }
B^{j}})C_{\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }}\right\} \frac
ue^{-u^{2}/2}}{\sqrt{(2\pi )^{3}}}\mathcal{\ }.
\end{eqnarray*
Als
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{1}(A_{u}(\widetilde{\beta
_{j}(x),S^{2}))\right\} =\sum_{\mathbb{\ell }=0}^{1}\left[
\begin{array}{c}
1+\mathbb{\ell } \\
\mathbb{\ell
\end{array
\right] \lambda ^{\mathbb{\ell }/2}\rho _{\mathbb{\ell }}(u)\mathcal{L}_{1
\mathbb{\ell }}(S^{2})=\left[
\begin{array}{c}
2 \\
\end{array
\right] \lambda ^{1/2}\rho _{1}(u)\mathcal{L}_{2}(S^{2})
\end{equation*
\begin{equation*}
=\frac{\pi }{2}\times 4\pi \times \left\{ \frac{\sum_{\mathbb{\ell }}b^{2}
\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }P_
\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}
)C_{\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }}\right\} ^{1/2}\frac{1}
\sqrt{2\pi }}\frac{e^{-u^{2}/2}}{\sqrt{2\pi }}
\end{equation*
\begin{equation*}
=\pi \left\{ \frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb
\ell }}\frac{2\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell }}^{\prime }(1)}{\sum_
\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{2\mathbb
\ell }+1}{4\pi }}\right\} ^{1/2}e^{-u^{2}/2}.
\end{equation*
Finall
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{2}(A_{u}(\widetilde{\beta
_{j}(x),S^{2}))\right\} =\rho _{0}(u)\mathcal{L}_{2}(S^{2})=\left\{ 1-\Phi
(u)\right\} 4\pi \text{ ,}
\end{equation*
which concludes the proof.
\end{proof}
\begin{remark}
Note that, in this settin
\begin{equation*}
\left[
\begin{array}{c}
2 \\
\end{array
\right] =\left[
\begin{array}{c}
2 \\
\end{array
\right] =1\text{ , }\left[
\begin{array}{c}
2 \\
\end{array
\right] =\frac{\pi }{2}\text{ .}
\end{equation*}
\end{remark}
In the case of spherical eigenfunctions, the previous Lemma takes the
following simpler form; the proof is entirely analogous, and hence omitted.
\begin{corollary}
For the field $\left\{ T_{\mathbb{\ell }}(.)\right\} ,$ we have that
\begin{equation}
\mathbb{E}\left\{ \mathcal{L}_{0}(A_{u}(\widetilde{T}_{\mathbb{\ell
}(.),S^{2}))\right\} =2\left\{ 1-\Phi (u)\right\} +\frac{\mathbb{\ell }
\mathbb{\ell }+1)}{2}\frac{ue^{-u^{2}/2}}{\sqrt{(2\pi )^{3}}}4\pi \text{ ,}
\label{sreekar}
\end{equation
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{1}(A_{u}(\widetilde{T}_{\mathbb{\ell
}(.),S^{2}))\right\} =\pi \left\{ \frac{\mathbb{\ell }(\mathbb{\ell }+1)}{2
\right\} ^{1/2}e^{-u^{2}/2},
\end{equation*
an
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{2}(A_{u}(\widetilde{T}_{\mathbb{\ell
}(.),S^{2}))\right\} =4\pi \times \left\{ 1-\Phi (u)\right\} \text{ .}
\end{equation*}
\end{corollary}
\begin{remark}
{Using the differential geometric definition of the Lipschitz-Killing
curvatures, it is easy to observe that}
\begin{equation*}
2\mathbb{E}\left\{ \mathcal{L}_{1}(A_{u}(\widetilde{T}_{\mathbb{\ell
}(.),S^{2})\right\} =\mathbb{E}\left\{ len(\partial A_{u}(\widetilde{T}_
\mathbb{\ell }}(.),S^{2})\right\} \text{ ,}
\end{equation*
where $len(\partial A_{u}(\widetilde{T}_{\mathbb{\ell }}(.),S^{2})$ is the
usual length of the boundary region of the excursion set, in the usual
Hausdorff sense, {which can also be expressed as $\mathcal{L}_{1}(\partial
A_{u}(T_{\mathbb{\ell }}(.),S^{2})$}. Henc
\begin{equation*}
\mathbb{E}\left\{ len(\partial A_{u}(\widetilde{T}_{\mathbb{\ell
}(.),S^{2})\right\} =2\pi \left\{ \frac{\mathbb{\ell }(\mathbb{\ell }+1)}{2
\right\} ^{1/2}e^{-u^{2}/2},
\end{equation*
which for $u=0$ fits with well-known results on the expected value of nodal
lines for random spherical eigenfunctions (see \cite{Wig2} and the
references therein). Likewis
\begin{equation}
\mathbb{E}\left\{ len(\partial A_{u}(\widetilde{\beta }_{j}(.),S^{2})\righ
\} =2\pi \left\{ \frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_
\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell }}^{\prime }(1
}{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{
\mathbb{\ell }+1}{4\pi }}\right\} ^{1/2}e^{-u^{2}/2}. \label{latter}
\end{equation
The expression (\ref{latter}) can be viewed as a weighted average of
\mathbb{E}\left\{ len(\partial A_{u}(T_{\mathbb{\ell }}(.),S^{2})\right\} $,
with weights provided by $w_{\mathbb{\ell }}:=b^{2}(\frac{\ell }{B^{j}})C_
\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi }.$ The sequence $w_{\mathbb
\ell }}$ is summable; in a heuristic sense, we can argue that it must hence
be eventually decreasing; \ we thus expect that, for $\mathbb{\ell }$ large
enoug
\begin{equation*}
\frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac
2\mathbb{\ell }+1}{4\pi }P_{\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell
}}b^{2}(\frac{\ell }{B^{j}})C_{\mathbb{\ell }}\frac{2\mathbb{\ell }+1}{4\pi
}\ll \frac{\mathbb{\ell }(\mathbb{\ell }+1)}{2}\text{ ,}
\end{equation*
and henc
\begin{equation*}
\mathbb{E}\left\{ len(\partial A_{u}(\widetilde{\beta }_{j}(.),S^{2})\righ
\} \ll \mathbb{E}\left\{ len(\partial A_{u}(\widetilde{T}_{\mathbb{\ell
}(.),S^{2})\right\} \text{ .}
\end{equation*
This fits with the heuristic understanding that the sequence of fields
\left\{ \beta _{j}(.)\right\} ,$ representing an average, is smoother than
the subordinating sequence $\left\{ T_{\mathbb{\ell }}(.)\right\} ;$ the
latter is thus expected to have rougher excursion sets, and hence greater
boundary regions. This heuristic argument can be made rigorous imposing some
regularity conditions on the behaviour of the angular power spectrum
\left\{ C_{\mathbb{\ell }}\right\} .$
\end{remark}
Much more explicit results can of course be obtained by setting a more
specific form for the behaviour of the angular power spectrum $\left\{ C_
\mathbb{\ell }}\right\} $ and the weighting kernel $b(.);$ for instance, in
a CMB related environment it is natural to consider the \emph{Sachs-Wolfe}
angular power spectrum $C_{\mathbb{\ell }}\sim G\ell ^{-\alpha },$ some
G>0, $ $\alpha >2,$ (see \cite{Durrer}{).}
\section{Gaussian subordinated fields}
\subsection{Local Transforms of $\protect\beta _{j}(.)$}
For statistical applications, it is often more interesting to consider
nonlinear transforms of random fields. For instance, in a CMB related
environment a lot of efforts have been spent to investigate local
fluctuations of angular power spectra; to this aim, moving averages of
squared wavelet/needlet coefficients are usually computed, see for instance
\cite{pietrobon1} and the references therein. Our purpose here is to derive
some rigorous results on the behaviour of these statistics.
To this aim, let us consider first the simple squared fiel
\begin{equation*}
H_{2j}(x):=H_{2}(\widetilde{\beta }_{j}(x))=\frac{\beta _{j}^{2}(x)}{\mathbb
E}\beta _{j}^{2}(x)}-1\text{ .}
\end{equation*
The expected value of Lipschitz-Killing curvatures for the excursion regions
of such fields is easily derived, indeed by the general Gaussian kinematic
formula we have, for $u\geq -1
\begin{eqnarray*}
\mathbb{E}\left\{ \mathcal{L}_{0}^{\widetilde{\beta
_{j}}(A_{u}(H_{2};S^{2}))\right\} &=&\sum_{k=0}^{2}(2\pi )^{-k/2}\mathcal{L
_{k}^{\widetilde{\beta }_{j}}(S^{2})\mathcal{M}_{k}^{\mathcal{N}}((-\infty ,
\sqrt{u+1})\cup (\sqrt{u+1},\infty )) \\
&=&\sum_{k=0}^{2}(2\pi )^{-k/2}\mathcal{L}_{k}^{\widetilde{\beta
_{j}}(S^{2})2\mathcal{M}_{k}^{\mathcal{N}}((\sqrt{u+1},\infty ))
\end{eqnarray*
\begin{equation*}
=4(1-\Phi (\sqrt{u+1}))+\frac{1}{2\pi }\frac{\sum_{\mathbb{\ell }}b^{2}
\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell
}P_{\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb
\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }}}\mathcal{L
_{2}(S^{2})\frac{e^{-(u+1)/2}}{\sqrt{2\pi }}2\sqrt{u+1}\text{ .}
\end{equation*
Likewis
\begin{eqnarray*}
\mathbb{E}\left\{ \mathcal{L}_{1}^{\widetilde{\beta
_{j}}(A_{u}(H_{2};S^{2}))\right\} &=&\sum_{k=0}^{1}(2\pi )^{-k/2}\left[
\begin{array}{c}
k+1 \\
\end{array
\right] \mathcal{L}_{k+1}^{\widetilde{\beta }_{j}}(S^{2})\mathcal{M}_{k}^
\mathcal{N}}((-\infty ,-\sqrt{u+1})\cup (\sqrt{u+1},\infty )) \\
&=&\mathcal{L}_{1}^{\widetilde{\beta }_{j}}(S^{2})\mathcal{M}_{0}^{\mathcal{
}}((-\infty ,-\sqrt{u+1})\cup (\sqrt{u+1},\infty )) \\
&&+(2\pi )^{-1/2}\frac{\pi }{2}\mathcal{L}_{2}^{\widetilde{\beta
_{j}}(S^{2})\mathcal{M}_{1}^{\mathcal{N}}((-\infty ,-\sqrt{u+1})\cup (\sqrt
u+1},\infty )) \\
&=&(2\pi )^{-1/2}\frac{\pi }{2}(4\pi \times \frac{\sum_{\mathbb{\ell }}b^{2}
\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell
}P_{\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb
\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }}}\mathcal{)}
\frac{e^{-(u+1)/2}}{\sqrt{2\pi }} \\
&=&2\pi (\frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})\frac
2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }}P_{\mathbb{\ell }}^{\prime }(1)}
\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell
+1}{4\pi }C_{\mathbb{\ell }}}\mathcal{)}e^{-(u+1)/2},
\end{eqnarray*
which implies for the Euclidean LK
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{1}(A_{u}(H_{2};S^{2}))\right\} =2\pi \left\{
\frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb
\ell }+1}{4\pi }C_{\mathbb{\ell }}P_{\mathbb{\ell }}^{\prime }(1)}{\sum_
\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}
4\pi }C_{\mathbb{\ell }}}\right\} ^{1/2}e^{-(u+1)/2},
\end{equation*
and therefor
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{1}(\partial A_{u}(H_{2};S^{2}))\right\} =4\pi
\left\{ \frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})\frac{
\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }}P_{\mathbb{\ell }}^{\prime }(1)}
\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell
+1}{4\pi }C_{\mathbb{\ell }}}\right\} ^{1/2}e^{-(u+1)/2}.
\end{equation*
Finall
\begin{eqnarray*}
\mathbb{E}\left\{ \mathcal{L}_{2}^{\widetilde{\beta
_{j}}(A_{u}(H_{2};S^{2}))\right\} &=&\mathcal{L}_{2}^{\widetilde{\beta
_{j}}(S^{2})\mathcal{M}_{0}^{\mathcal{N}}((-\infty ,-\sqrt{u+1})\cup (\sqrt
u+1},\infty )) \\
&=&4\pi \left\{ \frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}
)\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }}P_{\mathbb{\ell }}^{\prime
}(1)}{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb
\ell }+1}{4\pi }C_{\mathbb{\ell }}}\right\} 2(1-\Phi (\sqrt{u+1}))
\end{eqnarray*
entailing a Euclidean LK
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{2}(A_{u}(H_{2};S^{2}))\right\} =4\pi \times
2(1-\Phi (\sqrt{u+1}))\text{ .}
\end{equation*
It should be noted that the tail decay for the Euler characteristic and the
boundary length is much slower than in the Gaussian case. This is consistent
with the elementary fact that polynomial transforms shift angular power
spectra at higher frequencies, hence yielding a rougher path behaviour.
Likewise, for cubic transforms we have
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{0}^{\widetilde{\beta }_{j}}(A_{u}(\widetilde
\beta }_{j}^{3}(x);S^{2}))\right\} =2(1-\Phi (\sqrt[3]{u}))+\frac{1}{2\pi
\frac{\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi }C_{\ell
}P_{\ell }^{\prime }(1)}{\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})\frac{2\ell +
}{4\pi }C_{\ell }}\mathcal{L}_{2}(S^{2})\frac{e^{-(\sqrt[3]{u})^{2}/2}}
\sqrt{2\pi }}\sqrt[3]{u}\text{ ,}
\end{equation*
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{1}^{\widetilde{\beta }_{j}}(A_{u}(\widetilde
\beta }_{j}^{3}(x);S^{2}))\right\} =\pi (\frac{\sum_{\ell }b^{2}(\frac{\ell
}{B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }P_{\ell }^{\prime }(1)}{\sum_{\ell
}b^{2}(\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }}\mathcal{)}e^{-
\sqrt[3]{u})^{2}/2},
\end{equation*
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{1}(\partial A_{u}(\widetilde{\beta
_{j}^{3}(x);S^{2}))\right\} =2\pi \left\{ \frac{\sum_{\ell }b^{2}(\frac{\ell
}{B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }P_{\ell }^{\prime }(1)}{\sum_
\mathbb{\ell }}b^{2}(\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }
\right\} ^{1/2}e^{-(\sqrt[3]{u})^{2}/2},
\end{equation*
and finall
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{2}^{\widetilde{\beta }_{j}}(A_{u}(\widetilde
\beta }_{j}^{3}(x);S^{2}))\right\} =4\pi \left\{ \frac{\sum_{\ell }b^{2}
\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }P_{\ell }^{\prime }(1)}
\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }
\right\} 2(1-\Phi (\sqrt[3]{u}))
\end{equation*
entailing an expected value for the excursion area given b
\begin{equation*}
\mathbb{E}\left\{ \mathcal{L}_{2}(A_{u}(\widetilde{\beta
_{j}^{3}(x);S^{2}))\right\} =4\pi (1-\Phi (\sqrt[3]{u}))\mbox{ .}
\end{equation*}
Similar results could be easily derived for higher order polynomial
transforms. However, although such findings may be useful for applications,
as motivated above we believe it is much more important to focus on
transforms that entail some form of local averaging, as these are likely to
be more relevant for practitioners. To this issue we devote the rest of this
section and a large part of the paper.
\subsection{Nonlocal Transforms of $\protect\beta _{j}(.)$}
We now consider the case of smoothed nonlinear functionals. We are
interested, for instance, in studying the LKCs for local estimates of the
angular power spectrum, which as mentioned before have already found many
important applications in a CMB related framework. To this aim, we
introduce, for every $x\in S^{2}
\begin{equation}
g_{j;q}(x):=\int_{S^{2}}K(\left\langle x,y\right\rangle )H_{q}(\widetilde
\beta }_{j}(y))dy\text{ ; } \label{filipo}
\end{equation
throughout the sequel, we shall assume the kernel $K:[0,1]\rightarrow
\mathbb{R}$ to be rotational invariant and smooth, e.g. we assume that the
following finite-order expansion holds:
\begin{equation}
K(\left\langle x,y\right\rangle )=\sum_{\ell }^{L_{K}}\frac{2\ell +1}{4\pi
\kappa (\ell )P_{\ell }(\left\langle x,y\right\rangle )\text{ , some fixed
L_{K}\in \mathbb{N}\text{ .} \label{kernexp}
\end{equation
Here, as before we write $H_{q}(.)$ for the Hermite polynomials. For $q=1,$
we just get the smoothed Gaussian proces
\begin{equation}
g_{j}(x):=g_{j;1}(x)=\int_{S^{2}}K(\left\langle x,y\right\rangle )\widetilde
\beta }_{j}(y)dy\text{ .} \label{kernel}
\end{equation
The practical importance of the analysis of fields such as $g_{j;q}(.)$ can
be motivated as follows. A crucial topic when dealing with cosmological data
is the analysis of isotropy properties. For instance, in a CMB related
framework a large amount of work has focussed on the possible existence of
asymmetries in the behaviour of angular power spectra or bispectra across
different hemispheres (see for instance \cite{pietrobon1}, \cite{rudjord2}).
In these papers, powers of wavelet coefficients at some frequencies $j$ are
averaged over different hemispheres to investigate the existence of
asymmetries/anisotropies in the CMB distribution; some evidence has been
reported, for instance, for power asymmetries with respect to the Milky Way
plane for frequencies corresponding to angular scales of a few degrees (such
effects are related in the Cosmological literature to widely debated
anomalies known as \emph{the Cold Spot }and \emph{the Axis of Evil}, see
\cite{bennett2012, planckIS} and the references therein). To investigate
these anomalies, statistics which can be viewed as discretized versions of
sup_{x\in S^{2}}g_{j;q}(x)$ have been evaluated; their significance is
typically tested against Monte Carlo simulations, under the null of
isotropy. Our results below will provide the first rigorous derivation of
asymptotic properties in this settings.
Our first Lemma is an immediate application of spherical Fourier analysis
techniques. For notational simplicity and without loss of generality, we
take until the end of this Section $\mathbb{E}\beta _{j}^{2}(x)=1,$ so we
need no longer distinguish between $\beta _{j}$ and $\widetilde{\beta }_{j}$.
\begin{lemma}
The field $g_{j}(x)$ is zero-mean, finite variance and isotropic, with
covariance functio
\begin{equation*}
\mathbb{E}\left\{ g_{j}(x_1)g_{j}(x_2)\right\} =\sum_{\mathbb{\ell }}b^{2}
\frac{\ell }{B^{j}})\kappa ^{2}(\ell )\frac{2\ell +1}{4\pi }C_{\ell }P_{\ell
}(\left\langle x_{1},x_{2}\right\rangle )\text{ .}
\end{equation*}
\end{lemma}
\begin{proof}
Note first tha
\begin{eqnarray*}
\mathbb{E}\left\{ g_{j}(x_1)g_{j}(x_2)\right\} &=&\mathbb{E}\left\{
\int_{S^{2}}K(\left\langle x_{1},y_{1}\right\rangle )\beta
_{j}(y_{1})dy_{1}\int_{S^{2}}K(\left\langle x_{2},y_{2}\right\rangle )\beta
_{j}(y_{2})dy_{2}\right\} \\
&=&\left\{ \int_{S^{2}\times S^{2}}K(\left\langle x_{1},y_{1}\right\rangle
)K(\left\langle x_{2},y_{2}\right\rangle )\mathbb{E}\left\{ \beta
_{j}(y_{1})\beta _{j}(y_{2})\right\} dy_{1}dy_{2}\right\}
\end{eqnarray*
\begin{equation*}
=\int_{S^{2}\times S^{2}}K(\left\langle x_{1},y_{1}\right\rangle
)K(\left\langle x_{2},y_{2}\right\rangle )\sum_{\ell }b^{2}(\frac{\ell }
B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }P_{\ell }(\left\langle
y_{1},y_{2}\right\rangle )\text{ .}
\end{equation*
Recall the reproducing kernel formul
\begin{eqnarray*}
\int_{S^{2}}P_{\ell }(\left\langle x_{1},y_{1}\right\rangle )P_{\ell
}(\left\langle y_{1},y_{2}\right\rangle )dy_{1} &=&\frac{4\pi }{2\ell +1
P_{\ell }(\left\langle x_{1},y_{2}\right\rangle )\text{ ,} \\
\int_{S^{2}}P_{\ell _{1}}(\left\langle x_{1},y_{1}\right\rangle )P_{\ell
_{2}}(\left\langle y_{1},y_{2}\right\rangle )dy_{1} &=&0\text{ , }\ell
_{1}\neq \ell _{2}\text{ ,}
\end{eqnarray*
whence
\begin{equation*}
\int_{S^{2}\times S^{2}}K(\left\langle x_{1},y_{1}\right\rangle
)K(\left\langle x_{2},y_{2}\right\rangle )\sum_{\ell }b^{2}(\frac{\ell }
B^{j}})\frac{2\ell +1}{4\pi }C_{\ell }P_{\ell }(\left\langle
y_{1},y_{2}\right\rangle )
\end{equation*
\begin{eqnarray*}
&=&\int_{S^{2}\times S^{2}}\sum_{\ell _{1}}\frac{2\ell _{1}+1}{4\pi }\kappa
(\ell _{1})P_{\ell _{1}}(\left\langle x_{1},y_{1}\right\rangle )\sum_{\ell
_{2}}\frac{2\ell _{2}+1}{4\pi }\kappa (\ell _{2})P_{\ell _{2}}(\left\langle
x_{2},y_{2}\right\rangle ) \\
&&\times \sum_{\ell }b^{2}(\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi }C_
\mathbb{\ell }}P_{\mathbb{\ell }}(\left\langle y_{1},y_{2}\right\rangle
)dy_{1}dy_{2}
\end{eqnarray*
\begin{eqnarray*}
&=&\int_{S^{2}}\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi
C_{\ell }\sum_{\ell _{1}}\kappa (\ell _{1})\sum_{\ell _{2}}\frac{2\ell _{2}+
}{4\pi }\kappa (\ell _{2})P_{\ell _{2}}(\left\langle
x_{2},y_{2}\right\rangle ) \\
&&\times \int_{S^{2}}\frac{2\ell _{1}+1}{4\pi }P_{\ell _{1}}(\left\langle
x_{1},y_{1}\right\rangle )P_{\ell }(\left\langle y_{1},y_{2}\right\rangle
)dy_{1}dy_{2}
\end{eqnarray*
\begin{equation*}
=\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})\kappa (\ell )\frac{2\ell +1}{4\pi
C_{\ell }\sum_{\ell _{2}}\frac{2\ell _{2}+1}{4\pi }\kappa (\ell
_{2})\int_{S^{2}}P_{\ell _{2}}(\left\langle x_{2},y_{2}\right\rangle
)P_{\ell }(\left\langle x_{1},y_{2}\right\rangle )dy_{2}
\end{equation*
\begin{equation*}
=\sum_{\ell }b^{2}(\frac{\ell }{B^{j}})\kappa ^{2}(\ell )\frac{2\ell +1}
4\pi }C_{\ell }P_{\ell }(\left\langle x_{1},x_{2}\right\rangle )\text{ ,}
\end{equation*
as claimed.
\end{proof}
The derivation of analogous results in the case of $q\geq 2$ requires more
work and extra notation. In particular, we shall need the Wigner's $3j$
coefficients, which are defined by (for $m_{1}+m_{2}+m_{3}=0,$ see \cite{VMK
, expression (8.2.1.5))
\begin{align*}
\left(
\begin{array}{ccc}
\mathbb{\ell }_{1} & \mathbb{\ell }_{2} & \mathbb{\ell }_{3} \\
m_{1} & m_{2} & m_{3
\end{array
\right) & :=(-1)^{\mathbb{\ell }_{1}+m_{1}}\sqrt{2\mathbb{\ell }_{3}+1}\left[
\frac{(\mathbb{\ell }_{1}+\mathbb{\ell }_{2}-\mathbb{\ell }_{3})!(\mathbb
\ell }_{1}-\mathbb{\ell }_{2}+\mathbb{\ell }_{3})!(\mathbb{\ell }_{1}
\mathbb{\ell }_{2}+\mathbb{\ell }_{3})!}{(\mathbb{\ell }_{1}+\mathbb{\ell
_{2}+\mathbb{\ell }_{3}+1)!}\right] ^{1/2} \\
& \times \left[ \frac{(\mathbb{\ell }_{3}+m_{3})!(\mathbb{\ell }_{3}-m_{3})
}{(\mathbb{\ell }_{1}+m_{1})!(\mathbb{\ell }_{1}-m_{1})!(\mathbb{\ell
_{2}+m_{2})!(\mathbb{\ell }_{2}-m_{2})!}\right] ^{1/2} \\
& \times \sum_{z}\frac{(-1)^{z}(\mathbb{\ell }_{2}+\mathbb{\ell
_{3}+m_{1}-z)!(\mathbb{\ell }_{1}-m_{1}+z)!}{z!(\mathbb{\ell }_{2}+\mathbb
\ell }_{3}-\mathbb{\ell }_{1}-z)!(\mathbb{\ell }_{3}+m_{3}-z)!(\mathbb{\ell
_{1}-\mathbb{\ell }_{2}-m_{3}+z)!}\text{,}
\end{align*
where the summation runs over all $z$'s such that the factorials are
non-negative. This expression becomes somewhat neater for
m_{1}=m_{2}=m_{3}=0,$ where we hav
\begin{equation*}
\left(
\begin{array}{ccc}
\mathbb{\ell }_{1} & \mathbb{\ell }_{2} & \mathbb{\ell }_{3} \\
0 & 0 &
\end{array
\right) =
\end{equation*
\begin{equation}
\left\{
\begin{array}{c}
0\text{ , for }\mathbb{\ell }_{1}+\mathbb{\ell }_{2}+\mathbb{\ell }_{3}\text{
odd} \\
(-1)^{\frac{\mathbb{\ell }_{1}+\mathbb{\ell }_{2}-\mathbb{\ell }_{3}}{2}
\frac{\left[ (\mathbb{\ell }_{1}+\mathbb{\ell }_{2}+\mathbb{\ell }_{3})/
\right] !}{\left[ (\mathbb{\ell }_{1}+\mathbb{\ell }_{2}-\mathbb{\ell
_{3})/2\right] !\left[ (\mathbb{\ell }_{1}-\mathbb{\ell }_{2}+\mathbb{\ell
_{3})/2\right] !\left[ (-\mathbb{\ell }_{1}+\mathbb{\ell }_{2}+\mathbb{\ell
_{3})/2\right] !}\left\{ \frac{(\mathbb{\ell }_{1}+\mathbb{\ell }_{2}
\mathbb{\ell }_{3})!(\mathbb{\ell }_{1}-\mathbb{\ell }_{2}+\mathbb{\ell
_{3})!(-\mathbb{\ell }_{1}+\mathbb{\ell }_{2}+\mathbb{\ell }_{3})!}{(\mathbb
\ell }_{1}+\mathbb{\ell }_{2}+\mathbb{\ell }_{3}+1)!}\right\} ^{1/2}\text{ }
\\
\text{for }\mathbb{\ell }_{1}+\mathbb{\ell }_{2}+\mathbb{\ell }_{3}\text{
even
\end{array
\right. . \label{appe}
\end{equation
It is occasionally more convenient to focus on Clebsch-Gordan coefficients,
which are related to the Wigner's by a simple change of normalization, e.g
\begin{equation}
C_{\ell _{1}m_{1}\ell _{2}m_{2}}^{\ell _{3}m_{3}}:=\frac{(-1)^{\ell
_{3}-m_{3}}}{\sqrt{2\ell _{3}+1}}\left(
\begin{array}{ccc}
\mathbb{\ell }_{1} & \mathbb{\ell }_{2} & \mathbb{\ell }_{3} \\
m_{1} & m_{2} & -m_{3
\end{array
\right) \text{ .} \label{cgdef}
\end{equation
Wigner's $3j$ coefficients are elements of unitary matrices which intertwine
alternative reducible representations of the group of rotations $SO(3),$ and
because of this emerge naturally in the evaluation of multiple integrals of
spherical harmonics, see for instance \cite{marpecbook}, Section 3.5.2. As a
consequence, they also appear in the covariances of nonlinear transforms;
for $q=2,$ we have indeed
\begin{lemma}
The field $g_{j;2}(x)$ is zero-mean, finite variance and isotropic, with
covariance functio
\begin{equation*}
\mathbb{E}\left\{ g_{j;2}(x_{1})g_{j;2}(x_{2})\right\} =
\end{equation*
\begin{equation*}
2\sum_{\mathbb{\ell }}\kappa ^{2}(\mathbb{\ell })\frac{2\mathbb{\ell }+1}
4\pi }\sum_{\mathbb{\ell }_{1}\mathbb{\ell }_{2}}b^{2}(\frac{\mathbb{\ell
_{1}}{B^{j}})b^{2}(\frac{\mathbb{\ell }_{2}}{B^{j}})\frac{(2\mathbb{\ell
_{1}+1)(2\mathbb{\ell }_{2}+1)}{4\pi }C_{\mathbb{\ell }_{1}}C_{\mathbb{\ell
_{2}}\left(
\begin{array}{ccc}
\mathbb{\ell } & \mathbb{\ell }_{1} & \mathbb{\ell }_{2} \\
0 & 0 &
\end{array
\right) ^{2}P_{\mathbb{\ell }}(\left\langle x_{1},x_{2}\right\rangle )\text{
.}
\end{equation*}
\end{lemma}
\begin{proof}
Note first tha
\begin{equation*}
\mathbb{E}\left\{ g_{j;2}(x_{1})g_{j;2}(x_{2})\right\} =\mathbb{E}\left\{
\int_{S^{2}}K(\left\langle x_{1},y_{1}\right\rangle )H_{2}(\beta
_{j}(y_{1}))dy_{1}\int_{S^{2}}K(\left\langle x_{2},y_{2}\right\rangle
)H_{2}(\beta _{j}(y_{2}))dy_{2}\right\}
\end{equation*
\begin{equation*}
=\int_{S^{2}\times S^{2}}K(\left\langle x_{1},y_{1}\right\rangle
)K(\left\langle x_{2},y_{2}\right\rangle )\mathbb{E}\left\{ H_{2}(\beta
_{j}(y_{1}))H_{2}(\beta _{j}(y_{2}))\right\} dy_{1}dy_{2}
\end{equation*
\begin{equation*}
=2\int_{S^{2}\times S^{2}}K(\left\langle x_{1},y_{1}\right\rangle
)K(\left\langle x_{2},y_{2}\right\rangle )\left\{ \sum_{\mathbb{\ell }}b^{2}
\frac{\mathbb{\ell }}{B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell
}P_{\mathbb{\ell }}(\left\langle y_{1},y_{2}\right\rangle )\right\}
^{2}dy_{1}dy_{2}
\end{equation*
\begin{eqnarray*}
&=&2\int_{S^{2}\times S^{2}}\sum_{\mathbb{\ell }_{1}}\frac{2\mathbb{\ell
_{1}+1}{4\pi }\kappa (\mathbb{\ell }_{1})P_{\mathbb{\ell }_{1}}(\left\langle
x_{1},y_{1}\right\rangle )\sum_{\mathbb{\ell }_{2}}\frac{2\mathbb{\ell
_{2}+1}{4\pi }\kappa (\mathbb{\ell }_{2})P_{\mathbb{\ell }_{2}}(\left\langle
x_{2},y_{2}\right\rangle ) \\
&&\times \sum_{\mathbb{\ell }_{3}\mathbb{\ell }_{4}}b^{2}(\frac{\mathbb{\ell
}_{3}}{B^{j}})b^{2}(\frac{\mathbb{\ell }_{4}}{B^{j}})\frac{2\mathbb{\ell
_{3}+1}{4\pi }\frac{2\mathbb{\ell }_{4}+1}{4\pi }C_{\mathbb{\ell }_{3}}C_
\mathbb{\ell }_{4}}P_{\mathbb{\ell }_{3}}(\left\langle
y_{1},y_{2}\right\rangle )P_{\mathbb{\ell }_{4}}(\left\langle
y_{1},y_{2}\right\rangle )dy_{1}dy_{2}\text{ .}
\end{eqnarray*
Now recall tha
\begin{equation*}
\int_{S^{2}}P_{\mathbb{\ell }_{1}}(\left\langle x_{1},y_{1}\right\rangle )P_
\mathbb{\ell }_{3}}(\left\langle y_{1},y_{2}\right\rangle )P_{\mathbb{\ell
_{4}}(\left\langle y_{1},y_{2}\right\rangle )dy_{1}
\end{equation*
\begin{equation*}
=\frac{(4\pi )^{3}}{(2\mathbb{\ell }_{1}+1)(2\mathbb{\ell }_{3}+1)(2\mathbb
\ell }_{4}+1)}
\end{equation*
\begin{equation*}
\times \int_{S^{2}}\sum_{m_{1}m_{2}m_{3}}Y_{\mathbb{\ell }_{1}m_{1}}(y_{1}
\overline{Y}_{\mathbb{\ell }_{1}m_{1}}(x_{1})Y_{\mathbb{\ell
_{3}m_{3}}(y_{1})\overline{Y}_{\mathbb{\ell }_{3}m_{3}}(y_{2})Y_{\mathbb
\ell }_{4}m_{4}}(y_{1})\overline{Y}_{\mathbb{\ell }_{4}m_{4}}(y_{2})dy_{1}
\end{equation*
\begin{equation*}
=(\frac{(4\pi )^{5}}{(2\mathbb{\ell }_{1}+1)(2\mathbb{\ell }_{3}+1)(2\mathbb
\ell }_{4}+1)})^{1/2}
\end{equation*
\begin{equation*}
\times \sum_{m_{1}m_{3}m_{4}}\left(
\begin{array}{ccc}
\mathbb{\ell }_{1} & \mathbb{\ell }_{3} & \mathbb{\ell }_{4} \\
m_{1} & m_{3} & m_{4
\end{array
\right) \left(
\begin{array}{ccc}
\mathbb{\ell }_{1} & \mathbb{\ell }_{3} & \mathbb{\ell }_{4} \\
0 & 0 &
\end{array
\right) \overline{Y}_{\mathbb{\ell }_{1}m_{1}}(x_{1})\overline{Y}_{\mathbb
\ell }_{3}m_{3}}(y_{2})\overline{Y}_{\mathbb{\ell }_{4}m_{4}}(y_{2})\text{ ,}
\end{equation*
Likewis
\begin{equation*}
\int_{S^{2}}P_{\mathbb{\ell }_{2}}(\left\langle x_{2},y_{2}\right\rangle
\overline{Y}_{\mathbb{\ell }_{3}m_{3}}(y_{2})\overline{Y}_{\mathbb{\ell
_{4}m_{4}}(y_{2})dy_{2}
\end{equation*
\begin{equation*}
=\frac{4\pi }{2\mathbb{\ell }_{2}+1}\int_{S^{2}}\sum_{m_{2}}\overline{Y}_
\mathbb{\ell }_{2}m_{2}}(y_{2})Y_{\mathbb{\ell }_{2}m_{2}}(x_{2})\overline{Y
_{\mathbb{\ell }_{3}m_{3}}(y_{2})\overline{Y}_{\mathbb{\ell
_{4}m_{4}}(y_{2})dy_{2}
\end{equation*
\begin{equation*}
=\sqrt{\frac{(4\pi )(2\mathbb{\ell }_{3}+1)(2\mathbb{\ell }_{4}+1)}{2\mathbb
\ell }_{2}+1}}\sum_{m_{2}}\left(
\begin{array}{ccc}
\mathbb{\ell }_{2} & \mathbb{\ell }_{3} & \mathbb{\ell }_{4} \\
m_{2} & m_{3} & m_{4
\end{array
\right) \left(
\begin{array}{ccc}
\mathbb{\ell }_{2} & \mathbb{\ell }_{3} & \mathbb{\ell }_{4} \\
0 & 0 &
\end{array
\right) Y_{\mathbb{\ell }_{2}m_{2}}(x_{2})\text{ .}
\end{equation*
Using the orthonormality properties of Wigner's $3j$ coefficients (see again
\cite{marpecbook}, Chapter 3.5), we hav
\begin{equation*}
\sum_{m_{3}m_{4}}\left(
\begin{array}{ccc}
\mathbb{\ell }_{1} & \mathbb{\ell }_{3} & \mathbb{\ell }_{4} \\
m_{1} & m_{3} & m_{4
\end{array
\right) \left(
\begin{array}{ccc}
\mathbb{\ell }_{2} & \mathbb{\ell }_{3} & \mathbb{\ell }_{4} \\
m_{2} & m_{3} & m_{4
\end{array
\right) =\frac{\delta _{m_{1}}^{m_{2}}\delta _{\mathbb{\ell }_{1}}^{\mathbb
\ell }_{2}}}{(2\mathbb{\ell }_{1}+1)}\text{ ,}
\end{equation*
whence we ge
\begin{equation*}
\mathbb{E}\left\{ g_{j;2}(x_{1})g_{j;2}(x_{2})\right\} =
\end{equation*
\begin{equation*}
2\sum_{\mathbb{\ell }}\kappa ^{2}(\mathbb{\ell })\frac{2\mathbb{\ell }+1}
4\pi }\sum_{\mathbb{\ell }_{1}\mathbb{\ell }_{2}}b^{2}(\frac{\mathbb{\ell
_{1}}{B^{j}})b^{2}(\frac{\mathbb{\ell }_{2}}{B^{j}})\frac{(2\mathbb{\ell
_{1}+1)(2\mathbb{\ell }_{2}+1)}{4\pi }C_{\mathbb{\ell }_{1}}C_{\mathbb{\ell
_{2}}\left(
\begin{array}{ccc}
\mathbb{\ell } & \mathbb{\ell }_{1} & \mathbb{\ell }_{2} \\
0 & 0 &
\end{array
\right) ^{2}P_{\mathbb{\ell }}(\left\langle x_{1},x_{2}\right\rangle )\text{
,}
\end{equation*
as claimed. As a special case, the variance is provided b
\begin{equation*}
\mathbb{E}g_{j;2}^{2}(x)=2\sum_{\mathbb{\ell }}\kappa ^{2}(\mathbb{\ell }
\frac{2\mathbb{\ell }+1}{4\pi }\sum_{\mathbb{\ell }_{1}\mathbb{\ell
_{2}}b^{2}(\frac{\mathbb{\ell }_{1}}{B^{j}})b^{2}(\frac{\mathbb{\ell }_{2}}
B^{j}})\frac{(2\mathbb{\ell }_{1}+1)(2\mathbb{\ell }_{2}+1)}{4\pi }C_
\mathbb{\ell }_{1}}C_{\mathbb{\ell }_{2}}\left(
\begin{array}{ccc}
\mathbb{\ell } & \mathbb{\ell }_{1} & \mathbb{\ell }_{2} \\
0 & 0 &
\end{array
\right) ^{2}\text{ .}
\end{equation*}
\end{proof}
\begin{remark}
Because the field $\left\{ g_{j;2}(.)\right\} $ has finite-variance and it
is isotropic, it admits itself a spectral representation. Indeed, it is a
simple computation to show that the corresponding angular power spectrum is
provided b
\begin{equation}
C_{\mathbb{\ell };j,2}:=2\kappa ^{2}(\mathbb{\ell })\sum_{\mathbb{\ell }_{1
\mathbb{\ell }_{2}}b^{2}(\frac{\mathbb{\ell }_{1}}{B^{j}})b^{2}(\frac
\mathbb{\ell }_{2}}{B^{j}})\frac{(2\mathbb{\ell }_{1}+1)(2\mathbb{\ell
_{2}+1)}{4\pi }C_{\mathbb{\ell }_{1}}C_{\mathbb{\ell }_{2}}\left(
\begin{array}{ccc}
\mathbb{\ell } & \mathbb{\ell }_{1} & \mathbb{\ell }_{2} \\
0 & 0 &
\end{array
\right) ^{2}, \label{colizzi}
\end{equation}
for $\mathbb{\ell }=1,2,...$ This result will have a great relevance for the
practical implementation of the findings in the next sections.
\end{remark}
\subsubsection{Higher-order transforms}
The general case of non linear transforms with $q\geq 3$ can be dealt with
analogous lines, the main difference being the appearance of multiple
integrals of spherical harmonics of order greater than 3, and hence
so-called higher order Gaunt integrals and convolutions of Clebsch-Gordan
coefficients. For brevity's sake, we provide only the basic details; we
refer to \cite{marpecbook} for a more detailed discussion on nonlinear
transforms of Gaussian spherical harmonics. Here, we simply recall the
definition of the multiple Gaunt integral (see \cite{marpecbook}, Remark
6.30 and Theorem 6.31), which is given b
\begin{equation*}
\mathcal{G}(\ell _{1},m_{1};...\ell _{q},m_{q};\ell ,m):=\int_{S^{2}}Y_{\ell
_{1}m_{1}}(x)...Y_{\ell _{q}m_{q}}(x)Y_{\ell m}(x)d\sigma (x)\text{ ,}
\end{equation*
where the coefficients $\mathcal{G}(\ell _{1},m_{1};...\ell _{q},m_{q};\ell
,m)$ can be expressed as multiple convolution of Wigner/Clebsch-Gordan terms
(see \ref{cgdef})
\begin{equation*}
\mathcal{G}(\ell _{1},m_{1};...\ell _{q},m_{q};\ell ,m)=(-1)^{m}\sqrt{\frac
(2\ell _{1}+1)...(2\ell _{q}+1)}{(4\pi )^{q-1}(2\ell +1)}}
\end{equation*
\begin{equation*}
\times \sum_{\lambda _{1}...\lambda _{q-2}}C_{\ell _{1}0\ell _{2}0}^{\lambda
_{1}0}C_{\lambda _{1}0\ell _{3}0}^{\lambda _{2}0}...C_{\lambda _{q-2}0\ell
_{q}0}^{\ell 0}\sum_{\mu _{1}...\mu _{q-2}}C_{\ell _{1}m_{1}\ell
_{2}m_{2}}^{\lambda _{1}\mu _{1}}C_{\lambda _{1}\mu _{1}\ell
_{3}m_{3}}^{\lambda _{2}\mu _{2}}...C_{\lambda _{q-2}\mu _{q-2}\ell
_{q}m_{q}}^{\ell m}\text{ .}
\end{equation*
Following also \cite{marpecbook}, eq. (6.40), let us introduce the shorthand
notatio
\begin{equation}
C_{\ell _{1}0\ell _{2}0...\ell _{q}0}^{\lambda _{1}...\lambda _{q-2}\ell
0}:=C_{\ell _{1}0\ell _{2}0}^{\lambda _{1}0}C_{\lambda _{1}0\ell
_{3}0}^{\lambda _{2}0}...C_{\lambda _{q-2}0\ell _{q}0}^{\ell 0}\text{ ,
\mathcal{C}(\ell _{1},...,\ell _{q},\ell ):=\sum_{\lambda _{1}...\lambda
_{q-2}}\left\{ C_{\ell _{1}0\ell _{2}0...\ell _{q}0}^{\lambda _{1}...\lambda
_{q-2}\ell 0}\right\} ^{2}. \label{colizzi2}
\end{equation
It should be noted that, from the unitary properties of Clebsch-Gordan
coefficient
\begin{equation*}
\sum_{\ell }\mathcal{C}(\ell _{1},...,\ell _{q},\ell )=\sum_{\lambda
_{1}...\lambda _{q-2}}\left\{ C_{\ell _{1}0\ell _{2}0}^{\lambda
_{1}0}\right\} ^{2}...\sum_{\ell }\left\{ C_{\lambda _{q-2}0\ell
_{q}0}^{\ell 0}\right\} ^{2}=...=1.
\end{equation*}
\begin{lemma}
For general $q\geq 3,$ the field $g_{j;q}(x)$ is zero-mean, finite variance
and isotropic, with covariance functio
\begin{equation*}
\mathbb{E}\left\{ g_{j;q}(x_{1})g_{j;q}(x_{2})\right\}
\end{equation*
\begin{equation*}
=q!\sum_{\mathbb{\ell }}\kappa ^{2}(\mathbb{\ell })\sum_{\mathbb{\ell
_{1}...\mathbb{\ell }_{q}}\mathcal{C}(\ell _{1},...,\ell _{q},\ell )\left[
\prod}_{k=1}^{q}b^{2}(\frac{\mathbb{\ell }_{k}}{B^{j}})\frac{2\mathbb{\ell
_{k}+1}{4\pi }C_{\mathbb{\ell }_{k}}\right] P_{\mathbb{\ell }}(\left\langle
x_{1},x_{2}\right\rangle ).
\end{equation*}
\end{lemma}
\begin{proof}
We hav
\begin{eqnarray*}
\mathbb{E}g_{j;q}^{2}(x) &=&\mathbb{E}\left\{
\int_{S^{2}}\int_{S^{2}}K(\left\langle x,y_{1}\right\rangle )K(\left\langle
x,y_{2}\right\rangle )H_{q}(\beta _{j}(y_{1}))H_{q}(\beta
_{j}(y_{2}))dy_{1}dy_{2}\right\} \\
&=&q!\int_{S^{2}}\int_{S^{2}}K(\left\langle x,y_{1}\right\rangle
)K(\left\langle x,y_{2}\right\rangle )\left\{ \sum_{\mathbb{\ell }}b^{2}
\frac{\ell }{B^{j}})\frac{2\ell +1}{4\pi }P_{\mathbb{\ell }}(\left\langle
y_{1},y_{2}\right\rangle )\right\} ^{q}dy_{1}dy_{2}\text{ .}
\end{eqnarray*
It is convenient here to view $T_{\mathbb{\ell }}(x),\beta _{j}(x)$ as
isonormal processes of the for
\begin{eqnarray*}
T_{\mathbb{\ell }}(x) &=&\int_{S^{2}}\sqrt{\frac{2\ell +1}{4\pi }C_{\mathbb
\ell }}}P_{\mathbb{\ell }}(\left\langle x,y\right\rangle )dW(y)\text{ ,} \\
\beta _{j}(x) &=&\int_{S^{2}}\sum_{\mathbb{\ell }}b(\frac{\ell }{B^{j}}
\sqrt{\frac{2\ell +1}{4\pi }C_{\mathbb{\ell }}}P_{\mathbb{\ell
}(\left\langle x,y\right\rangle )dW(y)\text{ ,}
\end{eqnarray*
where $dW(y)$ denotes a Gaussian white noise measure on the sphere, whenc
\begin{equation*}
H_{q}(\beta _{j}(x))
\end{equation*
\begin{eqnarray*}
&=&\sum_{\mathbb{\ell }_{1}...\mathbb{\ell }_{q}}b(\frac{\ell _{1}}{B^{j}
)...b(\frac{\ell _{q}}{B^{j}})\sqrt{\prod\limits_{i=1}^{q}\left\{ \frac
2\ell _{i}+1}{4\pi }C_{\mathbb{\ell }_{i}}\right\} } \\
&&\times \int_{S^{2}\times ...\times S^{2}}P_{\mathbb{\ell
_{1}}(\left\langle x,y_{1}\right\rangle )...P_{\mathbb{\ell
_{q}}(\left\langle x,y_{q}\right\rangle )dW(y_{1})...dW(y_{q})
\end{eqnarray*
an
\begin{eqnarray*}
g_{j;q}(z) &=&\int_{S^{2}}\sum_{\ell }\kappa (\ell )\frac{2\ell +1}{4\pi
P_{\ell }(\left\langle z,x\right\rangle )\sum_{\mathbb{\ell }_{1}...\mathbb
\ell }_{q}}b(\frac{\ell _{1}}{B^{j}})...b(\frac{\ell _{q}}{B^{j}})\sqrt
\prod\limits_{i=1}^{q}\left\{ \frac{2\ell _{i}+1}{4\pi }C_{\mathbb{\ell
_{i}}\right\} } \\
&&\times \int_{S^{2}\times ...\times S^{2}}P_{\mathbb{\ell
_{1}}(\left\langle x,y_{1}\right\rangle )...P_{\mathbb{\ell
_{q}}(\left\langle x,y_{q}\right\rangle )dW(y_{1})...dW(y_{q})dx\text{ .}
\end{eqnarray*
It follows easily tha
\begin{equation*}
\mathbb{E}\left\{ g_{j;q}(z_{1})g_{j;q}(z_{2})\right\} =
\end{equation*
\begin{eqnarray*}
&=&\int_{S^{2}\times S^{2}}\sum_{\ell _{1}\ell _{2}}\frac{2\ell _{1}+1}{4\pi
}\kappa (\ell _{1})\frac{2\ell _{2}+1}{4\pi }\kappa (\ell _{2})P_{\ell
_{1}}(\left\langle z_{1},x_{1}\right\rangle )P_{\ell _{2}}(\left\langle
z_{2},x_{2}\right\rangle ) \\
&&\times \sum_{\mathbb{\ell }_{1}...\mathbb{\ell }_{q}}b^{2}(\frac{\ell _{1
}{B^{j}})...b^{2}(\frac{\ell _{q}}{B^{j}})\sqrt{\prod\limits_{i=1}^{q}\lef
\{ \frac{2\ell _{i}+1}{4\pi }C_{\mathbb{\ell }_{i}}\right\} }P_{\mathbb{\ell
}_{1}}(\left\langle x_{1},x_{2}\right\rangle )...P_{\mathbb{\ell
_{q}}(\left\langle x_{1},x_{2}\right\rangle )dx_{1}dx_{2}\text{ .}
\end{eqnarray*
Now write
\begin{equation*}
\frac{(2\mathbb{\ell }_{1}+1)...(2\mathbb{\ell }_{q}+1)}{(4\pi )^{q}}P_
\mathbb{\ell }_{1}}(\left\langle x_{1},x_{2}\right\rangle )...P_{\mathbb
\ell }_{q}}(\left\langle x_{1},x_{2}\right\rangle )
\end{equation*
\begin{equation*}
=\sum_{m_{1}...m_{q}}Y_{\mathbb{\ell }_{1}m_{1}}(x_{1})...Y_{\mathbb{\ell
_{q}m_{q}}(x_{1})\overline{Y}_{\mathbb{\ell }_{1}m_{1}}(x_{2})...\overline{Y
_{\mathbb{\ell }_{q}m_{q}}(x_{2})
\end{equation*
so tha
\begin{equation*}
\frac{(2\mathbb{\ell }_{1}+1)...(2\mathbb{\ell }_{q}+1)}{(4\pi )^{q}
\int_{S^{2}\times S^{2}}P_{\ell _{1}}(\left\langle z_{1},x_{1}\right\rangle
)P_{\ell _{2}}(\left\langle z_{2},x_{2}\right\rangle )P_{\mathbb{\ell
_{1}}(\left\langle x_{1},x_{2}\right\rangle )...P_{\mathbb{\ell
_{q}}(\left\langle x_{1},x_{2}\right\rangle )dx_{1}dx_{2}
\end{equation*
\begin{equation*}
=\sum_{\mu _{1}\mu _{2}}\sum_{m_{1}...m_{q}}\mathcal{G}(\mathbb{\ell
_{1},m_{1};...\mathbb{\ell }_{q},m_{q};\ell _{1},\mu _{1})\mathcal{G}
\mathbb{\ell }_{1},m_{1};...\mathbb{\ell }_{q},m_{q};\ell _{2},\mu
_{2})\left\{ \frac{4\pi }{2\ell +1}Y_{\ell _{1}\mu _{1}}(z_{1})\overline{Y
_{\ell _{2}\mu _{2}}(z_{2})\right\}
\end{equation*
\begin{equation*}
=\frac{4\pi }{2\ell +1}\sum_{\mu _{1}\mu _{2}}Y_{\ell _{1}\mu _{1}}(z_{1}
\overline{Y}_{\ell _{2}\mu _{2}}(z_{2})\delta _{\ell _{1}}^{\ell _{2}}\delta
_{\mu _{1}}^{\mu _{2}}=P_{\ell _{1}}(\left\langle z_{1},z_{2}\right\rangle
\text{ .}
\end{equation*
The general case $q\geq 3$ hence yields (see also \cite{marpecbook}, Theorem
7.5 for a related computation
\begin{equation*}
\mathbb{E}g_{j;q}^{2}(x)=
\end{equation*
\begin{equation*}
q!\sum_{\mathbb{\ell }}\kappa ^{2}(\mathbb{\ell })\sum_{\mathbb{\ell }_{1}..
\mathbb{\ell }_{q}}\mathcal{C}(\ell _{1},...,\ell _{q},\ell )b^{2}(\frac
\mathbb{\ell }_{1}}{B^{j}})...b^{2}(\frac{\mathbb{\ell }_{q}}{B^{j}})\frac{
\mathbb{\ell }_{1}+1}{4\pi }...\frac{2\mathbb{\ell }_{q}+1}{4\pi }C_{\mathbb
\ell }_{1}}...C_{\mathbb{\ell }_{q}}\text{ ,}
\end{equation*
an
\begin{equation*}
\mathbb{E}\left\{ g_{j;q}(x)g_{j;q}(y)\right\}
\end{equation*
\begin{equation*}
=q!\sum_{\mathbb{\ell }}\kappa ^{2}(\mathbb{\ell })\sum_{\mathbb{\ell
_{1}...\mathbb{\ell }_{q}}\mathcal{C}(\ell _{1},...,\ell _{q},\ell )b^{2}
\frac{\mathbb{\ell }_{1}}{B^{j}})...b^{2}(\frac{\mathbb{\ell }_{q}}{B^{j}}
\frac{2\mathbb{\ell }_{1}+1}{4\pi }...\frac{2\mathbb{\ell }_{q}+1}{4\pi }C_
\mathbb{\ell }_{1}}...C_{\mathbb{\ell }_{q}}P_{\mathbb{\ell }}(\left\langle
x_{1},x_{2}\right\rangle )\text{ ,}
\end{equation*
as claimed.
\end{proof}
\begin{remark}
It is immediately checked that the angular power spectrum of $g_{j;q}(y)$ is
given by (see (\ref{colizzi2})
\begin{equation}
C_{\mathbb{\ell };j,q}:=q!\frac{4\pi }{2\ell +1}\kappa ^{2}(\mathbb{\ell
)\sum_{\mathbb{\ell }_{1}...\mathbb{\ell }_{q}}\mathcal{C}(\ell
_{1},...,\ell _{q},\ell )\prod\limits_{k=1}^{q}\left[ b^{2}(\frac{\mathbb
\ell }_{k}}{B^{j}})\frac{2\mathbb{\ell }_{k}+1}{4\pi }C_{\mathbb{\ell }_{k}
\right] \text{ }. \label{porteaperte2}
\end{equation
As a special case, for $q=2$ we recover the previous result (\ref{colizzi}
\begin{equation}
C_{\mathbb{\ell };j,2}=2\kappa ^{2}(\mathbb{\ell })\sum_{\mathbb{\ell }_{1
\mathbb{\ell }_{2}}b^{2}(\frac{\mathbb{\ell }_{1}}{B^{j}})b^{2}(\frac
\mathbb{\ell }_{2}}{B^{j}})\frac{(2\ell _{1}+1)(2\ell _{2}+1)}{4\pi }C_
\mathbb{\ell }_{1}}C_{\mathbb{\ell }_{2}}\left(
\begin{array}{ccc}
\mathbb{\ell } & \mathbb{\ell }_{1} & \mathbb{\ell }_{2} \\
0 & 0 &
\end{array
\right) ^{2} \label{powspe}
\end{equation
\begin{equation*}
=2\kappa ^{2}(\mathbb{\ell })\frac{4\pi }{2\ell +1}\sum_{\mathbb{\ell }_{1
\mathbb{\ell }_{2}}\mathcal{C}(\ell _{1},\ell _{2},\ell )b^{2}(\frac{\mathbb
\ell }_{1}}{B^{j}})b^{2}(\frac{\mathbb{\ell }_{2}}{B^{j}})\frac{(2\ell
_{1}+1)}{4\pi }\frac{(2\ell _{2}+1)}{4\pi }C_{\mathbb{\ell }_{1}}C_{\mathbb
\ell }_{2}},
\end{equation*
becaus
\begin{equation*}
\mathcal{C}(\ell _{1},\ell _{2},\ell )=\left\{ C_{\ell _{1}0\ell
_{2}0}^{\ell 0}\right\} ^{2}=(2\ell +1)\left(
\begin{array}{ccc}
\mathbb{\ell } & \mathbb{\ell }_{1} & \mathbb{\ell }_{2} \\
0 & 0 &
\end{array
\right) ^{2}.
\end{equation*}
\end{remark}
\section{Weak Convergence}
In this Section, we provide our main convergence results. It must be
stressed that the convergence we study here is in some sense different from
the standard theory as presented, for instance, by \cite{Billingsley}, but
refers instead to the broader notion developed by \cite{davydov}, \cit
{aristotilediaconis}, see also \cite{dudley}, chapter 11.
We start first from the following Conditions:
\begin{condition}
\textbf{\label{ConditionB} }The angular power spectrum has the for
\begin{equation*}
C_{\ell }=G(\ell )\ell ^{-\alpha },\text{ }\ell =1,2,...,
\end{equation*
where $\alpha >2$ and $G(.)$ is such that
\begin{eqnarray*}
0 &<&c_{0}\leq G(.)\leq d_{0}, \\
\left\vert \frac{d^{r}}{dx^{r}}G(x)\right\vert &\leq &c_{r}\ell ^{-r},\text{
}r=1,2,...,M\in \mathbb{N}\text{ .}
\end{eqnarray*}
\end{condition}
\begin{condition}
\textbf{\label{ConditionA} }The Kernel $K(\cdot )$ and the field $\left\{
\beta _{j}(\cdot )\right\} $ are such that, for all $j=1,2,3,...
\begin{equation*}
Var\left\{ \int_{S^{2}}K(\left\langle x,y\right\rangle )H_{q}(\widetilde
\beta }_{j}(y))dy\right\} =\sigma _{j}^{2}B^{-2j}\mbox{ , for all }j=1,2,...
\end{equation*
and there exist positive constants $c_{1},c_{2}$ such that $c_{1}\leq \sigma
_{j}^{2}\leq c_{2}$ .
\end{condition}
These assumptions are mild and it is easy to find many physical examples
such that they are fulfilled. In particular, Condition \ref{ConditionB} is
fulfilled when $G(\ell )=P(\ell )/Q(\ell )$ and $P(\ell ),Q(\ell )>0$ are
two positive polynomials of the same order. In the now dominant Bardeen's
potential model for the angular power spectrum of the Cosmic Microwave
Background radiation (which is theoretically justified by the so-called
inflationary paradigm for the Big Bang Dynamics, see e.g., \cite{Durrer},
\cite{dode2004}) one has $C_{\ell }\sim (\ell (\ell +1))^{-1}$ for the
observationally relevant range $\ell \leq 5\times 10^{3}$ (the decay becomes
faster at higher multipoles, in view of the so-called Silk damping effect,
but these multipoles are far beyond observational capacity). This is clearly
in good agreement with Condition \ref{ConditionB}. On the other hand,
assuming that Condition \ref{ConditionB} holds and taking for instance
K(\left\langle x,y\right\rangle )\equiv 1,$ (e.g., focussing on the integral
of the field $\left\{ H_{q}(\beta _{j}(y))\right\} $)$,$ Condition \re
{ConditionA} has been shown to be satisfied by \cite{cammar}.
Indeed, it is readily checked that $\left\{ H_{q}(\beta
_{j}(y))\right\} $ is a polynomial of order $\simeq 2^{q(j+1)}$
and we can hence consider the following heuristic argument: we
have
\begin{eqnarray*}
\int_{S^{2}}K(\left\langle x,y\right\rangle )H_{q}(\widetilde{\beta
_{j}(y))dy &=&\int_{S^{2}}H_{q}(\widetilde{\beta }_{j}(y))dy \\
&=&\sum_{k\in \mathcal{X}_{j}}H_{q}(\widetilde{\beta }_{j}(\xi
_{jk}))\lambda _{jk}\mbox{ ,}
\end{eqnarray*
where $\left\{ \xi _{jk},\lambda _{jk}\right\} $ are a set of
cubature points and weights (see \cite{npw1}, \cite{bkmpBer}); indeed, because the
\beta _{j}(.)$ are band-limited (polynomial) functions, this
Riemann sum approximations can be constructed to be exact, with
weights $\lambda _{jk}$
of order $\simeq B^{-2j}.$ It is now known that under Condition \re
{ConditionB}, it is possible to establish a fundamental
uncorrelation inequality which will play a crucial role in our proof below, see also \cit
{bkmpAoS}, \cite{spalan}. \cite{mayeli}; indeed, we have that for
any $M\in
\mathbb{N}$, there exist a constant $C_{M}$ such tha
\begin{equation*}
Cov\left\{ H_{q}(\widetilde{\beta }_{j}(\xi _{jk_{1}})),H_{q}(\widetilde
\beta }_{j}(\xi _{jk_{2}}))\right\} \leq \frac{C_{M}q!}{\left\{
1+B^{j}d(\xi _{jk_{1}},\xi _{jk_{2}})\right\} ^{qM}}\mbox{ ,}
\end{equation*
entailing that the terms $H_{q}(\beta _{j}(\xi _{jk}))$ can be
treated as
asymptotically uncorrelated, for large $j.$ Henc
\begin{eqnarray*}
Var\left\{ \sum_{k\in \mathcal{X}_{j}}H_{q}(\widetilde{\beta
}_{j}(\xi _{jk}))\lambda _{jk}\right\} &\simeq &\sum_{k\in
\mathcal{X}_{j}}Var\left\{
H_{q}(\widetilde{\beta }_{j}(\xi _{jk}))\right\} \lambda _{jk}^{2} \\
&\simeq &C_{q}\sum_{k\in \mathcal{X}_{j}}\lambda _{jk}^{2}\simeq
C_{q}B^{-2j},
\end{eqnarray*
because $\sum_{k\in \mathcal{X}_{j}}\lambda _{jk}\simeq 4\pi $.
For instance, for $q=2$ we obtai
\begin{equation*}
Var\left\{ \int_{S^{2}}(\widetilde{\beta }_{j}^{2}(y)-1)dy\right\}
=Var\left\{ \int_{S^{2}}^{2}\widetilde{\beta
}_{j}^{2}(y)dy\right\}
\end{equation*
\begin{eqnarray*}
&=&\frac{Var\left\{ \sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}
)(2\mathbb{\ell }+1)\widehat{C}_{\mathbb{\ell }}\right\} }{\left\{ \sum_
\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})(2\mathbb{\ell }+1)C_
\mathbb{\ell }}\right\} ^{2}}=\frac{\sum_{\mathbb{\ell }}b^{4}(\frac{\mathbb
\ell }}{B^{j}})(2\mathbb{\ell }+1)^{2}Var(\widehat{C}_{\mathbb{\ell }})}
\left\{ \sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}})(2\mathbb
\ell }+1)C_{\mathbb{\ell }}\right\} ^{2}} \\
&=&\frac{2\sum_{\mathbb{\ell }=B^{j-1}}^{B^{j+1}}b^{4}(\frac{\mathbb{\ell }}
B^{j}})(2\mathbb{\ell }+1)C_{\mathbb{\ell }}^{2}}{\left\{
\sum_{\mathbb{\ell
}}b^{2}(\frac{\mathbb{\ell }}{B^{j}})(2\mathbb{\ell }+1)C_{\mathbb{\ell
}\right\} ^{2}}\simeq \frac{B^{j(2-2\alpha )})}{\left\{
B^{j(2-\alpha )}\right\} ^{2}}\simeq B^{-2j}\text{ ,}
\end{eqnarray*
as claimed.
\subsection{Finite-dimensional distributions}
The general technique we shall exploit to establish the Central Limit
Theorem is based upon sharp bounds on normalized fourth-order cumulants.
Note that, in view of results from \cite{nourdinpeccati}, this will actually
entail a stronger form of convergence, more precisely in total variation
norm (see \cite{nourdinpeccati}).
We start by recalling that the field $\left\{ \beta _{j}(.)\right\} $ can be
expressed in terms of the isonormal Gaussian process, e.g. as a stochastic
integral
\begin{equation*}
\beta _{j}(y):=\sum_{\mathbb{\ell }}b(\frac{\mathbb{\ell }}{B^{j}})T_
\mathbb{\ell }}(y)=\sum_{\mathbb{\ell }}b(\frac{\mathbb{\ell }}{B^{j}})\sqrt
\frac{(2\mathbb{\ell }+1)C_{\ell }}{4\pi }}\int_{S^{2}}P_{\mathbb{\ell
}(\left\langle y,z\right\rangle )W(dz)\text{ ,}
\end{equation*
where $W(A)$ is a white noise Gaussian measure on the sphere, which satisfie
\begin{equation*}
\mathbb{E}W(A)=0\text{, }\,\,\,\mathbb{E}\left\{ W(A)W(B)\right\}
=\int_{A\cap B}dz,\,\,\,\text{ for all }A,B\in \mathcal{B}(S^{2})\text{ .}
\end{equation*
It thus follows immediately that the transformed process $\left\{
H_{q}(\beta _{j}(.))\right\} $ belongs to the $q$-th order Wiener chaos, see
\cite{nourdinpeccati}, \cite{noupebook} for more discussion and detailed
definitions. Let us now recall the definition of the \emph{total variation}
distance between the laws of two random variables $X$ and $Z,$ which is
given b
\begin{equation*}
d_{TV}(X,Z)=\sup_{A\in \mathcal{B}(\mathbb{R})}\left\vert \Pr (W\in A)-\Pr
(X\in A)\right\vert .
\end{equation*
When $Z$ is a standard Gaussian and $X$ is a zero-mean, {unit variance}
random variable which belongs to the $q$-th order Wiener chaos of a Gaussian
measure, the following remarkable inequality holds for the total variation
distance
\begin{equation*}
d_{TV}(X,Z)\leq \sqrt{\frac{q-1}{3q}cum_{4}(X)}\text{ ,}
\end{equation*
see again \cite{nourdinpeccati}, \cite{noupebook} for more discussion and a
full proof.
Let us now introduce an isotropic zero-mean Gaussian process $f_{j;q},$ with
the same covariance function as that of $g_{j;q}.$ Our next result will
establish the asymptotic convergence of the finite-dimensional distributions
for $g_{j;q}$ and $f_{j;q}.$ In particular, we have:
\begin{lemma}
For any fixed vector $(x_{1},...,x_{p})$ in $S^{2},$ we have tha
\begin{equation*}
d_{TV}\left( \left( \frac{g_{j;q}(x_{1})}{\sqrt{Var(g_{j;q})}},...,\frac
g_{j;q}(x_{p})}{\sqrt{Var(g_{j;q})}}\right) ,\left( \frac{f_{j;q}(x_{1})}
\sqrt{Var(g_{j;q})}},...,\frac{f_{j;q}(x_{p})}{\sqrt{Var(g_{j;q})}}\right)
\right) =o(1),
\end{equation*
as $j\rightarrow \infty .$
\end{lemma}
\begin{proof}
For notational simplicity, we shall focus on the univariate case; also,
without loss of generality we normalize $\beta _{j}(.)$ to have unit
variance. In this case, the Nourdin-Peccati inequality (\cite{nourdinpeccati
, \cite{noupebook}) can be restated as
\begin{equation}
d_{TV}\left( \frac{g_{j;q}}{\sqrt{Var(g_{j;q})}},N(0,1)\right) \leq \sqrt
\frac{q-1}{3q}cum_{4}(\frac{g_{j;q}}{\sqrt{Var(g_{j;q})}})}\text{ .}
\label{baglioni}
\end{equation
In view of (\ref{baglioni}), for the Central Limit Theorem to hold we shall
only need to study the limiting behaviour of the normalized fourth-order
cumulant of $g_{j;q}.$ Let us then conside
\begin{equation*}
cum_{4}\left\{ g_{j}(x)\right\} =
\end{equation*
\begin{equation*}
\int_{\left\{ S^{2}\right\} ^{\otimes 4}}K(\left\langle x,y_{1}\right\rangle
)...K(\left\langle x,y_{4}\right\rangle )cum_{4}\left\{ H_{q}(\beta
_{j}(y_{1})),...,H_{q}(\beta _{j}(y_{4}))\right\} dy_{1}...dy_{4}
\end{equation*
\begin{eqnarray*}
&\leq &C_{1}\sup_{r=1,...,q-1}\int_{\left\{ S^{2}\right\} ^{\otimes
4}}\left\vert K(\left\langle x,y_{1}\right\rangle )...K(\left\langle
x,y_{4}\right\rangle )\right\vert \left\vert \rho (y_{1},y_{2})\right\vert
^{q-r} \\
&&\times \left\vert \rho (y_{2},y_{3})\right\vert ^{r}\left\vert \rho
(y_{3},y_{4})\right\vert ^{q-r}\left\vert \rho (y_{4},y_{1})\right\vert
^{r}dy_{1}...dy_{4}
\end{eqnarray*
\begin{equation*}
\leq C_{2}\sup_{r=1,...,q-1}\int_{\left\{ S^{2}\right\} ^{\otimes
4}}\left\vert \rho (y_{1},y_{2})\right\vert ^{q-r}\left\vert \rho
(y_{2},y_{3})\right\vert ^{r}\left\vert \rho (y_{3},y_{4})\right\vert
^{q-r}\left\vert \rho (y_{4},y_{1})\right\vert ^{r}dy_{1}...dy_{4}\text{ ,}
\end{equation*
wher
\begin{equation*}
\rho (y_{1},y_{2})=\frac{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}
B^{j}})\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }}P_{\mathbb{\ell
}(y_{1},y_{2})}{\sum_{\mathbb{\ell }}b^{2}(\frac{\mathbb{\ell }}{B^{j}}
\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell }}}\leq \frac{C_{M}}{\left\{
1+B^{j}d(y_{1},y_{2})\right\} ^{M}}\text{ ,}
\end{equation*
in view of (\ref{ConditionB}) and the uncorrelation inequality provided by
\cite{bkmpAoS}, see also \cite{spalan}. \cite{mayeli}. Now standard
computations yiel
\begin{equation*}
\int_{S^{2}}\left\vert \rho (y_{1},y_{2})\right\vert ^{r}dy_{2}\leq
\int_{S^{2}}\left\vert \rho (y_{1},y_{2})\right\vert dy_{2}\leq \int_{S^{2}
\frac{C_{M}}{\left\{ 1+B^{j}d(y_{1},y_{2})\right\} ^{M}}dy_{2}
\end{equation*
\begin{equation*}
\leq \int_{y_{2}:d(y_{1},y_{2})\leq 2B^{-j}}\frac{C_{M}}{\left\{
1+B^{j}d(y_{1},y_{2})\right\} ^{M}}dy_{2}+\int_{y_{2}:d(y_{1},y_{2})\geq
2B^{-j}}\frac{C_{M}}{\left\{ 1+B^{j}d(y_{1},y_{2})\right\} ^{M}}dy_{2}\leq
CB^{-2j}\text{ .}
\end{equation*
Hence
\begin{equation*}
C_{2}\sup_{r=1,...,q-1}\int_{\left\{ S^{2}\right\} ^{\otimes 4}}\left\vert
\rho (y_{1},y_{2})\right\vert ^{q-r}\left\vert \rho (y_{2},y_{3})\right\vert
^{r}\left\vert \rho (y_{3},y_{4})\right\vert ^{q-r}\left\vert \rho
(y_{4},y_{1})\right\vert ^{r}dy_{1}...dy_{4}\leq CB^{-6j}
\end{equation*
and
\begin{equation*}
cum_{4}\left\{ \frac{g_{j;q}(x)}{\sqrt{Var(g_{j;q}(x))}}\right\} =O(B^{-2j}
\text{ ,}
\end{equation*
entailing that for every fixed $x\in S^{2}$,
\begin{equation*}
d_{TV}\left( \frac{g_{j;q}}{\sqrt{Var(g_{j;q})}},N(0,1)\right) =O(B^{-2j}
\text{ ,}
\end{equation*
and hence the univariate Central Limit Theorem, as claimed. The proof in the
multivariate case is analogous and hence omitted for the sake of brevity.
\end{proof}
\subsection{Tightness}
We now focus on asymptotic tightness for both sequences $\left\{
g_{j;q}\right\} $ and $\left\{ f_{j;q}\right\}.$ We shall exploit the
following criterion from \cite{Kallenberg}:
\begin{proposition}
(\cite{Kallenberg}) Let $g_{j}:M\rightarrow D$ be a sequence of stochastic
processes, where $M$ is compact and $D$ is complete and separable. Then
g_{j}\Rightarrow g$ if $g_{j}\rightarrow _{f.d.d.}g$ and (tightness
condition)
\begin{equation*}
\lim_{h\rightarrow 0}\lim \sup_{j\rightarrow \infty }\mathbb{E
(\sup_{d(x,y)\leq h}\left\vert g_{j}(x)-g_{j}(y)\right\vert \wedge 1)=0\text{
.}
\end{equation*}
\end{proposition}
We are hence able to establish the following
\begin{lemma}
For every $q\in \mathbb{N},$ the sequence $\left\{ g_{j;q}\right\} $ and
\left\{ f_{j;q}\right\} $ are tight.
\end{lemma}
\begin{proof}
Write $\left\{ a_{\mathbb{\ell }m}(f_{j;q})\right\} $ for the spherical
harmonic coefficients of the fields $\left\{ f_{j;q}\right\} .$ For any
x_{1},x_{2}\in S^{2},$ we hav
\begin{eqnarray*}
\mathbb{E}\left\{ \sup_{d(x_{1},x_{2})\leq \delta }\left\vert
f_{j;q}(x_{1})-f_{j;q}(x_{2})\right\vert \right\} &=&\mathbb{E}\left\{
\sup_{d(x_{1},x_{2})\leq \delta }\left\vert \sum_{\mathbb{\ell }m}a_{\mathbb
\ell }m}(f_{j;q})\left\{ Y_{\mathbb{\ell }m}(x_{1})-Y_{\mathbb{\ell
m}(x_{2})\right\} \right\vert \right\} \\
&\leq &\sum_{\mathbb{\ell }m}\left\{ \mathbb{E}\left\vert a_{\mathbb{\ell
m}(f_{j;q})\right\vert \right\} \left\{ \sup_{d(x_{1},x_{2})\leq \delta
}\left\vert \left\{ Y_{\mathbb{\ell }m}(x_{1})-Y_{\mathbb{\ell
m}(x_{2})\right\} \right\vert \right\} \text{ .}
\end{eqnarray*
No
\begin{equation*}
\sup_{d(x_{1},x_{2})\leq \delta }\left\vert \left\{ Y_{\mathbb{\ell
m}(x_{1})-Y_{\mathbb{\ell }m}(x_{2})\right\} \right\vert \leq c\ell
^{2}\delta \text{ ,}
\end{equation*
an
\begin{equation*}
\sum_{\mathbb{\ell }m}\left\{ \mathbb{E}\left\vert a_{\mathbb{\ell
m}(f_{j;q})\right\vert \right\} \leq \sum_{\mathbb{\ell }m}\sqrt{\left\{
\mathbb{E}\left\vert a_{\mathbb{\ell }m}(f_{j;q})\right\vert ^{2}\right\}
=\sum_{\mathbb{\ell }}(2\mathbb{\ell }+1)\sqrt{C_{\mathbb{\ell }}(f_{j;q})}
\end{equation*
and because $K(.)$ is compactly supported in harmonic space (and hence,
again, a finite-order polynomial
\begin{equation*}
\leq \left\{ \sum_{\mathbb{\ell }}^{L_{K}}(2\mathbb{\ell }+1)\right\} ^{1/2
\sqrt{\sum_{\mathbb{\ell }}^{L_{K}}(2\mathbb{\ell }+1)C_{\mathbb{\ell
}(f_{j;q})}\leq O(L_{K})\text{ ,}
\end{equation*
whenc
\begin{equation*}
\mathbb{E}\left\{ \sup_{d(x_{1},x_{2})\leq \delta }\left\vert
f_{j;q}(x_{1})-f_{j;q}(x_{2})\right\vert \right\} \leq CL_{K}^{3}\delta
\text{ ,}
\end{equation*
for some $C>0,$ uniformly over $j,$ and thus the result follows. The proof
for $\left\{ g_{j;q}\right\} $ is analogous.
\end{proof}
\subsection{Asymptotic Proximity of Distributions}
Our discussion above shows that the finite-dimensional distributions of the
non-Gaussian sequence of random fields $\left\{ g_{j;q}\right\} $ converge
to those of the Gaussian sequence $\left\{ f_{j;q}\right\} ;$ moreover, both
sequences are tight. However, the finite-dimensional distributions of
neither processes converge to a well-defined limit. In view of this
situation, we need a broader notion of convergence than the one envisaged in
standard treatment such as \cite{Billingsley}; this extended form of
convergence is provided by the notion of \emph{Asymptotic Proximity, }or
\emph{Merging,} of distributions, as discussed for instance by \cite{davydov
, \cite{aristotilediaconis}, \cite{dudley}, and others.
\begin{definition}
(\emph{Asymptotic Proximity of Distribution }\cite{davydov}, \cit
{aristotilediaconis}, \cite{dudley}) Let $g_{n},f_{n}$ be two sequences of
random elements in some metric space $(X,\rho ),$ possibly defined on two
different probability spaces. We say that the laws of $g_{n},f_{n}$ are
\emph{asymptotically merging}, or $\emph{asymptotically}$ $\emph{proximal}$,
(denoted as $g_{n}\Rightarrow f_{n}$) if and only if as $n\rightarrow \infty
\begin{equation*}
\left\vert \mathbb{E}h(g_{n})-\mathbb{E}h(f_{n})\right\vert \rightarrow
\text{ ,}
\end{equation*
for all continuous and bounded functionals $h\in \mathcal{C}_{b}(X,\mathbb{R
)$.
\end{definition}
As discussed by \cite{davydov}, it is possible to provide a characterization
of asymptotic proximity that extends in a natural way to standard weak
convergence results.
\begin{theorem}
Assume that $g_{j;q},f_{j;q}\in \mathcal{C}_{b}(K,S),$ where $K$ is compact
and $S$ is complete and separable. Then $g_{j;q},f_{j;q}$ are asymptotically
proximal if and only if they are both tight and their finite-dimensional
distribution converge, i.e. for all $n\geq 1,$ $x_{1},...,x_{n}\in K,$ we
have that
\begin{equation*}
\left\vert \Pr \left\{ (g_{j;q}(x_{1}),...,g_{j;q}(x_{n}))\in A\right\} -\Pr
\left\{ (f_{j;q}(x_{1}),...,f_{j;q}(x_{n}))\in A\right\} \right\vert
\rightarrow 0\text{ ,}
\end{equation*
for all $A\in \mathcal{B}(\mathbb{R}^{n})$.
\end{theorem}
In view of the results provided in the previous subsection, we have hence
established that
\begin{theorem}
As $j\rightarrow \infty $
\begin{equation*}
g_{j;q}\Longrightarrow f_{j;q}\text{ ,}
\end{equation*
i.e. for all $h=h:\mathcal{C}(S^{2},\mathbb{R})\rightarrow \mathbb{R},$ $h$
continuous and bounded, we hav
\begin{equation*}
\left\vert \mathbb{E}h(g_{j;q})-\mathbb{E}h(f_{j;q})\right\vert \rightarrow
\text{ .}
\end{equation*}
\end{theorem}
As a simple application of the asymptotic proximity result, we hav
\begin{equation*}
\mathbb{E}\left\{ \frac{\sup g_{j;q}}{1+\sup g_{j;q}}\right\} \rightarrow
\mathbb{E}\left\{ \frac{\sup f_{j;q}}{1+\sup f_{j;q}}\right\} \text{ .}
\end{equation*
It should be noted that asymptotically proximal sequences do not enjoy all
the same properties as in the standard weak convergence case. For instance,
it is known that the Portmanteau Lemma does not hold in general, i.e. it is
not true that, for every Borel set such that $\Pr \left\{ g_{j}\in \partial
A\right\} =\Pr \left\{ f_{j}\in \partial A\right\} =0,$ we hav
\begin{equation*}
\left\vert \Pr \left\{ g_{j}\in A\right\} -\Pr \left\{ f_{j}\in A\right\}
\right\vert \rightarrow 0\text{ .}
\end{equation*}
As a counterexample, it is enough to consider the sequences $f_{j}=-j^{-1}$
and $g_{j}=j^{-1}.$ However, it is indeed possible to obtain more stringent
characterizations when the subsequences are asymptotically Gaussian. We have
the following
\begin{proposition}
For every $A\in \mathcal{B(}\mathbb{R)},$ we have tha
\begin{equation*}
\left\vert \Pr \left\{ \sup_{x\in S^{2}}g_{j;q}(x)\in A\right\} -\Pr \left\{
\sup_{x\in S^{2}}f_{j;q}\in A\right\} \right\vert \rightarrow 0\text{ .}
\end{equation*}
\end{proposition}
\begin{proof}
We shall argue again by contradiction. Assume that there exists a
subsequence $j_{n}^{\prime }$ such that for some $\varepsilon >0
\begin{equation}
\left\vert \Pr \left\{ \sup_{x\in S^{2}}g_{j_{n}^{\prime };q}(x)\in
A\right\} -\Pr \left\{ \sup_{x\in S^{2}}f_{j_{n}^{\prime };q}\in A\right\}
\right\vert >\varepsilon \text{ .} \label{igor}
\end{equation
By relative compactness, there exists a subsequence $j_{n}^{\prime \prime }$
and a limiting process $g_{\infty ;q}$ such tha
\begin{equation*}
\left\vert \Pr \left\{ \sup_{x\in S^{2}}g_{j_{n}^{\prime \prime };q}(x)\in
A\right\} -\Pr \left\{ \sup_{x\in S^{2}}g_{\infty ;q}\in A\right\}
\right\vert \rightarrow 0\text{ .}
\end{equation*
Likewise, consider $\left\{ j_{n}^{\prime \prime \prime }\right\} \subset
\left\{ j_{n}^{\prime \prime }\right\} ;$ again by relative compactness
there exist $f_{\infty ;q}$ such that $f_{j_{n}^{\prime \prime \prime
};q}\Rightarrow f_{\infty ;q}$ and hence
\begin{equation*}
\left\vert \Pr \left\{ \sup_{x\in S^{2}}f_{j_{n}^{\prime \prime \prime
};q}(x)\in A\right\} -\Pr \left\{ \sup_{x\in S^{2}}f_{\infty ;q}\in
A\right\} \right\vert \rightarrow 0\text{ .}
\end{equation*
Note that $f_{\infty ;q},g_{\infty ;q}$ are isotropic and continuous
Gaussian random fields, whence the supremum is necessarily a continuous
random variable, whence no problems with non-zero boundary probabilities
arise. Now the finite-dimensional distributions are a determining class,
whence the two Gaussian processes $f_{\infty ;q},g_{\infty ;q}$ must
necessarily have the same distribution. Hence
\begin{equation*}
\left\vert \Pr \left\{ \sup_{x\in S^{2}}f_{j_{n}^{\prime \prime \prime
};q}(x)\in A\right\} -\Pr \left\{ \sup_{x\in S^{2}}g_{j_{n}^{\prime \prime
\prime };q}(x)\in A\right\} \right\vert \rightarrow 0\text{ ,}
\end{equation*
yielding a contradiction with \ref{igor}.
\end{proof}
This result immediately suggests two alternative ways to achieve the
ultimate goal of this paper, e.g. the evaluation of excursion probabilities
on the non-Gaussian sequence of random fields $\left\{ g_{j;q}\right\} .$ On
one hand, it follows immediately that these probabilities may be evaluated
by simulations, by simply sampling realizations of a Gaussian field with
known angular power spectrum; for $q=2$, for example, $f_{j;q}$ is simply a
Gaussian process with angular power spectrum given by (\ref{powspe}). There
exist now very efficient techniques, based on packages such as HealPix (\cit
{HEALPIX}), for the numerical simulation of Gaussian fields with a given
power spectra; here the only burdensome step can be the numerical evaluation
of expressions like (\ref{powspe}), but this is in any case much faster and
simpler than the Monte Carlo evaluation of smoothed non-Gaussian fields.
Therefore our result has an immediate applied relevance.
One can try, however, to be more ambitious than this, and verify whether
these excursion probabilities can indeed be evaluated analytically, rather
than by Gaussian simulations. This is in fact the purpose of the next, and
final, Section.
\section{Asymptotics for the excursion probabilities}
The purpose of this final Section is to show how the previous weak
convergence results allow for very neat characterizations of excursion
probabilities, even in non-Gaussian circumstances. In particular, if we
consider the normalized version $\tilde{g}_{j;q}$ of the field $g_{j;q}$,
such that $\tilde{g}_{j;q}(x)=\left( Var(g_{j;q}(x))\right)^{-1/2}g_{j;q}(x)
$, making it a unit variance random field, then the following can be proved.
\begin{theorem}
\label{thm:exc:prob} There exists constants $\alpha >1$ and $\mu ^{+}>0$
such that, for $u>\mu ^{+}
\begin{equation*}
\limsup_{j\rightarrow \infty }\left\vert \Pr \left\{ \sup_{x\in S^{2}}\tilde
g}_{j;q}(x)>u\right\} -\left\{ 2(1-\Phi (u))+2u\phi (u)\lambda
_{j;q}\right\} \right\vert \leq \exp \left( -\frac{\alpha u^{2}}{2}\right)
\mbox{ ,}
\end{equation*
where (see (\ref{porteaperte2}))
\begin{equation}
\lambda _{j;q}=\frac{\sum_{\mathbb{\ell }=1}^{L}\frac{2\mathbb{\ell }+1}
4\pi }C_{\mathbb{\ell };j,q}P_{\mathbb{\ell }}^{\prime }(1)}{\sum_{\mathbb
\ell }=1}^{L}\frac{2\mathbb{\ell }+1}{4\pi }C_{\mathbb{\ell };j,q}}\mbox{ .}
\label{porteaperte}
\end{equation}
\end{theorem}
In order to establish the above Theorem, we shall need to fine tune Theorem
14.3.3 of \cite{RFG} to our needs. Let us begin with writing $f_{j;q}$ as a
mean zero Gaussian random field on $S^{2}$ whose covariance function matches
with that of $\tilde{g}_{j;q};$ note that in the previous sections, $f_{j;q}$
denoted the Gaussian random field whose mean and covariance matched with
that of $g_{j;q}$, whereas in this section, the new $f_{j;q}$ is basically
the normalized version of the old one so as to have unit variance. Then, for
each $x_{0}\in S^{2}$, define
\begin{eqnarray*}
\widehat{f}_{j;q}^{x_{0}}(x) &=&\frac{1}{1-\rho (x,x_{0})}\Big\
f_{j;q}(x)-\rho (x,x_{0})f_{j;q}(x_{0}) \\
&&-Cov(f_{j;q}(x),\frac{\partial }{\partial \vartheta }f_{j;q}(x_{0}))Var
\frac{\partial }{\partial \vartheta }f_{j;q}(x))\frac{\partial }{\partial
\vartheta }f_{j;q}(x) \\
&&-Cov(f_{j;q}(x),\frac{\partial }{\sin \vartheta \partial \phi
f_{j;q}(x_{0}))Var(\frac{\partial }{\sin \vartheta \partial \phi }f_{j;q}(x)
\frac{\partial }{\sin \vartheta \partial \phi }f_{j;q}(x)\Big\},
\end{eqnarray*
where $\rho (x,x_{0})=E\left( f_{j;q}(x)f_{j;q}(x_{0})\right) $. Next
define,
\begin{equation*}
\mu _{j}^{+}=\sup_{x_{0}}E\left( \sup_{x\neq x_{0}}\widehat{f
_{j;q}^{x_{0}}(x)\right)
\end{equation*
and,
\begin{equation*}
\sigma _{j}^{2}=\sup_{x_{0}}\sup_{x\neq x_{0}}Var(\widehat{f
_{j;q}^{x_{0}}(x)).
\end{equation*}
Then, under the previous regularity conditions on the kernel $K$, we have
the following Proposition, whose proof is deferred to the Appendix.
\begin{proposition}
\label{claim:lip} Under the assumption that the kernel $K$ appearing in the
definition of $\tilde{g}_{j;q}$ is of the form (\ref{kernexp}), the field
\widehat{f}_{j;q}^{x_{0}}$ satisfies the following:
\begin{equation}
\mathbb{E}(\widehat{f}_{j;q}^{x_{0}}(x_{2})-\widehat{f
_{j;q}^{x_{0}}(x_{1}))^{2}\leq c(L_{K},q)|x_{2}-x_{1}|,
\label{eqn:claim:lip}
\end{equation
where the constant $c(L_{K},q)$ depends on $q$ and $\mathbb{\ell }$, but
does not depend on $j$.
\end{proposition}
We are now in a position to provide the following
\begin{proposition}
\label{prop:unif:exc:prob} There exists constants $\alpha >1$ and $\mu
^{+}>0 $ such that, for $u>\mu ^{+}$
\begin{equation*}
\left\vert \Pr \left\{ \sup_{x\in S^{2}}f_{j;q}(x)>u\right\} -\mathbb{E
\mathcal{L}_{0}(A_{u}(f_{j;q},S^{2})\right\vert \leq \exp (-\frac{\alpha
u^{2}}{2})\mbox{,}
\end{equation*
uniformly over $j,$ where $\mathbb{E}\mathcal{L
_{0}(A_{u}(f_{j;q},S^{2}))=2(1-\Phi (u))+2u\phi (u)\lambda _{j;q}$.
\end{proposition}
\begin{proof}
From p.371 of \cite{RFG}, we know that for $u\geq \mu _{j}^{+}
\begin{eqnarray}
&&\left\vert \Pr \left\{ \sup_{x\in S^{2}}f_{j;q}(x)>u\right\} -\mathbb{E
\mathcal{L}_{0}(A_{u}(f_{j;q},S^{2})\right\vert \notag
\label{eqn:proposition} \\
&\leq &K\,u\,e^{-\frac{(u-\mu _{j}^{+})^{2}}{2}\left( 1+\frac{1}{2\sigma
_{j}^{2}}\right) }\sum_{i=0}^{2}\left\{ \mathbb{E}\left\vert \det_{i}\left(
-\nabla ^{2}f_{j;q}-f_{j;q}I_{2}\right) \right\vert ^{2}\right\} ^{1/2
\mbox{,}
\end{eqnarray
where $I_{2}$ is the $2\times 2$ identity matrix, $\det_{i}$ of a matrix is
the sum over all the $i$-minors of the matrix under consideration, and $K$
is a constant not depending on $j$. Note that the expression on p.371 of
\cite{RFG} also involves an integral over the parameter space with the
metric induced by the second order spectral moment. However under (\re
{kernexp}) this integral is easily seen to be uniformly bounded with respect
to to $j,$ so that we can we get rid of it by invoking the isotropy of the
field $f_{j;q}$ and its derivatives, and absorbing the arising constant into
$K$ upfront.
Our goal is to get a uniform bound for the right hand side of
\eqref{eqn:proposition}. Clearly, $\sum_{i=0}^{2}\mathbb{E}\left\vert
\det_{i}\left( -\nabla ^{2}f_{j;q}-f_{j;q}I_{2}\right) \right\vert ^{2}$ is
bounded above by a universal constant, largely because of the finite
expansion for the kernel $K(\cdot ,\cdot )$ used to define the field
g_{j;q} $.
Next, to get a uniform bound for $\mu_{j}^{+}$, we shall resort to the
standard techniques of estimating supremum of a Gaussian random field using
metric entropy. \textit{En passant,} a uniform bound on $\sigma _{j}^{2}$ is
also obtained.
Applying Theorem 1.4.1 of \cite{RFG} to the random fields $f_{j;q}$, and
assuming that the equation \eqref{eqn:claim:lip} is satisfied, we conclude
that the metric entropy of $f_{j;q}$ can be uniformly bounded above, which
in turn implies that $\mu ^{+}:=\sup_{j}\mu _{j}^{+}<\infty $.
Similarly, a uniform upper bound on $\sigma _{j}^{2}$ is also obtained; for
the sake of brevity, we shall present its proof in the Appendix. Thus for
u\geq \sup_{j}\mu _{j}^{+}$,
\begin{equation*}
\left\vert \Pr \left\{ \sup_{x\in S^{2}}f_{j;q}(x)>u\right\} -\mathbb{E
\mathcal{L}_{0}(A_{u}(f_{j;q},S^{2}))\right\vert \leq K\,u\,e^{-\frac{\alpha
(u-\mu ^{+})^{2}}{2}}\mbox{,}
\end{equation*
where the $K$ appearing here is different from the earlier one. Finally,
note that the linear term $u$ can also be ignored by choosing a smaller
\alpha $ in the exponent, which completes the proof of the theorem.
\end{proof}
\begin{proof}[Proof of Theorem \protect\ref{thm:exc:prob}]
From the results of previous section, we know that for any fixed $u$
\begin{equation*}
\lim_{j\rightarrow \infty }\left\vert \Pr \left\{ \sup_{x\in S^{2}}\tilde{g
_{j;q}(x)>u\right\} -\Pr \left\{ \sup_{x\in S^{2}}f_{j;q}(x)>u\right\}
\right\vert =0\mbox{.}
\end{equation*
Combining this with Proposition \ref{prop:unif:exc:prob}, we get the desired
result.
\end{proof}
|
2,869,038,154,752 | arxiv | \section{Introduction}
Conductance\cite{ChungSpectraBook} is a well studied and important measure of network resilience in the face of edge failures or attacks. Combinatorial conductance or edge based conductance\cite{SinclairJerrum,ChungSpectraBook} is defined as
\begin{eqnarray*}
\Phi(G) &= \min_{S \subset V, Vol(S) \le Vol(V)/2} \{ \frac{|Cut(S,V-S)|}{Vol(S)} \} \\
&= \min_{S \subset V, Vol(S) \le Vol(V)/2} \{ \frac{|Cut(S,V-S)|}{\delta_S|S|} \}
\end{eqnarray*}
where $|Cut(S,V-S)|$ is the size of the cut separating $S$ from $V-S$, $Vol(S)$ is the sum of the degrees of vertices in $S$, and $\delta_S$ is the \emph{average} degree of vertices in $S$. A reason for the importance of conductance is in the intimate relationship it shares with spectral gap and mixing time via \emph{Cheeger's inequality} \cite{ChungSpectraBook,SinclairJerrum}:
\begin{thm}\label{thm:cheeger}
Given a connected $d$-regular graph $G = (V,E)$, let $\lambda_2$ denote the second largest eigenvalue of $G$'s normalized adjacency matrix. Then,
\begin{equation}\label{eqn:cheeger}
\frac{\Phi(G)^2}{2} \le 1 - \lambda_2 \le 2\Phi(G)
\end{equation}
\end{thm}
Given the general inapproximability of conductance to any constant degree, Cheeger's inequality nonetheless yields good asymptotic bounds on the conductance of certain infinite graph families via spectral gap. An important characterization in this regard has been that of \emph{expander families} which are $d$-regular families of graphs that maintain excellent resilience properties despite keeping constant degree $d$ \cite{Alon86}. And, although conductance implicitly refers to edge based resilience, it is known to accurately represent node-based resilience for regular graphs as well.
However, conductance neither captures nor approximates node-based resilience in heterogeneous degree graphs, the most extremal example of which is the star family of graphs. Star graphs exhibit maximal conductance while being minimally resilient to node attacks, particularly that targeting the single central node and resulting in solely isolated nodes. With the need for a measure that behaves similarly to conductance in the case of regular graphs while also appropriately capturing node based resilience for heterogeneous degree graphs, the notion of \emph{vertex attack tolerance} (VAT, denoted by $\tau(G)$) was introduced by the authors in \cite{CASResSF,MattaBE14} and defined as:
\begin{equation*}
\tau(G) = \min_{S \subset V, S \neq \emptyset} \{\frac{|S|}{|V-S-C_{max}(V-S)|+1 } \}
\end{equation*}
where $C_{max}(V-S)$ is the largest connected component in $V-S$. In addition to comprehensive comparisons with other resilience measures for arbitrary degree graphs, the following upper bound was presented for VAT in terms of conductance in the case of $d$-regular graphs:\cite{MattaBE14}
\begin{thm}\label{thm:vatcond}
For any non-trivial $d$-regular graph $G = (V,E)$, if $\Phi(G) \le \frac{1}{d^2}$ then $\tau(G) < d\Phi(G)$.
\end{thm}
Note that due to both conductance and VAT being normalized measures in $(0,1]$, this can be unconditionally phrased as $\tau(G) < d^2\Phi(G)$ for $d$-regular non-trivial graphs $G$.
The primary contribution of the present work is to present a matching lower bound for VAT in terms of conductance for $d$-regular graphs when the conductance is not too high. Namely, we present and prove the following theorem:
\begin{thm}\label{thm:condvat}
For any non-trivial $d$-regular graph $G = (V,E)$, $\Phi(G) < d\tau(G)$.
\end{thm}
Applying Cheeger's inequality to both the previous bound and our new bound results in the following corollary:
\begin{cor}\label{cor:cheegervat}
Given a connected $d$-regular graph $G = (V,E)$, let $\lambda_2$ denote the second largest eigenvalue of $G$'s normalized adjacency matrix. Then,
\begin{equation}
\frac{\tau(G)^2}{2d^4} \le 1 - \lambda_2 \le 2d\tau(G)
\end{equation}
Furthermore, if $\Phi(G) \le \frac{1}{d^2}$, then
\begin{equation}
\frac{\tau(G)^2}{2d^2} \le 1 - \lambda_2 \le 2d\tau(G)
\end{equation}
\end{cor}
A secondary contribution of this work is to generalize VAT in two ways, the first via parametrization and the second via generalization to node-weighted graphs. We first present this secondary contribution and then prove our primary contribution.
\section{Definitions and Preliminaries}
Throughout this work we assume that the graph $G = (V,E)$ whose resilience is under consideration is connected and undirected. The resilience of any disconnected graph is otherwise zero. \footnote{To non-trivially define the resilience of a disconnected network one may restrict consideration to the largest connected component WLOG.} And, for any directed graph we may instead consider the underlying undirected graph without edge directionality. Therefore, in the context of this work, we refer to connected, undirected graphs $G = (V,E)$ with more than one node ($|V| \ge 2$) as \emph{non-trivial}.
Set-vertex attack tolerance is denoted as \cite{MattaBE14}
\begin{equation*}
\tau_S(G) =\frac{|S|}{|V-S-C_{max}(V-S)|+1 }
\end{equation*}
so that clearly $\tau(G) = \min_{S \subset V} \tau_S(G)$ and correspondingly
\begin{equation*}
S(\tau(G)) = argmin_{S \subset V} \tau_S(G)
\end{equation*}
Similary for set-conductance:
\begin{equation*}
\Phi_S(G) = \frac{|Cut(S,V-S)|}{\delta_S|S|}
\end{equation*}
so that clearly $\Phi(G) = \min_{S \subset V, |Vol(S)| \le |Vol(V)|/2} \Phi_S(G)$ and correspondingly
\begin{equation*}
S(\Phi(G)) = argmin_{S \subset V, |Vol(S)| \le |Vol(V)|/2} \tau_S(G)
\end{equation*}
Let us denote the subgraph of a graph $G = (V,E)$ that is induced by a vertex set $S \subset V$ as $G_S = (S, E_S)$.
The \emph{normalized adjacency matrix} of a graph $G = (V,E)$ is the $|V|$ by $|V|$ matrix $A$ where $A_{u,v} = 0$ if $\{ u,v \} \notin E$ and $A_{u,v} = \frac{1}{d_u}$ where $d_u$ is the degree of $u$ otherwise if $\{ u,v \} \in E$. Note that the normalized adjacency matrix of a graph is identical to the probability transition matrix, or Markov chain, of the natural random walk.
\subsection{A Useful Generalization}
A particularly useful generalization of vertex attack tolerance that we introduce here which allows for linear weighting parameters $\alpha, \beta$ is the following, which we call $(\alpha,\beta)$-vertex attack tolerance:
\begin{equation*}
\tau_{\alpha,\beta}(G) = \min_{S \subset V, S \neq \emptyset} \{\frac{\alpha|S|+\beta}{|V-S-C_{max}(V-S)|+1 } \}
\end{equation*}
Clearly VAT is identical to $(1,0)$-VAT. We will later see that $\tau_{1,1}$, which nominally appears quite similar to VAT, will also be helpful.
We also wish to introduce one further generalization of vertex attack tolerance which allow us to discuss a type of vertex-weighted graph $G$ which has costs $c(x)$ and values $v(x)$ associated with each vertex $x \in V$. Specifically, the most general graph context concerned here is that of undirected, connected graphs $G = (V, E, c, v)$ such that $c, v$ are positive real-valued functions on the vertex set $V$. If $c$ is specified without $v$ being specified, then we will assume that $c = v$, and similarly for the case that $v$ is specified without $c$ being specified. If neither $c$ nor $v$ are specified, then the assumption is that $c(x) = v(x) = 1$ for all $x \in V$. Having clarified the context of such cost-value node-weighted graphs $G$, the following is the appropriate generalization of VAT:
\begin{equation*}
\tau(G) = \min_{S \subset V, S \neq \emptyset} \{\frac{\sum_{x \in S} c(x)}{1+ \sum_{y \in V} v(y) - \sum_{y \in S+C_{max}(V-S)} v(y)} \}
\end{equation*}
Note that this generalization is consistent with the original definition of VAT in the originally considered case of unweighted graphs $G$. Moreover, applying other variations and generalizations to cost-value weighted graphs is straightforward. As an example, we may consider $(\alpha,\beta)$-VAT of cost-value node weighted graphs:
\begin{equation*}
\tau(G) = \min_{S \subset V, S \neq \emptyset} \{\frac{\alpha(\sum_{x \in S} c(x))+\beta}{1+ \sum_{y \in V} v(y) - \sum_{y \in S+C_{max}(V-S)} v(y)} \}
\end{equation*}
\subsection{Mathematical Preliminaries}
Let us note the following preliminary observation:
\begin{rem}\label{rem:vatbound}
For nontrivial $G = (V,E)$, $0 < \tau(G) \le 1$, and $C_{max}(V-S(\tau(G))) \neq \emptyset$.
\end{rem}
The first bound follows from the non-emptiness of $S$ by definition of $\tau$, in addition to the fact that, for any vertex $v \in V$, $\tau(G) \le \tau_{\{ v \} }(G) = \frac{1}{|V - \{ v \} - C_{max}|+1} \le 1$. The non-emptiness of the largest remaining connected component follows from the fact that the only way that $C_{max}$ can be empty is by taking $S = V$, but such $S$ cannot achieve as low a set-VAT as that achieved by a single node, and therefore cannot be the set corresponding to VAT.
The following similar bound is well-known for conductance:
\begin{rem}\label{rem:condbound}
For nontrivial $G = (V,E)$, $0 < \Phi(G) \le 1$.
\end{rem}
For the proofs to come, the following inequality will prove useful:
\begin{equation}\label{eqn:fracmid}
\forall a,b,x,y > 0, \; \frac{a}{x} < \frac{b}{y} \; \rightarrow \; \frac{a}{x} < \frac{a+b}{x+y} < \frac{b}{y}
\end{equation}
Even more useful is a corollary of this inequality that follows by induction:
\begin{cor}\label{cor:fracseries}
Let $n > 0$ be a natural number, and for each natural number $i$ from $1$ to $n$, let positive numbers $a_i$ and $b_i$ be given. Moreover, let $c$ be any real number that satisfies $c \le \min_{1 \le i \le n} \frac{a_i}{b_i}$. Then, the following is true:
\begin{equation}
c \le \frac{\sum_{1 \le i \le n} a_i}{\sum_{1 \le i \le n} b_i}
\end{equation}
\end{cor}
Additionally, we note the following lemma regarding conductance in $d$-regular graphs\cite{MattaBE14}:
\begin{lem}\label{lem:sconn}
Given a connected, undirected $d$-regular graph $G = (V,E)$, there exists a set $S$ such that $S = S(\Phi(G))$ and the induced subgraph $G_S$ is connected.
\end{lem}
\section{Lower-Bounding VAT}
\begin{proof}[\textbf{Proof of Theorem \ref{thm:condvat}}]
Let $S = S(\tau(G))$, which is non-empty by definition. Furthermore, let $T = C_{max}(V-S)$, which is also non-empty by non-triviality of $G$ and Remark \ref{rem:vatbound}. Consider the set conductance of $T$, namely $\Phi_T(G)$. There are two situations which we must consider separately:
\begin{enumerate}
\item $|T| > \frac{|V|}{2}$
\item $|T| \le \frac{|V|}{2}$
\end{enumerate}
Let us first consider the first situation. In that case, $\Phi_T(G) = \frac{|Cut(T,V-T)|}{d|V-T|}$. However, because all edges of $Cut(T,V-T)$ must be adjacent to $S$, we know that $|Cut(T,V-T)| \le d|S|$. Moreover, clearly, $|V-T| \ge |V-S-T+1|$. Combining these facts, we obtain that:
\begin{equation}
\Phi(G) \le \Phi_T(G) \le \frac{d|S|}{|V-S-T+1|} = d\tau(G)
\end{equation}
This finishes the proof for the case that $T$ is the majority of the nodes. Therefore, let us finally consider the case that $T$ does not comprise the majority of the vertices:
Let us denote $q$ as the number of connected components of the induced subgraph $G_{V-S-T}$, and, furthermore, denote each such remaining component by $C_i, \forall 1 \le i \le q$. Because $T$ itself is the \emph{largest} connected component of induced subgraph $G_{V-S}$ by definition, and $T < \frac{|V|}{2}$ by assumption, we know further that each $|C_i| < \frac{|V|}{2}$. It will be convenient to denote $C_{q+1} = T$. Therefore,
\begin{equation}\label{eqn:ccphi}
\forall 1 \le i \le q+1, \Phi_{C_i}(G) = \frac{|Cut(C_i,V-C_i)|}{d|C_i|}
\end{equation}
Now, note the following facts due to the action of $S$ and the definition of connected components:
\begin{enumerate}
\item $d|S| \ge |Cut(T,V-T)| + \sum_{i=1}^q |Cut(C_i,V-C_i)| = \sum_{i=1}^{q+1} |Cut(C_i,V-C_i)|$
\item $|V-S-T+1| \le |V-S| = |T| + \sum_{i=1}^q |C_i| = \sum_{i=1}^{q+1} |C_i|$
\end{enumerate}
The second fact is clear from the definition of connected components as a partition. The first fact follows from the action of $S$: All edges crossing the boundary of a connected component must be adjacent to $S$. Combining the two facts, we obtain that
\begin{equation}
\tau(G) = d\frac{|S|}{d|V-S-T+1|} \ge \frac{\sum_{i=1}^{q+1} |Cut(C_i,V-C_i)|}{\sum_{i=1}^{q+1} d|C_i|}
\end{equation}
Now, note that each term of the sum in the numerator is a numerator of $\Phi_{C_i}(G)$ whereas each term of the sum in the denominator is a corresponding denominator of $\Phi_{C_i}(G)$. Applying Corollary \ref{cor:fracseries} with $c = \Phi(G)$, with $a_i = |Cut(C_i,V-C_i)|$, and with $b_i = d|C_i|$, we obtain that $\tau(G) \ge \Phi(G)$, completing the proof.
\end{proof}
\bibliographystyle{plain}
|
2,869,038,154,753 | arxiv | \section{Introduction}
Packing problems form an important class of combinatorial optimization problems that have been well studied under numerous variants \cite{Specht,MOSTOFAAKBAR20061259,10.5555/1206604,Stracquadanio}. It is a classic type of NP-hard problems, for which there is no deterministic algorithm to find exact solutions in polynomial time unless \emph{$P = NP$}. Also, there are numerous applications in the industry, such as shipping industry~\cite{THAPATSUWAN2012737,TOFFOLO2017526}, manufacturing materials \cite{WASCHER20071109,Maddaloni123}, advertisement placement \cite{Freund2004,DAWANDE2005}, loading problems \cite{BIRGIN200519,JUNQUEIRA201274}, and more exotic applications like origami folding \cite{Demaine2010CirclePF,An2018AnEA}.
Packing problems are well studied since 1832 Farkas {et al.} \cite{farkas,10.5555/1206604} investigated the occupying rate (density) of packing circle items in a bounded equilateral triangle bin, and since then tremendous improvements have been made~\cite{Graham95densepackings,lubachevsky2004dense}.
In the past three decades, most researches focus on the single container packing. The container is either in square, circle, rectangle, or polygon shape \cite{LOPEZ2011512}, while the items can be rectangles, circles, triangles, or polygons.
As one of the most classic packing problem, the circle packing problem (CPP)
is mainly concerned with packing circular items in a container.
Researchers have proposed various methods for finding feasible near-optimal packing solutions \cite{hifi2007,cave2011,dosh,zeng2016iterated}, which fall into two types: constructive optimization approach and global optimization approach.
The construction approach places the circle items one by one appropriately in the container based on a heuristic that defines the building rules to form a feasible solution. Most researches of this category either fix the position of the container's dimension and pack the items sequentially satisfying the constraints \cite{Dickinson2011}, or adjust the size of the container using a constructive approach \cite{zeng2016iterated}. Representative approaches include the Maximum Hole Degree (MHD) based algorithms \cite{Huang20031,HUANG2006,Huang2005}, among which Huang {et al.}~\cite{Huang20031} came up with two greedy algorithms: "B.10" places the circle items based on MHD, while "B.15" strengthens the solution with a self-look-ahead search strategy. Another approach called Pruned Enriched Rosenbluth Method (PERM) \cite{LU20081742,PhysRevE.68.021113,PhysRevE.72.016704} is a population control algorithm incorporating the MHD strategy. There are also other heuristics such as the Best Local Position (BLP) based approaches~\cite{hifi2004,hifi2007,hifi2008,hifi2009}, which selects the best feasible positions to place the items among other positions that minimizes the size of the container.
On the other hand, the global optimization technique \cite{CAS} tries to solve the packing problem by improving the solution iteratively based on an initial solution, which is subdivided into two types. The first type is called the quasi-physical quasi-human algorithm \cite{quasi2016, quasi2017, HE201826}, which is mostly motivated by some physical phenomenon, or some wisdom observed in human activities~\cite{HE201567, He2016PackingUC}. The second type is called the meta-heuristic optimization, mainly built by defining an evaluation function that employs a trade-off of randomisation and local search that directs and re-models the basic heuristic to generate feasible solutions. The meta-heuristic searches an estimation in the solution space closing to the global optimum. Representative algorithms include the hybrid algorithm \cite{ZHANG20051941} that combines the simulated annealing and Tabu search \cite{glover1989tabu, glover1990tabu}. Recently another hybrid algorithm was proposed by combining Tabu search and Variable Neighborhood Descent, and yield state-of-the-art results~\cite{ZENG2018196}.
In this work, we address a new variant of packing problem called the two-dimensional circle bin packing problem (CBPP)~\cite{dosh}.
Given a collection of circles specified by their radii, we are asked to pack all items into a minimum number of identical square bins. A packing is called feasible if no circles overlap with each other or no circle be out of the bin boundary. The CBPP is a new type of geometric bin packing problem, and it is related to the well-studied 2D bin packing problem~\cite{LodiMV99,BansalLS-FOCS05}, which consists in packing a set of rectangular items into a minimum number of identical rectangular bins.
This manuscript is an extended version with significant improvement on the algorithm of our previous conference publication~\cite{dosh}, among which we first introduce this problem and propose a Greedy Algorithm with Corner Occupying Action (GACOA) to construct a feasible dense layout~\cite{dosh}. In this paper, we further strengthen the packing quality and propose an Adaptive Large Neighborhood Search (ALNS) algorithm. ALNS first calls GACOA to construct an initial solution, then iteratively perturbs the current solution by randomly selecting any two used bins and unassigning circles that intersect a random picked region in each of the selected bin. Then we use GACOA to pack the outside circles back into the bin in order to form a complete solution.
The complete solution is accepted if the update layout increases the objective function or the decrease on the objective function is probabilistically allowed under the current annealing temperature. Note that the objective function is not the number of bins used but is defined to assist in weighing the performance to reach the global optimum of the new candidate solution.
Computational numerical results show that ALNS always outperforms GACOA in improving the objective function, and sometimes ALNS even outputs packing patterns with less number of bins.
In this work, we make three main contributions:
1) we design a new form of objective function, embedding the number of containers used and the maximum difference between the containers with the highest density and the box with the lowest density. The new objective function can help identify the quality of the assignment, especially in the general case with the same number of bins.
2) we propose a method for local search on the complete assignment solution. We select two bins randomly and generate a rectangular area for each bin with equal area. All the circles that intersect the rectangular area were unassigned and the remaining circles form the new partial solution.
3) we modify the conditions for receiving the new partial solution. The previous local neighborhood search algorithm only accepts new partial solutions with larger objective functions. However, it is not conductive to the global optimum to some extent. We apply the idea of simulated annealing to this new algorithm so that partial solutions with lower objective function values can also be accepted with a variable possibility
The remaining of this paper is organised as follows. Section~\ref{sec:section2} introduces the mathematical constraints for the given problem, Section~\ref{sec:section3} presents the two frameworks used for the development of our algorithm. Section~\ref{sec:section4} further describes the objective function as well as the experimental setup. All the algorithms are computationally experimented and the results presented in Section~\ref{sec:section5}. Finally, Section~\ref{sec:section6} concludes with recommendations for future work.
\section{Problem Formulation}
\label{sec:section2}
Given a set of $n$ circles where item $C_{i}$ is in radius $r_i$ and $n$ identical square bins with side length $L$ (w.l.g. for any circle $C_{i}$, $2 \cdot r_i \leq L$), the CBPP problem is to locate the center coordinates of each $C_{i}$ such that any item is totally inside a container and there is no overlapping between any two items. The goal is to minimize the number of used bins, denoted as $K$ $(1\le K\le n)$.
A feasible solution to the CBPP is a partition of the items into sets $\mathcal{S} = \langle S_{1}, S_{2}, \dots , S_{K} \rangle$ for the bins, and the packing constraints are satisfied in each bin. An optimal solution is the one in which $K$, the number of bins used, cannot be made any smaller. A summary of the necessary parameters is given in Table \ref{tab:f1}
\begin{table}[width=1.0\linewidth,cols=2,pos=h]
\caption{Parameter regulation}
\centering
\begin{tabular*}{\tblwidth}{@{} |c|L|@{} }\hline
\toprule
Parameter & Description\\
\hline
$n$ & number of circles\tabularnewline
\hline
$C_{i}$ & the $i$-th circle\tabularnewline
\hline
$r_{i}$ & radius of the $i$-th circle\tabularnewline
\hline
$\left(x_{i},y_{i}\right)$ & center coordinates of $C_{i}$\tabularnewline
\hline
$b_{k}$ & the bin that $C_{i}$ is assigned, $1\le k\le K$\tabularnewline
\hline
$L$ & side length of square bin\tabularnewline
\hline
$I_{ik}$ & indicator of the placement of $C_{i}$ into $b_{k}$\tabularnewline
\hline
$B_k$ & indicator of the use of bin $b_k$\tabularnewline
\hline
$d_{ij}$ & distance between $(x_{i},y_{i})$ and $(x_{j},y_{j})$\tabularnewline
\bottomrule
\end{tabular*}
\label{tab:f1}
\end{table}
Assume that the bottom left corner of each bin $b_{k}$ is placed at $(0,0)$ in it's own coordinate system.
we formulate the CBPP as a constraint optimization problem.
\begin{equation}\textcolor{white}{.......................}
\sum_{k=1}^{n} I_{ik}=1,\label{eq:1}
\end{equation}
where
\begin{equation}
I_{ik}\in \{0,1\}, ~~~i,k\in\{1,\ldots,n\},
\label{eq:2}
\end{equation}
which implies that each circle is packed exactly once. Further, if a bin $b_{k}$ is used, then
\begin{equation}
B_{k}=\left\{ \begin{array}{ll} 1,\,\textrm{if}\,\sum_{i=1}^{n}I_{ik}>0,\,i,k\in \{1,\ldots, n\},\\0,\,\textrm{otherwise}.\end{array}\right.\label{eq:3} \end{equation} And, finally, for circles that are in the same bin, $I_{ik}=I_{jk}=1$, and $i,j,k\in \{1,\ldots,n\}$, no overlap is allowed, implying that
\begin{equation}
d_{ij}=\sqrt{(x_{i}-x_{j})^{2}+(y_{i}-y_{j})^{2}}\ge (r_{i}+r_{j})I_{ik}I_{jk}.\label{eq:4}
\end{equation}
Specifically, let the circles be ordered by their radii so that ${\it r_{1}}\ge {\it r_{2}}, \ldots \ge {\it r_{n}}$, $r_i \in \mathcal{R}^+$.
To ensure that no item passes across the boundary of the bin, we ask that
\begin{equation}
r_i \leq x_i \leq L-r_i, ~~ r_i \leq y_i \leq L-r_i
\label{eq:5}
\end{equation}
Conditions (\ref{eq:1})--(\ref{eq:4}) along with (\ref{eq:5}) are the constraints for CBPP.
The overall goal of CBPP is to use as few bins as possible to pack the $n$ circles, which is
\begin{equation} \textcolor{white}{.......................}\min K=\sum_{k=1}^{n}B_k.\label{eq:6}
\end{equation}
\section{General Search Framework}
\label{sec:section3}
In this section, we introduce two general optimization search frameworks for constraint optimization problem that we will use for the development of our CBPP algorithm. The two frameworks are Large Neighborhood Search (LNS), and a variation of the well-studied simulated annealing process~\cite{Kirkpatrick}, Adaptive Large Neighborhood Search (ALNS)~\cite{gendreau2010handbook}. A particular advantage of ALNS is the capacity to move the iterated solution out of the local optimum.
\subsection{Large neighborhood search}
Large neighborhood search (LNS) is a technique to iteratively solve constraint optimization problems \cite{gendreau2010handbook,LARSSON2018103}. At each iteration, the goal is to find a more promising candidate solution to the problem, and traverse a better search path through the solution space.
\newline\textbf{Definition 1. (Constraint optimization problem, COP)}. A constraint optimization problem $P=\left\langle n,D^{n},\mathcal{C},f\right\rangle $ is defined by an array of $n$ variables that can take values from a given domain $D^{n}$, subject to a set of constraints $\mathcal{C}$. $f$ is the objective function to measure the performance of assignment. An assignment is an array of values $\mathbf{a}^{n}\in D^{n}$. A constraint $c\in \mathcal{C}: D^{n} \rightarrow \{0,1\}$ is a predicate that decides whether an assignment $\mathbf{a}^{n}\in D^{n}$ is locally valid. A solution to $P$ is an assignment $\mathbf{a}^{n}$ that is locally valid for all constraints in $\mathcal{C}$, i.e. $c\left(\mathbf{a}^{n}\right)$ is true for all $c\in \mathcal{C}$. The optimal solution of $P$ is a solution that maximizes the objective function $f$.
In a nutshell, LNS starts from a non-optimal solution $\mathbf{a}^{n}$ and iteratively improves the solution until reaching an optimal or near-optimal solution.
The main ingredient in LNS is an effective algorithm for completing a partial solution.
\newline\textbf{Definition 2. (Partial solution)}. A partial solution $\left\langle k,\mathbf{a}^{k},I\right\rangle $ is an assignment $\mathbf{a}^{k}$ to a subset of $k$ variables (with their indexes $I$) of a constraint optimization problem $P$. Completing a partial solution means finding an assignment for the remaining variables $\left\{ a_{i}|i\notin I\right\} $ so that $\mathbf{a}^{n}$ is a solution to $P$.
At each iteration, LNS relaxes and repairs the solution by randomly
generating and then completing a partial solution (Alg.~\ref{alg:LNS-Algorithm}). If the new assignment $\mathbf{b}^n$ has higher objective function value, the previous assignment $\mathbf{a}^n$ will be replaced.
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetAlgoLined
\SetKw{KwBreak}{break}
\Input{A COP $P=\left\langle n,D^{n},\mathcal{C},f\right\rangle $, \\
number of iteration steps $N$}
\Output{A near-optimal solution $\mathbf{a}^{n}$}
$\mathbf{a}^{n}\leftarrow$
{compute\_initial\_solution}($P$) \\
\For{$i\leftarrow 1 $ \KwTo $N$}{
$I\leftarrow $ generate\_partial\_solution($\mathbf{a}^{n}$)\;
$\mathbf{b}^{n}\leftarrow $
complete\_partial\_solution ($\mathbf{a}^{n},I,f$)\;
\If{$f\left(\mathbf{b}^{n}\right)>f\left(\mathbf{a}^{n}\right)$}{
$\mathbf{a}^{n}\leftarrow \mathbf{b}^{n}$\;
}
}
\caption{Large neighborhood search}
\label{alg:LNS-Algorithm}
\end{algorithm}
\subsection{Adaptive large neighborhood search}
The new solution $\mathbf{b}^{n}$ obtained by complete\linebreak\_partial\_solution of LNS should be better than the previous solution $\mathbf{a}^{n}$ in order to be accepted (lines 5-7). But, in some sense, this approach actually limits the search for finding a global optimal solution. Thus, we consider an adaptive version of LNS, called ALNS~\cite{gendreau2010handbook}, obtained by altering LNS to allow for stepping to worse solutions. This depends on a predefined stochastic annealing schedule \cite{gendreau2010handbook,Kirkpatrick}, thus allowing for the solution search space to break out of local optima. We give a generic description of the ALNS algorithm in Alg.~\ref{alg:ALNS}, and a more complete explanation of the framework will be presented in Section \ref{sec:section4}.
\begin{algorithm}[h]
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetAlgoLined
\SetKw{KwBreak}{break}
\Input{A COP $P=\left\langle n,D^{n},\mathcal{C},f\right\rangle$,\\
number of iteration steps $N$}
\Output{A near-optimal solution $\mathbf{a}^{n}$ }
$\mathbf{a}^{n}\leftarrow $ compute\_initial\_solution($P$)\;
$\Theta \leftarrow$ initial temperature\;
$\theta \leftarrow \Theta$\;
\For{$i\leftarrow 1 $ \KwTo $N$}{
$I\leftarrow $ generate\_partial\_solution ($\mathbf{a}^{n}$)\;
$\mathbf{b}^{n}\leftarrow $
complete\_partial\_solution($\mathbf{a}^{n},I,f$)\;
\If{\textrm{{acceptMove}}($\mathbf{b}^{n}, \mathbf{a}^{n}, \theta$)}{
$\mathbf{a}^{n}\leftarrow \mathbf{b}^{n}$\;
}
$\theta = \theta - \Theta/N$\;
}
\caption{Adaptive large neighborhood search}
\label{alg:ALNS}
\end{algorithm}
\vspace{-1em}
\section{ALNS for CBPP}
\label{sec:section4}
\subsection{Domain and objective function}
For the CBPP as a constraint optimization problem \linebreak (COP), we define its domain $D^{n}$ as follows. For each circle $C_i$, its assignment variables include $\langle x_{i},y_{i},I_{i1},...,I_{in} \rangle$. \linebreak We simplify the notation to $a_{ik} = \left\langle x_{i},y_{i},b_k\right\rangle$ if $I_{ik} = 1$. The corresponding domain $D= \mathbb{R}^{2}\times \left\{ 1,\ldots,n\right\}$ defines all possible assignments of a circle in the bin--coordinate space. Since each circle $C_{i}$ is constrained by an indicator function to be put only in a single bin, we abuse the notation slightly, simply use $a_{i}$ to denote $a_{ik}$, and place an emphasis on the circles rather than the containers, and each of the components where $i \in \{1,\ldots,n\}$ is a 3-tuple denoted as $\linebreak \left\langle x_{i},y_{i},b_k\right\rangle$. Thus, when referring to a solution to $P$, we write ${\mathbf a}^{n}$, and $D^{n}=D\times D\times \ldots \times D$ corresponds to the domain for the $n$ tuple variables. So ${\mathbf a}^{n}$ denotes a possible packing.
Let $L^2$ be the area of a bin, the density of $b_k$ of a packing is defined as
\begin{equation}\textcolor{white}{.......................}
d_{k}({\mathbf a}^{n})=\frac{1}{L^2}\sum_{C_{i}\in S_{k}}\pi r_{i}^{2} I_{ik}.\label{eq:7}
\end{equation}
where $S_{k}$ is the set of items assigned to $b_k$.
Let $K$ be the number of used bins for a solution $\mathbf{a}^{n}$, so
\begin{equation} \textcolor{white}{.......................}
K =\sum_{k=1}^{n}B_k.\label{eq:9}
\end{equation}
\begin{equation*}
\begin{split}
\text{Let~~} d_{\min} &= \min \{d_{k}({\mathbf a}^{n})|1\le k\le K\},~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ \\
d_{\max} &= \max \{d_{k}({\mathbf a}^{n})|1\le k\le K\}.
\end{split}
\end{equation*}
Here $d_{min}$ denotes the density of the sparsest bin and $d_{max}$ the density of the densest bin.
We define a useful objective function, which will form part of our algorithm as
\begin{equation}\textcolor{white}{..............}
f({\mathbf a}^{n})=-{K}+d_{max}-d_{min}.\label{eq:8}
\end{equation}
The larger the value of $f(\cdot)$ is, the better the packing is, since an increase in $f(\cdot)$ corresponds to a denser packing as circles move out of lower density bins.
In order to clarify the process for taking a complete candidate solution ${\mathbf a}^{n}$ for $P$ to a partial solution, the following formal definitions are required.
Note that $0\le d_{max}-d_{min} \le1 $, this term is used for regularization. It implies that using fewer bins is preferable, that a difference in the number of bins is enough to compare two candidate solutions. With the same number of used bins in different solutions, we focus on the fullest bin and the emptiest bin on each candidate solution. The more dense the fullest bin is, the less wasted space is. The more sparse the emptiest bin is, the more concentrated the remaining still-reserved space is, which means it would be easier for assigning subsequent circles. So, the difference in density between the fullest bin and the emptiest bin determines the quality of each candidate solution.
\subsection{Construct initial solution}
\label{sec:cis}
An initial solution can be quickly constructed by our greedy algorithm GACOA (Alg.~\ref{alg:GACOA-Algorithm}).
\begin{equation}
\text{compute\_initial\_solution}(P)= GACOA(L,\{{C_{i}}|1\leq i \leq n \})
\label{eq:12}
\end{equation}
For each circle, GACOA computes a set of candidate positions by greedily moving on to the next bin if a circle cannot be packed in any of the previous bins. In particular, each circle is packed according to the following criteria.
\newline\textbf{Definition 5. (Candidate packing position)}.
A candidate-packing position of a circle in a bin is any position that places the circle tangent to a) any two packed items, or b) a packed item and the border of the bin, or c) two perpendicular sides of the bin (i.e. the corner).
\newline\textbf{Definition 6. (Feasible packing position)}. A packing position of a circle in a bin is feasible if it does not violate any constraints: circles do not
overlap and be fully contained in a bin. (See Eq. (\ref{eq:1})--(\ref{eq:5}) for detailed constraints).
\newline\textbf{Definition 7. (Quality of packing position)}. The distance between the feasible packing position and the border of the bin is given by
\begin{equation} q\left(x,y\right)= \left\{\min\left(d_{x},d_{y}\right),\max\left(d_{x},d_{y}\right)\right\}.\label{eq:7.1}
\end{equation}
where $d_{x}$ (resp. $d_{y}$) is a distance between the center of the circle and the closer side of the bin in the horizontal (resp. vertical) direction.
For a circle in the current target bin, all feasible positions in the bin are sorted in dictionary order of $q(x,y)$.
The smaller, the better.
We call an action that places a circle onto one of its candidate packing positions a Corner Occupying Action (COA).
GACOA works by packing circles one by one in the decreasing order of their radii. Each circle considers the target bin in the increasing order of the bin index $k$.
For bin $b_k$, a feasible candidate packing position with the highest quality is selected, i.e. a feasible assignment that maximizes $q(x,y)$ will be executed. Thus, the best feasible COA is selected that favours positions closer to the border of the bin.
If there is no feasible assignment in bin $b_k$, we will try to pack in the next bin $b_{k+1}$. The pseudo code is given in Alg.~\ref{alg:GACOA-Algorithm}.
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetAlgoLined
\SetKw{KwBreak}{break}
\Input{Bin side length $L$,
a set of $n$ circles $\{C_i | 1\leq i\leq n\}$ with radii $r_1,\ldots,r_n$~ $(r_i \geq r_{i+1})$}
\KwResult{For each circle $C_i$, find a bin $b_k$, and place the circle center at $\left(x_i,y_i\right)$}
$K\leftarrow 0$\
\For{$i\leftarrow 1$ \KwTo $n$}{
\For{$k\leftarrow 1$ \KwTo $n$}{
$S_{k}\leftarrow \emptyset$\;
\While{true}{
\If{$k > K$}{$K = k$}
$S_{k}\leftarrow$ Feasible packing positions for $C_i$\;
\If{$S_k \neq \emptyset$}{\KwBreak}
$k \leftarrow k+1$\
}
A best packing position from $S_{k}$ is selected according $q(x,y)$\;
$\left(x_i,y_i\right)\leftarrow \arg\max_{\left(x,y\right)\in S} q\left(x,y\right)$\;
Execute this packing position to pack circle $C_i$ into bin $b_k$\;
}
}
\caption{GACOA}\label{alg:GACOA-Algorithm}
\end{algorithm}
Partial solutions are generated from a complete solution by selecting
two bins at random and do perturbations. We randomly select a rectangular area of equal size in each bin, all circles that intersect the two rectangular areas will be taken out and added to the remain2assign set, and the complete solution becomes a partial solution.
The unassigned circles will be reassigned based on the partial solution. This is equivalent to perturbing a complete solution that has \linebreak reached a local optimum.
Let function random\_ints$(m,M)$ returns $m$ distinct integers randomly selected from set $M$. Let function random\_real $(R)$ returns a random real number $0 < r\leq R$.
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetAlgoLined
\SetKw{KwBreak}{break}
\Input{A (complete) solution $\mathbf{a}^n$}
\Output{A partial solution given by the indexes to keep $I$}
\tcp{Select two bins randomly}
$\left(k_1, k_2\right)\leftarrow$ random\_ints($2, \left\{1,\ldots,{K})\right\}$) \\
\tcp{Select a rectangular area in the first bin}
$rect_{1}\leftarrow\mathrm{sample\_rects}(b_{k_1})$ \\
\tcp{Select a rectangular area in the second bin}
$rect_{2}\leftarrow\mathrm{sample\_rects}(b_{k_2})$ \\
$remain2assign\leftarrow\bigcup_{j\in\left\{ 1,2\right\}}\{ i|\left\langle x_{i},y_{i},b_{k_j}\right\rangle \in\mathbf{a}^{n},I_{ik_j}=1
~\bigwedge \mathbf {intersects}(C_i,rect_j)==True \} $\\
$I \leftarrow I / remain2assign$
\caption{Generate partial solution}
\label{alg:PartiaSolution-Algorithm}
\end{algorithm}
Alg.~\ref{alg:SampleRects-Algorithm} randomly selects a circle in the non-empty bin, and then randomly generates a rectangular area with the circle center be the center of the area. This guarantees that at least one circle will intersect the generated rectangular area (Here we simply use the envelope rectangle of the circle to check its intersection with the area). In most cases, more than one circle items intersect the rectangular area and will be unassigned at each iteration.
We choose to select two bins and generate one rectangular area for each bin. Only one bin can be sampled at a time, but when the partial solution and unassigned circle set are continued to be placed in the future, in the worst case, it will be put back as it is to obtain the same complete solution as before. The worst instance is that the previous partial solution did not leave enough free space to allocate the unassigned circles except for the generated area. If there is only one bin and one rectangular area, the unassigned circles are very likely to be put back into the previous generated rectangular area during the iteration, which means that this iteration process has no effect and does not help jump out of the local optimum. Thus, two bins are selected for each iteration so that the unassigned circles have more free space to be allocated. Even in the worst case, the algorithm will try to exchange the circles in the two rectangular areas, which ensures that there will be some disturbance per iteration. Of course, an alternative way is to sample only one bin and generate two rectangular areas for the bin, but two rectangular areas in different bins can increase the randomness.
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetAlgoLined
\SetKw{KwBreak}{break}
\Input{Index of bin $k$; side length of bin $L$}
\Output{rectangle area Rects}
\tcp{Each rectangle is represented as a bottom-left point and a top-right point}
$w\leftarrow\mathbf{random\_real}(L) $\tcp{the width of rectangle area is w}
$h\leftarrow\mathbf{random\_real}(L) $ \tcp{the height of rectangle area is h}
let $l_x \leftarrow 0$, $l_y \leftarrow 0$
\tcp{ where $(l_x,l_y)$ is the coordinate of bottom-left point}
\If{($ !b_k.empty()$) }{
$i \leftarrow\mathbf{random\_ints}(1,\left\{ i\left|\left\langle x_{i},y_{i},b_{k}\right\rangle \in\mathbf{a}^{n}\right.\right\})$\\
$l_x \leftarrow C_i.x-0.5w$\\
$l_y \leftarrow C_i.y-0.5h$
}
$Rects =$ make\_ pair(point($l_x,l_y$),point($l_x+w,l_y+h$))
\caption{\label{alg:SampleRects-Algorithm}sample\_rects}
\end{algorithm}
\begin{algorithm}
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetAlgoLined
\SetKw{KwBreak}{break}
\Input{circle $C_i$, $rect_k$ with coordinate tuple\\
<$l_x$, $l_y$, $l_x+w$, $l_y+h$>}
\Output{true or false}\tcp{returns if $C_i$ intersects the rectangle}
\If{$C_i.x-C_i.r\ge l_x+w$}{
return false \tcp{non-intersect}
}
\If{$C_i.y-C_i.r\ge l_y+h$}{
return false \tcp{non-intersect}
}
\If{$C_i.x+C_i.r\le l_x$}{
return false \tcp{non-intersect}
}
\If{$C_i.y+C_i.r\le l_y$}{
return false \tcp{non-intersect}
}
return true \tcp{intersect}
\caption{\label{alg:Intersects-Algorithm}intersects}
\end{algorithm}
\label{sec:gps}
\subsection{Complete a partial solution}
\label{sec:cps}
Completing a partial solution is performed efficiently by the {GACOA}
algorithm (Alg.~\ref{alg:GACOA-Algorithm}) restricted to the bins
$b_{k1},b_{k2}$ from which circles were unassigned in the previous
step.
\begin{equation}
\begin{aligned}
\textrm{{complete\_partial\_solution}}\left(\mathbf{a}^{n},I\right) =~~~~~~~~~~~~~~~~~~~~~~\\~~~~~\textrm{{GACOA}}\left(L,\left\{ C_{i}|i \in remain2assign\right\} \right)\label{eq:13}
\end{aligned}
\end{equation}
$I$ is the partial solution generated by \linebreak generate\_partial\_solution(). GACOA is used to complete the partial solution.
\subsection{Acceptance metric}
\label{sec:acc}
A solution is accepted if it increases the objective function or
if the decrease in the objective function is probabilistically allowed
given the current annealing temperature $\theta$. This is the well-known simulated annealing move acceptance criteria \cite{Kirkpatrick},
\begin{equation}
\begin{aligned}
\textrm{{acceptMove}}(\mathbf{b}^n,\mathbf{a}^n,\theta) =
f(\mathbf{b}^n)>f(\mathbf{a}^n)~~~~~~~~~~~~~~\\
\vee{random\_real}(1)\leq e^{\frac{f(\mathbf{b}^{n})-f(\mathbf{a}^{n})}{\theta}}\label{eq:14}
\end{aligned}
\end{equation}
\subsection{ALNS for CBPP}
The complete algorithm for solving CBPP requires various steps from the above. The ALNS procedure is started using the initial solution along with the temperature $\Theta$ and the number of iterations $N$. The initial solution is then broken using Alg.~\ref{alg:PartiaSolution-Algorithm} and re-completed using the new solution filled by GACOA. This procedure outputs a new candidate solution, which is then either accepted or rejected based on the acceptance metric with simulated annealing. At which point, if the iteration limit has not reached, the process restarts using the new solution as an input
An illustration of the entire parameter setup flow of ALNS algorithm is shown in Figure~\ref{fig:flow}.
\begin{figure*}[pos=H]
\centering
\includegraphics[width=\textwidth, scale=0.8]{ALNS_flowchart_v18.png}
\caption{The parameter setup flow of ALNS.}
\label{fig:flow}
\end{figure*}
\begin{comment}
A complete summary of the framework is in Table~\ref{tab:ALNS circular}.
\begin{figure}[pos=H]
\begin{tabular}{ |c||l| }
\hline
\multicolumn{2}{|c|}{ALNS Framework of CBPP} \tabularnewline
\hline
\hline
\multirow{5}{*}{\sc Initial Phase} & \\
& \tabitem ${\mathbf a}^{n}$; initial complete solution; $C_{i}$ into bin $b_{i}$ for $i\in \{1,\ldots, n\}$ \\
& \tabitem $\Theta$; initial temperature $\theta=\Theta$\\
& \tabitem $N$; total number of permitted iterations\\
&\\
\hline
\hline
\multirow{5}{*}{\sc Iterative Phase}
&\\
& \tabitem break current solution ${\mathbf a}^{n}$; \noun{GeneratePartial Solution} \\
&\tabitem \noun{CompletePartialSolution}; obtain ${\mathbf b}^{n}$\\
&\tabitem evaluate objective function \noun{AcceptsMove}; ${\mathbf a}^{n}$, ${\mathbf b}^{n}$, $\theta$ as inputs \\
&\tabitem accept or reject solution based on \noun{AcceptsMove}\\
&\\
\hline
\hline
\multirow{4}{*}{\sc Post Processing Phase} & \\
&\tabitem update $\theta\rightarrow \theta-\Theta/N$\\
&\tabitem set ${\mathbf a}^{n}\lefttarrow {\mathbf b}^{n}$ (accept) or set ${\mathbf a}^{n}\rightarrow {\mathbf a}^{n}$ (reject)\\
& \tabitem increase iteration counter while $\le N$\\
&\\
\hline
\end{tabular}
\centering
\caption{Summary of the ALNS Framework for CBPP}
\label{tab:ALNS circular}
\end{figure}
\end{comment}
\begin{figure*}[pos=H]
\centering
\includegraphics[scale=0.50]{newspike.png}
\caption{ALNS vs. GACOA.}
\label{fig:my_label2}
\end{figure*}
\begin{table*}
\hfill{}
\begin{tabular}{|c|c||c||c|c|c|c|c|c||c|c|}
\hline
$n_{0}$ & $n$ & alg. & bin 1 & bin 2 & bin 3 & bin 4 & bin 5 & bin 6 & $f$ & $f_{A}-f_{G}$\tabularnewline
\hline
\hline
\multirow{2}{*}{8} & \multirow{2}{*}{40} & {A} & 0.83 & 0.80 & 0.76 & 0.74 & 0.65 & - & -5.18 & \multirow{2}{*}{0.08}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & G & 0.83 & 0.74 & 0.71 & 0.76 & 0.73 & - & -5.10 & \tabularnewline
\hline
\multirow{2}{*}{9} & \multirow{2}{*}{45} & {A} & 0.81 & 0.81 & 0.80 & 0.76 & 0.75 & - & -5.81 & \multirow{2}{*}{1.26}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & G & 0.76 & 0.72 & 0.75 & 0.75 & 0.72 & 0.21 & -6.55 & \tabularnewline
\hline
\multirow{2}{*}{10} & \multirow{2}{*}{50} & A & 0.83 & 0.82 & 0.79 & 0.78 & 0.76 & 0.08 & -6.75 & \multirow{2}{*}{0.09}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & {G} & 0.84 & 0.78 & 0.80 & 0.75 & 0.72 & 0.18 & -6.66 & \tabularnewline
\hline
\multirow{2}{*}{11} & \multirow{2}{*}{55} & A & 0.84 & 0.82 & 0.80 & 0.78 & 0.77 & - & -5.84 & \multirow{2}{*}{1.20}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & {G} & 0.83 & 0.78 & 0.77 & 0.75 & 0.69 & 0.19 & -6.64 & \tabularnewline
\hline
\multirow{2}{*}{12} & \multirow{2}{*}{60} & A & 0.84 & 0.81 & 0.80 & 0.80 & 0.80 & - & -5.84 & \multirow{2}{*}{1.23}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & G & 0.84 & 0.79 & 0.78 & 0.72 & 0.69 & 0.23 & -6.61 & \tabularnewline
\hline
\multirow{2}{*}{13} & \multirow{2}{*}{65} & A & 0.83 & 0.81 & 0.81 & 0.81 & 0.80 & 0.05 & -6.78 & \multirow{2}{*}{0.25}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & {G} & 0.83 & 0.79 & 0.76 & 0.72 & 0.69 & 0.31 & -6.52 & \tabularnewline
\hline
\multirow{2}{*}{14} & \multirow{2}{*}{70} & A & 0.83 & 0.82 & 0.82 & 0.81 & 0.79 & 0.09 & -6.74 & \multirow{2}{*}{0.21}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & {G} & 0.84 & 0.82 & 0.77 & 0.71 & 0.71 & 0.32 & -6.52 & \tabularnewline
\hline
\multirow{2}{*}{15} & \multirow{2}{*}{75} & A & 0.83 & 0.83 & 0.82 & 0.81 & 0.81 & 0.05 & -6.78 & \multirow{2}{*}{0.25}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & G & 0.85 & 0.81 & 0.76 & 0.70 & 0.70 & 0.32 & -6.53 & \tabularnewline
\hline
\multirow{2}{*}{16} & \multirow{2}{*}{80} & A & 0.84 & 0.84 & 0.82 & 0.81 & 0.79 & 0.08 & -6.76 & \multirow{2}{*}{0.18}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & G & 0.85 & 0.82 & 0.81 & 0.73 & 0.71 & 0.26 & -6.58 & \tabularnewline
\hline
\multirow{2}{*}{17} & \multirow{2}{*}{85} & A & 0.85 & 0.83 & 0.82 & 0.81 & 0.80 & 0.11 & -6.74 & \multirow{2}{*}{0.14}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & {G} & 0.85 & 0.82 & 0.79 & 0.75 & 0.75 & 0.26 & -6.60 & \tabularnewline
\hline
\multirow{2}{*}{18} & \multirow{2}{*}{90} & A & 0.85 & 0.83 & 0.82 & 0.81 & 0.81 & 0.11 & -6.74 & \multirow{2}{*}{0.17}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & {G} & 0.85 & 0.80 & 0.80 & 0.79 & 0.71 & 0.29 & -6.57 & \tabularnewline
\hline
\multirow{2}{*}{19} & \multirow{2}{*}{95} & A & 0.85 & 0.83 & 0.82 & 0.81 & 0.81 & 0.12 & -6.73 & \multirow{2}{*}{0.21}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & G & 0.84 & 0.81 & 0.80 & 0.77 & 0.69 & 0.32 & -6.52 & \tabularnewline
\hline
\multirow{2}{*}{20} & \multirow{2}{*}{100} & A & 0.84 & 0.83 & 0.82 & 0.81 & 0.80 & 0.12 & -6.72 & \multirow{2}{*}{0.17}\tabularnewline
\cline{3-10} \cline{4-10} \cline{5-10} \cline{6-10} \cline{7-10} \cline{8-10} \cline{9-10} \cline{10-10}
& & G & 0.86 & 0.81 & 0.78 & 0.77 & 0.71 & 0.31 & -6.55 & \tabularnewline
\hline
\end{tabular}\hfill{}
\caption{Experimental results on the fixed benchmarks with square bins when $r_i = i$. The average improvement is 0.19. }
\label{tab:fixed-square}
\end{table*}
\begin{figure*}[pos=H]
\centering
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{alns-cbpp_square_fixed_9_45.pdf}
\caption{Packing layouts generated by ALNS algorithm}
\label{fig:tae11}
\end{subfigure}
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{dosh-cbpp_square_fixed_9_45.pdf}
\caption{Packing layouts generated by GACOA algorithm}
\label{fig:ptae41}
\end{subfigure}
\caption{Solution for the fixed benchmark when $n_{0}=9$ and $n=45$.}
\label{fig:layout}
\end{figure*}
\begin{table*}
\hfill{}%
\begin{tabular}{|c|c||c||c|c|c|c|c||c|c|}
\hline
$n_{0}$ & \multicolumn{1}{c|}{$n$} & alg. & bin 1 & bin 2 & bin 3 & bin 4 & bin 5 & $f$ & $f_{A}-f_{G}$\tabularnewline
\hline
\hline
\multirow{2}{*}{8} & \multirow{2}{*}{34} & A & 0.83 & 0.80 & 0.74 & 0.73 & 0.24 & -4.41 & \multirow{2}{*}{0.09}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.83 & 0.74 & 0.74 & 0.70 & 0.33 & -4.5 & \tabularnewline
\hline
\multirow{2}{*}{9} & \multirow{2}{*}{30} & A & 0.81 & 0.80 & 0.77 & 0.44 & - & -3.63 & \multirow{2}{*}{0.18}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.78 & 0.75 & 0.71 & 0.59 & - & -3.81 & \tabularnewline
\hline
\multirow{2}{*}{10} & \multirow{2}{*}{37} & A & 0.81 & 0.80 & 0.79 & 0.76 & - & -3.95 & \multirow{2}{*}{0.4}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.83 & 0.78 & 0.71 & 0.67 & 0.18 & -4.35 & \tabularnewline
\hline
\multirow{2}{*}{11} & \multirow{2}{*}{38} & A & 0.83 & 0.81 & 0.79 & 0.00 & - & -3.17 & \multirow{2}{*}{0.04}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.82 & 0.81 & 0.78 & 0.03 & - & -3.21 & \tabularnewline
\hline
\multirow{2}{*}{12} & \multirow{2}{*}{43} & A & 0.83 & 0.81 & 0.80 & 0.29 & - & -3.46 & \multirow{2}{*}{0.12}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.82 & 0.79 & 0.72 & 0.40 & - & -3.58 & \tabularnewline
\hline
\multirow{2}{*}{13} & \multirow{2}{*}{44} & A & 0.83 & 0.81 & 0.78 & 0.43 & - & -3.6 & \multirow{2}{*}{0.17}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.82 & 0.75 & 0.69 & 0.59 & - & -3.77 & \tabularnewline
\hline
\multirow{2}{*}{14} & \multirow{2}{*}{48} & A & 0.84 & 0.82 & 0.80 & 0.40 & - & -3.56 & \multirow{2}{*}{0.15}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.82 & 0.77 & 0.73 & 0.53 & - & -3.71 & \tabularnewline
\hline
\multirow{2}{*}{15} & \multirow{2}{*}{53} & A & 0.85 & 0.83 & 0.81 & 0.32 & - & -3.47 & \multirow{2}{*}{0.11}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.85 & 0.81 & 0.72 & 0.43 & - & -3.58 & \tabularnewline
\hline
\multirow{2}{*}{16} & \multirow{2}{*}{55} & A & 0.85 & 0.82 & 0.81 & 0.46 & - & -3.61 & \multirow{2}{*}{0.14}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.84 & 0.78 & 0.73 & 0.59 & - & -3.75 & \tabularnewline
\hline
\multirow{2}{*}{17} & \multirow{2}{*}{53} & A & 0.84 & 0.82 & 0.80 & 0.32 & - & -3.48 & \multirow{2}{*}{0.15}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.82 & 0.79 & 0.73 & 0.45 & - & -3.63 & \tabularnewline
\hline
\multirow{2}{*}{18} & \multirow{2}{*}{72} & A & 0.84 & 0.83 & 0.83 & 0.72 & - & -3.88 & \multirow{2}{*}{0.36}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.84 & 0.81 & 0.78 & 0.70 & 0.08 & -4.24 & \tabularnewline
\hline
\multirow{2}{*}{19} & \multirow{2}{*}{58} & A & 0.83 & 0.83 & 0.81 & 0.07 & - & -3.24 & \multirow{2}{*}{0.12}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.83 & 0.76 & 0.76 & 0.19 & - & -3.36 & \tabularnewline
\hline
\multirow{2}{*}{20} & \multirow{2}{*}{78} & A & 0.84 & 0.84 & 0.81 & 0.67 & - & -3.83 & \multirow{2}{*}{0.01}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.86 & 0.82 & 0.80 & 0.69 & - & -3.83 & \tabularnewline
\hline
\end{tabular}\hfill{}
\caption{\label{tab:random-square}Experimental results on the random benchmarks for $r_i = i$. The average improvement is 0.12.}
\end{table*}
\begin{figure*}[pos=H]
\centering
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{alns-cbpp_square_random_18_72.pdf}
\caption{Packing layouts generated by ALNS algorithm}
\label{fig:tae1}
\end{subfigure}
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{dosh-cbpp_square_random_18_72.pdf}
\caption{Packing layouts generated by GACOA algorithm}
\label{fig:ptae42}
\end{subfigure}
\caption {\label{fig:layout18}} {Solution for the random benchmark $n_{0}=18$ and $n=72$.}
\end{figure*}
%
\begin{table*}
\hfill{}%
\begin{tabular}{|c|c||c||c|c|c|c||c|c|}
\hline
$n_{0}$ & $n$ & alg. & bin 1 & bin 2 & bin 3 & bin 4 & $f$ & $f_{A}-f_{G}$\tabularnewline
\hline
\hline
\multirow{2}{*}{8} & \multirow{2}{*}{40} & A & 0.83 & 0.83 & 0.76 & 0.30 & -3.47 & \multirow{2}{*}{0.21}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.80 & 0.76 & 0.70 & 0.47 & -3.67 & \tabularnewline
\hline
\multirow{2}{*}{9} & \multirow{2}{*}{45} & A & 0.83 & 0.82 & 0.79 & 0.33 & -3.5 & \multirow{2}{*}{0.19}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.82 & 0.73 & 0.70 & 0.51 & -3.69 & \tabularnewline
\hline
\multirow{2}{*}{10} & \multirow{2}{*}{50} & A & 0.83 & 0.81 & 0.80 & 0.38 & -3.55 & \multirow{2}{*}{0.15}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.84 & 0.76 & 0.69 & 0.54 & -3.7 & \tabularnewline
\hline
\multirow{2}{*}{11} & \multirow{2}{*}{55} & A & 0.84 & 0.81 & 0.81 & 0.40 & -3.56 & \multirow{2}{*}{0.09}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.83 & 0.78 & 0.77 & 0.48 & -3.65 & \tabularnewline
\hline
\multirow{2}{*}{12} & \multirow{2}{*}{60} & A & 0.83 & 0.83 & 0.81 & 0.39 & -3.56 & \multirow{2}{*}{0.07}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.83 & 0.80 & 0.77 & 0.46 & -3.63 & \tabularnewline
\hline
\multirow{2}{*}{13} & \multirow{2}{*}{65} & A & 0.84 & 0.83 & 0.80 & 0.40 & -3.56 & \multirow{2}{*}{0.10}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.83 & 0.82 & 0.72 & 0.49 & -3.66 & \tabularnewline
\hline
\multirow{2}{*}{14} & \multirow{2}{*}{70} & A & 0.85 & 0.82 & 0.81 & 0.36 & -3.51 & \multirow{2}{*}{0.08}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.84 & 0.79 & 0.78 & 0.43 & -3.59 & \tabularnewline
\hline
\multirow{2}{*}{15} & \multirow{2}{*}{75} & A & 0.84 & 0.83 & 0.82 & 0.37 & -3.53 & \multirow{2}{*}{0.08}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.85 & 0.80 & 0.75 & 0.46 & -3.61 & \tabularnewline
\hline
\multirow{2}{*}{16} & \multirow{2}{*}{80} & A & 0.86 & 0.83 & 0.81 & 0.38 & -3.52 & \multirow{2}{*}{0.12}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.85 & 0.81 & 0.73 & 0.49 & -3.64 & \tabularnewline
\hline
\multirow{2}{*}{17} & \multirow{2}{*}{85} & A & 0.85 & 0.84 & 0.81 & 0.39 & -3.54 & \multirow{2}{*}{0.13}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.83 & 0.79 & 0.77 & 0.50 & -3.67 & \tabularnewline
\hline
\multirow{2}{*}{18} & \multirow{2}{*}{90} & A & 0.87 & 0.82 & 0.80 & 0.41 & -3.54 & \multirow{2}{*}{0.09}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.85 & 0.80 & 0.76 & 0.48 & -3.63 & \tabularnewline
\hline
\multirow{2}{*}{19} & \multirow{2}{*}{95} & A & 0.85 & 0.82 & 0.82 & 0.40 & -3.55 & \multirow{2}{*}{0.05}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.86 & 0.82 & 0.77 & 0.45 & -3.59 & \tabularnewline
\hline
\multirow{2}{*}{20} & \multirow{2}{*}{100} & A & 0.86 & 0.82 & 0.82 & 0.41 & -3.55 & \multirow{2}{*}{0.05}\tabularnewline
\cline{3-8} \cline{4-8} \cline{5-8} \cline{6-8} \cline{7-8} \cline{8-8}
& & G & 0.87 & 0.80 & 0.77 & 0.47 & -3.6 & \tabularnewline
\hline
\end{tabular}\hfill{}
\caption{\label{tab:fixed-circle}Experimental results on the fixed benchmarks when $r_{i}=\sqrt{i}$. The average improvement of 0.22.}.
\end{table*}
\begin{figure*}[pos=H]
\centering
\begin{subfigure}[b]{0.95\textwidth}
\centering
\includegraphics[width=\textwidth]{alns840.pdf}
\caption{Packing layouts generated by ALNS algorithm}
\label{fig:tae3}
\end{subfigure}
\begin{subfigure}[b]{0.95\textwidth}
\centering
\includegraphics[width=\textwidth]{gacoa840.pdf}
\caption{Packing layouts generated by GACOA algorithm}
\label{fig:ptae}
\end{subfigure}
\caption{Solution for the fixed benchmark when $n_{0}=8$ and $n=40$ when $r_{i}=\sqrt{i}$.}
\label{fig:figxx}
\end{figure*}
\begin{table*}
\hfill{}%
\begin{tabular}{|c|c||c||c|c|c|c|c||c|c|}
\hline
$n_{0}$ & $n$ & alg. & bin 1 & bin 2 & bin 3 & bin 4 & bin 5 & $f$ & $f_{A}-f_{G}$\tabularnewline
\hline
\hline
\multirow{2}{*}{8} & \multirow{2}{*}{37} & A & 0.84 & 0.83 & 0.67 & - & - & -2.83 & \multirow{2}{*}{0.41}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.80 & 0.75 & 0.74 & 0.04 & - & -3.24 & \tabularnewline
\hline
\multirow{2}{*}{9} & \multirow{2}{*}{35} & A & 0.82 & 0.81 & 0.42 & - & - & -2.6 & \multirow{2}{*}{0.19}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.77 & 0.72 & 0.56 & - & - & -2.79 & \tabularnewline
\hline
\multirow{2}{*}{10} & \multirow{2}{*}{45} & A & 0.83 & 0.82 & 0.81 & 0.37 & - & -3.54 & \multirow{2}{*}{0.17}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.83 & 0.76 & 0.69 & 0.54 & - & -3.71 & \tabularnewline
\hline
\multirow{2}{*}{11} & \multirow{2}{*}{44} & A & 0.84 & 0.81 & 0.81 & 0.39 & - & -3.55 & \multirow{2}{*}{0.10}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.83 & 0.78 & 0.77 & 0.48 & - & -3.65 & \tabularnewline
\hline
\multirow{2}{*}{12} & \multirow{2}{*}{52} & A & 0.84 & 0.82 & 0.80 & - & - & -2.96 & \multirow{2}{*}{0.69}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.83 & 0.78 & 0.77 & 0.48 & - & -3.65 & \tabularnewline
\hline
\multirow{2}{*}{13} & \multirow{2}{*}{71} & A & 0.85 & 0.83 & 0.81 & 0.76 & - & -3.91 & \multirow{2}{*}{0.38}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.83 & 0.83 & 0.75 & 0.70 & 0.12 & -4.29 & \tabularnewline
\hline
\multirow{2}{*}{14} & \multirow{2}{*}{55} & A & 0.84 & 0.83 & 0.75 & - & - & -2.91 & \multirow{2}{*}{0.27}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.85 & 0.80 & 0.76 & 0.03 & - & -3.18 & \tabularnewline
\hline
\multirow{2}{*}{15} & \multirow{2}{*}{53} & A & 0.83 & 0.83 & 0.71 & - & - & -2.88 & \multirow{2}{*}{0.41}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.84 & 0.74 & 0.65 & 0.13 & - & -3.29 & \tabularnewline
\hline
\multirow{2}{*}{16} & \multirow{2}{*}{48} & A & 0.83 & 0.82 & 0.60 & - & - & -2.77 & \multirow{2}{*}{0.11}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.81 & 0.75 & 0.69 & - & - & -2.88 & \tabularnewline
\hline
\multirow{2}{*}{17} & \multirow{2}{*}{72} & A & 0.84 & 0.82 & 0.81 & 0.30 & - & -3.46 & \multirow{2}{*}{0.13}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.82 & 0.77 & 0.77 & 0.41 & - & -3.59 & \tabularnewline
\hline
\multirow{2}{*}{18} & \multirow{2}{*}{66} & A & 0.82 & 0.81 & 0.80 & 0.12 & - & -3.3 & \multirow{2}{*}{0.09}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.82 & 0.78 & 0.75 & 0.21 & - & -3.39 & \tabularnewline
\hline
\multirow{2}{*}{19} & \multirow{2}{*}{75} & A & 0.84 & 0.82 & 0.81 & 0.24 & - & -3.4 & \multirow{2}{*}{0.04}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.84 & 0.81 & 0.78 & 0.28 & - & -3.44 & \tabularnewline
\hline
\multirow{2}{*}{20} & \multirow{2}{*}{58} & A & 0.81 & 0.81 & 0.79 & 0.06 & - & -3.25 & \multirow{2}{*}{0.10}\tabularnewline
\cline{3-9} \cline{4-9} \cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & G & 0.78 & 0.78 & 0.76 & 0.14 & - & -3.36 & \tabularnewline
\hline
\hline
\end{tabular}\hfill{}
\caption{\label{tab:random-circle2}Experimental results on the random benchmarks for $r_{i}=\sqrt{i} $. The average improvement is 0.23.}
\end{table*}
\begin{figure*}[pos=H]
\centering
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{figuretop1random.PNG}
\caption{Packing layouts generated by ALNS algorithm}
\label{fig:tae4}
\end{subfigure}
\begin{subfigure}[b]{0.7\textwidth}
\centering
\includegraphics[width=\textwidth]{figurebottom1random.PNG}
\caption{Packing layouts generated by GACOA algorithm}
\label{fig:ptae4}
\end{subfigure}
\caption {\label{fig:figxx2}} {Solution for the random benchmark with $n_{0}=13$ and $n=71$ when $r_{i}=\sqrt{i}$.}
\end{figure*}
%
~
\begin{table*}[pos=H]
\hfill{}%
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
& \multicolumn{4}{c|}{$r_i = i$} & \multicolumn{4}{c|}{$r_{i}=\sqrt{i} $}\tabularnewline
\hline
\hline
& \multicolumn{2}{c|}{fixed} & \multicolumn{2}{c|}{random} & \multicolumn{2}{c|}{fixed} & \multicolumn{2}{c|}{random}\tabularnewline
\hline
$n_0$ & $n$ & $t$ & $n$ & $t$ & $n$ & $t$ & $n$ & $t$\tabularnewline
\hline
8 & 40 & 82 & 34 & 59 & 40 & 53 & 37 & 42\tabularnewline
\hline
9 & 45 & 72 & 30 & 63 & 45 & 87 & 35 & 61\tabularnewline
\hline
10 & 50 & 74 & 37 & 69 & 50 & 76 & 45 & 59\tabularnewline
\hline
11 & 55 & 91 & 38 & 160 & 55 & 88 & 44 & 82\tabularnewline
\hline
12 & 60 & 103 & 43 & 120 & 60 & 107 & 52 & 81\tabularnewline
\hline
13 & 65 & 116 & 44 & 127 & 65 & 137 & 71 & 142\tabularnewline
\hline
14 & 70 & 135 & 48 & 148 & 70 & 152 & 55 & 145\tabularnewline
\hline
15 & 75 & 154 & 53 & 177 & 75 & 182 & 53 & 210\tabularnewline
\hline
16 & 80 & 179 & 55 & 189 & 80 & 180 & 48 & 163\tabularnewline
\hline
17 & 85 & 204 & 53 & 179 & 85 & 202 & 72 & 227\tabularnewline
\hline
18 & 90 & 222 & 72 & 339 & 90 & 216 & 66 & 197\tabularnewline
\hline
19 & 95 & 247 & 58 & 209 & 95 & 234 & 75 & 261\tabularnewline
\hline
20 & 100 & 279 & 78 & 366 & 100 & 255 & 58 & 251\tabularnewline
\hline
\end{tabular}\hfill{}
\caption{Runtimes for ALNS execution in all benchmarks. }
\label{tab:Runtime}
\end{table*}
\section{Computational Experiments}
\label{sec:section5}
To evaluate the competency of the proposed approach, we implemented the ALNS for CBPP using the Visual C++ programming language. All results were generated by setting $N=2\times10^{6}$ (Alg.~\ref{alg:ALNS}) and obtained in a computer with Intel Core i7-8550U CPU @ 1.80GHz.
We generated benchmarks based on two groups of instances downloaded from \url{www.packomonia.com} for Single Circle Packing Problem (SCCP): $r_i=i$ for wide variation instances and $r_{i}=\sqrt{i}$ for smaller variation instances. On the packomania website, we use the range seed (number of circles for SCCP) from 8 to 20, and
generate sets of benchmarks as follows:
For each square bin, the best solution found in \cite{packomania} for the SCPP was used to fix the bin size $L$ as the best known record ${L_{best~known~record}}$ and we generate two sets of benchmarks called ``fixed" and ``random".Let $S=\{ C_{i}|1\leq i$ $\leq n \}$ be the set of circles packed in the current best solution for SCPP . The fixed set of CBPP benchmarks contains exactly $5$ copies of each circle packing for SCPP. The rand set of CBPP benchmarks contains a random number ($2\leq r\leq5$) of copies of each circle (i.e., the number of copies of each circle varies across the same benchmark) packed for SCPP.
In the following tables we get $52 \times 2$ generated instances from the two groups of instances, we compare the number of bins and the objective value for the best solution obtained by our search algorithm ALNS with the solution obtained from our constructive algorithm GACOA. They contain, for each original number of circles ($n_{0}$) and actual number of replicated circles in the CBPP benchmark ($n$), two rows showing the bin densities for ALNS (A) and GACOA (G) algorithm. The last two columns contain the final value of the objective function obtained in each algorithm and the relative improvement of ALNS over GACOA. Figure~\ref{fig:my_label2} shows in depth the typical behavior in comparison of ALNS and GACOA on the two benchmark instances. The objective function ($f$) in the y-axis under the number of circles in the x-axis. We can observe the packing occupying rate of ALNS is higher than that of GACOA. An in-depth analysis is explained in Section~\ref{sec:section5.1} and Section~\ref{sec:section5.2}.
\subsection{Comparison on $r=i$}
\label{sec:section5.1}
We run ALNS and GACOA on the benchmark instances of $r=i$. The number of unequal circle items ranges from 8 to 20 seeds for both fixed and random square settings. Table \ref{tab:fixed-square} shows the computational results of the fixed copies of 5 for each $n_0$. Table \ref{tab:random-square} displays the random number of copies of the circle items ranging from $2\leq r\leq5$ of the corresponding seed. From Table \ref{tab:fixed-square} we can observe that when $n=45, 55,$ or $ 60$ on the fixed benchmark, ALNS utilises 5 bins, while GACOA uses 6 bins to pack the circle items. Figure \ref{fig:layout} illustrates the packing layout for $n=45$. On the other hand, for the random instances in Table \ref{tab:random-square} when $n=37$ or $72$, ALNS also minimizes the number of bins by packing circles in 4 bins while GACOA packs in 5 bins. Figure \ref{fig:layout18} illustrates the packing layout for $n=72$. In summary, from the results of all the instances, ALNS returns feasible results, and has an overall average improvement of $19\%$ for the fixed benchmarks and $12\%$ for the random benchmarks when compared to GACOA.
\subsection{Comparison on $r_{i}=\sqrt{i}$}
\label{sec:section5.2}
Similarly we also run ALNS and GACOA on the benchmark instances of $r_{i}=\sqrt{i}$, which contains a smaller variation of radii. Table \ref{tab:fixed-circle} contains fixed instances while Table \ref{tab:random-circle2} contains random instances. The instances also range from 8 to 20 seeds. As an illustration, we select $n_0=8$ and $n=40$ from Table \ref{tab:fixed-circle} and illustrate the layout in Figure \ref{fig:figxx}. The occupying rate for the first three bins is higher for ALNS when compared to that of GACOA, showing that most circle items have maximally occupied each bin's area. The fourth last bin's density of the packed items for ALNS is lower than that of GACOA, indicating that ALNS contains fewer packed circle items in the last bin. On the random benchmarks in Table \ref{tab:random-circle2} for $n_0$ at instance 8, 12, 13, 14 and 15, we also observe that ALNS uses one lesser bin in total when compared to GACOA. The average improvement of ALNS over GACOA is $22\%$ for the fixed instances and $23\%$ for the random instances, respectively.
\subsection{Further discussion}
The run-times for ALNS at each benchmark are shown in Table \ref{tab:Runtime} (column of $t$ in seconds). We see that ALNS computes the instances efficiently in less than 300 seconds for 100 items. By comparison, as a greedy algorithm, GACOA completes the calculation in microseconds on any of these benchmarks. Such property facilitates the high efficiency of the ALNS algorithm.
In summary, for all the generated instances, we compare the number of bins and the objective value for the best solution obtained by ALNS algorithm with solutions obtained by GACOA. The results clearly show that ALNS consistently improves the objective function value as compared to GACOA over all sets of benchmarks, and it was even able to reduce the number of bins used in some benchmarks.\newpage The results show that our objective function does guide the ALNS algorithm to search for dense packing and promote reducing the number of bins used.
\section{Conclusions}
\label{sec:section6}
We address a new type of packing problem, two dimensional circle bin packing problem (2D-CBPP), and propose an adaptive local search algorithm for solving this NP-Hard problem. The algorithm adopts a simulated annealing search on our greedy constructive algorithm.
The initial solution is built by the greedy algorithm. Then during the search, we generate a partial solution by randomly selecting rectangular areas in two bins and remove the circle items that intersect the areas. And we implement our greedy algorithm for completing partial solutions during the search. To facilitate the search, we design a new form of objective function, embedding the number of containers used and the maximum gap of the densities of different containers. A new solution is conditionally accepted by simulated annealing, completing one iteration of the search.
Despite to all the improvements in this work, it is highly noted that the proposed problem is indeed challenging for combinatorial optimization heuristics and future researches are needed to get better solutions and generate high quality benchmarks. Implementing an adaptive local neighborhood search seems to be an attractive meta-heuristic to adopt. we would like to explore the idea to other circle bin packing problems, and extend our approach in addressing three dimensional circle bin packing problem which is more challenging with many applications that deserves proper attention.
\section*{Acknowledgement}
This work was supported by the National Natural Science Foundation of China (Grant No. U1836204, U1936108) and the Fundamental Research Funds for the Central Universities (2019kfyXKJC021).
\bibliographystyle{elsarticle-num}
|
2,869,038,154,754 | arxiv | \section{Introduction}
Nowadays it is strongly believed that the universe is experiencing
an accelerated expansion. The observation data confirm it such
as type Ia supernovae \cite{R3} in associated with large scale
structure \cite{R4} and Cosmic Microwave Background anisotropies
\cite{R5} have provided main evidence for this cosmic
acceleration. In order to explain why the cosmic acceleration
happens, many theories have been proposed. The standard
cosmological model (SCM) furnishes an accurate description of the
evolution of the universe, in spite of its success, the SCM
suffers from a series of problems such as the initial
singularity, the cosmological horizon, the flatness problem, the
baryon asymmetry and the nature of dark energy and dark matter,
although inflation partially or totally answers some of these
problems. Inflation theory was first proposed by Guth in 1981
\cite{gu}. Inflation is a period of accelerated expansion in the
early universe, it occurs when the energy density of the universe
is dominated by the potential energy of some scalar field called
inflaton. Currently all successful inflationary scenarios are
based on the use of weakly interaction scalar fields. Scalar
fields naturally arise in particle physics including string theory
and these can act as candidates for dark energy. So far a wide
verity of scalar field dark energy models have been proposed.
These include quintessence \cite{R6}, K-essence \cite{R7},
tachyon \cite{R8}, phantoms \cite{R9}, ghost condensates
\cite{R10} and so forth. Two main reason for use of scalar fields
to explain inflation are natural homogeneity and isotropy of such
fields and its ability to imitate a slowly decaying cosmological
constant \cite{R1}. However, no scalar field has ever been
observed, and designing models by using unobserved scalar fields
undermine their predictability and falsifiability, despite the
recent precision data. The latest theoretical developments (string
landscape) offer too much freedom for model-building, so higher
spin fields generically induce a spatial anisotropy and the
effective mass of such fields usually of the order of the Hubble
scale and the slow-roll inflation dose not occurs \cite{R11}.
Then an immediate question is, can we do Cosmology without scalar
fields? The authors of \cite{R1,R2} have shown that a successful
vector inflation can be simultaneously surmounted in a natural
way, and isotropy of the vector field condensate be achieved
either in the case of triplet of mutually orthogonal vector field
\cite{R12}. In spite of inflation success in explaining the
present state of the universe, it dose not solve the crucial
problem of the initial singularity \cite{R13}. The existence of
an initial singularity is disturbing, because the space-time
description breaks down "there". Non-singular universes have been
recurrently present in the scientific literature. Bouncing model
is one of them that was first proposed by Novello and Salim
\cite{R14} and Mlnikov and Orlov \cite{R15} in the late 70's. At
the end of the 90's the discovery of the acceleration of the
universe brought back to the front the idea that $\rho+3p$ could
be negative, which is precisely one of the conditions needed for
cosmological bounce in GR, and contributed to the revival of
nonsingular universes. Bouncing universe are those that go from
an era of acceleration collapse to an expanding era without
displaying a singularity \cite{R16}. Necessary conditions
required for a successful bounce during the contracting phase,
the scale factor $a(t)$ is decreasing , i.e. $\dot{a} < 0$, and
in the expanding phase we have $\dot{a} > 0$. At the bouncing
point, $\dot{a} = 0$, and around this point $\ddot{a} > 0$ for a
period of time. Equivalently in the bouncing cosmology the Hubble
parameter H runs across zero from $\dot{H} < 0$ to ${H}> 0$ and
$H = 0$ at the bouncing point. A successful bounce
requires around this point.\\
The remainder of the paper is as follows. In section 2 and 3, we
will consider vector field action where proposed in Refs.
\cite{R1,R2} and study bouncing solution of this model. In
section 4 and 5,
we will reconstruct physical quantities for this model and also will plot the corresponding graphs. Finally we will
apply three parametrization and compare them for this model.\\
\section{Vector field foundation}
We consider a massive vector field, which is non-minimally coupled
to gravity, \cite{R1,R2}. The action is given by
\begin{equation}\label{E1}
S=\int d^{4}x \sqrt{-g}~\left(\frac{1}{16 \pi G}R-\frac{1}{4}
F_{\mu\nu}F^{\mu \nu}-\frac{1}{2}m^2 U_{\mu}U^{\mu}+\frac{1}{2}\xi R
U_{\mu}U^{\mu}\right),
\end{equation}
where $F_{\mu\nu}=\partial_{\mu}U_{\nu}-\partial_{\nu}U_{\mu}$, and
$\xi$ is a dimensionless parameter for non-minimal coupling. We note
that, the non-minimal coupling of vector field is same with
conformal coupling of a scalar field in case $\xi=1/6$. We adopt FRW
universe with the metric signature of $(-+++)$.\\
The equations of motion are given by
\begin{eqnarray}\label{E2}
R_{\mu\nu}-\frac{1}{2}R g_{\mu\nu}={8 \pi
G}\bigg[F_{\mu\alpha}F^{\alpha\nu}-\frac{1}{4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}+\
(m^2 - \xi R )U_{\mu}U_{\nu}\nonumber\\-\frac{1}{2}g_{\mu\nu}(m^2 -
\xi R )U_{\alpha}U^{\alpha}\ - \xi g_{\mu\nu}U_{\alpha}U^{\alpha} +
\xi (\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\hbox{$\rlap{$\sqcup$}\sqcap$})U_{\alpha}U^{\alpha}
\bigg],
\end{eqnarray}
\begin{equation}\label{E3}
\nabla_{\nu}F^{\nu\mu} - m^2U^{\mu} + \xi R U^{\mu}=0.
\end{equation}
Where the right hand side of equation (\ref{E2}) is the
energy-momentum tensor of the vector field $U_i$. The variation
of the action with respect to $U_i$ yields the following
equations of motion,
\begin{equation}\label{E4}
\frac{1}{a^2}\nabla ^2 U_0 - \frac{1}{a^2}\partial_i
\dot{U_i}-m^2U_0 +\xi R U_0 =0,
\end{equation}
\begin{equation}\label{E5}
\ddot{U_i}+\frac{\dot{a}}{a}(\dot{U_i}-
\partial_iU_0)-\partial_i\dot{U_i}+\frac{1}{a^2}(\partial_i(\partial_kU_k)-\nabla^2 U_i)
+ m^2U_i -\xi RU_i=0,
\end{equation}
Where $a$ is the scale factor, the dot denotes the derivative with
respect to the cosmic time and the summation over repeated spatial
indices is satisfied. By considering the quasi-homogeneous vector
field $(\partial _iU_\alpha = 0)$ and Eq. (\ref{E4}) imply $U_0 =
0$, so that from Eq. (\ref{E5}) we obtain
\begin{equation}\label{E6}
\ddot{U_i}+ H\dot{U_i}-6\xi(\dot{H}+2H^2+\frac{k}{a^2})U_i+m^2U_i=0.
\end{equation}
By using acceleration relation $\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}(\rho +3p)$ we achieve as,
\begin{equation}\label{E7}
\dot{H}+H^2=\frac{-4\pi G}{a^2}\left(2\dot{U_i^2}-4(1+6\xi)H
U_i\dot{U_i}+6\xi U_i^2H^2 -m^2U_i^2 -\frac{2k}{a^2}\xi
U_i^2\right),
\end{equation}
where $H=\frac{\dot{a}}{a}$, $R = 6(\dot{H}^2+2H^2+\frac{k}{a})$
and $R^0_0=\dot{H}+H^2$ are Hubble 's parameter, Ricci scalar and
first component of Ricci tensor, respectively. As we know, a
dynamical vector field has generally a preferred direction, and
to introduce such a vector field may not be consistent with the
isotropy of the universe. In fact, the energy-momentum tensor of
the vector field $U_\mu$ has anisotropic components. However,
the anisotropic part of the energy-momentum tensor can be
eliminated by introducing a triplet of mutually orthogonal vector
fields. In that case, we obtain the energy density $\rho$ and the
pressure $p$ of the vector fields
\begin{equation}\label{E8}
\rho=\frac{1}{a^2}\left[\frac{3}{2}\dot{U_i}^2-3(1+6\xi)HU_i\dot{U_i}+9\xi
U_i^2H^2 +\frac{3}{2}m^2U_i^2-\frac{9k\xi}{a^2}U_i^2\right],
\end{equation}
\begin{equation}\label{E9}
p=\frac{1}{a^2}\left[\frac{3}{2}\dot{U_i}^2-3(1+6\xi)HU_i\dot{U_i}+9\xi
U_i^2H^2 -\frac{3}{2}m^2U_i^2+\frac{3k\xi}{a^2}U_i^2\right].
\end{equation}
Now to introduce a change of variable $\phi_i=\frac{U_i}{a}$ (for
more detail see Ref. [1]), equation (\ref{E4}) make change to,
\begin{equation}\label{E10}
\ddot{\phi_i}+3H\dot{\phi}+\left(m^2+(1-6\xi)(\dot{H}+2H^2)-\frac{6\xi
k}{a^2}\right)\phi_i=0.
\end{equation}
Then we consider $\xi=1/6$ and obtain the basic equations of motion
for a curved universe in terms of $\phi_i$,
\begin{equation}\label{E11}
\ddot{\phi_i}+3H\dot{\phi}+\left(m^2-\frac{k}{a^2}\right)\phi_i=0,
\end{equation}
\begin{equation}\label{E12}
H^2+\frac{k}{a^2}=4\pi G(\dot{\phi^2_i}+m^2 \phi^2_i
-\frac{k}{a^2}\phi^2 _i ),
\end{equation}
\begin{equation}\label{E13}
\dot{H}+H^2=-4\pi G(2\dot{\phi^2_i}-m^2 \phi^2_i),
\end{equation}
One can see where equations of motion of vector field is reduced to
minimally coupled massive scalar fields. So energy density $\rho$
and the pressure $p$ for the vector fields are derived in terms of
$\phi_i$ in case $\xi=1/6$ in the form,
\begin{equation}\label{E14}
\rho=\frac{3}{2}\dot{\phi_i}^2+\frac{3}{2}m^2 \phi^2_i
-\frac{3k}{2a^2}\phi^2_i,
\end{equation}
\begin{equation}\label{E15}
p=\frac{3}{2}\dot{\phi_i^2}-\frac{3}{2}m^2 \phi^2_i
+\frac{k}{2a^2}\phi^2_i,
\end{equation}
\\
Now we are going to consider behavior of the different values of
parameter $\xi$ for vector field. We solve numerically Eq.
(\ref{E6}) for $K=0, +1, -1$ which implies the flat, close and open
universe respectively. The Fig.1 shows graph of the vector field
with respect to time in all of cases $K$. One can see where vector
field has oscillation behavior and the magnitude slowly decrease
with respect to time evolution. Also by increasing the parameter
$\xi$, the magnitude of vector field will increase, but the period
of oscillation is constant. We note that negative values of $\xi$
actually is the same of above result.
\begin{figure}[th]
\centerline{\epsfig{file=1a.eps,scale=.25}\epsfig{file=1b.eps,scale=.25}\epsfig{file=1c.eps,scale=.25}}
\vspace*{8pt} \caption{Graphs of vector fields in term of time. The
solid, dash and doted lines represent $\xi=1$ ,$\frac{1}{6}$ and
$0.5$ respectively.}
\end{figure}
As above mention we suggest following solution for $U_i(t)$
\begin{equation}\label{E20}
U_i(t)=\sqrt{A}e^{-\gamma t}\cos (mt+\theta),
\end{equation}
where the parameter $A$ describes the oscillating amplitude of the
field with dimension of $[mass]^2$. Also $A$ is relation with the
parameter $\xi$, this solution implies the damping magnitude of the
oscillating vector field.
\section{Bouncing behavior}
We will start with a detailed examination on the necessary
conditions required for a successful bounce. During the
contracting phase, the scale factor $a(t)$ is decreasing, i.e.,
$\dot{a}<0$, and in the expanding phase we have $\dot{a}>0$. At
the bouncing point $\dot{a}=0$ and around this point $\ddot{a}>0$
for a period of time. Equivalently in the bouncing cosmology the
Hubble parameter $H$ runs across zero from $H<0$ to $H>0$ and $H =
0$ at the bouncing point. A successful bounce requires around
this point
\begin{equation}\label{E16}
\dot{H}=-4\pi G(\rho + p)+\frac{k}{a^2} > 0.
\end{equation}
At the point where the bounce occurs, Eqs. (\ref{E8}) and (\ref{E9})
reduce to
\begin{equation}\label{E17}
\rho_b=\frac{3}{2a^2}(\dot{U_i}^{2}+m^2U^2_i)-\frac{9k}{a^4}\xi
U_i^2,
\end{equation}
\begin{equation}\label{E18}
p_b=\frac{3}{2a^2}(\dot{U_i}^{2}-m^2U^2_i)+\frac{3k}{a^4}\xi
U_i^2,
\end{equation}
On the other hand, a successful bounce from Eqs. (\ref{E6}),
(\ref{E7}) and (\ref{E16}) obtain in the form,
\begin{equation}\label{E19}
\dot{U_i}^2 < \frac{1}{2}m^2U^2_i+\frac{k}{a^2}\xi U_i^2.
\end{equation}
This result is similar to slow roll inflation. This means that one
requires a flat potential where give rise to a point bounce for
the model of vector field. From conditions (\ref{E16}),
(\ref{E19}) it is clear that if we have bouncing solutions in
open universe, then we have such behaviour for flat and closed
universe as well.
Now we solve above equation
numerically by different value of $\xi$ on the curved universe
that is plotted in Fig. 2.\\\\
\begin{figure}[th]
\centerline{\epsfig{file=2a.eps,scale=.25}\epsfig{file=2b.eps,scale=.25}\epsfig{file=2c.eps,scale=.25}}
\vspace*{8pt} \caption{The graphs of the Hubble's parameter for $\xi=1/6$, $4\pi G=1$, $m=1$ and $k=0,+1,-1$ by choosing $\phi(0)=1$, $\dot{\phi}(0)=0.1$, $a(0)=1$ and $H(0)=0.01$.}
\end{figure}
\begin{figure}[th]
\centerline{\epsfig{file=3a.eps,scale=.25}\epsfig{file=3b.eps,scale=.25}\epsfig{file=3c.eps,scale=.25}}
\vspace*{8pt} \caption{The graphs of the scale factor for $\xi=1/6$, $4\pi G=1$, $m=1$ and $k=0,+1,-1$ by choosing $\phi(0)=1$, $\dot{\phi}(0)=0.1$, $a(0)=1$ and $H(0)=0.01$.}
\end{figure}
One can see the Hubble parameter $H$ running across zero in any
three cases of $k$. In all cases of $k$, we have $H<0$ to $H>0$
where implies to go from collapse era to an expanding era, and
this result will not change for the different values $\xi$ in all
of $k$. Also in Fig.3, we can see the behaviour of scale
factor in terms of time for different values of $k$.
It is clear that during the contracting phase, the scale factor $a(t)$ is decreasing,
i.e., $\ddot{a}<0$, and in the expanding phase we have $\ddot{a}>0$, so the point where
$\ddot{a}=0$ is bouncing point. \\
Therefore, in the vector field dominated universe we have a
successful bouncing point in close and flat universe but a
turn-around point in open universe. The bounce can be attributed
to the negative-energy matter, which dominates at small values of
$a$ and create a significant enough repulsive force so that a big
crunch is avoided.
\section{Reconstruction}
Now we are going to present a reconstruction process for vector
field in the curved universe by $\xi=1/6$ . In this section,
potential and kinetic energy are reconstructed with respect to
redshirt $z$. Also we obtain the EoS in term of $z$. After that
three type parametrization are represented for the EoS. By using
it we consider cosmology solutions such as the Eos, the
deceleration parameter and vector field. The stability condition
of this system is described by quantity of the sound speed. We
rewrite Eqs. (\ref{E14}) and (\ref{E15}) in term of the effective
potential energy $\hat{V}$ and the effective kinetic energy
$\hat{K}$ as the following form,
\begin{equation}\label{E21}
\rho=\frac{3}{2}\dot{\phi_i}^2+\frac{3}{2}m^2 \phi^2_i
-\frac{3k}{2a^2}\phi^2_i=3\hat{K}+3\hat{V},
\end{equation}
\begin{equation}\label{E22}
p=\frac{3}{2}\dot{\phi_i}^2-\frac{3}{2}m^2 \phi^2_i
+\frac{k}{2a^2}\phi^2_i=3\hat{K}-3\hat{V}-\frac{k}{a^2},
\end{equation}
\begin{equation}\label{E23}
\rho +p=6\hat{K}-\frac{k}{a^2}.
\end{equation}
Then we can write the Friedmann equations as following
\begin{equation}\label{E24}
3M^2_p (H^2+\frac{k}{a^2})=\rho_m+\rho=\rho_m+3\hat{K}+3\hat{V},
\end{equation}
\begin{equation}\label{E25}
2M^2_p(\dot{H}-\frac{k}{a^2})=-\rho_m-\rho-p=-\rho_m-6\hat{K}+\frac{k}{a^2},
\end{equation}
where $\rho_m$ is the energy density of dust matter. Also from Eqs.
(\ref{E21}) and (\ref{E22}), we obtain relationship between the Eos
with $\hat{V}$ and $\hat{K}$ as,
\begin{equation}\label{E26}
\omega=\frac{p}{\rho}=\frac{3\hat{K}-3\hat{V}-\frac{k}{a^2}}{3\hat{K}+3\hat{V}}=-1+\frac{2-\frac{k}{3a^2\hat{K}}}{1+\frac{\hat{V}}{\hat{K}}}.
\end{equation}
We obviously have
\begin{eqnarray}\label{E26-1}
\hat{V}+3\hat{K}>\frac{k}{3a^2} \Longrightarrow \omega > -1,\nonumber\\
\hat{V}+3\hat{K}<\frac{k}{3a^2} \Longrightarrow \omega < -1,\nonumber\\
\hat{V}+\hat{K}=\frac{k}{3a^2} \Longrightarrow \omega=-1.
\end{eqnarray}
By using Eqs. (\ref{E24}) and (\ref{E25}) we can write
\begin{equation}\label{E27}
\hat{K}=\frac{-\rho_m}{6}-\frac{M^2_p}{3}(\dot{H}-\frac{k}{3a^2})+\frac{k}{6a^2},
\end{equation}
\begin{equation}\label{E28}
\hat{V}=\frac{M^2_p}{3}(3H^2+\dot{H}+\frac{2k}{a^2})-\frac{\rho_m}{6}-\frac{k}{6a^2}.
\end{equation}
As in the present model, the dark energy fluid does not couple to
the background fluid, the expression of the energy density of dust
matter in respect of redshift $z$ is \cite{R17},
\begin{equation}\label{E29}
\rho_m=3M^2_pH^2_0\Omega_{m_0}(1+z)^3,
\end{equation}
where $\Omega_{m_0}$ is the ratio density parameter of matter fluid
and the subscript $0$ indicates the present value of the
corresponding quantity. By using the equation $1+z=\frac{a_0}{a}$
($a_0$ is quantity given at the present epoch) and its differential
form in following have,
\begin{equation}\label{E30}
\frac{d}{dt}=-H(1+z)\frac{d}{dz}.
\end{equation}
To introduce a new variable $r$ as,
\begin{equation}\label{E31}
r=\frac{H^2}{H^2_0},
\end{equation}
we rewrite the equation of motion of vector field against $z$ as,
\begin{eqnarray}\label{E31}
2r(1+z)^2 U^{\prime \prime}_i+2r(1+z)(1+H^2_0)U^{\prime}_i-r^\prime
(1+z)^2U^{\prime}_i\nonumber\\+r^\prime (1+z) U_i-r
U_i+\frac{2m^2}{H^2_0}U_i-\frac{2k}{a^2_0 H^2_0}(1+z)^2 U_i=0,
\end{eqnarray}
$\hat{K}$, $\hat{V}$ can be rewrite as following
\begin{equation}\label{E33}
\hat{K}=-\frac{1}{2}M^2_p H^2_0\Omega_{m0}(1+z)^3+\frac{1}{6}M^2_p
H^2_0r^\prime(1+z)+\frac{k}{6 a_0^2}(1+z)^2,
\end{equation}
\begin{equation}\label{E34}
\hat{V}=M^2_pH^2_0r
+\frac{2k}{3a_0^2}(1+z)^2-\frac{1}{6}M^2_pH^2_0(1+z)r^\prime-\frac{1}{2}M^2_pH^2_0\Omega_{m0}(1+z)^3-\frac{k}{6a_0^2}(1+z)^2.
\end{equation}
By using Eqs. (\ref{E26}), (\ref{E27}) and (\ref{E28}) we obtain
following expression for the EoS,
\begin{equation}\label{E35}
\omega=\frac{(1+z)r^\prime-3r +\frac{k (1+z)^2}{a_0^2 H^2_0
M^2_p}(M^2_p-2) }{3r-3\Omega_{m0}(1+z)^3 +\frac{k (1+z)^2}{a_0^2
H^2_0M^2_p}(M^2_p+2)}.
\end{equation}
Then we obtain following equation for $r(z)$
\begin{eqnarray}\label{E38}
r(z)=\Omega_{m0}(1+z)^3+(1-\Omega_{m0})e^{\beta(z)}\nonumber
\\+\alpha_0 \int^z_0
\left[\omega(\tilde{z})(2+M^2_p)+(2-M^2_p)\right]
(1+\tilde{z})e^{-\beta(\tilde{z})}d\tilde{z}.
\end{eqnarray}
where $\beta(z)=\int^z_0
\frac{3w(\tilde{z})}{1+\tilde{z}}d\tilde{z}$ and
$\alpha_0=\frac{k}{a_0^2 H^2_0 M^2_p}$.
\begin{equation}\label{E38}
r(z)=\Omega_{m0}(1+z)^3+(1-\Omega_{m0})e^{3\int^z_0}\frac{1+w(\tilde{z})}{1+\tilde{z}}d\tilde{z}.
\end{equation}
Also we have following expression for deceleration parameter q
\begin{equation}\label{E39}
q(z)=-1-\frac{\dot{H}}{H^2}=\frac{(1+z)r^\prime -2r}{2r}.
\end{equation}
Now we consider the stability of this model by use the
hydrodynamic analogy and judge on stability by examining the
value of the sound speed. Of course this is a simple approach,
the perturbations in vector inflation are much richer than in
hydrodynamic model, see recent interesting works in
\cite{{pel1},{pel2}}. The sound speed can be obtained by the
following equation,
\begin{equation}\label{E36}
c^2_s=\frac{p^\prime}{\rho^\prime}=\frac{-2r^\prime+(1+z)r^{\prime\prime}+2\frac{k
(1+z)}{a_0^2 H^2_0 M^2_p}(M^2_p-2)
}{-9\Omega_{m0}(1+z)^2+3r^\prime+2\frac{k(1+z)}{a_0^2 H^2_0
M^2_p}(M^2_p+2)},
\end{equation}
in order to deal the stability of our model, the sound speed must
become $c^2_s\geq0$, so we can obtain from above equation following
condition
\begin{equation}\label{E37}
r(z)\geq \omega_{m0}(1+z)^3 -\frac{16 k a^2_0}{H^2_0(1+z)^2}.
\end{equation}
\section{Parametrization}
Now we consider the three different forms of parametrization as
following and compare them together.\\
\textbf{Parametrization 1:} First Parametrization has
proposed by Chevallier and Polarski \cite{R18},and Linder
\cite{R19}, where the EoS of dark energy in term of redshift $z$ is
given by,
\begin{equation}\label{E40}
\omega(z)=\omega_0 +\frac{\omega_a z}{1+z}.
\end{equation}
\textbf{Parametrization 2:} Another the EoS in term of redshift z
has proposed by Jassal, Bagla and Padmanabhan \cite{R20} as,
\begin{equation}\label{E41}
\omega(z)=\omega_0 +\frac{\omega_b z}{(1+z)^2}.
\end{equation}
\textbf{Parametrization 3:} Third parametrization has proposed by
Alam, Sahni and Starobinsky \cite{R21}. They take expression of r in
term of $z$ as followoing,
\begin{equation}\label{E41}
r(z)=\Omega_{m0}(1+z)^3+A_0+A_1(1+z)+A_2(1+z)^2.
\end{equation}
By using the results of Refs. \cite{R22}, \cite{R23}, \cite{R24} and
\cite{R25}, we get coefficients of parametrization $1$ as
$\Omega_{m0}=0.29$, $\omega_0= -1.07$ and $\omega_a=0.85$,
coefficients of parametrization $2$ as $\Omega_{m0}=0.28$,
$\omega_0=-1.37$ and $\omega_b =3.39$ and coefficients of
parametrization $3$ as $\Omega_{m0}=0.30$, $A_0 = 1$, $A1 = -0.48$
and $A2 = 0.25$. The evolution of $\omega(z)$ and $q(z)$ are plotted
in Fig. 3. Also, using Eqs. (\ref{E33}) and (\ref{E34}) and the
three parametrization, the evolutions of $\hat{K(z)}$ and
$\hat{V(z)}$ are shown in Fig. 5 and Fig. 6 respectively. We note
that graphs simply represent only in the flat universe.
\begin{figure}[th]
\centerline{\epsfig{file=4a.eps,scale=.3}\epsfig{file=4b.eps,scale=.3}}
\vspace*{8pt} \caption{Graphs of the EoS and deceleration parameter
in respect of redshift $z$. The solid, dot and dash lines represent
parametrization 1, 2 and 3, respectively.}
\end{figure}
\begin{figure}[th]
\centerline{\epsfig{file=5.eps,scale=.4}} \vspace*{8pt}
\caption{Graphs of the reconstructed $\hat{K}$ in respect of
redshift $z$. The solid, dot and dash lines represent
parametrization 1, 2 and 3, respectively.}
\end{figure}
\begin{figure}[th]
\centerline{\epsfig{file=6.eps,scale=.4}} \vspace*{8pt}
\caption{Graphs of the reconstructed $\hat{V}$ in respect of
redshift $z$. The solid, dot and dash lines represent
parametrization 1, 2 and 3, respectively.}
\end{figure}
From Figs. 4, 5 and 6, we can see parametrization $1$ and $3$ are
same nearly and have slightly different from parametrization $2$.
The EoS for any parametrization show in Fig. (4) so that they
running cross to $-1$. Acceleration for all of parametrization
shows to tend to the positive value. The $\hat{K}$ and $\hat{V}$
increase for parametrization $1$ and $3$, but in parametrization
$2$ increase (decrease) for the $\hat{V}$ ($\hat{K}$). One can
see that parametrization 1 and 3 satisfy condition
$\hat{K}+\hat{V}>0$ and parametrization 2 satisfy condition
$\hat{K}+\hat{V}=Constant$. In order to by Eq. (\ref{E26-1}), we
have $\omega>-1$ ($\omega=-1$) when parametrization 1 and 3
(parametrization 2). This is mean that
parametrization 2 is better than others parametrization.\\
\begin{figure}[th]
\centerline{\epsfig{file=7.eps,scale=.35}} \vspace*{8pt}
\caption{Graphs of the reconstructed $U_i$ in respect of redshift
$z$. The solid, dot and dash lines represent parametrization 1, 2
and 3, respectively.}
\end{figure}
In Fig. 6, we can see the variation of the vector field against
redshift $z$. One is obviously showed slightly difference between
all of parametrization.\\\\
\newpage
\section{Conclusion}
In this paper, we have studied the bouncing solution in curved
universe which proposed by the model of a massive vector field,
$U_i$, non-minimally coupled to gravity. For our purpose we have
derived the corresponding energy density, pressure and Friedmann
equation for this model. Also we have obtained the bouncing
condition as Eq.(\ref{E19}). From this condition, and also
essential condition (\ref{E16}), it is clear that if we have
bouncing solutions in open universe, then we have such behaviour
for flat and closed universe as well. After we plot the Hubble
parameter in term of time by figures 2, for different $k$, we
understood that our model predict the bouncing behavior for all
cases of $k$. Fro these figures one can see that the Hubble
parameter $H$ running across zero in any three cases of $k$. In
all cases of $k$, we have $H<0$ to $H>0$ where implies to go from
collapse era to an expanding era, and this result will not change
for the different values $\xi$ in all of $k$. After that in
figures 3 we have shown that during the contracting phase, the
scale factor $a(t)$ is decreasing,
i.e., $\ddot{a}<0$, and in the expanding phase we have $\ddot{a}>0$, so the point where
$\ddot{a}=0$ is bouncing point, and this figure is consistent
with the results of Fig 2.\\
After that we have investigated an interesting method as the
reconstruction of the non-minimally coupled massive vector field
model with the action (\ref{E1}). Our aim was to see whether the
non-minimal coupling vector field can actually reproduce required
values of observable cosmology, such as evolution of the EoS and
the deceleration parameter in respect to the redshift $z$. We
have reconstructed our model in three different forms of
parametrization for massive vector field. In Fig. 4 we have found
the EoS crossing $-1$ in all of parametrization. The variation of
reconstructed kinetic and potential energy against $z$ have
plotted in Figs. 5 and 6, where the parametrization 2 in addition
is better than two others parametrization because
$\hat{K}+\hat{V}=Constant$. Also we have investigated the
stability of this system and have obtained a condition by the
sound speed in all of curvatures. Finally we note that
reconstructed physical quantities have just executed in flat
universe and one is suggested for open and close universe as
future work.\\\\
\section{Acknowledgment}
The authors are indebted to the anonymous referee for
his/her comments that improved the paper drastically.
|
2,869,038,154,755 | arxiv | \section{Introduction}
The risks that algorithms---including those embedded in AI systems---can raise concerns relating to bias are well recognized.
Addressing algorithmic bias requires an understanding of normative conceptions of fairness, and how they can be quantitatively measured. Moreover, algorithm designers and users will need to make informed decisions regarding which one or more of multiple possible fairness measures should be used for system assessments.
While fairness measures in relation to issues such as testing have been a topic of academic and broader interest for decades \cite{Hutchinson:2019:YTF:3287560.3287600}, recent years have seen rapidly growing interest among researchers within and beyond the technical community in the issue of \textit{algorithmic} fairness. The proliferation of papers describing fairness measures has spurred examination of the mathematical relationships among them.
For example, Kleinberg \textit{et al.}\@ \cite{1609.05807} considered three fairness measures (\textit{calibration within groups}, \textit{balance for the positive class}, and \textit{balance for the negative class}), and showed that ``except in highly constrained special cases, there is no method that can satisfy these three conditions simultaneously.'' Chouldechova \cite{1703.00056} has also written on the incompatibilities between fairness criteria
and has observed that \textit{test-fairness}, a measure ``originating in the field of educational and psychological testing'' can, when applied in the context of recidivism prediction, ``lead to considerable disparate impact when recidivism prevalence differs across groups.'' In another paper examining criminal risk assessments \cite{doi:10.1177/0049124118782533}, Berk \textit{et al.}\@ have observed that it is generally ``impossible to maximize accuracy and fairness at the same time, and impossible simultaneously to satisfy all kinds of fairness.''
Verma \textit{et al.}\@ \cite{Verma:2018:FDE:3194770.3194776} have ``collect[ed] the most prominent definitions of fairness for the algorithmic classification problem, explain[ed] the rationale behind these definitions, and demonstrate[d] each of them on a single unifying case-study.'' In a paper titled ``Fairness Through Awareness'' \cite{Dwork:2012:FTA:2090236.2090255}, Dwork \textit{et al.}\@, considered ``fairness in classification, where individuals are classified \ldots and the goal is to prevent discrimination against individuals based on their membership in some group, while maintaining utility for the classifier.'' Selbst \textit{et al.}\@ \cite{Selbst:2019:FAS:3287560.3287598}, have considered the broader societal context of seeking fair algorithms, and have argued that attempts to ``produce fairness-aware learning algorithms, and to intervene at different stages of a decision-making pipeline to produce `fair' outcomes'' can be ``dangerously misguided when they enter the societal context that surrounds decision-making systems.''
While the papers cited above are drawn from technical publications, algorithmic fairness is also receiving rapidly growing attention in the legal scholarship. Examples include Mayson \cite{Mayson} who has argued that ``[a]lgorithmic risk assessment has revealed the inequality inherent in all prediction, forcing us to confront a problem much larger than the challenges of a new technology,'' MacCarthy \cite{MacCarthy}, who ``describe[d] and assesse[d] various group and individual statistical standards of fairness, including the mathematical conflict between the two that requires organizations to choose which measure to satisfy,'' and Hellman \cite{hellman}, who has explored the role of ``parity in the ratio of false positives to false negatives.''
Against this backdrop, the present paper offers several new contributions. First, as many previous publications each only consider a relatively small subset of fairness measures, differences in terminology and definitions can make comparisons among a broader group of measures difficult to perform. To address this, we describe some of the most widely cited fairness measures using a common mathematical and notational framework. Second and more substantively, we derive and discuss a set of mathematical relationships that facilitate comparisons among these metrics. The remainder of this paper is organized as follows. Section 2 presents metrics using a common mathematical framework and also aims to address some of the terminology variations across authors. Section 3 details the relationships between various metric pairs, including consideration of conditions under which they become mutually incompatible as well as trade-offs involved in selecting one metric over the other. Section 4 offers a discussion and conclusions.
\section{Fairness metrics}
\subsection{Definitions}
In the initial portion of the discussion herein we focus on binary predictions and their relationship to binary outcomes. We subsequently consider metrics that generate a continuous-valued score. In the binary context we assume that prediction involves a classifier that generates a binary prediction $\widetilde{y}$ based on a feature vector $x$ containing the data corresponding to a particular individual. If the distribution of $\widetilde{y}$ over $x$ is the same as that of the binary outcome $y$, we have perfect prediction. Hence, given a sufficiently large sample, the fairness and other attributes of the predictor can be evaluated by comparing predictions $\widetilde{y}$ with outcomes $y$.
We denote individuals for whom $y=1$ as being members of the ``positive'' class and individuals for whom $y=0$ as being members of the ``negative'' class. For instance, if an algorithm predicts whether students will pass a test, $y$ will be 1 for the students who pass (and who are therefore members of the positive class) and 0 for students who fail (and are thus members of the negative class). While in this scenario the ``positive'' outcome $y=1$ corresponds to the desirable outcome, this will not always be the case. To take another example, in evaluating whether a parolee commits a new crime within a given time frame, $y=1$ can be used to denote the ``positive'' (but obviously undesirable) outcome that the parolee has committed a new crime.
We also assume that the dataset can be divided
into different groups based on attributes such as race, gender, etc., that may be of interest when evaluating whether an algorithm is biased. While there can be any number of such groups in different contexts, we will restrict our discussion to scenarios in which it is possible to identify two groups of individuals, with group membership indicated using a binary variable $G$. Finally, for some of the metrics we discuss, it will be necessary to consider
a real-valued score $s$ $\in [0,1]$ that is computed for each individual which can optionally then be subject to thresholding to generate a binary prediction $\widetilde{y}$.
In sum, we use the following notation:
\begin{itemize}
\itemsep-2pt
\item $x$: feature vector
\item $G$: binary group inde
\item $\widetilde{y}$: binary prediction
\item $y$: binary outcome
\item $s$: classifier score
\end{itemize}
We address some of the metrics that have received significant recent attention in the literature and/or that we believe present opportunities for more detailed comparative analysis. We do not claim to consider all possible fairness metrics (for example, we do not address \textit{fairness through awareness} as proposed by Dwork \textit{et al.}\@ \cite{Dwork:2012:FTA:2090236.2090255}).
To help clarify the discussion, in what follows we will sometimes refer to the following example involving loans. Consider two groups of people, which we will denote as the orange group (denoted group 0) and the blue group (group 1). The orange group has 60 members while the blue group has 40 members. If all the members of both groups were given loans, 40 members of the orange group and 20 members of the blue group would repay on time---these are the positive outcomes. However, among these 60 positive outcomes, we assume that the model predicted that only 28 of the orange group and 8 of the blue group will repay. These are the true positives for the two groups. Thus, the model correctly predicted the positive outcome of 36 individuals.
Similarly, if all members of both groups were given loans, 20 members of the orange group and 20 members of the blue group would default on the loan---these are the negative outcomes. From among these 40 negative outcomes, we assume that the model correctly predicted 12 of the defaulters in the orange group and 16 of the defaulters in the blue group. These are the true negatives for the two groups. In the aggregate, the model correctly predicted the negative outcome for 28 individuals. The table below summarizes these predictions and outcomes. We also add that, to make the illustration of the concepts more tractable, this example and the other numerical examples in this paper use small sample sizes (and in doing so implicitly assume statistical significance despite those small sample sizes). Of course, in a real scenario, to make reliable inferences on statistical outcomes and therefore on fairness assessments based on those outcomes, the sample sizes would need to be much larger.
\begin{table}[H]
\centering
\caption{Illustrative Example}
\begin{tabular}{|c|L|L|L|}\hline
& Orange Group & Blue Group & Total \\\hline
True positives & 28 & 8 & 36\\\hline
False negatives & 12 & 12 & 24 \\\hline
True negatives & 12 & 16 & 28\\\hline
False positives & 8 & 4 & 12 \\ \hline
Positives & 40 & 20 & 60 \\\hline
Negatives & 20 & 20 & 40 \\ \hline
Total & 60 & 40 & 100\\\hline
\end{tabular}
\end{table}
\subsection{Equalized odds and equality of opportunity}
A predictor satisfies \textit{equalized odds} if both the true positive rate (TPR) and (separately) the false positive rate (FPR) are the same across groups.
More formally, equalized odds requires that the group-specific TPR satisfy $p(\widetilde{y}=1|y=1,G=0) = p(\widetilde{y}=1|y=1,G=1)$ and that the group-specific FPR satisfy $p(\widetilde{y}=1|y=0,G=0) = p(\widetilde{y}=1|y=0,G=1)$. \cite{NIPS2016_6374} Since, the metric demands that the error rates be the same across groups, Chouldechova \cite{1703.00056} describes this metric using the term ''error rate balance''.
In the example, for the orange and blue groups, the TPRs are $0.7$ and $0.4$ respectively, and the FPRs are $0.2$ and $0.4$ respectively. Thus, the example fails to satisfy equalized odds.
It is also worth noting that a related and less stringent metric, \textit{equality of opportunity} \cite{NIPS2016_6374}, can be defined by requiring only that the TPR be equal across groups, with no requirement imposed on the FPR. Thus, equalized odds implies equality of opportunity, though not vice versa.
\\
\subsection{Statistical parity}\label{sec:statparity}
\textit{Statistical parity} \cite{1703.00056}, \cite{Verma:2018:FDE:3194770.3194776} (sometimes referred to as \textit{group fairness} \cite{Dwork:2012:FTA:2090236.2090255} or \textit{demographic parity} \cite{mehrabi2019survey}, \cite{NIPS2017_6995}) is achieved when members of both groups are predicted to belong to the positive class at the same rate. Mathematically, this means satisfying $p(\widetilde{y}=1|G=0)=p(\widetilde{y}=1|G=1)$.
Notably, this metric gives no consideration to the outcomes $y$. Therefore, when the base rates $p(y|G)$ differ across the two groups, statistical parity rules out the perfect predictor.
In the orange/blue group example, 36 of the 60 orange group members and 12 of the 40 blue group members were predicted to belong to the positive class. Since $p(\widetilde{y}=1|G=0)=36/60 = 0.6$ is not the same as $p(\widetilde{y}=1|G=1) =12/40 = 0.3$, the example does not satisfy statistical parity.
\subsection{Predictive parity}
Consistent with Chouldechova \cite{1703.00056}, Verma \textit{et al.}\@ \cite{Verma:2018:FDE:3194770.3194776}, and MacCarthy \cite{MacCarthy} we consider that \textit{predictive parity} is satisfied when the positive predictive value(PPV) is the same for both groups. PPV is defined as the probability that individuals \textit{predicted} to belong to the positive class \textit{actually} belong to the positive class. Mathematically, predictive parity therefore requires $p(y=1|\widetilde{y}=1,G=0) = p(y=1|\widetilde{y}=1,G=1)$.
We note that some authors
define predictive parity in a more constrained manner, requiring not only parity for PPV, but also for its counterpart, negative predictive value (NPV), which requires additionally satisfying $p(y=0|\widetilde{y}=0,G=0) = p(y=0|\widetilde{y}=0,G=1)$. Mayson \cite{Mayson} uses the term ``overall predictive parity'' to describe a predictor with equality across groups in both PPV and NPV, while Berk \textit{et al.}\@ \cite{doi:10.1177/0049124118782533} call this ``conditional use accuracy equality''.
In the loan example, the model predicted a total of 48 members across both groups to belong to the positive class. However, only 36 out of these were correct predictions; the rest were false positives. Hence, the overall PPV is $36/48=0.75$. Taken separately, the model predicted a total of 36 members of the orange group and a total of 12 members of the blue group to belong to the positive class. From these predictions, however, only 28 of the orange group and 8 of the blue group were correct. Therefore, the PPV for the orange group is $p(y=1|\widetilde{y}=1,G=0) =28/36 = 7/9 = 0.77$ and for the blue group is $p(y=1|\widetilde{y}=1,G=1)=8/12 = 0.66$. Since these values differ, the model from our example does not satisfy predictive parity.
The fairness metrics discussed above can be evaluated using knowledge only of binary predictions and outcomes. By contrast, we now discuss a set of metrics involving explicit generation of a continuous-valued score $s$. Optionally, the score can serve as the input to a thresholding function that outputs a binary prediction, though scores can also be used directly, without any thresholding.
\subsection{Calibration}
An algorithm is \textit{calibrated} if for all scores $s$, the individuals who have the same score have the same probability of belonging to the positive class, regardless of group membership. \cite{1703.00056} \cite{Corbett-Davies:2017:ADM:3097983.3098095} Mathematically, this is expressed through $p(y=1|S=s,G=0) = p(y=1|S=s,G=1)$
\ This metric has been termed \textit{test-fairness} by Chouldechova \cite{1703.00056}, Verma \textit{et al}.\@ \cite{Verma:2018:FDE:3194770.3194776}, and Mehrabi \textit{et al.}\@ \cite{mehrabi2019survey} and as \textit{matching conditional frequencies} by Hardt \textit{et al.}\@ \cite{NIPS2016_6374}
There is another related metric termed \textit{well-calibration} \cite{Verma:2018:FDE:3194770.3194776} or \textit{calibration within groups} \cite{1609.05807}\cite{doi:10.1177/0049124118782533} that imposes an additional, more stringent condition. In order for a model to be well-calibrated (or to have calibration within groups), individuals assigned score \textit{s} must have probability $s$ of belonging to the positive class.
If this condition is satisfied, then test-fairness will also automatically be satisfied, though the reverse does not hold.
The difference between calibration and well-calibration is simply one of mapping; the scores of a calibrated predictor can, using a suitable transformation, be converted to scores satisfying well-calibration.
\subsection{Balance for positive/negative class} \label{balance}
Kleinberg \textit{et al.}\@ \cite{1609.05807} have noted that when the average score \textit{s} for all individuals constituting the group-specific positive class is the same for both groups of interest, it can be said that there exists \textit{balance for the positive class}. Similarly, \textit{balance for the negative class} is satisfied when the average score \textit{s} for members of the negative class are equal, regardless of group membership. Mathematically this is expressed in terms of expected values. For the negative class, balance requires $\EX[s|y=0,G=0] = \EX[s|y=0,G=1]$, and for the positive class balance requires $\EX[s|y=1,G=0] = \EX[s|y=1,G=1]$.
As Pleiss \textit{et al.}\@ \cite{Pleiss:2017:FC:3295222.3295319} have explained, this can be viewed as a generalization of the equalized odds metric to non-binary cases. To see this, note that when the score $s$ can take on only the two values 0 and 1, the score itself is the prediction and the term $\EX[s|y=0,G]$ then represents the false positive rate and the term $\EX[s|y=1,G]$ represents the true positive rate. And, as noted above, under equalized odds the TPR and FPR are equal across groups.
\section{Comparisons and trade-offs between metrics}
Different fairness metrics formalize varying intuitive notions of fairness. This raises the question of the conditions under which more than one metric can be simultaneously satisfied, and relatedly, the ways in which different metrics might be in tension.
We will analyze metrics under the assumption that the ``base rate'' differs across groups. The base rate of a group is the ratio of people in the group who belong to the positive class ($y=1$) to the total number of people in that group. Thus, having non-equal base rates across groups means that $p(y=1|G=0) \neq p(y=1|G=1)$.
In the subsequent discussion, to simplify the equations we will use the following terms as defined here:
\begin{itemize}
\item $\text{TPR}_g=p(\widetilde{y}=1|y=1,G=g), g \in \{0,1\}$ --- the group-specific true positive rates.
\item $\text{FPR}_g=p(\widetilde{y}=1|y=0,G=g), g \in \{0,1\}$ --- the group-specific false positive rates.
\item $\text{PPV}_g=p(y=1|\widetilde{y}=1,G=g), g \in \{0,1\}$ --- the group-specific positive predictive value.
\end{itemize}
\subsection{Statistical parity, equalized odds and predictive parity}
Trade-offs among statistical parity, equalized odds and predictive parity have received significant attention in algorithmic fairness literature under a variety of formulations. Chouldechova \cite{1703.00056} articulates the tradeoffs between predictive parity and equalized odds empirically. Kleinberg \textit{et al.}\@ \cite{1609.05807}, while not directly referring to these terms, give a closely related result, writing that ``the calibration condition and the balance conditions for the positive and negative classes" are ``in general incompatible with each other; they can only be simultaneously satisfied in certain highly constrained cases.'' Berk\textit{ et al.}\@ \cite{doi:10.1177/0049124118782533} discuss incompatibility among metrics by taking several cases and examining the trade-offs in those scenarios.
We offer a common mathematical framework to examine trade-offs among statistical parity, equalized odds and predictive parity. We provide proofs regarding the combination of all three of these metrics and also explore conditions under which it may be possible to simultaneously satisfy two metrics. To provide an initial framing, it is interesting to note that using the basic probability relation $p(A,B) = p(A|B)p(B) = p(B|A)p(A)$, the respective probability distributions associated with each of these three metrics can be expressed as follows:
\begin{flalign} \label{eq:1}
\begin{aligned}
p(y,\widetilde{y}|G) = \underbrace{p(y|\widetilde{y},G)}_{\text{Predictive Parity}} \ \times \ \underbrace{p(\widetilde{y}|G)}_{\text{Statistical Parity}} = \underbrace{p(\widetilde{y}|y,G)}_{\text{Equalized Odds}} \ \times \ \ \underbrace{p(y|G)}_{\text{Base Rate}}
\end{aligned}
\end{flalign}
\subsubsection{All three?}
This section considers the feasibility of satisfying all three metrics under the assumption of unequal base rates. We start by assuming a predictor that satisfies both statistical parity and equalized odds, and then examining if it can also satisfy predictive parity. From equation \ref{eq:1}, we have:
\begin{flalign} \label{eq:2}
\begin{aligned}
p(y=1|\widetilde{y}=1,G) = \frac{p(\widetilde{y}=1|y=1,G) \times p(y=1|G)}{p(\widetilde{y}=1|G)}
\end{aligned}
\end{flalign}
Since the predictor satisfies equalized odds, the TPR must be the same across groups and therefore, we denote $\text{TPR}_0 = \text{TPR}_1 = \text{TPR}$.
And since the predictor by definition also satisfies statistical parity, $p(\widetilde{y}=1|G=0)=p(\widetilde{y}=1|G=1)=p(\widetilde{y}=1)$ Imposing these conditions and taking the difference of the PPV values of the two groups gives:
\begin{flalign} \label{eq:3}
\begin{aligned}
p(y=1|\widetilde{y}=1,G=0) - p&(y=1|\widetilde{y}=1,G=1) = \frac{\text{TPR}[p(y=1|G=0) - p(y=1|G=1)]}{p(\widetilde{y}=1)}
\end{aligned}
\end{flalign}
Predictive parity requires that the PPV be equal across both groups, and therefore that the difference on the left side of the above equation be zero, which in turn can only occur when the base rates across the two groups are equal as indicated by the right side of the equation. Note that equalized odds requires \textit{both} the TPR and the FPR to be the same. However, by demonstrating the incompatibility of just TPR in this section, we provide a sufficient proof for the incompatibility of equalized odds in the given scenario.
Thus, when the two groups have unequal base rates, satisfying all three of statistical parity, predictive parity and equalized odds is impossible. This remains true even if the predictor is perfect, as a perfect predictor can not (when the base rates are unequal) satisfy statistical parity.
\subsubsection{Statistical parity and predictive parity}
We now consider conditions under which a predictor can satisfy both statistical and predictive parity.
Recall that when statistical parity holds, we have $p(\widetilde{y}=1|G=0) = p(\widetilde{y}=1|G=1)=p(\widetilde{y}=1)$.
Taking the difference in PPV across the two groups gives:
\begin{flalign}\label{eq:4}
\begin{aligned}
p(y=1|\widetilde{y}=1,G=0) - p&(y=1|\widetilde{y}=1,G=1) = \frac{\text{TPR}_0p(y=1|G=0) - \text{TPR}_1p(y=1|G=1)}{p(\widetilde{y}=1)}
\end{aligned}
\end{flalign}
Under predictive parity the left side of the equation must be zero, which in turn requires that
the ratio of the true positive rates of the two groups
be the reciprocal of the ratio of the base rates, i.e.:
\begin{flalign}\label{eq:5}
\begin{aligned}
\frac{\text{TPR}_0}{\text{TPR}_1} = \frac{p(\widetilde{y}=1|y=1,G=0)}{p(\widetilde{y}=1|y=1,G=1)} = \frac{p(y=1|G=1)}{p(y=1|G=0)} = \frac{\text{Base Rate of Group 1}}{\text{Base Rate of Group 0}}
\end{aligned}
\end{flalign}
Thus, while statistical and predictive parity can be simultaneously satisfied even with different base rates, the utility of such a predictor is limited when the ratio of the base rates differs significantly from 1, as this forces the true positive rate for one of the groups to be very low.
As mentioned above, the definition of predictive parity used here, consistent with \cite{1703.00056}, only requires different groups to have the same PPV. However, if we were to consider the ``overall predictive parity'' \cite{Mayson}, and require a predictor to also have the same NPV across groups, the system would be overconstrained and it would not generally be possible to simultaneously satisfy statistical parity and predictive parity.
\subsubsection{Equalized odds and predictive parity}
Chouldechova (2017) \cite{1703.00056} observes that "predictive parity is incompatible with error rate balance when prevalence differs across groups," (p. 5). (As noted above, Chouldechova uses ``error rate balance'' to describe what we refer to here as equalized odds.) We explore this incompatibility
in more detail. As before when equalized odds and predictive parity are satisfied, we have $\text{TPR}_0=\text{TPR}_1$, $\text{FPR}_0=\text{FPR}_1$, and $\text{PPV}_0=\text{PPV}_1$. Noting that
\[p(\widetilde{y}=1|G) = \sum_y p(\widetilde{y}=1|y,G) p(y|G) = p(\widetilde{y}=1|y=1,G)p(y=1|G) + p(\widetilde{y}=1|y=0,G)p(y=0|G)\]
\begin{flalign}\label{eq:6}
\begin{aligned}
\implies p(\widetilde{y}=1|G) = \text{TPR}_0p(y=1|G) + \text{FPR}_0p(y=0|G)
\end{aligned}
\end{flalign}
and considering equation \ref{eq:1}, we can write:
\[p(\widetilde{y}=1|y=1,G=0)p(y=1|G=0) = p(y=1|\widetilde{y}=1,G=0) [\text{TPR}_0 p(y=1|G=0) + \text{FPR}_0p(y=0|G=0)]\]
\[ \implies \text{TPR}_0p(y=1|G=0) = \text{PPV}_0 [\text{TPR}_0 p(y=1|G=0) + \text{FPR}_0p(y=0|G=0)] \]
\[ \text{TPR}_0 p(y=1|G=0) = \text{PPV}_0 [\text{TPR}_0 p(y=1|G=0) + \text{FPR}_0(1-p(y=1|G=0))] \]
\begin{flalign}\label{eq:7}
\begin{aligned}
\implies p(y=1|G=0) = \frac{\text{PPV}_0\text{FPR}_0}{\text{PPV}_0\text{FPR}_0 + (1-\text{PPV}_0)\text{TPR}_0}
\end{aligned}
\end{flalign}
\begin{flalign}\label{eq:11}
\begin{aligned}
\text{Likewise, }p(y=1|G=1) = \frac{\text{PPV}_1\text{FPR}_1}{\text{PPV}_1\text{FPR}_1 + (1-\text{PPV}_1)\text{TPR}_1}
\end{aligned}
\end{flalign}
But since $\text{TPR}_0=\text{TPR}_1$, $\text{FPR}_0=\text{FPR}_1$ and $\text{PPV}_0=\text{PPV}_1$, equations 7 and 8 will be identical, so base rates for groups 1 and 2 will be the same: $p(y=1|G=0) = p(y=1|G=1)$. Hence, in the absence of perfect prediction, the base rates have to be equal for both equalized odds and predictive parity to simultaneously hold. When perfect prediction is achieved, equations \ref{eq:7} and \ref{eq:11} take on the indefinite form $0/0$ so therefore do not convey anything definitive about base rates in that scenario.
We also note that the metric equal opportunity (a less strict counterpart to equalized odds that requires only equal TPR across groups) is compatible with predictive parity. This is evident from equations \ref{eq:7} and \ref{eq:11} when the condition $\text{FPR}_0=\text{FPR}_1$ is removed, thereby allowing equalized opportunity and predictive parity to be simultaneously satisfied even with unequal base rates. However, achieving this condition with unequal base rates will require that the FPR differs across the groups. When the difference between the base rates is large, the variation between group-specific FPRs may have to be significant which may reduce suitability for some applications. Hence, while equal opportunity and predictive parity are compatible in the presence of unequal base rates, practitioners should consider the cost (in terms of FPR difference) before attempting to simultaneously achieve both. A similar analysis is possible when we considering parity in negative predictive value instead of positive predictive value, i.e. equal opportunity and parity in NPV are compatible, but only at the cost of variation between group-specific true negative rates (TNRs).
\subsubsection{Equalized odds and statistical parity}
We now consider if a predictor can simultaneously satisfy equalized odds and statistical parity. As before, for $\text{TPR} = \text{TPR}_0 = \text{TPR}_1$ (equal TPR) and $\text{FPR} = \text{FPR}_0 =\text{FPR}_1$ (equal FPR):
\[ p(\widetilde{y}=1|G)= \text{TPR}[p(y=1|G)] + \text{FPR}[p(y=0|G)]\\ \]
\begin{flalign}\label{eq:8}
\begin{aligned}
\implies p(\widetilde{y}=1|G=0)-p(\widetilde{y}=1|G=1) = &\text{TPR}[p(y=1|G=0)-p(y=1|G=1)] \\ +& \text{FPR}[p(y=0|G=0)-p(y=0|G=1)]\\
\end{aligned}
\end{flalign}
\begin{flalign}\label{eq:8}
\begin{aligned}
\implies p(\widetilde{y}=1|G=0)-p(\widetilde{y}=1|G=1) = (\text{TPR}-\text{FPR})[p(y=1|G=0)-p(y=1|G=1)]\\
\end{aligned}
\end{flalign}
Statistical parity requires the left side of equation \ref{eq:8} to be zero. For the equation to hold, this means the right side must also be zero, which can only occur when either $\text{TPR}=\text{FPR}$ or $p(y=1|G=0)=p(y=1|G=1)$. The latter case, however, violates the assumption that the base rates are different. Therefore to have both statistical parity and equalized odds, the only possibility is to have $\text{TPR}=\text{FPR}$, i.e. the false positive rate and the true positive rate have to be equal. Thus, while simultaneously achieving statistical parity and equalized odds is mathematically possible, it is not particularly useful since the goal is typically to develop a predictor in which the TPR is significantly higher than the FPR.
\subsection{Predictive parity and calibration}
There has been some confusion in the literature regarding the relationship between these two metrics. Chouldechova \cite{1703.00056} has correctly mentioned that ``While predictive parity and calibration look like very similar criteria, well-calibrated scores can fail to satisfy predictive parity at a given threshold,'' However, in other papers, the discussion on these two metrics sometimes gives less attention to the specifics of how they relate than we think is merited.
Given this backdrop, in the present section we explain the difference between predictive parity and calibration
with a mathematical derivation and an example.
It is possible to view calibration as a generalization of predictive parity to the non-binary setting.
The score $s$ discussed in relation to calibration is generally continuous-valued. However, in the special case in which it is limited to the two values 0 and 1 and therefore becomes the prediction itself, achieving calibration is the same as achieving equality across groups in both PPV and NPV. Thus, this satisfies both predictive parity (due to equality across groups in PPV) as well as ''overall predictive parity'' (due to equality across groups in both PPV and NPV).
Of course, in general the score $s$ is not binary. A continuous-valued score can be binarized through a thresholding operation to generate
a binary prediction $\widetilde{y}$. However, it is \textit{not} the case that thresholding a calibrated score in this manner necessarily leads to predictive parity.
To prove this, consider a threshold $s_{th} \in [0,1]$, such that $\forall s>s_{th}, \widetilde{y}=1$ and $\widetilde{y}=0$ otherwise. Hence, the distribution relevant to predictive parity $p(y=1|\widetilde{y}=1,G)$ can be expressed
$p(y=1|s>s_{th},G)$. Using this we can write:
\begin{flalign}
\begin{aligned}
p(y,s>s_{th}|G) = \int_{s_{th}}^1 \underbrace{p(y|s,G)}_{\text{calibration term}}p(s|G)ds
\end{aligned}
\end{flalign}
\begin{flalign}\label{eqn:12}
\begin{aligned}
\implies \underbrace{p(y|s>s_{th},G)}_{\text{predictive parity term}} = \frac{\int_{s_{th}}^1 p(y|s,G)p(s|G)ds}{\int_{s_{th}}^1 p(s|G)ds}
\end{aligned}
\end{flalign}
The above equation relates predictive parity to calibration, showing that even
when the calibration term $p(y|s,G)$ is the same for both groups, the probability distribution of the score, expressed in equation \ref{eqn:12} through $p(s|G)$, can vary across groups in a way that causes predictive parity not to be satisfied.
To make this more intuitive, we will consider a special case where there are only two score values $s_1$ and $s_2$ above the threshold $s_{th}$ such that $p(s|G) \neq 0$. In other words, all individuals who receive risk scores above the threshold have the possibility of receiving one of only two scores, $s_1$ or $s_2$. Hence, $p(s>s_{th}|G) = p(s=s_1|G)+p(s=s_2|G)$.
Under this special case equation \ref{eqn:12} reduces to:
\begin{flalign}\label{eqn:13}
\begin{aligned}
p(y=1|\widetilde{y}=1,G) = \frac{p(y=1|s=s_1, G)p(s=s_1|G) + p(y=1|s=s_2, G)p(s=s_2|G)}{p(s=s_1|G)+p(s=s_2|G)}
\end{aligned}
\end{flalign}
Using this scenario, consider an example in which we have 100 people in each of two groups: orange and blue (this is a new example, unrelated to the example using orange and blue groups introduced earlier in the paper). Consider further an algorithm that only gives one of three possible scores (0.25, 0.5 or 0.75) to every individual's loan application. Suppose that scores are being binarized using a threshold of 0.49, such that any individual with a score above 0.49 is deemed to belong to the positive class. In this example, this would mean there are two possible scores (0.5 and 0.75) that can lead to a positive prediction. This is illustrated in Table \ref{tab:ppvscal} given below.
\begin{table}[H]
\centering
\caption{Predictive Parity and Calibration Example}
\begin{tabular}{|c|L|L|L|}\hline
Score & Orange Group & Blue Group & Prediction after threshold with $s_{th}=0.49$ \\\hline
0.25 & 40 (16) & 40 (16) & Negative\\\hline
0.5 & 20 (10) & 40 (20) & Positive\\\hline
0.75 & 40 (30) & 20 (15) & Positive \\\hline
Total & 100(56) & 100(51) & \\\hline
\end{tabular}
\label{tab:ppvscal}
\end{table}
The first column represents the score that the model assigned. In the second and third columns, the numbers outside the parentheses convey the number of people in the group assigned that score. The numbers in parentheses represent the number of people from those assigned that score who actually belong to the positive class.
In this example the predictor is calibrated, since given a score, the fraction of people who actually belong to the positive class is independent of the group. For example, for score 0.5, $10/20=0.5=50\%$ of the people in the orange group with that score and $20/40=0.5=50\%$ of the people in the blue group with that score belong to the positive class.
Does this model satisfy predictive parity? Choosing 0.49 as the threshold gives a total of 60 positive predictions for both the orange group and the blue group. However, of the people with scores greater than 0.49, only 40 members in the orange group and 35 members of the blue group are actually in the positive class, resulting in a PPV of $40/60=0.66$ for the orange group and $35/60=0.583$ for the blue group. Thus, while the predictor is calibrated, choosing a threshold of 0.49 does not lead to a set of binary predictions that satisfy predictive parity.
It is also interesting to note that if all persons who had a score of 0.25 are instead given a score of 0.4, the model will not only be calibrated (because, as before, people with the same score have the same probability of belonging to the positive class) but also \textit{well}-calibrated (because, due to this change in scoring, for all scores the score itself would give the probability of belonging to the positive class). However, this change in score would have no impact on the thresholding example above, illustrating that even a well-calibrated model does not, after applying a threshold to produce binary predictions, necessarily satisfy predictive parity.
This discussion can be further generalized. Consider the comparison between statistical parity and calibration given by Kleinberg \textit{et al.}\@ \cite{1609.05807}. Kleinberg \textit{et al.}\@ give an alternate definition of statistical parity which is independent of the choice of threshold, defining it as the condition in which both groups have the same average score.
They then prove that when there are unequal base rates and imperfect prediction, statistical parity is incompatible with well-calibration.
In one sense, the Kleinberg \textit{et al.}\@ definition of statistical parity can be understood as a generalization of the binary case. This is because in the special case where the score itself is binary and represents the prediction, the Kleinberg \textit{et al.}\@ definition reduces to the common (i.e., binary) definition of statistical parity that we have provided in section \ref{sec:statparity}. However, when there is \textit{both} a score $s$ \textit{and} a binary prediction $\widetilde{y}$ obtained by thresholding the score, the fact that there was statistical parity prior to thresholding in accordance with the definition from Kleinberg \textit{et al.}\@ does not necessarily imply that there will be statistical parity of the binary predictions in accordance with \ref{sec:statparity}. Stated another way, if, prior to thresholding, the average score $s$ is the same across two groups, it does not necessarily follow that thresholding will produce predictions $\widetilde{y}$ in which $p(\widetilde{y}|G=0)=p(\widetilde{y}|G=1)$.
Under the standard definition of statistical parity (in \ref{sec:statparity})
it is possible to simultaneously satisfy both statistical parity and calibration.
Consider, again, the example in table \ref{tab:ppvscal}, where despite different base rates, when a threshold of 0.49 is applied to binarize the calibrated scores, it also satisfies statistical parity
(for both groups of 100, 60 individuals are predicted to belong to the positive class). The base rates are also different since a total of 56 people from the orange group and 51 people from the blue group belong to the positive class. Again, as before, if all persons who had a score of 0.25 were instead given a score of 0.4, the model will be well-calibrated and still satisfy statistical parity for the threshold of 0.49. Also, the prediction is clearly imperfect. Thus, statistical parity (as we have defined it in section \ref{sec:statparity} above) is not necessarily incompatible with having calibration (and well-calibration) even when the base rates are different and there is imperfect prediction.
This underscores the importance of being attentive to the difference between fairness metrics designed for use in relation to scores $s$ and those intended for use with binary predictions. While it is straightforward to convert scores to binary values through thresholding, the ease of the conversion masks important complexities regarding the extent to which fairness metrics might be met after the thresholding process. In the example above, the scores were calibrated, and upon application of a threshold of 0.49, the binary predictions exhibited statistical parity. However, the same underlying distribution of scores, had they been subject to a threshold of 0.55, would have yielded predictions \textit{without} statistical parity. More
generally, we believe that there is room for---and a need for---fairness metrics in relation to both continuous scores $s$ as well as binary predictions $\widetilde{y}$. But the benefits offered by having a greater number of tools for examining fairness must be balanced with an awareness of the issues that can arise when working across these two domains.
\section{Discussion and conclusions}
Several conclusions arise from the discussion above. With respect to metrics such as statistical parity, equalized odds and predictive parity that evaluate fairness by comparing binary predictions with binary outcomes, pairwise combinations of metrics can be simultaneously satisfied only under very limited conditions, if at all. For example, when the base rates are different, satisfying both statistical parity and predictive parity requires that the ratio of group-specific true positive rates be the inverse of the ratio of the base rates. When the ratio of the base rates does not deviate too far from 1, this constraint can be met while also preserving high true positive rates for both groups. The question of how far a deviation from 1 in the base rate ratio (and therefore in the inverse of the TPR ratio) would still be acceptable would of course be context dependent. For large (or small) base rate ratios, the resulting predictor would necessarily have a low true positive rate for one of the groups. We also showed that equalized odds and predictive parity are incompatible when the base rates across two groups differs. In addition, we showed that, given unequal base rates, equalized odds and statistical parity can only be simultaneously met in the rather impractical case where the true positive and false positive rates are equal. The above results illustrate that, for unequal base rates, a given pair of metrics can be 1) mathematically incompatible, 2) mathematically compatible but under constraints that are problematic from a policy standpoint, or 3) mathematically compatible under constraints that are consistent with positive policy outcomes.
With respect to more generalized predictors that generate a (non-binary) score, we explored the relationship between calibration (computed with respect to score that can in general be continuous) and predictive parity when that score is subject to thresholding to generate a binary prediction. We observed the value of utilizing fairness metrics designed for use in relation to scores, while also emphasizing some of the complexities involved in working across continuous and binary domains. In particular, we noted that calibrated scores, after thresholding, may still generate predictions that satisfy statistical parity, though threshold choice can play an important role in determining whether or not such conditions are satisfied.
In closing, we note that in addition to the above considerations, fairness metric selection can also be guided by the observability of each statistic in practice. For example, in the context of loan approvals, loans will typically only be given to the subset of applicants who are predicted to repay. When this occurs, it will be impossible to observe statistics such as false negatives, true negatives, negative predictive value, true positive rate, and true negative rate. By contrast, the positive predictive value will be readily observable, suggesting that a metric such as predictive parity would be easier to evaluate in practice.
\bibliographystyle{ieeetran}
|
2,869,038,154,756 | arxiv | \section*{\mbox{ } }\end{document}
\section{Introduction} \label{sec:intro}
We present spectrally resolved imaging of the \ciiw\ spectral line
from the luminous \sgrb\ region in our Galactic center. Understanding
the distribution of \cii\ intensity and its relationship with star
formation in galactic nuclei is important for understanding our
galaxy, as well as for interpreting observations of galaxies
throughout the universe. Velocity-resolved spectroscopy allows
separation of different physical components and investigation of
dynamical and physical states of the interstellar medium, and
structural localization at the sub-beam level.
The \cii\ fine structure line from singly-ionized carbon is a powerful
probe of the interstellar medium. A major cooling line, it is often
the brightest spectral indicator of star formation within our Galaxy
and in others \citep[e.g.,][and references therein]{crawford85,
stacey91, boselli02, delooze14, pineda14, herreracamus15}. Even at
energies below the hydrogen ionization edge, soft ultraviolet
radiation ionizes atomic carbon in a range of environments: neutral
diffuse material, the warm ionized interstellar medium, and warm and
dense molecular gas. Collisions with electrons, atomic hydrogen, and
molecular hydrogen populate the transition's upper
level. Photochemistry models (e.g., \citealt{tielens85, vandishoeck88,
wolfire90, sternberg95, kaufman99}) support interpretation of line
intensities as functions of physical conditions. Extreme conditions
in ultra-luminous galactic nuclei (ULIRGs) suppress \cii\ emission,
however (\citealt{fischer99} and references therein).
\sgrb\ is one of the most luminous regions within the Galaxy's Central
Molecular Zone (CMZ), a region of particularly dense, warm, and
turbulent molecular clouds in the inner few hundred parsecs of the
Galaxy (reviews by \citealt{guesten86} and \citealt{morris96} contain
overviews of the CMZ and its basic properties; \citealt{mills17}
provides recent updates). \sgrb\ contains extended emission regions
and high-density stellar clusters embedded in dense envelopes, making
it an excellent model for close examination of star formation within
galactic nuclei. Its proximity (we use 8\,kpc; \citealt{reid14} find
the distance to \sgrbt\ is 7.9\,kpc, and \sgrbt\ appears to be in
front of the main \sgrb\ cloud) allows detailed studies at many
wavelengths. The \sgrb\ region and some of its components were first
studied in single-dish radio continuum mapping of the Galactic center
\citep{kapitzky74, mezger76, downes80}. These observations identified
three main sources: G\,0.5--0.0 (\sgrbo), a bright peak \gsix, and a
yet brighter region G\,0.7--0.0 (\sgrbt). The designations were based
simply on radio continuum brightness peaks, and did not necessarily
identify physically isolated or connected regions. Subsequent
observations at higher spatial resolutions showed that the individual
sources contain multiple components. Interferometric radio continuum
imaging \citep{mehringer92, lang10} showed arcs and bars of emission
across \sgrbo, compact sources in \gsix, and very bright compact cores
within \sgrbt. In this paper we associate the \gsix\ and \sgrbo\
regions with extended \cii-bright emission that surrounds those
sources and extends further to positive latitude, since these all
appear to be parts of the same general structure, while the \sgrbt\
cores \sgrbt(M) and (N), and perhaps (S), seem to be quite distinct
foreground objects.
While the brightest material surrounding the dense young stellar
clusters in \sgrbt's cores have been of great interest for many years,
the larger region encompassing \sgrbt, \sgrbo, \gsix, and their
diffuse envelope emission, has received much less attention.
\citet{goi04} identified extended far-IR fine structure line emission
around \sgrbo\ and \sgrbt\ in their {\em ISO} satellite cross-cuts of
the region with the LWS grating spectrometer. \citet{simpson18b} and
\citet{simpson21} imaged the \sgrbo\ region in mid-and far-IR lines to
investigate conditions in the ionized and neutral material.
\citet{santamaria21} used the {\em Herschel}-SPIRE FTS to make a broad
spectral survey of the region around the luminous \sgrbt\ star
formation cores. None of these observations had sufficient spectral
resolution to separate components within the beams by velocity,
however.
Single pointings toward the most luminous core, \sgrbt(M), found \cii\
and \oisw\ absorption from material from the Galactic plane in {\em
ISO} LWS Fabry-Perot observations \citep{goi04}, and {\em
Herschel}-HIFI made high spectral resolution observations in a much
smaller beam as part of the HEXOS key project that showed absorption
near the center of the emission line \citep{bergin10, moeller21}.
\citet{langer17} observed \cii\ emission in strip maps toward the edge
of the CMZ with {\em Herschel}-HIFI. While their observations had the
high spectral resolution needed to identify different physical
components, their strips close to \sgrb\ did not cross any of its main
clouds.
Here we report full two-dimensional spectroscopic imaging of the
\sgrb\ complex with 0.55\,pc and 1\,km\,s$^{-1}$\
resolutions. Section~\ref{sec:obs} describes our observations,
Sec.~\ref{sec:results} is a summary of the main results, and
Sec.~\ref{sec:discus} discusses implications for the Galactic center
and interpretation of observations of external galaxies.
To set a preliminary framework for discussion, Fig.~\ref{fig:sketch}
is a sketch of the physical structure of the region. We explore our
logic for reaching this view in Sec.~\ref{ssec:physstruct}, but in
brief: \cii\ velocities and its spatial distribution indicate that the
\sgrb\ region is a coherent structure that physically incorporates
\sgrbo\ and \gsix, then continues past the Galactic latitude of the
\sgrbt\ cores. The bright \sgrbt\ star formation cores appear to be
associated with the dark dust lane that crosses in front (closer to
Earth) of the main \sgrb\ cloud, which may or may not be physically
connected to the main \cii-emitting region. Lack of \cii\
self-absorption across the \sgrb\ body places the \cii\ and \oi\ at or
close to the surface of a larger background (further from Earth) cloud
or clouds visible in molecular and long-wavelength dust emission.
Loosely-distributed stars in and near the surface provide UV photons
to ionize, excite, and heat the gas and dust in the surface layer.
Sec.~\ref{sec:summary} provides brief summaries of our main findings.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.45\textwidth]
{sketch.pdf}
\caption{Schematic ``top view'' sketch of the \sgrb\ region showing
the relative positions of the region's components, to no particular
scale, cut by a plane notionally defined by the Earth and the
position-velocity cut line in Fig.~\ref{fig:sgrbregion}. The
separation between the dust filament and the background cloud is
unknown, as emphasized by a gap in the sketch, although the velocity
of the filament and cloud are close. Our perspective is from the
bottom of the page.
\label{fig:sketch}}
\end{figure}
Far-infrared observations have ties to both infrared and radio
conventions, so we quote intensities $I$ in units either energy-based
(e.g., $[\mathrm{erg~s^{-1}~cm^{-2}~sr^{-1}}]$) or velocity-integrated
brightness temperatures (e.g., [K\,km\,s$^{-1}$]) depending on context. The two
are related by
\begin{equation}
I = \int I_\nu \, \mathrm{d}\nu = \frac{2k}{\lambda^3}
\int T_B \, \mathrm{d}{\rm v}
\label{eq:intens}
\end{equation}
through the Rayleigh-Jeans expansion that defines brightness
temperature $T_B$ from specific intensity $I_\nu$, and the Doppler
relationship between frequency and velocity.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]
{ciiOverview.pdf}
\caption{
\ciiw\ integrated intensity image of the \sgrb\ region covering
14--120\,km\,s$^{-1}$. The image covers $\Delta \ell = 0.48\degr$ in Galactic
longitude and $\Delta b = 0.32\degr$ in longitude, centered at $\ell =
0.598\degr$, $b = -0.057\degr$. For a distance of 8\,kpc, these
angles correspond to $67.0 \times 44.7$\,pc, sampled in a 0.55\,pc
beam. The main \cii\ emission region is about 0.24\degr\ long and
0.11\degr\ wide (34\,pc$\times 15$\,pc). Contour levels show
intensities of 175 to 550\,K\,km\,s$^{-1}$\ in steps of 75\,K\,km\,s$^{-1}$\ within
14.1$^{\prime\prime}$\ FWHM beams. Circles with letter labels denote positions
for the spectra in Fig.~\ref{fig:spectralstack}; crosses mark the
positions of \sgrbt(N), (M), and (S). The dotted straight line shows
the path of the position-velocity cut in Fig.~\ref{fig:pv}. The
yellow ellipse near the top of the image and circle to the bottom
right show the extents of compact emission sources at $+124$ and
$-59$\,km\,s$^{-1}$\ (Sec.~\ref{sec:compact}).
\label{fig:sgrbregion}}
\end{figure*}
\section{Observations} \label{sec:obs}
Our results are part of our larger \cii\ spectral data cube
\citep{wholecmz} of the Central Molecular Zone. Our larger cube
covers $1.5\degr$ in longitude and $0.32\degr$ in latitude. The map
center for this entire CMZ project was \ra{17}{47}{24.60},
\dec{-28}{23}{16.3}. Data presented here were obtained during two
observing campaigns flying from New Zealand in 2017 June and July and
2018 June, yielding the most positive Galactic longitude ($\ell$)
$3 \times 2$ ``tiles'' that formed the basic structure of the imaging
project. Each tile covered $560 \times 560$ arcseconds. Together,
these tiles covered an area $\Delta\ell = 0.48\degr$ by
$\Delta b = 0.32\degr$ centered at $\ell = 0.598\degr$,
$b = -0.057\degr$.
Each tile was mapped in total power on-the-fly mode in rotated Right
Ascension-Declination frames corresponding to Galactic coordinates.
Individual integrations were recorded every 0.3\,sec, with spacing of
7$^{\prime\prime}$\ along the scan direction; final images were convolved to
15$^{\prime\prime}$\ FWHM Gaussian beams on a 6.9$^{\prime\prime}$\ rectangular grid. The rms
pointing accuracy was 2$^{\prime\prime}$. Tile offset positions allowed for one
row (column) of overlap between neighboring tiles. Each tile was
observed in orthogonal directions to reduce striping and residual
scanning structure. We used two reference ``off'' positions: a
relatively nearby position with some expected line contamination
during mapping (\ra{17}{47}{41.3}, \dec{-28}{35}{00}), and a distant
field far from the Galactic plane (\ra{17}{55}{03.9},
\dec{29}{23}{02}) to measure and correct the contamination in the
closer ``off'' position.
We observed the 1.901\,THz ($\lambda$157.74\,$\mu$m) \cii\
$^2P_{3/2} - ^2P_{1/2}$ and the 4.748\,THz ($\lambda$63.18\,$\mu$m) \oi\
\mbox{ } $^{3}P_{1} - ^{3}P_{2}$ fine structure transitions with the
upgraded German Receiver for Astronomy at Terahertz Frequencies
(\grt\footnote{\grt\ is a development by the Max-Planck-Institut f\"ur
Radioastronomie and the I.~Physikalisches Institut of the
Universit\"at zu K\"oln, in cooperation with the DLR Institut f\"ur
Optische Sensorsysteme.}) \citep{risacher18} on the Stratospheric
Observatory For Infrared Astronomy (SOFIA, \citealt{young12}). Rapid
imaging of \cii, with auxiliary data in \oi\ (the observing strategy
was geared to the brighter \cii\ line), was possible with \grt's dual
frequency focal plane array configured for parallel observations with
both the low frequency arrays (LFA) and high frequency array (HFA).
Main beam sizes were 14.1$^{\prime\prime}$\ for the LFA, and 6.3$^{\prime\prime}$\ for the
HFA. Both arrays use niobium nitride hot electron bolometer mixers,
pumped with either a solid-state source (LFA) or quantum cascade laser
(HFA) local oscillator. The LFA has seven dual-polarized pixels in a
hexagonal arrangement with 31.8$^{\prime\prime}$\ radial spacing around a central
pixel. The HFA has the same symmetry, centered on the LFA's central
pixel, with single-polarized pixels on 13.8$^{\prime\prime}$\ radial spacing. Fast
Fourier Transform Spectrometers (FFTS4G, updated from
\citealt{klein12}) produced spectra across the 0-4\,GHz intermediate
frequency bands, with 32k channels binned for 1\,km\,s$^{-1}$\ velocity
resolution.
Raw data were amplitude calibrated with the $kosma\_kalibrate$
software package (versions 2017.08 and 2018.07) following
\citet{guan12}. Line temperatures are on a $T_{\rm MB}$ scale, with
estimated absolute uncertainty below 20\%. Small but repeatable
receiver instabilities produced spectral baseline structure that we
removed using a family of baseline structures derived from differences
between nearby ``off'' spectra \citep{higgins11,
kester14, higgins21}. This method retains information in broad lines
and is superior to partly subjective low-order polynomial fits.
Further processing used the GILDAS packages CLASS and GREG.
\section{Results}\label{sec:results}
\subsection{\cii\ emission from the main body of the \sgrb\ cloud \label{ssec:ciispect}}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.45\textwidth]{cii_13co_c18o_body.pdf}
\caption{ Lineshape comparisons between \cii, $^{13}$C$^{}$O\ $J=2-1$, and $^{}$C$^{18}$O\
$J=2-1$ \citep{apexco} emission averaged over the \sgrb\ region
indicated by the lowest (175\,K\,km\,s$^{-1}$) contour in
Fig.~\ref{fig:sgrbregion}. Line of sight \cii\ absorption, or
emission in the reference position, accounts
for the lack of flux near 0\,km\,s$^{-1}$. \label{fig:ciico}}
\end{figure}
\begin{figure*}[!ht]
\centering
\includegraphics[width=\textwidth]{CII_channel_70mue.pdf}
\caption{ \cii\ channel maps as images, in 20\,km\,s$^{-1}$\ wide velocity
bins. The wedge to the right defines the respective colors between 0 and 20
K km/s. Contours are 70\,$\mu$m\ continuum levels from \citet{molinari16}
at 2, 4, 10 and 40\% of the field's peak intensity at \sgrbt(M). A
false-color image of the latter is in the top left panel, on a log
scale to emphasize lower level emission, with positions of \sgrbt(M)
and (N) marked by black stars.
\label{fig:chanmaps}}
\end{figure*}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.7\textwidth]{pvcut.pdf}
\caption{ Position-velocity cut along a line marked by the dotted line
in Fig.~\ref{fig:sgrbregion}, with positions projected onto Galactic
longitude. The cut passes through the \cii\ peak at region A, near
region D at \gsix, and continues through \sgrbt(M) at region J, and
past the peak at region M. Brightness temperatures above 15\,K
saturate at the lightest color in this plot. \cp\ in the Galactic
plane along the line of sight to the \sgrbt(M) continuum source
accounts for the deep absorption at $\ell \sim 0.67\degr$ across 0
to 20\,km\,s$^{-1}$. Absorption shows as negative because \sgrbt(M)'s
continuum offset was removed in spectral baseline fitting.
\label{fig:pv}}
\end{figure*}
\begin{figure*}[t]
\centering
\includegraphics[width=0.7\textwidth]{plotSpectraStack.pdf}
\caption{ \cii\ spectra in single beams from the positions and areas
marked in Fig.~\ref{fig:sgrbregion}. The spectra are normalized to
unity by peak temperatures, which are given after each label. Light
vertical lines mark 50, 70, and 90\,km\,s$^{-1}$\ to assist in comparing velocity
shifts. Parameters for two-component Gaussian fits to emission are
in Table~\ref{tab:fits}.
\label{fig:spectralstack}}
\end{figure*}
\begin{deluxetable*}{crrrrrrrrrr}[t]
\tabletypesize{\scriptsize}
\tablecaption{Two-component Gaussian fit results to the main
components of the
spectra of Fig.~\ref{fig:spectralstack}. Letters refer to positions
in Fig.~\ref{fig:sgrbregion}. Line parameter columns give main-beam
brightness temperature, line center, line width, and intensity of
the two main components. The component with the larger peak
brightness temperature is listed first; dashes show that a second
very broad component was likely residual baseline
structure. Horizontal lines represent some separation by region.
\sgrbt(M) is at position J.
\label{tab:fits}}
\tablewidth{0pt}
\tablecolumns{9}
\tablehead{ \colhead{Pos.} &
\colhead{$\ell$} &
\colhead{$b$} &
\colhead{$T_1$} &
\colhead{$v_1$} &
\colhead{$\Delta v_1$} &
\colhead{$I_1$} &
\colhead{$T_{2}$} &
\colhead{$v_2$} &
\colhead{$\Delta v_2$} &
\colhead{$I_2$} \vspace{-3mm} \\
&
\colhead{[deg]} &
\colhead{[deg]} &
\colhead{[K]} &
\colhead{[\,km\,s$^{-1}$]} &
\colhead{[\,km\,s$^{-1}$]} &
\colhead{[K\,km\,s$^{-1}$]} &
\colhead{[K]} &
\colhead{[\,km\,s$^{-1}$]} &
\colhead{[\,km\,s$^{-1}$]} &
\colhead{[K\,km\,s$^{-1}$]} }
\startdata
A & 0.483 & -0.057 & $13.4 \pm 0.9$ & $42.2 \pm 0.4$ & $49.3 \pm 1.6$ & $701 \pm 54$ & $8.5 \pm 0.9$ & $45.7 \pm 0.3$ & $19.7 \pm 1.8$ & $177 \pm 25$ \\
B & 0.507 & -0.033 & $20.1 \pm 0.3$ & $46.6 \pm 0.1$ & $32.6 \pm 0.5$ & $699 \pm 14$ & --- & --- & --- & --- \\
C & 0.549 & -0.054 & $7.6 \pm 0.2$ & $53.1 \pm 0.2$ & $31.7 \pm 1.1$ & $257 \pm 12$ & --- & --- & --- & --- \\
\hline
D & 0.599 & -0.047 & $16.8 \pm 0.3$ & $50.3 \pm 0.2$ & $32.8 \pm 0.8$ & $587 \pm 17$ & $6.3 \pm 0.2$ & $84.4 \pm 0.8$ & $57.0 \pm 3.2$ & $380 \pm 25$ \\
E & 0.606 & -0.067 & $7.2 \pm 0.3$ & $82.3 \pm 0.2$ & $25.6 \pm 1.1$ & $195 \pm 11$ & $5.3 \pm 0.1$ & $61.8 \pm 0.9$ & $93.4 \pm 2.4$ & $523 \pm 20$ \\
\hline
F & 0.629 & -0.068 & $3.6 \pm 0.3$ & $61.2 \pm 0.7$ & $69.8 \pm 3.8$ & $265 \pm 28$ & $1.9 \pm 0.3$ & $64.2 \pm 0.7$ & $21.2 \pm 4.0$ & $43 \pm 11$ \\
G & 0.652 & -0.043 & $2.3 \pm 0.9$ & $65.7 \pm 2.4$ & $67.3 \pm 13.2$ & $166 \pm 72$ & $1.7 \pm 0.9$ & $68.8 \pm 1.8$ & $24.6 \pm 11.7$ & $43 \pm 31$ \\
H & 0.657 & -0.036 & $2.1 \pm 0.9$ & $66.8 \pm 1.5$ & $21.1 \pm 9.2$ & $47 \pm 29$ & $1.8 \pm 0.9$ & $69.0 \pm 3.3$ & $65.5 \pm 18.5$ & $126 \pm 72$ \\
I & 0.665 & -0.023 & $1.5 \pm 0.2$ & $70.8 \pm 2.9$ & $49.4 \pm 11.0$ & $80 \pm 20$ & $0.6 \pm 0.3$ & $42.6 \pm 5.4$ & $32.1 \pm 18.4$ & $20 \pm 15$ \\
J & 0.666 & -0.034 & $7.5 \pm 2.2$ & $67.8 \pm 0.7$ & $66.4 \pm 5.6$ & $530 \pm 160$ & $-8.2 \pm 2.1$ & $70.0 \pm 0.3$ & $37.3 \pm 3.5$ & $-326 \pm 90$ \\
\hline
K & 0.666 & -0.058 & $3.7 \pm 0.1$ & $75.8 \pm 1.2$ & $60.3 \pm 4.5$ & $236 \pm 20$ & $3.3 \pm 0.2$ & $44.1 \pm 0.8$ & $34.2 \pm 2.9$ & $121 \pm 14$ \\
L & 0.674 & -0.065 & $1.9 \pm 0.4$ & $45.0 \pm 1.8$ & $68.1 \pm 9.7$ & $139 \pm 35$ & --- & --- & --- & --- \\
M & 0.683 & -0.034 & $10.6 \pm 0.5$ & $67.3 \pm 0.2$ & $19.0 \pm 0.9$ & $215 \pm 14$ & $5.0 \pm 0.4$ & $67.3 \pm 0.6$ & $65.9 \pm 3.5$ & $354 \pm 36$ \\
\hline
N & 0.640 & -0.092 & $11.7 \pm 0.2$ & $82.4 \pm 0.2$ & $44.9 \pm 0.9$ & $559 \pm 15$ & $4.2 \pm 0.2$ & $39.5 \pm 0.6$ & $40.0 \pm 2.4$ & $180 \pm 14$
\enddata
\end{deluxetable*}
Figures~\ref{fig:sgrbregion} and \ref{fig:ciico} provide overviews of
the spatial and spectral characteristics of \cii\ emission toward the
\sgrb\ region. Spatial and velocity continuity in \cii\ indicate that
the \sgrbo\ and the \gsix\ \hii\ region are part of a larger structure
that extends to positive Galactic longitude (\pell) to behind and
beyond \sgrbt. As seen in \cii, the distinction between these
different regions appears to be rooted in the resolution of historical
single-dish continuum surveys \citep[e.g.]{mezger76}, rather than the
separation of physically disconnected structures.
Figure~\ref{fig:sgrbregion} is an image of \ciiw\ line intensity in
K\,\,km\,s$^{-1}$\ integrated over $14$ to $120$\,km\,s$^{-1}$. For a characteristic
distance of 8\,kpc to the main \sgrb\ cloud, the \sgrb\ \cii\ region
stretches approximately 34\,pc along Galactic longitude $\ell$, and
15\,pc in latitude $b$. The bright emission contained within the
175\,K\,km\,s$^{-1}$\ (lowest) contour around the main body, encompasses
465\,pc$^2$.
All of the main \cii\ emission peaks are associated with bright radio
continuum emission, although the converse is not always true. The
brightest \cii\ regions, marked by circles A and B, are within the
radio continuum sources named the ``ionized rim'' and ``ionized bar,''
respectively, by \citet{mehringer92}. The peak at $\ell = 0.60\degr$
(region D) is the \gsix\ \hii\ region, with its associated bright arc
visible below (region E is in the arc). Crosses in the figure
indicate the locations of the luminous compact star forming cores
\sgrbt(N), \sgrbt(M) (region J), and \sgrbt(S). As we discuss in
Sec.~\ref{ssec:missing}, none of the cores emit detectable \cii\
emission.
Figure~\ref{fig:ciico} compares the \cii, $^{13}$C$^{}$O $J = 2-1$, and $^{}$C$^{18}$O\
$J = 2-1$ \citep{apexco} spectra averaged over the main body of the
\sgrb\ cloud indicated by the 175\,K\,km\,s$^{-1}$\ contour at the center of
Fig.~\ref{fig:sgrbregion}. The \cii\ line peaks near $50$\,km\,s$^{-1}$, with
emission from about $-30$ to $+130$\,km\,s$^{-1}$. There appear to be two main
components, and a two-component Gaussian fit gives good estimates of
these, separating emission into a main component centered at
$46.0 \pm 1.0$\,km\,s$^{-1}$, with $5.3 \pm 0.03$\,K peak temperature and
$57.3 \pm 2.8$\,km\,s$^{-1}$\ width; and another at $85.5 \pm 0.27$\,km\,s$^{-1}$, with
$1.9 \pm 0.06$\,K peak and $71.5 \pm 0.80$\,km\,s$^{-1}$\ width. Absorption
from the Galactic plane blocks \cii\ emission around 0\,km\,s$^{-1}$, masking a
likely wing to lower velocity. The $\sim$50\,km\,s$^{-1}$\ component is
associated molecular and ionized gas \citep{apexco, mehringer92}, and
contains 69\% of the total \cii\ flux in this decomposition. The
subsidiary peak at near $90$\,km\,s$^{-1}$\ is considerably more prominent in
extended CO emission than in \cii.
Overall agreement between \cii, $^{13}$C$^{}$O, and $^{}$C$^{18}$O\ velocity structures
near $+50$\,km\,s$^{-1}$\ indicates that \cii\ and molecular emission are
associated to a large degree. All of these lines, as well as the
H\,110$\alpha$ radio recombination line \citep{mehringer92}, peak
close to $+50$\,km\,s$^{-1}$, indicating that the \cii\ emission is associated
with the main column of molecular gas and its ionized surface.
Comparison of the CO lines suggests that the line shapes are dominated
by large-scale motions, and that these lines are not very optically
thick on average.
The $+90$\,km\,s$^{-1}$\ \cii\ emission is present across much of the image. It
is likely associated with extended, moderately excited, molecular
material previously identified by \citet{vogel87} toward \sgrbt.
Excitation and velocity suggest that the emission is generally
associated with the Galactic center rather than the Galactic disk.
APEX CO and isotopologue $J = 2-1$ data cubes show an emission
component around $+90$\,km\,s$^{-1}$\ that stretches across the entire CMZ.
Figures~\ref{fig:chanmaps} and \ref{fig:pv} show that the velocity
field across the bright body of \sgrb\ has overall coherence in a
smooth increase in velocity with increasing Galactic longitude, along
with a great deal of fine structure. Linewidths of 30 to above
50\,km\,s$^{-1}$\ are common across the region in individual 0.55\,pc beams,
indicating that the \cii-emitting material shares in the broad
linewidths typical for the Galactic center's dynamic molecular clouds.
Figure~\ref{fig:chanmaps} is a set of velocity channel maps in 20\,km\,s$^{-1}$\
bins, with the 70\,$\mu$m\ continuum image and contours for comparison.
The $+50$\,km\,s$^{-1}$\ \cii\ component is brightest toward \sgrbo's radio
continuum and 70\,$\mu$m\ peaks \citep{mehringer92, lang10, molinari16},
with additional lower-level emission around this velocity across the
entire \cii-bright region. The channel maps also show a very wide
linewidth in the emission arc wrapping around \gsix\ to negative
latitude ($-b$). The arc is visible at a variety of wavelengths, and
may be material swept up and compressed by winds from \gsix's central
luminosity sources. In addition to extended $+90$\,km\,s$^{-1}$\ emission,
$+90$\,km\,s$^{-1}$\ \cii\ is also bright along a North-South band of wide
linewidth emission near the \gsix\ \hii\ region at $\ell = 0.60\degr$.
Overall velocity trends are clear in Figure~\ref{fig:pv}, a
position-velocity diagram along a line cut through a bright \cii\ peak
(region A), and through and beyond \sgrbt(M) (region J). The mean
velocity of the main component of \cii\ emission across the entire
\sgrb\ body follows a smooth trajectory in the sense of Galactic
rotation, velocity increasing with $\ell$. Lineshapes broaden and
become more complex at \gsix\ and beyond. Bright 158\,$\mu$m\ continuum
associated with a North-South dust lane and perhaps the \sgrbt\ cores
at $\ell \approx 0.66\degr$ causes strong absorption from material in
the Galactic plane at $+7$\,km\,s$^{-1}$. A faint streamer stretching in
velocity from $+50$ to over 110\,km\,s$^{-1}$\ from $\ell \approx 0.53\degr$ to
$0.55\degr$ in Fig.~\ref{fig:pv} is from line wings with velocities
above 100\,km\,s$^{-1}$\ visible in spectra between \sgrbo\ and \gsix.
Overall velocity trends are clear, but the velocity field is complex
in detail. Figure~\ref{fig:spectralstack} explores the complexity by
comparing lineshapes from 15$^{\prime\prime}$\ diameter samples across \sgrb, with
locations keyed to the letters in Fig.~\ref{fig:sgrbregion}. Our
convention is to set baselines to zero brightness, so line absorption
against continuum appears as negative-going structure. Double
Gaussian component fits to the lineshapes provide good representative
summaries of the main emission; fit parameters are in
Table~\ref{tab:fits}.
\sgrb's most intense emission, in regions marked A and B, is
associated with the bright \sgrbo\ ``rim'' and ``arc'' radio continuum
regions near $\ell = 0.50\degr$. Their \cii\ velocities of 44 and
47\,km\,s$^{-1}$\ agree well with H\,110$\alpha$ recombination line velocities
\citep{mehringer92}. Position A's linewidths are 49 and 20\,km\,s$^{-1}$, but B
is better fit by a single 33\,km\,s$^{-1}$\ FWHM component. Both peaks A and B
have correspondence in 70\,$\mu$m\ continuum structures, but none in
160\,$\mu$m.
Continuing to positive longitude, a spectrum sampling the body of the
cloud at position C has a 32\,km\,s$^{-1}$\ FWHM component at 53\,km\,s$^{-1}$. The main
\cii\ line component from the \gsix\ \hii\ region at position D has a
center velocity of 50\,km\,s$^{-1}$\ and width 33\,km\,s$^{-1}$\ FWHM, with a distinct
second component at the substantially higher velocity of 84\,km\,s$^{-1}$, with
57\,km\,s$^{-1}$\ FWHM. As is typical for all but the \sgrbt\ cores, \cii\ and
$^{}$C$^{18}$O\ lineshapes share similarities, indicating that the spectrum
identifies two separate structures rather than a single broad line
with foreground absorption at an intermediate velocity. \gsix\ has
what appears to be a partial shell, appearing as an arc toward
negative latitude; the spectrum at position E is representative along
the arc, with a 26\,km\,s$^{-1}$\ FWHM peak near 80\,km\,s$^{-1}$\ matching the component
in \gsix\ itself, and a very broad (93\,km\,s$^{-1}$\ FWHM) wing to lower
velocity, centered at 62\,km\,s$^{-1}$.
Regions F through I sample emission along the dust lane that crosses
most of \sgrb's body. The \sgrbt(M), (N), and (S) cores are all close
to the \pell\ (eastern) side of this high column density dust
structure that matches the size and shape of the \cii-dark notch and
\cii-dim stripe visible in Fig.~\ref{fig:sgrbregion}. As we discuss
in Sec.~\ref{ssec:comparisons}, this approximately North-South strip
is a clear feature in infrared continuum images, becoming increasingly
darker and longer from 70\,$\mu$m\ to 8\,$\mu$m\ wavelengths. At 160\,$\mu$m, which
is representative of continuum underlying the 158\,$\mu$m\ \cii\ line, the
lane is bright at its northern end, encompassing regions G-J and N,
with peaks at the \sgrbt\ cores, fading in brightness to the south
before dropping to the typical cloud brightness level just beyond
region F. All \cii\ spectra along this \cii-dim and IR-dark lane have
similar lineshapes dominated by a 65 to 70\,km\,s$^{-1}$\ FWHM component
centered near 67\,km\,s$^{-1}$. All but the southernmost position F have
sufficient 158\,$\mu$m\ continuum to cause absorption from about 0 to
20\,km\,s$^{-1}$\ in the Galactic plane. \cii\ emission's lineshape similarity
and run of peak line brightness with dust column density suggest that
the spectra originate in a generally cool dust lane that may be
optically thick to \cii\ from emission behind it. It appears that the
material producing the notch and dark lane are between us and the
continuous body of \sgrb\ otherwise highlighted in \cii.
The most dramatic spectrum is at position J, coincident with the
maximum absorption toward \sgrbt(M) and the peak of 1.3\,mm continuum
measured by ALMA \citep{sanchezmonge17}. Ignoring Galactic plane
absorption from $-60$ to $+$20\,km\,s$^{-1}$\ visible against \sgrbt(M)'s
intense continuum, emission in this spectrum is best represented as a
66\,km\,s$^{-1}$\ FWHM line centered at 68\,km\,s$^{-1}$, with a 37\,km\,s$^{-1}$\ FWHM absorption
to approximately zero brightness centered at 70\,km\,s$^{-1}$. We discuss this
feature and its implications in detail in Secs.~\ref{ssec:physstruct}
and \ref{ssec:missing}. While \cii\ absorption from the Galactic
plane is strongest against \sgrbt(M), and is slightly enhanced by
increased 158\,$\mu$m\ continuum toward \sgrbt(N) and (S), it extends along
the entire 160\,$\mu$m-bright (and \cii-faint) North-South strip visible in
Fig.~\ref{fig:sgrbregion}. With the exception of the 70\,km\,s$^{-1}$\
absorption notch, the emission lineshape toward J is very similar to
surrounding positions including I, which is in the darkest part of the
dark lane at shorter wavelengths, and brightest at 160\,$\mu$m.
Velocity components near 70 and 50\,km\,s$^{-1}$\ are present throughout the
regions near \sgrbt(M) and to larger longitudes (see also
Fig.~\ref{fig:chanmaps}). The bright spot at position M, near
\sgrbt(M), has both 19 and 66\,km\,s$^{-1}$\ FWHM lines centered at
67\,km\,s$^{-1}$. Further to $+\ell$, region K has components centered at both
44 and 76\,km\,s$^{-1}$, and region L peaks at 45\,km\,s$^{-1}$. Sampling to $-b$, the
bright region K has two velocity peaks, one at 76\,km\,s$^{-1}$\ with 60\,km\,s$^{-1}$\
FWHM, and the other at 44\,km\,s$^{-1}$\ with 34\,km\,s$^{-1}$\ FWHM. The run of nearby
spectra shows that these are separate emission peaks, and not the
residuals from absorption against a broader line at intermediate
velocity. As with other \cii\ peaks, this region is close to a local
peak of 20\,cm \citep{lang10} and 70\,$\mu$m\ continua, but has no 160\,$\mu$m\
\citep{molinari16} enhancement.
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.65\textwidth]{oiCII.pdf}
\caption{
Integrated \cii\ intensity (top left) and \cii\ and \oi\ spectra
averaged over black rectangles in bright regions, labeled in
increasing order with longitude (right to left in image). The \oi\
brightness temperature scale has been multiplied by
(157.74\,$\mu$m//63.18\,$\mu$m)$^3 = 15.6$ to allow direct comparison of
intensities. The intensity ratio is $I(\mbox{\oi})/I(\mbox{\cii})
\approx 0.3$ (see Table~\ref{tab:oiIntens}), in agreement with typical
values for unresolved spectra across the region \citep{goi04}. This
implies similar underlying physical conditions in the
\oi-emitting material across the source, on average.
\label{fig:ciioispect}}
\end{figure*}
\subsection{Additional \cii\ components}\label{sec:compact}
The \cii\ data cube reveals a number of isolated clouds at velocities
other than those that are typical for \sgrb's body. The most prominent are:
A distinct 40$^{\prime\prime}$\ diameter source with a 9.2\,K peak, 6.5\,km\,s$^{-1}$\ FWHM
line toward \ra{17}{47}{18.6}, \dec{-28}{40}{22}
($\ell = 0.417\degr, b = -0.178\degr$). At an LSR velocity of
$-58.7$\,km\,s$^{-1}$, this source is likely associated with the so-called
3\,kpc expanding arm. The source appears as a bright spot at the
southern end of a small $^{13}$C$^{}$O\ $J = 2-1$ filamentary cloud with the
same linewidth and velocity. A lack of a radio continuum counterpart
in the \citet{lang10} 20\,cm continuum data implies the absence of an
\hii\ region, in line with the general lack of compact \hii\ regions
in the 3\,kpc arm. Other similarly narrow line components in the
\cii\ data cube fall close in velocity to the Galactic plane
absorption, but are most likely remnants of the wings of broader
self-absorbed lines.
A small elliptical cloud with size 37$^{\prime\prime}$\ by 77$^{\prime\prime}$\ at PA
$330\degr$ and peak $T_{MB} = 1.6$\,K, velocity 124\,km\,s$^{-1}$, and width
36\,km\,s$^{-1}$\ is centered at \ra{17}{46}{55.5}, \dec{-28}{18}{56} ($\ell =
0.679\degr$, $b = 0.078\degr$). $^{13}$C$^{}$O\ and $^{}$C$^{18}$O\ $J = 2-1$ in the area show
mainly emission peaking at about $+40$\,km\,s$^{-1}$. The bulk of \cii\ here is
from material with very low CO emissivity: while $^{13}$C$^{}$O\ at velocities
near $128$\,km\,s$^{-1}$\ is present in patches throughout the region, it
matches only the edge of the higher-velocity \cii\ wing, and $^{}$C$^{18}$O\ is
not clearly present.
\subsection{\oi\ spectroscopy\label{ssec:oi} }
Figure~\ref{fig:ciioispect} shows the \grt\ velocity-resolved \oisw\
spectra from the brightest \cii\ regions. \oi\ emission is mostly at
the $\sim$50\,km\,s$^{-1}$\ velocity of the \sgrb\ cloud, and shares peak
velocities with \cii\ even when multiple components are visible in the
\cii\ line. Coincidence between \cii\ and \oi\ indicates that \cii\
is associated with neutral gas, and that emission from photodissociation
regions (PDRs) may dominate in the brighter, narrower-line \cii\
component. The peaked \oi\ lineshapes suggest that the lines
are not self-absorbed, so \oi\ intensities are accurate measures of
\oi's contribution to gas cooling.
Comparing line intensities gives
$I(\mbox{\oi})/I(\mbox{\cii}) \approx 0.3$ (Table~\ref{tab:oiIntens}).
Ratios from Table~\ref{tab:oiIntens} are equivalent to the typical
value of 0.3 from unresolved {\em ISO}-LWS spectra in $\sim$80$^{\prime\prime}$\
beams \citep{goi04} across the \sgrb\ cloud. In our data, only the
brightest \cii\ regions show the weaker \oi\ clearly. Our mapping
strategy was driven by the signal to noise ratio for the \cii\ line;
residual spectral baseline structure in the \oi\ spectra prevented
extraction of lower-level \oi\ emission averaged over larger
regions. The similar intensity ratio with the {\em ISO} data suggests
that \oi\ scales with \cii\ in fainter regions as well, however,
pointing to a component of \cii\ with similar physical conditions and
associated with neutral gas sufficiently dense
($n \gtrsim {\rm few} \times 10^3$\percmcu) to excite \oi\ across the
entire bright region.
\begin{deluxetable}{rrrrrrrr}[!ht]
\tabletypesize{\scriptsize}
\tablecaption{\cii\ and \oi\ intensities and intensity ratios for the
regions indicated in black rectangles in Fig.~\ref{fig:ciioispect}.
Center positions and region sizes are in Galactic coordinates.
Intensities are in units of $10 ^{-3} \;
[\mathrm{erg~s^{-1}~cm^{-1}~sr^{-1}}]$.
\label{tab:oiIntens}}
\tablewidth{0pt}
\tablecolumns{8}
\tablehead{
\colhead{Peak} &
\colhead{$\ell$} &
\colhead{$b$} &
\colhead{Size} &
\colhead{Vel.} &
\colhead{$I$(\cii)} &
\colhead{$I$(\oi)} &
\colhead{$I$(\oi)/} \vspace{-3mm} \\
\colhead{} &
\colhead{[deg]} &
\colhead{[deg]} &
\colhead{[arcsec]} &
\colhead{[km/s]} &
\colhead{ } &
\colhead{ } &
\colhead{$I$(\cii)}
}
\startdata
1 & 0.482 & $-0.053$ & $ 70 \times 25 $ & 25--60 & 2.99 & 0.87 & 0.29 \\
2 & 0.511 & $-0.029$ & $ 40 \times 30 $ & 40--60 & 1.96 & 0.57 & 0.29 \\
3 & 0.531 & $-0.042$ & $ 40 \times 20 $ & 35--60 & 1.93 & 0.64 & 0.33 \\
4 & 0.596 & $-0.049$ & $ 30 \times 30 $ & 40--60 & 1.41 & 0.42 & 0.30 \\
5 & 0.603 & $-0.068$ & $ 60 \times 30 $ & 75--90 & 0.75 & 0.15 & 0.21
\enddata
\end{deluxetable}
\begin{figure*}[!ht]
\centering
\includegraphics[height=8.4in]{comparisons.pdf}
\caption{
Integrated intensity images of various tracers toward \sgrb. \cii\
contours from Fig.~\ref{fig:sgrbregion} are on images of 20\,cm radio
continuum \citep{lang10}, APEX CO isotopologues $J=2-1$ integrated
over $\pm 120$\,km\,s$^{-1}$\ \citep{apexco}, {\em Herschel/PACS} 160\,$\mu$m\ and
70\,$\mu$m\ continuum \citep{molinari16}, and 21\,$\mu$m\ {\it MSX} and 8\,$\mu$m\
{\em Spitzer} \citep{price01, stolovy06} continuum, all sampled to the
\cii\ beamsize. Color stretches are linear, trimmed to emphasize
structure. Galactic latitude and longitude axes are in degrees.
\cii, 70\,$\mu$m, and 20\,cm intensities correspond closely, with the
exception of the \sgrbt\ cores at $\ell \approx 0.68\degr$, while the
background cloud traced in CO and 160\,$\mu$m\ is much larger than the
\cii\ region, and extends considerably further to
$+\ell$. Mid-infrared images generally emphasize compact sources.
\label{fig:Integ}}
\end{figure*}
\section{Discussion} \label{sec:discus}
With 12\% of the \cii\ emission from only 6\% of the area in our large
scale map of the entire CMZ \citep{wholecmz}, the \sgrbo\ region is a
notable contributor to the entire \cii\ line flux emitted by the
Galactic center region. The \sgrbo\ region is second only to the
\sgra-Arches region as the brightest source of \cii\ across the entire
CMZ, with the difference due to \sgrb's smaller extent rather than
lower surface brightness. Our data cube with sub-parsec spatial and
1\,km\,s$^{-1}$\ spectral resolution over an area of 3000\,pc$^2$ provides
context to understand the \cii\ excitation within the Galactic center,
as well as serving as a case study for interpreting \cii\ emission
from other galactic nuclei.
\subsection{Following the UV from young stars \label{ssec:comparisons}}
Ultraviolet radiation from young stars produces bright \cii\ and heats
dust as the surfaces of nuclear molecular clouds intercept and convert
essentially all UV to longer wavelengths. \cii\ is therefore expected
to be a good spectroscopic tracer of star formation in obscured
regions throughout the universe (e.g., \citealt{stacey91, stacey10,
delooze14, pineda14, herreracamus15}). SOFIA-\grt's resolution and
the Galactic center's proximity allow us to examine the relationship
between \cii\ and far-IR luminosities within a galactic nucleus in
detail. Imaging of \sgrb\ is at a scale between small isolated clouds
conducive to modeling and scales where relationships between the
far-UV from star formation and indirect tracers on square kiloparsec
scales hold (e.g., \citealt{calzetti10, kennicutt12}).
Figure~\ref{fig:Integ} shows comparisons of a number of potential
indirect tracers of far-UV. The panels all show \cii\ (contours) with
integrated intensity images of 20\,cm radio continuum \citep{lang10},
APEX CO isotopologues $J=2-1$ integrated over $\pm 120$\,km\,s$^{-1}$\
\citep{apexco}, {\em Herschel}/PACS 160\,$\mu$m\ and 70\,$\mu$m\ continuum
\citep{molinari16}, and {\it MSX} 21\,$\mu$m\ and {\it Spitzer} 8\,$\mu$m\
\citep{price01, stolovy06} continuum, all sampled to the \cii\
beamsize. Visually, the 20\,cm and 70\,$\mu$m\ distributions appear most
similar to \cii's morphology. $^{13}$C$^{}$O, $^{}$C$^{18}$O, and 160\,$\mu$m\ are more
extended, especially to $+\ell$. The mid-IR images have yet different
distributions, more concentrated toward the \cii\ peaks.
To put the spatial comparisons on a quantitative basis,
Figure~\ref{fig:biplot} shows the amplitudes of the first two
components in a principal components analysis (PCA) decomposition of
this set of images. PCA is a standard tool of multivariate analysis,
long used for multi-spectral imaging, either in different wavebands or
across spectra (\citealt{frieden91, ungerects97, heyer97} among
others, contain tutorials and examples). PCA constructs an orthogonal
set of basis vectors from the data themselves, here the concatenated
and normalized columns of each image. Any image can be reconstructed
from a linear combination of the new basis vectors. After arranging
the principal components (PCs) in decreasing order of variation, the
first principal component (PC1) gives the most common vector (or
image); orthogonality ensures that the second PC (PC2) shows the
largest difference from PC1 and is independent from it; PC3 shows the
largest independent differences from both PC1 and PC2; and so on.
Higher principal moments typically contain little information other
than noise. Vectors with similar principal component amplitudes
highlight similar structure within the vectors, or equivalently, in
the images.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.3\textwidth]{plotBiplot.pdf}
\caption{
Amplitudes of the first two Principal Component vectors, PC1 and PC2,
from a decomposition of continuum and integrated intensity images for 8
different tracers. Three groupings show spatially related distributions
from the large-scale cloud ($^{}$C$^{18}$O, 160\,$\mu$m, and $^{13}$C$^{}$O), a tight cluster
tracing UV radiation from the \sgrb\ body (\cii, 20\,cm, and 70\,$\mu$m),
and the mid-IR tracing UV and compact sources (21\,$\mu$m, 8\,$\mu$m). The
decomposition excludes a small area immediately covering the \sgrbt\
cores, which has little effect on the result; see text.
\label{fig:biplot}}
\end{figure}
Figure~\ref{fig:biplot} is a plot of the amplitudes of the PC1 and PC2
vectors from the 8 images. The first two principal components contain
80\% of the total variation among the images, so considering these two
PCs reveals the major commonalities and differences between the
images. All the vectors in Fig.~\ref{fig:biplot} have a positive PC1,
indicating a similar overall structure, but their PC2s, showing
deviations from the common structure, form three distinct groups. The
uppermost group contains $^{}$C$^{18}$O\ $J = 2-1$, 160\,$\mu$m, and $^{13}$C$^{}$O\ $J =
2-1$. The middle group shows a very tight cluster of \cii, VLA
20\,cm, and 70\,$\mu$m\ emission. Infrared 21\,$\mu$m\ and 8\,$\mu$m\ images form a
looser third group. This confirms the visual impression from
Fig.~\ref{fig:Integ}: while all images trace a roughly similar
structure, the top group shows more extended emission than the others
(Fig.~\ref{fig:Integ}), tracing large background cloud emission, while
the bottom group has high contrast between extended emission and
bright compact sources. With a large PC1 compared with a smaller and
negative PC2, the middle group of \cii, 20\,cm, and 70\,$\mu$m\ is more
typical of the global emission, although with somewhat more of the
compact than the extended emission. For full sensitivity to the
extended emission, this decomposition blanked a region $\Delta \ell =
130$$^{\prime\prime}$\ by $\Delta b = 100$$^{\prime\prime}$\ centered at \ra{17}{47}{21},
\dec{-28}{28}{03} ($\ell, b = 0.669, -0.038$) in all data sets. This
region covers the \sgrbt\ cores and their immediate surroundings,
while retaining more than 95\% of the total flux in all of the images.
Without blanking, the middle set of vectors spreads to some extent,
while still retaining their grouping, but there is no effect on the
ordering, and little effect on the top and bottom groups.
\begin{figure*}[t]
\centering
\includegraphics[width=\textwidth]{sgrb_stacked_cii_fir70.pdf}
\caption{
Intensity-intensity correlations of integrated \cii\ intensity vs.\
160\,$\mu$m, 70\,$\mu$m, 21\,$\mu$m, and 8\,$\mu$m\ continuum intensities across the
\sgrb\ region. Intensity units are cgs,
$[\mathrm{erg~s^{-1}~cm^{-1}~sr^{-1}}]$. The \cii\ emission has been
integrated from 14 to 120\,km\,s$^{-1}$. Each point (in green) is averaged over
an independent 21\,arcsecond square region, with contours showing the
density of points (in blue) at 0.3, 0.6, and 0.9 of the peak density
of points. The straight lines are linear fits to the density of data
points, weighted by the density to discriminate against outlier
points. Table~\ref{tab:slopeInt} contains slopes and
intercepts. Scales are all 2.5 by 2.5 dex, and the horizontal dashed line
marks the 175\,K\,\,km\,s$^{-1}$\ \cii\ intensity cut for the \sgrbo\ body
indicated by the lowest contour line in Fig.~\ref{fig:sgrbregion}.
\label{fig:ciiDust}}
\end{figure*}
Another quantitative view of the spatial relationships between
different wavebands is the spatial correlation of \cii\ with dust
continuum in bands frequently used to trace star formation.
Figure~\ref{fig:ciiDust} shows such correlations. Each point in the
plots represents an average over an independent 21$^{\prime\prime}$\ region with
signal detected above $3\sigma$ in both tracers. Contour lines show
the density of points, with the straight line giving a linear fit to
the density of data points, weighted by the density of points to
discriminate against outliers. Table~\ref{tab:slopeInt} contains fit
results. In all panels, the sprays of points to higher infrared
intensities lacking strong correlation with \cii\ intensities are from
compact and IR-luminous regions with unrelated \cii\ emission. The
correlation is tightest for \cii-70\,$\mu$m, panel (b), with the same
relationship within the body of \sgrb\ and in the fainter regions
outside that.
\begin{deluxetable}{rrrr}[!ht]
\vspace{5mm}
\tabletypesize{\scriptsize}
\tablecaption{Power laws and intercepts from linear fits to intensity
correlation data of Fig.~\ref{fig:ciiDust}, of form
$\log_{10}(\{I(\mbox{\cii})\}) = a \log_{10}\{I({\rm IR})\} + b$.
Uncertainties are 68\% confidence intervals from bootstrap
iterations of weighted fits. The last column is the Pearson
correlation coefficient $r$. \label{tab:slopeInt}}
\tablewidth{0pt} \tablecolumns{3}
\tablehead{
\colhead{IR band} &
\colhead{$a$} &
\colhead{$b$} &
\colhead{r}
}
\startdata
160\,$\mu$m & $ 0.52 \pm 0.08 $ & $ -2.44 \pm 0.10 $ & 0.44 \\
70\,$\mu$m & $ 0.98 \pm 0.05 $ & $ -1.72 \pm 0.06 $ & 0.89 \\
21\,$\mu$m & $ 0.73 \pm 0.08 $ & $ -1.93 \pm 0.13 $ & 0.72 \\
8\,$\mu$m & $ 0.78 \pm 0.07 $ & $ -1.94 \pm 0.10 $ & 0.66
\enddata
\end{deluxetable}
Correlation is poor for 160\,$\mu$m-\cii, panel (a), as the bright
continuum luminosity is concentrated around the \sgrbt\ cores and the
northern end of the dark lane, while \cii\ is bright across the \sgrb\
body. The brightest \cii\ regions do not have corresponding peaks at
160\,$\mu$m, and vice versa. At lower intensities, 160\,$\mu$m\ traces the
column density of the larger background cloud that is also apparent in
CO isotopologues. Some hint of two clusters of points in the
correlation plot may be attributable to separation between bright and
background emission.
The best-calibrated mid-IR tracer for obscured star formation over
large scales, {\em Spitzer} 24\,$\mu$m\ \citep{calzetti10}, had saturated
regions that prevented comparisons over the entire field, so we show
{\it MSX} 21\,$\mu$m\ data as an alternative in panel (c). The distribution
is tadpole-shaped, showing a general relationship between much of the
emission at the two wavelengths, with a flatter power law indicating
low correlation between bright 21\,$\mu$m\ emission from the compact
sources and \cii. The 8\,$\mu$m-\cii\ correlation is in panel (d). The
broader distributions about the mean compared with 70\,$\mu$m\ reflect the
differences in spatial distributions. A much tighter correlation
between 8\,$\mu$m\ and \cii\ intensities in the Orion region
\citep{pabst17} is attributed to UV excitation of both 8\,$\mu$m\ bands of
polycyclic aromatic hydrocarbons (PAHs) and \cii\ in PDRs. Unlike
those sources, which are largely isolated and extended diffuse clouds,
the \sgrb\ field contains a mix of extended and compact sources, and
is affected by absorption and emission from the $\sim$8\,kpc of
Galactic plane along the line of sight.
Figure~\ref{fig:ciiHeating} provides a view of the relationship
between \cii\ and 70\,$\mu$m\ intensities across \sgrb. Using the same
data and fitting method as for Fig.~\ref{fig:ciiDust}, we find a slope
of $ -0.01 \pm 0.07 $, indicating that radiation and physical
conditions toward \sgrb\ do not appreciably suppress the conversion of
UV to \cii. \cii\ and 70\,$\mu$m\ trace UV deposition at the cloud surface
equally well across \sgrb's body, with the exception of the brightest
emission associated with the \sgrbt\ cores. 70\,$\mu$m\ emission is near
the peak of a graybody dust emission curve (Sec.~\ref{sec:physcond})
and is tightly correlated with \cii, making it an accurate proxy for
the total FIR intensity from the \cii-emitting regions in \sgrb. A
scaling factor between 70\,$\mu$m\ and FIR intensities would change axis
offsets but not the relationship's slope in Fig.~\ref{fig:ciiHeating},
indicating that the heating efficiency, measured as the ratio of \cii\
to FIR intensities, vs. FIR intensity, is also nearly constant.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.25\textwidth]{sgrb_eff_fir70.pdf}
\caption{ \cii\ emission efficiency, measured as the ratio of \cii\ to
70\,$\mu$m\ intensities, vs.\ 70\,$\mu$m\ intensity. Data and fitting method
are the same as in Fig.~\ref{fig:ciiDust}, yielding a slope of
$ -0.01 \pm 0.07 $, or negligible dependence on 70\,$\mu$m\ intensity.
Since 70\,$\mu$m\ is a proxy for FIR intensity within the Galactic center
PDRs (Sec.~\ref{sec:physcond}), and a scaling factor would affect
both axes equally, this plot also shows that the heating efficiency
is independent of FIR intensity.
\label{fig:ciiHeating}}
\end{figure}
All of the comparisons above show a close connection between \cii\ and
70\,$\mu$m\ continuum, reinforcing the interpretation that they both arise
from UV depositing energy at cloud edges. The comparisons also show
that 20\,cm VLA imaging at $\sim$15$^{\prime\prime}$\ resolution toward the
Galactic center \citep{mehringer92, lang10} resolves out the Galactic
plane emission accounting for about half of the total flux toward the
center in single-dish observations \citep{law08}, while retaining good
sensitivity to the free-free emission from structures in the center.
Agreement between the far-IR and 20\,cm distributions also confirms
that the synchrotron contribution toward \sgrb\ at 20\,cm is small.
The other bands we considered do not pick out the far-UV as cleanly.
Emission at 160\,$\mu$m, $^{13}$C$^{}$O, and $^{}$C$^{18}$O\ are products of temperature and
column density, and trace combinations of large columns of cooler
material in background clouds as well as warm surfaces. Emission from
8\,$\mu$m\ and 21\,$\mu$m\ in the Galactic center is weighted toward bright,
compact regions, with dust absorption obscuring the most luminous
cores, and line of sight material in the Galactic plane causing some
confusion for extended emission. These results favoring \cii\ and
70\,$\mu$m\ as UV tracers likely hold in other galactic nuclei as well. A
fortunate combination of circumstances for 20\,cm radio continuum
(i.e., the fraction of resolved large-scale emission, emission size
scale, and lack of significant non-thermal emission) may not apply
elsewhere in the Galactic center or in other galactic nuclei, however.
\subsection{Physical conditions in the \cii\ emission regions \label{sec:physcond}}
In this section we examine the emission from the bright body of \sgrb\
contained within the lowest contour in Fig.~\ref{fig:sgrbregion} to
derive overall properties. A key question we approach is the fraction
of \cii\ emitted from \hii\ regions compared with photodissociation
regions (PDRs). This is of intrinsic interest in understanding the
emission from the CMZ, and it is also a guide to interpreting \cii\
emission from external galaxies.
We set a lower limit on the column density and mass of singly ionized
carbon in the bright \cii\ region of \sgrb's body by assuming that the
ions are in collisional equilibrium with their surroundings ($n >
n_{crit}$ of a few thousand cm$^{-3}$ in molecular hydrogen gas, or an
order of magnitude lower in ionized hydrogen plasma;
\citealt{launay77, flower77, wiesenfeld14}); that the temperature is
well above the level energy (91\,K); and that the emission is
optically thin. From expression (A2) in \citet{crawford85} (see
also \citealt{goldsmith12}), the measured \cii\ intensity of the bright
region outlined in Fig.~\ref{fig:sgrbregion} is $2.00\times 10^{-3}$
erg\,cm$^{-2}$\,s$^{-1}$\,sr$^{-1}$, corresponding to a \cp\ column
density $N_\mathrm{C^+} = 1.16\times 10^{18}$\percmsq\ and a mass
$m_\mathrm{C^+} = 50.0$\msun. Following \citet{genzel90} to allow
direct comparisons with other Galactic center results, we take a
[C]/[H] abundance ratio of $3\times 10^{-4}$ and assume all carbon is
in \cp, deriving a hydrogen column density $N_\mathrm{H} = 3.86\times
10^{21}$\percmsq\ and a mass $m_\mathrm{H} = 1.39 \times 10^4$\msun.
These values are comparable with those \citet{genzel90} found for the
Arches region of the Galactic center. A lower gas-phase carbon
abundance, such as that found along diffuse lines of sight in the
Galactic disk \citep{sofia04}, or with carbon in forms other than \cp,
would proportionally increase these estimates.
We use the \citet{lang10} 20\,cm VLA and \citet{law08} 21\,cm Green
Bank Telescope radio continuum data over the same region as the bright
\cii\ emission within the lowest contour of Fig.~\ref{fig:sgrbregion}
to constrain the \cii\ contribution from \hii\ regions. We estimate
that about one half of the 20\,cm flux averaged over the entire region
is from the foreground Galactic plane by averaging across longitude
over the latitudes toward \sgrb. Comparing VLA and GBT fluxes toward
\sgrb, we estimate that the VLA image contains 0.54 of the total
\sgrb\ flux, spatially resolving out the rest. \citet{mehringer92}
report continuum and recombination line observations of compact
regions across \sgrbo\ and \gsix, which cover much of the \sgrb\
complex area. They estimate that the 20\,cm emission is optically
thin with little contribution from synchrotron radiation in this
region, an estimate confirmed in multi-frequency single-dish
observations by \citet{law08}, and give typical electron temperatures
of $T_e \approx 4800$\,K for the compact regions. With this electron
temperature and the \citet{lang10} flux density corrected for power
resolved out by the interferometer, the emission measure across
\sgrb's bright region is $EM = 1.72\times 10^{5}$\,pc\,cm$^{-6}$. The
relationship between emission measure, column density $N$, and
particle density $n$ is $N \sim EM/n$, allowing an estimate of
$N$. Infrared fine-structure lines from the \sgrb\ \hii\ region
provide estimates of electron density of about 300\percmcu\
(\citealp{goi04, simpson18b}; this is also sufficient to thermalize
any \cii\ line emission from the \hii\ region). Combining these
values, we find $N(\mathrm{H^+}) = 1.9 \times 10^{21}$\percmsq, for a
total ionized mass of $m_\mathrm{H^+} = 5.7 \times 10^3$\,\msun\ in
the region. These densities imply that the ionized gas has a
characteristic line of sight depth of $s \approx N/n = 2$\,pc, which
seems plausible as a relatively thin layer over the region's surface.
Taking the values above, and given the uncertainties in the continuum
contribution from the Galactic plane, the H$^+$ filling factor, and
the atomic carbon abundance, we find a representative ratio of
$N(\mathrm{H^+})$/$N(\mathrm{H}) \approx 0.5$. Most corrections push
this ratio to lower values: Although the \cii\ lines are still in the
``effectively optically thin'' regime found by \citet{goldsmith12},
there is some evidence for \cii\ optical depths of a few within the
CMZ \citep{wholecmz}, and substantial \cii\ optical depths would
decrease the ratio. A lower gas-phase carbon abundance or increased
electron density would also decrease the ratio, as would synchrotron
contributions to the radio flux. Ionized hydrogen columns increase
sub-linearly with $T_e$.
Both PDRs and \hii\ regions emit \cii\ line radiation, but \oi\ is
excited in PDRs and destroyed in \hii\ regions. PDR models combined
with observed \cii, \oi, and far-infrared continuum therefore provide
another constraint on the fraction of \cii\ emission from neutral
material.
We measure the \cii\ and \oi\ intensities directly, but we must
separate the PDR's far-infrared dust intensity from cooler and warmer
cloud components. Since 70\,$\mu$m\ is very similar in spatial distribution
to \cii\ (Sec.~\ref{ssec:comparisons}), we estimate $I$(FIR) by scaling
the intensity falling within the 70\,$\mu$m\ PACS filter passband to the
full 8--500\,$\mu$m\ FIR band from a
\begin{equation}
I_\lambda(\lambda, \, T) \propto \left(1 -
\exp\left[\left(\lambda/\lambda_o\right)^{-\beta}\right]\right)\,
B_{\lambda}(\lambda, \, T)
\label{eq:graybody}
\end{equation}
graybody. \citet{goi04} used a two-component version of this
expression to fit far-IR continuum fluxes in the region. They found
cool and warm dust components, with warm dust component temperatures
of 30--38\,K and $1 \leq \beta \leq 1.5$ in the bodies of the massive
clouds in their 80$^{\prime\prime}$\ beams.
For the warmer dust associated with the PDRs, we prefer a temperature
of 45\,K $\beta = 1.5$, based on the tight similarity of the 70\,$\mu$m\
and \cii\ images and match to the 70\,$\mu$m/21\,$\mu$m\ flux ratio, including a
correction for PAH emission \citep{draine07}.
With this combination of temperature and $\beta$, a fraction 0.245 of
the total 8 to 500\,$\mu$m\ far-IR flux falls in the PACS 70\,$\mu$m\ filter
bandwidth, or $I({\rm FIR}) = I$(70\,$\mu$m)/0.245. The observed 70\,$\mu$m\
mean intensity of 0.11 erg\,s$^{-1}$\,cm$^{-2}$\,sr$^{-1}$ translates
to $I$(FIR) = 0.45 erg\,s$^{-1}$\,cm$^{-2}$\,sr$^{-1}$ from the PDR
regions. This value is insensitive to modest changes in temperature
or $\beta$ because the 70\,$\mu$m\ band is on the Rayleigh-Jeans side close
to the graybody peak. For $\beta = 1.5$, $T = 35$\,K gives a fraction
of 0.262; while at $T = 45$\,K, $\beta$ of 1.25 to 1.75 gives 0.248 to
0.239. Our derived intensity of $I$(\cii)/$I$(FIR)$= 0.46$\% is in
line with ratios of 0.1 to 1\% for starbursts \citep{stacey91} and
$0.48 \pm 0.12$\% for a large sample of disk galaxies \citep{smith17}.
We conclude that the 70\,$\mu$m\ flux can be scaled to provide a good
measure of the total FIR intensity for PDR modeling.
Figure~\ref{fig:pdrgrid} shows our data as contours along predictions
of models from the updated PhotoDissociation Region Toolbox
\citep{pound08, kaufman06, kaufman99}. This model's face-on geometry
matches that of the \sgrb\ region. Given the changing physical
conditions across \sgrb, we use center-of-error fluxes as guides to
identify boundaries on representative physical conditions, with more
detailed modeling left to future work. Since \oi\ is only associated
with the $+50$\,km\,s$^{-1}$\ clouds, we take only the $+50$\,km\,s$^{-1}$\ \cii\ emission
component in the comparison, for $I(\mbox{\cii}) = 1.38\times 10^{-3}$
erg\,cm$^{-2}$\,s$^{-1}$\,sr$^{-1}$. We use the typical intensity
ratio $I(\mbox{\oi})/I(\mbox{\cii}) = 0.3$ from
Table~\ref{tab:oiIntens} and \citet{goi04}. $I({\rm FIR})$ has been
derived above. This set of parameters does not produce a model
solution (the solid curves for ratios in the figure do not touch).
\begin{figure}[!ht]
\centering
\includegraphics[width=0.45\textwidth]{pdrGrid2020d.pdf}
\caption{ Observed quantities on theoretical PDR results
from PhotoDissociation Region Toolbox models. Lines in the plot are
contours that correspond to the specific intensities or ratios we
measure or derive in a value vs.\ radiation field strength vs.\
particle density contour plot. (See \citealp{kaufman99} for individual
examples and physical explanations.) The unmodified observed
quantities (solid lines) are inconsistent with the models. Model
solutions are possible where lines cross, bounded where 58\% (dashed
lines, triangle) and 76\% (dash-dot lines, circle) of the \cii\
intensity is assigned to \hii\ regions.
\label{fig:pdrgrid}}
\end{figure}
Using the total observed \cii\ flux for modeling includes radiation
from \hii\ regions as well as PDRs, however. Finding successful PDR
model solutions with reduced \cii\ flux allows us to deduce the
relative amounts of \cii\ from PDRs and \hii\ regions. Ratioed
quantities in the models set a lower limit of 58\% of the \cii\ flux
from \hii\ regions (red and blue dashed ratio lines in
Fig.~\ref{fig:pdrgrid} first touch, marked by a triangle) for
$n(H) \approx 10^{2}$\percmcu\ and a radiation field of
G$_o \approx 10^{2.5}$\,Habing units. Matching the absolute \cii\ and
\oi\ intensities in this case would require multiple optically thin
PDRs along the line of sight (intensities add, but ratios remain
approximately constant). Bringing ratios and the \cii\ intensity
together for a single PDR (dash-dot lines cross, marked by dot) sets
an approximate upper limit of 76\% of \cii\ from \hii\ regions of
$n_{\rm H} = 10^{2.8}$\percmcu\ and G$_o = 10^{3.2}$\,Habing units.
These densities and fields are in agreement with those deduced from a
different approach for assessing the \hii\ region contribution, but a
very similar PDR model, using the \cii\ and upper-state 145\,$\mu$m\ \oi\
lines \citet{simpson21}. Since the line intensity falls rapidly with
decreasing density below the critical density of about
$10^{3.5}$\percmcu, bright emission regions are very likely from
material with density above or near this. Neither solution matches
the intensity in the ground-state 63\,$\mu$m\ \oi\ line, a frequent problem
between observation and theory for \oi, although the discrepancy could
be reduced by geometries not considered in the model. Still, with the
predicted $I(\mbox{\oi}) \gtrsim I(\mbox{\cii})$, \oi\ is a major gas
coolant in the PDRs, even as \cii\ remains the dominant coolant for
the region as a whole.
The smaller bright regions identified in Fig.~\ref{fig:ciioispect} and
Table~\ref{tab:oiIntens} require even more \cii\ from \hii\ regions
and have an even larger discrepancy with $I$(\oi). Many of these
small bright regions could be structures seen edge-on that would not
fit the model's geometrical assumptions, and may have higher densities
as well.
Taking all information together, we find that approximately 50\% of
the \cii\ emission is from \hii\ regions, with the other half from
PDRs. The proton column density comparison from \cii\ and 20\,cm
radio continuum intensities is more precise; PDR modeling indicates
that this is reasonable, with bounds of about 58\% to 76\%. These
fractions agree well with values from \hii\ regions obtained from
ratios of \nii/\cii\ intensities in samples across the Galaxy, in the
Arches region of the Galactic center, in the Carina nebula, and
averaged over the nuclei of the galaxies NGC\,891 and IC\,342
\citep{goldsmith15, garcia21, oberst06, stacey10b, roellig16}.
Accounting for \cii\ emission from both PDRs and \hii\ regions is
necessary when interpreting \cii\ emission from galactic nuclei.
\subsection{Physical structure of the \sgrb\ region \label{ssec:physstruct}}
Figure~\ref{fig:sketch} provided an introductory overview of our
understanding of the \sgrb\ region; here we summarize and then explore
the information that led us to those conclusions. \cii\ emission
stretching from $0.45 \lesssim \ell \lesssim 0.70$ is continuous in
brightness and velocity (Sec.~\ref{ssec:ciispect};
Figs.~\ref{fig:sgrbregion} and \ref{fig:pv}), implying that the entire
region we imaged is physically connected. The \cii-bright area
encompasses \sgrbo\ and \gsix\ as brighter regions, and at least some
of the source in the \sgrbt\ region. As we discuss below, the
luminous star formation cores \sgrbt(N), (M), and (S) are essentially
invisible in \cii, and are likely somewhat physically separate from
the extended region, although not entirely so.
\cii\ velocities closely match the velocity field seen in the
H110$\alpha$ hydrogen recombination line, associating \cii\ with the
\hii\ region or regions seen at radio and in mid-infrared spectral
lines \citep{mehringer92, goi04, simpson18b}. At the same time,
brighter \cii\ regions are coincident with \oi\ 63 and 145\,$\mu$m\
emission peaks (Sec.~\ref{ssec:oi}; Fig.~\ref{fig:ciioispect};
\citealt{simpson21}), implying that these regions are edge-on PDRs,
consistent with some exciting stars being mixed with the \cii-emitting
material. The absence of large-scale intensity gradients in \cii\ or
70\,$\mu$m\ brightness also indicates that the exciting sources are
distributed, rather than being concentrated in one or a few compact
sources. A similar case can be made at 70\,$\mu$m: despite \sgrbt(M)'s
luminosity, it contributes only 5\% of the total 70\,$\mu$m\ intensity
across the larger \sgrb\ region, and it is not at the center of a
large-scale intensity gradient.
The general velocity field traced along the \cii\ and \hii\ surface
also matches that of the molecular cloud along the same lines of
sight, linking the \cii\ to molecular clouds as well as \hii\ regions
(Fig.~\ref{fig:ciico}, \citealt{mehringer92}). Molecular and dust
emission is more extended than the \cii-bright region in Galactic
longitude, and even more to increasing longitude
(Fig.~\ref{fig:Integ}). Lack of \cii\ self-absorption features
indicates that the region bright in \cii\ and \hii\ is on the near
(Earth) side of this cloud or clouds. The velocity field becomes
increasingly complex with increasing longitude beyond \gsix.
Of all the cores, only \sgrbt(M) has an absorption feature other than
those from the Galactic plane. Absorption is from a source smaller
than the beam that is coincident with the \sgrbt(M) continuum peak.
Figure~\ref{fig:sgrbtmspect} shows this 70\,km\,s$^{-1}$\ feature in a 16$^{\prime\prime}$\
FWHM beam (blue line, duplicated spectrum J of
Fig.~\ref{fig:spectralstack}). Both visual comparison of the spectra
in Fig.~\ref{fig:sgrbtmspect} and the two-component Gaussian fit shows
that the absorption is at a different velocity than the emission peak.
From Table~\ref{tab:fits}, the absorption is centered at
$70.0\pm 0.3$\,km\,s$^{-1}$, while the emission component is centered at
$67.3\pm 0.2$\,km\,s$^{-1}$. The absorption is shifted somewhat from the
65\,km\,s$^{-1}$\ cm-wave H$_2$CO and \hi\ absorptions \citep{mehringer95,
lang10}, although \citet{qin08} find multiple velocities in their
SMA imaging of submillimeter H$_2$CO absorption lines, with the
strongest absorptions at 68 and 76\,km\,s$^{-1}$. Considering velocity
centroids, it seems most probable that the \cii\ absorption is
associated with a compact \hii\ region within the \sgrbt(M) complex.
\citet{depree96} found that their sub-source B within the \sgrbt(M)
complex, one of the two brightest sources at 1.3\,cm, has its
H66$\alpha$ emission velocity peak at 71\,km\,s$^{-1}$\ (34\,km\,s$^{-1}$\ FWHM). Both
line center and width are in good agreement with the \cii\ absorption
parameters.
The \cii\ absorption dips close to the continuum level, either because
the \cii\ is thermalized at the dust temperature, or more likely
because surrounding emission in the beam's wings adds flux at the
absorption velocity to diminish a deeper absorption. This latter
effect is apparent in Fig.~\ref{fig:sgrbtmspect}, where the absorption
feature has completely disappeared in the 30$^{\prime\prime}$\ beam.
The difference in \cii\ emission and absorption linewidths firmly
place \sgrbt(M) on the Earth side of \sgrb's extended \cii\ emission.
If the \sgrbt(M) continuum source were behind the extended \sgrb\
cloud, then the \cii\ absorption would cover the velocity range
typical of that cloud (as it does for the Galactic plane features),
but the absorption is considerably narrower. Instead, \sgrbt(M) is
most likely embedded in the dark lane seen against \sgrb's extended
\cii\ emission (see Fig.~\ref{fig:ciiDust}). Similar cm-wave
absorption line center velocities against the \sgrbt\ cores' continua
(e.g., \citealt{mehringer93, lang10, mills18}) and spatial association
of the three \sgrbt\ cores and other star formation along the dark
dust lane \citep{ginsburg18} suggest that the molecular absorption
velocity is from the high column of material that forms the dark lane,
and that all of the cores are within the lane. The dark dust lane may
well be the physical manifestation of the ``moderate density
envelope'' \citet{huettemeister95} and others suggest surrounds the
star-forming cores.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.45\textwidth]{plotSgrB2Mspectra.pdf}
\caption{ \cii\ and $^{}$C$^{18}$O\ $J=2-1$ spectra toward \sgrbt(M), showing
the absorption velocity relative to the typical region's velocity
structure. The overlay compares \cii\ emission averaged over in
16$^{\prime\prime}$\ and 60$^{\prime\prime}$\ circular regions and $^{}$C$^{18}$O\ $J=2-1$ over
60$^{\prime\prime}$. All spectra are on the same brightness temperature scale.
Vertical lines indicate 60 and 70\,km\,s$^{-1}$.
\label{fig:sgrbtmspect}}
\end{figure}
A lack of obvious \cii\ or \hii\ emission from the lane's outer
regions suggests that it is either some distance from the stars
irradiating the \sgrb\ molecular cloud behind it, or that the dust has
sufficient optical depth to absorb most of the \cii\ emitted by the
side of the lane between the \sgrb\ stars and the Earth. Some UV must
be present to produce the \cp\ causing \sgrbt(M)'s 70\,km\,s$^{-1}$\ absorption
feature, but since absorption features are missing toward \sgrbt(N) or
(S), the material must be local to structure within \sgrb(M), as
suggested by the \citet{depree96} H66$\alpha$ velocity measurements.
Given the close association between \sgrbt(M) in velocity and
projected distance to other star formation across the overall \sgrb\
region, it seems likely that the \sgrbt\ cores and dust lane, while
somewhat separate from the larger \sgrb\ emission region, are still
physically related, and are not completely separate objects.
\subsection{``Missing'' \cii\ flux from the \sgrbt\ cores \label{ssec:missing}}
Figure~\ref{fig:sgrbtmspect} compares area-averaged \cii\ emission in
16$^{\prime\prime}$\ FWHM and arcminute-scale beams centered on \sgrbt(M). In
emission averaged over 1$^{\prime}$\ diameter areas, \cii\ and $^{}$C$^{18}$O\ have
single-component fit peak velocities of 64 and 62\,km\,s$^{-1}$, with 61\,km\,s$^{-1}$\ in
a $25^{\prime\prime} \times 25^{\prime\prime}$ area for H110$\alpha$
\citealt{mehringer93}; the absorption feature is distinctly displaced,
at 70\,km\,s$^{-1}$. Central velocity comparisons show that the emission
component of the 16$^{\prime\prime}$\ spectrum is characteristic of the area
surrounding \sgrbt(M), rather than of the source seen in absorption.
Lineshape similarities away from the absorption feature
in Fig.~\ref{fig:sgrbtmspect} further imply that the core is a minor
contributor to the region's \cii\ luminosity at most. Indeed, it has
no obviously detectable \cii\ emission.
The insignificant amount of \cii\ emission from \sgrbt(M) and the
other luminous cores is especially striking, since they are regions
with extreme star formation densities. A low \cii/FIR intensity ratio
compared with that in galactic star formation regions and starburst
galaxies (e.g., Sec.~\ref{ssec:comparisons}, \citealt{stacey91,
stacey10, delooze14, pineda14, herreracamus15, herreracamus18i}), is
reminiscent of the ``\cp\ deficit'' first discovered in (ultra)
luminous infrared galaxies, or (U)LIRGs, by \citet{fischer99}. With
an FIR luminosity of at least a ${\rm few} \times 10^6$\lsun (e.g.,
\citealt{odenwald84, lis91, gordon93, schmiedeke16}) and a maximum
linear size of the star formation region of about 1\,pc (e.g.,
\citealt{ginsburg18}), a conservative lower limit on its luminosity
density exceeds a ${\rm few} \times 10^{12}$\lsun\,kpc$^{-2}$. Even
allowing for substantial beam dilution observations averaging multiple
similar regions spread over large scales in distant galaxies, this is
in the range of beam-averaged luminosity surface brightnesses for
ULIRGs of $> 10^{10}$\,M$_\odot$\,kpc$^{-2}$ (e.g.,
\citealt{herreracamus18i}).
This comparison of \sgrbt(M) with (U)LIRGs is not new (e.g.,
\citealp{goi04, kamenetzky14, santamaria21}); our contribution to the
discussion is the combination of velocity resolution and spatial
dynamic range that definitively shows that the emitted \cii\ from the
cores themselves have negligible intensity, even as the surrounding
area provides \cii\ more typical of extended star forming regions.
A number of explanations have been advanced to explain the low
\cii/FIR ratio with increasing FIR: high optical depth, dust-bounded
\hii\ regions that emit strong FIR but little \cii, intense UV fields
that destroy \cp, grain charging that reduces the number of
photoelectrons exciting PDRs, and others (\citealt{luhman98,
malhotra01}; see also recent summaries in \citealt{herreracamus18ii}
and \citealt{santamaria21} for more details and references). In their
study of a wide range of IR-luminous galactic nuclei,
\citet{herreracamus18ii} found that the dominant mechanism suppressing
\cii\ in most (U)LIRGs is a reduction in photoelectric heating
efficiency as the ionization parameter increases; they note that
optical depth may be important as well. Comparisons of \cii\ emission
intensities within and just beyond the edge of the dark lane near
\sgrbt(M) provide estimates of lane attenuation factors of 3--5 for
\cii\ emission. Density gradients around the cores themselves will
increase the attenuation toward the most luminous regions, so
attenuation alone can quite plausibly reduce any \cii\ intensity from
dense cores by more than an order of magnitude.
To the extent that \sgrb\ is representative of regions around extreme
star formation in galactic nuclei, we must be seeing a mixture of
extended and compact star formation in external galaxies, with
weighting toward extended and relatively unobscured regions, and away
from the densest star formation cores. Proximity and a wealth of data
at many wavelengths all make \sgrbt(M) a valuable local region to
continue detailed investigations into the physical mechanisms
operating within (U)LIRG star formation regions.
\vspace{10mm}
\subsection{Implications for Galactic center star formation
models \label{ssec:starformation}}
There is debate about the origin of star formation across \sgrb.
\citet{mehringer92} proposed that \sgrbo's extended ionization comes
from a dissipating cluster of mostly O stars that formed earlier than
the clusters and compact \hii\ regions in \gsix\ and \sgrbt.
\citet{goi04} and \citet{simpson18b} also found that late-O stars are
consistent with most of the amount and degree of ionized material in
the \hii\ region that lies between the \sgrb\ background dust and
molecular cloud and the Earth.
The \sgrbt\ cores' more recent burst of star formation suggests some
kind of trigger, most probably associated with a special position
along the region's orbit. The existence of the dark lane and its
association with the cores suggests that shock-driven propagating star
formation across \sgrb\ is at play. Narrow dust lanes are common
signs of shock-concentrated material in other galaxies with nuclear
bars and in interacting galactic nuclei.
Star formation triggered by cloud-cloud collisions near \sgrbt\ had
been suggested by \citet{hasegawa94}, among others over the years,
supported by observations of shock-tracing SiO (e.g.,
\citealt{huettemeister95, armijos20, santamaria21}). The \cii\ p-v
diagram (Fig.~\ref{fig:pv}) shows that linewidths broaden
substantially with increasing longitude from \gsix, possibly signaling
accelerated gas motions from interactions between the \sgrb\ region
and the material to yet higher $\ell$ traced in CO and 160\,$\mu$m\
emission (Fig.~\ref{fig:Integ}). \citet{simpson18b} also concluded
that shocks or other mechanisms are needed to produce the very
energetic radiation needed to explain the high excitation species they
observe in the \sgrb\ \hii\ region. An alternative to shocks that
they propose, a tidal tail of hot stars from the dissolving Quintuplet
cluster, as \citet{habibi14}'s modeling might suggest, seems a less
attractive explanation than shocks because it requires a group of
stars with just the right geometry to excite the \sgrb\ area of the
molecular cloud.
There are two popular dynamical models for triggered star formation in
gas orbiting the Galactic center: One proposes that gas cloud
compression by tidal interactions at orbital pericenter (e.g.,
\citealt{kruijssen15, barnes17}) dominates. The other proposes
cloud-cloud collisions close to the apocenters (e.g.,
\citealt{binney91, sormani20, tress20}) as the chief mechanism.
The first model accounts for the spatial differences in star formation
between the extended star formation in \sgrbo\ and the highly
concentrated clusters of the \sgrbt\ cores by proposing that the two
regions could be on very different positions along their orbits. In
this model, \sgrbt\ is on the near side of its orbit around the center
after a relatively recent encounter with the central mass
concentration that compressed it and triggered star formation.
\sgrbo, on a similar orbit, passed the center well before \sgrbt, and
has now passed its apocenter and is returning toward the center on the
far side of its orbit. Its stars have disrupted their birthplaces and
are drifting apart, accounting for the more extended region. The
positional alignment between \sgrbo\ and the \sgrbt\ cores is by
chance. As \citet{simpson1819} and \citet{simpson18b} point out,
however, one problem with this explanation is that the difference in
stellar ages between \sgrbo\ and \sgrbt\ is larger than orbital
timescales predict. If the \sgrbt\ cores are indeed associated with
the dark lane, another problem is in explaining how this long, narrow
structure would be produced and then persist over a substantial
fraction of its orbit around the Galactic center. Typical velocity
widths in molecular absorptions features toward the \sgrbt\ cores
imply a dark lane lifetime well under $10^6$\,yr, considerably shorter
than the orbital time since periapse.
In the second model, \sgrbo\ and \sgrbt\ are physically related, and
close to an apocenter of their orbit. Cloud-cloud collisions between
individual gas clouds or streams flowing from the outer galaxy into
the central region along $x_1$ orbits \citep{x1x2} and clouds on the
$x_2$ orbits around the center \citep{sormani20, tress20}, or
interactions between gas on cusped or crossing orbits
\citep{jenkins94} of the $x_2$ orbits themselves, then trigger star
formation. Association of vigorous star formation with the dark dust lane,
and the increasingly complex \cii\ velocity field with increasing
latitude across \sgrb\ are qualitatively consistent with this
model. Such a model for \sgrb, which accommodates different stages of
star formation across the associated regions that comprise \sgrb, is
very appealing.
\section{Summary \label{sec:summary}}
Our large-scale, fully velocity-resolved spectroscopic imaging of
\cii\ has revealed:
\begin{enumerate}[noitemsep
\item With 12\% of the total \cii\ flux from 6\% of the area of our
large-scale image covering all of the \cii-bright Central Molecular
Zone, \sgrb\ is a major contributor to the entire Galactic center's
\cii\ luminosity.
\item The \sgrb\ region extends as a continuous, coherent structure
that encompasses the luminous components identified in early radio
continuum maps of the Galactic center. \sgrb\ starts near $\ell
\approx 0.44\degr$, contains \sgrbo\ ($\ell \approx 0.48\degr$),
runs along the Galactic plane past \gsix, and continues behind and
beyond \sgrbt\ ($\ell \approx 0.66\degr$) to $\ell \gtrsim
0.72\degr$. This is a span of some 34\,pc in longitude, with width
of some 15\,pc in latitude. Fainter emission extends further,
particularly to lower $\ell$ toward Sgr\,A. Many of the region's
components appear to be part of the same physical structure,
although the \sgrbt\ region appears as a foreground region that may
or may not have a physical connection to the larger region.
\item The spectra of \cii\ in the \sgrb\ region show two main velocity
components at $\sim$50\,km\,s$^{-1}$\ and $\sim$90\,km\,s$^{-1}$ , with additional
emission from $-30$ to $+130$\,km\,s$^{-1}$. The dominant emission is
centered at $\sim$50\,km\,s$^{-1}$, in agreement with the region's
velocity in molecular and radio recombination lines. The secondary
component at $\sim$90\,km\,s$^{-1}$\ is mainly associated with extended,
moderately excited, molecular material. Individual components have
typical linewidths of 30--60\,km\,s$^{-1}$\ FWHM, indicating emission from a
highly turbulent medium. A linear gradient of the mean velocity
along \sgrb's body runs from $\sim$40 to $\sim$70\,km\,s$^{-1}$\ in the same
sense as Galactic rotation.
\item Emission across the \sgrb\ region is spatially complex. Arcs,
ridges, and other structures have nearly exact counterparts in
70\,$\mu$m\ and 20\,cm continua \citep{molinari16, mehringer92, lang10}.
Other tracers share the same general distribution, but have
significant differences from \cii\ and each other. 160\,$\mu$m\
continuum and CO line emission highlight extended emission from
extended background clouds, while 21\,$\mu$m\ and 8\,$\mu$m\ continua
highlight compact sources and structure.
\item The absence of obvious self-absorption in \cii\ spectra, spatial
agreement with 70\,$\mu$m\ and 20\,cm radio continuum, and disparity with
the more extended CO and 160\,$\mu$m\ spatial distributions, indicate
that the \cii\ is emitted from the near surface of a larger
molecular cloud or cloud complex.
\item Comparison of a variety of tracers indicates that 70\,$\mu$m, \cii,
and 20\,cm continua are all excellent tracers of the UV flux
produced by young stars across the \sgrb\ region. Velocity
information in \cii\ is invaluable for separating physical
components along the line of sight. Longer-wavelength UV tracers
are especially important for the Galactic center and other edge-on
observations of galactic nuclei where even infrared obscuration is
influential.
\item Velocity resolved \cii/\oi\ flux ratios from a sample of bright
regions are close to those from large scale measurements
\citep{goi04, simpson21}, suggesting that PDRs are present across
the entire \sgrb\ region. The lineshape we measure for the \oisw\
line is similar to the $+50$\,km\,s$^{-1}$\ \cii\ component, indicating that
it is mainly associated with \sgrb\ itself, rather than other clouds
along the line of sight. Its lineshape also indicates that the \oi\
line is not significantly affected by self-absorption, and can be a
significant gas coolant.
\item PDR modeling places bounds of approximately 58\% to 76\% of
\cii\ flux from \hii\ regions toward \sgrb, with \cii\ and radio
continuum fluxes suggesting a fraction of more than 50\%. These
values agree with those obtained with an independent method using
\nii/\cii\ intensity ratios. Emission from both PDRs and \hii\
regions are important in interpreting \cii\ emission from galactic
nuclei.
\item Distributed star formation is common across \sgrb. The vast
majority of UV illumination comes from sources other than the
\sgrbt\ embedded cores, which produce only 5\% of
the 70\,$\mu$m\ flux across the region.
\item In \cii, the \sgrbt\ cores appear only as enhanced \cii\
absorption against their 158\,$\mu$m\ \sgrbt(M) continuum. \sgrbt(M) may
be an analog of the intense star formation regions in ULIRGs, which
exhibit a ``\cp\ deficit'' (e.g., \citealt{fischer99}) when compared
to their star formation rates. This region is an excellent local
source to study the phenomenon in detail.
\item The \sgrbt(M), (N), and (S) cores appear to be objects within
with a dark dust lane in front of the larger \sgrb\ region.
Velocity and positional coincidence suggest that the cores and lane
are still dynamically associated with the larger region, even as
they are somewhat separated from it.
\item Taken together, our results support triggered star formation
models that invoke local cloud-cloud collisions close to apocenters
of orbits in a barred potential (e.g., \citealt{binney91,
sormani20}), rather than models that invoke tidal compression at
pericenter passage close to a central mass concentration (e.g.,
\citealt{kruijssen15, barnes17}).
\end{enumerate}
\acknowledgments We thank the many people who have made this joint
U.S.-German project possible, including the SOFIA observatory staff
and Science Mission Operations former directors E.\ Young and H.\
Yorke. We also thank M.\ Wolfire and M.\ Pound for their insights and
assistance with PDR modeling, and C.\ Lang for access to her 20\,cm
data. We thank an anonymous referee for suggestions and requests that
substantially improved this paper.
This work is based on observations made with the NASA/DLR
Stratospheric Observatory for Infrared Astronomy (SOFIA). SOFIA is
jointly operated by the Universities Space Research Association,
Inc. (USRA), under NASA contract NNA17BF53C, and the Deutsches SOFIA
Institut (DSI) under DLR contract 50 OK 0901 to the University of
Stuttgart. Financial support for this work was provided by NASA
through awards 05-0022 and 06-0173 issued by USRA, the
Max-Planck-Institut f\"ur Radioastronomie and the Deutsche
Forschungsgemeinschaft (DFG) through the SFB~956 program award to
MPIfR and the Universit\"at zu K\"oln. R.S.\ acknowledges support by
the French ANR and the German DFG through the project ``GENESIS''
(ANR-16-CE92-0035-01/DFG1591/2-1).
This research made use of {\em Spitzer} data from the NASA/IPAC
Infrared Science Archive, which is operated by the Jet Propulsion
Laboratory, California Institute of Technology, under contract with
the National Aeronautics and Space Administration; from the {\it
Herschel} Science Archive, which is maintained by ESA at the
European Space Astronomy Centre; and data products from the Midcourse
Space Experiment, whose data processing was funded by the Ballistic
Missile Defense Organization with additional support from NASA Office
of Space Science.
\vspace{3mm}
\facilities{SOFIA (\grt)} \citep{young12, risacher18}
\software{GILDAS (\citealt{pety05, 2013ascl.soft05010G}),
SAOImageDS9 (\citealt{joye03, 2000ascl.soft03002S}),
R (\citealt{r}),
PhotoDissociation Region Toolbox (\citealt{pound08, kaufman06})
\&
Baseline correction using splines
(\url{https://github.com/KOSMAsubmm/kosma\_gildas\_dlc};
\citealt{higgins11, kester14, higgins21})
}
\bibliographystyle{aasjournal}
|
2,869,038,154,757 | arxiv | \section{Introduction}
In this paper we address the question of obtaining a priori estimates, positivity, boundary behavior, Harnack inequalities, and regularity for a suitable class of weak solutions of nonlinear nonlocal diffusion equations of the form:\vspace{-2mm}
\begin{equation}\label{FPME.equation}
\partial_t u+\mathcal{L}\,F (u)=0 \qquad\mbox{posed in }Q=(0,\infty)\times \Omega\,,\vspace{-2mm}
\end{equation}
where $\Omega\subset \mathbb{R}^N$ is a bounded domain with $C^{1,1}$ boundary, $N\ge 2$\footnote{Our results work also in dimension $N=1$ if the fractional exponent (that we shall introduce later) belongs to the range $0<s<1/2$. The interval $1/2\le s<1$ requires some minor modifications that we prefer to avoid in this paper.}, and $\mathcal{L}$ is a linear operator representing diffusion of local or nonlocal type, the prototype example being the fractional Laplacian (the class of admissible operators will be precisely described below).
Although our arguments hold for a rather general class of nonlinearities $F:\mathbb{R} \to \mathbb{R}$, for the sake of simplicity we shall focus on the model case $F(u)=u^m$ with $m>1$.
The use of nonlocal operators in diffusion equations reflects the need to model the presence of long-distance effects not included in evolution driven by the Laplace operator, and this is well documented in the literature. The physical motivation and relevance of the nonlinear diffusion models with nonlocal operators has been mentioned in many references, see for instance \cite{AC,BV2012,BV-PPR1,DPQRV1,DPQRV2,Vaz2014}.
Because $u$ usually represents a density, all data and solutions are supposed to be nonnegative. Since the problem is posed in a bounded domain we need boundary or external conditions that we assume of Dirichlet type.
This kind of problems has been extensively studied when $\mathcal{L}=-\Delta$ and $F(u)=u^m$, $m>1$,
in which case the equation becomes the classical Porous Medium Equation \cite{VazBook,DK0, DaskaBook, JLVmonats}.
Here, we are interested in treating nonlocal diffusion operators, in particular fractional Laplacian operators. Note that, since we are working on a bounded domain, the concept of fractional Laplacian operator admits several non-equivalent versions, the best known being the Restricted Fractional Laplacian (RFL), the Spectral Fractional Laplacian (SFL), and the Censored Fractional Laplacian (CFL);
see Section \ref{ssec.examples} for more details. We use these names because they already appeared in some previous works \cite{BSV2013, BV-PPR2-1}, but we point out that RFL is usually known as the Standard Fractional Laplacian, or plainly Fractional Laplacian, and CFL is often called Regional Fractional Laplacian.
The case of the SFL operator with $F(u)=u^m$, $m>1$, has been already studied by the first and the third author in \cite{BV-PPR1,BV-PPR2-1}. In particular, in \cite{BV-PPR2-1} the authors presented a rather abstract setting where they were able to treat not only the usual fractional Laplacians but also a large number of variants that will be listed below for the reader's convenience. Besides, rather general increasing nonlinearities $F$ were allowed. The basic questions of existence and uniqueness of suitable solutions for this problem were solved in \cite{BV-PPR2-1} in the class of `weak dual solutions', an extesion of the concept of solution introduced in \cite{BV-PPR1} that has proved to be quite flexible and efficient. A number of a priori estimates (absolute bounds and smoothing effects) were also derived in that generality.
Since these basic facts are settled, here we focus our attention on the finer aspects of the theory, mainly sharp boundary estimates and decay estimates. Such upper and lower bounds will be formulated in terms of the first eigenfunction $\Phi_1$ of $\mathcal{L}$, that under our assumptions will satisfy $\Phi_1\asymp \dist(\cdot, \partial\Omega)^\gamma$ for a certain characteristic power $\gamma\in (0,1]$ that depends on the particular operator we consider. Typical values are $\gamma=s$ (SFL), $\gamma=1$ (RFL), and $\gamma=s-1/2$ for $s>1/2$ (CFL), cf. Subsections \ref{ssec.examples} and \ref{sec.examples}. As a consequence, we get various kinds of local and global Harnack type inequalities.
It is worth mentioning that some of the boundary estimates that we obtain for the parabolic case are essentially elliptic in nature. The study of this issue for stationary problems is done in a companion paper \cite{BFV-Elliptic}. This has the advantage that many arguments are clearer, since the parabolic problem is more complicated than the elliptic one. Clarifying such difference is one of the main contributions of our present work.
Thanks to these results, in the last part of the paper we are able to prove both interior and boundary regularity, and to find the large-time asymptotic behavior of solutions.
Let us indicate here some notation of general use. The symbol $\infty$ will always denote $+\infty$. Given $a,b$, we use the notation $a\asymp b$ whenever there exist universal constants $c_0,c_1>0$ such that $c_0\,b\le a\le c_1 b$\,. We also use the symbols $a\vee b=\max\{a,b\}$ and $a\wedge b=\min\{a,b\}$.
We will always consider bounded domains $\Omega$ with boundary of class $C^{2}$. In the paper we use the short form `solution' to mean `weak dual solution', unless differently stated.
\subsection{Presentation of the results on sharp boundary behaviour}\label{ssec.results.boundary}
\noindent $\bullet$ A basic principle in the paper is that the sharp boundary estimates depend not only on $\mathcal{L}$ but also on the behavior of the nonlinearity $F(u)$ near $u=0$, i.e., in our case, on the exponent $m>1$. The elliptic analysis performed in the companion paper \cite{BFV-Elliptic} combined with some standard arguments will allow us to prove that, in {\em all} cases, $u(t)$ approaches the separate-variable solution ${\mathcal U}(x,t)=t^{-\frac1{m-1}}S(x)$ in the sense that
\begin{equation}\label{asymp.intro}
\left\|t^{\frac{1}{m-1}}u(t,\cdot)- S\right\|_{\LL^\infty(\Omega)}\xrightarrow{t\to\infty}0,
\end{equation}
where $S$ is the solution of the elliptic problem
(see Theorems \ref{Thm.Elliptic.Harnack.m} and \ref{Thm.Asympt.0}).
The behaviour of the profile $ S(x)$ is shown to be, when $2sm\ne \gamma(m-1)$,
\begin{equation}\label{as.sep.var}
S(x)\asymp \Phi_1(x)^{\sigma/m}, \qquad \sigma:=\min\left\{1,\frac{2sm}{\gamma(m-1)}\right\}.
\end{equation}
Thus, the behavior strongly depends on the new parameter $\sigma$, more precisely, on whether this parameter is equal to $1$ or less than $1$. As we shall see later, $\sigma$ encodes the interplay between the ``elliptic scaling power'' $2s/(m-1)$, the ``eigenfunction power'' $\gamma$, and the ``nonlinearity power'' $m$. When $2sm =\gamma(m-1)$ we have $\sigma=1$, but a logarithmic correction appears:
\begin{equation}\label{as.sep.var.1}
S(x)\asymp \Phi_1(x)^{1/m}\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}\,.
\end{equation}
\noindent $\bullet$ This fact and the results in \cite{BFR} prompted us to look for estimates of the form
\begin{equation}\label{intro.1b}
c_0(t)\frac{\Phi_1^{\sigma/m}(x_0)} {t^{\frac1{m-1}}}
\le u(t,x_0) \le c_1\frac{ \Phi_1^{\sigma/m}(x_0)} {t^{\frac1{m-1}}}\qquad \text{for all $t>0$, $x_0\in\Omega$,}
\end{equation}
where $c_0(t)$ and $c_1$ are positive and independent of $u$, eventually with a logarithmic term appearing when $2sm =\gamma(m-1)$, as in \eqref{as.sep.var.1}. We will prove in this paper that the upper bound holds for the three mentioned Fractional Laplacian choices, and indeed for the whole class of integro-differential operators we will introduce below, cf. Theorem \ref{thm.Upper.PME.II}. Also, separate-variable solutions saturate the upper bound.
The issue of the validity of a lower bound as in \eqref{intro.1b} is instead much more elusive.
A first indication for this is the introduction of a function $c_0(t)$ depending on $t$, instead of a constant. This wants to reflect the fact that the solution may take some time to reach the boundary behaviour that is expected to hold uniformly for large times. Indeed, recall that in the classical PME \cite{Ar-Pe,JLVmonats, VazBook}, for data supported away from the boundary, some `waiting time' is needed for the support to reach the boundary.
\medskip
\noindent $\bullet$ As proved in \cite{BFR}, the stated lower bound holds for the RFL with $c_0(t)\sim (1\wedge t)^{m/(m-1)}.$
In particular, in this nonlocal setting, infinite speed of propagation holds.
Here, we show that this holds also for the CFL and a number of other operators, cf. Theorem \ref{Thm.lower.B}.
Note that for the RFL and CFL we have $2sm>\gamma(m-1)$, in particular $\sigma=1$ which simplifies formula \eqref{intro.1b}.\\ A combination of an upper and a lower bound with matching behaviour (with respect to $x$ and $t$) will be called a {\sl Global Harnack Principle}, and holds for all $t>0$ for these operators, cf. Theorems \ref{thm.GHP.PME.I} and \ref{thm.GHP.PME.II}.
\medskip
\noindent $\bullet$ When $\mathcal{L}$ is the SFL, we shall see that the lower bound may fail. Of course, solutions by separation of variables satisfy the matching estimates in \eqref{intro.1b} (eventually with a extra logarithmic term in the limit case, as in \eqref{as.sep.var.1}), but it came as a complete surprise to us that for the SFL the situation is not the same for ``small'' initial data. More precisely:
\noindent(i) We can prove that the following bounds always hold for all times:
\begin{equation}\label{intro.1}
c_0\left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\Phi_1(x_0)} {t^{\frac1{m-1}}}
\le u(t,x_0) \le c_1\frac{ \Phi_1^{\sigma/m}(x_0)} {t^{\frac1{m-1}}}\,,
\end{equation}
(when $2sm=\gamma(m-1)$, a logarithmic correction $\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}$ appears in the right hand side),
cf. Theorem \ref{thm.Lower.PME}.
These are non-matching estimates.
\noindent(ii) For $2sm>\gamma(m-1)$, the sharp estimate \eqref{intro.1b} holds for any nonnegative nontrivial solution for large times $t \geq t_*$, cf. Theorem \ref{thm.Lower.PME.large.t}.
\noindent(iii) \textbf{Anomalous boundary behaviour.} Consider now the SFL with $\sigma<1$ (resp. $2sm=\gamma(m-1)$).\footnote{Since for the SFL $\gamma=1$, we have $\sigma<1$ if and only if
$$
0<s<s_*:=\frac{m-1}{2m}<\frac{1}{2}.
$$
Note that $s_*\to 0$ as we tend to the linear case $m=1$, so this exceptional regime dooes not appear for linear diffusions, both fractional and standard.}
In this case we can find initial data for which the upper bound in \eqref{intro.1} is not sharp. Depending on the initial data, there are several possible rates for the long-time behavior near the boundary. More precisely:
\begin{enumerate}
\item[(a)] When $u_0\leq A\,\Phi_1$, then $u(t)\le F(t)\Phi_1^{1/m} \ll \Phi_1^{\sigma/m}$ (resp. $\Phi_1^{1/m} \ll \Phi_1^{1/m}\left(1+|\log\Phi_1 |\right)^{1/(m-1)}$) for all times, see Theorem \ref{prop.counterex}. In particular
\begin{equation}\label{limit.intro}
\lim_{x\to \partial\Omega}\frac{u(t,x)}{\Phi_1(x)^{\sigma/m}}= 0 \quad \Bigl(\text{resp.} \lim_{x\to \partial\Omega}\frac{u(t,x)}{\Phi_1(x)^{1/m}\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}}= 0\Bigr)
\end{equation}
for any $t>0.$
\item[(b)] When $u_0\leq A\,\Phi_1^{1-2s/\gamma}$ then $u(t)\le F(t)\Phi_1^{1-2s/\gamma}$ for small times, see Theorem \ref{thm.Upper.PME.III}. Notice that when $\sigma<1$ we have always $1-\frac{2s}{\gamma}>\sigma/m$.
This sets a limitation to the improvement of the lower bound, which is confirmed by another result: In Theorem \ref{prop.counterex2} we show that lower bounds of the form $u(T,x)\geq \underline{\kappa}\Phi_1^\alpha(x)$ for data $u_0(x)\le A\Phi_1(x)$ are possible only for $\alpha\ge 1-2s/\gamma$.
\item[(c)] On the other hand, for ``large'' initial data, Theorem \ref{thm.GHP.PME.II} shows that the desired matching estimates from above and below hold.
\end{enumerate}
After discovering this strange boundary behavior, we looked for numerical confirmation. In Section \ref{sec.numer} we will explain the numerical results obtained in \cite{numerics}. Note that, if one looks for universal bounds independent of the initial condition, Figures 2-3 below seem to suggest that the bounds provided by \eqref{intro.1} are optimal for all times and all operators.
\medskip
\noindent $\bullet$ The current interest in more general types of nonlocal operators led us to a more general analysis where the just explained alternative has been extended to a wide class of integro-differential operators, subject only to a list of properties that we call (A1), (A2), (L1), (L2), (K2), (K4); a number of examples are explained in Section \ref{sec.hyp.L}. These general classes appear also in the study of the elliptic problem \cite{BFV-Elliptic}.
\subsection{Asymptotic behaviour and regularity}
Our quantitative lower and upper estimates admit a formulation as local or global Harnack inequalities. They are used at the end of the paper to settle two important issues.
\noindent\textbf{Sharp asymptotic behavior. }Exploiting the techniques in \cite{BSV2013},
we can prove a sharp asymptotic behavior for our nonnegative and nontrivial solutions when the upper and lower bound have matching powers. Such sharp results hold true for a quite general class of local and nonlocal operators. A detailed account is given in Section \ref{sec.asymptotic}.
\smallskip
\noindent\textbf{Regularity. }By a variant of the techniques used in \cite{BFR},
we can show interior H\"older regularity.
In addition, if the kernel of the operator satisfies some suitable continuity assumptions, we show that solutions are classical in the interior and are H\"older continuous up to the boundary if the upper and lower bound have matching powers.
We refer to Section \ref{sect.regularity} for details.
\section{General class of operators and their kernels}\label{sec.hyp.L}
The interest of the theory developed here lies both in the sharpness of the results and in the wide range of applicability. We have just mentioned the most relevant examples appearing in the literature, and more are listed at the end of this section. Actually, our theory applies to a general class of operators with definite assumptions, and this is what we want to explain now.
Let us present the properties that have to be assumed on the class of admissible operators.
Some of them already appeared in \cite{BV-PPR2-1}.
However, to further develop our theory, more hypotheses need to be introduced.
In particular, while \cite{BV-PPR2-1} only uses the properties of the Green function, here we shall make some assumptions also on the kernel of $\mathcal{L}$ (whenever it exists). Note that assumptions on the kernel $K$ of $\mathcal{L}$
are needed for the positivity results, because we need to distinguish between the local and nonlocal cases. The study of the kernel $K$ is performed in Subsection \ref{ss2.2}.
For convenience of reference, the list of used assumptions is (A1), (A2), (K2), (K4), (L1), (L2).
The first three are assumed in all operators $\mathcal{L}$ that we use.
\medskip
\noindent $\bullet$ {\bf Basic assumptions on $\mathcal{L}$.}
The linear operator $\mathcal{L}: \dom(\mathcal{L})\subseteq\LL^1(\Omega)\to\LL^1(\Omega)$ is assumed to be densely defined and sub-Markovian, more precisely, it satisfies (A1) and (A2) below:
\begin{enumerate}
\item[(A1)] $\mathcal{L}$ is $m$-accretive on $\LL^1(\Omega)$;
\item[(A2)] If $0\le f\le 1$ then $0\le \ee^{-t\mathcal{L}}f\le 1$.
\end{enumerate}
Under these assumption, in \cite{BV-PPR2-1}, the first and the third author proved existence, uniqueness, weighted estimates, and smoothing effects.
\medskip
\noindent $\bullet$ {\bf Assumptions on the kernel.}
Whenever $\mathcal{L}$ is defined in terms of a kernel $K(x,y)$ via the formula
\begin{equation*}
\mathcal{L} f(x)=P.V.\int_{\mathbb{R}^N} \big(f(x)-f(y)\big)\,K(x,y)\,{\rm d}y\,,
\end{equation*}
assumption (L1) states that there exists $\underline{\kappa}_\Omega>0$ such that
\[\tag{L1}
\inf_{x,y\in \Omega}K(x,y)\ge \underline{\kappa}_\Omega>0\,.
\]
We note that condition holds both for the RFL and the CFL, see Section \ref{ssec.examples}.
\noindent- Whenever $\mathcal{L}$ is defined in terms of a kernel $K(x,y)$ and a zero order term via the formula
\[
\mathcal{L} f(x)=P.V.\int_{\mathbb{R}^N} \big(f(x)-f(y)\big)\,K(x,y)\,{\rm d}y + B(x)f(x),
\]
assumptions (L2) states that
\[\tag{L2}
K(x,y)\ge c_0\p(x)\p(y),\quad c_0>0, \qquad\mbox{and}\quad B(x)\ge 0,
\]
where, from now on, we adopt the notation $\delta(x):=\dist(x, \partial\Omega)$.
This condition is satisfied by the SFL in a stronger form, see Section \ref{ss2.2} and Lemma \ref{Lem.Spec.Ker}.
\medskip
\noindent $\bullet$ {\bf Assumptions on $\mathcal{L}^{-1}$.} In order to prove our quantitative estimates, we need to be more specific about the operator $\mathcal{L}$. Besides satisfying (A1) and (A2), we will assume that it has a left-inverse $\mathcal{L}^{-1}: \LL^1(\Omega)\to \LL^1(\Omega)$ that can be represented by a kernel ${\mathbb G}$ (the letter ``G'' standing for Green function) as
\[
\mathcal{L}^{-1}[f](x)=\int_\Omega {\mathbb G}(x,y)f(y)\,{\rm d}y\,,
\]
where ${\mathbb G}$ satisfies the following assumption, for some $s\in (0,1]$:
There exist constants $\gamma\in (0,1]$ and $c_{0,\Omega},c_{1,\Omega}>0$ such that, for a.e. $x,y\in \Omega$,
\[\tag{K2}
c_{0,\Omega}\,\p(x)\,\p(y) \le {\mathbb G}(x,y)\le \frac{c_{1,\Omega}}{|x-y|^{N-2s}}
\left(\frac{\p(x)}{|x-y|^\gamma}\wedge 1\right)
\left(\frac{\p(y)}{|x-y|^\gamma}\wedge 1\right).
\]
(Here and below we use the labels (K2) and (K4) to be consistent with the notation in \cite{BV-PPR2-1}.)
Hypothesis (K2) introduces an exponent $\gamma$ which is a characteristic of the operator and will play a big role in the results.
Notice that defining an inverse operator $\mathcal{L}^{-1}$ implies that we are taking into account the Dirichlet boundary conditions. See more details in Section 2 of \cite{BV-PPR2-1}.
\medskip
\noindent - The lower bound in (K2) is weaker than the known bounds on the Green function for many examples under consideration; indeed, the following stronger estimate holds in many cases:
\[\tag{K4}
{\mathbb G}(x,y)\asymp \frac{1}{|x-y|^{N-2s}}
\left(\frac{\p(x)}{|x-y|^\gamma}\wedge 1\right)
\left(\frac{\p(y)}{|x-y|^\gamma}\wedge 1\right)\,.
\]
\noindent\textbf{Remarks. }(i) The labels (A1), (A2), (K1), (K2), (K4) are consistent with the notation in \cite{BV-PPR2-1}. The label (K3) was used to mean hypothesis (K2) written in terms of $\Phi_1$ instead of $\p$.\\
(ii) In the classical local case $\mathcal{L}=-\Delta$, the Green function ${\mathbb G}$ satisfies (K4) only when $N\geq 3$, as the formulas slightly change when $N=1,2$. In the fractional case $s \in (0,1)$
the same problem arises when $N=1$ and $s \in [1/2,1)$. Hence, treating also these cases would require a slightly different analysis based on different but related assumptions on ${\mathbb G}$. Since our approach is very general, we expect it to work also in these remaining cases without any major difficulties. However, to simplify the presentation, from now on we assume that
$$
\text{either $N\geq 2$ and $s\in(0,1),\qquad$ or $N=1$ and $s \in (0,1/2)$.}
$$
\noindent\textbf{The role of the first eigenfunction of $\mathcal{L}$. }We have shown in \cite{BFV-Elliptic} that, under assumption (K1), the operator $\mathcal{L}$ is compact, it has a discrete spectrum, and a first nonnegative bounded eigenfunction $\Phi_1$; assuming also (K2), we have that
\begin{equation}\label{Phi1.est}
\Phi_1(x)\asymp \p(x)=\dist(x,\partial\Omega)^\gamma\qquad\mbox{for all }x\in \overline{\Omega}.
\end{equation}
Hence, $\Phi_1$ encodes the parameter $\gamma$ that takes care of describing the boundary behavior. We recall that we are assuming that the boundary of $\Omega$ is smooth enough, for instance $C^{1,1}$.
\noindent\textbf{Remark. }We note that our assumptions allow us to cover all the examples of operators described in Sections \ref{ssec.examples} and \ref{sec.examples}.
\subsection{Main examples of operators and properties}
\label{ssec.examples}
When working in the whole $\mathbb R^N$, the fractional Laplacian admits different definitions that can be shown to be all equivalent. On the other hand, when we deal with bounded domains, there are at least three different operators in the literature, that we call the Restricted (RFL), the Spectral (SFL) and the Censored Fractional Laplacian (CFL). We will show below that these different operators exhibit quite different behaviour, so the distinction between them has to be taken into account.
Let us present the statement and results for the three model cases, and we refer to Section \ref{sec.examples} for further examples. Here, we collect the sharp results about the boundary behavior, namely the Global Harnack inequalities from Theorems \ref{thm.GHP.PME.I}, \ref{thm.GHP.PME.II}, and \ref{thm.GHP.PME.III}.
\medskip
\noindent\textit{The parameters $\gamma$ and $\sigma$.} The strong difference between the various operators $\mathcal{L}$ is reflected in the different boundary behavior of their nonnegative solutions. We will often use the exponent $\gamma$, that represents the boundary behavior of the first eigenfunction $\Phi_1 \asymp \dist(\cdot,\partial\Omega)^\gamma$, see~\cite{BFV-Elliptic}.
Both in the parabolic theory of this paper and the elliptic theory of paper \cite{BFV-Elliptic}
the parameter $\sigma=\min\left\{1, \frac{2sm}{\gamma(m-1)} \right\}$ introduced in \eqref{as.sep.var} plays a big role.
\subsubsection{The RFL} We define the fractional Laplacian operator acting on a bounded domain by using the integral representation on the whole space in terms of a hypersingular kernel, namely
\begin{equation}\label{sLapl.Rd.Kernel}
(-\Delta_{\mathbb{R}^N})^{s} g(x)= c_{N,s}\mbox{
P.V.}\int_{\mathbb{R}^N} \frac{g(x)-g(z)}{|x-z|^{N+2s}}\,dz,
\end{equation}
where $c_{N,s}>0$ is a normalization constant, and we ``restrict'' the operator to functions that are zero outside $\Omega$. We denote such operator by $\mathcal{L}=(-\Delta_{|\Omega})^s$\,, and call it the \textit{restricted fractional Laplacian}\footnote{In the literature this is often called the fractional Laplacian on domains, but this simpler name may be confusing when the spectral fractional Laplacian is also considered, cf. \cite{BV-PPR1}. As discussed in this paper, there are other natural versions.} (RFL). The initial and boundary conditions associated to the fractional diffusion equation \eqref{FPME.equation} read
$u(t,x)=0$ in $(0,\infty)\times\mathbb{R}^N\setminus \Omega$ and $u(0,\cdot)=u_0$. As explained in \cite{BSV2013}, such boundary conditions can also be understood via the Caffarelli-Silvestre extension, see \cite{Caffarelli-Silvestre}. The sharp expression of the boundary behavior for RFL has been investigated in \cite{RosSer}. We refer to \cite{BSV2013} for a careful construction of the RFL in the framework of fractional Sobolev spaces, and \cite{BlGe} for a probabilistic interpretation.
This operator satisfies the assumptions (A1), (A2), (L1), and also (K2) and (K4) with $\gamma=s<1$. Let us present our results in this case. Note that we have $\sigma=1$ for all $0<s < 1$, and Theorem \ref{thm.GHP.PME.I} shows the sharp boundary behavior for all times, namely for all $t>0$ and a.e. $x\in \Omega$ we have
\begin{equation}\label{thm.GHP.PME.I.Ineq.00}
\underline{\kappa}\, \left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\dist(x,\partial\Omega)^{s/m}}{t^{\frac{1}{m-1}}}
\le \, u(t,x) \le \overline{\kappa}\, \frac{\dist(x,\partial\Omega)^{s/m}}{t^{\frac{1}{m-1}}}\,.
\end{equation}
The critical time $t_*$ is given by a weighted $\LL^1$ norm, namely $t_*:= \k_* \|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$,
where $\k_*>0$ is a universal constant. Moreover, solutions are classical in the interior and we prove sharp H\"older continuity up to the boundary.
These regularity results have been first obtained in \cite{BFR}; we give here different proofs valid in the more general setting of this paper. See Section \ref{sect.regularity} for further details.
\subsubsection{The SFL } Starting from the classical Dirichlet Laplacian $\Delta_{\Omega}$ on the domain $\Omega$\,, the so-called {\em spectral definition} of the fractional power of $\Delta_{\Omega}$ may be defined via a formula in terms of the semigroup associated to the Laplacian, namely
\begin{equation}\label{sLapl.Omega.Spectral}
\displaystyle(-\Delta_{\Omega})^{s}
g(x)= \frac1{\Gamma(-s)}\int_0^\infty
\left(e^{t\Delta_{\Omega}}g(x)-g(x)\right)\frac{dt}{t^{1+s}}=\sum_{j=1}^{\infty}\lambda_j^s\, \hat{g}_j\, \varphi_j(x)\,,
\end{equation}
where $(\lambda_j,\varphi_j)$, $j=1,2,\ldots$, is the normalized spectral sequence of the standard Dirichlet Laplacian on $\Omega$\,, $
\hat{g}_j=\int_\Omega g(x)\varphi_j(x)\,{\rm d}x$, and $\|\varphi_j\|_{\LL^2(\Omega)}=1$\,.
We denote this operator by $\mathcal{L}=(-\Delta_{\Omega})^s$\,, and call it the \textit{spectral fractional Laplacian} (SFL) as in \cite{Cabre-Tan}. The initial and boundary conditions associated to the fractional diffusion equation \eqref{FPME.equation} read
$u(t,x)=0$ on $(0,\infty)\times\partial\Omega$ and $u(0,\cdot)=u_0$. Such boundary conditions can also be understood via the Caffarelli-Silvestre extension, see \cite{BSV2013}. Following ideas of \cite{SV2003}, we use the fact that this operator admits a kernel representation,
\begin{equation}\label{SFL.Kernel}
(-\Delta_{\Omega})^{s} g(x)= c_{N,s}\mbox{
P.V.}\int_{\Omega} \left[g(x)-g(z)\right]K(x,z)\,dz + B(x)g(x)\,,
\end{equation}
where $K$ is a singular and compactly supported kernel, which degenerates at the boundary, and $B\asymp \dist(\cdot,\partial\Omega)^{-2s}$ (see \cite{SV2003} or Lemma \ref{Lem.Spec.Ker} for further details).
This operator satisfies the assumptions (A1), (A2), (L2), and also (K2) and (K4) with $\gamma=1$. Therefore, $\sigma$ can be less than $1$, depending on the values of $s$ and $m$.
As we shall see, in our parabolic setting, the degeneracy of the kernel is responsible for a peculiar change of the boundary behavior of the solutions (with respect to the previous case) for small and large times. Here, the lower bounds change both for short and large times,
and they strongly depend on $\sigma$ and on $u_0$: we called this phenomenon \textit{anomalous boundary behaviour }in Subsection \ref{ssec.results.boundary}. More precisely, Theorem \ref{thm.GHP.PME.III} shows that for all $t>0$ and all $x\in \Omega$ we have
\begin{equation}\label{thm.GHP.PME.III.Ineq.00}
\underline{\kappa}\,\left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\dist(x,\partial\Omega) }{t^{\frac{1}{m-1}}}
\le \, u(t,x) \le \overline{\kappa}\, \frac{\dist(x,\partial\Omega)^{\sigma/m}}{t^{\frac1{m-1}}}
\end{equation}
(when $2sm=\gamma(m-1)$, a logarithmic correction $\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}$ appears in the right hand side).
Such lower behavior is somehow minimal, in the sense that it holds in all cases. The basic asymptotic result (cf. \eqref{asymp.intro} or Theorem \ref{Thm.Asympt.0.1}) suggests that the lower bound in \eqref{thm.GHP.PME.III.Ineq.00} could be improved by replacing $\dist(x,\partial\Omega)$ with $\dist(x,\partial\Omega)^{\sigma/m}$, at least for large times. This is shown to be true for $\sigma=1$ (cf. Theorem \ref{thm.Lower.PME.large.t}), but it is false for $\sigma<1$ (cf. Theorem \ref{prop.counterex}), since there are ``small'' solutions with non-matching boundary behaviour for all times, cf. \eqref{limit.intro}.
It is interesting that, in this case, one can appreciate the interplay between the ``elliptic scaling power'' $2s/(m-1)$ related to the invariance of the equation $\mathcal{L} S^m=S$ under the scaling $S(x)\mapsto \lambda^{-2s/(m-1)}S(\lambda x)$, the ``eigenfunction power'' $\gamma=1$,
and the ``nonlinearity power'' $m$, made clear through the parameter $\sigma/m$.
Also in this case, thanks to the strict positivity in the interior, we can show interior space-time regularity of solutions, as well as sharp boundary H\"older regularity for large times whenever upper and lower bounds match.
\subsubsection{The CFL} In the simplest case, the infinitesimal operator of the censored stochastic processes has the form
\begin{equation}
\mathcal{L} g(x)=\mathrm{P.V.}\int_{\Omega}\frac{g(x)-g(y)}{|x-y|^{N+2s}}\,{\rm d}y\,,\qquad\mbox{with }\frac{1}{2}<s<1\,.
\end{equation}
This operator has been introduced in \cite{bogdan-censor} (see also \cite{Song-coeff} and \cite{BV-PPR2-1} for further details and references).
In this case $\gamma=s-1/2<2s$, hence $\sigma=1$ for all $1/2<s < 1$,
and Theorem \ref{thm.GHP.PME.I} shows that for all $t>0$ and $x\in \Omega$ we have
\[
\underline{\kappa}\, \left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\dist(x,\partial\Omega)^{(s-1/2)/m}}{t^{\frac{1}{m-1}}}
\le \, u(t,x) \le \overline{\kappa}\, \frac{\dist(x,\partial\Omega)^{(s-1/2)/m}}{t^{\frac{1}{m-1}}}\,.
\]
Again, we have interior space-time regularity of solutions, as well as sharp boundary H\"older regularity for all times.
\medskip
\subsubsection{Other examples}
There a number of examples to which our theory applies, besides the RFL, CFL and SFL, since they satisfy the list of assumptions listed in the previous section. Some are listed in the last Section \ref{sec.comm}, see more detail in \cite{BV-PPR2-1}.
\section{Reminders about weak dual solutions}\label{sec.results}
\medskip
We denote by $\LL^p_{\Phi_1}(\Omega)$ the weighted $\LL^p$ space $\LL^p(\Omega\,,\, \Phi_1\,{\rm d}x)$, endowed with the norm
\[
\|f\|_{\LL^p_{\Phi_1}(\Omega)}=\left(\int_{\Omega} |f(x)|^p\Phi_1(x)\,{\rm d}x\right)^{\frac{1}{p}}\,.
\]
\medskip
\noindent{\bf Weak dual solutions: existence and uniqueness.}
We recall the definition of weak dual solutions used in \cite{BV-PPR2-1}. This is expressed in terms of the inverse operator $\mathcal{L}^{-1}$, and encodes the Dirichlet boundary condition. This is needed to build a theory of bounded nonnegative unique solutions to Equation \eqref{FPME.equation} under the assumptions of the previous section. Note that in \cite{BV-PPR2-1} we have used the setup with the weight $\p=\dist(\cdot,\partial\Omega)^\gamma$, but the same arguments generalize immediately to the weight $\Phi_1$; indeed under assumption (K2), these two setups are equivalent.
\begin{defn}\label{Def.Very.Weak.Sol.Dual} A function $u$ is a {\sl weak dual} solution to the Dirichlet Problem for Equation \eqref{FPME.equation} in $(0,\infty)\times \Omega$ if:
\begin{itemize}[leftmargin=*
\item $u\in C((0,\infty): \LL^1_{\Phi_1}(\Omega))$\,, $u^m \in \LL^1\left((0,\infty):\LL^1_{\Phi_1}(\Omega)\right)$;
\item The identity
\begin{equation}
\displaystyle \int_0^\infty\int_{\Omega}\mathcal{L}^{-1} u \,\dfrac{\partial \psi}{\partial t}\,\,{\rm d}x\,{\rm d}t
-\int_0^\infty\int_{\Omega} u^m\,\psi\,\,{\rm d}x \,{\rm d}t=0
\end{equation}
holds for every test function $\psi$ such that $\psi/\Phi_1\in C^1_c((0,\infty): \LL^\infty(\Omega))$\,.
\item A {weak dual} solution to the Cauchy-Dirichlet problem {(CDP)} is a weak dual solution to the homogeneous Dirichlet Problem for equation \eqref{FPME.equation} such that $u\in C([0,\infty): \LL^1_{\Phi_1}(\Omega))$ and $u(0,x)=u_0\in \LL^1_{\Phi_1}(\Omega)$.
\end{itemize}\end{defn}
This kind of solution has been first introduced in \cite{BV-PPR1}, cf. also \cite{BV-PPR2-1}.
Roughly speaking, we are considering the weak solution to the ``dual equation'' $\partial_t U=- u^m$\,, where $U=\mathcal{L}^{-1} u$\,, posed on the bounded domain $\Omega$ with homogeneous Dirichlet conditions. Such weak solution is obtained by approximation from below as the limit of the unique mild solution provided by the semigroup theory (cf. \cite{BV-PPR2-1}),
and it was used in \cite{Vaz2012} with space domain $\mathbb{R}^N$ in the study of Barenblatt solutions. We call those solutions \textit{minimal weak dual solutions, }and it has been proven in Theorems 4.4 and 4.5 of \cite{BV-PPR2-1} that such solutions exist and are unique for any nonnegative data $u_0\in\LL^1_{\Phi_1}(\Omega)$. The class of weak dual solutions includes the classes of weak, mild and strong solutions, and is included in the class of very weak solutions. In this class of solutions the standard comparison result holds.
\medskip
\noindent {\bf Explicit solution.} When trying to understand the behavior of positive solutions with general nonnegative data, it is natural to look for solutions obtained by separation of variables. These are given by\vspace{-1mm}
\begin{equation}\label{friendly.giant}
\mathcal{U}_T(t,x):=(T+t)^{-\frac{1}{m-1}}S(x)\,,\qquad T\geq 0,\vspace{-1mm}
\end{equation}
where $S$ solves the elliptic problem\vspace{-1mm}
\begin{equation}\label{Elliptic.prob}
\left\{\begin{array}{lll}
\mathcal{L} S^m= S & ~ {\rm in}~ (0,+\infty)\times \Omega,\\
S=0 & ~\mbox{on the boundary.}
\end{array}
\right.\vspace{-1mm}
\end{equation}
The properties of $S$ have been thoroughly studied in the companion paper \cite{BFV-Elliptic}, and we summarize them here for the reader's convenience.\vspace{-2mm}
\begin{thm}[Properties of asymptotic profiles]\label{Thm.Elliptic.Harnack.m}
Assume that $\mathcal{L}$ satisfies (A1), (A2), and (K2). Then there exists a unique positive solution $S$ to the Dirichlet Problem \eqref{Elliptic.prob} with $m>1$. Moreover, let $\sigma$ be as in \eqref{as.sep.var}, and assume that:\\
- either $\sigma=1$ and $2sm\ne \gamma(m-1)$;\\
- or $\sigma<1$ and (K4) holds.\\
Then there exist positive constants $c_0$ and $c_1$ such that the following sharp absolute bounds hold true for all $x\in \Omega$:
\begin{equation}\label{Thm.Elliptic.Harnack.ineq.m.1}
c_0 \Phi_1(x)^{\sigma/m}\le S(x)\le c_1 \Phi_1(x)^{\sigma/m}\,.
\end{equation}
When $2sm= \gamma(m-1)$ then, assuming (K4), for all $x\in \Omega$ we have
\begin{equation}\label{Thm.Elliptic.Harnack.ineq.m.1.log}
c_0 \Phi_1(x)^{1/m}\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}\le S(x)\le c_1 \Phi_1(x)^{1/m}\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}\,.\vspace{-1mm}
\end{equation}
\end{thm}
\noindent\textbf{Remark. }As observed in the proof of Theorem \ref{Thm.Asympt},
by applying Theorem \ref{thm.GHP.PME.I} to the separate-variables solution $t^{-\frac{1}{m-1}}S(x)$ we deduce that
\eqref{Thm.Elliptic.Harnack.ineq.m.1} is still true when $\sigma<1$ if, instead of assuming (K4),
we suppose that $K(x,y)\le c_1|x-y|^{-(N+2s)}$ for a.e. $x,y\in \mathbb{R}^N$
and that $\Phi_1\in C^\gamma(\Omega)$.
\medskip
When $T=0$, the solution $\mathcal U_0$ in \eqref{friendly.giant} is commonly named ``Friendly Giant'', because it takes initial data $u_0\equiv +\infty$ (in the sense of pointwise limit as $t\to0$) but is bounded for all $t>0$. This term was coined in the study of the standard porous medium equation.
In the following Sections \ref{sect.upperestimates} and \ref{Sec.Lower} we will state and prove our general results concerning upper and lower bounds respectively. These sections are the crux of this paper. The combination of such upper and lower bounds will then be summarized in Section \ref{sect.Harnack}. Consequences of these results in terms of asymptotic behaviour and regularity estimates will be studied in Sections \ref{sec.asymptotic} and \ref{sect.regularity} respectively.\vspace{-2mm}
\section{Upper boundary estimates}\label{sect.upperestimates}
We present a general upper bound that holds under the sole assumptions (A1), (A2), and (K2),
hence valid for all our examples.
\begin{thm}[Absolute boundary estimates]\label{thm.Upper.PME.II}
Let (A1), (A2), and (K2) hold.
Let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to $u_0\in \LL^1_{\Phi_1}(\Omega)$,
and let $\sigma$ be as in \eqref{as.sep.var}. Then, there exists a computable constant $k_1>0$,
depending only on $N, s, m $, and $\Omega$, such that for all $t\ge 0$ and all $x\in \Omega$
\begin{equation}\label{thm.Upper.PME.Boundary.F}
u(t,x) \le \frac{k_1}{t^{\frac1{m-1}}}\,
\left\{
\begin{array}{ll}
\Phi_1(x_0)^{\sigma/m} &\text{if $\gamma\ne 2sm/(m-1)$},\\
\Phi_1(x_0)^{1/m}\big(1+|\log \Phi_1(x_0)|\big)^{1/(m-1)}&\text{if $\gamma=2sm/(m-1)$}.\\
\end{array}
\right.
\end{equation}
\end{thm}
This absolute bound proves a strong regularization which is independent of the initial datum. It improves the absolute bound in \cite{BV-PPR2-1} in the sense that it exhibits a precise boundary behavior. The estimate gives the correct behaviour for the solutions $\mathcal{U}_T$ in \eqref{friendly.giant} obtained by separation of variables, see Theorem \ref{Thm.Elliptic.Harnack.m}. It turns out that the estimate will be sharp for all nonnegative, nontrivial solutions in the case of the RFL and CFL. We will also see below that the estimate is not always the correct behaviour for the SFL when data are small, as explained in the Introduction
(see Subsection \ref{ssec.upper.small.data}, and Theorem \ref{prop.counterex} in Section \ref{Sec.Lower}).
\noindent\textit{Proof of Theorem \ref{thm.Upper.PME.II}. }
This subsection is devoted to the proof of Theorems \ref{thm.Upper.PME.II}. The first steps are based on a few basic results of \cite{BV-PPR2-1} that we will also be used in the rest of the paper.
\noindent\textsc{Step 1. Pointwise and absolute upper estimates}\label{sec.upper.partI}
\noindent\textit{Pointwise estimates. } We begin by recalling the basic pointwise estimates which are crucial in the proof of all the upper and lower bounds of this paper.\vspace{-1mm}
\begin{prop}[\cite{BV-PPR1,BV-PPR2-1}]\label{prop.point.est}
It holds\vspace{-1mm}
\begin{equation}\label{thm.NLE.PME.estim.0}
\int_{\Omega}u(t,x){\mathbb G}(x , x_0)\,{\rm d}x\le \int_{\Omega}u_0(x){\mathbb G}(x , x_0)\,{\rm d}x \qquad\mbox{for all $t> 0$\,.}
\end{equation}
Moreover, for every $0< t_0\le t_1 \le t$ and almost every $x_0\in \Omega$\,, we have
\begin{equation}\label{thm.NLE.PME.estim}
\frac{t_0^{\frac{m}{m-1}}}{t_1^{\frac{m}{m-1}}}(t_1-t_0)\,u^m(t_0,x_0)
\le \int_{\Omega}\big[u(t_0,x)-u({t_1},x)\big]{\mathbb G}(x , x_0)\,{\rm d}x \le (m-1)\frac{t^{\frac{m}{m-1}}}{t_0^{\frac{1}{m-1}}}\,u^m(t,x_0)\,.
\end{equation}
\end{prop}
\textit{Absolute upper bounds. } Using the estimates above, in Theorem 5.2 of \cite{BV-PPR2-1}
the authors proved that solutions corresponding to initial data $u_0\in\LL^1_{\Phi_1}(\Omega)$ satisfy\vspace{-1mm}
\begin{equation}\label{thm.Upper.PME.Absolute.F}
\|u(t)\|_{\LL^\infty(\Omega)}\le \frac{K_1}{t^{\frac{1}{m-1}}}\qquad\mbox{for all $t>0$\,,}
\end{equation}
with a constant $K_1$ independent of $u_0$. For this reason, this is called ``absolute bound''.
\noindent\textsc{Step 2. Upper bounds via Green function estimates. }The proof of Theorem \ref{thm.Upper.PME.II} requires the following general statement (see \cite[Proposition 6.5]{BFV-Elliptic}):\vspace{-2mm}
\begin{lem}\label{Lem.Green.2aaa}
Let (A1), (A2), and (K2) hold, and let $v:\Omega\to \mathbb R$ be a nonnegative bounded function.
Let $\sigma$ be as in \eqref{as.sep.var}, and assume that, for a.e. $x_0\in\Omega$,
\begin{equation}\label{Lem.Green.2.hyp.aaa}
v(x_0)^m\le \kappa_0\int_{\Omega} v(x){\mathbb G}(x,x_0)\,{\rm d}x.
\end{equation}
Then, there exists a constant $\overline{\kappa}_\infty>0$, depending only on $s,\gamma, m, N,\Omega$, such that the following bound holds true for a.e. $x_0\in \Omega$:
\begin{equation}\label{Lem.Green.2.est.Upper.aaa}
\int_{\Omega} v(x){\mathbb G}(x,x_0)\,{\rm d}x\le \overline{\kappa}_\infty\kappa_0^{\frac{1}{m-1}}
\left\{
\begin{array}{ll}
\Phi_1(x_0)^\sigma &\text{if $\gamma\ne 2s m/(m-1)$},\\
\Phi_1(x_0)\big(1+|\log \Phi_1(x_0)|^{\frac{m}{m-1}}\big)&\text{if $\gamma= 2s m/(m-1)$}.\\
\end{array}
\right.
\end{equation}\vspace{-2mm}
\end{lem}
\noindent {\sc Step 3. End of the proof of Theorem \ref{thm.Upper.PME.II}. }
We already know that $u(t)\in \LL^\infty(\Omega)$ for all $t>0$ by \eqref{thm.Upper.PME.Absolute.F}. Also, choosing $t_1=2t_0$ in \eqref{thm.NLE.PME.estim} we deduce that, for $t \ge 0$ and a.e. $x_0\in \Omega$,
\begin{equation}\label{Upper.PME.Step.1.2.c}
u^m(t,x_0) \le \frac{2^{\frac{m}{m-1}}}{t}\int_{\Omega}u(t,x){\mathbb G}(x , x_0)\,{\rm d}x\,.
\end{equation}
The above inequality corresponds exactly to hypothesis \eqref{Lem.Green.2.hyp.aaa} of Lemma \ref{Lem.Green.2aaa} with the value $\kappa_0=2^{\frac{m}{m-1}}t^{-1}$. As a consequence, inequality \eqref{Lem.Green.2.est.Upper.aaa} holds,
and we conclude that for a.e. $x_0\in\Omega$ and all $t>0$
\begin{equation}\label{thm.Upper.PME.Boundary.2}
\int_{\Omega}u(t,x){\mathbb G}(x , x_0)\,{\rm d}x
\le \frac{\overline{\kappa}_\infty 2^{\frac{m}{(m-1)^2}}}{t^{\frac{1}{m-1}}}\left\{
\begin{array}{ll}
\Phi_1(x_0)^\sigma &\text{if $\gamma\ne \frac{2sm}{m-1}$},\\
\Phi_1(x_0)\big(1+|\log \Phi_1(x_0)|^{\frac{m}{m-1}}\big)&\text{if $\gamma=\frac{2sm}{m-1}$}.\\
\end{array}
\right.
\end{equation}
Hence, combining this bound with \eqref{Upper.PME.Step.1.2.c}, we get
$$
u^{m}(t,x_0)
\le \frac{k_1^{m}}{t^{\frac{m}{m-1}}}\left\{
\begin{array}{ll}
\Phi_1(x_0)^\sigma &\text{if $\gamma\ne \frac{2sm}{m-1}$},\\
\Phi_1(x_0)\big(1+|\log \Phi_1(x_0)|^{\frac{m}{m-1}}\big)&\text{if $\gamma=\frac{2sm}{m-1}$}.\\
\end{array}
\right.
$$
This proves the upper bounds \eqref{thm.Upper.PME.Boundary.F} and concludes the proof.\qed
\subsection{Upper bounds for small data and small times}\label{ssec.upper.small.data}
As mentioned in the Introduction, the above upper bounds may not be realistic when $\sigma<1$. We have the following estimate for small times if the initial data are sufficiently small.
\begin{thm}\label{thm.Upper.PME.III}
Let $\mathcal{L}$ satisfy (A1), (A2), and (L2). Suppose also that $\mathcal{L}$ has a first eigenfunction $\Phi_1\asymp \dist(x,\partial\Omega)^\gamma$\,, and assume that $\sigma<1$. Finally, we assume that for all $x,y\in \Omega$
\begin{equation}\label{Operator.Hyp.upper.III}\begin{split}
K(x,y)\le \frac{c_1}{|x-y|^{N+2s}}
\left(\frac{\Phi_1(x)}{|x-y|^\gamma }\wedge 1\right)
\left(\frac{\Phi_1(y)}{|x-y|^\gamma }\wedge 1\right)
&\quad\mbox{ and }\quad B(x)\le c_1\Phi_1(x)^{-\frac{2s}{\gamma}}\,.
\end{split}\end{equation}
Let $u\ge 0$ be a weak dual solution to (CDP) corresponding to $u_0\in\LL^1_{\Phi_1}(\Omega)$.
Then, for every initial data $u_0\leq A\,\Phi_1^{1-2s/\gamma}$ for some $A>0$, we have
$$
u(t)\leq \frac{\Phi_1^{1-\frac{2s}{\gamma}}}{[A^{1-m} -\tilde C t]^{m-1}}
\qquad \text{on }[0,T_A],\qquad\text{where } T_A:=\frac{1}{\tilde CA^{m-1}},
$$
and the constant $\tilde C>0$, that depends only on $N,s,m, \lambda_1, c_1$, and $\Omega$.
\end{thm}
\noindent\textbf{Remark. }This result applies to the SFL. Notice that when $\sigma<1$ we have always $1-\frac{2s}{\gamma}>\sigma/m$\,, hence in this situation small data have a smaller behaviour at the boundary than the one predicted in Theorem \ref{thm.Upper.PME.II}. This is not true for ``big'' data, for instance for solution obtained by separation of variables, as already said.
\noindent\textit{Proof of Theorem \ref{thm.Upper.PME.III}. }
In view of our assumption on the initial datum, namely $u_0\le A\,\Phi_1^{1-2s/\gamma}$, by comparison it is enough to prove that
the function
\[
\overline{u}(t,x)=F(t)\Phi_1(x)^{1-\frac{2s}{\gamma}},\qquad F(t)=\frac{1}{[A^{1-m} -\tilde Ct]^{m-1}}
\]
is a supersolution (i.e., $\partial_t\overline{u}\ge -\mathcal{L}\overline{u}^m$) in $(0,T_A)\times \Omega$
provided we choose $\tilde C$ sufficiently large.
To this aim, we use the following elementary inequality, whose proof is left to the interested reader:
for any $\eta>1$ and any $M>0$ there exists $\widetilde{b}=\widetilde{b}(M)>0$ such that letting $\widetilde{\eta}:=\eta\wedge 2$
\begin{equation}\label{supersol.1.step3}
a^\eta-b^\eta\le \eta\,b^{\eta-1}(a-b)+ \widetilde{b} |a-b|^{\widetilde{\eta}}\,,\qquad\mbox{ for all $0\le a,b\le M$.}
\end{equation}
We apply inequality \eqref{supersol.1.step3} to $a=\Phi_1(y)$ and $b=\Phi_1(x)$, $\eta=m(1-\frac{2s}{\gamma})$, noticing that $\eta>1$ if and only if $\sigma<1$\,, and we obtain (recall that $\Phi_1$ is bounded)
\begin{equation*
\begin{split}
\overline{u}^m(t,y)-\overline{u}^m(t,x)&= F(t)^m \left(\Phi_1(y)^{m(1-\frac{2s}{\gamma})}-\Phi_1(x)^{m(1-\frac{2s}{\gamma})} \right)
= F(t)^m \left(\Phi_1(y)^\eta-\Phi_1(x)^\eta \right)\\
&\le \eta\,F(t)^m\Phi_1(x)^{\eta-1}\left[\Phi_1(y)-\Phi_1(x)\right]
+\widetilde{b}\, F(t)^m\left|\Phi_1(y)-\Phi_1(x)\right|^{\widetilde{\eta}}\\
&\le \eta\,F(t)^m\Phi_1(x)^{\eta-1} \left[\Phi_1(y)-\Phi_1(x)\right]
+ \widetilde{b}\,F(t)^m c_\gamma^{\widetilde{\eta}} |x-y|^{\widetilde{\eta}\gamma},
\end{split}
\end{equation*}
where in the last step we have used that $\left|\Phi_1(y)-\Phi_1(x)\right|\le c_\gamma |x-y|^\gamma$.
Since $B\le c_1\Phi_1^{-2s/\gamma}$,
\[\begin{split}
\int_{\mathbb{R}^N}\left[\Phi_1(y)-\Phi_1(x)\right]K(x,y)\,{\rm d}y
&=-\mathcal{L} \Phi_1(x) + B(x)\Phi_1(x)
\le -\lambda_1\Phi_1(x) + c_1\Phi_1(x)^{1-\frac{2s}{\gamma}}\,, \\
\end{split}\]
Thus, recalling that $\eta,\widetilde{\eta}>1$ and that $\Phi_1$ is bounded, it follows
\begin{equation}\label{supersol.2}\begin{split}
-\mathcal{L}[\overline{u}^m](x) &=\int_{\mathbb{R}^N}\left[ \overline{u}^m(t,y)-\overline{u}^m(t,x)\right]K(x,y)\,{\rm d}y + B(x)\overline{u}^m(t,x)\\
&\le \eta\,F(t)^m\Phi_1(x)^{\eta-1} \left[-\lambda_1\Phi_1(x) + c_1\Phi_1(x)^{1-\frac{2s}{\gamma}}\right] + B(x)F(t)^m \Phi_1^{\eta}(x)\\
&+ \widetilde{b}\,c_\gamma^{\widetilde{m}}\,F(t)^m \int_{\mathbb{R}^N}|x-y|^{\widetilde{\eta}\gamma}K(x,y)\,{\rm d}y\\
&\le \widetilde{c}F(t)^m \left(\Phi_1(x)^{\eta-\frac{2s}{\gamma}} + \int_{\mathbb{R}^N}|x-y|^{\widetilde{\eta}\gamma}K(x,y)\,{\rm d}y\right)
\end{split}
\end{equation}
Next, we claim that, as a consequence of \eqref{Operator.Hyp.upper.III})
\begin{equation}\label{supersol.3}
\int_{\mathbb{R}^N}|x-y|^{\widetilde{\eta}\gamma}K(x,y)\,{\rm d}y
\le c_4\Phi_1(x)^{1-\frac{2s}{\gamma}}\,.
\end{equation}
Postponing for the moment the proof of the above inequality, we first show how conclude: combining \eqref{supersol.2} and \eqref{supersol.3} we have
\[
-\mathcal{L}\overline{u}^m \le c_5 F(t)^m \Phi_1(x)^{1-\frac{2s}{\gamma}}= F'(t)\Phi_1(x)^{1-\frac{2s}{\gamma}}= \partial_t \overline{u}
\]
where we used that
$F'(t)=c_5 F(t)^{\widetilde{m}}$
provided $\Tilde C=c_5(m-1)$. This proves that $\overline{u}$ is a supersolution in $(0,T)\times \Omega$.
Hence the proof is concluded once we prove inequality \eqref{supersol.3}; for this, using hypothesis \eqref{Operator.Hyp.upper.III} and choosing $r=\Phi_1(x)^{1/\gamma}$ we have
\[\begin{split}
\int_{\mathbb{R}^N}&|x-y|^{\widetilde{\eta}\gamma}K(x,y)\,{\rm d}y
\le c_1\int_{B_r(x)}\frac{1}{|x-y|^{N+2s-\widetilde{\eta}\gamma}}\,{\rm d}y+c_1\Phi_1(x)\int_{\Omega\setminus B_r(x)}\frac{1}{|x-y|^{N+2s+\gamma-\widetilde{\eta}\gamma}} \,{\rm d}y\\
&\le c_2 r^{\widetilde{\eta}\gamma-2s}+c_1\frac{\Phi_1(x)}{r^{2s}}
\int_{\Omega\setminus B_r(x)}\frac{1}{|x-y|^{N+\gamma-\widetilde{\eta}\gamma}}\,{\rm d}y
= c_2 r^{\widetilde{\eta}\gamma-2s}+ c_3\frac{\Phi_1(x)}{r^{2s}} \le c_4\Phi_1(x)^{1-\frac{2s}{\gamma}}\,,
\end{split}\]
where we used that $\widetilde{\eta}\gamma-2s>0$ and $\tilde\eta>1$.\qed
\noindent{\bf Remark.} For operators for which the previous assumptions hold with $B\equiv 0$, we can actually prove a better upper bound for ``smaller data'', namely:
\begin{cor}\label{thm.Upper.PME.IV}Under the assumptions of Theorem \ref{thm.Upper.PME.III}, assume that moreover $B\equiv 0$ and $u_0\leq A\,\Phi_1$ for some $A>0$. Then, we have
$$
u(t)\leq \frac{\Phi_1}{[A^{1-m} -\tilde C t]^{m-1}}
\qquad \text{on }[0,T_A],\qquad\text{where } T_A:=\frac{1}{\tilde CA^{m-1}},
$$
and the constant $\tilde C>0$, that depends only on $N,s,m, \lambda_1, c_1$, and $\Omega$.
\end{cor}
\noindent {\bf Proof.~}We have to show that $\overline{u}(t,x)=F(t)\Phi_1(x)$ is a supersolution: we essentially repeat the proof of Theorem \ref{thm.Upper.PME.III} with $\gamma=m$ (formally replace $1-2s/\gamma$ by $1$), taking into account that $B\equiv 0$ and $u_0\leq A\,\Phi_1$.\qed
\section{Lower bounds}\label{Sec.Lower}
This section is devoted to the proof of all the lower bounds summarized later in the main Theorems \ref{thm.GHP.PME.I}, \ref{thm.GHP.PME.II}, and \ref{thm.GHP.PME.III}. The general situation is quite involved to describe, so we will separate several cases and we will indicate for which examples it holds for the sake of clarity.
\medskip
\noindent $\bullet$ \textbf{Infinite speed of propagation: universal lower bounds. }First, we are going to quantitatively establish that all nonnegative weak dual solutions of our problems are in fact positive in $\Omega$ for all $t>0$. This result is valid for all nonlocal operators considered in this paper.
\begin{thm}\label{thm.Lower.PME}
Let $\mathcal{L}$ satisfy (A1), (A2), and (L2). Let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to $u_0\in \LL^1_{\Phi_1}(\Omega)$. Then there exists a constant $\underline{\kappa}_0>0$ such that the following inequality holds:
\begin{equation}\label{thm.Lower.PME.Boundary.1}
u(t,x)\ge \underline{\kappa}_0\,\left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\Phi_1(x)}{t^{\frac{1}{m-1}}}\qquad\mbox{for all $t>0$ and a.e. $x\in \Omega$}\,.
\end{equation}
Here $t_*=\k_*\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$, and the constants $\underline{\kappa}_0$ and $\k_*$ depend only on $N,s,\gamma, m, c_0,c_1$, and $\Omega$\,.
\end{thm}
Notice that, for $t
\geq t_*$, the dependence on the initial data disappears from the lower bound,
as inequality reads
$$
u(t)\geq \underline{\kappa}_0 \frac{\Phi_1}{t^{\frac{1}{m-1}}}\qquad \forall\,t \geq t_*,
$$
where $\underline{\kappa}_0$ is an absolute constant.
Assumption (L2) on the kernel $K$ of $\mathcal{L}$ holds for all examples mentioned in Section \ref{sec.examples}.
Clearly, the power in this lower bound does not match the one of the general upper bounds of Theorem \ref{thm.Upper.PME.II}, hence we can not expect these bounds to be sharp. However, when $\sigma<1$, for small times and small data and when $B\equiv 0$, the lower bounds \eqref{thm.Lower.PME.Boundary.1} match the upper bounds of Corollary \ref{thm.Upper.PME.IV}, hence they are sharp. Theorem \ref{thm.Lower.PME} shows that, even in the ``worst case scenario'', there is a quantitative lower bound for all positive times, and shows infinite speed of propagation.
\noindent$\bullet$ \textbf{Matching lower bounds I. }Actually, in many cases the kernel of the nonlocal operator satisfies a stronger property, namely $\inf_{x,y\in \Omega}K(x,y)\ge \underline{\kappa}_\Omega>0$ and $B\equiv 0$, in which case we can actually obtain sharp lower bounds for all times. Here we do not consider the potential logarithmic correction that may appear in ``critical case'' $2sm= \gamma(m-1)$: indeed, as far as examples are concerned, the next Theorem applies to the RFL and the CFL, for which $2sm> \gamma(m-1)$.
\begin{thm}\label{Thm.lower.B}
Let $\mathcal{L}$ satisfy (A1), (A2), and (L1).
Furthermore, suppose that $\mathcal{L}$ has a first eigenfunction $\Phi_1\asymp \dist(x,\partial\Omega)^\gamma$\,. Let $\sigma$ be as in \eqref{as.sep.var} and assume that:\\
- either $\sigma=1$;\\
- or $\sigma<1$, $K(x,y)\le c_1|x-y|^{-(N+2s)}$ for a.e. $x,y\in \mathbb{R}^N$,
and $\Phi_1\in C^\gamma(\overline{\Omega})$.\\
Let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to $u_0\in \LL^1_{\Phi_1}(\Omega)$. Then there exists a constant $\underline{\kappa}_1>0$ such that the following inequality holds:
\begin{equation}\label{Thm.B.lower.bdd}
u(t,x)\ge \underline{\kappa}_1 \left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\Phi_1(x)^{\sigma/m}}{t^{\frac{1}{m-1}}} \qquad\mbox{for all $t>0$ and a.e. $x\in \Omega$}\,,
\end{equation}
where $t_*=\k_*\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$. The constants $\k_*$ and $\underline{\kappa}_1$ depend only on $N,s,\gamma, m, \underline{\kappa}_\Omega,c_1,\Omega,$ and $\|\Phi_1\|_{C^\gamma(\Omega)}$.
\end{thm}
\noindent\textbf{Remarks. }(i) As in the case of the Theorem \ref{thm.Lower.PME}, for large times the dependence on the initial data disappears from the lower bound and we have absolute lower bounds.
\noindent(ii) The boundary behavior is sharp when $2sm\ne \gamma(m-1)$ in view of the upper bound from Theorem \ref{thm.Upper.PME.II}.
\noindent(iii) This theorem applies to the RFL and the CFL, but not to the SFL (or, more in general, spectral powers of elliptic operators), see Sections \ref{ssec.examples} and \ref{sec.hyp.L}. In the case of the RFL, this result was obtained in Theorem 1 of \cite{BFR}.
\medskip
We have already seen the example of the separate-variables solutions \eqref{friendly.giant}
that have a very definite behavior at the boundary $\partial \Omega$. The analysis of general solutions leads to completely different situations for
$\sigma=1$ and $\sigma<1$.
\medskip
\noindent $\bullet$ \textbf{Matching lower bounds II. The case $\sigma=1$. }When $\sigma=1$ we can establish a quantitative lower bound near the boundary that matches the separate-variables behavior for large times (except in the case $2sm= \gamma(m-1)$ where the result is false, see Theorem \ref{prop.counterex} below). We do not need the assumption of non-degenerate kernel, so SFL can be considered.
\begin{thm}\label{thm.Lower.PME.large.t}
Let $(A1)$, $(A2)$, and $(K2)$ hold, and let $\sigma=1$. Let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to $u_0\in \LL^1_{\Phi_1}(\Omega)$. There exists a constant $\underline{\kappa}_2>0$ such that
\begin{equation}\label{thm.Lower.PME.Boundary.large.t}
u(t,x) \ge \underline{\kappa}_2\,\frac{\Phi_1(x)^{1/m}}{t^{\frac{1}{m-1}}}\qquad\mbox{for all $t\ge t_*$ and a.e. $x\in \Omega$}\,.
\end{equation}
Here, $t_*=\k_*\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$, and the constants $\k_* $ and $\underline{\kappa}_2$ depend only on $N,s,\gamma, m $, and $\Omega$\,.
\end{thm}
\noindent {\bf Remarks.} (i) At first sight, this theorem may seem weaker than the previous positivity result. However, this result has wider applicability since it holds under the only assumption (K2)
on ${\mathbb G}.$ In particular it is valid in the local case $s=1$, where the finite speed of propagation makes it impossible to have global lower bounds for small times.
\noindent (ii) When $\mathcal{L}=-\Delta$ the result has been proven in \cite{Ar-Pe} and \cite{JLVmonats}
by quite different methods. On the other hand, our method is very general and immediately applies to the case when $\mathcal{L}$ is an elliptic operator with $C^1$ coefficients, see Section \ref{sec.examples}.
\noindent (iii) This result fixes a small error in Theorem 7.1 of \cite{BV-PPR1} where the power $\sigma$ was not present.
\medskip
\noindent$\bullet$ \textbf{The anomalous lower bounds with small data. }As shown in Theorem \ref{thm.Lower.PME},
the lower bound $u(t)\gtrsim \Phi_1$ is always valid. We now discuss the possibility of improving this bound.
Let $S$ solve the elliptic problem \eqref{Elliptic.prob}.
It follows by comparison whenever $u_0
\geq \epsilon_0 S$ with $\epsilon_0>0$ then $u(t)\geq \frac{S}{(T_0+t)^{1/(m-1)}}$, where $T_0=\epsilon_0^{1-m}$.
Since $S\asymp \Phi_1^{\sigma/m}$ under (K4) (up to a possible logarithmic correction in the critical case, see Theorem \ref{Thm.Elliptic.Harnack.m}), there are initial data for which the lower behavior is dictated by $\Phi_1(x)^{\sigma/m}t^{-1/(m-1)}$.
More in general,
as we shall see in Theorem \ref{Thm.Asympt.0}, given any initial datum $u_0
\in \LL^1_{\Phi_1}(\Omega)$ the function $v(t,x):=t^{\frac{1}{m-1}}u(t)$ always converges to $S$ in $\LL^\infty(\Omega)$ as $t\to \infty$, independently of the value of $\sigma$.
Hence, one may conjecture that there should exist a waiting time $t_*>0$ after which the lower behavior is dictated by $\Phi_1(x)^{\sigma/m}t^{-1/(m-1)}$, in analogy with what happens for the classical porous medium equation.
As we shall see, this is actually {\em false} when $\sigma<1$ or $2sm= \gamma(m-1)$.
Since for large times $v(t,x)$ must look like $S(x)$ in uniform norm away from the boundary (by the interior regularity that we will prove later), the contrasting situation for large times could be described as `dolphin's head' with the `snout' flatter than the `forehead'. As $t\to\infty$ the forehead progressively fills the whole domain.
The next result shows that, in general, we cannot hope to prove that $u(t)$
is larger than $\Phi_1^{1/m}$. In particular, when $\sigma<1$ or $2sm= \gamma(m-1)$, this shows that the behavior $u(t)\asymp S$ cannot hold.
\begin{thm}\label{prop.counterex}
Let (A1), (A2), and (K2) hold, and $u\ge 0$ be a weak dual solution to the (CDP) corresponding to a nonnegative initial datum $u_0\in \LL^1_{\Phi_1}(\Omega)$.
Assume that $u_0(x)\le C_0\Phi_1(x)$ a.e. in $\Omega$ for some $C_0>0$. Then
there exists a constant $\hat\kappa$, depending only $N,s,\gamma, m $, and $\Omega$, such that
$$
u(t,x)^m \leq C_0\hat\kappa \frac{\Phi_1(x)}{t}\qquad\mbox{for all $t>0$ and a.e.
$x\in \Omega$}\,.
$$
In particular, if $\sigma<1$ (resp. $2sm= \gamma(m-1)$), then
$$
\lim_{x\to \partial\Omega}\frac{u(t,x)}{\Phi_1(x)^{\sigma/m}}= 0 \quad \Bigl(\text{resp.} \lim_{x\to \partial\Omega}\frac{u(t,x)}{\Phi_1(x)^{1/m}\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}}= 0\Bigr)\qquad \text{for any $t>0$.}
$$
\end{thm}
The proposition above could make one wonder whether the sharp general lower bound
could be given by $\Phi_1^{1/m}$, as in the case $\sigma=1$.
Recall that, under rather minimal assumptions on the kernel $K$
associated to $\mathcal{L}$, we have a universal lower bound for $u(t)$ in terms of $\Phi_1$
(see Theorem \ref{thm.Lower.PME}).
Here we shall see that, under (K4), the bound $u(t)\gtrsim \Phi_1^{1/m}$
is false for $\sigma<1$.
\begin{thm}\label{prop.counterex2}
Let (A1), (A2), and (K4) hold, and let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to a nonnegative initial datum $u_0 \leq C_0 \Phi_1$ for some $C_0>0$.
Assume that there exist constants $\underline{\kappa},T,\alpha>0$ such that
$$
u(T,x)\geq \underline{\kappa}\Phi_1^\alpha(x)\qquad\mbox{for a.e. $x\in \Omega$}\,.
$$
Then $\alpha \geq 1-\frac{2s}{\gamma}$.
In particular $\alpha>\frac{1}{m}$ if $\sigma<1$.
\end{thm}
We devote the rest of this section to the proof of the above results, and to this end we collect in the first two subsections some preliminary lower bounds and results about approximate solutions.
\subsection{Lower bounds for weighted norms}\label{Sect.Weighted.L1}
Here we prove some useful lower bounds for weighted norms, which follow from the $\LL^1$-continuity for ordered solutions in the version proved in Proposition 8.1 of \cite{BV-PPR2-1}.
\begin{lem}[Backward in time $\LL^1_{\Phi_1}$ lower bounds]\label{cor.abs.L1.phi} Let $u$ be a solution to (CDP) corresponding to the initial datum $u_0\in\LL^1_{\Phi_1}(\Omega)$. For all
\begin{equation}\label{L1weight.PME.estimates.2}
0\le \tau_0\le t\le \tau_0+ \frac{1}{\big(2\bar K\big)^{1/(2s\vartheta_{\gamma})}\|u(\tau_0)\|_{\LL^1_{\Phi_1}(\Omega)}^{m-1}}
\end{equation}
we have
\begin{equation}\label{L1weight.PME.estimates.3}
\frac{1}{2}\int_{\Omega}u(\tau_0,x)\Phi_1(x)\,{\rm d}x\le \int_{\Omega}u(t,x)\Phi_1(x)\,{\rm d}x\,,
\end{equation}
where $\vartheta_{\gamma}:=1/[2s+(N+\gamma)(m-1)]$ and $\bar K>0$ is a computable constant.
\end{lem}
\noindent\textsl{Proof of Lemma \ref{cor.abs.L1.phi}. }We recall the inequality of Proposition 8.1 of \cite{BV-PPR2-1}, adapted to our case: for all $0\le \tau_0\le \tau,t$ we have
\begin{equation}\label{L1weight.contr.estimates.1}
\int_{\Omega} u(\tau,x) \Phi_1(x)\,{\rm d}x
\le \int_{\Omega} u(t,x) \Phi_1(x)\,{\rm d}x
+ \bar K\|u(\tau_0)\|_{\LL^1_{\Phi_1}(\Omega)}^{2s(m-1) \vartheta_{\gamma}+1}\,\left|t-\tau\right|^{2s\vartheta_{\gamma}} \,.
\end{equation}
Choosing $\tau=\tau_0$ in the above inequality, we get
\begin{equation}\label{cor.0.1}\begin{split}
\left[1- K_9\|u(\tau_0)\|_{\LL^1_{\Phi_1}(\Omega)}^{2s(m-1) \vartheta_{\gamma}}\,\left|t-\tau_0\right|^{2s\vartheta_{\gamma}}\right]\int_{\Omega}u(\tau_0,x)\Phi_1(x)\,{\rm d}x &\le \int_{\Omega}u(t,x)\Phi_1(x)\,{\rm d}x\,.
\end{split}
\end{equation}
Then \eqref{L1weight.PME.estimates.3} follows from \eqref{L1weight.PME.estimates.2}\,.\qed
We also need a lower bound for $\LL^p_{\Phi_1}(\Omega)$ norms.
\begin{lem}\label{lem.abs.lower}
Let $u$ be a solution to (CDP) corresponding to the initial datum $u_0\in\LL^1_{\Phi_1}(\Omega)$. Then the following lower bound holds true for any $t\in [0,t_*]$ and $p\ge 1$:
\begin{equation}\label{lem1.lower.bdd}
c_2 \left(\int_{\Omega}u_0(x)\Phi_1(x)\,{\rm d}x \right)^p \le \int_{\Omega}u^p(t,x)\Phi_1(x)\,{\rm d}x
\end{equation}
Here $t_*=c_*\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$, where $c_2, c_*>0$ are positive constants that depend only on $N,s,m,p,\Omega$.
\end{lem}
The proof of this Lemma is an easy adaptation of the proof of Lemma 2.2 of \cite{BFR}\,, so we skip it. Notice that $c_*$ has explicit form given in {\rm \cite{BV-PPR1,BV-PPR2-1, BFR}}, while the form of $c_2$ is given in the proof of Lemma 2.2 of \cite{BFR}.
\subsection{Approximate solutions}
\label{sect:approx sol}
To prove our lower bounds, we will need a special class of approximate solutions $u_\delta$. We will list now the necessary details. In the case when $\mathcal{L}$ is the Restricted Fractional Laplacian (RFL) (see Section \ref{sec.examples}) these solutions have been used in the Appendix II of \cite{BFR}, where complete proofs can be found; the proof there holds also for the operators considered here. The interested reader can easily adapt the proofs in \cite{BFR} to the current case.
Let us fix $\delta>0$ and consider the problem:
\begin{equation}\label{probl.approx.soln}
\left\{
\begin{array}{lll}
\partial_t v_\delta=-\mathcal{L}\left[(v_\delta+\delta)^m -\delta^m \right]& \qquad\mbox{for any $(t,x)\in (0,\infty)\times \Omega$}\\
v_\delta(t,x)=0 &\qquad\mbox{for any $(t,x)\in (0,\infty)\times (\mathbb{R}^N\setminus\Omega)$}\\
v_\delta(0,x)=u_0(x) &\qquad\mbox{for any $x\in\Omega$}\,.
\end{array}
\right.
\end{equation}
Next, we define
\[
u_\delta:=v_\delta+\delta.
\]
We summarize here below the basic properties of ${u_\delta}$.
Approximate solutions ${u_\delta}$ exist, are unique, and bounded for all $(t,x)\in (0,\infty)\times\overline{\Omega}$ whenever $0\le u_0\in \LL^1_{\Phi_1}(\Omega)$\,.
Also, they are uniformly positive: for any $t \geq 0$,
\begin{equation}\label{approx.soln.positivity}
{u_\delta}(t,x)\ge \delta>0 \qquad\mbox{for a.e. $x \in \Omega$.}
\end{equation}
This implies that the equation for ${u_\delta}$ is never degenerate in the interior,
so solutions are smooth as the linear parabolic theory with the kernel $K$
allows them to be (in particular, in the case of the fractional laplacian, they are $C^\infty$ in space and $C^1$ in time).
Also, by a comparison principle,
for all $\delta>\delta'>0$ and $t \geq 0$,
\begin{equation}\label{approx.soln.comparison1}
{u_\delta}(t,x)\ge u_{\delta'}(t,x) \qquad\mbox{for $x \in \Omega$}
\end{equation}
and
\begin{equation}\label{approx.soln.comparison}
{u_\delta}(t,x)\ge u(t,x) \qquad\mbox{for a.e. $x \in \Omega$\,.}
\end{equation}
Furthermore, they converge in $\LL^1_{\Phi_1}(\Omega)$ to $u$ as $\delta \to 0$:
\begin{equation}\label{Lem.0}
\|{u_\delta}(t)-u(t)\|_{\LL^1_{\Phi_1}(\Omega)}\le \|{u_\delta}(0)-u_0\|_{\LL^1_{\Phi_1}(\Omega)}=\delta\,\|\Phi_1\|_{\LL^1(\Omega)}\,.
\end{equation}
As a consequence of \eqref{approx.soln.comparison1} and \eqref{Lem.0}, we deduce that ${u_\delta}$ converge pointwise to $u$
at almost every point: more precisely, for all $t\geq 0,$
\begin{equation}\label{limit.sol}
{u}(t,x)=\lim_{\delta\to 0^+}{u_\delta}(t,x)\qquad\mbox{for a.e. $x \in \Omega$\,.}
\end{equation}
\subsection{Proof of Theorem \ref{thm.Lower.PME}}The proof consists in showing that
\[
u(t,x) \geq \underline{u}(t,x):=k_0\, t \,\Phi_1(x)
\]
for all $t\in [0,t_*]$, where the parameter $k_0>0$ will be fixed later.
Note that, once the inequality $u
\geq \underline{u}$ on $[0,t_*]$ is proved, we conclude as follows: since
$t\mapsto t^{\frac{1}{m-1}}\,u(t,x)$ is nondecreasing in $t>0$ for a.e. $x\in \Omega$
(cf. (2.3) in \cite{BV-PPR2-1}) we have
$$u(t,x)\ge \left( \frac{t_*}{t}\right)^{\frac{1}{m-1}}u(t_*,x) \geq k_0\, t_* \left( \frac{t_*}{t}\right)^{\frac{1}{m-1}}\,\Phi_1(x)\qquad \text{ for all $t\ge t_*$\,.}$$
Then, the result will follow $\underline{\kappa}_0=k_0t_*^{\frac{m}{m-1}}$
(note that, as we shall see below, $k_0t_*^{\frac{m}{m-1}}$ can be chosen independently of $u_0$).
Hence, we are left with proving that $u
\geq \underline{u}$ on $[0,t_*]$.
\noindent$\bullet~$\textsc{Step 1. }\textit{Reduction to an approximate problem. }Let us fix $\delta>0$ and consider the approximate solutions ${u_\delta}$ constructed in Section \ref{sect:approx sol}.
We shall prove that ${u_\delta}
\geq \underline{u}$ on $[0,t_*]$, so that the result will follow by the arbitrariness of $\delta$.
\noindent$\bullet~$\textsc{Step 2. }\textit{We claim that $\underline{u}(t,x)< {u_\delta}(t,x)$ for all $0\le t\le t_*$ and $x\in \Omega$, for a suitable choice of $k_0>0$\,.} Assume that the inequality $\underline{u}<{u_\delta}$ is false in $[0,t_*]\times \overline\Omega$, and let $(t_c,x_c)$ be the first contact point between $\underline{u}$ and ${u_\delta}$. Since ${u_\delta}=\delta>0=\underline{u}$ on the lateral boundary, $(t_c,x_c)\in(0,t_*]\times\Omega$\,. Now, since $(t_c,x_c)\in (0,t_*]\times\Omega$ is the first contact point, we necessarily have that
\begin{equation}\label{contact.1}
{u_\delta}(t_c,x_c)= \underline{u}(t_c,x_c)\qquad\mbox{and}\qquad
{u_\delta}(t,x)\ge \underline{u}(t,x)\,\quad\forall t\in [0,t_c]\,,\;\; \forall x\in\overline{\Omega}\,.
\end{equation}
Thus, as a consequence,
\begin{equation}\label{contact.2}
\partial_t {u_\delta}(t_c,x_c)\le \partial_t \underline{u}(t_c,x_c)=k_0\,\Phi_1(x_c)\,.
\end{equation}
Next, we observe the following Kato-type inequality holds: for any nonnegative function $f$,
\begin{equation}\label{Kato.ineq}
\mathcal{L}(f^m)\le mf^{m-1}\mathcal{L} f.
\end{equation}
Indeed, by convexity, $f(x)^m-f(y)^m \le m [f(x)]^{m-1}(f(x)-f(y))$, therefore
\begin{equation*
\begin{split}
\mathcal{L}(f^m)(x)&=\int_{\mathbb{R}^N}[f(x)^m-f(y)^m]\,K(x,y)\,{\rm d}y + B(x)f(x)^m \\
&\le m [f(x)]^{m-1}\int_{\mathbb{R}^N}[f(x)-f(y)]\,K(x,y)\,{\rm d}y + B(x)f(x)^m \\
&= m [f(x)]^{m-1}\left[\int_{\mathbb{R}^N}[f(x)-f(y)]\,K(x,y)\,{\rm d}y + B(x)f(x) \right] - (m-1)B(x)f(x)^m\\
&\le m [f(x)]^{m-1}\mathcal{L} f(x)\,.
\end{split}
\end{equation*}
As a consequence of \eqref{Kato.ineq}, since $t_c\le t_*$ and $\Phi_1$ is bounded,
\begin{multline}\label{contact.2b}
\mathcal{L}(\underline{u}^m)(t,x)\le m\underline{u}^{m-1}\mathcal{L}(\underline{u})=m [k_0 t \Phi_1(x)]^{m-1}\,k_0 t \mathcal{L}(\Phi_1)(x)\\
=m \lambda_1[k_0 t \Phi_1(x)]^m \le \k_1 (t_*k_0)^m\Phi_1(x)\,,
\end{multline}
Then, using \eqref{contact.2} and \eqref{contact.2b}, we establish an upper bound for $-\mathcal{L}(u_\delta^m-\underline{u}^m)(t_c,x_c)$ as follows:
\begin{equation}\label{contact.3}
-\mathcal{L}[u_\delta^m- \underline{u}^m](t_c,x_c)=\partial_t {u_\delta}(t_c,x_c)+\mathcal{L}(\underline{u}^m)(t_c,x_c)
\le k_0\, \left[1+\k_1t_*^mk_0^{m-1}\right]\Phi_1(x_c).
\end{equation}
Next, we want to prove lower bounds for $-\mathcal{L}(u_\delta^m-\psi^m)(t_c,x_c)$, and this is the point where the nonlocality of the operator enters, since we make essential use of hypothesis (L2). We recall that by \eqref{contact.1} we have $u_\delta^m(t_c,x_c)=\underline{u}^m(t_c,x_c)$, so that assumption (L2) gives
\begin{equation*}\begin{split}
-\mathcal{L} &\left[u_\delta^m-\underline{u}^m\right](t_c,x_c)= -\mathcal{L} \left[u_\delta^m-\underline{u}^m\right](t_c,x_c) + B(x_c)[u_\delta^m(t_c,x_c)-\underline{u}^m(t_c,x_c)]\\
&=-\int_{\mathbb{R}^N}\left[\big(u_\delta^m(t_c,x_c)-u_\delta^m(t_c,y)\big)-\big(\underline{u}^m(t_c,x_c)-\underline{u}^m(t_c,y)\big)\right]K(x_c,y)\,{\rm d}y\\
&=\int_{\Omega}\left[u_\delta^m(t_c,y)-\underline{u}^m(t_c,y)\right]K(x_c,y)\,{\rm d}y
\ge c_0\Phi_1(x_c)\int_{\Omega}\left[u_\delta^m(t_c,y)-\underline{u}^m(t_c,y)\right]\Phi_1(y)\,{\rm d}y\,,\\
\end{split}
\end{equation*}
from which it follows (since $\underline{u}^m= [k_0 t \Phi_1(x)]^m \le \k_2(t_*k_0)^m $)
\begin{equation}\label{contact.44}\begin{split}
-\mathcal{L} &\left[u_\delta^m-\underline{u}^m\right](t_c,x_c)\\
&\ge c_0\Phi_1(x_c)\int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y)\,{\rm d}y-c_0\Phi_1(x_c)\int_{\Omega}\underline{u}^m(t_c,y)\Phi_1(y)\,{\rm d}y.\\
&\ge c_0\Phi_1(x_c)\int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y)\,{\rm d}y-c_0\Phi_1(x_c)\k_3\,(t_*k_0)^m.
\end{split}
\end{equation}
Combining the upper and lower bounds \eqref{contact.3} and \eqref{contact.44} we obtain
\begin{equation}\label{contact.5}\begin{split}
c_0\Phi_1(x_c)\int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y)\,{\rm d}y &\le k_0\, \left[1+ (\k_1+\k_3)t_*^m k_0^{m-1} \right]\Phi_1(x_c)\,.
\end{split}
\end{equation}
Hence, recalling \eqref{lem1.lower.bdd}, we get
$$
c_2 \left(\int_{\Omega}u_0(x)\Phi_1(x)\,{\rm d}x \right)^m\le \int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y)\,{\rm d}y
\le \frac{k_0}{c_0}\, \left[1+ (\k_1+\k_3)t_*^mk_0^{m-1} \right].
$$
Since $t_*=\k_*\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$, this yields
$$
c_2\k_*^{\frac{m}{m-1}}t_*^{-\frac{m}{m-1}} \le \frac{k_0}{c_0}\, \left[1+ (\k_1+\k_3)t_*^mk_0^{m-1} \right]
$$
which gives the desired contradiction provided we choose $k_0$ so that $\underline{\kappa}_0:=k_0t_*^{\frac{m}{m-1}}$ is universally small.
\qed
\subsection{Proof of Theorem \ref{Thm.lower.B}. }The proof proceeds along the lines of the proof of Theorem \ref{thm.Lower.PME}, so we will just briefly mention the common parts.
We want to show that
\begin{equation}\label{lower.barrier}
\underline{u}(t,x):=\kappa_0\,t\, \Phi_1(x)^{\sigma/m}\,,
\end{equation}
is a lower barrier for our problem on $[0,t_*]\times \Omega$ provided $\kappa_0$ is small enough.
More precisely, as in the proof of Theorem \ref{thm.Lower.PME}, we aim to prove that $\underline{u}<{u_\delta}$ on $[0,t_*]$, as the lower bound for $t \geq t_*$ then follows by monotonicity.
Assume by contradiction that the inequality $\underline{u}(t,x)< u_{\delta}(t,x)$
is false inside $[0,t_*]\times \overline\Omega$.
Since $\underline{u}<{u_\delta}$ on the parabolic boundary, letting $(t_c,x_c)$ be the first contact point, we necessarily have that $(t_c,x_c)\in (0,t_*]\times\Omega$. The desired contradiction will be obtained by combining the upper and lower bounds (that we prove below) for the quantity $-\mathcal{L}\left[u_\delta^m- \underline{u}^m\right](t_c,x_c)$ , and then choosing $\kappa_0>0$ suitably small. In this direction, it is convenient in what follows to assume that
\begin{equation}\label{k0.first.cond}
\k_0\le 1\wedge t_*^{-\frac{m}{m-1}}\qquad\mbox{so that}\qquad \k_0^{m-1}t_*^m\le 1\,.
\end{equation}
\noindent\textit{Upper bound. }We first establish the following upper bound: there exists a constant $\overline{A}>0$ such that
\begin{equation}\label{contact.up}
-\mathcal{L}\left[u_\delta^m- \underline{u}^m\right](t_c,x_c)\le\partial_t {u_\delta}(t_c,x_c)+\mathcal{L}\underline{u}^m(t_c,x_c)\le \overline{A} \,\kappa_0\,.
\end{equation}
To prove this,
we estimate $\partial_t {u_\delta}(t_c,x_c)$ and $\mathcal{L}\underline{u}^m(t_c,x_c)$ separately.
First we notice that, since $(t_\delta,x_\delta)$ is the first contact point, we have
\begin{equation}\label{contact.up.1}
{u_\delta}(t_\delta,x_\delta)= \underline{u}(t_\delta,x_\delta)\qquad\mbox{and}\qquad
{u_\delta}(t,x)\ge \underline{u}(t,x)\,\quad\forall t\in [0,t_\delta]\,,\;\; \forall x\in\Omega\,.
\end{equation}
Hence, since $t_\delta\le t_* $,
\begin{equation}\label{contact.up.2}
\partial_t {u_\delta}(t_\delta,x_\delta)\le \partial_t \underline{u}(t_\delta,x_\delta)=\kappa_0\,\Phi_1(x)^{\sigma/m}
\le \kappa_0\, \|\Phi_1\|_{\LL^\infty(\Omega)}^{\sigma/m} = A_1\,\kappa_0 \,,
\end{equation}
where we defined $A_1:= \|\Phi_1\|_{\LL^\infty(\Omega)}^{\sigma/m}$. Next we estimate $\mathcal{L}\underline{u}^m(t_c,x_c)$, using the Kato-type inequality \eqref{Kato.ineq}\,, namely $\mathcal{L}[u^m]\le mu^{m-1}\mathcal{L} u$\,. This implies
\begin{equation}\label{contact.up.3}\begin{split}
\mathcal{L}[\underline{u}^m](t,x)&\le m\underline{u}^{m-1}(t,x)\, \mathcal{L} \underline{u}(t,x)= m (\kappa_0\, t)^m\,\Phi_1(x)^{\frac{\sigma (m-1)}{m}} \mathcal{L} \Phi_1^\sigma(x)\\
&\le m (\kappa_0\, t_*)^m\,\|\Phi_1\|_{\LL^\infty(\Omega)}^{\frac{\sigma (m-1)}{m}} \left\| \mathcal{L} \Phi_1^\sigma\right\|_{\LL^\infty(\Omega)}
:= A_2\,\kappa_0\,.
\end{split}
\end{equation}
Since $\kappa_0^{m-1}\, t_*^m\leq 1$ (see \eqref{k0.first.cond}),
in order to prove that $A_2$ is finite it is enough to bound $\left\| \mathcal{L} \Phi_1^\sigma\right\|_{\LL^\infty(\Omega)}$.
When $\sigma=1$ we simply have $\mathcal{L}\Phi_1=-\lambda_1\Phi_1$, hence $A_2\leq m \lambda_1 \|\Phi_1\|_{\LL^\infty(\Omega)}^{2-1/m }$. When $\sigma<1$,
we use the assumption $\Phi_1 \in C^\gamma(\Omega)$ to estimate
\begin{equation}\label{contact.up.3a}
|\Phi_1^\sigma(x)-\Phi_1^\sigma(y)|\le |\Phi_1 (x)-\Phi_1 (y)|^\sigma \le C|x-y|^{\gamma\sigma}\qquad\forall\,x,y\in \Omega\,.
\end{equation}
Hence, since $\gamma\sigma=2sm/(m-1)>2s$ and $K(x,y)\le c_1|x-y|^{-(N+2s)}$, we see that
\[
\begin{split}
\left|\mathcal{L} \Phi_1^\sigma(x)\right|&=\left|\int_{\mathbb{R}^N}\left[\Phi_1^\sigma(x)-\Phi_1^\sigma(y) \right]K(x,y)\,{\rm d}y\right|\\
&\le \int_{\Omega}|x-y|^{\gamma\sigma} K(x,y)\,{\rm d}y + C\|\Phi_1\|_{\LL^\infty(\Omega)}^\sigma\int_{\mathbb{R}^N\setminus B_1} |y|^{-(N+2s)}\,{\rm d}y<\infty,
\end{split}
\]
hence $A_2$ is again finite.
Combining \eqref{contact.up.2} and \eqref{contact.up.3}, we obtain \eqref{contact.up} with $\overline{A}:=A_1+A_2$.
\noindent\textit{Lower bound. } We want to prove that there exists $\underline{A}>0$ such that
\begin{equation}\label{contact.low}
-\mathcal{L}\left[u_\delta^m- \underline{u}^m\right](t_c,x_c)\ge\frac{\underline{\kappa}_\Omega}{\|\Phi_1\|_{\LL^\infty(\Omega)}} \int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y) \,{\rm d}y-\underline{A}\,\kappa_0\,.
\end{equation}
This follows by (L1)
and \eqref{contact.up.1} as follows:
\begin{equation}\label{contact.4aaaaa}\begin{split}
-\mathcal{L} &\left[u_\delta^m-\underline{u}^m\right](t_c,x_c)\\
&=-\int_{\mathbb{R}^N}\left[\big(u_\delta^m(t_c,x_c)-u_\delta^m(t_c,y)\big)-\big(\underline{u}^m(t_c,x_c)-\underline{u}^m(t_c,y)\big)\right]K(x,y)\,{\rm d}y\\
&=\int_{\Omega}\left[u_\delta^m(t_c,y)-\underline{u}^m(t_c,y)\right]K(x,y)\,{\rm d}y\\
&\ge \underline{\kappa}_\Omega \int_{\Omega}\left[u_\delta^m(t_c,y)-\underline{u}^m(t_c,y)\right] \,{\rm d}y
\ge \frac{\underline{\kappa}_\Omega}{\|\Phi_1\|_{\LL^\infty(\Omega)}} \int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y) \,{\rm d}y-\underline{A}\,\kappa_0,
\end{split}\end{equation}
where in the last step we used that $\underline{u}^m(t_c,y)= [\k_0 t \Phi_1^{\sigma/m}(y)]^m \le \k_2(\k_0 t_*)^m $ and $\k_0^{m-1}t_*^m\le 1$ (see \eqref{k0.first.cond}).
\noindent\textit{End of the proof.} The contradiction can be now obtained by joining the upper and lower bounds \eqref{contact.up} and \eqref{contact.low}.
More precisely, we have proved that
\[
\int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y) \,{\rm d}y\le \frac{\|\Phi_1\|_{\LL^\infty(\Omega)}}{\underline{\kappa}_\Omega}(\overline{A}+\underline{A})\,\kappa_0 :=\overline{\kappa}\,\k_0,
\]
that combined with the lower bound \eqref{lem1.lower.bdd} yields
$$
c_2 \left(\int_{\Omega}u_0(x)\Phi_1(x)\,{\rm d}x \right)^m\le \int_{\Omega}u_\delta^m(t_c,y)\Phi_1(y)\,{\rm d}y
\le \overline{\kappa}\,\k_0.
$$
Setting $\k_0 := \left(1\wedge \frac{c_2}{\overline{\kappa}}\right)t_*^{-m/(m-1)}$ we obtain the desired contradiction.\qed
\subsection{Proof of Theorem \ref{thm.Lower.PME.large.t}.~}
We first recall the upper pointwise estimates \eqref{thm.NLE.PME.estim}: for all $0\le t_0\le t_1 \le t $ and a.e. $x_0\in \Omega$\,, we have that
\begin{equation}\label{Lower.PME.Step.1.1}
\int_{\Omega}u(t_0,x){\mathbb G}(x , x_0)\,{\rm d}x - \int_{\Omega}u({t_1},x){\mathbb G}(x , x_0)\,{\rm d}x \le (m-1)\frac{t^{\frac{m}{m-1}}}{t_0^{\frac{1}{m-1}}} \,u^m(t,x_0)\,.
\end{equation}
The proof follows by estimating the two integrals on the left-hand side separately.
We begin by using the upper bounds \eqref{thm.Upper.PME.Boundary.2} to get
\begin{equation}\label{Lower.PME.Step.1.2}
\int_{\Omega}u(t_1,x){\mathbb G}(x, x_0)\,{\rm d}x\le \overline{\kappa} \frac{\Phi_1(x_0)}{t_1^{\frac{1}{m-1}}}\qquad\mbox{for all $(t_1,x)\in(0,+\infty)\times\Omega$}\,.
\end{equation}
Then we note that, as a consequence of (K2) and Lemma \ref{cor.abs.L1.phi},
\begin{equation}\label{Lower.PME.Step.1.4}
\int_{\Omega}u(t_0,x){\mathbb G}(x, x_0)\,{\rm d}x\ge \underline{\kappa}_\Omega \Phi_1(x_0)\int_{\Omega}u(t_0,x)\Phi_1(x)\,{\rm d}x
\ge \frac{\underline{\kappa}_\Omega}{2}\Phi_1(x_0)\int_{\Omega}u_0(x)\Phi_1(x)\,{\rm d}x
\end{equation}
provided $t_0\le \frac{\tau_0}{\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{m-1}}$.
Combining \eqref{Lower.PME.Step.1.1}, \eqref{Lower.PME.Step.1.2}, and \eqref{Lower.PME.Step.1.4}, for all $t\ge t_1\ge t_0\ge 0$ we obtain
$$
u^m(t,x_0)
\ge \frac{t_0^{\frac{1}{m-1}}}{m-1}\left(\frac{\underline{\kappa}_\Omega}{2}\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}
- \overline{\kappa} t_1^{-\frac{1}{m-1}}\right)\frac{\Phi_1(x_0)}{t^{\frac{m}{m-1}}}\,.
$$
Choosing
\[
t_0:=\frac{\tau_0}{\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{m-1}}\le t_1:=t_*=\frac{\k_*}{\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{m-1}}
\qquad\mbox{with}\qquad
\k_*\ge \tau_0 \vee \left(\frac{\underline{\kappa}_\Omega}{4\overline{\kappa}}\right)^{m-1}
\]
so that
$
\frac{\underline{\kappa}_\Omega}{2}\|u_0\|_{\LL^1_{\Phi_1}(\Omega)} - \overline{\kappa} t_1^{-\frac{1}{m-1}}
\ge \frac{\underline{\kappa}_\Omega}{4} \|u_0\|_{\LL^1_{\Phi_1}(\Omega)},$
the result follows.\qed
\subsection{Proof of Theorems \ref{prop.counterex} and \ref{prop.counterex2}}
\noindent\textbf{Proof of Theorem \ref{prop.counterex}. }
Since $u_0\leq C_0 \Phi_1$ and $\mathcal{L} \Phi=\lambda_1 \Phi_1$, we have
$$
\int_\Omega u_0(x) {\mathbb G}(x,x_0)\,{\rm d}x\leq C_0\int_\Omega \Phi_1(x){\mathbb G}(x,x_0)\,{\rm d}x=C_0 \mathcal{L}^{-1}\Phi_1(x_0)=\frac{C_0}{\lambda_1}\Phi_1(x_0).
$$
Since $t\mapsto \int_\Omega u(t,y){\mathbb G}(x,y)\,{\rm d}y$ is decreasing (see \eqref{thm.NLE.PME.estim.0}), it follows that
\begin{equation}
\label{eq:initial data.2}
\int_\Omega u(t,y){\mathbb G}(x_0,y)\,{\rm d}y\le \frac{C_0}{\lambda_1}\Phi_1(x_0)\qquad\mbox{for all }t\ge 0.
\end{equation}
Combining this estimate with \eqref{Upper.PME.Step.1.2.c} concludes the proof.
\qed
\noindent\textbf{Proof of Theorem \ref{prop.counterex2}. }
Given $x_0 \in \Omega$, set $R_0:={\rm dist}(x_0,\partial\Omega)$.
Since ${\mathbb G}(x,x_0) \gtrsim |x-x_0|^{-(N-2s)}$ inside $B_{R_0/2}(x_0)$ (by (K4)),
using our assumption on $u(T)$ we get
$$
\int_\Omega {\mathbb G}(x,x_0) u(T,x)\,{\rm d}x\gtrsim
\int_{B_{R_0/2}(x_0)} \frac{ \Phi_1(x)^\alpha}{|x-x_0|^{N-2s}}
\gtrsim \Phi_1(x_0)^\alpha R_0^{2s}.
$$
Recalling that $\Phi_1(x_0)\asymp R_0^\gamma$, this yields
$$
\Phi_1(x_0)^{\alpha+\frac{2s}{\gamma}}\lesssim \int_\Omega {\mathbb G}(x,x_0) u(T,x)\,{\rm d}x.
$$
Combining the above inequality with \eqref{eq:initial data.2} gives
\[
\Phi_1(x_0)^{\alpha+\frac{2s}{\gamma}} \lesssim \Phi_1(x_0) \qquad \forall\,x_0\in \Omega,
\qquad\mbox{which implies}\qquad \alpha \geq 1-\frac{2s}{\gamma}\,.
\]
Noticing that $1-\frac{2s}{\gamma}>\frac{1}{m}$ if and only if $\sigma<1$, this concludes the proof.\qed
\section{Summary of the general decay and boundary results}\label{sect.Harnack}
In this section we present as summary of the main results, which can be summarized in various forms of upper and lower bounds, which we call Global Harnack Principle, (GHP) for short. As already mentioned, such inequalities are important for regularity issues (see Section \ref{sect.regularity}), and they play a fundamental role in formulating the sharp asymptotic behavior (see Section \ref{sec.asymptotic}). The proof of such GHP is obtained by combining upper and lower bounds, stated and proved in Sections \ref{sect.upperestimates} and \ref{Sec.Lower} respectively. There are cases when the bounds do not match, for which the complicated panorama described in the Introduction holds. As explained before, as far as examples are concerned, the latter anomalous situation happens only for the SFL.
\begin{thm}[Global Harnack Principle I]\label{thm.GHP.PME.I}
Let $\mathcal{L}$ satisfy (A1), (A2), (K2), and (L1). Furthermore, suppose that $\mathcal{L}$ has a first eigenfunction $\Phi_1\asymp \dist(x,\partial\Omega)^\gamma$\,. Let $\sigma$ be as in \eqref{as.sep.var} and assume that $2sm\ne \gamma(m-1)$ and:\\
- either $\sigma=1$;\\
- or $\sigma<1$, $K(x,y)\le c_1|x-y|^{-(N+2s)}$ for a.e. $x,y\in \mathbb{R}^N$,
and $\Phi_1\in C^\gamma(\overline{\Omega})$.\\
Let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to $u_0\in \LL^1_{\Phi_1}(\Omega)$. Then, there exist constants $\underline{\kappa},\overline{\kappa}>0$, so that the following inequality holds:
\begin{equation}\label{thm.GHP.PME.I.Ineq}
\underline{\kappa}\, \left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\Phi_1(x)^{\sigma/m}}{t^{\frac{1}{m-1}}}
\le \, u(t,x) \le \overline{\kappa}\, \frac{\Phi_1(x)^{\sigma/m}}{t^{\frac1{m-1}}}\qquad\mbox{for all $t>0$ and all $x\in \Omega$}\,.
\end{equation}
The constants $\underline{\kappa},\overline{\kappa}$ depend only on $N,s,\gamma, m, c_1,\underline{\kappa}_\Omega,\Omega$, and $\|\Phi_1\|_{C^\gamma(\Omega)}$\,.
\end{thm}
\noindent {\sl Proof.~}We combine the upper bound \eqref{thm.Upper.PME.Boundary.F}
with the lower bound \eqref{Thm.lower.B}. The expression of $t_*$ is explicitly given in Theorem \ref{Thm.lower.B}. \qed
\medskip
\noindent\textbf{Degenerate kernels. }When the kernel $K$ vanishes on $\partial\Omega$, there are two combinations of upper/lower bounds that provide Harnack inequalities, one for small times and one for large times.
As we have already seen, there is a strong difference between the case $\sigma=1$ and $\sigma<1$.
\begin{thm}[Global Harnack Principle II]\label{thm.GHP.PME.II}
Let (A1), (A2), and (K2) hold. Let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to $u_0\in \LL^1_{\Phi_1}(\Omega)$. Assume that:\\
- either $\sigma=1$ and $2sm\ne \gamma(m-1)$;\\
- or $\sigma<1$, $u_0\geq \underline{\kappa}_0\Phi_1^{\sigma/m}$ for some $\underline{\kappa}_0>0$, and (K4) holds.\\
Then there exist constants $\underline{\kappa},\overline{\kappa}>0$ such that the following inequality holds:
$$
\underline{\kappa}\, \frac{\Phi_1(x)^{\sigma/m}}{t^{\frac{1}{m-1}}}
\le \, u(t,x) \le \overline{\kappa}\, \frac{\Phi_1(x)^{\sigma/m}}{t^{\frac1{m-1}}}\qquad\mbox{for all $t\ge t_*$ and all $x\in \Omega$}\,.
$$
When $2sm= \gamma(m-1)$, assuming (K4)
and that $u_0\geq \underline{\kappa}_0\Phi_1\left(1+|\log\Phi_1 |\right)^{1/(m-1)}$ for some $\underline{\kappa}_0>0$, then for all $t\ge t_*$ and all $x\in \Omega$
\begin{multline*}
\underline{\kappa}\, \frac{\Phi_1(x)^{1/m}}{t^{\frac{1}{m-1}}}\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}
\le \, u(t,x)
\le \overline{\kappa}\, \frac{\Phi_1(x)^{1/m} }{t^{\frac1{m-1}}}\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}\,.
\end{multline*}
The constants $\underline{\kappa},\overline{\kappa}$ depend only on $N,s,\gamma, m, \underline{\kappa}_0,\underline{\kappa}_\Omega$, and $\Omega$.
\end{thm}
\noindent {\sl Proof.~}In the case $\sigma=1$, we combine the upper bound \eqref{thm.Upper.PME.Boundary.F} with the lower bound \eqref{thm.Lower.PME.Boundary.large.t}. The expression of $t_*$ is explicitly given in Theorem \ref{thm.Lower.PME.large.t}.
When $\sigma<1$, the upper bound is still given \eqref{thm.Upper.PME.Boundary.F},
while the lower bound follows by comparison with the solution $S(x)(\underline{\kappa}_0^{1-m}+t)^{-\frac{1}{m-1}}$, recalling that $S\asymp \Phi_1^{\sigma/m}$ (see Theorem \ref{Thm.Elliptic.Harnack.m}).
\qed
\noindent\textbf{Remark. }Local Harnack inequalities of elliptic/backward type follow as a consequence of Theorems \ref{thm.GHP.PME.I} and \ref{thm.GHP.PME.II}, for all times and for large times respectively, see Theorem \ref{thm.Harnack.Local}.
\medskip
Note that, for small times, we cannot find matching powers for a global Harnack inequality (except for some special initial data), and such result is actually false for $s=1$ (in view of the finite speed of propagation).
Hence, in the remaining cases, we have only the following general result.
\begin{thm}[{Non matching upper and lower bounds}]\label{thm.GHP.PME.III}
Let $\mathcal{L}$ satisfy (A1), (A2), (K2), and (L2).
Let $u\ge 0$ be a weak dual solution to the (CDP) corresponding to $u_0\in \LL^1_{\Phi_1}(\Omega)$. Then, there exist constants $\underline{\kappa},\overline{\kappa}>0$, so that the following inequality holds when $2sm\ne \gamma(m-1)$:
\begin{equation}\label{thm.GHP.PME.III.Ineq}
\underline{\kappa}\,\left(1\wedge \frac{t}{t_*}\right)^{\frac{m}{m-1}}\frac{\Phi_1(x)}{t^{\frac{1}{m-1}}}
\le \, u(t,x) \le \overline{\kappa}\, \frac{\Phi_1(x)^{\sigma/m}}{t^{\frac1{m-1}}}\qquad\mbox{for all $t>0$ and all $x\in \Omega$}.
\end{equation}
When $2sm=\gamma(m-1)$, a logarithmic correction $\left(1+|\log\Phi_1(x) |\right)^{1/(m-1)}$ appears in the right hand side.
\end{thm}
\noindent {\sl Proof.~} We combine the upper bound \eqref{thm.Upper.PME.Boundary.F} with the lower bound \eqref{thm.Lower.PME.Boundary.1}. The expression of $t_*$ is explicitly given in Theorem \ref{thm.Lower.PME}.\qed
\noindent\textbf{Remark. }As already mentioned in the introduction, in the non-matching case, which in examples can only happen for spectral-type operators, we have the appearance of an \textit{anomalous behaviour of solutions }corresponding to ``small data'': it happens for all times when $\sigma<1$ or $2sm=\gamma(m-1)$, and it can eventually happen for short times when $\sigma=1$.
\section{Asymptotic behavior}\label{sec.asymptotic}
An important application of the Global Harnack inequalities of the previous section concerns the sharp asymptotic behavior of solutions. More precisely, we first show that for large times all solutions behave like the separate-variables solution $\mathcal{U}(t,x)={S(x)}\,{t^{-\frac{1}{m-1}}}$
introduced at the end of Section \ref{sec.results}.
Then, whenever the (GHP) holds, we can improve this result to an estimate in relative error.
\begin{thm}[Asymptotic behavior]\label{Thm.Asympt.0}
Assume that $\mathcal{L}$ satisfies (A1), (A2), and (K2), and let $S$ be as in Theorem \ref{Thm.Elliptic.Harnack.m}. Let $u$ be any weak dual solution to the (CDP).
Then, unless $u\equiv 0$,
\begin{equation}\label{Thm.Asympt.0.1}
\left\|t^{\frac{1}{m-1}}u(t,\cdot)- S\right\|_{\LL^\infty(\Omega)}\xrightarrow{t\to\infty}0\,.
\end{equation}
\end{thm}
\noindent {\sl Proof.~}The proof uses rescaling and time monotonicity arguments, and it is a simple adaptation of the proof of Theorem 2.3 of \cite{BSV2013}. In those arguments, the interior $C^{\alpha}_{x}(\Omega)$ continuity is needed to improve the $\LL^1(\Omega)$ convergence to $\LL^\infty(\Omega)$, but the interior H\"older continuity is guaranteed
by Theorem \ref{thm.regularity.1}(i) below. \qed
We now exploit the (GHP) to get a stronger result.
\begin{thm}[Sharp asymptotic behavior]\label{Thm.Asympt}
Under the assumptions of Theorem \ref{Thm.Asympt.0}, assume that $u\not\equiv 0$.
Furthermore,
suppose that either the assumptions of Theorem \ref{thm.GHP.PME.I} or of Theorem \ref{thm.GHP.PME.II} hold.
Set $\mathcal{U}(t,x):=t^{-\frac{1}{m-1}}S(x)$.
Then there exists $c_0>0$ such that, for all $t\ge t_0:=c_0\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$, we have
\begin{equation}\label{conv.rates.rel.err}
\left\|\frac{u(t,\cdot)}{\mathcal{U}(t,\cdot)}-1 \right\|_{\LL^\infty(\Omega)} \le \frac{2}{m-1}\,\frac{t_0}{t_0+t}\,.
\end{equation}
We remark that the constant $c_0>0$ only depends on $N,s,\gamma, m, \underline{\kappa}_0,\underline{\kappa}_\Omega$, and $\Omega$.
\end{thm}
\noindent\textbf{Remark. }This asymptotic result is sharp, as it can be checked by considering $u(t,x)=\mathcal{U}(t+1,x)$.
For the classical case, that is $\mathcal{L}=\Delta$, we recover the classical results of \cite{Ar-Pe,JLVmonats} with a different proof.
\noindent {\sl Proof.~}Notice that we are in the position to use of Theorems \ref{thm.GHP.PME.I} or \ref{thm.GHP.PME.II}, namely we have
$$
u(t) \asymp t^{-\frac1{m-1}}S=\mathcal{U}(t,\cdot) \qquad\mbox{for all $t\ge t_*$}\,,
$$
where the last equivalence follows by Theorem \ref{Thm.Elliptic.Harnack.m}. Hence, we can rewrite the bounds above saying that there exist $\underline{\kappa},\overline{\kappa}>0$ such that
\begin{equation}\label{GHP.ASYM}
\underline{\kappa}\, \frac{S(x)}{t^{\frac1{m-1}}}
\le \, u(t,x) \le \overline{\kappa}\, \frac{S(x)}{t^{\frac1{m-1}}}\qquad\mbox{for all $t\ge t_*$ and a.e. $x\in \Omega$}\,.
\end{equation}
Since $t_*=\kappa_*\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)}$,
the first inequality implies that
$$
\frac{S}{(t_*+t_0)^{\frac{1}{m-1}}}\leq \underline{\kappa}\, \frac{S}{t_*^{\frac1{m-1}}} \leq u(t_*)
$$
for some $t_0=c_0\|u_0\|_{\LL^1_{\Phi_1}(\Omega)}^{-(m-1)} \geq t_*$.
Hence, by comparison principle,
$$
\frac{S}{(t+t_0)^{\frac{1}{m-1}}}
\le \, u(t)\qquad\mbox{for all $t\ge t_*$.}
$$
On the other hand, it follows by \eqref{GHP.ASYM} that $u(t,x) \leq \mathcal U_T(t,x):=S(x)(t-T)^{-\frac1{m-1}}$ for all $t \geq T$ provided $T$ is large enough.
If we now start to reduce $T$, the comparison principle combined with the upper bound
\eqref{thm.Upper.PME.Boundary.F} shows that $u$ can never touch $\mathcal U_T$ from below in $(T,\infty)\times \Omega$. Hence we can reduce $T$ until $T=0$, proving that $u \leq \mathcal U_0$ (for an alternative proof, see Lemma 5.4 in
\cite{BSV2013}).
Since $t_0 \geq t_*$,
this shows that
$$
\frac{S(x)}{(t+t_0)^{\frac{1}{m-1}}}
\le \, u(t,x) \le \frac{S(x)}{t^{\frac1{m-1}}}\qquad\mbox{for all $t\ge t_0$ and a.e. $x\in \Omega$}\,,
$$
therefore
$$
\biggl|1-\frac{u(t,x)}{\mathcal{U}(t,x)}\biggr| \leq 1-\biggl(1-\frac{t_0}{t_0+t}\biggr)^{\frac{1}{m-1}}\leq \frac{2}{m-1}\frac{t_0}{t_0+t}\qquad\mbox{for all $t\ge t_0$ and a.e. $x\in \Omega$},
$$
as desired.\qed
\section{Regularity results}\label{sect.regularity}
\noindent In order to obtain the regularity results, we basically require the validity of a Global Harnack Principle, namely Theorems \ref{thm.GHP.PME.I}, \ref{thm.GHP.PME.II}, or \ref{thm.GHP.PME.III}, depending on the situation under study. For some higher regularity results, we will eventually need some extra assumptions on the kernels.
For simplicity we assume that $\mathcal{L}$ is described by a kernel, without any lower order term. However, it is clear that the presence of lower order terms does not play any role in the interior regularity.
\begin{thm}[Interior Regularity]\label{thm.regularity.1}
Assume that
\begin{equation*}
\mathcal{L} f(x)=P.V.\int_{\mathbb{R}^N}\big(f(x)-f(y)\big)K(x,y)\,{\rm d}y+B(x)f(x)\,,
\end{equation*}
with
$$
K(x,y)\asymp |x-y|^{-(N+2s)}\quad \text{ in $B_{2r}(x_0)\subset\Omega$},
\qquad
K(x,y)\lesssim |x-y|^{-(N+2s)}\quad \text{ in $\mathbb R^N\setminus B_{2r}(x_0)$}.
$$
Let $u$ be a nonnegative bounded weak dual solution to problem (CDP) on $(T_0,T_1)\times\Omega$, and assume that there exist $\delta,M>0$ such that
\begin{align*}
&0<\delta\le u(t,x)\qquad\mbox{for a.e. $(t,x)\in (T_0,T_1)\times B_{2r}(x_0)$,}\\
&0\leq u(t,x)\leq M\qquad\mbox{for a.e. $(t,x)\in (T_0,T_1)\times \Omega$.}
\end{align*}\vspace{-7mm}
\begin{enumerate}
\item[(i)] Then $u$ is {H\"older continuous in the interior}. More precisely, there exists $\alpha>0$ such that, for all $0<T_0<T_2<T_1$,
\begin{equation}\label{thm.regularity.1.bounds.0}
\|u\|_{C^{\alpha/2s,\alpha}_{t,x}((T_2,T_1)\times B_{r}(x_0))}\leq C.
\end{equation}
\item[(ii)] Assume in addition $|K(x,y)-K(x',y)|\le c |x-x'|^\beta\,|y|^{-(N+2s)}$ for some $\beta\in (0,1\wedge 2s)$ such that $\beta+2s$ is not an integer. Then {$u$ is a classical solution in the interior}. More precisely, for all $0<T_0<T_2<T_1$,
\begin{equation}\label{thm.regularity.1.bounds.1}
\|u\|_{C^{1+\beta/2s, 2s+\beta}_{t,x}((T_2,T_1)\times B_{r}(x_0))}\le C.
\end{equation}
\end{enumerate}
\end{thm}
The constants in the above regularity estimates depend on the solution only through the upper and lower bounds on $u$. This bounds can be made quantitative by means of local Harnack inequalities, of elliptic and forward type, which follows from the Global ones.
\begin{thm}[Local Harnack Inequalities of Elliptic/Backward Type]\label{thm.Harnack.Local}
Under the assumptions of Theorem \ref{thm.GHP.PME.I},
there exists a constant $\hat H>0$, depending only on $N,s,\gamma, m, c_1,\underline{\kappa}_\Omega,\Omega$, such that for all balls $B_R(x_0)$ such that $B_{2R}(x_0)\subset\Omega$:
\begin{equation}\label{thm.Harnack.Local.ell}
\sup_{x\in B_R(x_0)}u(t,x)\le \frac{\hat H}{\left(1\wedge\frac{t}{t_*}\right)^{\frac{m}{m-1}}}\, \inf_{x\in B_R(x_0)}u(t,x)\qquad\mbox{for all }t>0.
\end{equation}
Moreover, for all $t>0$ and all $h>0$ we have the following:
\begin{equation}\label{thm.Harnack.Local.ForwBack}
\sup_{x\in B_R(x_0)}u(t,x)\le \hat H\left[\left(1+\frac{h}{t}\right) \left(1\wedge\frac{t}{t_*}\right)^{-m}\right]^{\frac{1}{m-1}}\, \inf_{x\in B_R(x_0)}u(t+h,x)\,.
\end{equation}
\end{thm}
\noindent {\sl Proof.~}
Recalling \eqref{thm.GHP.PME.I.Ineq}, the bound \eqref{thm.Harnack.Local.ell}
follows easily from the following Harnack inequality for the first eigenfunction, see for instance \cite{BFV-Elliptic}:
$$
\sup_{x\in B_R(x_0)} \Phi_1(x)\le H_{N,s,\gamma,\Omega} \inf_{x\in B_R(x_0)} \Phi_1(x).
$$
Since $u(t,x)\leq (1+h/t)^{\frac{1}{m-1}}u(t+h,x)$, by the time monotonicity of $t\mapsto t^{\frac{1}{m-1}}\,u(t,x)$,
\eqref{thm.Harnack.Local.ForwBack} follows.\qed
\noindent\textbf{Remark. }The same result holds for large times, $t\ge t_*$ as a consequence of Theorem \ref{thm.GHP.PME.II}. Already in the local case $s=1$\,, these Harnack inequalities are stronger than the known two-sided inequalities valid for solutions to the Dirichlet problem for the classical porous medium equation, cf. \cite{AC83,DaskaBook,DiB88,DiBook,DGVbook}, which are of forward type and are often stated in terms of the so-called intrinsic geometry. Note that elliptic and backward Harnack-type inequalities usually occur in the fast diffusion range $m<1$ \cite{BGV-Domains, BV, BV-ADV, BV2012}, or for linear equations in bounded domains \cite{FGS86,SY}.
For sharp boundary regularity we need a GHP with matching powers, like Theorems \ref{thm.GHP.PME.I} or \ref{thm.GHP.PME.II}, and when $s>\gamma/2$, we can also prove H\"older regularity up to the boundary. We leave to the interested reader to check that the presence of an extra term $B(x)u^m(t,x)$ with $0\leq B(x)\leq c_1 {\rm dist}(x,\partial\Omega)^{-2s}$ (as in the SFL) does not affect the validity of the next result. Indeed, when considering the scaling in \eqref{eq:scaling}, the lower term scales as $\hat B_r u_r^m$ with $0\leq \hat B_r\leq c_1$ inside the unit ball $B_1$.
\begin{thm}[H\"older continuity up to the boundary]\label{thm.regularity.2}
Under assumptions of Theorem $\ref{thm.regularity.1}$(ii),
assume in addition that $2s>\gamma$
Then {$u$ is H\"older continuous up to the boundary}. More precisely, for all $0<T_0<T_2<T_1$ there exists a constant $C>0$ such that
\begin{equation}\label{thm.regularity.2.bounds}
\|u\|_{C^{\frac{\gamma}{m\vartheta},\frac{\gamma}{m}}_{t,x}((T_2,T_1)\times{\Omega})}\le C
\qquad\mbox{with }\quad\vartheta:=2s-\gamma\left(1-\frac{1}{m}\right)\,.
\end{equation}
\end{thm}
\noindent\textbf{Remark. } Since we have $u(t,x)\asymp \Phi_1(x)^{1/m}\asymp\dist(x,\partial\Omega)^{\gamma/m}$ (note that $2s>\gamma$ implies that $\sigma=1$ and that $2sm\ne \gamma(m-1)$), the spacial H\"older exponent is sharp, while the H\"older exponent in time is the natural one by scaling.
\subsection{Proof of interior regularity}
The strategy to prove Theorem \ref{thm.regularity.1} follows the lines of \cite{BFR} but with some modifications.
The basic idea is that, because $u$ is bounded away from zero and infinity, the equation is non-degenerate and we can use parabolic regularity for nonlocal equations to obtain the results. More precisely, interior H\"older regularity will follow by applying $C^{\alpha/2s,\alpha}_{t,x}$ estimates of \cite{FK} for a ``localized'' linear problem.
Once H\"older regularity is established, under an H\"older continuity assumption on the kernel we can use the Schauder estimates proved in \cite{Schauder} to conclude.
\subsubsection{Localization of the problem }\label{sssect.localization}
Up to a rescaling, we can assume $r=2$, $T_0=0$, $T_1=1$. Also, by a standard covering argument,
it is enough to prove the results with $T_2=1/2$.
Take a cutoff function $\rho\in C^\infty_c(B_4)$ such that $\rho\equiv 1$ on $B_3$,
$\eta\in C^\infty_c(B_2)$ a cutoff function such that $\eta\equiv 1$ on $B_1$,
and define $v=\rho u$. By construction $u=v$ on $(0,1)\times B_3$. Since $\rho\equiv 1$ on $B_3$, we can write the equation for $v$ on the small cylinder $(0,1)\times B_1$ as
$$
\partial_t v(t,x) = -\mathcal{L} [v^m](t,x) +g(t,x)= - L_a v(t,x) + f(t,x) +g(t,x)
$$
where
$$
L_a[v](t,x):=\int_{\mathbb{R}^N}\big(v(t,x)-v(t,y)\big)a(t,x,y)K(x,y)\,{\rm d}y\,,
$$
\begin{align*}
a(t,x,y) &:= \frac{v^m(t,x)-v^m(t,y)}{v(t,x)-v(t,y)}\eta(x-y)+ \big[1-\eta(x-y)\big]\\
&=m\eta(x-y)\int_0^1 \left[(1-\lambda)v(t,x) +\lambda v(t,y) \right]^{m-1} {\rm d}\lambda + \big[1-\eta(x-y)\big]\,,
\end{align*}
\[
f(t,x):=\int_{\mathbb{R}^N\setminus B_1(x)}\bigl(v^m(t,x)-v^m(t,y)-v(t,x)+v(t,y)\bigr)[1-\eta(x-y)] K(x,y)\,{\rm d}y\,,
\]
and
$$
g(t,x):= -\mathcal{L}\left[(1-\rho^m)u^m\right](t,x)=\int_{\mathbb{R}^N\setminus B_3}(1-\rho^m(y))u^m(t,y) K(x,y)\,{\rm d}y
$$
(recall that $(1-\rho^m)u^m\equiv 0$ on $(0,1)\times B_3$).
\subsubsection{H\"older continuity in the interior}\label{sssect.Calpha}
Set $b:=f+g$, with $f$ and $g$ as above. It is easy to check that, since $K(x,y)\lesssim |x-y|^{-(N+2s)}$, $b \in \LL^\infty((0,1)\times B_1)$.
Also, since $0<\delta\leq u \leq M$ inside $(0,1)\times B_1$, there exists $\Lambda>1$ such that
$\Lambda^{-1}\leq a(t,x,y)\leq \Lambda$ for a.e. $(t,x,y)\in (0,1)\times B_1\times B_1$ with $|x-y|\leq 1$.
This guarantees that the linear operator $L_a$ is uniformly elliptic,
so we can apply the results in \cite{FK} to ensure that
\[\|v\|_{C^{\alpha/2s,\alpha}_{t,x}((1/2,1)\times B_{1/2})}\leq C\bigl(\|b\|_{L^\infty((0,1)\times B_1)}+\|v\|_{L^\infty((0,1)\times\mathbb{R}^N)}\bigr)\]
for some universal exponent $\alpha>0$. This proves Theorem \ref{thm.regularity.1}(i).
\subsubsection{Classical solutions in the interior}
Now that we know that $u\in C^{\alpha/2s,\alpha}((1/2,1)\times B_{1/2})$,
we repeat the localization argument above with cutoff functions $\rho$ and $\eta$ supported inside $(1/2,1)\times B_{1/2}$ to ensure that $v:=\rho u$ is H\"older continuous in $(1/2,1)\times\mathbb{R}^N$. Then,
to obtain higher regularity we argue as follows.
Set $\beta_1:=\min\{\alpha,\beta\}$.
Thanks to the assumption on $K$ and Theorem \ref{thm.regularity.1}(i), it is easy to check that $K_a(t,x,y):=a(t,x,y)K(x,y)$
satisfies
$$
|K_a(t,x,y)-K_a(t',x',y)| \leq C\left(|x-x'|^{\beta_1}+|t-t'|^{\beta_1/2s}\right)|y|^{-(N+2s)}
$$
inside $(1/2,1)\times B_{1/2}$.
Also, $f,g \in C^{\beta_1/2s,\beta_1}((1/2,1)\times B_{1/2})$.
This allows us to apply the Schauder estimates from \cite{Schauder} (see also \cite{CKriv})
to obtain that
$$
\|v\|_{C^{1+\beta_1/2s,2s+\beta}_{t,x}((3/4,1)\times B_{1/4})}
\leq C\bigl(\|b\|_{C^{\beta/2s,\beta}_{t,x}((1/2,1)\times B_{1/2})}+\|v\|_{{C^{\beta/2s,\beta}_{t,x}((1/2,1)\times\mathbb{R}^N)}}\bigr).
$$
In particular, $u\in C^{1+\beta_1/2s,2s+\beta_1}((3/4,1)\times B_{1/8})$.
In case $\beta_1=\beta$ we stop here. Otherwise we set
$\alpha_1:=2s+\beta$ and we repeat the argument above with $\beta_2:=\min\{\alpha_1,\beta\}$
in place of $\beta_1$. In this way, we obtain that $u\in C^{1+\beta_1/2s,2s+\beta_1}((1-2^{-4},1)\times B_{2^{-5}})$. Iterating this procedure finitely many times, we finally obtain that
$$
u\in C^{1+\beta/2s,2s+\beta}((1-2^{-k},1)\times B_{2^{-k-1}})
$$
for some universal $k$. Finally, a covering argument completes the proof
of Theorem \ref{thm.regularity.1}(ii).
\subsection{Proof of boundary regularity}
The proof of Theorem \ref{thm.regularity.2} follows by scaling and interior estimates.
Notice that the assumption $2s>\gamma$ implies that $\sigma=1$,
hence $u(t)$ has matching upper and lower bounds.
Given $x_0 \in \Omega$, set $r=\textrm{dist}(x_0,\partial\Omega)/2$
and define
\begin{equation}
\label{eq:scaling}
u_r(t,x):=r^{-\frac{\gamma}{m}}\,u\left(t_0+r^\vartheta t,\,x_0+rx\right)\,, \qquad\mbox{with}\qquad\vartheta:=2s-\gamma\left(1-\frac{1}{m}\right)\,.
\end{equation}
Note that, because $2s>\gamma$, we have $\vartheta>0$.
With this definition, we see that $u_r$ satisfies the equation $\partial_t u_r+\mathcal{L}_r u_r^m=0$ in $\Omega_r:=(\Omega-x_0)/r$, where
\[
\mathcal{L}_r f(x)=P.V.\int_{\mathbb{R}^N}\big(f(x)-f(y)\big)K_r(x,y)\,{\rm d}y\,,\qquad K_r(x,y):=r^{N+2s}K(x_0+rx,x_0+ry)\,.
\]
Note that, since $\sigma=1$, it follows by the (GHP) that $u(t)\asymp \dist(x,\partial\Omega)^{\gamma/m}$. Hence,
\[
0<\delta\leq u_r(t,x)\leq M, \qquad\mbox{for all $t\in[r^{-\vartheta}T_0,r^{-\vartheta}T_1]$ and $x\in B_1$,}
\]
with constants $\delta,M>0$ that are independent of $r$ and $x_0$.
In addition, using again that $u(t)\asymp \dist(x,\partial\Omega)^{\gamma/m}$, we see that
\[
u_r(t,x)\leq C\bigl(1+|x|^{\gamma/m}\bigr)
\qquad \text{for all $t\in[r^{-\vartheta}T_0,r^{-\vartheta}T_1]$ and $x\in \mathbb{R}^N$}.
\]
Noticing that
$
u_r^m(t,x) \leq C(1+|x|^{\gamma})
$
and that $\gamma<2s$ by assumption, we see that the tails
of $u_r$ will not create any problem. Indeed, for any $x \in B_1$,
$$
\int_{\mathbb R^N\setminus B_2}u_r^m(t,y)K_r(x,y)^{-(N+2s)}\,dy\leq C\int_{\mathbb R^N\setminus B_2}|y|^\gamma |y|^{-(N+2s)}\,dy\leq \bar C_0,
$$
where $\bar C_0$ is independent of $r$.
This means that we can localize the problem using cutoff functions as done in Section \ref{sssect.localization}, and the integrals defining the functions $f$ and $g$
will converge uniformly with respect to $x_0$ and $r$.
Hence, we can apply Theorem \ref{thm.regularity.1}(ii) to get
\begin{equation}
\label{eq:interior ur}
\|u_r\|_{C^{1+\beta/2s,2s+\beta}([r^{-\vartheta}T+1/2,r^{-\vartheta}T+1]\times B_{1/2})} \leq C
\end{equation}
for all $T \in [T_0,T_1-r^{-\vartheta}]$.
Since $\gamma/m<2s+\beta$ (because $\gamma<2s$),
it follows that
$$
\|u_r\|_{L^\infty([r^{-\vartheta}T+1/2,r^{-\vartheta}T+1],C^{\gamma/m}(B_{1/2})} \leq \|u_r\|_{C^{1+\beta/2s,2s+\beta}([r^{-\vartheta}T+1/2,r^{-\vartheta}T_0+1]\times B_{1/2})} \leq C.
$$
Noticing that
$$
\sup_{t \in [r^{-\vartheta}T+1/2,r^{-\vartheta}T+1]}[u_r]_{C^{\gamma/m}(B_{1/2})}
=
\sup_{t \in [T+r^{\vartheta}/2,r^{-\vartheta}T+r^{-\vartheta}]}[u]_{C^{\gamma/m}(B_{r}(x_0))},
$$
and that $T \in [T_0,T_1-r^{-\vartheta}]$ and $x_0$ are arbitrary, arguing as in \cite{RosSer}
we deduce that, given $T_2\in (T_0,T_1)$,
\begin{equation}
\label{eq:holder x}
\sup_{t \in [T_2,T_1]}[u]_{C^{\gamma/m}(\Omega)}
\leq C.
\end{equation}
This proves the global H\"older regularity in space.
To show the regularity in time, we start again from \eqref{eq:interior ur} to get
$$
\|\partial_tu_r\|_{\LL^\infty([r^{-\vartheta}T+1/2,r^{-\vartheta}T+1]\times B_{1/2})} \leq C.
$$
By scaling, this implies that
$$
\|\partial_tu\|_{\LL^\infty([T+r^{\vartheta}/2,r^{-\vartheta}T+r^{-\vartheta}]\times B_{r}(x_0))} \leq Cr^{\frac{\gamma}{m}-\vartheta},
$$
and by the arbitrariness of $T$ and $x_0$ we obtain (recall that $r=\textrm{dist}(x_0,\partial\Omega)/2$)
\begin{equation}
\label{eq:ut}
|\partial_tu(t,x)|\leq C\textrm{dist}(x,\partial\Omega)^{\frac{\gamma}{m}-\vartheta}
\qquad \forall\, t \in [T_2,T_1],\,x \in \Omega.
\end{equation}
Note that $\frac{\gamma}{m}-\vartheta=\gamma-2s<0$ by our assumption.
Now, given $t_0,t_1 \in [T_2,T_1]$ and $x \in \Omega$, we argue as follows:
if $|t_0-t_1|\leq \textrm{dist}(x,\partial\Omega)^\vartheta$ then we
use \eqref{eq:ut} to get (recall that $\frac{\gamma}{m}-\vartheta<0$)
$$
|u(t_1,x)-u(t_0,x)|\leq C\textrm{dist}(x,\partial\Omega)^{\frac{\gamma}{m}-\vartheta}|t_0-t_1|\leq C|t_0-t_1|^\frac{\gamma}{m\vartheta}.
$$
On the other hand, if $|t_0-t_1|\geq \textrm{dist}(x,\partial\Omega)^\vartheta$,
then we use \eqref{eq:holder x} and the fact that $u$ vanishes on $\partial \Omega$
to obtain
$$
|u(t_1,x)-u(t_0,x)| \leq |u(t_1,x)|+|u(t_0,x)| \leq C\textrm{dist}(x,\partial\Omega)^{\frac{\gamma}{m}}\leq C|t_0-t_1|^\frac{\gamma}{m\vartheta}.
$$
This proves that $u$ is $\frac{\gamma}{m\vartheta}$-H\"older continuous in time,
and completes the proof of Theorem \ref{thm.regularity.2}. \qed
\section{Numerical evidence}\label{sec.numer}
After discovering the unexpected boundary behavior, we looked for numerical confirmation.
This has been given to us by the authors of \cite{numerics}, who exploited the analytical tools developed in this paper to support our results by means of accurate numerical simulations. We include here some of these simulations, by courtesy of the authors. In all the figures we shall consider the Spectral Fractional Laplacian, so that $\gamma=1$ (see Section \ref{ssec.examples} for more details).
We take $\Omega=(-1,1)$, and we consider as initial datum the compactly supported function $u_0(x)=e^{4-\frac{1}{(x-1/2)(x+1/2)}}\chi_{|x|< 1/2}$ appearing in the left of Figure \ref{fig1}.
In all the other figures, the solid line represents either \ $\Phi_1^{1/m}$ or $\Phi_1^{1-2s}$,
while the dotted lines represent \ $t^{\frac{1}{m-1}}u(t)$ for different values of $t$, where $u(t)$ is the solution starting from $u_0$.
These choices are motivated by Theorems \ref{thm.Lower.PME.large.t} and \ref{prop.counterex2}.
Since the map $t\mapsto t^{\frac{1}{m-1}}\,u(t,x)$ is nondecreasing for all $x \in \Omega$
(cf. (2.3) in \cite{BV-PPR2-1}),
the lower dotted line corresponds to an earlier time with respect to the higher one.
\vspace{-3mm}
\begin{figure}[H]\label{fig1}
\hspace{-1cm}
\includegraphics[width=260pt]{Initial_Condition2.jpg}\hspace{-8mm}\includegraphics[width=260pt]{m2s05t1t5SS1m.jpg}
\vspace{-8mm}\caption{ \footnotesize
On the left, the initial condition $u_0$. On the right, the solid line represents $\Phi_1^{1/m}$, and the dotted lines represent $t^{\frac{1}{m-1}}u(t)$ at $t=1$
and $t=5$. The parameters are $m=2$ and $s=1/2$, hence $\sigma=1$.
While $u(t)$ appears to behave as $\Phi_1\asymp \dist(\cdot,\partial\Omega)$ for very short times, already at $t=5$ it exhibits the matching boundary behavior predicted by Theorem \ref{thm.Lower.PME.large.t}.
}
\end{figure}\vspace{-4mm}
\begin{figure}[H]\label{fig2}
\hspace{-1.2cm}
\includegraphics[width=260pt]{m4s075t30t150SS1m.jpg}\hspace{-7mm} \includegraphics[width=260pt]{m4s02t150t600SS1m.jpg}\vspace{-4mm}
\caption{ \footnotesize
In both pictures, the solid line represents $\Phi_1^{1/m}$.
On the left, the dotted lines represent $t^{\frac{1}{m-1}}u(t)$ at $t=30$
and $t=150$, with parameters $m=4$ and $s=3/4$ (hence $\sigma=1$).
In this case $u(t)$ appears to behave as $\Phi_1\asymp \dist(\cdot,\partial\Omega)$ for quite some time, and only around $t=150$ it exhibits the matching boundary behavior predicted by Theorem \ref{thm.Lower.PME.large.t}.
On the right, the dotted lines represent $t^{\frac{1}{m-1}}u(t)$ at $t=150$ and $t=600$ with parameters $m=4$ and $s=1/5$ (hence $\sigma=8/15<1$).
In this case $u(t)$ seems to exhibit a linear boundary behavior even after long time (this linear boundary behavior is a universal lower bound for all times by Theorem
\ref{thm.Lower.PME}).
The second picture may lead one to conjecture that, in the case $\sigma<1$ and $u_0\lesssim \Phi_1$, the behavior $u(t)\asymp \Phi_1$ holds for all times.
However, as shown in Figure 3, there are cases when $u(t)\gg \Phi_1^{1-2s}$ for large times.}
\end{figure}\vspace{-4mm}
\begin{figure}[H]\label{fig3}
\hspace{-1.2cm}
\includegraphics[width=260pt]{m2s01t4t25SS1-2s.jpg}\hspace{-7mm}\includegraphics[width=260pt]{m2s01t40t150SS1-2s.jpg}
\vspace{-8mm}
\caption{ \footnotesize
In both pictures we use the parameters $m=2$ and $s=1/10$ (hence $\sigma=2/5<1$), and the solid line represents $\Phi_1^{1-2s}$.
On the left, the dotted lines represent $t^{\frac{1}{m-1}}u(t)$ at $t=4$
and $t=25$, on the right we see $t=40$ and $t=150$.
Note that $u(t)\asymp \Phi_1$ for short times.
Then, after some time, $u(t)$ starts looking more like $\Phi_1^{1-2s}$, and
for large times ($t=150$) it becomes much larger than $\Phi_1^{1-2s}$.
}
\end{figure}
Comparing Figures 2 and 3, it seems that when $\sigma<1$ there is no hope to find a universal behavior of solutions for large times. In particular, the bound provided by \eqref{intro.1} seems to be optimal.
\section{Complements, extensions and further examples}\label{sec.comm}
\noindent {\bf Elliptic versus parabolic.} The exceptional boundary behaviors we have found for some operators and data came as a surprise to us, since the solution to the corresponding ``elliptic setting'' $\mathcal{L} S^m= S$ satisfies $S\asymp \Phi_1^{\sigma/m}$ (with a logarithmic correction when
$2sm\ne \gamma(m-1)$), hence separate-variable solutions always satisfy \eqref{intro.1b} (see \eqref{friendly.giant} and Theorem \ref{Thm.Elliptic.Harnack.m}).
\medskip
\noindent\textbf{About the kernel of operators of the spectral type. }\label{ss2.2}
In this paragraph we study the properties of the kernel of $\mathcal{L}$. While in some situations $\mathcal{L}$ may not have a kernel (for instance, in the local case), in other situations that may not be so obvious from its definition. In the next lemma it is shown in particular that the SFL, defined by \eqref{sLapl.Omega.Spectral}, admits representation of the form \eqref{SFL.Kernel}. We state hereby the precise result, mentioned in \cite{AbTh} and proven in \cite{SV2003} for the SFL.
\begin{lem}[Spectral Kernels]\label{Lem.Spec.Ker}
Let $s \in (0,1)$, and let $\mathcal{L}$ be the $s^{th}$-spectral power of a linear elliptic second order operator $\mathcal A$, and let $\Phi_1 \asymp \dist(\cdot,\partial\Omega)^\gamma$ be the first positive eigenfunction of $\mathcal A$. Let $H(t,x,y)$ be the Heat Kernel of $\mathcal A$ , and assume that it satisfies the following bounds: there exist constants $c_0,c_1,c_2>0$ such that
for all $0<t\le1$
\begin{equation}\label{Bounds.HK.s=1.t<1}
c_0\left[\frac{\Phi_1(x)}{t^{\gamma/2}}\wedge 1\right] \left[\frac{\Phi_1(y)}{t^{\gamma/2}}\wedge 1\right]\frac{\ee^{-c_1\frac{|x-y|^2}{t}}}{t^{N/2}}\le H(t,x,y)\le c_0^{-1}\left[\frac{\Phi_1(x)}{t^{\gamma/2}}\wedge 1\right] \left[\frac{\Phi_1(y)}{t^{\gamma/2}}\wedge 1\right]\frac{\ee^{-\frac{|x-y|^2}{c_1\,t}}}{t^{N/2}}
\end{equation}
and
\begin{equation}\label{Bounds.HK.s=1.t>1}
0\leq H(t,x,y) \leq c_2 \Phi_1(x)\Phi_1(y)\qquad\mbox{for all $t\ge 1$. }
\end{equation}
Then the operator $\mathcal{L}$ can be expressed in the form
\begin{equation}\label{A.op.kern.1}
\mathcal{L} f(x)=P.V.\int_{\mathbb{R}^N} \big(f(x)-f(y)\big)\,K(x,y)\,{\rm d}y + B(x)u(x)
\end{equation}
with a kernel $K(x,y)$ supported in $\overline{\Omega}\times\overline{\Omega}$ satisfying
\begin{multline}\label{Lem.Spec.Ker.bounds}
K(x,y)\asymp\frac{1}{|x-y|^{N+2s}}
\left(\frac{\Phi_1(x)}{|x-y|^\gamma }\wedge 1\right)
\left(\frac{\Phi_1(y)}{|x-y|^\gamma }\wedge 1\right)\quad\mbox{and}\quad B(x)\asymp \Phi_1(x)^{-\frac{2s}{\gamma}}
\end{multline}
\end{lem}
The proof of this Lemma follows the ideas of \cite{SV2003}; indeed assumptions of Lemma \ref{Lem.Spec.Ker} allow to adapt the proof of \cite{SV2003} to our case with minor changes.
\smallskip
\noindent {\bf Method and generality. }Our work is part of a current effort aimed at extending the theory of evolution equations of parabolic type to a wide class of nonlocal operators, in particular operators with general kernels that have been studied by various authors (see for instance \cite{EJT, DPQR, Ser14}).
Our approach is different from many others: indeed, even if the equation is nonlinear, we concentrate on the properties of the inverse operator $\mathcal{L}^{-1}$ (more precisely, on its kernel given by the Green function ${\mathbb G}$), rather than on the operator $\mathcal{L} $ itself. Once this setting is well-established and good linear estimates for the Green function are available, the calculations and estimates are very general. Hence, the method is applicable to a very large class of equations, both for Elliptic and Parabolic problems, as well as to more general nonlinearities than $F(u)=u^m$ (see also related comments in the works \cite{BV-PPR1, BSV2013, BV-PPR2-1}).
\smallskip
\noindent {\bf Finite and infinite propagation.} In all cases consider in the paper for $s<1$ we prove that the solution becomes strictly positive inside the domain at all positive times. This is called {\sl infinite speed of propagation}, a property that does not hold in the limit $s=1$ for any $m>1$ \cite{VazBook} (in that case, finite speed of propagation holds and a free boundary appears). Previous results on this infinite speed of propagation can be found in \cite{BFR, DPQRV2}. We recall that infinite speed of propagation is typical of the evolution with nonlocal operators representing long-range interactions, but it is not true for the standard porous medium equation, hence a trade-off takes place when both effects are combined; all our models fall on the side of infinite propagation, but we recall that finite propagation holds for a related nonlocal model called ``nonlinear porous medium flow with fractional potential pressure'', cf. \cite{CV1}.
\smallskip
\noindent {\bf The local case.} Since $2sm>\gamma(m-1)$ when $s=1$ (independently of $m>1$), our results give a sharp behavior in the local case after a ``waiting time''.
Although this is well-known for the classical porous medium equation, our results apply also to the case uniformly elliptic operator in divergence form with $C^1$ coefficients, and yield new results in this setting. Actually one can check that, even when the coefficients are merely measurable, many of our results are still true and they provided universal upper and lower
estimates.
At least to our knowledge, such general results are
completely new.
\subsection{Further examples of operators}\label{sec.examples}
Here we briefly exhibit a number of examples to which our theory applies, besides the RFL, CFL and SFL already discussed in Section \ref{sec.hyp.L}. These include a wide class of local and nonlocal operators. We just sketch the essential points, referring to \cite{BV-PPR2-1} for a more detailed exposition.
\medskip
\noindent\textbf{Censored Fractional Laplacian (CFL) and operators with more general kernels. }As already mentioned in Section \ref{ssec.examples}, assumptions (A1), (A2), and (K2) are satisfied with $\gamma=s-1/2$. Moreover, it follows by \cite{bogdan-censor, Song-coeff} that we can also consider operators of the form:
\[
\mathcal{L} f(x)=\mathrm{P.V.}\int_{\Omega}\left(f(x)-f(y)\right)\frac{a(x,y)}{|x-y|^{N+2s}}\,{\rm d}y\,,\qquad\mbox{with }\frac{1}{2}<s<1\,,
\]
where $a(x,y)$ is a symmetric function of class $C^1$ bounded between two positive constants. The Green function ${\mathbb G}(x,y)$ of $\mathcal{L}$ satisfies the stronger assumption (K4), cf. Corollary 1.2 of~\cite{Song-coeff}.
\medskip
\noindent\textbf{Fractional operators with more general kernels. }Consider integral operators of the form\vspace{-1mm}
\[
\mathcal{L} f(x)=\mathrm{P.V.}\int_{\mathbb{R}^N}\left(f(x)-f(y)\right)\frac{a(x,y)}{|x-y|^{N+2s}}\,{\rm d}y\,,
\]
where $a$ is a measurable symmetric function, bounded between two positive constants, and satisfying
\[
\big|a(x,y)-a(x,x)\big|\,\chi_{|x-y|<1}\le c |x-y|^\sigma\,,\qquad\mbox{with }0<s<\sigma\le 1\,,
\]
for some $c>0$ (actually, one can allow even more general kernels, cf. \cite{BV-PPR2-1, Kim-Coeff}).
Then, for all $s\in (0, 1]$, the Green function ${\mathbb G}(x,y)$ of $\mathcal{L}$ satisfies (K4) with $\gamma=s$\,, cf. Corollary 1.4 of \cite{Kim-Coeff}.
\medskip
\noindent\textbf{Spectral powers of uniformly elliptic operators. }Consider a linear operator $\mathcal A$ in divergence form,
\[
\mathcal A=-\sum_{i,j=1}^N\partial_i(a_{ij}\partial_j)\,,
\]
with uniformly elliptic $C^1$ coefficients. The uniform ellipticity allows one to build a self-adjoint operator on $\LL^2(\Omega)$ with discrete spectrum $(\lambda_k, \phi_k)$\,. Using the spectral theorem, we can construct the spectral power of such operator as follows
\[
\mathcal{L} f(x):=\mathcal A^s\,f(x):=\sum_{k=1}^\infty \lambda_k^s \hat{f}_k \phi_k(x),\qquad\mbox{where }\qquad \hat{f}_k=\int_\Omega f(x)\phi_k(x)\,{\rm d}x
\]
(we refer to the books \cite{Davies1,Davies2} for further details), and the Green function satisfies (K2) with $\gamma=1$\,, cf. \cite[Chapter 4.6]{Davies2}. Then, the first eigenfunction $\Phi_1$
is comparable to $\dist(\cdot, \partial\Omega)$.
Also, Lemma \ref{Lem.Spec.Ker} applies (see for instance \cite{Davies2}) and allow us to get sharp upper and lower estimates for the kernel $K$ of $\mathcal{L}$, as in \eqref{Lem.Spec.Ker.bounds}\,.
\noindent\textbf{Other examples. } As explained in Section 3 of \cite{BV-PPR2-1}, our theory may also be applied to: (i) Sums of two fractional operators; (ii) Sum of the Laplacian and a nonlocal operator kernels; (iii) Schr\"odinger equations for non-symmetric diffusions; (iv) Gradient perturbation of restricted fractional Laplacians.
Finally, it is worth mentioning that our arguments readily extend to operators on manifolds
for which the required bounds hold.
\vskip .5cm
\noindent{\bf Acknowledgments. }M.B. and J.L.V. are partially funded by Project MTM2011-24696 and MTM2014-52240-P(Spain). A.F. has been supported by NSF Grants DMS-1262411 and DMS-1361122, and by the ERC Grant ``Regularity and Stability in Partial Differential Equations (RSPDE)''. M.B. and J.L.V. would like to acknowledge the hospitality of the Mathematics Department of the University of Texas at Austin, where part of this work has been done. J.L.V. was also invited by BCAM, Bilbao. We thank an anonymous referee for pointing out that Lemma \ref{Lem.Spec.Ker} was proved in \cite{SV2003} and mentioned in \cite{AbTh}.
\addcontentsline{toc}{section}{~~~References}
|
2,869,038,154,758 | arxiv | \section{Introduction}
The gravitational redshift (GRS) effect in clusters of galaxies
is a feature in any metric theory of gravity. It is
caused by the spatial variation of the gravitational potential;
light traveling from deeper in the potential of a cluster
is expected to be redshifted, compared
to light originating from the outskirts of the
cluster~\cite{2004ApJ...607..164K,*1983A&A...118...85N,*1995A&A...301....6C}.
The GRS effect has the potential to constrain
theories in which there are long-range non-gravitational forces acting on dark
matter, modifying gravity on cluster
scales~\cite{1993ApJ...403L...5G,*2004PhRvD..70l3511G,*2010PhRvD..81f3521K,*2007PhRvL..98q1302F}.
The effect was first measured by Wojtak, Hansen \& Hjorth (WHH)~\cite{2011Natur.477..567W},
a study which was subsequently repeated
(with minor modifications) by Dominguez-Romero \mbox{\etal \cite{MNL2:MNL21326}}.
WHH used 125k~spectroscopic redshifts,
taken from the seventh data release~\cite{Abazajian:2008wr}
of the Sloan Digital Sky Survey (SDSS)~\cite{2011AJ....142...72E},
matched to 7.8k~clusters from the GMBCG cluster catalog~\cite{2010ApJS..191..254H}.
They used the brightest cluster galaxies (BCGs) as a proxy
for the centers of clusters. They assumed that, in general, BCGs
have relatively small velocity dispersions
compared to other bound galaxies,
and reside close to the bottom of the gravitational potential.
WHH divided their galaxy sample into four bins, based on the transverse distance between cluster-galaxies
and respective BCGs, \ensuremath{r_{\mrm{gc}}}\xspace, extending up to~${6~\mrm{Mpc}}$. In each bin, they calculated the
line-of-sight velocity of galaxies in the rest-frame of the BCG,
\begin{equation}
\ensuremath{v_{\mrm{gc}}}\xspace = c \; \frac{z_{\mrm{gal}} - z_{\mrm{BCG}} }{1 + z_{\mrm{BCG}}} \;,
\label{eq_vGcDef}
\end{equation}
where $z_{\mrm{BCG}}$ and $z_{\mrm{gal}}$ respectively stand for the redshift
of BCGs and of associated galaxies, and $c$ is the speed of light.
The stacked \ensuremath{v_{\mrm{gc}}}\xspace-distributions of galaxies from the entire sample of clusters
were fitted with a phenomenological model, using a Markov chain Monte Carlo (MCMC) program.
The derived mean of the distributions was interpreted as the GRS signal.
On average, cluster-galaxies were found to have a
redshift difference relative to corresponding BCGs,
amounting to a velocity difference, $\ensuremath{\Delta v_{\mrm{gc}}}\xspace\approx-7~\ensuremath{\mrm{km/s}}\xspace$.
WHH calculated the prediction for the signal from general relativity (GR), as well as from
modified theories of gravity~\cite{2004PhRvD..70d3528C,*1983ApJ...270..365M,*2004PhRvD..70h3509B}.
They first derived the GRS profile of a single cluster in the weak field limit,
\begin{equation}
\Delta_{1}(\ensuremath{r_{\mrm{gc}}}\xspace) = \frac{2}{c \; \Sigma(\ensuremath{r_{\mrm{gc}}}\xspace)}\int\limits_{\ensuremath{r_{\mrm{gc}}}\xspace}^{\infty}
\left[\Phi(r) - \Phi(0) \right]
\frac{ \rho(r) \; r }{ \sqrt{\smash[b]{r^2 - \ensuremath{r_{\mrm{gc}}}\xspace^2}} } \mrm{d}r \;,
\label{eq_whhGR}
\end{equation}
where $\Phi$ is the gravitational potential, and $\rho$ and $\Sigma$ are respectively the
three-dimensional and surface-density profiles of galaxies. They then convolved $\Delta_{1}$ with
the distribution of cluster masses in their sample, estimated from
the observed velocity dispersion profile, using stacked NFW models~\cite{1997ApJ...490..493N}.
Subsequent works, notably those of Zhao~\etal\cite{2013PhRvD..88d3013Z} and of Kaiser~\cite{2013MNRAS.435.1278K},
modified the theoretical prediction; these took into account effects such as
the so-called transverse Doppler shift and
surface brightness modulation. The added
corrections were found to be of the same order of magnitude as the GRS
signal, some inducing redshifts and some blueshifts. Summed together, the prediction
of Kaiser is of a relatively flat dependence of \ensuremath{\Delta v_{\mrm{gc}}}\xspace on \ensuremath{r_{\mrm{gc}}}\xspace,
with a mean value of~$-9$ (GR only)
or~$-12~\ensuremath{\mrm{km/s}}\xspace$ (GR and kinematic effects).
The purpose of this study is to revise the measurement of WHH.
In the next sections we describe the analysis in detail, following up with our results
and conclusions.
\section{Methodology}
\subsection{Dataset}
We used spectroscopic redshifts derived from the tenth data release (DR10)~\cite{dr10.1307.7735} of the SDSS,
including measurements taken with the Baryon Oscillation Spectroscopic Survey (BOSS)~\cite{boss.1208.0022},
occupying the redshift range, $0.05$~to~$0.6$.
We associated the DR10 data with galaxy clusters, using
the catalog of Wen, Han \& Liu (WHL)~\cite{2012ApJS..199...34W}.
The WHL sample includes~${\sim130\mrm{k}}$
clusters, detected using a friends-of-friends algorithm, based on photometric data.
The virial radius of a cluster is commonly approximated by \ensuremath{r_{200}}\xspace,
the radius within which the mean density of a cluster is
200~times that of the critical density of the universe~\cite{peebles1993principles,*peacock1999cosmological}.
The latter is additionally
used to define $m_{200}$, the cluster mass within \ensuremath{r_{200}}\xspace.
The WHL catalog is nearly complete for clusters with masses, ${m_{200}>\powA{2}{14}\ensuremath{M_{\odot}}\xspace}$,
and redshifts, ${z<0.5}$, and is~${\sim75\%}$ complete for ${m_{200}>\powA{0.6}{14}\ensuremath{M_{\odot}}\xspace}$
and ${z<0.42}$.
Cluster mass is estimated using a scaling relation between mass and
optical richness.
The latter was estimated by WHL using x-ray or weak-lensing
methods, and is given in ${\mrm{Eq.\;2}}$ of~\cite{2012ApJS..199...34W}.
The derived average cluster mass in our selected cluster sample
is ${m_{200} = \powA{1.3}{14}\ensuremath{M_{\odot}}\xspace}$.
This is commensurate with the mean value of cluster masses in the WHH
dataset, ${m_{200} = \powA{1.6}{14}\ensuremath{M_{\odot}}\xspace}$, allowing us to directly
compare the results of their measurement with our own.
In the initial stage of the analysis, the
spectroscopic redshifts were subjected to various quality cuts,
ensuring \eg that the uncertainty on the redshift is below~\powB{-4}, and that the
confidence in the likelihood-fit of the redshift is high.
We then matched galaxy spectra to BCG positions,
keeping only those clusters for which the BCG had a corresponding spectrum.
Additionally, each cluster had to contain
at least one galaxy within transverse distance, ${\ensuremath{r_{\mrm{gc}}}\xspace<6~\mrm{Mpc}}$,
and velocity, ${\left| \ensuremath{v_{\mrm{gc}}}\xspace \right| < 4,000~\ensuremath{\mrm{km/s}}\xspace}$.
Conversion from angular to physical distances was performed using a flat \ensuremath{\Lambda}CDM\xspace cosmology, with
${\Omega_{\mrm{m}} = 0.307}$ and the Hubble
constant, ${\mrm{H}_{0} = 67.8~\mrm{km}\;\mrm{s}^{-1}\;\mrm{Mpc}^{-1}}$~\cite{2014A&A...571A..16P}.
The initial selection left us
with~${31\mrm{k}}$ clusters and~${426\mrm{k}}$ associated galaxies.
Following the selection procedure discussed in the next section, we were left with
${60\mrm{k}}$ galaxies and ${12\mrm{k}}$ clusters, having ${\ensuremath{r_{\mrm{gc}}}\xspace \lesssim 3~\mrm{Mpc}}$. An
additional ${25\mrm{k}}$ galaxies and ${5\mrm{k}}$ clusters were
used for systematic checks.
\subsection{Fitting procedure}
WHH employed a MCMC program to fit the velocity distribution to the phenomenological model,
\begin{equation}
f(\ensuremath{v_{\mrm{gc}}}\xspace) = p_{\mrm{cl}} \cdot f_{\mrm{Gauss}}(\ensuremath{v_{\mrm{gc}}}\xspace) + \left( 1 - p_{\mrm{cl}} \right)\cdot f_{\mrm{Lin}}(\ensuremath{v_{\mrm{gc}}}\xspace) \;,
\label{eq_vGcFitModel}
\end{equation}
where $f_{\mrm{Gauss}}$ is a
convolution of two Gaussian distributions, having a common mean value, \ensuremath{\Delta v_{\mrm{gc}}}\xspace,
and $f_{\mrm{Lin}}$ is a linear function.
The quasi-Gaussian
contribution represents galaxies bound to clusters.
It accounts for the
intrinsic non-Gaussianity of velocity distributions of individual clusters,
and for the variation in cluster masses in the sample.
The linear part of the model represents a uniform background of interlopers (line-of-sight
galaxies which are not gravitationally bound to the cluster).
The fraction of bound galaxies,
$p_{\mrm{cl}}$, is a free parameter of the MCMC program, which is marginalized over,
as are the two coefficients of $f_{\mrm{Lin}}$,
the width of the two Gaussian functions, and the relative normalization of the two.
Scaling the separation between galaxies and associated BCGs by \ensuremath{r_{200}}\xspace
takes advantage of the self-similarity of clusters;
we therefore used \ensuremath{r_{\mrm{gc}}}\xspace-bins defined in units of \ensuremath{r_{200}}\xspace.
We fitted the \ensuremath{v_{\mrm{gc}}}\xspace-distribution with \autoref{eq_vGcFitModel} in each bin using \texttt{MultiNest}\xspace,
a Bayesian inference tool employing importance nested
sampling~\cite{2009MNRAS.398.1601F, *2008MNRAS.384..449F, *2013arXiv1306.2144F}.
The fits for the various \ensuremath{r_{\mrm{gc}}}\xspace-bins
were found to be compatible with the data, scoring better than~$99\%$ in K-S tests.
The observed velocity dispersions were of the order of
several hundred~\ensuremath{\mrm{km/s}}\xspace, more than 50~times larger than the
GRS signal, \ensuremath{\Delta v_{\mrm{gc}}}\xspace. As the signal was difficult to confirm visually,
we also computed the ratio between the integral of the negative and of
the positive parts of the \ensuremath{v_{\mrm{gc}}}\xspace-distribution. The latter is a model-independent
measure of the magnitude of the signal. It was shown to correlate well with
the derived value of \ensuremath{\Delta v_{\mrm{gc}}}\xspace, validating that the \texttt{MultiNest}\xspace fitting procedure
is not biased.
In addition, we wrote a simple Metropolis-Hastings MCMC program
and cross-checked the fit-results.
\subsection{Sample composition and systematic tests}
Our baseline dataset may be utilized in various ways to perform the measurement. One of the
main sources of ambiguity is that we are interested in galaxies which are
several~Mpc away from the corresponding BCGs.
On average, the distance between close pairs of clusters in our dataset
corresponds to~${2.3 \ensuremath{r_{200}}\xspace}$, where for
the bulk of the cluster sample, ${0.8<\ensuremath{r_{200}}\xspace<1.2~\mrm{Mpc}}$.
Many galaxies are therefore likely to be associated with multiple clusters, depending on
their extent and separation.
We nominally define a pair of overlapping
clusters as having a transverse separation, ${r_{\mrm{cc}}<4\ensuremath{r_{200}}\xspace}$,
and a velocity difference, ${\left|v_{\mrm{cc}} \right| < 4,000~\ensuremath{\mrm{km/s}}\xspace}$.
We tested several galaxy selection schemes, with different restrictions
on overlapping configurations.
One possible selection procedure is to exclude all overlapping cluster pairs from
the analysis. Another option is to exclude all but one member from any
configuration of overlapping clusters.
Alternatively, we may choose not to take
into account cluster overlaps at all.
We then accept only those galaxies that have only one cluster association, effectively
performing exclusive selection on galaxies instead of on clusters.
In order to check the dependence of the signal on the composition of our dataset, we
ran the analysis on subsets of the data.
Of these, tests involving
BOSS-BCGs associated with SDSS galaxies
revealed a systematic positive bias of a few~\ensuremath{\mrm{km/s}}\xspace.
So as to understand this effect, we
define the quantity, ${\ensuremath{\delta m^{\mrm{gc}}_{r}}\xspace = m_{r}^{\mrm{g}} - m_{r}^{\mrm{c}}}$,
where $m_{r}^{\mrm{c}}$ and $m_{r}^{\mrm{g}}$ respectively stand for
the $r$-band magnitude of a BCG,
and that of the brightest matched galaxy within one \ensuremath{r_{200}}\xspace of the BCG.
Positive \ensuremath{\delta m^{\mrm{gc}}_{r}}\xspace values correspond to BCGs which are indeed found to be
the brightest source within the area of a cluster.
We observed on average, ${\ensuremath{\delta m^{\mrm{gc}}_{r}}\xspace = -0.3}$
for configurations in which the two surveys were mixed.
The implication of this is that for this sub-sample, it is likely that BCGs
were misidentified in the cluster catalog.
As a result, selected BCGs were less likely to represent the bottom of the
gravitational potential well of clusters, effectively suppressing the GRS signal.
One should also keep in mind that the difference between SDSS and BOSS
redshifts is almost an order of magnitude smaller than the uncertainties on the
redshifts. It is therefore possible that the bias
originates \eg from changes made in the template-fitting procedure between data-releases.
Another important systematic is the treatment of clusters with high
galaxies-multiplicities,
which we denote by \ensuremath{n_{\mrm{gal}}}\xspace. These configurations are subject to two types of bias.
The first is due to the fact that \ensuremath{v_{\mrm{gc}}}\xspace depends on the redshifts of both a
galaxy and the corresponding BCG; consequently, an error
in the redshift of a given BCG affects all matched galaxy-velocities in a correlated way.
The second effect that we observed, was that the value of \ensuremath{\delta m^{\mrm{gc}}_{r}}\xspace,
while generally positive, tends to decrease as \ensuremath{n_{\mrm{gal}}}\xspace increases.
We, therefore, concluded that clusters become more susceptible to misidentification of the BCG
with growing multiplicities.
In order to mitigate these effects, we down-weighted the contribution
of clusters with high multiplicities in the \ensuremath{v_{\mrm{gc}}}\xspace-distribution. We found that
this change mainly affected the signal for low values of \ensuremath{r_{\mrm{gc}}}\xspace.
An additional possible source of bias is the uncertainty associated
with individual spectroscopic redshifts. We checked that there
was no correlation between these, and the corresponding values of \ensuremath{v_{\mrm{gc}}}\xspace.
\section{Results}
\begin{figure*}[htb]
\begin{center}
\begin{minipage}[c]{0.385\textwidth}
\subfloat[]{\label{nSpecClstFIG}\includegraphics[trim=9.5mm 80mm 25mm 0mm,clip,width=1.\textwidth]{figures/nSpecClst.eps}}
\end{minipage}\hfill
\begin{minipage}[c]{0.58\textwidth}
\subfloat[]{\label{deltavGcFIG}\includegraphics[trim=3mm 0mm 20mm 0mm,clip,width=1.\textwidth]{figures/deltaV.eps}}
\end{minipage}\hfill
\caption{
\Subref{nSpecClstFIG}:~Dependence of the number of galaxies, \ensuremath{n_{\mrm{gal}}}\xspace,
of the number of associated clusters, \ensuremath{n_{\mrm{clst}}}\xspace, and of the
ratio, ${\ensuremath{n_{\mrm{gal}}}\xspace/\ensuremath{n_{\mrm{clst}}}\xspace}$, on the separation between BCGs
and associated galaxies, \ensuremath{r_{\mrm{gc}}}\xspace. Bins of \ensuremath{r_{\mrm{gc}}}\xspace are defined by a sliding window with
a width of ${0.5 \ensuremath{r_{200}}\xspace}$, where each data-point is placed in the center-position of
the corresponding bin. \\
%
\Subref{deltavGcFIG}:~ Dependence of the signal of the GRS, \ensuremath{\Delta v_{\mrm{gc}}}\xspace,
on \ensuremath{r_{\mrm{gc}}}\xspace, where the width of the sliding window is denoted by $w_{\mrm{sw}}$.
The shaded areas around the two
nominal results (circles and squares) correspond
to the variations in the signal due to the systematic tests described
in the text, combined with the uncertainty on the model-fit.
On average, ${\ensuremath{\Delta v_{\mrm{gc}}}\xspace = {-11^{+7}_{-5}~\ensuremath{\mrm{km/s}}\xspace}}$ for ${1 < \ensuremath{r_{\mrm{gc}}}\xspace/\ensuremath{r_{200}}\xspace < 2.5}$.
The third dataset (triangles)
includes configurations in which SDSS and BOSS redshifts are mixed together.
The bold lines represent the GR predictions of Kaiser~\cite{2013MNRAS.435.1278K},
with and without his added kinematic effects, as indicated; finally, the crosses
represent the measurement of WHH.
The top axis specifies the median
value and the width of the distribution of \ensuremath{r_{\mrm{gc}}}\xspace (in Mpc) for four bins of
width ${0.5 \ensuremath{r_{200}}\xspace}$, centered at
${\left( 0.5,\;1,\;1.5\;\mrm{and}\;2 \right) \ensuremath{r_{200}}\xspace}$.
} \label{nSpecClst_deltavGcFIG}
\end{center}
\end{figure*}
We estimated \ensuremath{\Delta v_{\mrm{gc}}}\xspace using a sliding window
for the transverse separation between galaxies and clusters. The sliding
window nominally had a width of~${0.5 \ensuremath{r_{200}}\xspace}$, and a step size of~${0.1 \ensuremath{r_{200}}\xspace}$.
For our primary selection, we elected to discard configurations in which SDSS and
BOSS spectra were mixed together.
This reduced the number of clusters and galaxies by~$16\%$ and~$11\%$, respectively.
We also excluded all overlapping cluster pairs,
further reducing these numbers by a respective~$34\%$ and~$46\%$.
The final dataset was composed
of ${12600}$ clusters and ${60626}$ matched galaxies. The
measurement was restricted to
transverse separation values below~${2.5 \ensuremath{r_{200}}\xspace}$.
The reason for this last condition may be inferred from \autoref{nSpecClstFIG}.
The figure shows the dependence on \ensuremath{r_{\mrm{gc}}}\xspace of the number of galaxies, \ensuremath{n_{\mrm{gal}}}\xspace,
of the number of associated clusters, \ensuremath{n_{\mrm{clst}}}\xspace, and of the
ratio, ${\ensuremath{n_{\mrm{gal}}}\xspace/\ensuremath{n_{\mrm{clst}}}\xspace}$. One may observe that for ${\ensuremath{r_{\mrm{gc}}}\xspace < 1.3 \ensuremath{r_{200}}\xspace}$,
the multiplicities of matched clusters and galaxies decrease; this is in accordance with the
expected trend for the surface density of galaxies in clusters
(see \eg figure~8 in~\cite{2005ApJ...633..122H}). However, for ${\ensuremath{r_{\mrm{gc}}}\xspace \gtrsim 2 \ensuremath{r_{200}}\xspace}$,
both the multiplicities and the galaxy-to-cluster ratio increase. This comes about
as galaxies at large \ensuremath{r_{\mrm{gc}}}\xspace have an increasingly higher probability
of being associated with another cluster. Such configurations therefore
tend to suppress the signal of the GRS, and should be rejected from the analysis.
The dependence of \ensuremath{\Delta v_{\mrm{gc}}}\xspace on \ensuremath{r_{200}}\xspace is presented in \autoref{deltavGcFIG}.
We find that on average, ${\ensuremath{\Delta v_{\mrm{gc}}}\xspace = {-11^{+7}_{-5}~\ensuremath{\mrm{km/s}}\xspace}}$ for ${1 < \ensuremath{r_{\mrm{gc}}}\xspace/\ensuremath{r_{200}}\xspace < 2.5}$,
with uncertainties given as 1~standard deviation of the average signal.
In physical scales, the measurement extends up to
galaxy-cluster transverse separations of~${\sim3~\mrm{Mpc}}$.
In addition to the primary result, we present a measurement with an increased
\ensuremath{r_{\mrm{gc}}}\xspace-bin width.
The two results are consistent within uncertainties.
We note that the set-up with the wider bins has a radial resolution
which is slightly too low to describe
the GRS effect at low values of \ensuremath{r_{\mrm{gc}}}\xspace. On the other hand, for high \ensuremath{r_{\mrm{gc}}}\xspace-values,
the increase in statistics in each bin seems to stabilize the result.
Finally, we also include a measurement of \ensuremath{\Delta v_{\mrm{gc}}}\xspace, in which SDSS and BOSS
spectra are used congruently.
This change incurs a systematic shift of a few~\ensuremath{\mrm{km/s}}\xspace, as discussed above.
The uncertainty on \ensuremath{\Delta v_{\mrm{gc}}}\xspace was derived from that on the \texttt{MultiNest}\xspace fit,
combined with the variations incurred
due to the following systematic checks:
changing the minimal number of matched galaxies in a cluster
between~1 and~7;
not down-weighting clusters with high galaxy multiplicities;
using different overlap-removal methods, as described above;
changing the values of the cluster overlap parameters, using
${3 < r_{\mrm{cc}}/\ensuremath{r_{\mrm{gc}}}\xspace < 5}$ and ${3000 < v_{\mrm{cc}} < 6000~\ensuremath{\mrm{km/s}}\xspace}$;
down-weighting galaxies with high spectroscopic redshift uncertainties;
randomly excluding a fraction of galaxies or clusters of a given
data sample.
Of these, the dominant systematic variation originated from changing $r_{\mrm{cc}}$, the threshold for
the transverse separation between clusters.
For comparison, \autoref{deltavGcFIG} also shows the results of WHH within the region of interest.
Our measurement is consistent with these, and has comparable uncertainty estimates.
However, we note that the significance of our result is smaller than
that of WHH, who quote a value, ${\ensuremath{\Delta v_{\mrm{gc}}}\xspace = -7.7 \pm 3.0~\ensuremath{\mrm{km/s}}\xspace}$.
The reason for this is that
WHH computed the integrated signal for all clusters within ${\ensuremath{r_{\mrm{gc}}}\xspace < 6~\mrm{Mpc}}$. They estimated
the uncertainty from that of their MCMC model-fit, which was mainly determined by the size
of their data sample.
In the case of the current analysis, the uncertainty is driven by our
systematic tests, rather than by the available number of clusters and galaxies.
Considering these, and the limited range of acceptance in \ensuremath{r_{\mrm{gc}}}\xspace, our
final relative uncertainty on \ensuremath{\Delta v_{\mrm{gc}}}\xspace is higher.
Calculating the GR predication for \ensuremath{\Delta v_{\mrm{gc}}}\xspace is beyond the scope of this study. However,
the range of cluster masses used in our analysis is comparable to that of the WHH sample.
We therefore refer to the corresponding estimate of Kaiser
of~$-9$ (GR only) or~${-12~\ensuremath{\mrm{km/s}}\xspace}$ (including kinematic effects)~\cite{2013MNRAS.435.1278K}.
Our results are in good agreement with this prediction for ${\ensuremath{r_{\mrm{gc}}}\xspace > \ensuremath{r_{200}}\xspace}$,
while at smaller values of \ensuremath{r_{\mrm{gc}}}\xspace, the profile of \ensuremath{\Delta v_{\mrm{gc}}}\xspace is steeper in the data.
Additionally, we observe that
it is not possible to distinguish between the GR predictions with and without the
kinematic corrections.
\section{Summary}
The gravitational redshift effect allows one to directly probe the gravitational
potential in clusters of galaxies. As such, it provides a fundamental test of
GR.
Following up on the analysis of Wojtak, Hansen \& Hjorth,
we present a new measurement with a larger dataset.
We use spectroscopic redshifts
taken with the SDSS and BOSS, and match them to the BCGs of clusters
from the catalog of Wen, Han \& Liu.
The analysis is based on extracting the GRS signal from the distribution
of the velocities of galaxies in the rest frame of corresponding BCGs.
We focus on optimizing the selection procedure
of clusters and of galaxies, and take into account multiple
possible sources of systematic biases not considered by WHH.
We find an average redshift of~${-11~\ensuremath{\mrm{km/s}}\xspace}$
with a standard deviation of~$+7$ and $-5~\ensuremath{\mrm{km/s}}\xspace$ for ${1 < \ensuremath{r_{\mrm{gc}}}\xspace/\ensuremath{r_{200}}\xspace < 2.5}$. The result is
consistent with the measurement of WHH.
However, our overall systematic uncertainty
is relatively larger than that of WHH, mainly due to
overlapping cluster configurations; the
significance of detecting the GRS signal in the current analysis
is therefore reduced in comparison.
Our measurement is in good agreement with the GR predictions.
Considering the current uncertainties, we can not distinguish between the
baseline GR effect and the recently proposed kinematic modifications.
With the advent of future spectroscopic surveys, such as
Euclid and DESI~\cite{2011arXiv1110.3193L,*desiRef},
we will have access to larger, more homogeneous datasets.
We expect that the new spectra will help to
reduce the systematic uncertainties on the measurement,
though dedicated target selection may be required.
Additionally, new data will facilitate novel techniques of
detecting the GRS signal, such as
the cross-correlation method suggested in~\cite{2013MNRAS.434.3008C}.
\section*{Acknowledgements}
We would like to thank
Jacob Bekenstein,
Jens Hjorth,
Pablo Jimeno,
Nick Kaiser,
John Peacock,
David Schlegel and
Radek Wojtak,
for the useful discussions regarding the nature of
spectroscopic redshifts, galaxy clusters and GRS.
O.L.\ acknowledges an Advanced European Research Council Grant, which
supports the postdoctoral fellowship of I.S.
This work uses publicly available data from the SDSS.
Funding for SDSS-III has been provided by the Alfred P.~Sloan Foundation,
the Participating Institutions, the National Science Foundation, and the U.S.\
Department of Energy Office of Science. The SDSS-III website
is \href{http://www.sdss3.org/}{http://www.sdss3.org/}.
\bibliographystyle{apsrev4-1} |
2,869,038,154,759 | arxiv | \section{Introduction} \label{s:introduction}
In the standard Cold Dark Matter (CDM) paradigm, the formation of
galaxies is driven by the growth of the large-scale structure of the
Universe and the formation of dark matter halos. Galaxies form by the
cooling and condensation of gas in the centers of the potential wells
of extended virialized dark matter halos
\citep{whiterees78, fallefstathiou1980, blumenthal1984}. In this picture,
galaxy properties, such as luminosity or stellar mass,
are expected to be tightly coupled to the depth of the halo potential
and thus to the halo mass.
There are various different approaches to link the properties of
galaxies to those of their halos. A first method attempts to derive
the halo properties from the properties of its galaxy population using
e.g. galaxy kinematics \citep{erickson1987, zaritsky1993, carlberg1996,
more2009a,more2009b}, gravitational lensing \citep{mandelbaum05,
mandelbaum06,cacciato2008}, or X-ray studies \citep{lin2003,linmohr2004}.
A second approach is to attempt to model the physics that shapes
galaxy formation \emph{ab initio} using either large numerical
simulations including both gas and dark matter \citep{katz1996,
springel2003} or semi-analytic models (SAMs) of galaxy formation
\citep[e.g.][]{kauffmann1993, cole1994,somerville1999}. In ``hybrid''
SAMs \citep[e.g.][]{croton06,bower2006}, dark matter ``merger trees''
are extracted from a dark matter only N-body simulation, and gas
processes are treated with semi-analytic recipes. An advantage of this
method is that high-resolution N-body simulations can track the
evolution of individual subhalos \citep{klypin1999, springel01} and
thus provide the precise positions and velocities of galaxies within a
halo. However, many of the physical processes involved in galaxy
formation (such as star formation and various kinds of feedback) are
still not well understood, and in many cases simulations are not able
to reproduce observed quantities with high accuracy.
With the accumulation of data from large galaxy surveys over the last
decade, a third method has been developed, which links galaxies to
halos using a statistical approach. The Halo Occupation
Distribution (HOD) formalism specifies the probability distribution
for a halo of mass $M$ to harbour $N$ galaxies with certain intrinsic
properties, such as luminosity, color, or type \citep[e.g.][]{peacock2000,
seljak2000, white2001, berlind2002}. More complex
formulations of this kind of modelling, such as the conditional
luminosity function (CLF) formalism \citep{yang03,vdb2003,yang04} have extended the HOD
approach. These methods have the advantage that they do not rely on
assumptions about the (poorly understood) physical processes that
drive galaxy formation. In this way, it is possible to constrain the
relationship between galaxy and halo properties (and thus, indirectly,
the underlying physics), and to construct mock catalogs that reproduce
in detail a desired observational quantity (such as the luminosity
function). One disadvantage of the classical HOD approach was that one
had to make assumptions about the distribution of positions and
velocities of galaxies within their host halos. In addition, the
results of the HOD modelling can be difficult to interpret in terms of
the underlying physics of galaxy formation.
In recent years, HOD models have been introduced that make use of
information about the positions, velocities and masses of halos and
subhalos extracted from a dissipationless N-body simulation. The
(sub)halo mass is then empirically linked to galaxy properties by
requiring that a statistical observational quantity (e.g. galaxy
luminosity function and/or galaxy two-point-correlation-function) is
reproduced. This is either done by assuming parameterized functions to
relate galaxy properties (such as luminosity) to halo mass or by
assuming a non-parametric monotonic relation. It has been shown that
these simple models reproduce galaxy clustering as a function of
luminosity over a wide range in redshift \citep{kravtsov2004,
tasitsiomi2004,tinker2005,valeostriker06,conroy06, shankar2006,
wang06,marin2008}.
Observationally, it is well known that galaxy clustering is a function
of spatial scale, galaxy properties (such as luminosity and type), and
redshift. Luminous (massive) galaxies are more strongly clustered than
less luminous (less massive) galaxies \citep{norberg2001,norberg2002,
zehavi2002,zehavi2005,li06}. One can split the galaxy two-point correlation
function (2PCF) into two separate parts: the one-halo and the two-halo terms.
The one-halo term, which dominates on small scales, depends strongly on the
galaxy distribution within the halo as well as the details of the HOD.
The two-halo term, which dominates on scales that are much larger than a typical halo, is
proportional to the auto-correlation of the halo population. In general the two terms are not
expected to combine to produce a featureless power-law, but generally show a
break or dip at the scale where the transition from the one-halo to the two-halo
term occurs \citep{zehavi2004}.
The extensive multi-wavelength spectrophotometric information that is
now available for large numbers of galaxies allows us to estimate
physical parameters of galaxies, such as stellar masses, instead of
relying on observational properties such as magnitudes \citep{bell2001,
kauffmann03,panter2004}. These estimates can even be
obtained --- with a proper measure of caution --- for high redshift
galaxies. Stellar mass estimates have been presented in the literature
for galaxies up to redshifts as high as $z\sim 6$
\citep{yan2006,eyles2007}, and stellar
mass function estimates have been presented up to $z\sim 5$
\citep{drory2005,fontana2006,elsner2008}.
The goal of our paper is to develop a ``Conditional Stellar
Mass Function'' (CMF) formalism, which is the stellar mass analog of the
CLF. The CMF yields the average number of galaxies with stellar masses
in the range $m\pm{\rm d}m$ as a function of the host halo mass $M$
and can be regarded as the stellar mass function (SMF) for halos of mass
$M$. We apply this formalism at low redshift and up to the highest
redshifts where reliable observational stellar mass estimates are
available ($0.1 \lesssim z \lesssim 4$). In this way, we derive a parameterized
relationship between dark matter halo mass and galaxy mass as a
function of redshift.
Using a parameterized relationship has several advantages. First, it
provides a convenient way for other researchers to make use of our
results and obtain an expression for stellar mass as a function of
halo mass. Second, it is straightforward to include scatter in the
relation, which is physically more realistic: one just has to choose a
number drawn from an assumed random distribution and add that to the
average relation. Finally, it is straightforward to treat central and
satellite galaxies separately and assume different relations between
stellar and halo mass for those populations. However, here we make
the assumption that both populations follow the same relation, which
has consequences for the clustering predictions of our model.
Using the CMF derived \emph{only} from constraints from the observed
SMF, we compute the predicted (projected) galaxy CF at $z\sim 0$ as a
function of stellar mass, and find good agreement with the
observational results of \citet{li06}.
Furthermore, we show that assuming central and satellite galaxies
follow the same relation between stellar and halo mass, adding the
clustering constraints does not tighten the constraints on
our model parameters; i.e., any model that satisfies the mass function
constraints will produce the correct clustering. Based on this result,
we use our redshift-dependent CMF results to \emph{predict} the
clustering as a function of stellar mass and redshift.
To date, observational measurements of clustering as a function of stellar mass
have only been published for $z\mathrel{\spose{\lower 3pt\hbox{$\sim$}}\raise 2.0pt\hbox{$<$}} 1$ \citep{meneux2008,meneux2009}.
We show that our model predictions agree very well with these measurements.
Very soon it will be possible to test our predictions for redshifts beyond
$z=1$ with the results from deep wide-field surveys (e.g. MUSYC, UKIDDS, etc).
We again present convenient fitting functions for the galaxy bias as a function
of both stellar mass and redshift. In a companion paper we will employ our
estimates of galaxy bias in order to compute the ``cosmic variance'',
the uncertainty in observational estimates of the volume density of
galaxies arising from the underlying large-scale density fluctuations.
This paper is organized as follows: in section \ref{s:simulation} we
describe the $N$-body simulation, the halo finding algorithm that was
used to obtain a halo catalogue and the treatment of `orphaned'
galaxies. Section \ref{s:galaxies} specifies our model: we motivate
the form of the stellar-to-halo mass (SHM) relation and constrain it by
requiring that the observed SMF is reproduced. The
clustering properties of galaxies are then inferred from those of the
halo population. We discuss the meaning of the parameters of the
SHM relation and demonstrate that clustering puts
only weak constraints on them. In section \ref{s:cmf} we introduce the
CMF, which describes how halos are occupied by galaxies, and compute
the occupation numbers. Section \ref{s:comparison} gives a comparison
between our results and several other models and observations. In
section \ref{s:redshift} we apply our method to higher redshifts and
determine the redshift dependence of the SHM
relation. We make predictions of the stellar mass dependent galaxy
CF at higher redshift which we use to compute the
galaxy bias. Finally, we summarize our methods and conclusions in
section \ref{s:conclusions}.
Throughout this paper we assume a $\Lambda$CDM cosmology with
($\Omega_m$,$\Omega_{\Lambda}$,$h$,$\sigma_8$,$n$) =
($0.26$,$0.74$,$0.72$,$0.77$,$0.95$). We employ a \citet{kroupa01}
initial mass function (IMF) and convert all stellar masses to this
IMF. In order to simplify the notation we will use the capital $M$ to
denote dark matter halo masses and the lower case $m$ to denote galaxy
stellar masses.
\section{The simulation and halo catalogs}
\label{s:simulation}
High-resolution dissipationless N-body simulations have shown that
distinct halos\footnote{We refer to virialized halos that are not
subhalos of another halo as ``distinct''.} contain subhalos which
orbit within the potential of their host halo. These subhalos were
distinct halos in the past, and entered the larger halo via merging
during the process of hierarchical assembly. We will refer to the
galaxy at the center of a distinct halo as a central galaxy, and the
galaxies within subhalos as ``satellites'', and we will use the term
`halo' to refer to the distinct halo for central galaxies and to the
subhalo in which the galaxy originally formed for satellite galaxies.
{\emph Ab initio} models of galaxy formation predict that the stellar
mass of a galaxy is tightly correlated with the depth of the potential
well of the halo in which it formed. For distinct halos, the relevant
mass is the virial mass at the time of observation. Subhalos, however,
lose mass while orbiting in a larger system as their outer regions are
tidally stripped. Stars are centrally concentrated and more tightly
bound than the dark matter, however, and so the stellar mass of a
galaxy which is accreted by a larger system probably changes only
slightly until most of the dark matter has been stripped
off. Therefore the subhalo mass at the time of observation is probably
not a good tracer for the potential well that shaped the galaxy
properties. A better tracer is the subhalo mass at the time that it
was accreted by the host halo, or its maxmimum mass over its
history\footnote{In an idealized situation, halo mass should increase
monotonically with time until the halo becomes a subhalo, at which
point the mass begins to decrease due to tidal stripping.}.
This was first proposed by \citet{conroy06}.
The population of dark matter halos used in this work is drawn from an
$N$-body simulation run with the simulation code {\small GADGET}-2
\citep{springel05a} on a SGI AltixII at the University Observatory Munich.
The cosmological parameters of the simulation are
chosen to match results from {\small WMAP}-3 \citep{spergel06} for a
flat $\Lambda$CDM cosmological model: $\Omega_m=0.26$,
$\Omega_{\Lambda}=0.74$, $h=H_0/(100$~km~s$^{-1}$~Mpc$^{-1})=0.72$,
$\sigma_8=0.77$ and $n=0.95$. The initial conditions were generated
using the GRAFIC software package \citep{bertschinger2001}. The
simulation was done in a periodic box with side length $100$ Mpc,
and contains $512^3$ particles with a particle mass of
$2.8\times 10^8\!\,{\rm M}_\odot$ and a force softening of $3.5$ kpc.
Dark matter halos are identified in the simulation using a
friends-of-friends (FoF) halo finder. Substructures inside the FoF
groups are then identified using the {\small SUBFIND} code described
in \cite{springel01}. For the most massive subgroup in a FoF group the
virial radius and mass are determined with a spherical overdensity
criterion: the density inside a sphere centered on the most bound
particle is required to be greater than or equal to the value
predicted by the spherical collapse model for a tophat perturbation in
a $\Lambda$CDM cosmology \citep{bryan98}. As discussed above, for
subhalos we use the maximum mass over its past history, which is
typically the mass when the halo was last a distinct halo and did not yet
overlap with its later host. Merger trees were constructed out of the
halo catalogs at 94 time-steps, equally spaced in expansion factor
($\Delta a=0.01$), based on the particle overlap of halos at different
time-steps.
Due to the finite mass resolution of the simulation
($M_{\rm min,halo}\simeq 10^{10} \!\,{\rm M}_\odot$), subhalos can no longer be
identified when their mass has dropped below this limit due to tidal
stripping. Since mass loss can be substantial (>90\%) this is
important even for fairly massive subhalos. A special treatment of
these so-called ``orphans'' is necessary. We determine the orbital
parameters at the last moment when a subhalo is identified in the
simulation and use them in the dynamical friction recipe of
\cite{boylan08}, which is applicable at radii $r<r_{vir}$. We
also tried an alternate recipe in which we make no explicit use of the
subhalo information, but apply the dynamical friction formula from the
time when the satellites first enter the host halo. We obtained very
similar subhalo mass functions and radial distributions with the
alternate recipe, confirming the self-consistency of the approach.
For the halo positions in the determination of CFs,
we use the coordinates of the most bound particle for distinct and
subhalos. For orphans, by definition, the position is not known, so we
follow the position of the most bound particle from the last time-step
when a subhalo was identified. Since the dynamical friction force
vanishes in the dark matter only simulation after a subhalo is dissolved,
yet not in reality when a galaxy is present at the center of the subhalo, the
distance to the center of the host halo might be slightly
overestimated with this prescription.
\section{Connecting galaxies and halos} \label{s:galaxies}
In this section we describe how we derive the relationship connecting
the stellar mass of a galaxy to the mass of its dark matter halo. In
the standard picture of galaxy formation, gas can only cool and form
stars if it is in a virialized gravitationally bound dark matter halo
\citep{whiterees78}. In this model the gas cooling rate, the star
formation rate and thus the properties of the galaxy depend
mainly on the virial mass of the host halo. Thus we expect the stellar
mass of a central galaxy to be strongly correlated with the virial mass of
the halo in which the galaxy formed. As we discussed in the last section,
this corresponds to the virial mass for central galaxies, and to the
maximum mass over the halo's history for satellite galaxies. In the
rest of this work, unless noted otherwise, the halo mass $M$ will
represent:
\begin{equation}
M = \begin{cases}
M_{\rm vir} & \text{for host halos}\\
M_{\rm max} & \text{for subhalos}
\end{cases}
\label{eqnmmax}
\end{equation}
Note that we have also experimented with instead using the present
mass for subhalos, and found that we were not able to reproduce the
galaxy clustering properties \citep[see also][]{conroy06}.
\subsection{The stellar-to-halo mass relation}\label{s:galaxymap}
In order to link the stellar mass of a galaxy $m$ to the mass of its
dark matter halo $M$ we need to specify the SHM
ratio. A direct comparison of the halo mass function $n(M)$ and the
galaxy mass function $\phi(m)$ helps to constrain the stellar-to-halo
mass function. If we assume that every host (sub) halo contains
exactly one central (satellite) galaxy and that each system has
exactly the same SHM ratio $m/M$, the galaxy stellar
mass function can be derived trivially from the halo mass function and
has the same features. The galaxy mass function derived for $m/M =
0.05$ is compared to the observed SDSS galaxy mass function in Figure
\ref{f:fig1}. The observed galaxy mass function is steeper for high
masses and shallower for low masses than the one derived from the halo
mass function. Thus, for a constant SHM ratio there
will inevitably be too many galaxies at the low and high mass end.
\begin{figure}
\centering
\plotone{fig1.eps}
\caption[Comparison between the halo mass function, observed and model
galaxy mass functions]{A comparison between the halo mass function
offset by a factor of 0.05 (dashed line), the observed galaxy mass
function (symbols), our model without scatter (solid line) and our model
including scatter (dotted line).We see that the halo
and the galaxy mass functions are different shapes,
implying that the stellar-to-halo mass ratio $m/M$ is not
constant. Our four parameter model for the halo mass dependent
stellar-to-halo mass ratio is in very good agreement with the
observations (both including and neglecting scatter).}
\label{f:fig1}
\end{figure}
This implies that the actual SHM ratio $m/M$ is not
constant, but increases with increasing mass, reaches a maximum around
$m^*$ and then decreases again. Hence we adopt the following
parametrization, similar to the one used in \citet{yang03}:
\begin{equation}
\frac{m(M)}{M} = 2 \left( \frac{m}{M}\right)_0 \left[\left(\frac{M}{M_1}\right)^{-\beta} + \left(\frac{M}{M_1}\right)^{\gamma}\right]^{-1}
\label{eqnmmap}
\end{equation}
It has four free parameters: the normalization of the stellar-to-halo
mass ratio $(m/M)_0$, a characteristic mass $M_1$, where the
SHM ratio is equal to $(m/M)_0$, and two slopes
$\beta$ and $\gamma$ which indicate the behavior of $m/M$ at the low
and high mass ends respectively. We use the same parameters for the
central and satellite populations, since -- unlike luminosity -- the
stellar mass of satellites changes only slightly after they are
accreted by the host halo.
Note that though both $\beta$ and $\gamma$ are expected to be
positive, they are not restricted to be so. The SHM
relation is therefore not necessarily monotonic.
\subsection{Constraining the free parameters}
\label{constraining}
Having set up the model we now need to constrain the four free
parameters $M_1$, $(m/M)_0$, $\beta$ and $\gamma$. To do this, we
populate the halos in the simulation with galaxies. The stellar masses
of the galaxies depend on the mass of the halo and are derived
according to our prescription (equation \ref{eqnmmap}). The positions
of the galaxies are given by the halo positions in the $N$-body
simulation.
Once the simulation box is filled with galaxies, it is straightforward
to compute the SMF $\Phi_{mod}(m)$. As we want to
fit this model mass function to the observed mass function
$\Phi_{obs}(m)$ by \citet{panter2007}, we choose the same stellar mass
range ($10^{8.5}-10^{11.85}~\!\,{\rm M}_\odot$) and the same binsize. The observed
SMF was derived using spectra from the Sloan Digital
Sky Survey Data Release 3 (SDSS DR3); see \citet{panter2004} for a description
of the method.
Furthermore it is possible to determine the stellar mass dependent
clustering of galaxies. For this we compute projected galaxy
CFs $w_{p,mod}(r_p,m_i)$ in several stellar mass
bins which we choose to be the same as in the observed projected
galaxy CFs of \citet{li06}. These were derived using
a sample of galaxies from the SDSS DR2 with stellar masses estimated
from spectra by \citet{kauffmann03}.
We first calculate the real space CF $\xi(r)$. In a
simulation this can be done by simply counting pairs in distance bins:
\begin{equation}
\label{corrfunc}
\xi(r_i) = \frac{dd(r_i)}{N_p(r_i)}-1
\end{equation}
where $dd(r_i)$ is the number of pairs counted in a distance bin and
$N_p(r_i) = 2 \pi N^2 r_i^2 \Delta r_i / L_{\rm box}^3$ where $N$ is the
total number of galaxies in the box. The projected CF
$w_p(r_p)$ can be derived by integrating the real space correlation
function $\xi(r)$ along the line of sight:
\begin{equation}
w_p(r_p) = 2 \int_{0}^{\infty} {\rm d}r_{||} \xi(\sqrt{r_{||}^2+r_p^2}) = 2 \int_{r_p}^{\infty} {\rm d}r \frac{r~\xi(r)}{\sqrt{r^2-r_p^2}} \;,
\end{equation}
where the comoving distance ($r$) has been decomposed into components
parallel ($r_{||}$) and perpendicular ($r_p$) to the line of
sight. The integration is truncated at $45$~Mpc. Due to the finite
size of the simulation box ($L_{\rm box}=100$~Mpc) the model correlation
function is not reliable beyond scales of
$r\sim0.1~L_{\rm box}\sim10$~Mpc.
In order to fit the model to the observations we use Powell's
directions set method in multidimensions (e.g. Press et al. 1992) to
find the values of $M_1$, $(m/M)_0$, $\beta$ and $\gamma$ that
minimize either $$ \chi_r^2 = \chi_r^2 (\Phi)= \frac{\chi^2(\Phi)}{N_{\Phi}}$$ (mass
function fit) or $$ \chi_r^2 = \chi_r^2 (\Phi)+\chi_r^2 (w_p) = \frac{\chi^2(\Phi)}{N_{\Phi}} +
\frac{\chi^2(w_p)}{N_r\;N_m}$$ (mass function and
projected CF fit)
with $N_{\Phi}$ and $N_{r}$ the number of data points
for the SMF and projected CFs, respectively, and $N_{m}$ the number
of mass bins for the projected CFs.
In this context $\chi^2(\Phi)$
and $\chi^2(w_p)$ are defined as:
\begin{eqnarray}
\chi^2(\Phi) &=& \sum_{i=1}^{N_{\Phi}} \left[ \frac{\Phi_{\rm mod}(m_i)-\Phi_{\rm obs}(m_i)}{\sigma_{\Phi_{\rm obs}(m_i)}} \right]^2\notag\\
\chi^2(w_p) &=& \sum_{i=1}^{N_m} \sum_{j=1}^{N_r} \left[ \frac{w_{p,\rm mod}(r_{p,j},m_i)-w_{p,\rm obs}(r_{p,j},m_i)}{\sigma_{w_{p,\rm obs}(r_{p,j},m_i)}} \right]^2\notag\;,
\end{eqnarray}
with $\sigma_{\Phi_{\rm obs}}$ and $\sigma_{w_{p,\rm obs}}$ the errors for the
SMF and projected CFs, respectively.
Note that for the simultaneous fit, by adding the reduced $\chi_r^2$,
we give the same weight to both data sets.
\begin{figure*}
\centering
\epsscale{1.15}
\plotone{fig2.eps}
\caption{Comparison between the model (lines) and observed (symbols
with errorbars) projected correlation functions. We show the model
results both including (solid) and excluding (dashed) orphan
galaxies. The models have been derived by fitting to the stellar mass
function only.}
\label{f:fig2}
\end{figure*}
\subsection{Estimation of parameter errors}\label{s:probdis}
In order to obtain estimates of the errors on the parameters, we need
their probability distribution prob($A\vert I$), where $A$ is the
parameter under consideration and $I$ is the given background
information. The most likely value of $A$ is then given by: $A_{\rm best}
={\rm max(prob(}A\vert I))$.
As we have to assume that all our parameters are coupled, we can only
compute the probability for a given set of parameters. This
probability is given by:
$${\rm prob(}M_1,(m/M)_0,\beta,\gamma\vert I) \propto \exp(-\chi^2)$$
In a system with four free parameters $A,B,C$ and $D$ one can
calculate the probability distribution of one parameter (e.g. $A$) if
the probability distribution for the set of parameters is known, using
marginalization:
\begin{eqnarray}
{\rm prob(}A\vert I) &=& \int_{-\infty}^{\infty}{\rm prob(}A,B\vert I) {\rm d}B\notag\\
&=& \int_{-\infty}^{\infty}{\rm prob(}A,B,C,D\vert I) {\rm d}B {\rm d}C {\rm d}D\notag
\end{eqnarray}
Once the probability distribution for a parameter is determined, one
can assign errors based on the confidence intervals. This is the
shortest interval that encloses a certain percentage $X$ of the area
under the posterior probability distribution. For the 1-sigma error $X = 68\%$
while for the 2-sigma error $X = 95\%$. Assuming that the probability distribution
has been normalized to have unit area we seek $A_1$ and $A_2$ such that
$$ \int_{0}^{A_1} {\rm prob(}A\vert I) {\rm d}A = \int_{A_2}^{\infty}
{\rm prob(}A\vert I) {\rm d}A = \frac{1-X}{2} .$$
Finally the parameter $A$ is given as $A = A_{\rm best}\text{
}_{-\sigma_{-}}^{+\sigma_{+}}$ with $\sigma_{+} = A_2 - A_{\rm best} $
and $\sigma_{-} = A_{\rm best} - A_1$.
The errors derived in this way only include sources that have been considered
when computing $\chi^2$. The calculation of the errors applies for uncorrelated
data points. Since in our case the data points are correlated the values of the errors
are slightly modified. Also errors caused by cosmic variance are not included.
\section{Fitting results}\label{s:results}
Here we present the results we obtain by fitting to the stellar mass
function only, and for the combined fit to the SMF
and the projected CF.
\subsection{The stellar mass function fit}
\begin{deluxetable}{lllllll}[hb!]
\tablecaption{Fitting results for Stellar-to-Halo Mass relationship}
\tablehead{
\colhead{} &
\colhead{$\log M_1$} &
\colhead{$(m/M)_0$} &
\colhead{$\beta$} &
\colhead{$\gamma$} &
\colhead{$\chi_r^2(\Phi)$} &
\colhead{$\chi_r^2(w_p)$}
}
\startdata
best fit & 11.884 & 0.02820 & 1.057 & 0.556 & 1.56 & 3.83\\
$\sigma^+$ & ~~0.030 & 0.00061 & 0.054 & 0.010 & &\\
$\sigma^-$ & ~~0.023 & 0.00053 & 0.046 & 0.004 & &\\
\enddata
\tablecomments{No scatter included. All masses are in units of $\!\,{\rm M}_\odot$}
\label{t:mfresults}
\end{deluxetable}
First we fit to the SDSS SMF and use the derived
best-fit parameters to calculate the model projected correlation
functions. Note that for now, we do not take into account
any possible scatter in the $m(M)$ relation. We will consider scatter
in \S\ref{s:scattersec}.
We see in Figure \ref{f:fig1} that our fit produces excellent
agreement with the observed SMF. Using the approach
described above we also compute the errors on the parameters. The
results are summarized in Table \ref{t:mfresults}.
Having derived the best-fit parameters, we can predict the projected
CFs. We present the results both including and not
including orphan galaxies, where we have fitted to the SMF for each case.
Figure \ref{f:fig2} shows a comparison between our model and the SDSS
projected correlation functions in five stellar mass bins ranging from
$\log m/\!\,{\rm M}_\odot=9.0$ to $\log m/\!\,{\rm M}_\odot=11.5$ with a binsize of $0.5 {\rm~dex}$.
The correlation function that has been derived without orphans is too low at
small scales and can be regarded as a lower limit. Neglecting these galaxies
results in an underprediction of satellite galaxy clustering. As on
small scales the projected CF depends mainly on
the one-halo term this results in the underprediction of $w_p(r_p)$.
This effect weakens for the clustering of more massive galaxies as they are
more likely to be central galaxies and thus not effected by tidal stripping at all.
The agreement with the observationally derived $w_p(r_p)$ for the
catalogue including orphaned galaxies is very good, which is also
reflected in the low value of $\chi_r^2(w_p) = 3.83$. Note that this
value has been calculated with the parameters from the mass function
fit given above and does not correspond to a fit to the projected CFs.
Note that we plot the projected CFs only up to $20\, {\rm
Mpc}$. Because of the finite box size, the clustering of host
halos and thus central galaxies is underpredicted at large scales
independent of mass. Additionally, due to the lack of
long-wavelength modes, massive halos and galaxies can be
underproduced leading to an underprediction of $w_p$ for the massive
objects, independent of scale. However, the latter effect is very
small, since the abundance of the massive halos in our simulation
agrees very well with the predicted average \citep{sheth1999}.
As a test we also used the present mass instead of the maximum mass for
subhalos. We then found that the projected CF was
underpredicted particularly on small scales. This effect is due to
tidal stripping of subhalos and is thus strongest at small scales
where the subhalo contribution dominates.
\subsection{The combined fit}\label{combinedfit}
\begin{figure}
\centering
\plotone{fig3.eps}
\caption{Sketch of the probability distributions for a simultaneous
fit. The solid line corresponds to $\chi^2(m)$ and the dotted line
to $\chi^2(w_p)$. The dashed line is the sum of both. Since
$\chi^2(w_p)$ is flat at the minimum, $\chi^2_{\rm tot}$ follows
$\chi^2(m)$ with an offset. The resulting probability distribution
does not change (after normalization).}
\label{f:fig3}
\end{figure}
We now investigate whether we can improve the agreement between the
model and the observed projected CFs by performing a
combined fit as described above. We obtain the same parameters as
those we derived from the fit to the SMF alone. This
seems surprising, but on further inspection we find that this is due
to $\chi^2(m)$ being a lot more sensitive to changes of the parameters
than $\chi^2(w_p)$. This means that if one changes the parameters a
little in order to improve the fit to the projected correlation
functions, one can get a slightly better agreement between the model
and the observed projected CFs only at the cost of a
large disagreement between the model and the observed stellar mass
functions. In other words: $\chi^2(w_p)$ is much flatter around its
minimum than $\chi^2(w_p)$, as shown in Figure \ref{f:fig3}.
This means that, assuming that both central and satellite
galaxies follow the same SHM relation, the model that matches the
SMF can reproduce the correct clustering. However, if subhalos have
a different SHM ratio there is an infinite number of solutions that
match the SMF but produce very different correlation functions. The
only way to constrain the SHM relations then is to take the
clustering data into account. By adopting different SHM relations
for central and satellite populations it is even possible to produce
a slightly better fit to the correlation functions \citep{wang06}.
On the other hand, if one wants to {\em predict} clustering as a
function of stellar mass (e.g. at higher redshift) then one has to
make an assumption about how the SHM ratios of central and satellite
galaxies are related. We made the very simple assumption, that the
relation between the stellar mass of central galaxies and the virial
mass of their host halo and the relation between the stellar mass of
satellite galaxies and the mass of the subhalo at the time of
accretion is the same, and have shown that this leads to very good
predictions for the mass dependent clustering. We conclude that under
this simple assumption we can use our model to predict clustering as a
function of stellar mass.
\subsection{The resulting stellar-to-halo mass relation} \label{masstomass}
\begin{figure}
\centering
\epsscale{1.15}
\plotone{fig4.eps}
\caption[The derived relation between stellar mass and halo mass]{The
derived relation between stellar mass and halo mass. The light
shaded area shows the $1\sigma$-region while the dark and light
shaded areas together show the $2\sigma$-region.
The upper panel shows the SHM relation while the lower panel shows the SHM ratio.}
\vspace{0.5cm}
\label{f:fig4}
\end{figure}
The upper panel of Figure \ref{f:fig4} shows the derived stellar mass
as a function of halo mass.
The light shaded area gives the 68\% confidence
interval while the dark and light shaded areas together give the
95\% confidence interval. These have been derived using a set
of different models computed on a mesh, as described in
\S\ref{s:probdis}.
For the SHM ratio we apply the same
procedure. The result is shown in the lower panel of Figure
\ref{f:fig4}. We see that the SHM ratio has the form
we expected: it increases with increasing halo mass, reaches its
maximum value around $M_1$ and then decreases again.
\subsection{Meaning of parameters and correlations}\label{s:parameters}
\begin{figure}
\centering
\epsscale{1.2}
\plotone{fig5.eps}
\caption{Correlations between the model parameters. The panels show
contours of constant $\chi^2$ (i.e. constant probability) for
the fit including contraints from the SMF only. The parameter
pairs are indicated in each panel.}
\vspace{0.5cm}
\label{f:fig5}
\end{figure}
We now explore the effects of changing each parameter in order to
understand how they affect the SMF.
\label{ratio}
If we keep $M_1$, $\beta$ and $\gamma$ fixed and only vary $(m/M)_0$,
this corresponds to changing the stellar mass of the galaxy that lives
inside each halo by a constant factor. This has no impact on the form
of the SMF. Its shape stays the same, while only the position on the
stellar mass axis changes. Due to the monotonic form of the SMF this
directly determines the value of the normalization $\phi^*$. For a
larger value of $(m/M)_0$ we get a larger value of $\phi^*$.
\label{m1}
Varying only $M_1$ we find that the shape of the SMF
changes drastically. For a higher $M_1$ than our best fit value, we
get too many massive galaxies and too few low mass galaxies, while for
a lower value of $M_1$ we get too few massive galaxies and too many
low mass galaxies. This is because $M_1$ is the characteristic mass
corresponding to the highest SHM ratio. In the
SMF, this corresponds to the knee and we get a
SMF which has its knee at the stellar mass
corresponding to $M_1$. For a larger $M_1$ the knee is shifted to a
higher stellar mass. Together, $M_1$ and the maximum stellar-to-halo
mass ratio $(m/M)_0$ determine the normalization of the stellar mass
function $\phi$ and the characteristic mass $m^*$.
\label{beta}
Changing $\beta$ affects mainly the low mass slope of the stellar mass
function. For larger values of $\beta$ the slope becomes shallower.
As $\beta$ influences mainly the slope of the low mass
end of the SMF, it is strongly related to the
parameter $\alpha$ of the Schechter function. A small value of $\beta$
corresponds to a high value of $\alpha$.
\label{gamma}
If we change $\gamma$, this mainly impacts the slope of the massive
end of the SMF. For larger values of $\gamma$ than for its best-fit
value the slope of the massive end becomes steeper. As $\gamma$
affects mainly the slope of the massive end of the SMF it is not
coupled to a parameter of the Schechter function though it is related
to the high-mass cutoff, assumed to be exponential in a Schechter
function.
Figure \ref{f:fig5} shows the contours of the two-dimensional probability
distributions for the parameters pairs. We see a
correlation between the parameters [$M_1,\gamma$] and
[$(m/M)_0,\gamma$] and an anti-correlation between [$\beta,\gamma$],
[$\beta,M_1$] and [$(m/M)_0,M_1$]. There does not seem to be a
correlation between [$\beta,(m/M)_0$].
\subsection{Introducing scatter}\label{s:scattersec}
\begin{figure}
\centering
\epsscale{1.15}
\plotone{fig6.eps}
\caption{Stellar mass as a function of halo mass with
$\sigma_m=0.15\, {\rm dex}$. The solid line corresponds to our model
without scatter while the points represent the model with scatter
(note that only 20\% of the total number of objects are
plotted). The relation between halo mass and the average stellar
mass for the model with scatter is shown by the dashed line.}
\label{f:fig6}
\end{figure}
Up until now we have assumed that there is a one-to-one, deterministic
relationship between halo mass and stellar mass. However, in nature,
we expect that two halos of the same mass $M$ may harbor galaxies with
different stellar masses, since they can have different halo concentrations,
spin parameters and merger histories.
For each halo of mass $M$, we now assign a stellar mass $m$ drawn from
a log-normal distribution with a mean value given by our previous
expression for $m(M)$ (Equation \eqref{eqnmmap}), with a variance of
$\sigma_m^2$. We assume that the variance is a constant for all halo
masses, which means that the percent deviation from $m$ is the same
for every galaxy. This is consistent with other halo occupation
models, semi-analytic models and satellite kinematics
\citep{cooray2006, vdb2007,more2009b}.
Assuming a value of $\sigma_m = 0.15{\rm~dex}$ and fitting the stellar
mass function only, we find the values given in Table
\ref{t:mfscatresults}. These values lie within the (2$\sigma$) error
bars of the best-fit values that we obtained with no scatter. The
largest change is on the value of $\gamma$, which controls the slope
of the SHM relation at large halo masses.
The SMF and the projected CFs for the model including scatter are
shown in Figures \ref{f:fig1} and \ref{f:fig2}, respectively, and show
very good agreement with the observed data.
\begin{deluxetable}{lllllll}[hb!]
\tablecaption{Fitting Results for for Stellar-to-Halo Mass relationship}
\tablehead{
\colhead{} &
\colhead{$\log M_1$} &
\colhead{$(m/M)_0$} &
\colhead{$\beta$} &
\colhead{$\gamma$} &
\colhead{$\chi_r^2(\Phi)$} &
\colhead{$\chi_r^2(w_p)$}
}
\startdata
best fit & 11.899 & 0.02817 & 1.068 & 0.611 & 1.42 & 4.21\\
$\sigma^+$ & ~~0.026 & 0.00063 & 0.051 & 0.012 & &\\
$\sigma^-$ & ~~0.024 & 0.00057 & 0.044 & 0.010 & &\\
\enddata
\tablecomments{Including scatter $\sigma_m=0.15$. All masses are in units of $\!\,{\rm M}_\odot$}
\label{t:mfscatresults}
\end{deluxetable}
In Figure \ref{f:fig6} we compare our model without scatter with the
model including scatter. We have also included the relation between
halo mass and the average stellar mass. Especially at the massive end
scatter can influence the slope of the SMF, since
there are few massive galaxies. This has an impact on $\gamma$ and as
all parameters are correlated scatter also affects the other
parameters. We thus see a difference between the model without scatter
and the most likely stellar mass in the model with scatter in Figure
\ref{f:fig6}.
\section{The conditional mass function} \label{s:cmf}
\begin{figure*}
\centering
\epsscale{1.11}
\plotone{fig7.eps}
\caption{The conditional mass function (CMF) predicted by our model at
$z=0$. We plot the derived SMFs (${\rm d}\tilde{n}_g / {\rm d}\log
m$) in a subsample of halo mass bins. The left panels show the CMF
for a model without scatter while the right panels show the CMF with
scatter of $\sigma_m=0.15$. The label in each panel is the range of
host halo mass $\log M/\!\,{\rm M}_\odot$. The stellar mass functions are
normalized such that a host halo contains exactly one central
galaxy. The total CMF consists of a central galaxy part (crosses)
and a satellite part (diamonds). The central part is described by a
lognormal distribution (solid line) and the satellite part is
described by a truncated Schechter function (dashed line) using the
parameters that were derived by a fit to the CMF. The dotted line
shows the completeness limit used in the fit to the satellite
contribution.}
\label{f:fig7}
\end{figure*}
In the previous section we derived a model that specifies the stellar
mass of a central galaxy as a function of the virial mass of its host
halo and the stellar mass of a satellite galaxy as a function of the
maximum mass of the subhalo in which it lives.
It has become common to represent the population of host halos by the
Halo Occupation Distribution (HOD). This includes the halo occupation
function $P(N\vert M)$ which is the probability distribution that a
halo of mass $M$ contains $N$ galaxies (of a specific type). A close
relative of the HOD is the ``conditional luminosity function''
\citep[CLF; e.g.][]{yang03,vdb2007,yang04}. It extends the halo
occupation function $P(N\vert M)$ (which gives only information about
the total number of galaxies per halo in a given luminosity range)
and yields the average number of galaxies with luminosities in the
range $L \pm {\rm d}L/2$ as a function of the virial mass $M$ of their
host halo.
We define its analog, the ``conditional mass function'' (CMF), or the
average number of galaxies with stellar masses in the range $m \pm
{\rm d}m/2$ as a function of the virial mass $M$ of their host
halo. This provides a direct link between the SMF
$\Phi(m)$ and the host halo mass function ${\rm d}n(M)/{\rm d}M$:
\begin{equation}
\Phi(m) = \int_0^\infty \Phi(m\vert M) \frac{{\rm d}n(M)}{{\rm d}M} \text{d}M
\end{equation}
A host halo of mass $M$ can contain a whole population of galaxies
with different stellar masses $m$. If we count the number of galaxies
living in host halos with a virial mass in the range $M \in [M_1,M_2]$
we can compute the SMF of the halo bin $[M_1,M_2]$:
\begin{equation}
\label{cmftosmf}
\tilde{\Phi}(m) = \int_{M_1}^{M_2} \Phi(m\vert M) \frac{{\rm d}n(M)}{{\rm d}M}
{\rm d}M \approx \Phi(m\vert \bar{M}) \Delta n
\end{equation}
The tilde over a function represents the fact that it is computed in a
halo mass bin. We have replaced the integral by a ``tophat'' with a
width of $\Delta n$ (number of host halos in the bin) and a height of
$\Phi(m\vert M_m)$, where $\bar{M}$ is the geometric mean of the
minimum and maximum halo masses bracketing the bin.
This equation allows us to put constraints on $\Phi(m\vert M)$ by
calculating $\tilde{\Phi}(m)/\Delta n$. We can then choose an adequate
parameterization of $\Phi(m\vert M)$ and fit these parameters to
$\tilde{\Phi}(m)/\Delta n$ in every halo mass bin. Finally we can
investigate the halo mass dependence of the parameters.
\subsection{Parameterization}
In order to specify the CMF $\Phi(m \vert M)$ we divide the galaxy
population into a central and a satellite part, as in the updated CLF
formalism \citep{zheng2005,zehavi2005,cooray2006,yang08,cacciato2008}.
The central part is $\Phi_c(m \vert M)$ and the satellite part is
$\Phi_s(m \vert M)$. Then the total CMF is the sum of both parts:
\begin{equation}
\Phi(m \vert M) = \Phi_c(m \vert M) + \Phi_s(m \vert M)
\end{equation}
Note that both $\Phi_c(m \vert M)$ and $\Phi_s(m \vert M)$ are
statistical functions and should not be regarded as the mass
functions of galaxies living in a given individual halo.
For the central population we expect the CMF to have a peak around the
stellar mass $m_c$ that corresponds to the host halo's virial mass $M$
in the SHM relation (equation
\ref{eqnmmap}). Due to the halo mass bin size this distribution gets
smeared out, because halos in the interval $[M_1,M_2]$ contain central
galaxies of stellar masses $m \in [m_1(M_1),m_2(M_2)]$. Thus
$\tilde{\Phi}(m)/\Delta n$ will be finite inside the interval
$[m_1(M_1),m_2(M_2)]$ and zero elsewhere with a normalization such
that the number of central galaxies per halo equals one. This can be
regarded as scatter $\sigma_{\rm bin}$ due to the binning. If we add
intrinsic scatter $\sigma_m$ to relation \eqref{eqnmmap}, we expect
$\Phi_c(m\vert M)$ to be a lognormal with a maximum around $m_c(M)$
and a variance of $\sigma_m^2$. To this scatter the binning scatter
$\sigma_{\rm bin}$ adds in quadrature (assuming that $\sigma_{\rm bin}$
and $\sigma_m$ are uncorrelated), resulting in a total scatter of
$\sigma_c^2=\sigma_m^2+\sigma_{\rm bin}^2$. For both cases ($\sigma_m=0$
and $\sigma_m\neq0$) we use a lognormal distribution:
\begin{equation}
\Phi_c(m \vert M) = \frac{1}{\sqrt{2\pi}\ln10\;m\;\sigma_c}
\exp\left[-\frac{\log^2(m/m_c)}{2\sigma_c^2}\right]\;,
\end{equation}
where the mean $m_c(M)$ and width $\sigma_c^2(M)$ are parameterized
functions of the halo mass M.
For the satellite population we adopt a Schechter function with a
steeper slope for the massive end. This is done by squaring the
argument of the exponential function in the Schechter function:
\begin{equation}
\Phi_s(m \vert M) = \frac{\Phi_s^*}{m_s} \left( \frac{m}{m_s}\right)^{\alpha_s}
\exp \left[ -\left( \frac{m}{m_s}\right)^2 \right]\;.
\end{equation}
Also here the parameters $\Phi_s^*(M)$, $m_s(M)$ and $\alpha_s(M)$ are
functions of the host halo mass $M$. They are the normalization, the
characteristic mass and the low mass slope of the satellite population
of host halos of mass $M$.
\subsection{Constraining the conditional mass function} \label{constraincmf}
We populate the halos and subhalos in our simulation with central and
satellite galaxies according to the prescription in section
\ref{s:galaxies}. Then we choose halo mass bins between $\log M/\!\,{\rm M}_\odot
= 10.2$ and $\log M/\!\,{\rm M}_\odot = 15.0$ with a bin size of $\Delta M =
0.4{\rm~dex}$. In every halo mass bin we seek all galaxies which live
in a host halo with a mass in that bin, which we divide between
central and satellite galaxies. For these populations we then compute
two seperate SMFs which we normalize
such that the number of central galaxies per host halo equals
one. This procedure then yields for every halo mass bin a central and
a satellite distribution $({\rm d}\tilde{n}_g / {\rm d}\log M) \Delta
n_h$.
Using equation \eqref{cmftosmf} we can now relate the stellar mass
function in a halo mass bin to the CMF:
\begin{eqnarray}
\frac{{\rm d}\tilde{n}_g(m)}{{\rm d}\log M}\frac{1}{\Delta n_h} &=&
\frac{\ln{10}}{\Delta n_h} \; M \; \frac{{\rm d}\tilde{n}_g(m)}{{\rm d}M} \nonumber\\
&=& \ln{10} \; M \; \frac{\tilde{\Phi}(m)}{\Delta n_h} \nonumber\\
&\approx& \ln{10} \; M \; \Phi(m\vert M)
\end{eqnarray}
Now we can fit the five parameters $m_c(M)$, $\sigma_c(M)$, $m_s(M)$,
$\Phi_s^*(M)$ and $\alpha_s(M)$ to the SMFs in each
halo bin. We compute and fit the central and the satellite parts
seperately.
The left panels of Figure \ref{f:fig7} show the CMF in a subsample of
halo mass bins running from $\log M/\!\,{\rm M}_\odot = 10.2 \pm 0.2 \text{ to }
15.0 \pm 0.2$, where we have not included intrinsic scatter in the
SHM relation. For the satellite part, only galaxies
with a mass above the completness limits for each halo mass bin
(as indicated in Figure \ref{f:fig7}) have been used in the fit.
In low-mass halos ($\log M/\!\,{\rm M}_\odot<11.0$) the contribution from
satellite galaxies is very small and the central contribution
dominates until $\log M/\!\,{\rm M}_\odot=12.0$. For massive halos ($\log
M/\!\,{\rm M}_\odot>13.0$) the satellite contibution dominates by number. The mean
of the lognormal fit to the central contribution also increases with
halo mass as stipulated by the model derived in
Section~\ref{s:galaxies}. The characteristic mass scale of the
satellite contribution also increases with halo mass meaning that the
most massive satellite galaxies have a mass which is comparable to the
mass of the central galaxy.
The scatter of the central contribution $\sigma_c(M)$ decreases with
halo mass. As we did not include any scatter in the model, this
scatter reflects the width ($0.4{\rm~dex}$) of the halo mass bins
($\sigma_{\rm bin}$). The halo mass dependence of $\sigma_c(M)$
arises because a fixed halo mass bin is mapped to a smaller galaxy mass bin
for larger halo mass due to the shape of the SHM relation.
Another feature of the CMF is the slope for low mass satellite galaxies
$\alpha_s(M)$ which becomes shallower with increasing halo mass.
\subsection{The parameters of the conditional mass function}\label{s:cmfpars}
In this section we investigate the halo mass dependence of the five
parameters of the CMF: $m_c(M)$, $\sigma_c(M)$, $m_s(M)$,
$\Phi_s^*(M)$ and $\alpha_s(M)$. They have been fixed by fitting to
the stellar mass functions in each halo mass bin. We introduce a
parameterization in order to describe the dependence on halo mass and
constrain these by a fit to each parameter. The results are presented
in Table \ref{t:cmfparameters}. This provides a complete description
of the CMF.
\begin{figure*}
\centering
\epsscale{1.15}
\plotone{fig8.eps}
\caption{The five parameters of the conditional mass function as a
function of halo mass. The crosses were derived from a fit to the
CMF in every halo mass bin (assuming no scatter in stellar-to-halo
mass relation). The solid line is a fit to the crosses using the
respective parameterization. The CMF parameters derived with a
scatter of $\sigma_m=0.15$ in the stellar-to-halo mass relation are
given by the diamonds. The left panels show the central
contribution: $m_c(M)$ (top), $\sigma_c(M)$ (middle) and an
illustration of the behavior of $\sigma_c(M)$ (bottom). The right
panels show the satellite contribution: $m_s(M)$ (top),
$\Phi_s^*(M)$ (middle) and $\alpha_s(M)$ (bottom). The dashed line
in the top right panel indicating $m_c(M)$ has been added for
comparison}
\label{f:fig8}
\end{figure*}
\begin{deluxetable}{lllll}
\tablecaption{Parameters of the CMF} \tablehead{ \colhead{} &
\colhead{$\sigma_m=0.0$} & \colhead{} & \colhead{$\sigma_m=0.15$} &
\colhead{} } \startdata $\log M_{1c}$ & 11.9347 & $\pm$ 0.0257 &
11.9008 & $\pm$ 0.0119 \\ $(m_c/M)_0$ & ~~0.0267 & $\pm$ 0.0006 &
~~0.0297& $\pm$ 0.0004 \\ $\beta_c$ & ~~1.0059 & $\pm$ 0.0332 &
~~1.0757& $\pm$ 0.0097 \\ $\gamma_c$ & ~~0.5611 & $\pm$ 0.0065 &
~~0.6310 & $\pm$ 0.0121 \\ \tableline $\log M_2$ & 11.9652 & $\pm$
0.1118 & 11.8045 & $\pm$ 0.0458 \\ $\sigma_{\infty}$ & ~~0.0569 &
$\pm$ 0.0052 & ~~0.1592& $\pm$ 0.0030 \\ $\sigma_1$ & ~~0.1204 & $\pm$
0.0191 & ~~0.0460& $\pm$ 0.0029 \\ $\xi$ & ~~6.3020 & $\pm$ 3.0720 &
~~4.2503& $\pm$ 0.9945 \\ \tableline $\log M_{1s}$ & 12.1988 & $\pm$
0.0878 & 12.0640& $\pm$ 0.0931 \\ $(m_s/M)_0$ & ~~0.0186 & $\pm$
0.0012 & ~~0.0198& $\pm$ 0.0015 \\ $\beta_s$ & ~~0.7817 & $\pm$ 0.0629
& ~~0.8097 & $\pm$ 0.0971 \\ $\gamma_s$ & ~~0.7334 & $\pm$ 0.0452 &
~~0.6910& $\pm$ 0.0390 \\ \tableline $-\log\Phi_0$ & 11.1622 & $\pm$
0.2874 & 10.8924& $\pm$ 0.4615 \\ $\lambda$ & ~~0.8285 & $\pm$ 0.0215
& ~~0.8032 & $\pm$ 0.0367 \\ \tableline $\log M_3$ & 12.5730 & $\pm$
0.1351 & 12.3646 & $\pm$ 0.0260 \\ $-\alpha_{\infty}$ & ~~1.3740 &
$\pm$ 0.0066 & ~~1.3676 & $\pm$ 0.0043 \\ $-\alpha_1$ & ~~0.0309 &
$\pm$ 0.0076 & ~~0.0524& $\pm$ 0.0051 \\ $\zeta$ & ~~4.3629 & $\pm$
2.6810 & ~~9.5727 & $\pm$ 6.8240 \\ \enddata
\tablecomments{The second
and third columns give the CMF parameters and their errors for a
model without scatter while the fourth and the fifth columns give
the CMF parameters and their errors for a model with a scatter of
$\sigma_m=0.15$. All quoted masses are in units of
$\!\,{\rm M}_\odot$ \vspace{0.3cm}}
\label{t:cmfparameters}
\end{deluxetable}
As we have already determined the mean relation between the stellar
mass of a galaxy and the mass of its halo, the form of $m_c(M)$ has to
be the same and can thus be decribed by equation \eqref{eqnmmap}:
\begin{equation}
\label{mcmap}
m_c(M) = 2 \; M \; \left( \frac{m_c}{M}\right)_0 \left[\left(\frac{M}{M_{1c}}\right)^{-\beta_c} + \left(\frac{M}{M_{1c}}\right)^{\gamma_c}\right]^{-1}
\end{equation}
This yields four parameters $(m_c/M)_0$, $M_{1c}$, $\beta_c$ and $\gamma_c$.
In the upper left panel of Figure \ref{f:fig8} $m_c(M)$ is plotted as
a function of halo mass. Note that by construction, it has the same form
as the SHM relation.
\label{sigmac}
The scatter of the central galaxy contribution is high for low halo
masses and decreases for more massive halos. The middle left panel of
Figure~\ref{f:fig8} shows $\sigma_c(M)$ as a function of halo mass. As
one can see, $\sigma_c(M)$ goes to a constant value both for low and
high halo masses while it decreases with halo mass. We therefore
choose the following parameterization:
\begin{equation}
\label{scmap}
\sigma_c(M) = \sigma_{\infty} + \sigma_1 \left[ 1-\frac{2}{\pi} \arctan\left( \xi \log \frac{M}{M_2} \right) \right]
\end{equation}
This yields four more parameters $\sigma_{\infty}$, $\sigma_1$, $\xi$
and $M_2$. Here, $\sigma_{\infty}$ sets the high mass limit of
$\sigma_c(M)$ while $\sigma_1$ sets the difference between the low and
high mass limits of $\sigma_c(M)$. The parameter $M_2$ determines the
mass scale at which the transition occurs and $\xi$ sets the
strength. For a large (small) value of $\xi$ the transition occurs in
a small (large) interval around $M_2$.
The specific shape of $\sigma_c(M)$ can be explained by the form of
the SHM relation (equation \ref{eqnmmap}). As we have not included any
scatter in this relation ($\sigma_m=0$), the width of the lognormal
function of the central galaxy distribution arises from the width of
the halo mass bin ($\sigma_c=\sigma_{\rm bin}$). A halo mass interval
$[M_1,M_2]$ contains only central galaxies with stellar masses of $m
\in [m_1(M_1),m_2(M_2)]$. The lower left panel of Figure~\ref{f:fig8}
illustrates this by showing how halo mass bins affect the bin size of
the stellar mass. If we choose the same bin size for low and high mass
halos, we get different bin sizes for low and high mass galaxies, due
to the changing slope of $m(M)$. Therefore the transition occurs where
the slope of $m(M)$ changes which is around $M_1$, so the value of
$M_2$ is very close to that value.
As Figure~\ref{f:fig7} shows that the satellite contribution falls off
around the mean mass of the central galaxy, we expect the
characteristic mass of the modified Schechter function $m_s(M)$ to
follow $m_c(M)$. We therefore describe $m_s(M)$ with the same function
we used for the parametrisation of $m_c(M)$:
\begin{equation}
\label{msmap}
m_s(M) = 2 \; M \; \left( \frac{m_s}{M}\right)_0 \left[\left(\frac{M}{M_{1s}}\right)^{-\beta_s} + \left(\frac{M}{M_{1s}}\right)^{\gamma_s}\right]^{-1}
\end{equation}
This function yields four parameters $(m_s/M)_0$, $M_{1s}$, $\beta_s$
and $\gamma_s$.
The upper right panel of Figure~\ref{f:fig8} plots $m_s(M)$ as a
function of halo mass.
We see that the shape is similar to that of $m_c(M)$.
Note that $m_s(M)$ is always lower than $m_c(M)$, while the deviation
increases with increasing halo mass. This implies that for high
halo masses the satellite contribution to the CMF falls off before the
mean mass of the central galaxy.
The normalization of the modified Schechter function is small for low
halo masses and increases with the mass of the host halo. The middle
right panel of Figure \ref{f:fig8} shows $\Phi_s^*(M)$ as a function of
halo mass. We see that $\Phi_s^*(M)$ can be described by a power law and
choose the following parametrisation:
\begin{equation}
\label{psmap}
\Phi_s^*(M) = \Phi_0 \left( \frac{M}{\!\,{\rm M}_\odot}\right)^{\lambda}
\end{equation}
We get two more parameters, $\Phi_0$ and $\lambda$. The normalization
of $\Phi_s^*(M)$ is given by $\Phi_0$ and the slope by $\lambda$.
The shape of $\Phi_s^*(M)$ implies that the probability for a host halo to
harbor satellite galaxies (in a given stellar mass range) increases with
increasing halo mass.
\label{alphas}
The slope of the modified Schechter function for the satellite
contribution becomes shallower for more massive
halos. The lower right panel of Figure~\ref{f:fig8} shows
$\alpha_s(M)$ as a function of halo mass and shows that $\alpha_s(M)$
goes to a constant value for both low and high halo masses.
Similar to $\sigma_c(M)$, we choose the parameterization:
\begin{equation}
\label{asmap}
\alpha_s(M) = \alpha_{\infty} + \alpha_1 \left[ 1-\frac{2}{\pi} \arctan\left( \zeta \log \frac{M}{M_3} \right) \right]
\end{equation}
This yields four more parameters $\alpha_{\infty}$, $\alpha_1$,
$\zeta$ and $M_3$. Here, $\alpha_{\infty}$ sets the high mass limit of
$\alpha_c(M)$ while $\alpha_1$ sets the difference between the low and
high mass limits of $\alpha_c(M)$. The mass scale at which this
transition occurs is determined by $M_3$ and $\zeta$ sets its
strength. The transition occurs in a small (large) interval around
$M_3$ for a large (small) value of $\zeta$.
\begin{figure*}
\centering
\epsscale{1.15}
\plotone{fig9.eps}
\caption{Occupation numbers as function of halo mass in stellar mass
bins, derived using the conditional mass function. The left, middle
and right panels show the average number of central, satellite and
total galaxies per halo, respectively.}
\label{f:fig9}
\end{figure*}
\subsection{The impact of scatter}
Until now, we have used the SHM relation \eqref{eqnmmap} without any
intrinsic scatter. In this section we investigate how the CMF and the
parameters change if we include a scatter $\sigma_m$ as described in
section \ref{s:scattersec}. This scatter is again assumed to be
constant with host halo mass.
The right panels of Figure~\ref{f:fig7} show the resulting CMF in a
subsample of halo mass bins for an intrinsic scatter of $\sigma_m =
0.15$. The central part is now no longer near-constant in the interval
$[m(M-\Delta M/2),m(M+\Delta M/2)]$ as in the left panels of
Figure~\ref{f:fig7} (where $\sigma_m = 0.0$) but has the form of a
lognormal with a broader distribution for bigger $\sigma_m$. As the
scatter has been taken from a lognormal distribution, the central
galaxy contribution to the CMF is distributed in the same way. Hence,
$\sigma_c(M)$ changes with respect to the model that does not include
artificial scatter. We notice that at the massive end the binning
scatter $\sigma_{\rm bin}^2$ and the intrinsic scatter $\sigma_m^2$ add to
the total scatter $\sigma_{\rm tot}^2$.
At the low mass end, however, the total scatter is less than what has
been obtained by using no intrinsic scatter. This shows that the two forms
of scatter do not add in quadrature and indicates that they are correlated.
We compare $m_c(M)$, $\sigma_c(M)$, $m_s(M)$, $\Phi_s^*(M)$ and
$\alpha_s(M)$ for $\sigma_m=0$ and $\sigma_m=0.15$ and show the
resulting parameters in Table \ref{t:cmfparameters} (columns four and
five) and in Figure~\ref{f:fig8}. The mean mass of the central galaxy
$m_c(M)$ does not change much if artificial scatter is introduced. The
most likely stellar mass of a central galaxy is still given by the
SHM relation, so the mean of the gaussian in
logarithmic space stays the same. Also the parameters of the satellite
population [$m_s(M)$,$\Phi_s^*(M)$ and $\alpha_s(M)$] do not change
significantly.
\subsection{The occupation numbers}
\begin{figure*}
\centering
\epsscale{1.15}
\plotone{fig10.eps}
\caption{Occupation numbers as function of halo mass for galaxy
samples with a stellar mass above a given threshold. The left,
middle and right panels show the average number of central,
satellite and total galaxies per halo, respectively.}
\label{f:fig10}
\end{figure*}
In order to compare our results to other HOD models it is useful to
compute the average number of galaxies per halo $\langle N\rangle$, as
this is the main prediction of the HOD approach. To compute $\langle
N\rangle(M)$ from the CMF we simply integrate $\Phi(m\vert M)$ over
the desired stellar mass range:
\begin{equation}
\langle N\rangle(M) = \int_{m_1}^{m_2} \Phi(m\vert M) {\rm d}m
\end{equation}
As we have divided $\Phi(m\vert M)$ into a central galaxy contribution
$\Phi_c(m\vert M)$ and a satellite galaxy contribution $\Phi_s(m\vert
M)$, we can compute seperate occupation numbers for central and
satellite galaxies:
\begin{eqnarray*}
\langle N\rangle(M) &=& \int_{m_1}^{m_2} \Phi_c(m\vert M) {\rm d}m + \int_{m_1}^{m_2} \Phi_s(m\vert M) {\rm d}m\notag\\
&=& \langle N_c\rangle(M) + \langle N_s\rangle(M)
\end{eqnarray*}
The average number of central galaxies per halo $\langle
N_c\rangle(M)$ is given by
\begin{equation}
\label{eqnocccen}
\langle N_c\rangle(M) = \frac{1}{2}\left[{\rm erf}(\eta_2)-{\rm erf}(\eta_1)\right] \;,
\end{equation}
with the error-function ${\rm erf}(x)$ and the integration boundaries
\begin{equation*}
\eta_1 = \frac{\log(m_1/m_c)}{\sqrt{2}\sigma_c} \quad \text{and} \quad \eta_2 = \frac{\log(m_2/m_c)}{\sqrt{2}\sigma_c} \;.
\end{equation*}
The average number of satellite galaxies per halo $\langle N_s\rangle(M)$ is
\begin{equation}
\label{eqnoccsat}
\langle N_s\rangle(M) = \frac{\Phi_s}{2}\left[\Gamma\left(\frac{\alpha_s}{2}+\frac{1}{2},\kappa_1\right)-\Gamma\left(\frac{\alpha_s}{2}+\frac{1}{2},\kappa_2\right)\right] \;,
\end{equation}
with the upper incomplete gamma function $\Gamma(a,x)$ and the
integration boundaries
\begin{equation*}
\kappa_1 = (m_1/m_s)^2 \quad \text{and} \quad \kappa_2 = (m_2/m_s)^2 \;.
\end{equation*}
Figure~\ref{f:fig9} shows the resulting occupation numbers for the
values of the CMF parameters that were derived in section
\ref{s:cmfpars} (using a scatter of $\sigma_m=0.15$).
The five lines in each panel correspond to different stellar mass bins.
The left panel shows the average number of central galaxies per halo
$\langle N_c\rangle(M)$ as a function of halo mass. In the middle
panel, the average number of satellite galaxies per halo $\langle
N_s\rangle(M)$ as a function of halo mass is shown. The right panel
plots the average number of all galaxies per halo $\langle
N_{\rm tot}\rangle(M)$ as a function of halo mass. A galaxy of a low
stellar mass can thus either be a central galaxy of a low mass halo,
or a satellite galaxy of a massive halo. It is not likely to live in a
halo of intermediate mass.
As it is common in the literature to plot occupation numbers not for
stellar mass intervals, but for galaxy samples with a mass above a
given threshold, we need to adjust equations \eqref{eqnocccen} and
\eqref{eqnoccsat}. The stellar mass threshold is then given by $m_1$
while $m_2 \to \infty$. This yields for the average number of central
galaxies
\begin{equation}
\langle N_c\rangle(M, m_1) = \frac{1}{2}\left[1 - {\rm erf}\left(\frac{\log(m_1/m_c)}{\sqrt{2}\sigma_c}\right)\right]\;,
\end{equation}
since ${\rm erf}(x\to\infty)\to 1$, and for the average number of satellite galaxies
\begin{equation}
\langle N_s\rangle(M, m_1) = \frac{\Phi_s}{2} \; \Gamma\left[\frac{\alpha_s}{2}+\frac{1}{2},\left( \frac{m_1}{m_s}\right)^2\right]\;
\end{equation}
since $\Gamma(a,x\to\infty)\to 0$.
Figure \ref{f:fig10} shows occupation numbers for different stellar
mass thresholds. The left panel shows the average number of central
galaxies per halo $\langle N_c\rangle(M)$ as a function of halo
mass. The middle panel plots the average number of satellite galaxies
per halo $\langle N_s\rangle(M)$ as a function of halo mass. It is
similar to the middle panel of Figure~\ref{f:fig9} while it is larger
at a given halo mass. In the right panel the average number of all
galaxies per halo $\langle N_{tot}\rangle(M)$ as a function of halo
mass is shown.
\section{Comparison}\label{s:comparison}
\subsection{Other HOD models}
\begin{deluxetable}{lllll}
\tablecaption{Comparison between different models} \tablehead{
\colhead{} & \colhead{$\log M_1$} & \colhead{$(m/M)_0$} &
\colhead{$\beta$} & \colhead{$\gamma$} } \startdata Our model &
11.884 & 0.0282 & 1.06 & 0.556\\ Non-Parametric & 11.766 & 0.0324 &
1.43 & 0.565\\ \citet{wang06} & 11.845 & 0.0319 & 1.42 &
0.710\\ Somerville SAM & 11.888 & 0.0276 & 0.98 & 0.629\\ Croton SAM &
11.742 & 0.0405 & 0.92 & 0.610\\ Yang GC & 12.067 & 0.0384 &
0.71 & 0.698\\
\enddata
\tablecomments{All quoted masses are in units of $\!\,{\rm M}_\odot$}
\label{t:paracomparison}
\end{deluxetable}
Numerous variations on halo occupation models have been presented in
the literature. In this section we describe some of the most popular
ones and compare them to our model. As many authors use different
initial mass functions and definitions of halo masses, we convert all
results to the conventions that we have used in this work (Kroupa IMF
and virial overdensity).
In the Non-Parametric model \citep{valeostriker06,conroy06,shankar2006},
galaxy properties, such as luminosity and stellar mass, are monotonically
related to the mass of dark matter halos.
Using the observed galaxy SMF, the most massive halo
is matched to the most massive galaxy:
\begin{equation}
n_g(>m_i) = n_h(>M_i)
\end{equation}
In this way, the observed SMF is automatically
reproduced. Applying this procedure and fitting the parameters of the
SHM relation to the result, we have derived the
values given in Table \ref{t:paracomparison}. These are in good
agreement with the parameters of our model, except for $\beta$. We
find that this is due to the shape of the SHM
ratio for low masses. For the Non-Parametric model, $m(M<M_1)$ can not
be perfectly described by a single power law, as is assumed in our
model.
Adding an additional parameter and assuming a fitting function with
five free parameters, we are able to fit the SHM relation predicted by
the non-parametric model quite precisely. The fifth parameter accounts
for the deviation from the power-law at high and low masses. Using the
parameterization
\begin{equation}
\label{eqnnonpar}
m(M)=m_0 \; \frac{(M/M_1)^{\gamma_1}}{\left[1+(M/M_1)^{\beta}\right]^{(\gamma_1-\gamma_2)/\beta}}
\end{equation}
we determine the values given in Table
\ref{t:paranonpar}. Figure~\ref{f:fig11} shows the results of four-
and five-parameter fits to the SHM relation derived via the
non-parametric method, compared with our usual model. In the range
where we applied the mass function fit, the non-parametric model lies
within our error-bars.
\begin{deluxetable}{llllll}
\tablecaption{Fit parameters for Equation~\eqref{eqnnonpar}}
\tablehead{
\colhead{ } &
\colhead{$\log m_0$} &
\colhead{$\log M_1$} &
\colhead{$\gamma_1$} &
\colhead{$\gamma_2$} &
\colhead{$\beta$}
}
\startdata
& 10.864 & 10.456 & 7.17 & 0.201 & 0.557\\
$\pm$ & ~~0.043 & ~~0.211 & 1.16 & 0.018 & 0.031\\
\enddata
\tablecomments{All masses are in units of $\!\,{\rm M}_\odot$}
\label{t:paranonpar}
\end{deluxetable}
In \citet{wang06} a model similar to ours is used to constrain the
SHM ratio. The halo catalogue is taken from the
Millennium simulation \citep{springel05b}; halos are identified using
a friends-of-friends group finder while substructure is found using
the {\small SUBFIND} algorithm of \citet{springel01}. As observational
constraints, the authors use a SMF which they
compute from the SDSS DR2 data using the mass estimates of
\citet{kauffmann03} and the projected CFs of
\citet{li06}.
The parameterization they use is similar to ours, with four free
parameters that can easily be converted to $M_1$, $(m/M)_0$, $\beta$
and $\gamma$ and an unconstrained scatter. These are fixed by
generating a grid of models and the best-fit model is defined as the
one for which $\chi^2=\chi^2(\Phi)+\chi^2(w_p)$ is minimal. They find
that their fit improves if they take a different set of parameters for
central and satellite galaxies. In Table~\ref{t:paracomparison} we
compare our best-fit parameters with their central galaxy best-fit
parameters which have been updated in \citet{wang07}. We show these
results in Figure~\ref{f:fig11}.
The values of $M_1$ and $(m/M)_0$ are in very good agreement with our
values, but the slopes are both higher, resulting in fewer massive and
fewer low mass galaxies. The reason for the difference in the low mass
end is the different simulation used. As the resolution of the
simulation in our model is higher, the low mass end can be constrained
more tightly. For the massive end the difference in $\gamma$ can be
explained by the additional unconstrained scatter that is used in
\citet{wang06}. As the mass function is steep at high masses and
shallow for low masses, a change in the scatter will influence the
number of massive galaxies strongly, while it will have only a small
effect on the low mass end. As the other three parameters $M_1$,
$(m/M)_0$ and $\beta$ are coupled to the Schechter function
parameters, there are two parameters to constrain the slope of the
massive end of the SMF. This degeneracy can cause
the difference in $\gamma$ between the two models. The fact that in
the Millennium simulation the cosmology is different to that of our
simulation also affects the value of the parameters.
\subsection{Gravitational lensing}
The relation between stellar mass and halo mass can be constrained
observationally using galaxy-galaxy lensing. Gravitational lensing
induces shear distortions of background objects around foreground
galaxies, allowing the mass of the dark matter halo to be
estimated. \citet{mandelbaum05,mandelbaum06} have used
SDSS data to calibrate the predicted signal from a halo model which
has been derived from a dissipationless simulation. They have
extracted the mean halo mass as a function of stellar mass. The
lensing data for combined early and late-type galaxies
(Mandelbaum, private communication)
are shown in Figure~\ref{f:fig11} and are in
excellent agreement with our model.
\begin{figure}
\centering
\epsscale{1.2}
\plotone{fig11.eps}
\caption{Comparison of the stellar-to-halo mass relation $m(M)$
between our model (solid line), models from other authors and
galaxy-galaxy lensing (symbols). The blue areas are the 1-$\sigma$
and 2-$\sigma$ levels and the error-bars on the symbols are the
2-$\sigma$ levels of the halo mass.}
\label{f:fig11}
\end{figure}
\subsection{Semi-analytic models}
\label{sams}
As we discussed in the introduction, semi-analytic models (SAMs) of
galaxy formation attempt to predict the relationship between dark halo
mass and stellar mass by a priori modelling of physical processes,
such as the growth of structure, cooling, star formation, and stellar
and AGN feedback. We compare our results with predictions from the
latest version of the semi-analytic models of \citet{somerville1999};
see \citet{somerville2008}. For this we compute the mean stellar mass
of central galaxies as a function of the mass of the host halo in halo
mass bins. The results are shown in Figure~\ref{f:fig11} and are in
good agreement with our model. This is not surprising, as the physical
parameters in the model of \citet{somerville2008} have been tuned to
match the observed stellar mass function at $z=0$.
In \citet{wang06} the authors use the semi-analytic model of
\citet{croton06} and link galaxy properties, such as the stellar mass,
to the mass of the halo in which the galaxy was last a central object
$M_{\rm infall}$. They fit the same four-parameter function that they
used for their empirical model (described above) to obtain the
parameter estimates from the SAM. We summarize these results in
Table~\ref{t:paracomparison}, and show them in Figure~\ref{f:fig11}.
The two slopes are in very good agreement with our results. However,
the normalization in the \citet{croton06} SAM is $\sim25\%$ higher and
the characteristic mass is $\sim25\%$ lower than what we found and
what \citet{wang06} find for their model. This is because the SAM of
\citet{croton06} does not produce a perfect fit to the observed
SMF.
\subsection{SDSS group catalogue}
\label{groupcat}
Another direct way of studying galaxy properties as a function of halo
mass is using the SDSS group catalogue presented in \citet{yang07}. In
this approach, galaxies are first linked together into ``groups''
using a friends-of-friends algorithm. Each group is then assigned a
total halo mass by matching to the theoretical dark matter halo mass
function. \citet{yang08} present the relation between the mean stellar
mass of the central galaxy and the host halo mass. We fit the
parameters of equation \ref{eqnmmap} to their relation and present the
results in Table~\ref{t:paracomparison}.
We note that the characteristic mass and the normalization derived
from the group catalogue are both higher than our model
parameters. The high mass slope of the SHM relation in the group
catalogue is shallower than that of our model. The low mass slope is
also shallower, however, the constraints on the low mass slope in the
group catalogue are weak, since the lowest halo masses are
$log(M/\!\,{\rm M}_\odot)\sim11.7$. This can also be seen in Figure~\ref{f:fig11}
where we show the SHM relation of the group catalogue for comparison.
\section{High Redshift} \label{s:redshift}
The discussion in the previous sections has focussed solely on the
present day universe. In this section we extend our analysis to higher
redshifts and derive the redshift dependence of the stellar-to-halo
mass relation. Having chosen a particular observed stellar mass
function at a given redshift, we can investigate how the parameters of
the SHM ratio change with time. This allows us to
learn about the evolution of galaxies. Also, with this information,
we can populate the $N$-body simulation snapshots with galaxies at
different redshifts using the appropriate redshift dependent
SHM relation, and then use the spatial information
from the simulation to compute the stellar mass dependent correlation
functions.
Since at the present time there are no high redshift ($z\mathrel{\spose{\lower 3pt\hbox{$\sim$}}\raise 2.0pt\hbox{$>$}} 1$)
clustering data as a function of stellar mass available, we fit the
four parameters of equation \eqref{eqnmmap} to the observed SMFs at
a given redshift. We argued in section \ref{combinedfit} that, under
the assumption that central and satellite galaxies follow the same
SHM relation, the SMFs provide much stronger constraints on the SHM
ratio than the clustering data. Thus we should be able to use our
model to predict clustering as a function of stellar mass at any
redshift.
\begin{figure*}
\centering
\epsscale{1.1}
\plotone{fig12.eps}
\caption{Comparison between the model and the observed stellar mass
functions for different redshifts. The observed stellar mass
functions are taken from \citet{drory2004} (for $z\leqslant 0.9$)
and from \citet{fontana2006} (for $z\geqslant 1.1$) and are
represented by the symbols. The model stellar mass functions have
been fitted to the observations and are represented by the solid
lines. The dashed lines are the theoretical mass function we obtain
from the redshift-dependent parameterization. The redshift is
indicated at the top of each panel.}
\label{f:fig12}
\end{figure*}
\begin{figure*}
\centering
\epsscale{1.15}
\plotone{fig13.eps}
\caption{Evolution of the stellar-to-halo mass relation parameters
with redshift. The symbols correspond to the derived values while
the solid line is a fit to the data. For $M_1$, $(m/M)_0$ and
$\gamma$ this is a power-law, while for $\beta$ it is a straight
line.}
\label{f:fig13}
\end{figure*}
\subsection{Which survey for which redshift}
In order to constrain the SHM relation we have to first select
observational stellar mass functions at the redshifts we want to
investigate. Because of the trade-off between surveying large areas
and obtaining deep samples, measurements of the SMF at high redshift
tend to suffer from limited dynamic range. Therefore it is important
to think about how the constraints on our four SHM function parameters
arise from the observations.
The characteristic mass $M_1$ and the maximum SHM
ratio $(m/M)_0$ mostly depend on galaxies and halos of intermediate
mass. The high mass slope $\gamma$ is fixed by the number of massive
galaxies since these live in the massive halos. On the other hand, the
low mass slope $\beta$ is set by the number of low mass galaxies since
these live in the low mass halos.
For a survey with a fixed area on the sky, the observed volume is
smaller for low redshifts $(z\lesssim1)$ than for high redshifts. In
order to compute the SMF at high galaxy masses, the
observed volume has to be relatively large, as massive galaxies are
rare. Thus for low redshifts one has to choose a wide survey (large
area) to determine the SMF for massive galaxies and
properly constrain $\gamma$. Constraining the SMF at
the low mass end requires a high level of completeness for low mass
galaxies, which are very faint objects. Hence we have to choose a deep
survey that can detect faint galaxies in order to constrain $\beta$.
Taking these considerations into account, we choose the stellar mass
functions presented in \citet{drory2004} to constrain the parameters
$M_1$, $(m/M)_0$ and $\gamma$ at low redshifts. The authors derive the
SMFs using MUNICS which is a wide area, medium-deep
survey selected in the $K$ band. The detection limit is $K\approx19.5$
and the subsample the authors use covers $0.28 {\rm~deg}^2$. We apply
our method using these mass functions and take the three parameters
from that analysis.
However, the MUNICS survey is not deep enough to detect galaxies that
are fainter than the characteristic mass of the SMF (the knee) and
thus is not sufficient to constrain the parameter $\beta$. To
constrain $\beta$ we choose the SMFs derived in
\citet{fontana2006}. This work is based on the GOODS-MUSIC sample, a
multicolor catalogue extracted from the survey conducted over the
Chandra Deep Field South. The catalogue is selected in the $z_{850}$
and $K$ bands, covers an area of $143.2 {\rm~arcmin}^2$, and is
complete to a typical magnitude of $K\approx23.5$. We apply our method
using the SMFs computed with the $z_{850}$ band selected sample and
take the parameter $\beta$ from that analysis.
For high redshift $(z\gtrsim1)$ we use the SMFs presented in
\citet{fontana2006} to constrain all four parameters. For high
redshifts, the volume of a redshift bin becomes large enough to sample
massive galaxies, and therefore the GOODS-MUSIC sample is sufficient
to constrain $\gamma$.
We convert all SMFs which use a Salpeter initial mass function to the
Kroupa/Chabrier initial mass function.
\subsection{Evolution of the parameters} \label{s:parevo}
\begin{deluxetable}{c cc cc ccc cc}
\tablecaption{\small Stellar-to-halo mass ratio parameters for different redshifts}
\tablehead{
\colhead{z} &
\colhead{\scriptsize $\log M_1$} &
\colhead{\scriptsize $\pm$} &
\colhead{\scriptsize $(m/M)_0$} &
\colhead{\scriptsize $\pm$} &
\colhead{\scriptsize $\beta$} &
\colhead{\scriptsize $-$} &
\colhead{\scriptsize $+$} &
\colhead{\scriptsize $\gamma$} &
\colhead{\scriptsize $\pm$}
}
\startdata
\scriptsize 0.0 & \scriptsize 11.88 & \scriptsize 0.02 & \scriptsize 0.0282 & \scriptsize 0.0005 & \scriptsize 1.06 & \scriptsize 0.05 & \scriptsize 0.05 & \scriptsize 0.56 & \scriptsize 0.00\\
\scriptsize 0.5 & \scriptsize 11.95 & \scriptsize 0.24 & \scriptsize 0.0254 & \scriptsize 0.0047 & \scriptsize 1.37 & \scriptsize 0.22 & \scriptsize 0.27 & \scriptsize 0.55 & \scriptsize 0.17\\
\scriptsize 0.7 & \scriptsize 11.93 & \scriptsize 0.23 & \scriptsize 0.0215 & \scriptsize 0.0048 & \scriptsize 1.18 & \scriptsize 0.23 & \scriptsize 0.28 & \scriptsize 0.48 & \scriptsize 0.16\\
\scriptsize 0.9 & \scriptsize 11.98 & \scriptsize 0.24 & \scriptsize 0.0142 & \scriptsize 0.0034 & \scriptsize 0.91 & \scriptsize 0.16 & \scriptsize 0.19 & \scriptsize 0.43 & \scriptsize 0.12\\
\scriptsize 1.1 & \scriptsize 12.05 & \scriptsize 0.18 & \scriptsize 0.0175 & \scriptsize 0.0060 & \scriptsize 1.66 & \scriptsize 0.26 & \scriptsize 0.31 & \scriptsize 0.52 & \scriptsize 0.40\\
\scriptsize 1.5 & \scriptsize 12.15 & \scriptsize 0.30 & \scriptsize 0.0110 & \scriptsize 0.0044 & \scriptsize 1.29 & \scriptsize 0.25 & \scriptsize 0.32 & \scriptsize 0.41 & \scriptsize 0.41\\
\scriptsize 1.8 & \scriptsize 12.28 & \scriptsize 0.27 & \scriptsize 0.0116 & \scriptsize 0.0051 & \scriptsize 1.53 & \scriptsize 0.33 & \scriptsize 0.41 & \scriptsize 0.41 & \scriptsize 0.41\\
\scriptsize 2.5 & \scriptsize 12.22 & \scriptsize 0.38 & \scriptsize 0.0130 & \scriptsize 0.0037 & \scriptsize 0.90 & \scriptsize 0.20 & \scriptsize 0.24 & \scriptsize 0.30 & \scriptsize 0.30\\
\scriptsize 3.5 & \scriptsize 12.21 & \scriptsize 0.19 & \scriptsize 0.0101 & \scriptsize 0.0020 & \scriptsize 0.82 & \scriptsize 0.72 & \scriptsize 1.16 & \scriptsize 0.46 & \scriptsize 0.21\\
\enddata
\tablecomments{For $M_1$, $(m/M)_0$ and $\gamma$ the errors are drawn
from a Gaussian and thus are symmetric (indicated by the symbol
$\pm$). For $\beta$ the errors are drawn from a lognormal
distribution and thus there is a lower error (indicated by the
symbol $-$) and an upper error (indicated by the symbol $+$). All
quoted masses are in units of $\!\,{\rm M}_\odot$}
\label{t:pararedshifttab}
\end{deluxetable}
Having selected the observational SMFs for a set of
different redshifts, we fit the four free parameters $M_1$, $(m/M)_0$,
$\beta$ and $\gamma$ to the observations. The errors on the parameters
are derived in a similar way as explained in section \ref{s:probdis},
but instead of using confidence intervals we have fitted a Gaussian to
the probability distributions of $M_1$, $(m/M)_0$ and $\gamma$ and a
lognormal to the probability distribution of $\beta$.
Figure~\ref{f:fig12} shows the observed and the model stellar mass
functions for different redshifts (indicated at the top of each
panel). The values of the resulting four parameters for the different
redshifts are shown in Table~\ref{t:pararedshifttab} and the redshift
evolution is plotted in Figure~\ref{f:fig13}. The characteristic mass
$M_1$ grows with increasing redshift, while the normalization of the
SHM ratio $(m/M)_0$ becomes smaller with increasing redshift. This
means that there is less stellar content in a halo of a given mass at
a higher redshift.
The high mass slope $\gamma$ can be constrained only weakly. This is
due to the limitation of the available galaxy surveys. As the area of
the survey is small, the volume in which galaxies are detected is
limited, and thus massive galaxies are very rare. This results in
large error bars for the SMF for massive galaxies
which propagate into the error bars of $\gamma$.
The situation improves slightly for higher redshifts as the volume of higher
redshift bins is larger and thus more massive galaxies can be observed.
The value of $\gamma$ decreases with increasing redshift. For higher
redshifts ($z>1$) the error bars on $\gamma$ become very large because
of the limited area covered by the available deep surveys (in this case, GOODS).
We leave it up to the reader to assess the reliability of our results
at $z>1$ based on our quoted error bars.
The low mass slope $\beta$ seems to increase with redshift until
$z\approx2$ and then drops to a low value. However, as the redshift
increases it becomes more and more difficult to observe low mass
galaxies which are very faint. Thus the high redshift values for
$\beta$ are not very well constrained and perhaps not to be fully
trusted. We therefore assume that $\beta$ grows with increasing
redshift.
As we explained in Section~\ref{beta}, $\beta$ is strongly related to
the parameter $\alpha$ of the Schechter function. A small value of
$\beta$ corresponds to a large absolute value of $\alpha$ while a
large value of $\beta$ results in a low absolute value of
$\alpha$. This would mean that for higher redshifts the stellar mass
function would become shallower, in contradiction with observations
(e.g. \citealt{fontana2006} show that the absolute value of $\alpha$
increases with redshift). However, one has to remember that the halo
mass function also changes with redshift and becomes steeper. Thus the
halo mass function steepens more than the SMF, so
$\beta$ has to increase in order to compensate.
With the derived parameter values it becomes possible to interpolate
and find the SHM ratio at any redshift. This is done
by choosing a redshift-parameterization for each of the parameters.
As $M_1$ and $(m/M)_0$ do not change much above a redshift of $z>1.5$
we choose power laws for the redshift dependence:
\begin{equation}
\log M_1(z) = \log M_1\vert_{z=0} \cdot(z+1)^{\mu} \;.
\end{equation}
and
\begin{equation}
\left( \frac{m}{M}\right)_0(z) = \left( \frac{m}{M}\right)_{z=0} \cdot(z+1)^{\nu} \;.
\end{equation}
with the normalizations $M_0$ and $(m/M)_{z=0}$ and the slopes $\mu$ and $\nu$.
To parameterize $\gamma$ over redshift, a linear dependence would lead
to a negative $\gamma$ at a certain redshift. Though this is not
forbidden, it leads to a SHM ratio which would be
increasing monotonically with halo mass which is inconsistent with
feedback processes at the massive end. Hence we also choose a
power-law parameterization for $\gamma$:
\begin{equation}
\gamma(z) = \gamma_0 \cdot(z+1)^{\gamma_1} \;.
\end{equation}
with the normalization $\gamma_0$ and the slope $\gamma_1$.
From Figure \ref{f:fig13} we are not able to infer whether $\beta$
converges to a constant value. Thus we adopt a simple linear
parameterization:
\begin{equation}
\beta(z) = \beta_1 \cdot z + \beta_0 \;.
\end{equation}
Note that we have also tried other parameterizations (constant
$\beta$, decreasing $\beta$) but could not reproduce the observed
stellar mass functions. Using the linear parameterization for $\beta$
and the power laws for the other parameters we were able to compute
stellar mass functions that are in good agreement with the observed
ones.
\begin{deluxetable}{l l l l l l l l l}
\tablecaption{Parameters for redshift dependent stellar-to-halo mass relation}
\tablehead{
\colhead{} &
\colhead{$M_1\vert_{z=0}$} &
\colhead{$\mu$} &
\colhead{$(m/M)_{z=0}$} &
\colhead{$\nu$} &
\colhead{$\gamma_0$} &
\colhead{$\gamma_1$} &
\colhead{$\beta_0$} &
\colhead{$\beta_1$}
}
\startdata
& 11.88 & 0.019 & 0.0282 & -0.72 & 0.556 & -0.26 & 1.06 & 0.17\\
$\pm$ & ~~0.01 & 0.002 & 0.0003 & ~0.06 & 0.001 & ~0.05 & 0.06 & 0.12\\
\enddata
\tablecomments{All quoted masses are in units of $\!\,{\rm M}_\odot$\\}
\label{t:redpars}
\end{deluxetable}
A fit to the derived values presented in Table~\ref{t:pararedshifttab}
yields the parameters given in Table~\ref{t:redpars}. As we do not
fully trust the derived values of $\beta$ for $z\gtrsim2$ we neglect
these two values and fit a line to the remaining values of $\beta$.
\subsection{The stellar-to-halo mass relation for different redshifts} \label{s:diffred}
\begin{figure}
\centering
\epsscale{1.15}
\plotone{fig14.eps}
\caption{Stellar mass as a function of halo mass for different
redshifts. The solid lines show different redshifts, which are
indicated at the top of the panels.}
\label{f:fig14}
\end{figure}
Having developed a redshift dependent model of the stellar-to-halo
mass relation we now test this model by computing interpolated stellar
mass functions for different redshifts. For this we use the method
described in section \ref{s:galaxies}. However, now we do not use the
parameters that have been derived at each redshift by fitting the
model to the observations but we use the eight parameters of the redshift
dependent SHM relation that have been derived in the previous section.
The resulting interpolated SMFs are compared to the
observations (and the fitted mass functions) in
Figure~\ref{f:fig12}. For $z\lesssim2$ we see excellent overall
agreement, the interpolated mass functions mostly overlap with the
error bars of the observations.
The SMFs for the high redshifts $z\gtrsim2$ are too low.
The deviations are largest at the low mass end. However, if we look at
Figure~\ref{f:fig12}, we see that $\beta$ is higher than the derived
value for the two highest redshifts which results in a low mass slope
that is too shallow.
\begin{figure*}
\centering
\epsscale{1.15}
\plotone{fig15.eps}
\caption{Correlation functions as a function of stellar mass at high
redshift. The different panels correspond to different redshifts,
which are given at the bottom of each panel. The different lines are
correlation functions for six stellar mass bins, which are given in
the upper left panel. The error-bars on the most massive sample are
from Poisson statistics. The correlation function of dark matter
particles (thick solid line) at the respective redshifts is
also shown for comparison.
At high redshift the correlation function of the massive samples is only shown
on large scales, since there is no relevant one-halo term.
}
\label{f:fig15}
\end{figure*}
To compare the relation at different redshifts, we use the redshift
dependent SHM relation with the eight parameters that have been
derived in the previous section. Figure~\ref{f:fig14} plots stellar
mass versus halo mass for different redshifts. The plot shows that at
a fixed low halo mass (e.g. $M=10^{11}\!\,{\rm M}_\odot$), galaxies that live in
such halos are more massive at low redshift ($m\sim10^9\!\,{\rm M}_\odot$ for
$z=0$) than galaxies that live in a halo of the same mass at a higher
redshift ($m\sim10^8\!\,{\rm M}_\odot$ for $z=2$). In contrast, massive halos
contain more massive galaxies at high redshift, while at low redshifts
the galaxies in massive halos have less mass. However, as halos also
become more massive over time, one cannot identify a halo of a certain
mass at high redshifts with a halo of the same mass at low
redshifts. Thus the fact that at a given (high) halo mass the mass of
the central galaxy is lower at present than at an earlier epoch does
not imply that individual galaxies lose mass during their
evolution. This only means that large halos accrete dark matter faster
than large galaxies grow in stellar mass, while the growth of low mass
halos is slower than that of the central galaxies they harbor
\citep[see also][]{conroy2008}. Because of its statistical nature, our
model is not suitable for following the evolution of an individual
galaxy through cosmic time. We also note that the SHM relation at the
massive end ($M\mathrel{\spose{\lower 3pt\hbox{$\sim$}}\raise 2.0pt\hbox{$>$}} 10^{13}\!\,{\rm M}_\odot$) undergoes very little evolution,
which has also been found by \citet{brown2008}.
\subsection{Clustering at higher redshift} \label{s:clustering}
\begin{figure}
\centering
\epsscale{1.15}
\plotone{fig16.eps}
\caption{Comparison between the model (lines) and observed (symbols)
projected correlation functions at $0.2<z<1.2$. The upper and the left
panels show the zCOSMOS data in three redshift bins while the lower
right panel shows the VVDS data. The different lines and symbols in
each panel are for different stellar mass bins and thresholds as indicated
in the panels.}
\label{f:fig16}
\end{figure}
Having determined the SHM relation as a function of redshift we are
now able to populate halos with galaxies at any redshift. We choose a
set of redshifts and populate the halos with galaxies, deriving the
stellar masses from the redshift dependent SHM relation. We divide
these galaxies into six samples of different stellar mass between
$\log m/\!\,{\rm M}_\odot=8.5$ and $11.5$. For each of these samples we compute
the real space CF $\xi(r)$ by counting pairs in distance bins
(equation \ref{corrfunc}). This leads to six CFs for every selected
redshift.
Figure~\ref{f:fig15} shows the CFs for six different redshifts as a
function of stellar mass. We also plot the correlation function of
dark matter at the respective redshifts for comparison. For all
redshifts we see that massive galaxies are clustered more strongly
than low mass galaxies. The higher the redshift, the more the CFs for
different stellar masses differ. For high redshift, there are very few
massive galaxies in our limited volume simulation box, and so the
error bars become larger.
At low redshift ($z\mathrel{\spose{\lower 3pt\hbox{$\sim$}}\raise 2.0pt\hbox{$<$}} 1$), observational measurements of
stellar mass dependent galaxy clustering have recently been
published using the VIMOS-VLT Deep Survey (VVDS) and the zCOSMOS
Survey \citep{meneux2008,meneux2009}. In order to compare our model
predictions to these data, we compute correlation functions for the
same stellar mass bins and thresholds and convert these to projected
correlation functions as described in section
\ref{constraining}. Figure \ref{f:fig16} plots the observed
projected correlation functions (symbols) and the model predictions
(lines) for different stellar mass bins or thresholds in three
redshift bins for the zCOSMOS Survey and one redshift bin for the
VVDS. There is good general agreement between the model and
observations. The zCOSMOS clustering amplitude agrees very well
with the model for $r_p<1{\rm~Mpc}$, but for $z<0.8$ deviates at
larger distances and becomes higher than the prediction. As
suggested by \citet{meneux2009}, this may be because the COSMOS
field represents an overdense volume at these redshifts. In
contrast, the VVDS clustering amplitudes are lower than those
predicted by our model, leading to the speculation that perhaps the
VVDS represents an underdense region.
\subsection{The galaxy bias} \label{bias}
\begin{figure}
\centering
\epsscale{1.05}
\plotone{fig17.eps}
\caption{The galaxy bias at a fixed scale ($\approx6{\rm~Mpc}$) as a
function of redshift for different stellar masses. The symbols have
been derived by averaging the bias over a distance interval while
the lines are fits to the symbols.}
\label{f:fig17}
\end{figure}
The bias of any object may be defined as the square root of the ratio
between the CF of the object $\xi_o(r)$ and the CF of dark matter
particles $\xi_{dm}(r)$:
\begin{equation}
\label{bias1}
b(r)=\sqrt{\frac{ \xi_o(r) }{ \xi_{dm}(r) }}\quad.
\end{equation}
Here we focus on the galaxy two-point CF
$\xi_{gg}(r,m,z)$, which in addition to the distance between the
galaxies also depends on the redshift and the stellar mass of the
galaxies:
\begin{equation}
\label{bias2}
b(r,m,z)=\sqrt{\frac{ \xi_{gg}(r,m,z) }{ \xi_{dm}(r,z) }}\quad.
\end{equation}
From our predicted galaxy CFs, we compute the bias for every redshift
and stellar mass by averaging between $r=2{\rm~Mpc}$ and
$10{\rm~Mpc}$, where $b(r)$ is roughly a constant (as one can see from
Figure~\ref{f:fig15}, the scale dependence of the bias is quite
weak). Figure~\ref{f:fig17} shows the redshift dependence of the bias.
The symbols represent the averaged value of the bias while the
solid lines correspond to a fit to the symbols. For this we have used
a power law form:
\begin{equation}
b(z)=b_0 (z+1)^{b_1}+b_2 \quad
\end{equation}
where the parameters $b_0$, $b_1$, and $b_2$ are functions of stellar
mass. The fit parameters are given in Table~\ref{t:biasfit}.
\begin{deluxetable}{lccc}
\tablecaption{Galaxy bias fit parameters}
\tablehead{
\colhead{$\log m_g$} &
\colhead{$b_0$} &
\colhead{$b_1$} &
\colhead{$b_2$}
}
\startdata
~~8.5~-~~~9.0 & 0.062 $\pm$ 0.017 & 2.59 $\pm$ 0.18 & 1.025 $\pm$ 0.062\\
~~9.0~-~~~9.5 & 0.074 $\pm$ 0.008 & 2.58 $\pm$ 0.26 & 1.039 $\pm$ 0.028\\
~~9.5~-~10.0 & 0.042 $\pm$ 0.003 & 3.17 $\pm$ 0.05 & 1.147 $\pm$ 0.021\\
10.0~-~10.5 & 0.053 $\pm$ 0.014 & 3.07 $\pm$ 0.17 & 1.225 $\pm$ 0.077\\
10.5~-~11.0 & 0.069 $\pm$ 0.014 & 3.19 $\pm$ 0.13 & 1.269 $\pm$ 0.087\\
11.0~-~11.5 & 0.173 $\pm$ 0.035 & 2.89 $\pm$ 0.20 & 1.438 $\pm$ 0.061\\
\enddata
\tablecomments{All quoted masses are in units of $\!\,{\rm M}_\odot$}
\label{t:biasfit}
\end{deluxetable}
This shows that the bias at a fixed stellar mass increases with
increasing redshift. Massive galaxies are biased more strongly
than galaxies of lower mass at any redshift. We find that the bias
of massive galaxies evolves more rapidly than that of low mass ones
\citep[cf.][]{white2007,brown2008}. Since the bias of massive halos evolves
more rapidly than that of low mass galaxies, this seems to be a
feature of any model in which the SHM relation is monotonically
increasing (i.e. the most massive galaxies reside in the most
massive halos).
\section{Conclusions}\label{s:conclusions}
The goal of this paper is to characterize the relationship between the
stellar masses of galaxies and the masses of the dark matter halos in
which they live at low and high redshift, and to make predictions of
stellar mass dependent galaxy clustering at high redshift.
We used a high-resolution $N$-body simulation and identified halos and
subhalos. Halos and subhalos were populated with central and satellite
galaxies using a parameterized SHM relation. For host halos the mass
was given by the virial mass $M_{\rm vir}$ while for subhalos we used
the maximum mass of the halo over its history $M_{\rm max}$ since we
expect the stellar mass of the satellite galaxy to be more tightly
linked to this quantity.
We described the ratio between stellar and halo mass by a function
with four free parameters, a low-mass slope $\beta$, a characteristic
mass $M_1$, a high-mass slope $\gamma$, and a normalization
$(m/M)_0$. We fit for the values of these parameters by requiring that
the observed galaxy SMF is reproduced. We find that
the SHM function has a characteristic peak at
$M_1\sim 10^{12}\!\,{\rm M}_\odot$, and declines steeply towards both smaller mass
($\beta \sim 1$) and less steeply towards larger mass halos ($\gamma \sim
0.6$). The physical interpretation of this behavior is the interplay
between the various feedback processes that impact the star formation
efficiency. Supernova feedback is more effective at reheating and
expelling gas in low mass halos, while AGN feedback is more effective
in high mass halos \citep[e.g.][]{shankar2006,croton06,bower2006,
somerville2008}. In this picture, the characteristic mass $M_1$ is the halo
mass where the efficiency of these two processes crosses.
We have thoroughly discussed the meaning of the parameters. We have
also investigated the effects on the SHM relation that arise from
introducing scatter to the relation. To do this we have added scatter
drawn from a lognormal distribution with a typical variance of
$\sigma_m=0.15$ to the SHM function. We showed that the impact of such
a scatter on three of the four parameters is negligible, with a small
but significant impact on the high-mass slope $\gamma$.
We showed that adding constraints from stellar mass dependent galaxy
clustering did not change the values of our best-fit parameters. Put
another way, the likelihood (here $\chi^2$) function for the
clustering constraint is much ``flatter'' than that for the mass
constraint, so adding the clustering constraint does not significantly
change the distribution for the most likely (best-fit) parameter
values. Fitting to the SMF only, we found that the
observed projected CFs of galaxies for five samples
of different stellar mass were reproduced well. This means that the
clustering properties of galaxies are predominantly driven by the
clustering of the halos and subhalos in which they reside. From this
we concluded that our model can predict clustering as a function of
stellar mass at any redshift.
In order to describe how galaxies of different masses populate host
halos, we introduced the conditional mass function $\Phi(m\vert M)$,
which yields the average number of galaxies with stellar masses in the
range $m\pm{\rm d}m/2$ that live in a distinct halo of mass $M$. It is
described by five parameters which are functions of halo mass. We
divided the conditional mass function into a contribution from central
galaxies (described by a lognormal distribution) and a contribution
from satellite galaxies (described by a modified Schechter
function). We computed the SMF in different halo
mass bins and fitted the five parameters in each bin. Introducing halo
mass dependent functions for every parameter and fitting these to the
derived values of the parameters in the halo mass bins, we determined
the halo mass dependence of the five parameters and thus fully
described the conditional mass function. We also computed the
occupation numbers of halos which give the average number of galaxies
of a given stellar mass that live inside a halo of mass $M$.
We compared the results for our SHM function with
those that have been derived using other approaches. These include
other halo occupation type models, gravitational lensing and
semi-analytic models. We showed that all methods yield consistent
SHM relations.
Using SMFs at higher redshifts, we applied our model
at earlier epochs of the universe. We thus constrained the
SHM relation at a given set of redshifts between
$z=0$ and $z\sim4$. This allowed us to study how the four parameters
of the SHM function depend on redshift. For each
parameter we introduced a redshift dependent function. We found that
the characteristic mass increases with redshift while the
normalization decreases with redshift. This indicates that there is
less stellar content in halos at higher redshifts.
As the halo mass function steepens more with redshift than the stellar
mass function, the low mass slope increases with redshift. We present
an eight parameter fitting function describing the redshift dependent
SHM relation.
Using the SHM relation that we derived in this way,
along with spatial information for halos from the $N$-body simulation,
we predicted the high-redshift real space CFs for
five stellar mass intervals. We find that for all redshifts, massive
galaxies are more clustered than galaxies of lower mass. Using the
real space CF of dark matter we calculated the
galaxy bias as a function of distance, redshift and stellar
mass. Averaging over spatial scale in an interval around
$r\approx6\,\,{\rm Mpc}$, we demonstrated that the galaxy bias increases
with redshift, and presented fitting formulae for the galaxy bias as a
function of stellar mass and redshift. In a companion paper
\citep{mostercv} we will use these bias results to present predictions
for the cosmic variance $\sigma_c$ for galaxies of different stellar
mass.
\acknowledgments{
We thank Benjamin Panter, Cheng Li, Adriano Fontana, Niv Drory,
Rachel Mandelbaum and Baptiste Meneux for providing their data in
electronic form. We also thank Eric Bell, Ramin Skibba, Risa Wechsler
and Charlie Conroy for helpful discussions and Zheng Zheng,
Baptiste Meneux, Micheal Brown and the referee for helpful comments
on the draft version of this paper.
This research was supported in part by the DFG cluster
of excellence `Origin and Structure of the Universe'.
}
|
2,869,038,154,760 | arxiv | \section{Introduction}
As a motivating example, we consider viral marketing in a social
network. In the standard version of the problem, the goal is to send
advertisements to influential members of a social network such that by
sending advertisements to only a few people our message spreads to a
large portion of the network. Previous work \cite{influentialnodes,
maximizingspread} has shown that, for many models of influence, the
influence of a set of nodes can be modelled as a submodular set
function. Therefore, selecting a small set of nodes with maximal
influence can be posed as a \emph{submodular function maximization}
problem. The related problem of selecting a minimal set of nodes to
achieve a desired influence is a \emph{submodular set cover} problem.
Both of these problems can be approximately solved via a simple greedy
approximation algorithm.
Consider a variation of this problem in which the goal is not to send
advertisements to people that are influential in the entire social
network but rather to people that are influential in a specific target
group. For example, our target group could be people that like
snowboarding or people that listen to jazz music. If the members of
the target group are unknown and we have no way of learning the
members of the target group, there is little we can do except assume
every member of the social network is a member of the target group.
However, if we assume the group has some known structure and that we
receive feedback from sending advertisements (e.g.\ in the form of ad
clicks or survey responses), it may be possible to simultaneously
discover the members of the group and find people that are influential
in the group.
We call problems like this \emph{learning and covering problems}. In
our example, the \emph{learning} aspect of the problem is discovering
the members of the target group (the people that like snowboarding),
and the \emph{covering} aspect of the problem is to select a small set
of people that achieve a desired level of influence in the target
group (the people to target with advertisements). Other applications
have similar structure. For example, we may want to select a small
set of representative documents about a topic of interest (e.g.\ about
linear algebra). If we do not initially know the topic labels for
documents, this is also a learning and covering problem.
We propose a new problem called \emph{interactive submodular set
cover} that can be used to model many learning and covering
problems. Besides addressing interesting new applications,
interactive submodular set cover directly generalizes submodular set
cover and exact active learning with a finite hypothesis class (query
learning) giving new insight into many previous theoretical results.
We derive and analyze a new algorithm that is guaranteed to perform
approximately as well as any other algorithm and in fact has the best
possible approximation ratio. Our algorithm considers simultaneously
the learning and covering parts of the problem. It is tempting to try
to treat these two parts of the problem separately for example by
first solving the learning problem and then solving the covering
problem. We prove this approach and other simple approaches may
perform much worse than the optimal algorithm.
\section{Background}
\subsection{Submodular Set Cover}
A submodular function is a set function satisfying a natural
diminishing returns property. We call a set function $F$ defined
over a ground set $V$ submodular iff for all $A \subseteq B \subseteq V$
and $v \in V \setminus B$
\begin{equation}
F(A + v) - F(A) \geq F(B + v) - F(B)
\label{submodeq}
\end{equation}
In other words, adding an element to $A$, a subset of $B$, results in
a larger gain than adding the same element to $B$. $F$ is called
modular if Equation \ref{submodeq} holds with equality. $F$ is
monotone non-decreasing if for all $A \subseteq B \subseteq V$,
$F(A) \leq F(B)$. Note that if $F$ is monotone non-decreasing
and submodular iff Equation \ref{submodeq} holds for all $v \in V$
(including $v \in B$).
\begin{prp} If $F_1(S), F_2(S), ... F_n(S)$ are all submodular, monotone
non-decreasing functions then $F_1(S) + F_2(S) + ... + F_n(S)$ is
submodular, monotone non-decreasing. \label{sumprp} \end{prp}
\begin{prp} For any function $f$ mapping set elements to real numbers
the function $F(S) \triangleq \max_{s \in S} f(s)$ is a submodular, monotone
non-deceasing function. \label{maxprp} \end{prp}
In the submodular set cover problem the goal is to find a set $S
\subseteq V$ minimizing a modular cost function $c(S) = \sum_{s \in S}
c(s)$ subject to the constraint $F(S)=F(V)$ for a monotone
non-decreasing submodular $F$.
\begin{leftbar}
\noindent \textbf{\large{Submodular Set Cover}} \\
\textbf{Given:} \
\begin{compactitem}
\item Ground set $V$
\item Modular cost function $c$ defined over $V$
\item Submodular monotone non-decreasing
objective function $F$ defined over $V$
\end{compactitem}
\textbf{Objective:} Minimize $c(S)$ such that $F(S) = F(V)$
\end{leftbar}
This problem is closely related to the problem of submodular function
maximization under a modular cost constraint $c(S) < k$ for a constant
$k$. A number of interesting real world applications can be posed as
submodular set cover or submodular function maximization problems
including influence maximization in social networks
\cite{maximizingspread}, sensor placement and experiment design
\cite{robust}, and document summarization \cite{documentsum}. In the
sensor placement problem, for example, the ground set $V$ corresponds
to a set of possible locations. An objective function $F(S)$ measures
the coverage achieved by deploying sensors to the locations
corresponding to $S \subseteq V$. For many reasonable definitions of
coverage, $F(S)$ turns out to be submodular.
Submodular set cover is a generalization of the set cover problem. In
particular, set cover corresponds to the case where each $v \in V$ is
a set of items taken from a set $\bigcup_{v \in V} v$. The goal is to
find a small set of sets $S \subseteq V$ such that $|\bigcup_{s \in S}
s| = |\bigcup_{v \in V} v|$. The function $F(S) = |\bigcup_{s \in S}
s|$ is monotone non-decreasing and submodular, so this is a submodular
set cover problem. As is the case for set cover, a greedy algorithm
has approximation guarantees for submodular set cover
\cite{analysisofgreedy}. In particular, if $F$ is integer valued,
then the greedy solution is within $H(\max_{v \in V} F(\lbrace v
\rbrace))$ of the optimal solution where $H(k)$ is the $k$th harmonic
number. Up to lower order terms, this matches the hardness of
approximation lower bound $(1 - o(1))\ln n$ \cite{thresholdsetcover}
where $n = |\bigcup_{v \in V} v| = F(V)$.
We note a variation of submodular set cover uses a constraint $F(S)
\geq \alpha$ for a fixed threshold $\alpha$. This variation does not
add any difficulty to the problem because we can always define a new
monotone non-decreasing submodular function $\hat{F}(S) = \min(F(S),
\alpha)$ \cite{robust, electrical} to convert the constraint $F(S) \geq
\alpha$ into a new constraint $\hat{F}(S) = \hat{F}(V)$. We can also
convert in the other direction from a constraint $F(S) = F(V)$ to
$F(S) \geq \alpha$ by setting $\alpha = F(V)$. Without loss of
generality or specificity, we use the variation of the problem with an
explicit threshold $F(S) \geq \alpha$.
\subsection{Exact Active Learning}
In the exact active learning problem we have a known finite hypothesis
class given by a set of objects $H$, and we want to identify an
initially unknown target hypothesis $h^* \in H$. We identify $h^*$ by
asking questions. Define $Q$ to be the known set of all possible
questions. A question $q$ maps an object $h$ to a set of valid
responses $q(h) \subseteq R$ with $q(h) \neq \emptyset$ where $R
\triangleq \bigcup_{q \in Q, h \in H} q(h)$ is the set of all possible
responses. We know the mapping for each $q$ (i.e.\ we know $q(h)$ for
every $q$ and $h$). Asking $q$ reveals some element $r \in q(h^*)$
which may be chosen adversarially (chosen to impede the learning
algorithm). Each question $q \in Q$ has a positive cost $c(q)$ defined
by the modular cost function $c$.
The goal of active learning is to ask a sequence of questions with
small total cost that identifies $h^*$. By identifying $h^*$, we mean
that for every $h \neq h^*$ we have received some response $r$ to a
question $q$ such that $r \notin q(h)$. Questions are chosen
sequentially so that the response from a previous question can be used
to decide which question to ask next. The problem is stated below.
\begin{leftbar}
\noindent \textbf{\large{Exact Active Learning}} \\
\textbf{Given:} \
\begin{compactitem}
\item Hypothesis class $H$ containing an unknown target $h^*$
\item Query set $Q$ and response set $R$
with $q(h) \subseteq R$ for $q \in Q$, $h \in H$
\item Modular query cost function $c$ defined over $Q$
\end{compactitem}
\textbf{Repeat:} Ask a question $\hat{q}_i \in Q$ and receive
a response $\hat{r}_i \in \hat{q}_i(h^*)$ \\
\textbf{Until:} $h^*$ is identified (for every $h \in H$ with $h \neq
h^*$ there is a $(\hat{q}_i, \hat{r}_i)$ with $\hat{r}_i \notin \hat{q}_i(h)$) \\
\textbf{Objective:} Minimize $\sum_i c(\hat{q}_i)$
\end{leftbar}
In a typical exact learning problem, $H$ is a set of different classifiers
and $h^*$ is a unique zero-error classifier. Questions in $Q$ can, for
example, correspond to label (membership) queries for data points. If
we have a fixed data set consisting of data points $x_i$, we can create
a question $q_i$ corresponding to each $x_i$ and set $q_i(h) = \lbrace
h(x_i) \rbrace$. Questions can also correspond to more complicated
queries. For example, a question can ask if any points in a set are
positively labelled. The setting we have described allows for
mixing arbitrary types of queries with different costs.
For a set of question-response pairs $\hat{S}$, define
the version space $V(\hat{S})$ to be the subset of $H$ consistent
with $\hat{S}$
\[ V(\hat{S}) \triangleq
\lbrace h \in H : \forall (q, r) \in \hat{S}, \ r \in q(h) \rbrace \]
In terms of the version space, the goal of exact active learning
is to ask a sequence of questions such that $|V(\hat{S})| = 1$.
We note that the assumption that $H$ and $Q$ are finite is not a
problem for many applications involving finite data sets. In
particular, if we have an infinite a hypothesis class (e.g.\ linear
classifiers with dimension $d$) and a finite data set, we can simply
use the effective hypothesis class induced by the data set
\cite{analysisofgreedyactive}. On the other hand, the assumption that
we have direct access to the target hypothesis (every $\hat{r}_i$ is in
$\hat{q}_i(h^*)$) and that the target hypothesis is in our hypothesis class
($h^* \in H$) is a limiting assumption. Stated differently, we assume
that there is no noise and that the hypothesis class is correct.
Building on previous work \cite{generaldim}, \citet{costcomp} showed
that a simple greedy active learning strategy is approximately optimal
in the setting we have described. The greedy strategy selects the
question which relative to cost distinguishes the greatest number of
hypotheses from $h^*$. \citet{costcomp} shows this strategy incurs no
more than $\ln |H|$ times the cost of any other question asking
strategy.
The algorithms and approximation factors for submodular set cover and
exact active learning are quite similar. Both are simple greedy
algorithms and the $\ln F(V)$ approximation for submodular set cover
is similar to the $\ln |H|$ approximation for active learning. These
similarities suggest these problems may be special cases of some other
more general problem. We show that in fact they are special cases of
a problem which we call interactive submodular set cover.
\section{Problem Statement}
We use notation similar to the exact active learning problem we
described in the previous section. Assume we have a finite hypothesis
class $H$ containing an unknown target hypothesis $h^* \in H$. We
again assume there is a finite set of questions $Q$, a question $q$
maps each object $h$ to a set of valid responses $q(h) \subseteq R$
with $q(h) \neq \emptyset$, and each question $q \in Q$ has a positive
cost $c(q)$ defined by the modular cost function $c$. We also again assume
that we know the mapping for each $q$ (i.e.\ we know $q(h)$ for every
$q$ and $h$). Asking $q$ reveals some adversarially chosen element $r
\in q(h^*)$. In the exact active learning problem the goal is to
identify $h^*$ through questions. In this work we consider a
generalization of this problem in which the goal is instead to satisfy
a submodular constraint that depends on $h^*$.
We assume that for each object $h$ there is a corresponding monotone
non-decreasing submodular function $F_h$ defined over subsets of $Q
\times R$ (sets of question-response pairs). We repeatedly ask a
question $\hat q_i$ and receive a response $\hat r_i$. Let the
sequence of questions be $\hat Q = (\hat q_1, \hat q_2, \dots)$
and sequence of responses be $\hat R = (\hat r_1, \hat r_2, \dots)$.
Define $\hat{S} = \bigcup_{\hat{q}_i \in \hat{Q}} \lbrace (\hat{q}_i,
\hat{r}_i) \rbrace$ to be the final set of question-response pairs
corresponding to these sequences. Our goal is to ask a sequence of
questions with minimal total cost $c(\hat{Q})$ which ensures
$F_{h^*}(\hat{S}) \geq \alpha$ for some threshold $\alpha$ without
knowing $h^*$ beforehand. We call this problem interactive submodular
set cover.
\begin{leftbar}
\noindent \textbf{\large{Interactive Submodular Set Cover}} \\
\textbf{Given:} \
\begin{compactitem}
\item Hypothesis class $H$ containing an unknown target $h^*$
\item Query set $Q$ and response set $R$
with known $q(h) \subseteq R$ for every $q \in Q$, $h \in H$
\item Modular query cost function $c$ defined over $Q$
\item Submodular monotone non-decreasing
objective functions $F_h$ for $h \in H$ defined over $Q \times R$
\item Objective threshold $\alpha$
\end{compactitem}
\textbf{Repeat:} Ask a question $\hat{q}_i \in Q$ and receive
a response $\hat{r}_i \in \hat{q}_i(h^*)$ \\
\textbf{Until:} $F_{h^*}(\hat{S}) \geq \alpha$
where $\hat{S} = \bigcup_i \lbrace (\hat{q}_i, \hat{r}_i) \rbrace$ \\
\textbf{Objective:} Minimize $\sum_i c(\hat{q}_i)$
\end{leftbar}
Note that although we know the hypothesis class $H$ and the
corresponding objective functions $F_h$, we do not initially know
$h^*$. Information about $h^*$ is only revealed as we ask questions
and receive responses to questions. Responses to previous questions
can be used to decide which question to ask next, so in this way the
problem is ``interactive.'' Furthermore, the objective function for
each hypothesis $F_h$ is defined over sets of question-response pairs
(as opposed to, say, sets of questions), so when asking a new question
we cannot predict how the value of $F_h$ will change until after we
receive a response. The only restriction on the response we receive
is that it must be consistent with the initially unknown target $h^*$.
It is this uncertainty about $h^*$ and the feedback we receive from
questions that distinguishes the problem from submodular set cover and
allows us to model learning and covering problems.
\subsection{Connection to Submodular Set Cover}
If we know $h^*$ (e.g.\ if $|H|=1$) and we assume $|q(h)|=1 $ $\forall
q \in Q, h \in H$ (i.e.\ that there is only one valid response to
every question), our problem reduces exactly to the standard
submodular set cover problem. Under these assumptions, we can compute
$F_{h^*}(\hat{S})$ for any set of questions without actually asking
these questions. \citet{robust} study a non-interactive version of
interactive submodular set cover in which $|q(h)|=1$ $\forall q \in Q,
h \in H$ and the entire sequence of questions must be chosen before
receiving any responses. This restricted version of the problem can
also be reduced to standard submodular set cover \citet{robust}.
\subsection{Connection to Active Learning}
\label{connectiontoactivesec}
Define
\begin{equation*}
F_{h}(\hat{S}) \triangleq F(\hat{S}) = |H \setminus V(\hat{S})|
\end{equation*}
where $V(\hat{S})$ is again the version space (the set of hypotheses
consistent with $\hat{S}$). This objective is the number of hypotheses
eliminated from the version space by $\hat{S}$.
\begin{lma} $F_{h}(\hat{S}) \triangleq |H \setminus V(\hat{S})|$
is submodular and monotone non-decreasing \label{vslma} \end{lma}
\begin{proof} To see this note that we can write $F_h$ as
$F_h(\hat{S}) = \sum_{h' \in H} \max_{(q, r) \in \hat{S}} f_{h'}((q, r))$
where $f_{h'}((q, r)) = 1$ if $r \notin q(h')$ and else
$f_{h'}((q, r)) = 0$. The result then follows
from Proposition \ref{sumprp} and Proposition \ref{maxprp}. \end{proof}
For this objective, if we set $\alpha = |H| - 1$ we get the standard
exact active learning problem: our goal is to identify $h^*$ using a
set of questions with small total cost. Note that in this case the
objective $F_h$ does not actually depend on $h$ (i.e. $F_h = F_{h'}$
for all $h, h' \in H$) but the problem still differs from standard
submodular set cover because $F_h(\hat{S})$ is defined over
question-response pairs.
Interactive submodular set cover can also model an approximate
variation of active learning with a finite hypothesis class and finite
data set. Define
\[ F_{h}(\hat{S}) \triangleq |H \setminus V(\hat{S})| (|X| - \kappa)
+ \sum_{h' \in V(\hat{S})} \min(|X| - \kappa, \sum_{x \in X} I(h'(x) = h(x))) \]
where $I$ is the indicator function, $X$ is a finite data set, and
$\kappa$ is an integer.
\begin{prp}$F_{h^*}(\hat{S}) = |H| (|X| - \kappa)$ iff all
hypotheses in the version space make at most $\kappa$ mistakes.\end{prp}
\begin{lma}
$F_{h}(\hat{S}) \triangleq |H \setminus V(\hat{S})| (|X| - \kappa)
+ \sum_{h' \in V(\hat{S})} \min(|X| - \kappa, \sum_{x \in X} I(h'(x) = h(x)))$
is submodular and monotone non-decreasing \end{lma}
\begin{proof} We can write $F_h$ as
$F_h(\hat{S}) = \sum_{h' \in H} \max_{(q, r) \in \hat{S}} f_h((q, r))$
where $f_{h'}((q, r)) = |X| - \kappa$ if $r \notin q(h')$
and else $f_{h'}((q, r))= \min(|X| - \kappa, \sum_{x \in X} I(h'(x) = h(x)))$.
The result then follows from Proposition \ref{sumprp} and
Proposition \ref{maxprp}. \end{proof}
For this objective, if we set $\alpha = |H|(|X| - \kappa)$ then our
goal is to ask a sequence of questions such that all hypotheses in the
version space make at most $\kappa$ mistakes. \citet{generaldim}
study a similar approximate query learning setting, and
\citet{querycollab} consider a slightly different setting where the
target hypothesis may not be in $H$.
\subsection{Connection to Adaptive Submodularity}
In concurrent work, \citet{adaptsubmod} show results similar to ours
for a different but related class of problems which also involve
interactive (i.e.\ sequential, adaptive) optimization of submodular
functions. What \citeauthor{adaptsubmod} call realizations correspond
to hypotheses in our work while items and states correspond to queries
and responses respectively. \citeauthor{adaptsubmod} consider both
average-case and worst-case settings and both maximization and
min-cost coverage problems. In contrast, we only consider worst-case,
min-cost coverage problems. In this sense our results are less
general.
However, in other ways our results are more general. The main greedy
approximation guarantees shown by \citeauthor{adaptsubmod} require
that the problem is \emph{adaptive submodular}; adaptive submodularity
depends not only on the objective but also on the set of possible
realizations and the probability distribution over these realizations.
In contrast we only require that for a fixed hypothesis the objective
is submodular. \citeauthor{adaptsubmod} call this pointwise
submodularity. Pointwise submodularity does not in general imply
adaptive submodularity (see the clustered failure model discussed by
\citeauthor{adaptsubmod}).
In fact, for problems that are pointwise modular but not adaptive
submodular, \citeauthor{adaptsubmod} show a hardness of approximation
lower bound of $\mathcal{O}(|Q|^{1-\epsilon})$; we note this does not
contradict our results as their proof is for average-case cost and uses
a hypothesis class with $|H|=2^{|Q|}$. \citeauthor{adaptsubmod} also
propose a simple non greedy approach with explicit explore and exploit
stages; this approach requires only a weaker assumption that the value
of the exploitation stage is adaptive submodular with respect to
exploration. However, it is not immediately obvious when this
condition holds, and it is also not clear how to apply this approach
to worst-case or min-cost coverage problems.
There are other smaller differences between our problem settings: we
let queries map hypotheses to \emph{sets} of valid responses (in
general $|q(h)|>1$) while \citeauthor{adaptsubmod} define realizations
as maps from items to \emph{single} states. Also, in our work we
allow for non uniform query costs (in general $c(q_i) \neq c(q_j)$)
while \citeauthor{adaptsubmod} require that every item has the same
cost (\citeauthor{adaptsubmod} do however mention that the extension
to non uniform costs is straightforward). We finally note that the
proof techniques we use are quite different.
Some other previous work has also considered interactive versions of
covering problems in an average-case model \cite{maximizingstochastic,
stochasticcovering}. The work of \citet{maximizingstochastic} is
perhaps most similar and considers a submodular function maximization
problem over independent random variables which are sequentially
queried. The setting considered by \citet{adaptsubmod} strictly
generalizes this setting. \citet{onlinemaximizing} study an online
version of submodular function maximization where a sequence of
submodular function maximization problems is solved. This problem is
related in that it also involves learning and submodular functions,
but the setting is very different than the one studied here where we
solve a single interactive problem as opposed to a series of
non-interactive problems.
\section{Example}
\label{examplesec}
\begin{figure}[t]
\begin{center}
\includegraphics[width=.2\columnwidth]{cartoonedit}
\end{center}
\caption{A cartoon example social network.}
\label{cartoon}
\end{figure}
In the advertising application we described in the introduction, the
target hypothesis $h^*$ corresponds to the group of people we want to
target with advertisements (e.g.\ the people that like snowboarding),
and the hypothesis class $H$ encodes our prior knowledge about $h^*$.
For example, if we know the target group forms a small dense subgraph
in the social network, then the hypothesis class $H$ would be the set
of \emph{all} small dense subgraphs in the social network. The query
set $Q$ and response set $R$ correspond to advertising actions and
feedback respectively, and finally the objective function $F_h$
measures advertising coverage within the group corresponding to $h$.
To make the discussion concrete, assume the advertiser sends a single
ad at a time and that after a person is sent an ad the advertiser
receives a binary response indicating if that person is in the target
group (i.e.\ likes snowboarding). Let $q_i$ correspond to sending an
ad to user $i$ (i.e.\ node $i$), and $q_i(h) = \lbrace 1 \rbrace$ if
user $i$ is in group $h$ and $q_i(h) = \lbrace 0 \rbrace$ otherwise.
For our coverage goal, assume the advertiser wants to ensure that
every person in the target group either receives an ad or has a friend
that receives an ad. We say a node is ``covered'' if it has received
an ad or has a neighbor that has received an ad. This is a variation of the
minimum dominating set problem, and we use the following objective
\[F_h(\hat{S}) \triangleq \sum_{v \in V_h} I \Big( v \in V_{\hat{S}} \mbox{ or }
\exists s \in V_{\hat{S}} : (v, s) \in E \Big) + |V \setminus V_h| \] where
$V$ and $E$ are the nodes and edges in the social network, $V_h$ is
the set of nodes in group $h$, and $V_{\hat{S}}$ is the set of nodes
corresponding to ads we have sent. With this objective
$F_{h^*}(\hat{S}) = |V|$ iff we have achieved the stated coverage goal.
\begin{lma}
$F_h(\hat{S}) = \sum_{v \in V_h} I ( v \in V_{\hat{S}} \mbox{ or }
\exists s \in V_{\hat{S}} : (v, s) \in E ) + |V \setminus V_h| $
is submodular and monotone non-decreasing.
\end{lma}
\begin{proof} We can write $F_h(\hat{S})$ as
$F_h(\hat{S}) = \sum_{v \in V} \max_{(\hat{q}, \hat{r}) \in \hat{S}}
f_v((\hat{q}, \hat{r}))$ where
$f_v((\hat{q}, \hat{r})) = 1$ if the action $\hat{q}$
covers $v$ or $v \notin V_h$ and
$f_v((\hat{q}, \hat{r})) = 0$ otherwise. The result then follows
from Proposition \ref{sumprp} and Proposition \ref{maxprp}. \end{proof}
Figure \ref{cartoon} shows a cartoon social network. For this
example, assume the advertiser knows the target group is one of the
four clusters shown (marked A, B, C, and D) but does not know which.
This is our hypothesis class $H$. The node marked $v$ is initially
very useful for learning the members of the target group: if we send
an ad to this node, no matter what response we receive we are
guaranteed to eliminate two of the four clusters (either $A$ and $B$
or $C$ and $D$). However, this node has only a degree of 2 and
therefore sending an ad to this node does not cover very many nodes.
On the other hand, the nodes marked $x$ and $w$ are connected to every
node in clusters $B$ and $D$ respectively. $x$ (resp.\ $w$) is
therefore very useful for achieving the coverage objective if the
target group is $B$ (resp.\ $D$). An algorithm for learning and
covering must choose between actions more beneficial for learning vs.\
actions more beneficial for covering (although sometimes an action can
be beneficial for both to a certain degree). The interplay between
learning and covering is similar to the exploration-exploitation
trade-off in reinforcement learning. In this example an optimal
strategy is to first send an ad to $v$ and then cover the remaining
two clusters using two additional ads for a worst case cost of 3.
A simple approach to learning and covering is to simply ignore
feedback and solve the covering problem for all possible target
groups. In our example application the resulting covering problem is
a simple dominating set problem for which we can use standard
submodular set cover methods. We call this the Cover All strategy.
This approach is suboptimal because in many cases feedback can make
the problem significantly easier. In our synthetic example, any
strategy not using feedback must use worst case cost of 4: four
ads are required to cover all of the nodes in the four clusters. Theorem
\ref{coverallgap} in
Section \ref{negresultsec} proves that in fact there are cases where
the best strategy not using feedback incurs exponentially greater cost
than the best strategy using feedback.
Another simple approach is to solve the learning problem first
(identify $h^*$) and then solve the covering problem (satisfy
$F_{h^*}(\hat{S})$). We can use, for example, query learning to solve
the learning problem and then use standard submodular set cover to
solve the covering problem. We call this the Learn then Cover
strategy. This approach turns out to match the optimal strategy in
the example given by Figure \ref{cartoon}. In this example the target
group can be identified using 2 queries by querying $v$ then $w$ if
the response is $1$ and $x$ if the response is $0$. After identifying
the target group, the target group can be covered in at most one
more query. However, this approach is not optimal for other instances
of this problem. For example, if we were to add an additional node
which is connected to every other node then the covering problem would
have a solution of cost 1 while the learning problem would still
require cost of 2. Theorem \ref{learnthencovergap} in Section
\ref{negresultsec} shows that in fact there are examples where solving
a learning problem is much harder than solving the corresponding
learning and covering problem. We therefore must consider other
methods for balancing learning and covering.
We note that this problem setup can be modified to allow queries to
have sometimes uninformative responses; this can be modeled by adding
an additional response to $R$ which corresponds to a ``no-feedback''
response and including this response in the set of allowable responses
($q(h)$) for certain query-hypothesis pairs . However, care must be
taken to ensure that the resulting problem is still interesting for
worst-case choice of responses; if we allow ``no-feedback'' responses
for every question-hypothesis pair, then the in the worst-case we will
never receive any feedback, so a worst case optimal strategy could
ignore all responses.
\section{Greedy Approximation Guarantee}
We are interested in approximately optimal polynomial time algorithms
for the interactive submodular set cover problem. We call a question
asking strategy correct if it always asks a sequence of questions such
that $F_{h^*}(\hat{S}) \geq \alpha$ where $\hat{S}$ is again the final
set of question-response pairs. A necessary and sufficient condition
to ensure $F_{h^*}(\hat{S}) \geq \alpha$ for worst case choice of
$h^*$ is to ensure $\min_{h \in V(\hat{S})} F_{h}(\hat{S}) \geq
\alpha$ where $V(\hat{S})$ is the version space. Then a simple
stopping condition which ensures a question asking strategy is correct
is to continue asking questions until $\min_{h \in V(\hat{S})}
F_{h}(\hat{S}) \geq \alpha$. We call a question asking strategy
approximately optimal if it is correct and the worst case cost
incurred by the strategy is not much worse than the worst case cost of
any other strategy.
As discussed informally in the previous section, it is important for a
question asking strategy to balance between learning (identifying
$h^*$) and covering (increasing $F_{h^*}$). Ignoring either aspect of
the problem is in general suboptimal (we show this formally in Section
\ref{negresultsec}). We propose a reduction which converts the
problem over many objective functions $F_h$ into a problem over a
single objective function $\bar{F}_{\alpha}$ that encodes the trade-off
between learning and covering. We can then use a greedy algorithm to
maximize this single objective, and this turns out to overcome the
shortcomings of simpler approaches. This reduction is inspired by the
reduction used by \citet{robust} in the non-interactive setting to
convert multiple covering constraints into a single covering
constraint.
Define
\[ \bar{F}_{\alpha}(\hat{S}) \triangleq (1/|H|) (\sum_{h \in
V(\hat{S})} \min(\alpha, F_h(\hat{S})) + \alpha |H \setminus
V(\hat{S})|) \] $\bar{F}_{\alpha} (\hat{S}) \geq \alpha$ iff
$F_h(\hat{S}) \geq \alpha$ for all $h \in V(\hat{S})$ so a question
asking strategy is correct iff it satisfies $\bar{F}_{\alpha}
(\hat{S}) \geq \alpha$. This objective balances the value of learning
and covering. The sum over $h \in V(\hat{S})$ measures progress
towards satisfying the covering constraint for hypotheses $h$ in the current
version space (covering). The second term $\alpha |H \setminus
V(\hat{S})|$ measures progress towards identifying $h^*$ through reduction in
version space size (learning).
Note that the objective does not make a hard distinction between
learning actions and covering actions. In fact, the objective will
prefer actions that both increase $F_h(\hat{S})$ for $h \in V(\hat{S})$
and decrease the size of $V(\hat{S})$. Crucially, $\bar{F}_{\alpha}$
retains submodularity.
\begin{lma}
$\bar{F}_{\alpha}$ is submodular and monotone non-decreasing
when every $F_h$ is submodular and monotone non-decreasing.
\end{lma}
\begin{proof} Note that the proof would be trivial if the sum
were over all $h \in H$. However, since the sum is over a subset of
$H$ which depends on $\hat{S}$, the result is not obvious.
We can write $\bar{F}_{\alpha}$ as
$\bar{F}_{\alpha}(\hat{S}) = (1/|H|) \sum_{h \in H} \hat{F}_{\alpha, h}(\hat{S})$
where we define $\hat{F}_{\alpha, h}(\hat{S}) \triangleq
I(h \in V(\hat{S})) \min(\alpha, F_h(\hat{S}))
+ I(h \notin V(\hat{S})) \alpha$. It is not hard to
see $\hat{F}_{\alpha, h}$ is monotone non-decreasing.
We show $\hat{F}_{\alpha, h}$ is also submodular and
the result then follows from Proposition \ref{sumprp}. Consider any
$(q, r) \notin B$ and $A \subseteq B \subseteq (Q \times R)$. We show
Equation \ref{submodeq} holds in three cases. Here we use
as short hand $\operatorname{Gain}(F, S, s) \triangleq F(S + s) - F(S)$.
\begin{compactitem}
\item If \textbf{$h \notin V(B)$} then
\[ \operatorname{Gain}(\hat{F}_{\alpha, h},A,(q, r))
\geq 0 = \operatorname{Gain}(\hat{F}_{\alpha, h},B,(q, r)) \]
\item If \textbf{$r \notin q(h)$} then
\[ \operatorname{Gain}(\hat{F}_{\alpha, h},A,(q, r))
= \alpha - \hat{F}_{\alpha, h}(A)
\geq \alpha - \hat{F}_{\alpha, h}(B)
= \operatorname{Gain}(\hat{F}_{\alpha, h},B,(q, r)) \]
\item If \textbf{$r \in q(h)$ and $h \in V(B)$} then
\begin{align*}
\operatorname{Gain}(\hat{F}_{\alpha, h},A,(q, r))
& = \min(F_h(A + (q, r)), \alpha) - \min(F_h(A), \alpha) \\
& \geq \min(F_h(B + (q, r)), \alpha) - \min(F_h(B), \alpha)
= \operatorname{Gain}(\hat{F}_{\alpha, h},B,(q, r))
\end{align*}
Here we used the submodularity of $\min(F_h(S), \alpha)$ \cite{electrical}.
\end{compactitem}
\end{proof}
\begin{algorithm}[t]
\caption{Worst Case Greedy}
\begin{algorithmic}[1]
\STATE $\hat{H} \Leftarrow H$
\STATE $\hat{S} \Leftarrow \emptyset$
\WHILE{$\bar{F}_{\alpha} (\hat{S}) < \alpha$}
\STATE $\hat{q} \Leftarrow \operatorname{argmax}_{q_i \in Q}
\min_{h \in V(\hat{S})} \min_{r_i \in q_i(h)}
(\bar{F}_{\alpha}(\hat{S} + (q_i, r_i)) -
\bar{F}_{\alpha}(\hat{S})) / c(q_i) $
\STATE Ask $\hat{q}$ and receive response $\hat{r}$
\STATE $\hat{S} \Leftarrow \hat{S} + (\hat{q}, \hat{r})$
\ENDWHILE
\end{algorithmic}
\label{greedyb}
\end{algorithm}
Algorithm \ref{greedyb} shows the worst case greedy algorithm which
at each step picks the question $q_i$ that maximizes the worst
case gain of $\bar{F}_{\alpha}$
\[ \min_{h \in V(\hat{S})} \min_{r_i \in q_i(h)}
(\bar{F}_{\alpha}(\hat{S} + (q_i, r_i)) - \bar{F}_{\alpha}(\hat{S})) /
c(q_i) \] We now argue that Algorithm \ref{greedyb} is an
approximately optimal algorithm for interactive submodular set cover.
Note that although it is a simple greedy algorithm over a single
submodular objective, the standard submodular set cover analysis
doesn't apply: the objective function is defined over
question-response pairs, and the algorithm cannot predict the actual
objective function gain until after selecting and commiting to a
question and receiving a response. We use an Extended Teaching
Dimension style analysis \cite{costcomp} inspired by previous work in
query learning. We are the first to our knowledge to use this kind of
proof for a submodular optimization problem.
Define an oracle (teacher) $T \in R^Q$ to be a function mapping
questions to responses. As a short hand, for a sequence of
questions $\hat{Q}$ define
\[
T(\hat{Q}) \triangleq \bigcup_{\hat{q}_i \in \hat{Q}}
\lbrace (\hat{q}_i, T(\hat{q}_i)) \rbrace
\]
$T(\hat{Q})$ is the set of question-response pairs received when $T$
is used to answer the questions in $\hat{Q}$. We now define a
quantity analogous to the General Identification Cost for exact active
learning \cite{costcomp}. Define the General Cover Cost, $GCC$
\[ GCC \triangleq \max_{T \in R^Q} (\min_{\hat{Q} :
\bar{F}_{\alpha}(T(\hat{Q})) \geq \alpha} c(\hat{Q})) \] $GCC$
depends on $H$, $Q$, $\alpha$, $c$, and the objective functions $F_h$,
but for simplicity of notation this dependence is suppressed. $GCC$
can be viewed as the cost of satisfying $\bar{F}_{\alpha}(T(\hat{Q}))
\geq \alpha$ for worst case choice of $T$ where the choice of $T$ is
\emph{known to the algorithm selecting $\hat{Q}$}. Here the worst
case choice of $T$ is over all mappings between $Q$ and $R$. There is
no restriction that $T$ answer questions in a manner consistent with
any hypothesis $h \in H$.
We first show that $GCC$ is a lower bound on the optimal worst case cost
of satisfying $F_{h^*}(\hat{S}) \geq \alpha$.
\begin{lma}If there is a correct question asking strategy for
satisfying $F_{h^*}(\hat{S}) \geq \alpha$ with worst
case cost $C^*$ then $GCC \leq C^*$. \label{lowerlma} \end{lma}
\begin{proof}Assume the lemma is false and there is a correct question
asking strategy with worst case cost $C^*$ and $GCC > C^*$. Using
this assumption and the definition of $GCC$, there is some oracle
$T^*$ such that
\[ \min_{\hat{Q} : \bar{F}_{\alpha}(T^*(\hat{Q})) \geq \alpha}
c(\hat{Q}) = GCC > C^* \]
When we use $T^*$ to answer questions, any sequence of
questions $\hat{Q}$ with total cost less than or equal to $C^*$
must have $\bar{F}_{\alpha}(\hat{S}) < \alpha$.
$\bar{F}_{\alpha}(\hat{S}) < \alpha$ in turn
implies $F_{h^*}(\hat{S}) < \alpha$ for some target hypothesis choice
$h^* \in V(\hat{S})$. This contradicts
the assumption there is a correct strategy with worst
case cost $C^*$.\end{proof}
We now establish that when $GCC$ is small, there must be a question
which increases $\bar{F}_{\alpha}$.
\begin{lma}For any initial set of questions-response pairs $\hat{S}$,
there must be a question $q \in Q$ such that
\[ \min_{h \in V(\hat{S})} \min_{r \in q(h)}
\bar{F}_{\alpha}(\hat{S} + (q, r)) -
\bar{F}_{\alpha}(\hat{S}) \geq
c(q) (\alpha - \bar{F}_{\alpha}(\hat{S})) / GCC \]
\label{progresslma}
\end{lma}
\begin{proof}
Assume the lemma is false and for every question $q$ there is some
$h \in V(\hat{S})$ and $r \in q(h)$ such that
\[
\bar{F}_{\alpha}(\hat{S} + (q, r)) -
\bar{F}_{\alpha}(\hat{S}) < c(q) (\alpha - \bar{F}_{\alpha}(\hat{S})) / GCC
\]
Define an oracle $T'$ which answers every question with a response
satisfying this inequality.
For example, one such $T'$ is
\[ T'(q) \triangleq
\operatorname{argmin}_{r} \bar{F}_{\alpha}(\hat{S} + (q, r))
- \bar{F}_{\alpha}(\hat{S}) \]
By the definition of $GCC$
\[
\min_{\hat{Q} :
\bar{F}_{\alpha}(T'(\hat{Q})) \geq \alpha} c(\hat{Q}))
\leq
\max_{T \in R^Q} (\min_{\hat{Q} :
\bar{F}_{\alpha}(T(\hat{Q})) \geq \alpha} c(\hat{Q})) = GCC
\]
so there must be a sequence of questions
$\hat{Q}$ with $c(\hat{Q}) \leq GCC$ such that
$\bar{F}_{\alpha}(T'(\hat{Q})) \geq \alpha$.
Because $\bar{F}_{\alpha}$ is monotone non-decreasing, we also know
$\bar{F}_{\alpha}(T'(\hat{Q}) \cup \hat{S}) \geq \alpha$. Using the
submodularity of $\bar{F}_{\alpha}$,
\begin{eqnarray*}
\bar{F}_{\alpha}(T'(\hat{Q}) \cup \hat{S})
& \leq & \bar{F}_{\alpha}(\hat{S}) + \sum_{q \in \hat{Q}}
(\bar{F}_{\alpha}(\hat{S} \cup \lbrace (q, T(q)) \rbrace) -
\bar{F}_{\alpha}(\hat{S})) \\
& < & \bar{F}_{\alpha}(\hat{S}) + \sum_{q \in \hat{Q}}
c(q) (\alpha - \bar{F}_{\alpha}(\hat{S})) / GCC \leq \alpha
\end{eqnarray*}
which is a contradiction.
\end{proof}
We can now show approximate optimality.
\begin{thm}
Assume that $\alpha$ is an integer and, for any $h \in H$, $F_h$ is an
integral monotone non-decreasing submodular function. Algorithm
\ref{greedyb} incurs at most $GCC (1 + \ln (\alpha n))$ cost.
\label{mainthm}
\end{thm}
\begin{proof}
Let $\hat{q}_i$ be the question asked on the $i$th iteration,
$\hat{S}_i$ be the set of question-response pairs after asking
$\hat{q}_i$ and $C_i$ be $\sum_{j \leq i} c(\hat{q}_j)$.
By Lemma \ref{progresslma}
\[ \bar{F}_{\alpha}(\hat{S}_i) - \bar{F}_{\alpha}(\hat{S}_{i-1})
\geq c(\hat{q}_i) (\alpha - \bar{F}_{\alpha}(\hat{S}_{i-1})) / GCC \]
After some algebra we get
\[ \alpha - \bar{F}_{\alpha}(\hat{S}_i)
\leq (\alpha - \bar{F}_{\alpha}(\hat{S}_{i-1})) (1 - c(\hat{q}_i) / GCC) \]
Now using $1-x < e^{-x}$
\[ \alpha - \bar{F}_{\alpha}(\hat{S}_i) \leq (\alpha -
\bar{F}_{\alpha}(\hat{S}_{i-1})) e^{-c(\hat{q}_i) / GCC} = \alpha
e^{-C_i / GCC} \] We have shown that the gap $\alpha -
\bar{F}_{\alpha}(\hat{S}_i)$ decreases exponentially fast with the
cost of the questions asked. The remainder of the proof proceeds by
showing that (1) we can decrease the gap to $1/|H|$ using questions
with at most $GCC \ln (\alpha |H|)$ cost and (2) we can decrease the
gap from $1/|H|$ to $0$ with one question with cost at most $GCC$.
Let $j$ is the largest
integer such that
$\alpha - \bar{F}_{\alpha}(\hat{S}_j) \geq 1/|H|$ holds. Then
\[
1/|H| \leq \alpha e^{-C_j / GCC}
\]
Solving for $C_j$ we get $C_j \leq GCC \ln (\alpha |H|)$. This
completes (1).
By Lemma \ref{progresslma}, $\bar{F}_{\alpha}(\hat{S}_i) <
\bar{F}_{\alpha}(\hat{S}_{i+1})$ (we strictly increase the objective
on each iteration). Because $\alpha$ is an integer and for every $h$
$F_h$ is an integral function, we can conclude
$\bar{F}_{\alpha}(\hat{S}_i) < \bar{F}_{\alpha}(\hat{S}_{i+1}) +
1/|H|$. Then $q_{j+1}$ will be the final question asked. By Lemma
\ref{progresslma}, $q_{j+1}$ can have cost no greater than $GCC$.
This completes (2). We can finally conclude the cost incurred by the
greedy algorithm is at most $GCC (1 + \ln(\alpha |H|))$
\end{proof}
By combining Theorem \ref{mainthm} and Lemma \ref{lowerlma} we get
\begin{cor}
For integer $\alpha$ and integral monotone non-decreasing submodular $F_h$,
the worst case cost of Algorithm \ref{greedyb} is within
$1 + \ln (\alpha |H|)$ of that of any other correct question asking strategy
\end{cor}
We have shown a result for integer valued $\alpha$ and objective
functions. We speculate that for more general non-integer objectives
it should be possible to give results similar to those for standard
submodular set cover \cite{analysisofgreedy}. These approximation
bounds typically add an additional normalization term.
\section{Negative Results}
\label{negresultsec}
\subsection{Na\"\i ve Greedy}
The algorithm we propose is not the most obvious approach to the
problem. A more direct extension of the standard submodular set cover
algorithm is to choose at each time step a question $q_i$ which has
not been asked before and that maximizes the worst case gain of
$F_{h^*}$. In other words, chose the question $q_i$ that maximizes
\[ \min_{h \in V(\hat{S})} \min_{r_i \in q_i(h)} (F_h(\hat{S} + (q_i,
r_i)) - F_h(\hat{S})) / c(q_i) \] This is in contrast to the method we
propose that maximizes the worst-case gain of $\bar{F}_{\alpha}$
instead of $F_h$. We call this strategy the Na\"\i ve Greedy Algorithm.
This algorithm in general performs much worse than the optimal
strategy. The counter example is very similar to that given by
\citet{robust} for the equivalent approach in the non-interactive
setting.
\begin{thm}
Assume $F_h$ is integral for all $h \in H$ and $\alpha$ is integer. The
Na\"\i ve Greedy Algorithm has approximation ratio at least
$\Omega(\alpha \max_i c(q_i) / \min_i c(q_i))$.
\end{thm}
\begin{proof}
Consider the following example with $|H| = 2$, $|Q| = \alpha + 2$, $|R|=1$
and $\alpha>1$. When $|R|=1$ responses reveal no information about
$h^*$, so the interactive problem is equivalent to the non-interactive
problem, and the objective function only depends on the set of
questions asked. Let $F_{h_1}$ and $F_{h_2}$ be modular
functions defined by
\begin{align*}
F_{h_1}(q_1) \triangleq \alpha &&
F_{h_1}(q_2) \triangleq 0 \\
F_{h_2}(q_1) \triangleq 0 &&
F_{h_2}(q_2) \triangleq \alpha
\end{align*}
and, for all $h$ and all $q_i$ with $i > 2$, $F_h(q_i) \triangleq
1$. The optimal strategy asks $q_1$ and $q_2$ (since $h^*$ is unknown
we must ask both). However, the worst-case gain of asking $q_1$ or
$q_2$ is zero while the gain of asking $q_i$ for $i > 2$ is $1 /
c(q_i)$. The Na\"\i ve Greedy Algorithm will then always ask every $q_i$
for $i > 2$ before asking $q_1$ and $q_2$ no matter how large $c(q_i)$
is compared to $c(q_1)$ and $c(q_2)$. By making $c(q_i)$ for $i > 2$
large compared to $c(q_1)$ and $c(q_2)$ we get the claimed approximation
ratio.
\end{proof}
\subsection{Learn then Cover}
The method we propose for interactive submodular set cover
simultaneously solves the learning problem and covering problem in
parallel, only solving the learning problem to the extent that it
helps solve the covering problem. A simpler strategy is to solve
these two problems in series (i.e.\ first identify $h^*$ using the
standard greedy query learning algorithm and second solve the
submodular set cover problem for $F_{h^*}$ using the standard greedy
set cover algorithm). We call this the Learn then Cover approach. We
show that this approach and in fact any approach that identifies $h^*$
exactly can perform very poorly. Therefore it is important to
consider the learning problem and covering problem simultaneously.
\begin{thm}Assume $F_h$ is integer for all $h$ and
that $\alpha$ is an integer. Any algorithm that exactly identifies $h^*$
has approximation ratio at least $\Omega(|H| \max_i c(q_i) / \min_i c(q_i))$.
\label{learnthencovergap}
\end{thm}
\begin{proof}
We give a simple example for which the learning problem
(identifying $h^*$) is hard
but the interactive submodular set cover problem (satisfying
$F_{h^*}(\hat{S}) \geq \alpha$) is easy.
For $i \in {1 ... |H|}$ let
$q_i(h_j) = \lbrace 1 \rbrace$ if $i = j$ and
$q_i(h_j) = \lbrace 0 \rbrace$ if $i \neq j$. For $i = |H| + 1$
let $q_i(h_j) = \lbrace 0 \rbrace$ for all $j$.
For worse case choice of $h*$, we need ask every question
$q_i$ for $i \in {1 ... |H|}$ in order to identify $h^*$.
However, if we define the objective to be
\[ F_{h}(\hat{S}) \triangleq I((q_{|H| + 1}, 0) \in \hat{S}) \]
for all $h$ with $\alpha = 1$,
the interactive submodular set cover problem is easy. To satisfy
$F_{h^*}(\hat{S}) \geq \alpha$ we simply need to ask question
$q_{|H| + 1}$. By making the cost of $q_{|H| + 1}$ small and
the cost of the other questions large, we get an approximation
ratio of at least $|H| \max_i c(q_i) / \min_i c(q_i)$.
\end{proof}
\subsection{Adaptivity Gap}
Another simple approach is to ignore feedback and solve the covering
problem for all $h \in H$. We call this the Cover All method. This
method is an example of a non-adaptive method: a non-adaptive (i.e.\
non interactive) method is any method that does not use responses to
previous questions in deciding which question to ask next. The
\emph{adaptivity gap} \cite{approxstochknap} for a problem
characterizes how much worse the best non-adaptive method can perform
as compared to the best adaptive method. For interactive
submodular set cover we define the adaptivity gap to be the maximum
ratio between the cost of the optimal non-adaptive strategy and the
optimal adaptive strategy. With this definition, we can show that, in
contrast to related problems \cite{maximizingstochastic} where the
adaptivity gap is a constant, the adaptivity gap for interactive
submodular set cover is quite large.
\begin{thm}
The adaptivity gap for interactive submodular set cover is at least
$\Omega(|H| / \ln |H|)$.
\label{coverallgap}
\end{thm}
\begin{proof}
The result follows directly from the connection to active learning
(Section \ref{connectiontoactivesec}) and in particular any example
of exact active learning giving an exponential speed up over
passive learning. A classic example is learning a threshold
on a line \cite{analysisofgreedyactive}. Let $|H| = 2^k$ for
some integer $k > 0$. Define the active learning objective as before
\[ F_{h}(\hat{S}) \triangleq |H \setminus V(\hat{S})| \] for all $h$.
The goal of the problem is to identify $h^*$. We define the
query set such that we can identify $h^*$ through binary search. Let
there be a query $q_i$ corresponding to each hypothesis $h_i$. Let
$q_i(h_j) = \lbrace 1 \rbrace$ if $i \leq j$ and $q_i(h_j) = \lbrace 0
\rbrace$ if $i > j$. Each $q_i$ can be thought of as a point on a
line with $h_i$ the binary classifier which classifies all points as
positive which are less than or equal to $q_i$. By asking question
$q_{2^{k-1}}$ we can eliminate half of $H$ from the version space. We
can then recurse on the remaining half of $H$ and identify $h^*$ in
$k$ queries. Any non-adaptive strategy on the other hand must perform
all $2^k$ queries in order to ensure $V(\hat{S})| = 1$ for worst case
choice of $h^*$.
\end{proof}
This result shows, even if we optimally solve the submodular set
cover problem, the Cover All method can incur exponentially greater
cost than the optimal adaptive strategy.
\subsection{Hardness of Approximation}
We show that the $1 + \ln (\alpha |H|)$ approximation factor achieved
by the method we propose is in fact the best possible up to the
constant factor assuming there are no slightly superpolynomial time
algorithms for NP. The result and proof are very similar to those for
the non-interactive setting \cite{robust}.
\begin{thm}
Interactive submodular set cover cannot be approximated within a
factor of $(1 - \epsilon) \max(\ln |H|, \ln \alpha)$ in polynomial
time for any $\epsilon > 0$ unless NP has $n^{O(\log \log n)}$ time
deterministic algorithms.
\end{thm}
\begin{proof}
We show the result by reducing set cover to interactive submodular
set cover in two different ways. In the first reduction, a set
cover instance of size $n$ gives an interactive submodular set cover
of with $|H|=1$ and $\alpha=n$. In the second reduction, a set cover instance
of size $n$ gives an interactive submodular set cover instance with $|H|=n$
and $\alpha = 1$. The theorem then follows from the result of
\citet{thresholdsetcover} which shows a set cover
cannot be approximated within a factor of $(1 - \epsilon) \ln n$
in polynomial time for any $\epsilon > 0$ unless NP has $n^{O(\log \log n)}$ time
deterministic algorithms.
Let $V$ be the set of sets defining the set cover problem. The ground
set is $\bigcup_{v \in V} v$. The goal of set cover is to find a
small set of sets $S \subseteq V$ such that $\bigcup_{s \in S} s =
\bigcup_{v \in V} v$. For both reductions we use $|R|=1$ (all
questions have only one response) and make each question in $Q$
correspond to a set in $V$. For a set of question-response pairs
$\hat{S}$ define $V_{\hat{S}}$ to be the subset of $V$ corresponding
to the questions in $\hat{S}$. For the first reduction with $|H|=1$,
we set the one objective function $F_h(\hat{S})
\triangleq |\bigcup_{v \in V_{\hat{S}}} v|$. With $\alpha=n$, we have
that $\bar{F}_{\alpha}(\hat{S}) = \alpha$ iff $V_{\hat{S}}$ forms a cover.
For the second reduction with $|H|=n$, define $F_{h_i}(\hat{S})$ for
the $i$th hypothesis $h_i$ to be $1$ iff the $i$th object in the
ground set of the set cover problem is covered by $V_{\hat{S}}$. More
formally $F_{h_i}(\hat{S}) \triangleq I(v_i \in V_{\hat{S}})$ where
$v_i$ is the $i$th item in the ground set (ordered arbitrarily). This
is similar to the first reduction except we have broken down the
objective into a sum over the ground set elements. With
$\alpha=1$, we then have that $\bar{F}_{\alpha}(\hat{S}) = \alpha$ iff
$V_{\hat{S}}$ forms a cover.
\end{proof}
The approximation factor we have shown for the greedy algorithm
is
\[ 1 + \ln(\alpha |H|) = 1 + \ln \alpha + \ln |H| < 1 + 2 \max(\ln |H|,
\ln \alpha) \] so our hardness of approximation result
matches up to the constant factor and lower order term.
\section{Experiments}
\begin{table*}[t]
\begin{small}
\begin{center}
\begin{tabular}{|l|r|r|r|}
\hline
Data Set / Hypothesis Class & Simultaneous Learning and Covering &
Learn then Cover & Cover All \\
\hline
Enron / Clusters & \textbf{156.64} & 161.81 & 3091.00 \\
Physics / Clusters & \textbf{175.97} & \textbf{177.88} & 3340.00 \\
Physics Theory / Clusters & \textbf{172.38} & \textbf{175.12} & 3170.00 \\
Epinions / Clusters & \textbf{774.81} & 779.23 & 15777.00 \\
Slashdot / Clusters & \textbf{709.30} & 715.39 & 15383.00 \\
\hline
Enron / Noisy Clusters & \textbf{179.00} & 231.03 & 3091.00 \\
Physics / Noisy Clusters & \textbf{186.13} & 225.02 & 3340.00 \\
Physics Theory / Noisy Clusters & \textbf{160.62} & 201.24 & 3170.00 \\
Epinions / Noisy Clusters & \textbf{788.52} & \textbf{788.06} & 15777.00 \\
Slashdot / Noisy Clusters & \textbf{804.87} & \textbf{804.86} & 15383.00 \\
\hline
\end{tabular}
\end{center}
\end{small}
\caption{Average number of queries required to find a dominating set in
the target group.}
\label{results}
\end{table*}
We tested our method on the interactive dominating set problem
described in Section \ref{examplesec}. In this problem, we are given
a graph and $H$ is a set of possibly overlapping clusters of nodes.
The goal is to find a small set of nodes which forms a dominating set
of an initially unknown target group $h^* \in H$. After selecting
each node, we receive feedback indicating if the selected node is in
the target group. Our proposed method (Simultaneous Learning and Covering)
simultaneously learns about the target group $h^*$ and finds a
dominating set for it. We compare to two baselines: a method which
first exactly identifies $h^*$ and then finds a dominating set for the
target group (Learn then Cover) and a method which simply ignores
feedback and finds a dominating set for the union of all clusters
(Cover All). Note that Theorem \ref{learnthencovergap} and Theorem
\ref{coverallgap} apply to Learn then Cover and Cover All
respectively, so these methods do not have strong theoretical
guarantees. However, we might hope however that for reasonable real
world problems they perform well. We use real world network data sets
with simple synthetic hypothesis classes designed to illustrate
differences between the methods. The networks are from Jure
Leskovec's collection of datasets available at
\url{http://snap.stanford.edu/data/index.html}. We convert all the
graphs into undirected graphs and remove self edges.
Table \ref{results} shows our results. Each reported result is the
average number of queries over 100 trials. Bolded results are the
best methods for each setting with multiple results bolded when
differences are not statistically significant (within $p=.01$ with a
paired t-test). In the first set of results (Clusters), we create $H$
by using the METIS graph partition package 4 separate times
partitioning the graph into 10, 20, 30, and 40 clusters. $H$ is the
combined set of 100 clusters, and these clusters overlap since they
are taken from 4 separate partitions of the graph. The target $h^*$
is chosen at random from $H$. With this hypothesis class, we've found
that there is very little difference between the Simultaneous Learning and
Covering and the Learn then Cover methods. The Cover All method performs
significantly worse because without the benefit of feedback it must
find a dominating set of the entire graph.
In the second set of results, we use a hypothesis class designed to
make learning difficult (Noisy Clusters). We start with $H$ generated as
before. We then add to $H$ 100 additional hypotheses which are each
very similar to $h^*$. Each of these hypotheses consists of the
target group $h^*$ with a random member removed. $H$ is then the
combined set of the 100 original hypotheses and these 100 variations
of $h^*$. For this hypothesis class, Learn then Cover performs
significantly worse than our Simultaneous Learning and Covering method
on 3 of the 5 data sets. Learn then Cover exactly identifies $h^*$,
which is difficult because of the many hypotheses similar to $h^*$.
Our method learns about $h^*$ but only to the extent that it is
helpful for finding a small dominating set. On the other two data
sets Learn then Cover and Simultaneous Learning and Covering are
almost identical. These are larger data sets, and we've found that
when the covering problem requires many more queries than the learning
problem, our method is nearly identical to Learn then Cover.
This makes sense since when $\alpha$ is large compared to the sum over
$F_h(\hat{S})$ the second term in $\bar{F}_{\alpha}$ dominates.
It is also possible to design hypothesis classes for which Cover All
outperforms Learn then Cover: we found this is the case when the
learning problem is difficult but the subgraph corresponding to the
union of all clusters in $H$ is small. In the appendix we give an
example of this. In all cases, however, our approach does about as
good or better than the best of these two baseline methods.
Although we use real world graph data, the hypothesis classes and
target hypotheses we use are very simple and synthetic, and as such these
experiments are primarily meant to provide reasonable examples in support
of our theoretical results.
\section{Future Work}
We believe there are other interesting applications which can be posed
as interactive submodular set cover. In some applications it may be
difficult to compute $\bar{F}_{\alpha}$ exactly because $H$ may be
very large or even infinite. In these cases, it may be possible to
approximate this function by sampling from $H$. It's also important
to consider methods that can handle misspecified hypothesis classes
and noise within the learning. One approach could be to extend
agnostic active learning \cite{agnostic} results to a similar
interactive optimization setting.
\begin{small}
|
2,869,038,154,761 | arxiv | \section{Introduction}
\label{sec:introduction}
The aligned understanding of a norm is an essential process for the interaction between different agents (human or artificial) in normative systems. Mainly because these systems take into consideration norms as the basis to specify and regulate the relevant behavior of the interacting agents \cite{jones1993characterisation}. This is especially important when we consider online communities in which different people with diverse profiles are easily connected with each other. In these cases, misunderstandings about the community norms may lead to interactions being unsuccessful. Thus, the goals of this research are: 1) to investigate the challenges associated with detecting when a norm is being violated by a certain member, usually due to a misunderstanding of the norm; and 2) to inform this member about the features of their action that triggered the violation, allowing the member to change their action to be in accordance with the understanding of the community, thus helping the interactions to keep running smoothly. To tackle these goals, our main contribution is to provide a framework capable of detecting norm violation and informing the violator of why their action triggered a violation detection.
The proposed framework is using data, that belongs to a specific community, to train a Machine Learning (ML) model that can detect norm violation. We chose this approach based on studies showing that the definition of what is norm violation can be highly contextual, thus it is necessary to consider what a certain community defines as norm violation or expected behavior \cite{al2019detection,chandrasekharan2019crossmod,van2003DeonticLogic}.
To investigate norm violations, this work is specifically interested in norms that govern online interactions, and we use the Wikipedia community as a testbed, focusing on the article editing actions. This area of research is not only important due to the high volume of interactions that happen on Wikipedia, but also for the proper inclusion and treatment of diverse people in these online interactions. For instance, studies show that, when a system fails to detect norm violations (e.g., hate speech or gender, sexual and racial discrimination), the interactions are damaged, thus impacting the way people interact in the community~\cite{KishonnaDiversity2018,mclean2019female}.
Previous works have dealt with norms and normative systems, proposing mechanisms for norm conflict detection \cite{NormConflict2018}, norm synthesis \cite{morales2018off}, norm violation on Wikipedia \cite{west2011multilingual,anandDeepLearningViolation2019} and other online communities, such as Stack Overflow \cite{cheriyan2020norm} and Reddit \cite{chandrasekharan2019crossmod}.\removeText{and Twitter \cite{kwokRacism2013}} However, our approach differs mainly in three points: 1) implementing an ML model that allows for the interpretation of the reasons leading to detecting norm violation; 2) incorporating a taxonomy to better explain to the violator which features of their actions triggered the norm violation, based on the results provided by our ML model; and 3) codifying actions in order to represent them through a set of features \change{acquired from previous knowledge about the domain}, which is necessary for the above two points\removeText{(the learning of the ML model and the taxonomy of action features)}. Concerning the last point, we note that our framework does not consider the action as is, but a representation of that action in terms of features and the relation of those features to norm violation \change{(as learned by the applied ML model)}. For the Wikipedia case, we represent the action of editing articles based on the categorization introduced in~\cite{west2011multilingual}, with features such as: the measure of profane words, the measure of pronouns, and the measure of Wiki syntax/markup (the details of these are later presented in Section \ref{subSec:wikipediaTaxonomy}).
To build our proposed framework, this work investigates the combination of two main algorithms: 1) the Logistic Model Tree, the algorithm responsible for classifying an article edit as a violation or not;\removeText{The dataset used for training is the vandalism on Wikipedia editions corpus} and 2) the K-Means, the clustering algorithm responsible for grouping the features that are most relevant for detecting a violation. The information about the relevant features is then used to navigate the taxonomy and get a simplified taxonomy of these relevant features.
Our experiments describe how the ML model was built based on the training data provided by Wikipedia, and the results of applying this model to the task of vandalism detection in Wikipedia's article edits illustrate how our approach can reach a precision of 78,1\% and a recall of 63,8\%. Besides, the results also show that our framework can provide information about the specific group of features that affect the probability of an action being considered a violation, and we make use of this information to provide feedback to the user on their actions.
The remainder of this paper is divided as follows. Section~\ref{sec:background} presents the basic mechanisms used by our proposed framework. Section~\ref{sec:framework} describes our framework, while Section~\ref{sec:useCase} presents its application to the Wikipedia edits use case, and Section~\ref{sec:experimentsResults} presents our experiment and its results. The related literature is presented in Section~\ref{sec:literatureReview}. We then describe our conclusions and future work in Section~\ref{sec:conclusion}.
\section{Background}
\label{sec:background}
This section aims to present the base concepts upon which this work is built. We first start with the description of the taxonomy\removeText{(Section~\ref{subSec:taxonomy})}, which we intend to use to formalize a community's knowledge about the features of the actions. Next, we describe the ML algorithms applied to build our framework. First, the Logistic Model Tree (LMT) algorithm\removeText{(Section~\ref{subSec:LMT})}, which is used to build the model responsible for detecting possible vandalism; and second, the K-Means algorithm\removeText{(Section~\ref{subSec:kmeans})}, responsible for grouping the features of the action that are most relevant for detecting violation.
\subsection{Taxonomy for Action Representation}
\label{subSec:taxonomy}
In the context of our work, an action (executed by a user in an online community) is represented by a set of features. Each of these features describes one aspect of the action being executed, i.e., the composing parts of the action. The goal of adopting this approach is to equip our system with an adaptive aspect, since by modelling an action as a set of features allows the system to deal with different kinds of actions (in different domains). For example, we could map the action of participating in an online meeting by features, such as: amount of time present in the meeting; volume of message exchange; and rate of interaction with other participants. Besides, in the context of norm violation, the proposed approach can use these features to explain which aspects of an action were problematic.
Defining an action through its features gives information about different aspects of the action that might have triggered a violation. However, it is still necessary to find a way to present this information to the violators. The idea is that this information must be provided in a human-readable way, allowing the users to understand what that feature means and how different features are related to each other. With these requirements in mind, we propose the use of a taxonomy to present this data.\removeText{A taxonomy is a way to classify similar concepts into groups, which has been used in different areas, specially in Biology~\cite{kirk96Taxonomy}.} This classification scheme provides relevant information about concepts of a complex domain in a structured way~\cite{kirk96Taxonomy}, thus handling the requirements of our solution.
We note that, in this work, the focus is not on {\it building} a taxonomy of features. Instead, we assume that the taxonomy is provided with their associated norms. Our system uses this taxonomy, navigating it to select the relevant features. The violator is then informed about the features (presented as a subsection of the larger taxonomy) that triggered the violation \change{detected by our model}.
\subsection{Logistic Model Tree}
\label{subSec:LMT}
With respect to the domain of detecting norm violations in online communities, interpreting the ML model is an important aspect to consider.\removeText{is an important aspect to consider. This need encourages the creation of a solution that offers a tool capable of explaining, to the members of a community, the reasons behind the model's decisions.} Thus, if a community is interested in providing the violator with information about the features of their action that are indicative of violation, then the proposed solution needs to work with a model that can correctly identify these problematic features.
\removeText{In ML, there are different approaches to create a model, }In this work, we are interested in\removeText{the concept of} supervised learning, which is the ML task of finding a map between the input and the output.\removeText{To do that, the model must be trained with labeled data, containing examples of input-output pairs~\cite{russell2002artificial}.} Several algorithms exist that implement the concepts of supervised learning, e.g., artificial neural networks and tree induction. We are most concerned with the ability of these algorithms to generate interpretable outputs, i.e., how the model explains the reasons for taking a certain decision. As such, the algorithm we chose that contains this characteristic is the tree induction algorithm
\removeText{\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figs/DT-PlayTenis.png}
\caption{An example of a Decision Tree, deciding to play tennis or not \cite{russell2002artificial}.}
\label{fig:dtPlayTenis}
\end{center}
\end{figure}}
The ability to interpret the tree induction model is provided by the way a path is defined in this technique (basically a set of {\em if-then} statements), which allows our model to find patterns in the data, present the path followed by the model and consequently provide the reasons that lead to that conclusion.
Although induction trees have been a popular approach to solve classification problems\removeText{ for a long time\cite{safavian1991survey,ghiasiDT2020}}, this algorithm also presents some disadvantages. This has prompted Landwehr et al.~\cite{landwehr2005logistic} to propose the Logistic Model Tree (LMT) algorithm, which adds logistic regression functions at the leaves of the tree.
In logistic regression, there are two types of variables: the independent and the dependent variables. The goal is to find a model able\removeText{to map the relationship between the independent and dependent variables, i.e., a function that} to describe the effects of the independent variables on the dependent ones. In our context, the output of the model is\removeText{in the interval [0,1], which is useful to indicate the probability that an event is going to happen, or in our context,} responsible for predicting the probability of an action being classified as norm violation.
Dealing with odds is an interesting \change{aspect} present in logistic regression, since the increase in a certain variable indicates how the odds changes for the classification output, in this case the odds indicate the effect of the independent variables on the dependent ones. Besides, another important aspect is the equivalence of the natural log of the odds ratio and the linear function of the independent variables, represented by equation \ref{eq:lnOdds}:
\begin{eqnarray}
\label{eq:lnOdds}
ln (\frac{p}{1-p}) & \gets & \beta_0 + \beta_1 x_1
\end{eqnarray}
where \(ln\) is the logarithm of the odds ratio, \(p\) [0,1] is\removeText{a value between 0 and 1,} the probability of an event occurring. \(\beta\) represents the parameters of the model, in our case the \change{weights for} features of the action. After calculating the natural logarithm, we can then use the inverse of the function to get our estimated regression equation:
\begin{eqnarray}
\label{eq:regressionEstimation}
\hat p & \gets \frac{\epsilon^{\beta_0 + \beta_1 x_1}}{1+\epsilon^{\beta_0 + \beta_1 x_1}}
\end{eqnarray}
where $\hat p$ is the probability estimated by the regression model.
With these characteristics of logistic regression, we can see how this technique can be used to highlight attributes (independent variables) that have \change{relevant} influence over the output of the classifier probability.
Landwehr et al.~\cite{landwehr2005logistic} demonstrate how neither of the two algorithms described above (Tree Induction and Logistic Regression) is better than the other.\removeText{One conclusion presented by~\cite{landwehr2005logistic} is that the performance of the methods depends upon the domain of investigation and the number of training data available. So, to tackle these issues, the LMT algorithm partitions the instance space in regions of constant class with estimations of the probabilities, while having a more gradual change, provided by the estimations from the logistic regression model.} Thus, to tackle the issues present in these two algorithms, LMT adds to the leaves of the tree a logistic regression function.
Figure~\ref{fig:lmtExample} presents the description of a tree generated by the LMT algorithm. With a similar process as the standard decision tree, the LMT algorithm obtains a probability estimation as follow: first, the feature is compared to the value associated with that node\removeText{(if the node contains a nominal attribute, with \(k\) values, then this node has \(k\) child nodes. But, if the value is numerical, then the feature is compared to a threshold, going left in the tree if the value is below the threshold and going right otherwise)}. This step is repeated until the algorithm has reached a leaf node, when the exploration is completed. Then the logistic regression function determines the probabilities for the class, as described by equation~\ref{eq:regressionEstimation}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\columnwidth]{figs/LMTExampleTree.png}
\caption{An example of a tree built by the LMT algorithm\cite{landwehr2005logistic}. \change{X\(_1\) and X\(_2\) are features present in the dataset. F\(_1\) and F\(_2\) are the equations found by the logistic regression model, describing the weights for each feature present in the training dataset.}}
\label{fig:lmtExample}
\end{center}
\end{figure}
\subsection{K-Means Clustering Method}
\label{subSec:kmeans}
K-Means is a clustering algorithm with the goal of finding a number \(K\) of clusters in the observed data, attempting to group the most `similar' data points together. This algorithm has been used successfully in different applications, such as\removeText{network intrusion detection~\cite{jianliangIntrusionKMeans2009} and} feature learning~\cite{RubaiatFeaturesKmeans2018} and computer vision~\cite{zheng2018image}. To achieve this goal, K-Means clusters the data using the nearest mean to the cluster center (calculating the squared Euclidean distance), thus reducing the variance within the group~\cite{rai2010survey}.
In this work, the K-Means algorithm can be used to group the features that may indicate an action as violation \change{(we use the features' weights multiplied by their input values as indication of relevance for the classification probability)}. First, after detecting a possible violation, the ML model provides the K-Means algorithm with the set of features present in the logistic regression and their associated \change{values (the input multiplied by the weight)}. Then, based on the values of these features, the algorithm is responsible for separating the features in two groups: 1) those \change{that our model found with highest values, i.g., the most relevant for the vandalism classification}; and 2) those \change{with the lowest values, e.g., less relevant for the vandalism classification}. Lastly, the output of K-Means informs the framework which are the most relevant features for detecting violations (i.e. the first group), which the framework can then use to navigate the taxonomy and present a selected simplified taxonomy of relevant features to the violator.
\section{Framework for Norm Violation Detection (FNVD)}
\label{sec:framework}
This section presents the main contribution of our work, the framework for norm violation detection (FNVD). The goal of this framework is to be deployed in a normative system so that when a violation is detected, the system can enforce the norms by, say, prohibiting the action.
The main component of our framework is the machine learning (ML) algorithm behind the detection of norm violations, specifically the LMT algorithm of Section~\ref{subSec:LMT}.
An important aspect to take into consideration, when using this algorithm, is the data needed to train the model. In \removeText{the context of }our work, the community must provide the definitions \removeText{and constitutions }of norm violations through a dataset that exemplifies actions that were previously labeled as norm violations.\removeText{Different communities use different methods to build this dataset, ranging from using only feedback from the community itself (e.g., from the moderators of the community) to hiring a service to classify the data (such as the Mechanical Turk).} Thus, here we are using data provided by Wikipedia, gathered using Mechanical Turk~\cite{Potthast2010OverviewOT}.
After defining the data source, our proposed approach essentially 1) collects the data \removeText{that will be }used to train the LMT model; 2) trains the LMT model to detect possible violations and to learn the action's features relevant to norm violations; and 3) when violations are detected, according to the LMT model's results, then the action responsible for the violation is rejected and the violator is informed about the features of their action \change{that triggered the model output}. Furthermore, in both cases (when actions are labelled as violating norms or not), we suggest that the framework collects feedback from the members of the community, which can then be used as new data to retrain the ML model. This is important as we strongly believe that communities and their members evolve, and what may be considered a norm violation today might not be in the future. For example, imagine a norm that states that hate speech is not allowed. Agreeing on the features of hate speech may change from one group of people to another and may also change over time. Consider the evaluation of the N-word, which is usually seen as a serious racial offense and can automatically be considered a text that violates the ``no hate speech" norm. However, imagine a community of African Americans frequently saluting each other with the phrase ``Wussup nigga'' and the ML model classifying their text as hate speech. Clearly, human communities do not always have one clear definition of concepts like hate speech, violation, freedom of speech, etc. The framework, as such, must have a mechanism to adapt to the views of the members of its community, as well as adapt to the views that may change over time. While we leave the adaptation part for future work, we highlight its need in this section, and prepare the framework to deal with such adaptions, as we illustrate in Figure~\ref{fig:frameworkDiagram}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figs/NormChange.png}
\caption{How the framework works when deployed in an online community.}
\label{fig:frameworkDiagram}
\end{center}
\end{figure}
To further clarify how our framework would act to detect a norm violation when deployed in a community, it is essential to explore the diagram in Figure~\ref{fig:frameworkDiagram}. Step~0 represents the training process of the LMT model, which is a fundamental part of our approach because it is in this moment that the rules for norm violation are specified. Basically, after training the model, our framework would have identified a set of rules that describe norm violation. We can portray these rules as a conjunction of two elements: 1) the tree that is built by the LMT algorithm on top of the collected data; and 2) the weights presented in the leaves of the tree. These weights are the parameters of the estimated regression equation that defines the probability of norm violation (depicted in Equation \ref{eq:regressionEstimation}). With the trained LMT model, the system starts monitoring every new action performed in the community (Step~1). In Step~2, the system maps the action to features that the community defined as descriptive of that action, which triggers the LMT model to start working to detect if that action is violating (or not) any of the norms. Step~3 presents the two different paths that can be executed by our system. If the action is detected as violating a norm (Condition~1), then we argue that the system must execute a sequence of steps to guarantee that the community norms are not violated: 1) the system does not allow the action to persist (i.e., action is not executed); 2) the system presents to the user information about which action features were the most relevant for our model to detect the norm violation, and the taxonomy of the relevant features is presented\removeText{. In the case of our work, these features can be seen as the composing parts of the actions, i.e., we are separating the actions into features (which are used to train our machine learning model, details in Section \ref{subSec:taxonomy})}; 3) the action is logged by the system, allowing other community members to give feedback about the edit attempt, thus providing the possibility of these members flagging the action as a non-violation. The feedback collected from the users can later be used to continuously train (Step~0) our LMT model (future work). However, if the executed action is not detected as violating a norm (Condition~2), then the system can proceed as follows. The action persists in the system (i.e., action is executed), and since any model may incorrectly classify some norm violation as non-violation, the system allows the members of the community to give feedback about that action, providing the possibility of flagging an already accepted action as a violation. Getting people's feedback on violations that go unnoticed by the model is a way to allow the system to adapt to new data (people's feedback) and update \change{the definitions of norm violations} by continuously training the LMT model (Step~0).
To obtain the relevant features for the norm violation classification (Condition~1), we use the K-Means algorithm\removeText{(Section \ref{subSec:kmeans})}. In our context, due to the estimated logistic regression equation, the LMT model provides the weights for each feature \change{multiplied by the value of these features for the action}. \change{This indicates} the influence of the features on the model's output (i.e., the probability of an action being classified as norm violation). With the weights \change{and specific values for the features}, \change{the K-Means algorithm can group the set of features that present the highest multiplied values, which are the ones we assume that contribute the most for the probability of norm violation}. Then, by searching the taxonomy using the group of relevant features, our system can provide the taxonomy structure of the features that trigger norm violation, this is useful due to the explanatory and interpretation characteristics of a taxonomy. The aim of providing this information is to clarify to the member of the community performing the action, what are the problematic aspects of their action \change{as learned by our model}.
\section{The Wikipedia Vandalism Detection Use Case}
\label{sec:useCase}
We focus on the problem of detecting vandalism in Wikipedia article edits. This use case is interesting because Wikipedia is an online community where norms such as `no vandalism' may have different interpretations by different people\removeText{, and norm violations can affect community experience}. In what follows, we first present the use case's domain, followed by the taxonomy used by our system, and finally, an illustration of how our proposed framework may be applied to this use case.
\subsection{Domain}
\label{subSec:domain}
Wikipedia~\cite{wiki}\removeText{\footnote{\url{https://en.wikipedia.org}}} is an online encyclopedia, containing articles about different subjects in any area of knowledge. It is free to read and edit, thus any person with a device connected to the internet can access it and edit\removeText{the content of} its articles. Due to the openness and collaborative structure of Wikipedia, the system is subject to different interpretations of what is the community's expectation concerning how content should be edited. To help address this issue, Wikipedia has compiled a set of rules, the Wikipedia norms~\cite{Potthast2010OverviewOT}, to maintain and organize its content.
Since we are looking for an automated solution for detecting norm violations by applying machine learning mechanisms, the availability of data becomes crucial. Wikipedia provides data on what edits are marked as vandalism, where vandalism annotations were provided by Amazon's Mechanical Turk. Basically, every article edit was evaluated by at least three people (Mechanical Turks) that decided whether the edit violates the `no vandalism' norm or not. In the context of our work, the actions performed by the members of the community are the Wikipedia users' attempts to edit articles, and the norm is {\em ``Do not engage in vandalism behavior"} (which we\removeText{sometimes} refer to as the `no vandalism' norm). It is this precise dataset that we have used to train the model that detects norm violations. We present an example of what is considered a vandalism in a Wikipedia article edit, where a user edited an article by adding the following text: {\em ``Bugger all in the subject of health ect."}
\subsection{Taxonomy Associated with Wikipedia's `No Vandalism' Norm}
\label{subSec:wikipediaTaxonomy}
An important step in our work is to map actions to features and then specify how they are linked to each other.\removeText{As illustrated earlier, we assume a taxonomy of these features are already provided with the norm. In our Wikipedia use case, however, we had to manually create this taxonomy.} We manually created a taxonomy to describe these features \change{ by separating them in categories that describe their relation with the action.\footnote{For the complete taxonomy, the reader can refer to \url{https://bit.ly/3sQFhQz}}} \change{In this work, we consider the 58 features described in~\cite{west2011multilingual} and 3 more that were available in the provided dataset: LANG\_MARKUP\_IMPACT, the measure of the addition/removal of Wiki syntax/markup; LANG\_EN\_PROFANE\_BIG and LANG\_EN\allowbreak\_PROFANE\_BIG\_IMPACT, the measure of addition/removal of English profane words. In the dataset, features ending with \_IMPACT are normalized by the difference of the article size after edition.} The main objective of this taxonomy is to help our system present to the violator an easy-to-read explanation of the reasons why their article edit was marked as violating a norm by our model, \change{specifying the features with highest influence to trigger this violation.}
To further explain our taxonomy approach, we present in Figure~\ref{fig:vandalismDetectionTaxonomy} the constructed taxonomy for Wikipedia's `no vandalism' norm. We observe that features can be divided in four main groups. The first is user's direct actions, which represent aspects of the user's article editing action, e.g., adding a text. This group is further divided in four sub-groups: a) written edition, which contains features about the text itself that is being edited by the user; b) comment on the edition, which contains features about the comments that users have left on that edition; c) article after edition, which contains features about how the edited article changed after the edition was completed; and d) time of edition, which contains features about the time when the user made their edition. The second group is the user's profile, general information about the user. The third is the page's history,\removeText{information about the page that the user is editing, e.g., } how the article changed with past editions. The last group is reversions, which is essentially information on past reversions.\footnote{A reversion is when an article is reverted back to a version before the vandalism occurred.} \change{In total, these groups have 61 features, but due to simplification purpose, Table~\ref{tab:ExampleFeatures} only presents a subset of those features. \removeText{that are part of these groups:}}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\columnwidth]{figs/SimplifiedGroupsTaxonomy.png}
\caption{Taxonomy associated with Wikipedia's `no vandalism' norm.}
\label{fig:vandalismDetectionTaxonomy}
\end{center}
\end{figure}
\begin{table}[h]
\scriptsize
\caption{Example of Features present in the taxonomy groups.}\label{tab:ExampleFeatures}
\centering
\begin{tabular}{l|llll}
\hline
\multicolumn{1}{c|}{{Group}} & {{Features}} \\
\hline
\textbf{Written Edition} & LANG\_ALL\_ALPHA; LANG\_EN\_PRONOUN\\
\textbf{Comment on Edition} & COMM\_LEN; COMM\_LEN\_NO\_SECT\\
\textbf{Article After Edition} & SIZE\_CHANGE\_RESULT; SIZE\_CHANGE\_CHARS\\
\textbf{Time of Edition} & TIME\_TOD; TIME\_DOW\\
\textbf{User's Profile} & HIST\_REP\_COUNTRY; USER\_EDITS\_DENSITY\\
\textbf{Page's History} & PAGE\_AGE; WT\_NO\_DELAY\\
\textbf{Reversions} & HASH\_REVERTED; HASH\_IP\_REVERT\\
\hline
\end{tabular}
\end{table}
\subsection{FNVD Applied to Wikipedia Vandalism Detection}
\label{subSec:frameworkWikipediaVandalism}
It this section, we first describe an example of how our framework can be configured to be deployed in the Wikipedia community. First, the community provides the features and the taxonomy describing that feature space (see Figure~\ref{fig:vandalismDetectionTaxonomy}).\removeText{This information is essential because the idea of the framework is not only to detect that an edition might be a vandalism, but also inform violators about the features that have contributed to the model classifying their edit as vandalism.} Then,\removeText{with this set of features and its taxonomy,} our framework trains the LMT model to classify norm violations based on the data provided (Step~0 of Figure~\ref{fig:frameworkDiagram}), which must contain examples of what that community understands as norm violation and regular behavior.
In the context of vandalism detection on Wikipedia, the relevant actions performed by the members of the community are the attempts to edit Wikipedia articles. Following the diagram in Figure~\ref{fig:frameworkDiagram}, when a user attempts to edit an article (Step~1), our system will analyze this edit. We note here that our proposed LMT model does not work with the action itself, but the features that describe it. As such, it is necessary to first find the features that represent the performed action\removeText{(the Wikipedia article edit)}. Thus, in Step~2\removeText{of Figure \ref{fig:frameworkDiagram}}, there is a pre-processing phase responsible for mapping actions to the features associated to the norm in question. For example, \change{an article about Asteroid was edited with the addition of the text {\em `` i like man!!"}. After getting this edition text, the system can compute the values (as described in \cite{west2011multilingual}) for the 61 features, which are used to calculate the vandalism probability}. \change{For brevity reasons, we only show the values for some of these features}:
\begin{enumerate}
\item LANG\_ALL\_ALPHA, the percentage of text which is alphabetic: 0,615385;
\item WT\_NO\_DELAY, the calculated WikiTrust score: 0,731452;
\item HIST\_REP\_COUNTRY, measure of how users from the same country as the editor behaved: 0,155146.
\end{enumerate}
\removeText{For that, first we have to go through the path of the tree, to determine the logistic regression equation that we are using to calculate the probability of vandalism, }
\change{After calculating the values for all features,} the LMT model can evaluate if this article edit is considered `vandalism' or not. In the case of detecting vandalism (Condition~1 of Figure~\ref{fig:frameworkDiagram}), the system does not allow the edition to be recorded on the Wikipedia article, and it presents to the violator two inputs. The first is the set of features of their edit that have the highest influence on the model's decision to detect the vandalism. \change{To get this set, after calculating the probability of vandalism \removeText{with the logistic regression equation }(as depicted in Equation~\ref{eq:regressionEstimation}), the LMT model provides the features that present a positive relationship with the \change{output}. These `positive features' are then used by K-Means to create the group with the most relevant ones (Table~\ref{tab:ExamplePositiveFeatures} presents an example of this process)}. The second input is the selected part of the taxonomy related to chosen set of features, providing further explanation of those features that triggered the norm violation. Additionally, the system will log the attempt to edit the article, which eventually may trigger feedback collection that can at a later stage be used to retrain our model.
\begin{table}[h]
\caption{\change{List of features that positively affects the probability of vandalism detection. Total Value is the multiplication between the feature's values and the features' weights\removeText{ (defined by the LMT model)}. The most relevant features, as found by K-Means, are marked with an (*).}}
\label{tab:ExamplePositiveFeatures}
\centering
\begin{tabular}{l|llll}
\hline
\multicolumn{1}{c|}{{Features}} & {{Total Value}} \\
\hline
\textbf{WT\_NO\_DELAY*} & 1.08254896\\
\textbf{HIST\_REP\_COUNTRY*} & 0.899847\\
\textbf{LANG\_ALL\_ALPHA*} & 0.7261543\\
\textbf{HASH\_REC\_DIVERSITY} & 0.15714292\\
\textbf{WT\_DELAYED} & 0.12748878\\
\textbf{LANG\_ALL\_CHAR\_REP} & 0.12\\
\textbf{HIST\_REP\_ARTICLE} & 0.093548\\
\hline
\end{tabular}
\end{table}
\removeText{Since our proposed approach navigates the taxonomy to get information about the features that are relevant for norm violation detection, an example demonstrating how this taxonomy is used in this case is interesting for clarification purposes. }\change{The features \(WT\_NO\_DELAY\), \(HIST\_REP\_COUNTRY\) and \(LANG\_ALL\allowbreak\_ALPHA\) were indicated by K-Means as the most relevant for the classification of vandalism}. With this information, our framework can search the taxonomy for the relevant features and then automatically retrieve the simplified taxonomy structure for these three specific features, as shown in Figure \ref{fig:featuresVandalismTaxonomy}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.6\columnwidth]{figs/PartTaxonomyKMeans.png}
\caption{Taxonomy for part of the features that were \change{most relevant for the vandalism classification. These features are then presented to the user with a descriptive text.}}
\label{fig:featuresVandalismTaxonomy}
\end{center}
\end{figure}
\removeText{Besides, a description text is also presented to the violator, explaining what that feature represents.}
However, in case the system classifies the article edit as `non-vandalism' (Condition~2 of Figure~\ref{fig:frameworkDiagram}), the Wikipedia article is updated according to the user's article edit and community members may provide feedback on this new article edit, which may later be used to retrain our model (as explained in Section~\ref{sec:framework}).
\section{Experiments and Results}
\label{sec:experimentsResults}
The goal of this section is to describe how the proposed approach was applied for detecting norm violation in the domain of Wikipedia article edits, with an initial attempt to improve the interactions in online communities. Then, we demonstrate and discuss the results achieved.
\subsection{Experiments}
\label{subSec:experiments}
\change{Data on vandalism detection in Wikipedia articles~\cite{west2011multilingual} were used for the experiments.}\removeText{Our experiments were conducted with data on vandalism detection in Wikipedia articles~\cite{west2011multilingual}}\removeText{provided by a competition the Wikipedia community organized to evaluate how different models deals with the automatization process of vandalism detection \cite{Potthast2010OverviewOT}} This dataset has 61 features and 32,439 instances for training (with 2,394 examples of vandalism editions and 30,045 examples of regular editions). The model was \change{trained with WEKA~\cite{weka} and} evaluated using 10 folds cross-validation.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\columnwidth]{figs/TreeModelVandalismDetection.png}
\caption{The built model for the vandalism detection, using Logistic Model Tree.}
\label{fig:LMTModelForVandalismDetection}
\end{center}
\end{figure}
\removeText{We built the model showed in Figure \ref{fig:LMTModelForVandalismDetection}, this was achieved by using the WEKA tool~\cite{weka}.\removeText{\footnote{\url{https://www.cs.waikato.ac.nz/ml/weka/}}} As described in Section \ref{subSec:LMT}, the leaves in the tree represent the estimated logistic equations that will be used to calculate the probability of a certain edition being classified as vandalism. In these equations, we have different \change{weight values} for each feature of our data.\footnote{Trained model available at:\url{https://bit.ly/3gBBkwP}}\removeText{, as shown in the list below:}}
\removeText{\begin{table}[h]
\caption{Some of the weights for features in the logistic regression equation.}\label{tab:weightsLMTModel}
\centering
\begin{tabular}{l|ll}
\hline
\multicolumn{1}{c|}{Feature} & \multicolumn{1}{c}{Value} \\
\hline
LANG\_EN\_PROFANE\_BIG\_IMPACT & 40.33\\
LANG\_EN\_PRONOUN\_IMPACT & 26.67\\
LANG\_ALL\_MARKUP\_IMPACT & -29.79\\
\hline
\end{tabular}
\end{table}
}
\removeText{
\begin{enumerate}
\item LANG\_EN\_PROFANE\_BIG\_IMPACT, value: 40,33
\item LANG\_EN\_PRONOUN\_IMPACT, value: 26,67
\item LANG\_ALL\_MARKUP\_IMPACT, value: -29,79
\end{enumerate}
The positive values indicate a positive relationship between the feature and the classification probability, and the negative values otherwise.
}
\subsection{Results}
\label{subSec:results}
The first important information to note is how the LMT model performs when classifying vandalism in Wikipedia editions. In Figure \ref{fig:LMTModelForVandalismDetection}, it is possible to see the model that was built to perform the classification task.\footnote{Trained model available at: \url{https://bit.ly/3gBBkwP}} The tree has four decision nodes and five leaves in total. Since the LMT model uses logistic regression at the leaves, the model has five different estimated logistic regression equations, each of these equations outputs' the probability of an edition being a vandalism.
\removeText{The LMT model performs well to classify the Wikipedia Vandalism dataset, since }The LMT model correctly classifies 96\% of instances in general. However, when we separate the results in two groups, vandalism editions and regular editions, it is possible to observe a difference in the model's performance. For the\removeText{instances that are } regular editions, the LMT model achieves a precision of 97,2\%, and a recall of 98,6\%. While for\removeText{the instances of} vandalism editions, the performance of the model drops, with a precision of 78,1\% and a recall of 63,8\%. This decrease\removeText{in the percentage for vandalism detection } can be explained by how the dataset was separated and the number of vandalism instances, which consequently leads to an unbalanced dataset. In the dataset, the total number of vandalism instances is 2,394 and the other 30,045 instances are of regular editions. A better balance between the number of vandalism editions and regular edition should improve our classifier, thus in the future we are exploring other \change{model configurations (e.g., ensemble models) to handle data imbalance}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\columnwidth]{figs/NumberOfOccurencesRelevantFeatures-Smaller.png}
\caption{Number of occurrences of relevant features in vandalism detection.}
\label{fig:GraphNumberOfOccurrences}
\end{center}
\end{figure}
The influence of each feature on determining the probability of a norm violation is provided by the LMT model \change{(as assumed in this work, feature influence is model specific, meaning that a different model can find a different set of relevant features)}.\removeText{In this regard, }\removeText{The proposed framework presents }\removeText{presenting the features that are most relevant when a vandalism is detect by our model.} The graph in Figure~\ref{fig:GraphNumberOfOccurrences} shows the number of times a feature is \change{classified as relevant by the built model}. Some features appear in most of the observations, indicating how important they are to detect\removeText{this kind of} vandalism. Future work shall investigate if this same behavior (some features present in the actions have more influence than other features to define the norm violation probability) can be detected in other domains
\removeText{An interesting characteristic from the Wikipedia Vandalism Detection data-set is }\removeText{We note that the feature}``LANG\_ALL\_ALPHA" recurrently appears as relevant when vandalism is detected. This happens because this feature presents, as estimated by the LMT model, a positive relationship with the norm violation, meaning that when a vandalism edition is detected, this feature is usually relevant for the classification.
\removeText{With the presentation of the relevant results, it is also important to demonstrate how a vandalism is detected by our system\removeText{, describing the relevant features that lead to this classification}. The text edition {\em `` They alos went to parties and drank tequila every saturday!”} is a vandalism edition. In our model, the estimated logistic regression equations calculated the relevant features, and the K-Means algorithm grouped these features together: LANG\_ALL\_ALPHA, WT\_NO\_DELAY, and HIST\_REP\_COUNTRY.
\removeText{
\begin{enumerate}
\item LANG\_ALL\_ALPHA;
\item WT\_NO\_DELAY;
\item HIST\_REP\_COUNTRY.
\end{enumerate}
}}
\section{Related Work}
\label{sec:literatureReview}
\done{BEFORE SUBMISSION: fix the right way to cite authors depending upon the manner they are present in the text.}
In this section, we present the most relevant works related to that reported in this paper. \change{Specifically, we reference the relevant literature that uses ML solutions to learn the meaning of a violation, then use that to detect violations in online communities.}\removeText{Specifically, we reference the literature relevant to the area of norm violation detection in online communities, with applications of Machine Learning (ML) solutions.}\removeText{We are interested in any sort of online communities, considering from question-answer websites to social medias.} In addition to the specific works presented below, it is also worth to mention a survey that studies a variety of research in the area, focusing on norm violation detection in the domains of hate speech and cyberbullying \cite{al2019detection}.
Also investigating norm violation in Wikipedia but using the dataset from the comments on talk page edits, Anand and Eswari~\cite{anandDeepLearningViolation2019} present a Deep Learning (DL) approach to classify a comment as abusive or not.\removeText{Besides, the proposed model also further classifies the violation in more specific terms, like toxic, obscene and identity hate.} Although the use of DL is an interesting approach to norm violation detection, we focus on offering interpretability, i.e., providing \change{features our model found as relevant} for the detection of norm violation. While the DL model in~\cite{anandDeepLearningViolation2019} does not provide such information.
The work by Cherian et al. \cite{cheriyan2020norm} explores norm violation on the Stack Overflow (SO) community. This violation is studied by analyzing the comments posted on the site, which can contain hate speech and abusive language. The authors state that the SO community could become less toxic by identifying and minimizing this kind of behavior, which they separate in two main groups: generic norms and SO specific norms.\removeText{Besides, \cite{cheriyan2020norm} proposes the idea of a recommendation system, responsible for informing the person violating the norm about rephrasing options for the comment being posted.} There are two important similarities between our works: 1) both studies use labeled dataset from the community, considering the relevant context; and 2) the norm violation detection workflow. The main difference is that we focus on the interpretation of the reasons that indicate a norm violation as detected by our model, providing information to the user so they can decide which specific features they are changing. This is possible because we are mapping the actions into features, while Cheriyan at al. \cite{cheriyan2020norm} work directly with the text from the comments, which allows them to focus on providing text alternatives to how the user should write their comment.
Chandrasekaran et al. \cite{chandrasekharan2019crossmod} build a system for comment moderation in Reddit, named Crossmod.\removeText{The authors work on top of previous work \cite{chandrasekharan2019hybrid}.} Crossmod is described as a sociotechnical moderation system designed using participatory methods (interview with Reddit moderators).\removeText{The system is intended to help moderating comments posted on a Reddit community (subreddit).} To detect norm violation, Crossmod uses a ML back-end, formed by an ensemble of classifiers\removeText{, specifically Crossmod has 108 classifiers}.\removeText{From these, 100 were trained in specific subreddits and 8 trained to detect general norm violation on Reddit.} Since there is an ensemble of classifiers, the ML back-end was trained using the concept of cross-community learning, which uses data from different communities to detect violation in a specific target community.\removeText{\cite{chandrasekharan2019crossmod} assumes that the moderation task depends on context, thus the main goal is not to automatically remove comments that are classified as violation, but rather give this information to moderators so they can decide the action to take.} Like our work, Crossmod uses labeled data from the community to train the classifiers and the norm violation detection workflow follows the same pattern. However, different from our approach, Chandrasekaran et al. \cite{chandrasekharan2019crossmod} use textual data directly, not mapping to features. Besides, Crossmod do not provide to the user information on the parts of the action that triggered the violation classifier. \removeText{so although they have cross-community learning, the classifiers focus on textual tasks.} \removeText{The focus on working directly with textual data differs. This differs from our approach of using the features of the actions, since not working directly with text gives our framework the flexibility to be deployed in communities with different kind of task. Besides, Crossmod do not provide to the user information on the parts of the action that triggered the violation classifier.}
Considering another type of ML algorithm, Di Capua et al. \cite{capuaBullying2016} build a solution based on Natural Language Processing (NLP) and Self-Oganizing Map (SOM) to automatically detect bullying behavior on social networks.\removeText{Although the solution was fine-tuned to work on Twitter, the approach was also applied to Youtube and Formspring.} The authors decided to use an unsupervised learning algorithm because they wanted to avoid the manual work of labeling the data, the assumption is that the dataset is huge and by avoiding manual labelling, they would also avoid imposing a priori bias about the possible classes. This differs from our assumptions since we regard the data/feedback from the community as the basis to deal with norm violation.
One interesting aspect about these studies is that they are either in the realm of hate speech or cyberbullying, which can be understood as a sub-group of norm violation by formalizing hate speech and cyberbullying in terms of norms that a community should adhere to. Researchers are interested in these fields mainly due to the damage that violating these norms can cause in the members of an online community, and due to the available data to study these communities.
\section{Conclusion and Future Work}
\label{sec:conclusion}
The proposed framework, combining machine learning (Logistic Model Trees and K-Means) and taxonomy exploration, is an initial approach on how to detect norm violations. In this paper, we focused on the issue of norm violation assuming violations may occur due to misunderstandings of norms originated by the diverse ways people interpret norms in an online community. To study norm violation, our work used a dataset from Wikipedia's vandalism edition, which contains data about Wikipedia article edits that were considered vandalism.
The framework described in this work is a first step towards detecting vandalism, and it provides relevant information about the problems (features) of the action that led to vandalism. Further investigation is still needed to get a measure of how our system would improve the interactions in an online community. The experiments conducted in our work show that our ML model has a precision of 78,1\% and a recall of 63,8\% when classifying data describing vandalism.
Future work is going to focus on the use of feedback from the community members to continuously train our ML model, as explained in Section~\ref{sec:framework}. The idea is to apply an online training approach to our framework, so when a community behavior changes, that would be taken to indicate a new view on the rules defining the norm, and our ML model should adapt to this new view.
Throughout this investigation, we have noticed that the literature mostly deals with norm violation that focus either on hate speech or cyberbullying. We aim that our approach can be applied to other domains (not only textual), thus we are planning to explore domains with different actions to analyze how our framework deals with a different context (since these domains would have a different set of actions to be executed in an online community)
\done{
This an idea for future work, but I don't think it fits this article now.
Another future direct that we plan to follow is to provide the problematic editions with personalized suggestion. Considering that we have access to profile information, we can then group the editions by certain profile groups, thus offering suggestions to different groups, rather than to the whole community. This is particularly interesting because the users may act in different ways and perhaps this way of behavior is not shared in a community level, only in a more profile specific level. Thus, when this group is interacting, the norm can be flexible, not being enforced as the general norm of the community.}
\section*{Acknowledgements}
This research has received funding from the European Union's Horizon 2020 FET Proactive project ``WeNet – The Internet of us'', grant agreement No 823783, as well as the RecerCaixa 2017 funded ``AppPhil'' project.
\bibliographystyle{splncs04}
\section{First Section}
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\section{Introduction}
\label{sec:introduction}
The aligned understanding of a norm is an essential process for the interaction between different agents (human or artificial) in normative systems. Mainly because these systems take into consideration norms as the basis to specify and regulate the relevant behavior of the interacting agents \cite{jones1993characterisation}. This is especially important when we consider online communities in which different people with diverse profiles are easily connected with each other. In these cases, misunderstandings about the community norms may lead to interactions being unsuccessful. Thus, the goals of this research are: 1) to investigate the challenges associated with detecting when a norm is being violated by a certain member, usually due to a misunderstanding of the norm; and 2) to inform this member about the features of their action that triggered the violation, allowing the member to change their action to be in accordance with the understanding of the community, thus helping the interactions to keep running smoothly. To tackle these goals, our main contribution is to provide a framework capable of detecting norm violation and informing the violator of why their action triggered a violation detection.
The proposed framework is using data, that belongs to a specific community, to train a Machine Learning (ML) model that can detect norm violation. We chose this approach based on studies showing that the definition of what is norm violation can be highly contextual, thus it is necessary to consider what a certain community defines as norm violation or expected behavior \cite{al2019detection,chandrasekharan2019crossmod,van2003DeonticLogic}.
To investigate norm violations, this work is specifically interested in norms that govern online interactions, and we use the Wikipedia community as a testbed, focusing on the article editing actions. This area of research is not only important due to the high volume of interactions that happen on Wikipedia, but also for the proper inclusion and treatment of diverse people in these online interactions. For instance, studies show that, when a system fails to detect norm violations (e.g., hate speech or gender, sexual and racial discrimination), the interactions are damaged, thus impacting the way people interact in the community~\cite{KishonnaDiversity2018,mclean2019female}.
Previous works have dealt with norms and normative systems, proposing mechanisms for norm conflict detection \cite{NormConflict2018}, norm synthesis \cite{morales2018off}, norm violation on Wikipedia \cite{west2011multilingual,anandDeepLearningViolation2019} and other online communities, such as Stack Overflow \cite{cheriyan2020norm} and Reddit \cite{chandrasekharan2019crossmod}.\removeText{and Twitter \cite{kwokRacism2013}} However, our approach differs mainly in three points: 1) implementing an ML model that allows for the interpretation of the reasons leading to detecting norm violation; 2) incorporating a taxonomy to better explain to the violator which features of their actions triggered the norm violation, based on the results provided by our ML model; and 3) codifying actions in order to represent them through a set of features \change{acquired from previous knowledge about the domain}, which is necessary for the above two points\removeText{(the learning of the ML model and the taxonomy of action features)}. Concerning the last point, we note that our framework does not consider the action as is, but a representation of that action in terms of features and the relation of those features to norm violation \change{(as learned by the applied ML model)}. For the Wikipedia case, we represent the action of editing articles based on the categorization introduced in~\cite{west2011multilingual}, with features such as: the measure of profane words, the measure of pronouns, and the measure of Wiki syntax/markup (the details of these are later presented in Section \ref{subSec:wikipediaTaxonomy}).
To build our proposed framework, this work investigates the combination of two main algorithms: 1) the Logistic Model Tree, the algorithm responsible for classifying an article edit as a violation or not;\removeText{The dataset used for training is the vandalism on Wikipedia editions corpus} and 2) the K-Means, the clustering algorithm responsible for grouping the features that are most relevant for detecting a violation. The information about the relevant features is then used to navigate the taxonomy and get a simplified taxonomy of these relevant features.
Our experiments describe how the ML model was built based on the training data provided by Wikipedia, and the results of applying this model to the task of vandalism detection in Wikipedia's article edits illustrate how our approach can reach a precision of 78,1\% and a recall of 63,8\%. Besides, the results also show that our framework can provide information about the specific group of features that affect the probability of an action being considered a violation, and we make use of this information to provide feedback to the user on their actions.
The remainder of this paper is divided as follows. Section~\ref{sec:background} presents the basic mechanisms used by our proposed framework. Section~\ref{sec:framework} describes our framework, while Section~\ref{sec:useCase} presents its application to the Wikipedia edits use case, and Section~\ref{sec:experimentsResults} presents our experiment and its results. The related literature is presented in Section~\ref{sec:literatureReview}. We then describe our conclusions and future work in Section~\ref{sec:conclusion}.
\section{Background}
\label{sec:background}
This section aims to present the base concepts upon which this work is built. We first start with the description of the taxonomy\removeText{(Section~\ref{subSec:taxonomy})}, which we intend to use to formalize a community's knowledge about the features of the actions. Next, we describe the ML algorithms applied to build our framework. First, the Logistic Model Tree (LMT) algorithm\removeText{(Section~\ref{subSec:LMT})}, which is used to build the model responsible for detecting possible vandalism; and second, the K-Means algorithm\removeText{(Section~\ref{subSec:kmeans})}, responsible for grouping the features of the action that are most relevant for detecting violation.
\subsection{Taxonomy for Action Representation}
\label{subSec:taxonomy}
In the context of our work, an action (executed by a user in an online community) is represented by a set of features. Each of these features describes one aspect of the action being executed, i.e., the composing parts of the action. The goal of adopting this approach is to equip our system with an adaptive aspect, since by modelling an action as a set of features allows the system to deal with different kinds of actions (in different domains). For example, we could map the action of participating in an online meeting by features, such as: amount of time present in the meeting; volume of message exchange; and rate of interaction with other participants. Besides, in the context of norm violation, the proposed approach can use these features to explain which aspects of an action were problematic.
Defining an action through its features gives information about different aspects of the action that might have triggered a violation. However, it is still necessary to find a way to present this information to the violators. The idea is that this information must be provided in a human-readable way, allowing the users to understand what that feature means and how different features are related to each other. With these requirements in mind, we propose the use of a taxonomy to present this data.\removeText{A taxonomy is a way to classify similar concepts into groups, which has been used in different areas, specially in Biology~\cite{kirk96Taxonomy}.} This classification scheme provides relevant information about concepts of a complex domain in a structured way~\cite{kirk96Taxonomy}, thus handling the requirements of our solution.
We note that, in this work, the focus is not on {\it building} a taxonomy of features. Instead, we assume that the taxonomy is provided with their associated norms. Our system uses this taxonomy, navigating it to select the relevant features. The violator is then informed about the features (presented as a subsection of the larger taxonomy) that triggered the violation \change{detected by our model}.
\subsection{Logistic Model Tree}
\label{subSec:LMT}
With respect to the domain of detecting norm violations in online communities, interpreting the ML model is an important aspect to consider.\removeText{is an important aspect to consider. This need encourages the creation of a solution that offers a tool capable of explaining, to the members of a community, the reasons behind the model's decisions.} Thus, if a community is interested in providing the violator with information about the features of their action that are indicative of violation, then the proposed solution needs to work with a model that can correctly identify these problematic features.
\removeText{In ML, there are different approaches to create a model, }In this work, we are interested in\removeText{the concept of} supervised learning, which is the ML task of finding a map between the input and the output.\removeText{To do that, the model must be trained with labeled data, containing examples of input-output pairs~\cite{russell2002artificial}.} Several algorithms exist that implement the concepts of supervised learning, e.g., artificial neural networks and tree induction. We are most concerned with the ability of these algorithms to generate interpretable outputs, i.e., how the model explains the reasons for taking a certain decision. As such, the algorithm we chose that contains this characteristic is the tree induction algorithm
\removeText{\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figs/DT-PlayTenis.png}
\caption{An example of a Decision Tree, deciding to play tennis or not \cite{russell2002artificial}.}
\label{fig:dtPlayTenis}
\end{center}
\end{figure}}
The ability to interpret the tree induction model is provided by the way a path is defined in this technique (basically a set of {\em if-then} statements), which allows our model to find patterns in the data, present the path followed by the model and consequently provide the reasons that lead to that conclusion.
Although induction trees have been a popular approach to solve classification problems\removeText{ for a long time\cite{safavian1991survey,ghiasiDT2020}}, this algorithm also presents some disadvantages. This has prompted Landwehr et al.~\cite{landwehr2005logistic} to propose the Logistic Model Tree (LMT) algorithm, which adds logistic regression functions at the leaves of the tree.
In logistic regression, there are two types of variables: the independent and the dependent variables. The goal is to find a model able\removeText{to map the relationship between the independent and dependent variables, i.e., a function that} to describe the effects of the independent variables on the dependent ones. In our context, the output of the model is\removeText{in the interval [0,1], which is useful to indicate the probability that an event is going to happen, or in our context,} responsible for predicting the probability of an action being classified as norm violation.
Dealing with odds is an interesting \change{aspect} present in logistic regression, since the increase in a certain variable indicates how the odds changes for the classification output, in this case the odds indicate the effect of the independent variables on the dependent ones. Besides, another important aspect is the equivalence of the natural log of the odds ratio and the linear function of the independent variables, represented by equation \ref{eq:lnOdds}:
\begin{eqnarray}
\label{eq:lnOdds}
ln (\frac{p}{1-p}) & \gets & \beta_0 + \beta_1 x_1
\end{eqnarray}
where \(ln\) is the logarithm of the odds ratio, \(p\) [0,1] is\removeText{a value between 0 and 1,} the probability of an event occurring. \(\beta\) represents the parameters of the model, in our case the \change{weights for} features of the action. After calculating the natural logarithm, we can then use the inverse of the function to get our estimated regression equation:
\begin{eqnarray}
\label{eq:regressionEstimation}
\hat p & \gets \frac{\epsilon^{\beta_0 + \beta_1 x_1}}{1+\epsilon^{\beta_0 + \beta_1 x_1}}
\end{eqnarray}
where $\hat p$ is the probability estimated by the regression model.
With these characteristics of logistic regression, we can see how this technique can be used to highlight attributes (independent variables) that have \change{relevant} influence over the output of the classifier probability.
Landwehr et al.~\cite{landwehr2005logistic} demonstrate how neither of the two algorithms described above (Tree Induction and Logistic Regression) is better than the other.\removeText{One conclusion presented by~\cite{landwehr2005logistic} is that the performance of the methods depends upon the domain of investigation and the number of training data available. So, to tackle these issues, the LMT algorithm partitions the instance space in regions of constant class with estimations of the probabilities, while having a more gradual change, provided by the estimations from the logistic regression model.} Thus, to tackle the issues present in these two algorithms, LMT adds to the leaves of the tree a logistic regression function.
Figure~\ref{fig:lmtExample} presents the description of a tree generated by the LMT algorithm. With a similar process as the standard decision tree, the LMT algorithm obtains a probability estimation as follow: first, the feature is compared to the value associated with that node\removeText{(if the node contains a nominal attribute, with \(k\) values, then this node has \(k\) child nodes. But, if the value is numerical, then the feature is compared to a threshold, going left in the tree if the value is below the threshold and going right otherwise)}. This step is repeated until the algorithm has reached a leaf node, when the exploration is completed. Then the logistic regression function determines the probabilities for the class, as described by equation~\ref{eq:regressionEstimation}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\columnwidth]{figs/LMTExampleTree.png}
\caption{An example of a tree built by the LMT algorithm\cite{landwehr2005logistic}. \change{X\(_1\) and X\(_2\) are features present in the dataset. F\(_1\) and F\(_2\) are the equations found by the logistic regression model, describing the weights for each feature present in the training dataset.}}
\label{fig:lmtExample}
\end{center}
\end{figure}
\subsection{K-Means Clustering Method}
\label{subSec:kmeans}
K-Means is a clustering algorithm with the goal of finding a number \(K\) of clusters in the observed data, attempting to group the most `similar' data points together. This algorithm has been used successfully in different applications, such as\removeText{network intrusion detection~\cite{jianliangIntrusionKMeans2009} and} feature learning~\cite{RubaiatFeaturesKmeans2018} and computer vision~\cite{zheng2018image}. To achieve this goal, K-Means clusters the data using the nearest mean to the cluster center (calculating the squared Euclidean distance), thus reducing the variance within the group~\cite{rai2010survey}.
In this work, the K-Means algorithm can be used to group the features that may indicate an action as violation \change{(we use the features' weights multiplied by their input values as indication of relevance for the classification probability)}. First, after detecting a possible violation, the ML model provides the K-Means algorithm with the set of features present in the logistic regression and their associated \change{values (the input multiplied by the weight)}. Then, based on the values of these features, the algorithm is responsible for separating the features in two groups: 1) those \change{that our model found with highest values, i.g., the most relevant for the vandalism classification}; and 2) those \change{with the lowest values, e.g., less relevant for the vandalism classification}. Lastly, the output of K-Means informs the framework which are the most relevant features for detecting violations (i.e. the first group), which the framework can then use to navigate the taxonomy and present a selected simplified taxonomy of relevant features to the violator.
\section{Framework for Norm Violation Detection (FNVD)}
\label{sec:framework}
This section presents the main contribution of our work, the framework for norm violation detection (FNVD). The goal of this framework is to be deployed in a normative system so that when a violation is detected, the system can enforce the norms by, say, prohibiting the action.
The main component of our framework is the machine learning (ML) algorithm behind the detection of norm violations, specifically the LMT algorithm of Section~\ref{subSec:LMT}.
An important aspect to take into consideration, when using this algorithm, is the data needed to train the model. In \removeText{the context of }our work, the community must provide the definitions \removeText{and constitutions }of norm violations through a dataset that exemplifies actions that were previously labeled as norm violations.\removeText{Different communities use different methods to build this dataset, ranging from using only feedback from the community itself (e.g., from the moderators of the community) to hiring a service to classify the data (such as the Mechanical Turk).} Thus, here we are using data provided by Wikipedia, gathered using Mechanical Turk~\cite{Potthast2010OverviewOT}.
After defining the data source, our proposed approach essentially 1) collects the data \removeText{that will be }used to train the LMT model; 2) trains the LMT model to detect possible violations and to learn the action's features relevant to norm violations; and 3) when violations are detected, according to the LMT model's results, then the action responsible for the violation is rejected and the violator is informed about the features of their action \change{that triggered the model output}. Furthermore, in both cases (when actions are labelled as violating norms or not), we suggest that the framework collects feedback from the members of the community, which can then be used as new data to retrain the ML model. This is important as we strongly believe that communities and their members evolve, and what may be considered a norm violation today might not be in the future. For example, imagine a norm that states that hate speech is not allowed. Agreeing on the features of hate speech may change from one group of people to another and may also change over time. Consider the evaluation of the N-word, which is usually seen as a serious racial offense and can automatically be considered a text that violates the ``no hate speech" norm. However, imagine a community of African Americans frequently saluting each other with the phrase ``Wussup nigga'' and the ML model classifying their text as hate speech. Clearly, human communities do not always have one clear definition of concepts like hate speech, violation, freedom of speech, etc. The framework, as such, must have a mechanism to adapt to the views of the members of its community, as well as adapt to the views that may change over time. While we leave the adaptation part for future work, we highlight its need in this section, and prepare the framework to deal with such adaptions, as we illustrate in Figure~\ref{fig:frameworkDiagram}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.0\columnwidth]{figs/NormChange.png}
\caption{How the framework works when deployed in an online community.}
\label{fig:frameworkDiagram}
\end{center}
\end{figure}
To further clarify how our framework would act to detect a norm violation when deployed in a community, it is essential to explore the diagram in Figure~\ref{fig:frameworkDiagram}. Step~0 represents the training process of the LMT model, which is a fundamental part of our approach because it is in this moment that the rules for norm violation are specified. Basically, after training the model, our framework would have identified a set of rules that describe norm violation. We can portray these rules as a conjunction of two elements: 1) the tree that is built by the LMT algorithm on top of the collected data; and 2) the weights presented in the leaves of the tree. These weights are the parameters of the estimated regression equation that defines the probability of norm violation (depicted in Equation \ref{eq:regressionEstimation}). With the trained LMT model, the system starts monitoring every new action performed in the community (Step~1). In Step~2, the system maps the action to features that the community defined as descriptive of that action, which triggers the LMT model to start working to detect if that action is violating (or not) any of the norms. Step~3 presents the two different paths that can be executed by our system. If the action is detected as violating a norm (Condition~1), then we argue that the system must execute a sequence of steps to guarantee that the community norms are not violated: 1) the system does not allow the action to persist (i.e., action is not executed); 2) the system presents to the user information about which action features were the most relevant for our model to detect the norm violation, and the taxonomy of the relevant features is presented\removeText{. In the case of our work, these features can be seen as the composing parts of the actions, i.e., we are separating the actions into features (which are used to train our machine learning model, details in Section \ref{subSec:taxonomy})}; 3) the action is logged by the system, allowing other community members to give feedback about the edit attempt, thus providing the possibility of these members flagging the action as a non-violation. The feedback collected from the users can later be used to continuously train (Step~0) our LMT model (future work). However, if the executed action is not detected as violating a norm (Condition~2), then the system can proceed as follows. The action persists in the system (i.e., action is executed), and since any model may incorrectly classify some norm violation as non-violation, the system allows the members of the community to give feedback about that action, providing the possibility of flagging an already accepted action as a violation. Getting people's feedback on violations that go unnoticed by the model is a way to allow the system to adapt to new data (people's feedback) and update \change{the definitions of norm violations} by continuously training the LMT model (Step~0).
To obtain the relevant features for the norm violation classification (Condition~1), we use the K-Means algorithm\removeText{(Section \ref{subSec:kmeans})}. In our context, due to the estimated logistic regression equation, the LMT model provides the weights for each feature \change{multiplied by the value of these features for the action}. \change{This indicates} the influence of the features on the model's output (i.e., the probability of an action being classified as norm violation). With the weights \change{and specific values for the features}, \change{the K-Means algorithm can group the set of features that present the highest multiplied values, which are the ones we assume that contribute the most for the probability of norm violation}. Then, by searching the taxonomy using the group of relevant features, our system can provide the taxonomy structure of the features that trigger norm violation, this is useful due to the explanatory and interpretation characteristics of a taxonomy. The aim of providing this information is to clarify to the member of the community performing the action, what are the problematic aspects of their action \change{as learned by our model}.
\section{The Wikipedia Vandalism Detection Use Case}
\label{sec:useCase}
We focus on the problem of detecting vandalism in Wikipedia article edits. This use case is interesting because Wikipedia is an online community where norms such as `no vandalism' may have different interpretations by different people\removeText{, and norm violations can affect community experience}. In what follows, we first present the use case's domain, followed by the taxonomy used by our system, and finally, an illustration of how our proposed framework may be applied to this use case.
\subsection{Domain}
\label{subSec:domain}
Wikipedia~\cite{wiki}\removeText{\footnote{\url{https://en.wikipedia.org}}} is an online encyclopedia, containing articles about different subjects in any area of knowledge. It is free to read and edit, thus any person with a device connected to the internet can access it and edit\removeText{the content of} its articles. Due to the openness and collaborative structure of Wikipedia, the system is subject to different interpretations of what is the community's expectation concerning how content should be edited. To help address this issue, Wikipedia has compiled a set of rules, the Wikipedia norms~\cite{Potthast2010OverviewOT}, to maintain and organize its content.
Since we are looking for an automated solution for detecting norm violations by applying machine learning mechanisms, the availability of data becomes crucial. Wikipedia provides data on what edits are marked as vandalism, where vandalism annotations were provided by Amazon's Mechanical Turk. Basically, every article edit was evaluated by at least three people (Mechanical Turks) that decided whether the edit violates the `no vandalism' norm or not. In the context of our work, the actions performed by the members of the community are the Wikipedia users' attempts to edit articles, and the norm is {\em ``Do not engage in vandalism behavior"} (which we\removeText{sometimes} refer to as the `no vandalism' norm). It is this precise dataset that we have used to train the model that detects norm violations. We present an example of what is considered a vandalism in a Wikipedia article edit, where a user edited an article by adding the following text: {\em ``Bugger all in the subject of health ect."}
\subsection{Taxonomy Associated with Wikipedia's `No Vandalism' Norm}
\label{subSec:wikipediaTaxonomy}
An important step in our work is to map actions to features and then specify how they are linked to each other.\removeText{As illustrated earlier, we assume a taxonomy of these features are already provided with the norm. In our Wikipedia use case, however, we had to manually create this taxonomy.} We manually created a taxonomy to describe these features \change{ by separating them in categories that describe their relation with the action.\footnote{For the complete taxonomy, the reader can refer to \url{https://bit.ly/3sQFhQz}}} \change{In this work, we consider the 58 features described in~\cite{west2011multilingual} and 3 more that were available in the provided dataset: LANG\_MARKUP\_IMPACT, the measure of the addition/removal of Wiki syntax/markup; LANG\_EN\_PROFANE\_BIG and LANG\_EN\allowbreak\_PROFANE\_BIG\_IMPACT, the measure of addition/removal of English profane words. In the dataset, features ending with \_IMPACT are normalized by the difference of the article size after edition.} The main objective of this taxonomy is to help our system present to the violator an easy-to-read explanation of the reasons why their article edit was marked as violating a norm by our model, \change{specifying the features with highest influence to trigger this violation.}
To further explain our taxonomy approach, we present in Figure~\ref{fig:vandalismDetectionTaxonomy} the constructed taxonomy for Wikipedia's `no vandalism' norm. We observe that features can be divided in four main groups. The first is user's direct actions, which represent aspects of the user's article editing action, e.g., adding a text. This group is further divided in four sub-groups: a) written edition, which contains features about the text itself that is being edited by the user; b) comment on the edition, which contains features about the comments that users have left on that edition; c) article after edition, which contains features about how the edited article changed after the edition was completed; and d) time of edition, which contains features about the time when the user made their edition. The second group is the user's profile, general information about the user. The third is the page's history,\removeText{information about the page that the user is editing, e.g., } how the article changed with past editions. The last group is reversions, which is essentially information on past reversions.\footnote{A reversion is when an article is reverted back to a version before the vandalism occurred.} \change{In total, these groups have 61 features, but due to simplification purpose, Table~\ref{tab:ExampleFeatures} only presents a subset of those features. \removeText{that are part of these groups:}}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.8\columnwidth]{figs/SimplifiedGroupsTaxonomy.png}
\caption{Taxonomy associated with Wikipedia's `no vandalism' norm.}
\label{fig:vandalismDetectionTaxonomy}
\end{center}
\end{figure}
\begin{table}[h]
\scriptsize
\caption{Example of Features present in the taxonomy groups.}\label{tab:ExampleFeatures}
\centering
\begin{tabular}{l|llll}
\hline
\multicolumn{1}{c|}{{Group}} & {{Features}} \\
\hline
\textbf{Written Edition} & LANG\_ALL\_ALPHA; LANG\_EN\_PRONOUN\\
\textbf{Comment on Edition} & COMM\_LEN; COMM\_LEN\_NO\_SECT\\
\textbf{Article After Edition} & SIZE\_CHANGE\_RESULT; SIZE\_CHANGE\_CHARS\\
\textbf{Time of Edition} & TIME\_TOD; TIME\_DOW\\
\textbf{User's Profile} & HIST\_REP\_COUNTRY; USER\_EDITS\_DENSITY\\
\textbf{Page's History} & PAGE\_AGE; WT\_NO\_DELAY\\
\textbf{Reversions} & HASH\_REVERTED; HASH\_IP\_REVERT\\
\hline
\end{tabular}
\end{table}
\subsection{FNVD Applied to Wikipedia Vandalism Detection}
\label{subSec:frameworkWikipediaVandalism}
It this section, we first describe an example of how our framework can be configured to be deployed in the Wikipedia community. First, the community provides the features and the taxonomy describing that feature space (see Figure~\ref{fig:vandalismDetectionTaxonomy}).\removeText{This information is essential because the idea of the framework is not only to detect that an edition might be a vandalism, but also inform violators about the features that have contributed to the model classifying their edit as vandalism.} Then,\removeText{with this set of features and its taxonomy,} our framework trains the LMT model to classify norm violations based on the data provided (Step~0 of Figure~\ref{fig:frameworkDiagram}), which must contain examples of what that community understands as norm violation and regular behavior.
In the context of vandalism detection on Wikipedia, the relevant actions performed by the members of the community are the attempts to edit Wikipedia articles. Following the diagram in Figure~\ref{fig:frameworkDiagram}, when a user attempts to edit an article (Step~1), our system will analyze this edit. We note here that our proposed LMT model does not work with the action itself, but the features that describe it. As such, it is necessary to first find the features that represent the performed action\removeText{(the Wikipedia article edit)}. Thus, in Step~2\removeText{of Figure \ref{fig:frameworkDiagram}}, there is a pre-processing phase responsible for mapping actions to the features associated to the norm in question. For example, \change{an article about Asteroid was edited with the addition of the text {\em `` i like man!!"}. After getting this edition text, the system can compute the values (as described in \cite{west2011multilingual}) for the 61 features, which are used to calculate the vandalism probability}. \change{For brevity reasons, we only show the values for some of these features}:
\begin{enumerate}
\item LANG\_ALL\_ALPHA, the percentage of text which is alphabetic: 0,615385;
\item WT\_NO\_DELAY, the calculated WikiTrust score: 0,731452;
\item HIST\_REP\_COUNTRY, measure of how users from the same country as the editor behaved: 0,155146.
\end{enumerate}
\removeText{For that, first we have to go through the path of the tree, to determine the logistic regression equation that we are using to calculate the probability of vandalism, }
\change{After calculating the values for all features,} the LMT model can evaluate if this article edit is considered `vandalism' or not. In the case of detecting vandalism (Condition~1 of Figure~\ref{fig:frameworkDiagram}), the system does not allow the edition to be recorded on the Wikipedia article, and it presents to the violator two inputs. The first is the set of features of their edit that have the highest influence on the model's decision to detect the vandalism. \change{To get this set, after calculating the probability of vandalism \removeText{with the logistic regression equation }(as depicted in Equation~\ref{eq:regressionEstimation}), the LMT model provides the features that present a positive relationship with the \change{output}. These `positive features' are then used by K-Means to create the group with the most relevant ones (Table~\ref{tab:ExamplePositiveFeatures} presents an example of this process)}. The second input is the selected part of the taxonomy related to chosen set of features, providing further explanation of those features that triggered the norm violation. Additionally, the system will log the attempt to edit the article, which eventually may trigger feedback collection that can at a later stage be used to retrain our model.
\begin{table}[h]
\caption{\change{List of features that positively affects the probability of vandalism detection. Total Value is the multiplication between the feature's values and the features' weights\removeText{ (defined by the LMT model)}. The most relevant features, as found by K-Means, are marked with an (*).}}
\label{tab:ExamplePositiveFeatures}
\centering
\begin{tabular}{l|llll}
\hline
\multicolumn{1}{c|}{{Features}} & {{Total Value}} \\
\hline
\textbf{WT\_NO\_DELAY*} & 1.08254896\\
\textbf{HIST\_REP\_COUNTRY*} & 0.899847\\
\textbf{LANG\_ALL\_ALPHA*} & 0.7261543\\
\textbf{HASH\_REC\_DIVERSITY} & 0.15714292\\
\textbf{WT\_DELAYED} & 0.12748878\\
\textbf{LANG\_ALL\_CHAR\_REP} & 0.12\\
\textbf{HIST\_REP\_ARTICLE} & 0.093548\\
\hline
\end{tabular}
\end{table}
\removeText{Since our proposed approach navigates the taxonomy to get information about the features that are relevant for norm violation detection, an example demonstrating how this taxonomy is used in this case is interesting for clarification purposes. }\change{The features \(WT\_NO\_DELAY\), \(HIST\_REP\_COUNTRY\) and \(LANG\_ALL\allowbreak\_ALPHA\) were indicated by K-Means as the most relevant for the classification of vandalism}. With this information, our framework can search the taxonomy for the relevant features and then automatically retrieve the simplified taxonomy structure for these three specific features, as shown in Figure \ref{fig:featuresVandalismTaxonomy}.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.6\columnwidth]{figs/PartTaxonomyKMeans.png}
\caption{Taxonomy for part of the features that were \change{most relevant for the vandalism classification. These features are then presented to the user with a descriptive text.}}
\label{fig:featuresVandalismTaxonomy}
\end{center}
\end{figure}
\removeText{Besides, a description text is also presented to the violator, explaining what that feature represents.}
However, in case the system classifies the article edit as `non-vandalism' (Condition~2 of Figure~\ref{fig:frameworkDiagram}), the Wikipedia article is updated according to the user's article edit and community members may provide feedback on this new article edit, which may later be used to retrain our model (as explained in Section~\ref{sec:framework}).
\section{Experiments and Results}
\label{sec:experimentsResults}
The goal of this section is to describe how the proposed approach was applied for detecting norm violation in the domain of Wikipedia article edits, with an initial attempt to improve the interactions in online communities. Then, we demonstrate and discuss the results achieved.
\subsection{Experiments}
\label{subSec:experiments}
\change{Data on vandalism detection in Wikipedia articles~\cite{west2011multilingual} were used for the experiments.}\removeText{Our experiments were conducted with data on vandalism detection in Wikipedia articles~\cite{west2011multilingual}}\removeText{provided by a competition the Wikipedia community organized to evaluate how different models deals with the automatization process of vandalism detection \cite{Potthast2010OverviewOT}} This dataset has 61 features and 32,439 instances for training (with 2,394 examples of vandalism editions and 30,045 examples of regular editions). The model was \change{trained with WEKA~\cite{weka} and} evaluated using 10 folds cross-validation.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=\columnwidth]{figs/TreeModelVandalismDetection.png}
\caption{The built model for the vandalism detection, using Logistic Model Tree.}
\label{fig:LMTModelForVandalismDetection}
\end{center}
\end{figure}
\removeText{We built the model showed in Figure \ref{fig:LMTModelForVandalismDetection}, this was achieved by using the WEKA tool~\cite{weka}.\removeText{\footnote{\url{https://www.cs.waikato.ac.nz/ml/weka/}}} As described in Section \ref{subSec:LMT}, the leaves in the tree represent the estimated logistic equations that will be used to calculate the probability of a certain edition being classified as vandalism. In these equations, we have different \change{weight values} for each feature of our data.\footnote{Trained model available at:\url{https://bit.ly/3gBBkwP}}\removeText{, as shown in the list below:}}
\removeText{\begin{table}[h]
\caption{Some of the weights for features in the logistic regression equation.}\label{tab:weightsLMTModel}
\centering
\begin{tabular}{l|ll}
\hline
\multicolumn{1}{c|}{Feature} & \multicolumn{1}{c}{Value} \\
\hline
LANG\_EN\_PROFANE\_BIG\_IMPACT & 40.33\\
LANG\_EN\_PRONOUN\_IMPACT & 26.67\\
LANG\_ALL\_MARKUP\_IMPACT & -29.79\\
\hline
\end{tabular}
\end{table}
}
\removeText{
\begin{enumerate}
\item LANG\_EN\_PROFANE\_BIG\_IMPACT, value: 40,33
\item LANG\_EN\_PRONOUN\_IMPACT, value: 26,67
\item LANG\_ALL\_MARKUP\_IMPACT, value: -29,79
\end{enumerate}
The positive values indicate a positive relationship between the feature and the classification probability, and the negative values otherwise.
}
\subsection{Results}
\label{subSec:results}
The first important information to note is how the LMT model performs when classifying vandalism in Wikipedia editions. In Figure \ref{fig:LMTModelForVandalismDetection}, it is possible to see the model that was built to perform the classification task.\footnote{Trained model available at: \url{https://bit.ly/3gBBkwP}} The tree has four decision nodes and five leaves in total. Since the LMT model uses logistic regression at the leaves, the model has five different estimated logistic regression equations, each of these equations outputs' the probability of an edition being a vandalism.
\removeText{The LMT model performs well to classify the Wikipedia Vandalism dataset, since }The LMT model correctly classifies 96\% of instances in general. However, when we separate the results in two groups, vandalism editions and regular editions, it is possible to observe a difference in the model's performance. For the\removeText{instances that are } regular editions, the LMT model achieves a precision of 97,2\%, and a recall of 98,6\%. While for\removeText{the instances of} vandalism editions, the performance of the model drops, with a precision of 78,1\% and a recall of 63,8\%. This decrease\removeText{in the percentage for vandalism detection } can be explained by how the dataset was separated and the number of vandalism instances, which consequently leads to an unbalanced dataset. In the dataset, the total number of vandalism instances is 2,394 and the other 30,045 instances are of regular editions. A better balance between the number of vandalism editions and regular edition should improve our classifier, thus in the future we are exploring other \change{model configurations (e.g., ensemble models) to handle data imbalance}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.8\columnwidth]{figs/NumberOfOccurencesRelevantFeatures-Smaller.png}
\caption{Number of occurrences of relevant features in vandalism detection.}
\label{fig:GraphNumberOfOccurrences}
\end{center}
\end{figure}
The influence of each feature on determining the probability of a norm violation is provided by the LMT model \change{(as assumed in this work, feature influence is model specific, meaning that a different model can find a different set of relevant features)}.\removeText{In this regard, }\removeText{The proposed framework presents }\removeText{presenting the features that are most relevant when a vandalism is detect by our model.} The graph in Figure~\ref{fig:GraphNumberOfOccurrences} shows the number of times a feature is \change{classified as relevant by the built model}. Some features appear in most of the observations, indicating how important they are to detect\removeText{this kind of} vandalism. Future work shall investigate if this same behavior (some features present in the actions have more influence than other features to define the norm violation probability) can be detected in other domains
\removeText{An interesting characteristic from the Wikipedia Vandalism Detection data-set is }\removeText{We note that the feature}``LANG\_ALL\_ALPHA" recurrently appears as relevant when vandalism is detected. This happens because this feature presents, as estimated by the LMT model, a positive relationship with the norm violation, meaning that when a vandalism edition is detected, this feature is usually relevant for the classification.
\removeText{With the presentation of the relevant results, it is also important to demonstrate how a vandalism is detected by our system\removeText{, describing the relevant features that lead to this classification}. The text edition {\em `` They alos went to parties and drank tequila every saturday!”} is a vandalism edition. In our model, the estimated logistic regression equations calculated the relevant features, and the K-Means algorithm grouped these features together: LANG\_ALL\_ALPHA, WT\_NO\_DELAY, and HIST\_REP\_COUNTRY.
\removeText{
\begin{enumerate}
\item LANG\_ALL\_ALPHA;
\item WT\_NO\_DELAY;
\item HIST\_REP\_COUNTRY.
\end{enumerate}
}}
\section{Related Work}
\label{sec:literatureReview}
\done{BEFORE SUBMISSION: fix the right way to cite authors depending upon the manner they are present in the text.}
In this section, we present the most relevant works related to that reported in this paper. \change{Specifically, we reference the relevant literature that uses ML solutions to learn the meaning of a violation, then use that to detect violations in online communities.}\removeText{Specifically, we reference the literature relevant to the area of norm violation detection in online communities, with applications of Machine Learning (ML) solutions.}\removeText{We are interested in any sort of online communities, considering from question-answer websites to social medias.} In addition to the specific works presented below, it is also worth to mention a survey that studies a variety of research in the area, focusing on norm violation detection in the domains of hate speech and cyberbullying \cite{al2019detection}.
Also investigating norm violation in Wikipedia but using the dataset from the comments on talk page edits, Anand and Eswari~\cite{anandDeepLearningViolation2019} present a Deep Learning (DL) approach to classify a comment as abusive or not.\removeText{Besides, the proposed model also further classifies the violation in more specific terms, like toxic, obscene and identity hate.} Although the use of DL is an interesting approach to norm violation detection, we focus on offering interpretability, i.e., providing \change{features our model found as relevant} for the detection of norm violation. While the DL model in~\cite{anandDeepLearningViolation2019} does not provide such information.
The work by Cherian et al. \cite{cheriyan2020norm} explores norm violation on the Stack Overflow (SO) community. This violation is studied by analyzing the comments posted on the site, which can contain hate speech and abusive language. The authors state that the SO community could become less toxic by identifying and minimizing this kind of behavior, which they separate in two main groups: generic norms and SO specific norms.\removeText{Besides, \cite{cheriyan2020norm} proposes the idea of a recommendation system, responsible for informing the person violating the norm about rephrasing options for the comment being posted.} There are two important similarities between our works: 1) both studies use labeled dataset from the community, considering the relevant context; and 2) the norm violation detection workflow. The main difference is that we focus on the interpretation of the reasons that indicate a norm violation as detected by our model, providing information to the user so they can decide which specific features they are changing. This is possible because we are mapping the actions into features, while Cheriyan at al. \cite{cheriyan2020norm} work directly with the text from the comments, which allows them to focus on providing text alternatives to how the user should write their comment.
Chandrasekaran et al. \cite{chandrasekharan2019crossmod} build a system for comment moderation in Reddit, named Crossmod.\removeText{The authors work on top of previous work \cite{chandrasekharan2019hybrid}.} Crossmod is described as a sociotechnical moderation system designed using participatory methods (interview with Reddit moderators).\removeText{The system is intended to help moderating comments posted on a Reddit community (subreddit).} To detect norm violation, Crossmod uses a ML back-end, formed by an ensemble of classifiers\removeText{, specifically Crossmod has 108 classifiers}.\removeText{From these, 100 were trained in specific subreddits and 8 trained to detect general norm violation on Reddit.} Since there is an ensemble of classifiers, the ML back-end was trained using the concept of cross-community learning, which uses data from different communities to detect violation in a specific target community.\removeText{\cite{chandrasekharan2019crossmod} assumes that the moderation task depends on context, thus the main goal is not to automatically remove comments that are classified as violation, but rather give this information to moderators so they can decide the action to take.} Like our work, Crossmod uses labeled data from the community to train the classifiers and the norm violation detection workflow follows the same pattern. However, different from our approach, Chandrasekaran et al. \cite{chandrasekharan2019crossmod} use textual data directly, not mapping to features. Besides, Crossmod do not provide to the user information on the parts of the action that triggered the violation classifier. \removeText{so although they have cross-community learning, the classifiers focus on textual tasks.} \removeText{The focus on working directly with textual data differs. This differs from our approach of using the features of the actions, since not working directly with text gives our framework the flexibility to be deployed in communities with different kind of task. Besides, Crossmod do not provide to the user information on the parts of the action that triggered the violation classifier.}
Considering another type of ML algorithm, Di Capua et al. \cite{capuaBullying2016} build a solution based on Natural Language Processing (NLP) and Self-Oganizing Map (SOM) to automatically detect bullying behavior on social networks.\removeText{Although the solution was fine-tuned to work on Twitter, the approach was also applied to Youtube and Formspring.} The authors decided to use an unsupervised learning algorithm because they wanted to avoid the manual work of labeling the data, the assumption is that the dataset is huge and by avoiding manual labelling, they would also avoid imposing a priori bias about the possible classes. This differs from our assumptions since we regard the data/feedback from the community as the basis to deal with norm violation.
One interesting aspect about these studies is that they are either in the realm of hate speech or cyberbullying, which can be understood as a sub-group of norm violation by formalizing hate speech and cyberbullying in terms of norms that a community should adhere to. Researchers are interested in these fields mainly due to the damage that violating these norms can cause in the members of an online community, and due to the available data to study these communities.
\section{Conclusion and Future Work}
\label{sec:conclusion}
The proposed framework, combining machine learning (Logistic Model Trees and K-Means) and taxonomy exploration, is an initial approach on how to detect norm violations. In this paper, we focused on the issue of norm violation assuming violations may occur due to misunderstandings of norms originated by the diverse ways people interpret norms in an online community. To study norm violation, our work used a dataset from Wikipedia's vandalism edition, which contains data about Wikipedia article edits that were considered vandalism.
The framework described in this work is a first step towards detecting vandalism, and it provides relevant information about the problems (features) of the action that led to vandalism. Further investigation is still needed to get a measure of how our system would improve the interactions in an online community. The experiments conducted in our work show that our ML model has a precision of 78,1\% and a recall of 63,8\% when classifying data describing vandalism.
Future work is going to focus on the use of feedback from the community members to continuously train our ML model, as explained in Section~\ref{sec:framework}. The idea is to apply an online training approach to our framework, so when a community behavior changes, that would be taken to indicate a new view on the rules defining the norm, and our ML model should adapt to this new view.
Throughout this investigation, we have noticed that the literature mostly deals with norm violation that focus either on hate speech or cyberbullying. We aim that our approach can be applied to other domains (not only textual), thus we are planning to explore domains with different actions to analyze how our framework deals with a different context (since these domains would have a different set of actions to be executed in an online community)
\done{
This an idea for future work, but I don't think it fits this article now.
Another future direct that we plan to follow is to provide the problematic editions with personalized suggestion. Considering that we have access to profile information, we can then group the editions by certain profile groups, thus offering suggestions to different groups, rather than to the whole community. This is particularly interesting because the users may act in different ways and perhaps this way of behavior is not shared in a community level, only in a more profile specific level. Thus, when this group is interacting, the norm can be flexible, not being enforced as the general norm of the community.}
\section*{Acknowledgements}
This research has received funding from the European Union's Horizon 2020 FET Proactive project ``WeNet – The Internet of us'', grant agreement No 823783, as well as the RecerCaixa 2017 funded ``AppPhil'' project.
\bibliographystyle{splncs04}
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2,869,038,154,762 | arxiv | \section{Introduction}
The problem of asymptotic completeness in quantum field theory (QFT) has been a subject of active research
over the last two decades, both on the relativistic \cite{Le08,DT10} and non-relativistic side \cite{Sp97,DG99,FGS04,DK11,GMR11}. However, all the results obtained so
far concern Wigner particles, i.e., excitations with a well-defined mass. The problem of a complete particle
interpretation in the presence of infraparticles, i.e., particles whose mass fluctuates due to the presence of other excitations,
appears to be open to date in all the models considered in the literature. In the present Letter we formulate a natural notion
of asymptotic completeness for two-dimensional massless relativistic QFT which remains meaningful in the presence of infraparticles.
We verify that a large class of chiral conformal field theories, containing theories of infraparticles, satisfies this property.
Since the seminal work of Schroer \cite{Sch63}, infraparticles have remained a prominent topic in mathematical physics.
Their importance relies on the fact that all the electrically charged particles, including the electron, turn out to be infraparticles \cite{Bu86}.
In models of non-relativistic QED scattering states of one electron and photons were successfully constructed by
Fr\"ohlich, Pizzo and Chen in \cite{CFP07}. In a more abstract framework of algebraic QFT a complementary approach to
scattering of infraparticles was proposed by Buchholz, Porrmann and Stein \cite{BPS91}. This theory of \emph{particle weights}
\cite{Po04.1,Po04.2,Dy10} does not aim at
scattering states, but rather provides an algorithm for a direct construction of (inclusive) collision cross-sections. Very recently the
theory of particle weights was applied to conformal field theories by the present authors \cite{DT11}. We found out that any chiral
conformal field theory in a charged, irreducible product representation describes infraparticles. We also checked that in some cases
these infraparticles have superselected velocity, similarly to the electron in QED. However, the question of complete particle interpretation
of these theories was not addressed in \cite{DT11}. We answer this question (affirmatively) in the present work.
This Letter is organized as follows: In Section~\ref{framework} we specify our framework and formulate a generalized concept
of asymptotic completeness (Definition~\ref{Asymptotic-completeness}). We remark that this is an implementation of ideas from
\cite{Bu87} in the setting of two-dimensional massless theories.
In Section~\ref{waves} we recall from \cite{Bu75} the scattering theory of waves, which are counterparts of Wigner particles in this setting. We show that any theory which has complete particle interpretation in the sense of waves satisfies also our generalized property of asymptotic completeness.
After this consistency check, we show in Section~\ref{infraparticles} that any chiral conformal field theory in an irreducible product representation
is asymptotically complete in the generalized sense.
As a corollary, we obtain in Section~\ref{rational} that any completely rational conformal field theory has the property
of generalized asymptotic completeness.
\vspace{0.5cm}
\noindent\bf Acknowledgements. \rm
W.D.\! would like to thank D.\! Buchholz, J.S.\! M\o ller, A.\! Pizzo, M.\!~Porrmann,
W.\! De Roeck and H.\! Spohn for interesting discussions on scattering theory.
A part of this work has been accomplished during the stay of Y.T. at Aarhus University.
He thanks J.S.\! M\o ller for his hospitality.
\section{The generalized concept of asymptotic completeness} \label{framework}
In this section we fix our framework, list the main definitions and facts relevant to
our investigation and formulate the generalized concept of asymptotic completeness.
We start with a variant of the Haag-Kastler postulates \cite{Ha} which we will use in this work:
\begin{definition}\label{two-dim-net}
A local net of von Neumann algebras on $\mathbb{R}^2$ is a pair $({\mfa},U)$ consisting of
a map $\mathcal{O}\mapsto {\mfa}(\mathcal{O})$ from the family of open, bounded regions of $\mathbb{R}^2$ to the family of von Neumann algebras on a Hilbert space
$\mathcal{H}$, and a strongly continuous unitary representation of translations $\mathbb{R}^2\ni \xi\mapsto U(\xi)$ acting on $\mathcal{H}$,
which are subject to the following conditions:
\begin{enumerate}
\item (isotony) If $\mathcal{O}_1 \subset \mathcal{O}_2$, then ${\mfa}(\mathcal{O}_1)\subset {\mfa}(\mathcal{O}_2)$.
\item (locality) If $\mathcal{O}_1 \perp \mathcal{O}_2$, then $[{\mfa}(\mathcal{O}_1),{\mfa}(\mathcal{O}_2)] = 0$, where $\perp$ denotes
spacelike separation.
\item (covariance) $U(\xi){\mfa}(\mathcal{O})U(\xi)^*={\mfa}(\mathcal{O}+\xi)$ for any $\xi\in\mathbb{R}^2$.
\item (positivity of energy) The joint spectrum of $U$ coincides with the closed forward lightcone~$V_+:=\{\,(\omega,\boldsymbol p} %{\mbox{\boldmath$p$})\in\mathbb{R}^2\,|\,\omega\geq |\boldsymbol p} %{\mbox{\boldmath$p$}|\,\}$.
\end{enumerate}
We also introduce the quasilocal $C^*$-algebra of this net ${\mfa}=\overline{\bigcup_{\mathcal{O}\subset\mathbb{R}^2}{\mfa}(\mathcal{O})}$.
\end{definition}
We assume that the spectrum of $U$ coincides with $V_+$ rather than being included in,
because we are interested in the scattering theory of massless particles. It is indeed automatic
for dilation-covariant theories or theories of waves (see Section \ref{waves}).
Our first task is to identify, in the above theoretical setting, observables which can be interpreted as
particle detectors. To this end, we have to list several definitions and results:
First, we recall that an observable $B\in {\mfa}$ is called almost-local,
if there exists a net of operators $\{\, B_r\in {\mfa}(\mathcal{O}_r)\,|\, r>0\,\}$, s.t.\! for any $k\in\mathbb{N}_0$
\begin{equation}
\lim_{r\to\infty} r^k\|B-B_r\|=0,
\end{equation}
where $\mathcal{O}_r=\{(t,\xb)\in\mathbb{R}^2\,|\, |t|+|\xb|<r\,\}$.
We also recall that the Arveson spectrum of an operator $B\in \mathfrak{A}$ w.r.t.\! the group of translation automorphisms
$\alpha_{\xi}(\,\cdot\,)=U(\xi)\,\cdot\,U(\xi)^*$, denoted by $\mathrm{Sp}^B \alpha$,
is the closure of the union of supports of the distributions
\begin{equation}
(\Psi_1|\tilde B(p)\Psi_2)=(2\pi)^{-1}\int_{\mathbb{R}^2} d x\,\varepsilon^{-ip\xi} (\Psi_1|B(\xi)\Psi_2) \label{energy-momentum-transfer}
\end{equation}
over all $\Psi_1,\Psi_2\in\mathcal{H}$, where $p=(\omega,\boldsymbol p} %{\mbox{\boldmath$p$})$, $\xi=(t,\xb)$, $p\xi=\omega t-\boldsymbol p} %{\mbox{\boldmath$p$}\xb$ and $B(\xi):=\alpha_{\xi}(B)$.
Let $E(\,\cdot\,)$ be the spectral measure of $U$. As shown in \cite{Ar82}, for any $B\in{\mfa}$ and any closed set $\Delta\subset\mathbb{R}^2$,
it holds that
\begin{eqnarray}
BE(\Delta)\mathcal{H}\subset E(\overline{\Delta+\mathrm{Sp}^B \alpha})\mathcal{H}. \label{Arveson}
\end{eqnarray}
Next, we introduce the lightline coordinates $\omega_{\pm}=\fr{\omega\pm \boldsymbol p} %{\mbox{\boldmath$p$}}{\sqrt{2}}$, $t_{\pm}=\fr{t\mp \xb}{\sqrt{2}}$
and define, for any $\delta>0$, the following subspaces of $\mathfrak{A}$:
\begin{eqnarray}
\mathcal L_{\pm,\delta}=\{\, B\in\mathfrak{A}\,|\, B\,\, \textrm{is almost-local and } \mathrm{Sp}^B\alpha\subset \{\, \omega_\pm\leq -\delta \,\} \textrm{ is compact }\}.
\end{eqnarray}
Following \cite{AH67,Bu90}, we construct particle detectors: For any $e_{\pm}>0$, $B_{\pm}\in \mathcal L_{\pm, e_{\pm}}$, $T\geq 1$
and $0<\eta<1$ we define
\begin{eqnarray}
Q^{T,\eta}_{\pm}(B_{\pm})=\int dt\,h_T(t) \int\, d\xb\, f_{\pm}^{\eta}(\xb/t) (B^*_{\pm}B_{\pm})(t,\xb), \label{as-obs-one}
\end{eqnarray}
where $h_T(t)=|T|^{-\varepsilon}h(|T|^{-\varepsilon}(t-T))$, $0<\varepsilon<1$ and $h\in C_0^{\infty}(\mathbb{R})$ is a non-negative function
s.t.\! $\int dt\, h(t)=1$ and $f_{\pm}^{\eta}\in C^{\infty}(\mathbb{R})$ have the following properties: $0\leq f_{\pm}^\eta \leq 1$,
$f_{+}^{\eta}(\xb)=1$ for $\xb\geq \eta$, $f_{+}^{\eta}(\xb)=0$ for $\xb\leq 0$, $f_-^{\eta}(\xb)=f_+^{\eta}(-\xb)$. Moreover,
$f_{\pm}^{\eta}(\xb)\nearrow \mathbf 1_{\mathbb{R}_{\pm}}(\xb)$ as $\eta\to 0$, where $\mathbf 1_{\mathbb{R}_{\pm}}$ are the characteristic functions of the sets $\mathbb{R}_{\pm}$. For large positive\footnote{In the present Letter we consider only outgoing configurations of
particles, since the incoming case is analogous. } $T$ and small $\eta$
these expressions can be interpreted as detectors sensitive to right-moving (in the $(+)$ case) and left-moving (in the $(-)$ case) particles.
The operators $Q^{T,\eta}_{\pm}(B_{\pm})$ are defined on the domain $\mathcal D=\bigcup_{n\in\mathbb{N}} E(\{(\omega,p)\,|\, \omega\leq n\})\mathcal{H}$
of vectors of bounded energy. This is a consequence of the following
abstract theorem due to Buchholz, which we will use frequently in this paper:
\begin{theoreme}[\cite{Bu90}]\label{H-A} Let $\mathbb{R}^s\ni \xb\mapsto U(\xb)$ be a group of unitaries on $\mathcal{H}$, $B\in B(\mathcal{H})$, $n\in \mathbb{N}$ and let $E_n$ be the orthogonal projection onto the intersection of the kernels of the $n$-fold products $B(\xb_1)...B(\xb_n)$ for arbitrary $\xb_1,\ldots,\xb_n\in\mathbb{R}^{s}$, where
$B(\xb)=U(\xb)BU(\xb)^*$.
Then there holds for each compact subset $K\subset \mathbb{R}^s$ the estimate
\begin{equation}
\left\|E_n\int_K d\xb\,(B^*B)(\xb)E_n\right\|\leq (n-1)\int_{\Delta K} d\xb\, \left\|[B^*,B(\xb)]\right\|,
\label{harmonic-analysis-bound}
\end{equation}
where $\Delta K=\{\, \xb-\boldsymbol y\,|\, \xb,\boldsymbol y\in K\,\}$.
\end{theoreme}
\noindent As noticed in \cite{Bu90}, if $B\in\mathcal L_{+} % {\mathrm{R} ,\delta}\cup \mathcal L_{-} %{\mathrm{L} ,\delta}$, then, for any compact set $\Delta$, the range of $E(\Delta)$ is contained in $E_n$ for sufficiently large $n$ due to relation (\ref{Arveson}). Exploiting almost-locality of $B$ one can replace $\Delta K$ on the
r.h.s.\! of (\ref{harmonic-analysis-bound}) with $\mathbb{R}$, obtaining a bound which is uniform in $K$. Then $E(\Delta)\int d\xb\,(B^*B)(\xb)E(\Delta)\in B(\mathcal{H})$
exists as a strong limit of integrals over compact subsets and
\begin{equation}
\int d\xb\, (B^*B)(\xb)E(\Delta)\in B(\mathcal{H}), \label{harmonic-analysis}
\end{equation}
since $\mathrm{Sp}^B\alpha$ is compact.
The existence of the limits $Q^{\mathrm{ out},\eta}_{\pm}\Psi:= \lim_{T\to\infty}Q^{T,\eta}_{\pm}(B_{\pm})\Psi$, $\Psi\in \mathcal D$, is not known in general.
If they exist for any $\Psi \in \mathcal D$, they define operators $Q^{\mathrm{ out},\eta}_{\pm}(B_{\pm})$ on $\mathcal D$.
We show that these operators are translation-invariant in Lemma \ref{lm:translation-invariance} (cf. Proposition~3.9 of \cite{Po04.1}):
\begin{equation}
U(x)Q^{\mathrm{ out},\eta}_{\pm}(B_{\pm})U(x)^*=Q^{\mathrm{ out},\eta}_{\pm}(B_{\pm}),\quad x\in\mathbb{R}^2. \label{translation-invariance}
\end{equation}
In particular, they preserve each spectral subspace of $U$. By the properties
of functions $f^{\eta}_{\pm}$ and (\ref{harmonic-analysis}), $\{Q^{\mathrm{ out},\eta}_{\pm}(B_{\pm}) \}_{\eta\in (0,1)}$ are monotonously
increasing (as $\eta\to 0$)
families of bounded operators on $\mathcal{H}(\Delta):=E(\Delta)\mathcal{H}$ which are uniformly bounded. Thus there exist the limits $Q^\mathrm{ out}_{\pm}(B_{\pm})
=\mathrm{s}\textrm{-}\lim_{\eta\to 0} Q^{\mathrm{ out},\eta}_{\pm}(B_{\pm} )$ as bounded operators on $\mathcal{H}(\Delta)$. Since $\Delta$ is an arbitrary compact set, $Q^\mathrm{ out}_{\pm}(B_{\pm})$ can be consistently defined as operators on $\mathcal D$, which also satisfy (\ref{translation-invariance}).
Keeping the above discussion in mind, we define the following subsets of $\mathcal L_{\pm,\delta}$:
\begin{align}
\hat{\mathcal L}_{\pm,\delta}=&\{\,B_{\pm}\in \mathcal L_{\pm,\delta}\,| \, Q^\mathrm{ out}_{\pm}(B_{\pm})\Psi:=\lim_{\eta\to 0}\lim_{T\to\infty}Q^{T,\eta}_{\pm}(B_{\pm})\Psi\,
\textrm{ exists for any } \Psi\in \mathcal D \}.
\end{align}
Every vector from the range of $Q^\mathrm{ out}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })$
(resp. $Q^\mathrm{ out}_{-} %{\mathrm{L} }(B_{-} %{\mathrm{L} })$) contains
an excitation moving to the right (resp. to the left), whose energy is larger than $e_{+} % {\mathrm{R} }/\sqrt{2}$
(resp. $e_{-} %{\mathrm{L} }/\sqrt{2}$), and possibly some other, unspecified excitations. The basis for this physical interpretation
of particle detectors is Proposition~\ref{detectors-on-waves}, stated below.
Now, for any $\varepsilon>0$, we define the following subset of the spectrum of $U$
\begin{eqnarray}
\Delta_{\varepsilon}(e_+} % {\mathrm{R} ,e_{-} %{\mathrm{L} })=\{\,(\omega_+,\omega_-)\in\mathbb{R}^2\,|\, e_{+} % {\mathrm{R} }\leq \omega_+\leq e_{+} % {\mathrm{R} }+\varepsilon, \, e_{-} %{\mathrm{L} }\leq \omega_-\leq e_{-} %{\mathrm{L} }+\varepsilon\,\}.
\end{eqnarray}
Let $\mathcal{H}_{\mathrm{c}}$ be the continuous subspace of the relativistic mass operator $H^2-\boldsymbol P ^2$, where $(H,\boldsymbol P )$ are the generators
of $U$.
Then, in view of the above discussion, every non-zero vector of the form
\begin{eqnarray}
\Psi^\mathrm{ out}_{\varepsilon}= Q^\mathrm{ out}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })Q^\mathrm{ out}_{-} %{\mathrm{L} }(B_{-} %{\mathrm{L} })\Psi,\quad \Psi\in E(\Delta_{\varepsilon}(e_+} % {\mathrm{R} ,e_{-} %{\mathrm{L} }))\mathcal{H}_{\mathrm{c}},\quad B_{\pm}\in \hat{\mathcal L}_{\pm,\delta} \label{scattering-states}
\end{eqnarray}
describes two `hard' massless excitations, the first moving to the right with energy $e_+} % {\mathrm{R} /\sqrt{2}$ and the
second moving to the left with energy $e_-} %{\mathrm{L} /\sqrt{2}$, as well as some unspecified `soft' massless particles,
whose total energy is less than $\sqrt{2}\varepsilon$.
Since the motion of massless excitations in two-dimensional Minkowski spacetime
is dispersionless, we expect that such two-body generalized scattering states span the entire subspace $\mathcal{H}_{\mathrm{c}}$. (In fact, two
excitations moving without dispersion in the same direction can be interpreted as one excitation). In view of
the above discussion, we define the generalized asymptotic completeness as follows:
\begin{definition}\label{Asymptotic-completeness} Suppose that for any $e_{+} % {\mathrm{R} },e_{-} %{\mathrm{L} },\varepsilon>0$
\begin{equation}
E(\Delta_{\varepsilon}(e_+} % {\mathrm{R} ,e_{-} %{\mathrm{L} }))\mathcal{H}_{\mathrm{c}}=
\mathrm{Span}\{\,Q^\mathrm{ out}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })Q^\mathrm{ out}_{-} %{\mathrm{L} }(B_{-} %{\mathrm{L} })E(\Delta_{\varepsilon}(e_+} % {\mathrm{R} ,e_{-} %{\mathrm{L} })) \mathcal{H}_{\mathrm{c}} \,|\, B_{\pm}\in \hat{\mathcal L}_{\pm, e_{\pm} } \, \}^{\mathrm{cl}},
\end{equation}
where ${\mathrm{cl}}$ means the closure.
Then we say that the theory has the property of generalized asymptotic completeness.
\end{definition}
In the present Letter we show that this property is a generalization of a more standard concept
of asymptotic completeness in the sense of waves (Section~\ref{waves}). We provide a large class
of examples which are not asymptotically complete in the sense of waves, but have the
generalized particle interpretation in the sense of Definition~\ref{Asymptotic-completeness}
(e.g. charged sectors of chiral conformal field theories). However, we do not expect that the
generalized asymptotic completeness holds in all theories satisfying the postulates from Definition~\ref{two-dim-net}.
It may fail in models with too many local degrees of freedom, as for example certain generalized free fields.
We refrain from giving concrete counterexamples here.
\section{Theories of waves}\label{waves}
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In this section we consider a local net of von Neumann algebras $({\mfa},U)$ in a vacuum representation. That is
we assume, in addition to the properties specified in Definition~\ref{two-dim-net}, the existence of
a unique (up to a phase) unit vector $\Omega\in\mathcal{H}$, which is invariant under $U$ and cyclic for $\mathfrak{A}$. Let
$\mathcal{H}_{\pm}=\ker(H\mp \boldsymbol P} %{{\bf P})$, where $(H,\boldsymbol P} %{{\bf P})$ are generators of $U$, and let $E_{\pm}$ be the corresponding orthogonal projections.
If each of the subspaces $\mathcal{H}_{\pm}$ contains some vectors orthogonal
to $\Omega$, then we say that the net $({\mfa},U)$ describes `waves', which are counterparts of Wigner particles in massless,
two-dimensional theories. A natural scattering theory for waves, developed by Buchholz in \cite{Bu75}, is outlined below.
We will show that theories which are asymptotically complete in the sense of this scattering theory have also the
property of generalized asymptotic completeness, formulated in Definition~\ref{Asymptotic-completeness} above.
Following \cite{Bu75}, for any $F\in{\mfa}$ and $T\geq 1$ we introduce the asymptotic field approximants:
\begin{eqnarray}
F_\pm(h_T)= \int h_T(t) F(t, \pm t) dt,
\end{eqnarray}
where $h_T$ is defined after formula~(\ref{as-obs-one}) above.
We recall the following result:
\begin{proposition}[\cite{Bu75}]\label{scattering-first-lemma} Let $F\in {\mfa}$. Then the limits
\begin{eqnarray}
\Phi_{\pm}^{\mathrm{ out}}(F):= \underset{T\to\infty}\mathrm{s}\textrm{-}\lim \, F_{\pm}(h_T) \quad \label{asymptotic-field}
\end{eqnarray}
exist and are called the (outgoing) asymptotic fields. They depend only on the respective vectors $\Phi_{\pm}^{\mathrm{ out}}(F)\Omega=E_{\pm}F\Omega$
and satisfy $[\Phi_{+}^{\mathrm{ out}}(F),\Phi_{-}^{\mathrm{ out}}(F')]=0$ for any $F,F'\in\mathfrak{A}$.
\end{proposition}
Now the scattering states are defined as follows: Since ${\mfa}$ acts irreducibly
on $\mathcal{H}$ (by the assumed uniqueness of the vacuum), for any $\Psi_{\pm}\in\mathcal{H}_{\pm}$ we can find $F_{\pm}\in{\mfa}$ s.t.\! $\Psi_{\pm}=F_{\pm}\Omega$ \cite{Sa}. The vectors
\begin{eqnarray}
\Psi_+\overset{\mathrm{ out}}{\times}\Psi_-=\Phi_{+}^{\mathrm{ out}}(F_+)\Phi_{-}^{\mathrm{ out}}(F_-)\Omega \label{wave-scattering-states}
\end{eqnarray}
are called the (outgoing) scattering states. By Proposition~\ref{scattering-first-lemma} they do not depend on the choice of
$F_{\pm}$ within the above restrictions. They have the following properties:
\begin{proposition}[\cite{Bu75}]\label{scattering-second-lemma} Let $\Psi_{\pm},\Psi_{\pm}'\in\mathcal{H}_{\pm}$. Then:
\begin{enumerate}
\item[(a)] $(\Psi_+\overset{\mathrm{ out}}{\times} \Psi_-,\Psi'_+\overset{\mathrm{ out}}{\times} \Psi'_-)=(\Psi_+,\Psi'_+)(\Psi_-,\Psi'_-)$,
\item[(b)] $U(\xi)(\Psi_+\overset{\mathrm{ out}}{\times} \Psi_-)=(U(\xi)\Psi_+)\overset{\mathrm{ out}}{\times} (U(\xi)\Psi_-)$, for $\xi\in\mathbb{R}^2$.
\end{enumerate}
\end{proposition}
If the states of the form~(\ref{wave-scattering-states}) span the entire Hilbert space, then we say that the theory
is {\bf asymptotically complete in the sense of waves}. In this case, the representation $U$
decomposes into a tensor product of representations of lightlike translations and
the spectrum of $U$ automatically coincides with $V_+$ by the theorem of Borchers \cite{Bo92}.
We will show below that any such theory is also asymptotically
complete in the sense of Definition~\ref{Asymptotic-completeness}. To this end we prove the following fact:
\begin{proposition}\label{detectors-on-waves} Let $B_{\pm}\in \mathcal L_{\pm, e_{\pm}}$ and let $\Psi_{\pm}\in\mathcal{H}_{\pm}$ be vectors of bounded
energy. Then
\begin{eqnarray}
\lim_{\eta\to 0}\lim_{T\to\infty}Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })(\Psi_+\overset{\mathrm{ out}}{\times} \Psi_-)&=&(E_+Q(B_{+} % {\mathrm{R} })\Psi_+)\overset{\mathrm{ out}}{\times} \Psi_-, \label{convergence-one}\\
\lim_{\eta\to 0}\lim_{T\to\infty}Q^{T,\eta}_{-} %{\mathrm{L} }(B_{-} %{\mathrm{L} }) (\Psi_+\overset{\mathrm{ out}}{\times} \Psi_-)&=&\Psi_+\overset{\mathrm{ out}}{\times} (E_-Q(B_{-} %{\mathrm{L} })\Psi_-),\label{convergence-two}
\end{eqnarray}
where $Q(B_{\pm}):=\int d\xb\,(B_{\pm}^*B_{\pm})(\xb)$ are operators defined on $\mathcal D$.
\end{proposition}
\noindent{\bf Proof. } We prove only equality (\ref{convergence-one}), as (\ref{convergence-two}) is analogous. Let $F_{\pm}\in\mathfrak{A}$
be s.t.\! $\Psi_{\pm}=F_{\pm}\Omega$. Since $\Psi_{\pm}$ have bounded energy, we can ensure, by smearing with suitable
test functions, that $\mathrm{Sp}^{F_{\pm}}\alpha$ are compact sets. Then it is clear that $\Psi_+\overset{\mathrm{ out}}{\times} \Psi_-=\Phi^{\tout}_+(F_+)\Phi^{\tout}_-(F_-)\Omega$
is a vector of bounded energy, and we can write
\begin{eqnarray}
Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} }) \Phi^{\tout}_+(F_+)\Phi^{\tout}_-(F_-)\Omega&=&[Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} }), \Phi^{\tout}_-(F_-)]\Phi^{\tout}_+(F_+)\Omega\nonumber\\
& &+\Phi^{\tout}_-(F_-)Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })\Phi^{\tout}_+(F_+)\Omega. \label{commuting-detectors}
\end{eqnarray}
Let us first consider the second term on the r.h.s.\! of (\ref{commuting-detectors}).
We define
\begin{eqnarray}
Q^T_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} }):=\int dt\,h_T(t) \int\, d\xb\, \mathbf{1}_{\mathbb{R}_+} (\xb/t) (B_+^*B_+)(t,\xb).
\end{eqnarray}
We note that $R_-^{T,\eta}(B_+):=Q^T_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })-Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })$ satisfies the
assumptions of Lemma~\ref{technical-commutators}. Consequently,
\begin{eqnarray}
\lim_{T\to\infty} R_-^{T,\eta}(B_+)\Phi^{\tout}_+(F_+)\Omega=
\lim_{T\to\infty} [R_-^{T,\eta}(B_+), F_+(h_T)]\Omega=0, \label{new-formula}
\end{eqnarray}
where we made use of the fact that $\sup_{T\in \mathbb{R}}\| R_-^{T,\eta}(B_+)E(\Delta)\|<\infty$
for any compact set $\Delta$. Now we compute
\begin{eqnarray}
Q^T_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })\Psi_+&=&\int dt\,h_T(t) \varepsilon^{iHt}\int_{\mathbb{R}_+}\, d\xb (B^*_{+}B_{+})(\xb)\varepsilon^{-i\boldsymbol P t}\Psi_+\nonumber\\
&=&\int dt\,h_T(t) \varepsilon^{i(H-\boldsymbol P )t}\int_{-t}^{\infty}\, d\xb (B^*_{+}B_{+})(\xb)\Psi_+\nonumber\\
&=&-\int dt\,h_T(t) \varepsilon^{i(H-\boldsymbol P )t}\int_{-\infty}^{-t}\, d\xb (B^*_{+}B_{+})(\xb)\Psi_+\nonumber\\
& &+\left(\int dt\,h_T(t) \varepsilon^{i(H-\boldsymbol P )t}-E_+\right)\int\, d\xb (B^*_{+}B_{+})(\xb)\Psi_+\nonumber\\
& &+E_+\int\, d\xb (B^*_{+}B_{+})(\xb)\Psi_+. \label{detector-on-sp-state}
\end{eqnarray}
Here in the first step we made use of the definition of $\mathcal{H}_+$. The second term on the r.h.s. above
tends to zero as $T\to \infty$ by the mean ergodic theorem.
Let us show that the first term on the r.h.s. of (\ref{detector-on-sp-state}) tends to zero as $T\to\infty$. This is a consequence
of the fact that
\begin{equation}
\lim_{t\to\infty}\int_{-\infty}^{-t}\, d\xb (B^*_{+}B_{+})(\xb)\Psi_+=0
\end{equation}
which follows from the discussion after Theorem~\ref{H-A} above. Thus we obtain
\begin{equation}
\lim_{T\to\infty}Q^T_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })\Psi_+=E_+Q(B_+)\Psi_+. \label{second-term}
\end{equation}
To conclude the proof, we still have to show that the first term on the r.h.s.\! of (\ref{commuting-detectors}) tends strongly to
zero as $T\to\infty$. This follows from the equality
\begin{equation}
\lim_{T\to\infty}\|E(\Delta)[Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} }), F_-(h_T)]E(\Delta')\|=0, \label{commutator-vanishing}
\end{equation}
valid for any compact sets $\Delta,\Delta'\subset V_+$, which is established in Lemma~\ref{technical-commutators}. In fact, let us consider separately the two terms forming the commutator
in (\ref{commuting-detectors}):
\begin{align}
Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })\Phi^{\tout}_-(F_-)\Phi^{\tout}_+(F_+)\Omega&=Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} }) F_-(h_T)\Phi^{\tout}_+(F_+)\Omega+o(1),\label{first-term-comm}\\
\Phi^{\tout}_-(F_-)Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })\Phi^{\tout}_+(F_+)\Omega&=\Phi^{\tout}_-(F_-)E_+Q(B_+) \Phi^{\tout}_+(F_+)\Omega+o(1)\nonumber \\
&=F_-(h_T)E_+Q(B_+) \Phi^{\tout}_+(F_+)\Omega+o(1)\nonumber\\
&=F_-(h_T)Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })\Phi^{\tout}_+(F_+)\Omega+o(1), \label{second-term-comm}
\end{align}
where $o(1)$ denotes terms tending in norm to zero as $T\to\infty$. In (\ref{first-term-comm}) we used the fact that
$\mathrm{Sp}^{F_{\pm}}\alpha$ are compact and relation (\ref{harmonic-analysis}) which gives $\sup_{T\in\mathbb{R} }\|Q^{T,\eta}_{+} % {\mathrm{R} }(B_{+} % {\mathrm{R} })E(\Delta)\|<\infty$
for any compact set $\Delta$. In the first and last step of (\ref{second-term-comm}) we exploited (\ref{second-term}) and (\ref{new-formula}).
>From (\ref{first-term-comm}), (\ref{second-term-comm}) and the compactness of $\mathrm{Sp}^{B_{+} % {\mathrm{R} }}\alpha$ we conclude that
(\ref{commutator-vanishing}) implies vanishing of the first term on the r.h.s.\! of (\ref{commuting-detectors}) as $T\to\infty$. $\Box$\medskip
Let us set $H_{\pm}=\fr{1}{\sqrt 2}(H\pm \boldsymbol P )$ and let $E_{\pm}(\,\cdot\,)$ be the spectral measures of $H_{\pm}|_{\mathcal{H}_{\pm}}$.
It is easily seen that the spectrum of $H_{\pm}|_{\mathcal{H}_{\pm}}$ is continuous, apart from an eigenvalue at zero. (In fact, if $\Psi_+$
is an eigenvector of $H_+|_{\mathcal{H}_+}$, then $(\Psi_+|A\Psi_+)=(e^{itH_+}\Psi_+|Ae^{itH_+}\Psi_+)=(e^{it\sqrt 2 \boldsymbol P }\Psi_+|Ae^{it\sqrt 2 \boldsymbol P }\Psi_+)=(\Psi_+|A(0,-\sqrt 2 t)\Psi_+)=\|\Psi_+\|^2(\Omega|A\Omega)$
for any $A\in\mathfrak{A}$ by the clustering property. Exploiting the fact that $\mathfrak{A}$ acts irreducibly on $\mathcal{H}$, we obtain that $\Psi_+$ is proportional to $\Omega$).
We note the following fact, whose proof relies on some ideas from the proof of Proposition~2.1 of \cite{BF82}:
\begin{lemma}\label{borchers-zero}
Let $\delta>0$, $\Psi_+\in \mathcal{H}_+$ and suppose that $B\Psi_+=0$ for
any $B\in\mathcal L_{+,\delta}$. Then $E_+([\delta,\infty))\Psi_+=0$. (An analogous result holds for $(+)$ replaced with $(-)$).
\end{lemma}
\noindent{\bf Proof. } Let us choose $b>a>\delta$, $0<\varepsilon<a-\delta$ and $c>0$. We choose functions $f_{\pm}\in S(\mathbb{R})$ s.t.\! $\mathrm{supp}\, \tilde f_+\subset (-\infty, -\delta ]$ is compact, $\tilde f_+(\omega_+)=1$ for $\omega_+\in [-b,-a+\varepsilon]$, $\mathrm{supp}\,\,\tilde f_-\subset [-2c,2c]$ and $\tilde f_-(\omega_-)=1$ for $\omega_-\in [-c,c]$. Let $f(x)=f_+(t_+)f_-(t_-)$.
Since $\tilde f(p)=\tilde f_+(\omega_+)\tilde f_-(\omega_-)$, we obtain that $A(f)=\int_{\mathbb{R}^2} dx\,A(x)f(x)$ is an element of $\mathcal L_{+,\delta}$ for any $A\in\mathfrak{A}(\mathcal{O})$, $\mathcal{O}\subset\mathbb{R}^2$. Thus, by assumption,
$A(f)\Psi_+=0$.
Making use of the fact that $\alpha_x(A(f))\in \mathcal L_{+,\delta}$ for any $x\in\mathbb{R}^2$,
we obtain that $U(x)A(f)U(x)^*\Psi_+=0$, hence $A(f)U(x)^*\Psi_+=0$ and consequently
\begin{equation}
E(\Delta_2)A(f)E(\Delta_1)\Psi_+=0,
\end{equation}
for any compact sets $\Delta_1$, $\Delta_2\subset \mathbb{R}^2$. Setting $\Delta_1=\{\,(\omega_+,\omega_-)\in\mathbb{R}^2\,|\,\omega_+\in [a,b],\omega_-\in [-c/2,c/2]\,\}$,
$\Delta_2=\{\, (\omega_+,\omega_-)\in\mathbb{R}^2 \,|\, \omega_+\in [0,\varepsilon], \omega_-\in [-c/2,c/2] \,\}$ and exploiting the properties
of $f$, we obtain that
\begin{equation}
E(\Delta_2)AE(\Delta_1)\Psi_+=0.
\end{equation}
As $\mathfrak{A}$ acts irreducibly on $\mathcal{H}$, (since we assumed the uniqueness of the vacuum vector), and $E(\Delta_2)\neq 0$, (which follows e.g.\! from the existence of the vacuum), we conclude that $E(\Delta_1)\Psi_+=E_+([a,b])\Psi_+=0$. Since the spectrum of $H_+|_{\mathcal{H}_+}$ is
continuous, apart from the eigenvalue at zero, we obtain that $E_+([\delta,\infty))\Psi_+=0$. $\Box$\medskip\\
Now we proceed to the main result of this section:
\begin{theoreme}\label{main-theorem-waves} Let $({\mfa},U)$ be a net of von Neumann algebras in a vacuum representation, which is asymptotically
complete in the sense of waves. Then it has the property of generalized asymptotic
completeness, stated in Definition~\ref{Asymptotic-completeness}.
\end{theoreme}
\noindent{\bf Proof. } First, we note that the continuous subspace $\mathcal{H}_{\mathrm{c}}$ of the relativistic mass operator $H^2-\boldsymbol P ^2$ is given by
\begin{equation}
\mathcal{H}_{\mathrm{c}}=\mathcal{H}_{+,\mathrm{c}}\overset{\mathrm{ out}}{\times}\mathcal{H}_{-,\mathrm{c}},
\label{continuous-spectrum}
\end{equation}
where $\mathcal{H}_{\pm,\mathrm{c}}=\mathcal{H}_{\pm}\cap\{\Omega\}^{\bot}$ are the continuous subspaces of $H_{\pm}|_{\mathcal{H}_{\pm}}$.
To justify this fact one notes that, as a consequence of
asymptotic completeness in the sense of waves, $\mathcal{H}_{\mathrm{c}}\subset \mathcal{H}_{+,\mathrm{c}}\overset{\mathrm{ out}}{\times}\mathcal{H}_{-,\mathrm{c}}$ and the only possible eigenvalue of $H^2-\boldsymbol P ^2$ is zero.
(Non-zero eigenvalues can easily be excluded with the help of the Haag-Ruelle scattering theory or by proceeding as in
Lemma~\ref{continuous-spectra} below).
Then it is readily checked that no vector from the subspace on the r.h.s.\! of~(\ref{continuous-spectrum}) can be a corresponding eigenvector.
Making use of (\ref{continuous-spectrum}) and of Proposition~\ref{scattering-second-lemma}, we obtain the following equality
\begin{eqnarray}
E(\Delta_{\varepsilon}(e_+} % {\mathrm{R} ,e_{-} %{\mathrm{L} }))\mathcal{H}_{\mathrm{c}}=E_+([e_{+} % {\mathrm{R} },e_{+} % {\mathrm{R} }+\varepsilon] )\mathcal{H}_{+,\mathrm{c}}\overset{\mathrm{ out}}{\times} E_-([e_{-} %{\mathrm{L} },e_{-} %{\mathrm{L} }+\varepsilon] )\mathcal{H}_{-,\mathrm{c}}.
\label{tensor-product-measures}
\end{eqnarray}
Now we note that any vector $\Psi\in E(\Delta)\mathcal{H}$, where $\Delta\subset \mathbb{R}^2$ is compact, can be expressed as
$\Psi=\sum_{m,n}c_{m,n}\Psi_{+,m}\overset{\mathrm{ out}}{\times} \Psi_{-,n}$, where $\Psi_{\pm,m}\in P(\Delta')\mathcal{H}_\pm$
form orthonormal systems, $\Delta'\subset \mathbb{R}^2$ is compact and $\sum_{m,n}|c_{m,n}|^2<\infty$. (See \cite[Lemma A.2]{DT11}).
Hence, Proposition~\ref{detectors-on-waves} and relation~(\ref{harmonic-analysis}) entail that for any $B_{\pm}\in\mathcal L_{\pm,e_{\pm}}$ and
$\Psi\in \mathcal D$ the limits $Q_{\pm}^\mathrm{ out}(B_\pm)\Psi=\lim_{\eta\to 0}\lim_{T\to\infty}Q^{T,\eta}_{\pm}(B_{\pm})\Psi$ exist. Consequently, it suffices to verify the following formula
\begin{equation}
E_+([e_{+} % {\mathrm{R} },e_{+} % {\mathrm{R} }+\varepsilon] )\mathcal{H}_{+,\mathrm{c}}=\mathrm{Span}\{\,E_+Q(B_+)E_+([e_{+} % {\mathrm{R} },e_{+} % {\mathrm{R} }+\varepsilon] )\mathcal{H}_{+,\mathrm{c}} \,|\, B_{+}\in \mathcal L_{+,e_{+}} \}^{\mathrm{cl}} \label{one-dim-zero}
\end{equation}
and its counterpart with $(+)$ replaced with $(-)$, whose proof is analogous.
(We recall that $Q(B_+)$ was defined in Proposition~\ref{detectors-on-waves}).
Since $E_+Q(B_+)E_+$ is invariant under spacetime translations,
it is obvious that the subspace on the r.h.s.\!
of (\ref{one-dim-zero}) is
contained in the subspace on the l.h.s. Let us now assume that the inclusion is proper i.e., we can
choose a non-zero vector $\Psi_+ \in (E_1-E_0)\mathcal{H}_{+,\mathrm{c}}$, where $E_1:=E_+([e_{+} % {\mathrm{R} },e_{+} % {\mathrm{R} }+\varepsilon] )$ and $E_0$ is
the orthogonal projection on the subspace on the r.h.s.\! of (\ref{one-dim-zero}).
By Lemma~\ref{borchers-zero}, there is an operator $B_+\in\mathcal L_{+,e_+}$
s.t.\! $B_+\Psi_+ \ne 0$. Then it is easy to see that $(\Psi_+|Q(B_+)\Psi_+) \ne 0$
which means that $(E_1-E_0)E_+Q(B_+) \Psi_+\ne 0$. Hence $E_+Q(B_+)\Psi_+ \neq 0$ and $E_+Q(B_+)\Psi_+\notin E_0\mathcal{H}_{+,\mathrm{c}}$,
which contradicts the definition of $E_0$. $\Box$\medskip
\section{Chiral nets and infraparticles}\label{infraparticles}
\setcounter{equation}{0}
In the previous section we showed that the generalized concept of particle interpretation, formulated
in Definition~\ref{Asymptotic-completeness}, is a consequence of a more standard notion of asymptotic
completeness in the sense of waves. We recall from \cite{DT10,DT11} that any chiral conformal field theory in a \emph{vacuum} product representation
is asymptotically complete in the sense of waves. Hence it is also asymptotically complete in the generalized sense.
It turns out that the range of validity of the generalized asymptotic completeness is
not restricted to theories of waves, but includes also some theories of \emph{infraparticles}. We say that a net
of von Neumann algebras $(\mathfrak{A},U)$ describes infraparticles, if one or both of the subspaces $\mathcal{H}_{\pm}$ are trivial,
but there exist non-zero particle detectors $Q_{\pm}^{\mathrm{ out}}(B_{\pm})$. As shown in \cite{DT11}, any chiral conformal
field theory in a \emph{charged} irreducible product representation describes infraparticles. In this section we show that these
theories of infraparticles are asymptotically complete in the generalized sense.
Let us now briefly recall the construction of chiral conformal field theories, focusing on these properties,
which are needed in our investigation. First, we recall the definition of a local net of von Neumann algebras on the real line:
\begin{definition}\label{circle-theory}
A local net of von Neumann algebras on $\mathbb{R}$ is a pair $({\cal A},V)$ consisting of
a map ${\cal I}\mapsto {\cal A}({\cal I})$ from the family of open, bounded subsets of $\mathbb{R}$ to the
family of von Neumann
algebras on a Hilbert space $\hilk$
and a strongly continuous unitary representation of translations $\mathbb{R}\ni \tau\mapsto V(\tau)$, acting on $\hilk$,
which are subject to the following conditions:
\begin{enumerate}
\item (isotony) If ${\cal I} \subset \mathfrak{J}$, then ${\cal A}({\cal I})\subset {\cal A}(\mathfrak{J})$.
\item (locality) If ${\cal I} \cap \mathfrak{J} = \varnothing$, then $[{\cal A}({\cal I}),{\cal A}(\mathfrak{J})] = 0$.
\item (covariance) $\beta_{\tau}({\cal A}({\cal I})):=V(\tau){\cal A}({\cal I})V(\tau)^* = {\cal A}({\cal I}+\tau)$ for any $\tau\in\mathbb{R}$.
\item (positivity of energy) The spectrum of $V$ coincides with $\mathbb{R}_+$.
\end{enumerate}
We also denote by ${\cal A}$ the quasilocal $C^*$-algebra of this net i.e., ${\cal A}=\overline{\bigcup_{{\cal I}\subset\mathbb{R}}{\cal A}({\cal I})}$. We assume that
it acts irreducibly on $\hilk$.
\end{definition}
Let $({\cal A}_\lambda,V_\lambda)$ and $({\cal A}_\rho,V_\rho)$ be two nets of von Neumann algebras on ${\mathbb R}$, acting on Hilbert spaces
$\hilk_\lambda$ and $\hilk_\rho$. To construct a local net $({\mfa}, U)$ on ${\mathbb R}^2$, acting on the tensor product space ${\cal H} = {\cal K}_\lambda\otimes {\cal K}_\rho$,
we identify the two real lines with the lightlines $I_{\pm}=\{\,(t,\xb)\in\mathbb{R}^2\,|\, \xb\mp t=0\,\}$ in ${\mathbb R}^2$.
Let us first specify the unitary representation of translations
\begin{equation}
U(t,\xb):= V_\lambda\left(\fr{1}{\sqrt{2}}(t-\xb)\right)\otimes V_\rho\left(\fr{1}{\sqrt{2}}(t+\xb)\right).
\label{unitary-representation}
\end{equation}
The spectrum of this representation coincides with $V_+$ due to property~4 from Definition~\ref{circle-theory}.
Any double cone $D\subset{\mathbb R}^2$ can be expressed as a product of intervals
on lightlines $D = {\cal I}\times \mathfrak{J}$. The corresponding local von Neumann algebra is given by ${\mfa}(D):= {\cal A}_\lambda({\cal I})\otimes{\cal A}_\rho(\mathfrak{J})$,
and for a general open region $\mathcal{O}$ we put ${\mfa}(\mathcal{O})=\bigvee_{D\subset \mathcal{O}}{\mfa}(D)$. The resulting net of von Neumann algebras $({\mfa},U)$,
which we call the chiral net, satisfies the properties stated in Definition~\ref{two-dim-net}. If both $\hilk_\lambda$ and $\hilk_\rho$ contains
translation invariant vectors, then we say that the net $({\mfa},U)$ is in a vacuum product representation. Otherwise we say that it is
in a charged product representation. These two cases will be treated on equal footing in the remaining part of this section.
We will show that any chiral net satisfies the generalized asymptotic completeness in the sense of Definition~\ref{Asymptotic-completeness}.
As a preparation, we prove the following two lemmas.
\begin{lemma}\label{borchers} Let $({\cal A},V)$ be a local net of von Neumann algebras on $\mathbb{R}$.
\begin{enumerate}
\item[(a)] For any $\delta>0$, we define the following subset of ${\cal A}$:
\begin{equation}
\mathcal L_{\delta}=\{\, A(f)\, |\, A\in{\cal A}({\cal I}), {\cal I}\subset \mathbb{R}, f\in S(\mathbb{R}), \mathrm{supp}\,\,\tilde f\subset (-\infty,-\delta]\,\,\,\, \mathrm{ compact } \,\},
\label{LLL}
\end{equation}
where $A(f):=\int d\tau\,\beta_{\tau}(A)f(\tau)$.
Suppose that $B\Psi=0$ for any $B\in \mathcal L_{\delta}$.
Then we have $E_{\#}([\delta,\infty))\Psi=0$, where $E_{\#}(\,\cdot\,)$ is the spectral measure of $V$.
\item[(b)] The spectrum of $V$ is absolutely continuous, apart from a possible eigenvalue at zero.
\end{enumerate}
\end{lemma}
\noindent{\bf Proof. } The argument below, which is a one-dimensional version of the proof of Lemma~\ref{borchers-zero},
relies on ideas from Proposition~2.1 and 2.2 of \cite{BF82}. To prove (a)
we choose $b>a>\delta$ and $0<\varepsilon<a-\delta$. We pick a function $f\in S(\mathbb{R})$ s.t.\! $\mathrm{supp}\, \tilde f\subset (-\infty, -\delta ]$ is compact
and $\tilde f(\omega)=1$ for $\omega\in [-b,-a+\varepsilon]$.
Then $A(f)\in \mathcal L_{\delta}$ and, by assumption, $A(f)\Psi=0$. Making use of the fact that $\beta_s(A(f))\in \mathcal L_{\delta}$ for any $s\in\mathbb{R}$,
we obtain
\begin{equation}
E_{\#}(\Delta_2)A(f)E_{\#}(\Delta_1)\Psi=0,
\end{equation}
for any compact sets $\Delta_1$, $\Delta_2$. Setting $\Delta_1=[a,b]$, $\Delta_2=[0,\varepsilon]$ and exploiting the properties
of the function $f$, we obtain that
\begin{equation}
E_{\#}(\Delta_2)AE_{\#}(\Delta_1)\Psi=0.
\end{equation}
Since ${\cal A}$ acts irreducibly on $\hilk$, we conclude that $E_{\#}([a,b])\Psi=0$. Thus we obtain that either $E_{\#}([\delta,\infty))\Psi=0$
or $V(s)\Psi=\varepsilon^{i\delta s}\Psi$ for all $s\in\mathbb{R}$.
Let us now exclude the latter possibility: By Lemma 2.2 of \cite{Bu90}, (stated as Theorem~\ref{H-A} above), we obtain that
\begin{equation}
\int ds\,(\Psi|\beta_s(B^*B)\Psi)<\infty
\end{equation}
for any $B\in\mathcal L_{\delta'}$, $\delta'>0$.
This is only possible if $B\Psi=0$ for all such $B$. Proceeding as in the first part of the proof, we conclude that $V(s)\Psi=\Psi$ i.e., $\delta=0$,
which is a contradiction. This concludes the proof of (a).
To show (b), we pick $0<\varepsilon<\delta$ and note that
\begin{equation}
\mathrm{Span}\{\, E_{\#}([\delta,\delta+\varepsilon]) B^*\Psi \,|\, B\in \mathcal L_{\delta},\, \Psi\in E_{\#}([0,\varepsilon])\hilk \,\}^{\mathrm{cl}}=E_{\#}([\delta,\delta+\varepsilon])\hilk.
\end{equation}
In fact, any vector from $E_{\#}([\delta,\delta+\varepsilon])\hilk$, which is orthogonal to the subspace on the l.h.s.\! is zero
by relation~(\ref{Arveson}) and part (a) of the present lemma. Next, by irreducibility, for any $A(f)\in \mathcal L_{\delta}$
where $\mathrm{supp}\, \tilde f \subset (-\infty, -\delta]$
we can find $A_n\in {\cal A}({\cal I}_n)$ s.t. $E_{\#}([\delta,\delta+\varepsilon])A(f)^*=\mathrm{s}\textrm{-}\lim_{n\to\infty} A_n(f)^*$. Consequently,
\begin{equation}
\mathrm{Span}\{\, B^*\Psi \,|\, B\in \mathcal L_{\delta},\, \Psi\in E_{\#}([0,\varepsilon])\hilk \,\}^{\mathrm{cl}}\supset E_{\#}([\delta,\delta+\varepsilon])\hilk.
\end{equation}
Now for any $B_1, B_2\in \mathcal L_{\delta}$ and $\Psi_1,\Psi_2\in E_{\#}([0,\varepsilon])\hilk$, we get
\begin{eqnarray}
|(B_1^*\Psi_1|V(s)B_2^*\Psi_2)|&=&|(\Psi_1|[B_1, B_2^*(s)] V(s)\Psi_2)|\nonumber\\
&\leq& \|\Psi_1\|\,\|\Psi_2\|\,\| [B_1, B_2^*(s)]\|,
\end{eqnarray}
which is a rapidly decreasing function of $s$. Making use of these facts and of the Plancherel theorem, one easily obtains that
$(\Psi|E_{\#}(\Delta)\Psi')=0$ for any $\Psi,\Psi'\in E_{\#}((0,\infty))\hilk$ and $\Delta\subset \mathbb{R}$ of zero Lebesgue measure. $\Box$\medskip
\begin{lemma}\label{continuous-spectra} Let $({\cal A}_\lambda,V_\lambda)$ and $({\cal A}_\rho,V_\rho)$ be two nets of von Neumann algebras on ${\mathbb R}$ acting on $\hilk_\lambda$ and $\hilk_\rho$,
respectively, and let $({\mfa},U)$ be the corresponding chiral net. Then
\begin{equation}
\mathcal{H}_{\mathrm{c}}=\hilk_{\lambda,\mathrm{c}}\otimes\hilk_{\rho,\mathrm{c}}, \label{spectrum}
\end{equation}
where $\mathcal{H}_{\mathrm{c}}$ is the continuous subspace of $H^2-\boldsymbol P ^2$ and $\hilk_{\lambda/\rho,\mathrm{c}}$ are the continuous subspaces
of $V_{\lambda/\rho}$.
\end{lemma}
\noindent{\bf Proof. } Let $T_{\lambda/\rho}$ be the generators of $V_{\lambda/\rho}$ and $E_{L/R}$ their spectral measures. We obtain from relation~(\ref{unitary-representation}) that
$\boldsymbol P =\fr{1}{\sqrt{2}}(T_\lambda\otimes I-I\otimes T_\rho)$,
$H=\fr{1}{\sqrt{2}}(T_\lambda\otimes I+I\otimes T_\rho)$ and $H^2-\boldsymbol P ^2=2(T_\lambda\otimes T_\rho)$. Thus it follows immediately from
Lemma~\ref{borchers}~(b) that $\mathcal{H}_{\mathrm{c}}\subset\hilk_{\lambda,\mathrm{c}}\otimes\hilk_{\rho,\mathrm{c}}$. To prove the opposite inclusion,
we have to show that the r.h.s.\! of (\ref{spectrum}) does not contain any eigenvectors of $H^2-\boldsymbol P ^2$. Let us therefore assume
that there exists $\Psi\in \hilk_{\lambda,\mathrm{c}}\otimes\hilk_{\rho,\mathrm{c}}$ s.t.\! $(H^2-\boldsymbol P ^2)\Psi=m^2\Psi$, $m\geq 0$. Then, for any $\Psi_{\lambda/\rho}\in \hilk_{\lambda/\rho,\mathrm{c}}$,
we can write
\begin{eqnarray}
|(\Psi|\Psi_\lambda\otimes\Psi_\rho)|
&=&|\int_{V_+} (\Psi| d E(q_\lambda,q_\rho)(\Psi_\lambda\otimes \Psi_\rho))|\nonumber\\
&=&|\int_{H_m} (\Psi| d E(q_\lambda,q_\rho)(\Psi_\lambda\otimes \Psi_\rho))|\nonumber\\
&\leq& \|\Psi\| (\Psi_\lambda\otimes \Psi_\rho| E(H_m)(\Psi_\lambda\otimes \Psi_\rho))^\fr{1}{2}, \label{spectral-measures}
\end{eqnarray}
where $d E( q_{\lambda},q_{\rho})=dE_{\lambda}(q_{\lambda})\otimes dE_{\rho}(q_{\rho})$ is the joint spectral measure of $(H,\boldsymbol P )$ expressed in the lightcone coordinates
$q_{\lambda}:=\fr{\omega+\boldsymbol p} %{\mbox{\boldmath$p$}}{\sqrt{2}}$, $q_{\rho}:=\fr{\omega-\boldsymbol p} %{\mbox{\boldmath$p$}}{\sqrt{2}}$. Here
$H_m=\{\,(q_{\lambda},q_{\rho})\in\mathbb{R}_+^2\,|\, q_{\lambda} q_{\rho}=m^2/2\}$ is the hyperboloid at mass $m$ (or the boundary of the lightcone in the case $m=0$) and the second equality in (\ref{spectral-measures}) follows
from the assumption that $\Psi$ is an eigenvector of $H^2-\boldsymbol P ^2$. The measure $(\Psi_\lambda\otimes \Psi_\rho| E(\,\cdot\,)(\Psi_\lambda\otimes \Psi_\rho))$
appearing in the last line of (\ref{spectral-measures})
is a product of Lebesgue absolutely continuous measures
by Lemma~\ref{borchers}~(b), hence it is also absolutely continuous.
Since $H_m$ has Lebesgue measure zero, the expression on the r.h.s. of (\ref{spectral-measures}) is zero.
Thus $\Psi=0$, which concludes the proof. $\Box$\medskip
\begin{remark} We note that the above lemma could be proven without exploiting the absolute continuity of the spectral measures.
In fact, for any two positive operators $T_\lambda, T_\rho$ with empty point spectrum,
the operator $T_\lambda\otimes T_\rho$ also has empty point spectrum. This follows from the elementary fact
that if $\mu_\lambda,\mu_\rho$ are two measures on ${\mathbb R}$ without an atomic part,
then the product measure $\mu_\lambda\times\mu_\rho$ of any hyperboloid is zero.
\end{remark}
\noindent Now we are ready to prove our main result:
\begin{theoreme} Any chiral net $({\mfa},U)$ satisfies generalized asymptotic completeness in the sense of Definition~\ref{Asymptotic-completeness}.
\end{theoreme}
\noindent{\bf Proof. } First, we obtain from Lemma~\ref{continuous-spectra},
\begin{equation}
E(\Delta_{\varepsilon}(e_+} % {\mathrm{R} ,e_-} %{\mathrm{L} ))\mathcal{H}_{\mathrm{c}}=E_\lambda([e_+} % {\mathrm{R} ,e_+} % {\mathrm{R} +\varepsilon])\hilk_{\lambda,\mathrm{c}}\otimes E_\rho([e_-} %{\mathrm{L} ,e_-} %{\mathrm{L} +\varepsilon])\hilk_{\rho,\mathrm{c}}, \label{spectral-measure}
\end{equation}
where $\hilk_{\lambda/\rho,\mathrm{c}}$ and $E_{\lambda/\rho}(\,\cdot\,)$ are the continuous subspaces and spectral measures of $V_{\lambda/\rho}$. Now let $\mathcal L_{\lambda/\rho,\delta}\subset {\cal A}_{\lambda/\rho}$ be sets defined
as in (\ref{LLL}). It is easy to see that
if $B_\lambda\in\mathcal L_{\lambda,e_{+} % {\mathrm{R} }}$ and $B_\rho\in\mathcal L_{\rho,e_{-} %{\mathrm{L} }}$, then
$B_\lambda\otimes I\in \mathcal L_{+} % {\mathrm{R} ,e_{+} % {\mathrm{R} }} $ and $I\otimes B_\rho\in \mathcal L_{-} %{\mathrm{L} ,e_{-} %{\mathrm{L} }}$.
Moreover, we obtain
\begin{eqnarray}
\lim_{\eta\to 0}\lim_{T\to\infty}Q^{T,\eta}_{+} % {\mathrm{R} }(B_\lambda\otimes I)(\Psi_\lambda\otimes\Psi_\rho)&=& (Q(B_\lambda)\Psi_\lambda)\otimes \Psi_\rho,\label{detector-convergence-one}\\
\lim_{\eta\to 0}\lim_{T\to\infty}Q^{T,\eta}_{-} %{\mathrm{L} }(I\otimes B_\rho)(\Psi_\lambda\otimes\Psi_\rho)&=&\Psi_\lambda\otimes (Q(B_\rho)\Psi_\rho), \label{detector-convergence-two}
\end{eqnarray}
where $Q(B_{\lambda/\rho}):=\int d\tau\, \beta^{(\lambda/\rho)}_{\tau/\sqrt{2}}(B_{\lambda/\rho}^*B_{\lambda/\rho})$,
$\Psi_{\lambda/\rho}\in E_{\lambda/\rho}(\Delta_{\lambda/\rho})\hilk_{\lambda/\rho}$ and $\Delta_{\lambda/\rho}\subset\mathbb{R}$ are compact subsets.
To verify relation~(\ref{detector-convergence-one}), we note that
\begin{eqnarray}
& &\lim_{T\to\infty}Q^{T,\eta}_{+} % {\mathrm{R} }(B_\lambda\otimes I)(\Psi_\lambda\otimes\Psi_\rho)\nonumber\\
& &\phantom{44444444}=\lim_{T\to\infty} \left(\int dt\,h_T(t)\int_{-\infty}^{\infty} ds\, f_+^{\eta}(1-s/t)
\beta^{(\lambda)}_{\tau/\sqrt{2}}(B_{\lambda}^*B_{\lambda})\Psi_\lambda\right)\otimes \Psi_\rho
\nonumber\\
& &\phantom{44444444}=(Q(B_\lambda)\Psi_\lambda)\otimes \Psi_\rho,
\end{eqnarray}
where in the last step we made use of the fact that the sequence $ a\mapsto \int_{-a}^a ds\, \beta^{(\lambda)}_{\tau/\sqrt{2}}(B_{\lambda}^*B_{\lambda})\Psi_\lambda$
converges, as $a\to\infty$, in the norm topology of $\hilk_\lambda$ (cf. Theorem~\ref{H-A}) and $t\mapsto (f_+^{\eta}(1-s/t)-1)$ converges to zero, as $t\to\infty$, uniformly in $s\in [-a,a]$.
Equality~(\ref{detector-convergence-two}) is proven analogously.
As a consequence of (\ref{spectral-measure}), (\ref{detector-convergence-one}) and (\ref{detector-convergence-two}), we obtain
\begin{eqnarray}
& &Q^\mathrm{ out}_{+} % {\mathrm{R} }(B_\lambda\otimes I)Q^\mathrm{ out}_{-} %{\mathrm{L} }(I\otimes B_\rho) E(\Delta_{\varepsilon}(e_+} % {\mathrm{R} ,e_-} %{\mathrm{L} ))\mathcal{H}_{\mathrm{c}}\nonumber\\
&=&Q(B_\lambda)E_\lambda([e_+} % {\mathrm{R} ,e_+} % {\mathrm{R} +\varepsilon])\hilk_{\lambda,\mathrm{c}}\otimes Q(B_\rho) E_\rho([e_-} %{\mathrm{L} ,e_-} %{\mathrm{L} +\varepsilon])\hilk_{\rho,\mathrm{c}},
\end{eqnarray}
where we used Lemma A.2 of \cite{DT11}, as in the discussion after formula~(\ref{tensor-product-measures}) above.
To conclude the proof, it suffices to show that
\begin{eqnarray}
E_\lambda([e_+} % {\mathrm{R} ,e_+} % {\mathrm{R} +\varepsilon])\hilk_{\lambda,\mathrm{c}}&=&\mathrm{Span}\{\, Q(B_\lambda)E_\lambda([e_+} % {\mathrm{R} ,e_+} % {\mathrm{R} +\varepsilon])\hilk_{\lambda,\mathrm{c}}\,|\, B_\lambda\in \mathcal L_{\lambda,e_+}\,\}^{\mathrm{cl}}, \label{one-dim-two}\\
E_\rho([e_-} %{\mathrm{L} ,e_-} %{\mathrm{L} +\varepsilon])\hilk_{\rho,\mathrm{c}}&=&\mathrm{Span}\{\, Q(B_\rho)E_\rho([e_-} %{\mathrm{L} ,e_-} %{\mathrm{L} +\varepsilon])\hilk_{\rho,\mathrm{c}}\,|\, B_\rho\in \mathcal L_{\rho,e_-} \,\}^{\mathrm{cl}}.
\end{eqnarray}
It is enough to prove the first equality above, as the second one is analogous. We proceed similarly as in the proof of Theorem~\ref{main-theorem-waves}:
By the translational invariance of $Q(B_\lambda)$ it is obvious that the subspace on the r.h.s.\! of (\ref{one-dim-two}) is
contained in the subspace on the l.h.s. Let us now assume that the inclusion is proper, i.e., we can
choose a non-zero vector $\Psi \in (E_1-E_0)\mathcal{H}_{\mathrm{c}}$, where $E_1:=E_\lambda([e_+,e_++\varepsilon])$ and $E_0$ is
the orthogonal projection on the subspace on the r.h.s.\! of (\ref{one-dim-two}).
By Lemma~\ref{borchers}, there is an operator $B_\lambda\in\mathcal L_{\lambda,e_+}$
s.t.\! $B_\lambda\Psi \ne 0$. Then it is easy to see that $(\Psi| Q(B_\lambda)\Psi) \ne 0$
which means that $(E_1-E_0)Q(B_\lambda) \Psi\ne 0$. Hence $Q(B_\lambda)\Psi \neq 0$ and $Q(B_\lambda)\Psi \notin E_0\hilk_{\lambda,\mathrm{c}}$,
which contradicts the definition of $E_0$. $\Box$\medskip
\section{Completely rational conformal nets}\label{rational}
\setcounter{equation}{0}
In this section we consider particle aspects of completely rational conformal nets,
whose definition is summarized below. This class contains massless two-dimensional
theories in a vacuum representation which are not asymptotically complete in the
sense of waves. Nevertheless, as we show below, they have the property of generalized
asymptotic completeness.
In the previous section we introduced the concept of a local net $({\cal A},V)$ of von Neumann algebras
on $\mathbb{R}$. Suppose that ${\cal A}$ extends to a local net on the circle $S^1$ (understood as a one-point compactification
of $\mathbb{R}$) and $V$ extends to a unitary representation of the universal covering of the M\"obius group $\overline{\rm\textsf{M\"ob}}$,
s.t.\! covariance still holds. Then we call the extension (resp. the original net) a \bf M\"obius covariant \rm net on $S^1$ (resp. on $\mathbb{R}$). Similarly,
if $V$ extends to a projective unitary representation of the group of orientation preserving diffeomorphisms of $S^1$, denoted
by ${\rm Diff}(S^1)$,
s.t.\! covariance still holds and $V(g)AV(g)^*=A$ if $A\in {\cal A}({\cal I})$ and $g\in{\rm Diff}(S^1)$ acts identically on ${\cal I}$,
then we say that the extension (resp. the original net) is a \bf conformal \rm net on $S^1$ (resp. on $\mathbb{R}$).
A conformal net $({\cal A},V)$ on $S^1$ is said
to be {\bf completely rational} \cite{KLM01} if the following conditions hold:
\begin{enumerate}
\item {\bf Split property.} For intervals ${\cal I}_1,{\cal I}_2 \subset S^1$, where
$\overline{{\cal I}_1} \subset {\cal I}_2$, there is a type I factor $\mathcal F$
such that ${\cal A}({\cal I}_1)\subset \mathcal{F} \subset {\cal A}({\cal I}_2)$.
\item {\bf Strong additivity.} For an interval ${\cal I}$ and ${\cal I}_1,{\cal I}_2$ which are made from ${\cal I}$ by
removing an interior point, it holds that ${\cal A}({\cal I}) = {\cal A}({\cal I}_1)\vee {\cal A}({\cal I}_2)$.
\item {\bf Finite $\mu$-index.} For disjoint intervals ${\cal I}_1,{\cal I}_2,{\cal I}_3,{\cal I}_4$ with a clockwise (or counterclockwise)
order and with the union dense in $S^1$, the Jones index of the inclusion
${\cal A}({\cal I}_1)\vee{\cal A}({\cal I}_3) \subset \left({\cal A}({\cal I}_2)\vee{\cal A}({\cal I}_4)\right)'$ is finite.
\end{enumerate}
Among the consequences, we recall that a completely rational net ${\cal A}$ has only
finitely many sectors and any (locally normal) representation of ${\cal A}$
(on a separable Hilbert space) can be decomposed into a direct sum of irreducible representations \cite{KLM01}.
Now let $({\mfa},U)$ be a local net of von Neumann algebras on ${\mathbb R}^2$ in a vacuum representation.
$({\mfa},U)$ is said to be {\bf M\"obius covariant} if the representation $U$ of translations extends to the group
$\overline{\rm\textsf{M\"ob}}\times\overline{\rm\textsf{M\"ob}}$ and the covariance still holds in the sense of local action
(see \cite{BGL93}).
If $U$ further extends to a projective unitary representation of
the group ${\rm Diff}(S^1)\times{\rm Diff}(S^1)$ which acts covariantly on the net, and it holds that
$U(g)AU(g)^* = A$ if $A \in {\mfa}({\cal O})$ and $g\in {\rm Diff}(S^1)\times{\rm Diff}(S^1)$ acts identically on $\mathcal{O}$, then the net ${\mfa}$ is said to be \bf conformal\rm.
See also \cite{KL04-2} for a general discussion on conformal nets on two-dimensional spacetime.
We define subgroups $\widetilde{G}_\lambda := \overline{\rm\textsf{M\"ob}}\times \{\iota\} \subset \overline{\rm\textsf{M\"ob}}\times\overline{\rm\textsf{M\"ob}}$
and $\widetilde{G}_\rho := \{\iota\}\times\overline{\rm\textsf{M\"ob}} \subset \overline{\rm\textsf{M\"ob}}\times\overline{\rm\textsf{M\"ob}}$,
where $\iota$ denotes the unit element in $\overline{\rm\textsf{M\"ob}}$.
Following \cite{Re00}, for any interval ${\cal I} \subset {\mathbb R}$,
we introduce the von Neumann algebra ${\cal A}_\lambda({\cal I}) = {\mfa}({\cal I}\times \mathfrak{J}) \cap U(\widetilde{G}_\rho)'$. This definition
does not depend on the choice of $\mathfrak{J}$, since the group $\overline{\rm\textsf{M\"ob}}$ acts transitively
on the set of intervals. Analogously, one defines ${\cal A}_\rho(\mathfrak{J}) := {\mfa}({\cal I}\times\mathfrak{J})\cap U(\widetilde{G}_\lambda)'$.
In this way we obtain two families of von Neumann algebras parametrized by intervals contained
in ${\mathbb R}$. It was shown by
Rehren that both ${\cal A}_\lambda$ and ${\cal A}_\rho$ extend to M\"obius covariant nets on the circle $S^1$
\cite[Section 2]{Re00}. If the net $({\mfa},U)$ is conformal, then both chiral components ${\cal A}_\lambda$ and ${\cal A}_\rho$ are nontrivial.
Indeed, they include the net generated by the diffeomorphisms of the form $g_\lambda\times {\rm id}$
and ${\rm id}\times g_\rho$, respectively. Such nets, generated by diffeomorphisms, are called the
Virasoro (sub)nets.
We say that a conformal net $({\mfa},U)$ on ${\mathbb R}^2$ is {\bf completely rational} if its chiral components
${\cal A}_\lambda,{\cal A}_\rho$ are completely rational. From the two nets ${\cal A}_\lambda$ and ${\cal A}_\rho$ we can construct the
chiral net ${\cal A}_\lambda\otimes{\cal A}_\rho$ as in the previous section, which can be naturally identified
with a subnet of ${\mfa}$. It is easy to see that the inclusion ${\cal A}_\lambda({\cal I})\otimes{\cal A}_\rho(\mathfrak{J}) \subset {\mfa}({\cal I}\times \mathfrak{J})$ is irreducible
(namely, the relative commutant is trivial). Indeed, any element in ${\mfa}({\cal I}\times\mathfrak{J})$ commutes with
diffeomorphisms supported outside ${\cal I}\times\mathfrak{J}$, which are contained in ${\cal A}_\lambda({\cal I}')\otimes{\cal A}_\rho(\mathfrak{J}')$,
where ${\cal I}'$ denotes the interior of the complement of ${\cal I}$ (in ${\mathbb R}$ or in $S^1$, which does not matter thanks to the
strong additivity).
By the strong additivity, an element in the relative commutant must commute with any diffeomorphism. Hence it must be a multiple of the identity, since ${\mfa}$ is in a vacuum representation. From this and the complete rationality,
it follows that the Jones index of the inclusion ${\cal A}_\lambda({\cal I})\otimes{\cal A}_\rho(\mathfrak{J}) \subset {\mfa}({\cal I}\times \mathfrak{J})$ is finite
\cite[Proposition 2.3]{KL04-1}. Thus the natural representation $\pi_{\mfa}$ of ${\cal A}_\lambda\otimes{\cal A}_\rho$ on
the Hilbert space ${\cal H}$ of ${\mfa}$ decomposes into a finite direct sum of irreducible representations.
If ${\cal A}_\lambda$ and ${\cal A}_\rho$ are both completely rational, then any irreducible representation of the
chiral net ${\cal A}_\lambda\otimes{\cal A}_\rho$ is a product representation
\cite[Lemma 27]{KLM01}. From this it follows that if ${\mfa}$ is completely rational,
then the Hilbert space ${\cal H}$ can be decomposed into a direct sum
of finitely many product representation spaces of ${\cal A}_\lambda$ and ${\cal A}_\rho$.
Thus the representation of the Virasoro subnets decomposes as well.
The representation $U$ of the spacetime translations can be obtained from
local diffeomorphisms, hence any representative $U(t,\xb)$ is contained in
$\bigcup_{{\cal I}\times \mathfrak{J}} {\cal A}_\lambda({\cal I})\otimes{\cal A}_\rho(\mathfrak{J})$.
According to the decomposition $\pi_{\mfa} = \bigoplus_i \pi_i$ of the natural inclusion representation of
${\cal A}_\lambda\otimes{\cal A}_\rho$, $U$ is decomposed into a direct sum $\bigoplus_i U_i$
and each $U_i$ implements the translations in the representation $\pi_i$.
In other words, we obtain a decomposition of $U$ which is consistent with the above decomposition
of ${\cal H}$.
In the previous sections we saw that any product representation of a
chiral net is asymptotically complete in the sense of
Definition \ref{Asymptotic-completeness}.
It is easy to check that the direct sum of asymptotically complete
representations is again asymptotically complete. Thus we obtain:
\begin{theoreme}\label{th:ac-in-crnet}
Any completely rational net represented on a separable Hilbert space is asymptotically complete in the sense of
Definition \ref{Asymptotic-completeness}.
\end{theoreme}
Recall that a two-dimensional conformal net is asymptotically complete in the sense of waves
if and only if it coincides with the chiral net ${\cal A}_\lambda\otimes{\cal A}_\rho$ \cite[Corollary 4.6]{Ta11}.
For a non-trivial
extension of a chiral net (see \cite{KL04-2} for examples and a classification result of
a certain class of conformal nets) asymptotic completeness in the
sense of waves fails, but generalized asymptotic completeness remains valid in the completely rational case
in view of the above theorem.
|
2,869,038,154,763 | arxiv | \section{Introduction}\label{sec:intro}
The ubiquitous Traveling Salesman problem \cite{lawler,gutin,applegate} is to find a tour of edges on a finite graph that returns to the initial vertex and has the shortest possible length. The Analyst's Traveling Salesman problem \cite{Jones-TST,Schul-survey} is to find a rectifiable curve that contains a finite or infinite, bounded set of points in a metric space that has the shortest possible length. The former problem always has a solution yet is computationally hard, while the latter problem may or may not have any solution at all. A sophisticated example from Geometric Measure Theory of a bounded point set that is not contained in any rectifiable curve is a \emph{Besicovitch irregular} set \cite{Bes28} (see \S \ref{sec:gmt} below); a trivial example is a solid square in the plane. Tests to decide which sets are contained in a rectifiable curve have been found in $\mathbb{R}^2$ \cite{Jones-TST}, $\mathbb{R}^N$ \cite{Ok-TST}, $l^2$ \cite{Schul-Hilbert}, the first Heisenberg group \cite{Li-Schul1,Li-Schul2}, Carnot groups \cite{Carnot-TSP,Li-TSP}, Laakso-type spaces \cite{Guy-Schul}, and in general metric spaces \cite{Hah05,Hah08,DS-metric}. Applications of Jones' and Okikiolu's solution of the Analyst's TSP in $\mathbb{R}^N$ have been given in Complex Analysis \cite{BJ,Bishop-dyn, Bishop-QC}, Dynamics and Probability \cite{wiggly, frontier}, Geometric Measure Theory \cite{BS1,BS3}, Harmonic Analysis \cite{Tolsa-P}, and Metric Geometry \cite{Naor-Peres,shortcuts}.
Let $E\subset\mathbb{R}^N$ be a nonempty set and let $Q\subset \mathbb{R}^N$ be a bounded set of positive diameter (such as a ball or a cube). Following \cite{Jones-TST}, the \emph{Jones beta number} $\beta_{E}(Q)$ is defined by
\begin{equation*} \beta_{E}(Q) := \inf_{\ell} \sup_{x\in E\cap Q}\frac{\mathop\mathrm{dist}\nolimits(x,\ell)}{\mathop\mathrm{diam}\nolimits{Q}}\in[0,1],\end{equation*}
where $\ell$ ranges over all straight lines in $\mathbb{R}^N$, if $E\cap Q\neq\emptyset$, and by $\beta_E(Q)=0$, if $E\cap Q=\emptyset$. Let $\Delta(\mathbb{R}^N)$ denote the family of dyadic cubes in $\mathbb{R}^N$,
\begin{equation*} \Delta(\mathbb{R}^N) := \{ [2^{k}m_1,2^{k}(m_1+1)]\times\cdots\times [2^{k}m_N,2^{k}(m_N+1)] : m_1,\dots,m_N, k\in \mathbb{Z}\}.\end{equation*} Given a cube $Q$ and a scaling factor $\lambda>0$, we let $\lambda Q$ denote the concentric dilate of $Q$ by $\lambda$.
\begin{atst}[{\cite{Jones-TST, Ok-TST}}] A bounded set $E\subset\mathbb{R}^N$ is contained in a rectifiable curve $\Gamma=f([0,1])$ if and only if
\begin{equation}\label{eq:betasum}
S_E:=\sum_{Q\in\Delta(\mathbb{R}^N)}\beta_{E}(3Q)^2 \mathop\mathrm{diam}\nolimits{Q} < \infty.
\end{equation}
More precisely, \begin{enumerate}
\item If $\Gamma$ is any curve containing $E$, then $\mathop\mathrm{diam}\nolimits E + S_E\lesssim_N \mathop\mathrm{length}\nolimits(\Gamma)$.
\item If $S_E<\infty$, then there exists a curve $\Gamma\supset E$ such that $\mathop\mathrm{length}\nolimits(\Gamma)\lesssim_N \mathop\mathrm{diam}\nolimits E+ S_E$.
\end{enumerate}
\end{atst}
We may refer to statements (1) and (2) as the \emph{necessary half} and \emph{sufficient half} of the Analyst's Traveling Salesman theorem, respectively. The theorem is valid if the length of a curve $\Gamma=f([0,1])$ is interpreted either as the 1-dimensional Hausdorff measure of the set $\Gamma$ or as the total variation of the parameterization $f$. A curious feature of the known proofs of the sufficient half of the Analyst's TST (see \cite{Jones-TST} or \cite{BS3}) is that a rectifiable curve $\Gamma$ containing the set $E$ is constructed as the limit of piecewise linear curves $\Gamma_k$ containing a $2^{-k}$-net for $E$ \emph{without constructing a parameterization} of $\Gamma_k$ or $\Gamma$. This aspect of the proof breaks the analogy with the classical TSP, in which one is asked to find a minimal tour of a graph.
In this paper, we provide a \emph{parametric proof} of the sufficient half of the Analyst's TST, which more closely parallels the classical TSP. Beyond its intrinsic interest, the method that we provide is important, because it allows us to establish multiscale tests to ensure that a bounded set of points in $\mathbb{R}^N$ is contained in a \emph{$(1/s)$-H\"older continuous curve} with $s\in(1,N)$. Rectifiable curves correspond precisely to the class of Lipschitz curves ($s=1$). Remarkably, in the H\"older Traveling Salesman theorem (see \S\S \ref{sec:statements} and \ref{sec:Holder}), we can replace approximation by lines in the definition of the Jones beta numbers with \emph{approximation by thin tubes}. For a self-contained statement of the ``parametric" Analyst's TST, see \S\ref{sec:Lip}. While our focus in this paper is primarily on sets, we are motivated by open questions about the structure of Radon measures \cite{BV,Badger-survey}. For applications of our H\"older Traveling Salesman theorems to fractional rectifiability of measures, see \S\ref{sec:gmt}.
\subsection{H\"older Traveling Salesman Theorem(s)}\label{sec:statements} A \emph{$(1/s)$-H\"older curve} $\Gamma$ in $\mathbb{R}^N$ is the image of a continuous map $f:[0,1]\rightarrow\mathbb{R}^N$ satisfying the H\"older condition, $$|f(x)-f(y)|\leq H|x-y|^{1/s}\quad\text{for all }x,y\in[0,1],$$ where $s\in[1,\infty)$ and $H$ is a finite constant independent of $x$ and $y$. A 1-H\"older curve is also called a \emph{Lipschitz curve} or a \emph{rectifiable curve}. While non-trivial rectifiable curves always have topological dimension 1 and asymptotically resemble a unique tangent line $\mathcal{H}^1$ almost everywhere, $(1/s)$-H\"older curves with $s>1$ exhibit a variety of more complicated behaviors. For example,
\begin{itemize}
\item an $m$-dimensional cube in $\mathbb{R}^N$ ($m\leq N$) is a $(1/m)$-H\"older curve;
\item the von Koch snowflake is a $\log_4(3)$-H\"older curve; and,
\item the standard Sierpi\'nski carpet is a $\log_8(3)$-H\"older curve.
\end{itemize}
In fact, Remes \cite{Remes} proved that any compact, connected self-similar set $K\subset\mathbb{R}^N$ of Hausdorff dimension $s$ that satisfies the open set condition is a $(1/s)$-H\"older curve. For related work on space-filling curves generated by graph-directed iterated function systems, see Rao and Zhang \cite{RZ16}.
Towards a H\"older version of the Analyst's Traveling Salesman theorem, the first and third authors proved in \cite{BV} as a test case that if $s>1$, $E\subset\mathbb{R}^N$ is bounded, and
\begin{equation*} \label{sum-no-beta} \sum_{\substack{Q\in \Delta(\mathbb{R}^N)\\ Q\cap E \neq \emptyset,\, \mathop\mathrm{side}\nolimits Q\leq 1}}(\mathop\mathrm{diam}\nolimits{Q})^s < \infty,\end{equation*}
then $E$ is contained in a $(1/s)$-H\"older curve. By establishing a parametric version of Jones' proof of the sufficient half of the Analyst's TST, we are able to obtain the following substantial improvement.
\begin{thm}[H\"older Traveling Salesman I]\label{thm:main2}
For all $N\geq 2$ and $s> 1$, there exists $\beta_0 \in (0,1)$ such that if $E\subset\mathbb{R}^N$ is bounded and
\begin{equation}\label{eq:thm2}
S^{s,+}_E:=\sum_{\substack{Q\in\Delta(\mathbb{R}^N)\\ \beta_E(3Q)\geq \beta_0}} (\mathop\mathrm{diam}\nolimits{Q})^s < \infty,
\end{equation}
then $E$ is contained in a $(1/s)$-H\"older curve. More precisely, $E\subset \Gamma=f([0,1])$ for some $(1/s)$-H\"older map $f:[0,1]\rightarrow\mathbb{R}^N$ with H\"older constant $H\lesssim_{N,s} \mathop\mathrm{diam}\nolimits E + (\mathop\mathrm{diam}\nolimits{E})^{1-s}S_E^{s,+}.$
\end{thm}
Condition (\ref{eq:thm2}) implies that at $\mathcal{H}^s$ almost every point, the set $E$ asymptotically lies in sufficiently \emph{thin tubes}. Theorem \ref{thm:main2} provides a sufficient test that identifies all subsets of some well-known H\"older curves such as snowflakes of small dimension. However, because of the richness of H\"older geometry, a condition using Jones beta numbers alone such as \eqref{eq:thm2} cannot be expected to hold for all subsets of every H\"older curve. Indeed \eqref{eq:thm2} fails when $E$ is a carpet or a square. For expanded discussion and related examples, see \textsection\ref{sec:notnec}.
Theorem 1.1 is a simplification of our main result, which is adapted to a nested sequence of separated sets in a finite or infinite-dimensional Hilbert space. See Theorem \ref{thm:main}.
To estimate the size of the constant $\beta_0$ in Theorem \ref{thm:main2}, see Lemma \ref{lem:flat2} and Remark \ref{rem:alpha-beta}. The following variant of Theorem \ref{thm:main2} is an immediate corollary, whose hypothesis does not require knowledge of $\beta_0$.
\begin{cor}[H\"older Traveling Salesman II] \label{cor:main} Suppose that $N\geq 2$, $s>1$, and $p>0$. If $E \subset \mathbb{R}^N$ is bounded and \begin{equation} \label{eq:sum-p} S_E^{s,p}:=\sum_{Q\in\Delta(\mathbb{R}^N)}\beta_{E}(3Q)^{p} (\mathop\mathrm{diam}\nolimits{Q})^s < \infty, \end{equation}
then $E$ is contained in a $(1/s)$-H\"older curve. More precisely, $E\subset \Gamma=f([0,1])$ for some $(1/s)$-H\"older map $f:[0,1]\rightarrow\mathbb{R}^N$ with H\"older constant \begin{equation*} H\lesssim_{N,s} \mathop\mathrm{diam}\nolimits E + \beta_0^{-p}(\mathop\mathrm{diam}\nolimits{E})^{1-s}S_E^{s,p},\end{equation*} where $\beta_0$ is the constant appearing in Theorem \ref{thm:main2}.
\end{cor}
A good exercise is to prove that any bounded set $E$ in $\mathbb{R}^N$ satisfying condition \eqref{eq:sum-p} with $s>1$ has zero $s$-dimensional Hausdorff measure. In \textsection\ref{sec:null}, we construct a countable, compact set $E$ (hence $\mathcal{H}^s(E)=0$) such that $E$ is not contained in any $(1/s)$-H\"older curve with $1\leq s<N$. Thus, Corollary \ref{cor:main} is nonvacuous.
\subsection{Overview of the proof of Theorem \ref{thm:main2}}\label{sec:overview}
In order to properly discuss the proof of Theorem \ref{thm:main2}, we quickly sketch the proof of the sufficient half of the Analyst's TST. The proof splits into three steps. In the first step, one uses the Jones beta numbers $\beta_E(3Q)$ (in particular, whether they are large or small) to construct a sequence of finite, connected graphs $G_k$ in $\mathbb{R}^N$ with straight edges that converge in the Hausdorff distance to a compact, connected set $G$ containing $E$. Each graph $G_k$ is obtained by refining $G_{k-1}$ and resembles a flat arc near points of $E$ that look flat at scale $2^{-k}$. In step two, one uses the structure of the graphs $G_k$ and the Pythagorean theorem to prove the existence of a constant $C>0$ such that
\begin{equation}\label{e:Jones-estimate} \mathcal{H}^1(G_{k+1}) \leq \mathcal{H}^1(G_{k}) + C \sum_{\substack{Q \in \Delta(\mathbb{R}^N) \\ \mathop\mathrm{side}\nolimits{Q}\simeq 2^{-k}}}\beta_E(3Q)^2 \mathop\mathrm{diam}\nolimits{Q}.\end{equation}
Condition (\ref{eq:betasum}) and {G}o\lambda \c{a}b's semicontinuity theorem (e.g.~see \cite{AO-curves}) ensure that $\mathcal{H}^1(G)\leq \liminf_{k\rightarrow\infty} \mathcal{H}^1(G_k) < \infty$. Thus, the first two parts of the proof yield a compact, connected set $G$ containing $E$ with $\mathcal{H}^1(G)<\infty$. The final step is to invoke Wa\.zewski's theorem to conclude \emph{existence} of a Lipschitz parameterization for $G$: if $G \subset\mathbb{R}^N$ is connected, compact, and $\mathcal{H}^1(G)<\infty$, then there exists a Lipschitz map $f:[0,1]\rightarrow\mathbb{R}^N$ such that $G=f([0,1])$ (see \cite[Theorem 4.4]{AO-curves} or \cite[Lemma 3.7]{Schul-Hilbert}). Note that the condition $\mathcal{H}^1(G)<\infty$ promotes connectedness of $G$ to local connectedness (because $G$ is a curve).
In the H\"older setting, there are at least two obstacles to following the approach above. First and foremost, a naive analogue of Wa\.zewski's theorem cannot hold for H\"older maps, since the condition $\mathcal{H}^s(G)<\infty$ does not imply a continuum is locally connected when $s>1$ (e.g.~the topologist's comb). What is more, even if $G$ is assumed to be an Ahlfors $s$-regular curve with finite $\mathcal{H}^s$ measure, we cannot conclude that $G$ is a $(1/s)$-H\"older curve; we provide examples in \textsection\ref{sec:regularcurve} using a theorem of Mart\'in and Mattila \cite{MM2000}. Another obstacle is the well-known failure of {G}o\lambda \c{a}b's semicontinuity theorem for Hausdorff measures $\mathcal{H}^s$ with $s>1$. Thus, in a proof of a H\"older Traveling Salesman theorem, estimating the Hausdorff measure of approximating sets has no direct use.
To overcome these obstacles, we reimagine the proof of the Analyst's TST, and in \S\ref{sec:curve}, give a procedure to construct a sequence of partitions $\{\mathscr{I}_k\}_{k\geq 0}$ of $[0,1]$ and a sequence of piecewise linear maps $\{f_k : [0,1] \to \mathbb{R}^N\}_{k\geq 0}$ that \emph{parameterize} approximating graphs $G_k$. Each map $f_k$ is built by carefully refining $f_{k-1}$ to ensure that $\|f_{k}-f_{k-1}\|_\infty\lesssim 2^{-k}$. This guarantees that the maps $f_k$ have a uniform limit $f$ whose image contains the Hausdorff limit of $G_k$ (and hence $E$). To prove that $f$ is H\"older continuous, one must estimate growth of the Lipschitz constants of the maps $f_k$ (see Appendix \ref{sec:lip-hold} for the basic method). In \textsection\ref{sec:mass}, we introduce a notion of \emph{mass} of intervals $I\in\mathscr{I}_k$, defined using the $s$-power of diameters of images $f_l(J)$ of intervals $J\subset I$, $l\geq k$. This lets us record estimates in the domain of the map rather than its image, and in \S\ref{sec:mass}, we provide a mass-centric analogue of \eqref{e:Jones-estimate} that is adapted to the H\"older setting. In turn, this lets us estimate the Lipschitz constants of the maps $f_k$ and complete the proof of the H\"older Traveling Salesman theorem in \S\ref{sec:Holder}. For completeness, we use our method to reprove and strengthen the sufficient half of the Analyst's TST in \S\ref{sec:Lip}.
\subsection{Wa\.zewski type theorem for flat continua}\label{sec:wazewski} The Hahn-Mazurkiewicz Theorem (e.g.~see \cite[Theorem 3.30]{hocking}) asserts that a set $E\subset\mathbb{R}^N$ is a continuous image of $[0,1]$ if and only if $E$ is compact, connected, and locally connected. The Wa\.zewski Theorem (for an attribution, see \cite{AO-curves}) asserts that $E\subset\mathbb{R}^N$ is a Lipschitz image of $[0,1]$ if and only if $E$ is compact, connected, and $\mathcal{H}^1(E)<\infty$. It is an easy exercise to check that every $(1/s)$-H\"older continuous image of $[0,1]$ is compact, connected, locally connected, and has $\mathcal{H}^s(E)<\infty$, but the converse fails when $s>1$ (see \S\ref{sec:regularcurve} below). This motivates the following, apparently open question: Is there a metric, geometric, and/or topological characterization of H\"older curves in $\mathbb{R}^N$?
The method of proof of the H\"older Traveling Salesman theorems leads to the following Wa\.zewski type theorem for flat continua. For the proof of Proposition \ref{thm:flat}, see \S\ref{sec:flatcontinua}. A set $E\subset\mathbb{R}^n$ is called \emph{Ahlfors $s$-regular} if there exist $0<c\leq C<\infty$ such that \begin{equation}\label{eq:regularity} cr^s \leq \mathcal{H}^s(E\cap B(x,r))\leq Cr^s\quad\text{for all $x\in E$ and $0<r\leq \mathop\mathrm{diam}\nolimits E$}.\end{equation}
We say that $E$ is \emph{lower (upper) Ahlfors $s$-regular} if the first (second) inequality in (\ref{eq:regularity}) holds for all $x\in E$ and $0<r\leq \mathop\mathrm{diam}\nolimits E$.
\begin{prop}\label{thm:flat}
There exists a constant $\beta_1 \in (0,1)$ such that if $s>1$ and $E\subset\mathbb{R}^N$ is compact, connected, $\mathcal{H}^s(E)<\infty$, $E$ is lower Ahlfors $s$-regular with constant $c$, and
\begin{equation}\label{e:mv}
\beta_{E}\left(\overline{B}(x,r)\right) \leq \beta_1\quad \text{for all }x\in E\text{ and }0<r\leq \mathop\mathrm{diam}\nolimits E, \end{equation}
then $E=f([0,1])$ for some injective $(1/s)$-H\"older continuous map $f:[0,1]\to\mathbb{R}^N$ with H\"older constant $H \lesssim_{s} c^{-1}\mathcal{H}^s(E)(\mathop\mathrm{diam}\nolimits E)^{1-s}$.
\end{prop}
Inclusion of lower Ahlfors regularity in the hypothesis of Proposition \ref{thm:flat} is justifiable, because it holds automatically when $s=1$, i.e.~ every non-trivial connected set is lower Ahlfors $1$-regular. When $s>1$, a non-trivial $(1/s)$-H\"older curve is not necessarily lower Ahlfors $s$-regular, and, in fact, could have zero $\mathcal{H}^s$ measure. Nevertheless, Mart\'in and Mattila \cite{MM1993} proved that if $\Gamma$ is a $(1/s)$-H\"older curve in $\mathbb{R}^N$ with $\mathcal{H}^s(\Gamma)>0$, then $$\liminf_{r\downarrow 0} \frac{\mathcal{H}^s(\Gamma\cap B(x,r))}{r^s}>0\quad \text{at $\mathcal{H}^s$-a.e.~$x\in\Gamma$}.$$ Even if it can be weakened, the lower regularity hypothesis in Proposition \ref{thm:flat} cannot be completely dropped: In \S\ref{sec:flatcurves-counterexample}, for any $s>1$ and $\beta_1 \in (0,1)$, we find a curve $E\subset\mathbb{R}^N$ with $\mathcal{H}^s(E) < \infty$ satisfying \eqref{e:mv} such that $E$ is not contained in a $(1/s)$-H\"older curve.
Sharp estimates on the Minkowski dimension of sets satisfying \eqref{e:mv} were provided by Mattila and Vuorinen \cite{MV}; for generalized \emph{Mattila-Vuorinen type sets}, see \cite{lsa}.
\subsection{Related Work} As noted above, one motivation for this paper is to develop tools to analyze the structure of Radon measures. See \S\ref{sec:gmt} for background and for an application of Corollary \ref{cor:main} to the fractional rectifiability of measures.
There is considerable interest in finding higher-dimensional analogues of the Analyst's Traveling Salesman theorem, for example finding a characterization of subsets of Lipschitz images of $[0,1]^2$. This problem is still open, but some positive steps were recently taken by Azzam and Schul \cite{AS-TST} for \emph{Hausdorff content lower regular sets}. Also see \cite{Villa}.
\subsection*{Acknowledgements} We would like to thank Mika Koskenoja for providing us with a copy of M. Remes' thesis \cite{Remes}. The first author would also like to thank Guy C. David for a useful conversation at an early stage of this project. We thank an anonymous referee for their close and careful reading of the initial manuscript.
\part{Proof of the H\"older Traveling Salesman Theorem}
In the first part of the paper, \S\S \ref{sec:prelim}--\ref{sec:Lip}, we establish several H\"older Traveling Salesman theorems, including Theorem \ref{thm:main2} and Theorem \ref{thm:main}. To start, in \S\ref{sec:prelim}, we introduce notation and essential concepts used in the proof, including nets, flat pairs, and variation excess. In \S\ref{sec:curve}, we present a refined version of Jones' Traveling Salesman construction, which takes a nested sequence $(V_k)_{k=0}^\infty$ of $\rho_kr_0$-separated sets, approximating lines $\{\ell_{k,v}:v\in V_k\}_{k=0}^\infty$, and associated errors $\{\alpha_{k,v}:v\in V_k\}_{k=0}^\infty$ and outputs a sequence of partitions $\mathscr{I}_k$ of $[0,1]$ and piecewise linear maps $f_k$ such that $f_k([0,1])\supset V_k$. In \S\ref{sec:mass}, we define and estimate a discrete $s$-variation of the maps $f_k$, which is adapted to the partitions $\mathscr{I}_k$ of the domain. When $s>1$, the total $s$-mass $\mathcal{M}_s([0,1])$ associated to the sequence of maps $f_k$ fills the role that 1-dimensional Hausdorff measure $\mathcal{H}^1$ plays in Jones' proof of the Analyst's TST. In \S\ref{sec:Holder}, we use the algorithm of \S\ref{sec:curve} and the mass estimates of \S\ref{sec:mass} to prove our main theorem (see Theorem \ref{thm:main}). Finally, in \S\ref{sec:Lip}, we use our method to obtain a stronger version of the sufficient half of the Analyst's Traveling Salesman theorem. The construction presented below can be carried out in any finite or infinite-dimensional Hilbert space.
\section{Preliminaries}\label{sec:prelim}
Given numbers $x,y\geq 0$ and parameters $a_1,\dots, a_n$, we may write $x \lesssim_{a_1,\dots,a_n} y$ if there exists a positive and finite constant $C$ depending on at most $a_1,\dots,a_n$ such that $x \leq C y$. We write $x \simeq_{a_1,\dots,a_n} y$ to denote $x \lesssim_{a_1,\dots,a_n} y$ and $y \lesssim_{a_1,\dots,a_n} x$. Similarly, we write $x\lesssim y$ or $x\simeq y$ to denote that the implicit constants are universal.
\subsection{Ordering flat sets}\label{sec:orderflat}
The following lemma shows that if a discrete set is sufficiently flat at the scale of separation, then there exists a natural linear ordering of its points. Estimates \eqref{e:graph1} and \eqref{e:graph2} are consequences of the Pythagorean theorem.
\begin{lem}[{\cite[Lemma 8.3]{BS3}}]\label{lem:approx}
Suppose that $V\subset\mathbb{R}^N$ is a $1$-separated set with $\operatorname{card}(V) \geq 2$ and there exist lines $\ell_1$ and $\ell_2$ and a number $\alpha \in (0,1/16]$ such that
\begin{equation*} \mathop\mathrm{dist}\nolimits(v,\ell_i) \leq \alpha \qquad\text{for all }v\in V\text{ and }i=1,2.\end{equation*}
Let $\pi_i$ denote the orthogonal projection onto $\ell_i$. There exist compatible identifications of $\ell_1$ and $\ell_2$ with $\mathbb{R}$ such that $\pi_1(v) \leq \pi_1(v')$ if and only if $\pi_2(v) \leq \pi_2(v')$ for all $v,v' \in V$. If $v_1$ and $v_2$ are consecutive points in $V$ relative to the ordering of $\pi_1(V)$, then \begin{equation}\label{e:graph1} \mathcal{H}^1([u_1,u_2]) \leq (1+ 3\alpha^2)\cdot \mathcal{H}^1([\pi_1(u_1),\pi_1(u_2)]) \quad\text{for all }[u_1,u_2]\subseteq[v_1,v_2].\end{equation}
Moreover, \begin{equation}\label{e:graph2} \mathcal{H}^1([y_1,y_2]) \leq (1+12\alpha^2)\cdot\mathcal{H}^1([\pi_1(y_1),\pi_1(y_2)])\quad\text{for all }[y_1,y_2]\subseteq\ell_2.\end{equation}
\end{lem}
Suppose that $V$, $\ell_1$, and $\pi_1$ are given as in Lemma \ref{lem:approx} and let $v,v_1,v_2 \in V$. Given an orientation of $\ell$ (that is, an identification of $\ell_1$ with $\mathbb{R}$), we say \emph{$v_1$ is to the left of $v_2$} and \emph{$v_2$ is to the right of $v_1$} if $\pi_{1}(v_1) < \pi_{1}(v_2)$. We say \emph{$v$ is between $v_1$ and $v_2$} if $\pi_1(v_1) \leq \pi_1 (v) \leq \pi_1(v_2)$ or $\pi_1(v_2) \leq \pi_1 (v) \leq \pi_1(v_1)$.
\begin{lem}\label{lem:var}
Suppose that $V\subset\mathbb{R}^N$ is a $\delta$-separated set with $\operatorname{card}(V) \geq 2$ and there exists a line $\ell$ and a number $\alpha\in (0,1/16]$ such that
$$\mathop\mathrm{dist}\nolimits(v,\ell) \leq \alpha\delta \quad\text{for all $v\in V$}.$$
Enumerate $V=\{v_1,\dots,v_n\}$ so that $v_{i+1}$ is to the right of $v_i$ for all $1\leq i\leq n-1$. Then \begin{equation} \label{e:var2} \sum_{i=1}^{n-1} |v_{i+1}-v_i|^s \leq (1+3\alpha^2)^s |v_1-v_n|^s\quad\text{for all }s\geq 1.\end{equation} Moreover, if $\operatorname{card}(V)\geq 3$, then
\begin{equation} \label{e:var3} \sum_{i=1}^{n-1}|v_{i+1}-v_i|^s \leq ((1+3\alpha^2)|v_1-v_n|-\delta)^s + \delta^s\quad\text{for all }s\geq 1.\end{equation}
\end{lem}
\begin{proof} Let $\pi$ denote the orthogonal projection onto $\ell$ and put $x_i:=\pi(v_i)$. Then $$|x_{i+1}-x_i|\leq |v_{i+1}-v_i| \leq (1+3\alpha^2)|x_{i+1}-x_i|\quad\text{for all } 1\leq i\leq n-1,$$ where the first inequality holds since projections are 1-Lipschitz and the second inequality holds by Lemma \ref{lem:approx}. Assume $s\geq 1$ and $\operatorname{card}(V)\geq 3$. Then
\begin{equation*}\begin{split} \sum_{i=1}^{n-1} \frac{|v_{i+1}-v_i|^s}{(1+3\alpha^2)^s} &\leq \left(\sum_{i=1}^{n-2} |x_{i+1}-x_i|^s\right)+|x_n-x_{n-1}|^s\\
&\leq \left(\sum_{i=1}^{n-2} |x_{i+1}-x_i|\right)^s +|x_n-x_{n-1}|^s\\
&= (|x_n-x_1|-|x_n-x_{n-1}|)^s + |x_n-x_{n-1}|^s\\
&\leq \left(|x_n-x_1| - \frac{\delta}{1+3\alpha^2}\right)^s + \left(\frac{\delta}{1+3\alpha^2}\right)^s\\
& \leq \left(|v_n-v_1|-\frac{\delta}{1+3\alpha^2}\right)^s + \left(\frac{\delta}{1+3\alpha^2}\right)^s,
\end{split}\end{equation*}
where the penultimate inequality holds because for any $M>0$, $\epsilon \in (0,M)$, and $s\geq 1$, the function $f(t) = t^s + (M-t)^s$ defined on $[\epsilon,M-\epsilon]$ attains its maximum at $t=\epsilon$. This establishes \eqref{e:var3}. Inequality \eqref{e:var2} follows from a similar (and easier) computation.
\end{proof}
\subsection{Nets, flat pairs, and variation excess} \label{sec:flat}
Let $(X,|\cdot|)$ denote the Hilbert space $l^2(\mathbb{R})$ of square summable sequences or the Euclidean space $\mathbb{R}^N$ for some $N\geq 2$.
Let $\mathscr{V}=\{(V_k,\rho_k)\}_{k\geq 0}$ be a sequence of pairs of nonempty finite sets $V_k$ in $X$ and numbers $\rho_k>0$. Assume that there exist $x_0 \in X$, $r_0>0$, $C^*\geq 1$, and $0<\xi_1\leq \xi_2<1$ such that $\mathscr{V}$ satisfies the following properties.
\begin{enumerate}
\item[(\textbf{V0})] When $k=0$, we have $\rho_0=1$. For all $k\geq 0$, we have $\xi_1 \rho_k \leq \rho_{k+1} \leq \xi_2 \rho_k$.
\item[(\textbf{V1})] When $k=0$, we have $V_0 \subset B(x_0,C^*r_0)$.
\item[(\textbf{V2})] For all $k\geq 0$, we have $V_{k} \subset V_{k+1}$.
\item[(\textbf{V3})] For all $k\geq 0$ and all distinct $v,v'\in V_k$, we have $|v-v'| \geq \rho_k r_0$.
\item[(\textbf{V4})] For all $k \geq 0$ and all $v\in V_{k+1}$, there exists $v'\in V_{k}$ such that $|v-v'| < C^*\rho_{k+1} r_0$.
\end{enumerate} With $C^*$ and $\xi_2$ given, define the associated parameter $$A^*:=\frac{C^*}{1-\xi_2}>C^*.$$
In addition to (V0)--(V4), assume that for each $k\geq 0$ and $v\in V_k$ we are given a number $\alpha_{k,v}\geq 0$ and a straight line $\ell_{k,v}$ in $X$ such that
\begin{equation}\tag{\textbf{V5}} \sup_{x\in V_{k+1}\cap B(v,30A^*\rho_k r_0)} \mathop\mathrm{dist}\nolimits(x,\ell_{k,v}) \leq \alpha_{k,v} \rho_{k+1}r_0. \end{equation}
We call the line $\ell_{k,v}$ an \emph{approximating line at $(k,v)$}.
The formulation of (V5) is motivated by \cite[Proposition 3.6]{BS3}. We remark that the number $\rho_{k+1}r_0$ appearing on the right hand side of (V5) is the scale of separation of points in $V_{k+1}$. While we allow $X=l^2(\mathbb{R})$, each $V_k$ can be identified with a subset of $\mathbb{R}^{N_k}$ for some increasing sequence $N_k$ if convenient, because each $V_k$ is finite and $V_k\subset V_{k+1}$.
\begin{lem}\label{rem:card}
Let $v\in V_k$ be such that $\alpha_{k,v} \leq 1/16$ and fix an orientation for $\ell_{k,v}$.
\begin{enumerate}
\item If $v'\in V_{k}\cap B(v,14A^*\rho_k r_0)$ is the first point to the right of $v$, then there exist fewer than $2+2.2C^*$ points of $V_{k+1}$ between $v$ and $v'$ (inclusive).
\item There exist fewer than $1.1C^*$ points of $V_{k+1}\cap B(v,C^*\rho_{k+1}r_0)$ to the right of $v$.
\end{enumerate}
\end{lem}
\begin{proof} The points in $V_{k+1}\cap B(v,30A^*\rho_k r_0 )$ are $\rho_{k+1}r_0$-separated and are linearly ordered by Lemma \ref{lem:approx}. Let $v'$ be the first point in $V_k\cap B(v,14A^*\rho_k r_0)$ to the right of $v$ and let $w_1,\dots,w_m$ denote the points in $V_{k+1}$ that lie between $v$ and $v'$ (inclusive). By (V4), each point $w_i$ belongs to $B(v,C^*\rho_{k+1} r_0)\cup B(v',C^*\rho_{k+1} r_0)$. Let $\pi_{k,v}$ denote the orthogonal projection onto $\ell_{k,v}$. By (V3) and \eqref{e:graph1}, $1.1|\pi(w_i)-\pi(w_j)| > |w_i-w_j|\geq \rho_{k+1}r_0$ for all distinct $i,j$, since $(1+3(1/16)^2)<1.1$. It follows that there are fewer than $1.1C^*$ points $w_i$ in $V_{k+1}\cap B(v,C^*\rho_{k+1} r_0)$ to the right of $v$ and fewer than $1.1C^*$ points $w_i$ in $V_{k+1}\cap B(v',C^*\rho_{k+1} r_0)$ to the left of $v'$. The first claim follows. A similar argument gives the second claim.
\end{proof}
\begin{definition}[flat pairs] \label{def:flatpairs} Fix a parameter $\alpha_0\in(0,1/16]$. For all $k\geq 0$, define $\mathsf{Flat}(k) $ to be the set of pairs $(v,v')\in V_k\times V_k$ such that
\begin{enumerate}
\item $\rho_k r_0\leq |v-v'| < 14A^* \rho_k r_0$,
\item $\alpha_{k,v} <\alpha_0$ and $v'$ is the first point in $V_k\cap B(v,14A^*\rho_k r_0 )$ to the left or to the right of $v$ with respect to ordering induced by $\ell_{k,v}$.
\end{enumerate} Define the corresponding set of geometric line segments $\mathscr{L}_k=\{[v,v']: (v,v')\in \mathsf{Flat}(k)\}$.
\end{definition}
Note that the collection $\mathsf{Flat}(k)$ of flat pairs is not symmetric in the sense that $(v,v') \in \mathsf{Flat}(k)$ does imply $(v',v)\in\mathsf{Flat}(k)$, because $\alpha_{k,v}$ does not control $\alpha_{k,v'}$.
\begin{lem}\label{lem:tridents}
Let $e_1$, $e_2$, $e_3$ be distinct elements of $\mathscr{L}_{k}$ for some $k\geq 0$.
\begin{enumerate}
\item Edges $e_1$ and $e_2$ intersect at most in a common endpoint.
\item Edges $e_1$, $e_2$ and $e_3$ do not have a common point.
\end{enumerate}
\end{lem}
\begin{proof} Let $e_1=[v_1,v_1']$, $e_2=[v_2,v_2']$, and $e_3=[v_3,v_3']$ represent distinct elements of $\mathscr{L}_k$, where $(v_i,v_i')\in \mathsf{Flat}(k)$ for all $i\in\{1,2,3\}$. If two or more of the edges intersect in a common point, say $\{e_i:i\in I_0\}$ for some $I_0\subset\{1,2,3\}$ with $\operatorname{card}(I_0)\geq 2$, then those edges are contained in $B(v_j,30A^*\rho_k r_0)$ for each $j\in I_0$, since each edge has diameter at most $14A^*\rho_k r_0$. Note that $V_k$ is a $\rho_k r_0$ separated set, $\mathop\mathrm{dist}\nolimits(v_i,\ell_{k,v_j})\leq \alpha_{k,v_j} \rho_{k+1}r_0< \alpha_{k,v_j}\rho_k r_0$ for all $i,j\in I_0$, and $\alpha_{k,v_j}\leq 1/16$. Thus, by Lemma \ref{lem:approx}, the vertices $\{v_i,v_i':i\in I_0\}$ are consistently linearly ordered according to their projections onto $\ell_{k,v_j}$ for each $j\in I_0$. Claims (1) and (2) follow immediately, since the segments in $\mathscr{L}_k$ emanating from a vertex $v\in V_k$ with $\alpha_{k,v}<\alpha_0$ are only drawn to the first vertex in $V_k \cap B(v,14A^*\rho_k r_0)$ to the left or right of $v$ with respect to the projection onto $\ell_{k,v}$.
\end{proof}
Given a pair $(v,v')\in \mathsf{Flat}(k)$, let $V_{k+1}(v,v')$ denote the set of all points $x \in V_{k+1}\cap B(v, 14A^*\rho_k r_0)$ such that $x$ lies between $v$ and $v'$ (including $v$ and $v'$).
\begin{definition}[variation excess] For all $s\geq 1$, for all $k\geq 0$, and for all $(v,v')\in\mathsf{Flat}(k)$, define the \emph{$s$-variation excess} $\tau_s(k,v,v')$ by $$\tau_s(k,v,v')|v-v'|^s = \max\left\{\left(\sum_{i=1}^{n-1} |v_{i+1}-v_i|^s\right)-|v-v'|^s,0\right\},$$ where $V_{k+1}(v,v')=\{v_1,\dots,v_n\}$ with $v_1=v$, $v_n=v'$, and $v_{i+1}$ is the first point to the right (or left) of $v_i$ for all $1\leq i\leq n-1$.\end{definition}
\begin{lem}\label{lem:monotone} For all $k\geq 0$ and $(v,v')\in\mathsf{Flat}(k)$, we have $\tau_1(k,v,v') \leq 3\alpha_{k,v}^2$.
\end{lem}
\begin{proof} Let $V_{k+1}(v,v')=\{v_1,\dots,v_n\}$, where $v_1=v$, $v_n=v'$, and $v_{i+1}$ is to the right of $v_i$ for all $1\leq i\leq n-1$. By Lemma \ref{lem:var}, with $s=1$, \begin{equation*}\sum_{i=1}^{n-1}|v_{i+1}-v_i| \leq (1+3\alpha_{k,v}^2)|v_n-v_1|=(1+3\alpha_{k,v}^2)|v-v'|.\end{equation*} Rearranging the inequality gives $\tau_1(k,v,v') \leq 3\alpha_{k,v}^2$.\end{proof}
We now demonstrate that when $s>1$, the variation excess $\tau_s(k,v,v')$ is zero whenever the set $V_{k+1}(v,v')$ lies in a sufficiently thin tube.
\begin{lem}[tube control] \label{lem:flat2} For all $s>1$, there exists $\epsilon_{s,C^*,\xi_1,\xi_2}\in(0,1/16]$ such that if $\alpha_{k,v} \leq \epsilon_{s,C^*,\xi_1,\xi_2}$, then $\tau_s(k,v,v')=0$ for all $(v,v')\in\mathsf{Flat}(k)$.\end{lem}
\begin{proof} Let $(v,v')\in \mathsf{Flat}(k)$ and enumerate $V_{k+1}(v,v')=\{v_1,\dots,v_n\}$ so that $v_1=v$, $v_n=v'$, and $v_{i+1}$ is to the right of $v_i$ for all $1\leq i\leq n-1$. If $n=2$, then $\sum_{i=1}^{n-1}|v_{i+1}-v_i|^s = |v-v'|^s$. Thus, suppose that $n\geq 3$. By Lemma \ref{lem:var}, with $\delta=\rho_{k+1}r_0$ and $\alpha=\alpha_{k,v}$,
\begin{align*}\sum_{i=1}^{n-1}|v_{i+1}-v_i|^s &\leq |v-v'|^s\left [ \left((1+3\alpha_{k,v}^2)-\frac{\rho_{k+1}r_0}{|v-v'|}\right)^s+\left(\frac{\rho_{k+1}r_0}{|v-v'|}\right)^s\right]\\ &\leq |v-v'|^s\left [ \left((1+3\alpha_{k,v}^2)-\frac{\xi_1}{14A^*}\right)^s+\left(\frac{\xi_1}{14A^*}\right)^s\right]=: A_{\alpha_{k,v}}|v-v'|^s,
\end{align*} because $\xi_2^{-1}\rho_{k+1} r_0 \leq |v-v'| \leq 14A^* \xi_1^{-1}\rho_{k+1}r_0$ and the function $$f(t)=((1+3\alpha_{k,v}^2)-t)^s + t^s\quad\text{on } [\xi_1/14A^*,\xi_2]$$ takes its maximum at $t=\xi_1/14A^*$. Since $s>1$, the coefficient $$A_\epsilon \rightarrow \left(1-\frac{\xi_1}{14A^*}\right)^s+\left(\frac{\xi_1}{14A^*}\right)^s<1\quad\text{as $\epsilon\to 0$}.$$ Thus, by continuity, there exists $\epsilon'>0$ such that $A_{\epsilon'}=1$. Let $\epsilon_{s,C^*,\xi_1,\xi_2}=\min\{\epsilon',1/16\}.$ Then $\sum_{i=1}^{n-1}|v_{i+1}-v_i|^s \leq |v-v'|^s$ whenever $\alpha_{k,v}\leq \epsilon_{s,C^*,\xi_1}$.
\end{proof}
\section{Traveling Salesman algorithm}\label{sec:curve}
For the rest of \S\S \ref{sec:curve} and \ref{sec:mass}, let $X$ denote the Hilbert space $l^2(\mathbb{R})$ or $\mathbb{R}^N$, let $(V_k)_{k\geq 0}$ be a sequence of sets in $X$ and let $\rho_k>0$ be a sequence of numbers satisfying (V0)--(V5) defined in \textsection\ref{sec:flat}. In addition, fix the parameter $\alpha_0\in(0,1/16]$ in Definition \ref{def:flatpairs}. For each integer $k\geq 0$, we will construct
\begin{enumerate}
\item two collections of pairwise disjoint, open intervals in $[0,1]$ denoted by $\mathscr{B}_k$ (called ``bridge intervals") and $\mathscr{E}_k$ (``edge intervals"),
\item two collections of pairwise disjoint, nondegenerate closed intervals in $[0,1]$ denoted by $\mathscr{F}_k$ (``frozen point intervals") and $\mathscr{N}_k$ (``non-frozen point intervals"), and
\item a continuous map $f_k :[0,1]\to X$
\end{enumerate}
that satisfy the following properties.
\begin{enumerate}
\item[\textbf{(P1)}] The four collections $\mathscr{B}_k$, $\mathscr{E}_k$, $\mathscr{F}_k$, $\mathscr{N}_k$ are mutually disjoint and for any $x\in[0,1]$ there exists unique interval $I$ contained in their union such that $x\in I$.
\item[\textbf{(P2)}] The map $f_k| I$ is affine on each $I\in\mathscr{E}_k\cup\mathscr{B}_k$ and the map $f_k|J$ is constant on each $J \in \mathscr{F}_k\cup\mathscr{N}_k$.
\item[\textbf{(P3)}] For all $I\in \mathscr{E}_k$, we have $ \mathop\mathrm{diam}\nolimits{f_k(I)} < 14A^*\rho_k r_0$.
\item[\textbf{(P4)}] The map $f_k| \bigcup\mathscr{E}_k$ is $2$-to-$1$; that is, for every $x\in \bigcup\mathscr{E}_k$ there exists a unique $x' \in \bigcup\mathscr{E}_k\setminus\{x\}$ such that $f_k(x) = f_k(x')$.
\item[\textbf{(P5)}] If $(v,v')\in \mathsf{Flat}(k)$, then there exists $I \in\mathscr{E}_k$ such that $f_k(I)$ joins $v$ with $v'$. Conversely, if $a$ and $b$ are endpoints of an interval $I\in \mathscr{E}_k$ and $\alpha_{k,f_{k}(a)} < \alpha_0$, then $(f_k(a),f_{k}(b))\in\mathsf{Flat}(k)$.
\item[\textbf{(P6)}] For each $I\in \mathscr{F}_k\cup \mathscr{N}_k$, the image $f_k(I) \in V_k$ and for each $v\in V_k$ there exists a unique $I\in\mathscr{N}_k$ such that $f_k(I) = v$.
\item[\textbf{(P7)}] If $J\in \mathscr{N}_k$ is such that $f_k(J)$ is an endpoint of $f_k(I)$ for some $I\in\mathscr{E}_k$, then there exists $I'\in \mathscr{E}_k$ (possibly $I'=I$) such that $f_k(I') = f_k(I)$ and $J\cap\overline{I'} \neq \emptyset$.
\end{enumerate}
\begin{lem}\label{lem:P5} Assume (P5) holds at stage $k\geq 0$. Let $a<b<a'<b'$ be such that $(a,b)$ and $(a',b')$ belong to $\mathscr{E}_{k}$, $f_k(b) = f_k(a')$, and $\alpha_{k,f_{k}(b)} \leq \alpha_0$. Then either
\begin{enumerate}
\item $f_k((a,b)) = f_k((a',b'))$ or
\item $f_{k}(b)$ lies between $f_k(a)$ and $f_k(b')$ with respect to order induced by $\ell_{k,f_k(b)}$.
\end{enumerate}
\end{lem}
\begin{proof} This is an immediate consequence of (P5) and Lemma \ref{lem:tridents}.\end{proof}
In \textsection\ref{sec:k=0}, we construct $\mathscr{E}_0$, $\mathscr{B}_0$, $\mathscr{N}_0$, $\mathscr{F}_0$, and $f_0$. In \textsection\ref{sec:hypotheses}, we formulate the inductive hypothesis. We construct the collections of intervals $\mathscr{E}_{k+1}$, $\mathscr{B}_{k+1}$, $\mathscr{N}_{k+1}$, $\mathscr{F}_{k+1}$, and $f_{k+1}$ in \textsection\textsection\ref{sec:bridges}--\ref{sec:noflatpoints}. We verify properties (P1)--(P7) in \textsection\ref{sec:concl}. Finally, in \textsection\ref{sec:choices}, we review choices in all choices of the algorithm.
\subsection{Step $0$}\label{sec:k=0}
Fix a point $v_0 \in V_0$. Let $G_0$ be the (not necessarily connected) graph with vertices $V_0$ and edges $\mathscr{L}_{0}$. Suppose that $G_0$ has components $G_0^{(1)}, \dots, G_0^{(l)}$ with $v_0\in G_0^{(1)}$.
\emph{Case 1.} Suppose that $V_0 = \{v_0\}$. Then set $\mathscr{E}_0 = \emptyset$, $\mathscr{B}_0 = \emptyset$, $\mathscr{N}_{0} = \{[0,1]\}$ and $\mathscr{F}_0 = \emptyset$. Define also $f_0:[0,1]\to X$ with $f_0(x) = v_0$ for all $x\in [0,1]$. Note that properties (P1)--(P7) are trivial in this case.
\emph{Case 2.} Suppose that $\operatorname{card}(V_0) \geq 2$ and that $l=1$, that is, $G_0$ is connected. We apply Proposition \ref{prop:graph} for $v_0$ with $\Delta=[0,1]$, $G=G_0$ and we obtain a collection of intervals $\mathcal{I}$ and a continuous map $g$. By Lemma \ref{lem:tridents}, each point $v\in V_0$ has valence at most $2$ in $G_0$ and there exists a component $J_v$ of $g^{-1}(v)$ such that if $e$ is an edge of $G_0$ that contains $v$ as an endpoint, then $e$ has a preimage $I\in\mathcal{I}$ such that $\overline{I}\cap J_v \neq \emptyset$. Let $\mathcal{N}$ be the collection of all such intervals $J_v$.
Set $\mathscr{E}_0 = \mathcal{I}$, $\mathscr{B}_0 = \emptyset$, $\mathscr{N}_0 = \mathcal{N}$, define $\mathscr{F}_0$ to be the components of $[0,1] \setminus\bigcup (\mathscr{E}_0\cup\mathscr{B}_0\cup\mathscr{N}_0)$, and let $f_0 = g$. Properties (P1)--(P7) follow from Proposition \ref{prop:graph}.
\emph{Case 3.} Suppose that $\operatorname{card}(V_0) \geq 2$ and that $l\geq 2$, that is, $G_0$ is disconnected. For each $j=2,\dots,l$ fix a vertex $u_j$ of $G_{0}^{(j)}$. Let $\{ I_1, \dots, I_{2l-2}\}$
be a collection of open intervals, enumerated according to the orientation of $[0,1]$, such that their closures are mutually disjoint and are contained in the interior of $[0,1]$. Let also $\{ J_{1},\dots, J_{2l-1}\}$
be the components of $I \setminus \bigcup_{j=1}^{2l-2}I_j$ enumerated according to the orientation of $[0,1]$. Applying Proposition \ref{prop:graph} for $G = G^{(1)}_0$, $v_0$ and $\Delta = J_{1}$, we obtain a family of open intervals $\mathcal{I}_1$, a map $g_1 : J_{1} \to G^{(1)}_0 $ and a family $\mathcal{N}_1$ of closed intervals. Similarly, for each $j=2,\dots,l$, applying Proposition \ref{prop:graph} for $G = G^{(j)}_0$, $u_j$ and $\Delta = J_{2j}$, we obtain a family of open intervals $\mathcal{I}_j$, a map $g_j : J_{2j} \to G^{(j)}_0 $ and a family $\mathcal{N}_j$ of closed intervals. There exists a continuous map $g:[0,1] \to X$ that extends the maps $g_j$ such that
\begin{enumerate}
\item $g(J_{2j+1}) = v_0$ for each $j\in \{1,\dots,l-1\}$;
\item $g|I_{j}$ is affine for each $j\in \{1,\dots,2l-2\}$ and $g(I_{2j-1}) = g(I_{2j}) = [u_j,v_0]$ for each $j\in \{1,\dots,l-1\}$.
\end{enumerate}
Set $\mathscr{E}_{0} = \bigcup_{j=1}^{l}\mathcal{I}_j$, $\mathscr{B}_{0} = \{I_{1},\dots, I_{2l-2}\}$, $\mathscr{N}_{0} = \bigcup_{j=1}^{l}\mathcal{N}_j$, define $\mathscr{F}_{0}$ to be the components of $[0,1] \setminus\bigcup (\mathscr{E}_0\cup\mathscr{B}_0\cup\mathscr{N}_0)$, and let $f_{0}|[0,1] = g$. Properties (P1)--(P7) follow from Proposition \ref{prop:graph}.
\subsection{Inductive hypothesis}\label{sec:hypotheses}
Suppose that for some $k\geq 0$ we have defined collections $\mathscr{B}_k$, $\mathscr{E}_k$ of open intervals in $[0,1]$, collections $\mathscr{F}_k$, $\mathscr{N}_k$ of nondegenerate closed intervals in $[0,1]$, and a continuous map $f_k :[0,1]\to X$, which satisfy properties (P1)--(P7).
We will define a new map $f_{k+1}:[0,1]\rightarrow X$ and new collections $\mathscr{B}_{k+1} , \mathscr{E}_{k+1}, \mathscr{F}_{k+1}, \mathscr{N}_{k+1}$,
\begin{align*}
\mathscr{B}_{k+1} &= \bigcup_{I\in\mathscr{B}_k\cup\mathscr{E}_k\cup\mathscr{F}_k\cup\mathscr{N}_k}\mathscr{B}_{k+1}(I), \quad &\mathscr{E}_{k+1} = \bigcup_{I\in\mathscr{B}_k\cup\mathscr{E}_k\cup\mathscr{F}_k\cup\mathscr{N}_k}\mathscr{E}_{k+1}(I),\\
\mathscr{F}_{k+1} &= \bigcup_{I\in\mathscr{B}_k\cup\mathscr{E}_k\cup\mathscr{F}_k\cup\mathscr{N}_k}\mathscr{F}_{k+1}(I), \quad &\mathscr{N}_{k+1} = \bigcup_{I\in\mathscr{B}_k\cup\mathscr{E}_k\cup\mathscr{F}_k\cup\mathscr{N}_k}\mathscr{N}_{k+1}(I),
\end{align*}
where $\mathscr{B}_{k+1}(I)$, $\mathscr{E}_{k+1}(I)$, $\mathscr{F}_{k+1}(I)$, $\mathscr{N}_{k+1}(I)$ are collections of intervals in $I$ that we define below. In particular:
\begin{itemize}
\item In \textsection\ref{sec:bridges}, we define the four collections and $f_{k+1}|I$ for $I\in\mathscr{B}_k$.
\item In \textsection\ref{sec:flatedges} and \textsection\ref{sec:noflatedges}, we define the four collections and $f_{k+1}|I$ for $I\in\mathscr{E}_k$.
\item In \textsection\ref{sec:fixed}, we define the four collections and $f_{k+1}|I$ for $I\in\mathscr{F}_k$.
\item In \textsection\ref{sec:flatpoints} and \textsection\ref{sec:noflatpoints}, we the four collections and $f_{k+1}|I$ for $I\in\mathscr{N}_k$.
\end{itemize}
\subsection{Step $k+1$: intervals in $\mathscr{B}_k$}\label{sec:bridges}
For any $I\in \mathscr{B}_k$ we set $\mathscr{B}_{k+1}(I) = \{I\}$, $\mathscr{E}_{k+1}(I) = \emptyset$, $\mathscr{F}_{k+1}(I) = \emptyset$, $\mathscr{N}_{k+1}(I) = \emptyset$, and we define $f_{k+1}|I = f_k|I$. In other words, bridge intervals are frozen and we make no changes on them.
\subsection{Step $k+1$: intervals in $\mathscr{E}_k$ with a at least one endpoint with flat image}\label{sec:flatedges}
Here we consider those intervals $I = (a_I,b_I) \in\mathscr{E}_{k}$ such that one of the $\alpha_{k,f_{k}(a_I)}$, $\alpha_{k,f_{k}(b_I)}$ is less than $\alpha_0$. If no such interval exists, we move to \textsection\ref{sec:noflatedges}. Assume now that such intervals exist. By (P4) and the induction step, such intervals come in pairs $\{I,I'\}$ where $f_k(I) = f_k(I')$ and $f_k(I)\cap f_k(J) = \emptyset$ for all $J\in\mathscr{E}_k\setminus\{I,I'\}$. Fix now such a pair $\{I,I'\}$. We choose one of the two intervals $I,I'$ to start with, say $I$.
Without loss of generality, assume that $\alpha_{k,f_k(a_{I})} < \alpha_0$. Let $\ell$ be the approximating line for $(k,f_k(a_I))$, oriented so that $f_k(a_{I})$ lies to the left of $f_k(b_{I})$. Let $V_{k+1,I}$ denote the points in $V_{k+1}\cap B(f_k(a_{I}),14A^* \rho_k r_0)$ that lie between $f_k(a_{I})$ and $f_k(b_{I})$ with respect to $\ell$, including $f_k(a_{I})$ and $f_k(b_{I})$. Enumerate $V_{k+1,I}$ from left to right,
\begin{equation*} V_{k+1,I} = \{v_1,\dots, v_l\}.\end{equation*} That is, $v_{i}$ lies to the left of $v_{i+1}$ for all $i\in\{1,\dots,l-1\}$, $v_1 = f_k(a_{I})$, and $v_l = f_k(b_{I})$.
\begin{rem}\label{rem:lleq6} By Lemma \ref{rem:card}, we have that $l<2+2.2C^*$.\end{rem}
Let $\{I_{1}, \dots, I_{l-1}\}$ be a collection of open intervals in $I$ with mutually disjoint closures, enumerated according to the orientation of $[0,1]$ so that the left endpoint of $I_1$ coincides with $a_{I}$ and the right endpoint of $I_{l-1}$ coincides with $b_{I}$. Let $\mathscr{N}_{k+1}(I)$ be the components of $[0,1]\setminus \bigcup_{i=1}^{l-1}I_i$ and $\mathscr{F}_{k+1}(I) = \emptyset$. Define
\begin{align*}
\mathscr{E}_{k+1}(I) &= \{I_i : |v_i-v_{i+1}| <14A^* \rho_{k+1}r_0\}, \\
\mathscr{B}_{k+1}(I) &= \{I_i : |v_i-v_{i+1}| \geq 14A^* \rho_{k+1}r_0\}.
\end{align*} Then define $f_{k+1}|\overline{I}$ continuously so that
\begin{enumerate}
\item $f_{k+1}$ is affine on each $J \in \mathscr{E}_{k+1}(I) \cup \mathscr{B}_{k+1}(I)$ and constant on each $J \in \mathscr{N}_{k+1}(I)\cup\mathscr{F}_{k+1}(I)$;
\item for each $j=1,\dots,l-1$, $f_{k+1}(\overline{I_j}) = [v_{j},v_{j+1}]$ mapping the left endpoint of $I_{j}$ onto $v_{j}$ and the right endpoint of $I_{j}$ onto $v_{j+1}$.
\end{enumerate}
See Figure \ref{fig:flatedges} for the image of $f_{k+1}(I)$.
Once we have defined the four families and $f_{k+1}$ for $I$, we work as follows for $I'$. First note that $V_{k+1,I}=V_{k+1,I'}$. Define $\psi_{I',I} : I' \to I$ to be the unique orientation-reversing linear map between $I'$ and $I$. Define
\begin{equation*} \mathscr{E}_{k+1}(I') = \{ \psi_{I',I}(J) : J \in \mathscr{E}_{k+1}(I)\} \quad\text{and}\quad \mathscr{B}_{k+1}(I') = \{ \psi_{I',I}(J) : J \in \mathscr{B}_{k+1}(I)\}\end{equation*}
This time, however, we set $\mathscr{F}_{k+1}(I')$ to be the components of $I' \setminus\bigcup_{i=1}^{l-1}I_i'$ and $\mathscr{N}_{k+1}(I') = \emptyset$. Define also $f_{k+1}|\overline{I'}$ continuously so that $f_{k+1}|I' = (f_{k+1}|I) \circ \psi_{I',I}$.
\begin{figure}
\includegraphics[width=.8\textwidth]{flatedges.png}
\caption{The image of $f_{k+1}(I)$: black arrows denote images of intervals in $\mathscr{E}_{k}\cup\mathscr{B}_k$, green arrows denote images of intervals in $\mathscr{E}_{k+1}$, and red arrows denote images of intervals in $\mathscr{B}_{k+1}$.}
\label{fig:flatedges}
\end{figure}
\begin{lem}\label{lem:2to1(1)}
For $i=1,2$, let $I_i\in\mathscr{E}_k$ be an interval with at least endpoint having flat image and let $I_i'\in \mathscr{E}_{k+1}(I_i)$. If $f_k(I_1)\neq f_k(I_2)$, then $f_{k+1}(I_1')\cap f_{k+1}(I_2') = \emptyset$.
\end{lem}
\begin{proof} Suppose that $I_1' \in \mathscr{E}_{k+1}(I_1)$, $I_2' \in \mathscr{E}_{k+1}(I_2) $, and $f_{k+1}(I_1')\cap f_{k+1}(I_2') \neq \emptyset$. Because $f_{k+1}(I_1')$ and $f_{k+1}(I_2')$ do not include their endpoints (since intervals in $\mathscr{E}_{k+1}$ are open), we conclude that $f_{k+1}(I_1')=f_{k+1}(I_2')$ by Lemma \ref{lem:tridents}. Now, the endpoints of the four intervals $f_{k}(I_1)$, $f_k(I_2)$, $f_{k+1}(I_1')$, and $f_{k+1}(I_2')$ lie in the $30A^*\rho_k r_0$ neighborhood of any flat endpoint of $f_{k}(I_1)$ or $f_{k}(I_2)$. In particular, the endpoints of the four intervals are linearly ordered by Lemma \ref{lem:approx}, and the endpoints of $f_{k+1}(I_i')$ lie between the endpoints of $f_{k}(I_i)$ by the construction of $\mathscr{E}_{k+1}(I_i)$. Because $f_{k+1}(I_1')=f_{k+1}(I_2')$, this forces $f_k(I_1)=f_k(I_2)$.
\end{proof}
\subsection{Step $k+1$: intervals in $\mathscr{E}_k$ with no endpoints with flat image}\label{sec:noflatedges}
Suppose that $I = (a_I,b_I) \in \mathscr{E}_k$ is such that $\alpha_{k,f_k(a_I)} \geq \alpha_0$ and $\alpha_{k,f_k(b_I)} \geq \alpha_0$. Then set $\mathscr{E}_{k+1}(I) = \emptyset$, $\mathscr{B}_{k+1}(I) = \{I\}$, $\mathscr{N}_{k+1}(I) = \emptyset$, $\mathscr{F}_{k+1}(I) = \emptyset$, and $f_{k+1}|I = f_k|I$. In other words, edge intervals with no endpoints with flat image become bridge intervals and remain bridge intervals for the rest of the construction.
\subsection{Step $k+1$: intervals in $\mathscr{F}_{k}$}\label{sec:fixed}
For any $I\in \mathscr{F}_k$ we set $\mathscr{E}_{k+1}(I) = \emptyset$, $\mathscr{B}_{k+1}(I) = \emptyset$, $\mathscr{F}_{k+1}(I) = \{I\}$, $\mathscr{N}_{k+1}(I) =\emptyset$ and we set $f_{k+1}|I = f_k|I$. In other words, frozen point intervals in $\mathscr{F}_k$ remain frozen for the rest of the construction.
\subsection{Step $k+1$: intervals in $\mathscr{N}_k$ with flat image}\label{sec:flatpoints}
We now consider the intervals $I \in \mathscr{N}_k$ for which $\alpha_{k,f_{k}(I)} < \alpha_0$. If no such interval exists we proceed to \textsection\ref{sec:noflatpoints}. Assume now that such intervals exist.
Let $I$ be such an interval and let $\ell$ be the approximating line for $(k,f_k(I))$. We consider three cases.
\subsubsection{Non-terminal vertices}\label{sec:nonterm} Suppose that there exist distinct $v,v' \in V_k\setminus \{f_k(I)\}$ such that $(f_k(I),v)$ and $(f_k(I),v')$ are in $\mathsf{Flat}(k)$.
By (P5), Lemma \ref{lem:P5} and the induction step, there exist $J, J' \in \mathcal{E}_k$ such that $f_k(I)$ and $v$ are the endpoints of $f_k(J)$, while $f_k(I)$ and $v'$ are the endpoints of $f_k(J')$. Hence, all points of $V_{k+1}$ between $v$ and $v'$ are contained in the image of $f_{k+1}(J \cup J')$ defined in \textsection\ref{sec:flatedges}. Set $\mathscr{E}_{k+1}(I) = \emptyset$, $\mathscr{B}_{k+1}(I) = \emptyset$, $\mathscr{N}_{k+1}(I) = \{I\}$, $\mathscr{F}_{k+1}(I) = \emptyset$ and $f_{k+1}|I = f_k|I$.
\subsubsection{1-sided terminal vertices}\label{sec:1-term} Suppose that there exists unique $v\in V_{k}\setminus \{f_{k}(I)\}$ such that $(f_k(I),v) \in \mathsf{Flat}(k)$. Fix an orientation for $\ell$ so that $v$ lies to the left of $f_k(I)$. As in \textsection\ref{sec:nonterm}, the points of $V_{k+1}$ that lie between $f_k(I)$ and $v$ are all contained in $f_{k+1}(J)$ for some $J\in \mathscr{E}_k$. Let $V_{k+1,I}$ denote the set that includes $f_k(I)$ and all points in $V_{k+1}\cap B(f_k(I),C^* \rho_{k+1} r_0)$ that lie to the right of $f_k(I)$. Enumerate $V_{k+1,I}= \{v_1,\dots,v_l\}$ from left to right. That is, $v_1 = f_k(I)$ and $v_l$ is the rightmost point of $V_{k+1,I}$
\begin{rem}\label{rem:lleq3}$\operatorname{card}{V_{k+1,I}}\leq 1+1.1C^*$ by Lemma \ref{rem:card}.\end{rem}
If $V_{k+1,I} = \{f_k(I)\}$, then set $\mathscr{E}_{k+1}(I) = \emptyset$, $\mathscr{B}_{k+1}(I) = \emptyset$, $\mathscr{N}_{k+1}(I) = \{I\}$, $\mathscr{F}_{k+1}(I) = \emptyset$ and $f_{k+1}|I = f_k|I$.
If $V_{k+1,I} \neq \{f_k(I)\}$, then let $G_{k+1,I}$ be the graph with vertices the points in $V_{k+1,I}$ and edges the segments $\{[v_1,v_2], \dots, [v_{l-1},v_l]\}$.
That is, $G_{k+1,I}$ forms a simple polygonal arc to the right of $f_k(I)$ joining $f_k(I)$ with $v_l$. Let $\mathcal{I}$ and $g$ be the collection and map given by Proposition \ref{prop:graph} for $\Delta = I$, $G=G_{k+1,I}$ and $v = f_k(I)$. For each $v'\in V_{k+1,I}$ fix a component of $I\setminus\bigcup\mathscr{I}_{k+1}(I)$ that is mapped onto $v'$ and let $\mathcal{N}$ be the collection of these components. Set $\mathscr{E}_{k+1}(I) = \mathcal{I}$, $\mathscr{B}_{k+1}(I) = \emptyset$, $\mathscr{N}_{k+1}(I) = \mathcal{N}$, and define $\mathscr{F}_{k+1}(I)$ to be the set of components of $I\setminus\bigcup(\mathscr{E}_{k+1}(I)\cup\mathscr{B}_{k+1}(I)\cup\mathscr{N}_{k+1}(I))$. Set $f_{k+1}|I = g$. See the left half of Figure \ref{fig:flatpoints} for the image of $f_{k+1}(I)$.
\subsubsection{2-sided terminal vertices}\label{sec:2-term} Suppose that there exists no point in $V_{k} \setminus\{f_k(I)\}$ such that $(f_k(I),v)\in\mathsf{Flat}(k)$. That is, $V_k\cap B(f_k(I),14A^* \rho_k r_0) = \{f_k(I)\}$. Set
\begin{equation*} V_{k+1,I} = V_{k+1}\cap B(f_k(I),C^* \rho_{k+1} r_0).\end{equation*} Fix an orientation for $\ell$ and enumerate $V_{k+1,I} = \{v_1,\dots,v_l\}$ from left to right.
\begin{rem}\label{rem:lleq5}
$\operatorname{card}(V_{k+1,I})\leq 1+ 2.2C^*$ by Lemma \ref{rem:card}.
\end{rem}
If $V_{k+1,I} =\{f_k(I)\}$, then set $\mathscr{E}_{k+1}(I) = \emptyset$, $\mathscr{B}_{k+1}(I) = \emptyset$, $\mathscr{N}_{k+1}(I) = \{I\}$, $\mathscr{F}_{k+1}(I) = \emptyset$ and $f_{k+1}|I = f_k|I$. If $V_{k+1,I} \neq \{f_k(I)\}$, then let $G_{k+1,I}$ be the graph with vertices the points in $V_{k+1,I}$ and edges the segments $\{[v_1,v_2], \dots, [v_{l-1},v_l]\}$. The remainder of the construction proceeds in the same way as in \textsection\ref{sec:1-term}. See the right half of Figure \ref{fig:flatpoints} for the image of $f_{k+1}(I)$.
\begin{figure}
\includegraphics[width=.8\textwidth]{flatpoints.png}
\caption{The image of $f_{k+1}(I)$: on the left, we have $f_{k+1}(I)$, where $I$ is as in \textsection\ref{sec:1-term}; on the right, we have $f_{k+1}(I)$, where $I$ is as in \textsection\ref{sec:2-term}.}
\label{fig:flatpoints}
\end{figure}
\begin{lem}\label{lem:2to1(2)}
Let $I_1,I_2\in \mathscr{N}_k$ be distinct intervals as in \textsection\ref{sec:flatpoints} and let $I_3 \in \mathscr{E}_k$ be an interval as in \textsection\ref{sec:flatedges}. If $I_i' \in \mathscr{E}_{k+1}(I_i)$ for $i=1,2,3$, then the segments $f_{k+1}(I_1')$, $f_{k+1}(I_2')$, and $f_{k+1}(I_3')$ are mutually disjoint.
\end{lem}
\begin{proof} This follows from similar arguments employed in the proof of Lemma \ref{lem:2to1(1)}.
\end{proof}
\subsection{Step $k+1$: intervals in $\mathscr{N}_k$ with non-flat image}\label{sec:noflatpoints}
In this final part of the algorithm, we define $\mathscr{E}_{k+1}(I)$, $\mathscr{B}_{k+1}(I)$, $\mathscr{N}_{k+1}(I)$, $\mathscr{F}_{k+1}(I)$ and $f_{k+1}|I$ for those $I\in\mathscr{N}_{k}$ such that $\alpha_{k,f_k(I)} \geq \alpha_0$. Let $\{I_1,\dots,I_n\} $ be an enumeration of such intervals. The construction in this case resembles that in Step $0$.
We start with $I_1$. Let $V_{k+1,I_1}$ be the set of points in $V_{k+1}\cap B(f_k(I_1),C^* \rho_{k+1} r_0)$ that are not images of some $I\in \mathscr{N}_{k+1}$ defined in \textsection\ref{sec:flatedges} and \textsection\ref{sec:flatpoints}. Let $\mathscr{L}_{k+1,I_1}$ be the set of edges in $\mathscr{L}_{k+1}$ that have an endpoint in $V_{k+1,I_1}$. Then define $\tilde{V}_{k+1,I_1}$ to be the union of $V_{k+1,I}$ and the set of all endpoints of edges in $\mathscr{L}_{k+1,I_1}$. By the triangle inequality, the set $\tilde{V}_{k+1,I_1}$ is subset of $B(v,15A^*\rho_{k+1} r_0)$. Finally, let $G_{k+1,I_1}$ denote the graph with vertices $\tilde{V}_{k+1,I_1}$ and with edges $\mathscr{L}_{k+1,I_1}$. We note that the graph $G_{k+1,I_1}$ may be connected or disconnected.
If $\tilde V_{k+1,I_1} = \{f_k(I_1)\}$, then we simply set $\mathscr{E}_{k+1}(I_1) = \emptyset$, $\mathscr{B}_{k+1}(I_1) = \emptyset$, $\mathscr{N}_{k+1}(I_1) = \{I_1\}$, $\mathscr{F}_{k+1}(I_1) = \emptyset$ and $f_{k+1}|I_1 = f_k|I_1$.
For the remainder of \S\ref{sec:noflatpoints}, let us assume that $\tilde V_{k+1,I_1}$ contains at least two points. Let $G_{k+1,I_1}^{(1)},\dots,G_{k+1,I_1}^{(l_1)}$ denote
the connected components of $G_{k+1,I_1}$, labeled so that $G_{k+1,I_1}^{(1)}$ is the component containing $f_k(I_1)$. There are two cases.
\subsubsection{Connected graph}\label{sec:conngraph} Suppose that $G_{k+1,I_1}$ is connected. Apply Proposition \ref{prop:graph} for $\Delta=I_1$, $G=G_{k+1,I_1}$ and $v=f_k(I_1)$ to obtain a collection of intervals $\mathcal{I}$ and a continuous map $g$.
If $v' \in V_{k+1,I_1}$, then $v'$ has valence at most 2 by Lemma \ref{lem:tridents}. Hence, by Proposition \ref{prop:graph}, there exists a component $J_{v'}$ of $g^{-1}(v')$ with the following property: \begin{quotation}If $v'$ is the endpoint of some $e \in \mathscr{L}_{k+1,I_1}$, then there exists $I \in \mathcal{I}$ such that $g(I) = e$ and $\overline{I}\cap J_{v'} \neq \emptyset$.\end{quotation} Let $\mathcal{N}$ be the collection of the fixed intervals $J_{v'}$ where $v' \in \tilde{V}_{k+1,I_1}$. Now define a set $\mathcal{E} \subseteq \mathcal{I}$ with two rules:
\begin{enumerate}
\item If $e\in \mathscr{L}_{k+1,I_1}$ has both its endpoints in $V_{k+1,I_1}$ then both components of $g^{-1}(e)$ are in $\mathcal{E}$.
\item If $e\in \mathscr{L}_{k+1,I_1}$ has one endpoint $v' \in V_{k+1,I_1}$ and another in $\tilde{V}_{k+1,I_1}\setminus V_{k+1,I_1}$, then only one component of $g^{-1}(e)$ (one that intersects $J_{v'}$) is in $\mathcal{E}$.
\end{enumerate}
Set $\mathscr{E}_{k+1}(I_1) = \mathcal{E}$, $\mathscr{B}_{k+1}(I_1) = \mathcal{I}\setminus\mathcal{E}$, $\mathscr{N}_{k+1}(I_1) = \mathcal{N}$, and define $\mathscr{F}_{k+1}(I_1)$ to be the set of components of $I_1 \setminus\bigcup(\mathscr{E}_{k+1}(I_1)\cup\mathscr{B}_{k+1}(I_1)\cup\mathscr{N}_{k+1}(I_1))$ and $f_{k+1}|I_1 = g$.
\subsubsection{Several components} Suppose that $l_1\geq 2$; that is, $G_{k+1,I_1}$ is disconnected. See Figure \ref{fig:noflatpoints} for the image of $f_{k+1}(I)$. We will add some edges which will make the graph connected and the preimage of these edges will be bridge intervals. To this end, for each $j\in \{2,\dots,l_1\}$ fix some point $v_j \in V_{k+1,I_1}\cap G_{k+1,I_1}^{(j)}$. Let $\{I_{1,1},\dots, I_{1,2l_1-2}\}$ be a collection of open intervals, enumerated according to the orientation of $[0,1]$, such that their closures are mutually disjoint and are contained in the interior of $I_1$. Let also $\{J_{1,1},\dots,J_{1,2l_1-1}\}$ be the components of $I_1 \setminus \bigcup_{j=1}^{2l_1-2}I_{1,j}$ enumerated according to the orientation of $[0,1]$.
Working as in \textsection\ref{sec:conngraph}, we obtain a family $\mathcal{I}_1$ of open intervals in $J_{1,1}$, a subset $\mathcal{E}_1 \subset \mathcal{I}_1$, a family $\mathcal{N}_1$ containing some components of $J_{1,1} \setminus \bigcup\mathcal{I}_1$ and a continuous map $g_1:J_{1,1} \to G_{k+1,I_1}^{(1)}$. Similarly, for each $j\in \{2,\dots,l_1\}$ we obtain a family $\mathcal{I}_{j}$ of open intervals in $J_{1,2(j-1)}$, a subset $\mathcal{E}_j \subset \mathcal{I}_{j}$, a family $\mathcal{N}_j$ containing some components of $J_{1,2(j-1)} \setminus \bigcup\mathcal{I}_j$ and a continuous map $g_j:J_{1,2(j-1)} \to G_{k+1,I_1}^{(j)}$.
There exists a continuous map $g:I_1 \to X$ that extends the maps $g_j$ such that
\begin{enumerate}
\item $g(J_{1,2j+1}) = f_k(I_1)$ for each $j\in \{1,\dots,l_1-1\}$;
\item $g|I_{1,j}$ is affine for all $j\in\{1,\dots,2l_1-1\}$ and $g(I_{1,2j-1}) = g(I_{1,2j}) = [v_j,f_k(I_1)]$ for all $j\in\{1,\dots,l_1-1\}$.
\end{enumerate}
Define edge intervals $\mathscr{E}_{k+1}(I_1) = \bigcup_{j=1}^{l_1}\mathcal{E}_j$ and bridge intervals $\mathscr{B}_{k+1}(I_1) = \bigcup_{j=1}^{2l_1-2}I_{1,j} \cup \bigcup_{j=1}^{l_1}(\mathcal{I}_j\setminus\mathcal{E}_j)$. Set $\mathscr{N}_{k+1}(I_1) = \bigcup_{j\in\{1,2,4,\dots,2l_1\}}\mathcal{N}_j$ and define $\mathscr{F}_{k+1}(I_1)$ to be the set of components of $I_1\setminus \bigcup(\mathscr{E}_{k+1}(I_1)\cup\mathscr{B}_{k+1}(I_1)\cup\mathscr{N}_{k+1}(I_1))$. Also, set $f_{k+1}|I_1 = g$.
\begin{figure}
\includegraphics[width=.5\textwidth]{nonflatpoints.png}
\caption{The image of $f_{k+1}(I)$: Blue segments represent edges in $\mathscr{L}_{k+1}$, black arrows represent images of intervals in $\mathscr{E}_k\cup\mathscr{B}_k$, green arrows represent images of intervals in $\mathscr{E}_{k+1}(I)$ and red arrows represent images of intervals in $\mathscr{B}_{k+1}(I)$.}
\label{fig:noflatpoints}
\end{figure}
\subsubsection{Inductive hypothesis} Inductively, suppose that for $i\in\{1,\dots,r-1\}$ we have defined $\mathscr{E}_{k+1}(I_i)$, $\mathscr{B}_{k+1}(I_i)$, $\mathscr{F}_{k+1}(I_i)$, $\mathscr{N}_{k+1}(I_i)$ and $f_{k+1}|I_i$. We work now for $I_r$. Let $V_{k+1,I_r}$ be the set of points in $V_{k+1}\cap B(f_k(I_1),C^* \rho_{k+1}r_0)$ that are not images of some $I\in \mathscr{N}_{k+1}$ defined in \textsection\ref{sec:flatedges} or in \textsection\ref{sec:flatpoints} or for the previous intervals $I_1,\dots,I_{r-1}$. Let $\mathscr{L}_{k+1,I_r}$ be the set of edges in $\mathscr{L}_{k+1}$ that have an endpoint in $V_{k+1,I_r}$ and let $\tilde{V}_{k+1,I_r}$ be the set of endpoints of edges in $\mathscr{L}_{k+1,I_r}$. Let now $G_{k+1,I_r}$ be the (not necessarily connected) graph with vertices the set $\tilde{V}_{k+1,I_r}$ and with edges the set $\mathscr{L}_{k+1,I_r}$. To continue, repeat the procedure carried out for $I_1$ \emph{mutatis mutandis}.
\begin{rem}\label{rem:neighbors}
By the choice of set $\mathcal{N}$ for $I_1$, it follows that if $I\in\mathscr{N}_{k+1}(I_1)$ and if $f_{k+1}(I)$ is the endpoint of $f_{k+1}(J)$ for some $J\in\mathscr{E}_{k+1}(I_1)$, then there exists $J'\in \mathscr{E}_{k+1}(I_1)$ (possibly $J'=J$) such that $f_{k+1}(J) = f_{k+1}(J')$ and $\overline{I}\cap\overline{J'} \neq \emptyset$. The same is true for all $I_j$.
\end{rem}
\begin{lem}\label{lem:2to1(3)}
Let $J_1\in \mathscr{N}_k$ be as in \textsection\ref{sec:noflatpoints}, let $J_2\in\mathscr{E}_k$ be as in \textsection\ref{sec:flatedges}, and let $J_3 \in \mathscr{N}_k$ be as in \textsection\ref{sec:flatpoints}. If $J_i' \in \mathscr{E}_{k+1}(J_i)$ for $i=1,2,3$, then the segments $f_{k+1}(J_1')$, $f_{k+1}(J_2')$, $f_{k+1}(J_3')$ are mutually disjoint.
\end{lem}
\begin{proof} By Lemma \ref{lem:2to1(2)} we know that $f_{k+1}(J_2')$ and $f_{k+1}(J_3')$ are disjoint. Fix an interval $J_1\in \mathscr{N}_k$ as in \textsection\ref{sec:noflatpoints}. Suppose that either $J_2 \in \mathscr{E}_k$ is as in \textsection\ref{sec:flatedges} or $J_2 \in \mathscr{N}_k$ is as in \textsection\ref{sec:flatpoints}. Let $J_1' \in \mathscr{E}_{k+1}(J_1)$ and $J_2' \in \mathscr{E}_{k+1}(J_2')$. By Lemma \ref{lem:tridents}, either $f_{k+1}(J_1')\cap f_{k+1}(J_2') \neq \emptyset$ or $f_{k+1}(J_1') = f_{k+1}(J_2')$. However, we have defined $\mathscr{L}_{k+1,J_1}$ as those elements in $\mathscr{L}_{k+1}$ that are not contained in $f_{k+1}(J)$, where $J \in \mathscr{E}_k$ is as in \textsection\ref{sec:flatedges} or $J \in \mathscr{N}_k$ is as in \textsection\ref{sec:flatpoints}. Thus, $f_{k+1}(J_1') \cap f_{k+1}(J_2') = \emptyset$.
\end{proof}
\subsection{Properties (P1)--(P7) for Step $k+1$}\label{sec:concl}
We have now defined $\mathscr{B}_{k+1}$, $\mathscr{E}_{k+1}$, $\mathscr{F}_{k+1}$, $\mathscr{N}_{k+1}$ and $f_{k+1}:[0,1]\rightarrow X$.
It remains to prove that $f_{k+1}$ is continuous and that properties (P1)--(P7) are satisfied by the new collections of intervals and $f_{k+1}$.
Properties (P1), (P2), and (P3) follow immediately from the construction.
\medskip
\textsc{Continuity of $f_{k+1}$.} By design, the map $f_{k+1}$ is continuous on every point interior to an interval in $\mathscr{E}_{k}\cup\mathscr{B}_{k}\cup\mathscr{N}_{k}\cup\mathscr{F}_{k}$. If $x$ is an endpoint of some interval in $\mathscr{E}_{k}\cup\mathscr{B}_{k}\cup\mathscr{N}_{k}\cup\mathscr{F}_{k}$, then $f_{k+1}(x) = f_k(x)$. Thus, continuity of $f_{k+1}$ at $x$ follows from continuity of $f_k$ at $x$.
\medskip
\textsc{Property (P6).} The first claim of (P6), that $f_{k+1}(I) \in V_{k+1}$ for all $I \in \mathscr{N}_{k+1}\cup\mathscr{F}_{k+1}$, is immediate from the construction. To check the second claim of (P6), fix $v\in V_{k+1}$. By (V4), there exists $v'\in V_k$ such that $|v-v'|< C^* \rho_{k+1} r_0$. By the inductive step, there exists $I\in \mathscr{N}_k$ such that $f_k(I) = v'$. There are two cases.
\emph{Case 1.} Suppose that $\alpha_{k,v'} < \alpha_0$. Then, following the discussion in \textsection\ref{sec:flatpoints}, either $v = f_{k+1}(J)$ for some $J \in \mathscr{N}_k(J')$ and $J'\in\mathscr{E}_k$ as in \textsection\ref{sec:flatedges}, or $v = f_{k+1}(J)$ for some $J \in \mathscr{N}_k(I)$.
\emph{Case 2.} Suppose that $\alpha_{k,v'} \geq \alpha_0$. Following the construction of the graph $G_{k+1,I}$ and the design of the map $f_{k+1}| I$, if $v$ is not the image of some $J \in \mathscr{N}_{k+1}(J')$, where $J' \in \mathscr{E}_k\cup\mathscr{B}_k\cup\mathscr{N}_k \setminus\{J'\}$, then there exists $J\in \mathscr{N}_{k+1}(I)$ such that $v = f_{k+1}(J)$.
\medskip
\textsc{Property (P4).} Fix $I\in\mathscr{E}_{k+1}$. There are three cases.
\emph{Case 1.} Suppose that $I\in \mathscr{E}_{k+1}(I_0)$ for some $I_0 \in \mathscr{E}_k$. By the inductive hypothesis, there exists unique $I_0' \in \mathscr{E}_k\setminus\{I_0\}$ such that $f_k(I_0')= f_k(I_0)$ while $f_k(I_0)\cap f_{k}(J)= \emptyset$ for all $J \in\mathscr{E}_k\setminus\{I_0,I_0'\}$. By construction, there exists $I'\in \mathscr{E}_{k+1}(I_0')$ such that $f_{k+1}(I) = f_{k+1}(I')$. Again by construction, $f_{k+1}(I)\cap f_{k+1}(J) = \emptyset$ for all $J \in \mathscr{E}_{k+1}(I_0)\cup\mathscr{E}_{k+1}(I_0') \setminus\{I,I'\}$. By Lemma \ref{lem:2to1(1)}, Lemma \ref{lem:2to1(2)} and Lemma \ref{lem:2to1(3)}, $f_{k+1}(I)$ does not intersect any $f_{k+1}(J)$ for any $J\in \mathscr{E}_{k+1}(J')$ and $J'\in\mathscr{E}_{k}\cup\mathscr{N}_{k}\setminus \{I_0,I_0'\}$.
\emph{Case 2.} Suppose that $I\in \mathscr{E}_{k+1}(I_0)$ for some $I_0 \in \mathscr{N}_k$ as in \textsection\ref{sec:flatpoints}. By construction, there exists $I' \in \mathscr{E}_{k+1}(I_0)\setminus\{I\}$ such that $f_{k+1}(I) = f_{k+1}(I')$ while $f_{k+1}(I)\cap f_{k+1}(J) = \emptyset$ for all $J\in\mathscr{E}_{k+1}(I_0)\setminus\{I,I'\}$. Moreover, by Lemma \ref{lem:2to1(2)} and Lemma \ref{lem:2to1(3)}, $f_{k+1}(I)$ does not intersect any $f_{k+1}(J)$ for any $J\in \mathscr{E}_{k+1}(J')$ and $J'\in\mathscr{E}_{k}\cup\mathscr{N}_{k}\setminus \{I_0\}$.
\emph{Case 3.} Suppose that $I \in \mathscr{E}_{k+1}(I_0)$ for some $I_0 \in \mathscr{N}_k$ as in \textsection\ref{sec:noflatpoints}. By Lemma \ref{lem:2to1(3)}, $f_{k+1}(I) \cap f_{k+1}(I') = \emptyset$ for all $I' \in \mathscr{E}_{k+1}(J)$ and all $J\in \mathscr{E}_k$ as in \textsection\ref{sec:flatedges} or $J\in \mathscr{N}_k$ as in \textsection\ref{sec:flatpoints}. By the construction of $f_{k+1}|I_0$, there are two possibilities.
\emph{Case 3a.} Suppose that both endpoints of $f_{k+1}(I_0)$ are in $V_{k_1,I_0}$. Then there exists an interval $I' \in \mathscr{E}_{k+1}(I_0)\setminus\{I\}$ such that $f_{k+1}(I)= f_{k+1}(I')$. On the other hand, $f_{k+1}(I) \not\in \mathscr{L}_{k+1,J}$ for any $J\in\mathscr{N}_{k}\setminus\{I_0\}$. Thus, by Lemma \ref{lem:tridents}, if $J' \in \mathscr{E}_{k+1}(J)$ and $J\in\mathscr{N}_{k}\setminus\{I_0\}$, then $f_{k+1}(J) \cap f_{k+1}(J') = \emptyset$.
\emph{Case 3b.} Suppose that only one endpoint of $f_{k+1}(I_0)$ is in $V_{k_1,I_0}$. In this case, by construction, $f_{k+1}(I)\cap f_{k+1}(J) = \emptyset$ for all $J \in \mathscr{E}_{k+1}(I_0)\setminus\{I\}$. Moreover, there exists unique $I_0' \in \mathscr{N}_k\setminus\{I_0\}$ as in \textsection\ref{sec:noflatpoints} such that $V_{k+1,I_0'}$ contains the other endpoint of $f_{k+1}(I)$. As with $I_0$, there exists unique $I'\in\mathscr{E}_{k+1}(I_0')$ such that $f_{k+1}(I') = f_{k+1}(I)$ while $f_{k+1}(J)\cap f_{k+1}(I) =\emptyset$ for all $J \in \mathscr{E}_{k+1}(I_0')$. Finally, by the construction and Lemma \ref{lem:tridents}, $f_{k+1}(I) \cap f_{k+1}(J) = \emptyset$ for all $J \in \mathscr{E}_{k+1}(J')$ and all $J' \in \mathscr{N}_k \setminus\{I_0,I_0'\}$ as in \textsection\ref{sec:noflatpoints}.
\medskip
\textsc{Property (P5).} To prove the first claim in (P5), fix $(v,v')\in \mathsf{Flat}(k+1)$. Let $v_0$ be the point of $V_k$ closest to $v$ and let $I_0 \in \mathscr{N}_k$ be such that $f_k(I_0) = v_0$. There are four cases.
\emph{Case 1.} Suppose that $\alpha_{k,v_0} < \alpha_0$ and $v_0$ is non-terminal (see \textsection\ref{sec:nonterm}). Then either both $v$ and $v'$ lie to the left of $v_0$ (with respect to $\ell-{k,v_0}$) or both lie to the right of $v_0$. In any case, $[v,v']$ is the preimage of some $I\in \mathscr{E}_{k+1}(J)$ under $f_{k+1}$ where $J \in \mathscr{E}_k$ and $f_{k}(J)$ is an edge with endpoint $f_k(I_0)$.
\emph{Case 2.} Suppose that $\alpha_{k,v_0} < \alpha_0$ and $v_0$ is 2-sided terminal (see \textsection\ref{sec:2-term}). Then $[v,v']$ is the preimage of some $I\in \mathscr{E}_{k+1}(I_0)$ under $f_{k+1}$.
\emph{Case 3.} Suppose that $\alpha_{k,v_0} < \alpha_0$ and $v_0$ is 1-sided terminal (see \textsection\ref{sec:1-term}). Then either both $v$ and $v'$ lie to the left of $v_0$ (with respect to $\ell_{k,v_0}$) or both lie to the right of $v_0$. Depending on their position, we work as in Case 1 or Case 2.
\emph{Case 4.} Suppose that $\alpha_{k,v_0} \geq \alpha_0$. By definition of graph $G_{k+1,I}$ in \textsection\ref{sec:noflatpoints}, the segment $[v,v']$ is the image of some $J\in \mathscr{E}_{k+1}$ under $f_{k+1}$.
\smallskip
To prove the second claim of (P5), fix $(a,b) \in\mathscr{E}_{k+1}$ such that one of its endpoints has flat image. Without loss of generality, assume $\alpha_{k+1,f_{k+1}(a)}< \alpha_0$.
\emph{Case 1.} Suppose that $(a,b) \in \mathscr{E}_{k+1}(I)$ for some $I\in\mathscr{N}_k$ as in \textsection\ref{sec:noflatpoints}. By construction of $f_{k+1}$ on such intervals, $(f_{k+1}(a),f_{k+1}(b)) \in \mathsf{Flat}(k+1)$.
\emph{Case 2.} Suppose that $(a,b) \in \mathscr{E}_{k+1}(I)$ for some $I\in\mathscr{N}_k$ as in \textsection\ref{sec:flatpoints}. By construction of $f_{k+1}$ on such intervals, no point of $V_{k+1}\cap B(f_{k+1}(a),14A^* \rho_{k+1}r_0)$, $$V_{k+1}\cap B(f_{k+1}(a),14A^* \rho_{k+1}r_0)\subset V_{k+1}\cap B(f_{k}(I),30A^*\rho_k r_0),$$ lies strictly between $f_{k+1}(a)$ and $f_{k+1}(b)$ with respect to $\ell_{k,f_k(I)}$. The same is true with respect to $\ell_{k+1,f_{k+1}(a)}$ by Lemma \ref{lem:approx}. Thus, $(f_{k+1}(a),f_{k+1}(b)) \in \mathsf{Flat}(k+1)$.
\emph{Case 3.} Suppose that $(a,b) \in \mathscr{E}_{k+1}(I)$ for some $I\in\mathscr{E}_k$ as in \textsection\ref{sec:flatedges}. The argument is similar to Case 2
\medskip
\textsc{Property (P7).} To check the final property, fix $J\in \mathscr{N}_{k+1}$ and choose $I\in \mathcal{E}_{k+1}$ such that $f_{k+1}(J)$ is an endpoint of $f_{k+1}(I)$. There are several cases.
\emph{Case 1.} Suppose $J\in\mathscr{N}_{k+1}(J_0)$ for some $J_0\in\mathscr{N}_k$ as in \textsection\ref{sec:noflatpoints}. Then there exists $I' \in \mathscr{E}_{k+1}(J_0)$ such that $f_{k+1}(I') = f_{k+1}(I)$. By the construction of $\mathscr{N}_{k+1}(J_0)$ in \S\ref{sec:noflatpoints}, $\overline{I'}\cap J \neq \emptyset$.
\emph{Case 2.} Suppose $J\in \mathscr{N}_{k+1}(J_0)$ for some $J_0 \in \mathscr{E}_k$ as in \textsection\ref{sec:flatedges}. By (P4), there exists $I' \in \mathscr{E}_{k+1}(J_0)$ such that $f_{k+1}(I') =f_{k+1}(I)$. The interval $I'$ satisfies $\overline{I'}\cap J \neq \emptyset$.
\emph{Case 3.} Suppose $J\in \mathscr{N}_{k+1}(J_0)$ for some $J' \in \mathscr{N}_k$ as in \textsection\ref{sec:flatpoints}. There are three subcases.
\emph{Case 3a.} Suppose that $f_{k+1}(J) \neq f_{k}(J_0)$. Then by the choice of $\mathscr{N}_{k+1}(J_0)$, there exists $I' \in \mathscr{E}_{k+1}(J_0)$ such that $f_{k+1}(I') = f_{k+1}(I)$ and $\overline{I'}\cap\overline{J} \neq \emptyset$.
\emph{Case 3b.} Suppose that $f_{k+1}(J) = f_{k}(J_0)$ and there exists $\tilde{I}\in\mathscr{E}_{k+1}(J_0)$ such that $f_{k+1}(\tilde{I})=f_{k+1}(I)$. As in Case 3a, the claim follows from the choice of $\mathscr{N}_{k+1}(J_0)$.
\emph{Case 3c.} Suppose that $f_{k+1}(J) = f_{k}(J_0)$ and there exists no $\tilde{I}\in\mathscr{E}_{k+1}(J_0)$ such that $f_{k+1}(\tilde{I})=f_{k+1}(I)$. In this case, $f_k(J_0)$ is the endpoint of $f_k(I_0)$ for some $I_0 \in \mathcal{E}_k$. By the inductive hypothesis and (P4), there exists at least one and at most two intervals $I_0' \in \mathscr{E}_{k}$ such that $f_k(I_0) = f_k(I_0')$ and $\overline{I_0'}\cap \overline{J_0} \neq \emptyset$. On one hand, if there is only one interval $I_0'$, then $J_0$ is as in \textsection\ref{sec:nonterm} and $J = J_0$. Hence there exists $I' \in I_0'$ such that $f_{k+1}(I') = f_{k+1}(I)$ and $\overline{I'} \cap J \neq \emptyset$. On the other hand, if there are two intervals $I_0', I_0''$, then one of them has a closure which intersects $J$, say $I_0'$. Then there exists $I' \in I_0'$ such that $f_{k+1}(I') = f_{k+1}(I)$ and $\overline{I'} \cap J \neq \emptyset$.
\subsection{Choices in the Traveling Salesman algorithm}\label{sec:choices}
In \S\S \ref{sec:flat} and \ref{sec:curve}, we made a series of implicit and explicit choices.
\begin{enumerate}
\item[(C0)] The choice of $\alpha_0\in(0,1/16]$ determines the set $\mathsf{Flat}(k)$ of flat pairs. The constant $14$ in the definition of $\mathsf{Flat}(k)$ is chosen to facilitate the estimates in \S\ref{sec:mass} (see (E2)), but has not been optimized. The constant 30 in the definition of $\alpha_{k,v}$ is chosen to be larger than $(1+3(1/16)^2)\cdot 2 \cdot 14$. For example, see the proof of Lemma \ref{lem:tridents}.
\item[(C1)] If $I, I'\in \mathscr{E}_k$ satisfy $f_k(I) = f_k(I')$ and are as in \textsection\ref{sec:flatedges} (that is, $I$ has at least one endpoint $x$ with $\alpha_{k,f_{k}(x)} < \alpha_0$), then either $\mathscr{N}_{k+1}(I)=\mathscr{F}_{k+1}(I') = \emptyset$ or vice-versa.
\item[(C2)] If $I \in \mathscr{N}_k$ is as in \textsection\ref{sec:2-term} (i.e.~ $\alpha_{k,f_{k}(I)} < \alpha_0$ and $V_{k}\cap B(f_k(I), 14A^* \rho_kr_0) = \{f_k(I)\}$), then there may exist up to two different ways to parameterize the graph $G_{k+1,I}$ therein.
\item[(C3)] If $I \in \mathscr{N}_k$ is as in \textsection\ref{sec:noflatpoints} (i.e.~ $\alpha_{k,f_{k}(I)} \geq \alpha_0$) and $G_{k+1,I}^{(1)},\dots,G_{k+1,I}^{(l)}$ are the graph components of the graph $G_{k+1,I}$ therein, then
\begin{enumerate}
\item[(C3a)] we choose the order in which we parameterize the graph components and
\item[(C3b)] in each graph component, there exists up to two choices of parameterization.
\end{enumerate}
Similar choices are made in the step $0$.
\item[(C4)] We get to choose the enumeration of intervals $I$ in $\mathscr{N}_k$ such that $\alpha_{k,f_k(I)} \geq \alpha_0$.
\end{enumerate}
The algorithm can be made more flexible by permitting four additional choices. Let $\tilde{\alpha}_0 \in (0,\alpha_0)$ and $\tilde{A}>14A^*$.
\begin{enumerate}
\item[(C5)] Suppose that $I\in\mathscr{N}_k$.
\begin{itemize}
\item If $\alpha_{k,f_k(I)} <\tilde{\alpha}_0$ then we treat $I$ as in \textsection\ref{sec:flatpoints}; i.e., we treat $f_k(I)$ as a flat vertex.
\item If $\alpha_{k,f_k(I)} \geq \alpha_0$ then we treat $I$ as in \textsection\ref{sec:noflatpoints}; i.e., we treat $f_k(I)$ as a non-flat vertex.
\item If $\alpha_{k,f_k(I)} \in [\tilde{\alpha}_0,\alpha_0)$ then we can either treat $I$ as in \textsection\ref{sec:flatpoints} or as in \textsection\ref{sec:noflatpoints}.
\end{itemize}
\item[(C6)] Suppose that $v\in V_k$ is chosen to be considered ``flat" by (C5). Let $\ell$ be the approximating line for $(k,v)$ and let $v' \in V_k$ be such that there exists no $v''\in V_k\cap B(v,\tilde{A} \rho_{k} r_0)$ such that $\pi_{\ell}(v'')$ is between $v$ and $v'$.
\begin{itemize}
\item If $|v-v'| < 14A^* \rho_kr_0$, then $(v,v')\in \mathsf{Flat}(k)$.
\item If $|v-v'| \geq \tilde{A} \rho_k r_0$, then $(v,v') \not\in \mathsf{Flat}(k)$.
\item If $|v-v'| \in [14A^* \rho_kr_0, \tilde{A}\rho_k r_0)$, then we are free to choose whether $(v,v')$ is contained in $\mathsf{Flat}(k)$ or not.
\end{itemize}
\item[(C7)] Similarly to (C6), suppose that $I\in \mathscr{E}_k$ is as in \textsection\ref{sec:flatedges}; i.e., $f_k(I)$ has at least one endpoint $x$ whose image is ``flat" by (C5). Let $\{v_1,\dots,v_l\}$ and $\{I_1,\dots,I_{l-1}\}$ be as in \textsection\ref{sec:flatedges}.
\begin{itemize}
\item If $|v_i-v_{i+1}| < 14A^* \rho_{k+1}r_0$, then we set $I_i\in\mathscr{E}_{k+1}$.
\item If $|v_i-v_{i+1}| \geq \tilde{A} \rho_{k+1}r_0$, then we set $I_i\in\mathscr{B}_{k+1}$.
\item If $|v_i-v_{i+1}| \in [14A^* \rho_{k+1} r_0, \tilde{A} \rho_{k+1}r_0)$, then we can choose in each instance whether $I_i\in \mathscr{E}_{k+1}(I)$ or $I_i\in \mathscr{B}_{k+1}(I)$.
\end{itemize}
\item[(C8)] Suppose that $\{I_1,\dots, I_n\}$ are the intervals in $\mathscr{N}_k$ that have a non-flat image. Suppose also that we have defined $f_{k+1}$ on $I_1,\dots,I_{r-1}$ and on intervals in $\mathscr{N}_k$ that have an image chosen to be flat. Let $v \in V_{k+1}$ be a point which is not the image of some $I\in\mathscr{N}_{k+1}(J)$, where $J\in\{I_1,\dots,I_{r-1}\}$ or $J\in\mathscr{N}_k$ is as in \textsection\ref{sec:flatpoints}.
\begin{itemize}
\item If $v \in B(f_k(I_r),14A^* \rho_k r_0)$, then $v\in V_{k+1,I_r}$.
\item If $v \in B(f_k(I_r),\tilde{A} \rho_k r_0)\setminus B(f_k(I_r),14A^* \rho_k r_0)$, then we may choose whether $v\in V_{k+1,I_r}$ or not.
\end{itemize}
\end{enumerate}
Note that the (C5) has subsequent implications on the treatment of intervals $I\in\mathscr{E}_k$. For instance, if both endpoints if $I$ have images chosen to be non-flat, then $I\in\mathscr{B}_{k+1}$; otherwise, we treat $I$ as in \textsection\ref{sec:flatedges}. Similarly, (C6) gives us the set $\mathscr{L}_{k}$ and together with (C8) affects the parametrization near non-flat vertices.
\begin{rem}[coherence] In the original, non-parametric Analyst's Traveling Salesman construction, Jones \cite{Jones-TST} required the coherence property (V2), i.e.~ $V_k\subset V_{k+1}$. The first author and Schul \cite{BS3} established a non-parametric Traveling Salesman construction, which replaced (V2) with the weaker property that for all $v\in V_k$, the set $$v'\in V_{k+1}\cap B(v,C^* \rho_k r_0)$$ is nonempty. This relaxation was crucial for the proof of the main result in \cite{BS3}, which characterized Radon measures in $\mathbb{R}^N$ that are carried by rectifiable curves. We would like to emphasize that in the parametric Traveling Salesman construction described above, we heavily rely on (V2). At this time, we do not know how to build a parameterization under the relaxed condition of \cite{BS3}.\end{rem}
\section{Mass of intervals}\label{sec:mass}
In this section, we use the construction of \textsection\ref{sec:curve}, to assign mass to intervals defined in \textsection\ref{sec:curve}. The total mass $\mathcal{M}_s$ on the domain of the maps fills the role that the Hausdorff measure $\mathcal{H}^1$ of the image plays in the proof of the sufficient half of the Analyst's TST given in \cite{Jones-TST} or \cite{BS3}. The main result of this section is Proposition \ref{prop:mass}, which bounds the total mass of $[0,1]$ by a sum involving the flatness approximation errors $\alpha_{k,v}$ and variation excess $\tau_{s}(k,v,v')$ defined in \S\ref{sec:flat}.
For each $k\geq 0$, set
\begin{equation*} \mathscr{I}_k := \mathscr{E}_k\cup\mathscr{B}_k\cup\mathscr{N}_k\cup\mathscr{F}_k \qquad\text{and}\qquad \mathscr{I} := \bigcup_{k\geq 0}\mathscr{I}_k.\end{equation*}
For each $I \in \mathscr{I}_k$, set
$\mathscr{I}_{k+1}(I) := \mathscr{E}_{k+1}(I)\cup\mathscr{B}_{k+1}(I)\cup\mathscr{N}_{k+1}(I)\cup\mathscr{F}_{k+1}(I)$.
\begin{rem}
If $I \in \mathscr{I}_k \cap \mathscr{I}_m$ for some $m\neq k$, then $f_k | I = f_m | I$.
\end{rem}
\subsection{Trees over intervals}\label{sec:trees}
Given $k\geq 0$ and $I \in \mathscr{I}_k$, we define a \emph{finite tree $T$ over $(k,I)$} to be a finite subset of $\bigcup_{m\geq 0}(\{m\}\times\mathscr{I}_m)$ satisfying the following three conditions.
\begin{enumerate}
\item The pair $(k,I) \in T$. If $(m,J) \in T$, then $m\geq k$ and $J\subset I$.
\item If $(m,J) \in T$ and there exists $J' \in \mathscr{I}_{m+1}(J)$ such that $(m+1,J')\cap T$, then $\{m+1\}\times\mathscr{I}_{m+1}(J)\subseteq T$.
\item If $(m,J)\in T$ for some $m>k$ and $J\in \mathscr{I}_m(J')$, then $(m-1,J')\in T$.
\end{enumerate}
The first condition says that the root of the tree is $(k,I)$ and its elements are descendants of $(k,I)$. The second condition says that if one child $(m+1,J')$ of $(m,J)$ is in $T$, then every child of $(m,J)$ is in $T$. The third condition says that if $(m,J)$ is in $T$, then all its ancestors up to $(k,I)$ are in $T$.
We extend this notion to the entire domain by defining a \emph{finite tree $T$} over $[0,1]$ to be a set of the form
\begin{equation*} T = \{[0,1]\} \cup \bigcup_{I\in\mathscr{I}_0 } T_I,\end{equation*}
where $T_I$ is a finite tree over $(0,I)$. A finite tree over $[0,1]$ may be thought to belong to step $k=-1$ of the construction.
Let $T$ be a finite tree over $(k,I)$. The \emph{boundary $\partial T$ of $T$} is defined by
\begin{equation*} \partial T := \{(m,J) \in T :(\{m+1\}\times\mathscr{I}_{m+1}(I)) \cap T = \emptyset\}.\end{equation*}
The \emph{depth $m(T)$ of $T$} is the integer defined by
\begin{equation*} m(T) := \max\{m\geq 0 : (m,J) \in T\} = \max\{k\geq 0 : (m,J) \in \partial T\}. \end{equation*}
If $m(T)\geq 1$, the \emph{parent tree $p(T)$} is defined by
\begin{equation*} p(T) := T \setminus (\{m(T)\}\times\mathscr{I}_{m(T)})\end{equation*}
Note that $m(p(T))=m(T)-1$.
\begin{rem}\label{rem:chainpartition}
If $T$ is a finite tree over $(k,I)$ and $\partial T = \{(k_1,J_1), \dots, (k_n,J_n)\}$, then the intervals $J_1,\dots,J_n$ partition $I$. That is, for all $x\in I$, there exists a unique $i\in\{1,\dots,n\}$ such that $x\in J_i$.
\end{rem}
\subsection{Mass of intervals}\label{sec:massinterval} For all $s\geq 1$, $k\geq 0$, and intervals $I\in\mathscr{I}_k$, define the \emph{$s$-mass $\mathcal{M}_s(k,I)$ of $(k,I)$} by
\[ \mathcal{M}_s(k,I) := \sup_{T} \sum_{(k',I')\in \partial T}(\mathop\mathrm{diam}\nolimits{f_{k'}(I')})^s \in [0,\infty],\]
where the supremum is taken over all finite trees over $(k,I)$. This notion extends to $[0,1]$ by assigning
\[ \mathcal{M}_s([0,1]) := \sum_{I \in \mathscr{I}_0 } \mathcal{M}_s(0,I) \in [0,\infty].\]
\begin{lem}\label{lem:sum}
Let $k\geq 0$ and $I\in\mathscr{I}_k$.
\begin{enumerate}
\item If $I \in \mathscr{B}_k$, then $\mathcal{M}_s(k,I) = (\mathop\mathrm{diam}\nolimits{f_k}(I))^s$.
\item If $I \in \mathscr{F}_k$, then $\mathcal{M}_s(k,I) = 0$.
\item $\mathcal{M}_s(k,I) \geq \sum_{I'\in \mathscr{I}_{k+1}(I)} \mathcal{M}_s(k+1,I')$.
\item If $I \in \mathscr{I}_k \cap \mathscr{I}_m$ for some $m\geq 0$, then $\mathcal{M}_s(k,I) = \mathcal{M}_s(m,I)$.
\end{enumerate}
\end{lem}
Before proving Lemma \ref{lem:sum}, we make two clarifying remarks. First, it is possible for an interval $I\in\mathscr{N}_k$ to have $\mathcal{M}_s(k,I) >0$ even though $\mathop\mathrm{diam}\nolimits f_k(I)^s=0$. This happens whenever $I\in\mathscr{N}_k$ and $\mathscr{E}_{k+1}(I)\cup\mathscr{B}_{k+1}(I)$ is non-empty. Second, Lemma \ref{lem:sum}(4) implies that given $I \in \mathscr{I}_k$, the mass $\mathcal{M}_s(k,I)$ is defined independently of the step of the construction in which $I$ appears. Nevertheless, we include the step $k$ in definition of the mass to improve exposition of the estimates in \textsection\ref{sec:totalmass} and \textsection\ref{sec:prooflemma}.
\begin{proof}[{Proof of Lemma \ref{lem:sum}}]
For the first claim, note that if $I\in\mathscr{B}_k$ and $m\geq k+1$, then $\mathscr{I}_m(I) = \{I\}$. Therefore, if $T$ is a finite tree over $I$ of depth $m$, then $\partial T = \{(m,I)\}$. Thus, $\mathcal{M}_s(k,I) = \sup_{m\geq k} (\mathop\mathrm{diam}\nolimits{f_m}(I))^s = (\mathop\mathrm{diam}\nolimits{f_k}(I))^s$.
For the second claim, note that if $I\in\mathscr{F}_k$ and $m\geq k+1$, then $\mathscr{I}_m(I) = \{I\}$ and $f_m(I)$ is a point. Therefore, if $T$ is a finite tree over $I$ of depth $m$, then $\partial T = \{(m,I)\}$. Thus, $\mathcal{M}_s(k,I) = \sup_{m\geq k}(\mathop\mathrm{diam}\nolimits{f_m}(I))^s=0$.
For the third claim, let us first assume that $\mathcal{M}_s(k+1,J) = \infty$ for some $J\in\mathscr{I}_{k+1}(I)$. Fix $M>0$ and find a finite tree $T_J$ over $(k+1,J)$ such that
\[ \sum_{(k',J') \in T_J} (\mathop\mathrm{diam}\nolimits{f_{k'}(J')})^s >M.\]
The collection $T = T_J\cup \{(k,I)\} \cup (\{k+1\}\times\mathscr{I}_{k+1}(I))$ is a finite tree over $(k,I)$ and $\partial T_J \subset \partial T$. Hence
\[ \mathcal{M}_s(k,I) \geq \sum_{(k',I')\in \partial T}(\mathop\mathrm{diam}\nolimits{f_{k'}(I')})^s \geq \sum_{(k',J')\in \partial T_J}(\mathop\mathrm{diam}\nolimits{f_{k'}(J')})^s >M.\]
We conclude that $\mathcal{M}_s(k,I)= \infty$.
Alternatively, assume that $\mathcal{M}_s(k+1,J)$ is finite for all $J\in\mathscr{I}_{k+1}(I)$. Fix $\epsilon>0$. For each interval $J\in\mathscr{I}_{k+1}(I)$, let $T_J$ be a finite tree over $(k+1,J)$ such that
\[ \sum_{(k',J')\in \partial T_J} (\mathop\mathrm{diam}\nolimits{f_{k'}(J')})^s > \mathcal{M}_s(k+1,J) - \frac{\epsilon}{\operatorname{card}(\mathscr{I}_{k+1}(I))}.\]
Then the collection $T = \{(k,I)\} \cup\bigcup_{J\in\mathscr{I}_{k+1}(I)} T_J$ is a finite tree over $(k,I)$ with $\partial T = \bigcup_{J\in\mathscr{I}_{k+1}(I)}\partial T_J $. Therefore,
\begin{align*}
\mathcal{M}_s(k,I) \geq \sum_{J\in\mathscr{I}_{k+1}(I)} \sum_{(k',J') \in \partial T_J} (\mathop\mathrm{diam}\nolimits{f_{k'}(J')})^s
> \sum_{J\in\mathscr{I}_{k+1}(I)} \mathcal{M}_s(k+1,J) - \epsilon.
\end{align*} The third claim follows by taking $\epsilon\downarrow 0$.
For the fourth claim, suppose that $I\in\mathscr{I}_k\cap\mathscr{I}_m$ for some $m$ and $k$, say without loss of generality that $m>k$. Because $I \in \mathscr{I}_k\cap\mathscr{I}_m$, we have $\mathscr{I}_n(I) = \{I\}$ for all $k\leq n \leq m$. Thus, iterating the third claim, $\mathcal{M}_s(k,I) \geq \mathcal{M}_s(m,I)$. For the opposite inequality, let $T$ be a finite tree over $(k,I)$. If $(m,I)\not\in T$, then $T = \{(k,I),\dots, (l,I)\}$ for some $k\leq l\leq m-1$ and we define $T'=\{(m,I)\}$. If $(m,I) \in T$ , then $T \supset \{(k,I),\dots,(m-1,I)\}$ and we set $T'=T\setminus \{(k,I),\dots,(m-1,I)\}$ so that $T'$ is a finite tree over $(m,I)$ and $\partial T = \partial T'$. In either case,
\[ \sum_{(k',I') \in \partial T} (\mathop\mathrm{diam}\nolimits{f_{k'}(I')})^s = \sum_{(k',I') \in \partial T'} (\mathop\mathrm{diam}\nolimits{f_{k'}(I')})^s\]
and it follows that $\mathcal{M}_s(k,I) \leq \mathcal{M}_s(m,I)$.
\end{proof}
When $s=1$, the 1-mass is comparable to the Hausdorff measure $\mathcal{H}^1$ of the image.
\begin{lem}\label{lem:massequiv}
For each $k\geq 0$ and each $I\in\mathscr{I}_k$,
\begin{equation*}
\mathcal{M}_1(k,I) = \lim_{m\to\infty} \sum_{\substack{J \in \mathscr{I}_m \\ J\subset I}} \mathop\mathrm{diam}\nolimits{f_m(J)} \geq \limsup_{m\to\infty} \mathcal{H}^1(f_m(I)).
\end{equation*} If there exists $n\in\mathbb{N}$ such that $f_m|\bigcup\mathscr{B}_k$ is at most $n$-to-1 for all $m\geq k$, then \begin{equation*} \mathcal{M}_1(k,I) \simeq_n \liminf_{m\to\infty} \mathcal{H}^1(f_m(I))\end{equation*}
\end{lem}
\begin{proof} Fix $k\geq 0$ and $I\in \mathscr{I}_k$.
By definition of the mass and (P2),
\[ \mathcal{M}_1(k,I) \geq \limsup_{m\to\infty}\sum_{\substack{J \in \mathscr{I}_m \\ J\subset I}} \mathop\mathrm{diam}\nolimits{f_m(J)} \geq \limsup_{m\rightarrow\infty} \mathcal{H}^1(f_m(I)). \]
To establish the other direction, let $T$ be a finite tree over $I$ of depth $m\geq k$ and enumerate with $\partial T = \{(k_1,I_1),\dots,(k_n,I_n)\}$, where each $k_i \leq m$. For each $i=1,\dots, n$, let $\mathcal{I}_i$ be the set of all intervals $J \in \mathscr{I}_{m}$ such that $J\subset I_i$. Then
\[ \sum_{\substack{J \in \mathscr{I}_{m} \\ J \subset I}} \mathop\mathrm{diam}\nolimits{f_{m}(J)} = \sum_{i=1}^n\sum_{J\in\mathcal{I}_i} \mathop\mathrm{diam}\nolimits{f_{m}(J)} \geq \sum_{i=1}^n\mathop\mathrm{diam}\nolimits{f_{k_i}(I_i)} = \sum_{(l,J)\in \partial T}\mathop\mathrm{diam}\nolimits{f_l(J)}. \]
Therefore,
\[ \sup_{m\geq k}\sum_{\substack{J \in \mathscr{I}_{m}\\ J\subset I}}\mathop\mathrm{diam}\nolimits{f_{m}(J)} \leq \mathcal{M}_1(k,I) = \sup_{T} \sum_{(l,J)\in \partial T}\mathop\mathrm{diam}\nolimits{f_l(J)} \leq \liminf_{m\to \infty}\sum_{\substack{J \in \mathscr{I}_{m} \\ J \subset I}} \mathop\mathrm{diam}\nolimits{f_{m}(J)}. \]
This shows that \[ \mathcal{M}_1(k,I) = \lim_{m\to\infty} \sum_{\substack{J \in \mathscr{I}_m \\ J\subset I}} \mathop\mathrm{diam}\nolimits{f_m(J)}.\]
By (P4), the maps $f_m|\bigcup\mathscr{E}_m$ are 2-to-1. In \S\ref{sec:curve}, we did not examine overlaps of images of bridge intervals. By modifying the algorithm, the overlap of images of bridge intervals can be made 2-to-1 (see the proof of Proposition \ref{prop:2-to-1Holder}). Nevertheless, suppose that we know the overlaps of images of bridge intervals is at most $n$-to-1 for some $n\geq 2$. Then \[ \mathcal{M}_1(k,I) = \lim_{m\to \infty}\sum_{\substack{J \in \mathscr{I}_{m} \\ J \subset I}} \mathop\mathrm{diam}\nolimits{f_{m}(J)} \leq \frac{1}{n}\liminf_{m\to\infty} \mathcal{H}^1(f_m(I)). \qedhere \]
\end{proof}
While Lemma \ref{lem:massequiv} does not hold when $s>1$, we always have the following comparison between the $s$-mass and the Hausdorff measure $\mathcal{H}^s$ of the closure of the points in $\bigcup_{k=0}^\infty V_k$.
\begin{lem}\label{lem:mass-measure}
For all $s\geq 1$, $\mathcal{H}^s\left(\overline{\bigcup_{k=0}^\infty V_k}\right)\lesssim_{s,C^*,\xi_2} \mathcal{M}_s([0,1]).$
\end{lem}
\begin{proof}Fix $\delta>0$ and choose $m\in\mathbb{N}$ sufficiently large such that $2C^*\xi_2^{m+1}r_0/(1-\xi_2)\leq \delta$. By (V0), (V2), and (V4), the collection
$\{ B(v,C^*\rho_{m+1}r_0/(1-\xi_2)) : v\in V_m\}$ is a cover of $\overline{\bigcup_{k=0}^{\infty} V_k}$ with elements of diameter at most $2C^*\rho_{m+1}r_0/(1-\xi_2)\leq 2C^*\xi_2^{m+1}r_0\leq \delta$. Let $T$ be the maximal finite tree over $[0,1]$ of depth $m$, i.e.~ $T=\bigcup_{k=0}^m\mathscr{I}_k$. Then
\begin{equation*}\begin{split} \mathcal{H}^s_\delta \left(\overline{\textstyle\bigcup_{k=0}^{\infty} V_k}\right) \leq \sum_{v\in V_m} \left(\frac{2C^*\rho_{m+1}r_0}{1-\xi_2}\right)^s &\leq \left(\frac{2C^*\xi_2}{1-\xi_2}\right)^s\sum_{(m,I)\in\partial T} (\mathop\mathrm{diam}\nolimits{f_m(I)})^s\\ &\lesssim_{s,C^*,\xi_2}\mathcal{M}_s([0,1]). \end{split}\end{equation*} Taking $\delta\downarrow 0$ completes the proof.
\end{proof}
We include Lemma \ref{lem:massequiv} and Lemma \ref{lem:mass-measure} for completeness. We will not use either lemma in any the estimates below.
\subsection{Terminal vertices and phantom mass}\label{sec:phantom}
Let $I \in \mathscr{N}_k$ be an interval such that $\alpha_{k,f_k(I)} < \alpha_0$. We classify $f_k(I)$ according to the arrangement of nearby points in $V_{k+1}$.
\begin{itemize}
\item If $I$ is as in \textsection\ref{sec:nonterm}, then $f_k(I)$ is called a \emph{non-terminal vertex in $V_k$}.
\item If $I$ is as in \textsection\ref{sec:1-term}, then $f_k(I)$ is called a \emph{1-sided terminal vertex in $V_k$}.
\item If $I$ is as in \textsection\ref{sec:2-term}, then $f_k(I)$ is called a \emph{2-sided terminal vertex in $V_k$}.
\end{itemize}
Motivated by \cite{Jones-TST} and \cite{BS3}, for each $k\geq 0$ we will define a set $\mathscr{P}_k\subset \{k\}\times\mathscr{N}_k$ and for each $(k,I)\in \mathscr{P}_k$ define a number $p_{k,I}>0$, which we call the \emph{phantom mass} at $(k,I)$. The phantom mass $p_{k,I}$ will let us pay for the length of edges between vertices in $V_{k+1}$ nearby $f_k(I)$ that do not lie between vertices in $V_k$ nearby $f_k(I)$ (i.e.~ the blue edges in Figure \ref{fig:flatpoints}). To start, define an auxiliary parameter $P$ depending only on $s$, $C^*$, and $\xi_2$ by requiring that $\left[P+2(1.1C^*)^s\right]\xi_2^s = P.$ That is, \begin{equation}\label{eq:Pdefinition} P= \frac{2(1.1C^*)^s}{1-\xi_2^s}.\end{equation}
For each $k\geq 0$, define
\[ \mathscr{P}_{k}= \{(k,I) : \text{$I \in \mathscr{N}_k$, $\alpha_{k,f_k(I)} < \alpha_0$ and $f_{k}(I)$ is $1$- or $2$-sided terminal in $V_k$}\}.\] For each $k\geq 0$ and $(k,I)\in\mathscr{P}_k$, assign
\[ p_{k,I} :=
\begin{cases}
2P\rho_k^sr_0^s, &\text{ if }f_k(I)\text{ is 2-sided terminal}\\
P\rho_k^sr_0^s, &\text{ if }f_k(I)\text{ is 1-sided terminal}.
\end{cases}\]
\begin{lem}\label{lem:flatpointsum} Let $I\in\mathscr{N}_k$ be an interval such that $\alpha_{k,f_k(I)}<\alpha_0$. If $f_k(I)$ is 1-sided terminal, then $$\sum_{J\in \mathscr{I}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits f_{k+1}(J))^s < 2(1.1C^*)^s\rho_{k+1}^sr_0^s.$$ If $f_k(I)$ is 2-sided terminal, then $$\sum_{J\in \mathscr{I}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits f_{k+1}(J))^s < 4(1.1C^*)^s \rho_{k+1}^sr_0^s.$$
\end{lem}
\begin{proof} Suppose $v=f_k(I)$ is 1-sided terminal and let $\{v_1,\dots,v_n\}$ be an enumeration of the points in $V_{k+1}\cap B(v, C^*\rho_{k+1}r_0)$ starting from $v_1=v$ and moving consecutively towards the terminal direction. Then $$\sum_{J\in \mathscr{I}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits f_{k+1}(J))^s = 2 \sum_{i=1}^n |v_{i+1}-v_i|^s \leq 2(1+3\alpha_{k,v}^2)^s |v_1-v_n|^s < 2(1.1)^s (C^*\rho_{k+1} r_0)^s$$ by Lemma \ref{lem:var}, since $1+3\alpha_{k,v}^2 \leq 1+3(1/16)^2 < 1.1$. The case that $v$ is 2-sided terminal follows from a similar computation.
\end{proof}
\subsection{Special bridge intervals}
Given $k\geq 1$, we define
\[ \mathscr{B}_k^*:= \{(k,I) : \text{$I \in \mathscr{B}_k(J)$, $J\in \mathscr{E}_{k-1}$ is as in \textsection\ref{sec:flatedges}}\}.\]
Recall from \textsection\ref{sec:flatedges} that if $(k,I) \in \mathscr{B}_k^*$, then $14A^*\rho_k r_0\leq \mathop\mathrm{diam}\nolimits{f_k(I)} <14 A^*\rho_{k-1} r_0$.
\begin{lem}\label{lem:bridgelength} If $I\in\mathscr{B}_k^*$, then $$\frac{1}{6}(\mathop\mathrm{diam}\nolimits f_k(I))^s \geq P\rho_{k}^s r_0^s.$$\end{lem}
\begin{proof} Because $\mathop\mathrm{diam}\nolimits f_k(I)\geq 14A^*\rho_kr_0$, it suffices to check that $(14A^*)^s \geq 6P$. Recalling the definition of $A^*$, we find that $$(14A^*)^s = \left(\frac{14C^*}{1-\xi_2}\right)^s \geq \frac{12(1.1C^*)^s}{1-\xi_2^s}=6P.$$ Here $1/(1-\xi_2)^s \geq 1/(1-\xi_2^s)$ because $0<\xi_2<1$ and $s\geq 1$.
\end{proof}
Let $T$ be a finite tree over $[0,1]$, let $m$ be the depth of $T$ and let $0\leq k \leq m$ be an integer.
Define
\[ \mathscr{B}_k^*(T) := \{(k,I) \in \mathscr{B}^*_k : \text{ there exists $(k,J)\in T\cap(\{k\}\times\mathscr{N}_k)$ such that $J\cap \overline{I}\neq \emptyset$} \}.\]
Although the sets $\mathscr{B}_k^*(T)$ are not necessarily subsets of $\partial T$, we show in the next lemma that each element in $\partial T$ generates at most two elements in $\bigcup_{k=1}^m\mathscr{B}_k^*(T)$.
\begin{lem}\label{lem:bridges*2}
Let $T$ be a finite tree over $[0,1]$ with depth $m\geq 1$ and let $k\leq m$.
\begin{enumerate}
\item For each $(k,I) \in \mathscr{B}_{k}^*(T)$, there exists a unique $(l,J) \in \partial T$ such that $I \subset J$. In fact, $J\in \mathscr{E}_l\cup\mathscr{B}_l$.
\item For each $(l,J) \in \partial T$ such that $J\in \mathscr{E}_l\cup\mathscr{B}_l$, there exist at most two distinct $(k,I) \in \bigcup_{i=0}^m\mathscr{B}_{i}^*(T)$ such that $I\subset J$.
\end{enumerate}
\end{lem}
\begin{proof}
The first claim of (1) follows immediately from Remark \ref{rem:chainpartition}. For the second claim, let $(k,I) \in \mathscr{B}_{k}^*(T)$. There are two cases.
\emph{Case 1.} If $(k,I) \in T$, then $\mathscr{I}_{n}(I) = \{I\}$ and $I\in\mathscr{B}_n$ for all $n > k$, since $I$ is a bridge interval. Therefore, $(l,I) \in \partial T$ for some $l\leq m$ and $I\in\mathscr{B}_l$.
\emph{Case 2.} Suppose now that $(k,I) \not\in T$.
If $J$ was in $\mathscr{N}_l\cup\mathscr{F}_l$, then the closure of $I$ would be contained in the interior of $J$ and $I$ would intersect only intervals $I' \in \mathscr{I}_{l'}$ for which $(l',I') \not\in T$, which is a contradiction.
For (2), fix $(l,J) \in \partial T$ with $J\in \mathscr{E}_l\cup\mathscr{B}_l$ and assume that $\{(k_i,J_i):i=1,2,3\}$ are distinct elements in $\bigcup_{j=0}^m\mathscr{B}_{j}^*(T)$ such that $J_i \subset J$ for $i=1,2,3$. Without loss of generality, assume that $J_2$ is between $J_1$ and $J_3$, in the orientation of $J$. Then there exists $l' \geq l+1$ and $J'\subseteq J$ such that $J' \in\mathscr{N}_{l'}$ and $(l',J') \in T$. But then $(l,J) \not\in \partial T$ which is a contradiction.
\end{proof}
We now impose an additional restriction on $\alpha_0$.
\begin{lem}\label{lem:flatdiam} For all $C^*$, $\xi_1$, and $\xi_2$, there exists $\alpha_1\in(0,1/16]$ such that if $\alpha_0\leq \alpha_1$, then for all $k\geq 0$, $(v,v')\in\mathsf{Flat}(k)$, and $y,y'\in V_{k+1}(v,v')$, we have $|y-y'| \leq |v-v'|$. \end{lem}
\begin{proof} Enumerate $V_{k+1}(v,v')=\{v_1,\dots,v_n\}$ from left to right, so that $v_1=v$ and $v_n=v'$. Let $y=v_l$ and $y'=v_m$ for some $1\leq l<m\leq n$. If $y=v$ and $y'=v'$, the conclusion is trivial. Thus, let us suppose that there exists at least one point to the left of $y$ or the right of $y'$, say without loss of generality that $l\geq 2$. Let $x_i=\pi_{\ell_{k,v}}$ for all $1\leq i\leq n$. Then, arguing as in the proof of Lemma \ref{lem:var},
$$\frac{|v_1-v_2| + |v_l-v_m|}{1+3\alpha_0^2} \leq |x_1-x_2| + |x_l-x_m| \leq |x_1-x_n|\leq |v-v'|.$$ Because $V_{k+1}$ is $\rho_{k+1}$ separated, \begin{equation*}\begin{split}|v_l-v_m| \leq (1+3\alpha_0^2) |v-v'| - |v_1-v_2| &\leq (1+3\alpha_0^2)|v-v'| - \rho_{k+1}r_0\\ &=|v-v'|\left(1+3\alpha_0^2 -\frac{\rho_{k+1}r_0}{|v-v'|}\right).\end{split}\end{equation*} Now $|v-v'| \leq 14A^*\rho_k r_0 \leq 14A^*\xi_1^{-1}\rho_{k+1} r_0$. Hence $|v_l-v_m|\leq |v-v'|$ provided that $$1+3\alpha_0^2 - \frac{\xi_1}{14A^*} \leq 1.$$ Thus, we can take \begin{equation}\label{alphabound} \alpha_1 := \min\left\{\frac{1}{16}, \left( \frac{\xi_1}{42 A^*} \right)^{1/2}\right\}.\qedhere\end{equation}\end{proof}
Together Lemma \ref{lem:bridges*2} and Lemma \ref{lem:flatdiam} yield the following result.
\begin{cor}\label{cor:bridges*} Assume $\alpha_0\leq \alpha_1$.
If $T$ is a finite tree over $[0,1]$ of depth $m$, then,
\[ \sum_{k=1}^{m}\sum_{(k,J) \in \mathscr{B}_{k}^*(T)} (\mathop\mathrm{diam}\nolimits{f_k(J)})^s \leq 2\sum_{(l,J) \in \partial T} (\mathop\mathrm{diam}\nolimits{f_l(J)})^s.\]
\end{cor}
\subsection{An upper bound for the total mass}\label{sec:totalmass}
Here we prove the following proposition which gives an upper bound for the mass of $[0,1]$ in terms of the variation excess $\tau_{s}(k,v,v')$ of flat pairs $(v,v')\in \mathsf{Flat}(k)$ defined in \textsection\ref{sec:flat}.
\begin{prop}\label{prop:mass} Assume that $\alpha_0\leq \alpha_1$ (see Lemma \ref{lem:flatdiam}). For all $s\geq 1$,
\[ \mathcal{M}_s([0,1]) \lesssim_{s,C^*,\xi_2} r_0^s + \sum_{k=0}^{\infty} \sum_{(v,v')\in\mathsf{Flat}(k)} \tau_{s}(k,v,v') \rho_{k}^s r_0^s + \sum_{k=0}^{\infty} \sum_{\substack{v\in V_k \\ \alpha_{k,v}\geq \alpha_0}} \rho_k^s r_0^s.\]
\end{prop}
The proof of Proposition \ref{prop:mass} reduces to proving the following lemma (cf. \cite[(9.4)]{BS3}). Recall the definition of the parent tree from \textsection\ref{sec:trees}.
\begin{lem}\label{lem:estimates} Let $k\geq 1$ be an integer, let $T$ be a finite tree over $[0,1]$ of depth $k+1$ and let $p(T)$ be the parent tree. There exists a constant $C>0$ depending only on $s$, $C^*$, and $\xi_2$ such that
\begin{align}\label{eq:estimates}\tag{E}
\begin{split}
&\sum_{(l,I)\in \partial T}(\mathop\mathrm{diam}\nolimits{f_l(I)})^s + \sum_{(k+1,I)\in \mathscr{P}_{k+1} \cap\partial T}p_{k+1,I} \\
&\quad\leq\sum_{(l,I)\in \partial p(T)}(\mathop\mathrm{diam}\nolimits{f_l(I)})^s + \sum_{(k,I)\in \mathscr{P}_{k}\cap \partial p(T)}p_{k,I} + \frac13\sum_{(k+1,I)\in \mathscr{B}_{k+1}^*(T)}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s\\
&\quad\ + C\sum_{\substack{v\in V_k \\ \alpha_{k,v}\geq \alpha_0}} \rho_{k}^sr_0^s+ C\sum_{\substack{w\in V_{k+1} \\ \alpha_{k+1,w}\geq \alpha_0}} \rho_{k+1}^sr_0^s + C\sum_{(v,v')\in\mathsf{Flat}(k)} \tau_{s}(k,v,v') \rho_k^s r_0^s.
\end{split}
\end{align}
\end{lem}
We prove Lemma \ref{lem:estimates} in \textsection\ref{sec:prooflemma}. Assuming that Lemma \ref{lem:estimates} holds, here is the proof of Proposition \ref{prop:mass}.
\begin{proof}[{Proof of Proposition \ref{prop:mass}}] Assume that $\alpha_1\leq \alpha_0$.
Let $T$ be a finite tree over $[0,1]$ of depth $m$. Set $T_m = T$ and for each $1\leq k\leq m$ (if any) set $T_{k-1} = p(T_{k})$. Note that $T_{0} = \{0\}\times\mathscr{I}_0$. By Lemma \ref{lem:estimates}, for all $1\leq k\leq m$ (if any),
\begin{align*}
&\sum_{(l,I) \in \partial T_{k}} (\mathop\mathrm{diam}\nolimits{f_l(I)})^s + \sum_{(k,I)\in \mathscr{P}_{k}\cap\partial T_{k}}p_{k,I} \\
&\quad\leq \sum_{(l,I) \in \partial T_{k-1}} (\mathop\mathrm{diam}\nolimits{f_l(I)})^s + \sum_{(k-1,I)\in \mathscr{P}_{k-1}\cap\partial T_{k-1}}p_{k-1,I}
+ \frac13\sum_{(k,I)\in \mathscr{B}_{k}^*(T_{k})}(\mathop\mathrm{diam}\nolimits{f_k(I)})^s\\
&\quad\ + C\sum_{\substack{v\in V_{k-1} \\ \alpha_{k-1,v}\geq \alpha_0}} \rho_{k-1}^s r_0^s+ C\sum_{\substack{w\in V_{k} \\ \alpha_{k,w}\geq \alpha_0}} \rho_{k}^s r_0^s + C\sum_{(v,v')\in\mathsf{Flat}(k-1)} \tau_{s}(k-1,v,v') \rho_{k-1}^s r_0^s.
\end{align*}
Iterating the latter inequality,
\begin{align*}
\sum_{(l,I) \in \partial T} (\mathop\mathrm{diam}\nolimits{f_l(I)})^s &\leq \sum_{I \in \mathscr{E}_0\cup \mathscr{B}_0} (\mathop\mathrm{diam}\nolimits{f_0(I)})^s + \sum_{I \in \mathscr{P}_0}p_{0,I}
+ \frac13\sum_{k=1}^{m}\sum_{I\in\mathscr{B}_k^*(T)} (\mathop\mathrm{diam}\nolimits{f_k(I)})^s \\
&\ + 2C\sum_{k=0}^{m}\sum_{\substack{v\in V_k \\ \alpha_{k,v}\geq \alpha_0}} \rho_{k}^s r_0^s+ C\sum_{k=1}^{m}\sum_{(v,v')\in\mathsf{Flat}(k)} \tau_{s}(k,v,v') \rho_{k}^s r_0^s.
\end{align*}
Since $\alpha_0\leq \alpha_1$, we obtain $\frac13\sum_{k=1}^{m}\sum_{I\in\mathscr{B}_k^*(T)} (\mathop\mathrm{diam}\nolimits{f_k(I)})^s \leq \frac23\sum_{(l,I) \in \partial T} (\mathop\mathrm{diam}\nolimits{f_l(I)})^s$ by Corollary \ref{cor:bridges*}. This is the only place in the proof of the Proposition that we use the restriction $\alpha_0\leq \alpha_1$.
Therefore,
\begin{align*}
\sum_{(l,I) \in \partial T} (\mathop\mathrm{diam}\nolimits{f_l(I)})^s &\leq 3\sum_{I \in \mathscr{E}_0\cup \mathscr{B}_0} (\mathop\mathrm{diam}\nolimits{f_0(I)})^s + 3\sum_{I \in \mathscr{P}_0}p_{0,I}\\
&\ + 6C\sum_{k=0}^{m}\sum_{\substack{v\in\ V_k\\ \alpha_{k,v}\geq \alpha_0}}\rho_{k}^s r_0^s + 3C\sum_{k=0}^{m-1}\sum_{(v,v')\in \mathsf{Flat}(k) }\tau_{s}(k,v,v') \rho_{k}^s r_0^s.
\end{align*}
There are now two alternatives. On one hand, suppose that $\alpha_{0,v}<\alpha_0$ for some $v\in V_0$. Then $V_{0}$ projects onto an $(1+3(1/16)^2)^{-1}r_0$ separated set in $\ell_{0,v}$ of diameter at most $2C^*r_0$ by Lemma \ref{lem:approx}. Hence $\operatorname{card}{V_0} \lesssim_{C^*} 1$ and $$\sum_{I \in \mathscr{E}_0\cup \mathscr{B}_0} (\mathop\mathrm{diam}\nolimits{f_0(I)})^s + \sum_{I \in \mathscr{P}_0}p_{0,I} \lesssim_{s,C^*,\xi_2} r_0^s.$$
On the other hand, suppose that $\alpha_{0,v}\geq \alpha_0$ for all $v\in V_0$. Then $$\sum_{I \in \mathscr{E}_0\cup \mathscr{B}_0} (\mathop\mathrm{diam}\nolimits{f_0(I)})^s + \sum_{I \in \mathscr{P}_0}p_{0,I} \lesssim_{s,C^*,\xi_2} \sum_{\substack{v\in V_0\\ \alpha_{0,v}\geq \alpha_0}} r_0^s$$ In either case, we arrive at
\begin{align*}
\sum_{(l,I)\in\partial T} (\mathop\mathrm{diam}\nolimits{f_l(I)})^s &\lesssim_{s,C^*,\xi_2} r_0^s+ \sum_{k=0}^{m-1}\sum_{(v,v')\in \mathsf{Flat}(k) }\tau_{s}(k,v,v') \rho_{k}^s r_0^s + \sum_{k=0}^{m}\sum_{\substack{v\in\ V_k\\ \alpha_{k,v}\geq \alpha_0}}\rho_{k}^s r_0^s \\
&\lesssim_{s,C^*,\xi_2} r_0^s+ \sum_{k=0}^{\infty}\sum_{(v,v')\in \mathsf{Flat}(k) }\tau_{s}(k,v,v') \rho_{k}^sr_0^s + \sum_{k=0}^{\infty}\sum_{\substack{v\in\ V_k\\ \alpha_{k,v}\geq \alpha_0}}\rho_{k}^s r_0^s.
\end{align*}
Since $T$ was an arbitrary tree over $[0,1]$, we obtain the desired bound on $\mathcal{M}_s([0,1])$.
\end{proof}
\subsection{Proof of Lemma \ref{lem:estimates}}\label{sec:prooflemma}
The proof is divided into five estimates (\ref{eq:est1}), (\ref{eq:est2}), (\ref{eq:est3}), (\ref{eq:est4}) and (\ref{eq:est5}), whose sum gives (\ref{eq:estimates}). Towards this end, we split the left hand side of (\ref{eq:estimates}) into smaller sums by making the following four decompositions.
Firstly, $\partial T$ can be partitioned as $\mathcal{E}_1\cup \mathcal{E}_2\cup \mathcal{E}_3\cup \mathcal{B}_1\cup \mathcal{B}_2\cup \mathcal{B}_3\cup\mathcal{B}_4 \cup \mathcal{F}\cup (\partial T \cap \partial p(T))$, where
\begin{align*}
\mathcal{E}_1 &= \{(k+1,I) \in \partial T : \text{$I \in \mathscr{E}_{k+1}(J)$ and $J\in\mathscr{N}_k$ is as in \textsection\ref{sec:noflatpoints}}\},\\
\mathcal{E}_2 &= \{(k+1,I) \in \partial T : \text{$I \in \mathscr{E}_{k+1}(J)$ and $J\in\mathscr{N}_k$ is as in \textsection\ref{sec:flatpoints}}\},\\
\mathcal{E}_3 &= \{(k+1,I) \in \partial T : \text{$I \in \mathscr{E}_{k+1}(J)$ and $J\in\mathscr{E}_k$ is as in \textsection\ref{sec:flatedges}}\},\\
\mathcal{B}_1 &= \{(k+1,I) \in \partial T : \text{$I \in\mathscr{B}_{k+1}(J)$ and $J \in \mathscr{N}_k$ is as in \textsection\ref{sec:noflatpoints}}\},\\
\mathcal{B}_2 &= \{(k+1,I) \in \partial T : \text{$I \in \mathscr{B}_{k+1}(J)$ and $J\in\mathscr{E}_k$ is as in \textsection\ref{sec:noflatedges}}\},\\
\mathcal{B}_3 &= \{(k+1,I) \in \partial T : \text{$I \in \mathscr{B}_{k+1}(J)$ and $J\in\mathscr{E}_k$ is as in \textsection\ref{sec:flatedges}}\},\\
\mathcal{B}_4 &= \{(k+1,I) \in \partial T : \text{$I \in \mathscr{B}_{k+1}(J)$ and $J\in\mathscr{B}_k$ is as in \textsection\ref{sec:bridges}}\},\\
\mathcal{F} &\subseteq \{k+1\}\times(\mathscr{N}_{k+1}\cup\mathscr{F}_{k+1}).
\end{align*}
Secondly, $\mathscr{P}_{k+1}\cap\partial T$ can be partitioned as $\mathcal{P}_1\cup \mathcal{P}_2\cup \mathcal{P}_3$, where
\begin{align*}
\mathcal{P}_1 &= \{(k+1,I) \in \partial T\cap\mathscr{P}_{k+1}: \text{$I \in \mathscr{N}_{k+1}(J)$ and $J\in \mathscr{N}_k$ is as in \textsection\ref{sec:noflatpoints}}\}, \\
\mathcal{P}_2 &= \{(k+1,I) \in \partial T\cap\mathscr{P}_{k+1}: \text{$I \in \mathscr{N}_{k+1}(J)$ and $J\in \mathscr{N}_k$ is as in \textsection\ref{sec:flatpoints}}\},\\
\mathcal{P}_3 &= \{(k+1,I) \in \partial T\cap\mathscr{P}_{k+1}: \text{$I \in \mathscr{N}_{k+1}(J)$ and $J\in \mathscr{E}_k$ is as in \textsection\ref{sec:flatedges}}\}.
\end{align*}
Thirdly, $ \partial p(T)$ can be partitioned as $\mathcal{E}_1'\cup \mathcal{E}_2' \cup \mathcal{F}' \cup (\partial T \cap \partial p(T))$, where
\begin{align*}
\mathcal{E}_1' &= \{(k,I) \in \partial p(T) \setminus \partial T : \text{$I \in \mathscr{E}_{k}$ is as in \textsection\ref{sec:noflatedges}}\},\\
\mathcal{E}_2' &= \{(k,I) \in \partial p(T) \setminus \partial T : \text{$I \in \mathscr{E}_{k}$ is as in \textsection\ref{sec:flatedges}}\},\\
\mathcal{B}_1' &= \{(k,I) \in \partial p(T) \setminus \partial T : \text{$I \in \mathscr{B}_{k}$ is as in \textsection\ref{sec:bridges}}\},\\
\mathcal{F}' &\subseteq \{k\}\times(\mathscr{N}_{k}\cup\mathscr{F}_{k}).
\end{align*}
Fourthly, set $\mathscr{B}^{\,**}_{k+1}(T)= \mathscr{B}_{k+1}^*(T)\sqcup\mathscr{B}_{k+1}^*(T)$. Then define collections $\mathcal{B}^*_1$ and $\mathcal{B}^*_2$ as follows. If $I\in \mathscr{N}_{k}$, $\alpha_{k,f_k(I)}<\alpha_0$, $\mathscr{I}_{k+1}(I)\subset\partial T$, and $f_k(I)$ is an endpoint of the image $f_{k+1}(J)$ of $(k+1,J)\in\mathscr{B}^*_{k+1}$, then we include a copy of $(k+1,J)$ from $\mathscr{B}^{\,**}_{k+1}(T)$ in $\mathcal{B}^*_1$. If $K\in\mathscr{N}_k$ and $f_k(K)$ is the endpoint of $f_{k+1}(I)$ for some $(k+1,I)\in\mathcal{B}_3$ that lies strictly between the endpoints of the image $f_k(J)$ of the associated edge interval $J\in\mathscr{E}_{k}$, then we include a copy of $(k+1,I)$ from $\mathscr{B}^{\,**}_{k+1}(T)$ in $\mathcal{B}^*_2$. Because each bridge has only two endpoints, we can choose the included copies so that $\mathcal{B}^*_1\cup\mathcal{B}^*_2\subset \mathscr{B}^{\,**}_{k+1}.$
Before proceeding to the estimates, we remark that
\[ \sum_{(k,I) \in \mathcal{F}'}(\mathop\mathrm{diam}\nolimits{f_k(I)})^s = \sum_{(k+1,I) \in \mathcal{F}}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s = 0,\] because $f_k$ is constant on each interval in $\mathscr{N}_k\cup\mathscr{F}_k$ by (P2).
\medskip
\underline{Estimate 1.} Here we deal with phantom masses and new intervals coming from some $I\in \mathscr{N}_k$ whose image is not flat. In particular, we will show that there exists $C_1$ depending only on $s$, $C^*$, and $\xi_2$ such that
\begin{equation} \label{eq:est1}\tag{E1}\begin{split}
\sum_{(k+1,I) \in \mathcal{E}_1\cup \mathcal{B}_1}&(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s + \sum_{(k+1,J) \in \mathcal{P}_1}p_{k+1,J}\\ & \leq C_1 \sum_{\substack{v\in V_k \\ \alpha_{k,v}\geq \alpha_0}} \rho_{k}^sr_0^s + C_1\sum_{\substack{w\in V_{k+1} \\ \alpha_{k+1,w}\geq \alpha_0}} \rho_{k+1}^sr_0^s.
\end{split}\end{equation}
Since
\[ \{I \in \mathscr{N}_k : \alpha_{k,f_k(I)}\geq \alpha_0 \text{ and }\mathscr{I}_{k+1}(I)\subset \partial T\} \subseteq \{I \in \mathscr{N}_k\cap \partial p(T) : \alpha_{k,f_k(I)}\geq \alpha_0\},\]
inequality (\ref{eq:est1}) follows from the inequality
\begin{equation}\label{eq:est1'}
\begin{split}
&\sum_{\substack{I \in \mathscr{N}_k \\ \{k+1\}\times\mathscr{I}_{k+1}(I) \subset \partial T \\ \alpha_{k,f_k(I)}\geq \alpha_0}} \left ( \sum_{J \in \mathscr{E}_{k+1}(I) \cup \mathscr{B}_{k+1}(I)}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s + \sum_{\substack{J \in \mathscr{N}_{k+1}(I) \\ (k+1,J) \in \mathscr{P}_{k+1}}} p_{k+1,J} \right )\\
&\qquad\leq C_1\sum_{\substack{I \in \mathscr{N}_k \\ \{k+1\}\times\mathscr{I}_{k+1}(I) \subset \partial T \\ \alpha_{k,f_k(I)}\geq \alpha_0}} \rho_{k}^sr_0^s
+ C_1\sum_{\substack{I \in \mathscr{N}_k \\ \{k+1\}\times\mathscr{I}_{k+1}(I) \subset \partial T \\ \alpha_{k,f_k(I)}\geq \alpha_0}} \sum_{\substack{J\in\mathscr{N}_{k+1}(I)\\ \alpha_{k+1,f_{k+1}(J)}\geq \alpha_0}} \rho_{k+1}^sr_0^s.
\end{split}
\end{equation}
To prove (\ref{eq:est1'}), fix any $I \in \mathscr{N}_k$ such that $\{k+1\}\times\mathscr{I}_{k+1}(I) \subset \partial T$ and $ \alpha_{k,f_k(I)}\geq \alpha_0$. Recall that $\mathcal{N}_{k+1}(I)$ is in one-to-one correspondence with $V_{k+1,I}$ defined in \S\ref{sec:noflatpoints}. There are now two possibilities.
On one hand, suppose that $\alpha_{k+1,w}<\alpha_0$ for some $w\in V_{k+1,I}$. Then $V_{k+1,I}$ projects onto an $(1+3(1/16)^2)^{-1}\rho_{k+1}r_0$ separated set in $\ell_{k+1,w}$ of diameter at most $2C^*\rho_{k+1}r_0$ by Lemma \ref{lem:approx}. Hence $\operatorname{card}{\mathscr{I}_{k+1}(I)} \lesssim_{C^*} 1$ and
\begin{equation*}\sum_{J \in \mathscr{E}_{k+1}(I) \cup \mathscr{B}_{k+1}(I)}(\mathop\mathrm{diam}\nolimits{f_{k+1}(J)})^s + \sum_{\substack{J \in \mathscr{N}_{k+1}(I) \\ (k+1,J) \in \mathscr{P}_{k+1}}} p_{k+1,J} \lesssim_{s,C^*,\xi_2} \rho_{k+1}^sr_0^s\lesssim_{s,C^*,\xi_2}\rho_{k}^sr_0^s. \end{equation*}
On the other hand, suppose that $\alpha_{k+1,w}\geq \alpha_0$ for all $w\in V_{k+1,I}$. Then \begin{equation*} \sum_{J \in \mathscr{E}_{k+1}(I) \cup \mathscr{B}_{k+1}(I)}(\mathop\mathrm{diam}\nolimits{f_{k+1}(J)})^s + \sum_{\substack{J \in \mathscr{N}_{k+1}(I) \\ (k+1,J) \in \mathscr{P}_{k+1}}} p_{k+1,J} \lesssim_{s,C^*,\xi_2} \sum_{J\in\mathscr{N}_{k+1}(I)} \rho_{k+1}^sr_0^s. \end{equation*}
Because the sets $V_{k+1,I}$ for different $I\in\mathscr{N}_k$ are pairwise disjoint (see \S\ref{sec:noflatpoints}), summing over all such $I$ yields (\ref{eq:est1'}).
\medskip
\underline{Estimate 2.} We estimate the new phantom masses and new intervals coming from some $I\in \mathscr{N}_{k}$ such that $\alpha_{k,f_k(I)} < \alpha_0$ and $\mathscr{I}_{k+1}(I) \subset \partial T$. In particular, we show that
\begin{equation}\label{eq:est2}\tag{E2} \begin{split}
&\sum_{(k+1,I)\in\mathcal{E}_2}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s + \sum_{(k+1,I) \in \mathcal{P}_2}p_{k+1,I}\\ &\qquad\leq \sum_{(k,I)\in \mathscr{P}_{k}\cap \partial p(T)}p_{k,I} + \frac16\sum_{(k+1,I) \in \mathcal{B}^*_1} (\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s.\end{split}
\end{equation} This estimate is responsible for the choice of the constant $14A^*$ appearing in the definition of $\mathsf{Flat}(k)$, and thus, for the constant $30A^*$ appearing in the definition of $\alpha_{k,v}$. Inequality (\ref{eq:est2}) is equivalent to
\begin{equation}\label{eq:est2*}
\begin{split}
\sum_{\substack{I \in \mathscr{N}_k\\ \{k+1\}\times\mathscr{N}_{k+1}(I) \subset \partial T \\ \alpha_{k,f_k(I)} < \alpha_0 }} &\left ( \sum_{J \in \mathscr{E}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s + \sum_{\substack{J \in \mathscr{N}_{k+1}(I) \\ (k+1,J) \in \mathscr{P}_{k+1}}}p_{k+1,J} \right )\\
&\leq \sum_{\substack{I \in \mathscr{N}_k\\ \{k+1\}\times\mathscr{N}_{k+1}(I) \subset \partial T \\ \alpha_{k,f_k(I)} < \alpha_0 }}p_{k,I} + \frac16 \sum_{(k+1,I') \in \mathcal{B}^*_1}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s.
\end{split}\end{equation}
To prove (\ref{eq:est2*}) fix $I \in \mathscr{N}_k$ such that $\{k+1\}\times\mathscr{N}_{k+1}(I) \subset \partial T$ and $\alpha_{k,f_k(I)} < \alpha_0$. There are nine cases (1a, 1b, 2a, 2b, 2c, 2d, 3a, 3b, 3c). We sincerely apologize to the reader.
For the first four cases (1a, 1b, 2a, 2b), we show that
\begin{equation}\label{eq:est2'}
\sum_{J \in \mathscr{E}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits{f_{k+1}(J)})^s + \sum_{\substack{J \in \mathscr{N}_{k+1}(I)\\ (k+1,J) \in \mathscr{P}_{k+1}}}p_{k+1,J} \leq p_{k,I} .
\end{equation}
\emph{Case 1.} Suppose $f_k(I)$ is 2-sided terminal in $V_k$.
\emph{Case 1a.} If $\mathscr{N}_{k+1}(I)=\{I\}$, then $f_{k+1}(I)$ is 2-sided terminal in $V_{k+1}$, $\mathscr{E}_{k+1}(I)=\emptyset$ and the new phantom mass $p_{k+1,I}=2P \rho_{k+1}^sr_0^s$ is dominated by the old phantom mass $p_{k,I}=2P\rho_{k}^sr_0^s$. Hence \eqref{eq:est2'} holds.
\emph{Case 1b.} Assume that $\mathscr{N}_{k+1}(I)$ contains at least two elements (see Figure \ref{fig:flatpoints} above). In this case, at most two elements of $\mathcal{N}_{k+1}(I)$ map to 1-sided terminal vertices in $V_{k+1}$. By Lemma \ref{lem:flatpointsum},
\begin{align*} p_{k+1,J_1}+p_{k+1,J_2} + \sum_{J\in\mathscr{I}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits f_{k+1}(J))^s & \leq 2P\rho_{k+1}^sr_0^s + 4(1.1C^*)^s\rho_{k+1}^s r_0^s\\
&\leq 2P \rho_k^sr_0^s = p_{k,I},\end{align*} because $P$ was chosen to be sufficiently large such that $$[P+2(1.1C^*)^s]\xi_2^s \leq P.$$
Thus, \eqref{eq:est2'} holds in this case, as well. \smallskip
\smallskip
\emph{Case 2.} Suppose that $f_k(I)$ is 1-sided terminal in $V_k$. Let $(v_1,v')$ be the unique element in $\mathsf{Flat}(k)$ with $v_1=f_k(I)$ and let $v_2$ be the first vertex in $V_{k+1}$ between $v_1$ and $v'$ in the direction going from $v_1$ to $v'$. By property (P7), we can find an interval $L$ in $\mathscr{E}_k$ such that $f_k(L)=(v_1,v')$ and $I\cap \overline{L}\neq\emptyset$. Let $K$ be an interval in $\mathscr{I}_{k+1}(L)$ such that $f_k(K)=(v_1,v_2)$. There will be four cases, depending on whether $\mathscr{N}_{k+1}(I)$ contains one or more elements and whether $K$ belongs to $\mathscr{E}_{k+1}(L)$ or $\mathscr{B}_{k+1}(L)$.
\emph{Case 2a.} Suppose that $\mathscr{N}_{k+1}(I)=\{I\}$ and $K\in\mathscr{E}_{k+1}(L)$. Then $f_{k+1}(I)$ is 1-sided terminal in $V_{k+1}$, $\mathscr{E}_{k+1}(I)=\emptyset$, and the new phantom mass $p_{k+1,I}=P\rho_{k+1}^sr_0^s$ is dominated by the old phantom mass $p_{k,I}=P\rho_k^sr_0^s$. Hence \eqref{eq:est2'} holds.
\emph{Case 2b.} Suppose that $\mathscr{N}_{k+1}(I)$ contains at least two elements (see Figure \ref{fig:flatpoints} above) and $K\in\mathscr{E}_{k+1}(L)$. Then at most one element of $\mathcal{N}_{k+1}(K)$ maps to a 1-sided terminal vertex in $V_{k+1}$. By Lemma \ref{lem:flatpointsum}, \begin{align*}
p_{k+1,J_1} + \sum_{J\in\mathscr{I}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits f_{k+1}(J))^s & \leq P\rho_{k+1}^sr_0^s + 2(1.1C^*)^s \rho_{k+1}^s r_0^s\\
& \leq P \rho_k^s r_0^s = p_{k,I},\end{align*} because $P$ was chosen to be sufficiently large such that $$[P+2(1.1C^*)^s]\xi_2^s \leq P.$$ Thus, \eqref{eq:est2'} holds, once again.
\emph{Case 2c.} Suppose that $\mathscr{N}_{k+1}(I)=\{I\}$ and $K\in\mathscr{B}_{k+1}(L)$. Then $f_{k+1}(I)$ is 2-sided terminal in $V_{k+1}$ and $\mathscr{E}_{k+1}(I)=\emptyset$. The new phantom mass that must be paid for is $p_{k+1,I}=2P\rho_{k+1}^sr_0^s$. In this case, we pay for one half of $p_{k+1,I}$ with $p_{k,I}=P\rho_{k}^sr_0^s$ and use Lemma \ref{lem:bridgelength} to pay for the other half of $p_{k+1,I}$ with $\frac16(\mathop\mathrm{diam}\nolimits f_{k+1}(K))^s$, where $K\in\mathscr{B}_{k+1}^*(T)$. That is, $$p_{k+1,I} \leq p_{k,I} + \frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K))^s.$$
\emph{Case 2d.} Suppose that $\mathscr{N}_{k+1}(I)$ contains at least two points and $K\in\mathscr{B}_{k+1}(L)$. Then $f_{k+1}(I)$ is 1-sided terminal in $V_{k+1}$, $\mathscr{E}_{k+1}(I)$ is nonempty, and up to one of the new vertices drawn could be 1-sided terminal in $V_{k+1}$, as well. In this case, $$p_{k+1,I}+\underbrace{p_{k+1,J_1} + \sum_{J\in \mathscr{E}_{k+1}(I)} (\mathop\mathrm{diam}\nolimits f_{k+1}(J))^s} \leq \frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K))^s+ p_{k,I}.$$ by Lemma \ref{lem:bridgelength}, Lemma \ref{lem:flatpointsum}, and the choice of $P$.
\smallskip
\emph{Case 3.} Suppose $f_k(I)$ is not terminal in $V_k$. Then $\mathscr{E}_{k+1}(I)=\emptyset$. It remains to pay for $p_{k+1,I}$ as needed. Let $L_1$, $L_2$, $K_1$, and $K_2$ be defined by analogy with $L$ and $K$ from Case 2, but corresponding to the two distinct flat pairs $(f_k(I),v')$ and $(f_k(I),v'')$. There are three cases, depending on whether $K_1$ and $K_2$ both edge intervals, one of $K_1$ or $K_2$ is an edge interval and the other is a bridge interval, or both $K_1$ and $K_2$ are bridge intervals.
\emph{Case 3a.} Suppose that $K_1$ belongs to $\mathscr{E}_{k+1}(L_1)$ and $K_2$ belongs to $\mathscr{E}_{k+1}(L_2)$. Then $f_{k+1}(I)$ is non-terminal in $V_{k+1}$. Hence $(k+1,I) \not\in\mathscr{P}_{k+1}$ and both sides of (\ref{eq:est2'}) are zero. In other words, there is nothing to pay for in Case 3a.
\emph{Case 3b.} Suppose that one of $K_1$ or $K_2$ is an edge interval and the other is a bridge interval, say without loss of generality that $K_1\in\mathscr{E}_{k+1}(L_1)$ and $K_2\in\mathscr{B}_{k+1}(L_2)$. Then $f_{k+1}(I)$ is 1-sided terminal in $V_{k+1}$ and $p_{k+1,I}=P\rho_{k+1}^sr_0^s \leq \frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K_2))^s$ by Lemma \ref{lem:bridgelength}.
\emph{Case 3c.} Suppose that $K_1$ belongs to $\mathscr{B}_{k+1}(L_1)$ and $K_2$ belongs to $\mathscr{B}_{k+1}(L_2)$. Then $f_{k+1}(I)$ is 2-sided terminal in $V_{k+1}$ and $$p_{k+1,I} = 2P\rho_{k+1}^sr_0^s \leq \ \frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K_1))^s + \frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K_2))^s.$$
Adding up the estimates in the nine cases, we obtain (\ref{eq:est2}).
\medskip
\underline{Estimate 3.} On one hand, $(k,I)\in \mathcal{B}_1'$ if and only if $(k+1,I)\in\mathcal{B}_4$ (see \S\ref{sec:bridges}). When $(k,I)\in \mathcal{B}_1'$, we have $\mathop\mathrm{diam}\nolimits{f_k(I)} = \mathop\mathrm{diam}\nolimits{f_{k+1}(I)}$. Thus,
$$\sum_{(k+1,I)\in\mathcal{B}_4}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s = \sum_{(k,I) \in \mathcal{B}_1'}(\mathop\mathrm{diam}\nolimits{f_{k}(I)})^s.$$
On the other hand, when both endpoints of the image of an edge interval are non-flat, the edge interval becomes a bridge interval and pays for itself (see \S\ref{sec:noflatedges}):
$$\sum_{(k+1,I)\in\mathcal{B}_2}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s = \sum_{(k,I) \in \mathcal{E}_1'}(\mathop\mathrm{diam}\nolimits{f_{k}(I)})^s.$$ All together, we have \begin{equation}\label{eq:est3}\tag{E3}
\sum_{(k+1,I)\in\mathcal{B}_2\cup\mathcal{B}_4}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s = \sum_{(k,I) \in \mathcal{E}_1'\cup\mathcal{B}_1'}(\mathop\mathrm{diam}\nolimits{f_{k}(I)})^s.
\end{equation}
\medskip
\underline{Estimate 4.} Next, we control the new phantom masses at endpoints of images $f_{k+1}(J)$ of bridge intervals $J\in\mathscr{B}_{k+1}(I)$ coming from some edge interval $I\in\mathscr{E}_k$ as in \S\ref{sec:flatedges} such that the endpoint lies between the endpoints of $f_k(I)$. Specifically, we show that
\begin{align}\label{eq:est4}\tag{E4}
\sum_{(k+1,I)\in \mathcal{P}_3}p_{k+1,I} \leq \frac16\sum_{(k+1,I)\in\mathcal{B}^*_2}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s.
\end{align}
Inequality (\ref{eq:est4}) is equivalent to
\begin{align}\label{eq:est4'}
\sum_{(k,I)\in\mathcal{E}_2'}\sum_{\substack{J\in \mathscr{N}_{k+1}(I)\\ (k+1,J) \in \mathscr{P}_{k+1}}}p_{k+1,J} \leq\frac16\sum_{(k+1,I)\in\mathcal{B}^*_2}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s.
\end{align}
To prove (\ref{eq:est4'}), fix $(k,I)\in\mathcal{E}_2'$ and $J\in \mathscr{N}_{k+1}(I)$ be such that $(k+1,J)\in\mathscr{P}_{k+1}$. Then $f_{k+1}(J)$ lies strictly between the endpoints of $f_k(I)$ and $f_{k+1}(J)$ is either 1- or 2-sided terminal in $V_{k+1}$. On one hand, if $f_{k+1}(J)$ is 1-sided terminal, then there exists precisely one element $(k+1,K)\in\mathcal{B}_2^*$ such that $f_{k+1}(J)$ is an endpoint of $f_{k+1}(K)$. In this case, $$p_{k+1,J} =P\rho_{k+1}^sr_0^s\leq \frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K))^s$$ by Lemma \ref{lem:bridgelength}. On the other hand, if $f_{k+1}(J)$ is 2-sided terminal, then there exist two elements $(k+1,K_1)$ and $(k+1,K_2)$ in $\mathcal{B}_2^*$ such that $f_{k+1}(J)$ is the common endpoint of $f_{k+1}(K_1)$ and $f_{k+1}(K_2)$. In this case, $$p_{k+1,J} = 2P\rho_{k+1}^sr_0^s \leq \frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K_1))^s+\frac{1}{6}(\mathop\mathrm{diam}\nolimits f_{k+1}(K_2))^s$$ by Lemma \ref{lem:bridgelength}.
\begin{rem} In Estimates 2 and Estimate 4, each endpoint of the image $f_{k+1}(I)$ of $(k+1,I)\in\mathcal{B}_1^*\cup \mathcal{B}_2^*$ is used once and each $f_{k+1}(I)$ has only two endpoints. Hence $$\frac16\sum_{(k+1,I)\in\mathcal{B}^*_1\cup \mathcal{B}^*_2}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s\leq \frac{1}{6}\sum_{(k+1,I)\in\mathscr{B}^{\,**}_{k+1}(T)}=\frac{1}{3}\sum_{(k+1,I)\in\mathscr{B}^{*}_{k+1}(T)}.$$
\end{rem}
\medskip
\underline{Estimate 5.} In this final estimate, we deal with new intervals in $\partial T$ coming from an edge interval in $\partial p(T)$ which has an endpoint with flat image. We will show that
\begin{align}\label{eq:est5}\tag{E5}
\sum_{(k+1,I)\in\mathcal{E}_3\cup\mathcal{B}_3} (\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s \leq \sum_{(k,I) \in \mathcal{E}_2'}(\mathop\mathrm{diam}\nolimits{f_{k+1}(I)})^s + (14A^*)^s\sum_{(v,v')\in\mathsf{Flat}(k)} \tau_{s}(k,v,v') \rho_k^sr_0^s.
\end{align}
For each $I \in \mathcal{E}_2'$, pick an endpoint $x_I$ of $I$ such that $\alpha_{k,f_k(x_I)} < \alpha_0$ and let $y_I$ be the other endpoint of $I$. Estimate (E5) follows immediately from
\begin{equation}\label{eq:est5'}
\begin{split}
&\sum_{(k,I) \in \mathcal{E}_2' } \sum_{J\in \mathscr{E}_{k+1}(I)\cup\mathscr{B}_{k+1}(I)}(\mathop\mathrm{diam}\nolimits{f_{k+1}(J)})^s \\
&\quad\leq \sum_{(k,I) \in \mathcal{E}_2'}\left((\mathop\mathrm{diam}\nolimits f_{k}(I)\right)^s + (14A^*)^s\tau_{s}(k,f_k(x_I),f_k(y_I)))\rho_k^sr_0^s.
\end{split}
\end{equation}
To show (\ref{eq:est5'}), fix $(k,I)\in \mathcal{E}_2'$ and enumerate $\mathscr{E}_{k+1}(I)\cup\mathscr{B}_{k+1}(I) = \{I_1,\dots, I_{n}\}$. Let $v=f_k(x_I)$ and let $v' = f_{k}(y_I)$. By definition of $\tau_s(k,v,v')$ and (P3), we have \begin{equation*}\begin{split}\sum_{i=1}^{n-1}(\mathop\mathrm{diam}\nolimits f_{k+1}(I_i))^s &\leq |v-v'|^s + \tau_s(k,v,v')|v-v'|^s \\ &\leq (\mathop\mathrm{diam}\nolimits f_k(I))^s + (14A^*)^2\tau_s(k,v,v')\rho_k^sr_0^s.\end{split}\end{equation*} Summing over all pairs $(k,I)\in\mathcal{E}_2'$, we obtain \eqref{eq:est5'}.
\medskip
Adding (\ref{eq:est1}), (\ref{eq:est2}), (\ref{eq:est3}), (\ref{eq:est4}), and (\ref{eq:est5}), we arrive at (\ref{eq:estimates}). This completes the proof of Lemma \ref{lem:estimates}.
\section{H\"older parametrization}\label{sec:Holder}
In \S\ref{sec:proof1}, we prove the following theorem, which is the paper's main result. Afterwards, in \S\ref{sec:cortothm}, we derive several corollaries, including Theorem \ref{thm:main2}. In \S\ref{sec:modifiedTST}, we state and prove a refinement of Theorem \ref{thm:main} that gives an essentially 2-to-1 curve. In \S\ref{sec:Carleson}, we show that replacing \eqref{eq:main-condition} with a Carleson type condition produces an upper Ahlfors regular curve.
Given parameters $C^*$, $\xi_1$, and $\xi_2$, let $\alpha_1$ be defined by \eqref{alphabound}. That is, \begin{equation*}\alpha_1 = \min\left\{\frac{1}{16}, \left( \frac{\xi_1(1-\xi_2)}{42 C^*} \right)^{1/2}\right\}.\qedhere\end{equation*}
\begin{thm}[H\"older Traveling Salesman with Nets] \label{thm:main}
Assume that $X=l^2(\mathbb{R})$ or $X=\mathbb{R}^N$ for some $N\geq 2$. Let $s\geq 1$, let $\mathscr{V}=(V_k,\rho_k)_{k\geq 0}$ be a sequence of finite sets $V_k$ in $X$ and numbers $\rho_k>0$ that satisfy properties (V0)--(V5) defined in \textsection\ref{sec:flat}. If $\alpha_0\in(0,\alpha_1]$ and
\begin{equation} \label{eq:main-condition} S^s_{\mathscr{V}} := \sum_{k=0}^{\infty}\sum_{(v,v')\in \mathsf{Flat}(k)}\tau_{s}(k,v,v') \rho_k^s+ \sum_{k=0}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} \geq \alpha_0}} \rho_k^s < \infty,\end{equation}
then there exists a $(1/s)$-H\"older map $f:[0,1] \to X$ such that $f([0,1])\supset \bigcup_{k\geq 0}V_k$ and the H\"older constant of $f$ satisfies $H \lesssim_{s,C^*,\xi_1,\xi_2} r_0 (1+ S^s_{\mathscr{V}})$.
\end{thm}
\subsection{Proof of Theorem \ref{thm:main} }\label{sec:proof1}
In this subsection, $w$ will always denote a finite word in the alphabet $\mathbb{N}=\{1,2,\dots\}$, including the empty word $\emptyset$. We denote the length of a word $w$ by $|w|$.
The conclusion holds trivially if $\bigcup_{k\geq 0}V_k$ is a singleton. Thus, in addition to $S^s_{\mathscr{V}}<\infty$, we may assume that $\bigcup_{k\geq 0}V_k$ contains at least two points. Because $\alpha_0\leq \alpha_1$, Proposition \ref{prop:mass} gives \begin{equation}\label{e:S-mass-bound} 0<\mathcal{M}_s([0,1]) \lesssim_{s,C^*,\xi_2} r_0^s(1+ S^s_{\mathscr{V}})<\infty.\end{equation}
To proceed, we start by renaming the intervals in $\{[0,1]\}\cup\mathscr{I}$. Denote $\Delta_{\emptyset} = [0,1]$ and write $\mathscr{I}_0= \{\Delta_1,\dots, \Delta_{n_{\emptyset}}\}$, enumerated according to the orientation of $[0,1]$. Inductively, suppose that for some word $w$ with $|w|=k$, we have defined $\Delta_{w} \in \mathscr{I}_{k}$. Suppose also that
\[ \mathscr{I}_{k+1}(\Delta_w) = \{J_1,\dots,J_{n_w}\},\]
enumerated according to the orientation of $[0,1]$. Then for each $i\in\{1,\dots,n_w\}$, denote $\Delta_{wi} = J_i$. Denote by $\mathcal{W}$ the set of all finite words with letters from $\mathbb{N}$ for which an interval $\Delta_{w}$ has been defined.
Next, we use the masses of intervals defined in \S\ref{sec:mass} to modify the length of intervals $\Delta_w$.
Define $\Delta'_{\emptyset} = [0,1]$. Let $\{\Delta'_{1},\dots,\Delta'_{n_{\emptyset}}\}$ be a partition of $\Delta'_{\emptyset}$, enumerated according to the orientation of $[0,1]$, satisfying
\begin{enumerate}
\item $\Delta_i'$ is open (resp.~ closed) if and only if $\Delta_i$ is open (resp.~ closed), and
\item $\mathop\mathrm{diam}\nolimits{\Delta'_{i}} = \mathcal{M}_s(0,\Delta_{i})/\mathcal{M}_s([0,1])$.
\end{enumerate}
These intervals exist, because $\mathcal{M}_s([0,1]) = \sum_{i=1}^{n_{\emptyset}}\mathcal{M}_s(0,\Delta_i)$.
Inductively, suppose that an interval $\Delta'_w\subset [0,1]$ has been defined for some $w\in\mathcal{W}$ such that
\[ \mathop\mathrm{diam}\nolimits{\Delta'_w} \geq \frac{\mathcal{M}_s(|w|-1,\Delta_w)}{\mathcal{M}_s([0,1])}\]
and $\Delta_{w}'$ is open (resp.~ closed) if and only if $\Delta_{w}$ is open (resp.~ closed). Let $\{\Delta'_{w1},\dots,\Delta'_{wn_w}\}$ be a partition of $\Delta'_w$, enumerated according to the orientation of $[0,1]$, satisfying
\begin{enumerate}
\item $\Delta_{wi}'$ is open (resp.~ closed) if and only if $\Delta_{wi}$ is open (resp.~ closed), and
\item $\mathop\mathrm{diam}\nolimits{\Delta'_{wi}} \geq \mathcal{M}_s(|w|,\Delta_{wi})/\mathcal{M}_s([0,1])$.
\end{enumerate}
This partition exists by Lemma \ref{lem:sum}(3).
Define the family $$\mathscr{E}_k' = \{\Delta'_w : \Delta_w \in \mathscr{E}_k\}$$ and similarly define the families $\mathscr{B}_k'$, $\mathscr{N}_k'$, $\mathscr{F}_k'$, and $\mathscr{I}_k'$.
For each $k\geq 0$, define a continuous map $F_k:[0,1] \to X$ by
\[ F_k|\Delta_{w}' = (f_k|\Delta_w)\circ \phi_w \qquad\text{for all }w\in \mathcal{W},\]
where $\phi_w$ is the unique increasing affine homeomorphism mapping $\Delta_w'$ onto $\Delta_w$ when $\Delta_w'$ is nondegenerate and $\phi_w$ maps to any point in $\Delta_w$ when $\Delta_w'$ is a singleton. (The latter possibility occurs only when $\Delta_w'$ belongs to $\mathscr{F}'_k$ or $\mathscr{N}'_k$.)
We now prove two auxiliary results for the sequence $(F_k)_{k\geq 0}$.
\begin{lem}\label{lem:Hold-conv1}
For all $k\geq 0$ and $x\in[0,1]$, we have $|F_{k+1}(x)-F_k(x)| \leq 30A^*\xi_2r_0\rho_{k}.$
\end{lem}
\begin{proof}
Fix $x\in [0,1]$ and $w\in \mathcal{W}$ such that $x\in\Delta_w$ and $|w|=k$. Let also $i\in\{1,\dots,n_{w}\}$ be such that $x\in\Delta_{wi}'$.
If $\Delta_w'\in \mathscr{B}_k'$, then $\Delta_w' = \Delta_{wi}'$, $\Delta_w = \Delta_{wi}$, and $F_k|\Delta_w' = F_{k+1}|\Delta_{wi}'$. We conclude that $|F_{k+1}(x) - F_k(x)| = 0$.
If $\Delta_w'\in \mathscr{F}_k'$, then $\Delta_w' = \Delta_{wi}' = \{x\}$, and $F_k(x) = F_{k+1}(x)$. Hence $|F_{k+1}(x) - F_k(x)| = 0$.
If $\Delta'_w\in \mathscr{E}_k'$, then $|F_{k+1}(x) - F_k(x)|\leq 2\mathop\mathrm{diam}\nolimits{F_k(\Delta_w')} = 2\mathop\mathrm{diam}\nolimits{f_k(\Delta_w)} \leq 28A^* \rho_{k+1}r_0$.
If $\Delta'_w\in \mathscr{N}_k'$, then $|F_{k+1}(x) - F_k(x)|\leq 2\mathop\mathrm{diam}\nolimits{f_{k}(\Delta_w)} \leq \mathop\mathrm{diam}\nolimits{\tilde{V}_{k+1,I}} \leq 30A^* \rho_{k+1}r_0$, where $\tilde V_{k+1,I}$ is a set defined in \S\ref{sec:noflatpoints}.
\end{proof}
\begin{lem}\label{lem:Hold-conv2}
For all $k\geq 0$ and $x,y \in[0,1]$,
\[|F_{k}(x) - F_k(y)| \leq \mathcal{M}_s([0,1]) r_0^{1-s}\rho_k^{1-s}|x-y|
\lesssim_{s,C^*,\xi_2} r_0 (1+ S^s_{\mathscr{V}}) \rho_k^{1-s} |x-y|.\]
\end{lem}
\begin{proof}
Fix $k\geq 0$ and $x,y \in [0,1]$. Without loss of generality, we may assume that $x<y$. We consider three cases. In the first two cases, the points $x$ and $y$ belong to the same interval $\Delta_w'$, $|w|=k$, while in the third case they belong to different intervals.
\emph{Case 1.} If $x,y \in \Delta_{w}' \in \mathscr{N}_k'\cup\mathscr{F}_k'$, then $|F_k(x)-F_k(y)| = 0|x-y|$, because the map $F_k|\Delta_w'$ is constant.
\emph{Case 2.} Suppose that $x,y \in \Delta_{w}' \in \mathscr{E}_{k}'\cup\mathscr{B}_k'$. Since $F_k|\Delta_w'$ is affine,
\begin{align*}
|F_k(x) - F_k(y)| &\leq \mathcal{M}_s([0,1])\frac{\mathop\mathrm{diam}\nolimits{f_k(\Delta_w)}}{\mathcal{M}_s(|w|-1,\Delta_w)} |x-y|\\
&\leq \mathcal{M}_s([0,1])\mathop\mathrm{diam}\nolimits{f_k(\Delta_w)}^{1-s}|x-y|\\
&\leq \mathcal{M}_s([0,1])r_0^{1-s}\rho_k^{1-s}|x-y|,
\end{align*} by (V3) and the assumption $s\geq 1$. Thus, by \eqref{e:S-mass-bound}, $$|F_k(x)-F_k(y)| \lesssim_{s,C^*,\xi_2} r_0(1+S_{\mathscr{V}}^s)\rho_k^{1-s}|x-y|.$$
\emph{Case 3.} Suppose that $x\in \Delta_{w}'$ and $y\in \Delta_{u}'$ for some $\Delta_w',\Delta_u' \in \mathscr{I}_k'$ with $\Delta_w' \cap \Delta_u' = \emptyset$. By the preceding cases and the Fundamental Theorem of Calculus,
\[ |F_k(x)-F_k(y)| \leq \int_x^y |\nabla F_k(t)| \, dt \leq \mathcal{M}_s([0,1])r_0^{1-s}\rho_k^{1-s}|x-y| \lesssim_{s,C^*,\xi_2} r_0(1+S_{\mathscr{V}}^s)\rho_{k}^{1-s}|x-y|.\qedhere\]
\end{proof}
We are now ready to prove Theorem \ref{thm:main}.
\begin{proof}[{Proof of Theorem \ref{thm:main}}]
Define $F:[0,1] \to X$ pointwise by
\[ F(x) := F_0(x) + \sum_{k=0}^{\infty} (F_{k+1}(x) - F_k(x)).\]
By Lemma \ref{lem:Hold-conv1}, $F$ is well defined and continuous in all $[0,1]$. By Lemma \ref{lem:Hold-conv1}, Lemma \ref{lem:Hold-conv2}, and Lemma \ref{l:LipHold} from the appendix, $F$ is $(1/s)$-H\"older continuous with H\"older constant
\[ H \leq \frac{1}{\xi_1}\left(\mathcal{M}_s([0,1])r_0^{1-s} + 60A^*r_0 \frac{\xi_2}{1-\xi_2}\right)\lesssim_{s,C^*,\xi_1,\xi_2} r_0 (1+ S^s_{\mathscr{V}}).\]
Finally, for any integer $k\geq 0$ and any integer $m\geq k$, we have $V_k \subset F_m([0,1])$. Therefore, $V_k \subset F([0,1])$ for all integers $k\geq 0$. \end{proof}
\subsection{Corollaries to Theorem \ref{thm:main} and Proof of Theorem \ref{thm:main2}}\label{sec:cortothm}
\begin{cor}[tube approximation] \label{cor:nets1}
For all $s>1$, $C^*\geq 1$, and $0<\xi_1<\xi_2<1$, there exists $\alpha^*>0$ with the following property. Assume that $X=l^2(\mathbb{R})$ or $X=\mathbb{R}^N$ for some $N\geq 2$. Let $\mathscr{V} = (V_k,\rho_k)_{k\geq 0}$ be a sequence of finite sets in $X$ and numbers $\rho_k>0$ satisfying properties (V0)--(V5) of \textsection\ref{sec:flat} with constants $C^*$, $\xi_1$, and $\xi_2$. If
\[S^{s,+}_{\mathscr{V}} := \sum_{k=0}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} \geq \alpha^*}} \rho_k^s < \infty,\]
then there exists a $(1/s)$-H\"older map $f:[0,1] \to X$ such that $\bigcup_{k\geq 0}V_k \subset f([0,1])$ and the H\"older constant of $f$ satisfies $H \lesssim_{s,C^*,\xi_1,\xi_2} r_0 (1+ S^{s,+}_{\mathscr{V}})$.
\end{cor}
\begin{proof}
By Lemma \ref{lem:flat2}, there exists $\epsilon_{s,C^*,\xi_1,\xi_2}\in(0,1/16]$ such that if $k\geq 0$, $v\in V_k$, and $\alpha_{k,v} \leq \epsilon_{s,C^*,\xi,\xi_2}$, then $\tau_{s}(k,v,v') = 0$ for all $(v,v')\in\mathsf{Flat}(k)$. Set $\alpha^*=\min\{\epsilon_{s,C^*,\xi_1,\xi_2},\alpha_1\}$ (a careful inspection shows $\epsilon_{s,C^*,\xi_1,\xi_2}$ is strictly smaller than $\alpha_1$). Thus, with $\alpha_0=\alpha^*$,
\[ \sum_{k=0}^{\infty}\sum_{(v,v')\in \mathsf{Flat}(k)}\tau_{s}(k,v,v') \rho_k^s + \sum_{k=0}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} \geq \alpha^*}} \rho_k^s = \sum_{k=0}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} \geq \alpha^*}} \rho_k^s< \infty.\] The conclusion follows immediately by Theorem \ref{thm:main}.
\end{proof}
\begin{cor}\label{cor:nets2} Alternatively, if
\[S^{s,p}_{\mathscr{V}} := \sum_{k=0}^{\infty}\sum_{v\in V_k} \alpha_{k,v}^p\, \rho_k^s < \infty\quad\text{for some }p>0,\]
then there exists a $(1/s)$-H\"older map $f:[0,1] \to X$ such that $\bigcup_{k\geq 0}V_k \subset f([0,1])$ and the H\"older constant of $f$ satisfies $H \lesssim_{s,C^*,\xi_1,\xi_2} r_0 (1+ (\alpha^*)^{-p}S^{s,p}_{\mathscr{V}})$.
\end{cor}
\begin{proof} Inspecting the definitions of the two sums, $S^{s,+}_{\mathscr{V}} \leq (\alpha^*)^{-p} S^{s,p}_{\mathscr{V}}$.\end{proof}
We now turn to the proof of Theorem \ref{thm:main2}.
\begin{rem}\label{rem:alpha-beta}
For all $x\in\mathbb{R}^N$ and $r>0$, a minimal dyadic cube $Q$ in $\mathbb{R}^N$ such that $x\in Q$ and $3Q$ contains $B(x,r)$ satisfies $\mathop\mathrm{diam}\nolimits 3Q \leq Cr$ for some $C=C(N)>0$.
\end{rem}
\begin{proof}[{Proof of Theorem \ref{thm:main2}}] Let $N\geq 2$ and $s>1$ be given. Fix $\beta_0>0$ to be specified below. Assume that $E\subset\mathbb{R}^N$ is a bounded set such that $$S_E^{s,+}=\sum_{\stackrel{Q\in\Delta(\mathbb{R}^N)}{\beta_E(3Q)\geq \beta_0}} (\mathop\mathrm{diam}\nolimits Q)^s<\infty.$$
Pick any $x_0 \in E$ and set $r_0=\mathop\mathrm{diam}\nolimits E$. Define $V_0 = \{x_0\}$. Assume that $V_k$ has been defined for some $k$. Choose a maximal $2^{-(k+1)}$-separated set in $E$ such that $V_{k+1}\supset V_k$. Then the sequence $\mathscr{V}=(V_k,2^{-k})_{k\geq 0}$ satisfies conditions (V0)--(V4) in \textsection\ref{sec:flat} with $C^*=2$ and $\xi_1=\xi_2 = 1/2$. Note that $$A^*=\frac{C^*}{1-\xi_2}=4,\quad 30A^*=120.$$ For all $k\geq 0$ and $v\in V_k$, let $Q_{k,v}$ be a minimal dyadic cube such that $v\in Q_{k,v}$ and $3Q_{k,v}$ contains $B(v,120\cdot 2^{-k}r_0)$ and choose $\ell_{k,v}$ be a line such that
\[ \sup_{x\in E \cap 3Q} \mathop\mathrm{dist}\nolimits(x, \ell_{k,v})=\beta_E(3Q) \mathop\mathrm{diam}\nolimits{3Q}.\] Then, by Remark \ref{rem:alpha-beta}, there exists $C=C(N)>0$ such that $$\alpha_{k,v}:=\frac{1}{2^{-{k+1}}r_0}\sup_{x\in V_{k+1}\cap B(v,120\cdot 2^{-k}r_0)} \mathop\mathrm{dist}\nolimits (x,\ell_{k,v}) \leq \frac{\mathop\mathrm{diam}\nolimits 3Q}{2^{-{k+1}}r_0} \beta_E(3Q) \leq 240C \beta_E(3Q).$$ We now specify that $\beta_0 = \alpha^*/240C$, where $\alpha^*$ is the constant from Corollary \ref{cor:nets1} and $C$ the constant from Remark \ref{rem:alpha-beta}. Because each dyadic cube $Q$ in $\mathbb{R}^N$ is associated to some $(k,v)$ at most $C(n)$ times, it follows that
\[S^{s,+}_{\mathscr{V}} \lesssim_N r_0^{-s}S^{s,+}_E < \infty.\]
Therefore, by Corollary \ref{cor:nets1}, there exists a $(1/s)$-H\"older m fap $f:[0,1] \to \mathbb{R}^N$ such that $\bigcup_{k\geq 0}V_k \subset f([0,1])$ and the H\"older constant of $f$ satisfies
\[H \lesssim_{N,s} r_0 (1+ S^{s,+}_{\mathscr{V}}) \lesssim_{N,s} \mathop\mathrm{diam}\nolimits{E} + \frac{S^{s,+}_E}{(\mathop\mathrm{diam}\nolimits{E})^{s-1}}.\]
Because $\bigcup_{k\geq 0}V_k$ is dense in $E$, the curve $f([0,1])$ also contains the set $E$.
\end{proof}
\subsection{A refinement of Theorem \ref{thm:main}}\label{sec:modifiedTST}
The parameterization in Theorem \ref{thm:main} can be made in such a way so that the sequence of maps $F_k$ obtained are \emph{essentially 2-to-1} in the sense of the following proposition.
\begin{prop}\label{prop:2-to-1Holder}
Let $\mathscr{V}=(V_k,\rho_k)_{k\geq 0}$ and $\alpha_0$ satisfy the hypothesis of Theorem \ref{thm:main} and let $x_0\in V_0$. There exists a sequence of piecewise linear maps $F_k:[0,1]\rightarrow X$ with the following properties.
\begin{enumerate}
\item For all $k\geq 0$, $F_k(0)=x_0=F_k(1)$.
\item For all $k\geq 0$, there exists $G_k\subset[0,1]$ such that $F_k|G_k$ is 2-1 and $F_k([0,1]\setminus G_k)$ is a finite set.
\item For all $k\geq 0$, $F_k([0,1])\supset V_k$; for all $x\in V_{k+1}$, $\mathop\mathrm{dist}\nolimits(x,F_k([0,1])) \leq C^*\rho_{k+1}r_0$.
\item For all $k\geq 0$, $\|F_k - F_{k+1}\|_{\infty} \lesssim_{C^*,\xi_2} \rho_{k+1}r_0$.
\item For all $k\geq 0$, the map $F_k$ is Lipschitz with $\mathop\mathrm{Lip}\nolimits(F_k) \lesssim_{s,C^*,\xi_1,\xi_2} r_0(1+S^s_{\mathscr{V}})\rho_k^{1-s}$.
\end{enumerate} The maps $F_k$ converge uniformly to a $(1/s)$-H\"older map $F:[0,1]\rightarrow X$ whose image contains $\bigcup_{k\geq 0}V_k$, the parameterization $F$ starts and ends at $x_0$ in the sense of (1), and the H\"older constant of $F$ satisfies $H\lesssim_{s,C^*,\xi_1,\xi_2} r_0(1+S^s_{\mathscr{V}})$.
\end{prop}
\begin{proof}
Following the algorithm of \textsection\ref{sec:curve}, we construct for each $k\geq 0$, four families $\mathscr{E}_k$, $\mathscr{B}_k$, $\mathscr{F}_k$, $\mathscr{N}_k$ of intervals in $[0,1]$ and a continuous piecewise linear map $f_k:[0,1] \to X$ that satisfy (P1)--(P7). In Step 0, we may assume that $f_0(0) = f_0(1) = x_0$. Thus, $f_k(0) = f_k(1) = x_0$ for all $k\geq 0$. Moreover, for all $x\in V_{k+1}$,
\[ \mathop\mathrm{dist}\nolimits(x,F_k([0,1])) \leq \mathop\mathrm{dist}\nolimits(x,V_k) < C^*r_0\rho_{k+1}\] by (V4). From the construction, $\|f_k-f_{k+1}\|_{\infty} \lesssim_{A^*}\rho_{k+1}r_0$. Hence $\|f_k-f_{k+1}\|_{\infty} \lesssim_{C^*,\xi_2}\rho_{k+1}r_0$. We have shown that the maps $f_k$ satisfy properties (1), (3) and (4).
As for property (2), we already know from (P4) that $f_k|\bigcup \mathscr{E}_k$ is 2-to-1. We proceed to modify $f_k$ on each $I\in\mathscr{B}_k$. From the algorithm in \textsection\ref{sec:curve}, recall that for each $I\in \mathscr{B}_k$, there exists unique $I' \in \mathscr{B}_k$, $I'\neq I$ such that $f_k(I) = f_k(I')$. Enumerate
\[\mathscr{B}_k = \{I_1, I_1',\dots,I_l, I_l'\},\]
where $f_k(I_j) = f_k(I_j')$. Starting with $I_1 = (a_1,b_1)$, define $\tilde{f}_k|I_1$ so that
\begin{enumerate}
\item[(a)] $\tilde{f}_k| I_1$ is piecewise linear and continuous, $\tilde{f}_k(a_1) = f_{k}(a_1)$ and $\tilde{f}_k(b_1) = f_{k}(b_1)$;
\item[(b)] $\mathcal{H}^1(\tilde{f}_k(I_1)) \leq |f_k(a_1) - f_k(b_1)| + \rho_{k}r_0$;
\item[(c)] $\tilde{f}_k(I_1) \cap \bigcup_{I \in \mathscr{E}_k} f_k(I)$ is a finite set.
\end{enumerate}
Let $\psi_1: I_1' \to I_1$ be the unique orientation-reversing linear map from $I_1'$ onto $I_1$. Then define $\tilde{f}_k|I_1' = (\tilde{f}_k|I_1)\circ \psi_1$. For induction, assume that we have defined $\tilde{f}_k$ on $$I_1,I_1',\dots, I_{r-1},I_{r-1}'.$$ Define $\tilde{f}_k|I_r$ as with $I_1$, only this time we require that the set
\[\tilde{f}_k(I_r) \cap\left (\bigcup_{i=1}^{r-1}\tilde{f}_k(I_i)\cup \bigcup_{i=1}^{r-1}\tilde{f}_k(I_i')\cup \bigcup_{I \in \mathscr{E}_k} f_k(I) \right)\]
be finite. Let $\psi_r: I_r' \to I_r$ be the unique orientation-reversing linear map from $I_r'$ onto $I_r$ and define $\tilde{f}_k|I_r' = (\tilde{f}_k|I_r)\circ \psi_r$. Extending $\tilde{f}_k|I = f_k|I$ for all $I\in\mathscr{E}_k\cup\mathscr{N}_k\cup\mathscr{F}_k$, we obtain a sequence $\tilde{f}_k$ of maps that satisfy properties (1)--(4).
The rest of the proof is similar to that of Theorem \ref{thm:main} and we only sketch the steps. Define the $\mathcal{M}_s$ for each $I \in \mathscr{I}_k$ and define the collections of intervals $\{\Delta_w\}$ and $\{\Delta_w'\}$. For each $w$, let $\phi_w: \Delta_w' \to \Delta_w$ be the unique affine homeomorphism from $\Delta_w'$ onto $\Delta_w$ and let $F_k|\Delta_w' = (\tilde{f}_k|\Delta_w)\circ\phi_w$. Although the maps $f_k$ are different from $\tilde{f}_k$, we have by (b) that $\mathop\mathrm{diam}\nolimits{f_k(I)} \simeq_{\xi_2}\mathop\mathrm{diam}\nolimits{\tilde{f}_k(I)}$ for all $k\geq 0$ and all $I\in\mathscr{I}_k$. Thus, Lemma \ref{lem:Hold-conv1} and Lemma \ref{lem:Hold-conv2} still hold with constants depending at most on $s$, $C^*$, $\xi_1$ and $\xi_2$. Therefore, the maps $F_k$ satisfy properties (1)--(5).
\end{proof}
\subsection{A Carleson condition for an upper Ahlfors $s$-regular curve}\label{sec:Carleson}
Replacing \eqref{eq:main-condition} in the main theorem with a Carleson-type condition ensures that the H\"older curve is upper Ahlfors regular. This answers a question posed to us by T. Orponen.
\begin{thm} \label{cor:carleson}
Assume that $X=l^2(\mathbb{R})$ or $X=\mathbb{R}^N$ for some $N\geq 2$. Let $s\geq 1$, let $\mathscr{V}=((V_k,\rho_k))_{k\geq 0}$ be a sequence of finite sets $V_k$ in $X$ and numbers $\rho_k>0$ that satisfy properties (V0)--(V4) defined in \textsection\ref{sec:flat}. Let $\Lambda \geq C^*$ and $\Lambda^*:=\Lambda/(1-\xi_2)$. Suppose for all $k\geq 0$ and $v\in V_{k+1}$, we are given a line $\ell_{k,v}$ and $\alpha_{k,v}\geq 0$ such that \begin{equation} \tag{$\widetilde{\mathrm{V5}}$} \sup_{x\in V_{k+1}\cap B(v,30\Lambda^*\rho_k r_0)} \mathop\mathrm{dist}\nolimits(x,\ell) \leq \alpha_{k,v} \rho_{k+1}.\end{equation} If $\Lambda\gg_{\xi_1,\xi_2} C^*$, $\alpha_0\in(0,\alpha_1]$, and there exists $M<\infty$ such that for all $j\geq 0$ and $w\in V_{j}$,
\begin{equation}\label{eq:carleson} S^s_{\mathscr{V}}(j,w) := \sum_{k=j}^{\infty}\sum_{\substack{(v,v')\in \mathsf{Flat}(k) \\ v,v'\in B(w,\Lambda \rho_{j}r_0)}} \tau_{s}(k,v,v') \rho_k^s+ \sum_{k=j}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} \geq \alpha_0 \\ v\in B(w,\Lambda \rho_{j}r_0)}} \rho_k^s \leq M \rho_{j}^s,\end{equation}
then there exists a $(1/s)$-H\"older map $f:[0,1] \to X$ such that $f([0,1])\supset \bigcup_{k\geq 0}V_k$ and the curve $f([0,1])$ is upper Ahlfors $s$-regular with constant depending on at most $s$, $C^*$, $\xi_1$, $\xi_2$, and $M$.
\end{thm}
\begin{proof} By (V0) and (V4), $\mathop\mathrm{excess}\nolimits\left(\textstyle\bigcup_{k=1}^\infty V_k, V_0\right) \leq \rho_1r_0+\rho_2r_0+\dots = \frac{r_0}{1-\xi_2}\leq A^*r_0,$ where $$\mathop\mathrm{excess}\nolimits(A,B)=\sup_{x\in A}\inf_{y\in B} |x-y|$$ whenever $A$ and $B$ are nonempty sets in $X$. Hence $\bigcup_{k=0}^\infty V_k \subset B(x_0,2A^*r_0)$ by (V1). Thus, there exists a $(1/s)$-H\"older map $f:[0,1]\rightarrow X$ such that $\Gamma:=f([0,1])\supset \bigcup_{k=0}^\infty V_k$ and the H\"older constant of $f$ satisfies $H_f \lesssim_{s,C^*,\xi_1,\xi_2} r_0(1+M)$ by Theorem \ref{thm:main}, since $S^s_\mathscr{V}=S^s_{\mathscr{V}}(0,w)\leq M$. In particular, $$\mathcal{H}^s(\Gamma) \leq H_f^s\, \mathcal{H}^1([0,1]) \lesssim_{s,C^*,\xi_1,\xi_2,M} r_0^s.$$
Let $x\in \Gamma$ and let $0<r\leq \mathop\mathrm{diam}\nolimits \Gamma$. Because $\bigcup_{k=0}^\infty V_k \subset B(x_0,2A^*r_0)$, we have $\mathop\mathrm{diam}\nolimits \Gamma \leq 4A^*r_0$. If $r\geq r_0$, then $$\mathcal{H}^s(\Gamma\cap B(x,r))\leq \mathcal{H}^s(\Gamma)\lesssim_{s,C^*,\xi_1,\xi_2,M} r_0^s \lesssim_{s,C^*,\xi_1,\xi_2,M} r^s.$$ Otherwise, $0<r<r_0$, say $\rho_{j+1}r_0\leq r < \rho_j r_0$ for some integer $j\geq 0$.
Choose $w\in V_j$ such that $|w-x|= \mathop\mathrm{dist}\nolimits(x, V_j)$. By Lemma \ref{lem:Hold-conv1}, (V0), and (V4), $$\mathop\mathrm{dist}\nolimits(x,f_j([0,1])) \leq \frac{30\xi_2}{1-\xi_2}A^*\rho_jr_0.$$ Because $\alpha_0\leq \alpha_1$, the longest line segment drawn between vertices in $V_j$ has length at most $14 A^* \rho_{k-1}r_0$. By (V0), it follows that $$\mathop\mathrm{excess}\nolimits(f_j([0,1]),V_j) \leq \frac{7}{\xi_1} A^*\rho_jr_0.$$ Thus, $\mathop\mathrm{dist}\nolimits(x,V_j)\lesssim_{\xi_1,\xi_2}A^* \rho_jr_0 \ll_{\xi_1,\xi_2} \Lambda\rho_jr_0.$ For all $k\geq 0$, define $$\widetilde{\rho_k} := \frac{\rho_{j+k}}{\rho_j}.$$ Define $\widetilde{V_0} := V_{j} \cap B(w,\Lambda\rho_jr_0)$. Then, for each $k\geq 1$, recursively define $\widetilde{V_{k}}$ to be set of all $x\in V_{j+k} \cap B(w,\Lambda\rho_jr_0)$ such that $\mathop\mathrm{dist}\nolimits(x,\widetilde{V}_{k-1})\leq \Lambda \rho_{j+k}r_0$. Then $\widetilde{\mathscr{V}}=(\widetilde{V_k},\widetilde{\rho_k})_{k\geq 0}$ satisfy (V0)--(V4) with respect to $\widetilde x_0=w$ and $\widetilde r_0=\rho_jr_0$, $\widetilde{C^*}=\Lambda$, $\widetilde{\xi_1}=\xi_1$, and $\widetilde{\xi_2}=\xi_2$. For all $k\geq 0$ and $v\in \widetilde{V}_k$, assign $\widetilde{\ell}_{k,v}:=\ell_{j+k,v}\quad\text{and}\quad \widetilde{\alpha}_{k,v}=\alpha_{j+k,v}.$ Then $\widetilde{\mathscr{V}}$ satisfies (V5) with respect to $\widetilde{\ell}_{k,v}$ and $\widetilde{\alpha}_{k,v}$ by ($\widetilde{\mathrm{V5}}$). Moreover, by \eqref{eq:carleson}, $$S_{\widetilde{\mathscr{V}}}^s=\sum_{k=0}^\infty \sum_{(v,v')\in\widetilde{\mathsf{Flat}(k)}} \widetilde{\tau}(k,v,v') \widetilde{\rho_k}^s+ \sum_{k=0}^\infty \sum_{\substack{v\in \widetilde{V_k} \\ \widetilde{\alpha_{k,v}}\geq \alpha_0}} \widetilde{\rho_k}^s = \frac{S_{\mathscr{V}}(j,w)}{\rho_j^s}\leq M.$$ Thus, by Theorem \ref{thm:main}, there is a $(1/s)$-H\"older map $g$ with H\"older constant $H_g \lesssim_{s,\Lambda,\xi_1,\xi_2} \widetilde{r_0}(1+M)$ such that $g([0,1])$ contains $\bigcup_{k\geq 0} \widetilde V_{k}.$ Because the algorithm in \S\ref{sec:curve} works locally in the image, $\mathop\mathrm{dist}\nolimits(x,V_j)\lesssim_{\xi_1,\xi_2}A^* \rho_jr_0 \ll_{\xi_1,\xi_2} \Lambda\rho_jr_0$, and $r<\rho_jr_0$, we can guarantee that $g([0,1])$ contains $f([0,1])\cap B(x,r)$ provided that $\Lambda$ is sufficiently large. Therefore, $$\mathcal{H}^s(\Gamma\cap B(x,r)) \leq H_g^s\, \mathcal{H}^1([0,1]) \lesssim_{s,\Lambda,\xi_1,\xi_2,} \widetilde{r_0}(1+M) \lesssim_{s,C^*,\xi_1,\xi_2,M} (\rho_jr_0)^s \lesssim_{s,C^*,\xi_1,\xi_2,M} r^s, $$ where the final inequality holds because $\rho_{j+1}r_0\leq r$.\end{proof}
\section{Lipschitz parameterization}\label{sec:Lip}
Using the method above, we obtain the following refinement of the sufficient half of the Analyst's TST in Hilbert space, which is originally due to Jones \cite{Jones-TST} in the Euclidean case and due to Schul \cite{Schul-Hilbert} in the infinite-dimensional case.
\begin{prop}[Sufficient half of the Analyst's Traveling Salesman with Nets]\label{prop:2-to-1Lip} Assume that $X=l^2(\mathbb{R})$ or $X=\mathbb{R}^N$ for some $N\geq 2$. Let $\mathscr{V}=(V_k,\rho_k)_{k\geq 0}$ be a sequence of finite sets $V_k$ in $X$ and numbers $\rho_k>0$ that satisfy properties (V0)--(V5) defined in \textsection\ref{sec:flat}. If
\begin{equation} S_{\mathscr{V}} := \sum_{k=0}^\infty \sum_{v\in V_k} \alpha_{k,v}^2 \rho_k < \infty,\end{equation}
then for every $x_0\in V_0$, we can find a sequence of piecewise linear maps $F_k:[0,1]\rightarrow X$ with the following properties.
\begin{enumerate}
\item For all $k\geq 0$, $F_k(0)=x_0=F_k(1)$.
\item For all $k\geq 0$, there exists $G_k\subset[0,1]$ such that $F_k|G_k$ is 2-1 and $F_k([0,1]\setminus G_k)$ is a finite set.
\item For all $k\geq 0$, $F_k([0,1])\supset V_k$; for all $x\in V_{k+1}$, $\mathop\mathrm{dist}\nolimits(x,F_k([0,1])) \leq C^*\rho_{k+1}r_0$.
\item For all $k\geq 0$, $\|F_k - F_{k+1}\|_{\infty} \lesssim_{C^*,\xi_2} \rho_{k+1}r_0$.
\item For all $k\geq 0$, the map $F_k$ is Lipschitz with $\mathop\mathrm{Lip}\nolimits(F_k) \lesssim_{C^*,\xi_1,\xi_2} r_0(1+ S_{\mathscr{V}})$.
\end{enumerate} The maps $F_k$ converge uniformly to a Lipschitz map $F:[0,1]\rightarrow X$ whose image contains $\bigcup_{k\geq 0}V_k$, the parameterization $F$ starts and ends at $x_0$ in the sense of (1), and the Lipschitz constant of $F$ satisfies $L\lesssim_{C^*,\xi_1,\xi_2} r_0(1+S_{\mathscr{V}})$.\end{prop}
\begin{proof} Let $\alpha_0=\alpha_1$ (see \eqref{alphabound}), which depends only on $C^*$, $\xi_1$, and $\xi_2$. If $(v,v')\in\mathsf{Flat}(k)$, then $\tau_1(k,v,v') \leq 3\alpha_{k,v}^2$ by Lemma \ref{lem:monotone}. Thus, by definition of $S_{\mathscr{V}}^1$ (see Theorem \ref{thm:main}), \begin{align*} S_{\mathscr{V}}^{1} &= \sum_{k=0}^{\infty}\sum_{(v,v')\in \mathsf{Flat}(k)}\tau_{1}(k,v,v') \rho_k+ \sum_{k=0}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} \geq \alpha_0}} \rho_k\\ &\leq 6\sum_{k=0}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} < \alpha_0}}\alpha_{k,v}^2 \rho_k+ \frac{1}{\alpha_0^2}\sum_{k=0}^{\infty}\sum_{\substack{v\in V_k\\ \alpha_{k,v} \geq \alpha_0}} \alpha_{k,v}^2\rho_k \leq \frac{1}{\alpha_0^2} S_{\mathscr{V}}.\end{align*} The conclusion now follows from Proposition \ref{prop:2-to-1Holder}.
\end{proof}
\part{Applications and Further Results}
In \S\ref{sec:gmt}, we give an application of the H\"older Traveling Salesman theorem to the geometry of measures. In particular, we obtain sufficient conditions for a pointwise doubling measure in $\mathbb{R}^N$ to be carried by $(1/s)$-H\"older curves, $s>1$. This extends the work \cite{BS1,BS3} by the first author and Schul, which characterizes 1-rectifiable Radon measures in $\mathbb{R}^N$ in terms of geometric square functions. In \S8, we use the method of Part I to obtain a Wa\.{z}ewski type theorem for flat continua, which we described above in \S\ref{sec:wazewski}. Finally, in \S\ref{sec:examples}, we present examples of H\"older curves and of sets that are not contained in any H\"older curve to highlight the rich geometry of sets in $\mathbb{R}^N$ and illustrate the strengths and limitations of our principal results.
\section{Fractional rectifiability of measures}\label{sec:gmt}
One goal of geometric measure theory is to understand the structure of a measure in $\mathbb{R}^N$ through its interaction with families of lower dimensional sets. For an extended introduction, see the survey \cite{Badger-survey} by the first author. In this section, we use the H\"older Traveling Salesman theorem to establish criteria for \emph{fractional rectifiability} of pointwise doubling measures in terms of $L^p$ Jones beta numbers. In particular, we extend part of the recent work of the first author and Schul \cite{BS3} on measures carried by rectifiable curves to measures carried by H\"older curves (see Theorem \ref{thm:Jsp}). The study of fractional (that is, non-integer dimensional) rectifiability of measures was first proposed by Mart\'in and Mattila \cite{MM1993,MM2000} and examined further by the first and third author \cite{BV}.
\subsection{Generalized rectifiability}
Let $\mathcal{A}$ be a nonempty family of Borel sets in $\mathbb{R}^N$ and let $\mu$ be a Borel measure on $\mathbb{R}^N$.
We say that $\mu$ is \emph{carried by} $\mathcal{A}$ if there exists a sequence $(A_i)_{i=1}^\infty$ of sets in $\mathcal{A}$ such that $\mu(\mathbb{R}^N\setminus\bigcup_i A_i)=0$. At the other extreme, we say that $\mu$ is \emph{singular to} $\mathcal{A}$ if $\mu(A)=0$ for all $A\in\mathcal{A}$. If $\mu$ is $\sigma$-finite, then $\mu$ can be uniquely written as the sum of a Borel measure $\mu_\mathcal{A}$ carried by $\mathcal{A}$ and a Borel measure $\mu^\perp_\mathcal{A}$ singular to $\mathcal{A}$ (e.g.~see the appendix of \cite{BV}). These definitions encode several commonly used notions of rectifiability of measures (see \cite{Badger-survey}).
Let $1\leq m\leq N-1$. Let $\mathcal{A}$ denote the family of Lipschitz images of $[0,1]^m$ in $\mathbb{R}^N$. We say that a Borel measure $\mu$ is \emph{$m$-rectifiable} if $\mu$ is carried by $\mathcal{A}$; we say that $\mu$ is \emph{purely $m$-unrectifiable} if $\mu$ is singular to $\mathcal{A}$. A Borel set $E\subset\mathbb{R}^n$ with $0<\mathcal{H}^m(E)<\infty$ is called $m$-rectifiable or purely $m$-unrectifiable if $\mathcal{H}^m\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } E$, the $m$-dimensional Hausdorff measure restricted to $E$, has that property. The classes of 1-rectifiable sets and purely 1-unrectifiable sets are also called \emph{Besicovitch regular} sets and \emph{Besicovitch irregular} sets, respectively, in reference to the pioneering investigations by Besicovitch \cite{Bes28,Bes38} into the geometry of 1-sets in the plane.
\begin{ex} Let $\Gamma_1,\Gamma_2,\dots$ be a sequence of rectifiable curves in $\mathbb{R}^N$ and let $a_1,a_2,\dots$ be a sequence of positive weights. Then the measure $\mu=\sum_i a_i\mathcal{H}^1\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} } \Gamma_i$ is 1-rectifiable. Note that if the closure of $\bigcup_i \Gamma_i$ is $\mathbb{R}^N$ and the weights are chosen so that $\sum_{i} a_i \mathcal{H}^1(\Gamma_i)=1$, we get a 1-rectifiable Borel probability measure $\mu$ whose support is $\mathbb{R}^N$. \end{ex}
\begin{ex} Let $C\subset\mathbb{R}$ be the middle halves Cantor set (formed by replacing $[0,1]$ with $[0,\frac14]\cup[\frac{3}{4},1]$ at iterating). Then $E=C\times C\subset\mathbb{R}^2$ is a Cantor set of Hausdorff dimension one, $0<\mathcal{H}^1(E)<\infty$, $E$ is Ahlfors 1-regular in the sense that $$\mathcal{H}^1(E\cap B(x,r))\simeq r\quad\text{for all $x\in E$ and $0<r\leq \mathop\mathrm{diam}\nolimits E$},$$ and $E$ is Besicovitch irregular (e.g.~see \cite{MM1993}). In particular, the set $E$ is compact and measure-theoretically one-dimensional, but $E$ is not contained in any rectifiable curve.\end{ex}
\subsection{$L^p$ Jones beta numbers and rectifiability}
Let $\mu$ be a Radon measure on $\mathbb{R}^N$, that is, a locally finite Borel regular measure, let $1\leq m\leq N-1$, let $p>0$, let $x\in\mathbb{R}^N$, let $r>0$, and let $L$ be an $m$-dimensional affine subspace of $\mathbb{R}^N$. We define
\begin{equation}\label{e:beta-p-line} \beta^{(m)}_p(\mu,x,r,L) := \left ( \int_{B(x,r)} \left (\frac{\mathop\mathrm{dist}\nolimits(z,L)}{r}\right )^p \frac{d\mu(z)}{\mu(B(x,r))} \right )^{1/p},\end{equation} \begin{equation}\label{e:beta-p}
\beta^{(m)}_p(\mu,x,r) := \inf_{L}\beta^{(m)}_p(\mu,x,r,L),\end{equation}
where the infimum is taken over all $m$-planes $L$ in $\mathbb{R}^N$. The quantity $\beta^{(m)}_p(\mu,x,r)$ is called the \emph{$m$-dimensional $L^p$ Jones beta number of $\mu$ in $B(x,r)$}. The $L^p$ Jones beta numbers were introduced by David and Semmes \cite{DS91,DS93} to study quantitative rectifiability of Ahlfors regular sets and boundedness of singular integral operators. The normalization of the measure in \eqref{e:beta-p-line} that we have chosen (i.e.~ dividing by $\mu(B(x,r))$) ensures that $\beta^{(m)}_p(\mu,x,r)\in[0,1]$ and $\beta^{(m)}_p$ is invariant under dilations $T_\lambda(z)=\lambda z$ in the sense that \begin{equation}\beta^{(m)}_p(\mu,x,r)=\beta^{(m)}_p(T_{\lambda}[\mu],\lambda x,\lambda r),\quad T_\lambda[\mu](E)=\mu(\lambda^{-1}E)\end{equation} for all $\mu$, $x\in\mathbb{R}^N$, $r>0$, and $\lambda>0$. By monotonicity of the integral, \begin{equation}\label{beta-double} s\mu(B(y,s))^{1/p}\beta_p^{(m)}(\mu,y,s)\leq r\mu(B(x,r))^{1/p}\beta_p^{(m)}(\mu,x,r)\quad\text{when }B(y,s)\subset B(x,r).\end{equation}
In a pair of papers, Tolsa \cite{Tolsa-n} and Azzam and Tolsa \cite{AT15} characterize $m$-rectifiable Radon measures $\mu$ on $\mathbb{R}^N$ with $\mu\ll\mathcal{H}^m$ in terms of $L^2$ Jones beta numbers. The restriction $\mu\ll\mathcal{H}^m$ is equivalent to the upper density bound $\limsup_{r\downarrow 0} r^{-m}\mu(B(x,r))<\infty$ $\mu$-a.e. (e.g.~see \cite[Chapter 6]{Mattila}) and implies that the Hausdorff dimension of the measure is at least $m$ (see \cite{measure-dimension}). The proof that \eqref{e:AS-Jones} implies the measure $\mu$ is $m$-rectifiable uses an intricate stopping time argument in conjunction with David and Toro's \emph{Reifenberg algorithm for sets with holes} \cite{DT} to construct bi-Lipschitz images of $\mathbb{R}^m$ inside $\mathbb{R}^N$ that carry $\mu$. For related developments, see \cite{ENV,Ghinassi}.
\begin{thm}[see {\cite{Tolsa-n}, \cite{AT15}}] \label{t:AT} Let $\mu$ be a Radon measure on $\mathbb{R}^N$. Assume that \begin{equation}\label{e:ubound} 0<\limsup_{r\downarrow 0} \frac{\mu(B(x,r))}{r^m}<\infty\quad\text{for $\mu$-a.e. $x\in\mathbb{R}^N$}.\end{equation} Then $\mu$ is $m$-rectifiable if and only if \begin{equation}\label{e:AS-Jones} \int_0^1 \beta^{(m)}_2(\mu,x,r)^2\,\frac{\mu(B(x,r))}{r^m}\,\frac{dr}{r}<\infty\quad\text{for $\mu$-a.e. $x\in\mathbb{R}^N$.}\end{equation}\end{thm}
In \cite{BS3}, the first author and Schul characterize $1$-rectifiable Radon measure $\mu$ on $\mathbb{R}^N$ in terms of $L^p$ Jones beta numbers and the lower density $\liminf_{r\downarrow 0} r^{-1}\mu(B(x,r))$. In contrast with Theorem \ref{t:AT}, the main theorem in \cite{BS3} does not require an \emph{a priori} relationship between the null sets of $\mu$ and $\mathcal{H}^1$, nor a bound on the Hausdorff dimension of $\mu$. To lighten the notation, we present Badger and Schul's theorem for pointwise doubling measures and refer the reader to \cite[Theorem A]{BS3} for the full result. Although the classes of measures satisfying \eqref{e:ubound} and \eqref{e:point-double} have no direct relationship with each other, \emph{a posteriori} an $m$-rectifiable measure satisfying \eqref{e:ubound} also satisfies \eqref{e:point-double}. The proof that \eqref{e:Jones-BS} implies the measure $\mu$ is 1-rectifiable uses a technical extension of the sufficient half of the Analyst's Traveling Salesman theorem. See \cite[Proposition 3.6]{BS3}.
\begin{thm}[see {\cite[Theorem E]{BS3}}] \label{t:BS} Let $\mu$ be a Radon measure on $\mathbb{R}^n$ and let $p\geq 1$. Assume that $\mu$ is pointwise doubling in the sense that \begin{equation}\label{e:point-double} \limsup_{r\downarrow 0} \frac{\mu(B(x,2r))}{\mu(B(x,r))} <\infty\quad\text{for $\mu$-a.e. $x\in\mathbb{R}^N$.}\end{equation} Then $\mu$ is 1-rectifiable if and only if \begin{equation}\label{e:Jones-BS}\int_0^1 \beta^{(1)}_p(\mu,x,r)^2\,\frac{r}{\mu(B(x,r))}\,\frac{dr}{r}<\infty\quad\text{for $\mu$-a.e. $x\in\mathbb{R}^N$.}\end{equation}
\end{thm}
\subsection{Sufficient conditions for fractional rectifiability}
The following theorem is an application of the H\"older Traveling Salesman theorem and generalizes the ``sufficient half'' of Theorem \ref{t:BS} (also see \cite[Theorem A]{BV}). The exponents $p$ and $q$ in the H\"older case ($s>1$) are less restrictive than in the Lipschitz case ($s=1$).
\begin{thm} \label{thm:Jsp} Let $\mu$ be a Radon measure on $\mathbb{R}^N$, let $s>1$, and let $p,q>0$. Then
\[
\mu\hbox{ {\vrule height .22cm}{\leaders\hrule\hskip.2cm} }\left\{x\in\mathbb{R}^n: \int_0^1 \beta^{(1)}_p(\mu,x,r)^q\frac{r^s}{\mu(B(x,r))}\,\frac{dr}{r}<\infty\text{ and } \limsup_{r\downarrow 0}\frac{\mu(B(x,2r))}{\mu(B(x,r))}<\infty \right\}\]
is carried by $(1/s)$-H\"older curves.
\end{thm}
At the core of the proof of Theorem \ref{thm:Jsp} is the following lemma.
\begin{lem}\label{lem:Jsp}
Let $\mu$ be a Radon measure in $\mathbb{R}^N$, and let $s>1$ and $p,q>0$ be fixed. Given $x_0\in\mathbb{R}^N$ and parameters $M>0$, $\theta>0$, and $P>0$, let $A$ denote the set of points $x\in B(x_0,1/2)$ such that
\begin{equation}\label{eq:Jsp1} \int_0^1 \beta^{(1)}_p(\mu,x,r)^q\frac{r^s}{\mu(B(x,r))}\,\frac{dr}{r} \leq M,\end{equation} \begin{equation}\label{double-3} \mu(B(x,2r)) \leq P\mu(B(x,r))\text{ for all }r\in (0,1],\end{equation} and let $A'$ denote the set of points in $A$ such that
\begin{equation}\label{eq:Jsp3}\mu(A\cap B(x,r)) \geq \theta\mu(B(x,r))\text{ for all }r\in (0,1].\end{equation}
Then $A'$ is contained in a $(1/s)$-H\"older curve $\Gamma=f([0,1])$ with H\"older constant depending on at most $N$, $s$, $p$, $q$, $M$, $P$, $\theta$, and $\mu(A)$.
\end{lem}
\begin{proof}
Let $\{A_k'\}_{k\geq 0}$ be a nested sequence of $2^{-k}$-nets in $A'$, so that the sets $V_k\equiv A_k'$ and scales $\rho_k=2^{-k}$ satisfy conditions (V0)--(V4) of \S\ref{sec:prelim} with parameters $r_0=1$, $C^*=2$, $\xi_1=\xi_2=1/2$. Note that $$A^*=\frac{C^*}{1-\xi_2}=4\quad\text{and}\quad 30A^*=120.$$ By \eqref{eq:Jsp1},
\begin{equation}\begin{split}\label{eq:meas3}
M \mu(A) &\geq \int_A\int_0^1 \beta^{(1)}_p(\mu,x,r)^q \frac{r^s}{\mu(B(x,r))} \frac{dr}{r} d\mu(x) \\ &= \sum_{k=9}^\infty \int_{2^{-(k+1)}}^{2^{-k}} (512r)^s \int_A \frac{\beta^{(1)}_p(\mu,x,512r)^q} {\mu(B(x,512r))}d\mu(x)\frac{dr}{r}
\end{split}\end{equation} where in the second line we used the change of variables $r\mapsto 512r$ (note $512=2^9$) and Tonelli's theorem. Now, the open balls $\{B(y,2^{-(k+1)}):y\in A_k'\}$ are pairwise disjoint, because the points in $A'_k$ are separated by distance at least $2^{-k}$. Thus, \begin{equation}\label{eq:combo1} M \mu(A) \geq \sum_{k=9}^{\infty}\int_{2^{-(k+1)}}^{2^{-k}}r^s\sum_{y\in A_k'}\underbrace{\int_{A\cap B(y,2^{-(k+1)})} \frac{\beta^{(1)}_p(\mu,x,512r)^q }{\mu(B(x,512r))} d\mu(x)}_{I(k,y,r)} \frac{dr}{r}.\end{equation}
Next, we bound $I(k,y,r)$ from below. Fix $k\geq 9$, $y\in A_k'$, and $r\in[2^{-(k+1)},2^{-k}]$. Suppose that $x\in A\cap B(y,2^{-(k+1)})$. Then \begin{equation} \mu(B(x,512r)) \leq P^2 \mu(B(x,128 r))\leq P^2 \mu(B(y,129r))\leq P^2 \mu(B(y,255\cdot 2^{-k}))\end{equation} by \eqref{double-3}. Since $B(y,255\cdot 2^{-k})\subset B(x,256\cdot 2^{-k})\subset B(x,512r)$, it follows that \begin{equation}\begin{split} \beta^{(1)}_p(y,255\cdot 2^{-k}) &\leq \left(\frac{512r}{255\cdot 2^{-k}}\right)\left(\frac{\mu(B(x,512r))}{\mu(B(y,255\cdot 2^{-k}))}\right)^{1/p} \beta_p^{(1)}(\mu,x,512r)\\ &\leq 3P^{2/p}\beta_p^{(1)}(\mu,x,512r) \end{split}\end{equation} by \eqref{beta-double}. Hence \begin{equation}I(k,y,r) \geq 3^{-q}P^{-2-2q/p}\frac{\beta_p^{(1)}(\mu,y,255\cdot 2^{-k})^q}{\mu(B(y,255\cdot 2^{-k}))}\int_{A\cap B(y,2^{-(k+1)})} d\mu(x). \end{equation} Invoking doubling again, $\mu(B(y,255 \cdot 2^{-k}))\leq P^{9} \mu(B(y,2^{-(k+1)}))$. Thus, by \eqref{eq:Jsp3}, \begin{equation}\frac{1}{\mu(B(y,255\cdot 2^{-k}))} \int_{A\cap B(y,2^{-(k+1)})} d\mu(x) \geq P^{-9}\frac{\mu(A\cap B(y,2^{-(k+1)}))}{\mu(B(y,2^{-(k+1)}))}\geq P^{-9}\theta.\end{equation} Therefore, \begin{equation} \label{eq:combo2} I(k,y,r) \geq 3^{-q}P^{-11-2q/p}\theta \, \beta_p^{(1)}(\mu,y,255\cdot 2^{-k})^q\end{equation}
Combining \eqref{eq:combo1} and \eqref{eq:combo2}, we obtain \begin{equation} 3^q P^{11+2q/p}\theta^{-1}M\mu(A) \geq \sum_{k=9}^\infty \left(\int_{2^{-(k+1)}}^{2^{-k}} r^s\frac{dr}{r}\right) \sum_{y\in A_k'} \beta_p^{(1)}(\mu,y,255\cdot 2^{-k})^q.\end{equation} In particular, we conclude that \begin{equation}\sum_{k=9}^\infty \sum_{y\in A_k'} \beta_p^{(1)}(\mu,y,255\cdot 2^{-k})^q\, 2^{-ks} \leq \frac{s}{1-2^{-s}}3^q P^{11+2q/p}\theta^{-1}M\mu(A)<\infty \end{equation} For each $k\geq 9$ and $y\in A'_k$, let $\ell_{k,v}$ be any line such that \begin{equation}\beta_p^{(1)}(\mu,y,255\cdot 2^{-k},\ell_{k,v})\leq 2\beta_p^{(1)}(\mu,y,255\cdot 2^{-k}).\end{equation} We will now bound the distance of points in $A'_{k+1}\cap B(y,120\cdot 2^{-k})$ to $\ell_{k,v}$. Fix any point $z\in A'_{k+1}\cap B(y,120\cdot 2^{-k})$ and let $t2^{-{k+1}}=\mathop\mathrm{dist}\nolimits(z,\ell_{k,v})$. Then \begin{equation}\begin{split} \beta_p(\mu,y,255\cdot 2^{-k},\ell_{k,v})^q &\geq\left(\frac{\frac{1}{2}t2^{-(k+1)}}{255\cdot 2^{-k}}\right)^q\left(\frac{\mu(B(z,\frac{1}{2}t2^{-(k+1)}))}{\mu(B(y,255\cdot 2^{-k}))}\right)^{q/p} \\ &\geq \left(\frac{t}{1020}\right)^q P^{-(q/p)\log_2(1920/t)} \geq \left(\frac{t}{1920}\right)^{q+(q/p)\log_2(P)},
\end{split} \end{equation} where in the second line we used doubling of $\mu$ at $z$. It follows that \begin{equation}\alpha_{k,v}:= 2^{k+1}\sup_{z\in A'_{k+1}\cap B(y,120\cdot 2^{-k})}
\mathop\mathrm{dist}\nolimits (z,\ell_{k,v}) \leq C(p,q,P) \beta_p(\mu,y,255\cdot 2^{-k},\ell_{k,v})^\eta,\end{equation} where $\eta[q+(q/p)\log_2(P)] = q$. Therefore, all together, \begin{equation} \sum_{k=9}^\infty \sum_{y\in A'_k} \alpha_{k,v}^{q+(q/p)\log_2(P)} 2^{-ks} \leq C(s,p,q,M,P,\theta,\mu(A))<\infty.\end{equation} Finally, by Corollary \ref{cor:nets2}, the set $A'$ is contained in the Hausdorff limit of $A_k'$ and this is contained in a $(1/s)$-H\"older curve $\Gamma=f([0,1])$ with H\"older constant depending on at most $s$, $p$, $q$, $M$, $P$, $\theta$, and $\mu(A)$. \end{proof}
Theorem \ref{thm:Jsp} follows from countably many applications of Lemma \ref{lem:Jsp} and a standard density theorem for Radon measures in $\mathbb{R}^N$. See the proof of \cite[Theorem 6.7]{BV}, where a similar argument is employed. We leave the details to the reader.
\section{H\"older parameterization of flat continua}\label{sec:flatcontinua}
The goal of this section is to prove Proposition \ref{thm:flat}, which we now restate.
\begin{prop}\label{thm:flat-statedagain}
There exists a constant $\beta_1 \in (0,1)$ such that if $s>1$ and $E\subset\mathbb{R}^N$ is compact, connected, $\mathcal{H}^s(E)<\infty$, $E$ is lower Ahlfors $s$-regular with constant $c$, and
\begin{equation}\label{e:mv-statedagain}
\beta_{E}\left(\overline{B}(x,r)\right) \leq \beta_1\quad \text{for all }x\in E\text{ and }0<r\leq \mathop\mathrm{diam}\nolimits E, \end{equation}
then $E=f([0,1])$ for some injective $(1/s)$-H\"older continuous map $f:[0,1]\to\mathbb{R}^N$ with H\"older constant $H \lesssim_{s} c^{-1}\mathcal{H}^s(E)(\mathop\mathrm{diam}\nolimits E)^{1-s}$.
\end{prop}
The proposition is trivial if $E$ is a singleton (in which case \eqref{e:mv-statedagain} is vacuous). Thus, we may assume that $E\subset\mathbb{R}^N$ is a \emph{continuum}; that is, $E$ is compact, connected, and contains at least two points. Furthermore, as the hypothesis and the conclusion are scale-invariant, we may assume without loss of generality that $\mathop\mathrm{diam}\nolimits E=1$. To complete the proof of the proposition, we mimic the proof of Theorem \ref{thm:main}, but with a few modifications. In \S\ref{sec:flatcontinua_algo}, we perform a simplified version of the algorithm in \textsection\ref{sec:curve}. Then, in \textsection\ref{sec:massflat}, we establish an upper bound on $\mathcal{M}_s([0,1])$ in terms of $c^{-1}\mathcal{H}^s(E)$, which fills the role that Proposition \ref{prop:mass} played for Theorem \ref{thm:main}. Equipped with this mass bound, the proof of Proposition \ref{thm:flat-statedagain} essentially follows by repeating the proof of Theorem \ref{thm:main} \emph{mutatis mutandis}.
At the core of the proof of Proposition \ref{thm:flat-statedagain} is the following property, which is satisfied by continua that are sufficiently flat at all locations and scales. We defer a proof of Lemma \ref{lem:flatcont} to \S\ref{sec:prop-continua}. An adventurous reader may wish to supply their own proof.
\begin{lem}\label{lem:flatcont}
Suppose that $E\subset\mathbb{R}^N$ is a continuum satisfying
\[ \beta_E\left(\overline{B}(x,r)\right) \leq 2^{-11} \qquad\text{for all $x\in E$ and $0<r\leq \mathop\mathrm{diam}\nolimits E$}.\]
Then for all distinct $x,y \in E$ and for all $z\in [x,y]$, there exists $z' \in E$ such that $\pi_{\ell_{x,y}}(z') = z$ and $|z-z'|\leq 2^{-4}|x-y|$, where $\ell_{x,y}$ denotes the line containing $x$ and $y$.
\end{lem}
\subsection{Traveling Salesman algorithm for flat continua}\label{sec:flatcontinua_algo}
Fix a constant $\beta_1>0$ (small) to be specified later. Let $E\subset\mathbb{R}^N$ be a continuum satisfying the hypothesis of Proposition \ref{thm:flat-statedagain}. Without loss of generality, we assume that $\mathop\mathrm{diam}\nolimits E=1$. Pick $x_0$ and $y_0$ such that $|x_0-y_0|=1$, set $r_0= 1$, $C^*=2$, and $\xi_1=\xi_2=1/2$. Then $$A^*=\frac{C^*}{1-\xi_2}=4,\quad 14A^*=56,\quad 30A^*=120.$$ Furthermore, the scales $\rho_k = 2^{-k}$ satisfy (V0).
Set $V_0=\{x_0,y_0\}$, $\alpha_{0,x_0}=4\beta_E\left(\overline{B}(x_0,1)\right)$ and $\alpha_{0,y_0}=4\beta_E\left(\overline{B}(y_0,1)\right)$, and let $\ell_{0,x_0}$ and $\ell_{0,y_0}$ be best fitting lines corresponding to $\beta_E$ on the closed balls $\overline{B}(x_0,1)$ and $\overline{B}(y_0,1)$, respectively.
Suppose that for some $k\geq 0$ and for all $0\leq j\leq k$, we have defined sets $V_j$, numbers $\alpha_{j,v}\geq 0$ for all $v\in V_j$, and lines $\ell_{j,v}$ for all $v\in V_j$ satisfying (V5). Choose $V_{k+1}$ to be any maximal $2^{-(k+1)}$-separated subset of $E$ such that $V_{k+1}\supset V_k$. For each $v\in V_{k+1}$, set $$\alpha_{k+1,v}:=\frac{2r_{k+1}}{2^{-(k+2)}}\beta_E\left(\overline{B}(v,r_{k+1})\right)\leq 480\beta_E\left(\overline{B}(v,r_{k+1})\right),$$ where $r_{k+1}:=\min\{120\cdot 2^{-(k+1)},1\}$, and let $\ell_{k+1,v}$ be a best fitting line corresponding to $\beta_E\left(\overline{B}(v,r_k)\right)$. The reader may check that the sequence of sets $(V_k)_{k=0}^\infty$ satisfy properties (V1)--(V5) in \ref{sec:flat}.
We now specify $\alpha_0=512\beta_1\leq 1/16$. This ensures that $\alpha_{k,v}<\alpha_0$ for all $k\geq 0$ and $v\in V_k$. Moreover, $\beta_1$ is sufficiently small that we may invoke Lemma \ref{lem:flatcont} for $E$.
With $\alpha_0$ fixed, carry out a modified version of the algorithm in \S\ref{sec:curve}, in which (P4), (P6), and (P7) are replaced by: \begin{enumerate}
\item[(P4')] $f_k|\bigcup{\mathscr{E}_k}$ is one-to-one.
\item[(P6')] For each $I\in\mathscr{N}_k\cup\mathscr{F}_k$, the image $f_k(I)\in V_k$. For every $v\in V_k$, there exists a unique interval $I\in\mathscr{N}_k\cup\mathscr{F}_k$ such that $v=f_k(I)$.
\item[(P7')] If $\mathscr{I}_k=\mathscr{E}_k\cup\mathscr{B}_k\cup\mathscr{N}_k\cup\mathscr{F}_k$ is enumerated according to the natural order in $[0,1]$, say $\mathscr{I}_k=\{I_1,\dots,I_{2l+1}\}$, then the intervals alternate between elements of $\mathscr{N}_k\cup \mathscr{F}_k$ and $\mathscr{E}_k$. (Thus, the family $\mathscr{B}_k=\emptyset$.) Moreover, $\operatorname{card} \mathscr{N}_k=2$ and $I_1,I_{2l+1}\in\mathscr{N}_k$. The vertices $f_k(I_1)$ and $f_{k}(I_{2l+1})$ are 1-sided terminal in $V_{k}$, while each other vertex $f_{k}(I_{2j+1})$ is non-terminal in $V_{k}$ for all $1\leq j\leq l-1$.
\end{enumerate}
We now sketch some steps in the modified algorithm.
\subsubsection{Step 0} Partition $[0,1]=[0,1/3]\cup (1/3,2/3) \cup [2/3,1]$ and assign $$\mathscr{E}_0=\{(1/3,2/3)\},\quad \mathscr{B}_0=\emptyset,\quad \mathscr{N}_0=\{[0,1/3], [2/3,1]\},\quad \mathscr{F}_0=\emptyset.$$ Also set $f_0([0,1/3])=x_0$ and $f_0([2/3,1])=y_0$, and define $f_0|(1/3,2/3)$ to be the restriction of the affine map which interpolates between $x_0$ and $y_0$. Verifying properties (P1), (P2), (P3), (P4'), (P5), (P6'), and (P7') is straightforward. We omit the details.
\subsubsection{Induction Step} Suppose that $\mathscr{E}_k$, $\mathscr{B}_k$, $\mathscr{N}_k$, $\mathscr{F}_k$, and $f_k$ have been defined and satisfy properties (P1), (P2), (P3), (P4'), (P5), (P6'), and (P7').
By (P7') and the induction assumption, the procedure in \S\ref{sec:bridges} is moot, because there are no bridge intervals.
Follow the procedure in \S\ref{sec:flatedges} for $I\in\mathscr{E}_k$ as written, except assign all closed subintervals $\mathscr{N}_{k+1}(I)\cup\mathscr{F}_{k+1}(I)$ generated in $I$ to $\mathscr{F}_{k+1}(I)$ instead of $\mathscr{N}_{k+1}(I)$. Also set $\mathscr{N}_{k+1}(I)=\emptyset$. Below, we check that $\mathscr{B}_{k+1}(I)=\emptyset$; see Lemma \ref{lem:TSPflat}.
Because $\alpha_{k,v}<\alpha_0$ for all $v\in V_k$, the procedure in \S\ref{sec:noflatedges} is moot.
Follow the procedure in \S\ref{sec:fixed} as written.
Replace the procedure in \S\ref{sec:flatpoints} as follows. \begin{quotation}
By property (P7'), only the intervals in $\mathscr{I}_k$ containing $0$ and $1$ belong to $\mathscr{N}_k$ and $f_k$ maps each of them onto a 1-sided terminal vertex in $V_k$.
Let $I$ be the interval in $\mathscr{N}_k$ containing $0$ and let $v=f_k(I)$. Let $(v,v')\in\mathsf{Flat}(k)$ be the unique flat pair with first coordinate $v$. Choose an orientation for $\ell_{k,v}$ so that $[v,v']$ lies on the right side of $v$. Enumerate the points in $V_{k+1}\cap B(f_k(I), C^*\rho_{k+1}r_0)$ on the left side of $v$ (including $v$) as $v_{l}, v_{l-1},\dots, v_1=v$, starting at the leftmost vertex and working right. The construction splits into three cases.
\emph{Case 1a.} If $l=1$, then no new points appeared to the left of $v$ and we set $\mathscr{N}_{k+1}(I)=I$ and $f_{k+1}|I=f_k|I$. Set $\mathscr{E}_{k+1}(I)=\mathscr{B}_{k+1}(I)=\mathscr{F}_{k+1}(I)=\emptyset$.
\emph{Case 1b.} If $l=2$, then one new point appeared to the left of $v$, the new point is 1-sided terminal in $V_{k+1}$ and $v$ is non-terminal in $V_{k+1}$. Subdivide $I=[0,a]=[0,\frac13a]\cup(\frac13a,\frac23a)\cup[\frac23a,a]$, set $\mathscr{N}_{k+1}(I)=[0,\frac13a]$, $\mathscr{E}_{k+1}(I)=(\frac13a,\frac23a)$, $\mathscr{F}_{k+1}(I)=[\frac23a,a]$, and $\mathscr{B}_{k+1}=\emptyset$. Define $f_{k+1}|I$ by assigning $f_{k+1}([0,\frac13a])=v_2$, $f_{k+1}|(\frac13a,\frac23a)$ to be the restriction of the affine map that interpolates between $v_2$ and $v_1$, and $f_{k+1}([\frac23a,a])=v_1$.
\emph{Case 1c.} If $l\geq 3$, then subdivide $I$ into $2l-1$ intervals, alternating between closed and open intervals. Assign the first interval to $\mathscr{N}_{k+1}(I)$ and the subsequent closed intervals to $\mathscr{F}_{k+1}(I)$. Assign the open intervals in $\mathscr{E}_{k+1}(I)$ and set $\mathscr{B}_{k+1}(I)=\emptyset$. The map $f_{k+1}|I$ is the piecewise linear map starting at $v_l$, connecting $v_l$ to $v_{l-1}$, \dots, connecting $v_2$ to $v_1$, which is constant on the intervals in $\mathscr{N}_{k+1}(I)\cup\mathscr{F}_{k+1}(I)$ and constant speed on the intervals in $\mathscr{E}_{k+1}(I)$.
Carry out a similar construction for the interval $J$ in $\mathscr{N}_k$ containing 1, but modified so that $\mathscr{N}_{k+1}(J)$ contains only one interval and that interval contains $1$.
\end{quotation}
Because $\alpha_{k,v}<\alpha_0$ for all $v\in V_k$, the procedure in \S\ref{sec:noflatpoints} is not used.
\begin{lem}\label{lem:TSPflat} For all $k\geq 0$, $\mathscr{B}_k=\emptyset$. Moreover, for all $I\in\mathscr{E}_k$, $\mathop\mathrm{diam}\nolimits f_k(I) < 3 \cdot 2^{-k}$.
\end{lem}
\begin{proof} We need to check that, for flat continua, the procedure in \S\ref{sec:flatedges} does not generate bridge intervals. Given $I\in\mathscr{E}_k$, let $v$ and $v'$ denote the endpoints of $f_k(I)$. Choose an orientation of $\ell_{k,v}$ so that $v'$ lies to the right of $v$. Enumerate $V_{k+1}(v,v')=\{v_1,\dots,v_n\}$, where $v_1=v$, $v_n=v'$, and $v_{i+1}$ is the first point to the right of $v_i$ for all $1\leq i\leq n-1$. Suppose for contradiction that $\mathscr{B}_{k+1}(I)\neq\emptyset$. Then $$|v_{j+1}-v_j|\geq 14 A^* \rho_{k+1}= 56\cdot 2^{-(k+1)}\quad \text{for some $1\leq j\leq n-1$.}$$ Let $x=(v_j+v_{j+1})/2$ denote the midpoint between $v_j$ and $v_{j+1}$. By Lemma \ref{lem:flatcont}, there exists $y\in E$ such that $|y-x|\leq (1/16)|v_{j+1}-v_j|$. Thus, $$\mathop\mathrm{dist}\nolimits(y, V_{k+1}) \geq \mathop\mathrm{dist}\nolimits(x,V_{k+1}) - |y-x| \geq \frac{7}{16}|v_{j}-v_{j-1}| > 24 \cdot 2^{-(k+1)}.$$ This contradicts our assumption that $V_{k+1}$ is a maximal $2^{-(k+1)}$-separated set for $E$. Therefore, $\mathscr{B}_{k+1}(I)=\emptyset$ for all $I\in\mathscr{E}_k$. The only other instances in the algorithm where bridge intervals could be generated are in the procedures in \S\S \ref{sec:noflatedges} and \ref{sec:noflatpoints}. However, since $\alpha_{k,v}<\alpha_0$ for all $k\geq 0$ and $v\in V_k$, these procedures were never used.
Similarly, suppose to get a contradiction that there exists $I\in \mathscr{E}_k$ such that $\mathop\mathrm{diam}\nolimits f_k(I) \geq 3 \cdot 2^{-k}$. Let $v$ and $v'$ denote the endpoints of $f_k(I)$, and let $x=(v+v')/2$ denote their midpoint. By Lemma \ref{lem:flatcont}, there exists $y\in E$ such that $|y-x|<(1/16)|v-v'|$. Then $$\mathop\mathrm{dist}\nolimits(y,V_k) \geq \mathop\mathrm{dist}\nolimits(x,V_k) - |x-y| \geq \frac{7}{16}|v-v'| \geq \frac{21}{16} 2^{-k}.$$ This contradicts our assumption that $V_{k+1}$ is a maximal $2^{-(k+1)}$-separated set for $E$.
\end{proof}
Verifying properties (P1), (P2), (P3), (P4'), (P5), (P6'), and (P7') for $\mathscr{E}_{k+1}$, $\mathscr{B}_{k+1}$, $\mathscr{N}_{k+1}$, $\mathscr{F}_{k+1}$, and $f_{k+1}$ is again routine. We leave the details to the reader.
\subsection{Mass estimate}\label{sec:massflat} Let $\mathcal{M}_s([0,1])$ be defined as in \S\ref{sec:mass}.
\begin{lem}\label{lem:flat3}
$\mathcal{M}_s([0,1]) \leq 48^s c^{-1}\mathcal{H}^s(E)$.\end{lem}
\begin{proof} Fix a finite tree $T$ over $[0,1]$ of depth $m$ (see \S\ref{sec:mass}) and suppose that
\[ \partial T = \{(k_1',J_1),(k_1,I_1),\dots,(k_l',J_l),(k_l,I_l),(k_{l+1}',J_{l+1})\},\]
enumerated according to the orientation of $[0,1]$ so that
\[ \{I_1,\dots,I_l\} \subseteq \bigcup_{k\geq 0}\mathscr{E}_k \qquad\text{and}\qquad\{J_1,\dots,J_{l+1}\} \subseteq \bigcup_{k\geq 0}(\mathscr{N}_k\cup\mathscr{F}_k).\]
The first interval $J_1\in\mathscr{N}_{k_1'}$ and the last interval $J_{l+1}\in\mathscr{N}_{k_{l+1}'}$, since they contain $0$ and $1$, respectively. The remaining intervals $J_i\in\mathscr{F}_{k'_i}$, because they do not contain $0$ or $1$.
For each $1\leq i\leq l$, let $x_i$ denote the midpoint of $f_{k_i}(I_i)$.
\begin{claim}\label{lem:edgesfar} One one hand, the set $E\cap B(x_i,(3/16)2^{-k_i})$ is nonempty for all $1\leq i\leq l$. On the other hand, the family $\left\{E\cap B(x_i, (1/4) 2^{-k_i}): 1\leq i\leq l\right\}$ is pairwise disjoint.\end{claim}
\begin{proof} Given $1\leq i\leq l$, let $v_i$ and $v_{i}'$ denote the endpoints of $f_{k_i}(I_i)$. By Lemma \ref{lem:TSPflat}, $$|v_i-v_i'| < 3\cdot 2^{-k_i}.$$ Hence there exists $z_i\in E\cap B(x_i, (1/16)|v_i-v_i'|) \subset E \cap B(x_i, (3/16)\cdot 2^{-k_i})$ by Lemma \ref{lem:flatcont}.
Suppose in order to reach a contradiction that there exists $$z\in E\cap B(x_i, (1/4) 2^{-k_i})\cap B(x_j, (1/4) 2^{-k_j})$$ for some $i\neq j$ with $0\leq k_i \leq k_j\leq m$.
\emph{Case 1}. Suppose that $k_j \geq k_i + 3$. Then $$|v_j-x_i| \leq |v_j-x_j|+|x_j-z|+|z-x_i| < \frac32\cdot 2^{-k_j} + \frac{1}{4} 2^{-k_j} + \frac14 \cdot 2^{-k_i}< \frac12 \cdot 2^{-k_i}$$ and similarly for $v_j'$. Because $\{v_j,v_j'\}\subset B(x_i,\frac{1}{2} 2^{-k_i})$, we conclude that $v_j$ and $v_j'$ lie between $v_i$ and $v_i'$ with respect to the linear ordering of $V_i\cap B(v_i,120\cdot 2^{-k_i})$. It follows that $I_j$ is contained in $I_i$, but $I_j\neq I_i$. This contradicts the assertion that $I_i\in\partial T$.
\emph{Case 2.} Suppose that $k_j \leq k_i + 2$. Then $$|v_i-v_j| \leq |v_i-x_i|+|x_i-z|+|z-x_j|+|x_j-v_j|< \frac32 2^{-k_i} + \frac14 \cdot 2^{-k_i} + \frac14 \cdot 2^{-k_j}
+ \frac32 2^{-k_j}<8\cdot 2^{-k_j},$$ $$|v_i'-v_j| \leq |v_i'-v_i|+|v_i-v_j| < 3\cdot 2^{-k_i} + 8\cdot 2^{-k_j} \leq 20 \cdot 2^{-k_j}.$$ In particular, $v_i$, $v_i'$, $v_j$, $v_j'$ belong to $V_j\cap B(v_j,120\cdot 2^{-k_j})$, which is linearly ordered by Lemma \ref{lem:var}, where $v_j$ and $v_{j+1}$ are consecutive points. Assume that $v_i$ and $v_i'$ both lie on the left or the right side of $[v_j,v_j']$, say without loss of generality that the appear from left to right as $v_j,v_j',v_i,v_i'$. Then $$|x_i-v_j'| \geq \frac{1}{2} |v_i'-v_i| \geq \frac{1}{2}\cdot 2^{-k_i} >\frac14\cdot 2^{-k_i}.$$ It follows that $B(v_j,\frac14\cdot{2^{-k_j}}) \cap B(v_i,\frac14\cdot2^{-k_i})$ is empty, which contradicts our assumption. Thus, one of $v_i$ or $v_i'$ lies on the left side of $[v_j,v_j']$ and the other lies on the right side. Then $I_j\subset I_i$. If $k_j \geq k_i+1$, then we reach the same contradiction as in Case 1. If $k_j=k_i$, then it follows that $I_j=I_i$, which contradicts our assumption that $i\neq j$.
\end{proof}
We now continue with the proof of Lemma \ref{lem:flat3}. By Claim \ref{lem:edgesfar}, we can find balls $B(z_i, (1/16) 2^{-k_i})$ centered in $E$ for all $1\leq i\leq l$, which are pairwise disjoint. Moreover, because $E$ is lower Ahlfors regular, we have $c[(1/16) 2^{-k_i}]^s \leq \mathcal{H}^s(E\cap B(z_i, (1/16) 2^{-k_i}))$ for all $1\leq i\leq l$. Therefore, by Lemma \ref{lem:TSPflat} and additivity of measures on disjoint sets, \begin{equation*}
\sum_{(k,I)\in\partial T} (\mathop\mathrm{diam}\nolimits{f_k(I)})^s = \sum_{i=1}^l (\mathop\mathrm{diam}\nolimits {f_{k_i}(I_i)})^s \leq \sum_{i=1}^l (3\cdot 2^{-k_i})^s \leq 48^s c^{-1}\mathcal{H}^s(E).\end{equation*} Because $T$ was an arbitrary finite tree over $[0,1]$, we obtain the corresponding inequality for the total mass $\mathcal{M}_s([0,1])$.
\end{proof}
\begin{cor}\label{cor:massest}
If $k\geq 0$, $I\in \mathscr{I}_k$, and $a$ is an endpoint of $I$, then
\[ \mathcal{M}_s(k,I) \leq 48^s c^{-1}\,\mathcal{H}^s(E\cap B(f_k(a),6\cdot 2^{-k})).\]
\end{cor}
\begin{proof} Let $T$ be a finite tree over $(k,I)$. By Lemma \ref{lem:TSPflat}, the image of any edge interval in $\partial T$ is contained in a ball centered at $f_k(a)$ of radius at most $$3\cdot 2^{-k}+3\cdot 2^{-(k+1)}+\dots=6\cdot 2^{-k}.$$ The conclusion follows by repeating the proof of Lemma \ref{lem:flat3}. \end{proof}
\subsection{Proof of Proposition {\ref{thm:flat-statedagain}}}
With Lemma \ref{lem:flat3} in hand, follow the proof of Theorem \ref{thm:main} in \S\ref{sec:proof1}, \emph{mutatis mutandis}. Construct families of intervals $\mathscr{E}_k'$, $\mathscr{N}_k'$, and $\mathscr{F}_k'$ as in \S\ref{sec:proof1}. We are free to specify the following additional constraints.
\begin{itemize}
\item If $I \in \mathscr{E}_k$, say $I \in \mathscr{E}_k(I_0)$ for some $I_0 \in \mathscr{E}_{k-1}\cup\mathscr{N}_{k-1}$, then the corresponding interval $I' \in \mathscr{E}_k'$ satisfies
\begin{equation}\label{eq:bound}
\mathop\mathrm{diam}\nolimits{I'} = \frac{\mathcal{M}_s(k,I)}{\mathcal{M}_s([0,1])} + \frac{1}{\operatorname{card}(\mathscr{E}_{k}(I_0))}\left ( \mathop\mathrm{diam}\nolimits{I_0} - \sum_{J \in \mathscr{I}_{k}(I_0)}\frac{\mathcal{M}_s(k,J)}{\mathcal{M}_s([0,1])} \right ).
\end{equation}
\item If $I \in \mathscr{F}_k$, then the corresponding interval $I' \in \mathscr{F}_k'$ satisfies $\mathop\mathrm{diam}\nolimits{I'} = 0$. That is, $I'$ is a singleton.
\item If $I \in \mathscr{N}_k$, then the corresponding interval $I' \in \mathscr{N}_k'$ satisfies $\mathop\mathrm{diam}\nolimits{I'} = \dfrac{\mathcal{M}_s(k,I)}{\mathcal{M}_s([0,1])}$.
\end{itemize}
\begin{lem}\label{lem:diamtozero}
For any $\epsilon>0$, there exists $k_0\geq 0$ such that $\mathop\mathrm{diam}\nolimits{I} \leq \epsilon$ for all $k\geq k_0$ and $I\in \mathscr{I}_k'$.
\end{lem}
\begin{proof}
Fix $\epsilon>0$. Note that if $J \in \mathscr{E}_l$, then $\operatorname{card}(\mathscr{E}_{l+1}(I)) \geq 2$ by Lemma \ref{lem:flatcont}, because $\mathop\mathrm{diam}\nolimits f_l(J) \geq 2^{-l}$ and $V_{l+1}$ is a maximal $2^{-(l+1)}$-separated set in $E$.
Suppose that $I \in \mathscr{I}_k$. Set $J_k = I$. Inductively, given $J_l \in \mathscr{I}_l$, let $J_{l-1}$ denote the unique interval in $\mathscr{I}_{l-1}$ with $J_l \subset J_{l-1}$. That is,
\[ I = J_k \subset J_{k-1} \subset \cdots \subset J_1 \subset J_0 = [0,1].\]
For each $i \in \{0,\dots,k\}$, let $J_i' \in \mathscr{I}_i'$ be the interval associated to $J_i$. We claim that for each $i \in \{0,\dots,k\}$
\begin{equation}\label{eq:bound2}
\mathop\mathrm{diam}\nolimits{J_i'} \leq \sum_{l=0}^{i} 2^{l-i}\frac{\mathcal{M}_s(l,J_l)}{\mathcal{M}_s([0,1])}
.\end{equation}
We prove (\ref{eq:bound2}) by induction. For $i=0$ the claim is clear. Assume that (\ref{eq:bound2}) is true for $0\leq i<k$. If $J_{i+i} \in \mathscr{F}_{i+1}(J_i)$, then $\mathop\mathrm{diam}\nolimits{J_{i+1}'}=0$ and (\ref{eq:bound2}) is clear. If $J_{i+i} \in \mathscr{F}_{i+1}(J_i)$, then $\mathop\mathrm{diam}\nolimits{J_{i+1}'} = \mathcal{M}_s(i+1,J_{i+1})/\mathcal{M}_s([0,1])$ and (\ref{eq:bound2}) is again clear. If $J_{i+i} \in \mathscr{E}_{i+1}(J_i)$, then by (\ref{eq:bound}) and the induction hypothesis,
\[ \mathop\mathrm{diam}\nolimits{J_{i+1}'} \leq \frac{\mathcal{M}_s(i+1,J_{i+1})}{\mathcal{M}_s([0,1])} + \frac{1}{2}\mathop\mathrm{diam}\nolimits{J_i'} \leq \sum_{l=0}^{i+1} 2^{l-(i+1)}\frac{\mathcal{M}_s(l,J_l)}{\mathcal{M}_s([0,1])}.\] This establishes \eqref{eq:bound2}.
Choose an integer $l_0$ sufficiently large so that $2^{-l_0}l_0 \leq \epsilon/2$. If $k\geq l_0$, then by (\ref{eq:bound2}) and the fact that $\mathcal{M}_s([0,1])\geq (\mathop\mathrm{diam}\nolimits E)^s=1$,
\[ \mathop\mathrm{diam}\nolimits{I'} \leq \sum_{l=0}^{l_0-1} 2^{l-k}\frac{\mathcal{M}_s(l,J_l)}{\mathcal{M}_s([0,1])} +\sum_{l=l_0}^k 2^{l-k}\frac{\mathcal{M}_s(l,J_l)}{\mathcal{M}_s([0,1])}
\leq \frac{\epsilon}{2}+ 2\,\mathcal{M}_s(l_0,J_{l_0}).\] Thus, by Corollary \ref{cor:massest}, $$\mathop\mathrm{diam}\nolimits {I'} \leq \frac{\epsilon}{2} + 2\,\frac{48^s}{c}\,\sup_{x\in E} \mathcal{H}^s(E\cap B(x, 6\cdot 2^{-l_0}))$$ whenever $k\geq l_0$.
Now, because $E$ is compact, $\mathcal{H}^s(E)<\infty$, and $\mathcal{H}^s$ has no atoms, $$\lim_{n\rightarrow\infty}\sup_{x\in E} \mathcal{H}^s(E \cap B(x, 6\cdot 2^{-n}))=0.$$ Hence, by choosing $l_0$ even larger if necessary, we can ensure that $$\sup_{x\in E} 2\,\frac{48^s}{c} \mathcal{H}^s(E\cap B(x, 6\cdot 2^{-l_0})) < \frac{\epsilon}{2}.$$ Therefore, $\mathop\mathrm{diam}\nolimits {I'} < \epsilon$ for all $k\geq l_0$ provided that $l_0$ is sufficiently large depending on $\epsilon$ and $E$.
\end{proof}
Following the proof of Theorem \ref{thm:main} in \S\ref{sec:proof1}, we obtain a sequence of maps $F_k:[0,1]\rightarrow \mathbb{R}^N$ and a $(1/s)$-H\"older continuous map $F:[0,1]\rightarrow\mathbb{R}^N$ satisfying the following properties.
\begin{enumerate}
\item For each $k\geq 0$, there exist (possibly degenerate) closed intervals $[0,a_k]$ and $[b_k,1]$ such that $F_k|[0,a_k]$ and $F|[0,b_k]$ are constant maps and $F(0),F(1)\in V_k$.
\item For each $k\geq 0$, the map $F_k|(a_k,b_k)$ is an injective piecewise linear map connecting $F_k(0)$ to $F_k(1)$ along line segments between points in $V_k$ of length at most $3\cdot 2^{-k}$.
\item If $x\in V_k\setminus\{F_k(0),F_k(1)\}$ for some $k\geq 0$, then $F_k^{-1}(x)=F_j^{-1}(x)$ for all $k\geq j$ (because intervals in $\mathscr{F}_j$ are frozen).
\item The maps $F_k$ converge uniformly to $F$ and $F([0,1])$ contains $\bigcup_{k=0}^\infty V_k$.
\item The H\"older constant of $F$ satisfies $$H \leq \frac{1}{\xi_1}\left(\mathcal{M}_s([0,1])r_0^{1-s} + 60A^* r_0 \frac{\xi_2}{1-\xi_2}\right)
\lesssim_s c^{-1}\mathcal{H}^s(E)(\mathop\mathrm{diam}\nolimits E)^{1-s}.$$\end{enumerate}
It remains to show that $F([0,1])=E$ and $F$ is injective.
On one hand, since each $V_k$ is a maximal $2^{-k}$-separated set in $E$,
$$F([0,1]) \supset \overline{\textstyle\bigcup_{k=0}^\infty V_k} = E$$
by (3). On the other hand, if $x\in F([0,1])$, say $x=F(t)$, then
$$\mathop\mathrm{dist}\nolimits(x,E) \leq \liminf_{k\rightarrow\infty} \mathop\mathrm{dist}\nolimits(F_k(t),E) =0$$
by (2). Thus, $F([0,1])=E$.
To check injectivity, we first establish a lemma. We say that an interval $I$ \emph{separates} two numbers $x < y$ if $x<z<y$ for all $z\in I$.
\begin{lem}\label{lem:inj}
Let $0 \leq x<y\leq 1$. If $k_0$ is the least integer $k\geq 0$ such that there exists an interval in $\mathscr{E}_k'$ separating $x$ and $y$, then $|F_k(x)-F_k(y)| \gtrsim 2^{-k_0}$ for all $k\geq k_0$.
\end{lem}
\begin{proof}Fix $0 \leq x<y\leq 1$, let $k_0$ be as in the statement of the lemma and let $k\geq k_0$. Let $I_0 \in \mathscr{E}_{k_0}$ be such that $I_0$ separates $x$ and $y$. The proof is divided into two cases.
\emph{Case 1.} Assume that $k\leq k_0 + 3$. By Remark \ref{rem:lleq5} and minimality of $k_0$, there exist at most $13$ intervals $I\in \mathscr{E}_{k_0}'$ separating $x$ from $y$. Therefore, there exist consecutive intervals $J_1,\dots,J_l \in \mathscr{I}_k$ such that $x, y \in \bigcup_{i=1}^l J_i$, $I_0 \subset \bigcup_{i=1}^l J_i$ and $l\leq 15$. Let $a$ be an endpoint of $I_0$. Since $\mathop\mathrm{diam}\nolimits{F_k(J_i)} \leq 3\cdot 2^{-k_0}$ for all $i\in\{1,\dots,l\}$, the points $F_k(x)$ and $F_k(y)$ are in $B := B(F_k(a), 45\cdot 2^{-k_0})$. Let $\ell$ be a best fitting line for $B$ and let $\pi$ be the orthogonal projection on $\ell$. Since $\beta_E(B) \leq 2^{-11}$, the points of $F_k(\bigcup_{i=1}^l J_i)\cap B$ are linearly ordered according to their projection on $\ell$. In particular, $|z-w| \leq 2|\pi(z)-\pi(w)|$ for all $z,w \in F_k(\bigcup_{i=1}^l J_i)\cap B$. Thus,
\[ |F_k(x) - F_k(y)| \geq |\pi_{\ell}(F_k(x)) - \pi_{\ell}(F_k(y))| \geq \mathop\mathrm{diam}\nolimits{\pi_{\ell}(F_k(I_0))} \geq \frac12\mathop\mathrm{diam}\nolimits{F_k(I_0)} \geq \frac12 2^{-k_0}.\]
\emph{Case 2.} Assume that $k\geq k_0+4$. For each integer $i\geq 0$, let $P_i(x,y)$ be the endpoints of intervals in $\mathscr{E}_{k_0+4+i}' $ lying between $x$ and $y$. Let $x_i$ (resp.~ $y_i$) be the leftmost (resp.~ rightmost) element of $P_i(x,y)$. For all $i\geq 0$,
\[ x \leq x_{i+1} \leq x_i < y_i \leq y_{i+1} \leq y, \]
and $I_0$ separates $x_0$ from $y_0$. As in Case 1, if $\ell$ is a best fitting line for $F_{k_0}(a)$, then
\[ |F_{k_0+3}(x_0) - F_{k_0+3}(y_0)| \geq |\pi_{\ell}(F_{k_0+3}(x_0)) - \pi_{\ell}(F_{k_0+3}(y_0))| \geq \mathop\mathrm{diam}\nolimits{\pi_{\ell}(F_{k_0}(I_0))} \geq \frac9{10}2^{-k_0}.\]
Now, each $x_{i+1}$ (resp.~ $y_{i+1}$) is contained in the closure of an interval in $\mathscr{E}_{k+4+i}'$ which has $x_i$ (resp.~ $y_i$) as an endpoint. This fact along with Lemma \ref{lem:TSPflat} yields
\begin{align*}
|F_{k_0+4+i}(x_i)-F_{k_0+5+i}(x_{i+1})| &\leq 3 \cdot 2^{-k_0-4-i},\\ |F_{k_0+4+i}(y_i)-F_{k_0+5+i}(y_{i+1})| &\leq 3 \cdot 2^{-k_0-4-i}.
\end{align*}
Therefore, by the triangle inequality,
\begin{align*}
|F_k(x) - F_k(y)| &\geq |F_{k_0+4}(x_0)-F_{k_0+4}(y_0)| - \sum_{i=0}^{\infty}|F_{k_0+4+i}(x_i)-F_{k_0+5+i}(x_{i+1})|\\
&\hspace{1in}- \sum_{i=0}^{\infty}|F_{k_0+4+i}(y_i)-F_{k_0+5+i}(y_{i+1})|\\
&\geq \frac9{10}2^{-k_0} - 6\cdot 2^{-k_0-4} - 6\cdot 2^{-k_0-4}.
\end{align*}
Hence $|F_k(x) - F_k(y)| \geq (3/20)2^{-k_0}$ and the proof is complete.
\end{proof}
Suppose that $x,y \in [0,1]$ with $x<y$. By Lemma \ref{lem:diamtozero}, there exist intervals $I\in \mathscr{E}'_k$ that separate $x$ and $y$ provided that $k$ is sufficiently large. If $k_0$ is the least such integer, then $|F(x) - F(y)| \gtrsim 2^{-k_0}>0$ by Lemma \ref{lem:inj}. This shows that $F$ is injective and completes the proof of Proposition \ref{thm:flat-statedagain}.
\subsection{Proof of Lemma {\ref{lem:flatcont}}} \label{sec:prop-continua}
We first give an auxiliary estimate.
\begin{lem}\label{lem:flat-twolines}
Let $E\subset\mathbb{R}^N$, $x\in E$, and $r>0$. If $y\in E\cap \overline{B}(x,r)$, $|y-x|\geq 64\beta_E(\overline{B}(x,r))r$, and $\ell_{x,y}$ is the line passing through $x$ and $y$, then $$\mathop\mathrm{dist}\nolimits(z,\ell_{x,y}) \leq 4\beta_E(\overline{B}(x,r))\left(1+\frac{1.1r}{|y-x|}\right)r\quad\text{for all }z\in E\cap B(x,r).$$
\end{lem}
\begin{proof} Let $z\in E\cap \overline{B}(x,r)$, $z\neq x$. Let $\ell$ be a best fitting line for $E$ in $\overline{B}(x,r)$. Then $\mathop\mathrm{dist}\nolimits(x,\ell)$, $\mathop\mathrm{dist}\nolimits(y,\ell)$, and $\mathop\mathrm{dist}\nolimits(z,\ell)$ are bounded above by $\beta_E(\overline{B}(x,r)) 2r$. Let $\ell_x=\ell-x$. Then $x\in\ell_x$ and $\mathop\mathrm{dist}\nolimits(y,\ell_x)$ and $\mathop\mathrm{dist}\nolimits(z,\ell_x)$ are bounded above by $2\beta_E(\overline{B}(x,r))2r$. If $y\in\ell_{x}$, then we have $\mathop\mathrm{dist}\nolimits(z,\ell_{x,y}) = \mathop\mathrm{dist}\nolimits(z,\ell_{x}) \leq 2\beta_E(\overline{B}(x,r))2r$ and we are done.
To continue, suppose that $y\not\in\ell_x$ and let $y'=\pi_{\ell_x}(y)$ and let $z'=\pi_{\ell_x}(z)$. Define $$w=x+\frac{|z'-x|}{|y'-x|}(y-x)\in\ell_{x,y}.$$
Since $z' \in \ell_x$ between $x$ and $y'$, we have that $z' = x+|z'-x||y'-x|^{-1}(y'-x)$. Therefore, $$\mathop\mathrm{dist}\nolimits(z',\ell_{x,y}) \leq |z'-w| = |y'-y|\frac{|z'-x|}{|y'-x|} \leq |y'-y|\frac{r}{|y'-x|}.$$ Thus, by the triangle inequality, $$\mathop\mathrm{dist}\nolimits(z,\ell_{x,y}) \leq |z'-z| + |y'-y|\frac{r}{|y'-x|} \leq 2\beta_E(\overline{B}(x,r))\left(1+ \frac{r}{|y'-x|}\right)2r.$$ Since $\mathop\mathrm{dist}\nolimits(y,\ell_x) \leq 2\beta_E(\overline{B}(x,r))2r \leq (1/16)|x-y|,$ we have $$1.1|y'-x| \geq (1+3(1/16)^2)|y'-x|\geq |y-x|$$ by Lemma \ref{lem:approx}, applied with $V=\{x,y\}$. Therefore, \begin{equation*}\mathop\mathrm{dist}\nolimits(z,\ell_{x,y}) \leq 2\beta_E(\overline{B}(x,r))\left(1+\frac{1.1r}{|y-x|}\right)2r. \qedhere\end{equation*}\end{proof}
We now give a proof of the key lemma.
\begin{proof}[Proof of Lemma {\ref{lem:flatcont}}] Without loss of generality, we may assume that $\mathop\mathrm{diam}\nolimits{E}=1$. Fix $x,y\in E$ and let $n\geq 0$ be the unique integer such that
\[ 2^{-(n+1)} < |x-y| \leq 2^{-n}.\]
\emph{Case 1.} Suppose that $n\in\{0,1\}$. Let $\ell$ be the line containing $x$ and $y$. Since $\beta_E(\overline{B}(x,1)) \leq 2^{-11}$ and since $|x-y|>2^{-2}$, by Lemma \ref{lem:flat-twolines} we have that
\begin{equation*}\begin{split} \sup_{w\in E}\mathop\mathrm{dist}\nolimits(w,\ell) = \sup_{w\in E\cap \overline{B}(x,1)}\mathop\mathrm{dist}\nolimits(w,\ell) &\leq \left( 1+ \frac{1.1}{|x-y|}\right ) 4\beta_E(\overline{B}(x,1)) \\ &\leq \frac{21.6}{2^{11}} \leq \frac{1}{2^{6}} < \frac{|x-y|}{2^4}.\end{split}\end{equation*}
Therefore, $E$ is contained inside the tube $T := \overline{B}(x,1) \cap B(\ell,2^{-4}|x-y|)$. Let $D$ be a closed $(N-1)$-ball centered at $z$, perpendicular to $\ell$ and of radius $2^{-4}|x-y|$. In other words, $D$ is the set of all points in $\overline{B}(z,2^{-6})$ whose projection on $\ell$ is $z$. Then $D$ cuts $ T$ into two pieces, one containing $x$ and another containing $y$. By connectedness of $E$, we must have $D\cap E \neq \emptyset$.
\emph{Case 2.} Suppose that $n\geq 2$. The procedure here is roughly the same as that in Case 1, with the difference that the tube $T$ is replaced by a more complicated set. By connectedness of $E$, for each $k\in \{1,\dots, n-1\}$, there exists a point $y_k \in B(x,2^{-k})\cap E$. For each $k\in \{1,\dots,n-1\}$ let $\ell_k$ be the line containing $x$ and $y_k$. Let also $\ell_n$ be the line containing $x$ and $y$.
Working as in Case 1, we can show that for each $k\in\{1,\dots,n\}$,
\[ E\cap \overline{B}(x,2^{-(k-1)}) \subset T_k := \overline{B}(x,2^{-(k-1)})\cap B(\ell_k,2^{-5} 2^{-(k-1)}).\]
Since $\mathop\mathrm{diam}\nolimits{E} =1$, we also have $E \subset T_1$. For each $k\in \{1,\dots,n-1\}$, let $T_{k,1}$, $T_{k,2}$ be the two components of $T_k \setminus \overline{B}(x, 2^{-k})$. Set
\[ T = T_{1,1}\cup\cdots\cup T_{n-1,1} \cup T_n \cup T_{n-1,2}\cup\cdots\cup T_{1,2}.\]
The sets $T_{1,1}, \dots, T_{n-1,1}, T_n , T_{n-1,2},\dots, T_{1,2}$ intersect at most in pairs. In particular,
\begin{enumerate}
\item if $i\in\{1,2\}$, then $T_{1,i}\cap T_{m,j} = \emptyset$ unless $m\in\{1,2\}$ and $j=i$;
\item if $k\in\{2,\dots,n-2\}$ (if any) and $i\in\{1,2\}$, then $T_{k,i} \cap T_{m,j} = \emptyset$ unless $m\in\{k-1,k,k+1\}$ and $j=i$;
\item if $i\in\{1,2\}$, then $T_{n-1,i}\cap T_{m,j} = \emptyset$ unless $m\in\{n-2,n-1\}$ and $j=i$;
\item $T_n \cap T_{m,j} = \emptyset$ unless $m=n-1$.
\end{enumerate}
As with Case 1, if $D$ is an $(N-1)$-ball centered at $z$, perpendicular to $\ell_n$ and of radius $2^{-4}2^{-n}$, then $D$ cuts $T_{n}$ into two pieces, one containing $x$ and another containing $y$. Consequently, $D$ cuts $T$ into two pieces, one containing $x$ and another containing $y$. By connectedness of $E$ and the fact that $E\subset T$, we must have $D\cap E \neq \emptyset$. \end{proof}
\section{Examples}\label{sec:examples}
In this section, we give examples of H\"older curves and of sets that are not contained in H\"older curves to illuminate Theorem \ref{thm:main2}, Proposition \ref{thm:flat}, and Theorem \ref{thm:main}.
\subsection{H\"older curves that are non-flat in all scales}\label{sec:notnec} First up, we show that condition (\ref{eq:thm2}) in Theorem \ref{thm:main2} is not necessary for a bounded set to be contained in a $(1/s)$-H\"older curve when $s>1$. In contrast, when $s=1$, condition \eqref{eq:betasum} in the Analyst's Traveling Salesman theorem is necessary and sufficient for a bounded set to be contained in a rectifiable curve.
Let $N\geq 2$ and $1\leq m \leq N-1$ be integers. Given a nonempty set $E \subset \mathbb{R}^N$ and an $N$-cube $Q\subset\mathbb{R}^N$ with $E\cap Q \neq \emptyset$, define the $m$-dimensional beta number
\[ \beta_{E}^{(m)}(Q) := \inf_{P} \sup_{x\in E\cap Q}\frac{\mathop\mathrm{dist}\nolimits(x,P)}{\mathop\mathrm{diam}\nolimits{Q}}\]
where the infimum is taken over all $m$-planes $P$ in $\mathbb{R}^N$. If $E\cap Q = \emptyset$, set $\beta_{E}^{(m)}(Q) = 0$. Note that $\beta_{E}^{(1)}(Q) =\beta_E(Q)$ as defined in \textsection\ref{sec:intro} and that $\beta_{E}^{(m)}(Q) \leq \beta_{E}^{(n)}(Q)$ whenever $m\geq n$.
\begin{prop}\label{ex:notnec}
For any $N\geq 2$ and any $s\in (1,N]$, there exists a $(1/s)$-H\"older curve $E\subset \mathbb{R}^N$ such that
\begin{equation}\label{eq:notnec}
\sum_{\substack{Q\in\Delta(\mathbb{R}^N)\\ \beta_{E}^{(N-1)}(3Q)\geq (6\sqrt{N})^{-1}}} (\mathop\mathrm{diam}\nolimits{Q})^s = \infty.
\end{equation}
\end{prop}
The construction splits into three cases. Before proceeding, we introduce some notation. Given a cube $Q\subset \mathbb{R}^N$, denote by $\Delta(Q)$ the set of dyadic cubes in $\Delta(\mathbb{R}^N)$ that are contained in $Q$. Moreover, given positive integers $m \leq N$, there exists a polynomial $P_{N,m}$ of degree $m$ with the following property: If $n\in\mathbb{N}$ and $\{Q_1,\dots,Q_{N^n}\}$ is a partition of $[0,1]^N$ into $N$-cubes of side-length $1/n$, then
\[ \operatorname{card}\{Q_i : \text{$Q_i$ intersects the $m$-skeleton of $\partial [0,1]^N$}\} = P_{N,m}(n).\]
Recall that if $I_1,\dots,I_N$ are nondegenerate compact intervals, and $Q=I_1\times\cdots\times I_N$ is an $N$-cube, then the $m$-skeleton of $Q$ is the union of sets $I_1'\times\cdots\times I_N'$ where $I_j' = I_j$ for $m$ indices $j$ and $I_j' = \partial I_j$ for the remaining $N-m$ indices $j$. Finally, we note that if $K$ is the set of vertices of a cube $Q$ in $\mathbb{R}^N$ and $P$ is an $(N-1)$-plane, then
\begin{equation}\label{eq:sierp4}
\mathop\mathrm{dist}\nolimits(x,P) \geq (2\sqrt{N})^{-1}\mathop\mathrm{diam}\nolimits{K} \qquad\text{for all $x\in K$}.
\end{equation}
\emph{Case 1: $s=N$.} We simply take $E=[0,1]^N$. It is well known that there exists a $(1/N)$-H\"older parametrization $f:[0,1]\to E$. On the other hand, by (\ref{eq:sierp4}),
\[\sum_{\substack{Q\in\Delta(\mathbb{R}^N)\\ \beta_{E}^{(N-1)}(3Q)\geq (2\sqrt{N})^{-1}}} (\mathop\mathrm{diam}\nolimits{Q})^N \geq \sum_{k=0}^{\infty}\sum_{\substack{Q\in\Delta([0,1]^N)\\ \mathop\mathrm{diam}\nolimits{Q} = \sqrt{N}2^{-k}}}(\mathop\mathrm{diam}\nolimits{Q})^N = \sum_{k=0}^{\infty} 2^{Nk}(\sqrt{N}2^{-k})^N = \infty.\]
\emph{Case 2: $s\in(1,N)\setminus \mathbb{N}$.} Let $m$ be the integer part of $s$. Since the degree of $P_{N,m}$ is strictly less than $s$ and strictly lager than $s-1$, we can fix $n\in\mathbb{N}$ such that
\begin{equation}\label{eq:sierp1}
n^s - (n-2)^s < P_{N,m}(n) < n^s - 1.
\end{equation}
By (\ref{eq:sierp1}) and the Intermediate Value Theorem, there exists $\lambda \in (\frac1{n},1-\frac2{n})$ such that
\begin{equation}\label{eq:sierp2}
P_{N,m}(n) n^{-s} + \lambda^s = 1.
\end{equation}
Partition $[0,1]^N$ into $N$-cubes of side-lengths $1/n$ and let $\{Q_i\}_{i=1}^{l}$ ($l=P_{N,m}(n)$) be those cubes that intersect the $m$-skeleton of $[0,1]^N$. Let also $Q_0 = [1/n, 1/n+\lambda]^N$. For each $i=0,\dots,l$, let $\phi_i$ be a similarity of $\mathbb{R}^N$ such that $\phi_i([0,1]^N) = Q_i$. Finally, define $E\subset \mathbb{R}^N$,
\[ E := \bigcap_{k=1}^{\infty} \bigcup_{i_1\cdots i_k \in \{0,\dots,l\}^k}\phi_{i_1}\circ\cdots\circ\phi_{i_k}([0,1]^N). \]
Since the maps $\{\phi_0,\dots,\phi_l\}$ satisfy the open set condition, $E$ is Ahlfors regular \cite{hutchinson}. By (\ref{eq:sierp2}) the Hausdorff dimension of $E$ is equal to $s$, so $E$ is $s$-regular.
\begin{lem}\label{lem:Eisconn}
The set $E$ is connected.
\end{lem}
\begin{proof}
Set $\mathcal{W} = \bigcup_{k\geq 0}\{0,\dots,l\}^k$ with the convention that $\{0,\dots,l\}^0$ is the empty word $\emptyset$ and $\phi_{\emptyset}$ is the identity map of $\mathbb{R}^N$ For each $w\in\mathcal{W}$, let $\mathcal{K}_w$ denote the $1$-skeleton of $\phi_w([0,1]^N)$. The proof is based now on two observations. First, by the choice of cubes $Q_1,\dots,Q_l$, it follows that $\mathcal{K}_w \subset E$ for all $w\in\mathcal{W}$. Second, $\mathcal{K}_w \cap \mathcal{K}_{wi} \neq \emptyset$ for all $w\in\mathcal{W}$ and $i\in\{0,\dots,l\}$.
Now fix $x\in E$. There exists a sequence of words $(w_n)_{n\geq 0}$ in $\mathcal{W}$ such that $w_0$ is the empty word, $w_{n+1} = w_ni_n$ with $i_n \in \{0,\dots,l\}$, and $x \in \bigcap_{n\geq 0}\phi_{w_n}(x)$. The set $\bigcup_{n\geq 0}\mathcal{K}_{w_n}$ is a path that joins $x$ with the origin. Hence $E$ is connected.
\end{proof}
By Lemma \ref{lem:Eisconn}, the fact that the Hausdorff dimension of $E$ is $s$, and Theorem 4.12 in \cite{Remes}, there exists a $(1/s)$-H\"older map $f:[0,1] \to \mathbb{R}^N$ such that $f([0,1]) = E$. It remains to show (\ref{eq:notnec}). We first prove a lemma.
\begin{lem}\label{lem:notnec}
If $Q \in \Delta([0,1]^N)$ is a dyadic cube that intersects $E$, then there exists a dyadic cube $Q'\subset 3Q$ such that $\mathop\mathrm{diam}\nolimits{Q'}\geq (3n)^{-1}\mathop\mathrm{diam}\nolimits{Q}$ and $\beta_{E}^{(N-1)}(3Q') \geq (6\sqrt{N})^{-1}$.
\end{lem}
\begin{proof}
Fix $x\in Q\cap E$ and let $i_1,i_2,\dots$ be a sequence of numbers in $\{0,\dots,l\}$ such that
\[ x\in \bigcap_{k=1}^{\infty}\phi_{i_1}\circ\cdots\circ\phi_{i_k}([0,1]^N).\]
Let $k_0$ be the smallest positive integer such that $\phi_{i_1}\circ\cdots\circ\phi_{i_{k_0}}([0,1]^N) \subset 3Q$ and define $K$ to be the set of vertices of $\phi_{i_1}\circ\cdots\circ\phi_{i_{k_0}}([0,1]^N)$. Since each $\phi_i$ has a scaling factor at least $1/n$, by minimality of $k_0$ we have that $\mathop\mathrm{diam}\nolimits{K} \geq (1/n)\mathop\mathrm{diam}\nolimits{Q}$.
Let $Q'$ be a dyadic cube in $\Delta(3Q)$ (possibly $Q'=Q$) of minimal diameter such that $K\subset 3Q'$. We claim that
\begin{equation}\label{eq:sierp3}
\frac{1}3\mathop\mathrm{diam}\nolimits{K} \leq \mathop\mathrm{diam}\nolimits{Q'} \leq \mathop\mathrm{diam}\nolimits{K}.
\end{equation}
The lower inequality is clear. If $\mathop\mathrm{diam}\nolimits{K} < \mathop\mathrm{diam}\nolimits{Q}$, then, since $K$ has edges parallel to the axes, $K$ is contained in $3Q_0$ for some dyadic cube $Q_0\subset 3Q$ with $\mathop\mathrm{diam}\nolimits{Q_0} = \frac12 \mathop\mathrm{diam}\nolimits{Q'}$, which is a contradiction. That establishes the upper inequality of (\ref{eq:sierp3}).
By (\ref{eq:sierp4}), (\ref{eq:sierp3}), and the fact that $K\subset E$,
\[ \beta_{E}^{(N-1)}(3Q') \geq \beta_{K}^{(N-1)}(3Q') \geq \frac{(2\sqrt{N})^{-1}\mathop\mathrm{diam}\nolimits{K}}{\mathop\mathrm{diam}\nolimits{3Q}} = \frac{(6\sqrt{N})^{-1}\mathop\mathrm{diam}\nolimits{K}}{\mathop\mathrm{diam}\nolimits{Q}} \geq (6\sqrt{N})^{-1}. \] This proves the lemma.
\end{proof}
By Ahlfors $s$-regularity of $E$, there exists a constant $C>1$ such that
\[ \operatorname{card}\{Q\in \Delta([0,1]^N) : \mathop\mathrm{diam}\nolimits{Q} = \sqrt{N}2^{-k} \text{ and } Q \cap E \neq \emptyset\} \geq C^{-1} 2^{sk}.\]
Fix a positive integer $k_0$ such that $2^{k_0} >3n$. For $k\in\mathbb{N}$, set
\[ \mathcal{Q}_k = \{Q\in \Delta([0,1]^N) : \mathop\mathrm{diam}\nolimits{Q} \in [\sqrt{N}2^{-k}, \sqrt{N}2^{-k-k_0}] \text{ and } \beta_{E}^{(N-1)}(3Q) \geq (6\sqrt{N})^{-1}\} .\]
By Lemma \ref{lem:notnec},
\begin{align*}
\operatorname{card}{\mathcal{Q}_k} \geq 3^{-N}\operatorname{card}\{Q\in \Delta([0,1]^N) : \mathop\mathrm{diam}\nolimits{Q} = \sqrt{N}2^{-k} \text{ and } Q \cap E \neq \emptyset\} \geq C^{-1}3^{-N} 2^{sk}.
\end{align*}
Therefore,
\begin{align*}
\sum_{\substack{Q\in\Delta(\mathbb{R}^N)\\ \beta_{E}^{(N-1)}(3Q) \geq (6\sqrt{N})^{-1}}} (\mathop\mathrm{diam}\nolimits{Q})^N &\geq \sum_{k=0}^{\infty}\sum_{Q \in \mathcal{Q}_{kk_0}}(\mathop\mathrm{diam}\nolimits{Q})^N \\
&\geq \sum_{k=0}^{\infty} C^{-1}3^{-N} 2^{skk_0} (\sqrt{N})^{s}2^{-s(k+1)k_0} = \infty.
\end{align*}
\emph{Case 3: $s\in\{2,\dots,N-1\}$.} Fix $n\in\mathbb{N}$ large enough so that $P_{N,s-1}(n) < n^s$. Partition $[0,1]^N$ into $N$-cubes with disjoint interiors and side-lengths $1/n$ and let $\{Q_1,\dots,Q_{l}\}$ ($l=n^s$) be a collection of such cubes so that the set $\bigcup_{k=1}^l Q_i$ is connected and contains the $(s-1)$-skeleton of $[0,1]^N$. The rest of the construction is similar to Case 2 and is left to the reader.
\subsection{Ahlfors regular curves without H\"older parametrizations}\label{sec:regularcurve} Next, for all $s>1$, we construct Ahlfors $s$-regular curves that are not contained in any $(1/s)$-H\"older curve. The basic strategy is take a disconnected set, which is not contained in a H\"older curve, and then extend the set to transform it into an $s$-regular curve. We call the curves that we construct ``Cantor ladders".
\begin{prop}
Let $N\in\mathbb{N}$ with $N\geq 2$, let $s\in (1,N)$, and let $m\in\mathbb{N}$ with $m\leq s$. There exists an Ahlfors $s$-regular curve $E\subset \mathbb{R}^N$, which is not contained in a $(m/s)$-H\"older image of $[0,1]^m$.
\end{prop}
We treat the cases $s\in \mathbb{N}$ and $s\not\in\mathbb{N}$ separately. Given $m\in\mathbb{N}$, let $\mathcal{W}_m$ be the set of finite words formed by the letters $\{1,\dots,m\}$ including the empty word $\emptyset$. We denote by $|w|$ the number of letters a word has with the convention $|\emptyset| = 0$.
\emph{Case 1.} Suppose that $s\in\{2,3,\dots,N-1\}$. Let $D_{\emptyset} = [0,1]^2$. Given a square $D_w \subset \mathbb{R}^2$ for some $w\in\mathcal{W}_4$, let $D_{w1}$, $D_{w2}$, $D_{w3}$, $D_{w4}$ be the four corner squares in $D_w$ with $\mathop\mathrm{diam}\nolimits{D_{wi}} = (1/4)\mathop\mathrm{diam}\nolimits{D_w}$. Let $\mathcal{C}_1$ be the Cantor set in $\mathbb{R}^2$ defined by
\[ \mathcal{C}_1 = \bigcap_{k=0}^{\infty}\bigcup_{\substack{w\in\mathcal{W}_4 \\ |w|=k}} D_w.\]
For each $i = 1,\dots, 2^{|w|}$, define $D_{w,i} = D_w\times\{(2i-1)2^{-|w|-1}\}$,
\[ K_2 = (\mathcal{C}_1\times[0,1]) \cup \bigcup_{w\in\mathcal{W}}\bigcup_{i=0}^{2^{|w|}-1} D_{w,i}\qquad\text{and}\qquad E = K_2\times [0,1]^{s-2} \times \{0\}^{N-s-1}.\]
Here and for the rest of \textsection\ref{sec:regularcurve}, we use the convention $A\times\{0\}^0 = A$.
\emph{Case 2.} Suppose that $s\in(1,N)\setminus \mathbb{N}$. Let $p = s-\lfloor s \rfloor$ be the fractional part of $s$. Let $I_{\emptyset} = [0,1]$. Given an interval $I_w = [a_w,b_w]$ for some $w\in\mathcal{W}_2$, let
\[ I_{w1} = [a_w, a_w+2^{-p}(b_w-a_w)] \qquad\text{and}\qquad I_{w2} = [b_w-2^{-p}(b_w-a_w),b_w].\]
Let $\mathcal{C}_p$ denote the Cantor set in $\mathbb{R}$ defined by
\[ \mathcal{C}_p = \bigcap_{k=0}^{\infty}\bigcup_{\substack{w\in\mathcal{W}_2 \\ |w|=k}} I_w.\]
Let $S$ be the bi-Lipschitz embedded image of $([0,1],|\cdot|^{\frac1{p+1}})$ into $\mathbb{R}^2$. For each $w\in\mathcal{W}_2$, let $S_w$ be a rescaled copy of $S$ whose endpoints are the right endpoint of $I_{w1}$ and the left endpoint of $I_{w2}$. For each $w\in\mathcal{W}_2$ and $i = 1,\dots, 2^{|w|}-1$, define
\[ S_{w,i} = S_w + (0,(2i-1)2^{-|w|-1})\]
and define
\[ K_{p+1} = (\mathcal{C}_{p}\times [0,1])\cup\bigcup_{w\in\mathcal{W}}\bigcup_{i=0}^{2^{|w|}} S_{w,i}\qquad\text{and}\qquad E = K_{p+1}\times [0,1]^{s-p-1} \times \{0\}^{N+p-s-1}.\]
Verification of the desired properties of $E$ is the same for the two cases, so we only treat Case 1. By Theorem 2.1 in \cite{MM2000}, there exists no $(m/s)$-H\"older map $f:[0,1]^m \to \mathbb{R}^N$ whose image contains $\mathcal{C}_1 \times [0,1]^{s-1}\times\{0\}^{N-s-1}$. We show that $E$ is a curve in \textsection\ref{sec:Eisacurve} and we prove $s$-regularity of $E$ in \textsection\ref{sec:Eisregular}.
\subsubsection{$E$ is a curve}\label{sec:Eisacurve}
By the Hahn-Mazurkiewicz theorem \cite[Theorem 3.30]{hocking}, to show that $E$ is a curve it is enough to show that $E$ is compact, connected, and locally connected.
For compactness, it is easy to see that $K_2 \subset [0,1]^3$, hence $E\subset [0,1]^N$. Moreover, as $|w|\to\infty$, the squares $D_{w,i}$ accumulate on $\mathcal{C}\times [0,1]$. Therefore, $K_2$ is closed. Consequently, $E$ is compact.
To settle both connectedness and local connectedness, we prove that there exists $C>1$ such that for all pairs of points $x,y \in E$ there exists a path joining $x$ with $y$ of diameter at most $C|x-y|^{1/2}$. Clearly, it suffices to show the claim for $K_2$ instead of $E$. Fix $x,y\in K_2$ and let $w_0$ be the word in $\mathcal{W}_4$ of maximum word-length such that the projections of $x$ and $y$ on $\mathbb{R}^2\times\{0\}$ are contained in $D_{w_0}$. This means that $|x-y| \geq \frac12 4^{-|w_0|}$. Choose $i_0 \in \mathbb{N}$ such that $\mathop\mathrm{dist}\nolimits(x,D_{w_0,i_0}) \leq 2\cdot 2^{-|w_0|}.$
If $x_0$ and $y_0$ are the projections of $x$ and $y$ onto $D_{w_0,i_0}$, respectively, then
\[ \max\{|x_0-x|, |y_0-y|\} \lesssim 2^{-|w_0|} + 4^{-|w_0|} + |x-y| \simeq 2^{-|w_0|}.\]
There exist sequences $(w_n)_{n\in\mathbb{N}}, (u_n)_{n\in\mathbb{N}}$ of words in $\mathcal{W}_4$ and sequences $(i_n)_{n\in\mathbb{N}}, (j_n)_{n\in\mathbb{N}}$ of positive integers such that
\begin{enumerate}
\item $|w_n| = |u_n| = |w_0|+n$;
\item the orthogonal projection of $x$ (resp.~ $y$) on $\mathbb{R}^2$ is contained in $D_{w_n}$ (resp.~ $D_{u_n}$);
\item there exists $x_n \in D_{w_n,i_n}$ such that
\[ \max\{ |x-x_n|, |y-y_n|\} \leq 4^{-|w_0|-n}\sqrt{2} + 2^{-|w_0|-n}.\]
\end{enumerate}
Properties (1) and (2) imply that $D_{w_0} \supsetneq D_{w_1} \supsetneq D_{w_2} \supsetneq \cdots$ and $D_{w_0} \supsetneq D_{u_1} \supsetneq D_{u_2} \supsetneq \cdots$, while property (3) implies that the Hausdorff distances
\[ \mathop\mathrm{dist}\nolimits_H(D_{w_n,i_n}, D_{w_{n+1},i_{n+1}})\lesssim 2^{-|w_0|-n} \quad\text{and}\quad \mathop\mathrm{dist}\nolimits_H(D_{u_n,j_n}, D_{u_{n+1},j_{n+1}}) \lesssim 2^{-|w_0|-n}.\]
Let $\gamma_0\subset K_2$ be the line segment joining $x_0$ with $y_0$. For each $n\geq 0$, let $z_n \in D_{w_n,i_n}$ be a corner point and let $z_n'$ be its projection on $D_{w_{n+1},i_{n+1}}$. Also, let $p_n \in D_{u_n,j_n}$ be a corner point and let $p_n'$ be its projection on $D_{u_{n+1},j_{n+1}}$. Consider the curve
\[ \gamma = \gamma_0 \cup \bigcup_{n\in\mathbb{N}}([x_n,z_n] \cup [z_n,z_n'] \cup [z_n',x_{n+1}])\cup \bigcup_{n\in\mathbb{N}}([y_n,p_n] \cup [p_n,p_n'] \cup [p_n',y_{n+1}]),\]
which is a subset of $K_2$ and joins $x$ with $y$. Then
\begin{align*}
\mathop\mathrm{diam}\nolimits{\gamma} &\lesssim \mathop\mathrm{diam}\nolimits{\gamma_0} + \sum_{n\geq 0}\mathop\mathrm{diam}\nolimits{\gamma_n} + \sum_{n\geq 0}\mathop\mathrm{diam}\nolimits{\sigma_n}\\
&\leq |x_0-y_0| + \sum_{n\geq 0}(|x_n-z_n| + |z_n-z_n'| + |z_n'-x_{n+1}|) \\
&\hspace{.2in}+ \sum_{n\geq 0}(|y_n-p_n| + |p_n-p_n'| + |p_n'-y_{n+1}|)\\
&\lesssim 4^{-|w_0|} + \sum_{n\geq 0}(2^{-|w_0|-n}+4^{-|w_0|-n} + 2^{-|w_0|-n}) \lesssim 2^{-|w_0|} \simeq |x-y|^{1/2}.
\end{align*}
\subsubsection{$E$ is $s$-regular.}\label{sec:Eisregular} We show $s$-regularity for $E$. Because the product of regular compact spaces of dimension $s_1$ and $s_2$ is $(s_1+s_2)$-regular, to show that $E$ is $s$-regular,
it suffices to show that $K_{2}$ is $2$-regular. Fix $x\in K_2$ and $r\in (0,\mathop\mathrm{diam}\nolimits{K_2})$.
We first show that
\begin{equation}\label{eq:lowreg}
\mathcal{H}^2(B(x,r)\cap K_2) \gtrsim r^2.
\end{equation}
If $x\in \mathcal{C}_1\times[0,1]$, then (\ref{eq:lowreg}) follows from the $2$-regularity of $\mathcal{C}_1\times[-1,1]$. If $x \in D_{w,i}$ and $r\leq 10 \mathop\mathrm{diam}\nolimits{D_w}$, then (\ref{eq:lowreg}) follows from the $2$-regularity of $D_{w,i}$. If $x \in D_{w,i}$ and $r\geq 10 \mathop\mathrm{diam}\nolimits{D_w}$, then there exists $z\in (\mathcal{C}_1\times[0,1])\cap B(x,r)$ such that $B(z,r/2) \subset B(x,r)$ and (\ref{eq:lowreg}) follows from the $2$-regularity of $B(z,r/2)\cap K_2$.
For the upper regularity of $K_2$, instead of working with balls $B(x,r)$, it is more convenient to use cubes
\[ Q(x,r) = x + [-r/2,r/2]^3 \qquad x\in K_2, \quad r>0.\]
Without loss of generality, we may assume that $r=4^{-k_0}$ for some $k_0\in\mathbb{N}$. For each $k\geq 0$, let
\[ \mathcal{D}_k(x,r)= \{D_{w,i} : Q(x,r)\cap D_{w,i} \neq \emptyset \text{ and }|w|=k\}.\]
Then by the $2$-regularity of $\mathcal{C}_1\times[-1,1]$, it suffices to show that
\begin{align*}
\sum_{k\geq 0}\sum_{D_{w,i}\in\mathcal{D}_k(x,r)} \mathcal{H}^2(Q(x,r)\cap D_{w,i}) \lesssim r^2.
\end{align*}
The following lemma will let us estimate the above sum. In the sequel, we denote by $m_0\geq 0$ the smallest integer for which there exists $D_{w,i}\in\mathcal{D}(x,r)$ with $|w|=m_0$.
\begin{lem}\label{lem:squareint}
Let $m_0\geq 0$ be the smallest integer for which $\mathcal{D}_{m_0}(x,r) \neq \emptyset$.
\begin{enumerate}
\item If $k>m_0$ and $\mathcal{D}_{k}(x,r) \neq \emptyset$, then $k\geq 2k_0$.
\item If $Q'$ is the projection of $Q(x,r)$ on $\mathbb{R}^2\times\{0\}$, then for all $k\geq 0$,
\[ \operatorname{card}{\{D_w : D_w\cap Q'\neq \emptyset \text{ and }|w|=k\}} \leq 1+4^{k}4^{-k_0}.\]
\item For each $w\in \mathcal{W}_4$,
\[ \operatorname{card}{\{ i : D_{w,i}\cap Q(x,r) \neq \emptyset\}} \leq 1+2^{|w|+1}4^{-k_0}.\]
\item For each $k\geq 0$, $\operatorname{card}{\mathcal{D}_k(x,r)} \leq (1+4^{k}4^{-k_0})(1+2^{k+1}4^{-k_0})$.
\item We have
\[ \sum_{D_{w,i}\in\mathcal{D}_{m_0}(x,r)}\mathcal{H}^2(D_{w,i}\cap Q(x,r)) \lesssim r^2.\]
\end{enumerate}
\end{lem}
\begin{proof}
For (1), recall that if $|w|>m_0$, then the vertical distance between $D_{w,i}$ and $D_{w_0,i_0}$ is at least $2^{-|w|}$. Since $r=4^{-k_0}$, the cube $Q(x,r)$ can not intersect any $D_{w,i}$, unless $4^{-k_0} \geq 2^{-|w|}$. Thus, $|w|\geq 2k_0$.
For (2), we first note that if $k\leq k_0$, then $Q'$ can intersect at most one square $D_w$ with $|w|=k$. We now use induction to show that for all $k\geq k_0$,
\[ \operatorname{card}{\{D_w : D_w\cap Q'\neq \emptyset \text{ and }|w|=k\}} \leq 4^{k}4^{-k_0}.\]
For $k=k_0$, it is true. Suppose that the claim is also true for some $k\geq k_0$. Then $Q'$ intersects $D_{w}$ with $|w|=k+1$ if and only if there exists $w'$ with $|w'|=k$ such that $Q\cap D_{w'} \neq \emptyset$ and $D_{w}\subset D_{w'}$. Since each square of generation $k$ contains $4$ squares of generation $k+1$,
\begin{align*}
\operatorname{card}{\{D_w : D_w\cap Q'\neq \emptyset \text{ and }|w|=k+1\}} &\leq 4\operatorname{card}{\{D_w : D_w\cap Q'\neq \emptyset \text{ and }|w|=k\}}\\
&\leq 4^{k+1}4^{-k_0}.
\end{align*}
For (3), fix $w\in\mathcal{W}_4$. Recall that the vertical height of $Q(x,r)$ is $2r=2\cdot4^{-k_0}$ and that the vertical distance between $D_{w,i}$ and $D_{w,j}$ with $i\neq j$ is at least $2^{-|w|}$. Therefore,
\begin{align*}
\operatorname{card}{\{ i : D_{w,i}\cap Q(x,r) \neq \emptyset\}} \geq 1 + (2r)/2^{-|w|} = 1+2^{|w|+1}4^{-k_0}.
\end{align*}
Claim (4) is immediate from (2) and (3).
It remains to show (5). On one hand, if $m_0> k_0$, then by (4), $\operatorname{card}(\mathcal{D}_{m_0}(x,r))=1$. Hence (5) follows from the $2$-regularity of squares $D_{w,i}$. On the other hand, if $m_0\leq k_0$, then by (4),
\[ \sum_{D_{w,i}\in\mathcal{D}_{m_0}(x,r)}\mathcal{H}^2(D_{w,i}\cap Q(x,r)) \leq \operatorname{card}(\mathcal{D}_{m_0}(x,r)) (4^{-m_0})^2 \lesssim 2^{-m_0} 4^{-2k_0} \leq r^2. \qedhere\]
\end{proof}
By Lemma \ref{lem:squareint}, we have
\begin{align*}
\sum_{D_{w,i}\in\mathcal{D}(x,r)} \mathcal{H}^2(B(x,r)\cap D_{w,i}) &\leq \sum_{D_{w,i}\in\mathcal{D}_{m_0}(x,r)} \mathcal{H}^2(D_{w,i}) + \sum_{k=2k_0}^{\infty}\sum_{D_{w,i}\in\mathcal{D}_k(x,r)} \mathcal{H}^2(D_{w,i})\\
&\lesssim r^2 + \sum_{k=2k_0}^{\infty} 2^{k}4^{-k_0}4^k4^{-k_0} 4^{-2k}.
\end{align*}
Finally,
\[ \sum_{k\geq 2k_0}^{\infty} 2^{k}4^{-k_0}4^k4^{-k_0} 4^{-2k} = 4^{-2k_0}\sum_{k\geq k_0} 2^{-k} \lesssim 4^{-3k_0} \lesssim r^2.\] Therefore, $K_2$ is $2$-regular.
\subsection{A compact countable set that is not contained in any H\"older cube}\label{sec:null}
\begin{prop}
For each $N\in\mathbb{N}$, $N\geq 2$, there exists a compact and countable set $E\subset\mathbb{R}^N$ with one accumulation point such that for any $m\in\{1,\dots,N-1\}$ and any $s\in[1,N/m)$, the set $E$ is not contained in a $(1/s)$-H\"older image of $[0,1]^m$.
\end{prop}
\begin{cor}
For each $N\in\mathbb{N}$, $N\geq 2$, there exists a compact and countable set $E\subset\mathbb{R}^N$ with one accumulation point such that $E$ is not contained in a rectifiable curve.
\end{cor}
For each integer $k\geq 0$, define $\mathcal{G}_{k}^0$ to be the union of all vertices of all dyadic cubes in $\mathbb{R}^N$ that are contained in $[0,1]^N$ and have side length $2^{-k}$. By a simple combinatorial argument, $\operatorname{card}(\mathcal{G}_{0}^k) = (2^k+1)^N$ for all $k\geq 0$.
Let $\phi_0$ be the identity map, and for each $k\geq 1$, define a map $\phi_k : \mathbb{R}^N \to \mathbb{R}^N$ by
\[ \phi_k(x) =(k+1)^{-2}x + \left (0,\dots,0,2\sum_{i=1}^{k} i^{-2} \right ).\]
Set $A := \sum_{i=1}^{\infty} i^{-2} =\pi^2/6$, and define the set
\[ E := \{(0,\dots,0,2A)\} \cup \bigcup_{k=0}^{\infty} \phi_k(\mathcal{G}_{k}^0).\]
The set $E$ is clearly countable. If $(x_1,\dots,x_N) \in E$, then $|x_i| \leq 1$ for all $i=1,\dots,N-1$ while $|x_N| \leq 2A$. Therefore, $E$ is bounded. Moreover, the only accumulation point of $E$ is the point $(0,\dots,0,2A)$ which is contained in $E$. Thus, $E$ is closed.
Next, we claim that
\begin{equation}\label{eq:dyadicnets}
|x-y| \geq 2^{-k}(k+1)^{-2} \qquad\text{for all $x \in \phi_k(\mathcal{G}_{k}^0)$ and all }y \in E\setminus\{x\}.
\end{equation}
Indeed, if $x,y\in\mathcal{G}_k^0$, then inequality (\ref{eq:dyadicnets}) is clear. Otherwise, $\mathop\mathrm{dist}\nolimits(\mathcal{G}_{k}^0, E\setminus \mathcal{G}_{k}^0) \geq (k+1)^{-2}$, and thus, (\ref{eq:dyadicnets}) holds again.
Suppose in order to get a contradiction that there exists a $(1/s)$-H\"older continuous map $f:[0,1]^m\rightarrow\mathbb{R}^N$ such that $E\subset f([0,1])$. Let $H$ be the H\"older constant of $f$. For each $k\geq 0$ and $x\in \mathcal{G}_{k}^0$, fix a point $w_{k,x}$ such that $f(w_{k,x}) = x$ and set
\[ B_{k,x} = B(w_{k,x}, \tfrac12 H^{-s}2^{-ks}(k+1)^{-2s}).\]
Inequality (\ref{eq:dyadicnets}) implies that the balls $B_{k,x}$ are mutually disjoint. Moreover, it is easy to see that each $B_{k,x}$ is contained in $[-1,2]^m$. Therefore,
\begin{align*}
1 \gtrsim_m\mathcal{H}^m([-1,2]^m) \geq \sum_{k=0}^{\infty}\sum_{x \in \mathcal{G}_{k}^0}\mathcal{H}^m(B_{k,x}) &\gtrsim_{H,s} \sum_{k=0}^{\infty}(2^k+1)^N\frac{2^{-skm}}{(k+1)^{2sm}} \simeq_{N} \sum_{k=0}^{\infty}\frac{2^{k(N-ms)}}{(k+1)^{2s}}.
\end{align*}
Since $N>ms$, the sum on the right hand side diverges and we reach a contradiction.
\subsection{Flat curves with finite $\mathcal{H}^s$ measure and no $(1/s)$-H\"older parametrizations}\label{sec:flatcurves-counterexample}
The following example shows that the assumption of lower $s$-regularity can not be dropped from Proposition \ref{thm:flat}.
\begin{prop}
For any $\beta_0 \in (0,1)$, there exists $s_0 \in (1,2)$ with the following property. For any $s\in (1,s_0)$ there exists a curve $E \subset \mathbb{R}^2$ such that
\begin{enumerate}
\item $\mathcal{H}^s(E) < \infty$ and
\item $\beta_{E}(Q) < \beta_0$ for all $Q\in\Delta(\mathbb{R}^N)$,
\end{enumerate}
but $E$ is not contained in any $(1/s)$-H\"older image of $[0,1]$.
\end{prop}
Before proceeding, we recall a well-known construction method for snowflakes in $\mathbb{R}^2$. Let $\textbf{p} = (p_0,p_1,\dots)$ be sequence of numbers in $[1/4,1/2)$. Let $\Gamma_0$ be the segment $[0,1]\times\{0\}$, oriented from $(0,0)$ to $(1,0)$. Assume that we have constructed an oriented polygonal arc $\Gamma_k$ with $4^k$ edges. Define $\Gamma_{k+1}$ to be the polygonal arc constructed by replacing each edge $e$ of $\Gamma_{k}$ by a rescaled and rotated copy of the oriented polygonal arc in Figure \ref{fig:figure1} with $p = p_k$, so that the new oriented arc lies to the left of $e$. A snowflake arc $\mathcal{S}_{\textbf{p}}$ is obtained by taking the limit of $\Gamma_{k}$, just as in the construction of the usual von Koch snowflake.
\begin{figure}[ht]
\includegraphics[scale=0.6]{Polygon.png}
\caption{}
\label{fig:figure1}
\end{figure}
\begin{rem}\label{rem:flatsnowflakes}
For any $\epsilon>0$, there exists $p^*>1/4$ (small) such that if a snowflake is built with parameters $1/4 \leq p_k \leq p^*$ for all $k\geq 0$, then $\beta_{\Gamma_k}(B(x,r)) \leq \epsilon r$ for all $k\geq 0$, $x\in\Gamma_k$, and $r>0$.
\end{rem}
Fix $\beta_0 \in (0,1)$. By the preceding remark, there exists $p^* \in (1/4,1/2)$ such that $\beta_{\mathcal{S}_{p_0}}(Q) < \beta_0$ for all $Q\in\Delta(\mathbb{R}^N)$. Set $\textbf{p}=(p^*,p^*,\dots)$, set $s_0 = -\log{4}/\log{p_0}$, and fix $s \in [1,s_0)$. It is well-known that there exists a $(1/s_0)$-bi-H\"older homeomorphism $\Phi : [0,1]\to \mathcal{S}_{\textbf{p}}$; e.g., see \cite{BoHei, RV}.
We now construct a self-similar Cantor set in $[0,1]$ in the following way. Let $I_{\emptyset} = [0,1]$. Assuming we have constructed $I_w = [a_w,b_w]$ for some $w\in\{1,2\}^n$, let
\[I_{w1} = [a_w, a_w + (b_w-a_w)2^{-s_0/s}] \quad\text{and}\quad I_{w2} = [b_w - (b_w-a_w)2^{-s_0/s}, b_w].\]
Define $E' = \bigcap_{n=0}^{\infty}\bigcup_{w\in\{1,2\}^n}I_w$. For each component $J$ of $[0,1]\setminus E'$, let $\gamma_J$ be the line segment joining the endpoints of $\Phi(J)$. Then define
\[ E = \Phi(E') \cup \bigcup_{J} \gamma_J,\]
where the union is taken over all components $J$ of $[0,1]\setminus E'$. Since $E'$ is $s/s_0$-regular and $\Phi$ is $(1/s_0)$-bi-H\"older,
\[\mathcal{H}^s(E) = \mathcal{H}^s(\Phi(E')) + \sum_{J}\mathcal{H}^s(\gamma_J) \leq C\mathcal{H}^{s/s_0}(E') < \infty .\]
Since $\Phi(E')\subset \mathcal{S}_{\textbf{p}}$ and $\gamma_J$ are line segments,
we have $\beta_{E}(Q) < \beta_0$ for all $Q\in\Delta(\mathbb{R}^N)$. Finally, by Theorem 2.1 in \cite{MM2000}, there does not exist a $(1/s)$-H\"older map $f:[0,1]\to \mathbb{R}^2$ whose image contains $\Phi(E')$ (and consequently $E$).
\subsection{Sharpness of exponent 1 in Theorem \ref{thm:main}}\label{sec:sharpnessof1}
To wrap up, we show that Theorem \ref{thm:main} does not hold if numbers $\tau_{s}(k,v,v')$ are replaced by $\tau_{s}(k,v,v')^p$ with $p>1$. When $s=1$, this follows from the necessary half of the Analyst's Traveling Salesman theorem. Thus, we may focus on the case $s>1$.
\begin{prop}
Let $p>1$, let $s>1$ be sufficiently close to $1$, and let $\alpha_0>0$ be sufficiently close to $0$. There exists a sequence of finite sets $\{(V_k,\rho_k)\}_{\geq 0}$ of numbers and finite sets in $\mathbb{R}^2$ satisfying (V0)--(V5) such that
\begin{equation}\label{eq:sharp}
\sum_{\substack{v\in V_k \\ \alpha_{k,v}\geq \alpha_0}}\rho_k^s r_0^s + \sum_{(v,v')\in \mathsf{Flat}(k)}\tau_s(k,v,v')^{p}\rho_k^sr_0^s < \infty
\end{equation}
but there does not exist a $(1/s)$-H\"older map $f:[0,1]\to \mathbb{R}^2$ such that $\bigcup_{k\geq 0}V_k \subset f([0,1])$.
\end{prop}
Let $s>1$ and $n_0 \in \mathbb{N}$ be constants to be specified below. Fix a number
\[ 0 < q < \min\{1/s, (p-1)/s\}. \]
For each $n\in \mathbb{N}$, let
\[ t_k = \sqrt{\frac{1}{4^{1/s}}\left ( 1 + \frac{1}{k+n_0} \right )^{2q} - \frac{1}{4}}.\]
Construct a sequence of polygonal arcs $\Gamma_k$ as in \textsection\ref{sec:flatcurves-counterexample} with parameters
\[ p_k = 1/4 + t_k^2.\]
We may assume that numbers $p_k$ are in $[1/4,1/2)$ by taking $n_0$ to be sufficiently large. For each $k\geq 0$, we define a finite set $V_k \subset \Gamma_k$ as follows. Define $V_0 := \{v_{0,1},v_{0,2}\}$, where $v_{0,1} = (0,0)$ and $v_{0,2} = (1,0)$. Suppose that for some $k\geq 0$ we have defined a set
\[ V_k = \{v_{k,1},\dots,v_{k,N_k}\},\qquad N_k = 2^{k}+1,\]
where points $v_{k,i}$ are enumerated according to the orientation of $\Gamma_k$.
For each $i=1,\dots, 2^k+1$, set $v_{k+1,2i-1} = v_{k,i}$, and assign $v_{k+1,2i}$ to the point of $\Gamma_{k+1}$ that lies between $v_{k+1,2i-1}$ and $v_{k+1,2i+1}$ and is equal distance to $v_{k+1,2i-1}$ and $v_{k+1,2i+1}$ (the peak of the triangle in Figure \ref{fig:figure1}). Define the quantities
\begin{equation}\label{eq:netparameters}
r_0 = 1, \ \ C^*=2, \ \ \xi_1 = 2^{-1/s},\ \ \xi_2 = \frac{1+2^{-1/s}}{2},\ \ \rho_0 = 1,\ \ \rho_k= 2^{-k/s}\frac{(k+1+n_0)^{q}}{(2+n_0)^{q}}.
\end{equation}
For each $k\geq 0$ and $v\in V_k$, define
\[ \alpha_{k,v} := \inf_{\ell} \sup_{x\in V_{k+1}\cap B(v,30A^*\rho_kr_0)} \frac{\mathop\mathrm{dist}\nolimits(x,\ell)}{\rho_{k+1}r_0},\]
where the infimum is taken over all lines $\ell$ in $\mathbb{R}^2$ and $A^*$ is as in \textsection\ref{sec:flat}. Let $\ell_{k,v}$ be a line $\ell$, which realizes the number $\alpha_{k,v}$.
\begin{lem}\label{lem:propnets}
There exist choices of $s$ and $n_0$ so that the following properties hold.
\begin{enumerate}
\item For all $k\geq 0$ and $i\in\{1,\dots N_k\}$, we have $|v_{k,i} - v_{k,i+1}| = \rho_k$.
\item The sequence $\{(V_k,\rho_k)\}_{k\geq 0}$ satisfies (V0)--(V5) with the parameters given in (\ref{eq:netparameters}).
\item For all $k\geq 0$ and $v\in V_k$, we have $\alpha_{k,v} \leq \alpha_0$, where $\alpha_0$ is as in Definition 2.4.
\item For all $k\geq 0$ and $i\in\{1,\dots,N_k\}$,
\[ \mathsf{Flat}(k) = \{(v_{k,i},v_{k,i+1}) : i=1,\dots,2^{k}\}\quad\text{and}\quad V_{k+1,i}(v_{k,i},v_{k,i+1}) = \{v_{k,i},v_{k+1,2i},v_{k,i}\}.\]
\end{enumerate}
\end{lem}
\begin{proof}
For (1), we work by induction. The claim is true for $k=0$ by the choice of points $v_{0,1}$ and $v_{0,2}$. Assume the claim is true for some $k\geq 0$. By the Pythagorean theorem,
\[ |v_{k+1,2i-1}-v_{k+1,2i}| = |v_{k,i} - v_{k+1,2i-1}| =(4^{-1}+t_1^2)^{1/2}|v_{k,i}-v_{k,i+1}| = (4^{-1}+t_1^2)^{1/2}\rho_{k} = \rho_{k+1}.\]
In similar fashion, one can compute $|v_{k+1,2i}-v_{k+1,2i+1}|$ and the proof of (1) is complete.
Claim (3) is immediate from Remark \ref{rem:flatsnowflakes} by taking $s$ sufficiently close to $1$ and $n_0$ sufficiently large.
For (V0), we have
\[ \frac{\rho_{k+1}}{\rho_k} = 2^{-1/s}\left ( \frac{k+2+n_0}{k+1+n_0} \right )^q.\]
Clearly, $\rho_{k+1} > \xi_1\rho_{k}$. On the other hand, since $2^{-1/s} < \xi_2 < 1$, if $n_0$ is sufficiently large, then $\rho_{k+1} \leq \xi_2\rho_k$. Properties (V1), (V2), and (V5) are immediately satisfied by our construction. For (V4), fix a point $v_{k+1,2i} \in V_{k+1}\setminus V_k$. By (1), we have $|v_{k+1,2i} - v_{k+1,2i+1}| = \rho_{k+1}$ and (V4) is satisfied.
For (V3), claim (3), and claim (4), we apply induction on $k$. For $k=0$ (V3) is immediate by the choice of parameters. For claim (3), we note that $\alpha_{0,v}=0$ for all $v\in V_0$, since $V_0$ contains only 2 points. For the same reason, claim (4) is satisfied when $k=0$.
To show (V3), we note by (3) that the closest point of $V_{k+1}$ to $v_{k+1,2i}$ are the points $v_{k,i}$ and $v_{k,i+1}$. Therefore,
\[ \min_{v\in V_{k+1} \setminus \{v_{k+1,2i}\}} |v-v_{k+1,2i}| = |v_{k+1,2i} - v_{k,i}| = \rho_{k+1}.\]
Similarly, by (3), the closest point of $V_{k+1}$ to $v_{k+1,2i+1}=v_{k,i+1}$ are the points $v_{k+1,2i}$ and $v_{k+1, 2(i+1)}$ (or only one point of these two if $i =0$ or $i = 2^{k}\}$) and the above inequality also applies.
Finally, to show (4), we apply (3) and the arguments in the proof of (V3). Namely, if $v_{k,i} \in V_k$ with $k\in\{2,2^k-1\}$, then $\alpha_{v_{k,i}} < \alpha_0$ and $v_{k,i}$ lies between points $v_{k,i-1}$ and $v_{k,i+1}$. Therefore,
\[ \mathsf{Flat}(k) = \{(v_{k,i},v_{k,i+1}) : i=1,\dots,2^{k}\}.\]
Furthermore, the only point of $V_{k+1}$ lying between $v_{k,i}$ and $v_{k,i+1}$ is $v_{k+1,2i}$. Thus,
\[ V_{k+1,i}(v_{k,i},v_{k,i+1}) = \{v_{k,i},v_{k+1,2i},v_{k,i}\}. \qedhere\]
\end{proof}
We now show that there does not exist a $(1/s)$-H\"older map $f:[0,1] \to \mathbb{R}^2$ whose image contains $\bigcup_{k\geq 0}V_k$. Contrary to the claim, assume that such a map $f$ exists and let $H$ be its H\"older constant. For each $v_{k,i}\in V_k$, fix $w_{k,i} \in [0,1]$ such that $f(w_{k,i})=v_{k,i}$. Then
\[ |w_{k,i}-w_{k,j}| \gtrsim_{H,s} |v_{k,i}-v_{k,j}|^s \gtrsim_{n_0,q,s} 2^{-k}(k+1)^{sq}.\]
Therefore,
\begin{align*}
1 \geq 2^k \min_{i=1,\dots,2^{k}+1}|w_{k,i}-w_{k,j}| \gtrsim_{H,s,n_0,q} (k+1)^{sq},
\end{align*}
which diverges as $k\to\infty$ and we reach a contradiction.
It remains to check (\ref{eq:sharp}). By Lemma \ref{lem:propnets}, it suffices to show that
\begin{align*}
\sum_{k=0}^{\infty}\sum_{i=1}^{2^{k}} \tau_s(k,v_{k,i},v_{k,i+1})^p |v_{k,i}-v_{k,i+1}|^s < \infty.
\end{align*}
By the Mean Value Theorem,
\begin{align*}
\tau_s(k,v_{k,i},v_{k,i+1}) &= \frac{|v_{k,i}-v_{k+1,2i}|^s + |v_{k+1,2i}-v_{k,i+1}|^s-|v_{k,i}-v_{k,i+1}|^s}{|v_{k,i}-v_{k,i+1}|^s}\\
&= 2\frac{(k+2+n_0)^{sq} - (k+1+n_0)^{sq}}{(k+2+n_0)^{sq}}\\
&\lesssim_{n_0,s,q}\frac1{k+1}.
\end{align*}
Finally, since $sq-p<-1$,
\begin{align*}
\sum_{k=0}^{\infty}\sum_{i=1}^{2^{k}} \tau_s(k,v_{k,i},v_{k,i+1})^p |v_{k,i}-v_{k,i+1}|^s &\lesssim_{n_0,s,q} \sum_{k=0}^{\infty} \frac{2^k}{ (k+1)^{p}}(2^{-k/s}(k+1)^q)^s\\
&= \sum_{k=0}^{\infty}(k+1)^{sq-p}<\infty.
\end{align*}
\addtocontents{toc}{\protect\vspace{10pt}}
|
2,869,038,154,764 | arxiv | \section{Introduction}
Throughout, let $F$ be a local non-archimedean field, $\OO$ its ring of integers, and let $G=\GL_r(F)$ be the general linear group of rank $r$
with center $Z$.
Let $\K=\GL_r(\OO)$ be the standard maximal compact subgroup of $G$ and for any $n\ge1$
let $K(n)=K_r(n)$ be the principal congruence subgroup, i.e.,
the kernel of the canonical map $\K\rightarrow\GL_r(\OO/\varpi^n\OO)$ where $\varpi$ is a uniformizer of $F$.
Denote by $\ball(n)$ the ball
\[
\ball(n)=\{g\in G:\norm{g},\norm{g^{-1}}\le q^n\}
\]
where $q$ is the size of the residue field of $F$ and $\norm{g}=\max\abs{g_{i,j}}_F$
where $g_{i,j}$ are the entries of $g$. Thus, $\{\ball(n):n\ge1\}$ is an open cover of $G$ by compact sets.
The purpose of this short paper is to give a new proof of the following result.\footnote{Throughout,
by a representation of $G$ we always mean a complex, smooth representation.}
\begin{theorem} \label{thm: main}
Let $(\pi,V)$ be a supercuspidal representation of $G$ and let $(\pi^\vee,V^\vee)$ be its contragredient.
Let $v\in V$, $v^\vee\in V^\vee$ and assume that $v$ and $v^\vee$ are fixed under $K(n)$ for some $n\ge1$.
Then the support of the matrix coefficient $g\mapsto (\pi(g)v,v^\vee)$ is contained in $Z\ball(c(r)n)$
where $c(r)$ is an explicit constant depending on $r$ only.
\end{theorem}
As explained in \cite{MR3001800}, the theorem is a direct consequence of the classification of irreducible
supercuspidal representations
by Bushnell--Kutzko \cite{MR1204652}, or more precisely, of the fact that every such representation is induced from
a representation of an open subgroup of $G$ which is compact modulo $Z$. This is in fact known for many other cases
of reductive groups over local non-archimedean fields and in these cases it implies the analogue
of Theorem \ref{thm: main}. We refer the reader to \cite{MR3001800} for more details.
In contrast, the proof given here is independent of the classification.
It is based on two ingredients.
The first, which is special to the general linear group,
is basic properties of local Rankin--Selberg integrals for $G\times G$ which were defined and studied by
Jacquet--Piatetski-Shapiro--Shalika. In particular, we use an argument of Bushnell--Henniart, originally used to give an upper
bound on the conductor of Rankin--Selberg local factors \cite{MR1462836}.
The second ingredient is Howe's result on the integrality of the formal degree with respect to a suitable Haar measure
\cite{MR0342645}, a result which was subsequently extended to any reductive group \cite{MR1702257, MR1159104, MR1471867}.
The two ingredients are linked by the fact, which also follows from properties of Rankin--Selberg integrals,
that the formal degree is essentially the conductor of $\pi\times\pi^\vee$,
a feature that admits a conjectural generalization for any reductive group \cite{MR2350057}.
(For another relation between formal degrees and support of matrix coefficients see \cite{MR1198303}.)
As explained in \cite{1504.04795, 1705.08191}, Theorem \ref{thm: main} is of interest for the problem of limit multiplicity.
I would like to thank Stephen DeBacker, Tobias Finis, Atsushi Ichino and Julee Kim for useful discussions
and suggestions. I am especially grateful to Guy Henniart for his input leading to Remark \ref{rem: Henn}.
\section{A variant for Whittaker functions}
It is advantageous to formulate a variant of Theorem \ref{thm: main} for the Whittaker model.
Throughout, fix a character $\psi$ of $F$ which is trivial on $\OO_F$ but non-trivial on $\varpi^{-1}\OO_F$.
Let $N$ be the subgroup of upper unitriangular matrices in $G$.
If $\pi$ is a generic irreducible representation of $G$, we write $\whit^\psi(\pi)$ for its Whittaker
model with respect to the character $\psi_N$ of $N$ given by $u\mapsto\psi(u_{1,2}+\dots+u_{r-1,r})$.
Recall that every irreducible supercuspidal representation of $G$ is generic \cite{MR0404534}.
Let $A$ be the diagonal torus of $G$. For all $n\ge1$ let $A(n)$ be the open subset
\[
A(n)=\{\diag(t_1,\dots,t_r)\in A:q^{-n}\le\abs{t_i/t_{i+1}}\le q^n,\ \ i=1,\dots,r-1\}.
\]
Clearly, $ZA(n)=A(n)$ and $A(n)$ is compact modulo $Z$.
\begin{theorem} \label{thm: main'}
There exists a constant $c=c(r)$ with the following property.
Let $\pi$ be an irreducible supercuspidal representation of $G$ with Whittaker model $\whit^\psi(\pi)$ and $n\ge1$.
Then the support of any $W\in\whit^\psi(\pi)^{K(n)}$ is contained in $NA(cn)\K$.
\end{theorem}
In order to prove Theorem \ref{thm: main'} we set some more notation.
For any function $W$ on $G$ let
\begin{align*}
M_W&=\sup\{\val(\det g):W(g)\ne0\text{ and }\norm{g_r}=1\},\\
m_W&=\inf\{\val(\det g):W(g)\ne0\text{ and }\norm{g_r}=1\},
\end{align*}
(including possibly $\pm\infty$) where $g_r$ is the last row of $g$ and $\norm{(x_1,\dots,x_r)}=\max\abs{x_i}$.
Recall that by a standard argument (cf. \cite[Proposition 6.1]{MR581582})
for any right $K(n)$-invariant and left $(N,\psi_N)$-equivariant function $W$ on $G$,
if $W(tk)\ne0$ for some $t=\diag(t_1,\dots,t_n)\in A$ and $k\in\K$ then $\abs{t_i/t_{i+1}}\le q^n$, $i=1,\dots,r-1$.
Hence, $m_W\ge -{r\choose2}n$.
For any $W\in\whit^\psi(\pi)$ let $\widetilde W\in\whit^{\psi^{-1}}(\pi^\vee)$ be given by
$\widetilde W(g)=W(w_r\,^tg^{-1})$
where $w_r=\left(\begin{smallmatrix}&&1\\&\iddots&\\1\end{smallmatrix}\right)$.
Let $t=t(\pi)$ be the order of the group of unramified characters $\chi$ of $F^*$ such that $\pi\otimes\chi\simeq\pi$.
Clearly, $t$ divides $r$.
Let $f=f(\pi\times\pi^\vee)\in\Z$ be the conductor of the pair $\pi\times\pi^\vee$ (see below).
Theorem \ref{thm: main'} is an immediate consequence of the following two results.
\begin{proposition} \label{prop: mwMW}
For any $0\ne W\in\whit^\psi(\pi)$ we have
\[
M_W+m_{\widetilde W}=r-t-f.
\]
In particular, if $W\in\whit^\psi(\pi)^{K(n)}$ then
\[
M_W\le{r\choose2}n+r-t-f.
\]
\end{proposition}
\begin{proposition} \label{prop: lwrbndf}
We have
\[
f\ge (r+1)r-2(t+v_q(t))
\]
where $v_q(t)$ is the maximal power of $q$ dividing $t$.
Moreover, $f$ is even if $q$ is not a square.
\end{proposition}
\begin{proof}[Proof of Proposition \ref{prop: mwMW}]
The argument is inspired by \cite{MR1462836}.
We may assume without loss of generality that $\pi$ is unitarizable.
For any $\Phi\in\mathcal{S}(F^r)$ consider the local Rankin--Selberg integral
\[
A^{\psi}(s,W,\Phi)=\int_{N\bs G}\abs{W(g)}^2\Phi(g_r)\abs{\det g}^s\ dg,
\]
a Laurent series in $x=q^{-s}$ which represents a rational function in $x$ \cite{MR701565}.
Note that if $\Phi(0)=0$ then $A^{\psi}(s,W,\Phi)$ is a Laurent polynomial in $x$ since $W$ is compactly
supported modulo $ZN$.
Also note that for any $\lambda\in F^*$ we have
\[
A^{\psi}(s,W,\Phi(\lambda\cdot))=\abs{\lambda}^{-rs}A^\psi(s,W,\Phi).
\]
Recall the functional equation (\cite[Theorem 2.7 and Proposition 8.1]{MR701565} together with \cite{MR1703873})
\[
q^{f(\frac12-s)}(1-q^{-ts})A^{\psi}(s,W,\Phi)=(1-q^{t(s-1)})A^{\psi^{-1}}(1-s,\widetilde W,\tilde\Phi)
\]
where
\[
\tilde\Phi(x)=\int_{F^n}\Phi(y)\psi(x\,^ty)\ dy
\]
is the Fourier transform of $\Phi$ and $f\in\Z$ is the conductor.
Now let $\Phi_0$ be the characteristic function of the standard lattice $\{\xi\in F^r:\norm{\xi}\le1\}$ and set
$A^{\psi}_0(s,W)=A^\psi(s,W,\Phi_0)$. Then $\tilde\Phi_0=\Phi_0$ and we obtain
\[
q^{f(\frac12-s)}(1-q^{-ts)})A^{\psi}_0(s,W)=
(1-q^{t(s-1)})A^{\psi^{-1}}_0(1-s,\widetilde W).
\]
Let $\Phi_1=\Phi_0-\Phi_0(\varpi^{-1}\cdot)$ be the characteristic function of the $\K$-invariant set
$\{\xi\in F^r:\norm{\xi}=1\}$ of primitive vectors. Then
$A^{\psi}_1(s,W):=A^\psi(s,W,\Phi_1)=(1-q^{-rs})A^{\psi}_0(s,W)$ and thus,
\[
q^{f(\frac12-s)}\frac{1-q^{-ts}}{1-q^{-rs}}A^{\psi}_1(s,W)=
\frac{1-q^{t(s-1)}}{1-q^{r(s-1)}}A^{\psi^{-1}}_1(1-s,\widetilde W).
\]
Recall that $A^\psi_1(s,W)$ is a Laurent polynomial $P^\psi_w(x)$ in $x=q^{-s}$. We get an equality of Laurent
polynomials
\begin{equation} \label{eq: polyeq}
q^{\frac12f}x^{f}\frac{1-x^{t}}{1-x^r}P^\psi_W(x)=
\frac{1-y^{t}}{1-y^r}P^{\psi^{-1}}_{\widetilde W}(y)
\end{equation}
where $y=q^{-1}x^{-1}$. Note that the fact that $P^\psi_W(x)$ is divisible by $\frac{1-x^r}{1-x^{t}}$
amounts to saying that the integral
\[
\int_{g\in N\bs G:\norm{g_r}=1,\val(\det g)\equiv a\pmod{r/t}}\abs{W(g)}^2\ dg
\]
is independent of $a$, which in turn follows from (and in fact, equivalent to) the fact that $W$ is orthogonal to
$W\abs{\det}^{\frac{2\pi\textrm{i}j}{r\log q}}$ unless $j$ is divisible by $r/t$.
Also note that $P^\psi_W$ has non-negative coefficients and the degree of $P^\psi_W$ is $M_W$.
Likewise, the degree of $P^\psi_W(y)$ as a Laurent polynomial in $x$ (i.e., the order of pole of $P^\psi_W$ at $0$)
is $-m_W$.
Comparting degrees in \eqref{eq: polyeq} we obtain Proposition \ref{prop: mwMW}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop: lwrbndf}]
By an argument based on properties of Rankin-Selberg integrals,
the formal degree, with respect to a suitable choice of Haar measure, is related to $f$ by the formula
\[
d_\pi=\frac{t}{r}q^{\frac12 f}\frac{1-q^{-1}}{1-q^{-t}}
\]
(\cite[Theorem 2.1]{MR3649356}).
Comparing it to the formal degree of the Steinberg representation $\St$ with respect to the same measure we get
\[
\frac{d_\pi}{d_{\St}}=t\cdot q^{t-{r+1\choose 2}+\frac12 f}\cdot\frac{q^r-1}{q^t-1}.
\]
On the other hand, $\frac{d_\pi}{d_{\St}}$ (which is independent of the choice of Haar measure)
is a (positive) integer (cf.~\cite{MR0342645}, \cite{MR607380}, \cite[Appendice 3]{MR743063}).
The lemma follows.
\end{proof}
\begin{remark} \label{rem: Henn}
As explained to me by Guy Henniart, the lower bound in Proposition \ref{prop: lwrbndf} is not sharp.
In fact, a precise formula for $f(\pi\times\pi^\vee)$, and more generally for
$f(\pi_1\times\pi_2)$ for an arbitrary pair of irreducible supercuspidal representations $\pi_i$
of $\GL_{n_i}(F)$, $i=1,2$ is given in \cite{MR1606410}.
The expression is in terms of the Bushnell--Kutzko description of supercuspidal representations.
Using this, one can sharpen Proposition \ref{prop: lwrbndf} as follows.
First note that $f(\pi\times\pi^\vee)$ is insensitive to twisting $\pi$ by a character.
Suppose that $\pi$ is minimal under twists, i.e., $f(\pi\times\chi)\ge f(\pi)$ for any character
$\chi$ of $F^*$ where $f(\pi)$ is the conductor of $\pi$.
Then by \cite[Lemma 3.5]{MR3711826} and its proof we have $f(\pi\times\pi^\vee)\ge r^2-t+\frac12r(f(\pi)-r)$
with equality if $f(\pi)=r$, i.e., if $\pi$ has level $0$ (that is, if $\pi$ has a non-zero vector invariant under $K(1)$,
in which case $t=r$). Thus, if $f(\pi)>r+1$ then we get $f(\pi\times\pi^\vee)\ge r(r+1)-t$.
On the other hand, we always have $f(\pi)\ge r$ \cite[(5.1)]{MR882297} and
if $f(\pi)=r+1$, i.e., if $\pi$ is epipelagic then $t=1$ and $f(\pi\times\pi^\vee)=(r-1)(r+2)$.
More generally, if $f(\pi)$ is coprime to $r$, i.e., if $\pi$ is a Carayol representation, then $t=1$ and
$f(\pi\times\pi^\vee)=(r-1)(f(\pi)+1)$. For instance, this follows from \cite[(6.1.1),(6.1.2)]{MR1606410} and [ibid., Theorem 6.5(i)]
where in its notation we have $n=e=d=r$ and $\mathfrak{c}(\beta_1)=m(r-1)$ --
cf. second paragraph (``minimal case'') of [ibid., p.~727] with $k=m$, $e(\gamma)=d=r$.
To conclude, for any supercuspidal $\pi$ we have
\begin{equation} \label{eq: sharp}
f(\pi\times\pi^\vee)\ge r(r+1)-2t
\end{equation}
with equality if and only if $\pi$ is a twist of either a representation of level $0$ or an epipelagic representation.
Alternatively, one could infer \eqref{eq: sharp} and the conditions for equality
from the local Langlands correspondence for $G$ (cf.~\cite{MR2730575}).
Details will be appear in the upcoming thesis of Kilic.
The results of \cite{MR1606410} also give that $f(\pi\times\pi^\vee)$ is even.
(Details will be given elsewhere.) In the Galois side this follows from a result of Serre \cite{MR0321908}.
I am grateful to Guy Henniart for providing me this explanation and allowing me to include it here.
For our purpose, the precise lower bound on $f(\pi\times\pi^\vee)$ is immaterial -- it is sufficient to have
the inequality $f\ge0$ (or even, $f\ge c'n$ for some fixed $c'$ depending only $r$).
The point is that we do not use either the local Langlands correspondence or the classification of supercuspidal
representations (but of course, we do use the non-trivial analysis of \cite{MR0342645}
which depends on \cite{MR0492088}).
Nonetheless, it would be interesting to prove \eqref{eq: sharp} (and perhaps the evenness of $f(\pi\times\pi^\vee)$)
without reference to the classification or to the local Langlands correspondence.
\end{remark}
\section{Proof of main result}
In order to deduce Theorem \ref{thm: main} from Theorem \ref{thm: main'} we make the argument of
\cite[Proposition 2.11]{MR3267120} effective in the case of $G=\GL_r$.
For any $t=\diag(t_1,\dots,t_r)\in A$ consider the compact open subgroup
\[
N(t)=N\cap t\K t^{-1}=\{u\in N:\val(u_{i,j})\ge\val(t_i)-\val(t_j)\text{ for all }i<j\}
\]
of $N$. Set
\[
t_0=\diag(\varpi,\varpi^2,\dots,\varpi^{2^{r-1}})\in A.
\]
\begin{proposition} \label{prop: effstab}
Let $f$ be a compactly supported continuous funciton on $G$.
Assume that $f$ is bi-invariant under $K_{r-1}(n)$ for some $n\ge1$.
Then
\[
\int_Nf(u)\psi_N(u)\ du=\int_{N(t_0^n)}f(u)\psi_N(u)\ du.
\]
In particular, $\int_Nf(u)\psi_N(u)\ du=0$ if $f$ vanishes on $\ball((2^{r-1}-1)n)$.
\end{proposition}
We first need some more notation.
Write $N=U\ltimes V$ where $U=N_{r-1}$ is the group of unitriangular matrices in $\GL_{r-1}$
embedded in $\GL_r$ in the upper left corner and $V$ is the unipotent radical
of the parabolic subgroup of type $(r-1,1)$, i.e., the $r-1$-dimensional abelian group
\[
V=\{u\in N:u_{i,j}=0\text{ for all }i<j<r\}.
\]
For any $\underline{\alpha}=(\alpha_1,\dots,\alpha_{r-1})\in F^{r-1}$ let $x(\underline{\alpha})$
be the $r\times r$-matrix that is the identity except for the $(r-1)$-th row which is $(\alpha_1,\dots,\alpha_{r-1}+1,0)$.
Of course $x$ is not a group homomorphism because of the diagonal entry.
For $t\in A$ write $U(t)=U\cap N(t)$ and $V(t)=V\cap N(t)$ so that $N(t)=U(t)\ltimes V(t)$.
The following is elementary.
\begin{lemma} \label{lem: aux123}
For any $n\ge1$ let $L(n)$ be the lattice of $F^{r-1}$ given by
\[
L(n)=\{(\alpha_1,\dots,\alpha_{r-1}):\val(\alpha_j)\ge n(2^{r-1}-2^{j-1}),\ j=1,\dots,r-1\}.
\]
Then
\begin{enumerate}
\item \label{part: canconj}
$x(\underline{\alpha}),x(\underline{\alpha})^u\in K_{r-1}(n)$ for any $\underline{\alpha}\in L(n)$ and $u\in U(t_0^n)$.
\item \label{part: charcan}
For any $v\in V$ we have
\[
\vol(L(n))^{-1}\int_{L(n)}\psi_N(v^{x(\underline{\alpha})})\ d\underline{\alpha}=
\begin{cases}\psi_N(v)&v\in V(t_0^n),\\0&\text{otherwise.}\end{cases}
\]
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Proposition \ref{prop: effstab}]
We prove the statement by induction on $r$. The case $r=1$ is trivial.
For the induction step, note that if $f$ is bi-invariant under $K_{r-1}(n)$ then
the function $h=\int_Vf(\cdot v)\psi_N(v)\ dv$ on $\GL_{r-1}$ is bi-$K_{r-2}(n)$-invariant
(since $\GL_{r-2}$ normalizes the character $\psi_N\rest_V$).
Therefore, by induction hypothesis we have
\begin{multline*}
\int_Nf(u)\psi_N(u)\ du=\int_Uh(u)\psi_N(u)\ du=
\int_{U(t_0^n)}h(u)\psi_N(u)\ du\\=\int_V\int_{U(t_0^n)}f(uv)\psi_N(uv)\ du\ dv.
\end{multline*}
Now we use Lemma \ref{lem: aux123}.
By part \ref{part: canconj}, Since $f$ is bi-$K_{r-1}(n)$-invariant, for any $\underline{\alpha}\in L(n)$
we can write the above as
\[
\int_V\int_{U(t_0^n)}f(ux(\underline{\alpha})vx(\underline{\alpha})^{-1})\psi_N(uv)\ du\ dv=
\int_V\int_{U(t_0^n)}f(uv)\psi_N(uv^{x(\underline{\alpha})})\ du\ dv.
\]
Averaging over $\underline{\alpha}\in L(n)$ and using part \ref{part: charcan},
we may replace the integration over $V$ by integration over $V(t_0^n)$. This yields the induction step.
\end{proof}
\begin{comment}
\begin{proposition} \label{prop: effstab}
Suppose that $f\in C_c(G//K(n))$ for some $n\ge1$. Then
\[
\int_Nf(u)\psi_N(u)\ du=\int_{N(t_0^n)}f(u)\psi_N(u)\ du.
\]
In particular, $\int_Nf(u)\psi_N(u)\ du=0$ if $f$ vanishes on $\ball((2^{r-1}-1)n)$.
\end{proposition}
We first need some more notation.
Write $N=C_1\cdots C_{r-1}$ where $C_j$ is the $j$-dimensional abelian group
\[
C_j=\{u\in N:u_{i,k}=0\text{ for all }i<k\ne j+1\}
\]
(i.e., the $j+1$-st column).
Also, let $N_i=C_1\cdots C_i$, $i=0,\dots,r-1$, i.e., the group of unitriangular matrices in $\GL_{i+1}$
embedded in $\GL_r$ in the upper left corner and $N^i=C_{i+1}\cdots C_{r-1}$ so that
$N=N_i\ltimes N^i$.
For any $i=1,\dots,r$ and $\underline{\alpha}=(\alpha_1,\dots,\alpha_i)\in F^i$ let $x_i(\underline{\alpha})$
be the matrix which is the identity except for the $i$-th row which is $(\alpha_1,\dots,\alpha_i+1,0,\dots,0)$.
Of course $x_i$ is not a group homomorphism because of the diagonal entry.
For $t\in A$ write
\[
N_i(t)=N_i\cap N(t),\ N^i(t)=N^i\cap N(t),\ C_i(t)=C_i\cap N(t),\ i=0,\dots,r-1.
\]
Note that
\[
N(t)=N_i(t)\ltimes N^i(t),\ N_i(t)=C_1(t)\cdots C_i(t),\ N^i(t)=C_{i+1}(t)\cdots C_{r-1}(t).
\]
The following is elementary.
\begin{lemma} \label{lem: aux123}
For any $i=1,\dots,r-1$ and $n\ge1$ let $R_i(n)$ be the lattice of $F^i$ given by
\[
R_i(n)=\{(\alpha_1,\dots,\alpha_i):\val(\alpha_j)\ge n(2^i-2^{j-1}),\ j=1,\dots,i\}.
\]
Then
\begin{enumerate}
\item \label{part: canconj}
$x_i(\underline{\alpha}),x_i(\underline{\alpha})^u\in K(n)$ for any $\underline{\alpha}\in R_i(n)$ and $u\in N_{i-1}(t_0^n)$.
\item \label{part: charcan}
For any $v\in C_i$ we have
\[
\vol(R_i(n))^{-1}\int_{R_i(n)}\psi_N(v^{x_i(\underline{\alpha})})\ d\underline{\alpha}=
\begin{cases}\psi_N(v)&v\in C_i(t_0^n),\\0&\text{otherwise.}\end{cases}
\]
\end{enumerate}
\end{lemma}
\begin{proof}[Proof of Proposition \ref{prop: effstab}]
We prove by induction that by indcution that for any $i=0,\dots,r-1$ we have
\[
\int_Nf(u)\psi_N(u)\ du=\int_{N_i(t_0^n)\ltimes N^i}f(u)\psi_N(u)\ du.
\]
The case $i=0$ is trivial while the case $i=r-1$ is the statement of the lemma.
For the induction step, suppose that $0\le i<r-1$ and write
\[
\int_{N_{i-1}(t_0^n)\ltimes N^{i-1}}f(u)\psi_N(u)\ du=
\int_{N_{i-1}(t_0^n)}\int_{N^{i-1}}f(uv)\psi_N(uv)\ du\ dv.
\]
By the $K(n)$-bi-invariance of $f$ and Lemma \ref{lem: aux123} part \ref{part: canconj}, for any $\underline{\alpha}\in R_i(n)$
we can write the above as
\[
\int_{N_{i-1}(t_0^n)}\int_{N^{i-1}}f(ux_i(\underline{\alpha})vx_i(\underline{\alpha})^{-1})\psi_N(uv)\ du\ dv=
\int_{N_{i-1}(t_0^n)}\int_{N^{i-1}}f(uv)\psi_N(uv^{x_i(\underline{\alpha})})\ du\ dv
\]
since $x_i(\underline{\alpha})$ normalizes $N^{i-1}$. Note that $x_i(\underline{\alpha})$ normalizes $\psi_N\rest_{N^i}$.
Therefore, we can write the above as
\begin{multline*}
\int_{N_{i-1}(t_0^n)}\int_{C_i}\int_{N^i}f(uvw)\psi_N(uv^{x_i(\underline{\alpha})}w^{x_i(\underline{\alpha})})\ du\ dv\ dw\\=
\int_{N_{i-1}(t_0^n)}\int_{C_i}\int_{N^i}f(uvw)\psi_N(uv^{x_i(\underline{\alpha})}w)\ du\ dv\ dw.
\end{multline*}
Averaging over $\underline{\alpha}\in R_i(n)$ and using Lemma \ref{lem: aux123} part \ref{part: charcan},
we may replace the integration over $C_i$ by integration over $C_i(t_0^n)$. This yields the induction step.
\end{proof}
\end{comment}
Let $\Pi_{\psi}=\ind_N^G\psi$. For $\varphi\in\Pi_{\psi}$ and $\varphi^\vee\in\Pi_{\psi^{-1}}$ let
\[
(\varphi,\varphi^\vee)_{N\bs G}=\int_{N\bs G}\varphi(g)\varphi^\vee(g)\ dg.
\]
Also set,
\[
A^\circ(n)=A\cap\ball(n)=\{\diag(t_1,\dots,t_r)\in A:q^{-n}\le\abs{t_i}\le q^n,\ \ i=1,\dots,r\}.
\]
Proposition \ref{prop: effstab}, together with the argument of \cite[Proposition 2.12]{MR3267120}, which
was communicated to us by Jacquet, yield the following.
\begin{corollary} \label{cor: suppindmc}
There exists a constant $c$, depending only on $r$ with the following property.
Assume that $\varphi\in\Pi_\psi^{K(n)}$ and $\varphi^\vee\in\Pi_{\psi^{-1}}^{K(n)}$
are both supported in $NA^\circ(n)\K$ for some $n\ge1$.
Then $(\Pi_\psi(\cdot)\varphi,\varphi^\vee)_{N\bs G}$ is supported in the ball $\ball(cn)$.
\end{corollary}
Indeed, the function $M(g)=(\Pi_\psi(\cdot)\varphi,\varphi^\vee)_{N\bs G}$ is clearly bi-$K(n)$-invariant.
Let $f$ be a compactly supported bi-$K(n)$-invariant function on $G$. By Fubini's theorem
\begin{multline*}
\int_GM(g)f(g)\ dg=\int_G\int_{N\bs G}\varphi(xg)\varphi^\vee(x)f(g)\ dg\ dx=
\int_{N\bs G}\int_G\varphi(xg)\varphi^\vee(x)f(g)\ dg\ dx\\=
\int_{N\bs G}\int_G\varphi(g)\varphi^\vee(x)f(x^{-1}g)\ dg\ dx=
\int_{N\bs G}\int_{N\bs G}\varphi^\vee(x)\varphi(y)K_f(x,y)\ dy\ dx
\end{multline*}
where
\[
K_f(x,y)=\int_Nf(x^{-1}ny)\psi_N(n)\ dn.
\]
From Proposition \ref{prop: effstab} we infer that there exists $c$, depending only on $r$, such that
if $f$ vanishes on $\ball(cn)$ then $K_f(x,y)=0$ for all $x,y\in\ball(n)$. The corollary follows.
Finally, we prove Theorem \ref{thm: main}.
\begin{proof}[Proof of Theorem \ref{thm: main}]
Let $\pi$ be a supercuspidal irreducible representation of $G$.
Suppose that $W\in\whit^{\psi}(\pi)^{K(n)}$ and $W^\vee\in\whit^{\psi^{-1}}(\pi^\vee)^{K(n)}$.
By Theorem \ref{thm: main'}, both $W$ and $W^\vee$ are supported in $NA(cn)\K$ for suitable $c$.
Upon modifying $c$, we may write $W(g)=\int_ZW_0(zg)\omega_{\pi}^{-1}(z)\ dz$
(with $\vol(Z\cap\K)=1$) where $W_0\in\Pi_{\psi}^{K(n)}$ is supported in $NA^\circ(cn)\K$.
For instance we can take $W_0=W\one_X$ where $X$ is the set
\[
X=\{g\in G:0\le\val(\det g)<r\}.
\]
Similarly, write $W^\vee(g)=\int_ZW_0^\vee(zg)\omega_{\pi}(z)\ dz$ where $W_0^\vee\in\Pi_{\psi^{-1}}^{K(n)}$
is supported in $NA^\circ(cn)\K$.
Then up to a scalar
\[
(\pi(g)W,W^\vee)=\int_{ZN\bs G}W(xg)W^\vee(x)\ dx=
\int_Z(\Pi_\psi(zg)W_0,W_0^\vee)_{N\bs G}\ dz.
\]
The result therefore follows from Corollary \ref{cor: suppindmc}.
\end{proof}
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\providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR }
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|
2,869,038,154,765 | arxiv | \section{Introduction}
\label{sec:intro}
Metric fixed point theory is a very extensive area of analysis with various applications.
Many of the most important nonlinear problems of applied mathematics reduce to
finding solutions of nonlinear functional equations which can be formulated
in terms of finding the fixed points of a given nonlinear operator of an infinite
dimensional function space $X$ into itself. There is a classical general existence
theory of fixed points for mappings satisfying a variety of contractive conditions.
The first basic result is the Banach contraction principle.
\begin{thm}[Banach \cite{Banach}]
\label{thm:Banach}
Let $X$ be a complete metric space and $T:X\to X$ be a strict contraction,
that is, there exists $\alpha$, $0<\alpha<1$, such that
\begin{equation}\label{eqn:Banach contraction}
\forall x,y\in X,\quad d(Tx,Ty) \leq \alpha d(x,y).
\end{equation}
Then $T$ has a unique fixed point $z$, and $T^nx\to z$, for every $x\in X$.
\end{thm}
The Banach contraction principle is fundamental in fixed point theory.
It has been extended to some larger classes of contractive mappings
by replacing the strict contractive condition \eqref{eqn:Banach contraction}
by weaker conditions of various types; see, for example,
\cite{Boyd-Wong-1969,
Kannan-1969,
Singh-1969,
Ciric-1971,
Bianchini-1972,
Sehgal-1972,
Chatterjea-1972,
Hardy-Rogers-1973,
Ciric-1974,
Caristi-1976,
Ekeland-1974,
Subrahmanyam-1974,
Ciric-1981}.
A comparative study of some of these results have been made
by Rhoades \cite{Rhoades-1977}.
There are thousands of theorems which assure the existence of a fixed point of
a self-map $T$ of a complete metric space $X$. These theorems can be categorized
into different types, \cite{Suzuki-new-type-of-fp-thm-2009}. One type, and perhaps
the most common one, is called the Leader-type \cite{Leader-1983}: the mapping $T$
has a unique fixed point, and the fixed point can always be found by using
Picard iterates $\set{T^nx}$, beginning with some initial choice $x$ in $X$.
Most of the theorems belong to Leader-type.
For instance, \'Ciri\'c in \cite{Ciric-1974} defined the class of \emph{quasi-contractions}
on a metric space $X$ consisting of all mappings $T$ for which there
exists $\alpha$, $0<\alpha<1$, such that $d(Tx, Ty) \leq \alpha \mathbf{m}(x,y)$, for every $x,y\in X$, where
\begin{equation*
\mathbf{m}(x,y)=\max\set{d(x, y), d(x, Tx), d(y, Ty), d(x, Ty), d(y, Tx)}.
\end{equation*}
The results presented in \cite{Ciric-1974} show that the condition of
quasi-contractivity implies all conclusions of Banach contraction principle.
We remark that \'Ciri\'c's quasi-contraction is considered as the most general
among contractions listed in \cite{Rhoades-1977}.
Another interesting generalization of Banach contraction principle was
given, in 1969, by Meir and Keeler \cite{Meir-Keeler-1969}.
They defined \emph{weakly uniformly strict contraction} mappings and proved
a fixed point theorem that generalized the fixed point theorem of Boyd and Wong \cite{Boyd-Wong-1969}
and extended the principle to wider classes of maps than those covered in \cite{Rhoades-1977}.
\begin{dfn}\label{dfn:Meir-Keeler-contraction}
A mapping $T$ on a metric space $X$ is said to be a \emph{Meir-Keeler contraction}
(or a \emph{weakly uniformly strict contraction} \cite{Meir-Keeler-1969})
if, for every $\epsilon>0$, there exists $\delta>0$ such that
\begin{equation*
\forall\, x,y\in X,\quad
\epsilon \leq d(x,y) < \epsilon+\delta\ \Longrightarrow\
d(Tx,Ty)<\epsilon.
\end{equation*}
\end{dfn}
\begin{thm}[Meir and Keeler \cite{Meir-Keeler-1969}]
\label{thm:Meir-Keeler}
Let $X$ be a complete metric space and let $T$ be a Meir-Keeler contraction
on $X$. Then $T$ has a unique fixed point $z$, and $T^nx\to z$, for every $x\in X$.
\end{thm}
The Meir and Keeler's generalized version of Banach contraction principle
initiated a lot of work in this direction and led to some important contribution
in metric fixed point theory; see, for example,
\cite{Maiti-Pal-1978,
Park-Rhoades-1981,
Rao-Rao-1985,
Jachymski-1995,
Cho-Murthy-Jungck-2000}.
The following theorem of \'Ciri\'c \cite{Ciric-1981} and Matkowski \cite[Theorem 1.5.1]{Kuczma}
generalizes the above Meir-Keeler fixed point theorem.
\begin{dfn}
\label{dfn:Ciric-Matkowski contraction}
A mapping $T$ on a metric space $X$ is said to be
a \emph{\'Ciri\'c-Matkowski contraction} if $d(Tx,Ty)<d(x,y)$ for
every $x,y\in X$ with $x\neq y$, and, for every $\epsilon>0$, there exists $\delta>0$ such that
\begin{equation}\label{eqn:Ciric-Matkowski-contraction}
\forall x,y\in X, \quad
\epsilon< d(x,y) < \epsilon+\delta \Longrightarrow d(Tx,Ty) \leq \epsilon.
\end{equation}
\end{dfn}
Obviously, the class of \'Ciri\'c-Matkowski contractions contains the class
of Meir-Keeler contractions. As it is mentioned in \cite[Proposition 1]{Jachymski-1995},
it is easy to see that condition \eqref{eqn:Ciric-Matkowski-contraction} in Definition \ref{dfn:Ciric-Matkowski contraction} can be replaced by the following:
\begin{equation*
\forall\, x,y\in X,\quad
d(x,y) < \epsilon+\delta\ \Longrightarrow\
d(Tx,Ty)\leq \epsilon.
\end{equation*}
\begin{thm}[\'Ciri\'c \cite{Ciric-1981}, Matkowski \cite{Kuczma}]
\label{thm:Ciric-Matkowski}
Let $X$ be a complete metric space and let $T$ be a \'Ciri\'c-Matkowski contraction
on $X$. Then $T$ has a unique fixed point $z$, and $T^nx\to z$, for every $x\in X$.
\end{thm}
In 1995, Jachymski \cite[Theorem 2]{Jachymski-1995} replaced the distance function $d(x, y)$ in
the \'Ciri\'c-Matkowski theorem by the following:
\begin{equation*
\mathbf{m}(x, y) = \max \set{d(x, y), d(x, T x), d(y, Ty), [d(x, Ty) + d(y, T x)]/2}.
\end{equation*}
As in \cite{Proinov-2006}, we refer to this result of Jachymski as
the Jachymski-Matkowski theorem because it is equivalent to a result
of Matkowski \cite[Theorem 1]{Matkowski-1980}.
In 2006, extending \'Ciri\'c's quasi-contraction to a very general setting,
Proinov \cite{Proinov-2006} obtained the following remarkable fixed point theorem
generalizing Jachymski-Matkowski theorem.
\begin{dfn}
\label{dfn:contractive+asymptotically-regular}
A self-map $T$ of a metric space $X$ is said to be
\emph{contractive} \cite{Edelstein-62} if $d(Tx, Ty) < d(x,y)$,
for all $x,y\in X$ with $x \neq y$; it is called
\emph{asymptotically regular} \cite{Browder-1966} if
$d(T^nx, T^{n+1}x)\to0$, for each $x\in X$.
\end{dfn}
\begin{thm}[Proinov \cite{Proinov-2006}]
\label{thm:Proinov}
Let $T$ be a continuous and asymptotically regular self-map
of a complete metric space $X$. Fix $\gamma\geq0$, and define
\begin{equation}\label{eqn:Proinov-m(x,y)}
\mathbf{m}(x,y) = d(x, y) + \gamma[d(x, T x) + d(y, Ty)].
\end{equation}
Suppose $d(Tx, Ty) < \mathbf{m}(x, y)$ for all $x, y\in X$ with $x\neq y$,
and, for any $\epsilon>0$, there exists $\delta>0$ such that $\mathbf{m}(x, y) < \delta+\epsilon$
implies $d(T x, Ty)\leq \epsilon$.
Then $T$ has a unique fixed point $z$, and the Picard iterates of $T$
converge to $z$.
\end{thm}
After establishing a technical lemma in section \ref{sec:technical-lemma},
we present, in section \ref{sec:fixed-point-theorems}, a fixed point theorem
that generalizes the Proinov's Theorem \ref{thm:Proinov}.
We shall also discuss asymptotic contractions of Meir-Keeler type
in section \ref{sec:asymptotic-contractions}. The significance
of our results is their simple proofs despite their generality.
\medskip \noindent
\textbf{Convention.}
Since we are mainly concerned with Picard iterates $\set{T^nx}_{n=0}^\infty$ of a given self-map $T$,
it is more convenient to take $\mathbb{N}_0=\mathbb{N}\cup\set{0}$ as the indexing set of all sequences
in this paper.
\bigskip
\section{A Technical Lemma}
\label{sec:technical-lemma}
The lemma we present in this section is fundamental in our discussion.
It provides a criterion for sequences in metric spaces to be Cauchy.
As a result, it can be easily verified that, if $T$ is a contraction
of Meir-Keeler type, then the Picard iterates of $T$
are Cauchy sequences.
\begin{lem}
\label{lem:technical-lemma}
Let $\set{x_n}$ be a sequence in a metric space. If $d(x_n,x_{n+1})\to0$,
then the following condition implies that $\{x_n\}$ is a Cauchy sequence.
\begin{itemize}
\item for every $\epsilon>0$, there exists a sequence $\set{\nu_n}$
of nonnegative integers such that, for any two subsequences
$\{x_{p_n}\}$ and $\{x_{q_n}\}$, if\/ $\limsup d(x_{p_n},x_{q_n})\leq \epsilon$,
then, for some $N$,
\begin{equation}\label{eqn:d(xpn+nun,xqn+nun)<=e}
d(x_{p_n+\nu_n},x_{q_n+\nu_n}) \leq \epsilon, \qquad (n\geq N).
\end{equation}
\end{itemize}
\end{lem}
It should be mentioned that the following proof of the lemma actually stems from the work of
Geraghty in \cite{Geraghty-73}.
\begin{proof}
Assume, towards a contradiction, that $\set{x_n}$ is not a Cauchy sequence.
Then, there exists $\epsilon>0$ such that
\begin{equation}\label{eqn:negation-of-Cauchy}
\forall k\in\mathbb{N},\ \exists\, p,q\geq k,
\quad d(x_p,x_q)>\epsilon.
\end{equation}
For this $\epsilon$, let $\{\nu_n\}$ be the sequence of
nonnegative integers given by the assumption.
Since $d(x_n,x_{n+1})\to0$, there exist positive integers $k_1<k_2<\dotsb$ such that
\[
d(x_\ell,x_{\ell+1})<\frac1{n(\nu_n+1)}, \qquad (\ell\geq k_n).
\]
For each $k_n+\nu_n$, by \eqref{eqn:negation-of-Cauchy}, there exist
integers $s_n$ and $t_n$ such that $t_n>s_n\geq k_n+\nu_n$ and
$d(x_{s_n},x_{t_n})>\epsilon$. We let $t_n$ be the smallest such integer
so that $d(x_{s_n},x_{t_n-1})\leq\epsilon$.
Take $p_n=s_n-\nu_n$ and $q_n=t_n-\nu_n$. Then $q_n>p_n \geq k_n$, and
\[
d(x_{p_n+\nu_n},x_{q_n+\nu_n})>\epsilon
\quad \text{and} \quad
d(x_{p_n+\nu_n},x_{q_n+\nu_n-1}) \leq \epsilon.
\]
Using triangle inequality, we have, for every $n$,
\begin{equation*}
d(x_{p_n},x_{q_n})
\leq \frac{\nu_n}{n(\nu_n+1)}+d(x_{p_n+\nu_n},x_{q_n+\nu_n-1}) +\frac{\nu_n+1}{n(\nu_n+1)}.
\end{equation*}
This implies that $\limsup d(x_{p_n},x_{q_n}) \leq \epsilon$. Since
$d(x_{p_n+\nu_n},x_{q_n+\nu_n})>\epsilon$, for every $n$,
we get a contradiction.
\end{proof}
\begin{rem}
In the above proof of the lemma, if we
apply the triangle inequality for the second time, we get
\[
\epsilon < d(x_{p_n+\nu_n},x_{q_n+\nu_n})
\leq \frac{\nu_n}{n(\nu_n+1)}+d(x_{p_n},x_{q_n}) +\frac{\nu_n}{n(\nu_n+1)}.
\]
This implies that $\epsilon\leq\liminf d(x_{p_n},x_{q_n})$.
Hence we have $d(x_{p_n},x_{q_n})\to\epsilon$.
\end{rem}
Using the following theorem, and its successive corollary, we will be able to give
very simple proofs for theorems mentioned in section \ref{sec:intro}.
\begin{thm}
\label{thm:main}
Let $X$ be a metric space and let $\{x_n\}$ be a sequence in $X$.
Suppose $\mathbf{m}$ is a nonnegative function on $X\times X$ such that,
for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$,
\begin{equation}\label{eqn:limsup m <= limsup d}
\limsup_{n\to\infty} \mathbf{m}(x_{p_n},x_{q_n})
\leq \limsup_{n\to\infty} d(x_{p_n},x_{q_n}).
\end{equation}
If $d(x_n,x_{n+1})\to0$, then the following condition implies that
$\set{x_n}$ is Cauchy.
\begin{itemize}
\item for every $\epsilon>0$, there exists a sequence $\set{\nu_n}$
of nonnegative integers such that, for any two subsequences
$\{x_{p_n}\}$ and $\{x_{q_n}\}$, if\/ $\limsup \mathbf{m}(x_{p_n},x_{q_n})\leq \epsilon$,
then, for some $N\in\mathbb{N}$,
\begin{equation*}
d(x_{p_n+\nu_n},x_{q_n+\nu_n}) \leq \epsilon, \qquad (n\geq N).
\end{equation*}
\end{itemize}
\end{thm}
\begin{proof}
Let $\epsilon>0$ and let $\{\nu_n\}$ be the sequence of nonnegative integers given by
the assumption. Let $\{x_{p_n}\}$ and $\{x_{q_n}\}$ be two subsequences with
$\limsup d(x_{p_n},x_{q_n})\leq \epsilon$. Then $\limsup \mathbf{m}(x_{p_n},x_{q_n})\leq \epsilon$,
and thus \eqref{eqn:d(xpn+nun,xqn+nun)<=e} holds. All conditions in
Lemma \ref{lem:technical-lemma} are fulfilled and so the sequence
is Cauchy.
\end{proof}
If $\{\nu_n\}$ is a constant sequence, e.g.\ $\nu_n=\nu$ for all $n$,
then we get the following result which is of particular importance.
\begin{cor}
\label{cor:main}
Suppose the function $\mathbf{m}$ satisfies \eqref{eqn:limsup m <= limsup d} and
$d(x_n,x_{n+1})\to0$. Then each of the following conditions implies that
$\set{x_n}$ is Cauchy.
\begin{enumerate}[\upshape(i)]
\item \label{item:cor:main:nu=0}
for every $\epsilon>0$ and for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$,
if \newline
$\limsup \mathbf{m}(x_{p_n},x_{q_n}) \leq \epsilon$, then, for some $N$,
\begin{equation*}
d(x_{p_n},x_{q_n}) \leq \epsilon, \qquad (n\geq N).
\end{equation*}
\item \label{item:cor:main:nu=1}
for every $\epsilon>0$ and for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$,
if \newline
$\limsup \mathbf{m}(x_{p_n},x_{q_n}) \leq \epsilon$, then, for some $N$,
\begin{equation*}
d(x_{p_n+1},x_{q_n+1}) \leq \epsilon, \qquad (n\geq N).
\end{equation*}
\item \label{item:cor:main:nu}
for every $\epsilon>0$, there exists $\nu\in\mathbb{N}$
such that, for any two subsequences
$\{x_{p_n}\}$ and $\{x_{q_n}\}$, if\/ $\limsup \mathbf{m}(x_{p_n},x_{q_n})\leq \epsilon$,
then, for some $N\in\mathbb{N}$,
\begin{equation*}
d(x_{p_n+\nu},x_{q_n+\nu}) \leq \epsilon, \qquad (n\geq N).
\end{equation*}
\end{enumerate}
\end{cor}
\bigskip
\section{Fixed Point Theorems
\label{sec:fixed-point-theorems}
In this section, using Theorem \ref{thm:main}, we present our fixed point theorem.
\begin{lem}
\label{lem:equiv-conditions-m-contractive-sequence}
If $\set{x_n}$ is a sequence in a metric space $X$ and
$\mathbf{m}$ is a nonnegative function on $X\times X$, then
the following statements are equivalent:
\begin{enumerate}[\upshape(i)]
\item \label{item:it-is-m-contractive-sequence}
for every $\epsilon>0$, there exists $\delta>0$ and $N\in\mathbb{N}_0$ such that
\begin{equation}\label{eqn:m-contractive-sequence}
\forall p,q\geq N, \quad
\mathbf{m}(x_p,x_q) < \epsilon+\delta \Longrightarrow d(x_{p+1},x_{q+1}) \leq \epsilon.
\end{equation}
\item \label{item:m-contractive-sequence-in-term-of-pnqn}
for every $\epsilon>0$, and for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$,
if \newline $\limsup \mathbf{m}(x_{p_n},x_{q_n})\leq \epsilon$ then,
for some $N$,
\[
d(x_{p_n+1},x_{q_n+1}) \leq \epsilon, \qquad (n\geq N).
\]
\end{enumerate}
\end{lem}
\begin{proof}
$\eqref{item:it-is-m-contractive-sequence}
\Rightarrow \eqref{item:m-contractive-sequence-in-term-of-pnqn}$:
Let $\epsilon>0$ and assume, for subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$,
we have $\limsup \mathbf{m}(x_{p_n},x_{q_n})\leq \epsilon$. There exists,
by \eqref{item:it-is-m-contractive-sequence},
some $\delta>0$ and $N\in\mathbb{N}_0$ such that \eqref{eqn:m-contractive-sequence}
holds. Take $N_1\in\mathbb{N}_0$ such that $\mathbf{m}(x_{p_n},x_{q_n}) < \epsilon+\delta$
for $n\geq N_1$. Therefore, we have $d(x_{p_n+1},x_{q_n+1}) \leq \epsilon$,
for $n>\max\set{N,N_1}$.
$\eqref{item:m-contractive-sequence-in-term-of-pnqn}
\Rightarrow \eqref{item:it-is-m-contractive-sequence}$:
Assume, to get a contradiction, that \eqref{item:it-is-m-contractive-sequence}
fails to hold. Then there exist $\epsilon>0$ and subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$
such that
\[
\mathbf{m}(x_{p_n},x_{q_n}) < \epsilon+\frac1n \quad \text{and} \quad
\epsilon < d(x_{p_n+1},x_{q_n+1}).
\]
This contradicts \eqref{item:m-contractive-sequence-in-term-of-pnqn}
because $\limsup \mathbf{m}(x_{p_n},x_{q_n}) \leq \epsilon$.
\end{proof}
\begin{dfn}
\label{dfn:m-contractive-sequence}
Let $X$ be a metric space and let $\mathbf{m}$ be a nonnegative function on
$X\times X$. A sequence $\set{x_n}$ in $X$ is said to be
\emph{$\mathbf{m}$-contractive} if it satisfies one (and hence all) of the conditions in
Lemma \ref{lem:equiv-conditions-m-contractive-sequence}.
\end{dfn}
The following is a direct consequence of Corollary \ref{cor:main} and
the above lemma.
\begin{thm}
\label{thm:m-contractive-sequences-are-Cauchy}
Let $X$ be a metric space, $\{x_n\}$ be a sequence in $X$, and
$\mathbf{m}$ be a nonnegative function on $X\times X$ satisfying \eqref{eqn:limsup m <= limsup d}.
If $\{x_n\}$ is $\mathbf{m}$-contractive and $d(x_n,x_{n+1})\to0$, then $\{x_n\}$ is Cauchy.
\end{thm}
\begin{cor}
\label{cor:m-contractive-orbits}
Let $T$ be a self-map of a metric space $X$, and $\mathbf{m}$ be a nonnegative function on $X\times X$.
Suppose there exists a point $x\in X$ such that
\begin{enumerate}[\upshape(i)]
\item for any $\epsilon>0$, there exist $\delta>0$ and $N\in\mathbb{N}_0$ such that
\begin{equation}\label{eqn:m-contractive-orbits}
\forall p,q\geq N, \quad
\mathbf{m}(T^px, T^qx) < \delta+\epsilon \Longrightarrow d(T^{p+1} x, T^{q+1}x)\leq \epsilon,
\end{equation}
\item condition \eqref{eqn:limsup m <= limsup d} holds for any two subsequences
$\{x_{p_n}\}$ and $\{x_{q_n}\}$ of\/ $\{T^nx\}$.
\end{enumerate}
If $d(T^nx,T^{n+1}x)\to0$, then $\set{T^nx}$ is a Cauchy sequence.
\end{cor}
The requirement that $d(x_n,x_{n+1})\to0$ is essential in Lemma \ref{lem:technical-lemma}
and its subsequent results. It can, however, be replaced by other conditions.
\begin{prop}
\label{prop:contractive-sequences}
Let $\{x_n\}$ be a sequence in a metric space $X$ such that
\begin{equation}\label{eqn:contractive-sequences}
\begin{cases}
d(x_{n+1},x_{n+2}) \leq d(x_n,x_{n+1}), & (n\in\mathbb{N}_0), \\
d(x_{n+1},x_{n+2}) < d(x_n,x_{n+1}), & (\text{if $x_n\neq x_{n+1}$}).
\end{cases}
\end{equation}
If $\mathbf{m}$ satisfies \eqref{eqn:limsup m <= limsup d} and $\{x_n\}$ is $\mathbf{m}$-contractive,
then $d(x_n,x_{n+1})\to0$ and, hence, $\{x_n\}$ is Cauchy.
\end{prop}
For instance, if $T:X\to X$ is contractive
then \eqref{eqn:contractive-sequences} holds for $x_n=T^nx$.
\begin{proof}
If $x_m=x_{m+1}$, for some $m$, then $x_n=x_{n+1}$ for all $n\geq m$,
and there is nothing to prove. Assume that $x_n\neq x_{n+1}$ for all $n$.
Then $d(x_{n+1},x_{n+2})< d(x_n,x_{n+1})$, for every $n$, and thus
$d(x_n,x_{n+1}) \downarrow \epsilon$, for some $\epsilon\geq0$.
If $\epsilon>0$, take $p_n=n$ and $q_n=n+1$ and we have, by \eqref{eqn:limsup m <= limsup d},
\[
\limsup_{n\to\infty} \mathbf{m}(x_n,x_{n+1})
\leq \limsup_{n\to\infty} d(x_n,x_{n+1})=\epsilon.
\]
Therefore, $d(x_{n+1},x_{n+2}) \leq \epsilon$ for $n$ large.
This is a contradiction since $\epsilon<d(x_n,x_{n+1})$ for all $n$.
So $\epsilon=0$ and $d(x_n,x_{n+1})\to0$.
\end{proof}
We are now in a position to state and prove our fixed point theorem.
\begin{thm}
\label{thm:fixed-point-theorem}
Let $T$ be an asymptotically regular self-map of a metric space $X$, and
$\mathbf{m}$ be a nonnegative function on $X\times X$.
Suppose
\begin{enumerate}[\upshape(i)]
\item \label{item:m-contractive-in-pre-fixed-point-theorem}
for any $\epsilon>0$, there exist $\delta>0$ and $N\in\mathbb{N}_0$ such that
\begin{equation}\label{eqn:m-contractive-in-fixed-point-theorem}
\forall x,y\in X, \quad
\mathbf{m}(T^Nx, T^Ny) < \delta+\epsilon \Longrightarrow d(T^{N+1}x,T^{N+1}y)\leq \epsilon,
\end{equation}
\item \label{item:m-satisfies-in-pre-fixed-point-theorem}
for every $x\in X$, condition \eqref{eqn:limsup m <= limsup d}
holds for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$
of the sequence $\{T^nx\}$.
\end{enumerate}
Then the Picard iterates of $T$ are Cauchy sequences.
Moreover, if $X$ is complete, $T$ is continuous, and $d(T^nx,T^ny)\to0$,
for all $x,y\in X$, then the Picard iterates converge to a unique fixed point
of $T$.
\end{thm}
\begin{proof}
All conditions in Corollary \ref{cor:m-contractive-orbits} are satisfied by every
point $x$ in $X$. Hence $\{T^nx\}$ is Cauchy, for every $x$.
If $X$ is complete, there is $z\in X$ such that $T^nx\to z$. If $T$ is continuous,
then $Tz=z$. If $d(T^nx,T^ny)\to0$, for every $x,y\in X$, then $T$ has
at most one fixed point.
\end{proof}
We remark that, by Corollary \ref{cor:m-contractive-orbits}, it is enough to impose
condition \eqref{eqn:m-contractive-in-fixed-point-theorem} on some orbit $\set{T^nx}$ as in
\eqref{eqn:m-contractive-orbits} to conclude that $\set{T^nx}$ is Cauchy.
\begin{example}
Let $T$ be a self-map of $X$, and consider the following functions:
\begin{align*}
\mathbf{m}_1(x,y) & = \max\set{d(x,Tx),d(y,Ty)}, \tag{Bianchini \cite{Bianchini-1972}}\\
\mathbf{m}_2(x,y) & = [d(x,Ty)+d(y,Tx)]/2, \tag{Chatterjea \cite{Chatterjea-1972}} \\
\mathbf{m}_3(x,y) & = \max\set{d(x,y),d(x,Tx),d(y,Ty)},
\tag{Maiti and Pal \cite{Maiti-Pal-1978}} \\
\mathbf{m}_4(x,y) & = \max\set{d(x,y),d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)},
\tag{Ciric \cite{Ciric-1974}} \\
\mathbf{m}_5(x,y) & = \max\set{d(x,y),d(x,Tx),d(y,Ty),[d(x,Ty)+d(y,Tx)]/2},
\tag{Jachymski \cite{Jachymski-1995}}\\
\mathbf{m}_6(x,y) & = d(x,y) + \gamma[d(x,Tx)+ d(y,Ty)],\ \text{where $\gamma\geq0$ is fixed}.
\tag{Proinov \cite{Proinov-2006}}
\end{align*}
Choose a point $x\in X$ and set $x_n=T^nx$, $n\in\mathbb{N}_0$.
If $d(x_n,x_{n+1})\to0$, then,
for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$, we have
\begin{equation*}
\begin{cases}
\limsup\limits_{n\to\infty}\mathbf{m}_1(x_{p_n},x_{q_n})=0, & \\[1ex]
\limsup\limits_{n\to\infty}\mathbf{m}_2(x_{p_n},x_{q_n})
\leq \limsup\limits_{n\to\infty} d(x_{p_n},x_{q_n}), & \\[1ex]
\limsup\limits_{n\to\infty} \mathbf{m}_i(x_{p_n},x_{q_n})
= \limsup\limits_{n\to\infty} d(x_{p_n},x_{q_n}), & \text{for $3\leq i \leq 6$}.
\end{cases}
\end{equation*}
\end{example}
We now state and prove a generalization of
Proinov's Theorem \ref{thm:Proinov}. First, a couple of notations:
For a subset $E$ of a metric space $X$, denote by
$\operatorname{diam} E$ the diameter of $E$. If $T$ is a self-map
of $X$ and $x\in X$, for every positive integer $s\in \mathbb{N}$,
let $\mathcal{O}_s(x)=\set{T^nx: 0 \leq n \leq s}$.
For positive integers $s,t\in\mathbb{N}$ and real numbers $\alpha,\beta\in[0,\infty)$, define
a function $\mathbf{m}$ on $X\times X$ as follows:
\begin{equation}\label{eqn:m(x,y)-in-generalized-Proinov}
\mathbf{m}(x,y) = d(x,y) +
\alpha \operatorname{diam} \mathcal{O}_s(x) + \beta \operatorname{diam} \mathcal{O}_t(y).
\end{equation}
\begin{thm}
\label{thm:Generalized-Proinov}
Let $X$ be a complete metric space, and $T$ be a continuous and
asymptotically regular self-map of $X$. Define $\mathbf{m}$ by
\eqref{eqn:m(x,y)-in-generalized-Proinov}, and suppose
\begin{enumerate}[\upshape(i)]
\item \label{item:d(Tx,Ty)<m(x,y)-in-generalized-Proinov}
$d(Tx,Ty)<\mathbf{m}(x,y)$, for every $x,y\in X$ with $x\neq y$,
\item \label{item:m-contractive-in-generalized-Proinov}
for any $\epsilon>0$, there exist $\delta>0$ and $N\in\mathbb{N}_0$ such that
\begin{equation}\label{eqn:m-contractive-in-generalized-Proinov}
\forall x,y\in X, \quad
\mathbf{m}(T^Nx, T^Ny) < \delta+\epsilon \Longrightarrow d(T^{N+1}x,T^{N+1}y)\leq \epsilon.
\end{equation}
\end{enumerate}
Then $T$ has a unique fixed point $z$, and the Picard iterates of $T$
converge to $z$.
\end{thm}
\begin{proof}
First, we prove that $T$ has at most one fixed point. If $Tx=x$ and
$Ty=y$ then $\mathcal{O}_s(x)=\{x\}$ and $\mathcal{O}_t(y)=\{y\}$ and thus
\[
\mathbf{m}(x,y)=d(x,y)=d(Tx,Ty).
\]
Condition \eqref{item:d(Tx,Ty)<m(x,y)-in-generalized-Proinov}
then implies that $x=y$.
Now, choose $x\in X$ and set $x_n=T^nx$, $n\in\mathbb{N}_0$.
Then $d(x_n,x_{n+1})\to0$ since $T$ is asymptotically regular. It is easy to see that,
for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$, we have
\[
\mathbf{m}(x_{p_n},x_{q_n})
\leq d(x_{p_n},x_{q_n})
+ \alpha \sum_{i=0}^{s-1} d(x_{p_n+i},x_{p_n+i+1})
+ \beta \sum_{j=0}^{t-1} d(x_{q_n+j},x_{q_n+j+1}).
\]
Since $d(x_n,x_{n+1})\to0$, we see that \eqref{eqn:limsup m <= limsup d}
holds. Hence, by Theorem \ref{thm:fixed-point-theorem}, the sequence
$\{T^nx\}$ is Cauchy and, since $X$ is complete,
it converges to some point $z\in X$.
Since $T$ is continuous, we have $Tz=z$.
\end{proof}
We next give an example to show that Theorem \ref{thm:Generalized-Proinov}
strictly extends Proinov's Theorem \ref{thm:Proinov}.
\begin{example}
Take $a_i=i$, for $0\leq i < 4$, let $r_0=0$, and $r_n=1/n$ for
$n\geq1$, and set $x_{4n+i}=a_i+r_n$. Let
\[
X=\set{x_{4n+i}:0\leq i<4,\,n\geq0}.
\]
Then, equipped with the Euclidean metric, $X$ is a complete metric space. Define
a mapping $T:X\to X$ by $T(x_\ell)=x_{2\ell}$. Define $\mathbf{m}(x,y)$ by setting $s=t=1$
and $\alpha=\beta=1$ in \eqref{eqn:m(x,y)-in-generalized-Proinov}, that is,
\[
\mathbf{m}(x,y)=d(x,y)+d(x,Tx)+d(y,Ty).
\]
(Note that $\mathbf{m}$ is also obtained from \eqref{eqn:Proinov-m(x,y)} by setting $\gamma=1$.)
First, we show that $T$ satisfies all conditions in Theorem \ref{thm:Generalized-Proinov}.
Clearly, $T$ is continuous and, for every $x_\ell,x_\nu\in X$, we have
\[
d(T^{n+2}x_\ell,T^{n+2}x_\nu)
= d(T^n x_{4\ell},T^n x_{4\nu})
= \Bigl|\frac1{2^{n+1}\ell}-\frac1{2^{n+1}\nu}\Bigr|\to0.
\]
It is a matter of calculation to see that $d(Tx,Ty)<\mathbf{m}(x,y)$, for all $x,y\in X$.
The following shows that $T$ satisfies \eqref{eqn:m-contractive-in-generalized-Proinov}
with $N=2$:
\begin{align*}
d(T^3x_\ell,T^3x_\nu)
& =|x_{8\ell}-x_{8\nu}|=\Bigl|\frac1{8\ell}-\frac1{8\nu}\Bigr|
= \frac12\Bigl|\frac1{4\ell}-\frac1{4\nu}\Bigr|\\
& = \frac12 |x_{4\ell}-x_{4\nu}| = \frac12 d(T^2x_\ell,T^2x_\nu).
\end{align*}
Next, we show that the following condition (in Proinov's theorem) is violated:
\begin{itemize}
\item for every $\epsilon>0$, there exist $\delta>0$ such that
\begin{equation*}
\forall x,y\in X, \quad
\mathbf{m}(x, y) < \delta+\epsilon \Longrightarrow d(Tx,Ty)\leq \epsilon.
\end{equation*}
\end{itemize}
Take $\epsilon=a_2-a_0$. Let $u_\ell=x_{4\ell}$ and $v_\ell=x_{4(\ell+1)+1}$.
Then
\begin{align*}
d(Tu_\ell,Tv_\ell)
& =|x_{4(2\ell)}-x_{4(2\ell+2)+2}| = a_2-a_0 + r_{2\ell}-r_{2\ell+2}>\epsilon,
\intertext{and}
\mathbf{m}(u_\ell,v_\ell)
& = |u_\ell-v_\ell|+|u_\ell-Tu_\ell|+|v_\ell-Tv_\ell| \\
& = |x_{4\ell}-x_{4(\ell+1)+1}|+|x_{4\ell}-x_{4(2\ell)}|+|x_{4(\ell+1)+1}-x_{4(2\ell+2)+2}|\\
& = (a_1-a_0+r_\ell-r_{\ell+1})+(r_\ell-r_{2\ell})+(a_2-a_1+r_{\ell+1}-r_{2\ell+2})\\
& = a_2-a_0 + 2r_\ell-r_{2\ell}-r_{2\ell+2}.
\end{align*}
If $\delta_\ell=2r_\ell-r_{2\ell}-r_{2\ell+2}$, we have
\[
\epsilon < d(Tu_\ell,Tv_\ell) < \mathbf{m}(u_\ell,v_\ell) \leq \epsilon+\delta_\ell.
\]
Since $\delta_\ell\to0$ as $\ell\to\infty$, we see that $T$
does not satisfy \eqref{eqn:m-contractive-in-generalized-Proinov}
for $\epsilon=a_2-a_0$.
\end{example}
We conclude this section by showing that the following theorem of
Geraghty \cite{Geraghty-73} is a special case of our fixe point theorem.
\begin{thm}[Geraghty \cite{Geraghty-73}]
\label{thm:Geraghty}
Let $T$ be a contractive self-map of a complete metric space $X$,
let $x\in X$, and set $x_n=T^n x$, $n\in\mathbb{N}_0$. Then $\set{x_n}$
converges to a unique fixed point of $T$ if and only if,
for any two subsequences $\set{x_{p_n}}$ and $\set{x_{q_n}}$, with
$x_{p_n}\neq x_{q_n}$, condition
\begin{equation*}
\lim_{n\to\infty}\frac{d(Tx_{p_n},Tx_{q_n})}{d(x_{p_n},x_{q_n})}=1,
\end{equation*}
implies $d(x_{p_n},x_{q_n})\to0$.
\end{thm}
\begin{prop}
\label{prop:G-contractive-property-is-a-general-one}
Let $\set{x_n}$ be a sequence in a metric space $X$ such that
$d(x_{p+1},x_{q+1}) \leq d(x_p,x_q)$, for all $p,q\in \mathbb{N}_0$.
Then the following statements are equivalent:
\begin{enumerate}[\upshape(i)]
\item \label{item:it-is-a-Geraghty-sequence}
for any two subsequences $\set{x_{p_n}}$ and $\set{x_{q_n}}$, with
$x_{p_n}\neq x_{q_n}$, condition
\begin{equation}\label{eqn:lim=1-in-Geraghty-equiv}
\lim_{n\to\infty}\frac{d(x_{p_n+1},x_{q_n+1})}{d(x_{p_n},x_{q_n})}=1,
\end{equation}
implies $d(x_{p_n},x_{q_n})\to0$.
\item \label{item:Geraghty-sequence-in-term-of-pnqn}
for every $\epsilon>0$, for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$,
if\newline $\limsup d(x_{p_n},x_{q_n})\leq \epsilon$ then
\begin{equation}\label{eqn:Geraghty-sequence-in-term-of-pnqn}
\limsup_{n\to\infty}d(x_{p_n+1},x_{q_n+1}) < \epsilon.
\end{equation}
\item \label{item:Geraghty-sequence-in-e-de-eta}
for every $\epsilon>0$, there exist $\delta>0$, $\eta\in(0,\epsilon)$, and $N\in\mathbb{N}_0$, such that
\begin{equation}\label{eqn:Geraghty-sequence-in-e-de-eta}
\forall p,q \geq N,\quad
d(x_p,x_q) < \epsilon + \delta \Longrightarrow
d(x_{p+1},x_{q+1}) \leq \eta.
\end{equation}
\end{enumerate}
\end{prop}
\begin{proof}
$\eqref{item:it-is-a-Geraghty-sequence}
\Rightarrow \eqref{item:Geraghty-sequence-in-term-of-pnqn}$:
Assume that, for two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$, we have
\[
\limsup_{n\to\infty} d(x_{p_n+1},x_{q_n+1})
=\limsup_{n\to\infty} d(x_{p_n},x_{q_n})>0.
\]
Since $d(x_{p_n+1},x_{q_n+1})\leq d(x_{p_n},x_{q_n})$ for all $n$,
by passing through subsequences, if necessary, we can assume that
\[
\lim_{n\to\infty} d(x_{p_n+1},x_{q_n+1})
=\lim_{n\to\infty} d(x_{p_n},x_{q_n})>0.
\]
Therefore, we get \eqref{eqn:lim=1-in-Geraghty-equiv}. Since
$d(x_{p_n},x_{q_n})$ does not converge to $0$, we conclude that the sequence
does not satisfy \eqref{item:it-is-a-Geraghty-sequence}.
$\eqref{item:Geraghty-sequence-in-term-of-pnqn}
\Rightarrow \eqref{item:Geraghty-sequence-in-e-de-eta}$:
Assume there is $\epsilon>0$ such that, for every $n\in\mathbb{N}_0$, there exist $p_n$ and $q_n$ with
$q_n>p_n\geq n$ such that
\[
d(x_{p_n},x_{q_n}) < \epsilon + \frac1n
\quad \text{and} \quad
\epsilon-\frac1n \leq d(x_{p_n+1},x_{q_n+1}).
\]
Then
\[
\epsilon \leq \limsup_{n\to\infty} d(x_{p_n+1},x_{q_n+1}) \leq \limsup_{n\to\infty} d(x_{p_n},x_{q_n}) \leq \epsilon.
\]
$\eqref{item:Geraghty-sequence-in-e-de-eta}
\Rightarrow \eqref{item:it-is-a-Geraghty-sequence}$:
Assume that, for two subsequences $\set{x_{p_n}}$ and $\set{x_{q_n}}$, condition \eqref{eqn:lim=1-in-Geraghty-equiv}
holds and also
\[
\epsilon=\limsup_{n\to\infty} d(x_{p_n},x_{q_n})>0.
\]
For this $\epsilon$, by \eqref{item:Geraghty-sequence-in-e-de-eta}, there
exist $\delta>0$, $\eta\in(0,\epsilon)$, and $N\in\mathbb{N}_0$, such that
\eqref{eqn:Geraghty-sequence-in-e-de-eta} holds true.
There is $N_1>N$ such that, for $n\geq N_1$, we have $d(x_{p_n},x_{q_n}) < \epsilon + \delta$
and thus $d(x_{p_n+1},x_{q_n+1})\leq \eta$. On the other hand \eqref{eqn:lim=1-in-Geraghty-equiv}
implies that, for every $r<1$, there is $N_2>N_1$ such that, for $n\geq N_2$,
\[
r d(x_{p_n},x_{q_n}) \leq d(x_{p_n+1},x_{q_n+1})\leq \eta.
\]
If $n\to\infty$ we get $r\epsilon \leq \eta$. If $r\to 1$, we get $\epsilon \leq \eta$
which is absurd.
\end{proof}
Now, to see why Theorem \ref{thm:Geraghty} follows from the results in this section,
take a point $x\in X$ and set $x_n=T^nx$, $n\in\mathbb{N}_0$. If $\set{x_n}$ satisfies
the condition in Theorem \ref{thm:Geraghty}, then, by Proposition
\ref{prop:G-contractive-property-is-a-general-one}, the sequence $\set{x_n}$ is
$d$-contractive (in the sense of Definition \ref{dfn:m-contractive-sequence}).
On the other hand, $T$ being contractive implies that $\set{x_n}$ satisfies
\eqref{eqn:contractive-sequences}. Hence, by Proposition \ref{prop:contractive-sequences},
we have $d(x_n,x_{n+1})\to0$. Theorem \ref{thm:m-contractive-sequences-are-Cauchy} now
implies that $\set{x_n}$ is a Cauchy sequence.
\bigskip
\section{Asymptotic Contractions of Meir-Keeler Type}
\label{sec:asymptotic-contractions}
In 2003, Kirk \cite{Kirk-2003} introduced the notion of asymptotic contraction
on a metric space, and proved a fixed-point theorem for such contractions
(see also \cite{Arandelovic-2005}). In 2006, Suzuki \cite{Suzuki-AC-2006}
introduced the notion of asymptotic contraction
of Meir-Keeler type, and proved a fixed-point theorem for such contractions, which is
a generalization of both Meir and Keeler's theorem \cite{Meir-Keeler-1969}
and Kirk's theorem \cite{Kirk-2003}. A year later, Suzuki \cite{Suzuki-AC-2007}
introduced the following notion of asymptotic contractions which is, in some sense, the final
definition of asymptotic contractions (see \cite[Theorem 6]{Suzuki-AC-2007}).
\begin{dfn}[Suzuki \cite{Suzuki-AC-2007}]
\label{dfn:ACF}
A mapping $T$ on a metric space $X$ is said to be an
\emph{asymptotic contraction of the final type} if
\begin{enumerate}[\upshape(i)]
\item $d(T^nx,T^ny)\to0$, for all $x,y\in X$,
\item for every $x\in X$ and $\epsilon>0$, there exist $\delta>0$ and $\nu\in\mathbb{N}$ such that
\begin{equation*}
\forall p,q\in\mathbb{N}, \quad
\epsilon<d(T^p x,T^q x) < \epsilon+\delta \Longrightarrow
d(T^{p+\nu} x,T^{q+\nu} x) \leq \epsilon.
\end{equation*}
\end{enumerate}
\end{dfn}
Then they proved the following result.
\begin{thm}[Suzuki \cite{Suzuki-AC-2007}]
\label{thm:Suzuki-AC-Cauchy}
Let $X$ be a metric space and let $T$ be an asymptotic contraction of the final type
on $X$. Then $\{T^nx\}$, for every $x$, is a Cauchy sequence.
\end{thm}
We present a short proof of the above theorem using the results in section \ref{sec:technical-lemma}.
But, first, let us make the following definition.
\begin{dfn}
\label{dfn:abtahi-ACF}
Let $T$ be a mapping on a metric space $X$, and $\mathbf{m}$ a nonnegative function on $X\times X$.
We call $T$ an \emph{asymptotic $\mathbf{m}$-contraction} if
\begin{enumerate}[\upshape(i)]
\item \label{item:d(Tnx,Tny)->0}
$d(T^nx,T^ny)\to0$, for all $x,y\in X$,
\item for every $x\in X$ and $\epsilon>0$, there exist $\delta>0$, $\nu\in\mathbb{N}$, and $N\in\mathbb{N}_0$
such that
\begin{equation*}
\forall p,q\geq N, \quad
\mathbf{m}(T^p x,T^q x) < \epsilon+\delta \Longrightarrow
d(T^{p+\nu} x,T^{q+\nu} x) \leq \epsilon.
\end{equation*}
\end{enumerate}
\end{dfn}
\begin{thm}
\label{thm:abtahi-AC-Cauchy}
Let $T$ be an asymptotic $\mathbf{m}$-contraction on $X$. If $\mathbf{m}$ satisfies
\eqref{eqn:limsup m <= limsup d}, for some $x\in X$, then $\{T^nx\}$ is a Cauchy sequence.
\end{thm}
\begin{proof}
Let $x_n=T^nx$, $n\in\mathbb{N}_0$. Then the following are equivalent (the proof is similar to that of Lemma
\ref{lem:equiv-conditions-m-contractive-sequence} and hence is omitted).
\begin{enumerate}[\upshape(i)]
\item for every $\epsilon>0$, there exist $\nu\in\mathbb{N}$, $\delta>0$ and $N\in\mathbb{N}_0$ such that
\[
\forall\, p,q \geq N,\quad
\mathbf{m}(x_p,x_q) < \epsilon+\delta \Longrightarrow
d(x_{p+\nu},x_{q+\nu}) \leq \epsilon,
\]
\item for every $\epsilon>0$, there exists $\nu\in\mathbb{N}$ such that,
for any two subsequences $\{x_{p_n}\}$ and $\{x_{q_n}\}$,
if\/ $\limsup \mathbf{m}(x_{p_n},x_{q_n})\leq \epsilon$, then, for some $N$,
\begin{equation*}
d(x_{p_n+\nu},x_{q_n+\nu}) \leq \epsilon, \qquad (n\geq N).
\end{equation*}
\end{enumerate}
Condition \eqref{item:d(Tnx,Tny)->0} in Definition \ref{dfn:abtahi-ACF}
implies that $d(x_n,x_{n+1})\to0$. Now part \eqref{item:cor:main:nu} of
Corollary \ref{cor:main} shows that $\{x_n\}$ is Cauchy.
\end{proof}
At first look, because of replacing the distance function $d(x,y)$ with
a more general function $\mathbf{m}(x,y)$, such as the one defined by \eqref{eqn:m(x,y)-in-generalized-Proinov},
it may seem that Definition \ref{dfn:abtahi-ACF} and
Theorem \ref{thm:abtahi-AC-Cauchy} are extensions of Suzuki's Definition \ref{dfn:ACF} and
Theorem \ref{thm:Suzuki-AC-Cauchy}, respectively. However, we have the following result which
confirms that Definition \ref{dfn:ACF} is the final definition of asymptotic contractions.
\begin{thm}
\label{thm:final}
Let $X$ be a complete metric space. If $T$ is an asymptotic $\mathbf{m}$-contraction,
for some $\mathbf{m}$ satisfying \eqref{eqn:limsup m <= limsup d} for all $x$, then $T$ is
an asymptotic contraction of the final type.
\end{thm}
\begin{proof}
For every $x\in X$, by Theorem \ref{thm:abtahi-AC-Cauchy}, the sequence $\set{T^nx}$ is Cauchy,
and, since $X$ is complete, there is $z\in X$ such that $T^nx\to z$. Since $d(T^nx,T^ny)\to0$,
for all $x,y\in X$, the point $z$ is unique. Now, Theorem 8 in \cite{Suzuki-AC-2007} shows
that $T$ is an asymptotic contraction of the final type.
\end{proof}
\bigskip
\section*{Acknowledgment}
The author expresses his sincere gratitude to the anonymous referee for his/her careful reading
and suggestions that improved the presentation of this paper.
|
2,869,038,154,766 | arxiv | \section{\label{sec:intro} Introduction}
The availability of free undulator tunnels at the European XFEL
facility offers a unique opportunity to build a beamline optimized
for coherent diffraction imaging of complex molecules, like proteins
and other biologically interesting structures. Crucial parameters
for such bio-imaging beamline are photon energy range, peak power,
and pulse duration \cite{HAJD}-\cite{SEIB}.
The highest diffraction signals are achieved at the longest
wavelength that supports a given resolution, which should be better
0.3 nm. With photon energy of about 3 keV one can reach a resolution
better than 0.3 nm with a detector designed to collect diffracted
light in all forward directions, that is at angles $2\theta <
\pi/2$. Higher photon energies up to about 13 keV give access to
absorption edges of specific elements used for phasing by anomalous
diffraction. The most useful edges to access are the K-edge of Fe
(7.2 keV) and Se (12.6 keV), \cite{BERG}. Access to the sulfur
K-edge (2.5 keV) is required too. Finally, the users of the
bio-imaging beamline also wish to investigate large biological
structures in the soft X-ray photon energy range down to the water
window (0.3 keV - 0.5 keV), \cite{BERG}.
Overall, one aims at the production of pulses containing enough
photons to produce measurable diffraction patterns, and yet short
enough to avoid radiation damage in a single pulse. This is, in
essence, the principle of imaging by "diffraction before
destruction" \cite{NEUT}. These capabilities can be obtained by
reducing the pulse duration to $5$ fs or less, and simultaneously
increasing the peak power to the TW power level or higher, at photon
energies between 3 keV and 5 keV, which are optimal for imaging of
macromolecular structures \cite{BERG}.
The requirements for a dedicated bio-imaging beamline are the
following. The X-ray beam should be delivered in ultrashort pulses
with TW peak power and within a very wide photon energy range
between 0.3 keV and 13 keV. The pulse duration should be adjustable
from 10 fs in hard X-ray regime to 2 fs - 5 fs in photon energy
range between 3 keV and 5 keV. At the European XFEL it will be
necessary to run all undulator beamlines at the same electron energy
and bunch charge. However, bio-imaging experiments should be
performed without interference with other main SASE1, SASE2
beamlines. This assumes the use of nominal electron energy and
electron beam distribution.
A key component of the bio-imaging beamline is the undulator source.
A basic concept for layout and design of the undulator system for a
dedicated bio-imaging beamline at the European XFEL was proposed in
\cite{OURCC}. All the requirements in terms of photon beam
characteristics can be satisfied by the use a very efficient
combination of self-seeding, fresh bunch, and undulator tapering
techniques \cite{TAP1}-\cite{BZVI}, \cite{HUAN}-\cite{WU}. A
combination of self-seeding and undulator tapering techniques would
allow to meet the design TW output power. The bio-imaging beamline
would be equipped with two different self-seeding setups, one
provide monochromatization in the soft X-ray range, and one to
provide monochromatization in the hard X-ray range. The most
preferable solution in the photon energy range for single
biomolecule imaging consists in using a fresh bunch technique in
combination with self-seeding and undulator tapering techniques. In
\cite{OURCC} it was shown how the installation of an additional
(fresh bunch) magnetic chicane behind the soft X-ray self-seeding
setup enables an output power in the TW level for the photon energy
range between 3 keV and 5 keV. Additionally, the pulse duration can
be tuned between 2 fs and 10 fs with the help of this chicane, still
operating with the nominal electron bunch distribution \cite{S2ER}.
The overall setup proposed in \cite{OURCC} is composed of four
undulators separated by three magnetic chicanes. The undulator parts
consist of 4,3,4 and 29 cells. Each magnetic chicane compact enough
to fit one 5 m-long undulator segment and the FODO lattice will not
be perturbed. The undulator system will be realized in a similar
fashion as other European XFEL undulators. In order to make use of
standard components we favor the use of SASE3 type of undulator
segments, which are optimized for the generation of soft X-rays. The
present layout of the European XFEL enables to accommodate such new
beamline. The previously proposed undulator source provides access
to a photon energy range between 3 keV and 5 keV only at the reduced
electron beam energy of 10.5 GeV. Although the 10.5 GeV is one of
the nominal electron energy, it may not be the preferable mode of
operation for SASE1, SASE2 beamline users. Note that the SASE3
undulator type would enable operation down to 0.7 keV at an electron
energy of 17.5 GeV. However, the delay of photons induced in the
grating monochromator (3 ps) and, consequently, the delay of the
electrons required in the magnetic chicane of the soft X-ray
self-seeding setup sets a limit to the electron energy.
This paper constitutes an update to the scheme proposed in
\cite{OURCC}. The present design assumes the use of the same 40
cells undulator system, with an improved design of both self-seeding
setups. To avoid any interference with other beamlines, we propose
to extend the photon energy range of the self-seeding setup with a
single crystal monochromator down to 3 keV \cite{OURCC2}. As a
result, the design electron energy can be increased up to 17.5 GeV
in the photon energy range most preferable for bio-imaging. This is
achieved exploiting 0.1 mm diamond crystals in symmetric Bragg
geometry. Based on the use C(111), C(220), and C(400) reflections
($\sigma$-polarization) it will be possible to cover the photon
energy range between 3 keV and 13 keV. In particular, we exploit
C(111) reflection ($\sigma$-polarization) in photon energy range
between 3 keV and 5 keV. Combination of self-seeding and fresh bunch
techniques, as in the case of the original design, has the advantage
that the pulse duration can be tuned between 2 fs and 10 fs.
The users of the bio-imaging beamline also wish to investigate their
samples around sulfur K-edge, i.e. in the photon energy range
between 2 keV and 3 keV. A solution suitable for this spectral range
constitutes a major challenge for self-seeding designers. In fact,
on the one hand crystals with right lattice parameters are difficult
to be obtained. On the other hand, grating monochromator throughput
is usually too low due to high absorption. As for the original
design we propose a method around this obstacle, which is based in
essence on a fresh bunch technique, and exploits a self-seeding
setup based on grating monochromator in the photon energy range
between 0.7 keV and 1 keV. It should be noted that due to extension
of the single crystal monochromator setup down to 3 keV, the maximal
photon energy of operation for the grating monochromator is reduced
from 1.7 keV in the original design down to 1 keV in the current
design.
Also, here we adopt an improved design of grating monochromator,
which was recently proposed for the soft X-ray self-seeding setup at
the LCLS \cite{FENG3}, substituting a previously proposed one
\cite{FENG, FENG2}. In this novel design the optical delay is
reduced down to below 1 ps. As a result, a self-seeding setup with
such grating monochromator allows for reduced constraints on the
magnetic chicane, and can operate at the European XFEL down to 0.7
keV at the highest nominal electron energy of 17.5 GeV. Such high
electron energy enables to increase the X-ray output peak power in
the most preferable photon energy range for bio-imaging experiments
up to 2 TW.
\section{Setup description}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-1.eps}
\caption{Design of the undulator system for the bio-imaging
beamline. The method exploits a combination of self-seeding, fresh
bunch, and undulator tapering technique. Each magnetic chicane
accomplishes three tasks by itself. It creates an offset for
monochromator or X-ray mirror delay line installation, it removes
the electron microbunching produced in the upstream undulator, and
it acts as a magnetic delay line. } \label{bio3f1}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-7.eps}
\caption{Compact grating monochromator originally proposed at SLAC
\cite{FENG3} for soft X-ray self-seeding setup. The chicane fits in
one European XFEL undulator undulator section (5 m). }
\label{bio3f7}
\end{figure}
Self-seeding is a promising approach to significantly narrow the
SASE bandwidth and to produce nearly transform-limited X-ray pulses
\cite{SELF}-\cite{WUFEL2}. In its simplest configuration, a
self-seeding setup in the hard X-ray regime consists of two
undulators separated by photon monochromator and electron bypass
beamline, typically a 4-dipole chicane. The two undulators are
resonant at the same radiation wavelength. The SASE radiation
generated by the first undulator passes through the narrow-band
monochromator, thus generating a transform-limited pulse, which is
then used as a coherent seed in the second undulator. Chromatic
dispersion effects in the bypass chicane smear out the microbunching
in the electron bunch produced by the SASE lasing in the first
undulator. Electrons and monochromatized photon beam are recombined
at the entrance of the second undulator, and the radiation is
amplified by the electron bunch in the second undulator, until
saturation is reached. The required seed power at the beginning of
the second undulator must dominate over the shot noise power within
the gain bandpass, which is order of a few kW.
Despite the unprecedented increase in peak power of the X-ray pulses
for SASE X-ray FELs (see e.g. \cite{LCLS2}), some applications,
including single biomolecule imaging, require still higher photon
flux. The most promising way to extract more FEL power than that at
saturation is by tapering the magnetic field of the undulator
\cite{TAP1}-\cite{LAST}. Also, a significant increase in power is
achievable by starting the FEL process from a monochromatic seed
rather than from noise \cite{OURY3}-\cite{WUFEL2}. Tapering consists
in a slow reduction of the field strength of the undulator in order
to preserve the resonance wavelength, while the kinetic energy of
the electrons decreases due to the FEL process. The undulator taper
could be simply implemented at discrete steps from one undulator
segment to the next. The magnetic field tapering is provided by
changing the undulator gap.
The setup suggested in this article constitutes an optimization of
the original proposal in \cite{OURCC} and is composed of five
undulator parts separated by four magnetic chicanes as shown in Fig.
\ref{bio3f1}. These undulators consist of $4$, $3$, $4$, $6$ and
$23$ undulator cells, respectively. Each magnetic chicane is compact
enough to fit one undulator segment. The installation of chicanes
does not perturb the undulator focusing system. The implementation
of the self-seeding scheme for soft X-ray would exploit the first
magnetic chicane. The second and third magnetic chicanes create an
offset for the installation of a single crystal monochromator or an
X-ray mirror delay line, and act as a magnetic delay line. Both
self-seeding setups should be compact enough to fit one undulator
module.
For soft X-ray self-seeding, the monochromator usually consists of a
grating \cite{SELF}. Recently, a very compact soft X-ray
self-seeding scheme has appeared, based on a grating monochromator
\cite{FENG3}. The proposed monochromator is composed of a toroidal
grating followed by three mirrors, and is equipped with an exit slit
only. The delay of the photons is about $1$ ps. The monochromator is
continuously tunable in the photon energy range between $0.3$ keV
and $1$ keV. The resolution is about $5000$. The transmission of
the monochromator beamline is close to $10 \%$. The magnetic chicane
delays the electron electron bunch accordingly, so that the photon
beam passing through the monochromator system recombines with the
same electron bunch. The chicane provides a dispersion strength of
about $0.6$ mm in order to match the optical delay and also smears
out the SASE microbunching generated in the first $4$ cells of the
undulator. It should be noted that in \cite{OSOF} we studied the
performance of a previous scheme of a grating monochromator for a
soft X-ray self-seeding setup \cite{FENG,FENG2}. For the present
investigation we consider the new scheme in \cite{FENG3}. The layout
of the bypass and of the monochromator optics is schematically shown
in Fig. \ref{bio3f7}.
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio5.eps}
\includegraphics[width=0.5\textwidth]{C400_T.eps}
\includegraphics[width=0.5\textwidth]{C400_Phi.eps}
\caption{X-ray optics for compact crystal monochromator originally
proposed in \cite{OURY5b} for a hard X-ray self-seeding setup, based
on the C(400) reflection ($\sigma$-polarization). Modulus and phase
of the transmissivity are shown in the two lower plots.}
\label{biof1}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-9.eps}
\includegraphics[width=0.5\textwidth]{C220_T.eps}
\includegraphics[width=0.5\textwidth]{C220_Phi.eps}
\caption{Schematic of single crystal monochromator for operation in
photon energy range between 5 keV and 7 keV. In this range the
C(220) reflection will be exploited. Modulus and phase of the
transmissivity are shown in the two lower plots.} \label{bioff3}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-8.eps}
\includegraphics[width=0.5\textwidth]{C111_T.eps}
\includegraphics[width=0.5\textwidth]{C111_Phi.eps}
\caption{Schematic of single crystal monochromator for operation in
photon energy range between 3 keV and 5 keV. In this range the
C(111) reflection will be exploited. Modulus and phase of the
transmissivity are shown in the two lower plots.} \label{bioff2}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio1.eps}
\caption{X-ray optical system for delaying the soft X-ray pulse with
respect to the electron bunch. Two distinct X-ray optical systems
can be installed within the second and the third magnetic chicane. }
\label{biof2}
\end{figure}
For hard X-ray self-seeding, a monochromator usually consists of
crystals in the Bragg geometry. A conventional 4-crystal, fixed exit
monochromator introduces optical delay of, at least, a few
millimeters, which has to be compensated with the introduction of an
electron bypass longer than one undulator module. To avoid this
difficulty, a simpler self-seeding scheme was proposed in
\cite{OURY5b}, which uses the transmitted X-ray beam from the single
crystal to seed the same electron bunch. Here we propose to use a
diamond crystal with a thickness of $0.1$ mm. Using the symmetric
C(400) Bragg reflection, it will be possible to cover the photon
energy range from $7$ keV to $9$ keV, Fig. \ref{biof1}. The range
between $5$ keV and $7$ keV can be covered with the C(220)
reflection, Fig. \ref{bioff3}, while the range between $3$ keV and
$5$ keV can be obtained using the C(111) reflection, Fig.
\ref{bioff2}.
One of the main technical problems for self-seeding designers is to
provide bio-imaging capabilities in $2$ keV - $3$ keV photon energy
range. Here we will use the same method already exploited in
\cite{OURCC} to get around this obstacle. Our solution is based in
essence on the fresh bunch technique \cite{BZVI} and exploits the
above described conservative design of self-seeding setup based on a
grating monochromator. The hardware requirement is minimal, and in
order to implement a fresh bunch technique it is sufficient to
install an additional magnetic chicane at a special position behind
the soft X-ray self-seeding setup. The function of this second
chicane is both to smear out the electron bunch microbunching, and
to delay the electron bunch with respect to the monochromatic soft
X-ray pulse produced in the second undulator. In this way, only half
of the electron bunch is seeded, and saturates in the third
undulator. Finally, the second half of the electron bunch, which
remains unspoiled, is seeded by the third harmonic of the
monochromatic radiation pulse generated in the third undulator,
which is also monochromatic. The final delay of the electron bunch
with respect to the seed radiation pulse can be obtained with a
third, hard X-ray self-seeding magnetic chicane, which in this mode
of operation is simply used to provide magnetic delay. The
monochromatic third harmonic radiation pulse used as seed for the
unperturbed part of the electron bunch is in the GW power level, and
the combination of self-seeding and fresh bunch technique is
extremely insensitive to non-ideal effects. The final undulator,
composed by $29$ cells, is tuned to the third harmonic frequency,
and is simply used to amplify the X-ray pulse up to the TW power
level.
In order to introduce a tunable delay of the photon beam with
respect to the electron beam, a mirror chicane can be installed
within the second magnetic chicane, as shown in Fig. \ref{biof2}.
The function of the mirror chicane is to delay the radiation in the
range between $0.7$ keV and $1$ keV relatively to the electron
bunch. The glancing angle of the mirrors is as small as $3$ mrad. At
the undulator location, the transverse size of the photon beam is
smaller than $0.1$ mm, meaning that the mirror length would be just
about $5$ cm. The single-shot mode of operation will relax the
heat-loading issues. The mirror chicane can be built in such a way
to obtain a delay of the radiation pulse of about $23~\mu$m. This is
enough to compensate a bunch delay of about $20~\mu$m from the
magnetic chicane, and to provide any desired shift in the range
between $0~\mu$m and $3~\mu$m. Note that for the European XFEL
parameters, $1$ nm microbunching is washed out with a weak
dispersive strength corresponding to an $R_{56}$ in the order of ten
microns. The dispersive strength of the proposed magnetic chicane is
more than sufficient to this purpose. Thus, the combination of
magnetic chicane and mirror chicane removes the electron
microbunching produced in the second undulator and acts as a tunable
delay line within $0~\mu$m and $3~\mu$m with the required choice of
delay sign.
\subsection{\label{sub:verysoft} Operation into the water window}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-2.eps}
\caption{Design of the undulator system for high power mode of
operation in the water window. The method exploits a combination of
self-seeding scheme with grating monochromator and an undulator
tapering technique. } \label{bio3f2}
\end{figure}
The five-undulator configuration in Fig. \ref{bio3f1} can be
naturally taken advantage of at different photon energies ranging
from soft to hard X-rays. Fig. \ref{bio3f2} shows the basic setup
for the high-power mode of operation in the soft X-ray wavelength
range. The second, the third and the fourth chicane are not used for
such regime, and must be switched off. After the first undulator (4
cells-long) and the grating monochromator, the output undulator
follows. The first section of the output undulator (consisting of
second and third undulator) is composed by $3$ untapered cells,
while tapering is implemented starting from the second cell of the
fourth undulator. The monochromatic seed is exponentially amplified
by passing through the first untapered section of the output
undulator. This section is long enough to allow for saturation, and
yields an output power of about $100$ GW. Such monochromatic FEL
output is finally enhanced up to $1$ TW in the second
output-undulator section, by tapering the undulator parameter over
the last cells after saturation. Under the constraints imposed by
undulator and chicane parameters it is only possible to operate at
the nominal electron beam energy of $10.5$ GeV. The setup was
optimized based on results of start-to-end simulations for a nominal
electron beam with 0.1 nC charge. Results were presented in
\cite{OSOF}, where we studied the performance of this scheme for the
SASE3 upgrade.
\subsection{Operation around the sulfur K-edge}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-3.eps}
\caption{Design of the undulator system for high power mode of
operation around the sulfur K-edge. The method exploits a
combination of self-seeding scheme with grating monochromator, fresh
bunch and undulator tapering techniques. } \label{bio3f3}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio12.eps}
\caption{Principle of the fresh bunch technique for the high power
mode of operation in the photon energy range between $2$ keV and $3$
keV. The second chicane smears out the electron microbunching and
delays the monochromatic soft X-ray pulse with respect to the
electron bunch of $6$ fs. In this way, half of of the electron bunch
is seeded and saturates in the third undulator.} \label{biof9}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio13.eps}
\caption{Principle of the fresh bunch technique for the high power
mode of operation in the photon energy range between $2$ keV and $3$
keV. The third magnetic chicane smears out the electron
microbunching and delays the electron bunch with respect to the
radiation pulse. The unspoiled part of electron bunch is seeded by a
GW level monochromatic pulse at third harmonic frequency. Tunability
of the output pulse duration can be easily obtained by tuning the
magnetic delay of the third chicane.} \label{biof10}
\end{figure}
Fig. \ref{bio3f3} shows the basic setup for high power mode of
operation in the photon energy range between 2 keV and 3 keV. The
first three chicanes are used for such regime, and must be switched
on, while the last fourth chicane is off. The third chicane is used
as a magnetic delay only, and the crystal must be removed from the
light path. We propose to perform monochromatization at photon
energies ranging between $0.7$ keV and $1$ keV with the help of a
grating monochromator, and to amplify the seed in the second
undulator up to the power level of $0.2$ GW. The second chicane
smears out the electron microbunching and delays the monochromatic
soft X-ray pulse of $2~\mu$m with respect to the electron bunch. In
this way, half of the electron bunch is seeded and saturates in the
third undulator up to $40$ GW. At saturation, the electron beam
generates considerable monochromatic radiation at the third harmonic
in the GW power level. The third magnetic chicane smears out the
electron microbunching and delays the electron bunch with respect to
the radiation of $2~\mu$m. Thus, the unspoiled part of the electron
bunch is seeded by the GW-level monochromatic pulse at the third
harmonic frequency, Fig. \ref{biof10}. The fourth, 29 cells-long
undulator is tuned to the third harmonic frequency (between $2$ keV
and $3$ keV), and is used to amplify the radiation pulse up to $1$
TW. The additional advantage of the proposed setup for bio-imaging
is the tunability of the output pulse duration, which is obtained by
tuning the magnetic delay of the third chicane. Simulations show
that the X-ray pulse duration can be tuned from $2$ fs to $5$ fs.
The production of such pulses is of great importance when it comes
to single biomolecule imaging experiments.
The soft X-ray background can be easily eliminated by using a
spatial window positioned downstream of the fourth undulator exit
\cite{OURCC}. Since the soft X-ray radiation has an angular
divergence of about $0.02$ mrad FWHM, and the slits are positioned
more than $100$ m downstream of the third undulator, the background
has much larger spot size compared with the $2$ keV - $3$ keV
radiation spot size, which is about 0.1 mm at the exit of the fourth
undulator. Therefore, the background radiation power can be
diminished of more than two orders of magnitude without any
perturbations of the main pulse.
With the monochromator design in \cite{FENG3}, it will be possible
to operate at an electron beam energy of $17.5$ GeV. The setup was
optimized based on results from start-to-end simulations for a
nominal electron bunch with a charge of $0.1$ nC. Results are
presented in the following Sections of this article. The proposed
undulator setup uses the electron beam coming from the SASE1
undulator. We assume that SASE1 operates at the photon energy of
$12$ keV, and that the FEL process is switched off for one single
dedicated electron bunch within each macropulse train. A method to
control the FEL amplification process is based on the betatron
switcher technique described in \cite{SWIT1,SWIT2}. Due to quantum
energy fluctuations in the SASE1 undulator, and to wakefields in the
SASE1 undulator pipe, the energy spread and the energy chirp of the
electron bunch at the entrance of the bio-imaging beamline
significantly increase compared with the same parameters at the
entrance of the SASE1 undulator. The dispersion strength of the
first chicane has been taken into account from the viewpoint of the
electron beam dynamics, because it disturbs the electron beam
distribution. The other two chicanes have tenfold smaller dispersion
strength compared with the first one. The electron beam was tracked
through the first chicane using the code Elegant \cite{ELEG}. The
electron beam distortions complicate the simulation procedure.
However, simulations show that the proposed setup is not
significantly affected by perturbations of the electron phase space
distribution, and yields about the same performance as in the case
for an electron beam without the tracking through the first chicane
(see below).
\subsection{Operation in the $3$ keV - $7$ keV photon energy range}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-4.eps}
\caption{Design of the undulator system for high power mode of
operation in the most preferable photon energy range for single
molecule imaging, between 3 keV and 5 keV. The method exploits a
combination of the self-seeding scheme with single crystal
monochromator, fresh bunch and undulator tapering techniques. }
\label{bio3f4}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-5.eps}
\caption{Design of the undulator system for high power mode of
operation in the photon energy range between 5 keV and 7 keV. The
method exploits a combination of self-seeding scheme with single
crystal monochromator, fresh bunch and undulator tapering
techniques. } \label{bio3f5}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-6.eps}
\caption{Design of the undulator system for high power mode of
operation in the photon energy range in the photon energy range
between 7 keV and 9 keV. The method exploits a combination of
self-seeding scheme with single crystal monochromator and an
undulator tapering technique. } \label{bio3f6}
\end{figure}
Starting with the energy range of $3$ keV it is possible to use a
single crystal monochromator instead of a grating monochromator at
an electron energy of $17.5$ GeV. Different crystal reflections and
different positions of the monochromator down the undulator enable
self-seeding for different spectral ranges.
For the range between 3 keV and 5 keV, Fig. \ref{bio3f4}, the first
chicane is not used and is switched off. After the first 7 cells the
electron and the photon beams are separated with the help of the
second magnetic chicane, and the C(111) reflection is used to
monochromatize the radiation. The seed is amplified in the next 4
cells. After that, the electron and the photon beam are separated
again by the third chicane, and an X-ray optical delay line allows
for the introduction of a tunable delay of the photon beam with
respect to the electron beam. The following 6 cells use only a part
of the electron beam as a lasing medium. A magnetic chicane follows,
which shifts the unspoiled part of the electron bunch on top of the
of the photon beam. In this way, a fresh bunch technique can be
implemented. Since the delays are tunable, the photon pulse length
can also be tuned. Finally, radiation is amplified into the last
$23$ tapered cells to provide pulses with about 2 TW power. The
photon energy range between 5 keV and 7 keV can be achieved
similarly, Fig. \ref{bio3f5}. The only difference is that now the
C(220) reflection is used, instead of the C(111).
It may be worth to point out the difference between the operation in
the $3$ keV - 7 keV range and the previously discussed range between
2 keV and 3 keV. In the 3 keV - 7 keV range we use seeding in
combination of a fresh bunch technique, but we do not exploit
harmonic generation. Moreover, the fresh bunch technique is only
used for tuning the duration of the radiation pulse.
\subsection{Operation in the $7$ keV - $9$ keV photon energy range}
The energy range between 7 keV and 9 keV can be achieved by
deactivating the first and the second magnetic chicane, thus letting
the SASE process building up the radiation pulse to be
monochromatized for 11 cells. After that, the third chicane is used
for the monochromator setup, which makes use of the C(400)
reflection. The last chicane is switched off, and the the output
undulator is long enough to reach $1$ TW power. The duration of the
output pulses is of about $10$ fs. If tunability of the pulse
duration is requested in this energy range, this is most easily
achieved by providing additional delay with the fourth magnetic
chicane installed behind the hard X-ray self-seeding setup.
\subsection{Operation around the selenium K-edge}
Finally, for the energy range between 9 keV and 13 keV, a
combination of self-seeding, fresh bunch tachnique and harmonic
generation is used. The undulator line is configured as for the
range between 3 keV and 5 keV, Fig. \ref{bio3f4}, the only
difference being that the final undulator segments are tuned at the
third harmonic of the fundamental thus enabling the 9 keV - 13 keV
energy range. As before, the first chicane is not used and is
switched off. After the first 7 cells the electron and the photon
beams are separated with the help of the second magnetic chicane,
and the C(111) reflection is used to monochromatize the radiation.
The seed is amplified in the next 4 cells. After that, the electron
and the photon beam are separated again by the third chicane, and an
X-ray optical delay line allows for the introduction of a tunable
delay of the photon beam with respect to the electron beam. The
second chicane smears out the electron microbunching and delays the
monochromatic soft X-ray pulse with respect to the electron bunch of
$6$ fs. In this way, half of of the electron bunch is seeded and
saturates in the following 6 cells. A magnetic chicane follows,
which shifts the unspoiled part of the electron bunch on top of the
of the photon beam. In this way, a fresh bunch technique can be
implemented. Since the delays are tunable, the photon pulse length
can also be tuned. Since third harmonic bunching is considerable,
the last $23$ tapered cells are tuned at the third harmonic of the
fundamental providing pulses with about 0.5 TW power.
\subsection{\label{subloc} Possible location of the bio-imaging
line}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-15.eps}
\caption{Original design of the European XFEL facility.}
\label{biof12}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-16.eps}
\caption{Current design of the European XFEL facility.}
\label{biof13}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=1.0\textwidth]{bio3-14.eps}
\caption{Schematic of the proposed extension of the European XFEL
facility.} \label{biof15}
\end{figure}
The original design of the European XFEL \cite{tdr-2006} was
optimized to produce XFEL radiation at $0.1$ nm, simultaneously at
two undulator lines, SASE1 and SASE2. Additionally, the design
included one FEL line in the soft X-ray range, SASE3, and two
undulator lines for spontaneous synchrotron radiation, U1 and U2,
Fig. \ref{biof12}. The soft X-ray SASE3 beamline used the spent
electron beam from SASE1, and the U1 and U2 beamlines used the spent
beam from SASE2. In fact, although the electron beam performance is
degraded by the FEL process, the beam can still be used in
afterburner mode in the SASE3 undulator, which will be equipped with
a $126$ m-long undulator system, for a total of $21$ cells.
After a first design report, the layout of the European XFEL
changed. In the last years after the achievement of the LCLS, and
the subsequent growth of interest in XFEL radiation by the
scientific community, it became clear that the experiments with XFEL
radiation, rather than with spontaneous synchrotron radiation, had
to be prioritized. In the new design, the two beamlines behind SASE2
are now free for future XFEL undulators installations, Fig.
\ref{biof13}.
Recently it was also realized that the amplification process in the
XFEL undulators can be effectively controlled by betatron FEL
switchers \cite{SWIT1,SWIT2}. The SASE3 undulator was then optimized
for generating soft X-rays. However, due to the possibility of
switching the FEL process in SASE1, it is possible to produce high
power SASE3 radiation in a very wide photon energy range between
$0.3$ keV and $13$ keV. The SASE3 beamline is now expected to
provide excellent performance, and to take advantage of its location
in the XTD4 tunnel, which is close to the experimental hall and has
sufficient free space behind the undulator for future expansion
($140$ m). After this section, the electron beam will be separated
from the photon beam and will be bent down to an electron beam dump,
Fig. \ref{biof12}. In the photon energy range between $3$ keV and
$13$ keV, the SASE3 beamline is now expected to provide even better
conditions for users than SASE1 and SASE2.
In this article we propose to build the bio-imaging beamline in the
XTD4 tunnel. The SASE3 undulator, which is composed by 21 cells, can
be installed from the very beginning in the free XTD3 tunnel, which
is shorter than the XTD4 tunnel but sufficiently long for such
installation, Fig. \ref{biof15}. The undulator can be placed within
the straight beam path that is defined by the last upstream and
first downstream dipole of the electron deflection system. The
limiting length given by these constraints is referred to as
"available length". For the XTD3 and the XTD4 tunnels this available
length respectively amounts to 255 m and 460 m, see \cite{DECKP}.
However, the electron beam optics requires sufficient space in front
and the above-mentioned dipoles. The length that fits these electron
optics restrictions is referred to as "potential length" and can be
actually used for installations. For the XTD3 and the XTD4 tunnels,
this potential length respectively amounts to 215 m and 400 m
\cite{DECKP}. It should be mentioned that the potential length of
the XTD4 tunnel is practically the same as the main SASE1 and SASE2
tunnels, and nicely fits with the undulator system for a dedicated
bio-imaging beamline. It offers thus a great potential for future
upgrades of this new beamline.
The bio-imaging beamline would support experiments carried out over
a rather wide photon energy range. It is therefore proposed that the
photon beam transport of the new beamline includes two lines. Line A
uses $0.5$m-long mirrors operating at a grazing angle of $2$ mrad.
This line is dedicated to the transport of X-ray radiation in the
photon energy range from $3$ keV up to $13$ keV. This would be
complementary to the Line B that is now optimized in the soft X-ray
range between $0.3$ keV and $3$ keV. The distance from the
40-cells-long undulator exit to the first mirror system will be only
of about $100$m\footnote{This is in contrast with SASE1 and SASE2
beamlines, where an opening angle of $0.003$ mrad at $3$ keV FEL
radiation leads to unacceptable mirror length of 2 m due to long
distance of about 500 m between the source and mirror system. For
these beamlines there is no possibility to use identical
configuration of mirrors within the photon energy range from 3 keV
to 13 keV. }.
\section{\label{spatio} Spatiotemporal transformation caused by the use of a single crystal
monochromator}
The development of self-seeding schemes in the hard X-ray
wavelength range necessarily involves crystal monochromators. Any
device like a crystal monochromator introduces spatiotemporal
deformations of the seeded X-ray pulse, which can be problematic for
seeding. The spatiotemporal coupling in the electric field relevant
to self-seeding schemes with crystal monochromators has been
analyzed in the frame of classical dynamical theory of X-ray
diffraction \cite{SHVID}. This analysis shows that a crystal in
Bragg reflection geometry transforms the incident electric field
$E(x,t)$ in the $\{x,t\}$ domain into $E(x- a t, t)$, that is the
field of a pulse with a less well-known distortion, first studied in
\cite{GABO}. The physical meaning of this distortion is that the
beam spot size is independent of time, but the beam central position
changes as the pulse evolves in time. Here we show in a simple
manner that, based on the use only Bragg law, we may arrive directly
to explanation of spatiotemporal coupling phenomena in the dynamical
theory of X-ray diffraction \cite{OURTILT}.
Let us consider an electromagnetic plane wave in the X-ray frequency
range incident on an infinite, perfect crystal. Within the
kinematical approximation, according to the Bragg law, constructive
interference of waves scattered from the crystal occurs if the angle
of incidence, $\theta_i$ and the wavelength, $\lambda$, are related
by the well-known relation
\begin{eqnarray}
\lambda = 2 d \sin \theta_i ~. \label{bragg}
\end{eqnarray}
assuming reflection into the first order. This equation shows that
for a given wavelength of the X-ray beam, diffraction is possible
only at certain angles determined by the interplanar spacings $d$.
It is important to remember the following geometrical relationships:
1. The angle between the incident X-ray beam and normal to the
reflection plane is equal to that between the normal and the
diffracted X-ray beam. In other words, Bragg reflection is a mirror
reflection, and the incident angle is equal to the diffracted angle
($\theta_i = \theta _D$).
2. The angle between the diffracted X-ray beam and the transmitted
one is always $2 \theta_i$. In other words, incident beam and
forward diffracted (i.e. transmitted) beam have the same direction.
We now turn our attention beyond the kinematical approximation to
the dynamical theory of X-ray diffraction by a crystal. An optical
element inserted into the X-ray beam is supposed to modify some
properties of the beam as its width, its divergence, or its spectral
bandwidth. It is useful to describe the modification of the beam by
means of a transfer function. The transmisivity curve - the
transmittance - in Bragg geometry can be expressed in the frame of
dynamical theory as
\begin{eqnarray}
T(\theta_i,\omega) = T(\Delta \omega + \omega_B \Delta \theta \cot
\theta_B ) ~, \label{reflectance}
\end{eqnarray}
where $\Delta \omega = (\omega - \omega_B)$ and $\Delta \theta =
(\theta_i - \theta_B)$ are the deviations of frequency and incident
angle of the incoming beam from the Bragg frequency and Bragg angle,
respectively. The frequency $\omega_B$ and the angle $\theta_B$ are
given by the Bragg law: $\omega_B \sin \theta_B = \pi c/d$. Here we
followed the usual procedure of expanding $\omega$ in a Taylor
series about $\omega_B$, so that
\begin{eqnarray}
\omega = \omega_B + (d \omega/d \theta)_B (\theta - \theta_B) + ...
~. \label{taylor}
\end{eqnarray}
Consider now a perfectly collimated, white beam incident on the
crystal. In kinematical approximation $T$ is a Dirac
$\delta$-function, which is simply represented by the differential
form of Bragg law:
\begin{eqnarray}
d \lambda/d \theta_i = \lambda \cot \theta_i ~. \label{difform}
\end{eqnarray}
In contrast to this, in dynamical theory the reflectivity width is
finite. This means that there is a reflected beam even when incident
angle and wavelength of the incoming beam are not related exactly by
Bragg equation. It is interesting to note that the geometrical
relationships 1. and 2. are still valid in the framework of
dynamical theory. In particular, reflection in dynamical theory is
always a mirror reflection. We underline here that if we have a
perfectly collimated, white incident beam, we also have a perfectly
collimated reflected and transmitted beam. Its bandwidth is related
with the width of the reflectivity curve. We will regard the beam as
perfectly collimated when the angular spread of the beam is much
smaller than the angular width of the transfer function $T$. It
should be realized that the crystal does not introduce an angular
dispersion similar to a grating or a prism. However, a more detailed
analysis based on the expression for the reflectivity, Eq.
(\ref{reflectance}), shows that a less well-known spatiotemporal
coupling exists. The fact that the reflectivity is invariant under
angle and frequency transformations obeying
\begin{eqnarray}
\Delta \omega + \omega_B \Delta \theta \cot \theta_B =
\mathrm{const}~ \label{transform}
\end{eqnarray}
is evident, and corresponds to the coupling in the Fourier domain
$\{k_x, \omega\}$. The origin of this relation is kinematical, in
the sense that it follows from Bragg diffraction. One might be
surprised that the field transformation derived in \cite{SHVID} for
an XFEL pulse after a crystal in the $\{x,t\}$ domain is given by
\begin{eqnarray}
E_{\mathrm{out}} (x,t) = FT[T(\Delta \omega,
k_x)E_{\mathrm{in}}(\Delta \omega, k_x)] = E(x- c t\cot \theta_B ,
t)~, \label{Eoutxt}
\end{eqnarray}
where $FT$ indicates a Fourier transform from the $\{k_x,\omega\}$
to the $\{x,t\}$ domain, and $k_x = \omega_B \Delta \theta/c$. In
general, one would indeed expect the transformation to be symmetric
in both the $\{k_x,\omega\}$ and in the $\{x,t\}$ domain due to the
symmetry of the transfer function\footnote{There is a breaking of
the symmetry in the diffracted beam in the $\{k_x,\omega\}$ domain.
While the symmetry is present at the level of the transfer function,
it is not present anymore when one considers the incident beam. We
point out that symmetry breaking in \cite{SHVID} is a result of the
approximation of temporal profile of the incident wave to a Dirac
$\delta$-function.}. However, it is reasonable to expect the
influence of a nonsymmetric input beam distribution. In the
self-seeding case, the incoming XFEL beam is well collimated,
meaning that its angular spread is a few times smaller than angular
width of the transfer function. Only the bandwidth of the incoming
beam is much wider than the bandwidth of the transfer function. In
this limit, we can approximate the transfer function in the
expression for the inverse temporal Fourier transform as a Dirac
$\delta$-function. This gives
\begin{eqnarray}
&& E_\mathrm{out}(x,t) = FT[R(\Delta \omega, k_x) E_\mathrm{in}
(\Delta \omega, k_x)]\cr && = \xi(t) \cdot \frac{1}{2\pi} \int d
k_x \exp(-i k_x c t \cot \theta_B) \exp(i k_x x)
E_\mathrm{in}(0,k_x)\cr && = \xi(t) b(x - c t\cot \theta_B) ~,
\label{eoutdel}
\end{eqnarray}
where we applied the Shift Theorem twice, and where
\begin{eqnarray}
\xi(t) = \frac{1}{2\pi} \int d Y \exp( i Y t) R(Y) \label{tempFT}
\end{eqnarray}
is the inverse temporal Fourier transform of the reflectivity curve.
The spatial shift given by Eq. (\ref{Eoutxt}) is proportional to
$\cot(\theta_B)$, and is maximal in the range for small $\theta_B$.
If we want to use long delay $c t \sim 15 \mu$m, a rms transverse
size of radiation pulse of about $15 \mu$m limits the maximum
acceptable value of $\cot(\theta_b)$ to $1-1.5$. Thus, the
spatiotemporal coupling is an issue, and efforts are necessarily
required to avoid distortion. It is worth mentioning that this
distortion is easily suppressed by the right choice of crystal
planes.
\section{FEL studies}
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5curr.eps}
\includegraphics[width=0.5\textwidth]{3p5emit.eps}
\includegraphics[width=0.5\textwidth]{3p5gam.eps}
\includegraphics[width=0.5\textwidth]{3p5siggam.eps}
\begin{center}
\includegraphics[width=0.5\textwidth]{3p5wake.eps}
\end{center}
\caption{Results from electron beam start-to-end simulations at the
entrance of the undulator system of the bio-imaging beamline
\cite{S2ER} for the hard X-ray case for the 17.5 GeV mode of
operation. (First Row, Left) Current profile. (First Row, Right)
Normalized emittance as a function of the position inside the
electron beam. (Second Row, Left) Energy profile along the beam,
lower curve. The effects of resistive wakefields along SASE1 are
illustrated by the comparison with the upper curve, referring to the
entrance of SASE1 (Second Row, Right) Electron beam energy spread
profile, upper curve. The effects of quantum diffusion along SASE1
are illustrated by the comparison with the lower curve, referring to
the entrance of SASE1. (Bottom row) Resistive wakefields in the
SASE3 undulator \cite{S2ER}.} \label{biof2f3}
\end{figure}
With reference to Fig. \ref{bio3f1} we performed feasibility studies
pertaining different energy ranges considered in this article. These
studies were performed with the help of the FEL code GENESIS 1.3
\cite{GENE} running on a parallel machine. Simulations are based on
a statistical analysis consisting of $100$ runs.
The main undulator parameters are reported in Table \ref{tt1}.
Operation is foreseen at two different energies: $10.5$ GeV and
$17.5$ GeV. The lower energy is used in the very soft X-ray regime,
between $0.3$ keV and $0.5$ keV. For this case, we refer to
\cite{OSOF} for a summary of the electron beam characteristics at
the entrance of the setup. The case for $17.5$ GeV instead is
summarized in Fig. \ref{biof2f3}, where we plot the results of
start-to-end simulations \cite{S2ER}.
\begin{table}
\caption{Undulator parameters}
\begin{small}\begin{tabular}{ l c c}
\hline & ~ Units & ~ \\ \hline
Undulator period & mm & 68 \\
Periods per cell & - & 73 \\
Total number of cells & - & 40 \\
Intersection length & m & 1.1 \\
Photon energy & keV & 0.3-13 \\
\hline
\end{tabular}\end{small}
\label{tt1}
\end{table}
\subsection{Soft X-ray photon energy range}
Production of soft X-rays with photon energies below $1$ keV is
enabled by configuring the setup as described in Fig. \ref{bio3f2}.
A feasibility study for this case has been already carried out in
\cite{OSOF}, to which we refer the reader for further details and
simulation results.
\subsection{Photon energy range between 2 keV and 3 keV}
We now turn to analyze the case described in Fig. \ref{bio3f3},
which pertains the energy range between $2$ keV and $3$ keV. The
electron beam energy here is $17.5$ GeV.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5sigx.eps}
\includegraphics[width=0.5\textwidth]{3p5sigy.eps}
\caption{Evolution of the horizontal (left plot) and vertical (right
plot) dimensions of the electron bunch as a function of the distance
inside the undulator. The plots refer to the longitudinal position
inside the bunch corresponding to the maximum current value.}
\label{sigma}
\end{figure}
The expected beam parameters at the entrance of the bio-imaging
beamline undulator, and the resistive wake inside the undulator are
shown in Fig. \ref{biof2f3}. The evolution of the transverse
electron bunch dimensions are plotted in Fig. \ref{sigma}. Since the
electron energy is fixed to $17.5$ GeV, both expected beam
parameters and evolution of the transverse beam dimensions before
the tapered part are valid for the different energy ranges treated
in the following Sections.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5pin.eps}
\includegraphics[width=0.5\textwidth]{2p5spin.eps}
\caption{Power and spectrum before the first magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{biof17}
\end{figure}
We begin our investigation by simulating the SASE power and spectrum
after the first $4$ undulator cells, that is before the first
magnetic chicane. Results are shown in Fig. \ref{biof17}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5pseed.eps}
\includegraphics[width=0.5\textwidth]{2p5spseed.eps}
\caption{Power and spectrum after the first magnetic chicane and
soft X-ray monochromator. Grey lines refer to single shot
realizations, the black line refers to the average over a hundred
realizations.} \label{biof18}
\end{figure}
The magnetic chicane is switched on, an the soft X-ray monochromator
is inserted. Assuming a monochromator efficiency of $10\%$, a
Gaussian line, and a resolving power of $5000$ we can filter the
incoming radiation pulse in Fig. \ref{biof17} accordingly, to obtain
the power and spectrum in Fig. \ref{biof18}. This power and spectrum
are used for seeding.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5curraft2ndc.eps}
\includegraphics[width=0.5\textwidth]{2p5gamaft2ndc.eps}
\begin{center}
\includegraphics[width=0.5\textwidth]{2p5siggamaft2ndc.eps}
\end{center}
\caption{Electron beam characteristics after the second magnetic
chicane. (First Row, Left) Current profile. (First Row, Right)
Energy profile along the beam. (Second Row) Electron beam energy
spread profile. } \label{biof19}
\end{figure}
Since we now deal with a conventional grating monochromator, the
photon pulse is delayed with respect to the electron pulse. In our
study case we assume a relatively large delay of about $1$ ps. In
order to compensate for such delay, one needs a chicane with a
relatively large dispersion strength $R_{56} \sim 0.6$ mm. In
principle we cannot neglect the effects of the chicane dispersion on
the electron bunch properties. We accounted for them with the help
of the code Elegant \cite{ELEG}, which was used to propagate the
electron beam distribution through the chicane. Results are shown in
Fig. \ref{biof19}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5poutall2.eps}
\includegraphics[width=0.5\textwidth]{2p5spoutall2.eps}
\caption{Power and spectrum at the fundamental harmonic after the
second chicane equipped with the X-ray optical delay line, delaying
the radiation pulse with respect to the electron bunch. Grey lines
refer to single shot realizations, the black line refers to the
average over a hundred realizations.} \label{biof20}
\end{figure}
Since the $R_{56}$ is large enough to wash out the electron beam
microbunching, we assume a fresh bunch at the entrance of the
following undulator part constituted by $3$ undulator cells. This
means that the results in Fig. \ref{biof19} are taken to generate a
new beam file to be fed into GENESIS. The electron bunch is now
seeded with the monochromatized radiation pulse in Fig.
\ref{biof18}, so that the seed is amplified in the $3$ undulator
cells following the chicane. After that, the electron beam is sent
through the second chicane, while the radiation pulse goes through
the X-ray optical delay line described in Fig. \ref{biof2}, where
the radiation pulse is delayed of about $6$ fs with respect to the
electron beam, as shown in Fig. \ref{biof9}. The power and spectrum
of the radiation pulse after the optical delay line are shown in
Fig. \ref{biof20}, where the combined effect of the optical delay
and of the magnetic chicane is illustrated. This results in an
overall delay of $6$ fs. Note that the use of the mirror chicane
also allows for an $R_{56}$ in the order of ten microns. Thus, the
combination of magnetic chicane and mirror chicane also allows for
removing the electron microbunching produced in the second
undulator.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5poutall4.eps}
\includegraphics[width=0.5\textwidth]{2p5spoutall4.eps}
\caption{Power and spectrum at the fundamental harmonic at the exit
of the third undulator and before the third magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{biof21}
\end{figure}
Besides allowing for the installation of the optical delay line,
which delays the radiation pulse of about half of the electron bunch
size, the second chicane also smears out the microbunching in the
electron bunch. As a result, at the entrance of the third undulator
part the electron bunch can be considered as unmodulated, and half
of it is seeded with the radiation pulse. The seeded half of the
electron bunch amplifies the seed in the third undulator part,
composed by four cells. The seeded part of the electron bunch is now
spent, and its quality has deteriorated too much for further lasing.
After that, electrons and radiation are separated once more going
through the third chicane. The hard X-ray self-seeding crystal is
out, and the chicane simply acts as a delay line for the electron
beam, which also smears out the microbunching. Power and spectrum
following the third chicane are shown in Fig. \ref{biof21}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5eledelay3h.eps}
\includegraphics[width=0.5\textwidth]{2p5spoutnotap3h.eps}
\caption{Power and spectrum at the third harmonic after the third
magnetic chicane. Grey lines refer to single shot realizations, the
black line refers to the average over a hundred realizations.}
\label{biof21}
\end{figure}
By tuning the third chicane in the proper way, one can superimpose
the radiation beam onto that part of the electron bunch that has not
been seeded in the third undulator part. This is fresh, and can lase
again in the fourth undulator part. Fig. \ref{biof21} shows the
effect of the magnetic chicane, which delays the electron bunch
relative to the radiation pulse in order to allow for the seeding of
the fresh part of the bunch. The radiation beam includes a relevant
third-harmonic content, just slightly below the GW level as can be
seen in Fig. \ref{biof21}, and is sufficient to act as a seed in the
last part of the undulator. The fourth undulator part is not tuned
at the fundamental harmonic, but rather at the third harmonic. This
allows to reach the photon energy range between $2$ keV and $3$ keV.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.5\textwidth]{2p5taplaw.eps}
\end{center}
\caption{Tapering law for the case $\lambda = 0.5$ nm.}
\label{biof22}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5sigx.eps}
\includegraphics[width=0.5\textwidth]{2p5sigy.eps}
\caption{Evolution of the horizontal (left plot) and vertical (right
plot) dimensions of the electron bunch as a function of the distance
inside the tapered part of the undulator at $\lambda = 0.5$ nm. The
plots refer to the longitudinal position inside the bunch
corresponding to the maximum current value. The quadrupole strength
varies along the undulator, and is tuned for optimum output.}
\label{sigmaaa}
\end{figure}
The fourth and last part of the undulator is composed by $29$ cells,
interrupted by a chicane which is switched off, and will be used in
different energy ranges. The last undulator part is partly tapered
post-saturation to allow for increasing the region where electrons
and radiation interact properly to the advantage of the radiation
pulse. Tapering is implemented by changing the $K$ parameter of the
undulator segment by segment according to Fig. \ref{biof22}. The
tapering law used in this work has been implemented on an empirical
basis, and the output has been optimized also by varying the
quadrupole strength as shown in Fig. \ref{sigmaaa}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5Pouttap.eps}
\includegraphics[width=0.5\textwidth]{2p5specouttap.eps}
\caption{Final output. Power and spectrum at the third harmonic
after tapering. Grey lines refer to single shot realizations, the
black line refers to the average over a hundred realizations.}
\label{biof23}
\end{figure}
The use of tapering together with monochromatic radiation is
particularly effective, since the electron beam does not experience
brisk changes of the ponderomotive potential during the slippage
process. The final output is presented in Fig. \ref{biof23} in terms
of power and spectrum. As one can see, simulations indicate an
output power of about $2$ TW.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5enouttap.eps}
\includegraphics[width=0.5\textwidth]{2p5varouttap.eps}
\caption{Final output. Energy and energy variance of output pulses
for the case $\lambda = 0.5$ nm. In the left plot, grey lines refer
to single shot realizations, the black line refers to the average
over a hundred realizations.} \label{biof24}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{2p5raddiv.eps}
\includegraphics[width=0.5\textwidth]{2p5radsize.eps}
\caption{Final output. X-ray radiation pulse energy distribution per
unit surface and angular distribution of the X-ray pulse energy at
the exit of output undulator for the case $\lambda = 0.5$ nm.}
\label{biof25}
\end{figure}
The energy of the radiation pulse and the energy variance are shown
in Fig. \ref{biof24} as a function of the position along the
undulator. The divergence and the size of the radiation pulse at the
exit of the final undulator are shown, instead, in Fig.
\ref{biof25}. In order to calculate the size, an average of the
transverse intensity profiles is taken. In order to calculate the
divergence, the spatial Fourier transform of the field is
calculated.
\subsection{Photon energy range between 3 keV and 5 keV}
We now consider generation of radiation in the photon energy between
3 keV and 5 keV, with reference to Fig. \ref{bio3f4}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5pin.eps}
\includegraphics[width=0.5\textwidth]{3p5specin.eps}
\caption{Power and spectrum before the second magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{biof173p5}
\end{figure}
For this mode of operation, the first chicane is switched off, so
that the first part of the undulator effectively consists of 7
uniform cells. We begin our investigation by simulating the SASE
power and spectrum after the first part of the undulator, that is
before the second magnetic chicane in the setup. Results are shown
in Fig. \ref{biof173p5}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5Pseed.eps}
\includegraphics[width=0.5\textwidth]{3p5spseed.eps}
\caption{Power and spectrum after the single crystal self-seeding
X-ray monochromator. A $100~\mu$m thick diamond crystal in Bragg
transmission geometry ( C(111) reflection, $\sigma$-polarization )
is used. Grey lines refer to single shot realizations, the black
line refers to the average over a hundred realizations. The black
arrow indicates the seeding region.} \label{biof1830p5}
\end{figure}
The second magnetic chicane is switched on, and the single-crystal
X-ray monochromator is set into the photon beam. For the 3 keV - 5
keV energy range we use a $100~\mu$m-thick diamond crystal in Bragg
transmission geometry. In particular, we take advantage of the
C(111) reflection, $\sigma$-polarization. The crystal acts as a
bandstop filter, and the output spectrum is plotted in Fig.
\ref{biof1830p5} (right). Due to the bandstop effect, the signal in
the time domain exhibits a long monochromatic tail, which is used
for seeding, Fig. \ref{biof1830p5} (left). To this purpose, the
electron bunch is slightly delayed by proper tuning of the magnetic
chicane to be superimposed to the seeding signal.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5cotgX.eps}
\includegraphics[width=0.5\textwidth]{3p5cotgY.eps}
\caption{Comparison between transverse profile of the seed field
intensity and transverse profile of the electron beam density at the
position used for seeding at $\lambda = 0.36$ nm. The plots refer to
the longitudinal position inside the bunch corresponding to the
maximum current value.} \label{compare}
\end{figure}
It should be remarked that, according to Section \ref{spatio},
spatio-temporal coupling induced by the crystal monochromator should
be accounted for in our study. In our simulations we calculate the
temporal profile of the wake by convolving the incoming radiation
pulse with the impulse response of the crystal. Since the incoming
radiation pulse has a finite length, it follows that the average
wake profile is different compared with the impulse response.
However, as concerns the inclusion of spatiotemporal coupling
effects, we based our analysis on the result discussed in Section
\ref{spatio}. This result was derived under the assumption that the
incoming radiation pulse is a Dirac $\delta$-function. In this case,
our wake in the time domain simply coincides with the impulse
response function of the crystal. The reason for such approximation
can be explained by comparing the typical delay associated with the
impulse response (about 20 $\mu$m) with the pulse duration (about 3
$\mu$m). From this comparison follows that we can neglect the length
of the incoming pulse with accuracy of about $10\% - 20\%$.
As already found in Section \ref{spatio}, the beam spot size is
independent of time, but the beam central position changes as the
pulse evolves in time. The transverse dependence of the electric
field, in fact, obeys Eq. (\ref{Eoutxt}). We account for this effect
automatically in our simulations. A comparison between the
transverse field profile used for seeding without accounting for the
spatio-temporal coupling is shown in Fig. \ref{compare}. This
feature is accounted for in calculations.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5opdelay.eps}
\includegraphics[width=0.5\textwidth]{3p5specdel.eps}
\caption{Power and spectrum after the third chicane equipped with
the X-ray optical delay line, delaying the radiation pulse with
respect to the electron bunch. Grey lines refer to single shot
realizations, the black line refers to the average over a hundred
realizations.} \label{biof2030p5}
\end{figure}
Following the seeding setup, the electron bunch amplifies the seed
in the following 4 undulator cells. After that, a third chicane is
used to allow for the installation of an x-ray optical delay line,
which retards the radiation pulse with respect to the electron
bunch. The power and spectrum of the radiation pulse after the
optical delay line are shown in Fig. \ref{biof2030p5}, where the
combined effect of the optical delay is illustrated.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5eldelay.eps}
\includegraphics[width=0.5\textwidth]{3p5speceldelay.eps}
\caption{Power and spectrum after the last magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{biof213p5}
\end{figure}
Due to the presence of the optical delay, only part of the electron
beam is used to further amplify the radiation pulse in the following
6 undulator cells. The electron beam part which has not lased is
fresh, and can be used for further lasing. In order to do so, after
amplification, the electron beam passes through the final magnetic
chicane, which delays the electron beam. The power and spectrum of
the radiation pulse after the last magnetic chicane are shown in
Fig. \ref{biof213p5}. By delaying the electron bunch, the magnetic
chicane effectively shifts forward the photon beam with respect to
the electron beam. Tunability of such shift allows the selection of
different photon pulse lengths. Moreover, an additional advantage of
brought by the use of a fresh bunch technique in this mode of
operation is the suppression of the intensity fluctuations of the
seed down to $40 \%$ in the nonlinear regime before the last
chicane. This suppression of fluctuations is useful in connection
with the application of the tapering technique in the last part of
the undulator. Fluctuations of the seed in the linear regime are, at
variance, close to $100 \%$.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.5\textwidth]{3p5taplaw.eps}
\end{center}
\caption{Tapering law for the case $\lambda = 0.36$ nm.}
\label{biof223p5}
\end{figure}
The last part of the undulator is composed by $23$ cells. It is
partly tapered post-saturation, to increase the region where
electrons and radiation interact properly to the advantage of the
radiation pulse. Tapering is implemented by changing the $K$
parameter of the undulator segment by segment according to Fig.
\ref{biof223p5}. The tapering law used in this work has been
implemented on an empirical basis.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5Pouttap.eps}
\includegraphics[width=0.5\textwidth]{3p5spouttap.eps}
\caption{Final output. Power and spectrum at the third harmonic
after tapering. Grey lines refer to single shot realizations, the
black line refers to the average over a hundred realizations.}
\label{biof233p5}
\end{figure}
The use of tapering together with monochromatic radiation is
particularly effective, since the electron beam does not experience
brisk changes of the ponderomotive potential during the slippage
process. The final output is presented in Fig. \ref{biof233p5} in
terms of power and spectrum. As one can see, simulations indicate an
output power of about $1.5$ TW.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5enouttap.eps}
\includegraphics[width=0.5\textwidth]{3p5varouttap.eps}
\caption{Final output. Energy and energy variance of output pulses
for the case $\lambda = 0.36$ nm. In the left plot, grey lines refer
to single shot realizations, the black line refers to the average
over a hundred realizations.} \label{biof243p5}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{3p5raddiv.eps}
\includegraphics[width=0.5\textwidth]{3p5radsize.eps}
\caption{Final output. X-ray radiation pulse energy distribution per
unit surface and angular distribution of the X-ray pulse energy at
the exit of output undulator for the case $\lambda = 0.36$ nm.}
\label{biof2530p5}
\end{figure}
The energy of the radiation pulse and the energy variance are shown
in Fig. \ref{biof243p5} as a function of the position along the
undulator. The divergence and the size of the radiation pulse at the
exit of the final undulator are shown, instead, in Fig.
\ref{biof2530p5}. In order to calculate the size, an average of the
transverse intensity profiles is taken. In order to calculate the
divergence, the spatial Fourier transform of the field is
calculated.
\subsection{Photon energy range between 5 keV and 7 keV}
Operation in the photon range between 5 keV and 7 keV will be
possible by configuring the bio-imaging beamline as in Fig.
\ref{bio3f5}. The configuration is very similar to the case for the
range between 3 keV and 5 keV. The only difference is that the
C(220) reflection is used instead of the C(111).
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{5pin.eps}
\includegraphics[width=0.5\textwidth]{5spin.eps}
\caption{Power and spectrum before the second magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{biof175}
\end{figure}
As before, the first chicane is switched off, so that the first
part of the undulator effectively consists of 7 uniform cells. We
begin our investigation by simulating the SASE power and spectrum
after the first part of the undulator, that is before the second
magnetic chicane in the setup. Results are shown in Fig.
\ref{biof175}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{5pseed220.eps}
\includegraphics[width=0.5\textwidth]{5spseed.eps}
\caption{Power and spectrum after the single crystal self-seeding
X-ray monochromator. A $100~\mu$m thick diamond crystal in Bragg
transmission geometry ( C(220) reflection, $\sigma$-polarization )
is used. Grey lines refer to single shot realizations, the black
line refers to the average over a hundred realizations. The black
arrow indicates the seeding region.} \label{biof183p5}
\end{figure}
The second magnetic chicane is switched on, and the single-crystal
X-ray monochromator is set into the photon beam. For the 5 keV - 7
keV energy range we use a $100~\mu$m-thick diamond crystal in Bragg
transmission geometry. In particular, we take advantage of the
C(220) reflection, $\sigma$-polarization, Fig. \ref{biof183p5}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{5pphdelay.eps}
\includegraphics[width=0.5\textwidth]{5spout1.eps}
\caption{Power and spectrum after the third chicane equipped with
the X-ray optical delay line, delaying the radiation pulse with
respect to the electron bunch. Grey lines refer to single shot
realizations, the black line refers to the average over a hundred
realizations.} \label{biof203p5}
\end{figure}
Following the seeding setup, the electron bunch amplifies the seed
in the following 4 undulator cells. After that, a third chicane is
used to allow for the installation of an x-ray optical delay line,
which delays the radiation pulse with respect to the electron bunch.
The power and spectrum of the radiation pulse after the optical
delay line are shown in Fig. \ref{biof203p5}, where the effect of
the optical delay is illustrated.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{5peledelay.eps}
\includegraphics[width=0.5\textwidth]{5spout3.eps}
\caption{Power and spectrum after the last magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{biof215}
\end{figure}
Due to the presence of the optical delay, only part of the electron
beam is used to further amplify the radiation pulse in the following
6 undulator cells. The electron beam part which is not used is
fresh, and can be used for further lasing. In order to do so, after
amplification, the electron beam passes through the final magnetic
chicane, which delays the electron beam. The power and spectrum of
the radiation pulse after the last magnetic chicane are shown in
Fig. \ref{biof215}. By delaying the electron bunch, the magnetic
chicane effectively shifts forward the photon beam with respect to
the electron beam. Tunability of such shift allows the selection of
different photon pulse length.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.5\textwidth]{5taplaw.eps}
\end{center}
\caption{Tapering law for the case $\lambda = 0.25$ nm.}
\label{biof225}
\end{figure}
The last part of the undulator is composed by $23$ cells. It is
partly tapered post-saturation, to increase the region where
electrons and radiation interact properly to the advantage of the
radiation pulse. Tapering is implemented by changing the $K$
parameter of the undulator segment by segment according to Fig.
\ref{biof225}. The tapering law used in this work has been
implemented on an empirical basis.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{5Pouttap.eps}
\includegraphics[width=0.5\textwidth]{5spouttap.eps}
\caption{Final output. Power and spectrum at the third harmonic
after tapering. Grey lines refer to single shot realizations, the
black line refers to the average over a hundred realizations.}
\label{biof235}
\end{figure}
The use of tapering together with monochromatic radiation is
particularly effective, since the electron beam does not experience
brisk changes of the ponderomotive potential during the slippage
process. The final output is presented in Fig. \ref{biof235} in
terms of power and spectrum. As one can see, simulations indicate an
output power of about $1.5$ TW.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{5eneouttap.eps}
\includegraphics[width=0.5\textwidth]{5varouttap.eps}
\caption{Final output. Energy and energy variance of output pulses
for the case $\lambda = 0.25$ nm. In the left plot, grey lines refer
to single shot realizations, the black line refers to the average
over a hundred realizations.} \label{biof245}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{5raddiv.eps}
\includegraphics[width=0.5\textwidth]{5radsize.eps}
\caption{Final output. X-ray radiation pulse energy distribution per
unit surface and angular distribution of the X-ray pulse energy at
the exit of output undulator for the case $\lambda = 0.25$ nm.}
\label{biof253p5}
\end{figure}
The energy of the radiation pulse and the energy variance are shown
in Fig. \ref{biof245} as a function of the position along the
undulator. The divergence and the size of the radiation pulse at the
exit of the final undulator are shown, instead, in Fig.
\ref{biof253p5}. In order to calculate the size, an average of the
transverse intensity profiles is taken. In order to calculate the
divergence, the spatial Fourier transform of the field is
calculated.
\subsection{Photon energy range between 7 keV and 9 keV}
We now consider the photon energy range between 7 keV and 9 keV. In
this case the beamline will be configured as in Fig. \ref{bio3f6}. A
feasibility study dealing with this energy range can be found in
\cite{OURCC}.
\subsection{Photon energy range between 9 keV and 13 keV}
Finally, we consider generation of radiation in the photon energy
between 9 keV and 13 keV. The undulator line will be configured as
in Fig. \ref{bio3f4}, the only difference now being that the final
$23$ cells will be tuned at the third harmonic of the fundamental,
thus allowing to reach the photon energy between 9 keV and 13 keV.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12Pin.eps}
\includegraphics[width=0.5\textwidth]{12SPin.eps}
\caption{Power and spectrum before the second magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{12biof173p5}
\end{figure}
For this mode of operation, the first chicane is switched off, so
that the first part of the undulator effectively consists of 7
uniform cells. We begin our investigation by simulating the SASE
power and spectrum after the first part of the undulator, that is
before the second magnetic chicane in the setup. Results are shown
in Fig. \ref{12biof173p5}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12Pseed1.eps}
\includegraphics[width=0.5\textwidth]{12SPseed1.eps}
\caption{Power and spectrum after the single crystal self-seeding
X-ray monochromator. A $100~\mu$m thick diamond crystal in Bragg
transmission geometry ( C(111) reflection, $\sigma$-polarization )
is used. Grey lines refer to single shot realizations, the black
line refers to the average over a hundred realizations. The black
arrow indicates the seeding region.} \label{12biof1830p5}
\end{figure}
The second magnetic chicane is switched on, and the single-crystal
X-ray monochromator is set into the photon beam. Since we want to
generate radiation in the 3 keV - 4 keV energy range, we use a
$100~\mu$m-thick diamond crystal in Bragg transmission geometry. In
particular, we take advantage of the C(111) reflection,
$\sigma$-polarization. The crystal acts as a bandstop filter, and
the output spectrum is plotted in Fig. \ref{12biof1830p5} (right).
Due to the bandstop effect, the signal in the time domain exhibits a
long monochromatic tail, which is used for seeding, Fig.
\ref{12biof1830p5} (left). To this purpose, the electron bunch is
slightly delayed by proper tuning of the magnetic chicane to be
superimposed to the seeding signal. The difference with respect to
Fig. \ref{biof1830p5} is in a slightly different frequency, since in
this Section we want to study the feasibility of production of $12$
keV radiation.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12cotgX.eps}
\includegraphics[width=0.5\textwidth]{12cotgY.eps}
\caption{Comparison between transverse profile of the seed field
intensity and transverse profile of the electron beam density at the
position used for seeding at $\lambda = 0.3$ nm. The plots refer to
the longitudinal position inside the bunch corresponding to the
maximum current value.} \label{12compare}
\end{figure}
As before it should be remarked that, according to Section
\ref{spatio}, spatio-temporal coupling induced by the crystal
monochromator should be accounted for in our study. A comparison
between the electron beam position and the photon beam position
after the crystal is shown in Fig. \ref{12compare}. The shift
difference is due to the spatio-temporal coupling induced by the
crystal, and must be accounted for in calculations. The longitudinal
position at which Fig. \ref{12compare} refers is that of the seeding
peak, compared to the lasing part of the bunch.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12Pdelayph.eps}
\includegraphics[width=0.5\textwidth]{12SPdelayph.eps}
\caption{Power and spectrum after the third chicane equipped with
the X-ray optical delay line, delaying the radiation pulse with
respect to the electron bunch. Grey lines refer to single shot
realizations, the black line refers to the average over a hundred
realizations.} \label{12biof2030p5}
\end{figure}
Following the seeding setup, the electron bunch amplifies the seed
in the following 4 undulator cells. After that, a third chicane is
used to allow for the installation of an x-ray optical delay line,
which retards the radiation pulse with respect to the electron
bunch. The power and spectrum of the radiation pulse after the
optical delay line are shown in Fig. \ref{12biof2030p5}, where the
effect of the optical delay is illustrated.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12Pdelayel3.eps}
\includegraphics[width=0.5\textwidth]{12SPdelayel3.eps}
\caption{Power and spectrum after the last magnetic chicane. Grey
lines refer to single shot realizations, the black line refers to
the average over a hundred realizations.} \label{12biof213p5}
\end{figure}
Due to the presence of the optical delay, only part of the electron
beam is used to further amplify the radiation pulse in the following
6 undulator cells. The electron beam part which has not lased is
fresh, and can be used for further lasing. In order to do so, after
amplification, the electron beam passes through the final magnetic
chicane, which delays the electron beam. During the previous
amplification, a consistent amount of bunching at the third harmonic
of the fundamental is produced. The power and spectrum of the
radiation pulse at the third harmonic after the last magnetic
chicane are shown in Fig. \ref{12biof213p5}. By delaying the
electron bunch, the magnetic chicane effectively shifts forward the
photon beam with respect to the electron beam. Tunability of such
shift allows the selection of different photon pulse lengths.
\begin{figure}[tb]
\begin{center}
\includegraphics[width=0.5\textwidth]{12Taplaw.eps}
\end{center}
\caption{Tapering law for the case $\lambda = 0.1$ nm.}
\label{12biof223p5}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12sigX.eps}
\includegraphics[width=0.5\textwidth]{12sigY.eps}
\caption{Evolution of the horizontal (left plot) and vertical (right
plot) dimensions of the electron bunch as a function of the distance
inside the the tapered part of the undulator at $\lambda = 0.1$ nm.
The plots refer to the longitudinal position inside the bunch
corresponding to the maximum current value. The quadrupole strength
is varied along the undulator axis in order to optimize the output.}
\label{12sigma}
\end{figure}
The last part of the undulator is composed by $23$ cells. It is
tuned at the third harmonic of the fundamental, i.e. around 12 keV
in our case, and partly tapered post-saturation, to increase the
region where electrons and radiation interact properly to the
advantage of the radiation pulse. Tapering is implemented by
changing the $K$ parameter of the undulator segment by segment
according to Fig. \ref{12biof223p5}. The tapering law used in this
work has been implemented on an empirical basis, and the output has
been optimized also by varying the quadrupole strength as shown in
Fig. \ref{12sigma}.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12Pout3.eps}
\includegraphics[width=0.5\textwidth]{12SPout3.eps}
\caption{Final output. Power and spectrum at the third harmonic
after tapering. Grey lines refer to single shot realizations, the
black line refers to the average over a hundred realizations.}
\label{12biof233p5}
\end{figure}
As usual, combining tapering with monochromatic radiation generation
is particularly effective, since the electron beam does not
experience brisk changes of the ponderomotive potential during the
slippage process. The final output is presented in Fig.
\ref{12biof233p5} in terms of power and spectrum. As one can see,
simulations indicate an output power of about 0.5 TW.
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12outvar.eps}
\includegraphics[width=0.5\textwidth]{12outene.eps}
\caption{Final output. Energy and energy variance of output pulses
for the case $\lambda = 0.1$ nm. In the left plot, grey lines refer
to single shot realizations, the black line refers to the average
over a hundred realizations.} \label{12biof243p5}
\end{figure}
\begin{figure}[tb]
\includegraphics[width=0.5\textwidth]{12divout.eps}
\includegraphics[width=0.5\textwidth]{12sizeout.eps}
\caption{Final output. X-ray radiation pulse energy distribution per
unit surface and angular distribution of the X-ray pulse energy at
the exit of output undulator for the case $\lambda = 0.1$ nm.}
\label{12biof2530p5}
\end{figure}
The energy of the radiation pulse and the energy variance are shown
in Fig. \ref{12biof243p5} as a function of the position along the
undulator. The divergence and the size of the radiation pulse at the
exit of the final undulator are shown, instead, in Fig.
\ref{12biof2530p5}. In order to calculate the size, an average of
the transverse intensity profiles is taken. In order to calculate
the divergence, the spatial Fourier transform of the field is
calculated.
\section{Conclusions}
The highest priority for bioimaging experiments at any advanced XFEL
facility is to establish a dedicated beamline for studying
biological objects at the mesoscale, including large macromolecules,
macromolecular complexes, and cell organelles. This requires 2 keV -
6 keV photon energy range and TW peak power pulses. However, higher
photon energies are needed to reach anomalous edges of commonly used
elements (such as Se) for anomalous experimental phasing. Studies at
intermediate resolutions need access to the water window
\cite{BERG}.
A conceptual design of a dedicated bio-imaging beamline based on the
self-seeding scheme developed for European XFEL was suggested in
\cite{OURCC}. The critical attribute of the proposed beamline,
compared with the baseline SASE1 and SASE2 beamlines, is a wider
photon energy range that spans from the water window up to the
K-edge of Selenium (12.6 keV). With the current design of the
European XFEL, the most preferable photon energy range between 3 keV
and 5 keV cannot be used for biological scattering experiments, but
the new proposed beamline could fill this gap operating at those
energies with TW peak power.
The first goal in developing a design for a dedicated bio-imaging
beamline is to make it satisfying all requirements. Once that is
done, the next step is to optimize the design, making it as simple
as possible. In order to improve the original design, here we
propose to extend the photon energy range of the self-seeding setup
with single crystal monochromator to lower photon energies down to 3
keV. An important aspect of this extension is that the self-seeding
scheme with single crystal monochromator is now routinely used in
generating of narrow bandwidth X-ray pulses at the LCLS
\cite{EMNAT}. It combines a potentially wide photon energy range
with a much needed experimental simplicity. Only one X-ray optical
element is needed, and no sensitive alignment is required. The range
of applicability of this novel method is a slightly limited, at
present, by the availability of a short pulse duration (of about 10
fs or less). However, this range nicely matches that for single
biomolecule imaging.
Optimization of the bio-imaging beamline is performed with extensive
start-to-end simulations, which also take into account effects such
as the spatiotemporal coupling caused by the single crystal
monochromator. One must keep this effect in mind when performing the
design of any self-seeding setup. The spatial shift is proportional
to $\cot(\theta_B)$, and is therefore maximal in the range for small
Bragg angles $\theta_B$. A Bragg geometry close to backscattering
(i.e. $\theta_B$ close to $\pi/2$) would be a more advantageous
option from this viewpoint, albeit with a decrease in the spectral
tunability \cite{SHVID, OURTILT}. It is worth mentioning that this
distortion is easily suppressed by the right choice of crystals
within the photon energy range between 3 keV and 9 keV. Here we
propose to use a set of three diamond crystals. For the C(111), the
C(220) and the C(400) Bragg reflections ($\sigma$-polarization), it
will be possible to respectively cover the photon energy ranges 3
keV - 5 keV, 5 keV - 7 keV, and 7 keV - 9 keV. Finding a solution
suitable for the spectral range between 9 keV and 13 keV is major
challenge due to the large value of $\cot(\theta_B)$ for the C(400)
reflection case. Fortunately, even in this case, this obstacle can
be overcome by using a fresh bunch technique, and exploiting the
self-seeding setup with a C(111) single crystal monochromator, which
is tunable in the photon energy range around 4 keV, in combination
with harmonic generation techniques.
The goal of the present optimized proposal for a dedicated
bio-imaging beamline presented here is to aim for experimental
simplification and performance improvement. The design electron
energy in the most preferable spectral range 3 keV - 5 keV is
increased up to 17.5 GeV. The peak power is shown to reach a maximum
value of 2 TW. The new design takes additional advantage of the fact
that 17.5 GeV is the most preferable operation energy for the SASE1
and the SASE2 beamlines. Because of this, the optimized beamline is
not sensitive to the parallel operation with other European XFEL
beamlines.
\section{Acknowledgements}
We are grateful to Massimo Altarelli, Reinhard Brinkmann, Henry
Chapman, Janos Hajdu, Viktor Lamzin, Serguei Molodtsov and Edgar
Weckert for their support and their interest during the compilation
of this work.
|
2,869,038,154,767 | arxiv | \section{Introduction and statement of main results}
Let $X$ and $Y$ be topological spaces.
Denote by $\CL(X)$ the family of all closed subsets of $X$,
and by $C(X,Y)$ the family of all continuous functions $f\colon X\to Y$.
Given a fixed family $\G\subseteq C(X,Y)$, let $R_\G$ be the binary relation defined by
$$R_\G=\{(f,E)\in C(X,Y)\times\CL(X)\!:(\exists g\in\G)\,f\r E=g\r E\}.$$
For $\F\subseteq C(X,Y)$ and $\E\subseteq\CL(X)$ denote
\begin{align*}
&E_\G(\F)=\{E\in\CL(X)\!:(\forall f\in\F)\,(f,E)\in R_\G\},\\
&F_\G(\E)=\{f\in C(X,Y)\!:(\forall E\in\E)\,(f,E)\in R_\G\}.
\end{align*}
The mappings
$$E_\G\colon\P(C(X,Y))\to\P(\CL(X))\quad\text{and}\quad F_\G\colon\P(\CL(X))\to\P(C(X,Y))$$
form a Galois connection between the partially ordered sets $(\P(C(X,Y)),\allowbreak\subseteq)$ and $(\P(\CL(X)),\subseteq)$.
This means that $E_\G$ and $F_\G$ are inclusion-reversing mappings such that for any $\F\subseteq C(X,Y)$ and $\E\subseteq\CL(X)$
one has $$\E\subseteq E_\G(\F)\quad\text{if and only if}\quad F_\G(E)\supseteq\F.$$
The compond mappings $$E_\G F_\G\colon\P(\CL(X))\to\P(\CL(X))\quad\text{and}\quad F_\G E_\G\colon\P(C(X,Y))\to\P(C(X,Y))$$
are closure operators on $\P(\CL(X))$ and $\P(C(X,Y))$, respectively.
Hence, for any $\E\subseteq\CL(X)$, $\E\subseteq E_\G F_\G(\E)$ and $E_\G F_\G(\E)=\E$ if and only if $\E=E_\G(\F)$ for some $\F\subseteq C(X,Y)$.
Similarly, for any $\F\subseteq C(X,Y)$, $\F\subseteq F_\G E_\G(\F)$ and $F_\G E_\G(\F)=\F$ if and only if $\F=F_\G(\E)$ for some $\E\subseteq\CL(X)$.
Let us denote
\begin{align*}
&\K_\G=\{\E\subseteq\CL(X)\!:E_\G F_\G(\E)=\E\}=\{E_\G(\F)\!:\F\subseteq C(X,Y)\},\\
&\L_\G=\{\F\subseteq C(X,Y)\!:F_\G E_\G(\F)=\F\}=\{F_\G(\E)\!:\E\subseteq\CL(X)\}
\end{align*}
the classes of all closed families with respect to the closure operators associated with the relation $R_\G$.
The families $\K_\G$ and $\L_\G$ are in a one-to-one correspondence and, when ordered by inclusion, form dually isomorphic complete lattices.
In fact, the mapping $\E\mapsto F_\G(\E)$ is an isomorphism $(\K_\G,\subseteq)\to(\L_\G,\supseteq)$ and its inverse is the mapping
$\F\mapsto E_\G(\F)$.
Moreover, the infimum in both lattices coincides with the set-theoretic intersection.
Let us note that in a complete lattice there exist the least and the greatest elements.
For a history of Galois connections and their applications we refer the reader to \cite{Erne}.
For more on their relations to complete lattices and formal concept analysis see \cite{Davey-Priestley}.
Let us note that Galois connections occur naturally in various settings; for some examples
related to analysis and topology see \cite{Lillemets} or \cite{Szasz}.
For applications of Galois connections in the theory of cardinal characteristics see \cite{Blass}.
Our study of restrictions of continuous functions was loosely motivated by classical
notions of Kronecker and Dirichlet sets from harmonic analysis, see, \cite{Rudin} and \cite{Korner}.
In the present paper we deal with the case $X=Y=\R$.
Our aim is to analyze the structure of the lattices $\K_\G$ and $\L_\G$ for certain simple
families $\G\subseteq C(\R,\R)$.
In Section~2 we characterize the elements of the lattices $\K_\G$ and $\L_\G$.
We prove that every element of $\K_\G$ is a hereditary family of closed sets and that each
hereditary family of closed sets is the least element of some lattice $\K_\G$.
We also find a family $\G$ such that $\K_\G$ is the lattice of all nonempty hereditary families
of closed sets.
In Sections 3--5 we describe the lattice $\K_\G$ for three families $\G$ determined
by a single continuous function $g$: the singleton $\{g\}$,
the family of all functions $f$ such that $f(x)<g(x)$ for all $x$,
and the family of all functions $f$ satisfying $f(x)\neq g(x)$ for all $x$.
In each case we characterize all families that yield the same lattice $\K_\G$.
\subsection{Notation and terminology.}
For $\F\subseteq C(\R,\R)$ and $E\subseteq\R$ we denote $\F\r E=\{f\r E\!:f\in\F\}$.
For $x\in\R$ we also denote $\F[x]=\{f(x)\!:f\in\F\}$.
If $\H\subseteq C(E,\R)$, we denote $[\H]=\{f\in C(\R,\R)\!:f\r E\in\H\}$.
We write $[h]$ instead of $[\{h\}]$ for $h\in C(\R,\R)$.
For $E\subseteq\R$ let $\CL(E)$ denote the family of all closed subsets of $E$.
To avoid ambiguity, we use notation $\CL(E)$ only if $E$ is closed; otherwise the family
of all subsets of $E$ that are closed in $\R$ is expressed by the term $\CL(\R)\cap\P(E)$.
Denote $\Eq_{f,g}=\{x\in\R\!:f(x)=g(x)\}$ for $f,g\in C(\R,\R)$.
Then for any $\F\subseteq C(\R,\R)$ and $\E\subseteq\CL(\R)$ we have
$E_\G(\F)=\bigcap_{f\in\F}\bigcup_{g\in\G}\CL(\Eq_{f,g})$ and
$\F_\G(\E)=\bigcap_{E\in\E}\bigcup_{g\in\G}[g\r E]$.
We shall identify a function $f\colon\R\to\R$ and its graph $\{(x,y)\in\R^2\!:f(x)=y\}$.
We say that a family $\G\subseteq C(\R,\R)$ is \emph{complete} if $g\in\G$
holds for every $g\in C(\R,\R)$ satisfying $g\subseteq\bigcup\G$.
A family $\G$ is \emph{connected} if for any $f,g\in\G$ and $x\neq y$ there exists
$h\in\G$ such that $h(x)=f(x)$ and $h(y)=g(y)$.
Let $\R^*=\R\cup\{-\infty,\infty\}$.
For $f,g\in C(\R,\R^*)$, define inequalities $f<g$ and $f\le g$
by $(\forall x\in\R)\,f(x)<g(x)$ and $(\forall x\in\R)\,f(x)\le g(x)$, respectively.
Further, let $(f,g)=\{h\in C(\R,\R)\!:f<h<g\}$ and
$[f,g\kern.5pt]=\{h\in C(\R,\R)\!:f\le h\le g\}$.
If there is no ambiguity we denote the constant function $f\colon x\mapsto z\in C(\R,\R^*)$ simply by $z$.
Let $\X$ be a family of subsets of a topological space.
We say that $\X$ is \emph{separated} if for any distinct sets $X,Y\in\X$
one can find disjoint open sets $U,V$ such that $X\subseteq U$, $Y\subseteq V$
and $(\forall Z\in\X)\ Z\subseteq U\,\lor\,Z\subseteq V$.
\subsection{Main results}
We will prove the following equalities and inclusions.
\begin{theorem}
Let $g\in C(\R,\R)$ and let\/ $\G\subseteq C(\R,\R)$ be nonempty.
\begin{enumerate}[\rm (3b)]
\item[\rm (1a)] $\K_{\{g\}}=\{\CL(E)\!:E\in\CL(\R)\}$.
\item[\rm (1b)] $\K_{\{g\}}\subseteq\K_\G$ if and only if\/ $\G[x]\neq\R$ for every $x\in\R$.
\item[\rm (1c)] $\K_{\{g\}}\supseteq\K_\G$ if and only if\/ $\G=[f,h\kern1pt]$
for some $f,h\in C(\R,\R^*)$.
\smallskip
\item[\rm (2a)] $\K_{(g,\infty)}=\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}$.
\item[\rm (2b)] $\K_{(g,\infty)}\subseteq\K_\G$ if and only if for every $x\in\R$ there exists $f\in C(\R,\R)$ such that
$E_\G(\{f\})=\{E\in\CL(\R)\!:x\notin E\}$.
\item[\rm (2c)] $\K_{(g,\infty)}\supseteq\K_\G$ if and only if\/ $\G$ is complete and connected.
\smallskip
\item[\rm (3a)] $\K_{(-\infty,g)\cup(g,\infty)}=\big\{\CL(\R)\cap\bigcup_{X\in\X}\!:\X\subseteq\P(\R)\text{ is separated\/}\big\}$.
\item[\rm (3b)] $\K_{(-\infty,g)\cup(g,\infty)}\subseteq\K_\G$ if and only if
$\CL(\R)\cap\P(\R\setminus\{x\})\in\K_\G$ for every $x\in\R$
and\/ $\CL(\R)\cap(\P(U)\cup\P(\R\setminus\cl U))\in\K_\G$ for every regular open set $U\subseteq\R$.
\item[\rm (3c)] $\K_{(-\infty,g)\cup(g,\infty)}\supseteq\K_\G$ if and only if\/ $\G=\bigcup_{i\in I}\G_i$
for some linearly ordered set $(I,<)$ and an indexed system
of complete connected families $\{\G_i\!:i\in I\}$
such that for every $i\in I$ there exist functions
$f_i,h_i\in C(\R,\R^*)$ satisfying
$$\bigcup_{j<i}\G_j\subseteq(-\infty,f_i),\quad
\G_i\subseteq(f_i,h_i)\quad\text{and}\quad
\bigcup_{j>i}\G_j\subseteq(h_i,\infty).$$
\end{enumerate}
\end{theorem}
First three statements are proved in Section~3 (Theorems~\ref{thm-singleton}, \ref{thm-contains-all-C(E)} and \ref{thm-is-contained-in-all-C(E)}),
statements (2a)--(2c) in Section~4 (Theorems~\ref{thm-strictly-below-1}, \ref{thm-strictly-below-equiv-1} and \ref{thm-strictly-below-equiv-2}),
and (3a)--(3c) in Section~5 (Theorems~\ref{thm-diff-equal}, \ref{thm-diff-2} and \ref{thm-diff-3}).
\section{The elements of lattices $\K_\G$ and $\L_G$}
We begin with two extremal cases.
\begin{proposition} \label{prop-1}
Let $\G\subseteq C(\R,\R)$.
\begin{enumerate}[\rm (1)]
\item $\K_\emptyset=\{\emptyset,\CL(\R)\}$, $\L_\emptyset=\{\emptyset,C(\R,\R)\}$.
\item $\K_{C(\R,\R)}=\{\CL(\R)\}$, $\L_{C(\R,\R)}=\{C(\R,\R)\}$.
\end{enumerate}
\end{proposition}
If $\G\neq\emptyset$ then every family $\E\in\K_\G$ (as well as every $\F\in\L_\G$)
is nonempty because it contains $\emptyset$.
Hence, $\emptyset\in\K_\G$ if and only if $\emptyset\in\L_\G$ if and only if $\G=\emptyset$.
\begin{proposition} \label{prop-2}
Let $\G\subseteq C(\R,\R)$, $\G\neq\emptyset$.
\begin{enumerate}[\rm (1)]
\item The least element of\/ $\K_\G$ is $E_\G(C(\R,\R))=\{E\in \CL(\R)\!:\G\r E=C(E,\R)\}$.
\item The least element of $\L_\G$ is $F_\G(\CL(\R))=F_\G(\{\R\})=\G$.
\end{enumerate}
\end{proposition}
It follows that the lattices $\K_\G$ and $\L_\G$ have at least two elements if and only if
$\G\neq C(\R,\R)$.
By Proposition~\ref{prop-2} (2), every family $\G\subseteq C(\R,\R)$ is the least element of $\L_\G$.
Now we are going to characterize families $\E\subseteq\CL(\R)$ that can be least elements of lattices
$\K_\G$.
We say that a family $\E\subseteq\CL(\R)$ is \emph{hereditary} if for any $D,E\in\CL(\R)$,
if $D\subseteq E$ and $E\in\E$ then $D\in\E$.
We show that the elements of lattices $\K_\G$ are exactly hereditary subfamilies of $\CL(\R)$
and every hereditary family $\E\subseteq\CL(\R)$ is the least element of some lattice $\K_\G$.
\begin{proposition} \label{prop-hereditary}
Let $\E\subseteq \CL(\R)$.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item There exists $\G\subseteq C(\R,\R)$ such that $\E\in \K_\G$.
\item $\E$ is hereditary.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1) \Rightarrow (2)$. If $\E\in \K_\G$ then $\E=E_\G(\F)$ for some $\F\subseteq C(\R,\R)$.
It follows from the definition that the family $E_\G(\F)$ is hereditary.
$(2) \Rightarrow (1)$. Fix $f\in C(\R,\R)$.
For every $E\in\E$, fix $g_E\in C(\R,\R)$ such that $\Eq_{f,g_E}=E$ and
let $\G=\{g_E\!:E\in\E\}$.
Then $\E=E_\G(\{f\})\in\K_\G$.
\end{proof}
\begin{lemma} \label{lem-incr-bij}
Let $I,J\subseteq\R$ be non-degenerate, bounded, closed intervals.
For every $n\in\omega$, let $x_n\in\int I$ be distinct and $A_n\subseteq J$ be dense in $J$.
Then there exists an increasing bijection $f\colon I\to J$ such that $(\forall n\in\omega)\,f(x_n)\in A_n$.
\end{lemma}
\begin{proof}
Let us note that any increasing bijection from $I$ to $J$ is necessarily continuous.
Define a sequence of increasing bijections $f_n\colon I\to J$ by induction as follows.
Let $f_0$ be linear.
For every $n$, let $a_0,\dots,a_{n+1}$ be the increasing enumeration of the set
$\{a,b\}\cup\{x_j\!:j<n\}$, where $I=[a,b]$.
Assume that $f_n\colon I\to J$ is an increasing bijection
which is linear on each interval $[a_j,a_{j+1}]$ and moreover the function $f_n-\frac{1}{2}f_0$
is strictly inceasing.
For every $j\le n$, if $x_n\notin(a_j,a_{j+1})$ then
let $f_{n+1}(x)=f_n(x)$ for every $x\in[a_j,a_{j+1}]$.
If $x_n\in(a_j,a_{j+1})$, let $f_{n+1}$ be defined linearly on intervals
$[a_j,x_n]$ and $[x_n,a_{j+1}]$, where $f_{n+1}(a_j)=f_n(a_j)$, $f_{n+1}(a_{j+1})=f_n(a_{j+1})$,
and $f_{n+1}(x_n)\in A_n$ is chosen so that
$f_{n+1}-\frac{1}{2}f_0$ is strictly increasing and for every $x\in(a_j,a_{j+1})$,
$\size{f_{n+1}(x)-f_n(x)}<2^{-n}$.
We obtain a uniformly convergent sequence of increasing bijections $f_n\colon I\to J$.
Its limit $f\colon I\to J$ is continuous and surjective.
Function $f-\frac{1}{2}f_0$, being a limit of a sequence of strictly increasing functions, is non-decreasing,
hence $f$ is strictly increasing.
For every $n$ we have $f(x_n)=f_{n+1}(x_n)\in A_n$.
\end{proof}
\begin{theorem}
Let $\E\subseteq\CL(\R)$ be hereditary.
Then there exists $\G\subseteq C(\R,\R)$ such that $\E$ is the least element of\/ $\K_\G$.
\end{theorem}
\begin{proof}
Fix disjoint countable dense sets $D_0,D_1\subseteq\R$ and
$h\colon\R\to\{0,1\}$ such that both $h^{-1}[\{0\}]$ and $h^{-1}[\{1\}]$ are dense.
Given a hereditary family $\E\subseteq\CL(\R)$, let $\G$ be
the family of all functions $g\in C(\R,\R)$ such that
$$(\forall E\in\CL(\R)\setminus\E)(\exists U\text{ open, }E\cap U\neq\emptyset)
(\forall x\in E\cap U)\ g(x)\notin D_{h(x)}.$$
We prove that $\E=E_\G(C(\R,\R))$.
Let us first show that for every $E\in\E$ and $f\in C(\R,\R)$ there exists $g\in[f\r E]$
such that $g(x)\notin D_{h(x)}$ for all $x\notin E$.
Without a loss of generality we may assume that the complement of $E$ is a disjoint union of
bounded open intervals and that the values of $f$ at the endpoints
of each of these intervals are different.
We can accomplish this by adding to $E$ an unbounded discrete set $Z\subseteq\R\setminus E$
dividing each interval adjacent to $E$, and suitably modifying the values of $f$ outside $E$
to ensure that $f(z)\notin D_{h(z)}$ for $z\in Z$ and $f(a)\neq f(b)$ for each interval
$[a,b]$ adjacent to $E\cup Z$.
Let $[a,b]$ be a closed interval adjacent to $E$.
Assume that $f(a)<f(b)$.
Let $I=[f(a),f(b)]$, $J=[a,b]$, $\{x_n\!:n\in\omega\}=(D_0\cup D_1)\cap\int I$, and for every $n$, let
$A_n=J\cap h^{-1}[\{i\}]$ where $i\in\{0,1\}$ is such that $x_n\notin D_i$.
Let $g_J\colon I\to J$ be the increasing bijection obtained in Lemma~\ref{lem-incr-bij}.
Its inverse $g_J^{-1}\colon [a,b]\to [f(a),f(b)]$ is an increasing bijection as well
and for every $x\in (a,b)$ we have $g_J^{-1}(x)\notin D_{h(x)}$.
Similarly, if $f(b)<f(a)$ then there exists a decreasing bijection
$g_J^{-1}\colon [a,b]\to [f(b),f(a)]$
such that $g_J^{-1}(x)\notin D_{h(x)}$ for all $x\in (a,b)$.
Define $g\colon\R\to\R$ by
$$g(x)=\begin{cases}f(x),&\text{if $x\in E$},\\
g^{-1}_J(x)&\text{if $J$ is a closed interval adjacent to $E$ and $x\in\int J$}.\end{cases}$$
Obviously, $g$ is continuous and $g(x)\notin D_{h(x)}$ for $x\notin E$.
We show that $g\in\G$.
Indeed, if $D\in\CL(\R)\setminus\E$ then $D\nsubseteq E$, since $\E$ is hereditary.
Let $U=\R\setminus D$.
Then $U$ is open, $E\cap U\neq\emptyset$, and $g(x)\notin D_{h(x)}$ for every $x\in E\cap U$.
We have shown that for every $E\in\E$ and $f\in C(\R,\R)$ there exists $g\in\G$ such that $g\r E=f\r E$,
hence $\E\subseteq E_\G(C(\R,\R))$.
To prove the opposite inclusion, let us take $E\in\CL(\R)\setminus\E$.
Let $\{x_n\!:n\in\omega\}$ be a countable dense subset of $E$, and
for every $n$, let $A_n=D_{h(x_n)}$.
By repeated use of Lemma~\ref{lem-incr-bij} we can find $f\in C(\R,\R)$ such that for $f(x_n)\in A_n$ for all $n$.
We show that $f\r E\notin\G\r E$ and hence $E\notin E_\G(C(\R,\R))$.
Let $g\in\G$ be arbitrary.
There exists an open set $U$ such that $E\cap U\neq\emptyset$ and $g(x)\notin D_{h(x)}$ for every $x\in E\cap U$.
Find $n$ such that $x_n\in E\cap U$.
Then $f(x_n)\in D_{h(x_n)}$ while $f(x_n)\neq g(x_n)$, hence $f\r E\neq g\r E$.
\end{proof}
\begin{theorem}
There exists $\G\subseteq C(\R,\R)$ such that\/ $\K_\G$ contains every nonempty hereditary family
$\E\subseteq\CL(\R)$.
\end{theorem}
\begin{proof}
Let $\{E_\alpha\!:\alpha<2^\omega\}$
be a one-to-one enumeration of all nonempty closed subsets of $\R$.
For each $\alpha<2^\omega$ fix some $x_\alpha\in E_\alpha$.
Using transfinite induction for $\alpha<2^\omega$ we define $y_\alpha\in\R$
and a sequence of functions $\{g_{\alpha,n}\!:n\in\omega\}\subseteq C(\R,\R)$.
We proceed as follows.
If $y_\beta$ and $g_{\beta,n}$ are defined for all $\beta<\alpha$ and $n\in\omega$,
find $y_\alpha\notin\{y_\beta\!:\beta<\alpha\}\cup\{g_{\beta,n}(x_\alpha)\!:\beta<\alpha,\,n\in\omega\}$.
Let $\{I_{\alpha,n}\!:n\in\omega\}$ be the family of all nonempty open intervals with rational endpoints
having nonempty intersection with $E_\alpha$.
For every $n$, there exists a function $g_{\alpha,n}\in C(\R,\R)$ such that
$g_{\alpha,n}(x)=y_\alpha$ if and only if $x\notin I_{\alpha,n}$,
and $g_{\alpha,n}(x_\beta)\neq y_\beta$ for all $\beta<\alpha$.
Let $\G=\{g_{\alpha,n}\!:\alpha<2^\omega,\,n\in\omega\}$.
For every $\alpha<2^\omega$, let $f_\alpha$ be the constant function with value $y_\alpha$.
We show that $f_\alpha\r E_\alpha\notin\G\r E_\alpha$.
If $g\in\G$ then $g=g_{\beta,n}$ for some $\beta<2^\omega$ and $n\in\omega$.
If $\beta<\alpha$ then $g_{\beta,n}(x_\alpha)\neq y_\alpha$ by the definition of $y_\alpha$.
Since $x_\alpha\in E_\alpha$, we have $f_\alpha\r E_\alpha\neq g\r E_\alpha$.
If $\beta=\alpha$ then there exists $x\in E_\alpha\cap I_{\alpha,n}$.
We have $g_{\alpha,n}(x)\neq y_\alpha$, hence $f_\alpha\r E_\alpha\neq g\r E_\alpha$.
Finally, if $\beta>\alpha$ then $g_{\beta,n}(x_\alpha)\neq y_\alpha$ by the definition of $g_{\beta,n}$.
Again, $f_\alpha\r E_\alpha\neq g\r E_\alpha$.
Let $\E$ be a nonempty hereditary family of closed subsets of $\R$.
Then $\emptyset\in\E$, hence each $E\in\CL(\R)\setminus\E$ is nonempty.
Denote $\F=\{f_\alpha\!:E_\alpha\in\CL(\R)\setminus\E\}$.
If $E\in\E$ and $f\in\F$ then $f=f_\alpha$ for some $\alpha<2^\omega$ such that $E_\alpha\in\CL(\R)\setminus\E$,
hence $E_\alpha\nsubseteq E$.
There exists $n$ such that $E\subseteq\R\setminus I_{\alpha,n}$.
By the definition of $g_{\alpha,n}$ we have $f_\alpha\r E=g_{\alpha,n}\r E$,
hence $f_\alpha\r E\in\G\r E$.
It follows that $E\in E_\G(\F)$, and we conclude that $\E\subseteq E_\G(\F)$.
Conversely, if $E\in\CL(R)\setminus\E$ then $E=E_\alpha$ for some $\alpha<2^\omega$.
Since $f_\alpha\r E_\alpha\notin\G\r E_\alpha$ and $f_\alpha\in\F$, we have $E\notin E_\G(\F)$.
It follows that $\E=E_\G(\F)$, hence $\E\in\K_\G$.
\end{proof}
\section{Results for family $\G=\{g\}$}
We show that if $\G$ is a singleton then the lattice $\K_\G$ is isomorphic to the complete lattice
$(\CL(\R),\subseteq)$ of all closed subsets of~$\R$.
Let us note that in $(\CL(\R),\subseteq)$ we have $\bigwedge\E=\bigcap\E$ and
$\bigvee\E=\cl\big(\bigcup\E\big)$, for any $\E\subseteq \CL(\R)$.
\begin{theorem} \label{thm-singleton}
Let $\G=\{g\}$, $g\in C(\R,\R)$.
Then $\K_\G=\{\CL(E)\!:E\in \CL(\R)\}$ and $\L_\G=\{[g\r E]\!:E\in \CL(\R)\}$.
\end{theorem}
\begin{proof}
It is clear that $(f,E)\in R_\G$ if and only if $E\subseteq\Eq_{f,g}$,
for any $f\in C(\R,\R)$ and $E\in \CL(\R)$.
Hence, for any $\F\subseteq C(\R,\R)$ we have
\begin{displaymath}
E_\G(\F)=
\bigcap_{f\in\F}\CL(\Eq_{f,g})=
\CL\left(\textstyle\bigcap\nolimits_{f\in\F}\Eq_{f,g}\right).
\end{displaymath}
Since for any set $E\in \CL(\R)$ there exists $f\in C(\R,\R)$ such that $E=\Eq_{f,g}$, we obtain that
$\K_\G=\{E_\G(\F)\!:\F\subseteq C(\R,\R)\}=\{\CL(E)\!:E\in \CL(\R)\}$.
For any $\E\subseteq \CL(\R)$ we also have
\begin{displaymath}
F_\G(\E)=
\bigcap_{E\in\E}[g\r E]=
\big[g\r \cl\big(\textstyle\bigcup\E\big)\big],
\end{displaymath}
hence $\L_\G=\{F_\G(\E)\!:\E\subseteq \CL(\R)\}=\{[g\r E]\!:E\in \CL(\R)\}$.
\end{proof}
It can be easily seen that if $\G=\{g\}$ then each element $\E\in \K_\G$ can be generated by a family
consisting of a single function $f$:
if $\E\in \K_\G$ then there exists $E\in \CL(\R)$ such that $\E=\CL(E)$,
for any $f\in C(\R,\R)$ satisfying $\Eq_{f,g}=E$ we then have $\E=E_\G(\{f\})$.
Similarly, each $\F\in \L_\G$ can be expressed as
$F_\G(\{E\})$ for some $E\in \CL(\R)$.
Moreover, this set $E$ is unique; if $D,E\in \CL(\R)$ are distinct then
$F_\G(\{D\})\neq F_\G(\{E\})$ by the normality of $\R$.
The next two results allows us to characterize families $\G\subseteq C(\R,\R)$
for which the lattice $\K_\G$ is the same as in Theorem~\ref{thm-singleton}.
\begin{theorem} \label{thm-contains-all-C(E)}
Let $\G\subseteq C(\R,\R)$ be nonempty.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\{\CL(E)\!:E\in\CL(\R)\}\subseteq\K_\G$.
\item The least element of $\K_\G$ is $\{\emptyset\}$.
\item For each $x\in\R$, $\G[x]\neq\R$.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1) \Rightarrow (2)$ is trivial.
$(2) \Rightarrow (3)$.
If $E_\G(C(\R,\R))=\{\emptyset\}$ then for each nonempty $E\in \CL(\R)$ there exists $f\in C(\R,\R)$ such that $f\r E\notin\G\r E$.
In particular, for every $x\in\R$ there exists $y\in\R$ such that
if $f(x)=y$ then $f\r\{x\}\notin\G\r\{x\}$, hence $y\notin\G[x]$.
$(3) \Rightarrow (1)$.
Fix a function $h\in\G$.
For every $E\in \CL(\R)$ and $x\notin E$ let us take $y\notin\G[x]$ and
a function $f_x\in [h\r E]$ such that $f_x(x)=y$.
Let $\F=\{f_x\!:x\notin E\}$.
If $D\in E_\G(\F)$ then for any $x\notin E$ we have $f_x\r D\in\G\r D$, hence $x\notin D$.
It follows that $D\subseteq E$ and thus $E_\G(\F)\subseteq\CL(E)$.
The opposite inclusion is clear, hence we obtain $\CL(E)=E_\G(\F)\in \K_\G$.
\end{proof}
\begin{theorem} \label{thm-is-contained-in-all-C(E)}
Let $\G\subseteq C(\R,\R)$ be nonempty.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\K_\G\subseteq\{\CL(E)\!:E\in\CL(\R)\}$.
\item There exist $h_1,h_2\in C(\R,\R^*)$ such that\/
$\G=\{g\in C(\R,\R)\!:h_1\le g\le h_2\}$.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1) \Rightarrow (2)$.
Denote $H=\bigcup\G$.
Let us first show that $H$ is a closed subset of $\R^2$.
Assume that $(x,y)\in\cl H$.
Since $\G$ is a nonempty family of continuous functions,
there exists in $H$ a sequence of points $\{(x_n,y_n)\!:n\in\omega\}$ converging to $(x,y)$ and such that
all $x_n$ are distinct.
Let $f\in C(\R,\R)$ be such that $f(x_n)=y_n$ for every $n$ and $f(x)=y$.
Then $\E=E_\G(\{f\})$ contains $\{x_n\}$ for every $n$.
Since $\E\in\K_\G$, it follows from 1 that $\cl\{x_n\!:n\in\omega\}\in\E$, hence $\{x\}\in\E$ and so $(x,y)\in H$.
We show that $\G[x]=\{g(x)\!:g\in\G\}$ is connected, for every $x\in\R$.
Otherwise we can find $g_1,g_2\in\G$ and $y\notin\G[x]$ such that $g_1(x)<y<g_2(x)$.
Since $H$ is closed, there exist $a,b,c,d\in\R$ such that $x\in(a,b)$, $y\in (c,d)$, and
$\big((a,b)\times (c,d)\big)\cap H=\emptyset$.
Let $f\in C(\R,\R)$ be such that $f(a)=g_1(a)$ and $f(b)=g_2(b)$.
For $\E=E_\G(\{f\})$ we have $\{a\}\in\E$, $\{b\}\in\E$, hence also $\{a,b\}\in\E$,
and thus there exists $g\in\G$ such that $g(a)=g_1(a)$ and $g(b)=g_2(b)$.
By Intermediate Value Theorem there is $z\in (a,b)$ such that $g(z)\in (c,d)$, which
contradicts the assumption that $\big((a,b)\times(c,d)\big)\cap H$ is empty.
So $\G[x]$ is a connected closed set, that is, a closed interval.
For every $x\in\R$ denote $h_1(x)=\inf\G[x]$ and $h_2(x)=\sup\G[x]$.
We show that $h_1,h_2$ are continuous.
If $y<h_1(x)$ then there exist $a,b,c,d\in\R$ such that $x\in(a,b)$, $y\in(c,d)$, and
$\big((a,b)\times(c,d)\big)\cap H=\emptyset$.
It follows $\big((a,b)\times(-\infty,d)\big)\cap H=\emptyset$,
otherwise one could find a contradiction using Intermediate Value Theorem, as before.
We can conclude that $h_1$ is lower semi-continuous.
Since $h_1$ is the infimum of a family of continuous functions, it is also upper semi-continuous,
and hence continuous.
A similar argument shows the continuity of $h_2$.
It remained to show that $\G=\{g\in C(\R,\R)\!:h_1\le g\le h_2\}$.
The inclusion from left to right is clear.
If $g\in C(\R,\R)$ is such that $h_1\le g\le h_2$, then for every $x\in\R$ we have $g(x)\in\G[x]$,
hence $\{x\}\in\E$ where $\E=E_\G(\{g\})$.
By (1), $\R=\bigcup\E\in\E$, hence $g\in\G$.
$(2) \Rightarrow (1)$.
Let $\E\in\K_\G$, that is, $\E=E_\G(\F)$ for some $\F\subseteq C(\R,\R)$, and let $E=\bigcup\E$.
Then $E=\{x\in\R\!:\F[x]\subseteq\G[x]\}$.
First, let us show that $E\in\CL(\R)$.
If $x\in\cl E$ then there exists a sequence $\{x_n\!:n\in\N\}$ in $E$ such that $x_n\to x$.
For any $y\in\F[x]$, let us take some $f\in\F$ such that $f(x)=y$.
For every $n$ we have $f(x_n)\in\F(x_n)\subseteq\G(x_n)$, hence $h_1(x_n)\le f(x_n)\le h_2(x_n)$.
By the continuity of $f$, $h_1$, and $h_2$ we obtain that $h_1(x)\le f(x)\le h_2(x)$, hence $y\in\G[x]$.
We have $\F[x]\subseteq\G[x]$, hence $x\in E$, and it follows that $E$ is closed.
We show that for every $f\in\F$ there exists $g\in\G$ such that $f\r E=g\r E$.
Fix some $f\in\F$.
For every $x\in E$ we have $f(x)\in\G[x]$, hence $h_1(x)\le f(x)\le h_2(x)$.
Let $g(x)=\min\{\max\{f(x),h_1(x)\},h_2(x)\}$, for all $x\in\R$.
Clearly, $g$ is continuous and $g\r E=f\r E$.
Since $\G$ is nonempty, we have $h_1\le h_2$, and hence also $h_1\le g\le h_2$.
By (2), we have $g\in\G$.
It follows that $E\in\E$, hence $\E=\CL(E)$.
\end{proof}
Now we can characterize those families $\G$ for which
$\K_\G=\{\CL(E)\!:E\in\CL(\R)\}$.
Let us recall that for $f,h\in C(\R,\R^*)$ we denoted $[f,h]=\{g\in C(\R,\R)\!:f\le g\le h\}$,
where $f\le g$ if and only if $(\forall x\in\R)\, f(x)\le g(x)$.
\begin{corollary}
Let $\G\subseteq C(\R,\R)$.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\K_\G=\{\CL(E)\!:E\in\CL(\R)\}$.
\item There exist $f,h\in C(\R,\R^*)$ such that $f\le h$,
$f^{-1}[\R]\cup h^{-1}[\R]=\R$, and $\G=[f,h]$.
\end{enumerate}
\end{corollary}
It follows that the same lattice $\K_\G$ is obtained for families $\G$ of the form
$(-\infty,g\kern.5pt]=\{f\in C(\R,\R)\!:f\le g\}$ and $[\kern.5ptg,\infty)=\{f\in C(\R,\R)\!:g\le f\}$.
\begin{corollary}
Let $g\in C(\R,\R)$.
Then $\K_{(-\infty,g]}=\K_{[g,\infty)}=\{\CL(E)\!:E\in\CL(\R)\}$.
\end{corollary}
\section{Results for family $\G=(g,\infty)$}
Recall that for $f,h\in C(\R,\R^*)$, $(f,h)=\{g\in C(\R,\R)\!:f<g<h\}$
where $f<g$ is a shorthand for $(\forall x\in\R)\,f(x)<g(x)$.
\begin{theorem} \label{thm-strictly-below-1}
Let $\G=(g,\infty)$ where $g\in C(\R,\R)$.
Then $\K_\G=\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}$.
\end{theorem}
\begin{proof}
If $\E\in\K_\G$ then $\E=E_\G(\F)$ for some $\F\subseteq C(\R,\R)$.
Denote $X=\bigcup\E$.
Clearly $\E\subseteq\CL(\R)\cap\P(X)$.
If $E\in\CL(\R)\cap\P(X)$ then for every $x\in E$ we have $x\in D$ for some $D\in\E$,
hence $f(x)>g(x)$ for all $f\in\F$.
For every $f\in\F$ there exists $f'\in\G$ such that $f'\r E=f\r E$;
it suffices to take $f'$ to be linear on each bounded interval adjacent to $E$,
and constant on unbounded ones, if there are any.
It follows that $E\in\E$, and we obtain $\E=\CL(\R)\cap\P(X)$,
hence $\K_\G\subseteq\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}$.
To prove the opposite, let $X\subseteq\R$.
Denote $\F=\{f_a\!:a\in\R\setminus X\}$, where $f_a(x)=g(x)+\size{x-a}$ for all $x\in\R$.
If $E\in E_\G(\F)$ then $f_a(x)>g(x)$ for all $x\in E$ and $a\in\R\setminus X$, hence $E\subseteq X$.
It follows that $E_\G(\F)\subseteq\CL(\R)\cap\P(X)$.
The opposite inclusion is clear since $\F\subseteq F_\G(\CL(\R)\cap\P(X))$.
We obtain that $\CL(\R)\cap\P(X)=E_\G(\F)\in\K_\G$, hence
$\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}\subseteq\K_\G$.
\end{proof}
A similar argument would prove the same result for the family $\G=(-\infty,g)$.
Nevertheless, it will also follow from Corollary~\ref{cor-strictly-below-together} below.
We will characterize those families $\G\subseteq C(\R,\R)$ for which
$\K_\G=\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}$.
Like in the previous section, we characterize both inclusions separately.
For $x\in\R$ denote $\A_x=\CL(\R)\cap\P(\R\setminus\{x\})=\{E\in\CL(\R)\!:x\notin E\}$.
\begin{theorem} \label{thm-strictly-below-equiv-1}
Let $\G\subseteq C(\R,\R)$ be nonempty.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}\subseteq\K_\G$.
\item $\{\A_x\!:x\in\R\}\subseteq\K_\G$.
\item For every $x\in\R$ there exists $f\in C(\R,\R)$ such that $E_\G(\{f\})=\A_x$.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1) \Rightarrow (2)$ is clear.
$(2) \Rightarrow (3)$.
For every $x\in\R$, we have $\{x\}\notin\A_x=E_\G(F_\G(\A_x))$,
hence there exists $f\in F_\G(\A_x)$ such that $f\r\{x\}\notin\G\r\{x\}$.
We have $\{f\}\subseteq F_\G(\A_x)$, hence $E_\G(\{f\})\supseteq E_\G(F_\G(\A_x))=\A_x$.
Conversely, if $E\in\CL(\R)\setminus\A_x$ then $x\in E$ and hence $f\r E\notin\G\r E$.
It follows that $E\notin E_\G(\{f\})$, and we obtain $E_\G(\{f\})\subseteq\A_x$.
$(3) \Rightarrow (1)$.
Let $X\subseteq\R$, $\E=\CL(\R)\cap\P(X)$, and $\F=F_\G(\E)$.
We will show that $E_\G(\F)=\E$.
If not, then there exists $E\in E_\G(\F)$ such that $E\nsubseteq X$.
Let $x\in E\setminus X$.
By (3) there exists $f\in C(\R,\R)$ such that $E_\G(\{f\})=\A_x$.
Since $\E\subseteq\A_x$, we have $F_\G(\E)\supseteq F_\G(\A_x)$, hence $f\in F_\G(\E)=\F$.
It follows that $E\in E_\G(\{f\})$, and we come to a contradiction.
\end{proof}
Note that the family $\{\A_x\!:x\in\R\}$ in condition (2) of Theorem~\ref{thm-strictly-below-equiv-1} is minimal.
Given $z\in\R$, let $\G$ be the family of all continuous functions $f$ such that
$f(x)>0$ for all $x\neq z$.
For every $y\in\R$, let $f_y(x)=\size{x-y}$.
If $y\neq z$ then $E_\G(\{f_y\})=\A_y$, hence $\A_y\in\K_\G$.
Since $F_\G(\A_z)=\{f\in C(\R,\R)\!:(\forall x\neq z)\,f(x)>0\}=\G$,
we obtain that $E_\G(F_\G(\A_z))=\CL(\R)$, hence $\A_z\notin\K_\G$.
To characterize all families $\G$ such that $\K_\G\subseteq\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}$
we need the following notion.
We say that a set $H\subseteq\R^2$ is \emph{functionally connected}
if for any two points $(x_1,y_1),(x_2,y_2)\in H$ such that $x_1\neq x_2$,
there exists a continuous function $h\colon[x_1,x_2]\to\R$ such that $h(x_1)=y_1$, $h(x_2)=y_2$,
and the graph of $h$ is included in $H$.
If $H$ is a functionally connected set then $\pr_1[H]$, the projection of $H$ to the first coordinate, is connected.
If $\pr_1[H]$ has at most one point then $H$ is functionally connected.
If $\pr_1[H]$ has more than one point and $H$ is functionally connected then $H$ must be pathwise connected.
A connected set need not to be functionally connected, a simple example is the unit circle $\{(x,y)\!:x^2+y^2=1\}$.
\begin{lemma} \label{lem-func-conn-1}
Let $a<b$ and let $H\subseteq\R^2$ be a functionally connected set such that $[a,b]\subseteq\pr_1[H]$.
Let $h\colon[a,b]\to\R$ be a continuous function such that $h\subseteq H$,
and let $u,v\in\R$ be such that points $(a,u),(b,v)\in H$.
Then for every open interval $J$ such that $u,v\in J$ and\/ $\rng(h)\subseteq J$,
there exists a continuous function $g\colon[a,b]\to J$ such that $g\subseteq H$, $g(a)=u$, and $g(b)=v$.
\end{lemma}
\begin{proof}
Let $a,b,H,h,u,v$, and $J$ be as above.
There exists a continuous function $f\colon[a,b]\to\R$
such that $f\subseteq H$, $f(a)=u$, and $f(b)=v$.
Let $a_2,b_2\in(a,b)$ be such that $a_2<b_2$ and $f(x)\in J$ for every $x\in[a,a_2]\cup[b_2,b]$.
Let us first prove that there exists some $a_1\in[a,a_2]$ and a continuous function
$f_1\colon[a,a_1]\to J$
such that $f_1\subseteq H$, $f_1(a)=f(a)$, and $f_1(a_1)=h(a_1)$.
This is clear if $f(x)=h(x)$ for some $x\in[a,a_2]$.
If this is not the case then the values $f(a)-h(a)$ and $f(a_2)-h(a_2)$ must have the same signs.
Without a loss of generality, assume that $f(a)>h(a)$ and $f(a_2)>h(a_2)$.
Let $f'\colon[a,a_2]\to\R$ be a continuous function such that $f'\subseteq H$,
$f'(a)=f(a)$, and $f'(a_2)=h(a_2)$.
Let $a_0=\max\{x\in[a,a_2]\!:f'(x)=f(x)\}$ and
$a_1=\min\{x\in[a_0,a_2]\!:f'(x)=h(x)\}$.
It follows that $f'(x)\in J$ for every $x\in[a_0,a_1]$, and we can define
$f_1(x)=f(x)$ for $x\in[a,a_0]$ and $f_1(x)=f'(x)$ for $x\in[a_0,a_1]$.
Then $f_1$ is as required.
Similarly, there exist $b_1\in[b',b]$ and a continuous function $f_2\colon[b_1,b]\to J$
such that $f_2\subseteq H$, $f_2(b_1)=h(b_1)$, and $f_2(b)=f(b)$.
Let $g(x)=f_1(x)$ for $x\in[a,a_1]$, $g(x)=h(x)$ for $x\in[a_1,b_1]$, and $g(x)=f_2(x)$ for $x\in[b_1,b]$.
Then $g$ has the required properties.
\end{proof}
\begin{lemma} \label{lem-func-conn-2}
Let $H\subseteq\R^2$ be a functionally connected set such that $\pr_1[H]=\R$.
Then for every $f\in C(\R,\R)$ and $E\in\CL(\R)$ such that $f\r E\subseteq H$,
there exists $g\in C(\R,\R)$ such that $g\r E=f\r E$ and $g\subseteq H$.
\end{lemma}
\begin{proof}
Let $H$, $f$ and $E$ be as assumed.
Let us note that for each point $(x,y)\in H$
there exists $g\in C(\R,\R)$ such that $g(x)=y$ and $g\subseteq H$.
For every closed interval $I$ adjacent to $E$ there exists a continuous function $g_I\colon\R\to\R$
such that $g_I\subseteq H$ and $g_I$ coincides with $f$ at the endpoints of $I$.
Let $g(x)=f(x)$ for $x\in E$ and $g(x)=g_I(x)$ for $x\in I$ if $I$ is a closed interval
adjacent to $E$.
We obtain a function $g\colon\R\to\R$ such that $g\r E=f\r E$ and $g\subseteq H$.
It remains to show that $g_I$ can be chosen so that $g$ is continuous.
This is clear if there are only finitely many such intervals, so we will assume the opposite.
Let $\{I_n\!:n\in\omega\}$ be one-to-one enumeration of all closed intervals adjacent to $E$.
We define intervals $g_n=g_{I_n}$ as follows.
If $I_n$ is unbounded then let $g_n\colon I_n\to\R$ be arbitrary continuous function
such that $g_n\subseteq H$ and $g_n$ coincides with $f$ at the only endpoint of $I_n$.
Assume that $I_n=[a_n,b_n]$.
For every continuous function $h\colon I_n\to\R$ denote $\osc(h)$ its oscillation, that is,
$\osc(h)=\max\{h(x)-h(y)\!:x,y\in I_n\}$.
Let $o_n=\inf\{\osc(h)\!:h\in\H_n\}$, where $\H_n$ is the family of all functions $h\in C(I_n,\R)$
such that $h\subseteq H$ and $h\r\{a_n,b_n\}=f\r\{a_n,b_n\}$.
Choose $g_n\in\H_n$ such that $\osc(g_n)\le o_n + 2^{-n}$.
We will prove that the function $g$ defined above is continuous at every point $z\in\R$.
Let us take a convergent sequence $z_k\to z$.
We will assume that this sequence is increasing, as it suffices to consider only one-sided
limits, and for decreasing sequences the proof is the same.
We may further assume that $z_k\in\R\setminus E$ for all $k$
since we have $g(x)=f(x)$ for $x\in E$ and $f$ is continuous at $z$.
For every $k$, let $n_k$ be such that $z_k\in I_{n_k}$.
If there exist $m,l$ such that $n_k=m$ for all $k>l$, then $g(z_k)=g_m(z_k)$ for all $k>l$, hence
$z\in\cl I_m$ and $g(z_k)\to g(z)$.
So we may assume that $n_k\to\infty$ and $z\in E$.
To prove that $g(z_k)\to g(z)$, it will suffice to show that $\osc(g_{n_k})\to 0$.
Fix some $h\in C(\R,\R)$ such that $h\subseteq H$ and $h(z)=g(z)$.
By Lemma~\ref{lem-func-conn-1}, for every $n$ we have $o_n\le\diam(f[I_n]\cup h[I_n])$.
Since both $f$ and $h$ is continuous at $z$, we have
$\diam(f[I_n]\cup h[I_n])\to 0$, hence
$\osc(g_{n_k})\le o_{n_k}+2^{-n_k}\to 0$.
\end{proof}
Recall that a family $\G\subseteq C(\R,\R)$ is said to be complete if $f\in\G$
for every $f\in C(\R,\R)$ such that $f\subseteq\bigcup\G$.
A complete family $\G\subseteq C(\R,\R)$ is connected if and only if $\bigcup\G$
is functionally connected.
\begin{theorem} \label{thm-strictly-below-equiv-2}
Let $\G\subseteq C(\R,\R)$ be nonempty.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\K_\G\subseteq\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}$.
\item $\G$ is a complete and connected family.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1) \Rightarrow (2)$.
Denote $H=\bigcup\G$.
Clearly, $\G\subseteq\{g\in C(\R,\R)\!:g\subseteq H\}$.
To prove the opposite inclusion, let $g\in C(\R,\R)$ be such that $g\subseteq H$.
Denote $\E=E_\G(\{g\})$.
For every $x\in\R$ we have $g(x)\in\G[x]$, hence $\{x\}\in\E$.
It follows that $\R\in\E$, hence $g\in\G$.
Thus, $\G=\{g\in C(\R,\R)\!:g\subseteq H\}$, hence $\G$ is complete.
It remains to show that $H$ is functionally connected.
Let $(x_1,y_1),(x_2,y_2)\in H$ and $x_1<x_2$.
Let $f\in C(\R,\R)$ be such that $f(x_1)=y_1$ and $f(x_2)=y_2$, and let $\E=E_\G(\{f\})$.
Since $\{x_1\},\{x_2\}\in\E$ and $\E\in\K_\G$, it follows that $\{x_1,x_2\}\in\E$.
Hence, there exists $g\in\G$ such that $g(x_1)=y_1$ and $g(x_2)=y_2$.
$(2) \Rightarrow (1)$.
Let $\E\in\K_\G$, that is, there exists $\F\subseteq C(\R,\R)$ such that $\E=E_\G(\F)$.
Let $X=\bigcup\E$.
We will show that $\E=\CL(\R)\cap\P(X)$.
Let us take $E\in\CL(\R)\cap\P(X)$ and an arbitrary $f\in\F$.
For every $x\in E$ we have $\{x\}\in\E$, hence $f(x)\in\G[x]$.
It follows that $f\r E\subseteq\bigcup\G$.
Since $\G$ is connected, by Lemma~\ref{lem-func-conn-2}
there exists $g\in C(\R,\R)$ such that $f\r E=g\r E$ and $g\subseteq\bigcup\G$.
Since $\G$ is complete, we have $g\in\G$.
This shows that $E\in E_\G(\F)$, so $\CL(\R)\cap\P(X)\subseteq\E$.
The opposite inclusion is clear.
\end{proof}
From Theorems \ref{thm-strictly-below-equiv-1} and \ref{thm-strictly-below-equiv-2}
we obtain the following characterization.
\begin{corollary} \label{cor-strictly-below-together}
Let $\G\subseteq C(\R,\R)$ be nonempty.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\K_\G=\{\CL(\R)\cap\P(X)\!:X\subseteq\R\}$.
\item $\G$ is a complete and connected family, and for every $x\in\R$
there exists a function $f\in C(\R,\R)$ such that $f\setminus\bigcup\G=f\r\{x\}$.
\end{enumerate}
\end{corollary}
\section{Results for family $(-\infty,g)\cup(g,\infty)$}
For $f,g\in C(\R,\R)$, if $f(x)\neq g(x)$ for every $x$ then either $f<g$ or $f>g$.
Hence, $\{f\in C(\R,\R)\!:(\forall x\in\R)\,f(x)\neq g(x)\}=(-\infty,g)\cup(g,\infty)$.
Recall that a family $\X$ of subsets of a topological space is said to be separated
if for every distinct $X,Y\in\X$ there exist disjoint open sets $U,V$ such that
$X\subseteq U$, $Y\subseteq V$ and $(\forall Z\in\X)\ Z\subseteq U\,\lor\,Z\subseteq V$.
\begin{theorem} \label{thm-diff-equal}
Let $g\in C(\R,\R)$, $\G=\{f\in C(\R,\R)\!:(\forall x\in\R)\,f(x)\neq g(x)\}$.
Then $$\K_\G=\left\{\CL(\R)\cap\bigcup_{X\in\X}\P(X)\!:\X\subseteq\P(\R)\text{ is separated\/}\right\}.$$
\end{theorem}
\begin{proof}
Let $\E\in\K_\G$, that is, $\E=E_\G(\F)$ for some $\F\subseteq C(\R,\R)$.
For $x\in\bigcup\E$, denote $\E_x=\{E\in\E\!:x\in E\}$,
and let $\X=\big\{\bigcup\E_x\!:x\in\bigcup\E\big\}$.
We will show that $\X$ is separated and $\E=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
Let us note that for every $x,y\in\R$, $\{x,y\}\in\E$ if and only if for every $f\in\F$,
$$(f(x)>g(x)\,\land\,f(y)>g(y))\,\lor\,(f(x)<g(x)\,\land\,f(y)<g(y)).$$
Hence, the relation $\sim$, defined by $x\sim y\Leftrightarrow\{x,y\}\in\E$, is an equivalence relation on $\bigcup\E$,
and $\X$ is the corresponding partition of $\bigcup\E$ into equivalence classes.
Let $X=\bigcup\E_x$ and $Y=\bigcup\E_y$ be distinct elements of $\X$.
Then $\{x,y\}\notin\E$, hence there exists $f\in\F$ such that $(f(x)-g(x))(f(y)-g(y))\le 0$.
We have $\{x\},\{y\}\in\E$, so $(f(x)-g(x))(f(y)-g(y))\neq 0$.
Without a loss of generality we may assume that $f(x)<g(x)$ and $f(y)>g(y)$.
Let $U=\{x\in\R\!:f(x)<g(x)\}$ and $V=\{x\in\R\!:f(x)>g(x)\}$.
Then $U,V$ are disjoint open sets such that $X\subseteq U$, $Y\subseteq V$.
Also, for every $z\in\bigcup\E$ we have $f(z)\neq g(z)$, hence $z\in U$ or $z\in V$.
Clearly, $z\in U$ implies $\E_z\subseteq U$, and similarly $z\in V$ implies $\E_z\subseteq V$,
hence the family $\X$ is separated.
We show that $\E=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
The inclusion from left to right follows from the definition of $\X$.
Conversely, if $E\in\CL(\R)\cap\P(X)$ for some $X\in\X$ then we have $x\sim y$ for all $x,y\in E$,
hence for every $f\in\F$ we have either $f<_E g$ or $g<_E f$,
where $f<_E g$ is a shorthand for $(\forall x\in E)\,f(x)<g(x)$.
It follows that $f\r E\in\G\r E$, thus $E\in E_\G(\F)=\E$.
We have proved that $\K_\G\subseteq\big\{\CL(\R)\cap\bigcup_{X\in\X}\!:\X\subseteq\P(\R)\text{ is separated\/}\big\}$.
Let $\E=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$
for some separated family $\X\subseteq\P(\R)$, and let $\F=F_\G(\E)$.
Then $f\in\F$ if and only if $f\in C(\R,\R)$ and $f\r E\in\G\r E$ for every $E\in\E$.
Hence, $\F=\{f\in C(\R,\R)\!:(\forall X\in\X)\,f<_X g\,\lor\,g<_X f\}$.
Let us further show that $E_\G(\F)=\E$.
Assume that $E\in\CL(\R)$ and $E\notin\E$.
Then either $E\nsubseteq\bigcup\X$ or there exist distinct sets $X,Y\in\X$ such that $E$ intersects both of them.
In the first case we take $z\in E\setminus\bigcup\X$ and $f\in C(\R,\R)$ such that $f(z)=g(z)$
and $f(x)>g(x)$ for all $x\neq z$.
Then $f\in\F$ but $f(z)\notin\G[z]$, hence $E\notin E_\G(\F)$.
In the second case let $U,V$ be disjoint open sets such that $X\subseteq U$, $Y\subseteq V$ and
$(\forall Z\in\X)\ Z\subseteq U\,\lor\,Z\subseteq V$.
There exists $f\in C(\R,\R)$ such that $U=\{x\in\R\!:f(x)>g(x)\}$ and $V=\{x\in\R\!:f(x)<g(x)\}$.
We have $f\in\F$ but $E\notin E_\G(\F)$.
It follows that $E_\G(\F)\subseteq\E$, hence the equality holds true, and thus $\E\in\K_\G$.
Hence, $\big\{\CL(\R)\cap\bigcup_{X\in\X}\P(X)\!:\X\subseteq\P(\R)\text{ is separated}\big\}\subseteq\K_\G$.
\end{proof}
If a family $\X\subseteq\P(\R)$ is separated then for every distinct sets $X,Y\in\X$
one can find regular open sets $U,V$ such that $X\subseteq U$, $Y\subseteq V$
and $(\forall Z\in\X)\ Z\subseteq U\,\lor\,Z\subseteq V$.
Indeed, if $U,V$ are disjoint open sets then their regularizations
$U'=\int(\cl U)$, $V'=\int(\cl V)$ satisfy $U\subseteq U'$, $V\subseteq V'$ and are disjoint as well.
We may also take $\R\setminus\cl U'$ instead of $V'$.
Let us recall that for $x\in\R$ we have denoted $\A_x=\CL(\R)\cap\P(\R\setminus\{x\})$.
For every open set $U\subseteq\R$ we also denote $\B_U=\CL(\R)\cap(\P(U)\cup\P(\R\setminus\cl U))$.
\begin{theorem} \label{thm-diff-2}
Let $\G\subseteq C(\R,\R)$ be nonempty.
The following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\big\{\CL(\R)\cap\bigcup_{X\in\X}\P(X)\!:\X\subseteq\P(\R)\text{ is separated\/}\big\}\subseteq\K_\G$.
\item $\{\A_x\!:x\in\R\}\cup\{\B_U\!:U\subseteq\R\text{ is regular open\/}\}\subseteq\K_\G$.
\item For any $x\in\R$ there exists $f\in C(\R,\R)$ such that $E_\G(\{f\})=\A_x$.
Moreover, for any $x,y\in\R$ and any regular open set $U\subseteq\R$
such that $x\in U$ and $y\notin\cl U$ there exists $f\in F_\G(\B_U)$ such that
$f\r\{x,y\}\notin\G\r\{x,y\}$.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1) \Rightarrow (2)$ is obvious.
$(2) \Rightarrow (3)$.
The first part of (3) follows from Theorem~\ref{thm-strictly-below-equiv-1}.
For the second part, let $U$ be a regular open set such that $x\in U$ and $y\notin\cl U$.
By (2), we have $\B_U\in\K_\G$, hence $E_\G(F_\G(\B_U))=\B_U$.
Since $\{x,y\}\notin\B_U$, there exists $f\in F_\G(\B_U)$ such that $f\r\{x,y\}\notin\G\r\{x,y\}$.
$(3) \Rightarrow (1)$.
Let $\X\subseteq\P(\R)$ be a separated family and let $\E=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
Denote $\F=F_\G(\E)$.
We show that $\E=E_\G(\F)$.
Let us take $E\in\CL(\R)$, $E\notin\E$.
Then either there exists $x\in E\setminus\bigcup\X$, or there exist $x,y\in E$ and distinct sets $X,Y\in\X$
such that $x\in X$ and $y\in Y$.
If $x\in E\setminus\bigcup\X$ then let $f\in F_\G(\A_x)$ be such that $f(x)\notin\G_x$.
Since $\E\subseteq\A_x$, we have $F_\G(\A_x)\subseteq F_\G(\E)$, hence $f\in\F$.
It follows that $\{x\}\notin E_\G(\F)$, hence $E\notin E_\G(\F)$.
If $x\in X$, $y\in Y$ for some distinct $X,Y\in\X$ then there exists an open regular set $U$
such that $X\subseteq U$, $Y\subseteq\R\setminus\cl U$ and each $Z\in\X$ is covered either by $U$
or by $\R\setminus\cl U$.
It follows that $\E\subseteq\B_U$.
By (3), there exists $f\in F_\G(\B_U)$ such that $f\r\{x,y\}\notin\G\r\{x,y\}$.
We have $f\in\F$, hence $\{x,y\}\notin E_\G(\F)$, so $E\notin E_\G(\F)$.
In both cases it follows that $E_\G(\F)=\E$, hence $\E\in\K_\G$.
\end{proof}
Let us note that the family $\H=\{\A_x\!:x\in\R\}\cup\{\B_U\!:U\text{ is regular open}\}$
in the second condition of Theorem~\ref{thm-diff-2} is not minimal.
Indeed, $\B_\emptyset=\B_\R=\CL(\R)$ is an element of $\K_\G$ for every $\G\subseteq C(\R,\R)$,
hence $\H\setminus\{\B_\emptyset\}\subseteq\K_\G\,\Leftrightarrow\,\H\subseteq\K_\G$
holds for every $\G$.
We do not know whether there exists a regular open set $U$ and
a family $\G\subseteq C(\R,\R)$ such that $\H\setminus\{\B_U\}\subseteq\K_\G$ and $\B_U\notin\K_\G$.
We also do not know whether one can find a minimal family $\M$
of nonempty hereditary families of closed sets
having the property that for every $\G$, if $\M\subseteq\K_\G$ then
$\CL(\R)\cap\bigcup_{X\in\X}\P(X)\in\K_\G$ for every separated family $\X$.
To characterize families $\G$ such that
$\CL(\R)\cap\bigcup_{X\in\X}\P(X)\in\K_\G$ holds for every separated family $\X$,
we need few more notions.
Given a fixed family $\G\subseteq C(\R,\R)$ and points $a=(a_1,a_2)$, $b=(b_1,b_2)$ in $\R^2$,
let us write $a\sim b$ if there exists a function $f\in\G$ such that $f(a_1)=a_2$ and $f(b_1)=b_2$.
Clearly, if $a\sim b$ and $a_1=b_1$ then $a=b$.
We say that family $\G$ is \emph{transitive} if for any points $a,b,c\in\R^2$ having distinct
first coordinates, if $a\sim b$ and $b\sim c$ then $a\sim c$.
We say that family $\G$ is \emph{sequential} if $a\sim b$ holds true whenever $a,b\in\bigcup\G$
and there exists a sequence of points $\{a_n\!:n\in\omega\}$ in $\bigcup\G$
such that $a_n\to a$, $a_n\sim b$, and the first coordinates of points $a$, $b$, and $a_n$
are distinct, for every $n$.
Let $(I,<)$ be a linearly ordered set, and for every $i\in I$, let $\G_i\subseteq C(\R,\R)$
be nonempty.
We say that indexed system $\{\G_i\!:i\in I\}$ is \emph{sliced} if for every $i\in I$
there exist functions $g^{-}_i,g^{+}_i\in C(\R,\R^*)$ such that
$$\bigcup_{j<i}\G_j\subseteq(-\infty,g^{-}_i),\quad
\G_i\subseteq(g^{-}_i,g^{+}_i)\quad\text{and}\quad
\bigcup_{j>i}\G_j\subseteq(g^{+}_i,\infty).$$
\begin{lemma} \label{lem-sliced-1}
Let $\{\G_i\!:i\in I\}$ be a sliced system.
Then for each $i\in I$ there exist functions $h^{-}_i,h^{+}_i\in C(\R,\R^*)$ such that
$\G_i\subseteq(h^{-}_i,h^{+}_i)$ and $h^{+}_i\le h^{-}_j$ whenever $i<j$.
\end{lemma}
\begin{proof}
For every $i$, let $g^{-}_i,g^{+}_i\in C(\R,\R^*)$ be such that
$\bigcup_{j<i}\G_j\subseteq(-\infty,g^{-}_i)$, $\G_i\subseteq(g^{-}_i,g^{+}_i)$,
and $\bigcup_{j>i}\G_j\subseteq(g^{+}_i,\infty)$.
It is clear that if $\bigcup_{j<i}\G_j\neq\emptyset$ then $g^{-}_i\in C(\R,\R)$
and, similarly, if $\bigcup_{j>i}\G_j\neq\emptyset$ then $g^{+}_i\in C(\R,\R)$.
Since each interval $(g^{-}_i(0),g^{+}_i(0))$ contains a rational number,
$I$ is at most countable.
For simplicity let us assume that $I$ is infinite.
In the finite case the proof will be the same.
Let $\{i(n)\!:n<\omega\}$ be a one-to-one enumeration of $I$.
By induction let us define
\begin{align*}
h^{-}_{i(n)}(x)=\max\big(\big\{h^{+}_{i(m)}(x)\!:m<n&\,\land\,i(m)<i(n)\big\}\cup\big\{g^{-}_{i(n)}(x)\big\}\big),\\
h^{+}_{i(n)}(x)=\min\big(\big\{h^{-}_{i(m)}(x)\!:m<n&\,\land\,i(m)>i(n)\big\}\cup\big\{g^{+}_{i(n)}(x)\big\}\big).
\end{align*}
It is clear that $h^{-}_i,h^{+}_i\in C(\R,\R^*)$, and $\G_i\subseteq(h^{-}_i,h^{+}_i)$.
Moreover, if $m<n$ then either $i(m)<i(n)$ and then $h^{+}_{i(m)}\le h^{-}_{i(n)}$ by the definition
of $h^{-}_{i(n)}$, or $i(m)>i(n)$ and then $h^{+}_{i(n)}\le h^{-}_{i(m)}$ by the definition of
$h^{+}_{i(n)}$.
Hence, $h^{+}_i\le h^{-}_j$ for any $i<j$.
\end{proof}
\begin{lemma} \label{lem-sliced-2}
Let $\G\subseteq C(\R,\R)$ be a complete, transitive and sequential family.
Then there exists a sliced system $\{\G_i\!:i\in I\}$ such that $\G=\bigcup_{i\in I}\G_i$ and
each $\G_i$ is complete and connected.
\end{lemma}
\begin{proof}
We may assume that $\G$ is nonempty.
Denote $H=\bigcup\G$.
For $a,b\in\R^2$, let us write $a\approx b$ if there exists $c$ such that $a\sim c\sim b$.
We prove that $\approx$ is an equivalence relation on $H$.
The symmetry and the reflexivity of $\approx$ is clear.
For the transitivity it suffices to prove that $a\sim b\sim c\sim d$ implies $a\approx d$.
Let $a=(a_1,a_2)$, $b=(b_1,b_2)$, $c=(c_1,c_2)$, $d=(d_1,d_2)$.
We may assume that $a\neq b\neq c\neq d$, hence $a_1\neq b_1\neq c_1\neq d_1$.
If $a_1\neq c_1$ then by transitivity of $\G$ we have $a\sim c$, and we are done.
A similar argument works if $b_1\neq d_1$, so we may assume that $a_1=c_1$ and $b_1=d_1$.
Without a loss of generality, let $a_1<b_1$.
Let $f\in\G$ be such that $f(b_1)=b_2$, and let $b'=(b'_1,b'_2)$ be such that
$b'_1>b_1$ and $b'_2=f(b'_1)$.
Then we have $a\sim b'$ and $b'\sim c$.
Since $b'_1\neq d_1$, it follows that $b'\sim d$, and thus $a\approx d$.
Let $\{H_i\!:i\in I\}$ be the partition of $H$ corresponding to the equivalence $\approx$,
and for every $i\in I$ let $\G_i=\{f\in\G\!:f\subseteq H_i\}$.
Let $f\in\G$ be arbitrary.
For all points $a,b\in f$ we have $a\approx b$, hence $f\subseteq H_i$ for some $i$.
It follows that $\G=\bigcup\{\G_i\!:i\in I\}$.
By the definition of $\approx$ and the completeness of $\G$, each $\G_i$ is connected and complete,
and we have $\bigcup\G_i=H_i$.
Clearly, if $\G_i\neq\G_j$ then
$(\forall f\in\G_i)(\forall g\in\G_j)\,f<g$ or $(\forall f\in\G_i)(\forall g\in\G_j)\,f>g$.
Thus there exists a linear order on $I$ so that
$i<j$ if and only if $f<g$ for all $f\in\G_i$ and $g\in\G_j$.
For every $i$ such that $\bigcup_{j>i}\G_j\neq\emptyset$,
let us define $h_1(x)=\inf\{\sup A_{x,\varepsilon}\!:\varepsilon>0\}$ and
$h_2(x)=\sup\{\inf B_{x,\varepsilon}\!:\varepsilon>0\}$,
where $A_{x,\varepsilon}=H_i\cap\big((x-\varepsilon,x+\varepsilon)\times\R\big)$ and
$B_{x,\varepsilon}=\bigcup_{j>i}H_j\cap\big((x-\varepsilon,x+\varepsilon)\times\R\big)$.
Then $h_1$ is upper semi-continuous, $h_2$ is lower semi-continuous,
and we have $f\le h_1\le h_2\le g$ for all $f\in\G_i$ and $g\in\bigcup_{j>i}\G_j$.
For $x\in\R$ denote $a=(x,h_1(x))$ and assume that $a\in H$.
Then there exists a sequence $\{a_n\!:n\in\omega\}$ in $H_i$ converging to $a$
and such that each $a_n$'s first coordinate is distinct from $x$.
Let $b\in H_i$ be such that for every $n$, first coordinates of $a$, $b$, and $a_n$ are distinct.
We have $a_n\sim b$ for every $n$.
Since $\G$ is sequential, it follows that $a\sim b$, hence $a\in H_i$.
Similarly, if $b=(x,h_2(x))$ and $b\in H$ then there exists $k\in I$ such that
$k=\min\{j\in I\!:j>i\}$, and $b\in H_k$.
It follows that if $h_1(x)=h_2(x)=y$ then $(x,y)\notin H$.
By a theorem of Michael (see~\cite{Engelking}, Exercise 1.7.15 (d)),
there exists a continuous function $h^{+}\colon\R\to\R$ such that $h_1\le h^{+}\le h_2$
and for every $x\in\R$, if $h_1(x)<h_2(x)$ then $h_1(x)<h^{+}(x)<h_2(x)$.
It follows that $f<h^{+}<g$ for any $f\in\G_i$ and $g\in\bigcup_{j>i}\G_j$.
A similar argument shows that if $\bigcup_{j<i}\G_j\neq\emptyset$ then
there exists $h^{-}\in C(\R,\R)$ such that $f<h^{-}<g$ for any $f\in\bigcup_{j<i}\G_j$ and $g\in\G_i$.
Hence, $\{\G_i\!:i\in I\}$ is a sliced system.
\end{proof}
\begin{theorem} \label{thm-diff-3}
Let $\G\subseteq C(\R,\R)$ be nonempty.
Then the following conditions are equivalent.
\begin{enumerate}[\rm (1)]
\item $\K_\G\subseteq\big\{\CL(\R)\cap\bigcup_{X\in\X}\P(X)\!:\X\text{ is separated\/}\big\}$.
\item There exists a sliced system $\{\G_i\!:i\in I\}$ of complete connected families
such that $\G=\bigcup_{i\in I}\G_i$.
\end{enumerate}
\end{theorem}
\begin{proof}
$(1) \Rightarrow (2)$.
Denote $H=\bigcup\G$.
If $g\in C(\R,\R)$ and $g\subseteq H$ then let $\E=E_\G(\{g\})$.
Since $\E\in\K_\G$, there exists a strictly separated family $\X\subseteq\P(\R)$ such that
$\E=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
For every $x\in\R$ we have $g(x)\in\G_[x]$, hence $\{x\}\in\E$, and thus $\bigcup\X=\R$.
For any disjoint open sets $U,V\subseteq\R$, if $U\cup V=\R$ then either $U=\R$ or $V=\R$.
It follows that $\X=\{\R\}$, hence $\R\in\E$ and thus $g\in\G$.
Hence, $\G=\{g\in C(\R,\R)\!:g\subseteq H\}$, so $\G$ is complete.
Let us show that $\G$ is transitive.
Let $a=(a_1,a_2)$, $b=(b_1,b_2)$, $c=(c_1,c_2)$ be such that $a_1,b_1,c_1$ are distinct,
and let $a\sim b\sim c$.
Since for any $(x,y)\in H$ there exists $g\in\G$ such that $g(x)=y$,
we can find $g,h\in\G$ such that $g(a_1)=a_2$, $g(b_1)=h(b_1)=b_2$, and $h(c_1)=c_2$.
Let $f\in C(\R,\R)$ be any function such that $f(a_1)=a_2$, $f(b_1)=b_2$, $f(c_1)=c_2$,
and let $\E=E_\G(\{f\})$.
Then $\{a_1,b_1\}\in\E$, $\{b_1,c_1\}\in\E$,
hence also $\{a_1,c_1\}\in\E$, and it follows that
there exists $f'\in\G$ such that $f'(a_1)=a_2$ and $f'(c_1)=(c_2)$.
Since $f'\subseteq H$, we have $f'\in\G$, hence $a\sim c$.
To show that $\G$ is also sequential, assume that $a,b\in H$ and there exists a sequence
$\{a_n\!:n\in\omega\}$ in $H$ such that $a_n\to a$ , $a_n\sim b$, and first coordinates
of points $a$, $b$, and $a_n$ are distinct, for every $n$.
We have to prove that $a\sim b$.
There exists a function $f\in C(\R,\R)$ such that $f(x)=y$, $f(u)=v$, and $f(x_n)=y_n$ for all $n$,
where $(x,y)=a$, $(u,v)=b$, and $(x_n,y_n)=a_n$.
Denote $\E=E_\G(\{f\})$.
Since $\E\in\K_\G$, by (1) there exists a separated family $\X\subseteq\P(\R)$
such that $\E=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
We have $\{x\}\in\E$ and $\{x_n,u\}\in\E$, for every $n$.
It suffices to show that $\{x,u\}\in\E$.
If this is not the case then there exist distinct sets $X,Y\in\X$ such that $x\in X$ and $u\in Y$.
Since $\X$ is separated, there exist disjoint open sets $U,V$ such that $X\subseteq U$, $Y\subseteq V$,
and $(\forall Z\in\X)\ Z\subseteq U\,\lor\,Z\subseteq V$.
Since $x_n\to x$, there exists $n$ such that $x_n\in U$.
But then $x_n\notin Y$, and this is in contradiction with $\{x_n,u\}\in\E$.
We have proved that $\G$ is complete, transitive and sequential.
Then condition~(2) follows from Lemma~\ref{lem-sliced-2}.
$(2) \Rightarrow (1)$.
Let $\{\G_i\!:i\in I\}$ be a sliced system of complete connected families such that
$\G=\bigcup_{i\in I}\G_i$.
By Lemma~\ref{lem-sliced-1}, for every $i$ there exist $h^{-}_i,h^{+}_i\in C(\R,\R^*)$ such that
$\G_i\subseteq(h^{-}_i,h^{+}_i)$ and $h^{+}_i\le h^{-}_j$ whenever $i<j$.
Let $f\in C(\R,\R)$ be arbitrary and let $\E=E_\G(\{f\})$.
For every $i$, denote $X_i=\{x\in\R\!:f(x)\in\G_x\,\land\,h^{-}_i(x)<f(x)<h^{+}_{i}(x)\}$,
and let $\X=\{X_i\!:i\in I\}$.
For every $i\in I$, let us take $U_i=\{x\in\R\!:h^{-}_i(x)<f(x)<h^{+}_i(x)\}$ and
$V_i=\{x\in\R\!:f(x)<h^{-}_i(x)\,\lor\,h^{+}_i(x)<f(x)\}$.
Then $U_i,V_i$ are disjoint open sets such that $X_i\subseteq U_i$ and
$X_j\subseteq V_i$ for every $j\neq i$.
It follows that $\X$ is a separated family.
We will prove that $\E=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
If $E\in\E$ then there exists $g\in\G$ such that $f\r E=g\r E$.
We have $g\in\G_i$ for some $i\in I$, and it easy to see that $E\subseteq X_i$.
We can conclude that $\E\subseteq\CL(\R)\cap\bigcup_{X\in X}\P(X)$.
To prove the opposite inclusion, assume that $E\notin\E$,
hence for every $g\in\G$ we have $f\r E\neq g\r E$.
If there exists $x\in E$ such that $f(x)\notin\G_x$ then $x\notin\bigcup\X$, and hence
$E\notin\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
Assume further that $E\subseteq\{x\in\R\!:f(x)\in\G_x\}$.
If there exists $i\in I$ such that $f\r E\subseteq\bigcup\G_i$, then by Lemma~\ref{lem-func-conn-2}
there exists $g\in\G_i$ such that $f\r E=g\r E$, which is impossible.
Hence, there exist $i\neq j$ and $x,y\in E$ such that
$(x,f(x))\in\bigcup\G_i$ and $(y,f(y))\in\bigcup\G_j$.
It follows that $x\in X_i$ and $y\in X_j$, hence $E\notin\CL(\R)\cap\bigcup_{X\in\X}\P(X)$.
We have proved that for every $f\in C(\R,\R)$ there exists a separated family $\X_f\subseteq\P(\R)$
such that $E_\G(\{f\})=\CL(\R)\cap\bigcup_{X\in\X_f}\P(X)$.
For arbitrary $\F\subseteq C(\R,\R)$, we have $E_\G(\F)=\bigcap_{f\in\F}E_\G(\{f\})$.
Let us take
$$\X=\Bigg\{\bigcap_{f\in\F}X_f\!:\langle X_f\!:f\in\F\rangle\in\prod_{f\in\F}\X_f\,\land\,
\bigcap_{f\in\F}X_f\neq\emptyset\Bigg\}.$$
We show that $\X$ is separated.
Let $\langle X_f\!:f\in\F\rangle,\langle Y_f\!:f\in\F\rangle\in\prod_{f\in\F}X_f$
be such that $\bigcap_{f\in\F}\neq\emptyset$, $\bigcap_{f\in\F}Y_f\neq\emptyset$,
and $\bigcap_{f\in\F}\neq\bigcap_{f\in\F}Y_f$.
Then there exists $h\in\F$ such that $X_h\neq Y_h$.
Since $\X_h$ is separated, there exist disjoint open sets $U,V\subseteq\R$ such that
$X_h\subseteq U$, $Y_h\subseteq V$, and $(\forall Z\in\X_h)\ Z\subseteq U\,\lor\,Z\subseteq V$.
It follows that $\bigcap_{f\in\F}X_f\subseteq U$ and $\bigcap_{f\in\F}Y_f\subseteq V$.
For every $\langle Z_f\!:f\in\F\rangle\in\prod_{f\in\F}\X_f$ we have
$Z_h\subseteq U\,\lor\,Z_h\subseteq V$, hence also
$\bigcap_{f\in\F}Z_f\subseteq U\,\lor\,\bigcap_{f\in\F}Z_f\subseteq V$.
For $E\in\CL(\R)$, we have $E\in E_\G(\F)$ if and only if
$(\forall f\in\F)(\exists X\in\X_f)\,E\subseteq X$ if and only if
$(\exists X\in\X)\,E\subseteq X$.
Hence, $E_\G(\F)=\CL(\R)\cap\bigcup_{X\in\X}\P(X)$ and condition~(1) follows.
\end{proof}
|
2,869,038,154,768 | arxiv | \section{Introduction}\label{introduction}
The issue of the initial conditions of the universe -- in particular, the degree to which they are ``unnatural'' or ``fine-tuned,'' and possible explanations thereof -- is obviously of central importance to cosmology, as well as to the foundations of statistical mechanics.
The early universe was a hot, dense, rapidly-expanding plasma, spatially flat and nearly homogeneous along appropriately chosen spacelike surfaces.%
\footnote{Our concern here is the state of the universe -- its specific configuration of matter and energy, and the evolution of that configuration through time -- rather than the coupling constants of our local laws of physics, which may also be fine-tuned.
I won't be discussing the value of the cosmological constant, or the ratio of dark matter to ordinary matter, or the matter/antimatter asymmetry.}
The question is, \emph{why} was it like that?
In particular, the thinking goes, these conditions don't seem to be what we would expect a ``randomly chosen'' universe to look like, to the extent that such a concept makes any sense.
In addition to the obvious challenge to physics and cosmology of developing a theory of initial conditions under which these properties might seem natural, it is a useful exercise to specify as carefully as possible the sense in which they don't seem natural from our current point of view.
Philosophers of science (and some physicists \cite{Penrose,Carroll:2004pn}) typically characterize the kind of fine-tuning exhibited by the early universe as being a state of \emph{low entropy}.
This formulation has been succinctly captured in David Albert's {\it Time and Chance} as ``The Past Hypothesis'' \cite{albert}.
A precise statement of the best version of the Past Hypothesis is the subject of an ongoing discussion (see {\it e.g.} \cite{earman2006past,wallace2011}).
But it corresponds roughly to the idea that the early universe -- at least the observable part of it, the aftermath of the hot Big Bang -- was in a low-entropy state with the right micro-structure to evolve in a thermodynamically sensible way into the universe we see today.
Cosmologists, following Alan Guth's influential paper on the inflationary-universe scenario \cite{Guth:1980zm}, tend to describe the fine-tuning of the early universe in terms of the horizon and flatness problems.\footnote{Guth also discussed the overabundance of magnetic monopoles predicted by certain grand unified theories. This problem was the primary initial motivation for inflation, but is model-dependent in a way that the horizon and flatness problems don't seem to be.}
The horizon problem, which goes back to Misner \cite{misner1969mixmaster}, is based on the causal structure of an approximately homogeneous and isotropic (Friedmann-Robertson-Walker) cosmological model in general relativity.
If matter and radiation (but not a cosmological constant) are the only sources of energy density in the universe, regions we observe using the cosmic background radiation that are separated by more than about one degree were never in causal contact -- their past light-cones, traced back to the Big Bang, do not intersect.
It is therefore mysterious how they could be at the same temperature, despite the impossibility in matter/radiation cosmology of any signal passing from one such point to another.
The flatness problem, meanwhile, was elucidated by Dicke and Peebles \cite{dicke1979big}.
Spatial curvature grows with respect to the energy density in matter or radiation, so it needs to be extremely small at early times so as not to be completely dominant today.
(For some history of the horizon and flatness problems, see \cite{brawer1995inflationary}.
Brawer notes that neither the horizon problem nor the flatness problem were considered of central importance to cosmology until inflation suggested a solution to them.)
The horizon and flatness problems and the low entropy of the early universe are clearly related in some way, but also seem importantly different.
In this essay I will try to clarify the nature of the horizon and flatness problems, and argue that they are \emph{not} the best way of thinking about the fine-tuning of the early universe.
The horizon problem gestures in the direction of a real puzzle, but the actual puzzle is best characterized as fine-tuning within the space of cosmological \emph{trajectories}, rather than an inability of the early universe to equilibrate over large distances.
This reformulation is important for the status of inflationary cosmology, as it makes clear that inflation by itself does not solve the problem (though it may play a crucial role in an ultimate solution).
The flatness problem, meanwhile, turns out to be simply a misunderstanding; the correct measure on cosmological trajectories predicts that all but a set of measure zero should be spatially flat.
Correctly describing the sense in which the early universe is fine-tuned helps us understand what kind of cosmological models physicists and philosophers should be endeavoring to construct.
\section{What Needs to be Explained}\label{what}
In order to understand the claim that the state of the universe appears fine-tuned, we should specify what features of that state we are talking about.
According to the standard cosmological model, the part of the universe we are able to observe is expanding, and emerged from a hot, dense Big Bang about fourteen billion years ago.%
\footnote{In classical general relativity such a state corresponds to a curvature singularity; in a more realistic but less well-defined quantum theory of gravity, what we call the Big Bang may or may not have been the absolute beginning of the universe, but at the least it is a moment prior to which we have no empirical access.}
The distribution of matter and radiation at early times was nearly uniform, and the spatial geometry was very close to flat.
We see the aftermath of that period in the cosmic microwave background (CMB), radiation from the time the universe became transparent about 380,000 years after the Big Bang, known as the ``surface of last scattering.''
This radiation is extremely isotropic; its observed temperature is 2.73~K, and is smooth to within about one part in $10^5$ across the sky \cite{Ade:2013ktc}.
Our observable region contains about $10^{88}$ particles, most of which are photons and neutrinos, with about $10^{79}$ protons, neutrons, and electrons, as well as an unknown number of dark matter particles.
(The matter density of dark matter is well-determined, but the mass per particle is extremely uncertain.)
The universe has evolved today to a collection of galaxies and clusters within a web of dark matter, spread homogeneously on large scales over an expanse of tens of billions of light years.
Note that our confidence in this picture depends on assuming the Past Hypothesis.
The most relevant empirical information concerning the smoothness of the early universe comes from the isotropy of the CMB, but that does not strictly imply uniformity at early times.
It is a geometric fact that a manifold that is isotropic around every point is also homogeneous; hence, the observation of isotropy and the assumption that we do not live in a special place together are often taken to imply that the matter distribution on spacelike slices is smooth.
But we don't observe the surface of last scattering directly; what we observe is the isotropy of the temperature of the radiation field reaching our telescopes here in a much older universe.
That temperature is determined by a combination of two factors: the intrinsic temperature at emission, and the subsequent redshift of the photons.
The intrinsic temperature at the surface of last scattering is mostly set by atomic physics, but not quite; because there are so many more photons than atoms, recombination occurs at a temperature of about 3\,eV, rather than the 13.6\,eV characterizing the ionization energy of hydrogen.
Taking the photon-to-baryon ratio as fixed for convenience, our observations therefore imply that the cosmological redshift is approximately uniform between us and the surface of last scattering in all directions on the sky.
But that is compatible with a wide variety of early conditions.
The redshift along any particular path is a single number; it is not possible to decompose it into ``Doppler'' and ``cosmic expansion'' terms in a unique way \cite{bunn2009}.
This reflects the fact that there are many ways to define spacelike slices in a perturbed cosmological spacetime.
For example, we can choose to define spacelike surfaces such that the cosmological fluid is at rest at every point (``synchronous gauge'').
In those coordinates there is no Doppler shift, by construction.
But the matter distribution at recombination could conceivably look very inhomogeneous on such slices; that would be compatible with our current observations as long as a direction-dependent cosmological redshift conspired to give an isotropic radiation field near the Earth today.
Alternatively, we could choose spacelike slices along which the temperature was constant; the velocity of the fluid on such slices could be considerably non-uniform, yet the corresponding Doppler effect could be canceled by the intervening expansion of space along each direction.
Such conspiratorial conditions seem unlikely to us, but they are more numerous (in the measure to be discussed below) in the space of all possible initial conditions.
Of course, we also know that most past conditions that lead to a half-melted ice cube in a glass of water look like a glass of liquid water at a uniform temperature, rather than the unmelted ice cube in warmer water we would generally expect.
In both cases, our conventional reasoning assumes the kind of lower-entropy state postulated by the Past Hypothesis.
With the caveat that the Past Hypothesis is necessary, let us assume that the universe we are trying to account for is one that was very nearly uniform (and spatially flat) at early times.
What does it mean to day that such a state is fine-tuned?
Any fine-tuning is necessarily a statement about one's expectations about what would seem natural or non-tuned.
In the case of the initial state of the universe, one might reasonably suggest that we simply have no right to have any expectations at all, given that we have only observed one universe.
But this is a bit defeatist.
While we have not observed an ensemble of universes from which we might abstract ideas about what a natural one looks like, we know that the universe is a physical system, and we can ask whether there is a sensible \emph{measure} on the relevant space of states for such a system, and then whether our universe seems generic or highly atypical in that measure.
In practice we typically use the classical Liouville measure, as we will discuss in Section~\ref{trajectories}.
The point of such an exercise is not to say ``We have a reliable theory of what universes should look like, and ours doesn't fit."
Rather, given that we admit that there is a lot about physics and cosmology that we don't yet understand, it's to look for clues in the state of the universe that might help guide us toward a more comprehensive theory.
Fine-tuning arguments of this sort are extremely important in modern cosmology, although the actual measure with respect to which such arguments are made are sometimes insinuated rather than expressly stated, and usually posited rather than being carefully derived under some more general set of assumptions.
The fine-tuning argument requires one element in addition to the state and the measure: a way to coarse-grain the space of states.
This aspect is often overlooked in discussions of fine-tuning, but it is again implicit.
Without a coarse-graining, there is no way to say that any particular state is ``natural'' or ``fine-tuned,'' even in a particular measure on the entire space of states.
Each individual state is just a point, with measure zero.
What we really mean to say is that states \emph{like} that state are fine-tuned, in the sense that the corresponding macrostate in some coarse-graining has a small total measure.
The coarse-graining typically corresponds to classifying states as equivalent if they display indistinguishable macroscopically observable properties.
It is usually not problematic, although as we will see it is necessary to be careful when we consider quantities such as the spatial curvature of the universe.
Given a measure and a coarse-graining, it is natural to think of the entropy of each state, defined by Boltzmann to be the logarithm of the volume of the corresponding macrostate.
Penrose has famously characterized the fine-tuning problem in these terms, emphasizing how small the entropy of the early universe was compared to the current entropy, and how small that is compared to how large the entropy could be \cite{Penrose}.
At early times, when inhomogeneities were small, we can take the entropy to simply be that of a gas of particles in the absence of strong self-gravity, which is approximately given by the number of particles; for our observable universe, that's about $10^{88}$.
Today, the entropy is dominated by supermassive black holes at the center of galaxies, each of which has a Bekenstein-Hawking entropy
\begin{equation}
S_{\mathrm{BH}} = \frac{A}{4G} = 4\pi G M^2
\end{equation}
in units where $\hbar = c = k_B = 1$.
The best estimates of the current entropy give numbers of order $10^{103}$ \cite{egan2010larger}.
Penrose suggests a greatest lower bound on the allowed entropy of our observable universe by calculating what the entropy would be if all of the matter were put into one gigantic black hole, obtaining the famous number $10^{122}$.
Since the universe is accelerating, distant galaxies will continue to move away from us rather than collapsing together to form a black hole, but we can also calculate the entropy \cite{Banks:2000fe,Banks:2001yp} of the de~Sitter phase to which the universe will ultimately evolve, obtaining the same number $10^{122}$.
(The equality of these last two numbers can be explained by the cosmological coincidence of the density in matter and vacuum energy; they are both of order $1/GH^2_0$, where $H_0$ is the current Hubble parameter.)
By any measure, the entropy of the early universe was fantastically smaller than its largest possible value, which seems to be a fine-tuning.%
\footnote{Even though the comoving volume corresponding to our observable universe was much smaller at early times, it was still the same physical system as the late universe, with the same space of states, governed by presumably-reversible dynamical laws.
It is therefore legitimate to calculate the maximum entropy of the early universe by calculating the maximum entropy of the late universe.}
Part of our goal in this essay is to relate this formulation to the horizon and flatness problems.
To close this section, we note that the early universe was in an extremely \emph{simple} state, in the sense that its ``apparent complexity'' is low: macroscopic features of the state can be fully characterized by a very brief description (in contrast with the current state, where a macroscopic description would still require specifying every galaxy, if not every star and planet).
But apparent complexity is a very different thing than entropy; the universe is simple at the earliest times, and will be simple again at much later times, while entropy increases monotonically \cite{aaronson}.
The simplicity of the early universe should not in any way be taken as a sign of its ``naturalness.''
A similar point has been emphasized by Price, who notes that any definition of ``natural'' that purportedly applies to the early universe should apply to the late universe as well \cite{Price:1993hr}.
The early universe appears fine-tuned because the macroscopic features of its state represent a very small region of phase space in the Liouville measure, regardless of how simple it may be.
\section{The Horizon Problem}\label{horizon}
\subsection{Defining the problem}
We now turn to the horizon problem as traditionally understood.
Consider a homogeneous and isotropic universe with scale factor $a(t)$, energy density $\rho(a)$, and a fixed spatial curvature $\kappa$. In general relativity, the evolution of the scale factor is governed by the Friedmann equation,
\begin{equation}
H^2 = \frac{8\pi G}{3}\rho - \frac{\kappa}{a^2},
\label{friedmann}
\end{equation}
where $H=\dot{a}/a$ is the Hubble parameter and
$G$ is Newton's constant of gravitation, and we use units where the speed of light is set equal to unity, $c=1$.
Often, the energy density will evolve as a simple power of the scale factor, $\rho \propto a^{-n}$. For example, ``matter'' to a cosmologist is any species of massive particles with velocities much less than the speed of light, for which $\rho_M \propto a^{-3}$, while ``radiation'' is any species of massless or relativistic particles, for which $\rho_R \propto a^{-4}$.
(In both cases, the number density decreases as the volume increases, $n\propto a^{-3}$; for matter the energy per particle is constant, while for radiation it diminishes as $a^{-1}$ due to the cosmological redshift.)
When the energy density of the universe is dominated by components with $n > 2$, $\dot a$ will be decreasing, and we say the universe is ``decelerating.''
In a decelerating universe the horizon size at time $t_*$ (the distance a photon can travel between the Big Bang and $t_*$) is approximately
\begin{equation}
d_{\mathrm{hor}}(t_*) \approx t_* \approx H_*^{-1}.
\end{equation}
We call $H^{-1}$ the ``Hubble distance'' at any given epoch.
Sometimes the Hubble distance is conflated with the horizon size, but they are importantly different; the Hubble distance depends only on the instantaneous expansion rate at any one moment in time, while the horizon size depends on the entire past history of the universe back to the Big Bang.
The two are of the same order of magnitude in universes dominated by matter and radiation, with the precise numerical factors set by the abundance of each component.
A textbook calculation shows that the Hubble distance at the surface of last scattering, when the CMB was formed, corresponds to approximately one degree on the sky today.
The horizon problem, then, is simple.
We look at widely-separated parts of the sky, and observe radiation left over from the early universe.
As an empirical matter, the temperatures we observe in different directions are very nearly equal.
But the physical locations from which that radiation has traveled were separated by distances larger than the Hubble distance (and therefore the horizon size, if the universe is dominated by matter and radiation) at that time.
They were never in causal contact; no prior influence could have reached both points by traveling at speeds less than equal to that of light.
Yet these independent regions seem to have conspired to display the same temperature to us.
That seems like an unnatural state of affairs.
We can formalize this as:
\begin{quote}
{\bf Horizon problem (causal version).}
If different regions in the early universe have non-overlapping past light cones, no causal influence could have coordinated their conditions and evolution.
There is therefore no reason for them to appear similar to us today.
\end{quote}
If that's as far as it goes, the horizon problem is perfectly well-formulated, if somewhat subjective.
The causal formulation merely points out that there is no reason for a certain state of affairs (equal temperatures of causally disconnected regions) to obtain, but it doesn't give any reason for expecting otherwise.
Characterizing this as a ``problem,'' rather than merely an empirical fact to be noted and filed away, requires some positive expectation for what we think conditions near the Big Bang \emph{should} be like: some reason to think that unequal temperatures would be more likely, or at least less surprising.
\subsection{Equilibration and entropy}
We can attempt to beef up the impact of the horizon problem by bringing in the notion of \emph{equilibration} between different regions.
Imagine we have a box of gas containing different components, or simply different densities.
If it starts from an inhomogeneous (low-entropy) configuration, given time the gas will typically equilibrate and attain a uniform distribution.
In that sense, an equal temperature across a hot plasma actually seems natural or likely.
But if we think of the early universe as such a box of gas, it hasn't had time to equilibrate.
That's a direct consequence of the fact that (in a matter/radiation dominated universe) the horizon size is much smaller than the scales over which we are currently observing.
Let's call this the ``equilibration'' version of the horizon problem.
\begin{quote}
{\bf Horizon problem (equilibration version).}
If different regions in the early universe shared a causal past, they could have equilibrated and come to the same temperature.
But when they do not, such as in a matter/radiation-dominated universe, equal temperatures are puzzling.
\end{quote}
The equilibration formulation of the horizon problem seems stronger than the causal version; it attempts to provide some reason why equal temperatures across causally disconnected reasons should be surprising, rather than merely noting their existence.
But in fact this version undermines the usual conclusions that the horizon problem is intended to justify, and invites a critique that has been advanced by Sheldon Goldstein \cite{goldsteinbox}.
I will try to argue that Goldstein's critique doesn't rebut the claim that the early universe is fine-tuned, though it does highlight the misleading nature of the equilibration version of the horizon problem.
The problem arises when we try to be more specific about what might count as ``natural'' initial conditions in the first place.
If we trace the Friedmann equation (\ref{friedmann}) backwards in time, we come to a singularity -- a point where the scale factor is zero and the density and Hubble parameter are infinite.
There is no reason to expect the equations of classical general relativity to apply in such circumstances; at the very least, quantum effects will become important, and we require some understanding of quantum gravity to make sensible statements.
Absent such a theory, there would be some justification in giving up on the problem entirely, and simply saying that the issue of initial conditions can't be addressed without a working model of quantum gravity.
(Presumably this was the feeling among many cosmologists before inflation was proposed.)
Alternatively, we could choose to be a bit more optimistic, and ask what kind of configurations would constitute natural initial conditions in the moments right after the initial singularity or whatever quantum phase replaces it.
Given the phase space describing the relevant degrees of freedom in that regime, we can coarse-grain into macrostates defined by approximately equal values of macroscopically observable quantities and define the Boltzmann entropy as the logarithm of the volume of the macrostate of which the microstate is an element, as discussed in Section~\ref{what}.
Calculating this volume clearly requires the use of a measure on the space of states; fortunately such a measure exists, given by Liouville in the classical case or by the Hilbert-space inner product in the quantum case.
Given that machinery, states with high Boltzmann entropy seem natural or generic or likely, simply because there are many of them; states with low Boltzmann entropy seem unnatural since they are relatively few, and suggest the need for some sort of explanation.%
\footnote{Note that the expectation of high entropy for the early universe, whether convincing or not, certainly has a different character than our expectation that long-lived closed systems will be high-entropy in the real world.
In the latter case, there is a dynamical mechanism at work: almost all initial conditions will evolve toward equilibrium.
In the case of the early universe, by contrast, we are actually making a statement about the initial conditions themselves, saying that high-entropy ones would be less surprising than low-entropy ones, since the former are more numerous.}
Let's apply this to a matter/radiation-dominated universe, in which different regions we observe in the CMB are out of causal contact.
Following Goldstein, we can draw an analogy with two isolated boxes of gas.
The boxes could be different sizes, and have never interacted.
All we know is that there is some fixed number of particles in the boxes, with some fixed energy.
Goldstein's observation is that, if we know nothing at all about the particles inside the two boxes, it should be \emph{completely unsurprising} if they had the same temperature.
The reasoning is simple: at fixed energy and particle number, there is more phase space corresponding to approximately equal-temperature configurations than to unequal-temperature ones.
Such configurations maximize the Boltzmann entropy.
Given the two boxes, some fixed number of particles, and some fixed total energy, a randomly-chosen point in phase space is very likely to feature equal temperatures in each box, even if they have never interacted.
Therefore, one might tentatively suggest, perhaps seeing equal temperatures in causally disconnected parts of the early universe isn't actually unlikely at all, and the horizon problem shouldn't be a big worry.
\subsection{Gravity and dynamics}
For isolated boxes of gas, this logic is surprising, but valid.
A randomly-selected state of two isolated boxes of gas is likely to have equal temperatures in each box, even in the absence of interactions.
For the early universe, however, the boxes turn out to not provide a very useful analogy, for a couple of (related) reasons: the importance of gravity, and the fact that we are considering time-dependent trajectories, not simply individual states.
Gravity plays the crucial role in explaining a peculiar feature of the early universe: it is purportedly a low-entropy state, but one in which the matter is radiating as a nearly-homogeneous black body, exactly as we are trained to expect from systems in thermal equilibrium.
The simple resolution is that, when the \emph{self-gravity} of a collection of particles becomes important, high-entropy configurations of specified volume and density are inhomogeneous rather than homogeneous.
Therefore, if the imaginary boxes that we consider are sufficiently large, we wouldn't expect the temperature to be uniform even inside a single box, much less between two boxes.
This can be seen in a couple of different ways.
One is to consider the Jeans instability: the tendency of sufficiently long-wavelength perturbations in a self-gravitating fluid to grow.
In a fluid with density $\rho$ and speed of sound $c_s$, modes are unstable to growth if they are larger than the Jeans length,
\begin{equation}
\lambda_J = \frac{c_s}{(G\rho)^{1/2}}.
\end{equation}
If the size of a box of gas is greater than the Jeans length for that gas, the distribution will fragment into self-gravitating systems of size $\lambda_J$ or smaller, rather than remaining homogeneous; in cosmology, that's the process of structure formation.
Given the fluid pressure $p$ as a function of the energy density, the speed of sound is defined by $c_s^2 = dp/d\rho$.
For a radiation bath it is $c_R = 1/\sqrt{3}$, while for collisionless nonrelativistic particles we have $c_M\approx 0$; a cosmological matter fluid is unstable to growth of inhomogeneities on all scales.%
\footnote{This story is complicated by the expansion of the universe, since the Hubble parameter acts as a friction term; in a purely radiation-dominated Friedmann-Robertson-Walker universe, density perturbations actually shrink as the universe expands.
But even when radiation dominates the energy density, there are still slowly-moving matter particles.
Given sufficient time in a hypothetical non-expanding universe, density perturbations in non-relativistic matter would grow very large.
This leads us to the discussion of trajectories rather than states, which we undertake in the next section.}
Classically, the only perfectly stable configuration of fixed energy in a fixed volume would be a black hole in vacuum, which is about as inhomogeneous as you can get.
Another way of reaching the same conclusion (randomly-chosen states of self-gravitating particles are likely to be inhomogeneous) is to examine phase-space volumes directly.
Consider a collection of particles with a given energy, interacting only through the inverse-square law of Newtonian gravity.
It is clear that the volume of phase space accessible to such a system is unbounded.
Keeping the energy fixed, we can send any number of particles to infinity (or arbitrarily large momentum) while compensating by moving other particles very close together, sending their mutual energy to minus infinity.
This is a real effect, familiar to researchers in galactic dynamics as the ``gravo-thermal catastrophe'' \cite{lynden-bell,nityananda2009gravitational}.
A galaxy populated by stars (or dark matter particles) interacting through Newtonian gravity will tend to become centrally condensed without limit, while ejecting other stars to infinity.
The entropy of such a system is unbounded, and there is no equilibrium configuration, but generic evolution is in the direction of greater inhomogeneity.
And, indeed, in semiclassical general relativity, the highest-entropy configuration of fixed energy in a fixed volume is generally a black hole when the energy is sufficiently high.%
\footnote{When the energy is too low, any black hole will have a higher Hawking temperature than its surroundings, and it will lose energy and shrink.
Its temperature will grow as the hole loses mass, and eventually it will evaporate completely away.
This is a reflection of the fact that black holes have negative specific heat.}
In the early universe, the Jeans length is generally less than the Hubble radius (although they are of the same order for purely-radiation fluids).
High-entropy states will generally be very inhomogeneous, in stark contrast with the intuition we have from boxes of gas with negligible self-gravity, in accordance with Penrose's analysis mention in Section~\ref{what}.
Hence, the equilibration version of the horizon problem is extremely misleading, if not outright incorrect.
In an alternative world in which particles could still gravitationally attract each other but the universe was not expanding, so that past light cones stretched forever and the horizons of any two points necessarily overlapped, our expectation would \emph{not} be for a smooth universe with nearly constant temperatures throughout space.
It would be for the opposite: a highly inhomogeneous configuration with wildly non-constant temperatures.
Whatever the fine-tuning problem associated with the early universe may be, it is not that ``distant regions have not had time to come into thermal equilibrium."
Indeed, the lack of time to equilibrate is seen to be a feature, rather than a bug: it would be even harder to understand the observed uniformity of the CMB if the plasma had had an arbitrarily long time to equilibrate.
From this perspective, the thermal nature of the CMB radiation is especially puzzling.
It cannot be attributed to ``thermal equilibrium,'' since the early plasma is not in equilibrium in any sense.%
\footnote{This is trivial, of course. Equilibrium states are time-independent, while the early universe is expanding and evolving. More formally, there is no timelike Killing vector.}
Sometimes an attempt is made to distinguish between ``gravitational" and ``non-gravitational'' degrees of freedom, and argue that the non-gravitational degrees of freedom are in equilibrium, while the gravitational ones are not.
This is problematic at best.
The relevant gravitational effect isn't one of degrees of freedom (which would be independently-propagating gravitational waves or gravitons), but the simple existence of gravity as a force.
One might try to argue that the primordial plasma looks (almost) like it would look in an equilibrium configuration in a universe where there was no force due to gravity, but it is unclear what force such an observation is supposed to have.%
\footnote{There has been at least one attempt to formalize this notion, by inventing a scenario in which the strength of gravity goes to zero in the very early universe \cite{Greene:2009tt}.
While intriguing, from the trajectory-centered point of view advocated in the next section even this model doesn't explain why our observed universe exhibits such an unlikely cosmological history.}
The same conclusion can be reached in a complementary way, by recalling that we are considering a highly time-dependent situation, rather than two boxes of gas at rest.
As mentioned in Section~\ref{what}, there is a great amount of freedom in choosing spatial hypersurfaces in a perturbed cosmological spacetime; for example, we could choose the temperature of the fluid as our time coordinate (as long as the fluid was not so inhomogeneous that constant-temperature surfaces were no longer spacelike).
Then, by construction, the temperature is completely uniform at any moment of time.
But we are not observing the CMB at the moment of recombination; the photons we see have been redshifted by a factor of over a thousand.
From that perspective, the question is not ``Why was the temperature uniform on spacelike hypersurfaces?", but rather ``Why is the redshift approximately equal in completely separate directions on the sky?"%
\footnote{Alternatively and equivalently, we could define spacelike hypersurfaces by demanding that each hypersurface be at a constant redshift factor from the present time.
On such surfaces the temperature would generically be very non-uniform.
In these coordinates the question becomes, ``Why did distant regions of the universe reach the recombination temperature at the same time?"}
This formulation suggests the importance of considering cosmological trajectories, which we turn to in Section~\ref{trajectories}.
\subsection{Inflation}
It is useful to see how inflation addresses the horizon problem.
At the level of the causal version, all inflation needs to do is invoke an extended period of accelerated expansion.
Then the past horizon size associated with a spacetime event becomes much larger than the Hubble radius at that time, and widely-separated points on the surface of last scattering can easily be in causal contact.
We can quantify the total amount of inflation in terms of the number of $e$-folds of expansion,
\begin{equation}
N_e=\int_{a_{{\rm i}}}^{a_{{\rm f}}}d\ln{a}=\int_{t_{{\rm i}}}^{t_{{\rm f}}}H\, dt.
\end{equation}
where the integral extends from the beginning to the end of the period of accelerated expansion ($\ddot a > 0$).
Generally, at least $N_e > 50$ $e$-folds of inflation are required to ensure that all the observed regions of the CMB share a causal patch.
Whether or not these regions have ``equilibrated'' depends on one's definitions.
During inflation, the energy density of the universe is dominated by the approximately-constant potential energy of a slowly-rolling scalar field.
The evolution is approximately adiabatic, until reheating when inflation ends and the inflaton energy converts into matter and radiation.
Perturbations generically shrink during the inflationary period, in accordance with intuition from the cosmic no-hair theorem \cite{Wald:1983ky}, which states that a universe dominated by a positive cosmological constant approach a de~Sitter geometry.
But it is not a matter of interactions between degrees of freedom in different regions sharing energy, as in a conventional equilibration process; perturbations simply decrease locally and independently as the universe expands.
Once accelerated expansion occurs, it is important that inflation \emph{end} in such a way that homogeneous and isotropic spatial slices persist after reheating (when energy in the inflaton field is converted into matter and radiation).
The success of this step is highly non-trivial; indeed, this ``graceful-exit problem'' was highlighted by Guth \cite{Guth:1980zm} in his original paper, which relied on bubble nucleation to enact the transition from a metastable false vacuum state to the true vacuum.
Unfortunately, there are only two regimes for such a process; if bubbles are produced rapidly, inflation quickly ends and does not last for sufficient $e$-folds to address the horizon problem, while if they are produced slowly, we are left with a wildly inhomogeneous universe where a few bubbles have appeared and collided while inflation continues in the false vacuum elsewhere.
An attractive solution to this dilemma came in the form of slow-roll inflation \cite{Linde:1981mu,Albrecht:1982wi}.
Here the field is not trapped in a false vacuum, but rolls gradually down its potential.
Slow-roll inflation can very naturally produce homogeneous and isotropic spatial slices, even if it proceeds for a very large number of $e$-folds.
Central to the success of this model is the fact that the rolling field acts as a ``clock,'' allowing regions that have been inflated to extreme distances to undergo reheating at compatible times \cite{Anninos:1991ma}.
It is thus crucially important that the universe during slow-roll inflation is \emph{not} truly in equilibrium, even though its evolution is is approximately adiabatic; the evolving inflaton field allows for apparent coordination among widely-separated regions.
Inflation, therefore, solves the puzzle raised by the horizon problem, in the following sense: given a sufficient amount of inflation, and a model that gracefully exits from the inflationary phase, we can obtain a homogeneous and isotropic universe in which distant points share a common causal past.
The success of this picture can obscure an important point: the conditions required to get inflation started in the first place are extremely fine-tuned.
This fine-tuning is often expressed in terms of entropy; a patch of spacetime ready to begin inflating has an enormously lower entropy than the (still quite low-entropy) homogeneous plasma into which it evolves \cite{Penrose,Carroll:2004pn}.
In this sense, inflation ``explains'' the fine-tuned nature of the early universe by positing an initial condition that is even more fine-tuned.
In the context of the horizon problem, this issue can be sharpened.
One can show that, in order for inflation to begin, we must not only have very particular initial conditions in which potential energy dominates over kinetic and gradient energies, but that the size of the patch over which these conditions obtain must be strictly larger than the Hubble radius \cite{Vachaspati:1998dy}.
In other words, even in an inflationary scenario, it is necessary to invoke smooth initial conditions over super-horizon-sized distances.
As compelling as inflation may be, it still requires some understanding of pre-inflationary conditions to qualify as a successful model.
Contemporary discussion of inflation often side-steps the problem of the required low-entropy initial condition by appealing to the phenomenon of eternal inflation: in many models, if inflation begins at all it continues without end in some regions of the universe, creating an infinite amount of spacetime volume \cite{Guth:2000ka}.
While plausible (although for some recent concerns see \cite{Boddy:2014eba}), this scenario raises a new problem: rather than uniquely predicting a universe like the kind we see, inflation predicts an infinite variety of universes, making prediction a difficult problem. I won't discuss this issue here, but for recent commentary see \cite{Ijjas:2013vea,Guth:2013sya,Linde:2014nna,Ijjas:2014nta}.
\section{Trajectories}\label{trajectories}
\subsection{Initial Conditions and Histories}
The horizon problem, as discussed in the previous section, might not seem like an extremely pressing issue.
The causal version is fairly weak, merely noting a state of affairs rather than offering any reason why we should expect the contrary, while the equilibration version is misleading -- the problem is not that distant regions are unable to equilibrate, it's that equilibration would have made things more inhomogeneous.
Nevertheless, there is no question that the early universe \emph{is} fine-tuned.
A better statement of the fine-tuning problem comes from considering cosmological trajectories -- histories of the universe over time -- rather than concentrating on initial conditions.
The relationship between initial conditions and trajectories is somewhat different in classical mechanics and quantum mechanics.
In classical mechanics, the space of trajectories of a system with a fixed phase space and time-independent Hamiltonian is isomorphic to the space of initial conditions, or indeed the space of possible conditions at any specified moment of time.
We can think of the history of a classical system as a path through a phase space $\Gamma$ with coordinates $\{q^i, p_i\}$, where the $\{q^i\}$ are canonical coordinates and the $\{p_i\}$ are their conjugate momenta, governed by a Hamiltonian ${\mathcal H}(q^i, p_i)$ (which for our purposes is taken to be time-independent).
Evolution is unitary (information-conserving), implying that the state at any one time plus the Hamiltonian is sufficient to determine the state at any prior or subsequent time.
Classically, however, trajectories (and time itself) can come to an end; that's what happens at spacelike singularities such as the Big Bang or inside a Schwarzschild black hole.
This reflects the great amount of freedom that exists in choosing the geometry of phase space, including the possible existence of singular points on the manifold.
It is therefore possible for some conditions to be truly ``initial,'' if they occur at the past boundary of the time evolution.
In quantum mechanics, there are no singularities or special points in the state space.
A general solution to the Schr\"odinger equation $\hat H |\psi\rangle = i\partial_t|\psi\rangle$ with a time-independent Hamiltonian can be written in terms of energy eigenstates as
\begin{equation}
|\psi(t)\rangle = \sum_n r_n e^{i(\theta_i - E_nt)}|E_n\rangle,
\end{equation}
where the constant real parameters $\{r_i, \theta_i\}$ define the specific state.
Each eigenstate merely rotates by a phase, singularities cannot arise, and time extends infinitely toward the past and future \cite{Carroll:2008yd}.%
\footnote{The Borde-Guth-Vilenkin (BGV) theorem \cite{Borde:2001nh} demonstrates that spacetimes with an average expansion rate greater than zero must be geodesically incomplete in the past (which is almost, but not quite, equivalent to saying there are singularities).
This has been put forward as evidence that the universe must have had a beginning \cite{Mithani:2012ii}; there are explicit counterexamples to this claim \cite{Aguirre:2001ks}, but such examples are arguably unstable or at least non-generic.
However, while the BGV theorem does not assume Einstein's equation or any other equations of motion, it only makes statements about classical spacetime.
It is therefore silent on the question of what happens when gravity is quantized.}
In contrast to the arbitrariness of classical phase space, the geometry of pure quantum states is fixed to be ${\mathbb C}P^n$, and evolution is always smooth \cite{kibble1979geometrization,Brody:1999cw,Bengtsson:2001yd}.
There is then no special ``initial'' condition; the state at any one time can be evolved infinitely far into the past and future.
When we come to quantum gravity, canonical quantization of general relativity suggests \cite{Wiltshire:1995vk} that the wave function of the universe may be an exact energy eigenstate with zero eigenvalue, as reflected in the Wheeler-DeWitt equation:
\begin{equation}
\hat{H}|\psi\rangle = 0,
\end{equation}
where the Hamiltonian $\hat H$ includes both gravitational and matter dynamics.
In that case there is no time evolution in the conventional sense, although time can conceivably be recovered as an effective concept describing correlations between a ``clock'' subsystem and the rest of the quantum state \cite{Page:1983uc,Banks:1984cw}.
It is nevertheless far from clear that the Wheeler-DeWitt equation is the proper approach to quantum gravity, or indeed that local spacetime curvature is the proper thing to quantize.
Evidence from the holographic principle \cite{Hooft:1993gx,Susskind:1994vu,Bousso:2002ju}, black hole complementarity \cite{Susskind:1993if}, the gauge-gravity correspondence \cite{Maldacena:1997re,Horowitz:2006ct}, the entanglement/spacetime connection \cite{Ryu:2006bv,swingle2009,VanRaamsdonk:2010pw,Maldacena:2013xja}, and thermodynamic approaches to gravity \cite{Jacobson:1995ab,Verlinde:2010hp} suggests that gravity can be thought of emerging from non-local degrees of freedom, only indirectly related to curved spacetime.
Given the current state of the art, then, it is safest to leave open the question of whether time is emergent or fundamental, and whether it is eternal or has a beginning.
Fortunately, this uncertainty over whether conditions can truly be initial does not prevent us from talking about cosmological fine-tuning.
For most of the history of the universe, many important cosmological quantities are well-described by classical dynamics.
This includes the expansion of the scale factor, as well as the evolution of perturbations, considered as modes in Fourier space of fixed comoving wavelength ({\it i.e.}, expanding along with the universe).%
\footnote{Inflation is an important exception.
During inflation itself, the state of the universe has essentially only one branch.
When inflation ends, reheating creates a large number of excited degrees of freedom, effectively ``measuring'' the state of the inflaton and causing the wave function to split into many branches \cite{Boddy:2014eba}.
This process explains how a perturbed post-inflationary universe can develop out of an unperturbed inflationary state.}
On small scales the dynamics are nonlinear and entropy-generating, due to a variety of processes such as star formation, supernovae, magnetic fields, and violent gravitational relaxation.
Consequently, the evolution at those wavelengths is not well-approximated by reversible equations on phase space.
This leaves us, however, with the dynamics on large scales -- in the present universe, more then ten million light-years across -- that can be treated as the Hamiltonian evolution of an autonomous set of degrees of freedom.
Therefore, we can circumvent conceptual problems raised by the idea of ``initial conditions'' by simply asking whether the trajectory of the large-scale universe since the aftermath of the Big Bang is natural, or fine-tuned, in the space of all such trajectories.
\subsection{The Canonical Measure}
Fortunately, it is possible to construct a preferred measure on the space of trajectories, which we can use to judge the amount of fine-tuning exhibited by our real universe.
We start by considering the measure on phase space itself, and use that to find a measure on the space of paths through phase space that represent solutions to the equations of motion.%
\footnote{This notion of a cosmological measure is completely separate from that which arises in what is sometimes called ``the measure problem in cosmology, which deals with the relative frequency of different kinds of observers in a multiverse.
See {\it e.g.} \cite{Winitzki:2006rn,Aguirre:2006ak,Linde:2008xf,SchwartzPerlov:2010ne,Freivogel:2011eg,Salem:2011qz}.}
In classical mechanics, there is a natural measure on phase space $\Gamma$, the Liouville measure.
To construct it in terms of coordinates $\{q^i\}$ and momenta $\{p_i\}$, we first write down the symplectic $2$-form on $\Gamma$,
\begin{equation}
\omega=\sum_{i=1}^{n}\mathrm{d}p_{i}\wedge\mathrm{d}q^{i}.
\label{symplectic}
\end{equation}
Note that the dimension of $\Gamma$ is $2n$.
The Liouville measure is then a $2n$-form given by
\begin{equation}
\Omega=\frac{\left(-1\right)^{n\left(n-1\right)/2}}{n!}\omega^{n}.
\label{liouville}
\end{equation}
All that matters for our current purposes is that such a measure exists, and is uniquely defined.
What makes the Liouville measure special is that it is conserved under time evolution.
That is, given states that initially cover some region $S\subset\Gamma$ and that evolve under Hamilton's equations to cover region $S^{\prime}$, we have
\begin{equation}
\int_{S}\Omega=\int_{S^{\prime}}\Omega.
\end{equation}
Classical statistical mechanics assumes that systems in equilibrium have a probability distribution in phase space that is uniform with respect to this measure, subject to appropriate macroscopic constraints.
Meanwhile, in connecting cosmology with statistical mechanics, we assume that the microstate of the early universe is chosen randomly from a uniform distribution in the Liouville measure, subject to the (severe) constraint that the macrocondition has the low-entropy form given by the Past Hypothesis.
Albert \cite{albert} calls this the ``Statistical Postulate".%
\footnote{Albert and Loewer have referred to the combined package of the dynamical laws, the Past Hypothesis, and the Statistical Postulate as the ``Mentaculus," from the Coen brothers film {\it A Serious Man} (see {\it e.g.} \cite{loewer2012emergence}).}
Of course one can question why probabilities should be uniform in \emph{this} measure rather than some other one.
Even if the Liouville measure is somehow picked out by the dynamics by virtue of being conserved under evolution, we are nevertheless free to construct any measure we like.
For our present purposes, this kind of question seems misguided.
As discussed in Section~\ref{what}, the point of fine-tuning arguments is to find clues that can guide us to inventing more comprehensive physical theories.
We are not arguing for some metaphysical principle to the effect that the universe \emph{should} be chosen uniformly in phase space according to the Liouville measure; merely that, given this measure's unique status as being picked out by the dynamics, states that look natural in this measure tell us very little, while states that look unnatural might reveal useful information.
(See \cite{Schiffrin:2012zf} for a critique of the use of cosmological measures in the way I am advocating here.)
In general, having a measure on phase space $\Gamma$ does not induce a natural measure on the space of trajectories $T$, which is one dimension lower.
(There is a natural map $\Gamma \rightarrow T$, which simply sends each point to the trajectory it is on; however, while differential forms can be pulled back under such maps, they cannot in general be pushed forward \cite{Carroll:2004st}.)
In the case of general relativity, Gibbons, Hawking and Stewart (GHS, \cite{Gibbons:1986xk}) showed that there is nevertheless a unique measure satisfying a small number of reasonable constraints: it is positive, independent of coordinate choices, respects the symmetries of the theory, and does not require the introduction of any additional structures.
GHS relied on the fact that general relativity is a constrained Hamiltonian system: because the metric component $g_{00}$ is not a propagating degree of freedom in the Einstein--Hilbert action, physical trajectories obey a constraint of the form $\mathcal{H}=\mathcal{H}_{\star}$, where $\mathcal{H}$ is the Hamiltonian and $\mathcal{H}_{\star}$ is a constant defining the constraint surface.
(In cosmological spacetimes we generally have $\mathcal{H}_{\star}=0$.)
The space $U$ of physical trajectories -- those obeying the Hamiltonian constraint -- is thus two dimensions lower than the full phase space $\Gamma$.
GHS construct a measure on $U$ by identifying the $n$th coordinate on phase space as time $t$, for which $\mathcal{H}$ is the conjugate variable.
The symplectic form (\ref{symplectic}) is then
\begin{equation}
\omega= \tilde{\omega}+\mathrm{d}\mathcal{H}\wedge\mathrm{d}t,
\end{equation}
where
\begin{equation}
\tilde{\omega}\equiv \sum_{i=1}^{n-1}\mathrm{d}p_{i}\wedge\mathrm{d}q^{i}.
\end{equation}
GHS show that the $\left(2n-2\right)$-form
\begin{equation}
\Theta=\frac{\left(-1\right)^{\left(n-1\right)\left(n-2\right)/2}}{\left(n-1\right)!}\tilde{\omega}^{n-1}
\label{GHSmeasure}
\end{equation}
is a unique measure satisfying their criteria.
The GHS measure (\ref{GHSmeasure}) has the attractive feature that it is expressed locally in phase space.
Therefore, it can be evaluated on the space of trajectories simply by choosing some transverse surface through which all trajectories (or all trajectories of interest for some purpose) pass, and calculating $\Theta$ on that surface; the results are independent of the surface chosen.
In cosmology, for example, we might choose surfaces of constant Hubble parameter, or constant energy density, and evaluate the measure at that time.
This feature has a crucial consequence: the total measure on some particular kind of trajectories, such as ones that are spatially flat or ones that are relatively smooth at some particular cosmological epoch, is completely independent of what the trajectories are doing at some other cosmological epoch.
Therefore, changing the dynamics in the early universe (such as modifying the potential for an inflaton field) cannot possible change the fraction of trajectories with certain specified properties in the late universe.
(Adding additional degrees of freedom can, of course, alter the calculation of the measure.)
New physics cannot change ``unnatural'' trajectories into ``natural'' ones.
At heart, there is not much conceptual difference between studying the purported fine-tuning of the universe in terms of the measure on trajectories and quantifying the low entropy of the early state.
There are relatively few initial conditions with low entropy, and the trajectories that begin from such conditions will have a small measure.
As discussed in Section~\ref{what}, in order to quantify fine-tuning, we generally need to specify a coarse-graining on the space of states as well as a measure.
In the language of trajectories, this corresponds to specifying macroconditions the trajectories must satisfy.
One benefit of the trajectory formalism is that it is relatively straightforward to ask questions that conditionalize over macroconditions specified at a different time; for example, we can talk about the fraction of the trajectories that are smooth at one time given that they are smooth at some other time.
Another benefit, and a considerable one, is that we can look at features of cosmic evolution that we truly understand without claiming to have full control over the space of states (as would be necessary to completely understand the entropy).
An objection to Penrose's argument is sometimes raised that we don't know enough about how to calculate the entropy in quantum gravity to make any statements at all; using the measure on classical trajectories allows us to make fine-tuning arguments while remaining wholly in a regime where classical general relativity should be completely valid.
\subsection{Flatness}\label{flatness}
An interesting, and surprisingly nontrivial, application of the GHS measure is to the flatness problem -- as we will see, it doesn't really exist.
(In this section and the next I am drawing on work from \cite{Carroll:2010aj}.)
Consider a Robertson-Walker universe with scale factor $a(t)$ and curvature parameter $\kappa$, obeying the Friedmann equation (\ref{friedmann}), with an energy density from components $\rho_i$ that each scale as a power law, $\rho_i = \rho_{i0} a^{-n_i}$ for some fixed $n_i$.
This includes the cases of nonrelativistic matter, for which we have $n_M=3$, and radiation, for which $n_R=4$.
Then we can define the corresponding density parameters $\Omega_i$, as well as an ``effective density parameter for curvature,'' via
\begin{equation}
\Omega_i\equiv \frac{8\pi G \rho_{i0} a^{-n_i}}{3H^2},\quad
\Omega_\kappa \equiv -\frac{\kappa}{a^2H^2}.
\label{densityparameters}
\end{equation}
The Friedmann equation then implies that $\sum_i\Omega_i + \Omega_\kappa = 1$.
The ratio of the curvature to one of the densities evolves with time as
\begin{equation}
\frac{\Omega_\kappa}{\Omega_i} \propto a^{n_i-2}.
\end{equation}
Whenever $n_i>2$, as for matter or radiation, the relative of importance of curvature grows with time.
The conventional flatness problem is simply the observation that, since the curvature is not very large today, it must have been extremely small indeed at early times.
Roughly speaking (since details depend on the amounts of matter, radiation, and vacuum energy, as well as their evolutions), we must have had $\Omega_\kappa/\Omega_{\mathrm{matter/radiation}} < 10^{-55}$ in the very early universe in order that the curvature not dominate today.
As we have argued, however, such a statement only has impact if the set of trajectories for which $\Omega_\kappa/\Omega_{\mathrm{matter/radiation}} < 10^{-55}$ in the very early universe is actually small.
It \emph{seems} small, since $10^{-55}$ is a small number.
But that just means that it would be small if trajectories were chosen uniformly in the variable $\Omega_\kappa/\Omega_{\mathrm{matter/radiation}}$, for which we have given no independent justification.
Clearly, this is a job for the GHS measure.
To make things quantitative, consider a Robertson-Walker universe containing a homogeneous scalar field $\phi(t)$ with potential $V(\phi)$.
(A scalar field is more directly applicable than a matter or radiation fluid, since the scalar has a Hamiltonian formulation; one can, however, consider scalar field theories that mimic the behavior of such fluids, so no real generality is lost.)
The scale factor will obey the Friedmann equation (\ref{friedmann}), with the energy density given by
\begin{equation}
\rho_\phi = \frac{1}{2}\dot\phi^2 + V(\phi).
\end{equation}
The dynamical coordinates for this model are $a$ and $\phi$, with conjugate momenta $p_a$ and $p_\phi$, as well as a Lagrange multiplier $N$ (the lapse function) that enforces the Hamiltonian constraint.
The lapse function is essentially the square root of the $00$ component of the metric; it is non-dynamical in general relativity, since no time derivatives of $g_{00}$ appear in the action.
Setting $8\pi G=1$ for convenience, the Einstein-Hilbert Lagrangian for the scale factor coupled to the scalar field is
\begin{equation}
\mathcal{L}=3\left(Na\kappa-\frac{a\dot{a}^{2}}{N}\right)+a^{3}\left[\frac{\dot{\phi}^{2}}{2N}-NV\left(\phi\right)\right].
\end{equation}
The canonical momenta, defined as $p_{i}=\partial\mathcal{L}/\partial\dot{q}^{i}$,
are
\begin{equation}
p_{N}=0,\qquad p_{a}=-6N^{-1}a\dot{a},\qquad \mathrm{and}\qquad p_{\phi}=N^{-1}a^{3}\dot{\phi}.\label{momenta}
\end{equation}
Performing a Legendre transformation, the Hamiltonian is
\begin{equation}
\mathcal{H} =N\left[-\frac{p_{a}^{2}}{12a}+\frac{p_{\phi}^{2}}{2a^{3}}+a^{3}V\left(\phi\right)-3a\kappa\right].
\label{Hamiltonian}
\end{equation}
The equation of motion for $N$ sets it equal to an arbitrary constant, which we can choose to be unity.
Varying the action with respect to $N$ gives the Hamiltonian constraint, $\mathcal{H}_{\star}=0$, which is equivalent to the Friedmann equation.
Setting $N=1$, the remaining phase space $\Gamma$ is four-dimensional.
The Hamiltonian constraint surface is three-dimensional, and the space of physical trajectories $U$ is two-dimensional.
The GHS measure can be written as the Liouville measure subject to the Hamiltonian constraint:
\begin{equation}
\Theta=\left.\left(\mathrm{d}p_{a}\wedge\mathrm{d}a+\mathrm{d}p_{\phi}\wedge\mathrm{d}\phi\right)\right|_{\mathcal{H}=0}.
\label{GHS-RW}
\end{equation}
The measure can be evaluated in expanding Friedmann-Robertson-Walker universes by integrating (\ref{GHS-RW}) over a specified transverse surface, for example by setting $H=H_*$.
The answer works out to be
\begin{align}
\mu &\equiv \int_{H=H_*}\Theta \\
&= \int_{H=H_*}\Theta_{a\phi}\, da\,d\phi \\
&= -6\int_{H=H_*}
\frac{3a^{3}H_*^2- a^3V + 2ak}
{\left(6a^{2}H_*^2- 2a^2V+ 6 k\right)^{1/2}}\,da\, d\phi .
\label{ghs3}
\end{align}
We can make this expression look more physically transparent by introducing the variable
\begin{equation}
\Omega_V\equiv \frac{V(\phi)}{3H^2},
\end{equation}
as well as the curvature density parameter defined in (\ref{densityparameters}).
The scale factor is strictly positive, so that integrating over all values of $\Omega_\kappa$ is equivalent to integrating over all values of $a$.
The measure is then
\begin{equation}
\mu = 3\sqrt{\frac{3}{2}}H_*^{-2}\int_{H=H_*}
\frac{1-\Omega_V - \frac{2}{3}\Omega_\kappa}
{|\Omega_\kappa|^{5/2}\left(1-\Omega_V -\Omega_\kappa\right)^{1/2}}\,d\Omega_\kappa\, d\phi.
\label{flatness1}
\end{equation}
This integral is divergent; it blows up as $\Omega_\kappa \rightarrow 0$, since the denominator includes a factor of $|\Omega_\kappa|^{5/2}$.
By itself, the divergence isn't surprising; the set of classical trajectories is non-compact.
The more interesting fact is \emph{where} it diverges -- for universes that are spatially flat ($\Omega_\kappa=0$), which is certainly a physically relevant region of parameter space.
This divergence was noted in the original GHS paper \cite{Gibbons:1986xk}, where it was attributed to ``universes with
very large scale factors'' due to a different choice of variables.
That characterization isn't very useful, since ``large scale factor'' is a feature along the trajectory of any open universe, rather than picking out a particular type of trajectory.
Later works \cite{Hawking:1987bi,Coule:1994gd,Gibbons:2006pa} correctly described the divergence as arising from nearly-flat universes.
Gibbons and Turok \cite{Gibbons:2006pa} advocated dealing with the infinity by discarding all flat universes by fiat, and concentrating on the non-flat universes.
Tam and I \cite{Carroll:2010aj} took the opposite view: what (\ref{flatness1}) is telling us is that almost every Robertson-Walker cosmology is spatially flat.
Rather than throwing such trajectories away, we should throw all of the others away and deal with flat universes.
What one wants, therefore, is a measure purely on the space of flat universes.
The procedure we advocated in \cite{Carroll:2010aj} for obtaining such a measure was faulty, as our suggested regularization gave a result that was not invariant under a choice of surface on which to evaluate the measure.
This problem was later solved in \cite{Remmen:2013eja}, which derived a measure on the phase space for flat universes by demanding that it obey Liouville's theorem, and \cite{Remmen:2014mia}, which used this to derive a measure on the space of trajectories.
Applying this to the case of inflation, we showed that one generically expects a very large amount of inflation for unbounded potentials such as $V(\phi) \propto \phi^2$, and relatively few $e$-folds for ``natural'' inflation \cite{Freese:1990rb} in which $V(\phi) \propto \cos(\phi)$.
From the point of view of fine-tuning, using the GHS measure completely alters our picture of the flatness problem.
We noted that the conventional formulation of the problem implicitly assumes a measure that is uniform in $\Omega_\kappa$, which seemed intuitively reasonable.
But in fact the measure in the vicinity of flat universes turns out to be proportional to $1/|\Omega_\kappa|^{5/2}$, which is a dramatic difference.
Rather than sufficiently flat universes being rare, they are actually generic.
We take this result to indicate that the flatness problem really isn't a problem at all; it was simply a mistake, brought about by considering an informal measure rather than one derived from the dynamics.
\subsection{Smoothness}
The surprising result that almost all universes are spatially flat might raise the hope that a careful consideration of the measure might also explain the smoothness of the universe: perhaps almost all cosmological trajectories are extremely smooth at early times.
Sadly, the opposite is true, as can be seen by extending the GHS measure to perturbed spacetimes \cite{Carroll:2010aj}.
This might seem like a difficult task, since there are many ways the universe can be perturbed.
But as long as the perturbations are small, every Fourier mode evolves independently according to a linear equation of motion.
Therefore, we can consider the measure on a mode-by-mode basis.
For linear scalar perturbations, the coupled dynamics of a matter/radiation fluid and the spacetime curvature in a background spacetime can be described by a single degree of freedom, the cosmological perturbation field $u(\vec{x},t)$, as discussed by Mukhanov, Feldman and Brandenberger \cite{Mukhanov:1990me}.
Given the action for this field, we can isolate the dynamical variables and construct the symplectic two-form on phase space, which can then be used to compute the measure on the set of solutions to Einstein's equations.
Various subtleties arise along the way, but the final answer is relatively straightforward.
Here I will just quote the results; calculations can be found in \cite{Carroll:2010aj}.
It is convenient to switch to conformal time,
\begin{equation}
\eta = \int a^{-1}dt.
\end{equation}
Derivatives with respect to $\eta$ are denoted by the superscript $'$, and $\widetilde{H} \equiv a'/a$ is related to the Hubble parameter $H=\dot{a}/a$ by $\widetilde{H} = aH$.
In Fourier space the cosmological perturbation field is a function $u(\vec{k}, t)$, where $\vec{k}$ is the comoving wave vector.
It is essentially a scaled version of the $00$ component of the metric perturbation, which is just the Newtonian gravitational potential:
\begin{equation}
\Phi = (\bar{\rho}+\bar{p})^{1/2}u,
\end{equation}
where $\bar\rho$ and $\bar{p}$ are the background energy density and pressure.
From that we can express the energy density perturbation in conformal Newtonian gauge,
\begin{equation}
\delta\rho = \frac{1}{4\pi G a^2}\left[\nabla^2\Phi - 3\widetilde{H}(\Phi' + \widetilde{H}\Phi)\right].
\end{equation}
The cosmological perturbation field obeys an equation of motion
\begin{equation}
u'' -c_s^2\nabla^2u - \frac{\theta''}{\theta} u = 0.
\label{ueq}
\end{equation}
Here, $c_s$ is the speed of sound in the matter/radiation fluid, and $\theta(\eta)$ is a time-dependent parameter given by
\begin{equation}
\theta =\frac{1}{a}\left[\frac{2}{3}\left(1 - \frac{\widetilde{H}'}{\widetilde{H}^2}\right)\right]^{-1/2}.
\end{equation}
This equation of motion can be derived from an action
\begin{equation}
S_u = \frac{1}{2}\int d^4x \left(u'^2 - c_s^2 \sum_i\partial_iu \partial_iu + \frac{\theta''}{\theta}u^2 \right).
\end{equation}
Defining the conjugate momentum $p_u = \partial{\mathcal L}/\partial u' = u'$, we can describe the
dynamics in terms of a Hamiltonian for an individual mode with wavenumber $k$,
\begin{equation}
\mathcal{H} = \frac{1}{2}p_u^2 + \frac{1}{2}\left(c_s^2k^2 - \frac{\theta''}{\theta}\right)u^2.
\label{pertham}
\end{equation}
This is simply the Hamiltonian for a single degree of freedom with a time-dependent
effective mass $m^2= c_s^2k^2 - \theta''/\theta$.
One convenient hypersurface in which we can evaluate the flux of trajectories is $\eta=
\eta_*=\mbox{constant}$.
A straightforward calculation shows that the measure evaluated on such a surface is simply
\begin{equation}
\mu = \int_{\eta=\eta_*} \,du\,dp_u.
\label{pertmeasure}
\end{equation}
In other words, the measure on a perturbation mode is completely uniform in the $\{u, p_u\}$ variables, much as we might have na\"ively guessed, and in stark contrast to the flatness problem.
All values for $u$ and $p_u$ are equally likely; there is nothing in the measure that would explain the small
observed values of perturbations at early times.
Hence, the observed homogeneity of our universe does imply considerable fine-tuning.
We can use this measure to roughly quantify how much fine-tuning is involved in the conventional assumption of a smooth universe near the Big Bang.
For purposes of convenience, \cite{Carroll:2010aj} asked a simple question: assuming that the universe had the observed amount of uniformity at the surface of last scattering ($\delta\rho/\bar{\rho}\leq 10^{-5}$), what fraction of the allowed trajectories were also smooth in the very early universe, say near the scale of grand unification when the energy density was $\rho \sim (10^{16}\,\mathrm{GeV})^4$?
The answer is, unsurprisingly, quite small.
For each individual mode, the chance that it was small at the GUT scale given that it is small at recombination is of order $10^{-66}$.
But there are many modes, and if any one of them is large then the universe is not truly smooth.
The total fraction of universes that were smooth at early times is just the product of the fractions corresponding to each mode.
Choosing reasonable bounds for the largest and smallest modes considered, the total fraction of trajectories that are smooth at early times works out to be
\begin{equation}
f(\textrm{smooth at GUT scale}|\textrm{smooth at recombination}) \approx 10^{-6.6\times 10^7}.
\label{fraction}
\end{equation}
This represents a very conservative estimate for the amount of fine-tuning involved in the standard cosmological model.
It might seem strange to ask a conditional question that assumes the universe was smooth at the time of last scattering, rather than directly inquiring about the fraction of universes that were smooth at recombination.
But that fraction is ill-defined, since the phase space of perturbations is unbounded.
We could have calculated the fraction of universes that are relatively smooth today that were also smooth at recombination, and obtained a similarly tiny number.
More importantly, we are interested in the fine-tuning necessary for the universe to obey the Past Hypothesis.
The reason why (\ref{fraction}) is such a small number is that most trajectories that are smooth at last scattering contain modes that were large at earlier times but decayed.
That is morally equivalent to trajectories that start with relatively high entropy, but that start with delicate correlations that cause the entropy to decrease as time passes.
All of conventional cosmology assumes that the early universe was not like that; (\ref{fraction}) quantifies the amount of fine-tuning implied by this assumption.
We can therefore conclude that the smoothness of the early universe does indeed represent an enormous amount of fine-tuning.
In reaching this conclusion, we made no reference to the causal structure nor to any thwarted attempts at equilibration.
Those considerations, which play a central role in formulating the horizon problem, are red herrings.
The real sense in which the early universe was fine-tuned is extremely simple: the overwhelming majority of cosmological trajectories, as quantified by the canonical measure, are highly nonuniform at early times, and we don't think the real universe was like that.
Clearly, the specific numerical value we obtain is not of central importance; what is certain is that the history of our actual universe does not look anything like it was chosen randomly.
\section{Discussion}\label{discussion}
I have argued that the traditional discussion of the fine-tuning of the early universe in terms of the horizon and flatness problems is misguided.
The flatness problem is based on an implicit use of an unjustified measure on the space of initial conditions; when a more natural measure is used, we see that almost all Robertson-Walker universes are spatially flat.
The difficulty with the horizon problem is not that it is incorrect, but that it is inconclusive, since we don't have a clear picture of how the situation would change if distant regions on the surface of last scattering actually had been in causal contact.
It is much more sensible to quantify fine-tuning in terms of the measure on the space of cosmological trajectories.
From that perspective, it is clear that the overwhelming majority of such trajectories, conditionalized on some reasonable requirement in the late universe, are wildly inhomogeneous at early times.
The fact that the actual early universe was not like that, as specified by the Past Hypothesis, gives us a clear handle on the kind of fine-tuning we are faced with.
As with most discussions of fine-tuning, in this paper I have prejudiced the discussion somewhat by assuming throughout that the universe is an approximately Robertson-Walker cosmology with a certain (large) amount of matter and radiation.
While we have seen that the universe is fine-tuned even given that framework, such an assumption is much stronger than what is required, for example, by the anthropic principle.
Life can certainly exist in universes with far fewer stars and galaxies than what we observe.
In the presence of a stable vacuum energy, the highest-entropy configuration for the universe to be in is empty de~Sitter space \cite{Carroll:2004pn}.
The worry there is that vacuum fluctuations give rise to an ensemble of freak ``Boltzmann brain'' observers \cite{Dyson:2002pf,Albrecht:2004ke,Bousso:2006xc}.%
\footnote{The problem is not that a de~Sitter scenario predicts that ``we should be Boltzmann brains."
Rather, we should consider ourselves to be one of any of the many observers that find themselves in precisely our current cognitive situation.
For most such observers, that cognitive situation -- {\it e.g.}, my current belief that there is a person named ``David Albert'' who wrote a book entitled {\it Time and Chance} -- is completely uncorrelated with the reality of their actual environment.
Such a situation is cognitively unstable, for reasons explained in related contexts in \cite{albert} (at least, to the best of my current recollection).}
As argued in \cite{Boddy:2014eba}, however, quantum fluctuations in de~Sitter space don't actually bring into existence decohered branches of the wave function containing such freak observers.
Nevertheless, it seems reasonable to think that the space of trajectories containing one person or one galaxy in an otherwise empty background has a much greater measure than the kind of universe in which we live, with over a hundred billion galaxies; at least, such a situation has a much higher entropy.
We are therefore still left with the fundamental cosmological question: ``Why don't we live in a nearly-empty de~Sitter space?''
Formulating cosmological fine-tuning in the language of the measure on trajectories puts the inflationary-universe scenario in its proper context.
A major original motivation of inflation was to solve the horizon and flatness problems.
Reformulating the issue raised by the horizon problem as a matter of the measure on cosmological trajectories brings the problem with inflation into sharp focus: the fact that most trajectories that are homogeneous at late times are highly non-homogeneous at early times is completely independent of physical processes in the early universe.
It depends only on the measure evaluated at relatively late times.
Inflation, therefore, cannot solve this problem all by itself.
Indeed, the measure reinforces the argument made by Penrose, that the initial conditions necessary for getting inflation to start are extremely fine-tuned, more so than those of the conventional Big Bang model it was meant to help fix.
Inflation does, however, still have very attractive features.
It posits an initial condition that, while very low-entropy, is also extremely simple, not to mention physically quite small.
(With inflation, our observable universe could have been one Planck length across at the Planck density; without inflation, the same patch was of order one centimeter across at that time.
That is an incredibly large volume, when considered in Planck units, over which to have initial homogeneity.)
Therefore, while inflation does not remove the need for a theory of initial conditions, it gives those trying to construct such a theory a relatively reasonable target to shoot for.
Of course, all of this discussion about fine-tuning and the cosmological measure would be completely pointless if we did have a well-formulated theory of initial conditions (or, better, of our cosmological history considered as a whole).
Ultimately the goal is not to explain why our universe appears unnatural; it's to explain why we live in this specific universe.
Making its apparent unnaturalness precise is hopefully a step toward achieving this lofty ambition.
\section*{Acknowledgments}
It is a pleasure to thank David Albert for inspiration and conversations over the years, Barry Loewer for his patience, Shelly Goldstein for useful discussions, my collaborators Heywood Tam and Grant Remmen for their invaluable insights, and Tim Maudlin for the interactions that proximately inspired this paper.
This research is funded in part by DOE grant DE-SC0011632, and by the Gordon and Betty Moore Foundation through Grant 776 to the Caltech Moore Center for Theoretical Cosmology and Physics.
\bibliographystyle{utphys}
|
2,869,038,154,769 | arxiv | \section*{Materials and Methods}
\subsection*{Experiment}
The experiments are performed in ultra-high vacuum at low temperature ($\approx$ 4.5 K) with an Omicron STM that is combined with an optical set-up adapted to detect light emitted at the STM tip-sample junction. The emitted photons are collected with a lens located on the STM head and then redirected out of the chamber through optical viewports. The light is then focused on an optical fibre coupled to a spectrograph itself connected to a low-noise liquid nitrogen cooled
CCD camera. The spectral resolution of the setup is $\approx$ 1 nm. Further details regarding the optical detection setup can be found in the Supplementary Materials of \cite{Chong2016} . \\
The STM tips are prepared by electrochemical etching of a tungsten wire in a NaOH solution. The tips are then sputtered with argon ions and annealed under UHV. To optimise their plasmonic response, the tips are eventually indented in clean Ag(111) to cover them with silver. The Ag(111) substrates are cleaned by simultaneous argon-ion sputtering and annealing. After this cleaning procedure, NaCl is evaporated on the Ag(111) substrate maintained at room temperature. Post-annealing at $\approx$ 370 K is performed to induce NaCl surface reorganization in bi- and tri-layers. Eventually, the 2-3 monolayer (ML) NaCl/Ag(111) sample is introduced into the STM chamber and cooled down to $\approx$ 4.5 K. H$_2$Pc and ZnPc molecules are then sublimed in very small quantities on the cold sample from a powder located in a quartz crucible. ZnPc--HPc$^-$ molecular dimers on NaCl are obtained by STM tip manipulation of ZnPc. To this end, the STM tip is first positioned at the edge of a ZnPc molecule at a bias $V$ = +2.5 V. In a second step, the tip-molecule distance is slowly reduced until a jump of current occurs, indicating a motion of the molecule. The procedure is reproduced until the desired structure in obtained. The d$I$/d$V$ maps of Fig. 2 in the main paper are recorded in constant current mode (closed feedback loop) with a voltage modulation of 50 mV at a DC voltage of 400 mV, whereas the d$I$/d$V$ spectra are recorded in constant height mode (open feed-back loop) with a voltage modulation of 20 mV.
\subsection*{DFT calculation of the ground-state electronic properties}
To analyze the molecular transport properties, the electronic structure of the molecule in a vacuum, and the distribution of the corresponding electron charge density in the neutral and charged species, we perform ground-state DFT calculations using the OCTOPUS code \cite{tancogne2020}. In OCTOPUS the electron density is represented on a real-space grid which does not constrain the localization of the density to predefined atomic orbitals and is therefore suitable to describe the ground-state electron densities of the charged molecules. The electron-ion interaction is modeled in the framework of the pseudopotential approximation. We use the Perdew-Zunger \cite{perdew1981} parametrization of the local density approximation (LDA) correlation and the Slater density functional for the LDA exchange functional \cite{dirac1930,slater1951}. We present further analysis of the molecular orbitals, electronic density and comparison with experimental ${\rm d}I/{\rm d}V$ images of the molecules with theory in Supplementary Fig. 1-3 and Supplementary Tab. 1.
\subsection*{Numerical analysis of the ground-state electron densities} We extract from OCTOPUS the ground-state electron densities and calculate the total electrostatic potential $\phi_{\rm tot}$ generated by the molecular charges to visualize the localization of the excess charge of the deprotonized molecules (as shown in Fig.\,\ref{fig2}). To that end we solve the Poisson equation:
\begin{align}
\Delta\phi_{\rm tot}=\frac{\rho_{\rm e}({\bf
r})}{\varepsilon_0}+\frac{\rho_{\rm ion}({\bf
r})}{\varepsilon_0},\label{seq:poisson}
\end{align}
with $\varepsilon_0$ being the vacuum permittivity. $\rho_{\rm e}({\bf
r})$ is the electron charge density of the valence electrons in the
singlet ground state of the molecule and $\rho_{\rm ion}({\bf r})$ is
the charge density of the positive nuclei screened by the core
electrons of the respective atoms. We represent the density of the
screened nuclei as a sum of Gaussian charge distributions:
\begin{align}
\rho_{\rm ion}({\bf r})=\sum_i \frac{Q_i}{(2\pi)^{3/2} \sigma_{\rm ion}^3}\exp{-\frac{|{\bf r}-{\bf R}_i|^2}{2\sigma_{\rm ion}^2}},
\end{align}
where $Q_i$ is the total charge of the screened ion $i$ at position ${\bf R}_i$. The charge distribution has a width $\sigma_{\rm ion}=0.05$\,nm. The electron charge density is extracted from the ground-state DFT calculations in the form of Gaussian cube files. The ground-state electron charge densities for H$_2$Pc and HPc$^-$ are shown in Supplementary Fig. 3c,d, respectively, alongside with the corresponding geometries of the molecules (H$_2$Pc in Supplementary Fig. 3a and HPc$^-$ in Supplementary Fig. 3b). Finally, we solve the Poisson equation [Eq.\,\eqref{seq:poisson}] numerically on a homogeneously spaced grid by a Fourier-based method.
\subsection*{TD-DFT calculations of the excited-state electronic properties}
To address the excited-state properties of the molecules we perform TD-DFT calculations as implemented in the software Gaussian 09 Revision D.01\cite{Gaussian}. The TD-DFT calculations (Fig. 3) are carried out with the B3LYP functional and the 6-311G(d,p) basis set. The convergence of 64 roots is asked for when calculating the excited singlet states at the ground state equilibrium geometries of all three compounds which are optimized using the same functional and basis set.
The dependence on a central fractional charge of the transition energy to the first excited singlet state of the doubly deprotonated Pc dianion (Fig. 3d) is calculated in the same way while keeping its equilibrium D4h geometry frozen.
\subsection*{DFT calculation of the geometry of the molecules on NaCl}
The geometrical structure of the molecules on NaCl was relaxed using Quantum Espresso\cite{giannozzi2009} which is a plane-wave pseudopotential DFT code suitable for the description of the extended substrate. The cubic supercell ($a=53.02457485$ \AA) included 100 Na and 100 Cl substrate atoms and the calculation was performed at the $\Gamma$ point.
The exchange and correlation terms were described using the local density approximation under the approach of Perdew and Zunger\cite{perdew1981}.
The projector augmented-wave pseudopotentials with core corrections were used to describe the electron-ion interaction\cite{blochl1994}. The energy cutoff and density cutoff were set to $50$ Ry and $500$ Ry, respectively.
A damped dynamics was used to perform the structural optimization of the system. The atoms were moved according to Newton's equation by using a Verlet algorithm\cite{verlet1967}. The structural optimization was stopped when two
subsequent total energy evaluations differed by less than $10^{-4}$ Ry and each force component was less than $10^{-3}$ Ry/bohr.
\subsection*{DFT and TD-DFT calculations of the molecular vibronic properties}
Finally, the vibronic intensities (Fig. 4b) are calculated as the square of the sum of the Franck and Condon and the Herzberg-Teller amplitudes for each relevant mode independently using Gaussian 09 Revision D.01. To this end, the results of the normal mode calculations are used to define equally spaced discrete distortions spanning the range between the classical turning points of each mode. Repeated TD-DFT calculations of the vertical transition energy and dipole moment to the lowest excited singlet are used to determine the mode displacements and the derivative of the transition dipole moments with respect to the dimensionless normal coordinate through a series of third order polynomial regressions. Since the third order terms are found to be negligible and since the changes in frequencies (from second order terms) between ground and excited states of all the modes that are identifiable in the experimental spectrum are all well below 5$\%$ and would not affect the Franck and Condon overlaps, Duschinsky rotations (mode mixing in the excited state) are ignored. The theoretical vibronic spectra presented are then calculated using the TD-DFT determined intensities by scaling the normal mode frequencies obtained analytically at the DFT level by a factor $0.96$ \cite{CCCBDB} and by broadening each line by a Lorentzian 20 cm$^{-1}$ wide at half maximum.
\section*{Acknowledgments}
The authors thank Virginie Speisser and Michelangelo Romeo for technical support. \textbf{Funding}: This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grant agreement No 771850) and the European Union's Horizon 2020 research and innovation program under the Marie Sk\l odowska-Curie grant agreement No 894434. The Labex NIE (Contract No. ANR-11-LABX-0058\_NIE), and the International Center for Frontier Research in Chemistry (FRC) are acknowledged for financial support. This work was granted access to the HPC resources of IDRIS under the allocation 2020-A0060907459 made by GENCI. \textbf{Author contributions}: All authors contributed jointly to all aspects of this work. \textbf{Competing
interests}: The authors declare no competing interests. \textbf{Data and
materials availability}: All data needed to evaluate the conclusions
in the paper are present in this paper and the supplementary
materials.
\section*{Supplementary materials}
Supplementary Figs. 1 to 13\\
Supplementary Table\\
Supplementary Notes 1 to 3\\
References
|
2,869,038,154,770 | arxiv | \section{Introduction}
The study of two-dimensional Conformal Quantum Field Theory (CQFT)
has received much attention in the last ten years, following the
pioneering
work of Belavin, Polyakov, and Zamolodchikov (BPZ) \cite{BPZ}.
Interest
in this subject has been promoted by the relation of 2D-CQFT's to
two-dimensional statistical systems near second-order
phase transitions as well as in their relevance in the study of
classical
solutions of string theories\footnote{See ref. \cite{Ibb} for an extensive
coverage and list of references.}.
Based on the general approach developed in
\cite{BPZ}, Friedan, Qiu and
Shenker (FQS) have shown \cite{FQS}
that unitarity of the representation of the
Virasoro algebra (VA) restricts the possible values of the
central charge
$c$, and the conformal dimensions $h$ of the fields to the range
$c\geq 0, h\geq0$, with $c$ and $h$ taking only the discrete values
\begin{equation}\label{1.1}
c=1-\frac{6}{(k+2)(k+3)},\quad k=1,2...\end{equation}
\begin{equation}\label{1.1b}
h_{p,q}^{(k)}=\frac{(p(k+3)-q(k+2))^2-1}{4(k+2)(k+3)}\end{equation}
if $0<c<1$. These values of $c$ characterize the so-called ``minimal
unitary models''. In particular, FQS have shown
that for the first four values
of $c$ in the series (\ref{1.1}),
$c=\frac{1}{2},\frac{7}{10},\frac{4}{5},\frac{6}{7}$,
the conformal dimensions given by (\ref{1.1b}) coincide with the
critical exponents of the Ising Model (IM),
Tricritical-IM, 3-state Potts
Model and Tricritcal Potts Model, respectively.
About the same time
it has further been shown \cite{H} that
the critical points of RSOS models
\cite{ABF} provide in fact particular realizations
of all members in the
discrete series.
The first attempt to obtain explicit Quantum Field theoretic
realizations
of these models was made in ref. \cite{GKO}, where the coset
construction
was used to obtain new representations of the Virasoro algebra (VA).
In particular
it was shown that the coset $G/H$, with $G=SU(2)_k\times SU(2)_1$ and
$H=SU(2)_{k+1}$, provide realizations of the
Virasoro algebra of minimal unitary models. The possibility of an
explicit
realization of these coset models by gauging WZW Lagrangians was also
suggested.
Since then, the search for Lagrangian formulations of the coset
construction
have received much attention \cite{GK}, \cite{KS},
\cite{BRS}, \cite{CMR}.
In refs. \cite{GK}, \cite{KS} it has been shown that the central
charges of suitably gauged WZW models coincide with the central
charges of the GKO coset models. The same conclusion has been
reached from the study of gauged fermionic models
\cite{BRS}, \cite{CMR}.
In \cite{KS} it was further shown that the energy-momentum tensor
of the gauged WZW theory coincides with the one
in the GKO construction
in the physical (BRST-invariant) subspace.
As far as the minimal unitary models are concerned,
some specific primary
fields were constructed in the bosonic formulation
by Gawedzki and Kupiainen (GK) \cite{GK} and a general
method for obtaining the other primaries has been given.
In the formulation
of \cite{GK} the 4-point functions of these primaries can be evaluated
using a decoupled picture. A study of the full operator content of
both bosonic and fermionic gauged theories,
including the relation between
their primary fields and the corresponding scaling
fields in statistical models
is, however, lacking.
In the present paper we partially fill this gap
and show how to realize
some of the primary fields, in the unitary minimal models
in the framework of
the fermionic coset construction \cite{BRS}, \cite{CMR}.
By representing
the partition function of the fermionic coset formulation
as a product of
``decoupled'' sectors, we show that the primary fields are given by
BRST-invariant composites of the decoupled fields.
We furthermore show
how to obtain the correlators of the primaries in this picture.
In the particular case of $k=1,c=\frac{1}{2}$
(corresponding to the IM)
our fermionic construction goes
beyond the bosonic one of GK as we give a
field-theoretic realization
of all operators (energy-, order-, disorder-operators
and Majorana
fermions) of the critical Ising model, as well as
of the corresponding
order-disorder algebra, in terms of BRST-invariant local composites of
decoupled fields involving massless bosons and Dirac fermions.
We further show that all
(mixed and unmixed) 4-point correlation functions agree
with the expected
results.
The paper is organized as follows:
In section 2 we present the partition
function for the GKO coset $(SU(2)_k\times SU(2)_1)/SU(2)_{k+1}$ in the
fermionic coset formulation, and show that it can be written as the
product of the partition functions of free Dirac fermions,
negative level WZW fields and ghosts.
The computation of the four-point function of
primaries in the ``decoupled picture'' requires the evaluation
of four-point correlators of negative level
$SU(n)$-WZW fields in arbitrary
representations.
For positive level $SU(n)$-WZW fields in the fundamental
representation the general solution
has been given by Knizhnik and Zamolodchikov (KZ) \cite{KZ}. The
computation of higher-spin 4-point correlators has, however,
been restricted to
$SU(2)$ WZW fields \cite{FZ}. Making use of the
equivalence between the
$SU(n)_N$ WZW theory and the fermionic coset $U(nN)/(SU(N)_n\times
U(1))$ \cite{NS}, we derive in section 3 a
``reduction formula'' allowing one
to calculate the correlators of primaries in terms of WZW correlators
belonging to $SU(2)$.
In section 4 we then specialize to the case of $k=1$, which is expected
to be equivalent to the critical two-dimensional Ising model \cite{IZ}.
We begin by identifying the primary fields corresponding to the energy
and spin operators and show that their respective 4-point functions
agree with known results \cite{KC}, \cite{LP}, \cite{BPZ}.
We next identify the disorder operator by requiring
the usual order-disorder
algebra and show that the 4-point function involving
both order and disorder
operators again agrees with known results \cite{BPZ}.
A brief summary of these results has been reported recently \cite{CR}.
The computations
of the correlators
are significantly simplified by the use of the reduction formula derived
in section 3. We conclude this section by identifying the
Onsager \cite{On} (Majorana) fermions.
In section 5 we discuss how the above construction of the primaries may be
generalized to the whole FQS series, and conclude in section 6.
\section{Fermionic Coset Realization of the Minimal Unitary Series}
\setcounter{equation}{0}
As was proposed in ref. \cite{P}, the fermionic
realization of the $G/H$ coset
model can be obtained from a free fermion Lagrangian with
symmetry $G$ $(G=
U(N),O(N),..)$ by freezing the degrees of freedom associated with the
subgroup $H$, imposing the conditions
\begin{equation}\label{2.1}
J_\mu^{(H)}|phys>=0\end{equation}
where $J_\mu^{(H)}$ are the conserved currents
associated with the group
$H$. In the path integral formalism (\ref{2.1})
is implemented by gauging
the $H$-symmetry of the fermionic Lagrangian.
As mentioned in the introduction, the minimal unitary series can be
represented by the coset models
\begin{equation}\label{2.2}
G/H=\frac{SU(2)_k\times SU(2)_1}{SU(2)_{k+1}}\end{equation}
One obtains a fermionic representation of
a model with symmetry group
$SU(2)_k\times SU(2)_1$ in terms of cosets,
by making use of the general
equivalence \cite{NS}
\begin{equation}\label{2.3}
SU(N)_k\hat=\frac{U(Nk)}{SU(k)_N\times U(1)}\end{equation}
where the l.h.s. stands for the realization of a
$SU(N)\ WZW$-model of
level $k$, and the r.h.s. is realized by a theory of
$N\times k$ free Dirac
fermions $\psi^{i\alpha}(i=1,...N$ and $\alpha=1,...,k)$, with the
$SU(k)$ currents $\psi^{\dagger i\alpha}\gamma_\mu
T^a_{\alpha\beta}\psi^{i\beta},
\ (T^a,a=1,...,k^2-1$ the $SU(k)$ generators) and the
$U(1)$ currents $\psi^{\dagger i\alpha}\gamma_\mu\psi^{i\alpha}$ freezed
by gauging the respective symmetry groups. In this way one obtains a
coset representation of the numerator of (\ref{2.2}), and one is led to
make the identification
\begin{equation}\label{2.4}
\frac{SU(2)_k\times SU(2)_1}{SU(2)_{k+1}}=
\frac{\frac{U(2k)}{SU(k)_2\times U(1)}
\times\frac{U(2)}{U(1)}}{SU(2)_{k+1}}\end{equation}
where the group $SU(2)_{k+1}$ is moded out by
further freezing the currents
$\psi^{\dagger i\alpha}\gamma_\mu t^a_{ij}\psi^{j\alpha}+
\chi^{\dagger i\alpha}\gamma_\mu t^a_{ij}\chi^j$, (with
$t^a$ the $SU(2)$ generators),
which satisfy
a Kac-Moody algebra of level $k+1$.
The identification (2.4)
receives support from the equality of central charges
as well as from the current
correlation functions in the corresponding Lagrangian
formulation \cite{GK}, \cite{CMR}.
According to the above prescriptions, the Lagrangian
realizing the r.h.s.
of (\ref{2.4}) is given by
\begin{eqnarray}\label{2.5}
{\cal L}^{(k)}&=&\frac{1}{\sqrt2\pi}\psi^{\dagger i\alpha}
\left((\raise.15ex\hbox{$/$}\kern-.57em\hbox{$\partial$}+i\raise.15ex\hbox{$/$}\kern-.57em\hbox{$a$})
\delta_{ij}\delta_{\alpha\beta}+i\raise.15ex\hbox{$/$}\kern-.57em\hbox{$B$}^a
T^a_{\alpha\beta}\delta_{ij}
+i\raise.15ex\hbox{$/$}\kern-.57em\hbox{$A$}^at^a_{ij}\delta_{\alpha\beta}\right)
\psi^{j\beta}+\nonumber\\
&+&\frac{1}{\sqrt2\pi}\chi^{\dagger i\alpha}\left((\raise.15ex\hbox{$/$}\kern-.57em\hbox{$\partial$}+i\raise.15ex\hbox{$/$}\kern-.57em\hbox{$b$})
\delta_{ij}+i\raise.15ex\hbox{$/$}\kern-.57em\hbox{$A$}^at^a_{ij}
\right)\chi^j\end{eqnarray}
where $i,j=1,2;\ \alpha,\beta=1,2,...,k$, and the gauge fields act
as Lagrange multipliers implementing the conditions (2.1).
In order to arrive at a ``decoupled'' description, we change variables by
writing
\begin{eqnarray}\label{2.6}
a\ =&i(\bar\partial h_a)h_a^{-1},\qquad \bar a\ =&i(\partial
\bar h_a)\bar h_a^{-1}\nonumber\\
b\ =&i(\bar\partial h_b)h_b^{-1},\qquad \bar b\ =&i(\partial
\bar h_b)\bar h_b^{-1}\nonumber\\
A\ =&i(\bar\partial g_A)g_A^{-1},\qquad \bar A\ =&i(\partial
\bar g_A)\bar g_A^{-1}\nonumber\\
B\ =&i(\bar\partial g_B)g_B^{-1},\qquad \bar B\ =&i(\partial
\bar g_B)\bar g_B^{-1}\nonumber\\
&&\nonumber\\
\psi_1\ =&h_ag_Ag_B\psi^{(0)}_1,\qquad
\psi^\dagger_2\ =&\psi^{(0)\dagger}_2
(h_ag_Ag_B)^{-1}\nonumber\\
\psi_2\ =&\bar h_a\bar g_A\bar g_B\psi^{(0)}_2,
\qquad\psi^\dagger_1\ =&\psi^{(0)\dagger}_1
(\bar h_a\bar g_A\bar g_B)^{-1}\nonumber\\
&&\nonumber\\
\chi_1\ =&h_bg_A\chi_1^{(0)},\qquad\chi_2^\dagger\ =
&\chi_2^{(0)^\dagger}
(h_bg_A)^{-1}
\nonumber\\
\chi_2\ =&\bar h_b\bar g_A\chi_2^{(0)},\qquad\chi_1^\dagger\ =&
\chi_1^{(0)^\dagger}
(\bar h_b\bar g_A)^{-1}\end{eqnarray}
where $A_\mu=A^a_\mu t^a$ and $B_\mu=B^a_\mu T^a$.
Under the gauge transformations $W_\mu\to GW_\mu G^{-1}+G\partial
_\mu G^{-1}$, with $W_\mu$ standing for $A_\mu,B_\mu,a_\mu$ and $b_\mu$,
the products
\begin{equation}\label{2.7}
\tilde g_I=g^{-1}_I\bar g_I,\ I=A,B;\quad\tilde h_i=h_i^{-1}\bar h_i,
\quad i=a,b\end{equation}
remain invariant. Parametrizing $\tilde h_i$ by
\begin{equation}\label{2.8} \tilde h_i=e^{-2\phi_i}\end{equation}
taking due account of the Jacobians of the respective transformations
\cite{PW}, \cite{Fi}
(see also \cite{CMR}), and factoring out the infinite gauge volume
$\Omega=\int {\cal D}g_A{\cal D}g_B\newline
{\cal D}h_a{\cal D} h_b$
one arrives at the following decoupled form for the partition
function associated with the Lagrangian (\ref{2.5}):
\begin{equation}\label{2.9}
Z_{\frac{SU(2)_k\times SU(2)_1}{SU(2)_{k+1}}}=Z_FZ_BZ_{WZW}Z_{gh}\end{equation}
where
\newpage
\begin{eqnarray}\label{2.10}
Z_F&=&\int{\cal D}\psi^{(0)}{\cal D}\psi^{(0)\dagger}
{\cal D}\chi^{(0)}{\cal D}\chi^{(0)\dagger}\exp
\left(-\frac{1}{\pi}\int(\psi^{(0)
\dagger}_2\bar\partial\psi_1+\psi^{(0)\dagger}_1
\partial\psi^{(0)}_2)\right)\cdot\nonumber\\
&&\qquad\cdot\exp\left(-\frac{1}{\pi}\int(\chi^{(0)\dagger}
_2\bar\partial\chi_1^{(0)}+\chi^{(0)\dagger}_1
\partial\chi^{(0)}_2)\right)\\
Z_B&=&\int{\cal D}\phi_a{\cal D}\phi_b
\exp\left(\frac{k}{\pi}\int \phi_a\Delta
\phi_a\right)\exp\left(\frac{1}{\pi}\int\phi_b\Delta\phi_b\right)
\nonumber\\
Z_{WZW}&=&\int{\cal D}\tilde g_A{\cal D}\tilde g_B
\exp((k+5)W[\tilde g_A])
\exp((2k+2)W[\tilde g_B])\nonumber\end{eqnarray}
and where $Z_{gh}$ is the partition functions of $SU(k),
\ SU(2),\ U(1)$
and $U(1)$ decoupled ghost fields. The
explicit form of the ghost partition function will not be required
at this time. $W[g]$ is the $WZW$ functional \cite{Wi}
\begin{eqnarray}\label{2.11}
W[g]&=&\frac{1}{16\pi}\int d^2 x tr(\partial_\mu g
\partial^\mu g^{-1})+\nonumber\\
&&+\frac{1}{24\pi}\int d^3 y\varepsilon_{ijk} tr(g^{-1}
\partial_i g g^{-1}\partial_j gg^{-1}\partial_k g).\end{eqnarray}
where the second integral is over the three-dimensional
ball with space time as boundary.
The total central charge is obtained by adding the
individual contributions. The central charge associated with a
$WZW$ field of level $K$
is given by the well-known formula \cite{KZ}
\begin{equation}\label{2.12}
c=\frac{ K\dim G}{K+C_V}\end{equation}
where $C_V$ is the Casimir of the corresponding symmetry group $G$ in
the fundamental representation: $f^{abc}f^{a'bc}=C_V\delta^{aa'}$.
Thus we have for the individual contributions in $Z_{WZW}$,
\begin{equation}\label{2.13}
c^{(A)}_{WZW}=\frac{3(k+5)}{k+3},\ c^{(B)}_{WZW}=
\frac{2(k+1)(k^2-1)}{k+2}.\end{equation}
Adding to these central charges the contributions
$c_F=2k+2,\ c_B=2$ and $c_{gh}=-2k^2-8$, we obtain (1.1),
thus giving support to the identification (\ref{2.4}).
Note that the $WZW$-sectors have negative levels $-2(k+1)$ and $-(k+5)$,
respectively, which taken by themselves
would imply the presence of negative norm states. Unitarity is, however,
restored by taking into account the other sectors.
Although the different sectors appear decoupled on the level of the
partition function (2.9), they are in fact coupled via the BRST
quantization conditions, the observables of the theory being required
to be BRST invariant \cite{KS}.
In terms of the variables of the gauged Lagrangian (\ref{2.5}),
this amounts to considering only gauge invariant composites of
these fields. In particular, the gauge invariant fermion fields
can be constructed in terms of the exponential of Schwinger
line-integrals as follows,
\begin{eqnarray}\label{2.14}
&&\hat\psi^{i\alpha}(x)= e^{-i\int_x^\infty dz^\mu a_\mu}
\left(Pe^{-i\int^\infty_x dz^\mu A_\mu}\right)_{ij}
\left( Pe^{-i\int^\infty_x dz^\mu B_\mu}\right)_{\alpha\beta}
\psi^{j\beta}(x)\nonumber\\
&&\hat\chi^i(x)=e^{-i\int^\infty_x dz^\mu b_\mu}
\left( Pe^{-i\int^\infty_x dz^\mu A_\mu}\right)_{ij}\chi^j(x)
\end{eqnarray}
where $P$ denotes ``path-ordering''.
In section 5 we shall show that some primaries of the FQS models
with conformal dimensions given by the Kac formula (\ref{1.1b})
can be constructed as suitable normal ordered products
of these fields. Furthermore, as we shall demonstrate for the
IM in section 4, the Schwinger line integrals in (2.14) play an
essential role in the realization of the order-disorder algebra.
\section{Reduction Formulae}
\setcounter{equation}{0}
In section 4 we shall be concerned with the calculation of four-point
functions of gauge-invariant local fermion bilinears such as
\begin{equation}\label{3.1}
\Phi^{(k)}_{2,2}=\hat\psi_2^{\dagger i\alpha}\hat\psi_2^{i\alpha}+
\hat\psi_1^{\dagger i\alpha}
\hat\psi_1^{i\alpha}=\psi_2^{\dagger i\alpha}\psi_2^{i\alpha}+
\psi_1^{\dagger i\alpha}
\psi_1^{i\alpha}\end{equation}
In terms of the decoupled fields, expression (\ref{3.1})
takes the form
\begin{eqnarray}\label{3.2}
&&\Phi_{2,2}^{(k)}=\psi_2^{(0)\dagger i\alpha}
\tilde g_A^{ij}\tilde g_B
^{\alpha\beta}e^{2\phi_a}\psi_2^{(0)j\beta}\nonumber\\
&&+\psi_1^{(0)\dagger i\alpha}\left(\tilde g_A^{-1}\right)^{ij}
\left(\tilde g_B^{-1}\right)^{\alpha\beta}
e^{-2\phi_a}\psi_1^{(0)j\beta}
\end{eqnarray}
The fields are understood to be normal-ordered.
Their conformal dimensions are, respectively
\begin{eqnarray}\label{3.3}
h_{\psi_\alpha}&=&\frac{1}{2}\nonumber\\
h_{e^{2\phi_1}}&=&h_{e^{-2\phi_1}}=-\frac{1}{4k}\end{eqnarray}
and \cite{KZ}
\begin{eqnarray}\label{3.4}
&&h_{\tilde g_A}=h_{\tilde g_A^{-1}}=-\frac{3}{4(k+3)}\nonumber\\
&&h_{\tilde g_B}=h_{\tilde g_B^{-1}}=-\frac{(k^2-1)}{2k(k+2)}\end{eqnarray}
Note that the conformal dimensions (\ref{3.4})
correspond to WZW-fields
$\tilde g_A$ and $\tilde g_B$ in the fundamental representation of
$SU(2)_{-(k+5)}$ and $SU(k)_{-2(k+1)}$, respectively.
The dimension of the composite $\Phi^{(k)}_{2,2}$
is simply given in terms
of the sum of the individual contributions,
\begin{equation}\label{3.5}
h_{2,2}^{(k)}=\frac{1}{2}-\frac{1}{4k}-\frac{3}{4(k+3)}
-\frac{(k^2-1)}{2k
(k+2)}=\frac{3}{4(k+2)(k+3)}\end{equation}
with the same result for (antiholomorphic) dimension
$\bar h_{2,2}^{(k)}$.
The result (\ref{3.5}) agrees with Kac's
formula (\ref{1.1b}) for $p=q=2$. This
suggests the identification of $\Phi_{2,2}^{(k)}$
with a primary field.
A typical contribution to the four-point function
of $\Phi_{2,2}^{(k)}$ is
given by
\begin{eqnarray}\label{3.6}
&&<(\psi^\dagger_2\psi_2)(1)(\psi^\dagger_1\psi_1)
(2)(\psi^\dagger_1\psi_1)(3)(\psi_2^\dagger\psi_2)(4) >=\\
&=&\frac{(\mu^2)^{\frac{1}{k}}}{16}
\left|\frac{z_{12}z_{13}z_{24}z_{34}}
{z_{14}z_{23}}\right|^{\frac{1}{k}}\left\lbrace
\frac{1}{|z_{12}z_{34}|^2}G_A(1,2,4,3)G_B(1,2,4,3)
\right.\nonumber\\
&&\qquad\qquad+\frac{1}{|z_{13}z_{24}|^2}G_A(1,3,4,2)
G_B(1,3,4,2)\nonumber\\
&&\qquad\qquad-\frac{1}{z_{12}z_{34}\bar z_{13}
\bar z_{24}}\hat G_A(1,3,4,2)
\hat G_B(1,3,4,2)\nonumber\\
&&\qquad\qquad\left.-\frac{1}{z_{13}z_{24}
\bar z_{12}\bar z_{34}}
\hat G_A(1,2,4,3)\hat G_B(1,2,4,3)\right\rbrace\nonumber\end{eqnarray}
where $z_{ij}=z_i-z_j,\bar z_{ij}=\bar z_i-\bar z_j$, and
\begin{eqnarray}
G_A(1,2,3,4)&=&<tr (\tilde g_A(1)\tilde g_A^{-1}(2))
tr(\tilde g_A(3)\tilde g_A^{-1}(4))>\label{3.7}\\
\hat G_A(1,2,3,4)&=&<tr (\tilde g_A(1)\tilde g_A^{-1}(2)
\tilde g_A(3)\tilde g_A^{-1}(4))>\label{3.8}\end{eqnarray}
with the corresponding definition for $G_B$ and $\hat G_B$ in terms
of $\tilde g_B$. The kinematic factors within the curly brackets in
(\ref{3.6}) arise from the contractions of the free fermions,
as given by the two-point
functions
\begin{eqnarray}\label{3.9}
<\psi^{(0)i_1}_1(1)\psi^{(0)\dagger i_2}_2(2)>&=&
\frac{1}{2}\frac{\delta^{i_1i_2}}
{z_{12}}\nonumber\\
<\psi^{(0)j_1}_2(1)\psi^{(0)\dagger j_2}_1(2)>&=&\frac{1}{2}
\frac{\delta^{j_1j_2}}
{\bar z_{12}}.\end{eqnarray}
and the overall factor multiplying (\ref{3.6}) arises
from the four-point
function of the exponentials $e^{\pm2\phi_1}$ upon using
\begin{eqnarray}\label{3.10}
<e^{-\alpha\phi_1(1)}e^{\beta\phi_1(2)}>&=&|\mu z_{12}
|^{\frac{\alpha^2}{4k}},
\quad\alpha=\beta\nonumber\\
&=&0,\quad \alpha\not=\beta\end{eqnarray}
where $\mu$ is an arbitrary infrared regulator.
The functions (3.7) and (3.8) and the corresponding ones
for $\tilde g_B$
may directly be obtained from the work of ref. \cite{KZ}, by continuing
the KZ result to the
negative levels in question. Instead we shall make use of the duality
between $SU(N)_n$ and $SU(n)_{-(N+2n)}$ following from (\ref{2.3})
\cite{NS} to obtain a
``reduction formula'' which will prove very useful further on.
We begin by making the identifications
\begin{eqnarray}\label{3.11}
&&g^{ij}=\mu^{-1}\psi_1^{\dagger j\alpha}\psi^{i\alpha}_1=\frac{1}{\mu}
\psi_1^{(0)\dagger_{j\alpha}}\tilde g_{\alpha\beta}e^{-2\phi}
\psi_1^{(0)i\beta}\nonumber\\
&&(g^{-1})^{ji}=\mu^{-1}\psi_2^{\dagger i\alpha}\psi^{j\alpha}_2=
\frac{1}{\mu}\psi_2^{(0)\dagger i\beta}
\tilde g_{\beta\alpha}^{-1}e^{2\phi}
\psi_2^{(0)j\alpha}\end{eqnarray}
where now
\begin{equation}\label{3.12}
<e^{-\alpha\phi(1)}e^{\alpha\phi(2)}>=|\mu z_{12}
|^{\frac{\alpha^2}{2nN}}.\end{equation}
Comparison of the respective four-point functions then leads to a
relation between the four-point function of $g\in SU(N)_n$ and $\tilde g
\in SU(n)_{-N-2n}$. With the identification (\ref{3.11}) we have
\begin{eqnarray}\label{3.13}
&&<g(1)g^{-1}(2)g^{-1}(3)g(4)>=\left(\frac{1}{\mu}\right)^4
<e^{-2\phi(1)}e^{2\phi(2)}e^{2\phi(3)}e^{-2\phi(4)}>\\
&&\times<(\psi^{(0)\dagger}_1\tilde g\psi_1^{(0)})(1)
(\psi^{(0)\dagger}_2\tilde g^{-1}
\psi_2^{(0)})(2)(\psi^{(0)\dagger}_2\tilde g^{-1}\psi_2^{(0)})(3)
(\psi^{(0)\dagger}_1\tilde g\psi_1^{(0)})(4)>\nonumber\end{eqnarray}
\begin{eqnarray}\label{3.14}
<g(1)g^{-1}(2)g^{-1}(3)g(4)>=&&\left(\frac{1}{\mu^2}
\right)^{2-\frac{2}{nN}}
\left|\frac{z_{12}z_{34}z_{13}z_{24}}{z_{143}z_{23}}
\right|^{\frac{2}{nN}}
\times\\
\times\frac{1}{16}\left\lbrace \right.&&I_1\bar I_1\frac{1}
{|z_{12}z_{34}|^2}<tr
(\tilde g(1)\tilde g^{-1}(2))tr(\tilde g^{-1}(3)\tilde g(4))>
\nonumber\\
+&&I_2\bar I_2\frac{1}{|z_{13}z_{24}|^2}<tr
(\tilde g(1)\tilde g^{-1}(3)tr(\tilde g^{-1}(2)\tilde g(4))>
\nonumber\\
-&&I_1\bar I_2\frac{1}{z_{13}z_{24}\bar z_{12}\bar z_{24}}<tr
(\tilde g(1)\tilde g^{-1}(3)\tilde g(4)\tilde g^{-1}(2))>\nonumber\\
-&&I_2\bar I_1 \left.\frac{1}{z_{12}z_{34}\bar z_{13}\bar z_{24}}<
tr(\tilde g(1)\tilde g(2)^{-1}\tilde g(4)\tilde g(3)^{-1})>\right\rbrace
\nonumber\end{eqnarray}
where $I_A,\bar I_B$ stand for the $SU(N)$ invariant tensors \cite{KZ}
\begin{eqnarray}\label{3.15}
&&I_1=\delta^{i_1i_2}\delta^{i_3i_4},\quad \bar I_1=\delta^{j_1j_2}
\delta^{j_3
j_4}\nonumber\\
&&I_2=\delta^{i_1i_3}\delta^{i_2i_4},\quad \bar I_2=\delta^{j_1j_3}
\delta^{j_2
j_4}.\end{eqnarray}
Following \cite{KZ}, we make the decomposition
\begin{equation}\label{3.16}
<g(1)g^{-1}(2)g^{-1}(3)g(4)>=|z_{14}z_{23}|^{-4h_g}
\sum_{A,B} I_A\bar I_B
G_{AB}(x,\bar x),\end{equation}
where
\begin{equation}\label{3.17}
h_g=\frac{N^2-1}{2N(N+n)},\end{equation}
and $x$, $\bar x$ are given by
\begin{equation}\label{3.18}
x=\frac{z_{12}z_{34}}{z_{14}z_{32}},\quad \bar x=\frac{\bar z_{12}
\bar z_{34}}
{\bar z_{14}\bar z_{32}}.\end{equation}
The $G_{AB}(x,\bar x)$ are the positive level
WZW blocks which have been explicitly calculated
in ref. \cite{KZ}
Comparison of both sides in eq. (\ref{3.10}) then yields
\begin{eqnarray}\label{3.19}
&&<tr(\tilde g(1)\tilde g^{-1}(2)) \ tr(\tilde g^{-1}(4)\tilde g(3))>=
\\
&&=16|\mu^2z_{14}z_{23}|^{\frac{2(n^2-1)}{n(N+n)}}\frac{|x|^2}
{|x(1-x)|\frac{2}{nN}}G_{11}(x,\bar x)\nonumber\\
&&\nonumber\\
&&<tr(\tilde g(1)\tilde g^{-1}(2)\tilde g(4)\tilde g^{-1}(3)
)>=\nonumber\\
&&=-16|\mu^2z_{14}z_{23}|^{\frac{2(n^2-1)}{n(N+n)}}\frac{x(1-\bar x)}
{|x(1-x)|\frac{2}{nN}}G_{21}(x,\bar x)\nonumber\end{eqnarray}
The remaining vacuum expectation values are obtained via the
substitution $x\to1-x$. The fact that only two WZW blocks ($G_{11}$
and $G_{21}$) are needed is a consequence of the bosonic character of
the WZW field, which implies
\begin{equation}\label{3.20}
G_{11}(1-x)=G_{22}(x),\ G_{12}(1-x)=G_{21}(x)\end{equation}
As we shall see in the following section, relations (\ref{3.16})
will prove
very useful. The real power of this reduction
technique will, however,
come into play when considering the four-point
function of other primaries,
which, as we shall see in section 5, require the
calculation of four-point
correlators of WZW fields belonging to representations
of $SU(2)_{-(k+5)}$ and
$SU(k)_{-2(k+1)}$, other than fundamental ones. This
poses the problem that
the four-point correlator of $SU(k)$-WZW fields $(k\not=2$) are
only known for the fundamental representation. Via the reduction
technique described above one can relate the correlator in a given
representation of $SU(k)_{-2(k+1)}$ to a correlator
in the transposed
representation of $SU(2)_k$. Since the four-point correlators of
WZW fields have been calculated for any integrable
(spin $j$, $j=0,1,...,\frac{k}{2}$) representation $SU(2)_k$, this
allows one to calculate
the four-point correlators of $\tilde g_B$ in any
representation. As for
the corresponding correlators involving $\tilde g_A$ belonging to
$SU(2)_{-(k+5)}$, they may be obtained from known $SU(2)_k$
correlators for
positive level $k$, or from (\ref{3.16}) in the case of $k=1$ (Ising
model). In short, this means that the correlators of the
primaries ${\Phi}
^{(k)}_{p,q}$ in the fermionic coset representation of minimal models
can in principle be calculated for arbitrary values of $k$ in
terms of $SU(2)$-WZW four-point functions, modulo
free-field correlators,
in analogy with the bosonic coset case.
\section{Fermionic Coset Description of Ising Model}
\setcounter{equation}{0}
As is well known, the critical Ising model can be described by a
continuum field theory of massless, free Majorana fermions
$\psi_M$ and $\bar\psi_M$.
\cite{IZ},
\cite{BS}. In this description the energy density operator
$\epsilon(x)$
is given by the local operator product of two Majorana fields.
While this
representation has proven to be useful, a corresponding
local representation
for the order (spin) operator $\sigma(x)$ and its dual, the disorder
operator $\mu(x)$, is lacking.
In this section we use the ideas developed previously in order to
construct an explicit local representation of the operators
$\epsilon(x),
\sigma(x)$ and $\mu(x)$
as suitable products of gauged Dirac fermions,
of conformal dimensions 1/2, 1/16, and 1/16 respectively, in agreement
with the critical exponents of the Ising model. We also show that the
corresponding four-point functions agree with results
obtained by other
methods \cite{KC}, \cite{LP}, \cite{BPZ}. A corresponding
realization of the Majorana fermions is also given.
\vspace{1cm}
\noindent {\it a) Energy and order operators}
\vspace{.5cm}
As was mentioned in the introduction, the critical Ising model
corresponds to a
conformal field theory with central charge $c=1/2$, that is, to $k=1$
in the series (\ref{1.1b}). As seen from (\ref{1.1b}), the operators
$\varepsilon$ and $\sigma$ can thus be associated with the primary
fields $\Phi^{(1)}_{2,1}$ and $\Phi^{(1)}_{2,2}$ of conformal
dimension $1/2$ and $1/16$, respectively. In terms of our coset
description, this leads us to make the gauge invariant ansatz
\begin{eqnarray}\label{4.1}
\Phi^{(1)}_{2,1}=\varepsilon&=&\frac{1}{\mu}\left[
\left(\hat\psi^{\dagger i}_2
\hat\chi^i_1+\hat\chi_2^{\dagger i}\hat\psi_1^i\right)
\left(\hat\chi_1^{\dagger i}\hat\psi^i_2+\hat\psi_1^{\dagger
i}\hat\chi^i_2\right)\right]\\
&=&-\frac{1}{\mu}\left(\psi_2^{(0)\dagger}:g_A^{-1}
g_A:\chi_1^{(0)}\right)\left(\chi_1^{(0)\dagger}:\bar g_A^{-1}\bar
g_A:\psi_2^{(0)}\right):e^{2\phi_a}::e^{-2\phi_b}:\nonumber\\
&&-\frac{1}{\mu}\left(\psi_2^{(0)\dagger}:g_A^{-1}
g_A:\chi_1^{(0)}\right)\left(\psi_1^{(0)\dagger}: \bar g_A^{-1}\bar
g_A:\chi_2^{(0)}\right):e^{\varphi_a}::e^{-\bar\varphi_a}
:\nonumber\\
&&\qquad\qquad:e^{-\varphi_b}::e^{\bar\varphi_b}:\nonumber\\
&&+(\psi\leftrightarrow \chi),\varphi_a\leftrightarrow\varphi_b,
\bar\varphi_a\leftrightarrow\bar\varphi_b)\nonumber\end{eqnarray}
for the energy operator, and
\begin{eqnarray}\label{4.2}
\Phi^{(1)}_{2,2}=\sigma&=&\frac{1}{\mu}\left(\hat\psi^{\dagger i}
\hat\psi^i+\hat\chi^{\dagger i}\hat\chi^i\right)\\
&=&\frac{1}{\mu}\left(\psi_2^{(0)\dagger}\tilde g_A
e^{2\phi_a}\psi_2^{(0)}+\psi_1^{(0)\dagger}e^{-2\phi_a}\tilde
g_A^{-1}\psi_1^{(0)}\right)+\left(\psi\to\chi,\phi_a,\to\phi_b\right)
\nonumber
\end{eqnarray}
for the order operator, where $\mu$ is a parameter of dimension
one, which we choose to coincide with the infrared regulator
in (3.9).
Recalling (3.2), and noting that $\exp(\pm 2\phi_b)$
has dimension $-1/4$, one checks that the two operators
$\varepsilon$ and $\sigma$ have the correct conformal
dimensions, once we make the identifications $:g_A^{-1}
g_A:=1$ and $:\bar g_A^{-1}\bar g_A:=1$ for the respective
normal ordered products.
For the fermionic coset formulation in question,
one has two different candidates for the $\Phi^{(1)}_{2,2}$
primary field.
The linear combination of bilinears in $\psi$ and $\chi$ in
(\ref{4.2}) is suggested by the known
operator product expansion of
$\sigma(x)\sigma(x+\varepsilon)$ \cite{BPZ,DFSZ}.
The specific form of $\epsilon$, on the other hand,
is suggested by the
usual identification of $\epsilon$ with the bilinear
$\psi_M\bar\psi_M$ of
Majorana spinors (see eqs. (4.28), (4.29)).
{}From (4.1) one sees that all the multipoint correlation functions
of $\varepsilon$ can be calculated explicitly,
once we set $:g_A^{-1}g_A:=:\bar g_A^{-1}\bar g_A:=1$.
For the four-point function a straight-forward calculation yields
\footnote{Here $Pf$ denotes the ``Pfaffian'', defined in general by\\
$Pf(A_{ij})=\sum_P(-1)^P A_{i,j_1}...A_{i_nj_n}$, the sum being
taken over all possible permutations $P$. Henceforth we make use
of the arbitrariness of the parameter $\mu$, in order to normalize
our correlators appropriately.},
\begin{eqnarray}\label{4.3}
\langle\varepsilon(1)\varepsilon(2)\varepsilon(3)
\varepsilon(4)\rangle&=&
\frac{1}{|z_{12}z_{34}|^2}\frac{|1-x+x^2|^2}{|1-x|^2}\nonumber\\
&=&\Bigl| Pf\left(\frac{1}{z_{ij}}\right)\Bigr|^2\end{eqnarray}
which agrees with the result obtained in the
Majorana formulation \cite{IZ}.
The evaluation of the four-point function of
$\sigma$ proceeds as in the
case of (\ref{3.4}). It is convenient to introduce the notation
\begin{eqnarray}\label{4.4}
\alpha:&=&\psi^{\dagger i}_1\psi^i_1,\ \alpha^{-1}:=
\psi^{\dagger i}_2\psi^i_2\nonumber\\
\tilde\alpha:&=&\chi^{\dagger i}_1\chi_1^i,\ \tilde\alpha^{-1}:
=\chi^{\dagger i}_2\chi^i_2\end{eqnarray}
Taking account of the selection rules contained in eqs. (\ref{3.6})
and (\ref{3.7}), we are left with 16 terms which, because of Bose
symmetry can each be reduced to the form
$\langle\alpha(1)\alpha^{-1}(2)\alpha^{-1}(3)\alpha(4)\rangle,\
\langle\tilde\alpha(1)\tilde\alpha^{-1}(2)\tilde\alpha^{-1}(3)
\tilde\alpha(4)\rangle$ or\\
$\langle\alpha(1)\alpha^{-1}(2)\tilde\alpha^{-1}
(3)\tilde\alpha(4)\rangle$, by suitable relabeling of the arguments.
The correlator $\langle\alpha(1)\alpha^{-1}(2)
\alpha^{-1}(3)\alpha(4)\rangle$
is given by (\ref{3.4}) with $k=1$ and $G_B=\hat G_B=1$,
corresponding to
the absence of the $B$-field in (2.5).
The four-point functions $G_A$ and
$\hat G_A$ of the $SU(2)_{-6}$ WZW field are
conveniently calculated from
the reduction formulae (\ref{3.16}), with $N=n=2$.
In that case $G_{11}$
and $G_{21}$ are the WZW blocks of the $SU(2)_2$ field \cite{KZ}:
\begin{eqnarray}\label{4.5}
G_{11}(x,\bar x)&=&\frac{|1-x|^{1/4}}{|x|^{3/4}}[F_1(x)F_1(\bar x)+
\frac{1}{4}|x|F_2(x)F_2(\bar x)]\nonumber\\
G_{21}(x,\bar x)&=&\frac{|1-x|^{1/4}}{|x|^{3/4}}
\left[\frac{1}{2}x F_3(x)F_1(\bar x)-\frac{1}{2}|x|
F_1(x)F_2(\bar x)\right].\end{eqnarray}
Here the $F_i$'s are the hypergeometric functions
\begin{eqnarray}\label{4.6}
F_1(x)&=&F(\frac{1}{4},-\frac{1}{4};\frac{1}{2};x)=\frac{1}
{2}(f_1(x)+f_2(x))
\nonumber\\
F_2(x)&=&F(\frac{1}{4},\frac{3}{4};\frac{3}{2};x)=\frac{1}
{{\sqrt x}}(f_1(x)-f_2(x))\nonumber\\
F_3(x)&=&F(\frac{5}{4},\frac{3}{4};\frac{3}{2};x)=-\frac{1}
{\sqrt{x(1-x)}}(f_2(x)-f_1(x))\nonumber\\
F_4(x)&=&F(\frac{1}{4},\frac{3}{4};\frac{1}{2};x)=
\frac{1}{2}\frac{1}{\sqrt{1-x}}(f_1(x)+f_2(x))\end{eqnarray}
with
\begin{equation}\label{4.7}
f_1(x)=\sqrt{1+{\sqrt x}},\ f_2(x)=\sqrt{1-\sqrt x}.\end{equation}
Note that $f_1$ and $f_2$ are solutions of the second order differential
equation arising from the null-vector condition for a
$\Phi^{(1)}_{2,2}$ field \cite{BPZ}.
{}From (4.5) and (\ref{3.16}) with $N=n=2$ we then obtain
\begin{equation}\label{4.8}
G_A(1,2,4,3)=8|\mu^2z_{14}z_{23}|^{3/4}\frac{|x|}{|x(1-x)|^{1/4}}
\left(f_1(x)f_1(\bar x)+f_2(x)f_2(\bar x)\right)\end{equation}
\begin{equation}\label{4.9}
\hat G_A(1,2,4,3)=-8|\mu^2z_{14} z_{23}|^{3/4}\frac{\sqrt{x(1-\bar
x)}}{|x(1-x)|^{1/4}}(f_1(x)f_2(\bar x)-f_2(x)f_1(\bar x)).
\end{equation}
Substitution of these results into (\ref{3.4}) (with $k=1$) then leads to
\begin{equation}\label{4.10}
\langle\alpha(1)\alpha^{-1}(2)\alpha^{-1}(3)\alpha(4)\rangle=
\frac{1}{|\mu^2 z_{14}z_{23}|^{1/4}}\frac{1}{|x(1-x)|^{1/4}}
(f_1(x)f_1(\bar x)+f_2(x)f_2(\bar x)).\end{equation}
It is remarkable that despite the appearance of two different
combinations of the $f_i$'s in (4.5) the final result (\ref{4.7})
can be written after a number of manipulations in terms of the
first one of the combinations.
The same result is evidently obtained for
$\langle\tilde\alpha(1)\tilde\alpha^{-1}(2)
\tilde\alpha^{-1}(3)\tilde\alpha(4)
\rangle$. Note that expression (4.7) has the remarkable
property of being invariant under the permutation of the arguments.
Hence all unmixed correlators of the above type are given by
the r.h.s. of (\ref{4.7}). This leaves us with the calculation of
the mixed correlator $\langle
\alpha(1)\alpha^{-1}(2)\tilde\alpha^{-1}(3)\tilde\alpha(4)\rangle$.
It is easy to see that this correlator only involves the trace (3.7).
Hence it involves the $f_i$'s only in the combination (4.8).
In fact one finds
\begin{equation}\label{4.11}
\frac{\langle\alpha(1)\alpha^{-1}(2)\tilde\alpha^{-1}(3)\tilde\alpha(4)
\rangle}{\langle\alpha(1)\alpha^{-1}(2)\alpha ^{-1}(3)\alpha(4)\rangle}=
\frac{1}{4}\end{equation}
Adding all contributions we thus finally obtain for the four-point
correlator of the order operator
\begin{eqnarray}\label{4.12}
&&\langle\sigma(1)\sigma(2)\sigma(3)\sigma(4)\rangle=\\
&&\frac{1}{|\mu^2 z_{14} z_{23}|^{1/4}}\frac{1}{|x(1-x)|^{1/4}}
\left(\sqrt{1+\sqrt x}\sqrt{1+\sqrt x}+\sqrt{1-\sqrt x}\sqrt{1-\sqrt{\bar
x}}\right)\nonumber\end{eqnarray}
where normalization constants have been absorbed into the arbitrary
mass scale $\mu$. This result agrees with the one obtained by \cite{BPZ}
using general conformal arguments.
\vspace{1cm}
\noindent{\it b) Disorder operator and dual algebra}
\vspace{.5cm}
A complete characterization of the Ising model must also include
the disorder operator $\mu$ of dimension $1/16$. This operator should
satisfy the equal-time dual algebra \cite{BS}
\begin{equation}\label{4.13}
\sigma(1)\mu(2)=e^{i\pi\theta(x_1-x_2)}\mu(2)\sigma(1)\end{equation}
where $x_i$ denotes the real part of $z_i$. This leads us to make the
following ansatz in terms of the gauge-invariant fermion fields (2.14):
\begin{equation}\label{4.14}
\mu(x)=\hat\psi^\dagger_2\hat\chi_2+\hat\chi^\dagger_1\hat\psi_1+(\psi
\leftrightarrow\chi).\end{equation}
The contribution of the non-abelian line integrals cancels as a
consequence of the underlying $SU(2)$-gauge invariance
of the bilinears,
as well as the
absence of singularities in $\langle \psi^{(0)^\dagger}_2
\chi_2^{\ (0)}\rangle$,
etc. We are thus formally left with
\begin{eqnarray}\label{4.15}
\mu(x)&=&\psi^\dagger_2 e^{i\int^\infty_x dz^\mu a_\mu}
e^{-i\int^\infty_x dz^\mu b_\mu}\chi_2+
+\chi^\dagger_1e^{i\int^\infty_x dz^\mu b_\mu}
e^{-i\int^\infty_x dz^\mu a_\mu}\psi_1\nonumber\\
&+& (\psi\leftrightarrow\chi,a_\mu
\leftrightarrow b_\mu).\end{eqnarray}
In terms of the decoupled fields (\ref{2.6}) we thus obtain
\begin{eqnarray}\label{4.16}
\mu(x)&=&\left(\psi_2^{(0)\dagger}\tilde g_A\chi_2^{(0)}
\right)e^{\varphi_a}
e^{\bar\varphi_b}+\left(\chi^{(0)\dagger}_1\tilde g_A^{-1}
\psi_1^{(0)}\right)
e^{-\varphi_a}e^{-\bar\varphi_b}+\nonumber\\
&&+(\psi\leftrightarrow\chi,\varphi_a\leftrightarrow
\varphi_b,\bar\varphi
\leftrightarrow\bar\varphi_b)\end{eqnarray}
where $(i=a,b)$
\begin{eqnarray}\label{4.17}
\varphi_i&=&\phi_i+i\int^\infty_x dz^\mu
\varepsilon_{\mu\nu}\partial_\nu\phi_i,
\nonumber\\
\bar\varphi_i&=&\phi_i-i\int^\infty_x
dz^\mu\varepsilon_{\mu\nu}\partial_\nu\phi_i,\end{eqnarray}
are respectively the holomorphic and anti-holomorphic
components of the fields $\phi_i$ parametrizing
$a_\mu$ and $b_\mu$. Using the (euclidean) equal-time commutator
\begin{equation}\label{4.18}
[\phi_i(x),\partial_0\phi_i(y)]_{ET}=-\frac{\pi}{2}\delta(x^1-y^1)\end{equation}
we have from (\ref{4.17})
\begin{eqnarray}\label{4.19}
&&[\varphi_i(x_1),\varphi_i(x_2)]_{ET}=\frac{-i\pi}{2}
\epsilon(x_1-x_2)
\nonumber\\
&&[\bar\varphi_i(x_1),\bar\varphi_i(x_2)]_{ET}=
\frac{-i\pi}{2}\epsilon(x_1-x_2)
\end{eqnarray}
\begin{equation}\label{4.20}
[\varphi_i(x_1),\bar\varphi_i(x_2)]_{ET}=\frac{i\pi}{2}\end{equation}
Making use of these commutation relations we obtain from (\ref{4.2})
and (\ref{4.16}) the equal-time duality relation (\ref{4.13}), as
required.
The evaluation of the 4-point function of the $\mu$-operator
proceeds along
the same lines as in the case of the 4-point function of the order
operator. The result is again given by the r.h.s. of
(\ref{4.12}), as
expected \cite{KC}.
We next calculate the mixed 4-point correlation function
$<\sigma(1)\mu(2)\sigma
(3)\mu
\newline
(4)>$. To this end we introduce in
addition to (4.4) the notation
\begin{eqnarray}\label{4.21}
\beta=\hat \psi_1^\dagger\hat\chi_1&&\quad\beta^{-1}=
\hat\chi_2^\dagger\hat\psi_2
\nonumber\\
\tilde\beta=\hat\chi_1^\dagger\hat\psi_1&&\quad\tilde\beta^{-1}=
\hat\psi_2^\dagger\hat\chi_2\end{eqnarray}
In the mixed case the evaluation is less straightforward than in the
case of the unmixed four-point functions, since the operators $\sigma$
and $\mu$ no longer commute.
The fermionic selection rules lead us to consider the evaluation
of 24 terms of the type $<\alpha(1)\beta(2)\alpha^{-1}
(3)\beta^{-1}(4)>,
<\alpha(1)\alpha^{-1}(2)\tilde\beta(3)\tilde\beta^{-1}(4)>,\\
<\alpha(1)\beta^{-1}(2)\tilde\alpha(3)\tilde\beta^{-1}(4)>,\
<\alpha^{-1}(1)\tilde\alpha^{-1}(2)\beta(3)\tilde\beta(4)>,$
as well as those resulting from the permutation of
the arguments, and the
exchange $\gamma\leftrightarrow\tilde \gamma$, where $\gamma$
stands generically
for the operators (\ref{4.4}) and (\ref{4.21}). This number can
be reduced by noting that two correlators related by the interchange
$\gamma\leftrightarrow\tilde \gamma$ are equal.
We outline the calculation
for the case of two typical terms.
\bigskip
i) Consider $<\alpha(1)\beta(2)\alpha^{-1}(3)\beta^{-1}(4)>$. From
(2.6) we have
\begin{eqnarray}\label{4.22}
&&<\alpha(1)\beta(2)\alpha^{-1}(3)\beta^{-1}(4)>=<e^{-(\varphi_a+
\bar\varphi_a)(1)}e^{-(\varphi_b+\bar\varphi_a)(2)}e^{(\varphi_a+
\bar\varphi_a)(3)}e^{(\varphi_b+\bar\varphi_a)(4)}>\nonumber\\
&&\\
&&\times <(\psi_1^{(0)\dagger}\tilde g_A\psi_1^{(0)}(1)
(\psi^{(0)\dagger}_1\tilde g_A\chi_1^{(0)})(2)(\psi_2^{(0)\dagger})
\tilde g_A^{-1}\psi_2^{(0)})(3)
(\chi_2^{(0)\dagger}\tilde g_A^{-1}\psi_2^{(0)})(4)>\nonumber\end{eqnarray}
where normal ordering with respect to the free bosons and
fermions is understood. Recalling (\ref{3.9}) and (\ref{3.10}) we find
\begin{eqnarray}\label{4.23}
&&\langle \alpha(1)\beta(2)\alpha^{-1}(3)\beta^{-1}(4)\rangle
=\mu^2\frac{e^{i\frac{\pi}{4}\eta}}{16}\left(\frac{z_{13}z_{24}\bar z
_{13}\bar z_{24}\bar z_{14}\bar z_{23}}{\bar z_{12}\bar z_{34}}
\right)^{1/2}\cdot\\
&&\cdot\left(\frac{1}{z_{13}z_{24}\bar z_{13}\bar z_{24}}G_A(1,3,2,4)
-\frac{1}
{z_{13}z_{24}\bar z_{14}\bar z_{23}}\hat G_A(1,3,2,4)
\nonumber\right)\end{eqnarray}
where the phase $\exp(i\pi\eta/4)$ arises from the
commutation relations
(\ref{4.20}). For the case in question $\eta=2$;
in general it takes the
values $\pm2$. Evaluation of (\ref{4.23}) shows that
it reduces, after a
number of manipulations\footnote{In particular one
makes use of $sgn (Im x)\frac{1}{|\mu^2 z_{14}z_{23}|^{1/4}}
\frac{1}{|x(1-x)|^{1/4}}(f_1(\dot x)f_1(\bar x)-f_2(x)f_2(\bar x))
=\frac{i}{|\mu^2z_{13}z_{24}|^{1/4}}\frac{1}{|y(1-y)|^{1/4}}
(f_1(y)f_2(\bar y)-f_2(y)f_1(\bar y))$ where $Im x$ stands for
``imaginary part of $x$'', and $y=x/(x-1)$.} to the
remarkably simple
result $(y=x/(x-1))$,
\begin{equation}\label{4.24}
<\alpha(1)\beta(2)\alpha^{-1}(3)\beta^{-1}(4)>=\frac{i}{
|\mu^2 z_{13}z_{24}|^{1/4}}\frac{1}{|y(1-y)|^{1/4}}
(f_1(y)f_2(\bar y)-f_2(y)f_1(\bar y))\end{equation}
this time involving another combination of the $f_i$'s in accordance
with the expected analiticity properties of the result.
(Compare with (4.8)). A numerical factor has again been
absorbed into the arbitrary
parameter $\mu$.
\bigskip
ii) We next consider $<\alpha(1)\beta^{-1}(2)\tilde\alpha(3)
\tilde\beta^{-1}(4)>$. One has this time
\begin{eqnarray}\label{4.25}
&&<\alpha(1)\beta^{-1}(2)\tilde\alpha(3)\tilde\beta^{-1}(4)>\nonumber\\
&&=<e^{-(\varphi_a(1)+
\bar\varphi_a(1))}e^{(\varphi_b(2)+\bar\varphi_a(2))}
(e^{-(\varphi_b(3)+
\bar\varphi_b(3))}e^{(\varphi_a(4)+\bar\varphi_b(4))}>
\nonumber\\
&&\times <\left(\psi_1^{(0)\dagger}\tilde
g_A\psi_1^{(0)}\right)(1)\left(\chi^{(0)\dagger}_2\tilde
g_A^{-1}\psi_2^{(0)}\right)(2)\left(\chi_1^{(0)\dagger}
\tilde g_A\chi_1^{(0)}\right)(3)
\left(\psi_2^{(0)\dagger}\tilde g_A^{-1}\chi_2^{(0)}\right)(4)>
\nonumber\\
&&=\frac{1}{16}e^{\frac{i\pi\eta}{4}}(z_{14}z_{23}\bar z_{12}\bar
z_{34})^{1/2}\frac{1}{z_{14}z_{23}\bar z_{12}\bar z_{34}}
\hat G_A(1,4,3,2)\end{eqnarray}
The phase in this case corresponds to $\eta=-2$.
Expression (\ref{4.25})
is seen to involve the $f_i$'s only in the combination of
(\ref{4.9}). In
fact one obtains
\begin{equation}\label{4.26}
\frac{<\alpha(1)\beta^{-1}(2)\tilde\alpha(3)\tilde\beta^{-1}(4)>}
{<\alpha(1)\beta(2)\alpha^{-1}(3)\beta^{-1}(4)>}=-1\end{equation}
Explicit calculation shows that up to numerical
factors, the same results are obtained for the remaining
terms contributing
to the mixed correlator. Absorbing again a normalization
constant into
the arbitrary scale parameter $\mu$, we finally have for the mixed
correlator $(y=x/(x-1))$
\begin{equation}\label{4.27}
<\sigma(1)\mu(2)\sigma(3)\mu(4)>=\frac{i}{|\mu^2z_{14}z_{23}|^{1/4}}
\frac{1}{|y(1-y)|^{1/4}}\left(f_1(y)f_2(\bar y)-
f_2(y)f_1(\bar y)\right)\end{equation}
This result agrees with the one obtained by BPZ using general
conformal arguments\footnote{There is a misprint in the relative
sign of the result of BPZ (eq. (I.39) of \cite{BPZ}).}.
\cite{BPZ}. This provides further support for our
ans\"atze (\ref{4.1}), (\ref{4.2}) and (\ref{4.14}).
\vspace{1cm}
\noindent{\it c) Realization of Onsager fermions}
\vspace{.5cm}
To complete our discussion of the Ising model, we give a realization
of the Onsager fermions $\psi_M(x)$ and $\bar\psi_M(x)$ \cite{On}
in the fermionic
coset framework. We identify these fermions with the gauge-invariant
composites
\begin{eqnarray}\label{4.28}
\psi_M&=&\hat\psi^\dagger_2\hat\chi_1+\hat\chi_2^\dagger\hat\psi_1
\nonumber\\
&=&\frac{1}{\sqrt\mu}\left(\psi_2^{(0)\dagger}
\chi_1^{(0)}:e^{-\varphi_a}
e^{\varphi_b}+\chi_2^{(0)\dagger}
\psi_1^{(0)}:e^{-\varphi_a}e^{\varphi_b}:\right)\end{eqnarray}
and
\begin{eqnarray}\label{4.29}
\bar\psi_M&=&\hat\psi^\dagger_1\hat\chi_2+\hat\chi_1^\dagger\hat\psi_2
\nonumber\\
&=&\frac{1}{\sqrt\mu}\left(\psi_1^{(0)\dagger}\chi_2^{(0)}:e^{-\bar
\varphi_a}e^{\bar\varphi_b}:+\chi_1^{(0)\dagger}
\psi_2^{(0)}:e^{\bar\varphi_a}e^{-\varphi_b}:\right)\end{eqnarray}
of dimensions $(\frac{1}{2},0)$ and $(0,\frac{1}{2})$,
respectively. This
assignment agrees with the usual representation of the energy operator
(\ref{4.1}) in terms of Majorana fermions, $\epsilon=\psi_M\bar\psi_M$
\cite{IZ}. Note that $\psi^\dagger_M=\bar\psi_M$, as required.
\section{Other statistical models in the minimal unitary series}
\setcounter{equation}{0}
Our fermionic coset formulation is expected to allow
for the realization of all the primaries in the FQS series as
products of the fundamental, gauge-invariant fields (2.14).
Let us illustrate
this for the primaries $\Phi^{(k)}_{p,p}$ and $\Phi^{(k)}_{p,p-1}$,
the conformal dimension of a general primary $\Phi^{(k)}_{p,q}$
being given by (1.2).
We take the bosonic representation in terms of
gauged WZW fields \cite{GK}
as a guideline for our construction. In ref.
\cite{GK} the primaries in
the FQS series are constructed in accordance with (2.2) in terms of
SU(2) WZW fields $g$ and $g'$ of level $k$ and 1, respectively,
in a gauged WZW theory.
This motivates us to consider as basic building blocks the bilinears
\cite{NS}
\begin{eqnarray}\label{5.1}
\hat g^{ij}&=&:\hat\psi_2^{i\alpha}\hat\psi_2^{\dagger j\alpha}:,
\quad \hat {g'}^{ij}=:\hat\chi_2^i\hat\chi^{\dagger j}_2:\nonumber\\
(\hat g^{-1})^{ij}&=&:\hat\psi_1^{i\alpha}\hat
\psi_1^{\dagger j\alpha}:, \quad
(\hat {g'}^{-1})^{ij}=:\hat\chi_1^i\hat\chi_1^{\dagger j}:\end{eqnarray}
Note that expressions (\ref{5.1}) involve the gauge-invariant fields
(2.14), so that $\hat g^{ij}$ and $(\hat g^{-1})^{ij}$
should be identified
with the gauge-invariant WZW fields of the gauged WZW action referred
to above.
According to ref. \cite{GK} we need to consider fields $\hat g_j$ in
the isospin $j$ representation of $SU(2)_k$ $(j=0,\frac{1}{2},1,..
\frac{k}{2})$. They can be constructed as the symmetrized direct
product of $2j$ fundamental fields (\ref{5.1}):
\begin{equation}\label{5.2}
\hat g_j=\left[:\underbrace{\hat g\otimes...\otimes\hat g:}_{2j\
{\rm times}}
\right]_{\cal S}\end{equation}
where ${\cal S}$ stands for the symmetrization with respect to left
and right
indices separately. We again restrict ourselves to the
isospin zero sector.
In this
sector we have for $\Phi^{(k)}_{p,p}$
\begin{equation}\label{5.3}
\Phi^{(k)}_{p,p}=Tr(\hat g_j+\hat g_j^{-1})=Tr(g_j+g_j^{-1})\end{equation}
where $p=2j+1$. The second equality in (\ref{5.3}) is a result of
the cancellation of the Schwinger line integrals upon
taking the trace.
In the notation of (2.4) the primary (\ref{5.3}) is given by
\begin{eqnarray}\label{5.4}
\Phi^{(k)}_{p, p}&=&g_j^{(0)^{i_1...i_r,k_1...k_r}}.
\left(\tilde g_A^{-1}\right)_j^{k_1...k_r,i_1...i_r} \nonumber\\
&&+\left(g^{(0)}_j\to g^{(0)-1}_j,\
\tilde g_A\to\tilde g_A^{-1}
\right)\end{eqnarray}
where
\begin{eqnarray}\label{5.5}
g^{(0)i_1...i_r,k_1...k_r}_j&=&
:e^{2r\phi_1}::\psi_2^{(0)\dagger k_1\gamma_1}...
\psi_2^{(0)\dagger k_\gamma\alpha_\gamma}::
\psi_2^{(0)^{i_1\alpha_1}}...\psi_2^{(0)i_r\alpha_r}:\nonumber\\
&&\cdot[:\tilde g^{\gamma_1\alpha_1}_B...
\tilde g^{\gamma_r\alpha_r}_B:]_{\cal A}
\nonumber\\
\left(\tilde g_A\right)_j^{k_1..k_r, i_1...i_r}&=&
\left[:\tilde g_A^{k_ji_1}...\tilde g_A^{k_ri_r}:\right]
_{\cal S}\end{eqnarray}
with $r=2j$, and where the superscript ``$(0)$'' in the l.h.s. means
that the diagonal
$SU(2)_{k+1}$ gauge field has been decoupled.
The field $g_j^{(0)}$ in
(\ref{5.5}) is the fermionic coset representation
of the $SU(2)_k$ WZW
field in the $j$-representation \cite{NS}.
Due to the Pauli principle the direct product of
$\tilde g_A$'s and
$\tilde g_B$'s project into the $(2j+1)$-dimensional symmetric
(${\cal S}$),
antisymmetric (${\cal A}$)
representation of $SU(2)$ and $SU(k)$, respectively,
with the corresponding
Casimirs given by \cite{NS}
\begin{eqnarray}\label{5.6}
C({\cal S})&=&j(j+1)\nonumber\\
C({\cal A})&=&(k+2)\left[j-\frac{j^2}{k}-\frac{j(j+1)}{k+2}\right]
\end{eqnarray}
In terms of these Casimirs, we have for the conformal dimension of the
corresponding products
\begin{eqnarray}\label{5.7}
h_{(\tilde g_A)_j}&=&\bar h_{(\tilde g_A)_j}=\frac{-j(j+1)}{k+3}
\nonumber\\
h_{\cal A}&=&\bar h_{\cal A}
=-\frac{C({\cal A})}{k+2}=-\left[j-\frac{j^2}{k}-\frac{j(j+1)}{k+2}
\right]\end{eqnarray}
Adding to $h_{\cal A}$ the contributions from the vertex
operator $(h_{e^{2r
\phi_1}}=\bar h_{e^{2r\phi_1}}=-\frac{j^2}{k})$ and the
free fermions, we
obtain
\begin{equation}\label{5.8}
h_{g_j(0)}=\frac{j(j+1)}{k+2}\end{equation}
We thus obtain $(p=2j+1)$
\begin{equation}\label{5.9}
h_{p,p}^{(k)}=h_{g_j^{(0)}}+h_{(\tilde g_A)_j}=
\frac{j(j+1)}{(k+2)(k+3)}
=\bar h_{p,p}\end{equation}
in agreement with the Kac formula (\ref{1.1b}).
On the other hand, in terms of the bilinears in $\chi_\alpha^i$, eq.
(\ref{5.2}), the Pauli principle only allows us
to construct the primary
\begin{eqnarray}\label{5.10}
\Phi^{(k)}_{k+1,k+1}&=&Tr(\hat {g'}+\hat {g'}^{-1})\nonumber\\
&=&Tr\left({g'}_{\frac{1}{2}}^{(0)}\tilde g_A+{g'}^{(0)-1}
_{\frac{1}{2}}
\tilde g_A^{-1}\right)\end{eqnarray}
where
\[{g'}^{(0)ij}_{\frac{1}{2}}=:e^{2\phi_2}\chi_2^{(0)i}
\chi_2^{(0)\dagger j}:\]
with dimension (see eq. (3.2))
\begin{equation}\label{5.11}
h_{k+1,k+1}^{(k)}=\bar h^{(k)}_{k+1,k+1}=\frac{k}{4(k+3)},\end{equation}
again in agreement with Kac's formula.
Note that the primary (\ref{5.10})
has the same dimension as the primary (\ref{5.4}) for the maximum
value of $j,j=\frac{k}{2}$. Note also that for $k=1$, these primaries
are just the components making up the order operator (\ref{4.1}) in
the Ising model.
Following again ref. \cite{GK}, we take for $\Phi^{(k)}_{p,p-1}$,
\begin{equation}\label{5.12}
\Phi^{(k)}_{p,p-1}={\rm Tr}\left\{\hat g_j(\hat g'\otimes
{1\!{\rm l}}_{j
-\frac{1}{2}})^{-1}+(\hat g'\otimes {1\!{\rm l}}
_{j-\frac{1}{2}})\hat g^{-1}_j
\right\},
\end{equation}
where, in terms of the bilinears (\ref{5.1}),
\begin{eqnarray}\label{5.13}
&&{\rm Tr} \left\{\hat g_j\left(\hat g'\otimes
{1\!{\rm l}}_{j-\frac{1}{2}}
\right)^{-1}\right\}=\nonumber\\
&&\left\{ \left[\left( \hat\psi_2^\dagger\hat\psi_2\right)^{i_1j_1}
\ldots\left(\hat\psi_2^\dagger\hat\psi_2\right)^{i_rj_r}\right]_{\cal S}
\left(
\hat\chi_1^{\dagger j_1}\hat\chi_1^{i_1}\delta^{j_2i_2}\ldots
\delta^{j_ri_r}\right)\right\},
\end{eqnarray}
where ${\cal S}$ again denotes symmetrization with
respect to the indices
$\{i_l\}$ and $\{j_l\}$, separately. Expressing the fermionic fields
in terms of the decoupled variables (\ref{2.6}), one finds,
\begin{eqnarray}\label{5.14}
\Phi^{(k)}_{p,p-1}&=&:e^{2r\phi_1}::e^{-2\phi_2}::
\psi_2^{(0)^\dagger k_1\beta_1}\ldots \psi_2^{(0)^\dagger j_r
\beta_r}:\nonumber\\
&\cdot&:\psi_2^{(0)j_1\alpha_1}\ldots\psi_2^{(0)j_r\alpha_r}:\left[
:\tilde g_B^{\beta_1 \alpha_1}\ldots
\tilde g_B^{\beta_r\alpha_r}:\right]
_{\cal A}\nonumber\\
&\cdot&\left[\tilde g_A^{k_2i_2}\ldots\tilde g_A^{k_ri_r}\right]_{\cal S}:
\chi_1^{(0)^\dagger j_1}\chi_1^{(0)k_1}:\delta^{j_2i_2}
\ldots\delta^{j_ri_r}.\end{eqnarray}
Note that we are left with the symmetrization of only
$r-1$ fields $g_A$.
This is a combined effect of the cancellation of the
Schwinger line integrals
and the use of $g_1g_1^{-1}=1,\bar g_1\bar g_1^{-1}=1$
within the normal
product. Using (3.3), (3.4) and (\ref{5.7}) one readily
checks that the
primary (\ref{5.12}) has the conformal dimension $(r=2j)$
\begin{eqnarray}\label{5.15}
h^{(k)}_{p,p-1}&=&-\frac{1}{4}-\frac{r^2}{4k}+\frac{1}{2}(r+1)
-\left[ \frac{r}{2}-\frac{r^2}{4k}-\frac{r(r+2)}{4(k+2)}\right]-
\frac{(r^2-1)}{4(k+3)}\nonumber\\
&&=\frac{(r+k+3)^2-1}{4(k+2)(k+3)}\end{eqnarray}
which is in agreement with Kac formula (\ref{1.1b}).
\vspace{1cm}
\noindent{\it Other primaries}
\vspace{.5cm}
The primaries considered so far were constructed as local products of
chiral bilinears of the gauge-invariant fermion fields (\ref{2.14}).
The construction of the remaining primaries will be
carried out in the decoupled picture, where the requirement of BRST
invariance is more easily realized.
It follows from the Lagrangian
(\ref{2.6}) that we have the BRST charges $Q_{U(1)}^{(a)},\
Q^{(b)}_{U(1)},\
Q^{(B)}_{SU(k)}$ and $Q^{(A)}_{SU(2)}$, associated with the gauge
fields $a_\mu,\ b_\mu,\ B_\mu$ and $A_\mu$, respectively.
Our construction will involve bilinears commuting with the first three
charges. Hence we need to consider only $Q:=Q^{(A)}_{SU(2)}$.
Following ref. \cite{He}, we have for the BRST charge $Q$,
\begin{equation}\label{5.16}
Q=\oint dz:\left[\eta^a(z)G^a(z)+\frac{1}{2} C_{abc}\eta^a\eta^b
{\cal P}^c\right]:\end{equation}
where $G^a(z)\approx 0$ are the first-class constraints
associated with
the diagonal $SU(2)$ gauge symmetry, $C_{abc}$ are the
structure constants of the corresponding constraint algebra,
\begin{equation}\label{5.17}
\left\{G^a(z),G^b(w)\right\}=C_{abc}G^c(z)\delta(z-w)\end{equation}
and
\begin{eqnarray}\label{5.18}
&&\{{\cal P}^a,\eta^b\}=-\delta^{ab}\nonumber\\
&&(\eta^a)^*=\eta^a,\ ({\cal P}^a)^*=-{\cal P}^a\end{eqnarray}
In order to deduce the first-class constraints $G^a\approx 0$ from the
decoupled partition function (2.9), we follow refs. \cite{P},
\cite{KS} by simultaneously gauging the sectors corresponding to
the free fermions (both $\psi$'s and $\chi$'s), $SU(2)_{-(k+5)}$
WZW field $\tilde g_A$, and $SU(2)$-ghost fields with an
external $SU(2)$ gauge field $W^a_\mu$. In terms of the variables
$w$ and $\bar w$ defined by
\begin{equation}\label{5.19}
W=i(\bar\partial w)w^{-1},\quad\bar W=i(\partial \bar w)
\bar w^{-1}\end{equation}
the dependence of the different sectors on the external gauge field
factors are as follows:
\begin{eqnarray}\label{5.20}
&&\int{\cal D}\psi{\cal D}\psi^\dagger\int{\cal D}
\chi{\cal D}\chi^\dagger
exp\{-\frac{1}{\sqrt2\pi}\int\psi^\dagger({\raise.15ex\hbox{$/$}\kern-.57em\hbox{$\partial$}}
+i{\raise.15ex\hbox{$/$}\kern-.57em\hbox{$W$}})\psi\}exp\{-\frac{1}{\sqrt2\pi}
\int\chi({\raise.15ex\hbox{$/$}\kern-.57em\hbox{$\partial$}}+i{\raise.15ex\hbox{$/$}\kern-.57em\hbox{$W$}})\chi\}\nonumber\\
&&\quad\qquad\qquad\qquad\qquad =Z_F\ e^{(k+1)W[w^{-1}w]}\nonumber\\
&&\nonumber\\
&&\int{\cal D}\eta{\cal D}{\cal P}\int{\cal D}
\bar\eta{\cal D}\bar{\cal P}
e^{-\int{\cal P}^a\bar{\cal D}^{ab}(w)\eta^b}e^{-\int\bar{\cal P}^a
{\cal D}^{ab}(\bar w) \bar\eta^b}\nonumber\\
&&\quad\qquad\qquad\qquad\qquad =e^{2C_VW[w^{-1}\bar w]}
\int{\cal D}\eta{\cal D}{\cal P}\int{\cal D}\bar\eta{\cal D}
\bar{\cal P}
e^{-\int{\cal P}^a\bar\partial\eta^a}e^{-\int\bar{\cal P}^a\partial
\bar\eta}\nonumber\\
&&\nonumber\\
&&\int{\cal D}\tilde g_A e^{(k+5)\tilde W[\tilde g_A,w,\bar w]}=
e^{-(k+5)W[w^{-1}\bar w]}
\int {\cal D}\tilde g_A
e^{(k+5)\tilde W[\tilde g_A]} \end{eqnarray}
where $\tilde W[\tilde g_A,w]$ stands for the gauged WZW action
\begin{equation}\label{5.21}
\tilde W[\tilde g_A,w,\bar w]=W[w^{-1}\tilde g_A\bar w]
-W[w^{-1}\bar w]\end{equation}
and use has been made of the invariance of the Haar measure.
Note that for the case in question $C_V=2$, so that the dependence of
the corresponding gauged partition function
$Z[w,\bar w]$ on the external field is seen to cancel, implying
\begin{equation}\label{5.22}
\frac{i^n}{Z[w]}\frac{\delta^n\ Z[w]}{\delta w^{a_1}(z_1)\ldots\delta
w^{a_n}(z_n)}\Bigm\vert_{w=0}=\langle J^{a_1}(z_1)\ldots J^{a_n}(z_n)
\rangle=0\end{equation}
where
\begin{equation}\label{5.23}
J^a(z)=\frac{1}{i}\frac{\delta}{\delta w^a(z)} S[w,\bar w]
\Bigm\vert_{w=0}\end{equation}
with $S[w,\bar w]$ the corresponding gauged action.
We are thus led to make the identification
\begin{equation}\label{5.24}
G^a(z)=J^a(z)=j^a_\psi(z)+j^a_\chi(z)+\tilde j^a(z)+j^a_{gh}(z)\end{equation}
where
\begin{eqnarray}\label{5.25}
j^a_\psi&=&\frac{1}{\sqrt2\pi}\psi_2^{(0)^\dagger}
t^a\psi_1^{(0)}\nonumber\\
j^a_\chi&=&\frac{1}{\sqrt2
\pi}\chi_2^{(0)^\dagger}t^a\chi_1^{(0)}\nonumber\\
\tilde j^a(z)&=&-\left(\frac{k+5}{2}\right)tr\left( t^a\tilde g_A
\partial\tilde g^{-1}_A\right)\nonumber\\
j^a_{gh}(z)&=&f^{abc}{\cal P}^b\eta^c.\end{eqnarray}
Analogous relations apply to the antiholomorphic part.
Alternatively, the BRST charge (\ref{5.14}) may also be obtained as the
Noether
charge associated with the invariance of the effective action in the
decoupled
picture \cite{Ta,Ba}.
The primaries $\Phi_{p,q}^{(k)}$ are constructed subject to the
requirement
that they commute with the BRST charge (\ref{5.16}).
For $p-q\geq 2$ this
construction will also have to include the currents (\ref{5.25})
as we shall
see below. Our starting point is the observation that
\begin{equation}\label{5.27}
h_{g_j^{(0)}}+h_{(\tilde g_A)_\ell}+\frac{(1-(-1)^{p-q})}{2}
h_{g^{\prime(0)}_{
\frac{1}{2}}}+N=h_{p,q}^{(k)}\end{equation}
where $j=(p-1)/2$ and $\ell=(q-1)/2$,
and $N$ is an {\it integer} given by
\begin{equation}\label{5.28}
N=\frac{1}{4}\left[(p-q)^2-\frac{(1-(-1)^{p-q})}{2}\right]\end{equation}
Eq. (\ref{5.27}) tells us that we can obtain the primaries
by taking Kac-Moody
descendants of level $N$, from composites $\phi_h$ of the fields
$g_j^{(0)},(\tilde g_A)_\ell$ and $g^{\prime(0)}_{\frac{1}{2}}$
(if $p-q$ is odd),
using the property
\begin{equation}\label{5.29}
h_{{\cal J}_{-N}\phi_h}=h+N\end{equation}
where
\begin{equation}\label{5.30}
{\cal J}_{-N}^a\phi_h(z)=\oint_{e_z}\frac{d\zeta}{2\pi i}\frac{{\cal
J}^a(\zeta)\phi_h(z)}
{(\zeta-z)^N}\end{equation}
with
\begin{equation}\label{5.31}
{\cal J}^a=j^a_\psi+\alpha j^a_\chi+\beta\tilde j^a.\end{equation}
The coefficients $\alpha$ and $\beta$ are then fixed by
the requirement of
BRST invariance of the primary (\ref{5.28}).
We shall illustrate the procedure for the case $p-q=2,$ that is $N=1$.
We restrict ourselves to the isospin-zero sector. Invariance under
$SU(2)_L\times
SU(2)_R$ leads one to consider
\begin{eqnarray}\label{5.32}
\Phi^{(k)}_{p,p-2}&=&tr :({\cal J} g^{(0)}_j\bar{\cal J}(\tilde
g^{-1}_A)_{j-1}): \ +c.c.
\nonumber\\
&\equiv&:\epsilon_{i_1\ell_1}\delta_{i_2\ell_2}{\cal J}_{\ell_1\ell
_2}(g^{(0)}_j)^{i_1i_2i_3
\ldots i_{2j},k_1k_2k_3\ldots k_{2j}}\nonumber\\
&&\cdot\delta_{k_1 n_1}\epsilon_{k_2n_2}\bar J_{n_1n_2}
(\tilde g_A^{-1})_{j-1}
^{k_3\ldots k_{2j},i_3\ldots i_{2j}}:
\end{eqnarray}
where ${\cal J}={\cal J}^at^a$. We now require
\begin{equation}\label{5.33}
Q\Phi^{(k)}_{p,p-2}(w,\bar w)|\Omega\rangle=
\left[ Q,\Phi^{(k)}_{p,p-2}
(w,\bar w)\right]|\Omega\rangle=0\end{equation}
where $|\Omega\rangle$ denotes the ground state.
This requires that the
short-distance expansion of $J^a(z)\Phi^{(k)}
_{p,p-2}(w,\bar w)$ be regular. However,
because of the normal product appearing in (\ref{5.32}),
the computation of the
commutator is more easily done by expanding $Q$
in terms of modes. From
(\ref{5.30}) we have
\begin{equation}\label{5.34}
:{\cal J}^a(z)\phi_h(z):={\cal J}^a_{-1}\phi_h(z).\end{equation}
With the mode expansion
\begin{equation}\label{5.35}
\eta^a(z)=\sum^\infty_{n=-\infty}z^n\eta^a_{-n},
J^a(z)=\sum^\infty_{n=-\infty}
z^n {\cal J}^a_{-n}\end{equation}
we have from eq. (5.16), (5.25), (5.30) and (5.31),
\begin{equation}\label{5.36}
\left[ Q,\Phi^{(k)}_{p,p-2}\right]|\Omega\rangle=\sum_{n=0}\
\eta^a_{-n}
\left[J^a_n,{\cal J}^b_{-1}\Phi^b\right]|\Omega\rangle=0\end{equation}
where
\begin{equation}\label{5.37}
\Phi^b:=tr\left(t^b g_j^{(0)}\bar {\cal J}_{-1}(\tilde
g_A^{-1})_{j-1}\right)+c.c.\end{equation}
and use thas been made of $\eta^a_{-n}|\Omega\rangle=0$, for $n<0$.
Associativity,
as well as the commutation relations
\begin{equation}\label{5.38}
[j^a_n,j^b_m]=f_{abc}j^c_{n+m}+\frac{K}{2} n\delta^{ab}
\delta_{n,-m}\end{equation}
for the currents (5.25) with level $K=k,1,-(k+5)$ and $2C_V=4$,
respectively,
gives
\begin{equation}\label{5.39}
\left[J_n^a,{\cal J}^b_{-1}\Phi^b\right]=\left[ f_{abc}{\cal
J}^c_{n-1}+n\delta^{ab}
\delta_{n,1}(k+\alpha-\beta(k+5))\right]\Phi^b+
{\cal J}^b_{-1}\left[J^a_n,
\Phi^b\right].\end{equation}
Now, for $n\geq0$,
\begin{equation}\label{5.40}
\left[ J^a_n\Phi^b\right]|\Omega\rangle=\delta_{n0}
J_0^a\Phi^b|\Omega\rangle\end{equation}
The r.h.s. of (\ref{5.40}) is evaluated by recalling
the explicit form of
$\Phi^b$ as given by (\ref{5.37}). We have from
(\ref{5.25}) and (\ref{5.5})
\begin{eqnarray}\label{5.41}
\left(j^a_\psi\right)_0 g_j^{(0)}&=&-t_j^a g_j^{(0)}\nonumber\\
\tilde j^a_0(\tilde g_A)_{j-1}&=&-t^a_{j-1}(\tilde g_A)_{j-1}\end{eqnarray}
where $t^a_\ell$ are the $SU(2)$ generators in the spin-$\ell$
representation,
\begin{equation}\label{5.42}
t^a_\ell=\frac{1}{(2\ell)!}\sum\left[\underbrace{t^a\otimes
1\!{\rm l} \otimes\ldots
\otimes 1\!{\rm l} }_{2\ell}\right]_{\cal S}\end{equation}
with $t^a\equiv t^a_{1/2}$, and $ 1\!{\rm l} $ the $2\times 2$
identity matrix; the
sum in (\ref{5.42}) runs over all possible permutations
of $t^a$ with the
identity matrices.
Using (\ref{5.41}) one has in the notation of (\ref{5.32}),
\begin{equation}\label{5.43}
J_0^a\Phi^b=-tr\left( t^b t^a_j g^{(0)}_j\bar{\cal J}
(\tilde g_A)^{-1}_{j-1}\right)
+tr\left(t^b g^{(0)}_j\bar{\cal J}((\tilde g_A)^{-1}_{j-1}
t^a_{j-1})\right).\end{equation}
One finds, after some algebra
\begin{equation}\label{5.44}
tr\left(t^bt^a_jg^{(0)}_j\bar{\cal J}(\tilde g_A)
^{-1}_{j^{-1}}\right)=
f_{abc}\Phi^c+tr\left(t^bg_j^{(0)}\bar{\cal J}\left((\tilde
g_A)^{-1}_{j-1}t^a_{j-1}
\right)\right)\end{equation}
Substituting (\ref{5.44}) into (\ref{5.43}) the contribution
of the second term in (\ref{5.43}) cancels,
leaving one with the expected
result
\begin{equation}\label{5.45}
J^a_0\Phi^b=-f_{bac}\Phi^c\end{equation}
Making use of this result we see that the operator (\ref{5.39}) will
annihilate the groundstate for all $n$, except $n=1$. For $n=1$ we
are left to compute
$f_{abc}J^c_0\Phi^b|\Omega\rangle$.
Proceeding as in (\ref{5.43}) and using (\ref{5.44}),
one evidently has
\begin{equation}\label{5.46}
J^a_0\Phi^b=f_{abc}\Phi^c+(\beta-1)tr\left(t^bg_j^{(0)}
\bar{\cal J}((\tilde
g_A)^{-1}_{j-1}t^a_{j-1})\right)\end{equation}
Contracting (\ref{5.46}) with $f_{abc}$, then replacing
$f_{abc}t^b$ in the
second term by $[t^c,t^a]$, and making use of
\begin{equation}\label{5.47} \sum^3_{a=1}t^a_{ij}t^a_{k\ell}=-
\frac{1}{2}(\delta_{i\ell}
\delta_{jk}-\frac{1}{2}\delta_{ij}\delta_{k\ell})\end{equation}
one obtains the remarkable result
\begin{equation}\label{5.48}
f_{abc}{\cal J}^c_0\Phi^b=\left[-c_V+(\beta-1)(j-1)\right]\Phi^a\end{equation}
Combining everything one thus finds
\begin{equation}\label{5.49}
\left[Q,\Phi^{(k)}_{p,p-2}\right]|\Omega\rangle=
\left(-c_V+(\beta-1)(j-1)
+(k+\alpha-\beta(k+5))\right)
\eta_{-1}^a\Psi^a|\Omega\rangle\end{equation}
BRST invariance thus requires
\begin{equation}\label{5.50}
\alpha=\beta(j-k-6)+c_V+(j-1)-k\end{equation}
This sample construction can be generalized to other primary
fields $\Phi_{p,q}^{(k)}$ along the lines discussed
following eq. (\ref{5.26}).
For $j=1$ $(p=3)$, expression (\ref{5.32}) can be written in the
compact form \cite{KZ} (recall (\ref{5.34})),
\begin{equation}\label{5.51}
\Phi^{(k)}_{3,1}={\cal J}^a_{-1}\bar{\cal J}^{\bar
a}_{-1}tr\left(:(g^{(0)})^{-1}
t^ag^{(0)}t^{\bar a}:\right)\end{equation}
where (see eq. (\ref{5.4}))
\begin{equation}\label{5.52}
\left(g^{(0)}\right)^{ij}:=\left(g^{(0)}_{1/2}\right)^{ij}=
:e^{2\phi}
:\psi_2^{(0)i\alpha}\psi_2^{(0)\dagger j
\gamma}g_B^{\gamma\alpha}\end{equation}
and use has been made of
\begin{equation}\label{5.53}
g^{(0)}_{ij}\epsilon_{ik}=-\left(g^{(0)}\right)^{-1}_{kj}
\epsilon_{ji}\end{equation}
\section{Conclusion}
In this paper we have shown how to obtain a fermionic coset realization
of the primaries in the minimal unitary models. In particular we have
been able to obtain in this framework an operator realization of all
the primaries in the Ising model, corresponding to the energy, order
and disorder operator, as well as the Onsager fermions.
The fermionic coset description played here a crucial role in the
realization
of the order-disorder algebra. The four-point functions of these
primaries were explicitly computed and shown to coincide with those
obtained from the representation theory of the Virasoro algebra
\cite{BPZ}.
This gave support to our identifications on operator level. In section 5
we then generalized the construction of the primaries $\Phi^{(1)}_{2,2},
\Phi^{(1)}_{2,1}$ of the
Ising model, to the primaries $\Phi^{(k)}_{p,q}$ with arbitrary
$k$ and $q=p,p-1$ and $p-2$. We have also indicated how the
construction would proceed for general $p-q>2$. In the general case
the evaluation of four-point functions will require the
knowledge of the four-point correlators of $SU(k)_{-2(k+1)}$ WZW fields.
In section 3 we have obtained a reduction formula reducing this
problem to the computation of the four-point function of $SU(2)_k$ WZW
fields, which are known \cite{FZ}.
We expect our general construction to prove useful for obtaining
a realization of the order-disorder algebra of other critical
statistical models. It may also prove useful in the study of statistical
models away from criticality.
\medskip
\noindent {\bf Acknowledgement:} One of the authors (D.C.C.) would
like to thank the Commission of the European Community for the
``Marie Curie Fellowship'', which made this collaboration possible.
|
2,869,038,154,771 | arxiv | \section{Introduction}
\label{introduction}
High-mass stars are responsible for the dynamical and chemical evolution of the interstellar medium and of their host galaxies by injecting heavier elements and energy in their surrounding environment by means of their strong UV emission and winds. Despite their importance, the processes that lead to the formation of high-mass stars are still not well understood \citep{Zinnecker07}.
Observations at high-angular resolution have confirmed a high degree of multiplicity for high-mass stars, suggesting these objects are not formed in isolated systems \citep{Grellmann13}. The same scenario is supported by three-dimensional simulations of high-mass star formation \citep{Krumholz09,Rosen16}. These objects are formed on a relatively short timescale ($\sim$10$^5$\,yr), requiring large accretion rates \citep[$\sim$10$^{-4}$\,M$_\odot$\,yr$^{-1}$,][]{Hosokawa09}.
Such conditions can only be achieved in the densest clumps in molecular clouds, with sizes of $\lesssim$\,1\,pc and masses of order 100-1000\,M$_\odot$ \citep{Bergin07}. These clumps are associated with large visual extinctions, thus observations at long wavelengths are required to study their properties and the star formation process.
After molecular hydrogen (H$_2$), which is difficult to observe directly in dense cold gas, carbon monoxide (CO) is the most abundant molecular species.
Thus, rotational transitions of CO are commonly used to investigate the physics and kinematics of star-forming regions (SFRs).
Traditionally, observations of CO transitions with low angular momentum quantum number $J$ from $J$\,=\,1--0 to 4--3 (here defined as low-$J$ transitions) have been used for this purpose (e.g. see \citealt{Schulz95}, \citealt{Zhang01} and \citealt{Beuther02}).
These lines have upper level energies, $E_{\rm{u}}$, lower than 55\,K and are easily excited at relatively low temperatures and moderate densities.
Therefore, low-$J$ CO lines are not selective tracers of the densest regions of SFRs, but are contaminated by emission from the ambient molecular cloud.
On the other hand, higher-$J$ CO transitions are less contaminated by ambient gas emission and likely probe the warm gas directly associated with embedded young stellar objects (YSOs).
In this paper we make use of the $J$\,=\,6--5 and 7--6 lines of CO, with $E_{\rm{u}}$\,$\sim$\,116\,K and 155\,K, respectively, and in the following we refer to them simply as mid-$J$ CO transitions.
Over the past decade the Atacama Pathfinder Experiment telescope \citep[APEX\footnote{Based on observations with the APEX telescope under programme IDs M-087.F-0030-2011, M-093.F-0026-2014 and M-096.F-0005-2015. APEX is a collaboration between the Max-Planck-Institut f\"ur Radioastronomie, the European Southern Observatory, and the Onsala Space Observatory.},][]{Gusten06} has enabled routine observations of mid-$J$ CO lines, while {\it Herschel} and SOFIA have opened the possibility of spectroscopically resolved observations of even higher-$J$ transitions ($J$=10--9 and higher, e.g. \citealt{Gomez12}; \citealt[][hereafter, $\rm{SJG13}$\xspace]{SanJose13}; \citealt{Leurini15,Mottram17}).
\citet[][hereafter, $\rm{vK09}$\xspace]{vanKempen09} and \citet{vanKempen09b} have shown the importance of mid-$J$ CO transitions in tracing warm gas in the envelopes and outflows of low-mass protostars. More recently, $\rm{SJG13}$\xspace used the High Frequency Instrument for the Far Infrared \citep[HIFI,][]{deGraauw10} on board of Herschel to study a sample of low- and high-mass star-forming regions in high-$J$ transitions of several CO isotopologues (e.g. CO, $^{13}$CO and C$^{18}$O $J$\,=\,10--9), finding that the link between entrained outflowing gas and envelope motions is independent of the source mass.
In this paper, we present CO\,(6--5) and CO\,(7--6) maps towards a sample of 99 high-mass clumps selected from the APEX Telescope Large Area Survey of the Galaxy (ATLASGAL), which has provided an unbiased coverage of the plane of the inner Milky Way in the continuum emission at 870\,$\mu$m \citep{Schuller09}. Complementary single-pointing observations of the CO\,(4--3) line are also included in the analysis in order to characterise the CO emission towards the clumps.
Section\,\ref{sec_obs_sample} describes the sample and Sect.\,\ref{observations} presents the observations and data reduction. In Sect.\,\ref{sec_COprofile} we present the distribution and extent of the mid-$J$ CO lines and their line profiles, compute the CO line luminosities and the excitation temperature of the gas, and compare them with the clump properties. In Sect.\,\ref{sec_discussion} we discuss our results in the context of previous works.
Finally, the conclusions are summarised in Sect.\,\ref{sec_summary}.
\section{Sample}
\label{sec_obs_sample}
ATLASGAL detected the vast majority of all current and future high-mass star forming clumps ($M_{\rm{clump}}>1000 {\rm{M}}_\odot$) in the inner Galaxy. Recently, \citet{2018MNRAS.473.1059U} completed the distance assignment for $\sim$ 97 per cent of the ATLASGAL sources and analysed their masses, luminosities and temperatures based on thermal dust emission, and discussed how these properties evolve.
Despite the statistical relevance of the ATLASGAL sample, detailed spectroscopic observations are not feasible on the whole sample.
Therefore, we defined the ATLASGAL Top 100 \citep[hereafter, $\rm{TOP100}$\xspace,][]{Giannetti14,Koenig15}, a flux-limited sample of clumps selected from this survey with additional infrared (IR) selection criteria to ensure it encompasses a full range of luminosities and evolutionary stages (from 70\,$\mu$m-weak quiescent clumps to $\rm{H\,\textsc{ii}}$\xspace regions).
The 99 sources analysed in this paper are a sub-sample of the original $\rm{TOP100}$\xspace \citep{Koenig15} and are classified as follows:
\begin{itemize}
\item Clumps which either do not display any point-like emission in the Hi-GAL Survey \citep{Molinari10} 70\,$\mu$m images and (or) only show weak, diffuse emission at this wavelength (hereafter, $\rm{70w}$\xspace, 14 sources);
\item Mid-IR weak sources that are either not associated with any point-like counterparts or the associated compact emission is weaker than 2.6\,Jy in the MIPSGAL survey \citep{Carey09} 24\,$\mu$m images (hereafter $\rm{IRw}$\xspace, 31 sources);
\item Mid-IR bright sources in an active phase of the high-mass star formation process, with strong compact emission seen in 8\,$\mu$m and 24\,$\mu$m images, but still not associated with radio continuum emission (hereafter $\rm{IRb}$\xspace, 33 sources);
\item Sources in a later phase of the high-mass star formation process that are still deeply embedded in their envelope, but are bright in the mid-IR and associated with radio continuum emission ($\rm{H\,\textsc{ii}}$\xspace regions, 21 sources).
\end{itemize}
\citet{Koenig15} analysed the physical properties of the $\rm{TOP100}$\xspace sample in terms of distance, mass and luminosity. They found that at least 85\% of the sources have the ability to form high-mass stars and that most of them are likely gravitationally unstable and would collapse without the presence of a significant magnetic field.
These authors showed that the $\rm{TOP100}$\xspace represents a statistically significant sample of high-mass star-forming clumps covering a range of evolutionary phases, from the coldest and quiescent 70\,$\mu$m-weak to the most evolved clumps hosting $\rm{H\,\textsc{ii}}$\xspace regions, with no bias in terms of distance, luminosity and mass among the different classes.
The masses and bolometric luminosities of the clumps range from $\sim$20 to 5.2$\times$10$^5$\,M$_\odot$\xspace and from $\sim$60 to 3.6$\times$10$^6$\,L$_\odot$\xspace, respectively. The distance of the clumps ranges between 0.86 and 12.6\,kpc, and 72 of the 99 clumps have distances below 5\,kpc. This implies that observations of the $\rm{TOP100}$\xspace at the same angular resolution sample quite different linear scales.
In Appendix\,\ref{appendix_co_tables}, Table\,\ref{tbl_observations_short}, we list the main properties of the observed sources.
We adopted the Compact Source Catalogue (CSC) names from \citet{Contreras13} for the $\rm{TOP100}$\xspace sample although the centre of the maps may not exactly coincide with those positions (the average offset is $\sim$5\farcs4, with values ranging between $\sim$0\farcs5--25\farcs8, see Table\,\ref{tbl_observations_short}).
In this paper, we investigate the properties of mid-$J$ CO lines for a sub-sample of the original $\rm{TOP100}$\xspace as part of our effort to observationally establish a solid evolutionary sequence for high-mass star formation.
In addition to the dust continuum analysis of \citet{Koenig15}, we further characterised the $\rm{TOP100}$\xspace in terms of the content of the shocked gas in outflows traced by SiO emission \citep{Csengeri16} and the ionised gas content \citep{Kim17}, the CO depletion \citep{Giannetti14}, and the progressive heating of gas due to feedback of the central objects \citep{Giannetti17,Tang17}.
These studies confirm an evolution of the targeted properties with the original selection criteria and strengthen our initial idea that the $\rm{TOP100}$\xspace sample constitutes a valuable inventory of high-mass clumps in different evolutionary stages.
\section{Observations and data reduction}
\label{observations}
\subsection{$\rm{CHAMP}^+$\xspace observations}
\label{sec_obs_champ}
Observations of the $\rm{TOP100}$\xspace sample were performed with the APEX 12-m telescope on the following dates of 2014 May 17-20, July 10, 15-19, September 9-11 and 20. The $\rm{CHAMP}^+$\xspace \citep{Kasemann06, Gusten08} multi-beam heterodyne receiver was used to map the sources simultaneously in the CO\,(6--5) and CO\,(7--6) transitions. Information about the instrument setup configuration is given in Table\,\ref{table:champ_setup}.
\begin{table*}[ht!]
\setlength{\tabcolsep}{5pt}
\caption{\label{table:champ_setup}Summary of the observations.}
\centering
\begin{tabular}{cccccccccccc}
\hline\hline
Trans. & $E_{\rm{u}}$ & Freq. & Instr. & $\eta_{\rm{mb}}$ & Beam & $\Delta V$ & $T_{\rm{sys}}$\xspace & \multicolumn{2}{c}{rms (K)} & Observed \\
& (K) & (GHz) & & & size (\arcsec) & ($\rm{km\,s^{-1}}$\xspace)& (K) & median & range & sources \\
\hline
CO\,(4--3) & 55 & 461.04 & FLASH$^+$ & 0.60 & 13.4 & 0.953 & 1398\,$\pm$\,761 & 0.35 & 0.12--1.50 & 98 \\
CO\,(6--5) & 116 & 691.47 & $\rm{CHAMP}^+$\xspace & 0.41 & 9.6 & 0.318 & 1300\,$\pm$\,250 & 0.21 & 0.07--0.75 & 99 \\
CO\,(7--6) & 155 & 806.65 & $\rm{CHAMP}^+$\xspace & 0.34 & 8.2 & 0.273 & 5000\,$\pm$\,1500 & 0.91 & 0.29--2.10 & 99 \\
\hline
\end{tabular}
\tablefoot{
The columns are as follows:
(1) observed transition;
(2) upper-level energy of the transition;
(3) rest frequency;
(4) instrument;
(5) main-beam efficiency ($\eta_{\rm{mb}}$);
(6) beam size at the rest frequency;
(7) spectral resolution;
(8) mean systemic temperature of the observations;
(9)-(10) median and range of the rms of the data of the single-pointing CO\,(4--3) spectra and the spectra extracted at central position of the CO\,(6--5) and CO\,(7--6) maps at their original resolution;
(11) number of observed sources per transition (AGAL301.136$+$00.226 was not observed in CO\,(4--3)).}
\end{table*}
\setlength{\tabcolsep}{6pt}
The $\rm{CHAMP}^+$\xspace array has 2\,$\times$\,7 pixels that operate simultaneously in the radio frequency tuning ranges 620-720\,GHz in the low frequency array (LFA) and the other half in the range 780-950\,GHz in the high frequency array (HFA), respectively.
The half-power beam widths ($\theta_{\rm{mb}}$\xspace) are 9\farcs0 (at 691\,GHz) and 7\farcs7 (807\,GHz), and the beam-spacing is $\sim$2.15\,$\theta_{\rm{mb}}$\xspace for both sub-arrays.
The observations were performed in continuous on-the-fly (OTF) mode and maps of 80\arcsec\,$\times$\,80\arcsec size, centred on the coordinates given in Table\,\ref{tbl_observations_short}, were obtained for each source.
The area outside of the central 60\arcsec$\times$60\arcsec\xspace region of each map is covered by only one pixel of the instrument, resulting in a larger rms near the edges of the map.
The sky subtraction was performed by observing a blank sky field, offset from the central positions of the sources by {600\arcsec} in right ascension.
The average precipitable water vapour (PWV) of the observations varied from 0.28 to 0.68\,mm per day, having a median value of 0.50\,mm.
The average system temperatures ($T_{\rm{sys}}$\xspace) ranged from 1050 to 1550\,K and 3500 to 6500\,K, at 691 and 807\,GHz, respectively.
Pointing and focus were checked on planets at the beginning of each observing session. The pointing was also checked every hour on Saturn and Mars, and on hot cores (G10.47+0.03~B1, G34.26, G327.3$-$0.6, and NGC6334I) during the observations.
Each spectrum was rest-frequency corrected and baseline subtracted using the ''Continuum and Line Analysis Single Dish Software'' (\texttt{CLASS}), which is part of the \texttt{GILDAS} software\footnote{\url{http://www.iram.fr/IRAMFR/GILDAS}}.
The data were binned to a final spectral resolution of 2.0\,$\rm{km\,s^{-1}}$\xspace in order to improve the signal-to-noise ratio of the spectra.
The baseline subtraction was performed using a first-order fit to the line-free channels outside a window of $\pm$\,100\,$\rm{km\,s^{-1}}$\xspace wide, centred on the systemic velocity, V$_{\rm lsr}$, of each source. We used a broader window for sources exhibiting wings broader than $\sim$\,80\,$\rm{km\,s^{-1}}$\xspace (AGAL034.2572+00.1535, AGAL301.136$-$00.226, AGAL327.393+00.199, AGAL337.406$-$00.402, AGAL351.244+00.669 and AGAL351.774$-$00.537, see Table\,\ref{tbl_intpropCO_fixbeam}).
Antenna temperatures ($T_{\rm{A}}^*$\xspace) were converted to main-beam temperatures ($T_{\rm{mb}}$\xspace) using beam efficiencies of 0.41 at 691\,GHz and 0.34 at 809\,GHz\footnote{\url{www3.mpifr-bonn.mpg.de/div/submmtech/heterodyne/champplus/champ\_efficiencies.16-09-14.html}}.
Forward efficiencies are 0.95 in all observations.
The gridding routine \texttt{XY\_MAP} in \texttt{CLASS} was used to construct the final datacubes. This routine convolves the gridded data with a Gaussian of one third of the beam telescope size, yielding a final angular resolution slightly coarser (9\farcs6 for CO\,(6--5) and 8\farcs2 for CO\,(7--6)) than the original beam size (9\farcs0 and 7\farcs7, respectively). The final spectra at the central position of the maps have an average rms noise of 0.20 and 0.87\,K for CO\,(6--5) and CO\,(7--6) data, respectively.
Figure\,\ref{fig:calibration_sources} presents the ratio of the daily integrated flux to the corresponding average flux for the CO\,(6--5) transition of each hot core used as calibrator as a function of the observing day.
The deviation of the majority of the points with respect to their average value is consistent within a $\pm$20\% limit; thus, this value was adopted as the uncertainty on the integrated flux for both mid-$J$ CO transitions.
On September 10, the observations of G327.3$-$0.6 showed the largest deviation from the average flux of the source (points at $y$\,$\sim$\,0.7 and $y$\,$\sim$\,0.5 in Fig.\,\ref{fig:calibration_sources}). For this reason, we associate an uncertainty of 30\% on the integrated flux of the sources AGAL320.881$-$00.397, AGAL326.661+00.519 and AGAL327.119+00.509, and of 50\% for sources AGAL329.066$-$00.307 and AGAL342.484+00.182, observed immediately after these two scans on G327.3$-$0.6.
\begin{figure}
\centering
\includegraphics[width=0.9\linewidth]{all_CO65_flux_date_txt.ps}\\[-1.0ex]
\caption{Ratio of the daily integrated flux to the average flux of the CO\,(6--5) transition for the calibration sources observed during the campaign.
A solid horizontal line is placed at 1.0 and the dashed lines indicate a deviation of 20\% from unity.}
\label{fig:calibration_sources}
\end{figure}
\subsection{$\rm{FLASH}^+$\xspace observations}
\label{sec_obs_flash}
The $\rm{FLASH}^+$\xspace \citep{Klein14} heterodyne receiver on the APEX telescope was used to observe the central positions of the CHAMP$^+$ maps in CO\,(4--3) on 2011 June 15 and 24, August 11 and 12. Table\,\ref{table:champ_setup} summarises the observational setup. The observations were performed in position switching mode with an offset position of {600\arcsec} in right ascension for sky-subtraction. Pointing and focus were checked on planets at the beginning of each observing session. The pointing was also regularly checked during the observations on Saturn and on hot cores (G10.62, G34.26, G327.3$-$0.6, NGC6334I and SGRB2(N)).
The average PWV varied from 1.10 to 1.55\,mm per day with a median value of 1.29\,mm.
The system temperatures of the observations ranged from 650 to 2150\,K.
The single-pointing observations were processed using \texttt{GILDAS/CLASS} software.
The data were binned to a final spectral resolution of 2.0\,$\rm{km\,s^{-1}}$\xspace and a fitted line was subtracted to stablish a straight baseline.
The antenna temperatures were converted to $T_{\rm{mb}}$\xspace by assuming beam and forward efficiencies of 0.60 and 0.95, respectively.
The resulting CO\,(4--3) spectra have an average rms noise of 0.36\,K.
The uncertainty on the integrated flux of $\rm{FLASH}^+$\xspace data was estimated to be $\sim$20\% based on the continuum flux of the sources observed during the pointing scans.
\subsection{Spatial convolution of the mid-$J$ CO data}
\label{sec_CO_convolution}
The CO\,(6--5) and CO\,(7--6) data were convolved to a common angular resolution of 13\farcs4, matching the beam size of the single-pointing CO\,(4--3) observations. The resulting spectra are shown in Appendix\,\ref{appendix_co_fixbeam}.
The median rms of the convolved spectra are 0.35\,K, 0.17\,K and 0.87\,K for the CO\,(4--3), CO\,(6--5) and CO\,(7--6) transitions, respectively. These values differ from those reported in Table\,\ref{table:champ_setup} for the $\rm{CHAMP}^+$\xspace data where the rms at the original resolution of the dataset is given.
Since our sources are not homogeneously distributed in distance (see Sect.\,\ref{sec_obs_sample}), spectra convolved to the same angular resolution of 13\farcs4 sample linear scales between 0.06 and 0.84\,pc.
In order to study the effect of any bias introduced by sampling different linear scales within the clumps, the CO\,(6--5) and CO\,(7--6) data were also convolved to the same linear scale, $\ell_{\rm{lin}}$\xspace, of $\sim$0.24\,pc, which corresponds to an angular size $\theta$ (in radians) of:
\begin{equation}
\theta = \tan^{-1}\left( \frac{\ell_{\mathrm{lin}}}{d} \right)
\label{eq_beamsize}
\end{equation}
\noindent that depends on the distance of the source. The choice of $\ell_{\rm{lin}}$\xspace is driven by the nearest source, AGAL353.066+00.452, for which the part of the map with a relatively uniform rms (see Sect.\,\ref{sec_obs_champ}) corresponds to a linear scale of $\sim$0.24\,pc.
Since we are limited by the beam size of the CO\,(6--5) observations ($\sim$10\arcsec), the same projected length can be obtained only for sources located at distances up to $\sim$5.0\,kpc.
This limit defines a sub-sample of 72 clumps (ten $\rm{70w}$\xspace, 20 $\rm{IRw}$\xspace, 26 $\rm{IRb}$\xspace and 16 $\rm{H\,\textsc{ii}}$\xspace regions).
The rest of the paper focuses on the properties of the full $\rm{TOP100}$\xspace sample based on the spectra convolved to 13\farcs4.
The properties of the distance-limited sub-sample differ from those of the 13\farcs4 data only for the line profile (see Sect.\,\ref{sec_intpropCO}). A detailed comparison between the CO line luminosity and the properties of the clumps for the distance-limited sample is presented in Appendix\,\ref{appendix_distlim}.
\subsection{Self-absorption and multiple velocity components}
\label{sec_selfabs}
The CO spectra of several clumps show a double-peak profile close to the ambient velocity (e.g. AGAL12.804$-$00.199, AGAL14.632$-$00.577, and AGAL333.134$-$00.431, see Fig.\,\ref{fig_fixbeam_gaussian_fit}).
These complex profiles could arise from different velocity components in the beam or could be due to self-absorption given the likely high opacity of CO transitions close to the systemic velocity.
To distinguish between these two scenarios, the 13\farcs4 CO spectra obtained in Sect.\,\ref{sec_CO_convolution} were compared to the C$^{17}$O\,(3--2) data from \citet{Giannetti14} observed with a similar angular resolution (19\arcsec). In the absence of C$^{17}$O observations (AGAL305.192$-$00.006, AGAL305.209$+$00.206 and AGAL353.066$+$00.452), the C$^{18}$O\,(2--1) profiles were used.
Since the isotopologue line emission is usually optically thin \citep[cf.][]{Giannetti14}, it provides an accurate determination of the systemic velocity of the sources and, therefore, can be used to distinguish between the presence of multiple components or self-absorption in the optically thick $^{12}$CO lines.
Thus, when C$^{17}$O or C$^{18}$O show a single peak corresponding in velocity to a dip in CO, we consider the CO spectra to be affected by self-absorption.
Otherwise, if also the isotopologue data show a double-peak profile, the emission is likely due to two different velocity components within the beam.
From the comparison with the CO isotopologues, we found 83 clumps with self-absorption features in the CO\,(4--3) line, 79 in the CO\,(6--5), 70 in the CO\,(7--6) transition.
These numbers indicate that higher-$J$ CO transitions tend to be less affected by self-absorption features when compared to the lower-$J$ CO lines.
Finally, only 15 objects do not display self-absorption features in any transitions.
The CO spectra affected by self-absorption features are flagged with an asterisk symbol in Table\,\ref{tbl_intpropCO_fixbeam}.
To assess the impact of self-absorption on the analysis presented in Sect.\,\ref{sec_CO_correlations}, in particular on the properties derived from the integrated flux of the CO lines, we compared the observed integrated intensity of each CO transition with the corresponding values obtained from the Gaussian fit presented in Sect.\,\ref{sec_gaussfit}.
This comparison indicated that self-absorption changes the offsets and the scatter of the data but not the slopes of the relations between the CO emission and the clump properties.
Then, we investigated the ratio between the observed and the Gaussian integrated intensity values as a function of the evolutionary classes of the $\rm{TOP100}$\xspace sample. We found that 95\% of the sources exhibit ratios between 0.7 and 1.0 for all three lines.
We also note a marginal decrease on the ratios from the earliest $\rm{70w}$\xspace class ($\sim$1.0) to $\rm{H\,\textsc{ii}}$\xspace regions ($\sim$0.8), indicating that self-absorption does not significantly affect the results presented in the following sections.
We further investigated the effects of self-absorption by studying the sub-sample of $\sim$15 sources not affected by self-absorption (that is, the sources that are not flagged with an asterisk symbol in Table\,\ref{tbl_intpropCO_fixbeam}) and verified that the results presented in the following sections for the full sample are consistent with those of this sub-sample, although spanning a much broader range of clump masses and luminosities. More details on the analysis of the robustness of the relations reported in Sect.\,\ref{sec_CO_correlations} are provided in Appendices\,\ref{appendix_distlim} and \ref{appendix_gaussfit}.
Five sources (see Appendix\,\ref{appendix_secondary}) show a second spectral feature in the $^{12}$CO transitions and in the isotopologue data of \citet{Giannetti14} shifted in velocity from the rest velocity of the source.
We compared the spatial distribution of the integrated intensity CO\,(6--5) emission with the corresponding ATLASGAL 870\,$\mu\rm{m}$\xspace images (see Fig.\,\ref{fig_laboca_secondary_components}) for these five clumps. We found that in all sources the morphology of the integrated emission of one of the two peaks (labelled as P2 in Tables\,\ref{tbl_gauss_fitting_co43_fixbeam} to \ref{tbl_gauss_fitting_co76_fixbeam}) has a different spatial distribution than the dust emission at 870\,$\mu\rm{m}$\xspace and, thus, is likely not associated with the $\rm{TOP100}$\xspace clumps.
These components are excluded from any further analysis in this paper.
\subsection{Gaussian decomposition of the CO profiles}
\label{sec_gaussfit}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{fitting_examples_4_samescale.ps}\\[-2.0ex]
\caption{Gaussian decomposition for the CO\,(6--5) line using up to 3 components.
Panel A: single Gaussian fit;
Panel B: two-components fit;
Panel C: two-components fit (channels affected by self absorption are masked);
Panel D: three-components fit (channels affected by self absorption are masked).
The spectra, fits and residuals were multiplied by the factor shown in each panel.
The Gaussian fits are shown in blue, green and yellow, ordered by their line width; the sum of all components is shown in red. The grey lines indicate the baseline and the dashed horizontal lines placed in the residuals correspond to the 3-$\sigma$ level. Self-absorption features larger than 5-$\sigma$ were masked out from the residuals.
}
\label{fitting_examples}
\end{figure*}
The convolved CO spectra were fitted using multiple Gaussian components.
The fits were performed interactively using the {\sc minimize} task in \texttt{CLASS/GILDAS}. A maximum number of three Gaussian components per spectrum was adopted.
Each spectrum was initially fitted with one Gaussian component: if the residuals had sub-structures larger than 3-$\sigma$, a second or even a third component was added.
In case of self-absorption (see Sect.\,\ref{sec_selfabs}), the affected channels were masked before performing the fit.
Any residual as narrow as the final velocity resolution of the data (2.0\,$\rm{km\,s^{-1}}$\xspace) was ignored.
In particular for CO\,(4--3), absorption features shifted in velocity from the main line are detected in several sources. These features are likely due to emission in the reference position, and were also masked before fitting the data. Examples of the line profile decomposition are given in Fig.\,\ref{fitting_examples}.
\begin{table}
\centering
\setlength{\tabcolsep}{3pt}
\caption{\label{table_gauss_ncomp}Results of the Gaussian fit of the CO spectra convolved to a common angular resolution of 13\farcs4.}
\begin{tabular}{l|ccc}
\hline
\hline
Transition & One comp. & Two Comp. & Three comp. \\
\hline
CO\,(4--3) & 27 (7,8,3,4) & 68 (4,13,21,4) & 3 (0,0,1,1) \\
CO\,(6--5) & 12 (6,4,1,1) & 58 (8,18,20,7) & 29 (0,7,12,7) \\
CO\,(7--6) & 35 (14,14,6,1) & 53 (0,16,24,10) & 10 (0,1,3,6) \\
\hline
\end{tabular}
\tablefoot{The columns are as follows: (1) CO transition, (2)--(4) number of sources fitted using (2) a single Gaussian component, (3) two and (4) three components. In each column, the values in parenthesis indicate the corresponding number of $\rm{70w}$\xspace, $\rm{IRw}$\xspace, $\rm{IRb}$\xspace and $\rm{H\,\textsc{ii}}$\xspace regions.}
\end{table}
\setlength{\tabcolsep}{6pt}
Each component was classified as narrow (N) or broad (B), adopting the scheme from \citet{SanJose13}.
According to their definition, the narrow component has a full-width at half maximum (FWHM) narrower than 7.5\,km\,s$^{-1}$, otherwise it is classified as a broad component. Results of the Gaussian fit are presented in Tables\,\ref{tbl_gauss_fitting_co43_fixbeam}--\ref{tbl_gauss_fitting_co76_fixbeam}.
In several cases, two broad components are needed to fit the spectrum.
For the CO\,(6--5) data, 29 of the profiles required 3 components and, thus, two or three components have received the same classification. In these cases, they were named as, for example, B1, B2; ordered by their width. The P2 features mark secondary velocity components not associated with $\rm{TOP100}$\xspace clumps (see Sect.\,\ref{sec_selfabs}).
As a consequence of high opacity and self-absorption, the Gaussian decomposition of the line profile can be somewhat dubious.
In some cases, and in particular for the CO\,(4--3) transition, the fit is unreliable (e.g.
AGAL305.192$-$00.006 and AGAL333.134$-$00.431 in Fig.\,\ref{fig_fixbeam_gaussian_fit}).
The sources associated with unreliable Gaussian decomposed CO profiles (32, 8 and 4 for CO\,(4--3), CO\,(6--5) and CO\,(7--6), respectively) are not shown in Tables\,\ref{tbl_gauss_fitting_co43_fixbeam} to \ref{tbl_gauss_fitting_co76_fixbeam} and their data are not included in the analysis presented in Sect.\,\ref{sec_COprofile}, as well as in that of the integrated properties of their line profiles (e.g. their integrated intensities and corresponding line luminosities, see Sect.\,\ref{sec_CO_correlations}).
The general overview of the fits are given in Table\,\ref{table_gauss_ncomp} and the statistics of FWHM of the narrow and broad Gaussian components are listed in Table\,\ref{table:co_components}.
The spectrum of each source with its corresponding decomposition into Gaussian components is presented in Fig.\,\ref{fig_fixbeam_gaussian_fit}.
\begin{table}
\centering
\setlength{\tabcolsep}{4pt}
\caption{\label{table:co_components}Statistics of the FWHM of the Gaussian components fitted on the CO line profiles convolved to a common angular resolution of 13\farcs4.}
\begin{tabular}{clc|cccc}
\hline\hline
& & & \multicolumn{4}{c}{FWHM (km\,s$^{-1}$)} \\
Transition & Class. & $N$ & Range & Mean & Median & $\sigma$ \\
\hline
\multirow{2}{*}{CO\,(4--3)} & Narrow & 28 & 3.23-7.47 & 6.27 & 6.57 & 1.14 \\
& Broad & 83 & 7.5-86.0 & 21.2 & 14.9 & 16.5 \\
\hline
\multirow{2}{*}{CO\,(6--5)} & Narrow & 48 & 2.55-7.48 & 5.69 & 6.91 & 1.36 \\
& Broad & 148 & 7.5-97.1 & 24.5 & 17.8 & 19.0 \\
\hline
\multirow{2}{*}{CO\,(7--6)} & Narrow & 32 & 2.00-7.38 & 5.90 & 6.32 & 1.25 \\
& Broad & 133 & 7.5-120.2 & 27.8 & 16.4 & 26.6 \\
\hline
\end{tabular}
\tablefoot{The table presents the statistics on the FWHM of the narrow and broad Gaussian velocity component, classified according to Sect.\,\ref{sec_gaussfit}. The columns are as follows: (1) referred CO transition; (2) classification of the Gaussian component; (3) number of fitted components per class ($N$); (4) the minimum and maximum value per class; (5) the mean, (6) the median, and (7) the standard deviation of the distribution.}
\end{table}
\setlength{\tabcolsep}{6pt}
\section{Observational results}
\label{sec_COprofile}
The whole sample is detected above a 3-$\sigma$ threshold in the single-pointing CO\,(4--3) data (source AGAL301.136$-$00.226 was not observed with $\rm{FLASH}^+$\xspace) and in the 13\farcs4 {CO\,(6--5)} and {CO\,(7--6)} spectra, with three $\rm{70w}$\xspace sources (AGAL030.893+00.139, AGAL351.571+00.762 and AGAL353.417$-$00.079) only marginally detected above the 3-$\sigma$ limit in CO\,(7--6).
In the rest of this section we characterise the CO emission towards the $\rm{TOP100}$\xspace sample through the maps of CO\,(6--5) (Sect.\,\ref{sec_maps}) and the analysis of the CO line profiles for the spectra convolved to 13\farcs4 (Sect.\,\ref{sec_intpropCO}).
In Sect.\,\ref{sec_CO_correlations}, we compute the CO line luminosities and compare them with the clump properties.
Finally, in Sect.\,\ref{sec_texc} we compute the excitation temperature of the gas.
\subsection{Extent of the CO emission}
\label{sec_maps}
In Fig.\,\ref{fig_co_maps} we present examples of the integrated intensity maps of the CO\,(6--5) emission as a function of the evolutionary class of the $\rm{TOP100}$\xspace clumps.
The CO\,(6--5) maps of the full $\rm{TOP100}$\xspace sample are presented in Appendix\,\ref{appendix_co_fixbeam} (see Fig.\,\ref{fig_fixbeam_gaussian_fit}).
\begin{figure}[ht!]
\centering
\includegraphics[width=\linewidth]{class_CO_bar.ps} \\ [-2ex]
\caption{Distribution of the CO\,(6--5) emission of four representative clumps of each evolutionary classes of the $\rm{TOP100}$\xspace clumps.
The CO contours are presented on top of the {\it Herschel}/PACS maps at 70\,$\mu$m.
Each 70\,$\mu\rm{m}$\xspace map is scaled in according to the colour bar shown in the corresponding panel.
The CO contours correspond to the emission integrated over the full-width at zero power (FWZP) of the CO\,(6--5) line, and the contour levels are shown from 20\% to 90\% of the peak emission of each map, in steps of 10\%.
The position of the CSC source from \citet{Contreras13} is shown as a $\times$ symbol.
The beam size of the CO\,(6--5) observations is indicated in the left bottom region.
}
\label{fig_co_maps}
\end{figure}
We estimated the linear size of the CO emission, $\Delta s$, defined as the average between the maximum and minimum elongation of the half-power peak intensity (50\%) contour level of the CO\,(6--5) integrated intensity (see Table\,\ref{table_co_extension}). The uncertainty on $\Delta s$ was estimated as the dispersion between the major and minor axis of the CO extent.
The linear sizes of the CO emission ranges between 0.1 and 2.4\,pc, with a median value of 0.5\,pc.
In order to investigate if $\Delta s$ varies with evolution, we performed a non-parametric two-sided Kolmogorov-Smirnov (KS) test between pairs of classes (i.e. $\rm{70w}$\xspace vs. $\rm{IRb}$\xspace; $\rm{IRw}$\xspace vs. $\rm{H\,\textsc{ii}}$\xspace).
The sub-samples were considered statistically different if their KS rank factor is close to 1 and associated with a low probability value, $p$, ($p$\,$\leq$\,0.05 for a significance $\geq$\,2\,$\sigma$).
Our analysis indicates that there is no significant change in the extension of CO with evolution (KS\,$\leq$\,0.37, $p$\,$\geq$\,0.05 for all comparisons).
The CO extent was further compared with the bolometric luminosity, $L_{\rm{bol}}$\xspace, and the mass of the clumps, $M_{\rm{clump}}$\xspace, reported by \citet{Koenig15}. The results are presented in Fig.\,\ref{fig_extension_co}.
$\Delta s$ shows a large scatter as a function of $L_{\rm{bol}}$\xspace while it increases with $M_{\rm{clump}}$\xspace ($\rho$\,=\,0.72, $p$\,$<$\,0.001 for the correlation with $M_{\rm{clump}}$\xspace\, $\rho$\,=\,0.42, $p$\,$<$\,0.001 for $L_{\rm{bol}}$\xspace, where the $\rho$ is the Spearman rank correlation factor and $p$ its associated probability).
This confirms that the extent of the CO emission is likely dependent of the amount of gas within the clumps, but not on their bolometric luminosity.
We derived the extent of the 70\,$\mu\rm{m}$\xspace emission ($\Delta s_{\rm{70\,\mu m}}$) towards the 70\,$\mu\rm{m}$\xspace-bright clumps by cross-matching the position of the $\rm{TOP100}$\xspace clumps with the sources from \citet{Molinari16}. Then, $\Delta s_{\rm{70\,\mu m}}$ was obtained by computing the average between the maximum and minimum FWHM reported on their work and the corresponding error was obtained as the standard deviation of the FWHM values.
The values are also reported in Table\,\ref{table_co_extension}.
Figure\,\ref{fig_extension_co_pacs} compares the extent of the CO\,(6--5) emission with that of the 70\,$\mu\rm{m}$\xspace emission towards the 70\,$\mu\rm{m}$\xspace bright clumps. The extent of the emission of CO\,(6--5) and of the 70\,$\mu\rm{m}$\xspace continuum emission are correlated (Fig.\,\ref{fig_extension_co_pacs}, $\rho$\,=\,0.67, $p$\,$<$\,0.001), and in the majority of cases, the points are located above the equality line, suggesting that the gas probed by the CO\,(6--5) transition tends to be more extended than the dust emission probed by the PACS data towards the 70\,$\mu\rm{m}$\xspace-bright clumps.
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{extension_vs_lbol.ps} \\[-0.9ex]
\includegraphics[width=0.95\linewidth]{extension_vs_mclump.ps} \\[-1.5ex]
\caption{Size of the CO\,(6--5) emission towards the $\rm{TOP100}$\xspace sample versus the bolometric luminosity (top) and the mass (bottom) of the sources.
The median values for each class are shown as open diamonds and their error bars correspond to the absolute deviation of the data from their median value.
The typical uncertainty is shown by the error bars on the bottom right of each plot.}
\label{fig_extension_co}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{CO_ext_vs_IR_ext_molinari16.ps} \\[-1.5ex]
\caption{Size of the CO\,(6--5) emission versus the size of the 70\,$\mu\rm{m}$\xspace emission from the {\it Herschel}-PACS images towards the 70\,$\mu\rm{m}$\xspace-bright clumps. The dashed line indicates $y$\,=\,$x$.
The median values for the 70\,$\mu\rm{m}$\xspace-bright classes are shown as open diamonds and their error bars correspond to the absolute deviation of the data from their median value.
The typical uncertainty, computed as the dispersion between the major and minor axis of the emission, is shown by the error bars on the bottom right of each plot.
}
\label{fig_extension_co_pacs}
\end{figure}
\subsection{Line profiles}
\label{sec_intpropCO}
In the majority of the cases, the CO profiles are well fit with two Gaussian components, one for the envelope, one for high-velocity emission (see Table\,\ref{table_gauss_ncomp}). A third component is required in some cases, in particular for the CO\,(6--5) data, which have the highest signal-to-noise ratio.
The majority of sources fitted with a single Gaussian component are in the earliest stages of evolution ($\rm{70w}$\xspace and $\rm{IRw}$\xspace clumps), suggesting that the CO emission is less complex in earlier stages of high-mass star formation.
We also detect non-Gaussian high-velocity wings likely associated with outflows in most of the CO\,(6--5) profiles.
A detailed discussion of the outflow content in the $\rm{TOP100}$\xspace sample and of their properties will be presented in a forthcoming paper (Navarete et al., in prep.).
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.4\linewidth]{avg_spectrum_d70.ps}
\includegraphics[width=0.4\linewidth]{avg_spectrum_d70_norm.ps} \\[-1.5ex]
\caption{Left: Average CO\,(4--3), CO\,(6--5) and CO\,(7--6) spectra convolved to {13\farcs4} beam of each evolutionary class scaled to the median distance of the whole sample ($d$\,=\,3.80\,kpc). Right: Same plot, but the average CO spectra are normalised by their peak intensity.
The baseline level is indicated by the solid grey line and the black dashed line is placed at 0\,km\,s$^{-1}$.
The FWZP of the profiles are shown in the upper right side of the panels (in $\rm{km\,s^{-1}}$\xspace units), together with the integrated intensities ($S_{\rm{int}}$\xspace, in K\,$\rm{km\,s^{-1}}$\xspace units) of the CO profiles shown in the left panels.
}
\label{fig_avgspc_fixbeam}
\end{figure*}
To minimise biases due to different sensitivities in the analysis of single spectra, we computed the average CO spectrum of each evolutionary class and normalised it by its peak intensity.
The spectra were shifted to 0\,$\rm{km\,s^{-1}}$\xspace using the correspondent $V_{\rm{lsr}}$\xspace given in Table\,\ref{tbl_observations_short}.
Then, the averaging was performed by scaling the intensity of each spectrum to the median distance of the sub-sample ($d$\,=\,3.26\,kpc for the distance-limited sample, $d$\,=\,3.80\,kpc for the full sample).
The resulting spectra of the 13\farcs4 dataset are shown in Fig.\,\ref{fig_avgspc_fixbeam} while those of the distance-limited sub-sample are presented in Appendix\,\ref{fig_avgspc_fixscale} (Fig.\,\ref{fig_avgspc_fixscale}).
While the 13\farcs4 data show no significant difference between the average profiles of $\rm{IRw}$\xspace and $\rm{IRb}$\xspace classes, in the distance-limited sub-sample the width (expressed through the full width at zero power, FWZP, to avoid any assumption on the profile) and the intensity of the CO lines progressively increase with the evolution of the sources (from $\rm{70w}$\xspace clumps towards $\rm{H\,\textsc{ii}}$\xspace regions) especially when the normalised profiles are considered.
The difference between the two datasets is due to sources at large distances (d\,>\,12\,kpc; AGAL018.606$-$00.074, AGAL018.734$-$00.226 and AGAL342.484+00.182) for which the observations sample a much larger volume of gas. The increase of line width with evolution is confirmed by the analysis of the individual FWZP values of the three CO lines, presented in Table\,\ref{tbl_intpropCO_fixbeam} (see Table\,\ref{table_limits_intpropCO} for the statistics on the full $\rm{TOP100}$\xspace sample).
\begin{table}
\centering
\caption{\label{table_limits_intpropCO}Statistics on the CO line profiles, convolved to a common angular resolution of 13\farcs4.}
\begin{tabular}{l|cccc}
\hline
\hline
& \multicolumn{4}{c}{FWZP ($\rm{km\,s^{-1}}$\xspace)} \\
& & & & Standard \\
Transition & Range & Mean & Median & deviation \\
\hline
CO\,(4--3) & 10-134 & 47 & 42 & 25 \\
CO\,(6--5) & 14-162 & 62 & 54 & 34 \\
CO\,(7--6) & 4-142 & 39 & 30 & 27 \\
\hline
\end{tabular}
\tablefoot{The table presents the FWZP of the CO lines.
The mid-$J$ CO lines were convolved to a common angular resolution of 13\farcs4. The columns are as follows: (1) referred CO line; (2) the minimum and maximum values, (3) the mean, (4) the median, and (5) the standard deviation of the distribution.}
\end{table}
Despite the possible biases in the analysis of the line profiles (e.g. different sensitivities, different excitation conditions, complexity of the profiles), our data indicate that the CO emission is brighter in late evolutionary phases.
The average spectra per class show also that the CO lines becomes broader towards more evolved phases likely due to the presence of outflows.
Our study extends the work of \citet{Leurini13} on one source of our sample, AGAL327.293$-$00.579. They mapped in CO\,(3--2), CO\,(6--5), CO\,(7--6) and in $^{13}$CO\,(6--5), $^{13}$CO\,(8--7) and $^{13}$CO\,(10--9) a larger area of the source than that presented here and found that, for all transitions, the spectra are dominated in intensity by the $\rm{H\,\textsc{ii}}$\xspace region rather than by younger sources (a hot core and an infrared dark cloud are also present in the area). They interpreted this result as an evidence that the bulk of the Galactic CO line emission comes from PDRs around massive stars, as suggested by \citet{Cubick08} for FIR line emission. Based on this, we suggest that the increase in mid-$J$ CO brightness in the later stages of the $\rm{TOP100}$\xspace is due to a major contribution of PDR to the line emission. We notice however that the increase of width and of intensity of the CO lines with evolution can also be due to an increase with time of multiplicity of sources in the beam.
\subsection{The CO line luminosities}
\label{sec_CO_correlations}
The intensity of the CO profiles ($S_{\rm{int}}$\xspace, in K\,km\,s$^{-1}$) was computed by integrating the CO emission over the velocity channels within the corresponding FWZP range.
Then, the line luminosity ($L_{\rm{CO}}$\xspace, in K\,km\,s$^{-1}$\,pc$^2$) of each CO line was calculated using Eq.\,2 from \citet{Wu05}, assuming a source of size equal to the beam size of the data (see Sect.\,\ref{sec_CO_convolution}).
The derived $L_{\rm{CO}}$\xspace values are reported in Table\,\ref{tbl_intpropCO_fixbeam}. The errors in the $L_{\rm{CO}}$\xspace values are estimated by error propagation on the integrated flux (see Sect.\,\ref{sec_obs_champ}) and considering an uncertainty of 20\% in the distance.
The median values of $L_{\rm{CO}}$\xspace, $L_{\rm{bol}}$\xspace, $M_{\rm{clump}}$\xspace and $L/M$\xspace, the luminosity-to-mass ratio, per evolutionary class are summarised in Table\,\ref{table_median_class}.
We also performed the same analysis on the data convolved to a common linear scale of 0.24\,pc (assuming the corresponding angular source size of 0.24\,pc to derive the line luminosity) and no significant differences in the slope of the trends were found. Therefore, the distance-limited sample will not be discussed any further in this section.
\begin{table*}[!]
\caption{\label{table_median_class}Median values per class of the clump and CO profile properties.}
\centering
\setlength{\tabcolsep}{4pt}
\begin{tabular}{l|cccc}
\hline\hline
Property & $\rm{70w}$\xspace & $\rm{IRw}$\xspace & $\rm{IRb}$\xspace & $\rm{H\,\textsc{ii}}$\xspace \\
\hline
$L_{\rm{bol}}$\xspace ($10^3$\,L$_\odot$) & 1.26$\pm$0.83 & 9.6$\pm$8.4 & 16.5$\pm$1.4 & 21.4$\pm$1.5 \\
$M_{\rm{clump}}$\xspace ($10^3$\,M$_\odot$) & 1.22$\pm$0.70 & 1.4$\pm$1.1 & 0.49$\pm$0.31 & 1.9$\pm$1.1 \\
$L/M$\xspace (L$_\odot$/M$_\odot$) & 2.58$\pm$0.93 & 9.0$\pm$6.8 & 40$\pm$23 & 76$\pm$28 \\
\hline
$L_{\rm{CO\,(4-3)}}$ (K\,km\,s$^{-1}$\,pc$^2$) & 9.8$\pm$8.5 & 30$\pm$23 & 21$\pm$15 & 119$\pm$58 \\
$L_{\rm{CO\,(6-5)}}$ & 5.1$\pm$4.0 & 16$\pm$12 & 19$\pm$12 & 51$\pm$44 \\
$L_{\rm{CO\,(7-6)}}$ & 4.7$\pm$3.6 & 11.8$\pm$8.6 & 14.8$\pm$9.7 & 48$\pm$45 \\
\hline
FWZP$_{\rm{CO\,(4-3)}}$ (km\,s$^{-1}$) & 24.0$\pm$6.0 & 34$\pm$12 & 52$\pm$16 & 62$\pm$18 \\
FWZP$_{\rm{CO\,(6-5)}}$ & 26.0$\pm$6.0 & 42$\pm$14 & 72$\pm$22 & 102$\pm$28 \\
FWZP$_{\rm{CO\,(7-6)}}$ & 12.0$\pm$2.0 & 24.0$\pm$6.0 & 38$\pm$14 & 66$\pm$20 \\
\hline
$T_{\rm{ex}}$\xspace(K) & 22.4$\pm$5.0 & 29.9$\pm$8.1 & 45$\pm$15 & 95$\pm$21 \\
\hline
\end{tabular}
\tablefoot{The median and the absolute deviation of the data from their median value are shown for the clump properties (bolometric luminosity, mass and luminosity-to-mass ratio), for the line luminosity and full width at zero power of the low-$J$ and mid-$J$ CO profiles convolved to the same angular size of 13\farcs4, and for the excitation temperature of the CO\,(6--5) emission.}
\end{table*}
\setlength{\tabcolsep}{6pt}
In Fig.\,\ref{fig_lco_correlation_fixbeam_lbol} we show the cumulative distribution function (CDF) of the line luminosities for the three CO transitions: $L_{\rm{CO}}$\xspace increases from $\rm{70w}$\xspace sources towards $\rm{H\,\textsc{ii}}$\xspace regions. Each evolutionary class was tested against the others by computing their two-sided KS coefficient (see Table\,\ref{table_lco_classes}).
The most significant differences are found when comparing the earlier and later evolutionary classes ($\rm{70w}$\xspace and $\rm{IRb}$\xspace, $\rm{70w}$\xspace and $\rm{H\,\textsc{ii}}$\xspace, $\rho$\,$\geq$\,0.66 for the CO\,(6--5) line), while no strong differences are found among the other classes (KS\,$\leq$\,0.5 and $p$\,$\geq$0.003 for the CO\,(6--5) transition).
These results indicate that, although we observe an increase on the CO line luminosity from $\rm{70w}$\xspace clumps towards $\rm{H\,\textsc{ii}}$\xspace regions, no clear separation is found in the intermediate classes ($\rm{IRw}$\xspace and $\rm{IRb}$\xspace, see also Table\,\ref{table_median_class}).
\begin{table}
\centering
\setlength{\tabcolsep}{3pt}
\caption{\label{table_lco_classes}Kolmogorov-Smirnov statistics of the CO line luminosity as a function of the evolutionary class of the clumps.}
\begin{tabular}{c|lll}
\hline
\hline
Classes & \multicolumn{1}{c}{CO\,(4--3)} & \multicolumn{1}{c}{CO\,(6--5)} & \multicolumn{1}{c}{CO\,(7--6)} \\
\hline
$\rm{70w}$\xspace-$\rm{IRw}$\xspace & 0.48, $p$\,=\,0.05 & 0.45, $p$\,=\,0.03 & 0.46, $p$\,=\,0.02 \\
$\rm{70w}$\xspace-$\rm{IRb}$\xspace & 0.35, $p$\,=\,0.25 & 0.66, $p$\,$<$\,0.001 & 0.66, $p$\,$<$\,0.001 \\
$\rm{70w}$\xspace-$\rm{H\,\textsc{ii}}$\xspace & 0.72, $p$\,=\,0.004 & 0.80, $p$\,$<$\,0.001 & 0.82, $p$\,$<$\,0.001 \\
$\rm{IRw}$\xspace-$\rm{IRb}$\xspace & 0.29, $p$\,=\,0.23 & 0.21, $p$\,=\,0.46 & 0.24, $p$\,=\,0.26 \\
$\rm{IRw}$\xspace-$\rm{H\,\textsc{ii}}$\xspace & 0.41, $p$\,=\,0.15 & 0.46, $p$\,=\,0.02 & 0.47, $p$\,=\,0.01 \\
$\rm{IRb}$\xspace-$\rm{H\,\textsc{ii}}$\xspace & 0.62, $p$\,=\,0.004 & 0.53, $p$\,=\,0.003 & 0.52, $p$\,=\,0.003 \\
\hline
\end{tabular}
\tablefoot{The rank KS and its corresponding probability ($p$) are shown for each comparison. A $p$-value of $<$\,0.001 indicate a correlation at 0.001 significance level. $p$-values of 0.05, 0.002 and $<$\,0.001 represent the $\sim$\,2, 3 and $>$\,3\,$\sigma$ confidence levels.}
\end{table}
\setlength{\tabcolsep}{6pt}
\begin{table}[h!]
\caption{\label{table_lco_fit}Parameters of the fits of $L_{\rm{CO}}$\xspace as a function of the clump properties.}
\centering
\begin{tabular}{cl|ccc}
\hline\hline
Transition & Property & $\alpha$ & $\beta$ & $\epsilon$ \\
\hline
& $L_{\rm{bol}}$\xspace & $-$0.86$^{+0.24}_{-0.22}$ & 0.55$\pm$0.05 & 0.41 \\
CO\,(4--3) & $M_{\rm{clump}}$\xspace & $-$1.37$^{+0.23}_{-0.19}$ & 0.92$\pm$0.06 & 0.34 \\
& $L/M$\xspace& $+$1.08$^{+0.12}_{-0.14}$ & 0.28$\pm$0.12 & 0.63 \\
\hline
& $L_{\rm{bol}}$\xspace & $-$1.33$^{+0.14}_{-0.13}$ & 0.63$\pm$0.03 & 0.25 \\
CO\,(6--5) & $M_{\rm{clump}}$\xspace & $-$1.58$^{+0.23}_{-0.22}$ & 0.92$\pm$0.07 & 0.37 \\
& $L/M$\xspace& $+$0.74$^{+0.09}_{-0.08}$ & 0.46$\pm$0.09 & 0.55 \\
\hline
& $L_{\rm{bol}}$\xspace & $-$1.64$^{+0.12}_{-0.11}$ & 0.68$\pm$0.03 & 0.22 \\
CO\,(7--6) & $M_{\rm{clump}}$\xspace & $-$1.64$^{+0.22}_{-0.24}$ & 0.92$\pm$0.08 & 0.43 \\
& $L/M$\xspace& $+$0.55$^{+0.10}_{-0.08}$ & 0.55$\pm$0.10 & 0.54 \\
\hline
\end{tabular}
\tablefoot{The fits were performed by adjusting a model with three free parameters in the form of $\log(y) = \alpha + \beta \log(x) \pm \epsilon$, where $\alpha$, $\beta$ and $\epsilon$ correspond to the intercept, the slope and the intrinsic scatter, respectively.}
\end{table}
We also plot $L_{\rm{CO}}$\xspace against the bolometric luminosity of the clumps (Fig.\,\ref{fig_lco_correlation_fixbeam_lbol}), their mass and their luminosity-to-mass ratio (Figs.\,\ref{fig_lco_correlation_fixbeam_mclump} and \ref{fig_lco_correlation_fixbeam_lmratio} for the CO\,(6--5) line). The $L/M$\xspace ratio is believed to be a rough estimator of evolution in the star formation process for both low- \citep{Saraceno96} and high-mass regimes \citep[e.g.][]{Molinari08}, with small $L/M$\xspace values corresponding to embedded regions where (proto-)stellar activity is just starting, and high $L/M$\xspace values in sources with stronger radiative flux and that have accreted most of the mass \citep{Molinari16,Giannetti17,2018MNRAS.473.1059U}. In addition, the $L/M$\xspace ratio also reflects the properties of the most massive young stellar object embedded in the clump \citep{Faundez04,Urquhart13}.
The fits were performed using a Bayesian approach, by adjusting a model with three free parameters (the intercept, $\alpha$, the slope, $\beta$, and the intrinsic scatter, $\epsilon$). In order to obtain a statistically reliable solution, we computed a total of 100\,000 iterations per fit. The parameters of the fits are summarised in Table\,\ref{table_lco_fit}.
The correlation between $L_{\rm{CO}}$\xspace and the clump properties was checked by computing their Spearman rank correlation factor and its associated probability ($\rho$ and $p$, respectively, see Table\,\ref{table_co_correlation}).
Since $L_{\rm{CO}}$\xspace with $L_{\rm{bol}}$\xspace and $M_{\rm{clump}}$\xspace have the same dependence on the distance of the source, a partial Spearman correlation test was computed and the partial coefficient, $\rho_{\rm p}$, was obtained (see Table\,\ref{table_co_correlation}).
\begin{table}[h!]
\caption{\label{table_co_correlation}Spearman rank correlation statistics for the CO line luminosity as a function of the clump properties towards the $\rm{TOP100}$\xspace sample.}
\centering
\setlength{\tabcolsep}{4pt}
\begin{tabular}{l|ccc}
\hline\hline
Property & CO\,(4--3) & CO\,(6--5) & CO\,(7--6) \\
\hline
\multirow{2}{*}{$L_{\rm{bol}}$\xspace} & 0.70, $p$\,$<$\,0.001; & 0.85, $p$\,$<$\,0.001; & 0.89, $p$\,$<$\,0.001; \\
& $\rho_p$\,=\,0.81 & $\rho_p$\,=\,0.91 & $\rho_p$\,=\,0.92 \\
\multirow{2}{*}{$M_{\rm{clump}}$\xspace} & 0.75, $p$\,$<$\,0.001; & 0.70, $p$\,$<$\,0.001; & 0.67, $p$\,$<$\,0.001; \\
& $\rho_p$\,=\,0.48 & $\rho_p$\,=\,0.55 & $\rho_p$\,=\,0.57 \\
$L/M$\xspace & 0.24, $p$\,=\,0.05 & 0.45, $p$\,$<$\,0.001 & 0.50, $p$\,$<$\,0.001 \\
\hline
\end{tabular}
\tablefoot{The rank $\rho$ and its corresponding probability ($p$) are shown for each comparison. A $p$-value of $<$\,0.001 indicate a correlation at 0.001 significance level. $p$-values of 0.05, 0.002 and $<$\,0.001 represent the $\sim$\,2, 3 and $>$\,3\,$\sigma$ confidence levels. For $L_{\rm{bol}}$\xspace and $M_{\rm{clump}}$\xspace, the partial correlation coefficient, $\rho_p$, is also shown.}
\end{table}
\setlength{\tabcolsep}{6pt}
In the right panel of Fig.\,\ref{fig_lco_correlation_fixbeam_lbol}, we show the CO line luminosity versus the bolometric luminosity of the $\rm{TOP100}$\xspace clumps. The plot indicates that $L_{\rm{CO}}$\xspace increases with $L_{\rm{bol}}$\xspace over the entire $L_{\rm{bol}}$\xspace range covered by the $\rm{TOP100}$\xspace clumps ($\sim$10$^2$-10$^6$\,L$_\odot$\xspace).
The Spearman rank test confirms that both quantities are well correlated for all CO lines ($\rho$\,$\geq$\,0.7, with $p$\,$<$\,0.001), even when excluding the mutual dependence on distance ($\rho_p$\,$\geq$\,0.81).
The results of the fits indicate a systematic increase in the slope of $L_{\rm{CO}}$\xspace versus $L_{\rm{bol}}$\xspace for higher-$J$ transitions: 0.55$\pm$0.05, 0.63$\pm$0.03 and 0.68$\pm$0.03 for the CO\,(4--3), CO\,(6--5) and CO\,(7--6), respectively. For the CO\,(6--5) and CO\,(7--6) lines, however, the slopes are consistent in within 2-$\sigma$.
Concerning the dependence of the CO luminosity on $M_{\rm{clump}}$\xspace (see Fig.\,\ref{fig_lco_correlation_fixbeam_mclump}), the partial correlation tests indicates that the distance of the clumps plays a more substantial role in the correlation found between $L_{\rm{CO}}$\xspace and $M_{\rm{clump}}$\xspace (0.48$\leq$\,$\rho_p$\,$\leq$\,0.57) than in the correlations found for $L_{\rm{CO}}$\xspace vs. $L_{\rm{bol}}$\xspace.
Finally, we do not find any strong correlation between the CO line luminosity and $L/M$\xspace ($\rho$\,$\leq$\,0.5 for all transitions) although the median $L_{\rm{CO}}$\xspace values per class do increase with $L/M$\xspace (Fig.\,\ref{fig_lco_correlation_fixbeam_lmratio}).
These findings are discussed in more detail in Sect.\,\ref{sec_discussion}.
\begin{figure*}[!]
\centering
\includegraphics[width=0.485\linewidth]{lco_cdf_all.ps}
\includegraphics[width=0.485\linewidth]{lco_lbol_highlighted.ps} \\[-1.5ex]
\caption{Left panels: Cumulative distribution function of the line luminosity of the CO\,(4--3) (upper panel), CO\,(6--5) (middle) and CO\,(7--6) emission (bottom) towards the $\rm{TOP100}$\xspace sample. The median values per class are shown as vertical dashed lines in their corresponding colours. Right panels: Line luminosity of the same CO $J$-transitions versus the bolometric luminosity of the $\rm{TOP100}$\xspace sources.
The median values for each class are shown as open diamonds and their error bars correspond to the absolute deviation of the data from their median value.
Data points highlighted in yellow indicate those sources from which no signs of self-absorption features where identified in the spectrum convolved to 13\farcs4.
The typical error bars are shown at the bottom right side of the plots. The black solid line is the best fit, the light grey shaded area indicates the 68\% uncertainty, and the dashed lines show the intrinsic scatter ($\epsilon$) of the relation.}
\label{fig_lco_correlation_fixbeam_lbol}
\end{figure*}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{lco_mclump_co65_highlighted.ps} \\[-1.5ex]
\caption{Same as the right panel of Fig.\,\ref{fig_lco_correlation_fixbeam_lbol}, but displaying the line luminosity of the CO\,(6--5) emission as a function of the mass of the clumps. Data points highlighted in yellow indicate those sources from which no signs of self-absorption features where identified in the spectrum convolved to 13\farcs4.}
\label{fig_lco_correlation_fixbeam_mclump}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth]{lco_lmratio_co65_highlighted.ps} \\[-1.5ex]
\caption{Same as the right panel of Fig.\,\ref{fig_lco_correlation_fixbeam_lbol}, but displaying $L_{\rm{CO}}$\xspace of the CO\,(6--5) line as a function of the $L/M$\xspace ratio of the $\rm{TOP100}$\xspace sources. Data points highlighted in yellow indicate those sources from which no signs of self-absorption features where identified in the spectrum convolved to 13\farcs4.
}
\label{fig_lco_correlation_fixbeam_lmratio}
\end{figure}
We further tested whether the steepness of the relations between $L_{\rm{CO}}$\xspace and the clump properties is not affected by self-absorption by selecting only those clumps which do not show clear signs of self-absorption (see Sect.\,\ref{sec_gaussfit}). This defines a sub-sample of 15 sources in the CO\,(4--3) line, 18 in the CO\,(6--5) line and 26 objects in the CO\,(7--6) transition. These sources are highlighted in Figs.\,\ref{fig_lco_correlation_fixbeam_lbol}, \ref{fig_lco_correlation_fixbeam_mclump} and \ref{fig_lco_correlation_fixbeam_lmratio}.
Then, we repeated the fit of the relations between $L_{\rm{CO}}$\xspace and the clump properties using these sub-samples, finding no significant differences in the slopes of the relations presented in Table\,\ref{table_lco_fit}.
The result of the fits for the sub-sample of sources with no signs of self-absorption in their 13\farcs4 spectra are summarised in Table\,\ref{table_lco_fit_nosabs} and the correlations between $L_{\rm{CO}}$\xspace and the clump properties are listed in Table\,\ref{table_co_correlation_nosabs}.
The correlations are systematically weaker due to the smaller number of points than those obtained for the whole $\rm{TOP100}$\xspace sample (see Table\,\ref{table_co_correlation}).
We found that the derived slopes for the relations between $L_{\rm{CO}}$\xspace and $L_{\rm{bol}}$\xspace increases from 0.58$\pm$0.09 to 0.71$\pm$0.07, from the CO\,(4--3) to the CO\,(7--6) transition.
Despite the larger errors in these relations, the slopes of the fits performed on these sources are not significantly different from those found for the whole $\rm{TOP100}$\xspace sample, confirming that at least the slopes of the relations found for the whole $\rm{TOP100}$\xspace sample are robust in terms of self-absorption effects.
In addition, similar results were also found for the relations between $L_{\rm{CO}}$\xspace and the mass of the clumps, while no strong correlation between $L_{\rm{CO}}$\xspace and $L/M$\xspace was found for this sub-sample.
\begin{table}[h!]
\caption{\label{table_lco_fit_nosabs}Parameters of the fits of $L_{\rm{CO}}$\xspace as a function of the clump properties for the $\rm{TOP100}$\xspace clumps that are not affected by self-absorption features.}
\centering
\begin{tabular}{cl|ccc}
\hline\hline
Transition & Property & $\alpha$ & $\beta$ & $\epsilon$ \\
\hline
& $L_{\rm{bol}}$\xspace & $-$0.95$^{+0.39}_{-0.39}$ & 0.58$\pm$0.09 & 0.56 \\
CO\,(4--3) & $M_{\rm{clump}}$\xspace & $-$1.60$^{+0.28}_{-0.27}$ & 1.00$\pm$0.09 & 0.31 \\
& $L/M$\xspace& $+$1.19$^{+0.23}_{-0.29}$ & 0.19$\pm$0.24 & 0.86 \\
\hline
& $L_{\rm{bol}}$\xspace & $-$1.54$^{+0.30}_{-0.29}$ & 0.68$\pm$0.08 & 0.43 \\
CO\,(6--5) & $M_{\rm{clump}}$\xspace & $-$1.92$^{+0.30}_{-0.28}$ & 1.05$\pm$0.09 & 0.38 \\
& $L/M$\xspace& $+$0.76$^{+0.24}_{-0.23}$ & 0.43$\pm$0.21 & 0.83 \\
\hline
& $L_{\rm{bol}}$\xspace & $-$1.71$^{+0.28}_{-0.29}$ & 0.71$\pm$0.07 & 0.33 \\
CO\,(7--6) & $M_{\rm{clump}}$\xspace & $-$1.95$^{+0.34}_{-0.32}$ & 1.02$\pm$0.11 & 0.41 \\
& $L/M$\xspace& $+$0.57$^{+0.22}_{-0.23}$ & 0.53$\pm$0.20 & 0.79 \\
\hline
\end{tabular}
\tablefoot{The fits were performed by adjusting a model with three free parameters in the form of $\log(y) = \alpha + \beta \log(x) \pm \epsilon$, where $\alpha$, $\beta$ and $\epsilon$ correspond to the intercept, the slope and the intrinsic scatter, respectively.}
\end{table}
\begin{table}[h!]
\caption{\label{table_co_correlation_nosabs}Spearman rank correlation statistics for the CO line luminosity as a function of the clump properties for the $\rm{TOP100}$\xspace clumps that are not affected by self-absorption features.}
\centering
\setlength{\tabcolsep}{4pt}
\begin{tabular}{l|ccc}
\hline\hline
Property & CO\,(4--3) & CO\,(6--5) & CO\,(7--6) \\
\hline
\multirow{2}{*}{$L_{\rm{bol}}$\xspace} & 0.56, $p$\,=\,0.03; & 0.81, $p$\,$<$\,0.001; & 0.83, $p$\,$<$\,0.001; \\
& $\rho_p$\,=\,0.54 & $\rho_p$\,=\,0.83 & $\rho_p$\,=\,0.89 \\
\multirow{2}{*}{$M_{\rm{clump}}$\xspace} & 0.72, $p$\,=\,0.002; & 0.73, $p$\,$<$\,0.001; & 0.79, $p$\,$<$\,0.001; \\
& $\rho_p$\,=\,0.24 & $\rho_p$\,=\,0.50 & $\rho_p$\,=\,0.45 \\
$L/M$\xspace & $-$0.02, $p$\,=\,0.95 & 0.39, $p$\,=\,0.09 & 0.30, $p$\,=\,0.11 \\
\hline
\end{tabular}
\tablefoot{The rank $\rho$ and its corresponding probability ($p$) are shown for each comparison. A $p$-value of $<$\,0.001 indicate a correlation at 0.001 significance level. $p$-values of 0.05, 0.002 and $<$\,0.001 represent the $\sim$\,2, 3 and $>$\,3\,$\sigma$ confidence levels. For $L_{\rm{bol}}$\xspace and $M_{\rm{clump}}$\xspace, the partial correlation coefficient, $\rho_p$, is also shown.}
\end{table}
\setlength{\tabcolsep}{6pt}
\subsection{The excitation temperature of the CO gas}
\label{sec_texc}
The increase of $L_{\rm{CO}}$\xspace with the bolometric luminosity of the source (see Fig.\,\ref{fig_lco_correlation_fixbeam_lbol}) suggests that the intensity of the CO transitions may depend on an average warmer temperature of the gas in the clumps due to an increase of the radiation field from the central source \citep[see e.g.][]{vanKempen09}.
To confirm this scenario, we computed the excitation temperature of the gas, $T_{\rm{ex}}$\xspace, and compared it with the properties of the clumps.
Ideally, the intensity ratio of different CO transitions well separated in energy (e.g. CO\,(4--3) and CO\,(7--6)) allows a determination of the excitation temperature of the gas.
However, most of the CO profiles in the $\rm{TOP100}$\xspace clumps are affected by self-absorption (see Sect.\,\ref{sec_selfabs}), causing a considerable underestimate of the flux especially in CO\,(4--3) and leading to unreliable ratios. Moreover, the CO\,(6--5) and CO\,(7--6) lines are too close in energy to allow a reliable estimate of the temperature.
Alternatively, the excitation temperature can be estimated using the peak intensity of optically thick lines.
From the equation of radiative transport, the observed main beam temperature ($T_{\rm{mb}}$) can be written in terms of $T_{\rm{ex}}$\xspace as:
\begin{equation}
T_{\rm{mb}} = \frac{h \nu}{k} \left[ J_\nu(T_{\rm{ex}})-J_\nu(T_{\rm{bg}})\right] \left[ 1 - exp\left(-\tau_\nu \right) \right]
\label{eq.line}
\end{equation}
\noindent where $J_\nu(T) = \left[{\rm exp}(h \nu / k T) -1\right]^{-1}$, $T_{\rm{bg}}$ is the background temperature and $\tau_\nu$ is the opacity of the source at the frequency $\nu$. In the following, we include only the cosmic background as background radiation.
Assuming optically thick emission ($\tau_\nu$\,$\gg$\,1), $T_{\rm{ex}}$ is given by:
\begin{equation}
T_{\rm{ex}} = \frac{h \nu / k}{{\rm ln}\left[ 1 + \frac{h \nu / k}{T_{\rm{mb}} + (h \nu / k) J_\nu(T_{\rm{bg}})} \right]}
\label{eq.tex}
\end{equation}
We computed $T_{\rm{ex}}$\xspace using the peak intensity of the CO\,(6--5) line from the Gaussian fit (Sect.\,\ref{sec_gaussfit}) and also from its maximum observed value. Since CO\,(6--5) may be affected by self-absorption, the maximum observed intensity likely results in a lower limit of the excitation temperature.
The values derived using both methods are reported in Table\,\ref{tbl_excitation_temperature}.
$T_{\rm{ex}}$\xspace derived from the peak intensity of the Gaussian fit ranges between 14 and 143\,K, with a median value of 35\,K. The analysis based on the observed intensity delivers similar results ($T_{\rm{ex}}$\xspace values range between 14 and 147\,K, with a median value of 34\,K).
The temperature of the gas increases with the evolutionary stage of the clumps and is well correlated with $L_{\rm{bol}}$\xspace ($\rho$\,=\,0.69, $p$\,$<$\,0.001, see Fig.\,\ref{fig_tex_lbol}). No significant correlation is found with $M_{\rm{clump}}$\xspace ($\rho$\,=\,0.09, $p$\,=\,0.37). On the other hand, the excitation temperature is strongly correlated with $L/M$\xspace ($\rho$\,=\,0.72, $p$\,$<$\,0.001), suggesting a progressive warm-up of the gas in more evolved clumps.
We further compared the $T_{\rm{ex}}$\xspace values obtained from CO with temperature estimates based on other tracers (C$^{17}$O\,(3--2), methyl acetylene, CH$_3$CCH, ammonia, and the dust, \citealt{Giannetti14,Giannetti17,Koenig15,Wienen12}).
All temperatures are well correlated ($\rho$\,$\geq$\,0.44, $p$,$<$\,0.001), however, the warm-up of the gas is more evident in the other molecular species than in CO \citep[cf. ][]{Giannetti17}.
\begin{table}
\centering
\setlength{\tabcolsep}{3pt}
\caption{\label{table_tex_classes}Kolmogorov-Smirnov statistics of the excitation temperature of the CO\,(6--5) line as a function of the evolutionary class of the clumps.}
\begin{tabular}{c|ll}
\hline
\hline
Classes & \multicolumn{1}{c}{Observed} & \multicolumn{1}{c}{Gaussian} \\
\hline
$\rm{70w}$\xspace-$\rm{IRw}$\xspace & 0.43, $p$\,=\,0.1 & 0.48, $p$\,=\,0.02 \\
$\rm{70w}$\xspace-$\rm{IRb}$\xspace & 0.83, $p$\,$<$\,0.001 & 0.76, $p$\,$<$\,0.001 \\
$\rm{70w}$\xspace-$\rm{H\,\textsc{ii}}$\xspace & 1.00, $p$\,$<$\,0.001 & 1.00, $p$\,$<$\,0.001 \\
$\rm{IRw}$\xspace-$\rm{IRb}$\xspace & 0.56, $p$\,$<$\,0.001 & 0.42, $p$\,=\,0.005 \\
$\rm{IRw}$\xspace-$\rm{H\,\textsc{ii}}$\xspace & 1.00, $p$\,$<$\,0.001 & 0.93, $p$\,$<$\,0.001 \\
$\rm{IRb}$\xspace-$\rm{H\,\textsc{ii}}$\xspace & 0.68, $p$\,=\,0.001 & 0.63, $p$\,=\,0.001 \\
\hline
\end{tabular}
\tablefoot{The rank KS and its corresponding probability ($p$) are shown for each comparison. A $p$-value of $<$\,0.001 indicate a correlation at 0.001 significance level. $p$-values of 0.05, 0.002 and $<$\,0.001 represent the $\sim$\,2, 3 and $>$\,3\,$\sigma$ confidence levels.}
\end{table}
\setlength{\tabcolsep}{6pt}
\begin{figure*}
\centering
\subfigure[]{
\includegraphics[width=0.485\linewidth]{bay_gtex_lbol_nocoeff.ps}
\label{fig_tex_lbol}}
\subfigure[]{
\includegraphics[width=0.485\linewidth]{bay_gtex_lmratio_nocoeff.ps}
\label{fig_tex_lmratio}} \\[-2.0ex]
\subfigure[]{
\includegraphics[width=0.485\linewidth]{bay_Gexc_lbol_nocoeff.ps}
\label{fig_gtex_lbol}}
\subfigure[]{
\includegraphics[width=0.485\linewidth]{bay_Gexc_lmratio_nocoeff.ps}
\label{fig_gtex_lmratio}} \\[-2.0ex]
\caption{Excitation temperature of the CO\,(6--5) line versus the bolometric luminosity of the $\rm{TOP100}$\xspace clumps (Left) and the luminosity-to-mass ratio (Right).
The excitation temperature was derived using the peak of the Gaussian fit of the CO profiles.
The median values for each class are shown as open diamonds and their error bars correspond to the median absolute deviation of the data from their median value.
The black solid line is the best fit, the light grey shaded area indicates the 68\% uncertainty, and the dashed lines show the intrinsic scatter ($\epsilon$) of the relation.
The best fit to the data is indicated by the filled black line.}
\label{fig_tex}
\end{figure*}
\section{Discussion}
\label{sec_discussion}
\subsection{Opacity effects}
\label{sec_opacity}
In Sect.\,\ref{sec_selfabs} we found that self-absorption features are present in most of the CO spectra analysed in this work.
To address this, we investigated the effects of self-absorption on our analysis and concluded that they are negligible since more than 80\% of the CO integrated intensities are recovered in the majority of the sources (Sect.\,\ref{sec_gaussfit}).
We also verified that the steepness of the relations between $L_{\rm{CO}}$\xspace and the clump properties is not affected by self-absorption (Sect.\,\ref{sec_CO_correlations}).
In addition, the CO lines under examination are certainly optically thick, and their opacity is likely to decrease with $J$.
Indeed, the comparison between $L_{\rm{CO}}$\xspace for different CO transitions and the bolometric luminosity of the clumps (see Sect.\,\ref{sec_CO_correlations}) suggests a systematic increase in the slope of the relations as a function of $J$ (see Table\,\ref{table_lco_fit} for the derived power-law indices for $L_{\rm{CO}}$\xspace versus $L_{\rm{bol}}$\xspace).
Such a steepening of the slopes with $J$ is even more evident when including the relation found by \citet{SanJose13} for CO\,(10--9) line luminosity in a complementary sample of the $\rm{TOP100}$\xspace. For the CO\,(10--9) transition, they derived, $\log(L_{\rm CO})$\,=\,($-$2.9$\pm$0.2)\,+\,(0.84$\pm$0.06)\,$\log(L_{\rm bol})$, which is steeper than the relations found towards lower-$J$ transitions reported in this work.
In Sect.\,\ref{sec_CO_highJ} we further discuss $\rm{SJG13}$\xspace results by analysing their low- and high-mass YSO sub-samples.
Our findings suggest that there is a significant offset between the sub-samples, leading to a much steeper relation between $L_{\rm{CO}}$\xspace and $L_{\rm{bol}}$\xspace when considering their whole sample. However, the individual sub-samples follow similar power-law distributions, with power-law indices of (0.70$\pm$0.08) and (0.69$\pm$0.21) for the high- and low-mass YSOs, respectively.
In Fig.\,\ref{fig_lco_powerlaw}, we present the distribution of the power-law indices of the $L_{\rm{CO}}$\xspace versus $L_{\rm{bol}}$\xspace relations, $\beta_{\rm J}$, as a function of their corresponding upper-level $J$ number, $J_{\rm up}$. We include also the datapoint from the $J_{\rm up}$\,=\,10 line for the high-mass sources of $\rm{SJG13}$\xspace (see also discussion in Sect.\,\ref{sec_CO_highJ}.
The best fit to the data, $\beta_{\rm J}$\,=\,(0.44$\pm$0.11)+(0.03$\pm$0.02)\,$J_{\rm up}$,
confirms that the power-law index $\beta_{\rm J}$ gets steeper with $J$.
The fact that the opacity decreases with $J$ could result in different behaviours of the line luminosities with $L_{\rm{bol}}$\xspace for different transitions. This effect was recently discussed by \citet{Benz16} who found that the value of the power-law exponents of the line luminosity of particular molecules and transitions depends mostly on the radius where the line gets optically thick.
In the case of CO lines, the systematic increase on the steepness of the $L_{\rm{CO}}$\xspace versus $L_{\rm{bol}}$\xspace relation with $J$ (see Table\,\ref{table_lco_fit}) suggests that higher $J$ lines trace more compact gas closer to the source and, thus, a smaller volume of gas is responsible for their emission.
Therefore, observations of distinct $J$ transitions of the CO molecule, from CO\,(4--3) to CO\,(7--6) (and even higher $J$ transitions, considering the CO\,(10--9) data from $\rm{SJG13}$\xspace), suggest that the line emission arises and gets optically thick at different radii from the central sources, in agreement with \citet{Benz16}.
\begin{figure}
\centering
\includegraphics[width=0.85\linewidth]{powerlaw.ps} \\[-1.5ex]
\caption{Power-law indices of the $L_{\rm{CO}}$\xspace versus $L_{\rm{bol}}$\xspace relations for different $J$ transitions as a function of the upper-level $J$ number.
The $\beta$ indices from Table\,\ref{table_lco_fit} (filled black circles) are plotted together with data from $\rm{SJG13}$\xspace (open grey circle, excluded from the fit) and the exponent derived for their high-luminosity sub-sample ($\ast$ symbol, see Fig.\,\ref{fig_lco_low_to_high_champres}).
The best fit is indicated by the dashed black line.}
\label{fig_lco_powerlaw}
\end{figure}
\subsection{Evolution of CO properties with time}
In Sect.\,\ref{sec_CO_correlations}, we showed that $L_{\rm{CO}}$\xspace does not correlate with the evolutionary indicator $L/M$\xspace. This result is unexpected if we consider that in $\rm{TOP100}$\xspace the evolutionary classes are quite well separated in $L/M$\xspace \citep[with median values of 2.6, 9.0, 40 and 76 for $\rm{70w}$\xspace, $\rm{IRw}$\xspace, $\rm{IRb}$\xspace and $\rm{H\,\textsc{ii}}$\xspace regions, respectively,][]{Koenig15}.
Previous work on SiO in sources with similar values of $L/M$\xspace as those of the $\rm{TOP100}$\xspace (e.g. \citealt{Leurini14} and \citealt{Csengeri16}) found that the line luminosity of low-excitation SiO transitions does not increase with $L/M$\xspace, while the line luminosity of higher excitation SiO lines (i.e. $J_{\rm{up}}>3$) seems to increase with time. Those authors interpreted these findings in terms of a change of excitation conditions with time which is not reflected in low excitation transitions. This effect likely applies also to low- and mid-$J$ CO lines with relatively low energies ($\le$\,155\,K); higher-$J$ CO transitions could be more sensitive to changes in excitation since they have upper level energies in excess of 300\,K (e.g. CO\,(10--9) or higher $J$ CO transitions).
This hypothesis is strengthened by our finding that the excitation temperature of the gas increases with $L/M$\xspace (see Fig.\,\ref{fig_tex_lmratio}).
This scenario can be tested with observations of high-$J$ CO transitions which are now made possible by the SOFIA telescope in the range $J_{\rm up}$\,=\,11--16. Also {\it Herschel}-PACS archive data could be used despite their coarse spectral resolution.
\subsection{Do embedded H\,{\sc{ii}} regions still actively power molecular outflows?}
\label{sec_hii}
In Sect.\,\ref{sec_intpropCO} we showed that $\rm{H\,\textsc{ii}}$\xspace regions have broad CO lines (see Fig.\,\ref{fig_avgspc_fixbeam}) likely associated with high-velocity outflowing gas. The $\rm{H\,\textsc{ii}}$\xspace sources in our sample are either compact or unresolved objects in the continuum emission at 5\,GHz.
Their envelopes still largely consist of molecular gas and have not yet been significantly dispersed by the energetic feedback of the YSOs. Our observations suggest that high-mass YSOs in this phase of evolution still power molecular outflows and are therefore accreting.
This result is in agreement with the recent study of \citet{Urquhart13b} and \citet{Cesaroni15} who suggested that accretion might still be present during the early stages of evolution of $\rm{H\,\textsc{ii}}$\xspace regions based on the finding that the Lyman continuum luminosity of several $\rm{H\,\textsc{ii}}$\xspace regions appears in excess of that expected for a zero-age main-sequence star with the same bolometric luminosity.
Such excess could be due to the so-called flashlight effect \citep[e.g.][]{Yorke99}, where most of the photons escape along the axis of a bipolar outflow.
Indeed \citet{Cesaroni16} further investigated the origin of the Lyman excess looking for infall and outflow signatures in the same sources. They found evidence for both phenomena although with low-angular resolution data. Alternatively, the high-velocity emission seen in CO in the $\rm{TOP100}$\xspace in this work and in SiO in the ultra-compact $\rm{H\,\textsc{ii}}$\xspace regions of \citet{Cesaroni16} could be associated with other younger unresolved sources in the clump and not directly associated with the most evolved object in the cluster. Clearly, high angular resolution observations (e.g. with ALMA) are needed to shed light on the origin of the high-velocity emission and confirm whether indeed the high-mass YSO ionising the surrounding gas is still actively accreting.
\subsection{CO line luminosities from low- to high-luminosity sources}
\label{sec_CO_highJ}
In this section, we further study the correlation between $L_{\rm{CO}}$\xspace and the bolometric luminosity of the clumps for different CO transitions to investigate the possible biases that can arise when comparing data with very different linear resolutions. This is important in particular when comparing galactic observations to the increasing number of extragalactic studies of mid- and high-$J$ CO lines \citep[e.g.][]{Weiss07,Decarli16}.
We use results from $\rm{SJG13}$\xspace for the high energy CO\,(10--9) line (with a resolution of $\sim 20$\arcsec) and from the CO\,(6--5) and CO\,(7--6) transitions observed with APEX by $\rm{vK09}$\xspace. The sources presented by $\rm{SJG13}$\xspace cover a broad range of luminosities (from $<$\,1\,L$_\odot$ to $\sim$\,10$^5$\,L$_\odot$) and are in different evolutionary phases. On the other hand, the sample studied by $\rm{vK09}$\xspace consists of eight low-mass YSOs with bolometric luminosities $\lesssim$\,30\,L$_\odot$\xspace.
To investigate the dependence of the line luminosity in different CO transitions on $L_{\rm{bol}}$\xspace from low- to high-mass star-forming clumps, we first divided the sources from $\rm{SJG13}$\xspace into low- ($L_{\rm{bol}}$\xspace$<$\,50\,L$_\odot$\xspace) and high-luminosity ($L_{\rm{bol}}$\xspace$>$\,50\,L$_\odot$\xspace) objects.
In this way and assuming the limit $L_{\rm{bol}}$\xspace=\,50\,L$_\odot$\xspace adopted by $\rm{SJG13}$\xspace as a separation between low- and high-mass YSOs, we defined a sub-sample of low-mass sources (the targets of $\rm{vK09}$\xspace for CO\,(6--5) and CO\,(7--6), and those of $\rm{SJG13}$\xspace with $L_{\rm{bol}}$\xspace$<$\,50\,L$_\odot$\xspace for CO $J$\,=\,10--9) and one of intermediate- to high-mass clumps (the $\rm{TOP100}$\xspace for the mid-$J$ CO lines and the sources from $\rm{SJG13}$\xspace with $L_{\rm{bol}}$\xspace$>$\,50\,L$_\odot$\xspace for CO\,(10--9)).
In the upper panels of Fig.\,\ref{fig_lco_low_to_high_champres} we compare our data with those of $\rm{vK09}$\xspace for the CO\,(6--5) and CO\,(7--6) transitions. We could not include the sources of $\rm{SJG13}$\xspace in this analysis because observations in the CO\,(6--5) or CO\,(7--6) lines are not available.
We calculated the CO line luminosity of their eight low-mass YSOs using the integrated intensities centred on the YSO on scales of $\sim$0.01\,pc (see their Table\,3).
In order to limit biases due to different beam sizes, we recomputed the CO luminosities from the central position of our map at the original resolution of the $\rm{CHAMP}^+$\xspace data (see Table\,\ref{table:champ_setup}), probing linear scales ranging from $\sim$0.04 to 0.6\,pc.
We performed three fits on the data: we first considered only the original sources of $\rm{vK09}$\xspace and the $\rm{TOP100}$\xspace separately, and then combined both samples.
The derived power-law indices of the CO\,(6--5) data are 0.59\,$\pm$\,0.25 and 0.59\,$\pm$\,0.04 for the low- and high-luminosity sub-samples, respectively.
Although the power-law indices derived for the two sub-samples are consistent within 1-$\sigma$, the fits are offset by roughly one order of magnitude (from $-2.75$ to $-1.54$\,dex), indicating that $L_{\rm{CO}}$\xspace values are systematically larger towards high-luminosity sources.
Indeed, the change on the offsets explains reasonably well the steeper power-law index found when combining both sub-samples (0.74\,$\pm$0.03).
Similar results are found for CO\,(7--6), although the difference between the offsets are slight smaller ($\sim$0.8\,dex).
The bottom panel of Fig.\,\ref{fig_lco_low_to_high_champres} presents the CO\,(10--9) luminosity for the $\rm{SJG13}$\xspace sample with the best fit of their low- and high-luminosity sources separately. The derived power-law indices are 0.69\,$\pm$\,0.21 and 0.70\,$\pm$\,0.08 for the low- and high-luminosity sub-samples, respectively. The fits are offset by roughly 0.3\,dex, which also explains the steeper slope of 0.84\,$\pm$\,0.06 found by $\rm{SJG13}$\xspace when fitting the two sub-samples simultaneously.
We interpret the shift in CO line luminosities between low- and high-luminosity sources as a consequence of the varying linear resolution and sampled volume of gas of the data across the $L_{\rm{bol}}$\xspace axis.
In high-mass sources, mid-$J$ CO lines trace extended gas (see the maps presented in Figs.\,\ref{fig_fixbeam_gaussian_fit} for the $\rm{TOP100}$\xspace) probably due to the effect of clustered star formation. Since the data presented in Fig.\,\ref{fig_lco_low_to_high_champres} are taken with comparable angular resolutions, the volume of gas sampled by the data is increasing with $L_{\rm{bol}}$\xspace because sources with high luminosities are on average more distant.
For the CO\,(10--9) data, the two sub-samples are likely differently affected by beam dilution. In close-by low-mass YSOs, the CO\,(10--9) line is dominated by emission from UV heated outflow cavities \citep{vanKempen10} and therefore is extended.
In high-mass YSOs, the CO\,(10--9) line is probably emitted in the inner part of passively heated envelope \citep{Karska14} and therefore could suffer from beam dilution.
This could explain the smaller offset in the CO\,(10--9) line luminosity between the low- and high-luminosity sub-samples.
$\rm{SJG13}$\xspace and \citet{Sanjose16} found a similar increase in the slope of the line luminosity of the CO\,(10--9) and of H$_2$O transitions versus $L_{\rm{bol}}$\xspace when including extragalactic sources (see Fig.\,14 of $\rm{SJG13}$\xspace and Fig.\,9 of \citealt{Sanjose16}).
These findings clearly outline the difficulties of comparing observations of such different scales and the problems to extend Galactic relations to extragalactic objects.
\begin{figure}[!ht]
\centering
\includegraphics[width=\linewidth]{clumpprop_lco_lbol_fullluminosity_midJ_d70.ps} \\[-2.0ex]
\caption{CO line luminosity as a function of the bolometric luminosity for the CO\,(6--5) (upper panel), CO\,(7--6) (middle) and CO\,(10--9) (bottom) transitions.
The fits were performed on the whole dataset (all points, shown with black contours), on the low- and high-luminosity sub-samples (points filled in red and blue, respectively).
The CO\,(6--5) and CO\,(7--6) data towards low-luminosity sources are from \citet{vanKempen09}; the CO\,(10--9) data are from \citet{SanJose13}. The dashed vertical line at $L_{\rm{bol}}$\xspace=\,50\,L$_\odot$\xspace marks the transition from low- to high-mass YSOs. The typical error bars are shown in the bottom right side of the plots.}
\label{fig_lco_low_to_high_champres}
\end{figure}
\section{Summary}
\label{sec_summary}
A sample of 99 sources, selected from the ATLASGAL 870\,$\mu\rm{m}$\xspace survey and representative of the Galactic population of star-forming clumps in different evolutionary stages (from 70\,$\mu\rm{m}$\xspace-weak clumps to $\rm{H\,\textsc{ii}}$\xspace regions), was characterised in terms of their CO\,(4--3), CO\,(6--5) and CO\,(7--6) emission.
We first investigated the effects of different linear resolutions on our data.
By taking advantage of our relatively high angular resolution maps in the CO\,(6--5) and CO\,(7--6) lines, we could study the influence of different beam sizes on the observed line profiles and on the integrated emission.
We first convolved the CO\,(6--5) and CO\,(7--6) data to a common linear size of $\sim$0.24\,pc using a distance limited sub-sample of clumps and then to a common angular resolution of 13\farcs4, including the single-pointing CO\,(4--3) data.
We verified that the results typically do not depend on the spatial resolution of the data, at least in the range of distances sampled by our sources.
The only difference between the two methods is found when comparing the average spectra for each evolutionary class: indeed, only when using spectra that sample the same volume of gas (i.e. same linear resolution) it is possible to detect an increase in line width from $\rm{70w}$\xspace clumps to $\rm{H\,\textsc{ii}}$\xspace regions, while the line widths of each evolutionary class are less distinct in the spectra smoothed to the same angular size due to sources at large distances ($>$12\,kpc).
This result is encouraging for studies of large samples of SF regions across the Galaxy based on single-pointing observations.
The analysis of the CO emission led to the following results:
\begin{enumerate}
\item All the sources were detected in the CO\,(4--3), CO\,(6--5) and CO\,(7--6) transitions.
\item The spatial distribution of the CO\,(6--5) emission ranges between 0.1 and 2.4\,pc. The sizes of mid-$J$ CO emission display a moderate correlation with the sub-mm dust mass of the clumps, suggesting that the extension of the gas probed by the CO is linked to the available amount of the total gas in the region. In addition, the CO\,(6--5) extension is also correlated with the infrared emission probed by the {\it Herschel}-PACS 70\,$\mu\rm{m}$\xspace maps towards the 70\,$\mu\rm{m}$\xspace-bright clumps.
\item The CO profiles can be decomposed using up to three velocity components. The majority of the spectra are well fitted by two components, one narrow (FWHM\,$<$\,7.5\,km\,s$^{-1}$) and one broad; 30\% of the sources need a third and broader component for the CO\,(6--5) line profile.
\item The FWZP of the CO lines increases with the evolution of the clumps (with median values of 26, 42, 72 and 94 for $\rm{70w}$\xspace, $\rm{IRw}$\xspace, $\rm{IRb}$\xspace clumps and $\rm{H\,\textsc{ii}}$\xspace regions, respectively, for the CO\,(6--5) transition).
$\rm{H\,\textsc{ii}}$\xspace regions are often associated with broad velocity components, with FWHM values up to $\sim$100\,$\rm{km\,s^{-1}}$\xspace. This suggests that accretion, resulting in outflows, is still undergoing in the more evolved clumps of the $\rm{TOP100}$\xspace.
\item The CO line luminosity increases with the bolometric luminosity of the sources, although it does not seem to increase neither with the mass nor with the $L/M$\xspace ratio of the clumps.
\item The dependence of the CO luminosity as a function of the bolometric luminosity of the source seems to get steeper with $J$. This likely reflects the fact that higher $J$ CO transitions are more sensitive to the temperature of the gas and likely arise from an inner part of the envelope.
These findings are quite robust in terms of self-absorption present in most of the $^{12}$CO emission.
\item The excitation temperature of the clumps was evaluated based on the peak intensity of the Gaussian fit of the CO\,(6--5) spectra.
We found that $T_{\rm{ex}}$\xspace increases as a function of the bolometric luminosity and the luminosity-to-mass ratio of the clumps, as expected for a warming up of the gas from $\rm{70w}$\xspace clumps towards $\rm{H\,\textsc{ii}}$\xspace regions.
The observed CO emission towards more luminous and distant objects likely originates from multiple sources within the linear scale probed by the size of beam (up to 0.84\,pc), thus, are systematically larger than the emission from resolved and nearby less luminous objects, from which the CO emission is integrated over smaller linear scales ($\sim$0.01\,pc).
We found that the line luminosity of the CO lines shows similar slopes as a function of the bolometric luminosity for low-mass and high-mass star-forming sources.
However, as a consequence, the distribution of the CO line luminosity versus the bolometric luminosity follows steeper power-laws when combining low- and high-luminosity sources.
\end{enumerate}
\begin{acknowledgements}
F.N. thanks to {\it Funda\c{c}\~{a}o de Amparo \`a Pesquisa do Estado de S\~{a}o Paulo} (FAPESP) for support through processes 2013/11680-2, 2014/20522-4 and 2017/18191-8.
%
T.Cs. acknowledges support from the \emph{Deut\-sche For\-schungs\-ge\-mein\-schaft, DFG\/} via the SPP (priority programme) 1573 'Physics of the ISM'.
%
We thank the useful comments and suggestions made by an anonymous referee that led to a much improved version of this work.
\end{acknowledgements}
\bibliographystyle{aa}
|
2,869,038,154,772 | arxiv | \section{Introduction}
\label{sec:intro}
In recent years, there have been efforts to apply generative models, e.g., StyleGAN2~\cite{karras2020analyzing}, for face image editing~\cite{song2022editing, alaluf2021HyperStyle, wang2021high, parmar2022spatially, tov2021designing, alaluf2021restyle, richardson2021encoding, xuyao2022, roich2021pivotal, nitzan2022mystyle, wei2022e2style} and restoration~\cite{wang2021towards,he2022gcfsr,zhu2022blind,yang2021gan}.
The basis of these applications is the GAN inversion.
The typical inversion strategy~\cite{richardson2021encoding, tov2021designing} is to train encoders to encode the face images into the latent of the pretrained generator and reconstruct the images via the generator.
However, such approaches can not result in artifact-free and precise reconstruction due to the inevitable information loss when translating the high-resolution image into the limited GAN latent space.
Some improved methods were proposed to further finetune the generator for image-specific reconstruction without losing editability~\cite{roich2021pivotal,nitzan2022mystyle}. Although these finetuning-based techniques improve the inversion accuracy of identity and style, the reconstruction of out-of-domain contents, e.g., the complex background, accessories, and hair, are needed to be improved since the capability of the latent and the generator is still constrained.
\footnotetext[1]{Corresponding author.}
\begin{figure}[tp]
\centering
\begin{tabular}{@{}c@{\hspace{1mm}}c@{}c@{}}
\scriptsize{Input \textbf{$\rightarrow$ +Age}} &
\scriptsize{(a) HyperStyle~\cite{alaluf2021HyperStyle}} &
\scriptsize{(b) HFGI$_{e4e}$~\cite{wang2021high}}
\\
\multicolumn{3}{@{}c@{}}{\includegraphics[width=0.97\linewidth]{Figures/Sec1/teaser1.pdf}}
\\
\scriptsize{(c) SAM~\cite{parmar2022spatially}} &
\scriptsize{(d) DiffCAM~\cite{song2022editing}} &
\scriptsize{(e) Ours}
\\
\multicolumn{3}{@{}c@{}}{\includegraphics[width=0.97\linewidth]{Figures/Sec1/teaser2.pdf}}
\\
\resizebox{0.333\linewidth}{!}{~} &
\resizebox{0.333\linewidth}{!}{~} &
\resizebox{0.333\linewidth}{!}{~}
\vspace{-1.7em}
\end{tabular}
\caption{\textbf{Out-of-domain GAN inversion and attribute manipulation.} Our method maintains out-of-domain objects in the original image (e.g., background, earrings) and generates high-fidelity texture in the attribute-edited image. Please zoom in for detail.}
\label{fig:teaser}
\vspace{-1em}
\end{figure}
To enhance the capability of pretrained GAN models, some works~\cite{wang2021high,wang2021towards, parmar2022spatially} strengthen the potential of out-of-domain GAN inversion by modulating the generator features with the input features extracted from source images.
Because the feature modulation operation breaks the pretrained GAN priors, such methods suffer from the fidelity-editability trade-off~\cite{wang2021high, shannon1959coding, tishby2015deep, he2022gcfsr, wang2021towards, yang2021gan}. The larger latent space increases the reconstruction quality but inevitably undermines the editability of the GAN inversion framework.
Furthermore, some recent works~\cite{song2022editing, parmar2022spatially} propose to disentangle the target image into different spatial areas. They refine the regions with low invertibility to improve the inversion fidelity. Such low invertibility regions are the parts that cannot be reconstructed well with the generator, so-called Out-Of-Domain (OOD) areas.
E.g., Song et al.~\cite{song2022editing} estimate a manipulation-aware mask with an attribute classifier. However, they ignore the geometrical misalignment between the inverted and the original images, and apply a deghosting module for result refinement, which leading to undesired artifacts in their results (Fig.~\ref{fig:teaser} (d)).
Parmar et al.~\cite{parmar2022spatially} train an invertibility mask prediction module with perceptual supervision. The invertibility masks are then used as guidance for GAN feature modulation. However, their prediction of invertibility is noisy and inconsistent with the face semantic parts.
Therefore, they adopt a segmentation model for refinement, but the invertibility in the same semantic area (e.g., occlusions on the face) could be inconsistent. Also, they manually design thresholds to filter out the OOD areas, which is hard to optimize for small objects (Fig.~\ref{fig:teaser} (c)).
In summary, existing invertibility estimation methods mainly adopt the reconstruction error as the reference to judge the OOD regions. However, they ignore that reconstruction errors also come from the In-Domain (ID) areas. Consequently, their predicted mask is noisy and unreliable.
In this paper, we propose a novel strategy for photo-realistic GAN inversion by decomposing the input images into OOD and ID areas with invertibility masks. We focus on the high-resolution ($1024^2$ pixels) GAN inversion on human face images and the downstream applications (e.g., Face editing).
Our basic idea is to reduce the reconstruction error of the ID areas and thus highlight the error of OOD regions. The reconstruction error of the ID area comes from both the textural and geometrical misalignment between the input image and the generated image. Although previous works improve the textural accuracy in the reconstruction by predicting or optimizing a better latent vector $w$, the geometrical misalignment is rarely discussed, which we believe is also important for invertibility estimation. Hence, we design an invertibility detector learned with an optical flow prediction module to reduce the influence of geometrical misalignment. The optical flow is computed between the features of the encoder and the generator, which is then applied to warp the generated features to alleviate their misalignment with the input features.
Comparing with feature modulation~\cite{wang2021high, parmar2022spatially, xuyao2022}, such warping will not break the fidelity of the generated textures.
Along the training, the reconstruction error of the ID area will be minimized, and the invertibility mask prediction will be gradually focused on the OOD regions.
The overall process needs no extra labels for the mask or flows.
Based on the invertibility prediction, we design an effective approach to composite the generated content with the out-of-domain input feature for a photo-realistic generation with high fidelity.
Our framework consists of three major parts: the encoder, the Spatial Alignment and Masking Module (SAMM), and the generator.
First, we extract features from the input image and predict its latent vector with a pretrained image-to-latent encoder~\cite{tov2021designing}. Second, we feed the latent vector into a pretrained StyleGAN2~\cite{karras2020analyzing} model for content generation, acquiring generated features.
Third, we estimate the optical flow and the invertibility mask between the input and the generated features at multiple resolutions.
Then, we warp the generated features with the computed flow, aiming to minimize the reconstruction error of ID regions.
Finally, we composite the input image with the warped generated content according to the invertibility mask.
Since only the spatial operation, i.e., warping, is applied to the generated features, we maintain their editability with existing GAN editing methods.
Combined with the artifact-free and precise inversion effects, our method has excellent superiority in reconstruction accuracy and editing fidelity over existing approaches.
In this paper, we adopt StyleGAN2 as the backbone for experiments, and extensive experimental results demonstrate that our method outperforms current state-of-the-art methods with higher reconstruction fidelity and better visual quality.
In summary, our contributions are listed as follows:
\begin{enumerate}
\item We propose a novel framework for out-of-domain GAN inversion on human face images by aligning and blending the generated image with the input image via optical flow and invertibility mask prediction.
\item We investigate the GAN invertibility with a novel Spatial Alignment and Masking Module, which is a new solution for invertibility decomposition.
\item Our proposed framework can produce photo-realistic results in both reconstruction and editing tasks. Experiments show that our framework outperforms existing methods in reconstruction accuracy and visual fidelity.
\end{enumerate}
\section{Related works}
GAN inversion aims to first encode a real-world image into a semantic-disentangled latent vector, then reconstruct the input image with a generator. It enables different downstream applications, e.g., face editing with labels~\cite{song2022editing, alaluf2021HyperStyle, wang2021high} or texts~\cite{patashnik2021styleclip,gal2022stylegan,abdal2022clip2stylegan}.
To tackle the non-trivial translation between image and latent vector, some efforts~\cite{richardson2021encoding, tov2021designing, alaluf2021restyle, hu2022style} have been made to design better encoders or predict the latent vector via iterative strategies~\cite{alaluf2021restyle,wei2022e2style}.
Bai et al.~\cite{bai2022high} investigate the padding space of the StyleGAN generator to increase the invertibility of the pretrained GAN model. Roich et al.~\cite{roich2021pivotal} and Nitzan et al.~\cite{nitzan2022mystyle} propose strategies to first encode images into latent vectors, then finetune the pretrained generator for specific images or people. More recently, Alaluf et al.~\cite{alaluf2021HyperStyle} adopt a hypernetwork~\cite{ha2016hypernetworks} to modulate the generator kernel for better GAN inversion.
However, such methods can not generate out-of-domain contents such as image-specific backgrounds or accessories.
\begin{figure}[t]
\centering
\includegraphics[width=1.0\linewidth]{Figures/Sec3/overview.pdf}
\caption{\textbf{An overview of our framework.} We extract features from the input image and align the generated features to the input features, which improves the in-domain reconstruction accuracy and eases the invertibility decomposition. Meanwhile, we predict the invertibility mask for feature and RGB space blending, which improves the out-of-domain reconstruction quality. }
\label{fig:overview}
\vspace{-1em}
\end{figure}
Meanwhile, some works~\cite{wang2021high, wang2021towards, xuyao2022, he2022gcfsr, zhu2022blind, Yang2021GPEN} add extra connections between the encoder and generator to enlarge the latent space, but such methods fall into a tricky dilemma of balancing the distortion-editability tradeoff~\cite{wang2021high, shannon1959coding}. A larger latent space helps increase the reconstruction precision but decreases the editability in the generation due to the increasing semantic entanglement problem. Recently, Parmar et al.~\cite{parmar2022spatially} proposed the SAM with an invertibility prediction module guided by LPIPS~\cite{zhang2018perceptual} loss to predict the spatial invertibility map for the input image. Nevertheless, their predictions are noisy and inconsistent with the input image. The predictions need to be smoothed with a pretrained segmentation network.
Furthermore, since SAM is an optimization-based method, it takes a long inference time to produce a high-fidelity result.
Moreover, Song et al.~\cite{song2022editing} propose DiffCAM to refocus the GAN inversion on the attributes to be manipulated. Instead of inversing the whole image, DiffCAM aims to find the edited area and blend the input image with the edited one. They adopt a semantic classifier (i.e., the facial attributes) as the prior for a rough editing mask prediction. However, due to the misalignment between the generated content and the input image, they suffer from undesired ghosting artifacts in their composition results. Thus, DiffCAM also needs to employ an extra ghosting removal module, while it can not fix the artifacts thoroughly.
Different from existing GAN inversion methods, we simultaneously improve the ID reconstruction via geometric alignment and out-of-domain reconstruction via image blending with the invertibility mask.
In our experiments, we compare the baseline methods' inversion and attribute manipulation performance with our framework. Our method outperforms existing works in reconstruction quality (Fig.~\ref{fig:inversion}) and can produce photo-realistic face editing results (Fig.~\ref{fig:editing_compare}) with off-the-shelf GAN manipulation approaches~\cite{shen2020interpreting, patashnik2021styleclip, harkonen2020ganspace}.
\section{Method}
In this paper, we propose a new framework for out-of-domain GAN inversion. The proposed framework can produce a precise invertibility mask to achieve excellent input and generated image composition.
The overview of our framework is shown in Fig.~\ref{fig:overview} and Fig.~\ref{fig:blending}. In the following section, we will introduce the overview of our framework in Sec.~\ref{sec:overview}, our proposed Spatial Alignment and Masking Module (SAMM) in Sec.~\ref{sec:SAMM}, and our training strategy in Sec.~\ref{sec:training}. Finally, we will introduce the inversion and editing pipeline with our proposed modules in Sec.~\ref{sec:blending}.
\subsection{Overview}
\label{sec:overview}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{Figures/Sec3/blend.pdf}
\vspace{-0.5em}
\caption{\textbf{Our blending mechanism.} With the predicted invertibility mask $m$, we blend the in-domain GAN inversion $\widehat{x_{in}}$ with the out-of-domain partition $x_{out}$ to produce the final result $\hat{x}$.}
\label{fig:blending}
\vspace{-1em}
\end{figure}
\noindent\textbf{Traditional GAN inversion.} The essential part of our framework is the estimation of the GAN invertibility on a pretrained StyleGAN2~\cite{karras2020analyzing}.
The StyleGAN2 generator is a multi-scale Convolutional-Neural-Network (CNN), where the results are generated gradually with enlarged spatial resolution.
Meanwhile, the image generation is controlled by the latent vector $w$ in the generation process. Recent works~\cite{richardson2021encoding, tov2021designing, alaluf2021restyle, wang2021high, parmar2022spatially, song2022editing} adopt the pretrained StyleGAN2 generator and aim to inverse the input RGB image into the well-disentangled $W^+ \in R^{512 \times 18}$ latent space proposed in \cite{richardson2021encoding} with an encoder $E$.
Given an input image $x \in R^{3 \times 1024 \times 1024}$, the GAN inversion of $x$ is to first encode $x$ into a latent vector $w \in W^+$ with the encoder $E$, then reconstruct the image $\hat{x}$ with the generator $G$, that:
\begin{equation}
\hat{x} = G(E(x)).
\label{eqn:gan_inversion}
\end{equation}
\noindent\textbf{GAN inversion with OOD decomposition.} Previous work~\cite{feng2021understanding} analyzed the GAN training strategy and discovered that $W$ space is a low-rank approximation for the target domain of generator $G$.
Thus, it is difficult to reconstruct $\hat{x}$ without distortion.
To solve this drawback, we tackle GAN inversion from the perspective of invertibility decomposition. Different from ~\cite{wang2021high, xuyao2022}, we intend to first decompose $x$ into the invertible partition $x_{in}$ and the OOD partition $x_{out}$ (e.g., accessories, tattoos) with an invertibility mask $m$, where
\begin{equation}
\label{eqn:decomp}
\begin{split}
x_{out} = x \cdot m, ~~
x_{in} = x \cdot (1-m).
\end{split}
\end{equation}
Then, we can reconstruct or edit $x_{in}$ via GAN inversion and preserve the OOD contents $x_{out}$ for better fidelity, that
\begin{gather}
\label{eqn:decomp_gan_inversion}
\widehat{x_{in}} =~ G(E(x)), \\
\label{eqn:blending}
\hat{x} =~ x_{out} + \widehat{x_{in}} \cdot (1 - m).
\end{gather}
The proposed framework consists of three major parts, i.e, the encoder $E$, the generator $G$, and SAMM.
With the pretrained and fixed $E$ and $G$, the SAMM estimates GAN invertibility by minimizing the reconstruction error of $\hat{x}$.
At the training stage, we adopt the pretrained image-to-latent encoder $E$ from \cite{tov2021designing} and train the SAMM for image-to-image reconstruction.
Please refer to the following sections for details.
\subsection{Spatial Alignment and Masking Module}
\label{sec:SAMM}
\noindent\textbf{The limitation of previous invertibility estimation.} Previous methods for invertibility mask prediction mainly adopt the reconstruction error as the supervision~\cite{parmar2022spatially}. However, the predictions are usually inaccurate since the subtle ID reconstruction error, which should not be considered as the OOD content, disturbs the prediction of invertibility.
We find these ID errors comes from the textural and geometrical misalignment between $x$ and $\widehat{x_{in}}$. While the textural misalignment (e.g. eyes or skin color) can be solve via latent space optimization~\cite{parmar2022spatially, alaluf2021HyperStyle, alaluf2021restyle, nitzan2022mystyle, roich2021pivotal}, it is hard to reconstruct delicate local structures (e.g., the boundary of the face or hair) in $x$ with $w$, especially when the resolution is high (e.g., $1024^2$ and above). In this paper, we mainly focus on minimizing the geometrical misalignment with spatial operations (i.e., optical flow).
\begin{figure}[t]
\centering
\includegraphics[width=0.9\linewidth]{Figures/Sec3/SAMM.pdf}
\vspace{-1em}
\caption{The spatial alignment and masking module.}
\label{fig:SAMM}
\vspace{-1em}
\end{figure}
\noindent\textbf{Our SAMM for invertibility estimation.} We train SAMM with an optical flow prediction module to remove the influence of the reconstruction error of the ID regions. The computed optical flow is utilized to fix the misalignment between $x$ and $\widehat{x_{in}}$, i.e., reducing the geometrical error.
Inspired by previous works done by Collins et al.~\cite{Collins20} and Chong et al.~\cite{chong2021stylegan} where they discovered that simple spatial operations in the GAN feature space maintain the realistic textures in the results.
We also find that because $w$ controls the semantic attributes of generation by modulating the variance of the generated feature maps channel-wisely, simple spatial warping in the feature space of $G$ does not break the attribute-disentangled property of $w$ in face generation and manipulation. I.e., we can decrease the geometrical reconstruction error for the ID areas without increasing the textural reconstruction error.
\noindent\textbf{The details of SAMM.} SAMM predicts not only the invertibility map but also the alignment flow on the extracted input features $f_i$ and the generated features $g_i$ in the $i$-th layer of the generation.
First, we encode the input image into the latent vectors $w$. Second, the generator produces the feature map $g_i$ conditioned on $w$ in the $i$-th layer ($i \in \{1, ..., L\}$). Then, as shown in Fig.~\ref{fig:SAMM}, we also extract feature map $f_i$ from the input image and feed both $g_i$ and $f_i$ into the SAMM for flow and mask prediction.
The prediction is conducted as an iterative manner ($N$ iterations) in each layer, as
\begin{equation}
\begin{aligned}
g_{i,0} &= g_i, \quad j \in \{1, ..., N\},\\
\delta_{i,j}^x, \delta_{i,j}^y, m_{i,j} &= {\rm SAMM} (f_i \oplus g_{i,j-1}),
\end{aligned}
\label{eqn:SAMM}
\end{equation}
\begin{equation}
\begin{aligned}
&\Delta_{i,j}^x, \Delta_{i,j}^y = \sum_{k=1}^j \delta_{i,k}^x, \sum_{k=1}^j \delta_{i,k}^y,
\end{aligned}
\label{eqn:flow}
\end{equation}
\begin{equation}
\begin{aligned}
\resizebox{0.8\linewidth}{!}{$
M_{i,j} = m_{i,j} \cdot M_{i,(j-1)} + M_{i,(j-1)} \cdot (1 - M_{i,(j-1)})
$},
\end{aligned}
\label{eqn:aligned_mask_0}
\end{equation}
\begin{equation}
\begin{aligned}
&g_{i,j} = \mathcal{GS} (g_i, \Delta_{ij}^x, \Delta_{ij}^y) \cdot M_{i,j} + g_i \cdot (1 - M_{i,j}),
\end{aligned}
\label{eqn:grid_sample}
\end{equation}
where $\delta_{i,j}^x$ and $\delta_{i,j}^y$ are the horizontal and vertical flow, $\oplus$ stands for concatenation operator, $m_{ij}$ is a mask, and $\mathcal{GS}$ is the grid sampling operation.
Although $m_{i,j}$ indicates differences between $g_{i,(j-1)}$ and $f_i$, both misalignments in the ID areas and the OOD contents are considered as differences.
To this, the iterative strategy can align $g_i$ to $f_i$ gradually with the alignment caused by $\mathcal{GS}$ in Eqn.~\eqref{eqn:grid_sample}. Moreover, in Eqn.~\eqref{eqn:aligned_mask_0}, we use the mask prediction from the last iteration to constrain the mask. Eqn.~\eqref{eqn:aligned_mask_0} promotes more areas to be refined by the flows via geometrical alignment.
Moreover, the iterative estimation of flows in Eqn.~\eqref{eqn:flow} improves the accuracy of flows.
The final output of SAMM in each layer is the mask $M_{i,N}$ and the flow $(\Delta_{i,N}^x, \Delta_{i,N}^y)$.
Furthermore, we also hope to obtain a consistent mask prediction among different layers of the generator. Following the strategy of Eqn.~\eqref{eqn:aligned_mask_0}, we have
\begin{equation}
\begin{aligned}
M_{i,N} =~&M_{i,N} \cdot {\rm \uparrow}(M_{(i-1), N}) \\
+{\rm \uparrow}(&M_{(i-1), N}) \cdot (1 - {\rm \uparrow}(M_{(i-1), N})),
\end{aligned}
\label{eqn:aligned_mask}
\end{equation}
where $\rm \uparrow(\cdot)$ is the bilinear upsampling function.
\subsection{Blending}
\label{sec:blending}
\noindent\textbf{Generate $\widehat{x_{in}}$.} Because $g_{i, N}$ is resampled from $g_i$, it maintains the local semantic attributes of $g_i$ and is spatially aligned to $f_i$. The aligned feature $g_{i, N}$ is then fed into the next styled-convolution layer $G_i$ and $g_{i+1} = G_i(g_{i, N}, w)$. And finally, $\widehat{x_{in}} = {\rm toRGB}(g_L)$, where $g_L$ is the last layer's feature map and ${\rm toRGB}$ is the output convolution in $G$. We visualize the generation process of $\widehat{x_{in}}$ in Fig.~\ref{fig:overview}.
In this paper, we align $g_i$ and $f_i$ in multiple resolution of $32^2, 64^2, 128^2$ and $256^2$. Also, we set $N=2$ and restrict the maximum value of $(\Delta_{i,N}^x, \Delta_{i,N}^y)$ to 0.08 in most of our experiments. For more details on our architecture, please refer to our supplement.
\noindent\textbf{Generate $m$.} In order to blend the inversed result $\widehat{x_{in}}$ and $x_{out}$ in the RGB domain (Eqn.~\eqref{eqn:blending}) for a photo-realistic result, it is crucial to find a blending mask $m$ which precisely distinguish the OOD contents only.
As is shown in Fig.~\ref{fig:mask}, we observe that when we set $\phi_{area}$ to a small value in the training stage, the high-intensity values in $M_{i,N}$ primarily gather around the OOD area (e.g., earrings, glasses). The mask of different resolutions focuses on slightly different areas because the target texture of each layer in the source generator is different. Usually, the lower layers of the source generator focus on larger structures, such as the rough shape of the face, and the higher layers focus on detailed structures, such as the skin texture and accessories, which is also observed in \cite{richardson2021encoding, tov2021designing, alaluf2021HyperStyle}.
Therefore, we could gather $M_{i,N}$ in each layer of generation for a final blending mask $m$ during inference. Instead of directly upscale and merging all $M_{i,N}$ to $m$, we consider that consistent high-intensity area as the OOD area. We adopt a merging function, which is similar to Eqn.~\eqref{eqn:aligned_mask}, sequentially merge $M_{i, N}$, $i\in \{1,...,L \}$, as
\begin{equation}
\label{eqn:aligned_mask2}
\begin{aligned}
\widetilde{M_{i,N}} =~&{\rm \Uparrow}(M_{i,N}) \cdot \widetilde{M_{(i-1), N}} \\
+&\widetilde{M_{(i-1), N}} \cdot (1 - \widetilde{M_{(i-1), N}}),
\end{aligned}
\end{equation}
where $\widetilde{M_{1, N}}={\rm \Uparrow}(M_{1, N})$, $\Uparrow$ means the up-sampling to the output resolution, i.e., 1024.
We have $m = \widetilde{M_{L,N}}$.
\noindent\textbf{Generate $\hat{x}$.} Our final output of GAN inversion result $\hat{x}$ is produced following Eqn.~\eqref{eqn:decomp_gan_inversion}. The blending process is visualized in Fig.~\ref{fig:blending}, where we blend $\widehat{x_{in}}$ and $x_{out}$ with $m$.
\subsection{Training Objectives}
\label{sec:training}
In Sec.~\ref{sec:SAMM}, we propose the SAMM to align the generated feature $g$ to $f$, where the invertibility is defined by $m$.
In this section, we demonstrate the loss functions to train SAMM.
First, we assume that
the subtle misalignment in $\widehat{x_{in}}$ can be fixed with simple spatial operations in Eqn.~\eqref{eqn:grid_sample}.
Hence, we train our framework by minimizing the reconstruction loss $L_{rec}$ on $\hat{x}$. Here we adopt the VGG perceptual loss~\cite{johnson2016perceptual} $L_{per}$, the MSE loss and the ArcFace~\cite{deng2018arcface} identity loss $L_{id}$ as our reconstruction objectives, that
\begin{equation}
L_{rec} = L_{per}(x, \hat{x}) + {\rm MSE}(x, \hat{x}) + L_{id}(x, \hat{x}),
\end{equation}
please refer to our supplement for detail definition of $L_{per}$ and $L_{id}$. Also, to make $\widehat{x_{in}}$ looks realistic, we keep the vanilla adversarial loss $L_{adv}$ for GAN model~\cite{karras2020analyzing} training. Here we skip the definition of $L_{adv}$ for simplicity. Refer to our supplement for more details.
\begin{figure*}[t]
\centering
\begin{tabular}{@{}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{}}
\scriptsize{Ground truth} &
\scriptsize{e4e~\cite{tov2021designing}} &
\scriptsize{HFGI$_{e4e}$~\cite{wang2021high}} &
\scriptsize{HyperStyle~\cite{alaluf2021HyperStyle}, iter=5} &
\scriptsize{SAM~\cite{parmar2022spatially}, iter=10} &
\scriptsize{FeatureStyle~\cite{xuyao2022}} &
\scriptsize{Ours}
\\
\includegraphics[width=0.14\linewidth]{Figures/inversion/3030/3030.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/3030/3030_e4e.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/3030/3030_hfgi.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/3030/3030_hyper.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/3030/3030_sam.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/3030/3030_fs.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/3030/3030_3.jpg}
\\
\includegraphics[width=0.14\linewidth]{Figures/inversion/7159/7159.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/7159/7159_e4e.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/7159/7159_hfgi.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/7159/7159_HyperStyle.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/7159/7159_sam.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/7159/7159_fs.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/7159/7159_3.jpg}
\\
\includegraphics[width=0.14\linewidth]{Figures/inversion/1178/1178.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/1178/1178_e4e.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/1178/1178_hfgi.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/1178/1178_HyperStyle.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/1178/1178_sam.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/1178/1178_fs.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/inversion/1178/1178_3.jpg}
\end{tabular}
\vspace{-1em}
\caption{Comparison of GAN inversion quality on CelebAMask-HQ~\cite{CelebAMask-HQ} testing dataset. Our method produces the best results in the hats and backgrounds since we skip the generation of the out-of-domain contents. Meanwhile, we eliminate the spatial misalignment in the generator features to better compose out-of-domain content with generated content without ghosting artifacts.}
\label{fig:inversion}
\vspace{-1em}
\end{figure*}
\begin{figure}[t]
\centering
\begin{tabular}{@{}c@{\hspace{2mm}}c@{\hspace{2mm}}c@{}}
\scriptsize{Input} & \scriptsize{Inversion} & \scriptsize{Narrow eyes}
\\
\includegraphics[width=0.31\linewidth]{Figures/inversion/821/821.jpg} &
\includegraphics[width=0.31\linewidth]{Figures/inversion/821/821_inversion.jpg} &
\includegraphics[width=0.31\linewidth]{Figures/inversion/821/821_edit_Narrow_Eyes.jpg}
\\
\multicolumn{3}{@{}c@{}}{\scriptsize{Predicted masks}}
\\
\multicolumn{3}{@{}c@{}}{\includegraphics[width=0.93\linewidth]{Figures/inversion/821/821_masks_inversion.jpg}}
\end{tabular}
\vspace{-1em}
\caption{The first row shows our GAN inversion and editing result with our method. From left to right, the second row shows the predicted masks $M_{iN}$ in the resolution of $32^2, 64^2, 128^3, 256^2$ and $m$ in the resolution of $1024^2$, respectively. }
\label{fig:mask}
\vspace{-1em}
\end{figure}
\subsubsection{Mask Regularization}
We hope to maximize the area of $x_{in}$ to better utilize the GAN invertibility for the follow-up applications (e.g., face attribute manipulation).
Thus, we train SAMM to produce $m$ with the maximum low-intensity area under the supervision of $L_{rec}$, since the region with low-intensity value in $M_{i, N}$ indicates only minor correction in $g_i$ is needed, which also implicitly defines ID areas in $x$.
Inspired by \cite{bae2022furrygan}, we use a regularization loss $L_{mask}$ on $m$. $L_{mask}$ consists of the binary regularization $L_{bin}$ and the area regularization $L_{area}$:
\begin{gather}
L_{bin}(M_{i, N}) = {\rm min}(M_{i, N}, (1-M_{i, N})), \\
\resizebox{0.85\linewidth}{!}{$L_{area}(M_{i, N}, \phi_{area, i}) = min(0, \phi_{area} - \frac{1}{|M_{i, N}|}\sum M_{i, N})$},
\end{gather}
where $\phi_{area, i}$ is the expect OOD size in the $i$-th layer,
and $|M_{i, N}|$ is the pixel count of mask $M_{i, N}$. Finally we have
\begin{equation}
\small
L_{mask} = \sum_{i=1}^{L} [\lambda_1 L_{bin}(M_{i, N}) + L_{area}(M_{i, N}, \phi_{area,i})],
\end{equation}
where $\lambda_1$ is the loss weight for $L_{bin}$.
In summary, our overall objective $L_{total}$ is:
\begin{equation}
L_{total} = L_{rec} + L_{adv} + L_{mask}.
\end{equation}
We found that if we loosen the $L_{mask}$ with a larger $\phi_{area}$ and a smaller $\lambda_1$, the predicted $m$ has a high value (close to 1) at most pixels except for the eyes and mouth area.
Consequently, the reconstruction error of $\widehat{x}$ can be optimized to be low, while the editability must be harmed. As we hope to maintain the off-the-shelf editability of styleGAN generator, in this paper, we set $\phi_{area} = 0.3$ for $32^2, 64^2$ masks and $\phi_{area} = 0.25$ for $128^2, 256^2$ masks.
\begin{figure*}[t]
\centering
\begin{tabular}{@{}c@{\hspace{2mm}}c@{\hspace{2mm}}c@{\hspace{2mm}}c@{\hspace{2mm}}c@{\hspace{2mm}}c@{\hspace{2mm}}c@{}}
\footnotesize{Input} & \footnotesize{Inversion} & \footnotesize{+Age} & \footnotesize{+Beard} & \footnotesize{-Smile} & \footnotesize{+Thick eyebrows}
\\
\includegraphics[width=0.15\linewidth]{Figures/inversion/2305/input.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2305/inversion.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2305/edit_Old.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2305/edit_Beard.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2305/edit_Smiling.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2305/2305_edit_Bushy_Eyebrows_5.jpg}
\\
\includegraphics[width=0.15\linewidth]{Figures/inversion/2720/2720.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2720/2720_inversion.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2720/2720_edit_Old.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2720/2720_edit_Beard.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2720/2720_edit_Mouth_Open.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2720/2720_edit_Bushy_Eyebrows_5.jpg}
\\
\includegraphics[width=0.15\linewidth]{Figures/inversion/2010/2010.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2010/2010_inversion.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2010/2010_edit_Old.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2010/2010_edit_Beard.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2010/2010_edit_Smiling.jpg} &
\includegraphics[width=0.15\linewidth]{Figures/inversion/2010/2010_edit_Bushy_Eyebrows.jpg}
\end{tabular}
\vspace{-1em}
\caption{Face manipulation results of our framework on CelebAMask-HQ~\cite{CelebAMask-HQ} dataset, please zoom in for detail.}
\label{fig:editing}
\vspace{-1em}
\end{figure*}
\begin{figure}[t]
\centering
\begin{tabular}{@{}c@{\hspace{2mm}}c@{\hspace{2mm}}c@{}}
\scriptsize{Input} &
\scriptsize{Inversion} &
\scriptsize{$\rightarrow$``Pink hair"}
\\
\includegraphics[width=0.31\linewidth]{Figures/editing/534/534.jpg} &
\includegraphics[width=0.31\linewidth]{Figures/editing/534/534_inversion.jpg} &
\includegraphics[width=0.31\linewidth]{Figures/editing/534/534_inversion_pink.jpg}
\\
\multicolumn{3}{@{}c@{}}{\scriptsize{Predicted masks}}
\\
\multicolumn{3}{@{}c@{}}{\includegraphics[width=0.93\linewidth]{Figures/editing/534/534_masks_inversion.jpg}}
\end{tabular}
\vspace{-0.5em}
\caption{\textbf{Edit with text guidance.} Our method also support off-the-shelf attribute manipulation with text guidance~\cite{patashnik2021styleclip} for in-domain area (i.e., hair color). }
\label{fig:clip_edit}
\vspace{-1.5em}
\end{figure}
\section{Experiments}
In this paper, we adopt the official checkpoints of e4e~\cite{tov2021designing} encoder $E$ and StyleGAN2~\cite{karras2020analyzing} (config-f) generator $G$ pretrained on the FFHQ~\cite{2018StyleGAN} dataset and fixed their parameters. Then, we train our SAMM with $E$ and $G$ on FFHQ for GAN inversion and minimize $L_{total}$.
Please refer to our supplement for more training details.
\begin{figure*}[tp]
\centering
\begin{tabular}{@{}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{}}
\scriptsize{Ground truth} &
\scriptsize{HFGI$_{e4e}$~\cite{wang2021high}} &
\scriptsize{HyperStyle~\cite{alaluf2021HyperStyle}} &
\scriptsize{SAM~\cite{parmar2022spatially}, iter=500} &
\scriptsize{FeatureStyle~\cite{xuyao2022}} &
\scriptsize{DiffCAM~\cite{song2022editing}} &
\scriptsize{Ours}
\\
\multicolumn{7}{c}{- Age}
\\
\includegraphics[width=0.14\linewidth]{Figures/editing/9785/9785.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9785/9785_hfgi.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9785/9785_hyper.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9785/9785_sam500.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9785/9785_fs.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9785/9785_diffcam.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9785/9785_edit_Old.jpg}
\\
\multicolumn{7}{c}{+ Smile}
\\
\includegraphics[width=0.14\linewidth]{Figures/editing/10241/10241.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/10241/10241_hfgi.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/10241/10241_hyper.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/10241/10241_sam500.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/10241/10241_fs.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/10241/10241_diffcam.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/10241/10241_edit_Smiling.jpg}
\\
\multicolumn{7}{c}{+ Thick Eyebrows}
\\
\includegraphics[width=0.14\linewidth]{Figures/editing/9418/9418.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9418/9418_hfgi.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9418/9418_hyper.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9418/9418_sam500.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9418/9418_fs.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9418/9418_diffcam.jpg} &
\includegraphics[width=0.14\linewidth]{Figures/editing/9418/9418_edit_Bushy_Eyebrows.jpg}
\end{tabular}
\vspace{-1em}
\caption{Comparison of attribute manipulation quality via GAN inversion. We apply same edit on the latent vector, and generate the editing results with baseline methods and our method. Our method produces the best result with well-preserved out-of-domain content and high-fidelity generated content. Please zoom in for detail.}
\label{fig:editing_compare}
\vspace{-1em}
\end{figure*}
\label{sec:eval}
\begin{table}[t]
\centering
\resizebox{1.0\linewidth}{!}{
\begin{tabular}{@{}l@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{\hspace{1mm}}c@{}}
\toprule
Method & PSNR$\uparrow$ & SSIM$\uparrow$ & LPIPS$\downarrow$ & FID$\downarrow$ & Runtime(s)$\downarrow$
\\
\midrule
e4e~\cite{tov2021designing} & 20.30 & 0.665 & 0.350 & 34.98 & 0.145\\
HFGI$_{e4e}$~\cite{wang2021high} & 23.66 & 0.724 & 0.285 & 22.79 & 0.159 \\
HyperStyle~\cite{alaluf2021HyperStyle}, iter=5 & 23.59 & 0.728 & 0.300 & 30.49 & 0.915 \\
SAM~\cite{parmar2022spatially}, iter=10 & 21.95 & 0.684 & 0.311 & 27.38 & 1.199\\
FeatureStyle~\cite{xuyao2022} & 25.24 & 0.717 & 0.188 & 16.86 & 0.181\\
\midrule
Ours$_{e4e}$, N=1 & 26.97 & 0.892 & 0.147 & 13.33 & 0.177\\
Ours$_{e4e}$ & \textbf{27.08} & \textbf{0.897} & \textbf{0.143} & \textbf{13.13} & 0.341\\
\bottomrule
\end{tabular}
}
\caption{Quantitative evaluation of GAN inversion quality on the first 1,000 images in the CelebAHQ-Mask~\cite{CelebAMask-HQ} testing dataset. The runtime is measured on a single nvidia RTX3090 GPU.}
\label{tab:quantitative_evaluation}
\vspace{-1em}
\end{table}
\noindent \textbf{Face inversion.}
Following previous works~\cite{wang2021high, song2022editing, parmar2022spatially, xuyao2022}, we evaluate our model and state-of-the-art baseline methods on the first 1,000 images in the testing partition of CelebAHQ-Mask~\cite{CelebAMask-HQ} dataset, assessing the inversion quality.
We measure the image reconstruction accuracy with PSNR and SSIM~\cite{wang2004image}, the perceptual distance with LPIPS~\cite{zhang2018perceptual}, and also the distribution distance between the reconstructed image and the source image that is represented with Fréchet inception distance~\cite{heusel2017gans} (FID).
For HyperStyle~\cite{alaluf2021HyperStyle}, we set the optimization iteration to 5. For SAM~\cite{parmar2022spatially}, it takes over 131 seconds to finish 1001 iterations of optimization to produce a result, which is too time-consuming for testing. To make a fair comparison, we set the optimization iteration of SAM to 10 in our experiment. Also, because DiffCAM~\cite{song2022editing} only works for attribute editing, we do not evaluate this model for face inversion.
As is shown in Tab.~\ref{tab:quantitative_evaluation}, our model outperforms previous methods, achieving the best restoration quality measure with different metrics.
Note that our method also maintains editability on the facial area.
We also visualize several inversion results in Fig.~\ref{fig:inversion} for comparison. Our method preserves meticulous details in the background, hats, and even the cigarette on the face by conducting the blending for the out-of-domain objects during the generation. Please refer to our supplement for more visual results.
\noindent \textbf{Attribute manipulation.}
Also, we have compared our method with state-of-the-art methods~\cite{wang2021high, alaluf2021restyle, alaluf2021HyperStyle, parmar2022spatially} on face attribute manipulation~\cite{harkonen2020ganspace,patashnik2021styleclip}.
In Fig.~\ref{fig:editing}, we apply off-the-shelf GAN editing approach~\cite{shen2020interfacegan, shen2020interfacegan} with our framework on CelebAHQ-Mask~\cite{CelebAMask-HQ} images, our method preserves the out-of-domain contents regardless of the editing direction. Moreover, as shown in Fig.~\ref{fig:clip_edit}, we also successfully perform a text-guided semantic editing on the hair color with the CLIP~\cite{radford2021learning,patashnik2021styleclip} model.
In Fig.~\ref{fig:editing_compare}, we compare our attribute editing performance with baseline approaches. Our framework produces high-fidelity editing results without undesired artifacts and provides the best editing quality against existing works.
Compared with ~\cite{alaluf2021HyperStyle, parmar2022spatially, wang2021high, xuyao2022}, our work preserves more details in microphone, hat, and background.
In addition, we notice an unnatural artifact in the results of DiffCAM~\cite{song2022editing}. Please see Fig.~\ref{fig:teaser} for a detailed comparison. We hypothesize such an artifact is introduced by their ghosting removal module. Instead, our SAMM module helps decrease the ID reconstruction error along with the generation of $\widehat{x_{in}}$. Thus we do not need a post-processing process after the blending. Furthermore, ours works better than ~\cite{song2022editing, parmar2022spatially} in cases with occlusions on faces. For more visual results, please refer to our supplement.
\subsection{Ablation Study}
\label{sec:ablation}
In this section, we conduct ablation study on our spatial alignment and masking module.
\noindent \textbf{Spatial alignment.}
To study the effectiveness of our spatial alignment module with the optical flow, we first skip the spatial alignment in the generation and visualize the results in Fig.~\ref{fig:abla_SA}. Apparently, without the spatial alignment, there are more reconstruction errors in $\widehat{x_{in}}$, which also causes more ghosting artifacts in the blending result (Fig.~\ref{fig:abla_SA} (a)). Also, we train a mask predictor without the spatial alignment module. The mask predictor trained with our full model highlights the OOD areas much more precisely (Fig.~\ref{fig:abla_SA} (d)) than the mask predictor trained without spatial alignment (Fig.~\ref{fig:abla_SA} (c)). Consequently, the editablity of our proposed framework is better preserved with spatial alignment since more areas in the input image are considered as the ID partition which could be manipulate with off-the-shelf GAN editing approaches. For more details, please refer to our supplement.
\begin{figure}[t]
\centering
\begin{tabular}{@{}c@{}c@{}c@{}}
\scriptsize{Input} &
\scriptsize{(a) Ours w/o spatial alignment} &
\scriptsize{(b) Ours}
\\
\multicolumn{3}{@{}c@{}}{\includegraphics[width=1.0\columnwidth]{Figures/Sec5/abla_sa.pdf}}
\\
\multicolumn{3}{@{}c@{}}{\scriptsize{(c) Predicted invertibility masks w/o spatial alignment}}
\\
\multicolumn{3}{@{}c@{}}{\includegraphics[width=1.0\columnwidth]{Figures/Sec5/250_masks_inversion_woSA.jpg}}
\\
\multicolumn{3}{@{}c@{}}{\scriptsize{(d) Predicted invertibility masks}}
\\
\multicolumn{3}{@{}c@{}}{\includegraphics[width=1.0\columnwidth]{Figures/Sec5/250_masks_inversion.jpg}}
\\
\resizebox{0.333\columnwidth}{!}{~} &
\resizebox{0.333\columnwidth}{!}{~} &
\resizebox{0.333\columnwidth}{!}{~}
\vspace{-1.7em}
\end{tabular}
\caption{\textbf{Ablation study on spatial alignment.} (a): we skip the grid sampling operation on $g_i$ and inverse the image with the latent vector and the OOD prediction. Compared to the result of our full model (b), the ghosting artifact is evident in (a). Also, we visualize the mask prediction without and with the spatial alignment module in (c) and (d), respectively. }
\label{fig:abla_SA}
\vspace{-2em}
\end{figure}
\noindent \textbf{Iterative alignment.}
In Sec.~\ref{sec:SAMM}, we introduce the iterative alignment strategy for flow and mask prediction. To study the influence of iterative alignment, we set $N=1$ and see a performance drop in the inversion quality (Tab.~\ref{tab:quantitative_evaluation}). Therefore, our iterative alignment strategy does decrease the reconstruction error in the results.
Also, the average intensity of the predicted masks $M_{i,N}$ of the $N=1$ model is $2.4\%$ larger than the $N=2$ model on our testing dataset. The higher intensity will block the manipulation on more ID areas.
We also conduct more experiments to investigate the usefulness of our loss function on the masks. Please refer to our supplement for details.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we propose a novel framework for out-of-domain GAN inversion via invertibility decomposition.
We design the SAMM to predict the invertibility mask and the optical flows along with the generation process.
Hence, we can separate the OOD area from the input image and inverse the ID area with the pretrained encoder and the generator, which facilitate attribute manipulation in the ID area with off-the-shelf approaches. Also, we align the generated features with the input features for better reconstruction accuracy in the ID area.
Finally, we blend the OOD contents with the generated image seamlessly. In our experiments, our framework produces photo-realistic results and is significantly superior to previous works in reconstruction accuracy and generation fidelity. We will extend our work to more target domains (e.g., Cars, Animals) in the future.
{\small
\bibliographystyle{ieee_fullname}
|
2,869,038,154,773 | arxiv | \section{Introduction}
Quantum field theory (QFT) is one of the most elegant and accurate frameworks which describes nature to a very high accuracy. QFTs admit deformations by operators which trigger a flow, known as the renormalization group (RG) flow, as we look into the theory at different scales \cite{Wilson}, \cite{Polchinski}. The fixed points of the RG-flow trajectory define conformal field theories (CFT) which are crucial in understanding aspects of condensed matter physics and string theory. However, it is equally significant to understand the RG flow away from the fixed points by deforming the QFT with a relevant or irrelevant operator. A relevant deformation stimulates the flow at lower energies (IR) while an irrelevant deformation drives the flow at higher energies (UV). Exploring the later is more difficult as it requires infinitely many counterterms in computing physical quantities in such a theory. However, a special kind of irrelevant deformation, known as the the $T\bar{T}$-deformation, has been introduced recently in two dimensions \cite{Zamolodchikov}, \cite{Smirnov} and has received wide attention in the past few years as the theory is solvable.
\\\\
The $T\bar{T}$-deformation is generated by the irrelevant operator $\det (T_{\mu\nu})$, the determinant of the energy-momentum tensor $T_{\mu\nu}$ of the theory, and thus can be considered in any generic QFT. At the classical level, the Lagrangian of the deformed theory itself looks very interesting and was obtained in \cite{Andrea}, \cite{Bonelli_2018}. For example, the Lagrangian of a $T\bar{T}$-deformed free massless boson is equivalent to the Nambu-Goto action for a string in three spacetime dimensions in the static gauge \cite{Andrea}. Although the deformation makes the theory non-local, often complicated and non-renormalizable, there are several remarkable features that make the deformed theory so compelling. One such feature is that the energy spectrum of the deformed theory can be derived non-perturbatively and in a compact form \cite{Zamolodchikov}. However, the spectrum appears to be sensitive to the sign of the $T\bar{T}$-coupling. For one of the signs, the highly excited states in the spectrum carry complex energies where the deformed theory becomes non-unitary. For this particular sign of the coupling, a holographic dual was proposed in \cite{McGough} and was further investigated in \cite{Kraus} - \cite{Caputa}. For the other sign of the coupling, the spectrum is real and the deformed theory is unitary. Moreover, at high energies, the density of states exhibit Hagedorn behaviour like the two dimensional little string theory (LST). In the context of holography, it was argued that certain single-trace $T\bar{T}$-deformation of two dimensional CFTs corresponds to a two dimensional vacuum of LST \cite{Giveon1}, see \cite{Giveon2}-\cite{Barbon} for related exciting works in this direction.
Another interesting aspect of the two dimensional $T\bar{T}$-deformed theories is the partition function - one can compute the partition function of the deformed theory on a torus or a cylinder or a disk and each of them satisfy a linear diffusion-type differential equation \cite{Cardy1}. In particular, $T\bar{T}$-deformation of a CFT was considered in \cite{Datta} and the torus partition function of the deformed CFT was obtained to be modular invariant despite the fact that the deformed theory is not conformal. See \cite{Aharony}, \cite{Jiang} for further exciting results concerning the partition functions in the $T\bar{T}$-deformed CFTs.
\\\\
Although the $T\bar{T}$-deformation can be defined in any two dimensional QFT, a special interest has been taken in the study of integrable quantum systems \cite{Smirnov}, \cite{Delfino}-\cite{Caselle}. Integrable QFTs contain infinite number of conserved charges. The reason for this interest is that a $T\bar{T}$ deformation preserves the integrable structure of the theory \cite{Smirnov}. In \cite{Smirnov}, it was also argued that the $T\bar{T}$-deformation modifies the $S$-matrix only by a CDD phase factor.
\\\\
Being an irrelevant deformation, the Lagrangian of the $T\bar{T}$-deformed theory is apparently non-renormalizable as it involves an infinite number of counterterms. However, it is the integrable structure of the theory that enables one to obtain a renormalized Lagrangian in such a naively non-renormalizable theory. The $T\bar{T}$-deformed integrable theory provides an infinite number of constraints which uniquely fix the all counterterms that appear in the theory. In \cite{Rosenhaus}, the authors considered a $T\bar{T}$-deformed free scalar field theory and showed how to derive the renormalized Lagrangian perturbatively by demanding that it should produce the correct $S$-matrix. Obtaining a renormalized Lagrangian unambiguously allows one to compute all physical quantities of interest. This motivates us to consider the $T\bar{T}$-deformed free massive Dirac fermion in two dimensions and to compute the renormalized Lagrangian of such a theory perturbatively. As we will see, the renormalized Lagrangian exists and is qualitatively very different from the classical Lagrangian. In particular, while the classical Lagrangian has only one scale, namely the bare coupling constant, the renormalized Lagrangian to second order reveals three different scales in the theory.
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In this paper, the perturbative renormalization of the $T\bar{T}$-deformed free massive Dirac fermion in two dimensions will be explicitly performed. This will be done using the LSZ reduction formula to compute the two-to-two $S$-matrix and introducing counterterms to cancel divergences. The paper is organized as follows: In section \ref{integrability} the basics of $T\bar{T}$-deformations as well as some related aspects of integrability will be reviewed. In section \ref{classicalLag} the $T\bar{T}$-deformed free massive Dirac fermion in two dimensions will be considered. In section \ref{renormalizedLag} the renormalized Lagrangian to second order in the $T\bar{T}$-coupling constant will be computed. Finally, in section \ref{Discussion} the main results and future directions will be discussed.
\section{{Integrability and $T\bar{T}$-deformation s}}
\label{integrability}
In this section several aspects of integrability, two-dimensional quantum field theories and $T\bar{T}$-deformations are discussed.
\vspace{2mm}
\\
In two dimensional integrable theories the $S$-matrix of any scattering process factorizes into two-to-two $S$-matrices and there is no particle production \cite{ZZ}-\cite{Bombardelli}.
Consider a two-to-two scattering process of particles with identical mass $m$ in two dimensions. The 2-momentum of the $i^{th}$ particle can be parameterized by its rapidity, $\theta_i$\footnote{Rapidity or the parameter of velocity is defined as, $\theta_i=\tanh^{-1}(v_i)$, where $v_i$ is the velocity of the $i$-th particle and the speed of light is set to $c=1$.}, as,
\begin{equation}
p_i^{\mu}=
\left(\begin{array}{lr}
m\cosh\theta_i\\
m\sinh\theta_i
\end{array}\right)\ .
\label{twomomenta}
\end{equation}
Identify $p_i^{0}=m \cosh\theta_i\equiv E_i(\theta_i)$ as the energy of the $i^{th}$ particle and $p_i^{1}=m \sinh\theta_i\equiv P_i(\theta_i)$ as the momentum of the $i^{th}$ particle.
\\
Two dimensional kinematics of the two-to-two scattering processes have the unique property that the incoming momenta of the particles equal the outgoing momenta of the particles. This fact and the Lorentz invariance implies that the $S$-matrix, denoted by $S(\theta)$, will only be a function of the difference of rapidities, $\theta=\theta_1-\theta_2$.
\vspace{2mm}
\\
In what follows the $T\bar{T}$-deformation of an integrable field theory will be considered. If a $T\bar{T}$-deformation is performed on an integrable field theory, the deformed theory is integrable as well \cite{Smirnov}. As we mentioned earlier, there is no particle production in a scattering process in an integrable theory and hence, unitarity demands
\begin{eqnarray}
|S(\theta)|^2=1\ .
\label{unitarity}
\end{eqnarray}
On the other hand, the crossing symmetry of the $S$-matrix\footnote{It is the symmetry of the $S$-matrix under the interchange of the $s$ and $u$-channels as will be shown later.} implies,
\begin{eqnarray}
S(\theta)=S(i\pi-\theta)\ .
\label{crossing}
\end{eqnarray}
The solution to (\ref{unitarity}) and (\ref{crossing}) is simple and given by the CDD factor,
\begin{eqnarray}
S_{\alpha}(\theta)=\frac{\sinh\theta-i\sin\alpha}{\sinh\theta+i\sin\alpha} \ ,
\label{CDD}
\end{eqnarray}
where $\alpha$ is a real parameter related to the coupling constant. The product of $S_{\alpha}(\theta)$ over $\alpha$, $\prod_{\alpha}S_{\alpha}(\theta)$, is also a solution to (\ref{unitarity}) and (\ref{crossing}).
\\
\\
An alternative solution for the CDD factor was recently considered by Smirnov and Zamolodchikov \cite{Smirnov}, which admits the following representation
\begin{eqnarray}
S_{\alpha}(\theta)= e^{i\alpha \sinh\theta} \ .
\label{CDDSZ}
\end{eqnarray}
The authors considered $T\bar{T}$-deformed theories in two dimensions\footnote{To be precise, the authors in \cite{Smirnov} considered integrable quantum field theories (IQFT) and deformed them by generic scalars $X_s$ such that the deformations preserve integrability. Being IQFT, these theories have an infinite number of conserved currents ($T_{s+1}(z), \Theta_{s-1}(z)$) and ($\bar{T}_{s+1}(z), \bar{\Theta}_{s-1}(z)$), where the index $s$ represents the spin of the corresponding fields thus labeling the currents. The scalars $X_s$ are defined in terms of these local currents. For example, $X_1$ is precisely the composite operator $T\bar{T}$ \hspace{1mm}considered in our paper.} where the deformation produces a one-parameter family of Lagrangians obeying the $T\bar{T}$ flow equation,
\begin{eqnarray}
\frac{\partial \mathcal{L}(\lambda)}{\partial \lambda}=-4\Big(T^{\lambda}(z)\bar{T}^{\lambda}(z)-\theta^{\lambda}(z)\bar\theta^{\lambda}(z)\Big)\ ,
\label{floweqn}
\end{eqnarray}
$\lambda$ being the $T\bar{T}$-coupling constant, $T^{\lambda}=T_{zz}^{\lambda}$, $\bar{T}^{\lambda}=T_{\bar{z} \bar{z}}^{\lambda}$ and $\theta^{\lambda}=\bar\theta^{\lambda}=T_{z\bar{z}}^{\lambda}$ are the components of the energy momentum tensor of the deformed theory and $\mathcal{L}(\lambda=0)$ is the Lagrangian of the undeformed theory.
It was argued that the deformed theory is integrable and the $S$-matrix of this theory can be expressed in a factorizable form, $\hat{S'}(\theta)S(\theta)$, where $S(\theta)$ is the CDD factor determined by unitarity and crossing-symmetry of the $S$-matrix,
\begin{eqnarray}
S(\theta)=e^{i\lambda m^2 \sinh\theta}\ .
\label{CDDSZ1}
\end{eqnarray}
The factor $\hat{S'}(\theta)$ signifies the presence of mass degeneracies in the spectrum and it satisfies the Yang-Baxter equation \cite{Bombardelli}. This typically fixes the “flavor” structure of the $S$-matrix. Simply, the $T\bar{T}$-deformation (\ref{floweqn}) corresponds to multiplying the $S$-matrix by the factor (\ref{CDDSZ1}).
\vspace{2mm}
\\
However, observe that (\ref{CDDSZ1}) grows exponentially at large imaginary momenta.
This behaviour is inconsistent with the analytic behaviour of $S$-matrices in a local QFT.
Nevertheless, rather than just throwing out these theories, one can try to understand such theories as QFTs coupled to gravity \cite{Gorbenko} - \cite{Conti2}.
In this paper, however, we will be restricted to low-energy regime of the $T\bar{T}$-deformed theories. In this regime these are quantum field theories and can be studied perturbatively in the $T\bar{T}$-coupling $\lambda$ around $\lambda=0$ \cite{Kraus}, \cite{Rosenhaus}, \cite{He1}-\cite{Dey} \footnote{For non-perturbative studies of the $T\bar{T}$-deformed theories, see \cite{Cardy2}, \cite{Haruna}.}.
\section{The $T\bar{T}$-deformed free massive Dirac fermion}
\label{classicalLag}
Consider the Euclidean action for the free massive Dirac fermion with mass $m$,
\begin{equation}
I_0=\int dx_1 dx_2\, \mathcal{L}_{0} =\int dx_1 dx_2\, \big(i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi -m \bar{\psi} \psi \big)~.
\label{freeaction}
\end{equation}
Performing a Wick rotation to the Euclidean action $x_1=i t$ and $x_2=x$, yields the Lorentzian action,
\begin{equation}
-I_0=-\int dx_1 dx_2\, \mathcal{L}_{0} =i\int dt dx\, \big(-i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi +m \bar{\psi} \psi \big)\ ,
\label{freeactionLor}
\end{equation}
where the $\gamma^{\mu}$s are the two dimensional gamma matrices satisfying the Clifford algebra,
\begin{equation}
\{\gamma^{\mu},\gamma^{\nu}\}=2\ \eta^{\mu\nu} \ \mathbb{I}
\label{Cliffordalgebra}
\end{equation}
and $\eta_{\mu\nu}$ is the two dimensional Minkowski metric. We represent the gamma matrices in terms of the Pauli matrices: $\gamma_0=\sigma_x$ and $\gamma_1=-i\sigma_y$.
\\\\
The equation of motion for the fermionic fields $\psi$ and $\bar{\psi}$ are given by,
\begin{eqnarray}
i \gamma^{\mu}{\partial}_{\mu}\psi-m\psi = 0\:\:\:\:\:\:\text{and}\:\:\:\:\:\:
i {\partial}_{\mu}\bar{\psi} \gamma^{\mu}+m\bar{\psi} =0.
\label{Diraceq}
\end{eqnarray}
To solve the Dirac equation (\ref{Diraceq}), make the ansatz, $\psi(x)=u(k)e^{-i k . x}$. Plugging the ansatz
into the Dirac equation and using the normalization $u^{\dagger}(k)u(k)=1$, one finds the normalized positive energy plane wave solution,
\begin{align}
u(k)=\left(\begin{array}{c}\sqrt{E_{+}+k^1}\\ \frac{m}{\sqrt{E_{+}+k^1}}\end{array}\right) \hspace{3mm}
= \hspace{2mm}\left(\begin{array}{c}
\sqrt{k^0+k^1}\\\sqrt{k^0-k^1}
\end{array}\right)
\label{+vesoln}
\end{align}
where, $E_{+}=k^0=\sqrt{(k^1)^2+m^2}$.
\\\\
Similarly, using $\psi(x)=v(k)e^{i k . x}$ one can obtain the negative energy plane wave solution,
\begin{align}
v(k)=\left(\begin{array}{c}-\sqrt{E_{-}+k^1}\\ \frac{m}{\sqrt{E_{-}+k^1}}\end{array}\right) \hspace{3mm}
= \hspace{2mm}\left(\begin{array}{c}
-\sqrt{k^0+k^1}\\\sqrt{k^0-k^1}
\end{array}\right)
\label{-vesoln}
\end{align}
where, $E_{-}=k^0=-\sqrt{(k^1)^2+m^2}$.
\\\\
Hence, the solution to the wave equation for the free massive Dirac fermion is,
\begin{eqnarray}
\psi(x)&=&\int \frac{d\vec{k}}{2\pi}\frac{1}{\sqrt{2E}}\left(a(\vec{k})u(\vec{k})e^{-ik\cdot x}+b^{\dagger}(\vec{k})v(\vec{k})e^{ik\cdot x}\right)\ ,
\end{eqnarray}
where $a^{\dagger}(\vec{k})$ and $a(\vec{k})$ are the fermion creation and annihilation operators respectively, while $b^{\dagger}(\vec{k})$ and $b(\vec{k})$ are the anti-fermion creation and annihilation operators respectively. The creation and annihilation operators satisfy the Clifford algebra,
\begin{eqnarray}
\{a(\vec{k_1}), a^{\dagger}(\vec{k_2})\}=2\pi \delta(\vec{k_1}-\vec{k_2})\:\:\:\:\:\:\text{and}\:\:\:\:\:\:
\{b(\vec{k_1}), b^{\dagger}(\vec{k_2})\}=2\pi \delta(\vec{k_1}-\vec{k_2})\ ,
\end{eqnarray}
while all other anti-commutators vanish. The Dirac adjoint of a spinor $\psi$ is defined as, $\bar{\psi}=\psi^{\dagger}\gamma^0$.
\\\\
Consider the $\bar{T}T$-deformation of the free massive Dirac fermion in two dimensional Euclidean spacetime,
\begin{equation}
I=\int d^{\,2}x\, \mathcal{L}(\lambda)=\int dx_1 dx_2\, \big(i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi -m \bar{\psi} \psi \big) + \lambda \int dx_1 dx_2\, \mathcal{O}_{T\bar{T}}\ ,
\label{deformedaction}
\end{equation}
where $\mathcal{O}_{T\bar{T}}$ is the local $T\bar{T}$-operator given by the determinant of the energy-momentum tensor,
\begin{equation}
\mathcal{O}_{T\bar{T}}=\det (T^{(\lambda)})=\frac{1}{2} \epsilon^{\mu\nu}\epsilon^{\rho\sigma}T_{\mu\nu}^{(\lambda)}T_{\rho\sigma}^{(\lambda)}
=\frac{1}{2}\Big[\big(T_{\mu}^{ \ \mu(\lambda)}\big)^2-T_{\mu\nu}^{(\lambda)}T^{\mu\nu(\lambda)}\Big]\ ,
\end{equation}
$\epsilon^{\mu\nu}$ the two-dimensional Levi-Civita tensor and $T_{\mu\nu}^{(\lambda)}$ the energy-momentum tensor of the finite-$\lambda$ theory.
\\
The canonical energy-momentum tensor of the undeformed theory is given by,
\begin{equation}
T_{\mu\nu (c)}^{(0)}=X_{\mu\nu}-\delta_{\mu\nu}(\Tr X -m \bar{\psi}\psi)\ ,
\label{canemtensor}
\end{equation}
where
\begin{eqnarray}
X_{\mu\nu}=\frac{i}{2} \big(\bar{\psi}\gamma_{\mu}{\partial}_{\nu}\psi
-{\partial}_{\nu}\bar{\psi} \gamma_{\mu}\psi\big)\ .
\label{Xmunu}
\end{eqnarray}
The above canonical energy-momentum tensor can be symmetrized using the Belinfante technique, yielding
\begin{eqnarray}
T_{\mu\nu}^{(0)}&=&\tilde{X}_{\mu\nu}-\delta_{\mu\nu}(\Tr X -m \bar{\psi}\psi)\ ,
\label{emtensor}
\end{eqnarray}
where
\begin{eqnarray}
\tilde{X}_{\mu\nu}&=&\frac{i}{2} \Big(\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi
-{\partial}_{(\mu}\bar{\psi} \gamma_{\nu)}\psi\Big) \nonumber\\
&=&\frac{i}{4} \big(\bar{\psi}\gamma_{\mu}{\partial}_{\nu}\psi+\bar{\psi}\gamma_{\nu}{\partial}_{\mu}\psi
-{\partial}_{\mu}\bar{\psi} \gamma_{\nu}\psi-{\partial}_{\nu}\bar{\psi} \gamma_{\mu}\psi\big)\ .
\label{Xtmunu}
\end{eqnarray}
The $T\bar{T}$-deformed Lagrangian of the free massive Dirac fermion can be obtained by solving the $T\bar{T}$ flow equation \cite{Bonelli_2018},
\begin{eqnarray}
\frac{\partial \mathcal{L}(\lambda)}{\partial{\lambda}}=\mathcal{O}_{T\bar{T}}\ ,
\label{floweqn1}
\end{eqnarray}
with the initial condition $\mathcal{L}(\lambda=0)=\mathcal{L}_0$.
\\
\\
Solving (\ref{floweqn1}) perturbatively in the $T\bar{T}$-coupling $\lambda$, one finds the $T\bar{T}$-deformed Lagrangian for the free massive Dirac fermion \cite{Bonelli_2018},
\begin{eqnarray}
\mathcal{L}(\lambda)&=& \big(i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi -m \bar{\psi} \psi \big)-\frac{\lambda}{2}\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{ \ \mu})^2+2m\bar{\psi}\psi \tilde{X}_{\mu}^{ \ \mu}-2m^2(\bar{\psi}\psi)^2\Big)\nonumber\\
&&+\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{ \ \mu})^2\Big)\ ,
\label{deformedLag1}
\end{eqnarray}
where $\tilde{X}_{\mu\nu}$ is given by (\ref{Xtmunu}). It is noteworthy to mention that the $T\bar{T}$-deformed Lagrangian (\ref{deformedLag1}) is exact in $\lambda$, all the higher order terms in $\lambda$ vanish identically due to the Grassmann nature of the fermion fields.\footnote{Although, terms proportional to $\tilde{X}^4$ can be present at third order in the $T\bar{T}$-coupling $\lambda$, the authors of \cite{Bonelli_2018} claimed that the $\mathcal{O}(\lambda^3)$ term vanishes by using Fierz identities. It is also possible to verify this claim directly by plugging the expression into MATHEMATICA.}
\\\\
Therefore, the $T\bar{T}$-deformed Lorentzian action of the free massive Dirac fermion is,
\begin{eqnarray}
-I=-\int dx_1 dx_2\, \mathcal{L} =i&\int& dt dx\, \Big[\big(-i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi +m \bar{\psi} \psi \big)+\frac{\lambda}{2}\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{ \ \mu})^2\nonumber\\
&+&2m\bar{\psi}\psi \tilde{X}_{\mu}^{ \ \mu}-2m^2(\bar{\psi}\psi)^2\Big)
-\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{ \ \mu})^2\Big) \Big] \nonumber\\
\label{deformedactionLor}
\end{eqnarray}
In what follows two-to-two scattering in the $T\bar{T}$-deformed free massive Dirac fermion theory will be considered. The $S$-matrix, $S(\theta)$, will be computed perturbatively to second order in the $T\bar{T}$-coupling $\lambda$. Finally the $S$-matrix will be compared with (\ref{CDDSZ1}) and the renormalized Lagrangian will be constructed.
\section{Renormalization of the $T\bar{T}$-deformed free massive Dirac fermion}
\label{renormalizedLag}
In section \ref{classicalLag}, the classical Lagrangian of the $T\bar{T}$-deformed free massive Dirac fermion was stated. In this section the renormalized Lagrangian will be computed to second order.
\vspace{2mm}
\\
In order to compute the renormalized Lagrangian similar methods to those found in \cite{Rosenhaus} will be used. First, the $S$-matrix will be computed using the classical Lagrangian giving
rise to UV divergences. Next, counterterms will be added to the Lagrangian in order to
cancel the divergences and ensure that the final $S$-matrix is given by (\ref{CDDSZ1}).
\subsection{The $S$-matrix}
\label{subsecSmatrix}
Consider the two-to-two scattering of a fermion and anti-fermion in the $T\bar{T}$-deformed free massive Dirac theory\footnote{One can also consider the other possible two-to-two scattering, namely, the fermion-fermion scattering. However, the resulting renormalized Lagrangian would be the same as it does not depend on the particular scattering process. Because of this fact only the fermion anti-fermion scattering process is considered.},
\begin{eqnarray}
f_1+\bar{f_2}\rightarrow f_3+\bar{f_4}\nonumber
\end{eqnarray}
where $f_1$ represents the incoming fermion with momentum $p_1$, $\bar{f_2}$ the incoming anti-fermion with momentum $p_2$, $f_3$ the outgoing fermion with momentum $p_3$ and $\bar{f_4}$ the outgoing anti-fermion with momentum $p_4$.
\\
Recall, due to the two dimensional kinematics of two-to-two scattering the momenta of the incoming particles equal the momenta of the outgoing particles. When considering fermion anti-fermion scattering the $0^{th}$-order $S$-matrix is just the identity. Hence, the momentum of the incoming fermion must equal the momentum of the outgoing fermion and the momentum of the incoming anti-fermion must equal the momentum of the outgoing anti-fermion\footnote{If scattering in a scalar theory is considered \cite{Rosenhaus}, one can choose $p_1=p_4$ and $p_2=p_3$ also. However, it is easy to see that if one works with fermions, we must have $p_1=p_3$ and $p_2=p_4$ to have a non-zero scattering matrix.}. In particular,
\begin{eqnarray}
p_1=p_3 \ \ \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \ \ \ p_2=p_4 .
\label{inoutmomenta}
\end{eqnarray}
By (\ref{twomomenta}) and (\ref{inoutmomenta}) the Mandelstam variables in the $(+,-)$ signature take the form,
\begin{eqnarray}
s&=&(p_1+p_2)^2=2m^2(1+\cosh\theta)\ , \nonumber \\
t&=&(p_1-p_3)^2=0 \ , \nonumber\\
u&=&(p_1-p_4)^2=2m^2(1-\cosh\theta)=4m^2-s\ .
\label{Mandelstam}
\end{eqnarray}
Observe that the Mandelstam variables can be related to each other by the transformations,
\begin{eqnarray}
u=s|_{\theta\rightarrow i\pi-\theta} \ \ \ \ \ \text{and} \ \ \ \ \ \ t=s|_{\theta\rightarrow i\pi}.
\label{crossingsymmetry}
\end{eqnarray}
Finally due to (\ref{inoutmomenta}), for the two-to-two scattering in an integrable theory the $S$-matrix, $S(\theta)$, is defined as,
\begin{eqnarray}
{}_{out} \langle p_3, p_4| p_1, p_2\rangle_{in}&=& (2\pi)^2 \delta(p_1 - p_3) \delta(p_2 - p_4) 2E(p_1) 2 E(p_2) S(\theta)\ ,
\end{eqnarray}
where the zeroth order $S$-matrix in the $T\bar{T}$-coupling is $S^{(0)}(\theta) = 1$.
The $S$-matrix $S(\theta)$ is related to the scattering amplitude $\mathcal{A}$ by \cite{Rosenhaus},
\begin{equation}
S(\theta)= \frac{\mathcal{A}}{ 4 m^2\sinh\theta}~.
\label{AtoS}
\end{equation}
However, before computing the $S$-matrix, express the deformed action (\ref{deformedactionLor}) in a simpler form. By the field redefinitions,
\begin{eqnarray}
\psi_{a}\rightarrow \psi_a+ \lambda \frac{m}{2} \psi_{a}(\bar{\psi}\psi)\:\:\:\:\:\text{and}\:\:\:\:\:
\bar{\psi}_{a}\rightarrow \bar{\psi}_a+ \lambda \frac{m}{2}(\bar{\psi}\psi) \bar{\psi}_{a}\ ,
\label{newfields}
\end{eqnarray}
where $a=\{1,2\}$ are the spinor indices, one can write the action (\ref{deformedactionLor}) as,
\begin{eqnarray}
-I=i\int dt dx\, &\Big[&\big(-i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi +m \bar{\psi} \psi \big)+\frac{\lambda}{2}\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\mu})^2\Big)\nonumber\\
&+&\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\mu})^2\Big)+\frac{7}{4}\lambda^2 m^2(\bar\psi\psi)^2 \ \tilde{X}_{\mu}^{\mu}-\frac{7}{4}\lambda^2 m^3 (\bar\psi\psi)^3
\Big].\nonumber\\
\label{deformedactionLor2}
\end{eqnarray}
For details see Appendix \ref{AppendixA}.
\\
\\
Observe that the new action no longer contains linear terms of the form $\bar{\psi}\psi \tilde{X}_{\mu}^{ \ \mu}$ and $(\bar{\psi}\psi)^2$, making it simpler to compute the one-loop bubble diagrams by decreasing the total number of terms. But, at second order, two new terms $(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{ \ \mu}$ and $(\bar{\psi}\psi)^3$ are generated which contribute to the tadpole diagrams at one-loop. However, these new terms give vanishing contributions as will be shown later.
\vspace{2mm}
\\
The two dimensional Dirac spinors $u(p_i)$, $\bar u(p_i)$, $v(p_i)$ and $\bar v(p_i)$ will arise in the computation of the $S$-matrix. Using (\ref{+vesoln}), (\ref{-vesoln}) and the fact that $p_3^{\mu}=p_1^{\mu}=(m \cosh \theta_1, \ m\sinh \theta_1)$ and $p_4^{\mu}=p_2^{\mu}=(m \cosh \theta_2, \ m\sinh \theta_2)$ the two dimensional Dirac spinors can be written as,
\begin{align}
u(p_1)=\sqrt{m}\left(\begin{array}{c}
\sqrt{\cosh\theta_1+\sinh\theta_1}\\\sqrt{\cosh\theta_1-\sinh\theta_1}
\end{array}\right)=u(p_3) \ , \hspace{4mm}
u(p_2)=\sqrt{m}\left(\begin{array}{c}
\sqrt{\cosh\theta_2+\sinh\theta_2}\\\sqrt{\cosh\theta_2-\sinh\theta_2}
\end{array}\right)=u(p_4) \nonumber\\
v(p_1)=\sqrt{m}\left(\begin{array}{c}
-\sqrt{\cosh\theta_1+\sinh\theta_1}\\\sqrt{\cosh\theta_1-\sinh\theta_1}
\end{array}\right)=v(p_3) \ , \hspace{4mm}
v(p_2)=\sqrt{m}\left(\begin{array}{c}
-\sqrt{\cosh\theta_2+\sinh\theta_2}\\\sqrt{\cosh\theta_2-\sinh\theta_2}
\end{array}\right)=v(p_4)\ .\nonumber\\
\label{uv}
\end{align}
The Dirac adjoints $\bar u(p_i)=u(p_i)^{\dagger} \gamma^0$ and $\bar v(p_i)=v(p_i)^{\dagger} \gamma^0$ can be written as,
\begin{align}
\bar u(p_1)=\sqrt{m}\left(\begin{array}{c}
\sqrt{\cosh\theta_1-\sinh\theta_1} \hspace{10mm} \sqrt{\cosh\theta_1+\sinh\theta_1}
\end{array}\right)=\bar u(p_3)\ , \nonumber\\
\bar u(p_2)=\sqrt{m}\left(\begin{array}{c}
\sqrt{\cosh\theta_2-\sinh\theta_2} \hspace{10mm} \sqrt{\cosh\theta_2+\sinh\theta_2}
\end{array}\right)=\bar u(p_4)\ , \nonumber\\
\bar v(p_1)=\sqrt{m}\left(\begin{array}{c}
\sqrt{\cosh\theta_1-\sinh\theta_1} \hspace{10mm} -\sqrt{\cosh\theta_1+\sinh\theta_1}
\end{array}\right)=\bar v(p_3)\ , \nonumber\\
\bar v(p_2)=\sqrt{m}\left(\begin{array}{c}
\sqrt{\cosh\theta_2-\sinh\theta_2} \hspace{10mm} -\sqrt{\cosh\theta_2+\sinh\theta_2}
\end{array}\right)=\bar v(p_4)\ .
\label{ubarvbar}
\end{align}
It is finally time to compute the $S$-matrix perturbatively up to second order in the $T\bar{T}$-coupling $\lambda$ using the redefined Lagrangian,
\begin{eqnarray}
\mathcal{L}(\lambda)=\mathcal{L}_0+\mathcal{L}_1(\lambda)+\mathcal{L}_2(\lambda)\ ,
\label{redefinedLag}
\end{eqnarray}
where,
\begin{eqnarray}
\mathcal{L}_0&=&-i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi +m \bar{\psi} \psi\ , \nonumber\\
\mathcal{L}_1(\lambda)&=&\frac{\lambda}{2}\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\Big)\ ,\nonumber\\
\mathcal{L}_2(\lambda)&=&\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{ \ \mu})^2\Big)+\frac{7}{4}\lambda^2 m^2(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{ \ \mu}-\frac{7}{4}\lambda^2 m^3(\bar{\psi}\psi)^3 .
\label{redefinedL0L1L2}
\end{eqnarray}
\vspace{1mm}
\subsubsection{First order $S$-matrix}
\begin{figure}[h!]
\centering \noindent
\includegraphics[width=8cm]{tree_level.pdf}
\caption{Tree-level diagram that contributes to the $S$-matrix at first order.
}
\label{treelevel diagram}
\end{figure}
At first order in the $T\bar{T}$-coupling $\lambda$, the $S$-matrix only gets a contribution from the tree-level diagrams of the form found in figure \ref{treelevel diagram}, where the vertex corresponds to the quartic couplings, $\tilde{X}_{\mu\nu} \tilde{X}^{\mu\nu}$ or $(\tilde{X}_{\mu}^{\ \mu})^2$. The total contribution to the tree-level amplitude of a fermion anti-fermion scatering process is,
\begin{eqnarray}
\mathcal{A}^{(1)}=i \frac{\lambda}{2}\langle 0 | b(p_4) a(p_3) \text{T}\big[\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle.
\label{treeamplitude}
\end{eqnarray}
In order to evaluate (\ref{treeamplitude}) compute the general contribution to the amplitude of four external fields with arbitrary indices for the fermion anti-fermion scattering process,
\begin{eqnarray}
\mathcal{A}^{(1)}_{\bar{\psi}_a\psi_b\bar{\psi}_c\psi_d}&=&\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}_a\psi_b\bar{\psi}_c\psi_d\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\ &=&\bar{u}_a(p_3)v_b(p_4)\bar{v}_c(p_2)u_d(p_1)+\bar{v}_a(p_2)u_b(p_1)\bar{u}_c(p_3)v_d(p_4)
-\bar{v}_a(p_2)v_b(p_4)\bar{u}_c(p_3)u_d(p_1)\nonumber\\
&&-\bar{u}_a(p_3)u_b(p_1)\bar{v}_c(p_2)v_d(p_4)
\label{generaltree}
\end{eqnarray}
where the four possible Wick contractions were performed, $u_a$, $v_a$, $\bar{u}_a$ and $\bar{v}_a$ are the Dirac spinors given by (\ref{uv}) and (\ref{ubarvbar}) and $a, b, c, d=\{1,2\}$ are the spinor indices.
\vspace{2mm}
\\
The total first order amplitude can be computed by plugging each vertex into (\ref{generaltree}) and adding the results together.
By (\ref{Xtmunu}), the quartic couplings $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$ and $(\tilde{X}_{\mu}^{\ \mu})^2$ can be expressed as,
\begin{align}
\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}
&=-\frac{1}{4} \Big(\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi
-2\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi
+{\partial}_{(\mu}\bar{\psi} \gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi\Big)
\label{TmunuSquare}\\
(\tilde{X}_{\mu}^{\ \mu})^2
&=-\frac{1}{4} \Big(\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \ \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi
-2\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \ {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi
+{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi \ {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi\Big)
\label{TmumuSquare}.
\end{align}
Using (\ref{generaltree}) the tree-level contribution from each of these terms in (\ref{TmunuSquare}) can be computed and thus the contribution from $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$ to the tree-level amplitude can be determined.
For example, the first term in (\ref{TmunuSquare}), $\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi$, contributes,
\begin{eqnarray}
&&\mathcal{A}_{\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi}^{(1)}\nonumber\\
&=&\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle \nonumber\\
&=&\gamma_{ab(\mu}\gamma_{cd}^{(\mu}\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}_a {\partial}_{\nu)}\psi_b\bar{\psi}_c{\partial}^{\nu)}\psi_d\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\
&=&\gamma_{ab(\mu}\gamma_{cd}^{(\mu} \Big[\bar{u}_a(p_3)v_b(p_4)\bar{v}_c(p_2)u_d(p_1) i p_{4\nu)} (-i)p_1^{\nu)}
+\bar{v}_a(p_2)u_b(p_1)\bar{u}_c(p_3)v_d(p_4)(-i)p_{1\nu)} i p_4^{\nu)}\nonumber\\
&&-\bar{v}_a(p_2)v_b(p_4)\bar{u}_c(p_3)u_d(p_1)i p_{4\nu)} (-i)p_1^{\nu)}
-\bar{u}_a(p_3)u_b(p_1)\bar{v}_c(p_2)v_d(p_4)(-i)p_{1\nu)} i p_4^{\nu)}\Big]\nonumber\\
&=& \bar{u}(p_1)\cdot \gamma_{(\mu}p_{2\nu)} \cdot v(p_2) \
\bar{v}(p_2)\cdot \gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
+\bar{v}(p_2)\cdot \gamma_{(\mu}p_{1\nu)} \cdot u(p_1) \
\bar{u}(p_1)\cdot \gamma^{(\mu}p_2^{\nu)}\cdot v(p_2)\nonumber\\
&&-\bar{v}(p_2)\cdot \gamma_{(\mu}p_{2\nu)} \cdot v(p_2) \
\bar{u}(p_1)\cdot \gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
-\bar{u}(p_1)\cdot \gamma_{(\mu}p_{1\nu)} \cdot u(p_1) \
\bar{v}(p_2)\cdot \gamma^{(\mu}p_2^{\nu)}\cdot v(p_2)\nonumber\\
&=& -2m^4\big(1+3\cosh \theta +2 \cosh 2\theta\big).
\label{tree1}
\end{eqnarray}
where on the third line the general amplitude, (\ref{generaltree}), was used, a factor of $\pm i$ appears whenever a derivative operator acts on $\psi$ or $\bar{\psi}$ \footnote{$+i$ appears if the derivative operator acting on $\psi(x)$ gives an outgoing momentum $p_3$ or $p_4$, while $-i$ appears if the derivative operator produces an ingoing momentum $p_1$ or $p_2$ upon acting on $\psi(x)$.}. (\ref{inoutmomenta}) was used to get the fourth equality and on the final line the mathematical expressions for the Dirac spinors (\ref{uv}) and (\ref{ubarvbar}) were plugged in.
\vspace{2mm}
\\
Similarly, the contributions from the other two terms in (\ref{TmunuSquare}) can be computed. The computation yields,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi}^{(1)}&=&\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle \nonumber\\
&=& -2m^4\big(1-\cosh \theta -2 \cosh 2\theta\big)
\label{tree2}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{{\partial}_{(\mu}\bar{\psi} \gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi}^{(1)}&=&\langle 0 | b(p_4) a(p_3) \text{T}\big[{\partial}_{(\mu}\bar{\psi} \gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle \nonumber\\
&=& -2m^4\big(1+3\cosh \theta +2 \cosh 2\theta\big)\ .
\label{tree3}
\end{eqnarray}
Adding together (\ref{tree1}), (\ref{tree2}) and (\ref{tree3}) according to (\ref{TmunuSquare}) gives the contribution of the quartic coupling $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$,
\begin{eqnarray}
\mathcal{A}_{\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}}^{(1)}&=&- \ \frac{1}{4}\Big(\mathcal{A}_{\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi}^{(1)}-2\mathcal{A}_{\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi}^{(1)}+\mathcal{A}_{{\partial}_{(\mu}\bar{\psi} \gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi}^{(1)}\Big)\nonumber\\
&=&4 m^4 (\cosh\theta+\cosh2\theta).
\label{treeamplitudeTmunuSq}
\end{eqnarray}
One can compute the contributions from the three terms found in (\ref{TmumuSquare}) to evaluate the contribution from $(\tilde{X}_{\mu}^{ \ \mu})^2$ to the tree level amplitude in a similar manner,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi}^{(1)}=\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle &=& -4m^4\big(1+\cosh \theta\big) \nonumber\\
\mathcal{A}_{\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{(1)}=\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle &=& 4m^4\big(1+\cosh \theta \big) \nonumber\\
\mathcal{A}_{{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi{\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{(1)}=\langle 0 | b(p_4) a(p_3) \text{T}\big[{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi{\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle &=& -4m^4\big(1+\cosh \theta\big). \nonumber\\
\label{tree4_5_6}
\end{eqnarray}
Adding the terms in (\ref{tree4_5_6}) according to (\ref{TmumuSquare}) yields the contribution from the quartic coupling $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{(\tilde{X}_{\mu}^{\ \mu})^2}^{(1)}&=&- \frac{1}{4}\Big(\mathcal{A}_{\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi}^{(1)}-2\mathcal{A}_{\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{(1)}+\mathcal{A}_{{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi{\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{(1)}\Big)\nonumber\\
&=&4 m^4 (1+\cosh\theta).
\label{treeamplitudeTmumuSq}
\end{eqnarray}
Substituting (\ref{treeamplitudeTmunuSq}) and (\ref{treeamplitudeTmumuSq}) into (\ref{treeamplitude}) gives the total tree-level amplitude,
\begin{eqnarray}
\mathcal{A}^{(1)}&=&i\frac{\lambda}{2}\Big[\mathcal{A}_{\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}}^{(1)}-\mathcal{A}_{(\tilde{X}_{\mu}^{\ \mu})^2}^{(1)}\Big]
=4 i \lambda m^4 \sinh ^2\theta.
\label{treeamplitudefinal}
\end{eqnarray}
Therefore, by (\ref{AtoS}) the first order $S$-matrix is,
\begin{eqnarray}
S^{(1)}(\theta)=\frac{\mathcal{A}^{(1)}}{4 m^2 \sinh \theta}= i \lambda m^2 \sinh\theta \ .
\label{treeSmatrix}
\end{eqnarray}
This result exactly matches with what one would expect from (\ref{CDDSZ1}) at linear order in the $T\bar{T}$-coupling $\lambda$.
\subsubsection{Second order $S$-matrix}
At second order the $S$-matrix gets contribution from one-loop diagrams. At one-loop, two types of diagrams can contribute to the second order amplitude: the first one is the tadpole diagram while the second one is the bubble diagram. Both of these diagrams will be computed in this section.
\subsubsection*{Contribution from tadpole diagrams}
The second order Lagrangian of the $T\bar{T}$-deformed free massive Dirac fermion can be written as,
\begin{eqnarray}
i\mathcal{L}_2(\lambda)&=&i\Big[\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{ \ \mu})^2\Big)+\frac{7}{4}\lambda^2 m^2(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{ \ \mu}-\frac{7}{4}\lambda^2 m^3(\bar{\psi}\psi)^3 \Big]
\label{L2L0}
\end{eqnarray}
where,
\begin{eqnarray}
\bar{\psi}\psi \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}
&=&-\frac{1}{4} \bar{\psi}\psi \Big(\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi
-2\bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi
+{\partial}_{(\mu}\bar{\psi} \gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi\Big)\ , \nonumber\\
\label{psibarpsiTmunuSquare}
\end{eqnarray}
\begin{eqnarray}
\hspace{-8mm}\bar{\psi}\psi(\tilde{X}_{\mu}^{ \ \mu})^2
&=&-\frac{1}{4} \bar{\psi}\psi \Big(\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \ \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi
-2\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \ {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi
+{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi \ {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi\Big)\ ,
\label{psibarpsiTmumuSquare}
\end{eqnarray}
\begin{eqnarray}
\hspace{-7.1cm}(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{\ \mu}
&=&\frac{i}{2} (\bar{\psi}\psi)^2 \big(\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi
-{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi\big) \ .
\label{psibarpsiSqTmumu}
\end{eqnarray}
Since the interaction vertices contain six fields, two internal fields must be contracted resulting in loops.
\begin{figure}[t!]
\centering \noindent
\includegraphics[width=8cm]{tadpole.pdf}
\caption{Tadpole diagram that contributes to the $S$-matrix at second order.}
\label{tadpole diagram}
\end{figure}
The sextic couplings give rise to the one-loop tadpole diagrams shown in figure \ref{tadpole diagram} and contribute to the second order $S$-matrix. The corresponding amplitude is given by,
\begin{eqnarray}
\mathcal{A}^{\text{tad}}&=&i\frac{\lambda^2}{2}\langle 0 | b(p_4) a(p_3) \text{T}\Big[m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{ \ \mu})^2\Big)+\frac{7}{2}m^2(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{ \ \mu}\nonumber\\
&& \hspace{4.3cm}-\frac{7}{2} m^3(\bar{\psi}\psi)^3 \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle.
\label{tadpoleamplitude}
\end{eqnarray}
Just as in the first order case one can evaluate (\ref{tadpoleamplitude}) by first deriving the general amplitude of a general sextic coupling,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi}_a\psi_b\bar{\psi}_c\psi_d\bar{\psi}_e\psi_f}^{\text{tad}}&=&\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}_a\psi_b\bar{\psi}_c\psi_d\bar{\psi}_e\psi_f\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\
&=&-\left\langle\psi_b(x)\bar{\psi}_a(x)\right\rangle\Big[\bar{u}_c(p_3)v_d(p_4)\bar{v}_e(p_2)u_f(p_1)
+\bar{v}_c(p_2)u_d(p_1)\bar{u}_e(p_3)v_f(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_c(p_2)v_d(p_4)\bar{u}_e(p_3)u_f(p_1)
-\bar{u}_c(p_3)u_d(p_1)\bar{v}_e(p_2)v_f(p_4)\Big]\nonumber\\
&&+\left\langle\psi_d(x)\bar{\psi}_a(x)\right\rangle\Big[\bar{u}_c(p_3)v_b(p_4)\bar{v}_e(p_2)u_f(p_1)+\bar{v}_c(p_2)u_b(p_1)\bar{u}_e(p_3)v_f(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_c(p_2)v_b(p_4)\bar{u}_e(p_3)u_f(p_1)-\bar{u}_c(p_3)u_b(p_1)\bar{v}_e(p_2)v_f(p_4)\Big]\nonumber\\
&&-\left\langle\psi_f(x)\bar{\psi}_a(x)\right\rangle\Big[\bar{u}_c(p_3)v_b(p_4)\bar{v}_e(p_2)u_d(p_1)
+\bar{v}_c(p_2)u_b(p_1)\bar{u}_e(p_3)v_d(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_c(p_2)v_b(p_4)\bar{u}_e(p_3)u_d(p_1)-\bar{u}_c(p_3)u_b(p_1)\bar{v}_e(p_2)v_d(p_4)\Big]\nonumber\\
&&+\left\langle\psi_b(x)\bar{\psi}_c(x)\right\rangle\Big[\bar{u}_a(p_3)v_d(p_4)\bar{v}_e(p_2)u_f(p_1)
+\bar{v}_a(p_2)u_d(p_1)\bar{u}_e(p_3)v_f(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_a(p_2)v_d(p_4)\bar{u}_e(p_3)u_f(p_1)-\bar{u}_a(p_3)u_d(p_1)\bar{v}_e(p_2)v_f(p_4)\Big]\nonumber\\
&&-\left\langle\psi_b(x)\bar{\psi}_e(x)\right\rangle\Big[\bar{u}_a(p_3)v_d(p_4)\bar{v}_c(p_2)u_f(p_1)+\bar{v}_a(p_2)u_d(p_1)\bar{u}_c(p_3)v_f(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_a(p_2)v_d(p_4)\bar{u}_c(p_3)u_f(p_1)-\bar{u}_a(p_3)u_d(p_1)\bar{v}_c(p_2)v_f(p_4)\Big]\nonumber\\
&&-\left\langle\psi_d(x)\bar{\psi}_c(x)\right\rangle\Big[\bar{u}_a(p_3)v_b(p_4)\bar{v}_e(p_2)u_f(p_1)
+\bar{v}_a(p_2)u_b(p_1)\bar{u}_e(p_3)v_f(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_a(p_2)v_b(p_4)\bar{u}_e(p_3)u_f(p_1)-\bar{u}_a(p_3)u_b(p_1)\bar{v}_e(p_2)v_f(p_4)\Big]\nonumber\\
&&+\left\langle\psi_f(x)\bar{\psi}_c(x)\right\rangle\Big[\bar{u}_a(p_3)v_b(p_4)\bar{v}_e(p_2)u_d(p_1)
+\bar{v}_a(p_2)u_b(p_1)\bar{u}_e(p_3)v_d(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_a(p_2)v_b(p_4)\bar{u}_e(p_3)u_d(p_1)-\bar{u}_a(p_3)u_b(p_1)\bar{v}_e(p_2)v_d(p_4)\Big]\nonumber\\
&&+\left\langle\psi_d(x)\bar{\psi}_e(x)\right\rangle\Big[\bar{u}_a(p_3)v_b(p_4)\bar{v}_c(p_2)u_f(p_1)+\bar{v}_a(p_2)u_b(p_1)\bar{u}_c(p_3)v_f(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_a(p_2)v_b(p_4)\bar{u}_c(p_3)u_f(p_1)-\bar{u}_a(p_3)u_b(p_1)\bar{v}_c(p_2)v_f(p_4)\Big]\nonumber\\
&&-\left\langle\psi_f(x)\bar{\psi}_e(x)\right\rangle\Big[\bar{u}_a(p_3)v_b(p_4)\bar{v}_c(p_2)u_d(p_1)
+\bar{v}_a(p_2)u_b(p_1)\bar{u}_c(p_3)v_d(p_4)\nonumber\\
&&\hspace{2.5cm}-\bar{v}_a(p_2)v_b(p_4)\bar{u}_c(p_3)u_d(p_1)-\bar{u}_a(p_3)u_b(p_1)\bar{v}_c(p_2)v_d(p_4)\Big]\nonumber\\
\label{generaltadpole}
\end{eqnarray}
where all possible Wick contractions of fermionic fields were performed. The internal propagators give rise to loops whose values are derived in Appendix \ref{AppendixB} and given by,
\begin{align}
\langle \psi_a(x)\bar{\psi}_b(x)\rangle&=N_0\delta_{ab} ,\label{tadpoleprop}\\
\langle \partial^\mu\psi_a(x)\bar{\psi}_b(x)\rangle&= -\langle \psi_a(x)\partial^\mu\bar{\psi}_b(x)\rangle=N_1\gamma^{\mu}_{ab} ,\label{prop2}\\
\langle \partial^\mu\psi_a(x)\partial^\nu\bar{\psi}_b(x)\rangle&=imN_1\eta^{\mu\nu}\delta_{ab}\label{prop3}\,
\end{align}
where,
\begin{eqnarray}
N_0&=&-\frac{m}{4\pi}\log(\frac{m^2}{\Lambda^2})\ ,
\label{N0}\\
N_1&=&\frac{i\Lambda^2}{8\pi}\Big[1+\frac{m^2}{\Lambda^2}\log(\frac{m^2}{\Lambda^2})\Big] \label{N1}\ ,
\end{eqnarray}
and $\Lambda$ is the UV cut-off.
\vspace{2mm}
\\
Using the general expression for the amplitude of a sextic coupling, (\ref{generaltadpole}), one can compute the contribution of $(\bar{\psi}\psi)^3$ to the amplitude,
\begin{eqnarray}
\mathcal{A}_{(\bar{\psi}\psi)^3}^{\text{tad}}&=&
\langle 0 | b(p_4) a(p_3) \text{T}\big[(\bar{\psi}\psi)^3\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\ &=&\delta_{ab}\delta_{cd}\delta_{ef}\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}_a\psi_b\bar{\psi}_c\psi_d\bar{\psi}_e\psi_f\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle=0
\label{tadpoleamplitudepsibarpsiCube}
\end{eqnarray}
where (\ref{tadpoleprop}), (\ref{uv}) and (\ref{ubarvbar}) were used to obtain the above result.
\vspace{2mm}
\\
The contribution to the amplitude due to the coupling $\bar{\psi} \psi \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$ can be evaluated by computing the contributions from each term of (\ref{psibarpsiTmunuSquare}).
The first term in (\ref{psibarpsiTmunuSquare}) contributes,
\begin{eqnarray}
&&\mathcal{A}_{\bar{\psi} \psi \bar{\psi} \gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi}^{\text{tad}}\nonumber\\
=&&\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi} \psi \ \bar{\psi} \gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle \nonumber\\
=&&\delta_{ab}\gamma_{cd(\mu}\gamma_{ef}^{(\mu}\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}_a\psi_b \bar{\psi}_c {\partial}_{\nu)}\psi_d\bar{\psi}_e{\partial}^{\nu)}\psi_f\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\
=&-&2N_0\Big[\bar{u}(p_3)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
+\bar{v}(p_2)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1) \ \bar{u}(p_3)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{v}(p_2)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
-\bar{u}(p_3)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1) \ \bar{v}(p_2)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\Big]\nonumber\\
&+&N_1\Big[\bar{u}(p_3)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
+\bar{v}(p_2)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot u(p_1) \ \bar{u}(p_3)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{v}(p_2)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
-\bar{u}(p_3)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot u(p_1) \ \bar{v}(p_2)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\Big]\nonumber\\
&-&N_1\Big[\bar{v}(p_2)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1)
+\bar{u}(p_3)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot u(p_1) \ \bar{v}(p_2)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{u}(p_3)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1)
-\bar{v}(p_2)\cdot \gamma_{(\mu}\gamma_{\nu)}\cdot u(p_1) \ \bar{u}(p_3)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4)\Big]\nonumber\\
&+&N_0\Big[\bar{u}(p_3)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
+\bar{v}(p_2)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1) \ \bar{u}(p_3)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{v}(p_2)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
-\bar{u}(p_3)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1) \ \bar{v}(p_2)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\Big]\nonumber\\
&-&N_0\Big[\bar{v}(p_2)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
+\bar{u}(p_3)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1) \ \bar{v}(p_2)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{u}(p_3)\cdot i\gamma_{(\mu}p_{4\nu)}\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
-\bar{v}(p_2)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1) \ \bar{u}(p_3)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\Big]\nonumber\\
&-&N_1 \ \Tr\big(\gamma_{(\mu}\gamma_{\nu)}\big)\Big[\bar{u}(p_3)\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
+\bar{v}(p_2)\cdot u(p_1) \ \bar{u}(p_3)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{v}(p_2)\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i)\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
-\bar{u}(p_3)\cdot u(p_1) \ \bar{v}(p_2)\cdot i\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\Big]\nonumber
\end{eqnarray}
\begin{eqnarray}
&+&N_1\Big[\bar{u}(p_3)\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma^{(\mu}\gamma^{\nu)}\gamma_{(\mu}p_{1\nu)}\cdot u(p_1)
+\bar{v}(p_2) \cdot u(p_1) \ \bar{u}(p_3)\cdot i \gamma^{(\mu}\gamma^{\nu)}\gamma_{(\mu}p_{4\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{v}(p_2)\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i) \gamma^{(\mu}\gamma^{\nu)}\gamma_{(\mu}p_{1\nu)}\cdot u(p_1)
-\bar{u}(p_3)\cdot u(p_1) \ \bar{v}(p_2)\cdot i \gamma^{(\mu}\gamma^{\nu)}\gamma_{(\mu}p_{4\nu)}\cdot v(p_4)\Big]\nonumber\\
&+&N_1\Big[\bar{u}(p_3)\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma_{(\mu}\gamma_{\nu)}\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
+\bar{v}(p_2) \cdot u(p_1) \ \bar{u}(p_3)\cdot i \gamma_{(\mu}\gamma_{\nu)}\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{v}(p_2)\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i) \gamma_{(\mu}\gamma_{\nu)}\gamma^{(\mu}p_1^{\nu)}\cdot u(p_1)
-\bar{u}(p_3)\cdot u(p_1) \ \bar{v}(p_2)\cdot i \gamma_{(\mu}\gamma_{\nu)}\gamma^{(\mu}p_4^{\nu)}\cdot v(p_4)\Big]\nonumber\\
&-&N_1\Tr\big(\gamma^{(\mu}\gamma^{\nu)}\big)\Big[\bar{u}(p_3)\cdot v(p_4) \ \bar{v}(p_2)\cdot (-i)\gamma_{(\mu}p_{1\nu)}\cdot u(p_1)
+\bar{v}(p_2) \cdot u(p_1) \ \bar{u}(p_3)\cdot i \gamma_{(\mu}p_{4\nu)}\cdot v(p_4)\nonumber\\
&&-\bar{v}(p_2)\cdot v(p_4) \ \bar{u}(p_3)\cdot (-i) \gamma_{(\mu}p_{1\nu)}\cdot u(p_1)
-\bar{u}(p_3)\cdot u(p_1) \ \bar{v}(p_2)\cdot i \gamma_{(\mu}p_{4\nu)}\cdot v(p_4)\Big] \,
\label{psibarpsiTmunuSquare1_step}
\end{eqnarray}
where $N_0$ and $N_1$ are given by (\ref{N0}) and (\ref{N1}) respectively. Evaluating (\ref{psibarpsiTmunuSquare1_step}) one sees that,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi} \psi \bar{\psi} \gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi}^{\text{tad}}=0.
\label{psibarpsiTmunuSquare1}
\end{eqnarray}
Notice that the last term in (\ref{psibarpsiTmunuSquare}) is the complex conjugate of the first term and will hence give a vanishing contribution to the amplitude as well,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi} \psi {\partial}_{(\mu}\bar{\psi}\gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi}\gamma^{\nu)}\psi}^{\text{tad}}=\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi} \psi {\partial}_{(\mu}\bar{\psi}\gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi}\gamma^{\nu)}\psi\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle=0.
\label{psibarpsiTmunuSquare3}
\end{eqnarray}
The second term in (\ref{psibarpsiTmunuSquare}) gives a non-zero contribution to the amplitude and can be computed in a similar way,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi} \psi \bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi}\gamma^{\nu)}\psi}^{\text{tad}}&=&\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi} \psi \bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi}\gamma^{\nu)}\psi\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\
&=&-\frac{m^3\Lambda^2}{\pi}\Big[1+4\big(2-\cosh\theta\big)\frac{m^2}{\Lambda^2}\log\frac{m^2}{\Lambda^2}\Big] \cosh^2\frac{\theta}{2}.
\label{psibarpsiTmunuSquare2}
\end{eqnarray}
Combining (\ref{psibarpsiTmunuSquare1}), (\ref{psibarpsiTmunuSquare3}) and (\ref{psibarpsiTmunuSquare2}) according to (\ref{psibarpsiTmunuSquare}) the total contribution from the sextic coupling $\bar{\psi}\psi \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$ is,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi}\psi \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}}^{\text{tad}}&=&- \ \frac{1}{4}\Big[\mathcal{A}_{\bar{\psi}\psi \bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi \bar{\psi}\gamma^{(\mu}{\partial}^{\nu)}\psi}^{\text{tad}}-2\mathcal{A}_{\bar{\psi}\psi \bar{\psi}\gamma_{(\mu}{\partial}_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi}^{\text{tad}}+\mathcal{A}_{\bar{\psi}\psi {\partial}_{(\mu}\bar{\psi} \gamma_{\nu)}\psi {\partial}^{(\mu}\bar{\psi} \gamma^{\nu)}\psi}^{\text{tad}}\Big]\nonumber\\
&=&-\frac{m^3\Lambda^2}{2\pi}\Big[1+4\big(2-\cosh\theta\big)\frac{m^2}{\Lambda^2}\log\frac{m^2}{\Lambda^2}\Big] \cosh^2\frac{\theta}{2}.
\label{tadpoleamplitudepsibpsiTmunuSq}
\end{eqnarray}
Following the same steps as in (\ref{psibarpsiTmunuSquare1_step}) one can compute the contribution from the sextic coupling $\bar{\psi}\psi(\tilde{X}_{\mu}^{\ \mu})^2$ by determining the contribution from each term in (\ref{psibarpsiTmumuSquare}),
\begin{eqnarray}
\mathcal{A}_{\bar{\psi}\psi\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi}^{\text{tad}}=\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}\psi\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle &=& 0\nonumber\\
\mathcal{A}_{\bar{\psi}\psi\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{\text{tad}}=\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}\psi\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle &=& \frac{m^3\Lambda^2}{\pi}\big(1+\cosh \theta \big) \nonumber\\
\mathcal{A}_{\bar{\psi}\psi{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi{\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{\text{tad}}=\langle 0 | b(p_4) a(p_3) \text{T}\big[\bar{\psi}\psi{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi{\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi\big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle &=& 0.
\label{tadpole4_5_6}
\end{eqnarray}
Combining the individual amplitudes according to (\ref{psibarpsiTmumuSquare}) the total contribution from the sextic coupling $\bar{\psi}\psi (\tilde{X}_{\mu}^{\ \mu})^2$ is,
\begin{eqnarray}
\mathcal{A}_{\bar{\psi}\psi (\tilde{X}_{\mu}^{\ \mu})^2}^{\text{tad}}&=&- \ \frac{1}{4}\Big[\mathcal{A}_{\bar{\psi}\psi\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi \bar{\psi}\gamma^{\nu}{\partial}_{\nu}\psi}^{\text{tad}}-2\mathcal{A}_{\bar{\psi}\psi\bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi {\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{\text{tad}}+\mathcal{A}_{\bar{\psi}\psi{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi{\partial}_{\nu}\bar{\psi} \gamma^{\nu}\psi}^{\text{tad}}\Big]\nonumber\\
&=&\frac{m^3\Lambda^2}{2\pi}\big(1+\cosh \theta \big)
\label{tadpoleamplitudepsibpsiTmumuSq}
\end{eqnarray}
Similarly, evaluating the contribution to the amplitude from each term in $(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{\ \mu}$, (\ref{psibarpsiSqTmumu}), yields a vanishing contribution,
\begin{eqnarray}
\mathcal{A}_{(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{\ \mu}}^{\text{tad}}&=&\ \frac{i}{2}\Big[\mathcal{A}_{(\bar{\psi}\psi)^2 \bar{\psi}\gamma^{\mu}{\partial}_{\mu}\psi}^{\text{tad}}-\mathcal{A}_{(\bar{\psi}\psi)^2{\partial}_{\mu}\bar{\psi} \gamma^{\mu}\psi}^{\text{tad}}\Big]=0
\label{tadpoleamplitudepsibpsiSqTmumu}
\end{eqnarray}
Finally, substituting (\ref{tadpoleamplitudepsibarpsiCube}), (\ref{tadpoleamplitudepsibpsiTmunuSq}), (\ref{tadpoleamplitudepsibpsiTmumuSq}) and (\ref{tadpoleamplitudepsibpsiSqTmumu}) into (\ref{tadpoleamplitude}) yields the contribution to the amplitude from the tadpole diagrams,
\begin{eqnarray}
\mathcal{A}^{\text{tad}}&=&i\frac{\lambda^2}{2}\Big[m\Big(\mathcal{A}_{\bar{\psi}\psi \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}}^{\text{tad}}-\mathcal{A}_{\bar{\psi}\psi (\tilde{X}_{\mu}^{\ \mu})^2}^{\text{tad}}\Big)+\frac{7}{2}m^2\mathcal{A}_{(\bar{\psi}\psi)^2 \tilde{X}_{\mu}^{\ \mu}}^{\text{tad}}-\frac{7}{2}m^3\mathcal{A}_{(\bar{\psi}\psi)^3}^{\text{tad}} \Big]\nonumber\\
&=&-i\frac{\lambda^2 m^6 }{4\pi}\cosh^2\frac{\theta}{2}\Big[3\frac{\Lambda^2}{m^2}+4\big(2-\cosh\theta\big)\log\frac{m^2}{\Lambda^2}\Big] \ .
\label{tadpoleamplitudefinal}
\end{eqnarray}
Converting from amplitude to $S$-matrix using (\ref{AtoS}) yields the tadpole contribution to the $S$-matrix,
\begin{eqnarray}
S^{\text{tad}}(\theta)=\frac{\mathcal{A}^{\text{tad}}}{4 m^2 \sinh \theta}= -i\frac{\lambda^2 m^4 }{32\pi}\coth\frac{\theta}{2}\Big[3\frac{\Lambda^2}{m^2}+4\big(2-\cosh\theta\big)\log\frac{m^2}{\Lambda^2}\Big] \ .
\label{tadpoleSmatrix}
\end{eqnarray}
\
\subsubsection*{Contribution from bubble diagrams}
\label{Bubble_Contribution}
The final and most complicated contribution to the $S$-matrix is the contribution that arises from the first order term in the Lagrangian squared,
\begin{eqnarray}
\frac{1}{2!}\left(i\mathcal{L}_1(\lambda)\right)^2=\frac{1}{2!}\Big(\frac{i\lambda}{2}\Big)^2\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\mu})^2\Big)^2
\label{L2NL0}.
\end{eqnarray}
These interaction vertices give rise to one-loop bubble diagrams of the form figure \ref{bubble diagram}.
\begin{figure}[h!]
\centering
\subfigure[$s$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{general_bubble_s.pdf}
\label{bubble diagram_s} }
\subfigure[$t$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{general_bubble_t.pdf}
\label{bubble diagram_t} }
\subfigure[$u$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{general_bubble_u.pdf}
\label{bubble diagram_u} }
\caption{\small The $s$, $t$ and $u$-channels of a typical bubble diagram that contribute to the $S$-matrix
at second order.}
\label{bubble diagram}
\end{figure}
\\\\
From (\ref{L2NL0}) there are three different kinds of bubble diagrams:
\begin{enumerate}
\item[\textbf{(a)}] both vertices contain $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$ interactions,
\item[\textbf{(b)}] one vertex contains interaction $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ while the other contains $(\tilde{X}_{\mu}^{\ \mu})^2$ and
\item[\textbf{(c)}] both vertices contain $(\tilde{X}_{\mu}^{\ \mu})^2$ interactions.
\end{enumerate}
For the bubble diagram case the amplitudes will be split up into $s$, $t$ and $u$-channel contributions to the amplitude, as shown in figure (\ref{bubble diagram}). As done in the previous cases, begin by writing down the general expressions for the amplitude where both vertices $x$ and $y$ contain arbitrary non-derivative quartic couplings. These expressions will be the building blocks for the computation of the amplitude from the bubble diagrams.
\vspace{2mm}
\\
In order to compute the general contributions of the $s$, $t$ and $u$-channels to the amplitude consider the case when the vertex $x$ involves a quartic coupling $\bar{\psi}_a(x)\psi_b(x)\bar{\psi}_c(x)\psi_d(x)$ and vertex $y$ contains another quartic coupling $\bar{\psi}_e(y)\psi_f(y)\bar{\psi}_g(y)\psi_h(y)$.
\\\\
The general contribution of the $s$-channel (figure \ref{bubble diagram_s}) to the amplitude is given by,
\begin{eqnarray}
\mathcal{B}_{abcdefgh}^{(s)}&=&\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}_e(y)\psi_f(y)\bar{\psi}_g(y)\psi_h(y)\bar{\psi}_a(x)\psi_b(x)\bar{\psi}_c(x)\psi_d(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\bar{v}_c(p_2)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{u}_e(p_3)v_f(p_4)G_{ha}(\xi+q)G_{bg}(q)-\bar{u}_e(p_3)v_h(p_4)G_{fa}(\xi+q)G_{bg}(q)\nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{ha}(\xi+q)G_{be}(q)+\bar{u}_g(p_3)v_h(p_4)G_{fa}(\xi+q)G_{be}(q)\Big)
\nonumber\\
&-&\bar{v}_a(p_2)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{u}_e(p_3)v_f(p_4)G_{hc}(\xi+q)G_{dg}(q)-\bar{u}_e(p_3)v_h(p_4)G_{fc}(\xi+q)G_{dg}(q)\nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{hc}(\xi+q)G_{de}(q)+\bar{u}_g(p_3)v_h(p_4)G_{fc}(\xi+q)G_{de}(q)\Big)
\nonumber\\
&+&\bar{v}_a(p_2)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{u}_e(p_3)v_f(p_4)G_{hc}(\xi+q)G_{bg}(q)-\bar{u}_e(p_3)v_h(p_4)G_{fc}(\xi+q)G_{bg}(q)\nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{hc}(\xi+q)G_{be}(q)+\bar{u}_g(p_3)v_h(p_4)G_{fc}(\xi+q)G_{be}(q)\Big)
\nonumber\\
&+&\bar{v}_c(p_2)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{u}_e(p_3)v_f(p_4)G_{ha}(\xi+q)G_{dg}(q)-\bar{u}_e(p_3)v_h(p_4)G_{fa}(\xi+q)G_{dg}(q)\nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{ha}(\xi+q)G_{de}(q)+\bar{u}_g(p_3)v_h(p_4)G_{fa}(\xi+q)G_{de}(q)\Big)\ ,
\label{schannelbubblegeneral}
\end{eqnarray}
where $\xi^2=(p_1+p_2)^2=s$ and $G_{ab}(q)=\frac{i(\gamma\cdot q+m)_{ab}}{q^2-m^2+i\epsilon}$ is the fermionic propagator in the free theory.
\\\\
Next consider the contribution of the $t$-channel (figure \ref{bubble diagram_t}) to the amplitude. Since $p_1=p_3$, the general $t$-channel contribution to the amplitude is given by,
\begin{eqnarray}
\mathcal{B}_{abcdefgh}^{(t)}&=&\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}_e(y)\psi_f(y)\bar{\psi}_g(y)\psi_h(y)\bar{\psi}_a(x)\psi_b(x)\bar{\psi}_c(x)\psi_d(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&\bar{u}_c(p_3)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{v}_g(p_2)v_h(p_4)G_{fa}(q)G_{be}(q)-\bar{v}_g(p_2)v_f(p_4)G_{ha}(q)G_{be}(q)\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fa}(q)G_{bg}(q)+\bar{v}_e(p_2)v_f(p_4)G_{ha}(q)G_{bg}(q)\Big)
\nonumber\\
&+&\bar{u}_a(p_3)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{v}_g(p_2)v_h(p_4)G_{fc}(q)G_{de}(q)-\bar{v}_g(p_2)v_f(p_4)G_{hc}(q)G_{de}(q)\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fc}(q)G_{dg}(q)+\bar{v}_e(p_2)v_f(p_4)G_{hc}(q)G_{dg}(q)\Big)
\nonumber\end{eqnarray}\begin{eqnarray}
&-&\bar{u}_a(p_3)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{v}_g(p_2)v_h(p_4)G_{fc}(q)G_{be}(q)-\bar{v}_g(p_2)v_f(p_4)G_{hc}(q)G_{be}(q)\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fc}(q)G_{bg}(q)+\bar{v}_e(p_2)v_f(p_4)G_{hc}(q)G_{bg}(q)\Big)
\nonumber\\
&-&\bar{u}_c(p_3)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{v}_g(p_2)v_h(p_4)G_{fa}(q)G_{de}(q)-\bar{v}_g(p_2)v_f(p_4)G_{ha}(q)G_{de}(q)\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fa}(q)G_{dg}(q)+\bar{v}_e(p_2)v_f(p_4)G_{ha}(q)G_{dg}(q)\Big)\ .
\label{tchannelbubblegeneral}
\end{eqnarray}
\\
Finally consider the contribution of the $u$-channel (figure \ref{bubble diagram_u}) to the amplitude. The general $u$-channel contribution to the amplitude is given by,
\begin{eqnarray}
\mathcal{B}_{abcdefgh}^{(u)}&=&\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}_e(y)\psi_f(y)\bar{\psi}_g(y)\psi_h(y)\bar{\psi}_a(x)\psi_b(x)\bar{\psi}_c(x)\psi_d(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&\Big(u_d(p_1)v_b(p_4)\bar{u}_e(p_3)\bar{v}_g(p_2)-u_d(p_1)v_b(p_4)\bar{u}_g(p_3)\bar{v}_e(p_2)-u_b(p_1)v_d(p_4)\bar{u}_e(p_3)\bar{v}_g(p_2)\nonumber\\
&+&u_b(p_1)v_d(p_4)\bar{u}_g(p_3)\bar{v}_e(p_2)\Big)\int\frac{d^2q}{(2\pi)^2}\Big(G_{hc}(\zeta-q)G_{fa}(q)-G_{ha}(\zeta-q)G_{fc}(q)\Big)\ ,\nonumber\\
\label{uchannelbubblegeneral}
\end{eqnarray}
where $\zeta^2=(p_1-p_4)^2=(p_1-p_2)^2=u$ as $p_2=p_4$.
\\
\\
Notice that a general $s$-channel amplitude (\ref{schannelbubblegeneral}) involves evaluation of the one-loop integral,
\begin{eqnarray}
I_{abcd}^{(s)}(\xi)&=&\int \frac{d^2 q}{(2\pi)^2}G_{ab}(\xi+q)G_{cd}(q)=\int \frac{d^2 q}{(2\pi)^2} \frac{i\big(\gamma\cdot (\xi+q)+m\big)_{ab}}{(\xi+q)^2-m^2} \frac{i\big(\gamma\cdot q+m\big)_{cd}}{q^2-m^2} \nonumber\\
&=&-\int \frac{d^2 q}{(2\pi)^2}\frac{\gamma^{\mu}_{ab}\gamma^{\nu}_{cd}(\xi+q)_{\mu}q_{\nu}+m\delta_{ab}\gamma^{\mu}_{cd}q_{\mu}+m\gamma^{\mu}_{ab}(\xi+q)_{\mu}\delta_{cd}+m^2\delta_{ab}\delta_{cd}}
{\big[(\xi+q)^2-m^2\big](q^2-m^2)}\nonumber\\
&=&-L^{(s)}(\xi)\Big(m\slashed{\xi}_{ab}\delta_{cd}+m^2\delta_{ab}\delta_{cd}\Big)-\Big(\slashed{\xi}_{ab}\slashed{L}^{(s)}_{cd}(\xi)+m\delta_{ab}\slashed{L}^{(s)}_{cd}(\xi)+m\slashed{L}^{(s)}_{ab}(\xi)\delta_{cd}\Big)\nonumber\\
&&-\gamma^{\mu}_{ab}\gamma^{\nu}_{cd}L^{(s)}_{\mu\nu}(\xi)\ ,
\label{I_abcd_s}
\end{eqnarray}
where $\ \ \ \xi^{\mu}=(p_1+p_2)^{\mu}$, $ \ \ \slashed{L}^{(s)}_{ab}=\gamma^{\mu}_{ab}L^{(s)}_{\mu}$ , $ \ \ \slashed{\xi}_{ab}=\gamma^{\mu}_{ab}\xi_{\mu}\ \ $ and
\begin{eqnarray}
L^{(s)}&=&\int \frac{d^2 q}{(2\pi)^2} \frac{1}{\big[(\xi+q)^2-m^2\big]\big(q^2-m^2\big)}=-\frac{\pi+i\theta}{4\pi m^2 \sinh\theta} \ , \nonumber\\
L^{(s)}_{\mu}&=&\int \frac{d^2 q}{(2\pi)^2} \frac{q_{\mu}}{\big[(\xi+q)^2-m^2\big]\big(q^2-m^2\big)}=\frac{\pi+i\theta}{8\pi m^2 \sinh\theta} \ \xi_{\mu} \ , \nonumber\\
L^{(s)}_{\mu\nu}&=&\int \frac{d^2 q}{(2\pi)^2} \frac{q_{\mu}q_{\nu}}{\big[(\xi+q)^2-m^2\big]\big(q^2-m^2\big)}\nonumber\\
&=&-\frac{i}{8\pi}\bigg[1+\log \frac{\Lambda^2}{m^2}+\big(i\pi-\theta\big)\tanh\frac{\theta}{2}\bigg]\eta_{\mu\nu}+\frac{i}{16\pi m^2\cosh^2\frac{\theta}{2}}\bigg[1+\big(i\pi-\theta\big)\coth\theta\bigg]\xi_{\mu}\xi_{\nu} \ .\nonumber\\
\label{L_s_trial}
\end{eqnarray}
The above integrals are derived in detail in Appendix \ref{AppendixC}.
\\
\\
Similar yet simpler integrals are involved in the evaluation of the general contribution to the amplitude of $t$-channel (\ref{tchannelbubblegeneral}). In the $t$-channel case the integrals are much simpler as both the internal propagators carry momentum $q$ since $p_1=p_3$ and take the form,
\begin{eqnarray}
I_{abcd}^{(t)}&=&\int \frac{d^2 q}{(2\pi)^2}G_{ab}(q)G_{cd}(q)\nonumber\\
&=&-m^2\delta_{ab}\delta_{cd}L^{(t)}-\Big(m\delta_{ab}\slashed{L}^{(t)}_{cd}+m\slashed{L}^{(t)}_{ab}\delta_{cd}\Big)-\gamma^{\mu}_{ab}\gamma^{\nu}_{cd}L^{(t)}_{\mu\nu}\ ,
\label{I_abcd_t}
\end{eqnarray}
where,
\begin{eqnarray}
L^{(t)}=\int \frac{d^2 q}{(2\pi)^2} \frac{1}{\big(q^2-m^2\big)^2}\ , \ \ \
L^{(t)}_{\mu}=\int \frac{d^2 q}{(2\pi)^2} \frac{q_{\mu}}{\big(q^2-m^2\big)^2}, \ \ \
L^{(t)}_{\mu\nu}=\int \frac{d^2 q}{(2\pi)^2} \frac{q_{\mu}q_{\nu}}{\big(q^2-m^2\big)^2}\nonumber.\\
\label{L_t_trial}
\end{eqnarray}
However, there is no need to explicitly compute $L^{(t)}$, $L^{(t)}_{\mu}$ and $L^{(t)}_{\mu\nu}$ as their values can easily be obtained from their $s$-channel counterparts (\ref{L_s_trial}). In section \ref{subsecSmatrix}, it was shown that the $t$-channel corresponds to $\theta\rightarrow i\pi$. Thus, one can simply substitute $\theta\rightarrow i\pi$ and $\xi\rightarrow 0$ into (\ref{L_s_trial}) to obtain the corresponding $t$-channel loop integrals $L^{(t)}$, $L^{(t)}_{\mu}$ and $L^{(t)}_{\mu\nu}$. Hence,
\begin{eqnarray}
L^{(t)}=\frac{i}{4\pi m^2} \ , \ \ \ \ \ \ \
L^{(t)}_{\mu}=0 \ , \ \ \ \ \ \ \
L^{(t)}_{\mu\nu}=\frac{i}{8\pi}\Big(1-\log\frac{\Lambda^2}{m^2}\Big)\eta_{\mu\nu}.
\end{eqnarray}
In the $u$-channel case (\ref{uchannelbubblegeneral}) one must compute the following loop-integral similar to the $s$-channel case,
\begin{eqnarray}
I_{abcd}^{(u)}(\zeta)&=&\int \frac{d^2 q}{(2\pi)^2}G_{ab}(\zeta-q)G_{cd}(q)\nonumber\\
&=&-L^{(u)}(\zeta)\Big(m\slashed{\zeta}_{ab}\delta_{cd}+m^2\delta_{ab}\delta_{cd}\Big)-\Big(\slashed{\zeta}_{ab}\slashed{L}_{cd}^{(u)}(\zeta)+m\delta_{ab}\slashed{L}_{cd}^{(u)}(\zeta)-m\slashed{L}_{ab}^{(u)}(\zeta)\delta_{cd}\Big)\nonumber\\
&&+\gamma^{\mu}_{ab}\gamma^{\nu}_{cd}L_{\mu\nu}^{(u)}(\zeta)\ ,
\label{I_abcd_u_trial}
\end{eqnarray}
where $\zeta^{\mu}=(p_1-p_4)^{\mu}=(p_1-p_2)^{\mu} \ \ \ $ and
\begin{eqnarray}
L^{(u)}&=&\int \frac{d^2 q}{(2\pi)^2} \frac{1}{\big[(\zeta-q)^2-m^2\big]\big(q^2-m^2\big)}\ , \nonumber\\
L^{(u)}_{\mu}&=&\int \frac{d^2 q}{(2\pi)^2} \frac{q_{\mu}}{\big[(\zeta-q)^2-m^2\big]\big(q^2-m^2\big)}\ , \nonumber\\
L^{(u)}_{\mu\nu}&=&\int \frac{d^2 q}{(2\pi)^2} \frac{q_{\mu}q_{\nu}}{\big[(\zeta-q)^2-m^2\big]\big(q^2-m^2\big)}\ .
\label{L_u_trial}
\end{eqnarray}
Similarly to the $t$-channel case, it was shown in section \ref{subsecSmatrix} that the $u$-channel corresponds to $\theta\rightarrow i\pi-\theta$. Hence, naively one can evaluate the $u$-channel loop-integrals $L^{(u)}$, $L^{(u)}_{\mu}$ and $L^{(u)}_{\mu\nu}$ directly from their $s$-channel counterparts (\ref{L_s_trial}) by replacing $\theta\rightarrow i\pi-\theta$.\footnote{Replacing $\theta$ by $i\pi-\theta$ is equivalent to replacing $\xi^2$ by $\zeta^2$.} However, notice that in the u-channel loop integrals (\ref{L_u_trial}) the functional dependence of the integrands on $\zeta$ is of the form $\zeta-q$, while in the $s$-channel loop integrals (\ref{L_s_trial}) the corresponding functional dependence is $\xi+q$.
Therefore, since the functional dependence of the integrands on $\zeta$ is $\zeta-q$, an extra minus sign must be included whenever the integrand is odd in $q$,
\begin{eqnarray}
L^{(u)}&=&i\frac{\theta}{4\pi m^2\sinh\theta}\ ,\nonumber\\
L^{(u)}_{\mu}&=&i\frac{\theta}{8\pi m^2\sinh\theta}\zeta_{\mu}\ ,\nonumber\\
L^{(u)}_{\mu\nu}&=&-\frac{i}{8\pi}\bigg[1+\log \frac{\Lambda^2}{m^2}-\theta\coth\frac{\theta}{2}\bigg]\eta_{\mu\nu}-\frac{i}{16\pi m^2\sinh^2\frac{\theta}{2}}\big(1-\theta\coth\theta\big)\zeta_{\mu}\zeta_{\nu}\ .
\end{eqnarray}
The derivations of the above integrals are given explicitly in Appendix \ref{AppendixC}.
\vspace{2mm}
\\
When the vertices have derivative interactions the amplitudes involve more complicated one-loop integrals,
\begin{eqnarray}
(I_{\mu_1\cdots\mu_n})_{abcd}(\xi)&=&\int \frac{d^2 q}{(2\pi)^2}\prod_{i=1}^{n} q_{\mu_i} G_{ab}(\xi+q)G_{cd}(q)\ ,
\end{eqnarray}
which requires the computation of integrals of the form,
\begin{eqnarray}
L_{\mu_1\cdots\mu_{n+2}}=\int \frac{d^2q}{2\pi^2} \frac{\prod_{i=1}^{n+2} q_{\mu_i}}{\big[(\xi+q)^2-m^2\big](q^2-m^2)}\label{L_i}
\end{eqnarray}
where $n\in\{0,...,4\}$.
A detailed discussion on the evaluation of these one-loop integrals can be found in Appendix \ref{AppendixC}.
At this point one can plug the one-loop integral expressions and vertices into MATHEMATICA to obtain the contribution to the amplitudes from the $s$, $t$ and $u$-channels.
\vspace{2mm}
\\
It is important, however, to mention that in the $S$-matrix computation for a $T\bar{T}$-deformed scalar \cite{Rosenhaus}, one does not need to calculate amplitudes for all the channels explicitly to compute the total amplitude. Rather, one can use the following trick: evaluate the s-channel amplitude explicitly, then consider the appropriate limits to obtain the $t$ and $u$-channel amplitudes. However, such a procedure can not be performed when there are non-trivial polarization vectors that depend on the momenta or rapidities of the external particles.
To compute the total amplitude of the $T\bar{T}$-deformed free massive Dirac fermionic theory one must take into account the external polarization vectors described by the Dirac spinors $u_a$, $v_a$, $\bar{u}_a$ and $\bar{v}_a$ which depend on the rapidities $\theta_i$ of the particles. Thus the naive substitution $\theta\rightarrow i\pi$ or $\theta\rightarrow i\pi-\theta$ into the final $s$-channel amplitude would produce incorrect results for the amplitude from the other channels and hence they must be explicitly computed.
\subsubsection*{(a) Both vertices contain interaction $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$:}
First consider the bubble diagrams shown in figure \ref{bubblediagram_twoderv} where both vertices contain quartic couplings $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$.
\begin{figure}[h!]
\centering
\subfigure[$s$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{s_twoderv.pdf}
\label{bubblediagram_twoderv_s} }
\subfigure[$t$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{t_twoderv.pdf}
\label{bubblediagram_twoderv_t} }
\subfigure[$u$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{u_twoderv.pdf}
\label{bubblediagram_twoderv_u} }
\caption{\small The $s$, $t$ and $u$-channel amplitudes when both interaction vertices are $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$.}
\label{bubblediagram_twoderv}
\end{figure}
The amplitude due to these kinds of bubble diagrams is given by,
\begin{eqnarray}
\mathcal{A}_{\text{a}}&=& \frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2\langle 0 | b(p_4) a(p_3) \text{T}\Big[\tilde{X}_{\mu\nu}(y)\tilde{X}^{\mu\nu}(y) \tilde{X}_{\rho\lambda}(x)\tilde{X}^{\rho\lambda}(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\
&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[
\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&+&\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \
{\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\nonumber\\
&+&4\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \ {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&+&{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\nonumber\\
&-&2{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&+&{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \ {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \ \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\ .
\label{TmunuPower4}
\end{eqnarray}
The $s$-channel amplitude (figure \ref{bubblediagram_twoderv_s}) can be computed by evaluating the contributions to the $s$-channel from each of the nine terms in (\ref{TmunuPower4}). By the general $s$-channel amplitude (\ref{schannelbubblegeneral}) the contribution from the first term in (\ref{TmunuPower4}), $\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)$, to the amplitude is given by,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_1)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.2cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\frac{\lambda^2}{128}\gamma_{ef(\mu}\gamma^{(\mu}_{gh}\gamma_{ab(\rho}\gamma^{(\rho}_{cd} \ \times\nonumber\\
&\bigg[&-\bar{v}_c(p_2)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}
\Big(\bar{u}_e(p_3)v_f(p_4)G_{ha}(\xi+q)G_{bg}(q)i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&-&\bar{u}_e(p_3)v_h(p_4)G_{fa}(\xi+q)G_{bg}(q) (-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)} \nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{ha}(\xi+q)G_{be}(q)i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&+&\bar{u}_g(p_3)v_h(p_4)G_{fa}(\xi+q)G_{be}(q)(-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\Big)
\nonumber\end{eqnarray}\begin{eqnarray}
&-&\bar{v}_a(p_2)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{u}_e(p_3)v_f(p_4)G_{hc}(\xi+q)G_{dg}(q) i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&-&\bar{u}_e(p_3)v_h(p_4)G_{fc}(\xi+q)G_{dg}(q)(-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{hc}(\xi+q)G_{de}(q)i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&+&\bar{u}_g(p_3)v_h(p_4)G_{fc}(\xi+q)G_{de}(q)(-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\Big)
\nonumber\\
&+&\bar{v}_a(p_2)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{u}_e(p_3)v_f(p_4)G_{hc}(\xi+q)G_{bg}(q)i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&-&\bar{u}_e(p_3)v_h(p_4)G_{fc}(\xi+q)G_{bg}(q)(-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{hc}(\xi+q)G_{be}(q)i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&+&\bar{u}_g(p_3)v_h(p_4)G_{fc}(\xi+q)G_{be}(q)(-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\Big)
\nonumber\\
&+&\bar{v}_c(p_2)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{u}_e(p_3)v_f(p_4)G_{ha}(\xi+q)G_{dg}(q)i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&-&\bar{u}_e(p_3)v_h(p_4)G_{fa}(\xi+q)G_{dg}(q)(-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&-&\bar{u}_{g}(p_3)v_f(p_4)G_{ha}(\xi+q)G_{de}(q)i p_{4\nu)}(-i)(\xi+q)^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&+&\bar{u}_g(p_3)v_h(p_4)G_{fa}(\xi+q)G_{de}(q)(-i)(\xi+q)_{\nu)}i p_4^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\Big)\bigg]\ ,
\label{twoderv_s1}
\end{eqnarray}
where $\xi^{\mu}=(p_1+p_2)^{\mu}$.
The above expression, (\ref{twoderv_s1}), involves the one-loop integrals $(I_{\mu})^{(s)}_{abcd}(\xi)$ and $(I_{\mu\nu})^{(s)}_{abcd}(\xi)$ whose values are given by (\ref{I1_abcd}) and (\ref{I2_abcd}), respectively. Plugging the values of the integrals and Dirac spinors, (\ref{uv}) and (\ref{ubarvbar}), into (\ref{twoderv_s1}) one finds the $s$-channel contribution from the first term in (\ref{TmunuPower4}) to be,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_1)}&=&-\frac{\lambda ^2 m^6}{768 \pi }\bigg[\frac{3}{2} \pi \csch \theta \Big(87+24 \cosh \theta +17 \cosh 2 \theta +12 \cosh 3 \theta +4 \cosh 4 \theta \Big)
\nonumber\\
&+&i \Big(95-6 \frac{\Lambda ^2}{m^2}-24\log\frac{m}{\Lambda}+6 \cosh \theta \big(1+25 \log \frac{m}{\Lambda }\big)-6 \cosh 2 \theta \big(3+2 \frac{\Lambda ^2}{m^2}-12 \log \frac{m}{\Lambda }\big)\nonumber\\
&-&2\cosh3\theta\big(5-12\log\frac{m}{\Lambda}\big)+\frac{3}{2} \theta \csch \theta \big(87+24 \cosh \theta +17 \cosh 2 \theta +12 \cosh 3 \theta \nonumber\\
&+&4 \cosh 4 \theta \big)\Big)\bigg]\ .
\label{twoderv_s1_final}
\end{eqnarray}
The contributions to the $s$-channel amplitude from the eight additional terms in (\ref{TmunuPower4}) can be obtained in a similar manner. Their values are given by,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_2)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.2cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big]a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\frac{\lambda ^2 m^6}{1536 \pi }\bigg[6\pi \csch \theta \Big(29+4 \cosh \theta +3 \cosh 2 \theta +8\cosh 3 \theta +4 \cosh 4 \theta \Big)\nonumber\\
&+&i \Big(98-6 \frac{\Lambda ^2}{m^2}+\cosh \theta \big(-47-42 \frac{\Lambda ^2}{m^2}+228 \log \frac{m}{\Lambda }\big)+ 2 \cosh 2\theta \big(-46-21\frac{\Lambda^2}{m^2}\nonumber
\end{eqnarray}
\begin{eqnarray}
&+&96\log \frac{m}{\Lambda }\big)+\cosh3\theta \big(-37+96\log\frac{m}{\Lambda}\big)+6\theta \csch \theta\big(29+4 \cosh \theta +3 \cosh 2 \theta \nonumber\\
&+&8\cosh 3 \theta +4 \cosh 4 \theta\big)
\Big)\bigg]\ ,
\label{twoderv_s2}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_3)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.2cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\frac{\lambda ^2 m^6}{1536 \pi }\bigg[3\pi \csch \theta \Big(87+24 \cosh \theta +17 \cosh 2 \theta +12\cosh 3 \theta +4 \cosh 4 \theta \Big)\nonumber\\
&+&i \Big(70-240\log\frac{m}{\Lambda}+2\cosh \theta \big(-74+15 \frac{\Lambda ^2}{m^2}+48 \log \frac{m}{\Lambda }\big)+ \cosh 2\theta \big(-82+144\log \frac{m}{\Lambda }\big)\nonumber\\
&+&4\cosh3\theta \big(-5+12\log\frac{m}{\Lambda}\big)+3\theta \csch \theta\big(87+24 \cosh \theta +17 \cosh 2 \theta +12\cosh 3 \theta \nonumber\\
&+&4 \cosh 4 \theta\big)
\Big)\bigg]\ ,
\label{twoderv_s3}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_4)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(s_2)}\ ,
\label{twoderv_s4}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_5)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.2cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\frac{\lambda ^2 m^6}{76800\pi }\bigg[600 \pi \csch \theta \Big(15-7 \cosh 2\theta +4 \cosh 3 \theta +4 \cosh 4 \theta \Big)
\nonumber\\
&+&i \Big(4252-120 \frac{\Lambda ^2}{m^2}-9600\log\frac{m}{\Lambda}+ \cosh \theta \big(1111-5640\frac{\Lambda^2}{m^2}-1350\frac{\Lambda^4}{m^4}+10800 \log \frac{m}{\Lambda }\big)\nonumber\\
&-&8 \cosh 2 \theta \big(986+615 \frac{\Lambda ^2}{m^2}-1200 \log \frac{m}{\Lambda }\big)
+\cosh3\theta\big(-3247+9600\log\frac{m}{\Lambda}\big)\nonumber\\
&+&600\theta \csch \theta \big(15-7 \cosh 2\theta +4 \cosh 3 \theta +4 \cosh 4 \theta \big)\Big)\bigg]\ ,
\label{twoderv_s5}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_6)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(s_2)}\ ,
\label{twoderv_s6}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_7)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(s_3)}\ ,
\label{twoderv_s7}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_8)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(s_2)}\ ,
\label{twoderv_s8}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s_9)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(s_1)}\ .
\label{twoderv_s9}
\end{eqnarray}
Combining the $s$-channel contributions, (\ref{twoderv_s1_final}) - (\ref{twoderv_s9}), gives the total $s$-channel amplitude from figure \ref{bubblediagram_twoderv_s},
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(s)}&=&2\Big(\mathcal{A}_{\text{a}}^{(s_1)}+\mathcal{A}_{\text{a}}^{(s_2)}+\cdots+\mathcal{A}_{\text{a}}^{(s_9)}\Big)\nonumber\\
&=&-\frac{\lambda ^2 m^6}{38400\pi }\bigg[9600 \pi \csch \theta \Big(10+2\cosh\theta +\cosh 2\theta +2 \cosh 3 \theta + \cosh 4 \theta \Big)
\nonumber\\
&+&i \Big(49852-2520 \frac{\Lambda ^2}{m^2}-38400\log\frac{m}{\Lambda}+ \cosh \theta \big(-21889-11040\frac{\Lambda^2}{m^2}-1350\frac{\Lambda^4}{m^4}\nonumber\\
&+&96000 \log \frac{m}{\Lambda }\big)
+24 \cosh 2 \theta \big(-1587-655 \frac{\Lambda ^2}{m^2}+3200 \log \frac{m}{\Lambda }\big)
+\cosh3\theta\big(-14647\nonumber\\
&+&38400\log\frac{m}{\Lambda}\big)
+9600\theta \csch \theta \big(10+2\cosh\theta +\cosh 2\theta +2 \cosh 3 \theta + \cosh 4 \theta \big)\Big)\bigg]\ .\nonumber\\
\label{twoderv_s_amplitude}
\end{eqnarray}
An extra multiplicative factor $2$ arises from the identical contribution of the diagram with the two vertices $x$ and $y$ exchanged. The same factor will be included while computing the $t$ and $u$-channel amplitudes.
\\\\
Next, the amplitude due to the same quartic interactions will be computed in the $t$-channel case (figure \ref{bubblediagram_twoderv_t}).
By the general $t$-channel amplitude (\ref{tchannelbubblegeneral}) the contribution from the first term in (\ref{TmunuPower4}), $\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)$, to the amplitude is given by,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_1)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.2cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-\frac{\lambda^2}{128}\gamma_{ef(\mu}\gamma^{(\mu}_{gh}\gamma_{ab(\rho}\gamma^{(\rho}_{cd} \ \times\nonumber\\
&\bigg[&
\bar{u}_c(p_3)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}
\Big(\bar{v}_g(p_2)v_h(p_4)G_{fa}(q)G_{be}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&-&\bar{v}_g(p_2)v_f(p_4)G_{ha}(q)G_{be}(q)(-i) q^{\nu)}i p_{4\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fa}(q)G_{bg}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&+&\bar{v}_e(p_2)v_f(p_4)G_{ha}(q)G_{bg}(q)(-i) q^{\nu)}i p_{4\nu)}(-i) q_{\lambda)}(-i)p_1^{\lambda)}\Big) \nonumber\end{eqnarray}\begin{eqnarray}
&+&\bar{u}_a(p_3)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{v}_g(p_2)v_h(p_4)G_{fc}(q)G_{de}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&-&\bar{v}_g(p_2)v_f(p_4)G_{hc}(q)G_{de}(q)i p_{4\nu)}(-i) q^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fc}(q)G_{dg}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\nonumber\\
&+&\bar{v}_e(p_2)v_f(p_4)G_{hc}(q)G_{dg}(q)i p_{4\nu)}(-i) q^{\nu)}(-i) p_{1\lambda)}(-i)q^{\lambda)}\Big)
\nonumber\\
&-&\bar{u}_a(p_3)u_d(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{v}_g(p_2)v_h(p_4)G_{fc}(q)G_{be}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i)q_{\lambda)}(-i) p_1^{\lambda)}\nonumber\\
&-&\bar{v}_g(p_2)v_f(p_4)G_{hc}(q)G_{be}(q) i p_{4\nu)}(-i) q^{\nu)}(-i)q_{\lambda)}(-i) p_1^{\lambda)}\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fc}(q)G_{bg}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i)q_{\lambda)}(-i) p_1^{\lambda)}\nonumber\\
&+&\bar{v}_e(p_2)v_f(p_4)G_{hc}(q)G_{bg}(q)i p_{4\nu)}(-i) q^{\nu)}(-i)q_{\lambda)}(-i) p_1^{\lambda)}\Big)
\nonumber\\
&-&\bar{u}_c(p_3)u_b(p_1)\int\frac{d^2q}{(2\pi)^2}\Big(\bar{v}_g(p_2)v_h(p_4)G_{fa}(q)G_{de}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i)p_{1\lambda)}(-i) q^{\lambda)}\nonumber\\
&-&\bar{v}_g(p_2)v_f(p_4)G_{ha}(q)G_{de}(q)i p_{4\nu)}(-i) q^{\nu)}(-i)p_{1\lambda)}(-i) q^{\lambda)}\nonumber\\
&-&\bar{v}_{e}(p_2)v_h(p_4)G_{fa}(q)G_{dg}(q)(-i) q_{\nu)}i p_4^{\nu)}(-i)p_{1\lambda)}(-i) q^{\lambda)}\nonumber\\
&+&\bar{v}_e(p_2)v_f(p_4)G_{ha}(q)G_{dg}(q)i p_{4\nu)}(-i) q^{\nu)}(-i)p_{1\lambda)}(-i) q^{\lambda)}\Big)
\bigg]\ .
\label{twoderv_t1}
\end{eqnarray}
The above expression, (\ref{twoderv_t1}), involves the one-loop integral $(I_{\mu\nu})^{(t)}_{abcd}$ given by (\ref{I_t_abcd}). Plugging the value of this integral and Dirac spinors into (\ref{twoderv_t1}) one finds the t-channel contribution from the first term in (\ref{TmunuPower4}) to be,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_1)}&=&-i\frac{\lambda ^2 m^6}{256 \pi }\bigg[1+ \frac{\Lambda ^2}{m^2}+6\log\frac{m}{\Lambda}- \cosh \theta \big(4+5\frac{\Lambda^2}{m^2}+28 \log \frac{m}{\Lambda }\big)+4\cosh 2 \theta \big(1+2 \log \frac{m}{\Lambda }\big)\bigg]\ .\nonumber\\
\label{twoderv_t1_final}
\end{eqnarray}
The contributions to the $t$-channel amplitude from the eight additional terms in (\ref{TmunuPower4}) can be obtained in a similar manner. Their values are given by,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_2)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.3cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda ^2 m^6}{256 \pi }\bigg[-2-2 \frac{\Lambda ^2}{m^2}-12\log\frac{m}{\Lambda}- 3\cosh \theta \big(1+\frac{\Lambda^2}{m^2}+6 \log \frac{m}{\Lambda }\big)+8\cosh 2 \theta \big(1+2 \log \frac{m}{\Lambda }\big)\bigg]\ ,\nonumber\\
\label{twoderv_t2}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_3)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.4cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda ^2 m^6}{256 \pi }\bigg[1+ \frac{\Lambda ^2}{m^2}+6\log\frac{m}{\Lambda}- \cosh \theta \big(5+4\frac{\Lambda^2}{m^2}+26 \log \frac{m}{\Lambda }\big)+4\cosh 2 \theta \big(1+2 \log \frac{m}{\Lambda }\big)\bigg]\ ,\nonumber\\
\label{twoderv_t3}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_4)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(t_2)}\ ,
\label{twoderv_t4}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_5)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.2cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&i\frac{\lambda ^2 m^6}{512\pi }\bigg[-8+10 \frac{\Lambda ^2}{m^2}+9\frac{\Lambda^4}{m^4}-48\log\frac{m}{\Lambda}+2 \cosh \theta \big(1+3\frac{\Lambda^2}{m^2}+12 \log \frac{m}{\Lambda }\big)\nonumber\\
&-&32\cosh 2\theta \big(1+2 \log \frac{m}{\Lambda }\big)\bigg]\ ,
\label{twoderv_t5}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_6)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(t_2)}\ ,
\label{twoderv_t6}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_7)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(t_3)}\ ,
\label{twoderv_t7}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_8)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(t_2)}\ ,
\label{twoderv_t8}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t_9)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(t_1)}\ .
\label{twoderv_t9}
\end{eqnarray}
Combining the $t$-channel contributions, (\ref{twoderv_t1_final}) - (\ref{twoderv_t9}), gives the total $t$-channel amplitude from figure \ref{bubblediagram_twoderv_t},
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(t)}&=&2\Big(\mathcal{A}_{\text{a}}^{(t_1)}+\mathcal{A}_{\text{a}}^{(t_2)}+\cdots+\mathcal{A}_{\text{a}}^{(t_9)}\Big)\nonumber\\
&=&i\frac{\lambda ^2 m^6}{256 \pi }\bigg[18 \frac{\Lambda ^2}{m^2}+9\frac{\Lambda ^4}{m^4}+2\cosh \theta \big(31+33\frac{\Lambda^2}{m^2}+192 \log \frac{m}{\Lambda }\big)-128\cosh 2 \theta \big(1+2 \log \frac{m}{\Lambda }\big)\bigg]\ .\nonumber\\
\label{twoderv_t_amplitude}
\end{eqnarray}
Lastly, the amplitude due to the same quartic interaction will be computed in the $u$-channel case (figure \ref{bubblediagram_twoderv_u}). By the general $u$-channel amplitude (\ref{uchannelbubblegeneral}), the contribution from the first term in (\ref{TmunuPower4}), $\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)$, to the amplitude is given by,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_1)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-\frac{\lambda^2}{128}\gamma_{ef(\mu}\gamma^{(\mu}_{gh}\gamma_{ab(\rho}\gamma^{(\rho}_{cd}\ \times\nonumber\\
&\Big[&u_d(p_1)v_b(p_4)\bar{u}_e(p_3)\bar{v}_g(p_2)i p_{4\lambda)}(-i)p_1^{\lambda)}
-u_d(p_1)v_b(p_4)\bar{u}_g(p_3)\bar{v}_e(p_2)i p_{4\lambda)}(-i)p_1^{\lambda)}\nonumber\\
&-&u_b(p_1)v_d(p_4)\bar{u}_e(p_3)\bar{v}_g(p_2)(-i)p_{1\lambda)}i p_4^{\lambda)}
+u_b(p_1)v_d(p_4)\bar{u}_g(p_3)\bar{v}_e(p_2)(-i)p_{1\lambda)}i p_4^{\lambda)}\Big]\nonumber\\
&\times&\int\frac{d^2q}{(2\pi)^2}\Big[G_{hc}(\zeta-q)G_{fa}(q)(-i)q_{\nu)}(-i)(\zeta-q)^{\nu)}
-G_{ha}(\zeta-q)G_{fc}(q)(-i)q_{\nu)}(-i)(\zeta-q)^{\nu)}\Big]\ ,\nonumber\\
\label{twoderv_u1}
\end{eqnarray}
where $\zeta^{\mu}=(p_1-p_4)^{\mu}=(p_1-p_2)^{\mu}$. The above expression, (\ref{twoderv_u1}), involves the one-loop integrals $(I_{\mu})_{abcd}^{(u)}(\zeta)$ and $(I_{\mu\nu})_{abcd}^{(u)}(\zeta)$ whose values are given by (\ref{I1_abcd_u}) and (\ref{I2_abcd_u}), respectively. Plugging the values of the integrals and Dirac spinors into (\ref{twoderv_u1}) one finds the $u$-channel contribution from the first term in (\ref{TmunuPower4}) to be,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_1)}&=&-i\frac{\lambda ^2 m^6}{128 \pi}\cosh^2\frac{\theta}{2}\big(1-4\cosh\theta\big)\bigg[3 \frac{\Lambda ^2}{m^2}+10\log\frac{m}{\Lambda}+3\theta\coth \frac{\theta}{2}+4\theta \sinh \theta\nonumber\\
&-&\cosh \theta \big(3-8\log \frac{m}{\Lambda }\big)\bigg]\ .
\label{twoderv_u1_final}
\end{eqnarray}
The contributions to the u-channel amplitude from the eight additional terms in (\ref{TmunuPower4}) can be obtained in a similar manner. Their values are given by,
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_2)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda ^2 m^6}{192 \pi}\cosh^2\frac{\theta}{2}\bigg[-18-24 \frac{\Lambda ^2}{m^2}+9\theta\coth \frac{\theta}{2}+24\theta \sinh 2\theta+\cosh \theta \big(25+42\frac{\Lambda^2}{m^2}+24\log \frac{m}{\Lambda }\big)\nonumber\\
&+&\cosh 2\theta \big(-25+48\log \frac{m}{\Lambda }\big)\bigg]\ ,
\label{twoderv_u2}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_3)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y)\bar{\psi}(y)\gamma^{(\mu}{\partial}^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.1cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda ^2 m^6}{19200 \pi}\cosh^2\frac{\theta}{2}\bigg[-1311+30 \frac{\Lambda ^2}{m^2}-675\frac{\Lambda^4}{m^4}+8700\log\frac{m}{\Lambda}+150\theta\coth\frac{\theta}{2}\big(1-4\cosh\theta\big)^2\nonumber\\
&+&2\cosh \theta \big(29+660\frac{\Lambda^2}{m^2}+1200 \log \frac{m}{\Lambda }\big)
+\cosh 2 \theta \big(-1447+2400\log \frac{m}{\Lambda }\big)\bigg]\ ,
\label{twoderv_u3}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_4)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda ^2 m^6}{64 \pi }\cosh \frac{\theta }{2} (-1+4 \cosh \theta )\bigg[\cosh \frac{3 \theta }{2}\big(-1+4\log \frac{m}{\Lambda}\big) +\cosh \frac{ \theta }{2}\big(1+\theta\coth\frac{\theta}{2}+4\theta\sinh\theta\big)\bigg]\ ,\nonumber\\
\label{twoderv_u4}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_5)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda ^2 m^6}{768\pi }\bigg[-43-12 \frac{\Lambda ^2}{m^2}+72\log\frac{m}{\Lambda}+24\theta\cosh^2\frac{\theta}{2}\coth\frac{\theta}{2}\big(3-4\cosh\theta\big)^2\nonumber\\
&+&4 \cosh \theta \big(7+3\frac{\Lambda^2}{m^2}-6 \log \frac{m}{\Lambda }\big)+\cosh 2\theta \big(31
+24 \frac{\Lambda^2}{m^2}\big)-8\cosh 3\theta \big(5-12 \log\frac{m}{\Lambda}\big)\bigg]\ ,\nonumber\\
\label{twoderv_u5}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_6)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma_{(\mu}{\partial}_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(u_2)}\ ,
\label{twoderv_u6}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_7)}&=&\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{4.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda ^2 m^6}{128\pi }\big(\cosh\frac{\theta}{2}+2\cosh\frac{3\theta}{2}\big)^2\big(\theta\coth\frac{\theta}{2}+2\log\frac{m}{\Lambda}\big)\ ,
\label{twoderv_u7}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_8)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(u_4)}\ ,
\label{twoderv_u8}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u_9)}&=&\frac{1}{2!}\big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{(\mu}\bar{\psi}(y) \gamma_{\nu)}\psi(y) {\partial}^{(\mu}\bar{\psi}(y) \gamma^{\nu)}\psi(y) \ \nonumber\\
&& \hspace{0.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}
\ \ = \ \ \mathcal{A}_{\text{a}}^{(u_1)}\ .
\label{twoderv_u9}
\end{eqnarray}
Combining the $u$-channel contributions, (\ref{twoderv_u1_final}) - (\ref{twoderv_u9}), gives the total $u$-channel amplitude from figure \ref{bubblediagram_twoderv_u},
\begin{eqnarray}
\mathcal{A}_{\text{a}}^{(u)}&=&2\Big(\mathcal{A}_{\text{a}}^{(u_1)}+\mathcal{A}_{\text{a}}^{(u_2)}+\cdots+\mathcal{A}_{\text{a}}^{(u_9)}\Big)\nonumber\\
&=&-i\frac{\lambda ^2 m^6}{9600 \pi }\cosh^2\frac{\theta}{2}\bigg[18011+7470 \frac{\Lambda ^2}{m^2}+675\frac{\Lambda ^4}{m^4}+6\cosh\theta\big(-3843-2620\frac{\Lambda^2}{m^2}+3200\log\frac{m}{\Lambda}\big)\nonumber\\
&+&\cosh 2\theta \big(14647-38400\log\frac{m}{\Lambda}\big)-9600 \big(\theta\coth\frac{\theta}{2}+4 \log \frac{m}{\Lambda }+2\theta\sinh2\theta\big)\bigg]\ .
\label{twoderv_u_amplitude}
\end{eqnarray}
Finally, adding together the contributions from the $s$, $t$ and $u$-channels to the amplitude, (\ref{twoderv_s_amplitude}), (\ref{twoderv_t_amplitude}) and (\ref{twoderv_u_amplitude}), one finds the total amplitude from the bubble diagrams where both vertices contain the quartic interaction $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$,
\begin{eqnarray}
\mathcal{A}_{\text{a}}&=&\mathcal{A}_{\text{a}}^{(s)}+\mathcal{A}_{\text{a}}^{(t)}+\mathcal{A}_{\text{a}}^{(u)}\nonumber\\
&=&-\frac{\lambda^2m^6}{19200\pi\sinh\theta}\bigg[4800\pi\Big(10+2\cosh\theta+\cosh2\theta+2\cosh3\theta+\cosh4\theta\Big)+i\Big(\big(34571\nonumber\\
&+&4860\frac{\Lambda^2}{m^2}\big)\sinh\theta
-\sinh2\theta\big(6659+9360\frac{\Lambda^2}{m^2}+9600\log\frac{m}{\Lambda}\big)+4800\big(8\theta-13\sinh\theta\log\frac{m}{\Lambda}\big)\nonumber\\
&-&\sinh3\theta\big(3163+7860\frac{\Lambda^2}{m^2}-14400\log\frac{m}{\Lambda}\big)
\Big)\bigg]\ .
\label{amplitude_twoderv}
\end{eqnarray}
\subsubsection*{(b) One vertex contains $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ while the other contains $(\tilde{X}_{\mu}^{\ \mu})^2$ :}
Consider the bubble diagrams shown in figure \ref{bubblediagram_onederv} where one vertex contains the quartic coupling $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ while the second vertex contains the quartic coupling $(\tilde{X}_{\mu}^{\ \mu})^2$.
\begin{figure}[h!]
\centering
\subfigure[$s$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{s_onederv.pdf}
\label{bubblediagram_onederv_s} }
\subfigure[$t$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{t_onederv.pdf}
\label{bubblediagram_onederv_t} }
\subfigure[$u$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{u_onederv.pdf}
\label{bubblediagram_onederv_u} }
\caption{\small The $s$, $t$ and $u$-channel amplitudes when one interaction vertex is $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ and the other vertex is $(\tilde{X}_{\mu}^{\ \mu})^2$.}
\label{bubblediagram_onederv}
\end{figure}
The amplitude due to these kinds of bubble diagrams is given by,
\begin{eqnarray}
\mathcal{A}_{\text{b}}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \langle 0 | b(p_4) a(p_3) \text{T}\Big[\big(\tilde{X}_{\mu}^{\ \mu}(y)\big)^2 \tilde{X}_{\rho\lambda}(x)\tilde{X}^{\rho\lambda}(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\
&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[
\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&+&\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \
{\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\nonumber\\
&+&4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&+&{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\nonumber\\
&-&2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\nonumber\\
&+&{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \
\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\ .
\label{TmumusqTmunusq}
\end{eqnarray}
The multiplicative factor of $2$ arises from the two identical types of cross-terms:
\vspace{2mm}
\\
$\big(\tilde{X}_{\mu}^{\ \mu}(y)\big)^2 \tilde{X}_{\rho\lambda}(x)\tilde{X}^{\rho\lambda}(x)$ \hspace{1mm} and \hspace{1mm} $\tilde{X}_{\mu\nu}(y)\tilde{X}^{\mu\nu}(y)\big(\tilde{X}_{\rho}^{\ \rho}(x)\big)^2 $.
\vspace{3mm}
\\
The $s$-channel contribution to the amplitude from figure \ref{bubblediagram_onederv_s} can be computed by evaluating the contributions from each of the above terms in (\ref{TmumusqTmunusq}) and adding them together. By the general s-channel amplitude (\ref{schannelbubblegeneral}), the contributions from all nine terms in (\ref{TmumusqTmunusq}) are,
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_1)}
&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16} \langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&\frac{\lambda^2 m^6}{192\pi\sinh\theta}\bigg[\frac{3\pi}{2}\big(29+10\cosh\theta+7\cosh2\theta+2\cosh 3\theta\big)
+i\Big(\sinh\theta\big(36-6\frac{\Lambda^2}{m^2}-24\log\frac{m}{\Lambda}\big)\nonumber\\
&+&\sinh\theta\cosh 2\theta \big(1+12\log\frac{m}{\Lambda}\big)+5\sinh2\theta(1+3\log\frac{m}{\Lambda})+\frac{3}{2}\theta\big(29+10\cosh\theta+7\cosh2\theta\nonumber\\
&+&2\cosh 3\theta\big)\Big)\bigg]\ ,
\label{onederv_s1_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_2)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&\frac{\lambda^2 m^6}{192\pi\sinh\theta}\bigg[3\pi\big(7+2\cosh\theta+5\cosh2\theta+2\cosh 3\theta\big)
+i\Big(13\sinh\theta-\sinh2\theta\big(7+\frac{3}{2}\frac{\Lambda^2}{m^2}\nonumber\\
&-&15\log\frac{m}{\Lambda}\big)
+24\sinh2\theta\cosh\theta \log\frac{m}{\Lambda}+3\theta\big(7+2\cosh\theta+5\cosh2\theta+2\cosh 3\theta\big)\Big)\bigg]\ , \nonumber\\
\label{onederv_s2_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_3)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times
{\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&\frac{\lambda^2 m^6}{128\pi\sinh\theta}\bigg[\pi\big(29+10\cosh\theta+7\cosh2\theta+2\cosh 3\theta\big)
+i\Big(8\sinh\theta\big(1-4\log\frac{m}{\Lambda}\nonumber\\
&+&\cosh2\theta\log\frac{m}{\Lambda}\big)+\sinh2\theta\big(-6+3\frac{\Lambda^2}{m^2}+4\log\frac{m}{\Lambda}\big)
+\theta\big(29+10\cosh\theta+7\cosh2\theta\nonumber\\
&+&2\cosh 3\theta\big)\Big)\bigg]\ ,
\label{onederv_s3_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_4)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ -2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&\frac{\lambda^2 m^6}{384\pi\sinh\theta}\bigg[6\pi\big(29+10\cosh\theta+7\cosh2\theta+2\cosh 3\theta\big)
+i\Big(\sinh\theta\big(115-24\frac{\Lambda^2}{m^2}\nonumber\end{eqnarray}\begin{eqnarray}
&-&144\log\frac{m}{\Lambda}+\cosh3\theta\big)
+\sinh\theta\cosh2\theta\big(13-6\frac{\Lambda^2}{m^2}+48\log\frac{m}{\Lambda}\big)+\sinh2\theta\big(\frac{13}{2}+42\log\frac{m}{\Lambda}\big)\nonumber\\
&+&6\theta\big(29+10\cosh\theta+7\cosh2\theta+2\cosh 3\theta\big)\Big)\bigg]\ ,
\label{onederv_s4_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_5)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&\frac{\lambda^2 m^6}{19200\pi\sinh\theta}\bigg[600\pi\big(7+2\cosh\theta+5\cosh2\theta+2\cosh 3\theta\big)
+i\Big(12\sinh\theta\big(177+30\frac{\Lambda^2}{m^2}\nonumber\\
&+&400\log\frac{m}{\Lambda}+\frac{151}{12}\cosh3\theta\big)
+48\sinh\theta\cosh2\theta\big(8-5\frac{\Lambda^2}{m^2}+100\log\frac{m}{\Lambda}\big)-\sinh2\theta\big(\frac{3343}{2}\nonumber\\
&+&540\frac{\Lambda^2}{m^2}+225\frac{\Lambda^4}{m^4}
-3000\log\frac{m}{\Lambda}\big)
+600 \ \theta\big(7+2\cosh\theta+5\cosh2\theta+2\cosh 3\theta\big)\Big)\bigg]\ ,\nonumber\\
\label{onederv_s5_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_6)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(s_4)}\ ,
\label{onederv_s6_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_7)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(s_3)}\ ,
\label{onederv_s7_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_8)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16} \langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(s_2)}\ ,
\label{onederv_s8_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s_9)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(s_1)}\ ,
\label{onederv_s9_final}
\end{eqnarray}
where the computational method that was employed was exactly the same as when $\mathcal{A}_{\text{a}}^{(s_1)}$ was computed in (\ref{twoderv_s1}).
\vspace{2mm}
\\
Combining the $s$-channel contributions to the amplitude, (\ref{onederv_s1_final}) - (\ref{onederv_s9_final}), gives the total $s$-channel contribution to the amplitude from the diagrams where one vertex contains the quartic coupling $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ while the other vertex contains the quartic coupling $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(s)}&=& 2\Big(\mathcal{A}_{\text{b}}^{(s_1)}+\mathcal{A}_{\text{b}}^{(s_2)}+\cdots+\mathcal{A}_{\text{b}}^{(s_9)}\Big)\nonumber\\
&=&\frac{\lambda^2m^6}{19200\pi\sinh\theta}\bigg[9600\pi\big(9+3\cosh\theta+3\cosh2\theta+\cosh 3\theta\big)+i\Big(9600 \ \theta\big(9+3\cosh\theta\nonumber\\
&+&3\cosh2\theta+\cosh3\theta\big)+12\sinh\theta\big(4147-470\frac{\Lambda^2}{m^2}-4800\log\frac{m}{\Lambda}\big)
-2\sinh2\theta\big(3347\nonumber\\
&-&60\frac{\Lambda^2}{m^2}+225\frac{\Lambda^4}{m^4}
-14400\log\frac{m}{\Lambda}\big)
+12\sinh3\theta\big(157-70\frac{\Lambda^2}{m^2}+1600\log\frac{m}{\Lambda}\big)+251\sinh4\theta\Big)\bigg]\ .\nonumber\\
\label{onederv_s_amplitude}
\end{eqnarray}
Just as in the computation of the first type of bubble diagram, (\ref{twoderv_s_amplitude}), a factor of $2$ arises in the first line of the above expression, (\ref{onederv_s_amplitude}), because the same contribution would be found if the vertices $x$ and $y$ were exchanged. A similar factor would appear while evaluating the $t$ and $u$-channel amplitudes as well.
\vspace{2mm}
\\
The contribution to the amplitude from the $t$-channel diagram, figure \ref{bubblediagram_onederv_t}, can be computed using the general $t$-channel amplitude, (\ref{tchannelbubblegeneral}). The nine terms in (\ref{TmumusqTmunusq}) contribute to the amplitude as,
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_1)}
&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16} \langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&i\frac{\lambda^2m^6}{64\pi}\Big[1+\frac{\Lambda^2}{m^2}+6\log\frac{m}{\Lambda}-\cosh\theta\big(2+\frac{\Lambda^2}{m^2}+8\log\frac{m}{\Lambda}\big)\Big]\ ,
\label{onederv_t1_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_2)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{64\pi}(2+\cosh\theta)\Big[1+\frac{\Lambda^2}{m^2}+6\log\frac{m}{\Lambda}\Big]\ ,
\label{onederv_t2_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_3)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times
{\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&i\frac{\lambda^2m^6}{64\pi}\Big[1+\frac{\Lambda^2}{m^2}+6\log\frac{m}{\Lambda}-\cosh\theta\big(1+2\frac{\Lambda^2}{m^2}+10\log\frac{m}{\Lambda}\big)\Big]\ ,
\label{onederv_t3_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_4)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ -2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{64\pi}(-2+3\cosh\theta)\Big[1+\frac{\Lambda^2}{m^2}+6\log\frac{m}{\Lambda}\Big]\ ,
\label{onederv_t4_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_5)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda^2m^6}{128\pi}\Big[8+14\frac{\Lambda^2}{m^2}+3\frac{\Lambda^4}{m^4}+48\log\frac{m}{\Lambda}+2\cosh\theta\big(1+3\frac{\Lambda^2}{m^2}+12\log\frac{m}{\Lambda}\big)\Big]\ ,
\label{onederv_t5_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_6)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(t_4)}\ ,
\label{onederv_t6_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_7)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(t_3)}\ ,
\label{onederv_t7_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_8)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16} \langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(t_2)}\ ,
\label{onederv_t8_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t_9)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.1cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{b}}^{(t_1)}\ ,
\label{onederv_t9_final}
\end{eqnarray}
where the computational method that was employed was exactly the same as was used to evaluate $\mathcal{A}_{\text{a}}^{(t_1)}$ in (\ref{twoderv_t1}).
\vspace{2mm}
\\
Combining the $t$-channel contributions to the amplitude, (\ref{onederv_t1_final}) - (\ref{onederv_t9_final}), gives the total $t$-channel contribution to the amplitude from the diagram where one vertex contains the quartic coupling $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ while the other vertex contains the quartic coupling $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(t)}&=& 2\Big(\mathcal{A}_{\text{b}}^{(t_1)}+\mathcal{A}_{\text{b}}^{(t_2)}+\cdots+\mathcal{A}_{\text{b}}^{(t_9)}\Big)\nonumber\\
&=&-i\frac{\lambda^2 m^6}{64\pi}\bigg[3\frac{\Lambda^4}{m^4}+6\frac{\Lambda^2}{m^2}+2\cosh\theta\Big(15+17\frac{\Lambda^2}{m^2}+96\log\frac{m}{\Lambda}\Big)\bigg]\ .
\label{onederv_t_amplitude}
\end{eqnarray}
The contribution to the amplitude of the $u$-channel diagram, figure \ref{bubblediagram_onederv_u}, can be computed using the general $u$-channel amplitude, (\ref{uchannelbubblegeneral}). The contributions from the nine terms in (\ref{TmumusqTmunusq}), to the amplitude, are given by,
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_1)}
&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16} \langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda^2m^6}{96\pi}\cosh^2\frac{\theta}{2} \ \big(1-4\cosh\theta\big)\Big[-4+3\frac{\Lambda^2}{m^2}+18\log\frac{m}{\Lambda}+\cosh\theta+3\theta\coth\frac{\theta}{2} \ \Big]\ ,\nonumber\\
\label{onederv_u1_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_2)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2m^6}{96\pi}\coth\frac{\theta}{2}\bigg[\frac{3}{2}\sinh\theta\big(1+2\frac{\Lambda^2}{m^2}\big)-2\cosh\frac{3\theta}{2}\sinh\frac{\theta}{2}\big(8-3\frac{\Lambda^2}{m^2}-24\log\frac{m}{\Lambda}\big)\nonumber\\
&+&\cosh\frac{5\theta}{2}\sinh\frac{\theta}{2}-3\theta\big(1-\cosh\theta-2\cosh2\theta\big)\bigg]\ ,
\label{onederv_u2_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_3)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times
{\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2m^6}{1600\pi}\bigg[25\theta\coth\frac{\theta}{2}\big(1+3\cosh\theta+2\cosh2\theta\big)+\cosh^2\frac{\theta}{2}\Big(-79+270\frac{\Lambda^2}{m^2}-75\frac{\Lambda^4}{m^4}\nonumber\\
&&+1100\log\frac{m}{\Lambda}
+2\cosh\theta\big(-119-60\frac{\Lambda^2}{m^2}+200\log\frac{m}{\Lambda}\big)+17\cosh2\theta\Big)\bigg]\ ,
\label{onederv_u3_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_4)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ -2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda^2m^6}{16\pi}\cosh^2\frac{\theta}{2} \ \big(1-4\cosh\theta\big)\Big[4\log\frac{m}{\Lambda}+\theta\coth\frac{\theta}{2} \ \Big]\ ,
\label{onederv_u4_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_5)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2m^6}{96\pi}\coth\frac{\theta}{2} \Big[\sinh\theta \ \big(\frac{1}{2}+6\frac{\Lambda^2}{m^2}\big)-6\theta\big(1-\cosh\theta-2\cosh2\theta\big)\nonumber\\
&&+\sinh\frac{\theta}{2}\cosh\frac{3\theta}{2}\big(-13+72\log\frac{m}{\Lambda}\big) \ \Big]\ ,
\label{onederv_u5_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_6)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2m^6}{16\pi}\cosh^2\frac{\theta}{2} \ \bigg[-1+6\frac{\Lambda^2}{m^2}+16\log\frac{m}{\Lambda}+\theta\big(4\sinh\theta+3\coth\frac{\theta}{2}\big)\nonumber\\
&&+\cosh\theta\big(-5+8\log\frac{m}{\Lambda}\big)\bigg]\ ,
\label{onederv_u6_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_7)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) \bar{\psi}(x)\gamma^{(\rho}{\partial}^{\lambda)}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda^2m^6}{32\pi}\cosh^2\frac{\theta}{2} \ \big(1-4\cosh\theta\big)\Big[2\log\frac{m}{\Lambda}+\theta\coth\frac{\theta}{2} \ \Big]\ ,
\label{onederv_u7_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_8)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16} \langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times \bar{\psi}(x)\gamma_{(\rho}{\partial}_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2m^6}{32\pi}\bigg[1+4\cosh\theta \ \log\frac{m}{\Lambda}-\cosh2\theta\big(1-4\log\frac{m}{\Lambda}\big)\nonumber\\
&&+\theta\big(5\sinh\theta+2\sinh2\theta+2\coth\frac{\theta}{2}\big)\bigg]\ ,
\label{onederv_u8_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u_9)}&=&-2\frac{1}{2!} \big(i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.5cm} \times {\partial}_{(\rho}\bar{\psi}(x) \gamma_{\lambda)}\psi(x) {\partial}^{(\rho}\bar{\psi}(x) \gamma^{\lambda)}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2m^6}{32\pi}\cosh^2\frac{\theta}{2} \ \bigg[3\frac{\Lambda^2}{m^2}+10\log\frac{m}{\Lambda}+\cosh\theta\big(-3+8\log\frac{m}{\Lambda}\big)+\theta\big(4\sinh\theta+3\coth\frac{\theta}{2}\big)\bigg]\ ,\nonumber\\
\label{onederv_u9_final}
\end{eqnarray}
where the computational method that was employed was exactly the same as when $\mathcal{A}_{\text{a}}^{(u_1)}$ was computed in (\ref{twoderv_u1}).
\vspace{2mm}
\\
Combining the $u$-channel contributions to the amplitude, (\ref{onederv_u1_final}) - (\ref{onederv_u9_final}), gives the total $u$-channel contribution to the amplitude from the diagram where one vertex contains the quartic coupling $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ while the other vertex contains the quartic coupling $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{b}}^{(u)}&=& 2\Big(\mathcal{A}_{\text{b}}^{(u_1)}+\mathcal{A}_{\text{b}}^{(u_2)}+\cdots+\mathcal{A}_{\text{b}}^{(u_9)}\Big)\nonumber\\
&=&-i\frac{\lambda^2m^6}{2400\pi}\cosh\frac{\theta}{2}
\bigg[4800 \ \theta\cosh\frac{3\theta}{2}\coth\frac{\theta}{2}+\cosh\frac{\theta}{2}\big(2063+3510\frac{\Lambda^2}{m^2}-225\frac{\Lambda^4}{m^4}+251\cosh2\theta\big)\nonumber\\
&+&\cosh\frac{\theta}{2}\cosh\theta\big(-7114+840\frac{\Lambda^2}{m^2}+28800\log\frac{m}{\Lambda}\big)\bigg]\ .
\label{onederv_u_amplitude}
\end{eqnarray}
Finally, adding together the contributions from the $s$, $t$ and $u$-channels to the amplitude, (\ref{onederv_s_amplitude}), (\ref{onederv_t_amplitude}) and (\ref{onederv_u_amplitude}), one finds the total amplitude from the bubble diagrams where one vertex contains the quartic coupling $\tilde{X}_{\rho\lambda}\tilde{X}^{\rho\lambda}$ while the other vertex contains quartic coupling $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{b}}&=&\mathcal{A}_{\text{b}}^{(s)}+\mathcal{A}_{\text{b}}^{(t)}+\mathcal{A}_{\text{b}}^{(u)}\nonumber\\
&=&\frac{\lambda^2m^6}{800\pi\sinh\theta}\Big[400\pi\big(9+3\cosh\theta+3\cosh2\theta+\cosh3\theta\big)-i\Big(\sinh\theta\big(-2047+930\frac{\Lambda^2}{m^2}\nonumber\\
&+&3600\log\frac{m}{\Lambda}\big)
+\sinh2\theta\big(\frac{91}{2}+570\frac{\Lambda^2}{m^2}+2400\log\frac{m}{\Lambda}\big)+2\sinh3\theta\big(-177+35\frac{\Lambda^2}{m^2}\nonumber\\
&+&200\log\frac{m}{\Lambda}\big)-3200 \ \theta\Big)\Big]\ .
\label{amplitude_onederv}
\end{eqnarray}
\subsubsection*{(c) Both vertices contain interactions $(\tilde{X}_{\mu}^{\ \mu})^2$ :}
Finally consider bubble diagrams shown in figure \ref{bubblediagram_noderv} where both vertices contain the quartic coupling $(\tilde{X}_{\mu}^{\ \mu})^2$ .
\begin{figure}[h!]
\centering
\subfigure[$s$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{s_noderv.pdf}
\label{bubblediagram_noderv_s} }
\subfigure[$t$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{t_noderv.pdf}
\label{bubblediagram_noderv_t} }
\subfigure[$u$-channel]{
\includegraphics[width=0.31\columnwidth,height=0.21\columnwidth]{u_noderv.pdf}
\label{bubblediagram_noderv_u} }
\caption{\small The $s$, $t$ and $u$-channel amplitudes when both of the interaction vertices are $(\tilde{X}_{\mu}^{\ \mu})^2$.}
\label{bubblediagram_noderv}
\end{figure}
The amplitude due to these kinds of bubble diagrams is given by,
\begin{eqnarray}
\mathcal{A}_{\text{c}}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\langle 0 | b(p_4) a(p_3) \text{T}\Big[\big(\tilde{X}_{\mu}^{\ \mu}(y)\big)^2 \big(\tilde{X}_{\rho}^{\ \rho}(x)\big)^2\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\nonumber\\
&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2 \frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[
\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\nonumber\\
&+&\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \
{\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \ \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\nonumber\\
&+&4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \ \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\nonumber\\
&-&2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\nonumber\\
&+&{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\nonumber\\
&-&2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\nonumber\\
&+&{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \ {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x) \
\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle\ .
\label{TmumuPower4}
\end{eqnarray}
The $s$-channel contribution to the amplitude from figure \ref{bubblediagram_noderv_s} can be computed by evaluating the contributions from each of the above terms in (\ref{TmumuPower4}) and adding the results together. By the general $s$-channel amplitude (\ref{schannelbubblegeneral}), the contributions from all nine terms in (\ref{TmumuPower4}) are,
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_1)}
&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\frac{\lambda^2 m^6}{384 \pi \sinh\theta}\bigg[3\pi\big(11+4\cosh\theta+\cosh2\theta\big)+i\Big(\sinh \theta \big(34-12\frac{\Lambda^2}{m^2}-48\log \frac{m}{\Lambda }\big) \nonumber\\
&+&2\sinh2\theta\big(4+3\log \frac{m}{\Lambda }\big)+3 \theta \big(11+4\cosh\theta+\cosh2\theta\big)\Big)\bigg]\ ,
\label{noderv_s1_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_2)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber
\end{eqnarray}\begin{eqnarray}
&=&-\frac{\lambda^2 m^6}{384 \pi \sinh\theta}\bigg[6\pi\big(11+4\cosh\theta+\cosh2\theta\big)+i\Big(\sinh \theta \big(53-18\frac{\Lambda^2}{m^2}-96\log \frac{m}{\Lambda }\big) \nonumber\\
&+&\sinh2\theta\big(13+6\log \frac{m}{\Lambda }\big)+3\cosh2\theta\sinh\theta+6 \theta \big(11+4\cosh\theta+\cosh2\theta\big)\Big)\bigg]\ ,
\label{noderv_s2_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_3)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\frac{\lambda^2 m^6}{384 \pi \sinh\theta}\bigg[3\pi\big(11+4\cosh\theta+\cosh2\theta\big)+i\Big(\sinh \theta \big(9-48\log \frac{m}{\Lambda }\big) +3\frac{\Lambda^2}{m^2}\sinh2\theta\nonumber\\
&+&\sinh3\theta+3 \theta \big(11+4\cosh\theta+\cosh2\theta\big)\Big)\bigg]\ ,
\label{noderv_s3_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_4)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ -2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{0.2cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(s_2)} \ ,
\label{noderv_s4_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_5)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}\nonumber\\
&=&-\frac{\lambda^2 m^6}{19200 \pi \sinh\theta}\bigg[600\pi\big(11+4\cosh\theta+\cosh2\theta\big)+i\Big(\sinh \theta \big(5948-2280\frac{\Lambda^2}{m^2}-9600\log \frac{m}{\Lambda }\big) \nonumber\\
&+&\sinh2\theta\big(\frac{2839}{2}-180\frac{\Lambda^2}{m^2}-75\frac{\Lambda^4}{m^4}+600\log \frac{m}{\Lambda }\big)+8\sinh\theta\cosh2\theta\big(26+15\frac{\Lambda^2}{m^2}\big)\nonumber\\
&+&17\sinh\theta\cosh3\theta
+600 \ \theta \big(11+4\cosh\theta+\cosh2\theta\big)\Big)\bigg]\ ,
\label{noderv_s5_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_6)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(s_2)}\ ,
\label{noderv_s6_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_7)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(s_3)}\ ,
\label{noderv_s7_final}
\end{eqnarray}\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_8)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(s_2)}\ ,
\label{noderv_s8_final}
\end{eqnarray}\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s_9)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(s)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(s_1)}\ ,
\label{noderv_s9_final}
\end{eqnarray}
where the employed computational method was exactly the same as when $\mathcal{A}_{\text{a}}^{(s_1)}$ was computed in (\ref{twoderv_s1}).
\vspace{1mm}
\\
Combining the $s$-channel contributions to the amplitude, (\ref{noderv_s1_final}) - (\ref{noderv_s9_final}), gives the total $s$-channel contribution to the amplitude from the diagrams where both vertices contain interactions $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(s)}&=& 2\Big(\mathcal{A}_{\text{c}}^{(s_1)}+\mathcal{A}_{\text{c}}^{(s_2)}+\cdots+\mathcal{A}_{\text{c}}^{(s_9)}\Big)\nonumber\\
&=&-\frac{\lambda^2m^6}{19200\pi\sinh\theta}\bigg[4800\pi\big(11+4\cosh\theta+\cosh2\theta\big)+i\Big(4800 \ \theta\big(11+4\cosh\theta+\cosh2\theta\big)\nonumber\\
&-&8\sinh\theta\big(-5111+1785\frac{\Lambda^2}{m^2}+9600\log\frac{m}{\Lambda}\big)
+2\sinh2\theta\big(4811+120\frac{\Lambda^2}{m^2}-75\frac{\Lambda^4}{m^4}\nonumber\\
&+&2400\log\frac{m}{\Lambda}\big)
+24\sinh3\theta\big(42+5\frac{\Lambda^2}{m^2}\big)+17\sinh4\theta\Big)\bigg]\ .
\label{noderv_s_amplitude}
\end{eqnarray}
Just as in the computation of the first and second type of bubble diagrams, (\ref{twoderv_s_amplitude}) and (\ref{onederv_s_amplitude}), a factor of $2$ arises in the first line of the above expression, (\ref{noderv_s_amplitude}), because the same contribution would be found if the vertices $x$ and $y$ were exchanged. A similar factor would appear while evaluating the $t$ and $u$-channel amplitudes as well.
\vspace{2mm}
\\
The contribution to the amplitude from the $t$-channel diagram, figure \ref{bubblediagram_noderv_t}, can be computed using the general $t$-channel amplitude (\ref{tchannelbubblegeneral}). The nine terms in (\ref{TmumuPower4}) yield the following contributions to the amplitude,
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_1)}
&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{64\pi}\bigg[1+\frac{\Lambda^2}{m^2}+6\log\frac{m}{\Lambda}-\cosh\theta\big(\frac{\Lambda^2}{m^2}+4\log\frac{m}{\Lambda}\big)\bigg]\ ,
\label{noderv_t1_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_2)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{64\pi}(2-\cosh\theta)\bigg[1+\frac{\Lambda^2}{m^2}+6\log\frac{m}{\Lambda}\bigg]\ ,
\label{noderv_t2_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_3)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{64\pi}\bigg[1+\frac{\Lambda^2}{m^2}+6\log\frac{m}{\Lambda}-\cosh\theta\big(1+2\log\frac{m}{\Lambda}\big)\bigg]\ ,
\label{noderv_t3_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_4)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ -2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{0.2cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(t_2)}\ ,
\label{noderv_t4_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_5)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{128\pi}\bigg[8+6\frac{\Lambda^2}{m^2}-\frac{\Lambda^4}{m^4}+48\log\frac{m}{\Lambda}-2\cosh\theta\big(1+3\frac{\Lambda^2}{m^2}+12\log\frac{m}{\Lambda}\big)\bigg]\ ,
\label{noderv_t5_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_6)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(t_2)}\ ,
\label{noderv_t6_final}
\end{eqnarray}\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_7)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(t_3)}\ ,
\label{noderv_t7_final}
\end{eqnarray}\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_8)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(t_2)}\ ,
\label{noderv_t8_final}
\end{eqnarray}\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t_9)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(t)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(t_1)}\ ,
\label{noderv_t9_final}
\end{eqnarray}
where the employed computational method was exactly the same as was used in evaluating $\mathcal{A}_{\text{a}}^{(t_1)}$ in (\ref{twoderv_t1}).
\vspace{2mm}
\\
Combining the $t$-channel contributions to the amplitude, (\ref{noderv_t1_final}) - (\ref{noderv_t9_final}), gives the total $t$-channel contribution to the amplitude from the diagram where both vertices contain interactions $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(t)}&=& 2\Big(\mathcal{A}_{\text{c}}^{(t_1)}+\mathcal{A}_{\text{c}}^{(t_2)}+\cdots+\mathcal{A}_{\text{c}}^{(t_9)}\Big)\nonumber\\
&=&-i\frac{\lambda^2 m^6}{64\pi}\bigg[32+30\frac{\Lambda^2}{m^2}-\frac{\Lambda^4}{m^4}+192\log\frac{m}{\Lambda}-2\cosh\theta\big(7+9\frac{\Lambda^2}{m^2}+48\log\frac{m}{\Lambda}\big)\bigg]\ .\nonumber\\
\label{noderv_t_amplitude}
\end{eqnarray}
The contribution to the amplitude of the $u$-channel diagram, figure \ref{bubblediagram_noderv_u}, can be computed using the general $u$-channel amplitude (\ref{uchannelbubblegeneral}). The contributions to the amplitude from the nine terms in (\ref{TmumuPower4}) are given by,
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_1)}
&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{96\pi}\cosh^2\frac{\theta}{2}\bigg[4-3\frac{\Lambda^2}{m^2}-18\log\frac{m}{\Lambda}-\cosh\theta-3\theta\coth\frac{\theta}{2}\bigg]\ ,
\label{noderv_u1_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_2)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{48\pi}\cosh^2\frac{\theta}{2}\bigg[5-6\frac{\Lambda^2}{m^2}-24\log\frac{m}{\Lambda}+\cosh\theta-3\theta\coth\frac{\theta}{2}\bigg]\ ,
\label{noderv_u2_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_3)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ \bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ \bar{\psi}(y)\gamma^{\nu}{\partial}_{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{4800\pi}\cosh^2\frac{\theta}{2}\bigg[279-270\frac{\Lambda^2}{m^2}+75\frac{\Lambda^4}{m^4}-1500\log\frac{m}{\Lambda}+2\cosh\theta\big(19+60\frac{\Lambda^2}{m^2}\big)\nonumber\\
&-&17\cosh2\theta-150 \ \theta\coth\frac{\theta}{2}\bigg]\ ,
\label{noderv_u3_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_4)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[ -2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda^2 m^6}{16\pi}\cosh^2\frac{\theta}{2}\bigg[4\log\frac{m}{\Lambda}+\theta\coth\frac{\theta}{2}\bigg]\ ,
\label{noderv_u4_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_5)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[4\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y)\ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&-i\frac{\lambda^2 m^6}{48\pi}\cosh^2\frac{\theta}{2}\bigg[5-6\frac{\Lambda^2}{m^2}-36\log\frac{m}{\Lambda}+\cosh\theta-6\theta\coth\frac{\theta}{2}\bigg]\ ,
\label{noderv_u5_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_6)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2\bar{\psi}(y)\gamma^{\mu}{\partial}_{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(u_2)}\ ,
\label{noderv_u6_final}
\end{eqnarray}\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_7)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{4.7cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ \bar{\psi}(x)\gamma^{\lambda}{\partial}_{\lambda}\psi(x) \Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}\nonumber\\
&=&i\frac{\lambda^2 m^6}{32\pi}\cosh^2\frac{\theta}{2}\bigg[2\log\frac{m}{\Lambda}+\theta\coth\frac{\theta}{2}\bigg]\ ,
\label{noderv_u7_final}
\end{eqnarray}
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_8)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[-2{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times \bar{\psi}(x)\gamma^{\rho}{\partial}_{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x)\gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(u_4)}\ ,
\label{noderv_u8_final}
\end{eqnarray}\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u_9)}&=&\frac{1}{2!} \big(-i\frac{\lambda}{2}\big)^2\frac{1}{16}\langle 0 | b(p_4) a(p_3) \text{T}\Big[{\partial}_{\mu}\bar{\psi}(y) \gamma^{\mu}\psi(y) \ {\partial}_{\nu}\bar{\psi}(y) \gamma^{\nu}\psi(y) \ \nonumber\\
&& \hspace{0.2cm} \times {\partial}_{\rho}\bar{\psi}(x) \gamma^{\rho}\psi(x) \ {\partial}_{\lambda}\bar{\psi}(x) \gamma^{\lambda}\psi(x)\Big] a^{\dagger}(p_1) b^{\dagger}(p_2)|0 \rangle^{(u)}
\ \ = \ \ \mathcal{A}_{\text{c}}^{(u_1)}\ .
\label{noderv_u9_final}
\end{eqnarray}
Combining the $u$-channel contributions to the amplitude, (\ref{noderv_u1_final}) - (\ref{noderv_u9_final}), gives the total $u$-channel contribution to the amplitude from the diagram where both vertices contain interactions $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{c}}^{(u)}&=& 2\Big(\mathcal{A}_{\text{c}}^{(u_1)}+\mathcal{A}_{\text{c}}^{(u_2)}+\cdots+\mathcal{A}_{\text{c}}^{(u_9)}\Big)\nonumber\\
&=&-i\frac{\lambda^2 m^6}{2400\pi}\cosh^2\frac{\theta}{2}\bigg[2179-2370\frac{\Lambda^2}{m^2}+75\frac{\Lambda^4}{m^4}-14400\log\frac{m}{\Lambda}+2\cosh\theta\big(119+60\frac{\Lambda^2}{m^2}\big)\nonumber\\
&-&17\cosh2\theta-2400 \ \theta\coth\frac{\theta}{2}\bigg]\ .
\label{noderv_u_amplitude}
\end{eqnarray}
Finally, adding together the contributions from the $s$, $t$ and $u$-channels to the amplitude, (\ref{noderv_s_amplitude}), (\ref{noderv_t_amplitude}) and (\ref{noderv_u_amplitude}), one finds the total amplitude from the bubble diagrams where both vertices contain interactions $(\tilde{X}_{\mu}^{\ \mu})^2$,
\begin{eqnarray}
\mathcal{A}_{\text{c}}&=&\mathcal{A}_{\text{c}}^{(s)}+\mathcal{A}_{\text{c}}^{(t)}+\mathcal{A}_{\text{c}}^{(u)}\nonumber\\
&=&-\frac{\lambda^2m^6}{2400\pi\sinh\theta}\bigg[600\pi\big(11+4\cosh\theta+\cosh2\theta\big)
-i\Big(\sinh\theta\big(-7586+1800\frac{\Lambda^2}{m^2}+9600\log\frac{m}{\Lambda}\big)\nonumber\\
&-&3\sinh\theta\cosh2\theta\big(101+20\frac{\Lambda^2}{m^2}\big)+\sinh2\theta\big(-\frac{3089}{2}+870\frac{\Lambda^2}{m^2}+4800\log\frac{m}{\Lambda}\big)-4800 \ \theta\Big)\bigg]\ .\nonumber\\
\label{amplitude_noderv}
\end{eqnarray}
\subsubsection*{Total bubble diagram contribution to the amplitude}
The total contribution to the amplitude from the bubble diagrams is the sum of (\ref{amplitude_twoderv}), (\ref{amplitude_onederv}) and (\ref{amplitude_noderv}),
\begin{eqnarray}
\mathcal{A}^{\text{bubble}}&=&\mathcal{A}_{\text{a}}+\mathcal{A}_{\text{b}}+\mathcal{A}_{\text{c}}\nonumber\\
&=&-2\lambda^2m^6\sinh^3\theta-i\frac{\lambda^2m^6}{4800\pi}\cosh^2\frac{\theta}{2}\bigg[27683+9240\frac{\Lambda^2}{m^2}-38400\log\frac{m}{\Lambda}\nonumber\\
&-&2\cosh\theta\big(10447+5940\frac{\Lambda^2}{m^2}-24000\log\frac{m}{\Lambda}\big)\bigg]\ .
\label{amplitude_bubble}
\end{eqnarray}
\subsubsection*{Total second order $S$-matrix}
Adding the tadpole contribution (\ref{tadpoleamplitudefinal}) and the bubble contribution (\ref{amplitude_bubble}) to the amplitude gives the total second order amplitude,
\begin{eqnarray}
\mathcal{A}^{(2)}&=&\mathcal{A}^{\text{tad}}+\mathcal{A}^{\text{bubble}}\nonumber\\
&=&-2\lambda^2m^6\sinh^3\theta-i\frac{\lambda^2m^6}{4800\pi}\cosh^2\frac{\theta}{2}\bigg[27683+12840\frac{\Lambda^2}{m^2}-19200\log\frac{m}{\Lambda}\nonumber\\
&-&2\cosh\theta\big(10447+5940\frac{\Lambda^2}{m^2}-19200\log\frac{m}{\Lambda}\big)\bigg]\ .
\label{amplitude_secondorder}
\end{eqnarray}
Therefore, by (\ref{AtoS}) the second order $S$-matrix is,
\begin{eqnarray}
S^{(2)}(\theta)&=&\frac{\mathcal{A}^{(2)}}{4 m^2 \sinh \theta}\nonumber\\
&=&-\frac{1}{2}\lambda^2m^4\sinh^2\theta-i\frac{\lambda^2m^4}{38400\pi}\coth\frac{\theta}{2}\bigg[27683+12840\frac{\Lambda^2}{m^2}-19200\log\frac{m}{\Lambda}\nonumber\\
&-&2\cosh\theta\big(10447+5940\frac{\Lambda^2}{m^2}-19200\log\frac{m}{\Lambda}\big)\bigg]\ .
\label{Smatrix_secondorder}
\end{eqnarray}
It is interesting to note that at second order, the real part of the amplitude comes only from the $s$-channel, while the $t$ and $u$-channels give purely imaginary contributions to the amplitude.
\subsection{Renormalized Lagrangian}
Now that the $S$-matrix has been computed the $T\bar{T}$-deformed theory can be renormalized by demanding that the $S$-matrix has the form (\ref{CDDSZ1}).
\vspace{2mm}
\\
Discarding the imaginary finite pieces of second order $S$-matrix (\ref{Smatrix_secondorder}) yields, \footnote{The finite pieces are not essential as one can change them by rescaling the cut-off, their choices are just equivalent to using different regularization schemes.}
\begin{eqnarray}
S^{(2)}(\theta)&=&-\frac{1}{2}\lambda^2m^4\sinh^2\theta-i\frac{\lambda^2m^4}{38400\pi}\coth\frac{\theta}{2}\bigg[12840\frac{\Lambda^2}{m^2}-19200\log\frac{m}{\Lambda}
-2\cosh\theta\big(5940\frac{\Lambda^2}{m^2}\nonumber\\
&-&19200\log\frac{m}{\Lambda}\big)\bigg]\nonumber\\
&=&-\frac{1}{2}\lambda^2m^4\sinh^2\theta-i\frac{\lambda^2 m^4}{\pi}\coth\frac{\theta}{2}\bigg[\frac{107-99\cosh\theta}{320}\frac{\Lambda^2}{m^2}
-\frac{1-2\cosh\theta}{2}\log\frac{m}{\Lambda}\bigg]\ .\nonumber\\
\label{Smatrix_secondorder_imaginary}
\end{eqnarray}
Observe how the first term in (\ref{Smatrix_secondorder_imaginary}) is exactly the expected second order $S$-matrix in an integrable field theory, but (\ref{Smatrix_secondorder_imaginary}) also contains an extra divergent imaginary part.
The $S$-matrix was computed using the classical (bare) Lagrangian (\ref{deformedLag1}), so counterterms must be added to the classical Lagrangian in order to get rid of the imaginary part of $S^{(2)}(\theta)$. In this process the Lagrangian will be perturbatively renormalized up to second order in the $T\bar{T}$-coupling $\lambda$.
\vspace{2mm}
\\
It is important to mention that the integrable structure of the $T\bar{T}$-deformed free massive Dirac fermion theory is what enables the theory to be renormalized perturbatively. In general, the scattering amplitude of a quantum field theory may contain logarithms of functions of Mandelstam variables, which involves $\theta$\footnote{(\ref{eqn2}) is an expression for the logarithm of a function of Mandelstam variables which appear while computing the one-loop momentum integrals in the scattering amplitude. When expressed in terms of rapidity, it yields terms involving $\theta$.}. However, local counterterms can never cancel a term involving $\theta$ and can only cancel terms involving powers of $\cosh^2\Big(\frac{\theta}{2}\Big)$\footnote{A polynomial in Mandelstam variable $s$ corresponds to a polynomial in $\cosh^2\Big(\frac{\theta}{2}\Big)$, when expressed in terms of the rapidity.}. Due to the integrable structure of the theory, our final amplitude does not contain terms involving $\theta$ although individual contributions to the amplitude have terms involving $\theta$. This is expected because the $S$-matrix of an integrable theory can not have branch cuts and can only have poles.
\vspace{2mm}
\\
Discarding the imaginary finite pieces of the second order amplitude (\ref{amplitude_secondorder}) yields,
\begin{eqnarray}
\mathcal{A}^{(2)}&=&-2\lambda^2m^6\sinh^3\theta-i\frac{8\lambda^2m^6}{\pi}\cosh^2\frac{\theta}{2}\bigg[\frac{107-99\cosh\theta}{320}\frac{\Lambda^2}{m^2}
-\frac{1-2\cosh\theta}{2}\log\frac{m}{\Lambda}\bigg]\nonumber\\
&=&-2\lambda^2m^6\sinh^3\theta-i\frac{8\lambda^2 m^6}{\pi}\bigg[\Big(\frac{103}{160}\frac{\Lambda^2}{m^2}-\frac{3}{2}\log\frac{m}{\Lambda}\Big)\cosh^2\frac{\theta}{2}+\Big(-\frac{99}{160}\frac{\Lambda^2}{m^2}\nonumber\\
&+&2\log\frac{m}{\Lambda}\Big)\cosh^4\frac{\theta}{2}\bigg]\ .
\label{amplitude_secondorder_imaginary}
\end{eqnarray}
Let the renormalized Lagrangian be,
\begin{eqnarray}
\mathcal{L}_{\text{ren}}(\lambda)&=&-i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi +m \bar{\psi} \psi +\frac{\lambda}{2}\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\Big)
+\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\Big)\nonumber\\
&+&\frac{7}{4}\lambda^2 m^2(\bar\psi\psi)^2 \ \tilde{X}_{\mu}^{\ \mu}-\frac{7}{4}\lambda^2 m^3 (\bar\psi\psi)^3+\alpha \lambda^2 (\tilde{X}_{\mu}^{\ \mu})^2+\beta \lambda^2 \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}+\mathcal{O}(\lambda^3)\ , \nonumber\\
\label{renormLag}
\end{eqnarray}
where $\alpha$ and $\beta$ are divergent coefficients which must be tuned to exactly cancel the imaginary divergent contributions to the second order $S$-matrix (\ref{Smatrix_secondorder_imaginary}).
\vspace{2mm}
\\
The term $\alpha \lambda^2 (\tilde{X}_{\mu}^{\ \mu})^2$ contributes to the amplitude as,
\begin{eqnarray}
\mathcal{A}_{\alpha}^{\text{count}}=(i\alpha \lambda^2 ) 4m^4(1+\cosh\theta)
=8i\alpha \lambda^2 m^4\cosh^2\frac{\theta}{2}\ .
\label{counteralpha}
\end{eqnarray}
While the term $\beta \lambda^2 \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$ contributes to the amplitude as,
\begin{eqnarray}
\mathcal{A}_{\beta}^{\text{count}}=(i\beta \lambda^2) 4m^4(\cosh\theta+\cosh2\theta)
=-8i\beta \lambda^2 m^4\big(3\cosh^2\frac{\theta}{2}-4\cosh^4\frac{\theta}{2}\big)\ .
\label{counterbeta}
\end{eqnarray}
Adding (\ref{counteralpha}) and (\ref{counterbeta}), gives the total contribution from the counterterms to the amplitude,
\begin{eqnarray}
\mathcal{A}^{\text{count}}=\mathcal{A}_{\alpha}^{\text{count}}+\mathcal{A}_{\beta}^{\text{count}}
=8i\lambda^2 m^4\big(\alpha-3\beta \big)\cosh^2\frac{\theta}{2}+32i\beta\lambda^2 m^4\cosh^4\frac{\theta}{2}\ .
\label{counterAmplitude}
\end{eqnarray}
The condition that the sum amplitude of the counterterms, (\ref{counterAmplitude}), must exactly cancel the imaginary part of (\ref{amplitude_secondorder_imaginary}) gives the conditions,
\begin{eqnarray}
m^4\big(\alpha-3\beta\big)&=&\frac{m^6}{\pi}\Big(\frac{103}{160}\frac{\Lambda^2}{m^2}-\frac{3}{2}\log\frac{m}{\Lambda}\Big)\ ,
\label{counterEq1}
\end{eqnarray}
\begin{eqnarray}
32\beta m^4&=&\frac{8m^6}{\pi}\Big(-\frac{99}{160}\frac{\Lambda^2}{m^2}+2\log\frac{m}{\Lambda}\Big)\ .
\label{counterEq2}
\end{eqnarray}
Solving (\ref{counterEq1}) and (\ref{counterEq2}) for $\alpha$ and $\beta$ gives,
\begin{eqnarray}
\alpha&=&\frac{23}{128\pi}\Lambda^2\ ,\\
\label{alphavalue}
\beta&=&\frac{m^2}{4\pi}\Big(-\frac{99}{160}\frac{\Lambda^2}{m^2}+2\log\frac{m}{\Lambda}\Big).
\label{betavalue}
\end{eqnarray}
Substituting the above values of $\alpha$ and $\beta$ into the renormalized Lagrangian (\ref{renormLag}) yields,
\begin{eqnarray}
\mathcal{L}_{\text{ren}}(\lambda)&=&-i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi +m \bar{\psi} \psi +\frac{\lambda}{2}\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\Big)
+\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\Big)\nonumber\\
&+&\frac{7}{4}\lambda^2 m^2(\bar\psi\psi)^2 \ \tilde{X}_{\mu}^{\ \mu}-\frac{7}{4}\lambda^2 m^3 (\bar\psi\psi)^3\nonumber\\
&+&\frac{23\Lambda^2}{128\pi} \lambda^2 (\tilde{X}_{\mu}^{\ \mu})^2+\frac{m^2}{4\pi}\Big(-\frac{99}{160}\frac{\Lambda^2}{m^2}+2\log\frac{m}{\Lambda}\Big) \lambda^2 \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}+\mathcal{O}(\lambda^3)\ .
\label{renormLagFinal}
\end{eqnarray}
However, the above renormalized Lagrangian is written in terms of the redefined fields. The renormalized Lagrangian can be written in terms of the original fields as,
\begin{eqnarray}
\mathcal{L}_{\text{ren}}(\lambda)&=&-i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi +m \bar{\psi} \psi
+\frac{\lambda}{2}\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2+2m\bar{\psi}\psi \tilde{X}_{\mu}^{\ \mu}-2m^2(\bar{\psi}\psi)^2\Big)\nonumber\\
&-&\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\Big)\nonumber\\
&+&\frac{23\Lambda^2}{128\pi} \lambda^2 (\tilde{X}_{\mu}^{\ \mu})^2+\frac{m^2}{4\pi}\Big(-\frac{99}{160}\frac{\Lambda^2}{m^2}+2\log\frac{m}{\Lambda}\Big) \lambda^2 \tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}+\mathcal{O}(\lambda^3) \ .
\label{renormLagFinalOriginal}
\end{eqnarray}
Finally, the renormalized Lagrangian of the $T\bar{T}$-deformed free massive Dirac fermion in two dimensional Euclidean spacetime is given by,
\begin{eqnarray}
\mathcal{L}_{\text{ren}}(\lambda)&=&i \bar{\psi} \gamma^{\mu}{\partial}_{\mu} \psi -m \bar{\psi} \psi -\frac{\lambda}{2}\Big(2m\bar{\psi}\psi \ \tilde{X}_{\mu}^{\ \mu}-2m^2(\bar{\psi}\psi)^2\Big)-\frac{g}{2}\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}+\frac{h}{2}(\tilde{X}_{\mu}^{\ \mu})^2\nonumber\\
&+&\frac{\lambda^2}{2}m\bar{\psi}\psi\Big(\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}-(\tilde{X}_{\mu}^{\ \mu})^2\Big)+\mathcal{O}(\lambda^3)\ ,
\label{renormLagFinal1}
\end{eqnarray}
where the renormalized couplings are given by,
\begin{eqnarray}
g&=&\lambda-\frac{\lambda^2 m^2}{2\pi}\Big(\frac{99}{160}\frac{\Lambda^2}{m^2}-2\log\frac{m}{\Lambda}\Big)\ ,\nonumber\\
h&=&\lambda-\frac{23\lambda^2}{64\pi}\Lambda^2\ .
\label{renormcouplings}
\end{eqnarray}
It is important to notice the major qualitative difference between the renormalized Lagrangian (\ref{renormLagFinal1}) and the classical Lagrangian (\ref{deformedLag1}). The classical Lagrangian (\ref{deformedLag1}) has only one scale, $\lambda$, which appears to be the coupling for all the quartic terms. However, the renormalized Lagrangian contains three different couplings, $\lambda$, $g$ and $h$. In the renormalized Lagrangian, the two quartic terms $\bar{\psi}\psi \ \tilde{X}_{\mu}^{\ \mu}$ and $(\bar{\psi}\psi)^2$ share the old classical coupling $\lambda$, where as the other two quartic terms $\tilde{X}_{\mu\nu}\tilde{X}^{\mu\nu}$ and $(\tilde{X}_{\mu}^{\ \mu})^2$ have very different couplings $g$ and $h$, respectively.
\section{Discussion}
\label{Discussion}
In this paper the $T\bar{T}$-deformed free massive Dirac fermion in two dimensions was studied. First, the Lagrangian of the deformed theory was stated and massaged into an easier form for amplitude calculations using a field redefinition. The two-to-two $S$-matrix of the fermion anti-fermion scattering process was computed to second order in the $T\bar{T}$-coupling $\lambda$. At first order, the $S$-matrix exactly matches the expected result for an integrable field theory (\ref{CDDSZ1}). However, at second order the $S$-matrix matches the expected result up to some divergent imaginary second order terms. Counterterms were added to the Lagrangian to cancel these divergent pieces and ensure that the final second order $S$-matrix agrees with the expected result (\ref{CDDSZ1}), in the process the renormalized Lagrangian was obtained. Amazingly, integrability allows the naively non-renormalizable theory to be renormalized perturbatively. The renormalized Lagrangian was qualitatively very different from the classical Lagrangian as there are three different coupling constants ($\lambda$, $g$ and $h$) in the renormalized Lagrangian, while in the classical case there is only one coupling constant ($\lambda$). Thus, the quantum integrability here leads to a more complicated renormalized Lagrangian than the classical one. This is not what always happens in an integrable theory, for example, the integrable $\sinh$-Gordon model gives rise to a renormalized Lagrangian which has the same functional form as the classical one \cite{Rosenhaus}. Further, the existence of the renormalized Lagrangian means that all quantities of the theory may now be computed using the standard QFT techniques. For example, one can compute the correlation functions of local fields perturbatively using the renormalized Lagrangian.
\\
\\
In this paper, renormalization was performed by computing the two-to-two $S$-matrix for the fermion anti-fermion scattering: $f_1+\bar{f}_2\rightarrow f_3+\bar{f}_4$, and adding counterterms to cancel the divergences. However, one may consider the other possible two-to-two scattering process in this theory, namely, the fermion-fermion scattering: $f_1+f_2\rightarrow f_3+f_4$. If this process had been chosen the same renormalized Lagrangian would be expected. The calculation of the $S$-matrix would involve an almost identical computation to what was done here except that only one plane wave solution to the Dirac equation, $u(k)$, would be present in the expressions.
\\
\\
The form of the renormalized Lagrangian is already exciting at second order in the $T\bar{T}$-coupling. It would be interesting to see how the renormalized Lagrangian looks at higher orders, because of the simple structure of the $S$-matrix the renormalized Lagrangian may have a simple and compact form. Similar to \cite{Rosenhaus}, the second order real contribution to the $S$-matrix came from the $s$-channel only. It would be useful to understand why this occurs and whether this is a general property of the $T\bar{T}$-deformed integrable theories.
\section*{Acknowledgement}
We would like to thank Michael Smolkin for suggesting the problem to us, for many insightful discussions throughout the work and useful comments on the draft. We are also thankful to Mikhail Goykhman and Lorenzo Di Pietro for valuable discussions. This research was supported by the Israeli Science Foundation Center of Excellence (grant No. 2289/18) and the Quantum Universe I-CORE program of the Israel Planning and Budgeting Committee (grant No. 1937/12).
\section*{Appendices}
|
2,869,038,154,774 | arxiv | \section{Introduction}
\label{sec:INTRODUCTION}
Large-scale datasets of mobile communication activity collected by network operators are becoming an important source of information for investigations of the spatiotemporal features of human dynamics. These datasets represent an important proxy of the movement patterns and social interactions of vast populations of millions of individuals, and allow investigating subscribers' endeavors at unprecedented scales.
Within this context, the study of population dynamics from mobile traffic data offers rich insights on human mobility laws~\cite{song2010limits}, disaster recovery~\cite{bagrow2011collective}, infective disease epidemics~\cite{bajardi2011human}, commuting patterns~\cite{yang14}, or urban planning~\cite{caceres12}. A comprehensive review is provided in~\cite{naboulsi16survey}. These studies have demonstrated how data collected by mobile network operators can effectively complement --or even replace-- traditional sources of demographic data, such as censuses and surveys. Indeed, classic data sources have major limitations in terms of compilation cost, scalability, and timely updating. Mobile traffic data can overcome all these issues, as it is relatively inexpensive to collect, covers large populations, and can be retrieved and analyzed in real-time or with minimum latency.
The features listed above are especially critical in the estimation of population densities. Extensive censuses of the populations living in urban and suburban regions are carried out at every few years by local authorities, at best. However, significant alterations of the static population distribution (\textit{i.e.}, the density of inhabitants based on their home locations) occur at shorter timescales, and cannot be tracked with conventional methods. Even more so when considering dynamic population distributions (\textit{i.e.}, the instantaneous density of inhabitants throughout, \textit{e.g.}, a day, based on their current position), which require repeated refreshing of the data during a same day.
Mobile network traffic data yields the potential to enable automated, near-real-time estimation of population density.
The correlation between the mobile communication activity and census data on static population density was first identified in~\cite{krings2009urban}. Since then, several works further investigated the possibility of exploiting telecommunication data, in isolation or mixed with other data sources, for the estimation of static and dynamic population distributions.
The current state-of-the-art methodology, presented in~\cite{douglass2015high}, leverages data from mobile traffic, map databases and satellite imagery to generate estimates of the static population with a correlation coefficient of 0.66 with ground-truth information.
In this paper, we build on the classical power relationship between mobile activity volume and population density~\cite{deville2014dynamic} and introduce a novel approach to the estimation of population. Tests with substantial real-world mobile traffic datasets prove that our model attains a significantly improved correlation with ground-truth data, typically in the range 0.80-0.87. Moreover:
\begin{itemize}
\item we design a solution based exclusively on metadata collected by the mobile network operator, avoiding complex and cumbersome data mixing;
\item we show that subscriber presence data inferred from the mobile communications of each user is a much better proxy of the population distribution than previously adopted metrics;
\item we introduce a number of original data filters that allow refining the population distribution estimates;
\item we evaluate our methodology in multiple urban scenarios, obtaining consistently good results;
\item we unveil the multivariate relationship between population density, subscriber presence and subscriber activity level;
\item we leverage our model to generate dynamic representations of the population distribution.
\end{itemize}
The document is organized as follows. Sec.\,\ref{sec:RELATEDWORK} reviews the literature related to our work. Sec.\,\ref{sec:DATASET} describes the reference original and derived datasets that we employ in our study. Sec.\,\ref{sec:NIGHTTIME} presents our proposed model for the estimation of static population distribution.
Sec.\,\ref{sec:DYNAMIC} shows model enhancements to deal with dynamic population densities.
Sec.\,\ref{sec:CONCLUSION} summarizes our work and draws conclusions.
\section{Related work}
\label{sec:RELATEDWORK}
There is wide agreement on the suitability of mobile network traffic data as a source of information for generic positioning analysis. Previous works have demonstrated that this type of data allows for the effective estimation of important places~\cite{csaji2013exploring}, the inference of trips among such places~\cite{bekhor2013evaluating}, and the derivation of origin-destination matrices by aggregating a large number of trips~\cite{calabrese2011estimating}.
As far as population distribution estimation is concerned, mobile communication data was first proposed as a proxy for the density of inhabitants in~\cite{ratti06}. Early evidences of the existence of an actual correlation between the mobile network activity and the underlying population density were presented in~\cite{krings2009urban}: the authors showed that city population sizes and the number of mobile customers follow similar distributions.
Subsequent works carried out more comprehensive evaluations. In~\cite{csaji2013exploring}, the home location of each subscriber was localized as the most frequently visited cell with a home profile (\textit{i.e.}, where the activity peak occurs in the evening). The density of home locations was then found to match very well --with a 0.92 correlation-- census data on nationwide population distribution. Similarly, excellent agreement between the overnight spatial density of mobile subscribers and that of nationwide static populations was found in~\cite{bekhor2013evaluating,calabrese2011estimating}. However, these results refer to a nationwide population, and the spatial granularity of the studies is counties or tracts (\textit{i.e.}, large regions comprising whole cities or macroscopic city neighborhoods).
Our focus is on intra-urban population distribution estimation: to that end, we downscale the study at the individual cell level, considering orders-of-magnitude higher accuracy and making the task much more challenging.
Citywide population estimation from mobile traffic data has been addressed by a limited number of works in the literature.
In~\cite{kang2012towards}, LandScan\texttrademark, a tool for ambient population estimation was employed to explore the relationship between the voice call activity and the underlying inhabitant density, at a 1-km$^2$ resolution. The authors found a weak correlation of 0.24, which improved to 0.45 by limiting the analysis to selected time intervals rather than considering the daily communication volume.
In~\cite{deville2014dynamic}, telecommunications data is mixed with a number of other sources, including information on Corine land use, OpenStreetMap infrastructure, satellite nightlights, and slope. This plethora of data is processed through a dasymetric modeling approach, resulting in high 0.92 correlation with census information. However, the correlation is a nationwide average, and the authors indicate that the accuracy is lower for the most densely populated areas, i.e., large cities. Indeed, in such areas, a normalized error of around 0.6 is measured in~\cite{deville2014dynamic}, whereas we obtain values below 0.06.
The approach in~\cite{douglass2015high} represents the current state-of-the-art in the estimation of cell-level population distribution from mobile traffic data. It performs random forest regression on the
median outgoing voice call volumes recorded on the whole urban region. This information is augmented with land use data extracted from OpenStreetMap (41\% of the spatial cells) and satellite imagery (59\% of spatial cells).
We improve this approach in a number of ways. First, we use subscriber \textit{presence} metadata, which we prove to be a more sensible metric than the outgoing voice call volumes used in~\cite{douglass2015high}: in fact, this is a distinctive element of our study, telling it apart from all previous works in the literature. Second, we include filters based on daytime, land use information --which we infer from the mobile traffic activity itself-- and outlying human dynamics:
these allow us to account for important phenomena, such as the heterogeneity of subscribers' behaviors over the day and during the night, the diversity of mobile service usage in residential and non-residential area, or the variations of mobile traffic activity during weekdays and weekends or holidays. Third, we extend our evaluation to multiple cities, demonstrating the general viability of the methodology.
As a result, our model achieves a significant improvement in terms of estimation accuracy: we find a correlation of 0.83 in the Milan urban scenario, whereas the solution in~\cite{douglass2015high} yielded a 0.66 correlation in the same region.
Moreover, our solution relies on data collected from a single source, \textit{i.e.}, the mobile network operator, and does not need additional external data.
\section{Datasets}
\label{sec:DATASET}
In this work, we leverage several datasets made available by Telecom Italia Mobile (TIM) within their 2015 Big Data Challenge.
Specifically, we focus on three major urban areas for which substantial data is available, i.e., the conurbations of Milan, Turin and Rome. For each city, we collect data describing the mobile traffic activity (Sec.\,\ref{sub:traffic}) and population distribution (Sec.\,\ref{sub:pop}), and we infer information on land use (Sec.\,\ref{sub:landuse}).
\subsection{Mobile network traffic}
\label{sub:traffic}
The telecommunication data covers the months of March and April, 2015, and describes the volume of traffic divided by type (voice calls in/out, text SMS in/out, and Internet), and the presence of subscribers.
All metrics are aggregated in time and space. In time, the data is totaled over 15-minute time intervals. In space, metrics are computed over an irregular grid tessellation, whose geographical cells have sizes ranging from 255$\times$325 m$^2$ to 2$\times$2.5 km$^2$. The number of cells is 1419 for Milan, 571 for Turin and 927 for Rome.
\begin{figure}[tb]
\centering
\includegraphics[width=0.9\linewidth]{figures/Presence.png}
\caption{Subscriber presence in the TIM Big Data Challenge.}
\label{fig:presence}
\vspace*{-7pt}
\end{figure}
While voice, text, and data volumes are directly computed from the recorded demand,
the presence information is the result of a simple preprocessing performed by the mobile network operator. Basically, each subscriber is associated to the geographical cell where he/she performed his/her last action (e.g., issued a call, received an SMS, and so on).
Therefore if the presence of a user is recorded in square $A$, then he performs an action in square $B$ at time $t_1$, and finally at time $t_2$ he interacts with the network in square $C$, her presence will be recorded as follows: for $t<t_1$, the presence of the user is recorded in $A$; at $t=t_1$, the presence of the user is moved from $A$ to $B$; for $t_1<t<t_2$, the presence of the user is recorded in $B$; at $t=t_2$, the user is moved from $B$ to $C$; for $t>t_2$, the presence of the user is recorded in $C$ until he performs an action in a different square. Fig.\,~\ref{fig:presence} shows an example of presence data preprocessing: the location of a specific user is detected every 15 minutes, if he entails at least one action during that time slot.
\subsection{Population distribution}
\label{sub:pop}
Population distribution data comes from the 2011 housing census in Italy by the national organization for statistics in Italy, ISTAT. It includes: population counts; a survey of structural attributes, update and review of municipal anagraphical lists; the number and structural features of houses and buildings.
Specifically, population counts are measured in terms of families, cohabitants, persons temporarily present, domiciles, and other types of lodging and buildings, for each administrative area.
In our study, we will consider such census data as a ground truth for the static population distribution in the reference urban regions.
To that end, we need to ensure spatial consistency between the mobile network traffic and the census data. We proceed as follows.
Let us denote as $U_j$ the total number of inhabitants in the administrative area $j$, and as $A_j$ its surface. The population density $\rho_i$ in a geographical cell $i$ defined in Sec.\,\ref{sub:traffic} is then computed as
\begin{equation}
\rho_i=\frac{1}{A_i}\sum_{j=1}^K U_j \frac{A_{i \cap j}}{A_j}
\label{eq:density}
\end{equation}
where $A_i$ is the surface of cell $i$, $K$ denotes the total number of administrative areas, and $A_{i \cap j}$ stands for the intersection surface of cell $i$ and administrative area $j$.
\subsection{Land use}
\label{sub:landuse}
Land use information is critical to the accurate estimation of population densities, as already demonstrated in the recent literature~\cite{douglass2015high}.
In this work, we leverage the telecommunication data itself to classify the geographical cells based on their primary land use. To that end, we employ MWS, which is the current state-of-the-art technique for land use detection from mobile traffic data~\cite{furno2015comparative}.
MWS computes, for each spatial cell of the target region, a mobile traffic signature, i.e., a compact representation of the typical dynamics of mobile communications in the considered cell. Specifically, MWS signatures are computed as the median voice call and texting activity in a cell recorded at every hour of the week. Signatures are then clustered based on their shape. This allows inferring a limited set of archetypal signatures, which are representative of distinct types of human activities. When applied to our reference datasets, MWS identifies seven major activity types: residential, office, transportation, touristic, university, shopping and nightlife. Clearly, all cells whose signatures are clustered into, e.g., the residential archetypal signature, can be classified as residential areas. Overall, this allows identifying land uses in our reference region.
It is important to stress that this approach is sensibly different from those used in previous works addressing population density estimation. In fact, we do not rely on external data sources such as, \textit{e.g.}, crowdsourced map databases or post-processed satellite imagery as, \textit{e.g.}, in~\cite{douglass2015high}. Instead, we resort solely to data collected by the mobile network operator. On the one hand, this makes the solution presented in this paper \textit{self-contained}, \textit{i.e.}, practicable by using just one type of dataset and without any need for dataset mixing -- whose blend of data of different natures is often cumbersome. On the other hand, land uses detected through mobile traffic data can be much more accurate and meaningful than those returned by other methods. As an example, the Milan conurbation territory is classified into buildings, vegetation, water, road, and railroads in~\cite{douglass2015high}: these categories are based on pure geographical features, and have small relation with the activity of individuals. Instead, the classes identified by MWS in the same region, listed above, map to a diversity of actual human endeavors, and, as such, have a stronger tie to the population density we aim at estimating. Our results will corroborate this intuition.
\section{Static population estimation}
\label{sec:NIGHTTIME}
The baseline population estimation model we adopt builds on previous results that demonstrated a consistent power relationship between the mobile network traffic activity $\sigma_i$ and the population density
$\rho_i$ at a same region $i$~\cite{douglass2015high, deville2014dynamic}. Thus
\begin{equation}
\rho_i =\alpha \sigma_i^\beta,
\label{eq:regression}
\end{equation}
where the parameters of $\alpha$ and $\beta$ represent the intercept (or the scale ratio) and slope (or the effect of $\rho_i$ on $\sigma_i$) of the model, respectively. By transforming the formula to a logarithmic scale, we obtain $log(\rho_i) = log(\alpha) + \beta log(\sigma_i)$. We can then use a regression model to estimate the parameters $\alpha$ and $\beta$ in the expression~(\ref{eq:regression}).
\begin{figure}[tb]
\centering
\includegraphics[width=0.67\linewidth]{figures/metrics.png}
\caption{Milan. ISTAT population density as function of the call volume, SMS volume, and subscriber presence.}
\label{fig:hetero}
\vspace*{-5pt}
\end{figure}
Unfortunately, the regression yields poor results if run on the raw telecommunication data, no matter the data type. Let us consider the data for Milan: the simple visualization in Fig.\,\ref{fig:hetero} unveils that the datasets described in Sec.\,\ref{sub:traffic} suffer from heterogeneity and heteroscedasticity. The plots illustrate the density of calls, SMS and presence\footnote{We omit Internet traffic data as it results in even sparser density maps. Indeed, it is widely acknowledged that mobile Internet data, affected by traffic generated by applications running in background, is less representative of human activity than voice and texting~\cite{furno2015comparative}.}
as a function of the associated population density in Sec.\,\ref{sub:pop}. The high variance between dependent and independent variables reveals the absence of homoscedasticity in the data. However, the classical regression models assume that there is no heteroscedasticity in the data, and, based on this assumption, infer the best linear unbiased estimators along with the lowest variance among all other unbiased estimators.
It follows that filtering the noisy data in Fig.\,\ref{fig:hetero} is a necessary step toward applying effective regression models. Next, we discuss data de-noising on multiple levels. We focus on the Milan case study, and then generalize the results of our approach by testing it in other city scenarios.
\subsection{Data type}
\label{sec:variablesfiltering}
\begin{figure}[tb]
\centering
\includegraphics[width=1.0\linewidth]{figures/1.png}
\caption{Milan. Top: correlation between different types of mobile network traffic data and the population census density, on a hourly basis. Bottom: zoom on presence metadata.}
\label{fig:type}
\vspace*{-9pt}
\end{figure}
The very first step we take is determining which type of mobile network traffic metadata is the most reliable for population estimation.
To that end, we study the correlation (i.e., the Pearson correlation coefficient) between the population census data and the different metrics presented in Sec.\,\ref{sub:traffic}, as recorded on a per-cell basis.
The top plot in Fig.\,\ref{fig:type} shows the correlation coefficient value obtained for incoming voice calls and text messages%
\footnote{Equivalent results were obtained for outgoing voice calls and text messages, and are not shown for the sake of brevity.},
and subscriber presence. The correlation is separated on a hourly basis to evidence the impact of daytime.
Our results are aligned with those in~\cite{douglass2015high}, which, however, only considered calls and SMS, and not the subscriber presence. As a result, their conclusion was that calls between 10 am and 11 am yield the strongest correlation with population density.
In fact, Fig.\,\ref{fig:type} shows that subscriber presence is a much fitter metric in that sense. The correlation values for presence metadata are consistently higher than those for voice calls and text SMS. Thus, our choice of data type is subscriber presence. We stress that we are the first to consider this type of mobile network traffic metadata for population estimation, which is often undisclosed by mobile operators for privacy reasons.
\subsection{Hour filtering}
\label{sec:timefiltering}
A second dimension on which to perform data filtering is the temporal one. As found in the literature~\cite{deville2014dynamic}, the correlation between mobile network traffic data and the population density varies over time. This can be verified in the top plot of Fig.\,\ref{fig:type}.
In fact, an even clearer picture of the phenomenon is provided by the zoomed bottom plot, which focuses on presence metadata.
The highest correlation coefficient is recorded at night, between 4 am and 5 am. We underscore that this is a very reasonable outcome, since the ISTAT population data refers to housing, and individuals are most likely at home overnight.
Thus, we pick subscriber presence during that specific hour as the mobile network traffic metadata for the regression model. Henceforth, $\sigma_i$ defined in~(\ref{eq:density}) describes the user presence in cell $i$ between 4 am and 5 am.
\subsection{Day filtering}
\label{sec:daysfiltering}
\begin{figure*}[!ht]
\centering
\includegraphics[width=0.8\linewidth]{figures/daytodayCoor.png}
\caption{Milan. Pearson coefficient of correlation for different mobile network traffic data, in March and April 2015, between 4 am and 5 am. A) Incoming call (range up to 0.8), B) Incoming SMS (range up to 0.8), C) Subscriber presence (range starts at 0.88). D) Average call density, with weekends highlighted in gray. E) Percentage of cells for which no information is available.}
\label{fig:day}
\vspace*{-14pt}
\end{figure*}
As mentioned in Sec.\,\ref{sub:traffic}, the mobile network traffic data we employ covers 60 days in March and April, 2015. An interesting question is if all these days should be considered in the regression model, or if there exists a more meaningful subset of days. Indeed, mobile network traffic data is clearly affected by the diverse activity patterns and social phenomena that may characterize different days.
The three top plots in Fig.\,\ref{fig:day} show the Pearson coefficient of correlation over the 60 days of our reference datasets. They refer to incoming voice calls (A), incoming SMS (B), and presence (C), and, consistently with the previous results, are computed over the hourly interval 4-5 am.
The plots confirm our previous conclusion on the relevance of subscriber presence as a proxy of population density: the correlation coefficient ranges between 0.88 and 0.94, i.e., much higher values than those attained by calls and SMS, in the range between 0.6 and 0.8.
No clear trend of the correlation coefficient is instead observed over the 60 days, for all data types.
A more insightful result is obtained by considering the average call density, i.e., the average number of calls (per km$^2$) recorded during a 15-minute slot in a spatial cell, between 4 am and 5 am (D). In this case, a remarkable weekly pattern appears, with weekdays and weekends (highlighted in gray) resulting in different behaviors. This recurrent phenomenon is an indicator that weekends may yeild different behaviors in terms of population presence (e.g., due to much increased nightlife), which could introduce a significant bias in our study. Thus, we filter out weekends from our data.
Further motivation for this choice comes from the bottom plot in Fig.\,\ref{fig:day}, showing the percentage of cells for which no data is available between 4 am and 5 am on each day (E). We remark that there exists a high percentage of missing data during weekends (denoted by red bars) and holidays (denoted by black bars, and corresponding to Good Friday and Easter): we thus filter out holidays as well from our data.
Finally, plots D and E in Fig.\,\ref{fig:day} also highlight an evident diversity between the month of March and April. We speculate that this is due to issues in the data collection process.
\subsection{Regression with RANSAC}
\label{sec:RANSACRegression}
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\linewidth]{figures/RANSAC.png}
\caption{Milan. RANSAC and linear regression on filtered subscriber presence data. Figure best viewed in color.}
\label{fig:ransac}
\vspace*{-14pt}
\end{figure}
In order to estimate the parameters $\alpha$ and $\beta$ in (\ref{eq:regression}), we employ the RANSAC regressor
on the filtered presence data. RANSAC estimates the parameters of a model from observations in an iterative manner, and automatically detects and excludes outlying points. Fig.\,\ref{fig:ransac} illustrates the fitting with the RANSAC regression model, in a log-log scale (solid red line).
The inliers and outliers detected in the data by RANSAC are shown in violet and gray, respectively. The estimated parameters are $\hat{\alpha}= 1.265$, $\hat{\beta}=0.979$.
The result underscores the quasi-linearity of the relation between subscriber presence and population, as also evidenced by the decent match with a linear fitting (dashed black line).
\begin{figure}[tb]
\centering
\includegraphics[width=0.95\linewidth]{figures/outlier_cells.png}
\caption{Milan. Geographical distribution of the cells that most frequently entail outliers detected by RANSAC.}
\label{fig7:outliers}
\vspace*{-7pt}
\end{figure}
It is interesting to have a closer look at the outliers detected by RANSAC. As a matter of fact, an analysis of these points reveals that outliers refer to a subset of cells, consistently over time. Thus, those cells yield some features that make their associated subscriber presence less related to their local population density.
A map of such cells is portrayed in Fig.\,\ref{fig7:outliers}.
It is evident that most outlying behaviors are encountered at the borders of the region. We speculate that this is an artifact of the spatial tessellation used in our reference datasets: clearly, the tessellation is a rough approximation of the cellular coverage, and it may well be that some border effect emerges, with an artificial (too high or too low) mobile network traffic associated to border cells.
We also observe cells with outlying behaviors within the geographical region under consideration in Fig.\,\ref{fig7:outliers}. We hypothesize that these are areas where the population density changed significantly over the last few years. As a matter of fact, the ISTAT census dates back to 2011, whereas the mobile network traffic data refers to 2015. It is likely that the distribution of the local population has undergone some evolution during the four years that separate the two datasets.
For instance, a confirmation to this consideration seems to come from the high density of outlying cells in the Southern periphery of Milan (in proximity of the South beltway): this is an expansion area of the city, and the mobile network traffic activity reflects a density of inhabitants in 2015 that is significantly higher than that recorded in 2011.
We deem this result to be an evidence of how mobile network traffic data could help updating population distribution maps, without any cost and with very limited latency.
\subsection{Model evaluation}
\label{assess}
The regression model allows computing an estimate of the static population $\hat{\rho}_i$ from the presence metadata $\sigma_i$, within each spatial cell $i$, as
\begin{equation}
\hat{\rho}_i =\hat{\alpha} \sigma_i^{\hat{\beta}}.
\label{eq:estimreg}
\end{equation}
The estimated population $\hat{\rho}_i$ can then be compared to the ground-truth data $\rho_i$ from the ISTAT census.
To that end, we use the determination coefficient $R^2$, and the Normalized Root Mean Square Error ($NRMSE$). The former provides a measure of the quality of the fitting of the estimates on ground-truth data.
The latter describes the fraction of the error between values predicted by the model and the values of census population data.
The $R^2$ coefficient is computed as
\begin{equation}\label{eq:r2}
R^2 = 1-\frac{\sum_{i=1}^N \left(\rho_i - \hat{\rho}_i\right)^2}{\sum_{i=1}^N \left(\rho_i-\bar{\rho}\right)^2},
\end{equation}
where $N$ denotes the number of cells in the spatial tessellation, and $\bar{\rho}$ is the overall average population density computed on all cells.
The $NRMSE$ facilitates the comparison of the model results in different contexts. It is defined as
\begin{equation}\label{eq:RMSE}
NRMSE = \frac{1}{\rho_{max}-\rho_{min}}\sqrt{\frac{\sum_{i=1}^N(\hat{\rho}_i-\rho_i)^2}{N}},
\end{equation}
where $\rho_{max}$ and $\rho_{min}$ are the maximum and minimum population densities recorded in the target region.
Since the model is trained on ISTAT census data, we adopt a two-fold cross-validation procedure, as follows.
For each test, we separate the ground-truth data into two subsets: two-thirds of the data are used as a training set, and the remaining one-third as a test set. Then, the training set is used to learn the model parameters, and the resulting model is evaluated against the test set.
\begin{figure}[tb]
\centering
\includegraphics[width=1\linewidth]{figures/regression3.png}
\caption{Milan. Model evaluation. A) $R^2$ for training and test sets, separated by land use. B) $R^2$ and $NRMSE$ of the model trained on residential land use on the other land uses.}
\label{fig:corrclus}
\vspace*{-7pt}
\end{figure}
The baseline result, in the Milan case study, is shown in Fig.\,\ref{fig:corrclus}. The top plot (A) shows the determination coefficient $R^2$ obtained with the training and test data. A first observation is that results are comparable for training and test data, which validates our model.
A second remark is that the results are separated by land use, an information that we extract from the mobile network traffic data itself, as detailed in Sec.\,\ref{sub:landuse}.
The rationale for this choice is that land uses affect the activity of individuals, possibly including their mobile communication habits. For instance, the mobile network traffic generated in a residential zone undergoes dynamics that are very different from those observed in a shopping area. In turn, this diversity can induce variations in the quality of the population estimation model across land uses. Plot (A) in Fig.\,\ref{fig:corrclus} shows precisely this effect: some land uses have higher $R^2$ than others. Specifically, residential zones display the highest fitting score, with $R^2=0.84$ and a $NRMSE$ of 0.046 (not shown). This is reasonable, given that in these areas the majority of the communication activity is generated by users in their homes.
We also verified up to which extent a model trained on residential land use can be used to estimate populations in areas characterized by different land uses.
The bottom plot (B) in Fig.\,\ref{fig:corrclus} shows the results in the Milan case study. Performance are good for commuting, touristic, and shopping zones, fair for office and nightlife zones, and bad for university areas.
We speculate that this phenomenon could be due to the presence of overnight mobile communication activities in university campuses (e.g., parties or measurement for research purposes) where nobody actually lives.
In all cases, university areas only represent a negligible minority of spatial cells, which allows concluding that a model trained on residential land use can be effectively used to estimate the population density in a whole urban region.
\subsection{Other city use cases}
\label{sub:cities}
\begin{table*}[tb]
\caption{$R^2$ and $NRMSE$ in different cities of Italy, for residential-only and mixed land use.\vspace*{-4pt}}
\centering
\ra{1.3}
\begin{tabular}{@{}lllllllllllllll@{}}
& \multicolumn{10}{c}{\textsc{Residential}} & \phantom{abc} & \multicolumn{2}{c}{\textsc{Mixed}} \\
\cmidrule{2-11} \cmidrule{13-14}
&& \multicolumn{5}{c}{Training} && \multicolumn{3}{c}{Test} &&& \\
\cmidrule{2-8} \cmidrule{10-11}
& ${\scriptstyle \hat{\alpha}}$ & ${\scriptstyle 95\% C.I.}$ & ${\scriptstyle \hat{\beta}}$ & ${\scriptstyle 95\% C.I.}$& &${\scriptstyle R^2}$ & ${\scriptstyle NRMSE}$ & &${\scriptstyle R^2}$ & ${\scriptstyle NRMSE}$ && ${\scriptstyle \bar{R^2}}$ & ${\scriptstyle \overline{NRMSE}}$ \\
\midrule
\textsc{Milan} &1.24&[1.03,1.37]&0.97 &[0.89,1.0]&&0.84 & 0.046 && 0.83 & 0.047 && 0.80 & 0.059 \\
\cdashline{1-14}[2pt/3pt]
\textsc{Turin} &0.94&[0.82,1.2]&0.99&[0.88,1.2]&& 0.80 & 0.052 && 0.80 & 0.053 &&0.76&0.065 \\
\cdashline{1-14}[2pt/3pt]
\textsc{Rome} & 0.75&[0.67,0.98]&1.03&[0.92,1.1]&& 0.87 & 0.035 && 0.87 & 0.035 &&0.84 &0.044 \\
\bottomrule
\label{tab:cities}
\vspace*{-23pt}
\end{tabular}
\end{table*}
All previous results refer to the Milan case study. We generalize our analysis by considering two other major cities in Italy, i.e., Rome and Turin.
For each urban case study, we adopt the aforementioned cross-validation procedure, separating training and test datasets. We then estimate the model parameters $\hat{\alpha}$ and $\hat{\beta}$: consistently with our previous results, this is performed by considering training data related to residential land use only.
The {\it Residential} portion of Tab.\,\ref{tab:cities} shows the results we obtain, in terms of parametrization of $\hat{\alpha}$ and $\hat{\beta}$ and quality of the estimation on the test set of remaining residential areas.
The right portion of the Tab.\,\ref{tab:cities}, denoted as {\it Mixed}, refers to the quality of the estimation on all areas, including those that are not residential in nature: it thus gives us information on the accuracy of a model trained on residential data only, when used on a complete urban region.
In this second case, we employ the following averaged quality metrics:
\begin{equation}
\bar{R^2}=\sum^L_{\ell=1} \frac{N_\ell}{N} R_\ell^2,
\end{equation}
\begin{equation}
\overline{NRMSE}=\sum^L_{\ell=1} \frac{n_\ell}{N} NRMSE_\ell,
\end{equation}
where $L$ is the number of different land uses, $N_\ell$ stands for number of spatial cells associated to a given land-use $\ell$, and $R_\ell^2$ (respectively, $NRMSE_\ell$) are the determination coefficient (respectively, normalized root mean square error) computed on land use $\ell$. Thus, these metrics provide an weighted average of the performance of the estimation over all land uses.
The results in Tab.\,\ref{tab:cities} are fairly close for all cities. Rome shows highest scores, with $R^2$=0.87 and $NRMSE$=0.035 for residential land use, and $R^2$=0.84 and $NRMSE$=0.044 for the overall case. Milan is a close second, and Turin performs slightly worse. Yet, the $R^2$ we measure is consistently higher than that observed in recent previous works, e.g.,~\cite{douglass2015high}.
\begin{figure}[tb]
\centering
\includegraphics[width=0.95\linewidth]{figures/corr_mat.png}
\caption{Cross-city test: models trained on data collected in cities along rows are used to estimate the population of cities along columns. Tables refer to $R^2$ and $NRMSE$.}
\label{fig:corr_mat}
\vspace*{-14pt}
\end{figure}
From Tab.\,\ref{tab:cities}, we observe that the parameters $\hat{\alpha}$ and $\hat{\beta}$ are not dramatically different across cities. We thus explore the possibility of estimating the population in an urban area using a model trained on data collected in another city.
Fig.\,\ref{fig:corr_mat} summarizes our findings.
Overall, we find high $R^2$ and low $NRMSE$ in all cases, which let us conclude that a cross-city estimation of population is possible.
This is an important observation, paving the road to the estimation of populations in cities for which mobile network traffic data is available, but where no ground-truth population distribution is provided.
\section{Dynamic population estimation}
\label{sec:DYNAMIC}
In the previous discussion, we have focused on static populations, considering the distribution of inhabitants based on their home locations.
However, an interesting possibility offered by mobile network traffic analysis is that of estimating dynamic populations, i.e., the variations in the distribution of inhabitants determined by their daily activities.
Clearly, this is an even more challenging task, due to the faster timescale at which such daily dynamics occur.
The main problem one has to cope with in estimating dynamic populations is the lack of ground truth data, which makes it impossible training a model such as that in (\ref{eq:estimreg}). At the same time, reusing the parameters $\hat{\alpha}$ and $\hat{\beta}$ computed for the static population may be risky, because there is no certainty that the relationship between the subscriber presence $\sigma_i$ and the static population remains valid for dynamic populations.
Our approach consists in drawing a novel multivariate relationship between the population distribution, the subscriber presence and the level of mobile communication activity of subscribers. This allows taking the constants $\alpha$ and $\beta$ out of the equation, and finding a unifying equation that can be used to estimate dynamic populations.
\subsection{Subscriber presence and activity level}
\label{sub:dyn-activity}
We start by discussing the interplay between the presence and the mobile communication activity level.
The latter is formally defined as the frequency with which a subscriber interacts with the mobile network.
\begin{comment}
The relation between the activity level $\lambda$ and the probability of a given user $j$ that remain stationary inside a given cell $i$ and been undetected during time-slot $[t, t+T]$ is defined as $P^{ndet}(T)$. In eq. \ref{eq:lambdai} we assume independence between the activities (i.e. call, sms, data) and an exponentially distributed interval between calls or sms. We expresses in eq. \ref{eq:lambdai} the relation between the activity level of a device and probability undetected inside the presence dataset but present in a given cell $i$. Nonetheless, it remains that the exact relation between the activities level of a given user and the probability $P^{ndet}$ is difficult to derive due to the complex interplay between the user's mobility and the user's behavior.
\begin{eqnarray}
P^{ndet}(T) & = & \prod_{a \in \mathcal{A}} P[ X_{a} > T ] = e^{\sum_{a \in \mathcal{A}}\lambda_{a} T} \label{eq:lambdai} \\
\text{where}\ \mathcal{A} & = & \{call, sms, data \} \nonumber
\end{eqnarray}
\noindent We estimate (cf Fig.~\ref{fig2:activ} A and B) the inter-arrival between an activity as $\lambda_{call}$ and $\lambda_{sms}$ as follows:
\begin{eqnarray}
\lambda_{call} & = & \frac{1}{CT} \sum_{i=0}^{C}\sum_{t=0}^{T}\frac{\mathcal{V}_{callin}^{i}(t) + \mathcal{V}_{callout}^{i}(t)}{\mathcal{V}_{presence}^{i}(t)} \\
\lambda_{sms} & = & \frac{1}{CT} \sum_{i=0}^{C}\sum_{t=0}^{T} \frac{\mathcal{V}_{smsin}^{i}(t) + \mathcal{V}_{smsout}^{i}(t)}{\mathcal{V}^{c}_{presence}(t)} \label{eq:activity}
\end{eqnarray}
\noindent where $\mathcal{V}(t)$ stands for the number of callin/smsin made per active device in a given cell $c$ during the time-slot $t$ of 15 minutes and $\mathcal{V}_{presence}^{i}(t)$ stands for the number of unique device counted in the cell $i$ in presence dataset during a 15 minutes time-slot $t$.
\end{comment}
\begin{figure}[tb]
\centering
\includegraphics[width=0.95\linewidth]{figures/lambda.png}\vspace*{-4pt}
\caption{Subscriber activity level for voice calls (A, top) and SMS (B, bottom) over daytime.\vspace*{-3pt}}
\label{fig2:activ}
\vspace*{-10pt}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=0.8\linewidth]{figures/3.png}
\caption{Subscriber activity level for calls, split by land use.\vspace*{-2pt}}
\label{fig3:activlanduse}
\vspace*{-10pt}
\end{figure}
Fig.\,\ref{fig2:activ} depicts the average level of activity per subscriber, as recorded in the data over the 24 hours of each day. It is expressed as the mean number of events per user ($\lambda$) observed in the data at every 60 minutes.
We remark a significant variation of activity, with minimum network usage at night and increased mobile communications during the working hours: this is consistent with previous analyses of mobile traffic dynamics.
Interestingly, differences across land uses are moderate, as shown on Fig.\,\ref{fig3:activlanduse}.
We conclude that the mobile communication activity is heterogeneous, and such a behavior emerges in time more than over land uses.
Such heterogeneity in the activity level has an impact on the correctness of the presence information. Let us consider again Fig.\,\ref{fig:presence}: the more often a mobile device sends or receives calls, SMS, and data packets, the more accurate is its localization in the presence dataset.
A legitimate question is then if the heterogeneous activity we discussed can be linked to the model parametrization, and explain -- in part or in full -- the diversity of values of $\hat{\alpha}$ and $\hat{\beta}$ observed in Sec.\,\ref{sec:NIGHTTIME}.
\subsection{Population estimation with activity level}
\label{sub:dyn-estimation}
\begin{figure}[tb]
\centering
\includegraphics[width=1\linewidth]{figures/regression2.png}\vspace*{-4pt}
\caption{Linear relationship between the activity level $\lambda$ and the model parameters $\hat{\alpha}$ and $\hat{\beta}$.}
\label{fig2:alphabeta}
\vspace*{-14pt}
\end{figure}
We thus investigate the existence of a connection between the level of activity of subscribers and the values of $\alpha$ and $\beta$ in (\ref{eq:regression}) that regulate the relationship between presence and population.
We do not have access to the real values, but to their estimations $\hat{\alpha}$ and $\hat{\beta}$.
We thus collect data in all cities that refer to the overnight period, i.e., from midnight to 8 am: in this period the ISTAT census information can be still considered a reliable ground truth, as most people will be at home.
We then plot a scatterplot of the activity level $\lambda$ versus the regression parameters $\hat{\alpha}$ and $\hat{\beta}$ obtained in these scenarios from (\ref{eq:estimreg}). The results are depicted in Fig.\,\ref{fig2:alphabeta}.
We find a striking linear relationship between $\lambda$ and both parameters. The coefficients of the linear models are indicated in the plots.
This result allows drawing a unifying multivariate model that links the population density to both the subscriber presence and the subscriber activity level.
We can then refine our estimation model as:
\begin{equation}
\hat{\rho}_i(\lambda_i,\sigma_i) =
(\hat{a}_\alpha \lambda_i+\hat{b}_\alpha) \cdot
\sigma_i^{(\hat{a}_\beta \lambda_i + \hat{b}_\beta)}.
\label{eq:dynamic}
\end{equation}
An important consideration is that (\ref{eq:dynamic}) substantiates the need for the power relationship between presence and population density in (\ref{eq:regression}) for a correct modeling of dynamic populations.
Also, the new parameters $\hat{a}_\alpha$, $\hat{b}_\alpha$, $\hat{a}_\beta$, $\hat{b}_\beta$ are valid for all scenarios, and are consistent across different times of the day.
We thus consider that the model in (\ref{eq:dynamic}) can be reliably used for the estimation of dynamic populations, given than the time series of the subscriber presence $\sigma_i$ and of the subscriber activity level $\lambda_i$ are available from mobile network traffic metadata.
\subsection{A case study in Milan}
\label{sub:dyn-milan}
\begin{figure}[tb]
\centering
\includegraphics[width=1\linewidth]{figures/density.png}
\caption{Dynamic distribution of Milan population. A) April 15 at noon. B) April 22 at 5 pm. C) April 19 at 10 pm. Left plots show the whole conurbation, right ones the city only. Figure best viewed in colors.}
\label{fig1:density}
\vspace*{-10pt}
\end{figure}
We provide a proof-of-concept of exploitation of the model in (\ref{eq:dynamic}), considering the case of Milan. Fig.\,\ref{fig1:density} illustrates the dynamic distribution of the population in Milan and its suburbs, as inferred from our model, at three sample time instants: April 15, 2015, at noon (A), April 22, 2015, at 5 pm (B) and April 19, 2015, at 10 pm (C). In each plot, colors denote the variation of population during the corresponding hour: from high in-flow of individuals moving to the cell (red) to high high out-flow of individuals leaving the cell (blue). There are also neutral cells where the population density does not vary during the considered time period (white).
The first two times refer to subsequent Wednesdays, at inherently different hours of the day. We observe that at noon the population tends to move towards the city outskirts (A, left plot), going back home for lunch. An opposite trend is observed at 5 pm, when there is a significant in-flow towards the city center or commercial areas (B, right plot).
April 19 is a Saturday, and our estimate of the dynamic population captures the movement of people towards residential areas outside Milan (C, left plot) as well as towards nightlife areas in the city (C, right plot). Interestingly, the model conveys well the crowd attracted by an important football match that took place that day at the main city stadium (C, right plot, dark red zone towards the West).
Although we cannot compare these population variations against a proper ground truth, we remark that the model entails very reasonable behaviors that match well those known to characterize the movements of inhabitants of the Milan conurbation.
\section{Conclusions}
\label{sec:CONCLUSION}
We introduced a novel approach to the estimation of population based on existing relationship between mobile activity volume and population density. Our solution builds exclusively on metadata collected by mobile network operators.
Our results demonstrate how our model allows for a reliable representation of static populations across different cities. It also outperforms previous proposals in the literature, thanks to the use of more appropriate metadata as well as proper data filtering based on time and land use.
\section*{Acknowledgements}
The authors would like to thank Angelo Furno for his help with extracting land use information. This work was supported by the French National Research Agency under grant ANR-13-INFR-0005 ABCD, by the EU FP7 ERA-NET program under grant CHIST-ERA-2012 MACACO, and by BPI-France through the FUI FluidTracks project.
\bibliographystyle{IEEEtran}
|
2,869,038,154,775 | arxiv | \section{Introduction}
\label{sec:intro}
The seminal paper \cite{Donaldson1998} of Donaldson-Thomas has
inspired a considerable amount of work related to gauge theory in
higher dimensions. Tian \cite{Tian2000} and Tao--Tian \cite{Tao2004}
made significant progress on important foundational analytical
questions. Recent work of Donaldson--Segal \cite{Donaldson2009} and
Haydys \cites{Haydys2011,Haydys2011a} shed some light on the shape of
the theories to be expected.
In this article we will focus on the study of gauge theory on
${\rm G}_2$-manifolds. These are $7$-manifolds equipped with a
torsion-free ${\rm G}_2$-structure. The ${\rm G}_2$-structure allows to
define a special class of connections, called ${\rm G}_2$--instantons.
These share many formal properties with flat connections on
$3$--manifolds and it is expected that there are ${\rm G}_2$--analogues of
those $3$--manifold invariants related to ``counting flat
connections'', i.e., the Casson invariant, instanton Floer homology,
etc.
So far non-trivial examples of ${\rm G}_2$--instantons are rather rare. By
exploiting the special geometry of the known ${\rm G}_2$--manifolds some
progress has been made recently. At the time of writing, there are
essentially two methods for constructing compact ${\rm G}_2$--manifolds in
the literature. Both yield ${\rm G}_2$--manifolds close to degenerate
limits. One is Kovalev's twisted connected sum construction
\cite{Kovalev2003}, which produces ${\rm G}_2$--manifolds with ``long
necks'' from certain pairs of Calabi-Yau 3-folds with asymptotically
cylindrical ends. A technique for constructing ${\rm G}_2$--instantons on
Kovalev's ${\rm G}_2$--manifolds has recently been introduced by Sá Earp
\cites{SaEarp2011,SaEarp2011a}. The other (and historically the
first) method for constructing ${\rm G}_2$--manifolds is due to Joyce and
is based on desingularising ${\rm G}_2$--orbifolds \cite{Joyce1996}. In this
article we introduce a method to construct ${\rm G}_2$--instantons on
Joyce's manifolds.
To setup the framework for our construction, let us briefly review the
geometry of Joyce's construction. Give $T^7$ the structure of a flat
${\rm G}_2$--orbifold and let $\Gamma$ be a finite group of
diffeomorphisms of $T^7$ preserving the flat ${\rm G}_2$--structure. Then
$Y_0=T^7/\Gamma$ is a flat ${\rm G}_2$--orbifold. In general, the singular
set $S$ of $Y_0$ can be quite complicated. For the purpose of this
article, we assume that each of the connected components $S_j$ of $S$
has a neighbourhood $T_j$ modelled on $(T^3\times {\bf C}^2/G_j)/H_j$.
Here $G_j$ is a non-trivial finite subgroup of ${\rm SU}(2)$ and $H_j$ is a
finite subgroup of isometries of $T^3\times {\bf C}^2/G_j$ acting freely on
$T^3$. We call ${\rm G}_2$--orbifolds $Y_0$ satisfying this condition
\emph{admissible}. Now, for each $j$, pick an ALE space $X_j$
asymptotic to ${\bf C}^2/G_j$ and an isometric action $\rho_j$ of $H_j$ on
$X_j$ which is asymptotic to the action of $H_j$ on ${\bf C}^2/G_j$. Given
such a collection $\{(X_j,\rho_j)\}$ of \emph{resolution data} for
$Y_0$ and a small parameter $t>0$, one can rather explicitly construct
a compact $7$--manifold $Y_t$ together with a ${\rm G}_2$--structure that
is close to being torsion-free. Joyce proved that this
${\rm G}_2$--structure can be perturbed slightly to yield a torsion-free
${\rm G}_2$--structure provided $t>0$ is sufficiently small.
We will introduce a notion of \emph{gluing data}
$\((E_0,\theta),\{(x_j,f_j)\},\{(E_j,A_j,\tilde\rho_j,m_j)\}\)$
compatible with resolution data $\{(X_j,\rho_j)\}$ for $Y_0$,
consisting of a flat connection $\theta$ on a $G$--bundle $E_0$ over
$Y_0$ and, for each $j$, a $G$--bundle $E_j$ together with an ASD
instanton $A_j$ framed at infinity as well as various additional data
satisfying a number of compatibility conditions. With this piece of
notation in place, the main result of this article is the following.
\begin{theorem}\label{thm:a}
Let $Y_0$ be an admissible flat ${\rm G}_2$--orbifold, let
$\{(X_j,\rho_j)\}$ be resolution data for $Y_0$ and let
$\((E_0,\theta),\{(x_j,f_j)\},\{(E_j,A_j,\tilde\rho_j,m_j)\}\)$ be
compatible gluing data. Then there is a constant $\kappa>0$ such that for
$t\in(0,\kappa)$ one can construct a $G$--bundle $E_t$ on $Y_t$
together with a connection $A_t$ satisfying
\begin{align*}
p_1(E_t)=-\sum_j k_j\,\PD[S_j]\quad\text{with}\quad
k_j=\frac{1}{8\pi^2}\int_{X_j} |F_{A_j}|^2
\end{align*}
and
\begin{align*}
\<w_2(E_t),\Sigma\>=\<w_2(E_j),\Sigma\>
\end{align*}
for each $\Sigma\in{}H_2(X_j)^{H_j}\subset{}H_2(Y_t)$. Here
$[S_j]\in{}H_3(Y_t)$ is the homology class induced by the
component $S_j$ of the singular set $S$.
If $\theta$ is regular as a ${\rm G}_2$--instanton and each of the $A_j$
is infinitesimally rigid, then for $t\in(0,\kappa)$ there is a small
perturbation $a_t\in\Omega^1(Y_t,\mathfrak g_{E_t})$ such that $A_t+a_t$ is
a regular ${\rm G}_2$--instanton.
\end{theorem}
The analysis involved in the proof of Theorem~\ref{thm:a} is similar
to unpublished work of Brendle on the Yang--Mills equation in
higher dimension \cite{Brendle2003} and Pacard--Ritoré's work on the
Allen--Cahn equation \cite{Pacard2003}, however our assumptions lead to
some simplifications. From a geometric perspective our result can be
viewed as a higher dimensional analogue Kronheimer's work on ASD
instantons on Kummer surfaces \cite{Kronheimer1991}.
Here is an outline of this article. To benefit the reader
Sections~\ref{sec:review}, \ref{sec:g2i}, \ref{sec:kummer} and
\ref{sec:asdale} contain some foundational material on
${\rm G}_2$--manifolds and ${\rm G}_2$--instantons as well as brief reviews of
Joyce's generalised Kummer construction and Kronheimer's and
Nakajima's work on ASD instantons on ALE spaces. In
Section~\ref{sec:approx} we define the notion of gluing data and prove
the first part of Theorem~\ref{thm:a}. We also introduce weighted
Hölder spaces adapted to the problem at hand and prove that the
connection $A_t$ is a good approximation to a ${\rm G}_2$--instanton. In
Section~\ref{sec:model} we set up the analytical problem underlying
the proof of Theorem~\ref{thm:a} and discuss the properties of a model
for the linearised problem. We complete the proof of
Theorem~\ref{thm:a} in Section~\ref{sec:deform}. A number of examples
with $G={\rm SO}(3)$ are constructed in Section~\ref{sec:ex}, before we
close our exposition with a short discussion of the relevance of our
work to computing a conjectural ${\rm G}_2$ Casson invariant and hint at
potential future applications in Section~\ref{sec:discuss}.
\paragraph{Acknowledgements} This article is the outcome of work
undertaken by the author for his PhD thesis at Imperial College
London, funded by the European Commission. I would like to thank my
supervisor Simon Donaldson for guidance and inspiration.
\section{Review of ${\rm G}_2$--manifolds}
\label{sec:review}
The Lie group ${\rm G}_2$ can be defined as the subgroup of elements of
${\rm GL}(7)$ fixing the $3$--form
\begin{align*}
\phi_0={\rm d} x^{123}+{\rm d} x^{145}+{\rm d} x^{167}+{\rm d} x^{246}-{\rm d}
x^{257}-{\rm d} x^{347}-{\rm d} x^{356}.
\end{align*}
Here ${\rm d} x^{ijk}$ is a shorthand for ${\rm d} x^i\wedge {\rm d} x^j \wedge
{\rm d} x^k$ and $x_1,\ldots,x_7$ are standard coordinates on ${\bf R}^7$. The
particular choice of $\phi_0$ is not important. In fact, any
non-degenerate $3$--form $\phi$ is equivalent to $\phi_0$ under a
change of coordinates, see e.g.~\cite{Salamon2010}*{Theorem~3.2}.
Here we say that $\phi$ is non-degenerate if for each non-zero vector
$u\in{\bf R}^7$ the $2$--form $i(u)\phi$ on ${\bf R}^7/\<u\>$ is symplectic.
${\rm G}_2$ naturally is a subgroup of ${\rm SO}(7)$. To see this, note that
any element of ${\rm GL}(7)$ that preserves $\phi_0$ also preserves the
standard inner product and the standard orientation of ${\bf R}^7$. This
follows from the identity
\begin{align*}
i(u)\phi_0\wedge i(v)\phi_0\wedge\phi_0=6g_0(u,v) \mathrm{vol}_0.
\end{align*}
In particular, every non-degenerate $3$--form $\phi$ on a
$7$--dimensional vector space naturally induces an inner product and
an orientation. As an aside we should point out here that
non-degenerate $3$--forms form one of two open orbits of ${\rm GL}(7)$ in
$\Lambda^3 ({\bf R}^7)^*$. For $\phi$ in the other open orbit, the above
formula yields an indefinite metric of signature $(3,4)$. In
particular, if we take $u=v$ to be a light-like vector, then
$i(u)\phi$ is not a symplectic form on ${\bf R}^7/\<u\>$.
A \emph{${\rm G}_2$--manifold} is $7$--manifold equipped with a
torsion-free ${\rm G}_2$--structure, i.e., a reduction of the structure
group of $TY$ from ${\rm GL}(7)$ to ${\rm G}_2$ with vanishing intrinsic
torsion. From the above it is clear that a ${\rm G}_2$--structure on $Y$
is equivalent to a $3$--form $\phi$ on $Y$, which is non-degenerate at
every point of $Y$. The condition to be torsion-free means that
$\phi$ is parallel with respect to the Levi--Civita connection
associated with the metric induced by $\phi$. This can also be
understood as follows. Let $\mathscr P\subset\Omega^3(Y)$ denote the
subspace of all non-degenerate $3$--forms $\phi$. Then there is a
\emph{non-linear} map $\Theta\mskip0.5mu\colon\thinspace \mathscr P \to \Omega^4(Y)$ defined by
\begin{align*}
\Theta(\phi)=*_\phi \phi.
\end{align*}
Here $*_\phi$ is the Hodge-$*$-operator coming from the inner product
and orientation induced by $\phi$. Now the ${\rm G}_2$--structure induced
by $\phi$ is torsion-free if and only if
\begin{align*}
{\rm d}\phi=0, \quad {\rm d}\Theta(\phi)=0.
\end{align*}
The proof of this fact is an exercise in linear algebra.
Since ${\rm G}_2$--manifolds have holonomy contained in ${\rm G}_2$ which can
also be viewed as the subgroup of ${\rm Spin}(7)$ fixing a non-trivial
spinor, ${\rm G}_2$-manifolds are spin and admit a non-trivial parallel
spinor. In particular, they are Ricci-flat. Moreover,
${\rm G}_2$--manifold naturally carry a pair of calibrations: the
\emph{associative calibration} $\phi$ and the \emph{coassociative
calibration} $\psi:=\Theta(\phi)$. This should be enough to
convince the reader, that ${\rm G}_2$--manifolds are interesting geometric
objects worthy of further study.
Simple examples of ${\rm G}_2$--manifolds are $T^7$ with the constant
torsion-free ${\rm G}_2$--structure $\phi_0$ and products of the form
$T^3\times\Sigma$, where $\Sigma$ is a hyperkähler surface, with the
torsion-free ${\rm G}_2$--structure induced by
\begin{align*}
\phi_0=\delta_1\wedge\delta_2\wedge\delta_3
+\delta_1\wedge\omega_1
+\delta_2\wedge\omega_2
-\delta_3\wedge\omega_3
\end{align*}
where $\delta_1,\delta_2,\delta_3$ are constant orthonormal $1$--forms
on $T^3$ and $\omega_1,\omega_2,\omega_3$ define the hyperkähler
structure on $\Sigma$. These examples all have holonomy strictly
contained in ${\rm G}_2$. Compact examples with full holonomy ${\rm G}_2$ are
much harder to come by. In Section~\ref{sec:kummer}, we will review
Joyce's generalised Kummer construction which produced the first
examples of ${\rm G}_2$--manifolds of this kind.
\section{Gauge theory on ${\rm G}_2$--manifolds}
\label{sec:g2i}
Let $(Y,\phi)$ be a ${\rm G}_2$--manifold and let $E$ be a $G$--bundle
over $Y$. Denote by $\psi:=\Theta(\phi)$ the corresponding
coassociative calibration. A connection $A\in\mathscr A(E)$ on $E$ is called
a \emph{${\rm G}_2$--instanton} if it satisfies the equation
\begin{align*}
F_A\wedge\psi=0.
\end{align*}
This equation is equivalent to $*(F_A\wedge\phi)=-F_A$ and thus
${\rm G}_2$--instantons are solutions of the Yang--Mills equation. In
fact, if $Y$ is compact, then ${\rm G}_2$-instantons are absolute minima
of the Yang--Mills functional.
From an analytical point of view the above equations are not
satisfying because they are not manifestly elliptic. This
issue is resolved by considering instead the equation
\begin{align*}
*(F_A\wedge\psi)+{\rm d}_A\xi=0
\end{align*}
where $\xi\in\Omega^0(Y,\mathfrak g_E)$. If $Y$ is compact (or under certain
decay assumptions) it follows from the Bianchi identity and
${\rm d}\psi=0$ that ${\rm d}_A\xi=0$. Thus this it is equivalent to the
${\rm G}_2$--instanton equation. Once gauge invariance is accounted for
the equation becomes elliptic. The gauge fixed linearisation at a
${\rm G}_2$--instanton $A$ is given by the operator
$L_A\mskip0.5mu\colon\thinspace\Omega^0(Y,\mathfrak g_E)\oplus\Omega^1(Y,\mathfrak g_E) \to
\Omega^0(Y,\mathfrak g_E)\oplus\Omega^1(Y,\mathfrak g_E)$ defined by
\begin{align*}
L_A=\begin{pmatrix}
0 & {\rm d}_A^* \\
{\rm d}_A & *\left(\psi\wedge{\rm d}_A\right)
\end{pmatrix}.
\end{align*}
It is easy to see that $L_A$ is a self-adjoint elliptic operator.
In particular, the virtual dimension of the moduli space
\begin{align*}
\mathscr M(E,\phi)=\left\{ A\in\mathscr A(E) : F_A\wedge\psi=0 \right\}/\mathscr G
\end{align*}
is zero. It is very tempting to try to define a \emph{${\rm G}_2$ Casson
invariant} by ``counting'' $\mathscr M=\mathscr M(E,\phi)$. If every
${\rm G}_2$--instanton $A$ on $E$ is \emph{regular}, this means that $L_A$
is an isomorphism, then $\mathscr M$ is a smooth zero-dimensional manifold,
i.e., a discrete set. If $\mathscr M$ was compact, then up to questions of
orientation we could indeed count $\mathscr M$.
Whether there is a rigorous general definition of a ${\rm G}_2$ Casson
invariant and whether it is, in fact, invariant under isotopies the
${\rm G}_2$--structure is an open question. A brief discussion of parts
of this circle of ideas can be found in Donaldson--Segal
\cite{Donaldson2009}*{Section~6}.
Simple examples of ${\rm G}_2$--instantons are flat connections and ASD
instantons on K3 surfaces $\Sigma$ pulled back to $T^3\times\Sigma$.
Moreover, the Levi--Civita connection on a ${\rm G}_2$--manifolds is a
${\rm G}_2$--instanton. We will construct non-trivial examples using
Theorem~\ref{thm:a} in Section~\ref{sec:ex}.
\section{Joyce's generalised Kummer construction}
\label{sec:kummer}
Consider $T^7$ together with a constant $3$--form $\phi_0$ defining a
${\rm G}_2$--structure and let $\Gamma$ be a finite group of
diffeomorphisms of $T^7$ preserving $\phi_0$. Then $Y_0=T^7/\Gamma$
naturally is a flat ${\rm G}_2$--orbifold. Denote by $S$ the singular set
of $Y_0$ and denote by $S_1, \ldots, S_k$ its connected components.
In general, $S$ can be rather complicated. In this article we make
the assumption that each $S_j$ has a neighbourhood isometric to a
neighbourhood of the singular set of $(T^3\times {\bf C}^2/G_j)/H_j$. Here
$G_j$ is a non-trivial finite subgroup of ${\rm SU}(2)$ and $H_j$ is a
finite subgroup of isometries of $T^3\times {\bf C}^2/G_j$ acting freely on
$T^3$. ${\rm G}_2$--orbifolds $Y_0$ if this kind will be called
\emph{admissible}.
Let $Y_0$ be an admissible flat ${\rm G}_2$--orbifold. Then
there is a constant $\zeta>0$ such that if we denote by $T$ the set of
points at distance less that $\zeta$ to $S$, then $T$ decomposes into
connected components $T_1, \ldots, T_k$, such that $T_j$ contains
$S_j$ and is isometric to $(T^3\times B^4_\zeta/G_j)/H_j$. On $T_j$
we can write
\begin{align*}
\phi_0=\delta_1\wedge\delta_2\wedge\delta_3
+\delta_1\wedge\omega_1
+\delta_2\wedge\omega_2
-\delta_3\wedge\omega_3
\end{align*}
where $\delta_1,\delta_2,\delta_3$ are constant orthonormal $1$--forms
on $T^3$ and $\omega_1,\omega_2,\omega_3$ can be written as
$\omega_1=e^{1}\wedge e^{2}+e^{3}\wedge e^{4}$, $\omega_2=e^{1}\wedge
e^{3}-e^{2}\wedge e^{4}$, $\omega_3=e^{1}\wedge e^{4}+e^{2}\wedge
e^{3}$ in an oriented orthonormal basis $(e^1,\ldots,e^4)$ for
$({\bf C}^2)^*$.
We will remove the singularity along $S_j$ by replacing ${\bf C}^2/G_j$
with an \emph{ALE space asymptotic to ${\bf C}^2/G_j$}. That is a
hyperkähler manifold
$(X,\tilde\omega_1,\tilde\omega_2,\tilde\omega_3)$ together with a
continuous map $\pi\mskip0.5mu\colon\thinspace X\to{\bf C}^2/G_j$ inducing a diffeomorphisms from
$X\setminus\pi^{-1}(0)$ to $({\bf C}^2\setminus\{0\})/G_j$ such that
\begin{align*}
\nabla^k(\pi_*\tilde\omega_i-\omega_i)=O(r^{-4-k}),
\end{align*}
as $r\to\infty$ for $i=1,2,3$ and $k\geq 0$. Here
$r\mskip0.5mu\colon\thinspace {\bf C}^2/G_j\to[0,\infty)$ denotes the radius function. Due to work of
Kronheimer \cites{Kronheimer1989,Kronheimer1989a} ALE spaces are very
well understood. Here is a summary of his main results.
\begin{theorem}[Kronheimer]\label{thm:ale}
Let $G$ be a non-trivial finite subgroup of ${\rm SU}(2)$. Denote by $X$
the manifold underlying the crepant resolution $\wtilde{{\bf C}^2/G
}$. Then for each three cohomology classes
$\alpha_1,\alpha_2,\alpha_3\in H^2(X,{\bf R})$ satisfying
\begin{align*}
(\alpha_1(\Sigma),\alpha_2(\Sigma),\alpha_3(\Sigma))\neq 0
\end{align*}
for each $\Sigma\in H_2(X,{\bf Z})$ with $\Sigma\cdot\Sigma=-2$ there is
a unique ALE hyperkähler structure on $X$ for which the cohomology
classes of the Kähler forms $[\omega_i]$ are given by $\alpha_i$.
\end{theorem}
There is always a unique crepant resolution of ${\bf C}^2/G$. It can be
obtained by a sequence of blow-ups. The exceptional divisor $E$ of
$X=\wtilde{{\bf C}^2/G}$ has irreducible components
$\Sigma_1,\ldots,\Sigma_k$. By the McKay correspondence
\cite{McKay1980}, these components form a basis of $H_2(X,{\bf Z})$ and the
matrix with coefficients $C_{ij}=-\Sigma_i\cdot\Sigma_j$ is the Cartan
matrix associated with the Dynkin diagram corresponding to $G$ in the
ADE classification of finite subgroups of ${\rm SU}(2)$.
As a consequence of the above, one can always find an ALE space $X_j$
asymptotic to ${\bf C}^2/G_j$. If $H_j$ is non-trivial, then we also
require an isometric action $\rho_j$ of $H_j$ on $X_j$ which is
asymptotic to the action of $H_j$ on ${\bf C}^2/G_j$. A collection
$\{(X_j,\rho_j)\}$ of this kind is called \emph{resolution data} for
$Y_0$.
Denote by $\pi_j\mskip0.5mu\colon\thinspace X_j\to{\bf C}^2/G_j$ the resolution map for $X_j$. Let
$\pi_{j,t}:=t\pi_j:X_j\to{\bf C}^2/G_j$ and let
\begin{align*}
\tilde T_{j,t}:=(T^3\times{}\pi_{j,t}^{-1}(B^4_{\zeta}/G_j))/H_j
\quad\text{and}\quad \tilde T_t:=\bigcup_j \tilde T_{j,t}.
\end{align*}
Then using $\pi_{j,t}$ we can replace each $T_j$ in $Y_0$ by $\tilde
T_{j,t}$ and thus obtain a compact $7$--manifold $Y_t$. The Betti
numbers of $Y_t$ can be read off easily from this construction. It is
easy to see that each of the components $S_j$ of the singular set $S$
give rise to a homology class $[S_j]\in{}H_3(Y_t)$ and that each
$\Sigma\in{}H_2(X_j)$ invariant under $H_j$ yields a homology class
$\Sigma\in{}H_2(Y_t)$. Moreover, one can work out fundamental group
$\pi_1(Y_t)$. This is very important, because ${\rm G}_2$--manifolds have
full holonomy ${\rm G}_2$ if and only if their fundamental group is finite.
On $\tilde T_{j,t}$ there is a ${\rm G}_2$--structure defined by
\begin{align*}
\tilde\phi_t=\delta_1\wedge\delta_2\wedge\delta_3
+t^2\delta_1\wedge\tilde\omega_1
+t^2\delta_2\wedge\tilde\omega_2
-t^2\delta_3\wedge\tilde\omega_3.
\end{align*}
If we view $Y_0 \setminus S$ as a subset of $Y_t$, then $\tilde\phi_t$
almost matches up with $\phi_0$ in the following sense.
Introduce a function $d_t\mskip0.5mu\colon\thinspace Y_t\to [0,\zeta]$ measuring the distance
from the ``exceptional set'' by
\begin{align*}
d_t(p)=
\begin{cases}
|\pi_{j,t}(y)|, & p=[(x,y)] \in \tilde T_{j,t} \\
\zeta, & p \in Y_t \setminus \tilde T_t.
\end{cases}
\end{align*}
Then on sets of the form $\{x \in T_t : d_t(x)\geq\epsilon\}$, we have
$\tilde\phi_t-\phi_0=O(t^4)$.
\begin{theorem}[Joyce]\label{thm:joyce}
Let $Y_0=T^7/\Gamma$ be a flat ${\rm G}_2$--orbifold and let
$\{(X_j,\rho_j)\}$ be resolution data for $Y_0$. Then there is a
constant $\kappa>0$ such that for $t\in(0,\kappa)$ there is a
torsion-free ${\rm G}_2$--structure $\phi_t$ on the $7$-manifold $Y_t$
constructed above satisfying the following conditions. For
$\alpha\in(0,1)$ there is a constant $c=c(\alpha)>0$ such that
\begin{align*}
\|\phi_t-\tilde\phi_t\|_{L^\infty} +
t^{\alpha}[\phi_t-\tilde\phi_t]_{C^{0,\alpha}} +
t\|\nabla(\phi_t-\tilde\phi_t)\|_{L^\infty} \leq ct^{3/2}
\end{align*}
holds on $\tilde{T}_{j,t}$. Moreover, for every $\epsilon>0$, there
is a constant $c=c(\alpha,\epsilon)>0$ such that
\begin{align*}
\|\phi_t-\phi_0\|_{C^{0,\alpha}}\leq c(\epsilon)t^{3/2}
\end{align*}
holds on $\{x\in{}Y_t:d_t(x)\geq\epsilon\}$.
\end{theorem}
To prove this, one first writes down a closed
$3$--form $\psi_t\in\Omega^3(Y_t)$ that interpolates between
$\tilde\phi_t$ and $\phi_0$ in the annulus
\begin{align*}
R_t:=\{x\in Y_t:d_t(x)\in[\zeta/4,\zeta/2]\}.
\end{align*}
This can be done reasonably explicitly using cut-off functions, since
\begin{align*}
t^2(\pi_{j,t})_*\tilde\omega_i = \omega_i + {\rm d}\varrho_i
\end{align*}
with $\nabla^k \varrho_i = O(r^{-3-k})$ for $k\geq 0$. If $t>0$ is
sufficiently small this will induce a ${\rm G}_2$--structure which is
torsion-free on the complement of $R_t$ and has small torsion on
$R_t$. The ansatz $\phi_t=\psi_t+{\rm d}\eta_t$ leads to the following
non-linear equation
\begin{align*}
{\rm d}\Theta(\psi_t+{\rm d}\eta_t)=0.
\end{align*}
The gauge fixed linearisation of this equation is the Laplacian on
$2$-forms $\Delta$. As $t$ tends to zero, the properties of $\Delta$
degenerate but at the same time $\psi_t$ becomes closer and closer to
being torsion-free. By careful scaling considerations Joyce
\cite{Joyce1996}*{Part I} was able to show that the equation can
always be solved provided $t>0$ is sufficiently small. We should
point out that the estimates we give here are rather stronger than
those given originally by Joyce. The stated estimate (which is still
suboptimal) can be established by a careful analysis of Joyce's
construction or, alternatively, by applying some of the ideas in this
article to the above problem. The details of this argument will be
carried out elsewhere.
\section{ASD instantons on ALE spaces}
\label{sec:asdale}
Let $\Gamma$ be a finite subgroup of ${\rm SU}(2)$ and let $X$ be an ALE
space asymptotic to ${\bf C}^2/\Gamma$. Fix a flat connection $\theta$
over $S^3/\Gamma$ and let $E$ be a $G$--bundle over $X$ asymptotic at
infinity to the bundle underlying $\theta$. We consider the space
$\mathscr A(E,\theta)$ of connections $A$ on $E$ that are asymptotic at
infinity to $\theta$ (at a certain rate) and the based gauge group
$\mathscr G_0$ of gauge transformations that are asymptotic at infinity to
the identity (again at a certain rate). The space
\begin{align*}
M(E,\theta)=\{A \in \mathscr A(E,\theta) : F_A^+=0\}/\mathscr G_0
\end{align*}
is called the \emph{moduli space of framed ASD instantons on $E$
asymptotic to $\theta$}. It does not depend on the precise choices
as long as the rate of decay to $\theta$ is not faster that $r^{-3}$
but still fast enough to ensure that $\|F_A\|_{L^2}<\infty$. If
$\rho\mskip0.5mu\colon\thinspace \Gamma\to G$ denotes the monodromy representation of $\theta$,
then the group $G_\rho=\{g\in G: g\rho g^{-1}=\rho\}$ acts on
$M(E,\theta)$. This can be thought of as the residual action of the
group gauge group $\mathscr G$ (consisting of gauge transformations with
bounded derivative).
\begin{theorem}[Nakajima \cite{Nakajima1990}*{Theorem~2.6}]
The moduli space $M(E,\theta)$ is a smooth hyperkähler manifold.
\end{theorem}
Formally, this can be seen as an infinite-dimensional instance of a
hyperkähler reduction \cite{Hitchin1987}. The space $\mathscr A(E,\theta)$
inherits a hyperkähler structure from $X$ and the action of the based
gauge group $\mathscr G_0$ has a hyperkähler moment map given by
$\mu(A)=F_A^+$. To make this rigorous one needs to setup a suitable
Kuranishi model for $M(E,\theta)$. This can be done using weighted
Sobolev spaces as in \cite{Nakajima1990}. The infinitesimal
deformation theory is then governed by the operator
$\delta_A\mskip0.5mu\colon\thinspace \Omega^1(X,\mathfrak g_E)\to\Omega^0(X,\mathfrak g_E)\oplus\Omega^+(X,\mathfrak g_E)$
defined by
\begin{align*}
\delta_A(a)=({\rm d}_A^*a, {\rm d}_A^+a).
\end{align*}
\begin{prop}\label{prop:decay}
Let $X$ be an ALE space, let $E$ be a $G$--bundle over $X$ and let
$A\in\mathscr A(E)$ be a finite energy ASD instanton on $E$. Then the
following holds.
\begin{enumerate}
\item If $a\in\ker\delta_A$ decays to zero at infinity, then
$a=O(r^{-3})$.
\item If $(\xi,\omega)\in\ker\delta_A^*$ decays to zero at infinity,
then $(\xi,\omega)=0$.
\end{enumerate}
\end{prop}
The last part of this proposition tells us that the deformation
theory is always unobstructed and hence $M(E,\theta)$ is a smooth
manifold. By the first part the tangent space of $M(E,\theta)$ at
$[A]$ agrees with the $L^2$ kernel of $\delta_A$ and thus the formal
hyperkähler structure is indeed well-defined.
\begin{proof}[Proof of Proposition \ref{prop:decay}]
The proof is based on a refined Kato inequality. Since
$\delta_Aa=0$, it follows that $|{\rm d}|a||\leq \gamma |\nabla_A a|$
with a constant $\gamma < 1$. To see that, recall that the Kato
inequality hinges on the Cauchy-Schwarz inequality $|\<\nabla_A
a,a\>|\leq |\nabla\alpha||\alpha|$. The equation $\delta_Aa=0$
imposes a linear constraint on $\nabla_A a$ incompatible with
equality in this estimate unless $\nabla_A a=0$. This shows that
one can find $\gamma<1$, such that a refined Kato inequality holds.
Indeed one can easily compute that $\gamma=\sqrt{3/4}$. Now, the
Weitzenböck formula implies that for $\sigma=2-1/\gamma^2=2/3$
\begin{align*}
(2/\sigma) \Delta |a|^\sigma &=
|a|^{\sigma-2}(\Delta|a|^2-2(\sigma-2)|{\rm d}|a||^2) \\
&\leq
|a|^{\sigma-2}(\Delta|a|^2+2|\nabla_A a|^2) \\
&=
|a|^{\sigma-2}\<a,\nabla_A^*\nabla_Aa\> \\
&=
|a|^{\sigma-2}(\<\delta_A^*\delta_A a,a\>+\<\{R,a\},a\>+\<\{F_A,a\},a\>) \\
&\leq O(r^{-4}) |a|^\sigma.
\end{align*}
Here $\{R,.\}$ and $\{F_A,\cdot\}$ denote certain actions of the
Riemannian curvature $R$ and the curvature $F_A$ of $A$
respectively. Since we only care about the behaviour of $|a|$ at
infinity, we can view it as function on ${\bf R}^4$ which satisfies
$\Delta_{{\bf R}^4} |a|^\sigma \leq O(r^{-4})$. Using the analysis in
\cite{Joyce2000}*{Section~8.3}, we can find $f$ on ${\bf R}^4$ with $f =
O(r^{-2})$ and $\Delta_{{\bf R}^4} f= \left(\Delta_{{\bf R}^4}
|a|^\sigma\right)^+$. Now, $g=|a|^\sigma-f$ is a decaying
subharmonic function on ${\bf R}^4$, so it decays at least like $r^{-2}$.
This follows from a maximum principle argument using a large
multiple of the Green's function as a barrier. Since $\sigma=2/3$,
it follows that $|a|$ decays like $r^{-3}$. This proves the first
part. For the second part, note that ${\rm d}_A^*{\rm d}_A\xi=0$. By part
one ${\rm d}_A \xi=O(r^{-3})$, thus integration by parts yields
${\rm d}_A\xi=0$, hence $\xi=0$. Similarly, one shows that $\omega=0$.
\end{proof}
The dimension of $M(E,\theta)$ can be computed using the following
index formula which can be proved using the Atiyah-Patodi-Singer index
theorem and the Atiyah-Bott-Lefschetz formula to compute the
contribution from infinity.
\begin{theorem}[Nakajima~\cite{Nakajima1990}*{Theorem~2.7}]\label{thm:index}
Let $\Gamma$ be a non-trivial finite subgroup of ${\rm SU}(2)$ and let
$\theta$ be a flat connection on a $G$--bundle over $S^3/\Gamma.$ Let
$X$ be an ALE space asymptotic to ${\bf C}^2/\Gamma$, let $E$ be a
$G$--bundle over $X$ asymptotic at infinity to the bundle supporting
$\theta$ and let $A$ be a finite energy ASD instanton on $E$
asymptotic to $\theta$. Then the dimension of the $L^2$--kernel of
$\delta_A$ is given by
\begin{align*}
\dim\ker\delta_A=-2\int_X p_1(\mathfrak g_E)
+\frac{2}{|\Gamma|}\sum_{g\in\Gamma\setminus\{e\}}
\frac{\chi_\mathfrak g(g)-\dim\mathfrak g}{2-\mathop{\mathrm{tr}}\nolimits g}.
\end{align*}
Here $p_1(\mathfrak g_E)$ is the Chern-Weil representative of the first
Pontryagin class of $E$ and $\chi_\mathfrak g$ is the character of the
representation $\rho\mskip0.5mu\colon\thinspace\Gamma \to G$ corresponding to $\theta$.
\end{theorem}
The space $M(E,\theta)$ is usually non-compact and often incomplete.
This is related to instantons wandering off to infinity and bubbling
phenomena. For details we refer the reader to \cite{Nakajima1990}.
Let $\hat X$ be a conformal compactification of $X$. (The point at
infinity will be an orbifold singularity, but this causes no trouble).
Since $A\in M(E,\theta)$ has finite energy it follows from Uhlenbeck's
removable singularities theorem \cite{Uhlenbeck1982}*{Theorem~4.1}
that $A$ extends to $\hat X$. Now radial parallel transport yields a
frame at infinity for $A$ in which we can write
\begin{align*}
A=\theta+a \quad\text{with}\quad \nabla^k a=O(r^{-3-k}).
\end{align*}
This will turn out to be crucial later on. The reader may find it
useful to think of $M(E,\theta)$ as a moduli space of ASD instantons
on $\hat X$ framed at the point at infinity (although equipped with a
non-standard metric).
There is a very rich existence theory for ASD instantons on ALE
spaces. Gocho--Nakajima \cite{Gocho1992} observed that for each
representation $\rho\mskip0.5mu\colon\thinspace \Gamma \to {\rm U}(n)$ there is a bundle $\mathcal R_\rho$
over $X$ together with an ASD instanton $A_\rho$ asymptotic to the
flat connection determined by $\rho$, and if $\sigma$ is a further
representation of $\Gamma$, then $A_{\rho\oplus\sigma} = A_\rho \oplus
A_\sigma$. Kronheimer--Nakajima \cite{Kronheimer1990} took this as
the starting point for an ADHM construction of ASD instantons on ALE
spaces. One important consequence of their work is the following
rigidity result.
\begin{lemma}[Kronheimer--Nakajima~\cite{Kronheimer1990}*{Lemma
7.1}]\label{lem:Rreg}
Each $A_\rho$ is infinitesimally rigid, i.e., the $L^2$--kernel
of $\delta_{A_\rho}$ is trivial.
\end{lemma}
By combining this result applied to the regular representation with
the index formula Kronheimer--Nakajima derive a geometric version of
the McKay correspondence \cite{Kronheimer1990}*{Appendix A}. Let
$\Delta(\Gamma)$ denote the Dynkin diagram associated to $\Gamma$ in
the ADE classification of the finite subgroups of ${\rm SU}(2)$. Each
vertex of $\Delta(\Gamma)$ corresponds to a non-trivial irreducible
representation. We label these by $\rho_1,\ldots,\rho_k$ and denote
the associated bundles by $\mathcal R_j$ and the associated ASD instantons by
$A_j$.
\begin{theorem}[Kronheimer--Nakajima]\label{thm:gmk}
The harmonic $2$--forms $c_1(\mathcal R_j)=\frac{i}{2\pi}\mathop{\mathrm{tr}}\nolimits{F_{A_j}}$ form
a basis of $L^2\mathcal H^2(X)\cong{}H^2(X,{\bf R})$ and satisfy
\begin{align*}
\int_X c_1(\mathcal R_i)\wedge c_1(\mathcal R_j)= -(C^{-1})_{ij}
\end{align*}
where $C$ is the Cartan matrix associated with $\Delta(\Gamma)$.
Moreover, there is an isometry $\kappa\in\mathrm{Aut}(H_2(X,{\bf Z}),\cdot)$ such
that $\{c_1(\mathcal R_j)\}$ is dual to $\{\kappa[\Sigma_j]\}$, where
$\Sigma_j$ are the irreducible components of the exceptional divisor $E$ of
$\wtilde{{\bf C}^2/\Gamma}$. If $X$ is isomorphic to
$\wtilde{{\bf C}^2/\Gamma}$ as a complex manifold, then $\kappa=\mathrm{id}$.
\end{theorem}
This results is very useful for computing the index of $\delta_A$
where $A$ is constructed out of ASD instantons of the form $A_\rho$
(by taking tensor products, direct sums, etc.).
Here is a simple example that we will be
used later on.
\begin{prop}\label{prop:rigid}
Let $X$ be an ALE space asymptotic to ${\bf C}^2/{\bf Z}_k$. Denote by
$\mathcal R_j=\mathcal R_{\rho_j}$ and $A_j=A_{\rho_j}$ the line bundle and
ASD instanton associated with $\rho_j\mskip0.5mu\colon\thinspace {\bf Z}_k \to U(1)$ defined by
$\rho_j(\ell)=\exp\(\frac{2\pi i}{k} j\ell\)$. For $n,m\in{\bf Z}_k$ let
$A_{n,m}$ be the induced ASD instanton on the ${\rm SO}(3)$--bundle
$E_{n,m}={\bf R}\oplus(\mathcal R_n^*\otimes \mathcal R_{n+m})$. Then $A_{n,m}$ is
infinitesimally rigid, asymptotic at infinity to the flat connection
over $S^3/{\bf Z}_k$ induced by $\rho_j$,
\begin{align*}
\frac{1}{8\pi^2}\int |F_{A_{n,m}}|^2=\frac{(k-m)m}{k}
\end{align*}
and
\begin{align*}
w_2(E_{n,m})=c_1(\mathcal R_{n+m})-c_1(\mathcal R_{n}) \in H^2(X,{\bf Z}_2).
\end{align*}
\end{prop}
\begin{proof}
To see that $A_{n,m}$ is infinitesimally rigid apply
Lemma~\ref{lem:Rreg} to $A_{n} \oplus A_{n+m}$. The statement about the second
Stiefel-Whitney class is obvious.
To compute the energy of $A_{n,m}$, it is enough to note that the
first term in the index formula is precisely twice the energy and
the second term is given by $\(-\frac2k\)$--times
\begin{align*}
-\sum_{g\neq e} \frac{\chi_\mathfrak g(g)-\dim\mathfrak g}{2-\mathop{\mathrm{tr}}\nolimits
g}=\sum_{j=1}^{k-1} \frac{1- \cos(2\pi mj/k)}{1-\cos(2\pi
j/k)}=(k-m)m.
\end{align*}
\end{proof}
\section{Approximate ${\rm G}_2$--instantons}
\label{sec:approx}
Let $Y_0$ be an admissible ${\rm G}_2$--orbifold, let $\{(X_j,\rho_j)\}$
be resolution data for $Y_0$. Let $\kappa$ be as in
Theorem~\ref{thm:joyce} and for $t\in(0,\kappa)$ denote by
$(Y_t,\phi_t)$ the ${\rm G}_2$ manifold obtained from Joyce's generalised
Kummer construction.
Let $\theta$ be a flat connection on a $G$--bundle $E_0$ over $Y_0$.
Then the monodromy of $\theta$ around $S_j$ induces a representation
$\mu_j\mskip0.5mu\colon\thinspace \pi_1(T_j)\cong({\bf Z}^3 \times G_j) \rtimes H_j \to G$ of the
orbifold fundamental group of $T_j$. If all $\mu_j|_{G_j}$ are
trivial there is a straightforward way to lift $E_0$ and $\theta$ up
to $Y_t$. In general, more input is required. A collection
$\((E_0,\theta),\{(x_j,f_j)\},\{(E_j,A_j,\tilde\rho_j,m_j)\}\)$
consisting of $E_0$ and $\theta$ as above as well for each $j$ the
choice of
\begin{itemize}
\item a point $x_j \in S_j$ together with a framing $f_j\mskip0.5mu\colon\thinspace
(E_0)_{x_j} \to G$ of $E_0$ at $x_j$,
\item a $G$-bundle $E_j$ over $X_j$ together with a framed ASD
instanton $A_j$ asymptotic at infinity to the flat connection on
$S^3/G_j$ induced by the representation $\mu_j|_{G_j}$,
\item a lift $\tilde\rho_j$ of the action $\rho_j$ of $H_j$ on $X_j$
to $E_j$ and
\item a homomorphism $m_j\mskip0.5mu\colon\thinspace {\bf Z}^3 \to \mathscr G(E_j)$
\end{itemize}
is called \emph{gluing data} compatible with the resolution data
$\{(X_j,\rho_j)\}$ if the following compatibility conditions are
satisfied:
\begin{itemize}
\item The action $\tilde\rho_j$ of $H_j$ on $E_j$ preserves $A_j$ and the
induced action on the fibre of $E_j$ at infinity is given by
$\mu_j|_{H_j}$.
\item The action of ${\bf Z}^3$ on $E_j$ given by $m_j$ preserves $A_j$ and
the induced action on the fibre of $E_j$ at infinity is given by
$\mu_j|_{{\bf Z}^3}$.
\item For all $h\in{}H_j$ and $g\in{\bf Z}^3$ we have
$\tilde\rho_j(h)m_j(g)\tilde\rho_j(h)^{-1}=m_j(hgh^{-1})$.
\end{itemize}
We should point out here that it is by far not always possible to
extend a choice of $(E_0,\theta)$ and $\{(E_j,A_j)\}$ to compatible
gluing data. This will become clear from the examples in
Section~\ref{sec:ex}. With this notation in place we can prove the
first part of Theorem~\ref{thm:a}.
\begin{prop}\label{prop:gdc}
Let $\((E_0,\theta),\{(x_j,f_j)\},\{(E_j,A_j,\tilde\rho_j,m_j)\}\)$
be gluing data compatible with $\{(X_j,\rho_j)\}$. Then for
$t\in(0,\kappa)$ one can construct a $G$--bundle $E_t$ on $Y_t$
together with a connection $A_t$ satisfying
\begin{align*}
p_1(E_t)=-\sum_j k_j\,\PD[S_j]\quad\text{with}\quad
k_j=\frac{1}{8\pi^2}\int_{X_j} |F_{A_j}|^2
\end{align*}
and
\begin{align*}
\<w_2(E_t),\Sigma\>=\<w_2(E_j),\Sigma\>
\end{align*}
for each $\Sigma\in{}H_2(X_j)^{H_j}\subset{}H_2(Y_t)$. Here
$[S_j]\in{}H_3(Y_t)$ is the cohomology class induced by the
component $S_j$ of the singular set $S$.
\end{prop}
\begin{proof}
The choices of $\tilde\rho_j$ and $m_j$ define a lift of the action
of ${\bf Z}^3\rtimes H_j$ on ${\bf R}^3\times X_j$ to the pullback of $E_j$ to
${\bf R}^3\times X_j$. Passing to the quotient yields a $G$--bundle over
$(T^3\times X_j)/H_j$ which we still denoted by $E_j$. It follows
from the compatibility conditions that the pullback of $A_j$ to
${\bf R}^3\times X_j$ passes to this quotient and defines a connection on
$E_j$ which we still denote by $A_j$.
Fix $t\in(0,\kappa)$. Let $R_{j,t}=R_t\cap\tilde T_{j,t}$ with
$R_t$ and $\tilde T_{j,t}$ defined as in Section \ref{sec:kummer}.
By the compatibility conditions the monodromy of $A_j$ along $S_j$
on the fibre at infinity matches up with the monodromy of $\theta$
along $E_0|_{S_j}$. Thus, by parallel transport the identification
of $(E_0)_{x_j}$ with the fibre at infinity of $E_j$ extends to an
identification of $E_0|_{R_{j,t}}$ with $E_j|_{R_{j,t}}$. Patching
$E_0$ and the $E_j$ via this identification yields the bundle $E_t$.
Under the identification of $E_0|_{R_{j,t}}$ with $E_j|_{R_{j,t}}$,
we can write
\begin{align*}
A_j=\theta+a_j \quad\text{with}\quad
\nabla^ka_j=t^{2+k}O(d_t^{-3-k})
\end{align*}
where $d_t\mskip0.5mu\colon\thinspace Y_t\to[0,\zeta]$ is the function measuring the distance
from the exceptional set introduced in Section~\ref{sec:kummer}.
Fix a smooth non-increasing function $\chi:[0,\zeta]\to[0,1]$ such
that $\chi(s)=1$ for $s\leq\zeta/4$ and $\chi(s)=0$ for
$s\geq\zeta/2$. Set $\chi_t:=\chi\circ d_t$. After cutting off
$A_j$ to $\theta+\chi_t\cdot a_j$ it can be matched with $\theta$
and we obtain the connection $A_t$ on the bundle $E_t$.
Using Chern-Weil theory one can compute $p_1(E_t)$. The statement
about $w_2(E_t)$ follows from the naturality of Stiefel-Whitney
classes.
\end{proof}
If $t$ is small, then the connection $A_t$ is a close to being a
${\rm G}_2$--instanton. In order to make this precise we introduce
weighted Hölder norms. It will become more transparent over the
course of the next two sections that these are well adapted to the
problem at hand. We define weight functions by
\begin{align*}
w_t(x) := t + d_t(x) \quad
w_t(x,y) := \min\{w_t(x),w_t(y)\}.
\end{align*}
For a Hölder exponent $\alpha\in(0,1)$ and a weight parameter
$\beta\in{\bf R}$ we define
\begin{align*}
[f]_{C^{0,\alpha}_{\beta,t}}
&:= \sup_{d(x,y) \leq w_t(x,y)}
w_t(x,y)^{\alpha-\beta} \frac{|f(x)-f(y)|}{d(x,y)^\alpha} \\
\|f\|_{L^{\infty}_{\beta,t}}
&:=\|w_t^{-\beta}f\|_{L^\infty} \\
\|f\|_{C^{k,\alpha}_{\beta,t}}
&:= \sum_{j=0}^k \|\nabla^j f\|_{L^{\infty}_{\beta-j,t}}
+ [\nabla^j f]_{C^{0,\alpha}_{\beta-j,t}}.
\end{align*}
Here $f$ is a section of a vector bundle on $Y_t$ equipped with a inner product
and a compatible connection. Note that, if
$\beta=\beta_1+\beta_2$, then
\begin{align*}
\|f\cdot{}g\|_{C^{k,\alpha}_{\beta,t}} \leq
\|f\|_{C^{k,\alpha}_{\beta_1,t}}\|g\|_{C^{k,\alpha}_{\beta_2,t}}.
\end{align*}
Also for $\beta>\gamma$ we have
\begin{align*}
\|f\|_{C^{k,\alpha}_{\beta,t}} \leq
t^{\gamma-\beta}\|f\|_{C^{k,\alpha}_{\gamma,t}}.
\end{align*}
\begin{prop}\label{prop:ag}
Let $A_t$ be as Proposition \ref{prop:gdc} and denote by
$\psi_t=\Theta(\phi_t)$ the coassociative calibration on
corresponding to $\phi_t$. Then for $\alpha\in(0,1)$ and
$\beta\geq-4$, there is a constant $c=c(\alpha,\beta)>0$ such that
for all $t\in(0,\kappa)$ the following estimate holds
\begin{align*}
\|F_{A_t}\wedge\psi_t\|_{C^{0,\alpha}_{\beta,t}}\leq
c(t^2+t^{-\beta-\alpha-\frac12}).
\end{align*}
\end{prop}
\begin{proof}
On $Y_t \setminus \tilde T_t$ the connection $A_t$ is flat. Thus we
can focus our attention to $\tilde T_{j,t}$.
By the definition of $A_t$ we have
\begin{align*}
F_{A_t}=(1-\chi_t)F_{A_j}
+{\rm d}\chi_t\wedge a_j
+\frac{\chi_t^2-\chi_t}2[a_j\wedge a_j].
\end{align*}
The last two terms in this expression are supported in $R_t$ and of
order $t^2$ in $C^{0,\alpha}$. Since the
coassociative calibration $\tilde\psi_t=\Theta(\tilde\phi_t)$ on the
model $\tilde{T}_{j,t}$ is given by
\begin{align*}
\tilde\psi_t=\frac12\omega_1\wedge\omega_1
+t^2\delta_2\wedge\delta_3\wedge\tilde\omega_1
+t^2\delta_3\wedge\delta_1\wedge\tilde\omega_2
-t^2\delta_1\wedge\delta_2\wedge\tilde\omega_3
\end{align*}
and $F_{A_j}\wedge\tilde \psi_t=0$, we are left with estimating the
size of $F_{A_j}\wedge(\psi_t-\tilde\psi_t)$. But by Theorem~\ref{thm:joyce}
\begin{align*}
\|\psi_t-\tilde\psi_t\|_{C^{0,\alpha}_{0,t}}\leq{}ct^{\frac32-\alpha}.
\end{align*}
Since $\|F_{A_j}\|_{C^{0,\alpha}_{-4,t}} \leq ct^{2}$, this
immediately implies the desired estimate.
\end{proof}
\section{A model operator on ${\bf R}^3\times \text{ALE}$}
\label{sec:model}
In order proof Theorem~\ref{thm:a} we need to find
$a_t\in\Omega^1(Y_t,\mathfrak g_{E_t})$ such that
\begin{align*}
F_{A_t+a_t}\wedge\psi_t=0,
\end{align*}
where $\psi_t=\Theta(\phi_t)$ is the coassociative calibration on
$Y_t$ provided that $t>0$ is small. As we explained in Section
\ref{sec:g2i}, this is equivalent to solving
\begin{align*}
*_t\left(F_{A_t+a_t}\wedge\psi_t\right) + {\rm d}_{A_t+a_t}\xi_t = 0
\end{align*}
for $\xi_t\in\Omega^0(Y_t,\mathfrak g_{E_t})$ and
$a_t\in\Omega^1(Y_t,\mathfrak g_{E_t})$. Here $*_t$ denotes the Hodge star
associated with $\phi_t$. There is no loss in additionally requiring
the gauge fixing condition ${\rm d}_{A_t}^*a_t = 0.$ The equation to be
solved can then be written as
\begin{align*}
L_{t}(\xi_t,a_t) + Q_t(\xi_t,a_t)
+ *_t\left(F_{A_t}\wedge\psi_t\right) = 0
\end{align*}
where $L_t=L_{A_t}$ is the linearised operator introduced in Section
\ref{sec:g2i} and the non-linear operator $Q_t$ is given by
\begin{align*}
Q_t(\xi,a) = \frac12 *_t\left([a\wedge
a]\wedge\psi_t\right) + [a,\xi].
\end{align*}
Once the linearisation $L_{A_t}$ is sufficiently well understood,
solving this equation is rather easy.
Away from the exceptional set $L_t$ is essentially equivalent to the
deformation operator $L_\theta$ associated with the flat connection
$\theta$. To gain better understanding of $L_t$ near the exceptional
set we introduce the following model. Let
$(X,\omega_1,\omega_2,\omega_3)$ be an ALE space and let $\delta_1,
\delta_2, \delta_3$ be a triple of constant $1$--forms on standard
${\bf R}^3$. Then the coassociative calibration $\psi$ corresponding to
the induced ${\rm G}_2$--structure on ${\bf R}^3 \times X$ is given by
\begin{align*}
\psi=\frac12 \omega_1\wedge\omega_1
+\delta_1\wedge\delta_2\wedge\omega_3
+\delta_2\wedge\delta_3\wedge\omega_1
-\delta_3\wedge\delta_1\wedge\omega_2.
\end{align*}
Let $E$ be a $G$--bundle over $X$ together with an ASD instanton $A$.
Without changing the notation we pullback $E$ and $A$ to ${\bf R}^3\times
X$. As in Section~\ref{sec:g2i} we denote by
$L_A\mskip0.5mu\colon\thinspace\Omega^0({\bf R}^3\times X,\mathfrak g_E)\oplus\Omega^1({\bf R}^3\times
X,\mathfrak g_E) \to \Omega^0({\bf R}^3\times X,\mathfrak g_E)\oplus\Omega^1({\bf R}^3\times
X,\mathfrak g_E)$ the deformation operator associated with $A$ defined by
\begin{align*}
L_A=\begin{pmatrix}
0 & {\rm d}_A^* \\
{\rm d}_A & *(\psi\wedge{\rm d}_{A})
\end{pmatrix}.
\end{align*}
Separating the parts of $L_A$ which differentiate in the direction of
${\bf R}^3$ from those that differentiate in the direction of $X$ yields
the following.
\begin{prop}
We can write $L_A=F+D_A$ where $F$ and $D_A$ are commuting formally
self-adjoint operators and $F^2=\Delta_{{\bf R}^3}$. In particular,
\begin{align*}
L_A^*L_A=\Delta_{{\bf R}^3}+D_A^*D_A.
\end{align*}
Under the identification $T_x^*{\bf R}^3\to \Lambda^+T_y^*X$ defined
by $\delta_1\mapsto\omega_1$, $\delta_2\mapsto\omega_2$,
$\delta_3\mapsto-\omega_3$ the operator $D_A$ takes the form
\begin{align*}
D_A=\begin{pmatrix}
0 & \delta_A \\
\delta_A^* & 0
\end{pmatrix}.
\end{align*}
\end{prop}
To understand the properties of $L_A$ we work with weighted
Hölder norms. We define weight functions by
\begin{align*}
w(x) := 1 + |\pi(x)|, \quad
w(x,y) := \min\{w(x),w(y)\}.
\end{align*}
Here $\pi\mskip0.5mu\colon\thinspace X\to{\bf C}^2/G$ denotes the resolution map associated with the ALE
space. For a Hölder exponent $\alpha\in(0,1)$ and a weight parameter
$\beta\in{\bf R}$ we define
\begin{align*}
[f]_{C^{0,\alpha}_{\beta}}
&:= \sup_{d(x,y) \leq w(x,y)}
w(x,y)^{\alpha-\beta} \frac{|f(x)-f(y)|}{d(x,y)^\alpha} \\
\|f\|_{L^{\infty}_{\beta}}
&:=\|w^{-\beta}f\|_{L^\infty} \\
\|f\|_{C^{k,\alpha}_{\beta}}
&:= \sum_{j=0}^k \|\nabla^j f\|_{L^{\infty}_{\beta-j}}
+ [\nabla^j f]_{C^{0,\alpha}_{\beta-j}}.
\end{align*}
Here $f$ is a section of a vector bundle on ${\bf R}^3\times{}X$ equipped
with a inner product and a compatible connection.
These norms are related to the ones introduced in
Section~\ref{sec:approx} as follows. In the situation of
Section~\ref{sec:approx} define $p_{j,t}:{\bf R}^3\times{}X_j\to\tilde{T}_{j,t}$ by
\begin{align*}
p_t(x,y)=[(tx,y)].
\end{align*}
This map pulls back the metric on $\tilde T_{j,t}$ associated with
$\tilde\phi_t$, that is $g_{\tilde\phi_t}=g_{T^3}\oplus t^2g_{X_j}$,
to $t^2(g_{{\bf R}^3}\oplus g_{X_j})$. Since the metric on $Y_t$ induced
by $\phi_t$ is uniformly equivalent to the one induced by
$\tilde\phi_t$ on $\tilde{T}_t$, this yields the following relation
\begin{align*}
\|p_{j,t}^*\alpha\|_{C^{k,\alpha}_{\beta}(p_t^{-1}(\tilde{T}_{j,t}))}
\sim t^{d+\beta}\|\alpha\|_{C^{k,\alpha}_{\beta,t}(\tilde{T}_{j,t})}
\end{align*}
for $d$--forms $\alpha$. Here $\sim$ means equivalent with a constant
independent of $t$. The relation between $L_{A_j}$ and the
deformation operator $L_t$ associated with $A_t$ is as follows. If
$(\eta,b)=L_t(\xi,a)$ for $\xi\in\Omega^0(Y_t,\mathfrak g_{E_t})$ and
$a\in\Omega^1(Y_t,\mathfrak g_{E_t})$, then
\begin{align*}
\|t(p_{j,t}^*\eta,t^{-1}p_{j,t}^*b)-L_{A_j}(p_{j,t}^*\xi,t^{-1}p_{j,t}^*a)\|_{C^{0,\alpha}_{\beta-1}}
\leq ct^{2}\left(\|p_{j,t}^*\xi\|_{C^{1,\alpha}_{\beta}} +
\|t^{-1}p_{j,t}^*a\|_{C^{1,\alpha}_{\beta}}\right)
\end{align*}
for some constant $c=c(\alpha,\beta)>0$ independent of $t$.
In this sense $L_{A_j}$ is a model for $L_t$ near the exceptional set.
\begin{prop}\label{prop:lk}
If $\beta\in(-3,0)$, then the kernel of $L_A\mskip0.5mu\colon\thinspace
C^{1,\alpha}_{\beta}\to C^{0,\alpha}_{\beta-1}$ consists of elements
of the $L^2$--kernel of $\delta_A$.
\end{prop}
This follows immediately from Proposition~\ref{prop:decay} and the
following lemma which we will prove in Appendix~\ref{app:liouville}.
\begin{lemma}\label{lem:liouville}
Let $E$ be a vector bundle of bounded geometry over a Riemannian
manifold $X$ of bounded geometry and suppose that $D\mskip0.5mu\colon\thinspace
C^\infty(X,E)\to{}C^\infty(X,E)$ is a bounded uniformly elliptic
operator of second order which is positive, i.e., $\<Da,a\>\geq 0$
for all $a \in W^{2,2}(X,E)$, and formally self-adjoint. If $a
\in C^\infty({\bf R}^n\times X,E)$ satisfies
\begin{align*}
(\Delta_{{\bf R}^n}+D)a=0
\end{align*}
and there are $s\in{\bf R}$ and $p\in(1,\infty)$ such that
$\|a(x,\cdot)\|_{W^{s,p}}$ is bounded independent of $x\in{\bf R}^n$,
then $a$ is constant in the ${\bf R}^n$--direction, that is
$a(x,y)=a(y)$.
\end{lemma}
If $A$ is not infinitesimally rigid, then although the kernel of $L_A$
is finite dimensional, there is an infinite dimensional ``effective
kernel'', that is to say, for each $\epsilon>0$ there is an infinite
dimensional space of elements of the form $\chi a$ where $\chi$ is a
smooth function on ${\bf R}^3$ and $a\in\ker\delta_A$ such that $\|L_A\chi
a\|_{C^{0,\alpha}_{\beta-1}} \leq \epsilon \|\chi
a\|_{C^{1,\alpha}_\beta}$. This problem can be remedied to some
extend by working orthogonal to elements of this form. But it clearly
shows, that one cannot expect be able to solve the non-linear equation
introduced above in general. On the other hand, if $A$ is
infinitesimally rigid, then it follows from Proposition~\ref{prop:lk}
that $L_A$ is injective. In fact, under this assumption one can show
that $L_A$ is invertible using the following Schauder estimate and the
observation that $L_A$ has no ``kernel at infinity'', which is
essentially what will be established in Case 3 of the proof of
Proposition~\ref{prop:2ndterm}.
\begin{prop}\label{prop:mse}
For $\alpha\in(0,1)$ and $\beta\in{\bf R}$ there is a constant
$c=c(\alpha,\beta)>0$, such that the following estimate holds
\begin{align*}
\|\xi\|_{C^{1,\alpha}_{\beta}} + \|a\|_{C^{1,\alpha}_{\beta}}
\leq c(\|L_A(\xi,a)\|_{C^{0,\alpha}_{\beta-1}}
+ \|\xi\|_{L^{\infty}_{\beta}} + \|a\|_{L^{\infty}_{\beta}}).
\end{align*}
\end{prop}
\begin{proof}
The desired estimate is local in the sense that is enough to prove
estimates of the form
\begin{align*}
\|\xi\|_{C^{1,\alpha}_{\beta}(U_i)} + \|a\|_{C^{1,\alpha}_{\beta}(U_i)}
\leq c(\|L_A (\xi,a)\|_{C^{0,\alpha}_{\beta-1}}
+ \|\xi\|_{L^{\infty}_{\beta}} + \|a\|_{L^{\infty}_{\beta}})
\end{align*}
with $c>0$ independent of $i$, where $\{U_i\}$ is an open cover of
${\bf R}^3\times X$.
Fix $R>0$ suitably large and set $U_0=\{(x,y)\in{\bf R}^3\times{}X :
|\pi(x)|\leq R\}.$ Then there clearly is a constant $c>0$ such that
the above estimate holds for $U_i=U_0$. Now pick a sequence
$(x_i,y_i)\in{\bf R}^3\times{}X$ such that $r_i:=|\pi(y_i)|\geq R$ and the
balls $U_i:=B_{r_i/8}(x_i,y_i)$ cover the complement of $U_0$. Now on
$U_i$, we have a Schauder estimate of the form
\begin{gather*}
\|\underline{a}\|_{L^\infty(U_i)} +
r_i^{\alpha}[\underline{a}]_{C^{0,\alpha}(U_i)} + r_i\|\nabla_A
\underline{a}\|_{L^\infty(U_i)} +
r_i^{1+\alpha}[\nabla_A \underline{a}]_{C^{0,\alpha}(U_i)} \\
\leq c\left(r_i\|L_A\underline a\|_{L^\infty(V_i)}
+ r_i^{1+\alpha}[L_A\underline{a}]_{C^{0,\alpha}(V_i)}
+ \|\underline{a}\|_{L^\infty(V_i)}\right)
\end{gather*}
where $V_i=B_{r_i/4}(x_i,y_i)$ and $\underline{a}=(\xi,a)$. The constant $c>0$
depends continuously on the coefficients of $L_A$ over $V_i$ and it is
thus easy to see that one can find a constant $c>0$ such that the
above estimate holds for all $i$. Since on $V_i$ we have $\frac12 r_i
\leq w \leq 2r_i$ multiplying the above Schauder estimate by
$r_i^{-\beta}$ yields the desired local estimate.
\end{proof}
\section{Deforming to genuine ${\rm G}_2$--instantons}
\label{sec:deform}
We continue with the assumptions of Section~\ref{sec:approx} and we
suppose that the connection $A_t$ on $E_t$ over $Y_t$ was constructed
using Proposition~\ref{prop:gdc} from a choice of compatible gluing
data $\((E_0,\theta),\{(x_j,f_j)\},\{(E_j,A_j,\tilde\rho_j,m_j)\}\)$.
In this section we will prove the following result. This finishes the
proof of Theorem \ref{thm:a}.
\begin{prop}\label{prop:a}
Assume that $\theta$ is regular as a ${\rm G}_2$--instanton and that
each $A_j$ is infinitesimally rigid. Then there is a constant $c>0$
such that for $t$ sufficiently small there is a small perturbation
$a_t\in\Omega^1(Y_t,\mathfrak g_{E_t})$ satisfying
\begin{align*}
F_{A_t+a_t}\wedge\psi_t=0
\end{align*}
and $\|a_t\|_{C^{1,1/2}_{-1,t}}\leq ct$. Here
$\psi_t=\Theta(\phi_t)$ denotes the coassociative calibration on
$Y_t$. Moreover, $A_t+a_t$ is a regular ${\rm G}_2$--instanton.
\end{prop}
As discussed in Section~\ref{sec:model} it is crucial to understand
the properties of the linearisation $L_{t}$. The key to proving
Proposition \ref{prop:a} is the following result.
\begin{prop}\label{prop:key}
Given $\alpha\in(0,1)$ and $\beta\in(-3,0)$, there is a constant
$c=c(\alpha,\beta)>0$ such that for $t$ sufficiently small the
following estimate holds
\begin{align*}
\|\xi\|_{C^{1,\alpha}_{\beta,t}} + \|a\|_{C^{1,\alpha}_{\beta,t}}
\leq c \|L_t(\xi,a)\|_{C^{0,\alpha}_{\beta-1,t}}.
\end{align*}
\end{prop}
Before we move on to prove this, let us quickly show how it is used to
establish Proposition \ref{prop:a}. Recall the following elementary
observation.
\begin{lemma}\label{lem:CM}
Let $X$ be a Banach space and let $T \mskip0.5mu\colon\thinspace X\to X$ be a smooth map
with $T(0)=0$. Suppose there is a constant $c>0$ such that
\begin{align*}
\|Tx-Ty\|\leq c(\|x\|+\|y\|)\|x-y\|.
\end{align*}
Then if $y\in X$ satisfies $\|y\|\leq
\frac{1}{10c}$, there exists a unique $x\in X$ with $\|x\|\leq
\frac{1}{5c}$ solving
\begin{align*}
x+Tx=y.
\end{align*}
Moreover, the unique solution satisfies $\|x\|\leq 2\|y\|$.
\end{lemma}
\begin{proof}[Proof of Proposition \ref{prop:a}]
By Proposition \ref{prop:key} $L_t$ injective and has closed range.
Since $L_t$ is formally self-adjoint, it follows from elliptic
regularity that $L_t$ is also surjective and thus invertible. Denote
its inverse by $R_t$.
If we set $(\xi_t,a_t)=R_t(\eta_t,b_t)$, then the equation we need
to solve becomes
\begin{align*}
(\eta_t,b_t) + Q_t(R_t(\eta_t,b_t)) = -*_t\(F_{A_t}\wedge\psi_t\).
\end{align*}
It follows from Proposition \ref{prop:key} with $\beta=-1$ and
$\alpha=\frac12$ that
\begin{align*}
&\|Q_t(R_t(\eta_1,b_1))-Q_t(R_t(\eta_2,b_2))\|_{C^{0,1/2}_{-2,t}} \\
&\quad \leq c \left(\|(\eta_1,b_1)\|_{C^{0,1/2}_{-2,t}} +
\|(\eta_2,b_2)\|_{C^{0,1/2}_{-2,t}}\right)
\|(\eta_1,b_1)-(\eta_2,b_2)\|_{C^{0,1/2}_{-2,t}}
\end{align*}
with a constant $c>0$ independent of $t$. Since by Proposition
\ref{prop:ag}
\begin{align*}
\|F_{A_t}\wedge\psi_t\|_{C^{0,1/2}_{-2,t}}\leq ct
\end{align*}
Lemma \ref{lem:CM} provides us with a unique
solution $(\eta_t,b_t)$ to the above equation provided $t$ is
small enough. Then
$(\xi_t,a_t)=R_t(\eta_t,b_t)\in C^{1,1/2}_{-1,t}$ is the desired
solution of
\begin{align*}
*_t\left(F_{A_t+a_t}\wedge\psi_t\right) + {\rm d}_{A_t+a}\xi_t = 0
\end{align*}
and satisfies $\|a_t\|_{C^{1,1/2}_{-1,t}}\leq ct$.
This solution is smooth by elliptic regularity. To see that
$A_t+a_t$ is a regular ${\rm G}_2$--instanton, note that
$\|R_tL_{A_t+a_t}-\mathrm{id}\|\leq ct$ and thus $L_{A_t+a_t}$ is invertible
for $t$ sufficiently small.
\end{proof}
Before embarking on the proof of Proposition \ref{prop:key}, it will
be helpful to make a few observations. On $Y_t\setminus\tilde T_{t}$
the operators $L_t$ and $L_{\theta}$ agree. For fixed $\epsilon>0$,
the norms introduced in Section \ref{sec:approx} considered on $\{ x
\in Y_t : d_t(x)\geq \epsilon >0 \}$ are uniformly equivalent to the
corresponding unweighted Hölder norms. Since $A_j$ is asymptotic to
$\theta$, for any given $\epsilon>0$ the restriction of $L_t$ to
$\{x\in Y_t:d_t(x)>\epsilon\}$ becomes arbitrarily close to $L_\theta$
restricted to $\{x\in Y_0:d(x,S)>\epsilon\}$ as $t$ goes to zero.
These observations combined with remarks preceding Proposition
\ref{prop:lk} yield the following Schauder estimate.
\begin{prop}\label{prop:se}
For $\alpha\in(0,1)$ and $\beta\in{\bf R}$ there is a constant
$c=c(\alpha,\beta)>0$ such that the following estimate holds for
\begin{align*}
\|a\|_{C^{1,\alpha}_{\beta,t}} + \|\xi\|_{C^{1,\alpha}_{\beta,t}}
\leq c (\|L_t (a,\xi)\|_{C^{0,\alpha}_{\beta-1,t}} +
\|a\|_{L^{\infty}_{\beta,t}} + \|\xi\|_{L^{\infty}_{\beta,t}}).
\end{align*}
\end{prop}
This reduces the proof of Proposition \ref{prop:key} to the following
statement.
\begin{prop}\label{prop:2ndterm}
For $\alpha\in(0,1)$ and $\beta\in(-3,0)$ there is a constant
$c=c(\alpha,\beta)>0$ such that the following
estimate holds for $t$ sufficiently small
\begin{align*}
\|\xi\|_{L^\infty_{\beta,t}}+\|a\|_{L^\infty_{\beta,t}}
\leq c\|L_t(\xi,a)\|_{C^{0,\alpha}_{\beta-1,t}}.
\end{align*}
\end{prop}
\begin{proof}
Suppose not. Then there exists a sequence $(\xi_i,a_i)$ and a
null-sequence $t_i$ such that
\begin{align*}
\|a_i\|_{L^{\infty}_{\beta,t_i}}+\|\xi_i\|_{L^{\infty}_{\beta,t_i}}=1
\quad\text{and}\quad
\|L_{t_i}(a,\xi)\|_{C^{0,\alpha}_{\beta-1,t_i}} \leq \frac1i.
\end{align*}
Hence by Proposition \ref{prop:se}
\begin{align*}
\|\xi_i\|_{C^{1,\alpha}_{\beta,t_i}}+\|a\|_{C^{1,\alpha}_{\beta,t_i}}
\leq 2c.
\end{align*}
Pick $x_i \in Y_{t_i}$ such that
\begin{align*}
w_{t_i}(x_i)^{-\beta}(|\xi_i(x_i)|+|a_i(x_i)|)=1.
\end{align*}
Without loss of generality one of the following three cases occurs.
We will rule out all of them, thus proving the proposition.
\setcounter{case}{0}
\begin{case}
The sequence $x_i$ accumulates on the regular part of $Y_0$: $\lim
d_{t_i}(x_i)>0$.
\end{case}
Let $K$ be a compact subset of $Y_0\setminus{}S$. We can view $K$
as a subset of $Y_t$. As $t$ goes to zero, the metric on $K$
induced from the metric on $Y_t$ converges to the
metric on $Y_0$, similarly we can identify $E_0|_K$ with $E_t|_K$
and via this identification $A_t$ converges to $\theta$ on $K$.
When restricted to $K$, $\xi_i$ and $a_i$ are uniformly bounded in
$C^{1,\alpha}$. Thus, by Arzelà-Ascoli, we can assume (after
passing to a subsequence) that $(\xi_i,a_i)$ converges to a limit
$(\xi,a)$ in $C^{1,\alpha/2}$. Since $K$ was arbitrary this yields
$(\xi,a) \in \Omega^0(Y_0\setminus
S,\mathfrak g_{E_0})\oplus\Omega^1(Y_0\setminus S,\mathfrak g_{E_0})$ satisfying
\begin{align*}
|\xi|+|a|< c \cdot d(\cdot,S)^{\beta}
\quad\text{and}\quad
|\nabla_\theta \xi|+|\nabla_\theta a|< c \cdot d(\cdot,S)^{\beta-1}
\end{align*}
as well as
\begin{align*}
L_\theta(\xi,a)=0.
\end{align*}
Since $\beta>-3$, the first condition implies that $(\xi,a)$
satisfies $L_\theta(\xi,a)=0$ in the sense of distributions on all
of $Y_0$. But then elliptic regularity implies that $(\xi,a)$ is
smooth. Because $\theta$ is assumed to be regular $(\xi,a)$ must be
zero. But, we can also assume that $x_i$ converges to $x \in
Y_0\setminus S$, and thus $|\xi|(x) + |a|(x)=d(x,S)^\beta \neq 0$.
This is a contradiction.
\begin{case}
The sequence $x_i$ accumulates on one of the ALE spaces: $\lim
d_{t_i}(x_i)/t_i<\infty$.
\end{case}
We can assume that $x_i$ converges to $x\in S_j$ for some $j$.
There is no loss in assuming that each $x_i$ is in $\tilde
T_{j,t_i}$. Pulling $(\xi_i,a_i)$ and $x_i$ back to ${\bf R}^3\times X$
via $p_{j,t}$ as in the remarks preceding Proposition \ref{prop:lk}
yields $(\tilde \xi_i, \tilde a_i) \in
\Omega^0({\bf R}^3\times{}X_j,\mathfrak g_E)\oplus\Omega^1({\bf R}^3\times{}X_j,\mathfrak g_E)$
as well as $\tilde x_i\in {\bf R}^3\times X_j$ satisfying
\begin{gather*}
\|\tilde\xi_i\|_{C^{1,\alpha}_{\beta}}
+\|\tilde a_i\|_{C^{1,\alpha}_{\beta}}\leq 4c \\
(1+|\pi_j(\tilde{x}_i)|)^{-\beta}
\left(|\tilde\xi_i(\tilde{x}_i)|+|\tilde{a}(\tilde{x}_i)|\right)
\geq \frac12 \\
\|L_{A_j}(\tilde\xi_i,\tilde{a}_i)\|_{C^{0,\alpha}_{\beta-1}} \leq
2/i.
\end{gather*}
There is a slight abuse of notation here in that
$(\tilde\xi_i,\tilde a_i)$ are not really defined over all of
${\bf R}^3\times X_j$ but just over an exhausting sequence of subsets.
This does not cause any trouble.
Again we can assume that there is $(\tilde \xi, \tilde a) \in
\Omega^0({\bf R}^3\times{}X_j,\mathfrak g_E)\oplus\Omega^1({\bf R}^3\times{}X_j,\mathfrak g_E))$
such that $(\tilde\xi_i,\tilde a_i)$ converges to $(\tilde
\xi,\tilde a)$ in $C^{1,\alpha/2}$ on compact subsets of ${\bf R}^3\times
X_j$. It follows that $(\tilde\xi,\tilde a)\in
C^{1,\alpha/2}_{\beta}$ satisfies
\begin{align*}
L_{A_j}(\tilde\xi,\tilde a)=0.
\end{align*}
But then it must be zero by Proposition \ref{prop:lk}, since
$\beta\in(-3,0)$ and each $A_j$ is infinitesimally rigid. On the
other hand by translation we can always arrange that the
${\bf R}^3$--component of $\tilde x_i$ is zero and thus we can view
$\tilde x_i$ as a point in $X_j$. Then the condition $\lim
d_{t_i}(x_i)/t_i<\infty$ translates into $\lim |\pi_j(\tilde
x_i)|<\infty$. Therefore, we can assume without loss of generality
that $\tilde x_i$ converges to some point $\tilde x\in X_j$. But
then $|\tilde\xi(\tilde{x})|+|\tilde{a}(\tilde{x})| \geq
\frac12(1+|\pi_j(\tilde x)|)^\beta > 0,$ which contradicts
$(\tilde\xi,\tilde{a})=0$.
\begin{case}
The sequence $x_i$ accumulates on one of the necks: $\lim
d_{t_i}(x_i)=0$ and $\lim d_{t_i}(x_i)/t_i=\infty$.
\end{case}
As in Case 2, we rescale to obtain $(\tilde\xi_i,\tilde{a}_i)$ and
$\tilde x_i$. Again we can assume by translation that the
${\bf R}^3$--component of $\tilde x_i$ is zero. Since $\lim
d_{t_i}(x_i)/t_i=\infty$, we have $\lim |\pi_j(\tilde x_i)|=\infty$.
Now fix a sequence $R_i>0$ going to infinity such that
$R_i/|\pi_j(\tilde x_i)|$ goes to zero. We can think of the sets
${\bf R}^3\times({\bf C}^2\setminus B_{R_i}^4)/G_j$ as subsets of
${\bf R}^3\times{}X_j$. Restricting $(\tilde\xi,\tilde{a}_i)$ to this
set and rescaling by $1/|\pi_j(\tilde x_i)|$ yields (with a small
abuse of notation and without relabelling) a sequence
$(\tilde{\xi}_i,\tilde{a}_i) \in
\Omega^0({\bf R}^3\times{}({\bf C}^2\setminus\{0\})/G_j) \oplus
\Omega^1({\bf R}^3\times{}({\bf C}^2\setminus\{0\})/G_j)$ and
$\tilde{x}_i\in{\bf C}^2\setminus\{0\}$ satisfying
\begin{gather*}
\|\tilde{\xi}_i\|_{C^{1,\alpha}_{\beta,0}}
+ \|\tilde{a}_i\|_{C^{1,\alpha}_{\beta,0}}\leq 8c \\
|\tilde{x}_j|^{-\beta}(|\tilde\xi_i(\tilde{x}_i)| +
|\tilde{a}_i(\tilde{x}_i)|)\geq \frac14 \\
\|L(\tilde\xi_i,\tilde{a}_i)\|_{C^{0,\alpha}_{\beta-1,0}} \leq
4/i.
\end{gather*}
Here the norms $\|\cdot\|_{C^{k,\alpha}_{\beta,0}}$ are defined like
those in Section \ref{sec:model} except with the weight function
now defined by $w(x,y)=|y|$ for $(x,y)\in{\bf R}^3\times {\bf C}^2/G_j$. The
operator $L$ is defined by
\begin{align*}
L(\xi,a)=({\rm d}^*a, {\rm d} \xi+*(\psi\wedge{\rm d} a))
\end{align*}
where
$\psi=\frac12 \omega_1\wedge\omega_1
+\delta_2\wedge\delta_3\wedge\omega_1
+\delta_3\wedge\delta_1\wedge\omega_2
-\delta_1\wedge\delta_2\wedge\omega_3$
with
$\omega_i\in\Omega^2({\bf C}^2)$ as given in Section \ref{sec:kummer}.
Passing to a limit via Arzelà-Ascoli as before yields
$(\tilde\xi,\tilde{a})\in
\Omega^0({\bf R}^3\times{}({\bf C}^2\setminus\{0\})/G_j) \oplus
\Omega^1({\bf R}^3\times{}({\bf C}^2\setminus\{0\})/G_j)$
satisfying
\begin{align*}
|\tilde\xi|+|\tilde{a}|< c r^{\beta}, \quad
|\nabla\tilde\xi|+|\nabla \tilde{a}|< c r^{\beta-1},
\end{align*}
where $r\mskip0.5mu\colon\thinspace{\bf C}^2/G_j\to[0,\infty)$ is the radius function, and
\begin{align*}
L(\tilde\xi,\tilde{a})=0.
\end{align*}
Again, since $\beta>-3$, $L(\tilde\xi,\tilde{a})=0$ actually holds
in the sense of distributions on all of ${\bf R}^3\times{}{\bf C}^2/G_j$. But
$L^*L=\Delta_{{\bf R}^3}+\Delta_{{\bf C}^2}$, so Lemma \ref{lem:liouville}
shows that $(\tilde\xi,\tilde{a})$ is invariant in the
${\bf R}^3$--direction. Thus we can think of the components of
$(\tilde\xi,\tilde{a})$ as harmonic functions on ${\bf C}^2$. Since
$\beta<0$, they decay to zero at infinity and thus vanish
identically. But we know that $|\tilde{x}_i|=1$ and thus (without
loss of generality) $\tilde{x_i}$ converges to a point
$\tilde{x}\in{\bf C}^2/G_j$ with $|\tilde{x}|=1$. At this point
$|\tilde\xi|(\tilde x)+|\tilde{a}|(\tilde x)\geq \frac14$,
contradicting $(\tilde\xi,\tilde{a})=0$.
\end{proof}
\section{Examples with $G={\rm SO}(3)$}
\label{sec:ex}
We construct a few examples of ${\rm G}_2$--instantons on the
${\rm G}_2$-manifolds from \cite{Joyce2000}*{Section 12.3 and 12.4}.
Consider the flat ${\rm G}_2$--structure on $T^7$ given by
\begin{align*}
\phi_0={\rm d} x^{123}+{\rm d} x^{145}+{\rm d} x^{167}+{\rm d} x^{246}-{\rm d} x^{257}-{\rm d}
x^{347}-{\rm d} x^{356}.
\end{align*}
Here ${\rm d} x^{ijk}$ is a shorthand for ${\rm d} x^i\wedge {\rm d} x^j \wedge
{\rm d} x^k$ and $x_1,\ldots,x_7$ are standard coordinates on
$T^7={\bf R}^7/{\bf Z}^7$. The ${\rm G}_2$--structure $\phi_0$ is preserved by
$\alpha,\beta,\gamma\in\mathrm{Diff}(T^7)$ defined by
\begin{align*}
\alpha(x_1,\ldots,x_7)&=
\left(x_1,x_2,x_3,-x_4,-x_5,-x_6,-x_7\right) \\
\beta(x_1,\ldots,x_7)&=
\left(x_1,-x_2,-x_3,x_4,x_5,\tfrac12-x_6,-x_7\right) \\
\gamma(x_1,\ldots,x_7)&=
\left(-x_1,x_2,-x_3,x_4,-x_5,x_6,\tfrac12-x_7\right).
\end{align*}
It is easy to see that $\Gamma=\<\alpha,\beta,\gamma\>\cong{\bf Z}_2^3.$
To understand the singular set $S$ of $T^7/\Gamma$ note that the only
elements of $\Gamma$ having fixed points are $\alpha,\beta,\gamma$.
The fixed point set of each of these elements consists of $16$ copies
of $T^3$. The group $\<\beta,\gamma\>$ acts freely on the set of
$T^3$ fixed by $\alpha$ and $\<\alpha,\gamma\>$ acts freely on the set
of $T^3$ fixed by $\beta$ while $\alpha\beta\in\<\alpha,\beta\>$ acts
trivially on the set of $T^3$ fixed by $\gamma$. It follows that $S$
consists of $8$ copies of $T^3$ coming from the fixed points of
$\alpha$ and $\beta$ and $8$ copies of $T^3/{\bf Z}_2$. Near the copies of
$T^3$ the singular set is modelled on $T^3\times {\bf C}^2/{\bf Z}_2$ while
near the copies of $T^3/{\bf Z}_2$ it is modelled on $(T^3\times
{\bf C}^2/{\bf Z}_2)/{\bf Z}_2$ where the action of ${\bf Z}_2$ on $T^3\times {\bf C}^2/{\bf Z}_2$
is given by
\begin{align*}
(x_1,x_2,x_3,\pm(z_1,z_2))\mapsto(x_1,x_2,x_3+\tfrac12,\pm(z_1,-z_2)).
\end{align*}
The 8 copies of $T^3$ can be desingularised by any choice of 8 ALE
spaces asymptotic to ${\bf C}^2/{\bf Z}^2$. To desingularise the copies of
$T^3/{\bf Z}^2$ we need to chose ALE spaces which admit an isometric action
of ${\bf Z}^2$ asymptotic to the action ${\bf Z}^2$ on ${\bf C}^2/{\bf Z}^2$ given by
$\pm(z_1,z_2)\mapsto\pm(z_1,-z_2)$. Two possible choices are the
resolution of ${\bf C}^2/{\bf Z}^2$ or a smoothing of ${\bf C}^2/{\bf Z}^2$.
See \cite{Joyce2000}*{pp.~313--314} for details.
We construct our examples on desingularisations of quotients of
$T^7/\Gamma$. To this end we define
$\sigma_1,\sigma_2,\sigma_3\in\mathrm{Diff}(T^7)$ by
\begin{align*}
\sigma_1(x_1,\ldots,x_7)&=
\left(x_1,x_2,\tfrac12+x_3,\tfrac12+x_4,\tfrac12+x_5,x_6,x_7\right)\\
\sigma_2(x_1,\ldots,x_7)&=
\left(x_1,\tfrac12+x_2,x_3,\tfrac12+x_4,x_5,x_6,x_7\right)\\
\sigma_3(x_1,\ldots,x_7)&=
\left(\tfrac12+x_1,x_2,x_3,x_4,\tfrac12+x_5,\tfrac12+x_6,x_7\right).
\end{align*}
The elements $\sigma_j$ commute with all elements of $\Gamma$ and thus
act on $T^7/\Gamma$. Moreover, the action is free.
\begin{example}
Let $A=\<\sigma_2,\sigma_3\>$. By analysing how $A$ acts on the
singular set of $T^7/\Gamma$ one can see that the singular set of
$Y_0=T^7/\(\Gamma \times A\)$ consists of one copy of $T^3$, denoted
by $S_1$, and $6$ copies of $T^3/{\bf Z}_2$, denoted by $S_2, \ldots,
S_7$. $S_1$ has a neighbourhood modelled on $T^3\times {\bf C}^2/{\bf Z}_2$
while $S_2,\ldots,S_6$ have neighbourhoods modelled on
$(T^3\times{\bf C}^2/{\bf Z}^2)/{\bf Z}^2$ where ${\bf Z}_2$ acts freely on $T^3$ and by
$\pm(z_1,z_2)\mapsto\pm(z_1,-z_2)$ on ${\bf C}^2/{\bf Z}^2$. As before, $S_1$
can be desingularised by a choice of any ALE space asymptotic to
${\bf C}^2/{\bf Z}_2$. $S_2\ldots,S_6$ can be desingularised by the
resolution of ${\bf C}^2/{\bf Z}_2$ or a smoothing of ${\bf C}^2/{\bf Z}_2$.
To compute the orbifold fundamental group $\pi_1(Y_0)$, note that it
is isomorphic to the fundamental group $\pi_1(Y_0\setminus{}S)$ of
the regular part of $Y_0$. Denote by $p\mskip0.5mu\colon\thinspace R^7\to{}Y_0$ the canonical
projection. Then $p\mskip0.5mu\colon\thinspace p^{-1}(Y_0\setminus{}S)\to{}Y_0\setminus{}S$ is
a universal cover. Up to conjugation, we can therefore identify
$\pi_1(Y_0)$ with the group of deck transformations:
\begin{align*}
\pi_1(Y_0)=\<\alpha,\beta,\gamma,\sigma_2,\sigma_3,
\tau_1,\ldots,\tau_7\>\subset{\rm GL}(7).
\end{align*}
Here we think of $\alpha,\beta,\gamma,\sigma_2,\sigma_3$ as elements
of ${\rm GL}(7)$ defined by the formulae above and $\tau_i$ translates
the $i$--th coordinate of ${\bf R}^7$ by $1$. The group $\pi_1(Y_0)$ is
a non-split extension
\begin{align*}
0\to{\bf Z}^7\to\pi_1(Y_0)\to\Gamma\times{}A\to 0.
\end{align*}
To work out the orbifold fundamental $\pi_1(T_j)$ of $T_j$, again up
to conjugation, one simply has to understand the subgroup of deck
transformations preserving a fixed component of
$p^{-1}(T_j)\subset{}p^{-1}(Y_0\setminus S)$. In this way one can
compute
\begin{gather*}
\pi_1(T_1)=\<\alpha,\tau_1,\tau_2,\tau_3\>\\
\pi_1(T_2)=\<\beta,\sigma_3\alpha,\tau_1,\tau_4,\tau_5\>\quad
\pi_1(T_3)=\<\tau_3\beta,\sigma_3\alpha,\tau_1,\tau_4,\tau_5\>\\
\pi_1(T_4)=\<\gamma,\alpha\beta,\sigma_2,\tau_4,\tau_6\>\quad
\pi_1(T_5)=\<\tau_3\gamma,\tau_3\alpha\beta,\sigma_2,\tau_4,\tau_6\>\\
\pi_1(T_6)=\<\tau_5\gamma,\tau_5\alpha\beta,\sigma_2,\tau_4,\tau_6\>\quad
\pi_1(T_7)=\<\tau_3\tau_5\gamma,\tau_3\tau_5\alpha\beta,\sigma_2,\tau_4,\tau_6\>.
\end{gather*}
Here $\tau_2$ does not appear explicitly in $\pi_1(T_j)$, for
$j=4,\ldots,7$, because $\sigma_2^2=\tau_2\tau_4$.
Denote by $V=\<a,b,c \,|\, a^2=b^2=c^2=1, ab=c\>={\bf Z}_2^2$ the Klein
four-group. $V$ can be thought of as a subgroup of ${\rm SO}(3)$. We
define $\rho\mskip0.5mu\colon\thinspace \pi_1(Y_0) \to V \subset {\rm SO}(3)$ by
\begin{gather*}
\beta,\gamma,\tau_1,\ldots,\tau_7 \mapsto 1 \\
\alpha \mapsto a \quad
\sigma_2\mapsto a \quad
\sigma_3\mapsto b.
\end{gather*}
To see that the flat connection $\theta$ induced by $\rho$ is regular as
a ${\rm G}_2$--instanton we use the following observation.
\begin{prop}\label{prop:fr}
A flat connection $\theta$ on a $G$--bundle $E_0$ over a flat
${\rm G}_2$--orbifold $Y_0$ corresponding to a representation
$\rho\mskip0.5mu\colon\thinspace\pi_1(Y_0)\to{\rm G}$ is regular as a ${\rm G}_2$--instanton if and
only if the induced representation of $\pi_1(Y_0)$ on
$\mathfrak g\oplus({\bf R}^7\otimes\mathfrak g)$ has no non-zero fixed vectors.
\end{prop}
\begin{proof}
Since $Y_0$ is flat as a Riemannian orbifold and $\theta$ is a
flat connection the Weitzenböck formula takes the form
\begin{align*}
L_\theta^*L_\theta=\nabla_\theta^*\nabla_\theta.
\end{align*}
Therefore, all infinitesimal deformations of $\theta$ are actually
parallel sections of the bundle
$\mathfrak g_{E_0}\oplus(T^*Y_0\otimes\mathfrak g_{E_0})$ and these are in one-to-one
correspondence with fixed vectors of the representation of
$\pi_1(Y_0)$ on $\mathfrak g\oplus{\bf R}^7\otimes\mathfrak g$.
\end{proof}
The elements $\sigma_2$ and $\sigma_3$ act trivially on ${\bf R}^7$
and their action on $\mathfrak{so}(3)$ has no common non-zero fixed vectors.
Therefore the action of $\pi_1(Y_0)$ on $\mathfrak g\oplus{\bf R}^7\otimes\mathfrak g$
has no non-zero fixed vector and thus $\theta$ is regular.
The monodromy representation $\mu_j|_{G_j}\mskip0.5mu\colon\thinspace G_j={\bf Z}_2\to{\rm SO}(3)$
associated with the flat connection $\theta$ is non-trivial only for
$j=1$. Let $A_1=A_{0,1}$ be the infinitesimally rigid ASD instanton
on $E_1=E_{0,1}$ given in Proposition \ref{prop:rigid}. For
$j=2,\ldots,6$, we choose $A_j$ to be the product connection on the
trivial bundle $E_j$. It is easy to see that this can be extended
to a collection of compatible gluing data independent of how the
$S_j$ are desingularised. Thus we obtain examples of
${\rm G}_2$--instantons on each of these desingularisations by appealing
to Theorem~\ref{thm:a}.
Note that any choice of resolution data for $T^7/\(\Gamma \times
A\)$ lifts to an $A$--invariant choice of resolution data for
$T^7/\Gamma$. We can then carry out Joyce's generalised Kummer
construction in a $A$--invariant way and lift up the
${\rm G}_2$--instanton constructed above. But we could have not
constructed this ${\rm G}_2$-instanton directly using
Theorem~\ref{thm:a}, since the lift of $\theta$ to $T^7/\Gamma$ is
not regular.
\end{example}
\begin{example}
Here is a more complicated example. Let $Y_0=T^7/\(\Gamma \times
A\)$ be as before. Define $\rho\mskip0.5mu\colon\thinspace \pi_1(Y_0)\to{}V\subset{\rm SO}(3)$ by
\begin{gather*}
\gamma,\tau_1,\ldots,\tau_7 \mapsto 1 \\
\alpha \mapsto a \quad
\beta \mapsto b \quad
\sigma_2 \mapsto b \quad
\sigma_3\mapsto a.
\end{gather*}
Again, the resulting flat connection $\theta$ is regular. For
$j=1,2,3$, let $A_j=A_{0,1}$ be the rigid ASD instanton on
$E_j=E_{0,1}$. For $j=4,\ldots,7$, let $A_j$ be the product
connection on the trivial bundle $E_j$. To be able to extend this
to compatible gluing data we need a lift $\tilde\rho_j$ of the
action of ${\bf Z}_2$ on $X_j$ to $E_j$ preserving $A_j$ and acting
trivially on the fibre at infinity for $j=2,3$. If $X_j$ is a
smoothing of ${\bf C}^2/{\bf Z}_2$, then the ${\bf Z}_2$ action on $X_j$ does lift
to $E_j$ preserving $A_j$. But the action does not lift if $X_j$ is
the resolution of ${\bf C}^2/{\bf Z}_2$. The reason for this is that in the
first case the action of ${\bf Z}_2$ on $H^2(X,{\bf R})$ is given by the
identity, while in the second case it acts via multiplication
by~$-1$, c.f.~\cite{Joyce2000}*{pp.~313--314}. Thus we can only
find compatible gluing data if we resolve both $S_2$ and $S_3$ using
a smoothing of ${\bf C}^2/{\bf Z}_2$.
Here is a small modification of this example. Define
$\rho\mskip0.5mu\colon\thinspace pi_1(Y_0)\to{}V\subset{\rm SO}(3)$ by
\begin{gather*}
\gamma,\tau_1,\ldots,\tau_7 \mapsto 1 \\
\alpha \mapsto a \quad
\beta \mapsto b \quad
\sigma_2 \mapsto b \quad
\sigma_3\mapsto c.
\end{gather*}
To find compatible gluing data, one simply has to compose
$\tilde\rho_j$ as above with multiplication by $b \in V \subset
\mathscr G(E_j)$, for $j=2,3$.
\end{example}
\begin{example}
We provide one more example. This is mainly to give the reader
something to play with. Also, it further illustrates the phenomenon
we already observed in the first
example.
Let $B=\<\sigma_1,\sigma_2,\sigma_3\>$, and $Y_0=T^7/\(\Gamma \times
B\)$. Then the singular set of $Y_0$ consists of $4$ copies of
$T^3/{\bf Z}^2$, denoted by $S_1,\ldots,S_4$, each of which has a
neighbourhood modelled on $(T^3\times{\bf C}^2/{\bf Z}^2)/{\bf Z}^2$ where ${\bf Z}_2$
acts freely on $T^3$ and by $\pm(z_1,z_2)\mapsto\pm(z_1,-z_2)$ on
${\bf C}^2/{\bf Z}^2$. The orbifold fundamental group $\pi_1(Y_0)$ is given
by
\begin{align*}
\pi_1(Y_0)=\<\alpha,\beta,\gamma,\sigma_1,\sigma_2,\sigma_3,\tau_1,\ldots,\tau_7\>\subset{\rm GL}(7).
\end{align*}
Up to conjugation the fundamental groups of the neighbourhoods $T_j$
of $S_j$ are given by
\begin{gather*}
\pi_1(T_1)=\<\alpha,\tau_4^{-1}\tau_5^{-1}\beta\sigma_1\sigma_2\sigma_3,\tau_1,\tau_2,\tau_3\>\quad
\pi_1(T_2)=\<\beta,\sigma_3\alpha,\tau_1,\tau_4,\tau_5\> \\
\pi_1(T_4)=\<\gamma,\alpha\beta,\sigma_2,\tau_4,\tau_6\> \quad
\pi_1(T_5)=\<\tau_3\gamma,\tau_3\alpha\beta,\sigma_2,\tau_4,\tau_6\>.
\end{gather*}
Define
$\rho\mskip0.5mu\colon\thinspace \pi_1(Y_0)\to V\subset{\rm SO}(3)$ by
\begin{gather*}
\alpha,\beta,\sigma_3,\tau_1,\ldots,\tau_7 \mapsto 1, \\
\sigma_1\mapsto a \quad
\sigma_2\mapsto b \quad
\gamma\mapsto b.
\end{gather*}
The induced flat connection $\theta$ is clearly regular. If both
$S_3$ and $S_4$ are desingularised using a resolution of
${\bf C}^2/{\bf Z}_2$ one can find compatible gluing data.
The resulting ${\rm G}_2$--instanton can be lifted to appropriate
$\sigma_1$--invariant desingularisations of $T^7/\(\Gamma\times
A\)$, but we could not have constructed it there directly, since the
lift of $\theta$ to $T^7/\(\Gamma \times A\)$ it is not regular.
\end{example}
The list of examples is far from exhaustive. We invite the reader to
try to produce further examples in order to develop a good
understanding for when compatible gluing data for a given flat
connection can be found and when not.
\section{Discussion}
\label{sec:discuss}
Assuming certain restrictions on the first Pontryagin class and the
second Stiefel-Whitney class of the underlying bundle we expect that
all ${\rm G}_2$--instantons on generalised Kummer constructions arise from
(a suitable generalisation) of our construction. In these cases one
could hope to make the ${\rm G}_2$ Casson invariant rigorously defined
and, in fact, computable in terms of algebraic data.
If the ${\rm G}_2$ Casson invariant can be rigorously defined, it is
natural to ask for applications. One (possibly naive) hope is that it
will provide a means of distinguishing ${\rm G}_2$--manifolds. It is
known that there are large sets of input for Joyce's generalised
Kummer construction which yield ${\rm G}_2$--manifolds that cannot be told
apart by simply looking at their fundamental group and their Betti
numbers. It is reasonable to expect that not all of those
${\rm G}_2$-manifolds are isomorphic. Our work might give some indication
as to which of those ${\rm G}_2$--manifolds can be distinguished from each
other by their ${\rm G}_2$ Casson invariant.
Recently, Kovalev-Nordström \cite{Kovalev2010} found examples of
${\rm G}_2$--manifolds that can be constructed using Joyce's as well as
Kovalev's method. It is an interesting question to what extend the
images of these constructions in the landscape of ${\rm G}_2$--manifolds
overlap. Building on Sá Earp's work one could hope to be able better
understand the ${\rm G}_2$ Casson invariant on ${\rm G}_2$--manifolds arising
from Kovalev's twisted connected sum construction. This is related to
a kind of ${\rm G}_2$--analogue of the Atiyah-Floer conjecture. Combining
our methods with Sá Earp's might, one day, shed some new light on the relation
between Joyce's and Kovalev's construction.
|
2,869,038,154,776 | arxiv | \section{Introduction}
The problem of minimizing automata and transition systems has been widely studied in the literature. Minimization involves finding the smallest equivalent structure, using an appropriate definition of equivalence, (e.g. language equivalence or simulation equivalence). In many software engineering applications, automata need to be minimized before complex operations such as model checking or test case generation can be carried out.
For different automata models and different notions of equivalence,
the complexity of the minimization problem can vary considerably. The survey \cite{Ber11} considers minimization algorithms for DFA up to language equivalence,
with time complexities varying between $\mathcal O(n^{2})$ and $\mathcal O(n\;log\;n)$.
Kripke structures represent a generalisation of DFA to allow non-determinism and multiple outputs. They
have been widely used to model concurrent and embedded systems.
An algorithm for mimimizing Kripke structures has been given in
\cite{BusGru03}.
In the presence of non-determinism, the complexity of minimization is quite high.
Minimization up to language equivalence requires exponential time, while
minimization up to a weaker simulation equivalence can be carried out in polynomial time
(see \cite{BusGru03}).
By contrast, we will show that {\em deterministic} Kripke structures can be efficiently minimized even up to language equivalence
with a worst case time complexity of $\mathcal O(kn \log_{2} n)$. For this, we generalise the concepts of
right language and Nerode congruence from DFA to deterministic Kripke structures. We then show how the
DFA minimization algorithm of \cite{Hop71} can be generalised to compute the Nerode congruence $\equiv$
of a deterministic Kripke structure $\mathcal{K}$.
The quotient Kripke structure $\mathcal{K} / \equiv$ is minimal and language equivalent to $\mathcal{K}$.
Our research \cite{MeinkeSindhu2011} into software testing has shown that this minimization algorithm
makes the problems of model checking and test case generation more tractable for large models.
The paper is organized as follows. In Section \ref{prem}, we introduce some mathematical pre-requisites.
In Section \ref{sec3}, we give a minimization algorithm for deterministic Kripke structures. In Section \ref{sec4}, we give
a correctness proof for this algorithm. In Section \ref{sec5} we provide a complexity analysis.
Finally, in Section \ref{sec6} we discuss some conclusions.
\section{Preliminaries}
\label{prem}
We assume familiarity with the basic concepts of deterministic finite automata (DFA).
A \emph{Kripke structure} is a generalisation of a DFA to allow multiple outputs and non-determinism.
A Kripke structure $\mathcal{K}$ over a finite set \emph{AP} of atomic propositions is a five tuple
$\mathcal{K}=\langle Q,\Sigma,\delta,q_{0},\lambda\rangle$, where \emph{Q}, is the set of states, $\Sigma=\{\sigma_{1},...,\sigma_{n}\}$ is a finite alphabet, $\delta\subseteq Q\times\Sigma\times Q$ is the transition relation for states, $q_{0}$ is the initial state of $\mathcal{K}$ and $\lambda:Q\rightarrow2^{AP}$ is a function to label states.
If $\vert \emph{AP} \vert = k$ we say that $\mathcal{K}$ is a $k$-bit Kripke structure.
We say that
$\mathcal{K}$ is \emph{deterministic} if
the relation $\delta$ is actually a function, $\delta: Q\times\Sigma\rightarrow Q$. We let $\delta^{*}:Q\times\Sigma^{*}\rightarrow Q$ denote the iterated state transition function where $\delta(q,\epsilon)=q$ and $\delta^{*}(q,\sigma_{1},...,\sigma_{n})=\delta(\delta^{*}(q,\sigma_{1},...,\sigma_{n-1}),\sigma_{n})$. Each property in \emph{AP }describes some local property of system states $q\in Q$.
It is convenient to redefine the labelling function $\lambda$ as \emph{}$\lambda:Q\to\mathbb{B}^{k}$ given an enumeration of the set \emph{AP}. Then the iterated output function $\lambda^{*}:Q\;\times\;\Sigma^{*} \rightarrow \mathbb{B}^{k}$ is given by $\lambda^{*}(q,\sigma_{1},...,\sigma_{n})=\lambda(\delta^{*}(q,\sigma_{1},...,\sigma_{n}))$. More generally for any $q \in Q$ define $\lambda_{q}^{*}(\sigma_{1},...,\sigma_{n})=\lambda^{*}(q,\sigma_{1},...,\sigma_{n})$. Given any $R\subseteq Q$ we write $\lambda(R)=\cup_{r\in R}\lambda(r)$. We let $q. \sigma$ denote $\delta(q,\sigma)$ and $R. \sigma$ denotes $\{r. \sigma \;\vert\; r \in R\}$ for $R\subseteq Q$.
We can represent a Kripke structure graphically in the usual way using a \emph{state transition diagram}. For example, a Kripke structure
with three bit labels in the output is shown in Fig \ref{fig:A-3-bit-Kripke}(A).
\begin{figure}
\begin{centering}
\includegraphics[width=0.7\textheight,height=0.35\textheight]{Kripke}
\par\end{centering}
\caption{\label{fig:A-3-bit-Kripke} 3-bit Kripke Structure $\mathcal{K}$}
\end{figure}
\subsection{Minimal DFA and minimal deterministic Kripke structures}
Let us consider a DFA $\mathcal{A}=\langle Q,\Sigma,\delta,q_{0},F\rangle$ . For each state $q\in Q$ of $\mathcal{A}$ there corresponds a
subautomaton of $\mathcal{A}$ rooted at $q$ which accepts the regular language
$\mathcal{L}_{q}(\mathcal{A}) \subseteq \Sigma^*$, consisting of just those words accepted by the subautomaton with $q$ as initial state.
Thus $\mathcal{L}_{q_0}(\mathcal{A})$ is the language accepted by $\mathcal{A}$.
The language $\mathcal{L}_{q}(\mathcal{A})$ is called either the \emph{future }of state \emph{q }or the \emph{right language }of \emph{q. }$\mathcal{A}$ is\emph{ minimal} if for each pair of distinct states \emph{$p,q\in Q$}, we have, $\mathcal{L}_{p}(\mathcal{A})\neq\mathcal{L}_{q}(\mathcal{A})$. For any regular language
$\mathcal{L} \subseteq \Sigma^*$ there is a smallest DFA (in terms of the number of states) accepting $\mathcal{L}$. This DFA is minimal, and is
unique up to isomorphism.
An equivalence relation $\equiv$ can be defined on
the states of a DFA by $p\equiv q$ if and only if $\mathcal{L}_{p}(\mathcal{A})=\mathcal{L}_{q}(\mathcal{A})$.
This relation is a congruence, i.e. if $p \equiv q$ then
$p. \sigma \equiv q. \sigma $ for all $\sigma \in\Sigma^{*}$. It is known as the \emph{Nerode congruence}.
Consider the quotient DFA $\mathcal{A} / \equiv$. This is the unique smallest DFA
which accepts the
regular language $\mathcal{L}_{q_0}(\mathcal{A})$.
The problem of minimizing a DFA $\mathcal{A}$ is therefore to compute its Nerode congruence, which will be the identity relation if, and only if $\mathcal{A}$ is a minimal automaton.
The problem of computing a minimal Kripke structure $\mathcal{K}$ is
an analogous but more general problem.
In this case, the right language $\mathcal{L}_{q}(\mathcal{K})$ associated with a state $q$ of $\mathcal{K}$ can be defined by
$$
\mathcal{L}_{q}(\mathcal{K}) =
\{ \hskip 3pt (\sigma_{1},...,\sigma_{n} , a ) \in \Sigma^* \times \mathbb{B}^{k}
\hskip 3pt \vert \hskip 3pt \lambda_{q}^{*}(\sigma_{1},...,\sigma_{n}) = a \hskip 3pt \} .
$$
As before, $\mathcal{K}$ is\emph{ minimal} if for each pair of distinct states \emph{$p,q\in Q$} we have, $\mathcal{L}_{p}(\mathcal{K})\neq\mathcal{L}_{q}(\mathcal{K})$.
There is again a smallest Kripke structure associated with a right language
$\mathcal{L} \subseteq \Sigma^* \times \mathbb{B}^{k}$. This Kripke structure is also minimal, and
unique up to isomorphism.
The Nerode congruence for a Kripke structure $\mathcal{K}$ is now defined by:
\begin{center}
$p\equiv q$ if and only if $\lambda_{p}^{*}(\sigma_{1},...,\sigma_{n})=\lambda_{q}^{*}(\sigma_{1},...,\sigma_{n})$ for all $(\sigma_{1},...,\sigma_{n})\in\Sigma^{*}$.
\par\end{center}
and
$\mathcal{K} / \equiv$ is the unique smallest Kripke structure
associated with the
right language $\mathcal{L}_{q_0}(\mathcal{K})$.
So the problem of minimising $\mathcal{K}$ is to compute this congruence.
\section{Kripke Structure Minimization
Algorithm}
\label{sec3}
\begin{algorithm}\label{KMN}
\caption{Kripke Structure Minimization}\label{alg1}
\KwIn{A deterministic Kripke structure $\mathcal{K}$ with no unreachable states and $k$ output bits.}
\KwOut{The Nerode congruence $\equiv$ for $\mathcal{K}$, i.e. equivalence classes of states for the
minimized structure $ \mathcal{K}_{min}$ behaviourally equivalent to $\mathcal{K}$.}
Create\nllabel{partition} an initial state partition
$P=\{B_{q}=\{q^{\prime}\in
Q\;\vert\;\lambda(q)=\lambda(q^{\prime})\}\;\vert\;
q\in Q\}$. Let $n=\vert P\vert$. Let $B_{1},...,B_{n}$ be an
enumeration of P.
\textbf{if} {$n=\vert Q\vert$} \textbf{then} \textbf{go to} line \ref{termination}.
\ForEach{$\sigma\in\Sigma$\nllabel{subpartition}}{
\For{$i\leftarrow 1$ \KwTo $n$ } {$B(\sigma, i)=\{q\in B_{i}\;\vert\; \exists r \in Q \;s.t\;
\delta(r,\sigma) = q\}.$
/*This constitutes the subset of states in block $B_i$ which have predecessors
through input $\sigma$. */}
}
$count=n+1$;
\ForEach{$\sigma\in\Sigma$}{\nllabel{initwait}
choose\nllabel{initwait1} all the subsets $B(\sigma, i)$ (excluding any empty subsets)
and put their block numbers $i$ on a waiting list (i.e. an
unordered
set) $W(\sigma)$ to be processed.
}
\BlankLine
Boolean splittable = true;
\While{splittable}{\nllabel{while}
\ForEach{$\sigma\in\Sigma$ }{
\ForEach{$i\in$ $W(\sigma)$}{
Delete i from $W(\sigma)$\nllabel{bodystart}
\For{$j\leftarrow 1$ \KwTo $count-1$ s.t. $\exists t \in B_{j}$
with $\delta(t,\sigma)\in B(\sigma,i)$}{
Create $B_{j}^{\prime}=\{t\in B_{j}\;\vert\;\delta(t,\sigma)\in
B(\sigma,i)\}$\nllabel{newBj}
\If{$B_{j}^{\prime}\subset B_{j}$}{
$ B_{count}=B_{j}-B_{j}^{\prime}$; \nllabel{block} $ B_{j}=B_{j}^{\prime}$
\ForEach{$\sigma\in\Sigma$}{
$B(\sigma,count)=\{q\in B(\sigma,j) \;\vert\; q \in B_{count} \}$;\nllabel{newacount}
$ B(\sigma,j)=\{q\in B(\sigma,j)
\;\vert
\; q \in B_{j}
\}$ \nllabel{newaj}
\If{$ j\notin W(\sigma)$ and\nllabel{if} $0<\vert B(\sigma,j)\vert\le\vert B(\sigma,count)\vert$}{$ W(\sigma)=W(\sigma)\cup\{j\}$}
\Else{ $W(\sigma)=W(\sigma)\cup\{count\}$\nllabel{endif}}
}
$ count=count+1$;
}
}
}
}
splittable = false;
\ForEach{$\sigma\in\Sigma$}{
\If {$W(\sigma) \not= \emptyset$}{splittable=true; \nllabel{bodyend}}
}
}
Return \nllabel{termination}partition blocks
$B_{1},...,B_{count}$.
\end{algorithm}
Algorithm \ref{KMN} presents an efficient algorithm to compute the Nerode congruence $\equiv$ of a
deterministic Kripke structure $\mathcal{K}$, which is the same as the state set of the associated quotient
Kripke structure $\mathcal{K} / \equiv$.
We demonstrate the behavior of this algorithm on a simple example
given in Fig.\ref{fig:A-3-bit-Kripke}(A) as follows.
The algorithm begins by inverting the state
transition table as shown in Fig.\ref{fig:A-3-bit-Kripke}(C). Then it creates four
initial blocks of states on the basis of unique bit labels which are:
$B_{1}=\{q_{0},q_{5}\}$, $B_{2}=\{q_{1,}q_{2}\}$,
$B_{3}=\{q_{3}\}$
and $B_{4} = \{q_{4}\}$. Next it is checked whether the number of blocks is equal to the number of states $\vert Q\vert$
of the given Kripke structure. This is not the case,
so the next step is to refine each partition block $B_{i}$ into subsets $B(\sigma , i)$ of states which have predecessors
via each input symbol of $\sigma \in \Sigma$.
This gives $B(a,1)=\{q_{0}\}$, $B(b,1)=\{q_{5}\}$, $B(a,2)=\{q_{1}\}$,
$B(b,2)=\{q_{1},q_{2}\}$, $B(a,3)=\{q_{3}\}$, $B(b,3)=\{q3\}$ ,
$B(a,4)=\{q_{4}\}$ and $B(b,4)=\{q_{4}\}$. The next step is to
initialize the waiting list $W(\sigma)$ for each symbol $\sigma \in \Sigma$ by
inserting the block numbers of all non-empty subpartition blocks $B(\sigma , i)$ created in the previous
step. We obtain $W(a)=\{1,2,3,4\}$ and $W(b)=\{1,2,4\}$.
Now the algorithm can refine the initial partition
$B_{1} , \ldots , B_{4}$
by iterating the loop on line \ref{while} until $W(\sigma)=\emptyset$ for all
$\sigma\in\Sigma$.
For $i=1$ and $a\in\Sigma$ we have $W(a)=\{2,3,4\}$ and
$B(a,1)=\{q_{0}\}$. We can see that $\delta(q_{1},a)=q_{0}\in
B(a,1)$
and $\delta(q_{2},a)=q_{0}\in B(a,1)$. But both $q_{1}$ and
$q_{2}$ are
in $B_{2}$. Therefore $B_{2}^{\prime}\not\subset B_{2}$ and
hence
no refinement of the partition is possible in this step.
We proceed
with the next iteration of the loop by deleting $i=2$ from
$W(a)$ so that $W(a)=\{3,4\}$. Now we have
$B(a,2)=\{q_{1}\}$.
We can see that $\delta(q_{0},a)=q_{1}\in B(a,2)$. Therefore we have $B_{1}^{\prime}=\{q_{0}\}$. Since $B_{1}^{\prime}\subset B_{1}$ we therefore split $B_{1}$ into
$B_{5}=B_{1}-B_{1}^{\prime}=\{q_{0}, q_{5}\}-\{q_{0}\}=\{q_5\}$ and $B_{1}=B_{1}^{\prime}=\{q_{0}\}$.
Next we update the subsets $B(\sigma, i)$ and we get
$B(a,1)=\{q_{0}\}$,
$B(b,1)=\{\}$, $B(a,5)=\{\}$ and $B(b,5)=\{q_{5}\}$. The
updated
waiting sets are then $W(a)=\{1,3,4\}$ and $W(b)=\{1,2,4,5\}$. Next
we choose $i=1$, $\sigma=a$ and $W(a)=\{3,4\}$ and we obtain
$B(a,1)=\{q_{0}\}.$
It can be seen that $\delta(q_{1},a)=q_{0}\in a(a,1)$ and
$\delta(q_{2},a)=q_{0}\in a(a,1)$.
Therefore $B_{2}^{\prime}=\{q_{1},q_{2}\}$, but
$B_{2}^{\prime}\not\subset B_{2}$
and hence no refinement of the partition is possible in this case.
We delete $i=3$ from $W(a)$ and obtain $W(a)=\{4\}$ and $B(a,3)=\{q_{3}\}$.
We then find that for $q_{4}\in B_{4}$, $\delta(q_{4},a)=q_{3}\in B(a,3)$. Therefore we have $B_{4}^{\prime}=\{q_{4}\}$. But $B_{4}^{\prime}\not\subset B_{4}$, so no refinement of the partition is possible in this case. Continuing in the same way it will be seen that there is no further refinement of the partition possible for $i=4$ and $\sigma=a$ and for $i=1,2,4,5$ and $\sigma=b$ both $W(a)$ and $W(b)$ become empty. We terminate with five blocks in the partition. These constitute the states of our minimized Kripke structure as shown in Fig \ref{fig:A-3-bit-Kripke}(B).
\section{Correctness of Kripke Structure Minimization}
\label{sec4}
In this section we give a rigorous but simple proof of the correctness of Algorithm \ref{KMN}.
By means of a new induction argument, we have simplified
the correctness argument compared with \cite{Ber11} and \cite{Hop71}.
First let us establish termination of the algorithm by using an appropriate well-founded ordering for the main loop variant.
\begin{defn}
\label{def:definition3}Consider any pair of finite sets of
finite sets $A=\{A_{1},...,A_{m}\}$
and $B=\{B_{1},...,B_{n}\}$. We define an ordering relation
$\leq$ on $A$ and $B$
by $A\leq B$ iff $\forall1\leq i\leq m$, $\exists1\leq j\leq n$
such that $A_{i}\subseteq B_{j}$. Define $A<B\iff A\leq B\;\&\;
A\neq B$.
Clearly $\leq$ is a reflexive, transitive relation. Furthermore $\leq$ is well-founded, i.e.
there are no infinite
descending chains $A_{1}>A_{2}>A_{3}...$ , since $\emptyset$ is
the smallest element under $\leq$.\end{defn}
\begin{propos}
Algorithm \ref{KMN} always terminates.
\end{propos}
\begin{pf}
We have two cases for the termination of the algorithm as a result of the partition formed on line \ref{partition} of the algorithm: (1) when $n=\vert Q\vert$, and (2) when $n<\vert Q\vert$.
Consider the case when $n=\vert Q\vert$ then each block in the partition corresponds to a state of the
given Kripke structure with a unique bit-label and hence in this case the algorithm will terminate on line \ref{termination} by providing the description of these blocks.
Now consider the case when $n<\vert Q\vert$. Then the waiting sets $W(\sigma)$ for all $\sigma\in\Sigma$ will be initialized on lines \ref{initwait}, \ref{initwait1} and the termination of the algorithm depends on proving the termination of the loop on line \ref{while}. Now $W(\sigma)$ is intialized by loading the block numbers of the split sets on line \ref{initwait1}. There are only two possiblities after any execution of the loop. Let $W_{m}(\sigma)$ and $W_{m+1}(\sigma)$ represent the state of the variable $W(\sigma)$ before and after one execution of the loop respectively at any given time. Then either $W_{m}(\sigma) = W_{m+1}(\sigma)\cup\{i\}$
and no splitting has taken place and \emph{i} is the deleted block number, or $W_{m}(\sigma)\cup\{j\}=W_{m+1}(\sigma)\cup\{i\}$ or $W_{m}(\sigma)\cup\{k\}=W_{m+1}(\sigma)\cup\{i\}$
where j and k represent the split blocks and one of them goes into $W_{m}(\sigma)$ if it has fewer incoming transitions. In either case $W_{m}(\sigma)>W_{m+1}(\sigma)$ by Definition \ref{def:definition3}. Therefore $W(\sigma)$ strictly decreases with each iteration of the loop
on line \ref{while}. Since the ordering $\leq$ is well-founded, Algorithm \ref{KMN} must terminate. \end{pf}
Now we only need to show that when Algorithm \ref{KMN} has terminated, it returns
the Nerode congruence $\equiv$ on states.
\begin{propos}
Let $P_{i}$ be the partition (block set) on the $ith$ iteration of Algorithm \ref{KMN}. For any blocks $B_{j}, B_{k} \in P_{i}$ and any states $p\in B_{j}, q\in B_{k}$ if $j\neq k$ then
$p\nequiv q$.\end{propos}
\begin{pf}
By induction on the number $i$ of times the loop on line \ref{while} is executed.
\vskip 6pt
\noindent\textbf{Basis:} Suppose $i=0$ then clearly the result holds because each block created at line \ref{partition} is distinguishable by the empty string $\epsilon$.
\vskip 6pt
\noindent\textbf{Induction Step:} Suppose $i=m>0$. Let us assume that the proposition holds after $m$ executions of the loop.
Consider any $B_{j},B_{k}\in P_{m}$. During the $m+1$th execution of the loop on line \ref{while} either block $B_{j}$ is split into $B_{j}^{\prime}$ and $B_{j}^{\prime\prime}$ or $B_{k}$ is split into $B_{k}^{\prime}$ and $B_{k}^{\prime\prime}$ but not both during one execution of the loop (due to line \ref{block}).
Consider the case when $B_{j}$ is split then for any $p \in B_{j}$, either $p \in B_{j}^{\prime}$ or $p \in B_{j}^{\prime\prime}$. But for any $p \in B_{j}$ and $q \in B_{k}$, $p \nequiv q$ by the induction hypothesis. Therefore, for $p \in B_{j}^{\prime}$ or $p\in B_{j}^{\prime\prime}$ $p \nequiv q$. Hence the proposition is true for $m+1$th execution of the loop in this case.
By symmetry the same argument holds when $B_{k}$ is split.
\end{pf}
The following Lemma gives a simple, but very effective way to understand Algorithm \ref{KMN}.
Note that this analysis is more like a temporal logic argument than a loop invariant approach.
This approach reflects the non-determinism inherent in the algorithm.
\begin{lem}\label{lm}
For any states $p , q \in Q$, if $p \not\equiv q$ and initially $p$ and $q$ are in the same block
$p , q \in B_{i_0}$ then eventually $p$ and $q$ are split into different blocks,
$p \in B_j$ and $q \in B_k$ for $j \not= k$.
\end{lem}
\begin{pf}
Suppose that $p \not\equiv q$ and that initially $p , q \in B_{i_0}$ for some block $B_{i_0}$.
Since $p \not\equiv q$ then for some $n \geq 0$, and
$\sigma_1 , \ldots , \sigma_n \in \Sigma$, $$
\lambda^* ( p , \sigma_1 , \ldots , \sigma_n ) \not=
\lambda^* ( q , \sigma_1 , \ldots , \sigma_n ) .
$$
We prove the result by induction on $n$.
\vskip 6pt
\noindent\textbf{Basis} Suppose $n = 0$, so that $\lambda ( p ) \not= \lambda ( q )$. By line \ref{partition},
$p \in B_p$ and $q \in B_q$ and $B_p \not= B_q$. So the implication holds vacuously.
\vskip 6pt
\noindent\textbf{Induction Step} Suppose $n > 0$ and for some $\sigma_1 , \ldots , \sigma_n \in \Sigma$,
$$
\lambda^* ( p , \sigma_1 , \ldots , \sigma_n ) \not=
\lambda^* ( q , \sigma_1 , \ldots , \sigma_n ) .
$$
\noindent{\bf(a)} Suppose initially $\delta(p, \sigma_{1})\in B(\sigma_{1}, \alpha)$ and $\delta(q, \sigma_{1}) \in B(\sigma_{1}, \beta)$ for $\alpha \neq \beta$.
Consider when $\sigma = \sigma_{1}$ on the first iteration of the loop on line \ref{while}. Clearly, $B(\sigma_{1}, \alpha ), B(\sigma_{1}, \beta ) \in W(\sigma)$ at this point. Choosing $i= \alpha$ and $j=i_{0}$ on this iteration then since $\delta(p,\sigma_{1})\in B(\sigma_{1},\alpha)$ we have
\begin{center}
$B_{i_{0}}^{\prime}=\{t\in B_{i_{0}}\;\vert\;\delta(t,\sigma_{1})\in B(\sigma_{1},\alpha)\}\subset B_{i_{0}}$
\end{center}
This holds because $q \in B_{i_0}$ but $\delta( q , \sigma_1 ) \in B( \sigma_1 , \beta )$ and
$B( \sigma_1 , \alpha ) \not= B( \sigma_1 , \beta )$ so
$B( \sigma_1 , \alpha ) \cap B( \sigma_1 , \beta ) = \emptyset$ and hence
$q \not\in B'_{i_0}$.
Therefore $p$ and $q$ are split into different blocks on the first iteration so that
$p \in B'_{i_0}$ and $q \in B_{i_0} - B'_{i_0}$.
By symmetry, choosing $i = \beta$ and $j = {i_0}$ then $p$ and $q$ are split on the first loop iteration with
$q \in B'_{i_0}$ and $p \in B_{i_0} - B'_{i_0}$.
\noindent{\bf (b)} Suppose initially
$\delta( p , \sigma_1 ) , \delta( q , \sigma_1 ) \in B( \sigma_1 , \alpha )$
for some $\alpha$. Now
\begin{center}
$
\lambda^* (\; \delta( p , \sigma_1 ) , \sigma_2 , \ldots , \sigma_n \; ) \not=
\lambda^* ( \;\delta( q , \sigma_1 ) , \sigma_2 , \ldots , \sigma_n \;) .
$
\end{center}
So by the induction hypothesis, eventually $\delta( p , \sigma_1 )$ and $\delta( q , \sigma_1 )$
are split into different blocks, $\delta( p , \sigma_1 )\in B_{\alpha}$ and
$\delta( p , \sigma_1 )\in B_{\beta}$. At that time one of $B_{\alpha}$ or $B_{\beta}$
is placed in a waiting set $W(\sigma)$. Then either on the same iteration of the loop on line \ref{while} or on the
next iteration, we can apply the argument of part (a) again to show that $p$ and $q$ are split into
different blocks.
\end{pf}
Observe that only one split block is loaded into $W(\sigma)$
on lines \ref{if}-\ref{endif}. From the proof
of Lemma \ref{lm} we can see that it does not matter logically which of these two blocks we insert into $W(\sigma)$. However,
by choosing the subset with fewest incoming transitions we can obtain a worst case time complexity
of order $O( kn \hskip 3pt log_2 \hskip 3pt n )$, as we will show.
\begin{corollary}
For any states $p , q \in Q$, if $p \not\equiv q$
then $p$ and $q$ are in different blocks when the algorithm terminates.
\end{corollary}
\begin{pf}
Assume that $p \not\equiv q$.
\noindent{\bf (a)} Suppose at line 3 that $n = \vert Q \vert$. Then initially, all blocks $B_i$ are singleton sets and
so trivially $p$ and $q$ are in different blocks when the algorithm terminates.
\vskip 0.12pt
\noindent{\bf (b)} Suppose at line 3 that $n < \vert Q \vert$.
\noindent{\bf (b.i)} Suppose that $p$ and $q$ are in different blocks initially.
Since blocks are never merged then the result holds.
\noindent{\bf (b.ii)} Suppose that $p$ and $q$ are in the same block initially.
Since $p \not\equiv q$ then the result follows by Lemma \ref{lm}.
\end{pf}
\section{Complexity Analysis}
\label{sec5}
Let us consider the worst-case time complexity of Algorithm \ref{KMN}.
\begin{propos}
If $\mathcal{K}$ has $n$ states and $\Sigma$ has $k$ input symbols then Algorithm \ref{KMN}
has worst case time complexity $O( kn \log_{2} n )$.
\end{propos}
\begin{pf}
Creating the initial block partition on line \ref{partition} requires at most $O(n)$ assignments.
The block subpartitioning in the loop on line \ref{subpartition} requires at most $O(kn)$ moves of states. Also the
the initialisation of the waiting lists $W(\sigma)$ in the loop on line \ref{initwait} requires at most $O(kn)$ assignments.
Consider one execution of the body of the loop starting on line \ref{while}, i.e. lines \ref{bodystart} - \ref{bodyend}. Consider any states
$p , \hskip 3pt q \in Q$ and suppose that $\delta ( p , \hskip 3pt \sigma ) = q$ for some $\sigma \in \Sigma$. Then the state $p$
can be: (i) moved into $B'_j$ (line \ref{newBj}), (ii) removed from $B_j$ (line \ref{block}), or (iii) moved into
$B( \sigma , \hskip 3pt i)$ or $B( \sigma , \hskip 3pt count)$ (lines \ref{newacount}, \ref{newaj}) if, and only if, a block $i$ is being
removed from $W(\sigma)$
such that $q \in B( \sigma , \hskip 3pt i)$ at that time.
(Such a block sub-partition $B( \sigma , \hskip 3pt i)$ can be termed a {\it splitter} of $q$.)
Now each time a block $i$ containing $q$ is removed from $W(\sigma)$ its size is less than half of the size when it
was originally entered into $W(\sigma)$, by lines \ref{if}-\ref{endif}.
So $i$ can be removed from $W(\sigma)$ at most $O( log_2 \hskip 3pt n )$ times.
Since there are at most $k$ values of $\sigma$ and $n$ values of $p$, then the total number of state moves
between blocks and block sub-partitions is at most $O( kn \hskip 3pt log_2 \hskip 3pt n )$.
\end{pf}
\section{Conclusions}
\label{sec6}
We have given an algorithm for the minimization of deterministic Kripke structures with
worst case time complexity $\mathcal O(kn \log_{2} n)$. We have analysed the correctness and performance
of this algorithm. An efficient implementation of this algorithm has been developed which confirms the run-time performance theoretically predicted in Section 5.
This research has been supported by the Swedish Research Council (VR), the Higher Education Commission of Pakistan (HEC), as well as EU projects HATS FP7-231620, and MBAT ARTEMIS JU-269335.
\bibliographystyle{elsarticle-harv}
|
2,869,038,154,777 | arxiv | \section{Introduction}
Recently, many studies discussed verification of maximally entangled states
from theory \cite{HMT,Ha09-2,HayaM15,PLM18,ZH4,Markham} to experiment \cite{MST,KSKWW,Bavaresco,FVMH,JWQ}.
However, their verification ensures only that the constructed state is close to
the maximally entangled state.
Therefore, it does not guarantee that
it is exactly the same as the maximally entangled state.
That is, such an experimental verification does not necessarily support the existence of the maximally entangled state.
Hence, it is impossible to experimentally verify
the \textit{standard entanglement structure} (SES)
of the composite system,
in which a state on the composite system
is given as a normalized positive semi-definite matrix on the tensor product space
even if the local systems are fully equal to standard quantum theory.
Furthermore,
a theoretical structure of quantum bipartite composite systems
is not uniquely determined
even if
we impose that the local subsystems are exactly the same as standard quantum subsystems
\cite{Janotta2014,Lami2017,Aubrun2020,Plavala2021,Arai2019,YAH2020,HK2022,ALP2019,ALP2021}.
This problem is recently studied in the modern operational approach of foundations of quantum theory, called \textit{General Probabilistic Theories} (GPTs) \cite{Janotta2014,Lami2017,Aubrun2020,Plavala2021,Arai2019,YAH2020,HK2022,ALP2019,ALP2021,
Kimura2010, Bae2016, PR1994,Pawlowski2009,Short2010,Barnum2012,Plavala2017,Matsumoto2018,Takagi2019,Yoshida2020,CDP2010,Spekkens2007,MAB2022,CS2015,CS2016,CS2015-2,
Muller2013,BLSS2017,Barnum2019,Janotta2013}.
GPTs start with fundamental probabilistic postulates to define states and measurements.
Even though the postulates and the mathematical definition of GPTs are physically reasonable,
GPTs cannot uniquely determine the model of the bipartite quantum composite system
even if the subsystems are equivalent to standard quantum systems.
For example, GPTs allow the model with no entangled states
as well as the model with ``strongly entangled'' states than the standard quantum system
in addition to the SES \cite{Janotta2014,Lami2017,Aubrun2020,Plavala2021,Arai2019,YAH2020,HK2022,ALP2019,ALP2021}.
Some studies of GPTs deal with the most general models satisfying fundamental probabilistic postulates \cite{PR1994,Pawlowski2009,Short2010,Barnum2012,Plavala2017,Matsumoto2018,Takagi2019,Yoshida2020,CDP2010,Spekkens2007,MAB2022,CS2015,CS2016,CS2015-2},
and they investigate physical or informational properties in general.
Because general models are sometimes quite different from standard quantum systems,
general properties behave unlike present experimental facts \cite{PR1994,Pawlowski2009,Short2010,Barnum2012,Plavala2017,Spekkens2007}.
While the above studies aim to investigate physical and informational properties in general models,
our interest is whether there exists a quantum-like model satisfying present experimental facts except for standard quantum systems.
That is, this paper aims to impose several conditions on GPTs for behaving in a similar way as quantum theory
and to exprole the existence of other GPTs to satisfy these conditions.
As the first condition,
this paper deals with a class of the general models
called
\textit{bipartite entanglement structures with local quantum systems} \cite{Janotta2014,Lami2017,Aubrun2020,Plavala2021,Arai2019,YAH2020,HK2022,ALP2019,ALP2021}(hereinafter, we simply call it \textit{Entanglement Structures} or ESs)
\footnote{
\textit{Entanglement} is a concept defined not only in quantum composite systems but also in general models
whose local subsystems are not necessarily equal to standard quantum systems \cite{ALP2019,ALP2021,CS2016,CS2015-2}.
However, our interest is a ``similar structure'' to standard quantum entanglement;
therefore, we impose that the local subsystems are equal to standard quantum systems, as we mentioned.
},
i.e.,
we deal with the composite models in GPTs with the assumption that their local systems are completely equivalent to standard quantum systems.
As many studies pointed out,
many models satisfy this condition \cite{Janotta2014,Lami2017,Aubrun2020,Plavala2021,Arai2019,YAH2020,HK2022,ALP2019,ALP2021},
and some models do not behave in a similar way as quantum theory \cite{Lami2017,Aubrun2020,Arai2019,YAH2020}.
Therefore, as present experimental facts,
this paper mainly focuses on two additional conditions, \textit{undistinguishability} and \textit{self-duality}, mentioned below.
The second condition, undistinguishability,
is introduced as the possibility of the verification of maximally entangled states with tiny errors.
The error probability of verification is upper bounded by using the trace norm
due to a simple inequality.
Therefore, we mathematically define $\epsilon$-undistinguishability as $\epsilon$-upper bound of a distance based on trace norm between the state space in an ES and the set of maximally entangled states.
If an ES satisfies undistinguishability with enough small errors,
it cannot be denied by physical experiments of verification of maximally entangled states
that our physical system might obey the structure (not standard one).
The third condition, self-duality,
is defined as the equality between the state space and the effect space in an ES.
This paper introduces self-duality as a saturated situation of \textit{pre-duality},
and we point out the correspondence between pre-duality and \textit{projectiviy}.
Projectivity is one of the postulates in standard quantum theory \cite{Neumann1932,Luders1951,Davies1970,Ozawa1984},
which ensures a measurement whose post-measurement states are given as the normalization of its effects.
Self-duality is a saturation of projectivity and a common property that classical and quantum theory possess.
Moreover, self-duality is important for deriving algebraic structures in physical systems.
When a self-dual model satisfies a kind
of strong symmetry, called homogeneity,
the state space is characterized by Jordan Algebras \cite{Jordan1934,Koecher1957,Barnum2019,BMA2020},
which leads to essentially limited types of models, including classical and quantum theory \cite{Jordan1934,Koecher1957}.
In summary, this paper aims to discuss whether there exists an ES with $\epsilon$-undistinguishability and self-duality other than the SES.
Such a structure cannot be distinguished by any verification of maximally entangled states with errors larger than $\epsilon$ and satisfies saturated projectivity.
Due to this physical similarity, we call an ES with $\epsilon$-undistinguishability and self-duality an \textit{$\epsilon$-Pseudo Standard Entanglement Structure ($\epsilon$-PSES)},
and our main question is whether there exists an $\epsilon$-PSES other than the SES, especially for small $\epsilon$.
Surprisingly,
we show that there exists infinitely many $\epsilon$-PSESs for any $\epsilon>0$.
In other words,
there exist infinite possibilities of ESs that cannot be distinguished from the SES by physical experiments of verification of maximally entangled states
even though the error of verification is extremely tiny and even though we impose projectivity.
In the next step, we explore the operational difference between PSESs and the SES in contrast to the physical similarity between $\epsilon$-PSESs and the SES.
For this aim, this paper focuses on the performance of perfect state discrimination,
and we show the infinite existence of $\epsilon$-PSESs
that have two perfectly distinguishable non-orthogonal states.
Perfect distinguishability in GPTs has been studied well
\cite{Arai2019,YAH2020,Kimura2010,Bae2016,Muller2013,BLSS2017,Barnum2019}.
For example,
while perfect distinguishablity is equivalent to orthogonality in quantum and classical theory,
the reference \cite{Muller2013} has implied that orthogonality is a sufficient condition for perfectly distinguishablity in any self-dual model under a specific condition.
Also, the reference \cite{Arai2019,YAH2020} has shown that a non-self-dual model of quantum composite systems has a distinguishable non-orthogonal pair of two states.
Therefore, it is interesting to consider
whether non-self-duality is necessary for non-orthogonal perfect distinguishability.
In this paper, we negatively solve this problem, i.e.,
we show that infinitely many $\epsilon$-PSESs with non-orthogonal distinguishability.
In other words,
some $\epsilon$-PSESs have superiority over the SES in perfect discrimination
even though $\epsilon$-PSESs cannot be distinguished from the SES by verification tasks with errors.
Finally, since the SES cannot be distinguished from ESs based on present experimental facts, $\epsilon$-undistinguishability and self-duality,
we focus on another condition, symmetry conditions. That is, we investigate what symmetry condition determines the SES.
Symmetric conditions cannot be observed directly,
but a symmetric condition plays an important role in characterizing models corresponding to Jordan Algebras out of general models of GPTs \cite{Muller2013,BLSS2017,Barnum2019}.
In this paper,
restricting the characterization to the class of ESs,
we determine the SES out of ESs by a condition about the global unitary group, which is smaller than the group in \cite{Muller2013,BLSS2017,Barnum2019}.
As a result,
we clarify that global unitary symmetry is an essential property of the SES.
The remaining part of this paper is organized as follows.
First, we introduce the mathematical definition of models and composite systems in GPTs,
and
we see non-uniqueness of models of the quantum composite systems,
i.e.,
any model satisfying the inclusion relation \eqref{eq:quantum} is regarded as the quantum composite systems in section~\ref{sect.definition}.
Next, we introduce ESs and the standard entanglement structure in section~\ref{sect.ses}.
In this section, we discuss the condition when an ES cannot be distinguished from the SES,
and we introduce $\epsilon$-undistinguishable condition.
Next, we introduce pre-duality and self-duality
as consequences of projectivity in section~\ref{sect.self-dual}.
Also, we introduce a PSES as an entanglement structure with self-duality and $\epsilon$-undistinguishable condition.
Section~\ref{sect.hierarchy} establishes a general theory for the construction of self-dual
models.
We show that any pre-dual model can be modified to a saturating model self-duality (theorem~\ref{theorem:sd}, theorem~\ref{theorem:hie1}).
In section~\ref{sect.construct}, we apply the above general theory to the quantum composite system.
We show the existence of infinitely many examples of PSESs (theorem~\ref{theorem:main}).
Also, we show that the PSESs have non-orthogonal perfectly distinguishable states (theorem~\ref{theorem:dist}) in section~\ref{sect.discrimination}.
Further, we discuss the characterization of the SES with group symmetric conditions in section~\ref{sect.symmetry}.
Finally, we
summarize our results
and give an open problem in section~\ref{sect.conclude}.
In this paper, detailed proofs of some results are written in appendix.
\begin{table*}[htb]
\caption{Notations}
\centering
\begin{tabular}{clc}
\hline
notation & meaning & equation \\ \hline \hline
$\mathcal{S}(\mathcal{K},u)$ & the state space of the model $\mathcal{K}$ with the unit $u$ &\eqref{def:state}\\
$\mathcal{E}(\mathcal{K},u)$ & the effect space of the model $\mathcal{K}$ with the unit $u$ &\eqref{def:eff} \\
$\mathcal{M}(\mathcal{K},u)$ & the measurement space of the model $\mathcal{K}$ with the unit $u$ &\eqref{def:mea} \\
$\mathcal{T}(\mathcal{H})$ & the set of all Hermitian matrices on a Hilbert space $\mathcal{H}$ &-\\
\multirow{2}{*}{$\mathcal{T}_+(\mathcal{H})$} & the set of all Positive semi-definite matrices&\multirow{2}{*}{-}\\
&\multicolumn{1}{r}{ on a Hilbert space $\mathcal{H}$ }&\\
$\mathcal{K}_1\otimes\mathcal{K}_2$ \quad& the tensor product of positive cones & \eqref{eq:tensor} \\
$\mathrm{SEP}(A;B)$\quad & the positive cone that has only separable states &\eqref{eq:sep}\\
$\mathrm{SES}(A;B)$\quad & the standard entanglement structure &\eqref{eq:SES}\\
$\mathrm{ME}(A;B)$\quad & the set of all maximally entangled states &-\\
\multirow{2}{*}{$D(\mathcal{K}\|\sigma)$\quad }& the distance between an entanglement structure $\mathcal{K}$ &\multirow{2}{*}{\eqref{def:distance1}}\\
&\multicolumn{1}{r}{ and a state $\sigma$}&\\
$D(\mathcal{K}_1\|\mathcal{K}_2)$\quad & the distance between entanglement structures $\mathcal{K}_1$ and $\mathcal{K}_2$ &\eqref{def:distance2}\\
\multirow{2}{*}{$D(\mathcal{K})$\quad} & the distance between an entanglement structure $\mathcal{K}$ &\multirow{2}{*}{\eqref{def:distance}}\\
&\multicolumn{1}{r}{ and the SES} &\\
$\tilde{\mathcal{K}}$\quad & a self-dual modification of pre-dual cone $\mathcal{K}$ \quad &-\\
$\mathrm{MEOP}(A;B)$\quad & the set of maximally entangled orthogonal projections&\eqref{eq:proj}\\
$\mathrm{NPM}_r(A;B)$\quad & a set of non-positive matrices &\eqref{def:NPM}\\
$\mathcal{K}_r(A;B)$\quad & a set of non-positive matrices with parameter $r$ &\eqref{def:Kr}\\
$r_0(A;B)$\quad & the parameter given in proposition~\ref{prop:construction1} &\eqref{def:r0}\\
\multirow{2}{*}{$\mathcal{P}_0(\vec{P})$\quad} & a family belonging to $\mathrm{MEOP}(A;B)$ &\multirow{2}{*}{\eqref{def:PE}}\\
&\multicolumn{1}{r}{ defined by a vector $\vec{P}\in\mathrm{MEOP}(A;B)$ }&\\
$\mathrm{GU}(A;B)$\quad & the group of global unitary maps&\eqref{eq:gu}\\
$\mathrm{LU}(A;B)$\quad & the group of local unitary maps&\eqref{eq:lu}\\
\multirow{2}{*}{$N(r;\{E_k\})$\quad} & a non-positive matrix with a parameter $r\ge0$&\multirow{2}{*}{\eqref{def:Nr}}\\
&\multicolumn{1}{r}{ and a family $\{E_k\}\in\mathrm{MEOP}(A;B)$ }&\\
\hline
\end{tabular}
\end{table*}
\section{GPTs and Composite systems}\label{sect.definition}
At the beginning, we simply introduce the concept of GPTs, which is a generalization of classical and quantum theory.
We consider a finite-dimensional general model that contains states and measurements.
Because any randomization of two states is also a state,
state space must be convex.
A measurement is an operation over a state to get an outcome $\omega$ with a certain probability
dependent on the given state and the way of the measurement.
Mathematically, this concept defines a measurement as
a family of functional from state space to $[0,1]$, whose output corresponds to the probability.
Also, any randomization of two measurements is also a measurement;
therefore, measurement space must be convex.
As a consequence of the above assumptions,
a model of GPTs is defined by the following mathematical setting.
Let $\mathcal{V}$ be a real vector space with an inner product $\langle,\rangle$.
We call $\mathcal{K}\subset\mathcal{V}$ a positive cone if $\mathcal{K}$ satisfies the following three conditions:
$\mathcal{K}$ is a closed convex set, $\mathcal{K}$ has an inner point, and $\mathcal{K}\cap(-\mathcal{K})=\{0\}$.
Also, we define the dual cone $\mathcal{K}^\ast$ for a positive cone $\mathcal{K}$ as $\mathcal{K}^\ast:=\{x\in\mathcal{V}\mid \langle x,y\rangle\ge0\ \forall y\in\mathcal{K} \}$.
Then, a model of GPTs is defined as a tuple $(\mathcal{V},\mathcal{K},u)$,
where $u$ is a fixed inner point in $\mathcal{K}^\ast$.
In a model of GPTs, the state space, the effect space, and the measurement space are defined as follows.
The state space $\mathcal{S}(\mathcal{K},u)$ of $(\mathcal{V},\mathcal{K},u)$ is defined as
\begin{align}\label{def:state}
\mathcal{S}(\mathcal{K},u):=\{\rho\in\mathcal{K}\mid\langle \rho,u\rangle=1\},
\end{align}
and an extremal point of $\mathcal{S}(\mathcal{K},u)$ is called a pure state.
Also, the effect space $\mathcal{E}(\mathcal{K},u)$ and the measurement space $\mathcal{M}(\mathcal{K},u)$ of $(\mathcal{V},\mathcal{K},u)$ is respectively defined as
\begin{align}
\mathcal{E}(\mathcal{K},u):&=\left\{e\in\mathcal{K}^\ast\mid 0\le\langle e,\ \rho\rangle\le1 \ \forall \rho\in\mathcal{S}(\mathcal{K},u)\right\},\label{def:eff}\\
\mathcal{M}(\mathcal{K},u):&=\left\{\{M_{\omega}\}_{\omega\in\Omega}\middle| M_\omega\in\mathcal{K}^\ast, \ \sum_{\omega\in\Omega} M_\omega=u\right\},\label{def:mea}
\end{align}
where $\Omega$ is the finite set of outcome.
Also, an extremal element $e\in\mathcal{E}(\mathcal{K},u)$ is called a pure effect.
Besides, the probability to get an outcome $\omega$ is given by $\langle \rho,M_\omega\rangle$ for a state $\rho\in\mathcal{S}(\mathcal{K},u)$ and a measurement $\{M_\omega\}_{\omega\in\Omega}\in\mathcal{M}(\mathcal{K},u)$.
Here, we remark the definition of a measurement.
A vector space and its dual space are mathematically equivalent when the dimension of vector space is finite.
In this paper, the effect space and the measurement space are defined as subsets of the original vector space for later convenience.
The above mathematical setting is a generalization of classical and quantum theory.
For example, the model of quantum theory is given by the model $(\mathcal{T}(\mathcal{H}),\mathcal{T}_+(\mathcal{H}),I)$,
where $\mathcal{T}(\mathcal{H})$, $\mathcal{T}_+(\mathcal{H})$, and $I$ are denoted as the set of all Hermitian matrices on Hilbert space $\mathcal{H}$, the set of all positive semi-definite (PSD) matrices on $\mathcal{H}$, and the identity matrix on $\mathcal{H}$, respectively.
Then, the state space $\mathcal{S}(\mathcal{T}_+(\mathcal{H}),I)$ and the measurement space $\mathcal{M}(\mathcal{T}_+(\mathcal{H}),I)$ are equal to the set of all density matrices and the set of all positive operator valued measures (POVMs), respectively.
This is because the dual $\mathcal{T}_+(\mathcal{H})^\ast$ is equal to itself.
This property $\mathcal{K}^\ast=\mathcal{K}$ is called self-duality, as we mention later.
In this way, the model $(\mathcal{T}(\mathcal{H}),\mathcal{T}_+(\mathcal{H}),I)$ is regarded as the model of quantum theory.
Next, we define a model of composite systems in GPTs.
We say that a model $(\mathcal{V},\mathcal{K},u)$ is a model of the composite system of two submodels $(\mathcal{V}_A,\mathcal{K}_A,u_A)$ and $(\mathcal{V}_B,\mathcal{K}_B,u_B)$
when the model $(\mathcal{V},\mathcal{K},u)$ satisfies the following three conditions:
(i) $\mathcal{V}=\mathcal{V}_A\otimes\mathcal{V}_B$,
(ii) $\mathcal{K}_A\otimes\mathcal{K}_B\subset\mathcal{K}\subset(\mathcal{K}_A^\ast\otimes\mathcal{K}_B^\ast)^\ast$,
and (iii) $u=u_A\otimes u_B$.
Here, the tensor product of two cones $\mathcal{K}_A\otimes\mathcal{K}_B$ is defined as
\begin{align}\label{eq:tensor}
\mathcal{K}_A\otimes\mathcal{K}_B:=\left\{\sum_k a_k\otimes b_k\middle| a_k\in\mathcal{K}_A,\ b_k\in\mathcal{K}_B\right\}.
\end{align}
This definition derives from the following physical reasonable assumption.
The composite system contains Alice's system $(\mathcal{V}_A,\mathcal{K}_A,u_A)$ and Bob's system $(\mathcal{V}_B,\mathcal{K}_B,u_B)$.
It is natural to assume that Alice and Bob can prepare local states $\rho_A\in\mathcal{S}(\mathcal{K}_A,u_A)$ and $\rho_B\in\mathcal{S}(\mathcal{K}_B,u_B)$ independently.
Consequently, the product state $\rho_A\otimes\rho_B$ is prepared in the composite system (figure~\ref{figure-composite}),
i.e.,
the global state space $\mathcal{S}(\mathcal{K},u_A\otimes u_B)$ contains the product state $\rho_A\otimes\rho_B$.
This scenario implies the inclusion $\mathcal{K}_A\otimes\mathcal{K}_B\subset\mathcal{K}$.
Similarly, the product effect $e_A\otimes e_B$ can also be prepared in the composite system.
This scenario also implies $\mathcal{K}_A^\ast\otimes\mathcal{K}_B^\ast\subset\mathcal{K}^\ast$,
which is rewritten as $\mathcal{K}\subset(\mathcal{K}_A^\ast\otimes\mathcal{K}_B^\ast)^\ast$.
We give another scenario that derives the definition of models of composite systems in appendix~\ref{append-com}.
\begin{figure}[t]
\centering
\includegraphics[width=8cm]{figure_composite_2.pdf}
\caption{
When local states $\rho_A$ and $\rho_B$ are prepared individually by Alice and Bob,
the product states $\rho_A\otimes\rho_B$ is prepared on composite system.
}
\label{figure-composite}
\end{figure}
\section{Entanglement structures and the standard entanglement structure}\label{sect.ses}
Now, let us consider the composite system of two quantum subsystems $(\mathcal{T}(\mathcal{H}_A),\mathcal{T}_+(\mathcal{H}_A),I_A)$ and $(\mathcal{T}(\mathcal{H}_B),\mathcal{T}_+(\mathcal{H}_B),I_B)$.
An entanglement structure,
i.e.,
a model of the composite system is given as $(\mathcal{T}(\mathcal{H}_A\otimes \mathcal{H}_B),\mathcal{K},I_{A;B})$ that satisfies
\begin{gather}
\mathrm{SEP}(A;B)\subset\mathcal{K}\subset\mathrm{SEP}^\ast(A;B),\label{eq:quantum}\\
\mathrm{SEP}(A;B):=\mathcal{T}_+(\mathcal{H}_A)\otimes\mathcal{T}_+(\mathcal{H}_B)\label{eq:sep}.
\end{gather}
The cone $\mathrm{SEP}(A;B)$ corresponds to the model that has only separable states,
but the model has beyond-quantum measurements that can discriminate non-orthogonal separable states \cite{Arai2019}.
Also, the cone $\mathrm{SEP}^\ast(A;B)$ corresponds to the model that has elements in $\mathrm{SEP}^\ast(A;B)\setminus\mathcal{T}_+(\mathcal{H}_A\otimes\mathcal{H}_B)$, which are regarded as more strongly entangled elements.
It is believed that actual quantum composite systems obey the model $\mathcal{T}_+(\mathcal{H}_A\otimes\mathcal{H}_B)$,
and an important aim of studies of GPTs is to characterize this model.
Hereinafter, we call this model \textit{standard entanglement structure} (SES),
and we use the notation
\begin{align}\label{eq:SES}
\mathrm{SES}(A;B):=\mathcal{T}_+(\mathcal{H}_A\otimes\mathcal{H}_B).
\end{align}
In this way, a model of composite systems is not uniquely determined in general,
i.e.,
there are many possible entanglement structures of the composite system in GPTs.
Next, to consider the experimental verification of a given model,
we introduce the distinguishability of two state spaces of two given models $\mathcal{K}_1$ and $\mathcal{K}_2$.
Because any Hermitian matrices $X$, $\rho$, and $\sigma$ satisfy the inequality
\begin{align}
\left|\Tr X\rho-\Tr X\sigma\right|\le\|X\|_\infty\|\rho-\sigma\|_1,
\end{align}
this paper estimates the error probability of verification tasks by trace norm,
where $\|\ \|_\infty$ is spectral norm.
Therefore,
given a state $\sigma\in\mathcal{S}(\mathcal{K}_2,u_2)$, the quantity
\begin{align}\label{def:distance1}
D(\mathcal{K}_1\|\sigma):= \min_{\rho \in \mathcal{S}(\mathcal{K}_1,u_1)}\| \rho-\sigma\|_1
\end{align}
expresses how well
the state $\sigma$ is distinguished from states in $\mathcal{K}_1$.
Optimizing the state $\sigma$, we consider the quantity
\begin{align}\label{def:distance2}
D(\mathcal{K}_1\|\mathcal{K}_2):=\max_{\sigma \in \mathcal{S}(\mathcal{K}_2,u_2)} D(\mathcal{K}_1\|\sigma),
\end{align}
which expresses the optimum distinguishability of the model $\mathcal{K}_2 $
from the model $\mathcal{K}_1$.
Hence, the quantity $D(\mathrm{SES}(A;B) \|\mathcal{K})$ expresses
how the standard model $\mathrm{SES}(A;B)$
can be distinguished from a model $\mathcal{K}$.
However, we often consider the verification of
a maximally entangled state
because a maximally entangled state is the furthest state from separable states.
In order to consider maximally entangled states,
we assume that $\dim(\mathcal{H}_A)=^dim(\mathcal{H}_B)=d$ in the following discussion.
When the range of the above maximization \eqref{def:distance2} is restricted to maximally entangled states,
the distinguishability of the standard model $\mathrm{SES}(A;B)$ from
the model $\mathcal{K}$ is measured by the following quantity:
\begin{align}\label{def:distance}
D(\mathcal{K})
:= &\max_{\sigma \in \mathrm{ME}(A;B) } D(\mathcal{K}\|\sigma),
\end{align}
where the set $\mathrm{ME}(A;B)$ is denoted as the set of all maximally entangled states on $\mathcal{H}_A\otimes\mathcal{H}_B$.
Given a model $\mathcal{K}$,
we introduce \textit{$\epsilon$-undistinguishable condition} as
\begin{align}\label{cd:epsilon}
D(\mathcal{K})\le\epsilon.
\end{align}
That is, if a model $\mathcal{K}$ satisfies $\epsilon$-undistinguishablity,
even when we pass the verification test for any maximally entangled state,
we cannot deny the possibility that our system is the model $\mathcal{K}$ (figure~\ref{figure-cverification}).
Clearly, there are many models satisfying this condition.
For example,
$\mathrm{SEP}^\ast$ satisfies it because $D(\mathrm{SEP}^\ast)=0$.
In other words, it is impossible to deny such a possibility without assuming an additional constraint for our model.
The aim of this paper is to examine whether there exists a natural condition to deny $\epsilon$-undistinguishablity.
As a natural condition, the next section introduces self-duality
via projective measurements.
\begin{figure}[t]
\centering
\includegraphics[width=12cm]{figure_verification.pdf}
\caption{
Even if the verifier's system is subject to an $\epsilon$-undistinguishable entanglement structure $\mathcal{K}\neq\mathrm{SES}(A;B)$,
the verifier achieves the verification task of a given maximally entangled state $\sigma$ with error $\epsilon$ by preparing a state $\rho'\in\mathcal{K}$ satisfying $\|\rho'-\sigma\|_1\le\epsilon$.
In this sense, such verification tasks can not distinguish the entanglement structures $\mathrm{SES}(A;B)$ and $\mathcal{K}$ when $\mathcal{K}$ satisfies $\epsilon$-undistinguishability.
}
\label{figure-cverification}
\end{figure}
\section{Projective measurement, self-duality, and pseudo standard entanglement structures}\label{sect.self-dual}
Next, we introduce pre-duality and self-duality via projective measurements.
In standard quantum theory,
there exists a measurement $\{e_i\}_{i\in I}$ such that the post-measurement state with the outcome $i$ is given as $e_i/\Tr e_i$ independently of the initial state when the effect $e_i$ is pure.
Such a measurement is called a projective measurement \cite{Neumann1932,Davies1970,Ozawa1984}.
The measurement projectivity is one of the postulates of standard quantum theory \cite{Neumann1932,Davies1970,Ozawa1984}.
Therefore, in this paper, we impose that any model $\mathcal{K}$ satisfies the following condition:
for any pure effect $e\in\mathcal{E}(\mathcal{K},u)$, there exists a measurement $\{e_i\}$ such that
an element $e_{i_0}$ is equal to $e$, and the post-measurement state is given as $\overline{e_{i_0}}:=e_{i_0}/\Tr e_{i_0}$.
Because any effect satisfies the condition $0\le\langle e,\rho\rangle\le1\ \forall \rho\in\mathcal{S}(\mathcal{K},u)$ in \eqref{def:eff},
the element $u-e$ also belongs to $\mathcal{E}(\mathcal{K},u)$,
which implies that
the family $\{e,u-e\}$ belongs to $\mathcal{M}(\mathcal{K},u)$ for any effect $e\in\mathcal{E}(\mathcal{K},u)$.
Also,
pure effects span the effect space $\mathcal{E}(\mathcal{K},u)$ with convex combination,
and the effect space $\mathcal{E}(\mathcal{K},u)$ generates the dual cone $\mathcal{K}^\ast$ with constant time.
Therefore, the existence of projective measurement implies the inclusion relation $\mathcal{K}\supset\mathcal{K}^\ast$.
In this paper, this property $\mathcal{K}\supset\mathcal{K}^\ast$ is called pre-duality.
\begin{figure}[t]
\centering
\includegraphics[width=4cm]{figure_projection.pdf}
\caption{
When a projective measurement $\{e_i\}$ is applied to the system with an initial state $\rho$,
we obtain an outcome $i$ and the corresponding post-measurement state $\overline{e_i}=e_i/\Tr e_i$ independent of the initial state $\rho$.
}
\label{figure-projection}
\end{figure}
Here, we remark on the relation between projectivity and repeatability.
Repeatability is a postulate of standard quantum theory, sometimes included in the projection postulate \cite{Neumann1932,Chefles2003,Buscemi2004,CY2014,CY2016,Perinotti2012,SSKH2021}.
Repeatability ensures that
the same effect is observed with probability 1 in the sequence of the same measurements,
and the effects do not change the post-measurement state (figure~\ref{figure-repeatable}).
\begin{figure}[t]
\centering
\includegraphics[width=6cm]{figure_repeatability_2.pdf}
\caption{When an initial state is measured by a measurement $\{e_i\}$ twice,
the post-measurement states of first and second measurement with an outcome $i$ are equivalent.
}
\label{figure-repeatable}
\end{figure}
In standard quantum theory,
repeatability is sometimes confounded with the above condition that any pure effects can constructs a projective measurement.
However, similarly to the reference \cite{Neumann1932,Chefles2003,Buscemi2004,CY2014,CY2016,Perinotti2012,SSKH2021} and the translated version \cite[Discussion]{Luders1951}, the above concept of repeatability implies the following more weaker condition than the existence of projective measurements.
Repeatability requires that the tuple of post-measurement states $\{\sigma_{e_i}\}_{i\in I}$ is perfectly distinguishable by the measurement $\{e_i\}_{\i\in I}$,
i.e.,
the equation $\Tr \sigma_{e_i} e_j=\delta_{i,j}$.
In other words,
repeatability requests the $|I|$ number of constraints for the post-measurement state $\sigma_{e_i}$.
On the other hand,
projectivity determines post-measurement states completely.
In other words,
projectivity requests the same number of constraints for the post-measurement state as the dimension of $\mathcal{K}^\ast$.
In general, the number of outcomes $|I|$ is smaller than the dimension of $\mathcal{K}^\ast$;
therefore, projectivity is a stronger postulate than repeatability in terms of the number of constraints.
Now, we consider pre-dual models of composite systems $(\mathcal{T}(\mathcal{H}_A\otimes \mathcal{H}_B),\mathcal{K},I_{A;B})$.
For example,
let us consider the model that contains only separable measurements.
In such a model, the dual cone is given as $\mathcal{K}^\ast=\mathrm{SEP}(A;B)$,
and therefore, the model satisfies $\mathcal{K}=\mathrm{SEP}(A;B)^\ast$ because the dual of a dual cone is equal to the original cone.
However, the state space $\mathcal{S}(\mathrm{SEP}(A;B)^\ast,I_{A;B})$ has excessive many states;
the state space $\mathcal{S}(\mathrm{SEP}(A;B)^\ast,I_{A;B})$ has not only all quantum states but also all entanglement witnesses with trace 1.
Then, there exist two state $\rho_1,\rho_2\in\mathcal{S}(\mathcal{K},I_{A;B})$ such that they satisfy $\Tr\rho_1\rho_2<0$.
Not only the case $\mathcal{K}=\mathrm{SEP}(A;B)^\ast$,
but also any pre-dual model has two states $\rho_1,\rho_2$ with $\Tr\rho_1\rho_2<0$ unless $\mathcal{K}=\mathcal{K}^\ast$.
In this way, pre-dual models have a gap between the state space and the effect space unless $\mathcal{K}=\mathcal{K}^\ast$.
In order to remove such a gap,
as the saturated situation of pre-duality,
we extend the measurement effect space and restrict the state space by modifying the cone to $\tilde{\mathcal{K}}$ with satisfying $\tilde{\mathcal{K}}\supset\tilde{\mathcal{K}}^\ast$.
Here, we denote the modified model as $(\mathcal{T}(\mathcal{H}_A\otimes \mathcal{H}_B),\tilde{\mathcal{K}},I_{A;B})$,
and we say that the model is \textit{self-dual} if the cone $\tilde{\mathcal{K}}$ satisfies $\tilde{\mathcal{K}}^\ast=\tilde{\mathcal{K}}$.
Self-duality denies entanglement structures whose effect space is strictly larger than state space,
for example, $\mathrm{SEP}^\ast(A;B)$.
In this paper,
in order to investigate quantum-like entanglement structures,
we consider the combination of $\epsilon$-undistinguishable condition and self-duality.
Hereinafter,
we say that an entanglement structure $\mathcal{K}$ is an \textit{$\epsilon$-pseudo standard entanglement structure} ($\epsilon$-PSES)
if $\mathcal{K}$ satisfies $\epsilon$-undistinguishable condition and self-duality.
A typical example of $\epsilon$-PSESs is, of course, the SES,
but
another example of $\epsilon$-PSESs is not known,
especially in the case when $\epsilon$ is very small.
If there exists another $\epsilon$-PSES for small epsilon,
the model cannot be distinguished from the SES by physical experiments of verification of maximally entangled states with small errors even though we impose projectivity.
In this paper,
we investigate the problem of whether there exists another example of $\epsilon$-PSESs,
especially another entanglement structure $\mathcal{K}$ with self-duality and \eqref{cd:epsilon}.
As a result,
we give an infinite number of examples of PSESs by applying a general theory given in the next section.
\section{Self-dual modification and hierarchy of pre-dual cones with symmetry under operations}\label{sect.hierarchy}
In this section,
we state general theories to show the existence of self-dual models satisfying \eqref{eq:quantum}.
The first result is that any pre-dual model can always be modified to a self-dual model.
\begin{theorem}[self-dual modification]\label{theorem:sd}
Let $\mathcal{K}$ be a pre-dual cone in $\mathcal{V}$.
Then, there exists a positive cone $\tilde{\mathcal{K}}$ such that
\begin{align}\label{eq:sd}
\mathcal{K}\supset\tilde{\mathcal{K}}=\tilde{\mathcal{K}}^\ast\supset\mathcal{K}^\ast.
\end{align}
\end{theorem}
A self-dual cone $\tilde{\mathcal{K}}$ to satisfy \eqref{eq:sd} is called a self-dual modification (SDM) of $\mathcal{K}$.
Here, we remark that the reference \cite{BF1976} has also shown a result essentially similar to Theorem~\ref{theorem:sd}.
In the reference \cite{BF1976}, a cone is defined as a closed convex set satisfying only the property that $rx\in\mathcal{C}$ for any $r\ge0$ and any $x\in\mathcal{C}$.
This paper assumes additional properties, $\mathcal{C}$ has non-empty interior and $\mathcal{C}\cap(-\mathcal{C})=\{0\}$.
Actually, we can easily modify the proof in \cite{BF1976} in our definition,
but this thesis gives another proof
for reader's convenience in appendix~\ref{append-2}.
Also, we remark on the difference between self-dual modification and self-dualization in \cite{Janotta2013}.
The reference \cite{Janotta2013} has shown that the state space and the effect space
of any model can be transformed by a linear homomorphism from one to another, where the effect space is considered as the subset of $\mathcal{V}^\ast$.
The result \cite{Janotta2013} can be interpreted in our setting as follows; The state space and effect space become equivalent by changing the inner product.
This process is called self-dualization in \cite{Janotta2013}.
However, our motivation is constructing models of
the composite system with keeping the inner product to be the product form of
the inner products in the models of the subsystems.
Therefore, the result \cite{Janotta2013} cannot be used for our purpose.
A given pre-dual cone does not uniquely determine SDM
because the proof of Theorem~\ref{theorem:sd} and the proof in \cite{BF1976} are neither constructive nor deterministic.
Indeed, even when two self-dual cones are self-dual modifications of different pre-dual cones, they are not necessarily different self-dual cones in general.
For example, when we have three different self-dual cones $\mathcal{K}_1,\mathcal{K}_2,\mathcal{K}_3$,
then $\mathcal{K}_1+\mathcal{K}_2$ and $\mathcal{K}_2+\mathcal{K}_3$ are pre-dual cones,
but $\mathcal{K}_2$ is regarded as a modification of $\mathcal{K}_1+\mathcal{K}_2$ and $\mathcal{K}_2+\mathcal{K}_3$.
Hence, the following two concepts are useful to clarify the difference among self-dual modifications.
\begin{definition}[$n$-independence]
For a natural number $n$, we say that a family of sets $\{\mathcal{K}_i\}_{i=1}^n$ is $n$-independent if no sets $\mathcal{K}_i\ (1\le i\le n)$ satisfy that $\mathcal{K}_i \subset \sum_{j\neq i} \mathcal{K}_j$.
Especially, we say that $\{\mathcal{K}_i\}_{i=1}^n$ is $n$-independent family of cones when any $\mathcal{K}_i$ is a positive cone.
\end{definition}
\begin{definition}[exact hierarcy with depth $n$]
For a natural number $n$, we say that pre-dual cone $\mathcal{K}$ has an exact hierarchy with depth $n$
if there exists a family of sets $\{\mathcal{K}_i\}_{i=1}^n$ such that
\begin{align}
\mathcal{K}\supset\mathcal{K}_1\supsetneq\mathcal{K}_2\supsetneq\cdots\supsetneq\mathcal{K}_n
\supset\mathcal{K}_n^\ast\supsetneq\cdots\supsetneq\mathcal{K}_1^\ast\supset\mathcal{K}^\ast.
\end{align}
Especially, we say that $\{\mathcal{K}_i\}_{i=1}^n$ is exact hierarchy of cones when any $\mathcal{K}_i$ is a positive cone.
\end{definition}
Then, as an extension of theorem~\ref{theorem:sd},
the following theorem shows the equivalence between
the existence of an $n$-independent family of self-dual cones
and the existence of an exact hierarchy of pre-dual cones with depth $n$.
\begin{theorem}\label{theorem:hie1}
Let $\mathcal{K}$ be a positive cone.
The following two statements are equivalent:
\begin{enumerate}
\item there exists an exact hierarchy of pre-dual cones $\{\mathcal{K}_i\}_{i=1}^n$ satisfying $\mathcal{K}\supset\mathcal{K}_i\supset\mathcal{K}^\ast$.
\item there exists an $n$-independent family of self-dual cones $\{\mathcal{L}_i\}_{i=1}^n$ satisfying that $\mathcal{L}_i$ is a self-dual modification of $\mathcal{K}_i$,
i.e.,
$\mathcal{L}_i$ is a self-dual cone satisfying $\mathcal{K}_i\supset\mathcal{L}_i\supset\mathcal{K}_i^\ast$.
\end{enumerate}
\end{theorem}
The proof of theorem~\ref{theorem:hie1} is written in appendix~\ref{append-hie}.
\section{Existence of infinite $\epsilon$-PSESs}\label{sect.construct}
In this section,
in order to discuss the existence of $\epsilon$-PSESs,
we apply theorem~\ref{theorem:hie1} to ESs on $\mathcal{H}_A\otimes\mathcal{H}_B$ with $\dim(\mathcal{H}_A)=\dim(\mathcal{H}_B)=d$ (due to the $\epsilon$-undistinguishable condition).
As a result, we show that there exist infinitely many exactly different $\epsilon$-PSESs.
First,
we denote
$\mathrm{MEOP}(A;B)$
as the set of all maximally entangled orthogonal projections on $\mathcal{H}_A\otimes\mathcal{H}_B$,
i.e.,
\begin{align}\label{eq:proj}
\begin{aligned}
&\mathrm{MEOP}(A;B):=\Bigl\{\vec{E}=\{\ketbra{\psi_k}{\psi_k}\}_{k=1}^{d^2} \Bigm\vert \braket{\psi_k|\psi_l}=\delta_{kl}, \\
&\ketbra{\psi_k}{\psi_k} :\mbox{ maximally entangled state on }\mathcal{H}_A\otimes\mathcal{H}_B\Bigr\}.
\end{aligned}
\end{align}
Now, we define the followin sets for the construction of PSESs.
\begin{definition}\label{definition:Kr}
Given a subset $\mathcal{P}\subset\mathrm{MEOP}(A;B)$ and a parameter $r\ge0$,
we define the following set of non-positive matrices:
\begin{align}
\mathrm{NPM}_r(\mathcal{P})
&:=\Bigl\{\rho=-\lambda E_1+(1+\lambda)E_2+\frac{1}{2}\sum_{k=3}^{d^2}E_k\Big|
0\le\lambda\le r,\ \vec{E}=\{E_k\}\in\mathcal{P}
\Bigr\}.\label{def:NPM}
\end{align}
Using the above set $\mathrm{NPM}_r(\mathcal{P})$,
given a parameter $r\ge0$,
we define the following two cones
$\mathcal{K}^{(0)}_r(\mathcal{P})$
and
$\mathcal{K}_r(\mathcal{P})$
as
\begin{align}
\mathcal{K}^{(0)}_r(\mathcal{P})
:&=\mathrm{SES}(A;B)+
\mathrm{NPM}_r(\mathcal{P})
,\\
\mathcal{K}_r(\mathcal{P})
:&=\left(
\mathcal{K}^{(0)\ast}_r(\mathcal{P})+\mathrm{NPM}_r(\mathcal{P})
\right)^\ast.\label{def:Kr}
\end{align}
\end{definition}
Then, the following proposition holds.
\begin{proposition}\label{prop:construction1}
Given $\mathcal{H}_A$, $\mathcal{H}_B$,
define a real number $r_0(A;B)$ as
\begin{align}\label{def:r0}
r_0(A;B):=\left(\sqrt{2d}-2\right)/4.
\end{align}
When two parameters $r_1$ and $r_2$ satisfy $r_2\le r_1\le r_0(A;B)$,
two cones
$\mathcal{K}_{r_1}(\mathcal{P})$
and
$\mathcal{K}_{r_2}(\mathcal{P})$
are pre-dual cones satisfying \eqref{eq:quantum}
and the inclusion relation
\begin{align}\label{eq:con-hie}
\mathcal{K}_{r_2}(\mathcal{P})\subsetneq\mathcal{K}_{r_1}(\mathcal{P}).
\end{align}
\end{proposition}
The proof of proposition~\ref{prop:construction1} is written in appendix~\ref{append-construction}.
Proposition~\ref{prop:construction1} guarantees that
$\mathcal{K}_r(\mathcal{P})$
is pre-dual for any $r\le r_0$.
Therefore, theorem~\ref{theorem:sd} gives a self-dual modification of
$\mathcal{K}_r(\mathcal{P})$
with \eqref{eq:quantum}.
Next, we calculate the value $D(\tilde{\mathcal{K}}_r(\mathcal{P}))$.
The following proposition estimates the value $D(\tilde{\mathcal{K}}_r(\mathcal{P}))$.
\begin{proposition}\label{prop:construction2}
Given a parameter $r$ with $0<r\le r_0(A;B)$
and a self-dual modification
$\tilde{\mathcal{K}}_r(\mathcal{P})$,
the following inequality holds:
\begin{align}\label{eq:est-distance}
D(\tilde{\mathcal{K}}_r(\mathcal{P}))
\le2\sqrt{\cfrac{2r}{2r+1}}.
\end{align}
\end{proposition}
The proof of proposition~\ref{prop:construction2} is written in appendix~\ref{append-construction}.
For the latter use, we define the parameter $\epsilon_r$ as
\begin{align}\label{def:er}
\epsilon_r:=2\sqrt{\cfrac{2r}{2r+1}}.
\end{align}
Proposition~\ref{prop:construction2} implies that the model
$\tilde{\mathcal{K}_r}(\mathcal{P})$
is an $\epsilon_r$-PSES with \eqref{eq:quantum}.
Also,
due to \eqref{eq:con-hie} in proposition~\ref{prop:construction1},
for an arbitrary number $n$,
an exact inequality
\begin{align}\label{eq:ri}
0<r_n<\cdots<r_1\le r_0(A;B)
\end{align}
gives an exact hierarchy of pre-dual cones
$\{\mathcal{K}_{r_i}(\mathcal{P})\}_{i=1}^n$
with \eqref{eq:quantum}.
Thus,
theorem~\ref{theorem:hie1} gives an independent family
$\{\tilde{\mathcal{K}}_{r_i}(\mathcal{P})\}$
with \eqref{eq:quantum},
and the distance
$D(\tilde{\mathcal{K}}_{r_i}(\mathcal{P}))$
is estimated as
\begin{align}
D(\tilde{\mathcal{K}}_{r_i}(\mathcal{P}))
\le2\sqrt{\cfrac{2r_i}{2r_i+1}}<2\sqrt{\cfrac{2r_1}{2r_1+1}}=\epsilon_{r_1}
\end{align}
by inequalities \eqref{eq:est-distance} and \eqref{eq:ri}.
In other words, the family
$\{\tilde{\mathcal{K}}_{r_i}(\mathcal{P})\}$
is an $n$-independent family of $\epsilon_{r_1}$-PSESs.
Because $n$ is arbitrary and $\epsilon_{r_1}\to0$ holds with $r_1\to0$,
we obtain the following theorem.
\begin{theorem}\label{theorem:main}
For any $\epsilon>0$,
there exists an infinite number of $\epsilon$-PSESs.
\end{theorem}
In other words,
there exist infinitely many ESs that cannot be distinguished from the SES by a verification of a maximally entanglement state with small errors
even if the ES is self-dual.
\section{Non-orthogonal discrimination in PSESs}
\label{sect.discrimination}
As the above,
there exist infinite $\epsilon$-PSESs except for the SES
even though $\epsilon$-PSESs are physically similar to the SES.
As the next step,
in this section,
we discuss the operational difference between $\epsilon$-PSESs and the SES in terms of informational tasks.
We focus on the difference between the behaviors of perfect discrimination in
$\tilde{\mathcal{K}}_r(\mathcal{P})$
and $\mathrm{SES}$.
As a result, we show that there exists non-orthogonal perfectly distinguishable states in
$\tilde{\mathcal{K}}_r(\mathcal{P})$ for a certain subset $\mathcal{P}\subset\mathrm{MEOP}(A;B)$.
In GPTs, perfect distinguishability is defined similarly to quantum theory as follows.
\begin{definition}[perfect distinguishablity]
Let $\{\rho_k\}_{k=1}^n$ be a family of states $\rho_k\in\mathcal{S}(\mathcal{K},u)$.
Then, $\{\rho_k\}_{k=1}^n$ are perfectly distinguishable
if there exists a measurement $\{M_k\}_{k=1}^n\in\mathcal{M}(\mathcal{K},u)$ such that
$\langle \rho_k,M_l\rangle=\delta_{kl}$.
\end{definition}
The reference \cite{Muller2013} has implied that orthogonality is a sufficient condition for perfectly distinguishablity in any self-dual model under a certain condition in the proof of its main theorem.
Also, the reference \cite{Arai2019} has shown that a model of quantum composite system with non-self-duality has a pair of two distinguishable non-orthogonal states.
Therefore, it is non-trivial problem whether there exists a self-dual model that has non-orthogonal distinguishable states.
In this section, we show that
any self-dual modification $\tilde{\mathcal{K}}_r(\mathcal{P})$ in section~\ref{sect.construct}
has a measurement to discriminate non-orthogonal states in $\tilde{\mathcal{K}}_r(\mathcal{P})$ perfectly
for a certain subset $\mathcal{P}\subset\mathrm{MEOP}(A;B)$.
First,
given a vector $\vec{P}=\{P_k\}_{k=1}^{d^2}\in\mathrm{MEOP}(A;B)$,
we define a vector $\vec{E_P}=\{P'_k\}_{k=1}^{d^2}\in\mathrm{MEOP}(A;B)$ as
\begin{align}\label{eq:E-E'}
P'_1:=P_2,\quad P'_2:=P_1, \quad P'_k=P_k \ (k\ge3).
\end{align}
Then,
given a vector $\vec{P}=\{P_k\}_{k=1}^{d^2}\in\mathrm{MEOP}(A;B)$,
we define a subset $\mathcal{P}_0(\vec{P})\subset\mathrm{MEOP}(A;B)$ as
\begin{align}\label{def:PE}
\mathcal{P}_0(\vec{P}):=\{\vec{P},\vec{E_P}\}.
\end{align}
Now, we consider perfect discrimination in a self-dual modification $\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{E}))$.
By the equations \eqref{def:NPM} and \eqref{def:PE},
the following two matrices belong to $\mathrm{NPM}_r(\mathcal{P}_0(\vec{E}))$ for any $\vec{E}$ and any $0\le\lambda\le r$:
\begin{align}
\begin{aligned}
\label{eq:measurement}
M_1(\lambda;\vec{P})&:=-\lambda P_1+(1+\lambda)P_2+\frac{1}{2}\sum_{k\ge3}P_k,\\
M_2(\lambda;\vec{P})&:=-\lambda P_1'+(1+\lambda)P_2'+\frac{1}{2}\sum_{k\ge3}P_k'
=(1+\lambda) P_1-\lambda P_2+\frac{1}{2}\sum_{k\ge3}P_k,
\end{aligned}
\end{align}
which implies that $M_i(\lambda;\vec{P})\in\mathcal{K}_r^{(0)\ast}(\mathcal{P}_0(\vec{P}))\subset\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$ for $i=1,2$.
Also, because of the equation \eqref{eq:E-E'},
the equation $M_1(\lambda;\vec{P})+M_2(\lambda;\vec{P})=I$ holds.
Therefore, the family $M(\lambda;\vec{P})=\{M_i(\lambda;\vec{P})\}_{i=1,2}$ is a measurement in $\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$ when $0\le\lambda\le r$.
Next, we choose a pair of distinguishable states by $M(\lambda;\vec{P})$.
Let $\ket{\psi_k}$ be a normalized eigenvector of $P_k$.
Then, we define two states $\rho_1,\rho_2$ as follows:
\begin{align}
\begin{aligned}
\rho_1:&=\ketbra{\phi_1}{\phi_1},\quad \rho_2:=\ketbra{\phi_2}{\phi_2},\\
\ket{\phi_1}:&=\sqrt{\cfrac{r}{2r+1}}\ket{\psi}_1+\sqrt{\cfrac{r+1}{2r+1}}\ket{\psi}_2,\\
\ket{\phi_2}:&=\sqrt{\cfrac{r+1}{2r+1}}\ket{\psi}_1+\sqrt{\cfrac{r}{2r+1}}\ket{\psi}_2.
\end{aligned}
\end{align}
Because of the relation $\vec{P}\in\mathrm{MEOP}(A;B)$, the projections $P_i$ and $P_j$ are orthogonal for $i\neq j$,
which implies,
the equations
\begin{align}\label{eq:psi1-2}
\braket{\psi_i|\psi_j}&=\delta_{i,j},\\
\braket{\psi_i|P_j|\psi_i}&=\delta_{i,j}.\label{eq:psi-E}
\end{align}
Therefore, the following relation holds for $i,j=1,2$:
\begin{align}
\Tr \rho_iM_j(r;\vec{P})=\delta_{i,j},
\end{align}
i.e.,
the states $\rho_1$ and $\rho_2$ are distinguishable by the measurement $M(\lambda;\vec{P})$.
Next, we show that $\rho_1,\rho_2\in\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$,
which is shown as follows.
Because of the equation
$\mathrm{NPM}_r(\mathcal{P}_0(\vec{P})):=\{M_i(\lambda;\vec{P}) | 0\le \lambda\le r,\ i=1,2\}$,
any extremal element $x\in\mathcal{K}_r^{(0)}(\mathcal{P}_0(\vec{P}))$ can be written as $x=\sigma+M_i(\lambda;\vec{P})$,
where $\sigma\in\mathrm{SES}(A;B)$, $0\le \lambda\le r$, $i=1,2$.
Besides, the following two inequalities hold:
\begin{align}
\Tr \rho_1 M_1(\lambda;\vec{P})&=-\lambda\cfrac{r}{2r+1}+(1+\lambda)\cfrac{r+1}{2r+1}
=(\lambda+r+1)\cfrac{1}{2r+1}\stackrel{(a)}{\ge}\cfrac{r+1}{2r+1}\ge0,\label{eq:rho1M1}\\
\Tr \rho_1 M_2(\lambda;\vec{P})&=-\lambda\cfrac{r+1}{2r+1}+(1+\lambda)\cfrac{r}{2r+1}=(-\lambda+r)\cfrac{1}{2r+1}\stackrel{(b)}{\ge}0.\label{eq:rho1M2}
\end{align}
The equations $(a)$ and $(b)$ are shown by the inequality $0\le \lambda \le r$.
Because the inequality $\Tr \rho_1\sigma\ge0$ holds for any $\sigma\in\mathrm{SES}(A;B)$,
we obtain $\Tr \rho_1 x\ge0$ for any $x\in\mathcal{K}_r^{(0)}(\mathcal{P}_0(\vec{P}))$,
which implies $\rho_1\in\mathcal{K}^{(0)\ast}_r(\mathcal{P}_0(\vec{P}))$.
Therefore, $\rho_1\in\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$ because of the inclusion relation $\mathcal{K}^{(0)\ast}_r(\mathcal{P}_0(\vec{P}))\subset \tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$.
The same discussion derives that $\rho_2\in\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$.
As a result,
we obtain a measurement and a distinguishable pair of two states by the measurement in
$\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$.
Finally, the following equality implies that $\rho_1$ and $\rho_2$ are non-orthogonal for $r>0$:
\begin{align}
\Tr\rho_1\rho_2
&=2\cfrac{r(r+1)}{(2r+1)^2}>0.\label{eq:orthogonal1}
\end{align}
That is to say,
$\rho_1$ and $\rho_2$ are perfectly distinguishable non-orthogonal states.
Here, we apply proposition~\ref{prop:construction2} for the case with
$\epsilon=2\sqrt{(2r)/(2r+1)}$.
Then,
$\tilde{\mathcal{K}}_r(\mathcal{P}_0(\vec{P}))$ is an $\epsilon$-PSES that contains a pair of two perfectly distinguishable states $\rho_1$ and $\rho_2$ with
\begin{align}\label{eq:orthogonal2}
\Tr\rho_1\rho_2
&\stackrel{(a)}{\ge}\cfrac{\epsilon^2(\epsilon^2+8)}{32}
\end{align}
if $r$ satisfies $\epsilon=2\sqrt{(2r)/(2r+1)}$.
The inequality $(a)$ is shown by simple calculation as seen in appendix~\ref{append-4-2} (proposition~\ref{prop:dist2}).
We summarize the result as the following theorem.
\begin{theorem}\label{theorem:dist}
For any $\epsilon>0$,
there exists an $\epsilon$-PSES that has a measurement and a pair of two perfectly distinguishable states $\rho_1,\rho_2$ with \eqref{eq:orthogonal2}.
\end{theorem}
In this way,
$\epsilon$-PSESs are different from the SES in terms of state discrimination.
This result implies the possibility that orthogonal discrimination can characterize the standard entanglement structure rather than self-duality.
In other words, we propose the following conjecture as a considerable statement, which is a future work.
\begin{conjecture}\label{conj1}
If a model of the quantum composite system $\mathcal{K}$ is not equivalent to the SES,
$\mathcal{K}$ has a pair of two non-orthogonal states discriminated perfectly by a measurement in $\mathcal{K}$.
\end{conjecture}
\section{Entanglement structures with group symmetry}\label{sect.symmetry}
As the above,
the SES cannot be distinguished from ESs based on present experimental facts, $\epsilon$-undistinguishability and self-duality.
In contrast to the experimental facts,
we investigate whether there exists other ES with group symmetric conditions than the SES.
As a result,
we clarify that some symmetric conditions characterize the SES uniquely.
In the general setting of GPTs,
symmetric properties of groups play an important role in restricting models to natural one \cite{Muller2013,BLSS2017,Barnum2019}.
In this paper,
for the investigation of an entanglement structure with certain properties,
we introduce the following symmetry (so called \textit{$G$-symmetry}) for a set $X$ under a subgroup $G$ of $\mathrm{GL}(\mathcal{V})$:
\begin{itemize}
\item[(S)] $G$-closed set $X$: $g(x)\in X$ for any $x \in X$ and any $g\in G$.
\end{itemize}
Also, we say that a set of families $\mathcal{X}$ is $G$-symmetric
if any element $g\in G$ and any family $\{X_\lambda\}_{\lambda\in\Lambda}\in\mathcal{X}$ satisfy
$\{g(X_\lambda)\}_{\lambda\in\Lambda}\in\mathcal{X}$.
With the case $\mathcal{V}=\mathcal{T}(\mathcal{H}_A\otimes\mathcal{H}_B)$,
a typical subgroup $G$ of $\mathrm{GL}(\mathcal{V})$ is the class of global unitary maps given as
\begin{align}
\mathrm{GU}(A;B):=&\{g\in\mathrm{GL}(\mathcal{T}(\mathcal{H}_A\otimes\mathcal{H}_B)) \mid g(\cdot):=U^\dag (\cdot) U,\nonumber\\
& U \ \mbox{is a unitary matrix on $\mathcal{H}_A\otimes\mathcal{H}_B$}\}.\label{eq:gu}
\end{align}
In terms of physics, the condition $\mathrm{GU}(A;B)$-symmetry means that any unitary map is regarded as a transformation from state to state, i.e., a time-evolution in GPTs.
In other words,
the condition $\mathrm{GU}(A;B)$-symmetry corresponds to the condition of global structure about time-evolutions.
However,
we remark that
$\mathrm{GU}(A;B)$-symmetry is an essential condition for the SES,
i.e.,
the assumption of $\mathrm{GU}(A;B)$-symmetry fixes entanglement structures to the standard one.
In fact, the following proposition also holds.
\begin{theorem}\label{prop:global2}
Assume that a model $\mathcal{K}$ satisfies \eqref{eq:quantum} and $\mathrm{GU}(A;B)$-symmetric.
Then, $\mathcal{K}=\mathrm{SES}(A;B)$.
\end{theorem}
The proof is written in appendix~\ref{append-5-1},
but we mention the essential fact to prove proposition~\ref{prop:global2}.
At first, we say that a group $G$ preserves entanglement structures
if any $g\in G$ and any $x\in\mathrm{SEP}^\ast(A;B)\setminus\mathrm{SEP}(A;B)$ satisfy
$g(x)\in\mathrm{SEP}^\ast(A;B)\setminus\mathrm{SEP}(A;B)$.
As we see in appendix~\ref{append-5-1},
the group $\mathrm{GU}(A;B)$ does not preserve entanglement structures,
and therefore,
$\mathrm{GU}(A;B)$-symmetry is not derived reasonably from local structures.
This is the essential reason why proposition~\ref{prop:global2} holds.
On the other hand,
we remark that
the local unitary group $\mathrm{LU}(A;B)$ defined as
\begin{align}
\begin{aligned}
\mathrm{LU}(A;B):=\{g\in\mathrm{GL}(\mathcal{T}(\mathcal{H}_A\otimes\mathcal{H}_B))&\mid g(\cdot):=(U_A^\dag\otimes U_B^\dag) (\cdot) (U_A\otimes U_B)\\
U_A,&U_B \ \mbox{are unitary matrices on $\mathcal{H}_A,\mathcal{H}_B$}\}\label{eq:lu}
\end{aligned}
\end{align}
preserves entanglement structures.
Similar to $\mathrm{GU}(A;B)$-symmetry,
the condition $\mathrm{LU}(A;B)$-symmetry means that any unitary map on local system is regarded as a time-evolution,
i.e.,
$\mathrm{LU}(A;B)$-symmetry corresponds to a local structure about time-evolutions.
Another important symmetric property is \textit{homogeneity}.
\begin{definition}
For a positive cone $\mathcal{K}$ in a vector space $\mathcal{V}$, define the set $\mathrm{Aut}(\mathcal{K})$ as
\begin{align}
\mathrm{Aut}(\mathcal{K}):=\{f\in\mathrm{GL}(\mathcal{V}) \mid f(\mathcal{K})=\mathcal{K}\}.
\end{align}
Then, we say that a positive cone $\mathcal{K}$ is homogeneous
if there exists a map $g\in\mathrm{Aut}(\mathcal{K})$ for any two elements $x,y\in \mathcal{K}^\circ$ such that $g(x)=y$.
\end{definition}
A positive cone with self-duality and homogeneity is called a \textit{symmetric cone},
which is essentially classified into finite kinds of cones including the SES \cite{Jordan1934,Koecher1957}.
As shown in the following theorem,
any symmetric cone with \eqref{eq:quantum} is restricted to the SES.
\begin{theorem}\label{prop:sym1}
Assume a symmetric cone $\mathcal{K}$ satisfies \eqref{eq:quantum}.
Then, $\mathcal{K}=\mathrm{SES}(A;B)$.
\end{theorem}
The proof is written in appendix~\ref{append-4-2}
However, theorem~\ref{prop:sym1} implies that $\mathrm{Aut}(\mathcal{K})$ is larger than $\mathrm{GU}(A;B)$ under the condition that $\mathcal{K}$ is a symmetric cone with \eqref{eq:quantum}.
\begin{proposition}\label{prop:sym2}
A symmetric cone $\mathcal{K}$ with \eqref{eq:quantum} satisfies that
$\mathrm{Aut}(\mathcal{K})\supset\mathrm{GU}(A;B)$.
\end{proposition}
Proposition~\ref{prop:sym2} is shown by theorem~\ref{prop:sym1} and the inclusion relation $\mathrm{Aut}(\mathrm{SES}(A;B))\supset\mathrm{GU}(A;B)$.
Since theorem~\ref{prop:sym1} requires symmetry of a larger group than
theorem~\ref{prop:global2} under the condition \eqref{eq:quantum},
we can conclude that
the assumption of theorem~\ref{prop:sym1} is a mathematically stronger condition than that of theorem~\ref{prop:global2}.
The reference \cite{Muller2013,BLSS2017,Barnum2019} discussed the relation between the symmetry and
the properties of cones.
Our result is different from their analysis as follows.
The reference \cite{Muller2013,BLSS2017,Barnum2019} assumes \textit{strong symmetry} on a cone,
which is the symmetry based on $\mathrm{Aut}(\mathcal{K})$.
As shown in the reference \cite{Muller2013,BLSS2017,Barnum2019},
the strong symmetry with an additional assumption implies
that the cone is a symmetric cone.
Therefore, the assumption in \cite{Muller2013,BLSS2017,Barnum2019} also implies the inclusion relation $\mathrm{Aut}(\mathcal{K})\supset\mathrm{GU}(A;B)$ under the condition \eqref{eq:quantum}.
In this way,
the conditions in the preceding study \cite{Muller2013,BLSS2017,Barnum2019} are stronger than $\mathrm{GU}(A;B)$-symmetry under the condition \eqref{eq:quantum}.
This paper aims to investigate whether there exist other quantum-like structures than the SES,
and we have considered conditions about the behavior of quantum systems.
On the other hand,
for the aim of the derivation of the SES from operational properties and local structures,
$\mathrm{GU}(A;B)$ is not suitable as the above.
Therefore,
it is also an important problem of whether $\mathrm{LU}(A;B)$-symmetry characterize the SES uniquely.
Here, we give the following two important examples:
\begin{enumerate}[(EI)]
\item $\Gamma(\mathrm{SES}(A;B))$ (where $\Gamma$ is the partial transposition map that transposes Bob's system)
\item $\mathcal{K}_r^\ast(\mathcal{P})$ (where $\mathcal{P}$ is an $\mathrm{LU}(A;B)$-symetric subset of $\mathrm{MEOP}(A;B)$)
\end{enumerate}
These two examples satisfy two of three conditions,
$\mathrm{LU}(A;B)$-symmetry, self-duality, and $\epsilon$-undistinguishablity.
That is,
the example (EI) is an $\mathrm{LU}(A;B)$-symmetric self-dual model (proposition~\ref{prop:gamma} in appendix~\ref{append-5-4}),
but (EI) does not satisfy $\epsilon$-undistinguishablity for enough small $\epsilon$.
Also,
the example (EII) satisfies $\mathrm{LU}(A;B)$-symmetry and $\epsilon$-undistinguishablity for any $\epsilon\ge0$, $r=\epsilon^2/(2(4-\epsilon))$, and $\mathrm{LU}(A;B)$-symmetric subset $\mathcal{P}\subset\mathrm{MEOP}(A;B)$ (because of proposition~\ref{prop:lu} in appendix~\ref{append-5-4}),
but (EII) is not self-dual.
A typical example of $\mathrm{LU}(A;B)$-symmetric subset $\mathcal{P}$ is $\mathrm{MEOP}(A;B)$.
On the other hand, no known example satisfies the above three conditions except for $\mathrm{SES}(A;B)$.
One might consider that a self-dual modification $\tilde{\mathcal{K}}_r(\mathcal{P})$ satisfies all of the above three condition
because (EII) is pre-dual as seen in section~\ref{sect.construct}.
However, theorem~\ref{theorem:sd} does not ensure that a self-dual modification $\tilde{\mathcal{K}}$ is $\mathrm{LU}(A;B)$-symmetric even if the pre-dual cone $\mathcal{K}$ is $\mathrm{LU}(A;B)$-symmetric.
Therefore, it remains open whether there exists a model that satisfies these three conditions and that is different from SES.
\section{Discussion}\label{sect.conclude}
In this paper,
we have discussed whether quantum-like structures exist except for the SES,
even if the structures satisfy experimental facts.
For this aim,
we have considered several conditions for behaving in a similar way as quantum theory.
As the first condition,
we have considered a class of general models, called ESs, whose bipartite local subsystems are equal to quantum systems.
Besides, as an experimental fact,
we have assumed $\epsilon$-undistinguishability
as the success of verification of any maximally entangled state with $\epsilon$ errors.
Also, as another experimental fact,
we have assumed self-duality via projectivity and pre-duality.
Then, we have investigated whether there exists an $\epsilon$-PSES,
i.e.,
self-dual entanglement structure with $\epsilon$-undistinguishability except for the SES.
For this aim,
we have stated general theory,
and we have shown the equivalence between the existence of an independent family of self-dual cones and an exact hierarchy of pre-dual cones.
Then, we have applied the general theories to the problem of whether there exists another model of $\epsilon$-PSESs different from the SES.
Surprisingly, we have shown the existence of an infinite number of independent $\epsilon$-PSESs.
In other words,
we have clarified the existence of infinite possibilities of ESs that cannot
be distinguished from the SES by physical experiments of verification of maximally
entangled states even though the error of verification is extremely tiny and even though
we impose projectivity.
Besides,
we have investigated the difference between the SES and PSESs.
As an operational difference,
we have shown that a certain measurement in our PSESs can discriminate non-orthogonal states perfectly.
Further,
we have explored the possibility of characterizing the SES by a condition about group symmetry.
We have revealed that global unitary symmetry derives the SES from all entanglement structures.
Also, we have shown that any entanglement structure corresponding to a symmetric cone is restricted to the SES.
In this way,
we have shown the existence infinite examples of $\epsilon$-PSESs,
and moreover, our examples are important in the sense that they are self-dual and non-homogeneous.
As we mentioned in the introduction,
the combination of self-duality and homogeneity leads limited types of models including classical and quantum theory.
On the other hand, the reference \cite{Levent2001} presented
only one example for a self-dual and non-homogeneous model
and a known example is limited to the above one.
Contrary to such preceding studies,
our results have presented infinite examples for self-dual and non-homogeneous models,
which implies a large variety of self-dual models.
This observation also implies that self-duality is not sufficient for the determination of the unique entanglement structure.
For the aim of determination of the unique entanglement structure, we have to impose some additional assumptions to our setting or find a new better assumption,
which denies infinitely many self-dual models of the composite systems.
Our result also has implied the possibility that conjecture~\ref{conj1} resolves the problem,
i.e.,
the combination of self-duality and orthogonal discrimination derives the standard entanglement structure.
Another possibility is occurred from symmetric condition for example $\mathrm{LU}(A;B)$-symmetry,
but it remains open to exist $\mathrm{LU}(A;B)$-symmetric $\epsilon$-PSESs for small $\epsilon$.
These are important open problems.
\section*{Acknowledgement}
HA is
supported by a JSPS Grant-in-Aids for JSPS Research Fellows No. JP22J14947, a JSPS Grant-in-Aids for Scientific Research (B) Grant No. JP20H04139, and a JSPS Grant-in-Aid for JST SPRING No. JPMJSP2125.
MH is supported in part by the National Natural Science Foundation of China (Grant No. 62171212) and
Guangdong Provincial Key Laboratory (Grant No. 2019B121203002).
\section*{References}
|
2,869,038,154,778 | arxiv |
\section{Introduction}
In 2015, \citeauthor{Berezhiani2015} proposed a new hypothesis that combines features of Cold Dark Matter ({\sc CDM}) and Modified Newtonian Dynamics ({\sc MOND}) \citep{Milgrom1983a,Milgrom1983b,Milgrom1983c,Bekenstein1984}: superfluid dark matter, hereafter {\sc SFDM}. In {\sc SFDM}, dark matter is composed of a light ($\sim$eV) scalar field which can condense to a superfluid. In the superfluid phase, phonons mediate a force which is similar to the force of {\sc MOND}. This hypothesis has since passed several observational tests \citep{Berezhiani2018, Hossenfelder2019, Hossenfelder2020}.
However, recently it was found that {\sc SFDM} needs about 20\% less baryonic mass than {\sc MOND} to fit the Milky Way rotation curve at $R \lesssim 25$ kpc \citep{Hossenfelder2020}.
To investigate whether this is a general trend, we fit {\sc SFDM} to the {\sc SPARC} data \citep{Lelli2016} with the stellar mass-to-light ratio $M/L_*$ as a fitting parameter.
\section{Models}
\label{sec:models}
{\sc SFDM} requires four parameters for which we use the
fiducial values from \citet{Berezhiani2018}, $m = 1\,\mathrm{eV}$, $\Lambda = 0.05\,\mathrm{meV}$, $\alpha = 5.7$, and $\beta = 2$.
We keep those parameters fixed, see also Appendix~\ref{sec:sfdm:params}.
The total acceleration inside the superfluid core of a galaxy is
$
\vec{a}_{\mathrm{tot}} = \vec{a}_\theta + \vec{a}_b + \vec{a}_{\mathrm{SF}} \,$,
where $\vec a_{\theta}$ is the acceleration created by the phonon force, $\vec a_{\rm SF}$ the acceleration stemming from the normal gravitational attraction of the superfluid, and $\vec a_b$ that stemming from the mass of the baryons.
The position-dependence of those accelerations is determined by the {\sc SFDM} equations of motion and the distribution of baryonic mass.
At a transition radius where the superfluid condensate is estimated to break down, one matches the superfluid core to an {\sc NFW} halo \citep{Berezhiani2018}.
We assume that all rotation curve data points are within the superfluid core; otherwise rotation curves will not be naturally {\sc MOND}-like.
From integrating the standard Poisson equation including the superfluid's energy density $\rho_{\mathrm{SF}}$ as a source term, one obtains $\hat{\mu}(\vec{x}) = \mu_{\mathrm{nr}} - m \phi_N(\vec{x})$, where $\mu_{\mathrm{nr}}$ is the chemical potential and $\phi_N(\vec{x})$ is the Newtonian gravitational potential.
The gradient of $\phi_N(\vec{x})$ gives $\vec{a}_b + \vec{a}_{\mathrm{SF}}$.
In the so-called no-curl approximation, one obtains the phonon force $\vec{a}_\theta$ as an algebraic function of $\vec{a}_b$ and $\varepsilon_*(\vec{x})$ (see Appendix~\ref{sec:appendix:estar}),
\begin{align}
\label{eq:estar}
\varepsilon_*(\vec x) := \frac{2 m^2}{\alpha M_{\mathrm{Pl}} |\vec{a}_b(\vec x)|} \frac{\hat{\mu} (\vec{x})}{m} \,,
\end{align}
where $M_{\rm Pl}$ is the Planck mass (it enters through Newton's constant).
The quantity $\varepsilon_*(\vec{x})$ controls how closely {\sc SFDM} resembles {\sc MOND}.
We will refer to $|\varepsilon_*| \ll 1$ as the {\sc MOND}-limit and to $|\varepsilon_*| = {\cal O}(1)$ as the pseudo-{\sc MOND} limit.
The relation between $\varepsilon_*$ and these limits of {\sc SFDM} was previously derived in \citet{Mistele2020}.
Details on the definition and rationale behind these limits are in Appendix~\ref{sec:appendix:models}.
Solutions of the equations of motion can be parameterized by one boundary condition, $\varepsilon:= \varepsilon_*(R_{\mathrm{ mid}})$, where $R_{\mathrm{mid}}:= (R_{\mathrm{min}} + R_{\mathrm{max}})/2$, and $R_{\mathrm{min}}$ ($R_{\mathrm{max}}$) is the smallest (largest) radius with a rotation curve data point. $\varepsilon$ quantifies how closely the phonon force resembles a {\sc MOND}-force in the middle of the observed rotation curve.
We will compare {\sc SFDM} to {\sc MOND} with one of the standard interpolation functions \citep{Lelli2017b}
\begin{align}
\nu_e(y) = \frac{1}{1 - e^{-\sqrt{y}}} \,,
\end{align}
where $y = |{\vec a}_b|/a_0$ and $a_0$ is the one free parameter in {\sc MOND}.
In {\sc SFDM} the interpolation function is slower to reach its limits for large and small $y$.
Also, usually $a_0$ is chosen smaller in SFDM compared to MOND to account for the presence of $a_{\mathrm{SF}}$ \citep{Berezhiani2018}.
For MOND, we adopt $a_0^{\mathrm{MOND}} \approx 1.2 \cdot 10^{-10}\,\mathrm{m}/\mathrm{s}^2$ from \citet{Lelli2017b}.
For SFDM, the fiducial parameters from \citet{Berezhiani2018} give
$ a_0^{\mathrm{SFDM}} \approx 0.87 \cdot 10^{-10}\,\mathrm{m}/\mathrm{s}^2$.
To check how sensitive our results are to the particular theoretical realization of {\sc SFDM} we include the two-field model from \cite{Mistele2020}.
In this two-field model, the phenomenology on galactic scales is similar to standard {\sc SFDM}, but it has the advantages that (a) it does not require ad-hoc finite-temperature corrections for stability, (b) its phonon force is always close to its {\sc MOND}-limit, and (c) the superfluid can remain in equilibrium much longer than galactic timescales.
Both models are described in more detail in Appendix~\ref{sec:appendix:models}.
\section{Data}
\label{ssec:data}
We take the observed rotation velocity $V_{\mathrm{obs}}$ directly from {\sc SPARC} \citep{Lelli2016}.
To find the best {\sc SFDM} fit, we then need the baryonic energy density $\rho_b(R,z)$ because it is a source for the equation of motion of the superfluid.
For this, we use updated high-resolution mass models including resolved gas surface density profiles for 169 of the 175 SPARC galaxies (Lelli 2021, private communication).
We exclude the 6 galaxies lacking radial profiles for the gas distribution.
These mass models provide surface densities $\Sigma$ for the bulge, the stellar disk, and the HI disk of each galaxy for a discrete set of positions.
We linearly interpolate the data points and assume constant density at radii smaller than the smallest radius in the series, and zero density at radii larger than the largest radius in the series. This gives a simple, data-compatible approximation for the density distribution at all radii.
For the bulge, we assume spherical symmetry and extract its energy density from its surface density by an Abel transform
\begin{align}
\rho_{\mathrm{bulge}}(r) = -\frac{1}{\pi} \int_0^{\infty} d\bar{r} \frac{\Sigma_{\mathrm{bulge}}'(\sqrt{\bar{r}^2+r^2})}{\sqrt{\bar{r}^2+r^2}} \,.
\end{align}
For the stellar disk, we assume a scale height $h_*$ \citep{Lelli2016}
\begin{align}
h_* = 0.196 \cdot (R_{\mathrm{disk}}[\mathrm{kpc}])^{0.633} \,\mathrm{kpc} \,,
\end{align}
where $R_{\mathrm{disk}}$ is the disk scale length from {\sc SPARC}. Again, we use a linear interpolation of the {\sc SPARC} surface brightness data points.
For the gas disk we do the same as for the stellar disk, except that in this case we assume a fixed scale height $h_g = 0.130\,\mathrm{kpc}$.
This is the same scale height used in \citet{Hossenfelder2020}.
We do not expect this choice of scale height to significantly affect the results.
To account for the non-HI gas,
we multiply the HI surface density by $1.4$ \citep{McGaugh2020}.
\section{Method}
In our fitting procedure, we keep $V_{\mathrm{obs}}$ and the fiducial model parameters of {\sc SFDM} fixed,
but we allow a common factor $Q_*$ to adjust the stellar disk and bulge $M/L_*$ relative to $(M/L_*)_{\mathrm{disk}} = 0.5$ and $(M/L_*)_{\mathrm{bulge}} = 0.7$,
\begin{eqnarray}
\rho_b(R,z) &=& \rho_{\mathrm{gas}}(R,z) \nonumber + 0.5 \cdot Q_* \cdot \rho_*(R,z) \nonumber \\
&+& 0.7 \cdot Q_* \cdot \rho_{\mathrm{bulge}}(\sqrt{R^2+z^2}) \,.
\end{eqnarray}
Using this total baryonic energy density, we solve the SFDM equations of motion for different boundary conditions. From that we then obtain the expected rotation curve.
In our fits, we require that
$\varepsilon_*(\vec x)$ (see Eq.\ (\ref{eq:estar})) is larger than an algebraic minimum value $\varepsilon_{\mathrm{min}}$ everywhere within the superfluid.
This minimum value is reached when $\rho_{\mathrm{SF}}$ vanishes and (for the case of $\beta=2$) is given by $\varepsilon_{\mathrm{min}} = - \sqrt{3/32} \approx - 0.31$. As mentioned in section \ref{sec:models}, we parametrize solutions with $\varepsilon = \varepsilon_*(R_{\mathrm{mid}})$.
After solving the SFDM equations of motion, we check whether all data points lie within the superfluid core.
It turns out that for 31 of the 169 galaxies this is not the case. However, the criterion for the exact value of the transition-radius to the {\sc NFW} halo is quite ad-hoc. We therefore do not discard these solutions, though we have checked that they do not alter the main conclusions, see also Appendix~\ref{sec:results:sfdm:thermal}.
Then we compare how good this rotation curve matches with the observed velocities, $V_{\mathrm{obs}}$, from {\sc SPARC}. For this, we define the best fit for each galaxy as that with the smallest $\chi^2$,
\begin{align}
\chi^2 = \frac{1}{N-f} \sum_{R} \frac{(V_{\mathrm{obs}}(R) - V_c(R))^2}{\sigma_{V_{\mathrm{obs}}}^2(R)} \,.
\end{align}
Here, $N$ is the number of data points in the galaxy, $f = 2$ is the number of fit parameters ($Q_*$ and $\varepsilon$), $\sigma_{V_{\mathrm{obs}}}$ is the uncertainty on the velocity $V_{\mathrm{obs}}$ from {\sc SPARC}, $V_c(R)$ is the calculated rotation curve in {\sc SFDM}, and the sum is over the data points at radius $R$.
We minimize $\chi^2$ for
\begin{align}
\label{eq:rangeQ}
10^{-2} \leq \, & Q_* \leq 15 \,, \\
\label{eq:rangeestar}
10^{-2} \leq \, & \left(\varepsilon - \varepsilon_{\mathrm{min}}\right) \leq 10^4 \,.
\end{align}
In our fit code, we scan values of $\log_{10}(Q_*)$ and $\log_{10}(\varepsilon - \varepsilon_{\mathrm{min}})$.
In the {\sc SPARC} data, the Newtonian acceleration due to gas sometimes points outwards from the galactic center, not towards it, because of a hole in the HI data, possibly due to a transition from atomic to molecular gas.
Usually, such a negative gas contribution is countered by the positive contributions from the stellar disk and the bulge and does not pose a problem. When this is not the case, there is technically no stable circular orbit so we cannot calculate a rotation curve.
When this happens, we omit those data points when calculating $\chi^2$.
As a cross-check and as a comparison for {\sc SFDM},
we also fit the radial acceleration relation ({\sc RAR}) to the {\sc SPARC} data, i.e. we fit the {\sc SPARC} data with {\sc MOND} assuming no curl term and the exponential interpolation function $\nu_e$ \citep{Lelli2017b}.
In this case, we have only one free fit parameter, $Q_*$, and consequently, when calculating $\chi^2$, we set $f=1$.
We describe our fitting and calculation methods in more detail in Appendix~\ref{sec:method}.
\section{Results}
\label{sec:results}
The result of our {\sc MOND} fit is similar to that of \citet{Li2018},
which also fitted the {\sc RAR} to {\sc SPARC} galaxies.
The major difference is that \citet{Li2018} used a {\sc MCMC} procedure with Gaussian priors, while we used a simple parameter scan to minimize $\chi^2$.
We also do not vary distance and inclination and do not separately vary the mass-to-light ratio of the stellar disk and the bulge. As a consequence of this simplified fitting procedure,
our distribution of best-fit $M/L_*$ has more outliers and looks less Gaussian than that of \citet{Li2018}.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{plots/short/fY-hist-SFDM.pdf}
\caption{
Histograms of the best-fit $Q_\ast$ values for the SFDM and MOND fits restricted to the $Q=1$ galaxies.
}
\label{fig:fYhists}
\end{figure}
Still, our median best-fit stellar mass-to-light ratios and the best-fit $\chi^2$ values are similar to those from \citet{Li2018}.
The median stellar disk $M/L_*$ is $0.39$.
When we restrict ourselves to galaxies with high quality data ($Q=1$), this becomes $0.47$, very similar to the $0.50$ from \citet{Li2018}.
We show the $\chi^2$ cumulative distribution function (CDF) in Fig.~\ref{fig:chi2cdfs} which is also in reasonable agreement with \citet{Li2018}.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{plots/short/chi2-cdf.pdf}
\caption{
Best-fit $\chi^2$ cumulative distribution functions for the $Q=1$ galaxies for different models.
}
\label{fig:chi2cdfs}
\end{figure}
In Fig.~\ref{fig:fYhists}, one sees that some galaxies end up at the minimum stellar mass-to-light ratio allowed in our fitting method, corresponding to $Q_* \approx 0.01$.
If we do not restrict ourselves to $Q=1$, this peak at $Q_* \approx 0.01$ is even more pronounced.
As discussed in Appendix~\ref{sec:rar}, this is an artifact of our fitting procedure and can be ignored in what follows.
\subsection{MOND vs SFDM}
\label{ssec:rar}
Fig.~\ref{fig:fYhists} shows the best-fit $Q_*$ for the 97 galaxies with $Q=1$.
Contrary to what one might naively expect from the Milky Way result \citep{Hossenfelder2019},
the {\sc SFDM} fits do not have significantly smaller $Q_*$ than the {\sc MOND} fits.
Indeed, the median $Q_*$ for the $Q=1$ galaxies is about $4\%$ larger than for {\sc MOND}.
\begin{figure}[t]
\centering
\includegraphics[width=.48\textwidth]{plots/estar-scatter.pdf}
\caption{
The best-fit $\varepsilon$ values versus the best-fit $Q_*$ values for the $Q=1$ galaxies.
For standard SFDM, the correlation coefficient is $r = 0.28$.
}
\label{fig:estarscatter}
\end{figure}
One reason for this is that for many galaxies the superfluid is not in the {\sc MOND}-limit $|\varepsilon| \ll 1$, as one sees from Fig. \ref{fig:estarscatter}.
We theoretically explain why going outside the MOND limit allows for larger $M/L_*$ in Appendix~\ref{sec:sfdm:mond}.
To confirm this, we did the fits again but required that the galaxies are in the {\sc MOND}-limit, $|\varepsilon| < 0.4$. For the rationale behind the precise value 0.4, please refer to Appendix~\ref{sec:nomondrequired}.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{plots/short/fY-hist-SFDM04.pdf}
\caption{
Same as Fig.~\ref{fig:fYhists} but for SFDM restricted to $|\varepsilon| < 0.4$.
}
\label{fig:fYhist04}
\end{figure}
As one can see from Fig.~\ref{fig:chi2cdfs}, the fits with the requirement $|\varepsilon| < 0.4$ are not much worse than those without.
The averaged $Q_*$ is now smaller than in {\sc MOND}; for the $Q=1$ galaxies, the median stellar disk $M/L_*$ is about $10\%$ smaller than for {\sc MOND}.
This confirms superfluids outside the MOND limit as one reason for the large $Q_*$ values in {\sc SFDM}, see also Appendix~\ref{sec:largeML}.
Another reason why {\sc SFDM} does not universally give smaller $Q_*$ than {\sc MOND} is that the best-fit $Q_*$ depends on the type of galaxy. In
{\sc SFDM}, $Q_*$ is systematically smaller for galaxies with relatively large accelerations $a_b$, but
not for those with small accelerations.
This can be seen for example in Fig.~\ref{fig:this}
which shows the best-fit $Q_*$ of each galaxy in {\sc SFDM} relative to the best-fit $Q_*$ for {\sc MOND} as a function of the observed asymptotic rotation velocity $V_{\mathrm{flat}}$.
A larger $V_{\mathrm{flat}}$ is associated with larger accelerations -- this is where {\sc SFDM} systematically gives smaller $Q_*$ than MOND.
There are similar trends for surface brightness and the gas fraction;
both also correlate with the accelerations $a_b$, see Appendix~\ref{sec:results:sfdm:MLtrends} for more details.
The reason for this trend is that the smaller $a_0$ value of SFDM makes the acceleration $\sqrt{a_0 a_b}$ smaller than in {\sc MOND}.
This acceleration $\sqrt{a_0 a_b}$ is dominant at small $a_b$, so that {\sc SFDM} needs more baryonic mass than MOND to get the same total acceleration (at least if $a_{\mathrm{SF}}$ is negligible).
This is explained in more detail in Appendix~\ref{sec:sfdm:less}
Such trends of the stellar $M/L_*$ with galaxy properties are not expected from stellar population synthesis models \citep{Schombert2019}.
This disfavors {\sc SFDM}, especially compared to {\sc MOND} which does not show such trends \citep{McGaugh2004}.
\begin{figure}
\centering
\includegraphics[width=.48\textwidth]{plots/short/ML-trends-relative-Vflat.pdf}
\caption{
The best-fit $Q_*$ values in SFDM restricted to the MOND limit ($|\varepsilon| < 0.4$) relative to those for MOND as a function of the observed flat rotation velocity $V_{\mathrm{flat}}$.
Gray arrows indicate two outliers with relatively large $Q_*/Q_*^{\mathrm{MOND}}$.
Some galaxies can barely satisfy the condition $|\varepsilon| < 0.4$ and therefore give a bad fit to the data.
Their best-fit $Q_*$ is meaningless and they are excluded in this plot.
Specifically, we exclude galaxies that have both $\varepsilon > 0.38$ and $\chi^2 > 100$.
We also show only the $Q=1$ galaxies.
}
\label{fig:this}
\end{figure}
\subsection{Tension with strong lensing}
Irrespective of the resulting $M/L_*$ values,
there is a price to pay for enforcing the {\sc MOND}-limit in {\sc SFDM}.
A {\sc MOND}-like rotation curve in the {\sc MOND}-limit $|\varepsilon_*| \ll 1$ can only be achieved by reducing the acceleration created by the gravitational pull of the superfluid.
As a result, the total dark matter mass in those galaxies, $M_{200}^{\mathrm{DM}}$, comes out to be quite small.
Here, $M_{200}^{\mathrm{DM}}$ is the dark matter mass within the radius $r_{200}$ where the mean dark matter density drops below $\rho_{200} = 200 \times 3 H^2/(8 \pi G)$ with the Hubble constant $H$.
We adopt $H = 67.3\,\mathrm{km}/(\mathrm{s} \cdot \mathrm{Mpc})$.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{plots/M200max-scat.pdf}
\caption{
The total baryonic mass $M_b$ versus the upper bound $M_{200,\mathrm{max}}^{\mathrm{DM}}/M_b$ of the ratio of the total dark matter mass $M_{200}^{\mathrm{DM}}$ and $M_b$ for the SPARC galaxies.
This is for $(M/L_*)_{\mathrm{disk}} = 0.5$ and $(M/L_*)_{\mathrm{bulge}} = 0.7$.
The upper bound comes from the condition that the rotation curve is in the proper MOND limit ($|\varepsilon| < 0.4$, blue circles) or at least the pseudo-MOND limit ($|\varepsilon| < 5$, red squares).
Also shown are the best-fit results from the strong lensing analysis \citet{Hossenfelder2019},
where we use their best-fit $M_{200}^{\mathrm{DM}}$ for the vertical axis.
}
\label{fig:M200max}
\end{figure}
A small $M_{200}^{\mathrm{DM}}$ is not a problem for fitting {\sc SFDM} to the observed rotation curves, but it is a problem if one \emph{also} wants to fit strong lensing data.
Indeed, \citet{Hossenfelder2019} previously found that {\sc SFDM} requires ratios $M_{200}^{\mathrm{DM}}/M_b \gtrsim 1000$ to fit strong lensing constraints, where $M_b$ is the total baryonic mass.
Requiring a rotation curve in the {\sc MOND}-limit $|\varepsilon_*| \ll 1$ for the {\sc SPARC} galaxies produces average masses at least an order of magnitude smaller.
To illustrate the problem with strong lensing, we have in Fig.~\ref{fig:M200max} plotted the (logarithm of) the total baryonic and the maximum possible total dark matter mass given our requirement $|\varepsilon| < 0.4$ in comparison to the values found in \citet{Hossenfelder2019}.
For this, we used $Q_* = 1$ for all {\sc SPARC} galaxies because the precise stellar mass-to-light ratio is irrelevant here.
`Maximum possible' here refers not only to the requirement $|\varepsilon| < 0.4$ but also to uncertainties in how to determine the radius where the superfluid core is matched to an {\sc NFW} halo:
We use the transition radius that gives the largest total dark matter mass.
See Appendix~\ref{sec:M200estar04} for details.
The best {\sc SFDM} fits to strong lensing data tend to have $M_b \gtrsim 10^{11}\,M_\odot$ and $M_{200}^{\mathrm{DM}}/M_b \gtrsim 1000$.
In contrast, despite our generous NFW matching procedure, the {\sc SPARC} galaxies with $M_b > 10^{11}\,M_\odot$ have $M_{200}^{\mathrm{DM}}/M_b < 10$ when restricted to have rotation curves in the {\sc MOND} limit $|\varepsilon| < 0.4$.
This is a stark contrast.
The {\sc SPARC} galaxies don't reach baryonic masses quite as large as the lensing galaxies from \citet{Hossenfelder2019}.
But from Fig.~\ref{fig:M200max} it seems clear that the trend goes into the wrong direction:
The larger the galaxy, the smaller the maximum possible $M_{200}^{\mathrm{DM}}/M_b$ (given $|\varepsilon| < 0.4$).
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{plots/M200max-atot-corrections-hist.pdf}
\caption{
Histogram of $a_{\mathrm{SF}}$ relative to $a_b + a_\theta$ at the last rotation curve data point at $R = R_{\mathrm{max}}$.
This is for the maximum possible total dark matter mass $M_{200}^{\mathrm{DM}}$ given the condition $|\varepsilon| < 0.4$ (blue) and $|\varepsilon| < 5$ (red).
We take $Q_* = 1$ for all galaxies and show only the galaxies with $M_b > 10^{11}\,M_\odot$ relevant for strong lensing.
}
\label{fig:M200maxVccorrections}
\end{figure}
To study this closer, we did another calculation in which we allow galaxies into the pseudo-{\sc MOND}-limit.
Concretely, we redid the maximum $M_{200}^{\mathrm{DM}}/M_b$ calculation with the requirement $|\varepsilon| < 5$.
The precise value 5 is again somewhat arbitrary.
We explain why this is a pragmatic choice in Appendix~\ref{sec:nomondrequired}.
We see from Fig.~\ref{fig:M200max} that in the pseudo-{\sc MOND}-limit galaxies with $M_b > 10^{11}\,M_\odot$ still have smaller total dark matter masses than what is required for strong lensing, although the problem is less severe than in the proper {\sc MOND}-limit.
The pseudo-{\sc MOND}-limit, however, is unsatisfactory for two reasons.
First, it relies sensitively on ad-hoc finite-temperature corrections of {\sc SFDM} that may be unphysical.
Second, the pseudo-MOND-limit has the disadvantage that the acceleration from the superfluid, $a_{\mathrm{SF}}$, can be significant.
If $a_{\mathrm{SF}}$ is significant, we do not naturally get the {\sc MOND}-type galactic scaling relations,
since then the superfluid boundary condition must be adjusted for each galaxy to get the correct total acceleration.
In this case, {\sc SFDM} loses its advantage over {\sc CDM} despite the phonon force being close to $\sqrt{a_0 a_b}$.
Fig.~\ref{fig:M200maxVccorrections} shows the size of $a_{\mathrm{SF}}$ relative to $a_b + a_\theta$ at the last rotation curve data point at $R = R_{\mathrm{max}}$,
assuming the maximum total dark matter masses from Fig.~\ref{fig:M200max}.
Indeed, for the pseudo-{\sc MOND} limit, $a_{\mathrm{SF}}$ is significant for the galaxies with $M_b > 10^{11}\,M_\odot$ relevant for strong lensing.
This is despite SFDM having a very cored dark matter profile.
Thus, also with the pseudo-MOND limit, we cannot naturally get MOND-like rotation curves and strong lensing at the same time.
\subsection{Two-field SFDM}
\label{sec:twofield}
\begin{figure}[t]
\centering
\includegraphics[width=.48\textwidth]{plots/short/fY-hist-two-field.pdf}
\caption{
Same as Fig.~\ref{fig:fYhists} but for two-field SFDM.
}
\label{fig:fYhisttwofield}
\end{figure}
For two-field {\sc SFDM}, the $Q_*$-distribution (Fig.\ \ref{fig:fYhisttwofield}) and the corresponding {\sc CDF} (Fig.\ \ref{fig:chi2cdfs}) are similar to those of standard {\sc SFDM}.
However, the two-field model is constructed so that it is easier for the phonon force to be close to the {\sc MOND}-like value $\sqrt{a_0 a_b}$.
For this reason, the best fits for two-field {\sc SFDM} all have $|\varepsilon_*| \ll 1$, as expected (Fig.~\ref{fig:estarscatter}).
Only for two galaxies (NGC6789, UGC0732) does $\varepsilon_*$ become larger than $0.1$.
Its largest value is $0.36$ for NGC6789.
See Appendix~\ref{sec:twofield:ML} for more details.
Thus, two-field {\sc SFDM} can easily have large dark matter masses and $|\varepsilon_*| \ll 1$ at the same time.
It does not have the same problem with strong lensing as the proper MOND limit $|\varepsilon_*| \ll 1$ of standard SFDM.
Two-field {\sc SFDM} does, however, still have a problem with strong lensing similar to the pseudo-{\sc MOND}-limit of standard {\sc SFDM}.
Large total dark matter masses imply that the rotation curve receives significant corrections from the superfluid's gravitational pull $a_{\mathrm{SF}}$.
This is despite two-field {\sc SFDM} having, like standard {\sc SFDM}, a very cored density profile.
For this reason, large total dark matter masses imply a rotation curve that is not naturally {\sc MOND}-like.
To illustrate this problem we depict in Fig.~\ref{fig:M200maxtf} the maximum possible total dark matter mass for the two-field model,
given the requirement that $a_{\mathrm{SF}}$ is at most $30\%$ as large as $a_b + a_\theta$ at the last rotation curve data point at $R=R_{\mathrm{max}}$, see Appendix~\ref{sec:appendix:twofieldlensing}.
The scatter in the distribution is smaller in the two-field model because it depends less on the details of the baryonic matter distribution, see Appendix~\ref{sec:appendix:twofieldlensing}.
As one can see, in the two-field model the discrepancy with the lensing data is weaker than for the proper {\sc MOND}-limit $|\varepsilon_*| \ll 1$ of standard {\sc SFDM}, but still present.
Avoiding this tension with the lensing data would require rotation curves that are even less {\sc MOND}-like.
\begin{figure}[t]
\centering
\includegraphics[width=0.48\textwidth]{plots/M200max-scat-two-field.pdf}
\caption{
Same as Fig.~\ref{fig:M200max} but for the two-field model and with the requirement that $a_{\mathrm{SF}}$ is at most $30\%$ as large as $a_b + a_\theta$ at the last rotation curve data point $R_{\mathrm{max}}$.
}
\label{fig:M200maxtf}
\end{figure}
\section{Conclusion}
We have found that it is difficult to reproduce the achievements of {\sc MOND} with the models that have so far been proposed for superfluid dark matter.
\begin{acknowledgements}
This work was supported by the DFG (German Research Foundation) under grant number HO 2601/8-1.
\end{acknowledgements}
\bibliographystyle{aa}
|
2,869,038,154,779 | arxiv | \section{Introduction}
\label{sec:intro}
In this paper we will consider the partition functions of five-dimensional $\mathcal{N} = 1$ gauge theories with an $\mathrm{SU}(2)$ R-symmetry on $\mathcal{M}_4 \times S^1$,
partially topologically twisted on the toric K\"ahler manifold $\mathcal{M}_4$ \cite{Witten:1988ze,Witten:1991zz}, with a view to holography. In particular, we are interested in evaluating through localization \cite{Pestun:2007rz}
the \emph{topologically twisted index} of a five-dimensional theory on $\mathcal{M}_4 \times S^1$, which is defined as the equivariant Witten index
\begin{equation}} \newcommand{\ee}{\end{equation}
Z_{\mathcal{M}_4 \times S^1}(\mathfrak{s}_I, y_I) = \Tr_{\mathcal{M}_4} (-1)^F e^{-\beta \{{\cal Q},\bar {\cal Q}\}} \prod_I y_I^{J_I} \, ,
\ee
of the topologically twisted theory on $\mathcal{M}_4$, where $\mathfrak{s}_I$ are magnetic fluxes on $\mathcal{M}_4$ that explicitly enter in the Hamiltonian and $y_I$ complexified fugacities for the flavor symmetries $J_I$ of the theory.
In three and four dimensions, the topologically twisted index has proven useful in checking dualities and in the microscopic counting for the entropy of a class of asymptotically AdS$_{4/5}$
black holes/strings \cite{Benini:2015noa,Benini:2015eyy}. Since one of the goals of this work is to extend the above analyses to higher dimensions, let us briefly review what is known in lower dimensions.
\subsection{The three- and four-dimensional indices}
The topologically twisted index of three-dimensional $\mathcal{N} = 2$ and four-dimensional $\mathcal{N} = 1$ gauge theories with an R-symmetry is the supersymmetric partition function on $\Sigma_{\mathfrak{g}_1} \times T^d$,
partially topologically $A$-twisted along the genus $\mathfrak{g}_1$ Riemann surface $\Sigma_{\mathfrak{g}_1}$, where $T^d$ is a torus with $d= 1, 2$, respectively.
The index can be computed in two different ways. It has been first derived by topological field theory arguments in \cite{Okuda:2012nx,Okuda:2013fea,Nekrasov:2014xaa,Gukov:2015sna}.
In this approach, further discussed and generalized in \cite{Okuda:2015yea,Gukov:2016gkn,Closset:2016arn,Closset:2017zgf,Dedushenko:2017tdw,Gukov:2017zao,Closset:2017bse,Closset:2018ghr},
the index is written as a sum of contributions coming from the Bethe vacua,
the critical points of the twisted superpotential of the two-dimensional theory obtained by compactifying on $T^d$.
The index has been also derived using localization in \cite{Benini:2015noa,Benini:2016hjo} and can be written as the contour integral
\begin{equation}
Z (\mathfrak{s}_I, y_I) =
\sum_{\mathfrak{m} \,\in\, \Gamma_\mathfrak{h}} \oint_\mathcal{C} Z_{\text{int}} (\mathfrak{m}, x; \mathfrak{s}_I , y_I) \, ,
\end{equation}
of a meromorphic differential form in variables $x$ parameterizing the Cartan subgroup and subalgebra of the gauge group, summed over
the lattice of gauge magnetic fluxes $\mathfrak{m}$ on $\Sigma_{\mathfrak{g}_1}$. Here $\mathfrak{s}_I $ and $y_I = e^{\mathbbm{i} \Delta_I}$ are, respectively, fluxes and fugacities for the global symmetries of the theory.
One remarkable application of the topologically twisted index is to understand the microscopic origin of the Bekenstein-Hawking entropy of asymptotically anti de Sitter (AdS) black holes.
In particular, the microscopic entropy of certain four-dimensional static, dyonic, BPS black holes \cite{Cacciatori:2009iz,DallAgata:2010ejj,Hristov:2010ri,Katmadas:2014faa,Halmagyi:2014qza},
which can be embedded in AdS$_4\times S^7$, has been calculated in this manner \cite{Benini:2015eyy,Benini:2016rke},
by showing that the ABJM \cite{Aharony:2008ug} twisted index, in the large $N$ limit, agrees with the area law for the black hole entropy ---
$S_{\rm BH} = A / ( 4 G_{\rm N})$ with $A$ being the horizon area and $G_{\rm N}$ the Newton's constant.
Specifically, the statistical entropy $S_{\text{BH}}$ as a function of the magnetic and electric charges $(\mathfrak{s} , \mathfrak{q})$ is given by the Legendre transform of the field theory twisted index $Z (\mathfrak{s} , \bar \Delta)$,
evaluated at its critical point $\bar \Delta_I$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{intro:I-extremization}
\mathcal{I} ( \mathfrak{s} , \bar \Delta ) \equiv \log Z (\mathfrak{s} , \bar \Delta) - \mathbbm{i} \sum_{I} \mathfrak{q}_I \bar \Delta_I = S_{\text{BH}} (\mathfrak{s} , \mathfrak{q}) \, .
\ee
This procedure was dubbed $\mathcal{I}$-extremization in \cite{Benini:2015eyy}.
These results have been generalized to black strings in five dimensions \cite{Hosseini:2016cyf,Hosseini:2018qsx,Hong:2018viz},
black holes with hyperbolic horizons \cite{Benini:2016hjo,Cabo-Bizet:2017jsl},
universal black holes \cite{Azzurli:2017kxo},%
\footnote{These can be embedded in all M-theory and massive type IIA compactifications, thus explaining the name universal.}
black holes in massive type IIA supergravity \cite{Hosseini:2017fjo,Benini:2017oxt},
M-theory black holes in the presence of hypermultiplets \cite{Bobev:2018uxk},
Taub-NUT-AdS/Taub-Bolt-AdS solutions \cite{Toldo:2017qsh},
and $N$ M5-branes wrapped on hyperbolic three-manifolds \cite{Gang:2018hjd}.%
\footnote{Other interesting progresses in this context include: computing the logarithmic correction to AdS$_4 \times S^7$ black holes \cite{Liu:2017vbl} (see also \cite{Liu:2017vll,Jeon:2017aif}),
evaluating the on-shell supergravity action for the latter black holes \cite{Halmagyi:2017hmw,Cabo-Bizet:2017xdr},
localization in gauged supergravity \cite{Hristov:2018lod} (see also \cite{Nian:2017hac}), relation between anomaly polynomial of $\mathcal{N} = 4$ super Yang-Mills (6D $\mathcal{N} = (2,0)$ theory)
and rotating, electrically charged, AdS black holes in five (seven) dimensions \cite{Hosseini:2017mds,Hosseini:2018dob}.} Another interesting general result is the Cardy behaviour
of the topologically twisted index of four-dimensional $\mathcal{N} = 1$ gauge theories that flow to an infrared (IR) two-dimensional $\mathcal{N} = (0,2)$ superconformal field theory (SCFT) upon twisted compactification on $\Sigma_{\mathfrak{g}_1}$ \cite{Hosseini:2016cyf}:
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{Cardy:4d:intro}
\log Z_{\Sigma_{\mathfrak{g}_1} \times T^2} \approx \frac{\mathbbm{i} \pi}{12 \tau} c_r (\mathfrak{s} , \Delta) \, ,
\ee
where $c_r (\mathfrak{s} , \Delta)$ is the trial right-moving central charge of the $\mathcal{N} = (0,2)$ SCFT \cite{Benini:2012cz} and $\tau$ is the modular parameter of the torus $T^2$.%
\footnote{Here we use the chemical potentials $\Delta_I / \pi$ to parameterize a trial R-symmetry of the theory.} This result is valid at high temperature,
$\tau\rightarrow \mathbbm{i} 0^+$, and is a consequence of the fact that the index computes the elliptic genus of the two-dimensional CFT.
Furthermore, it has been shown in \cite{Benini:2015eyy,Hosseini:2016cyf} that, in the large $N$ limit, one particular Bethe vacuum dominates the partition function.
It has also been found in \cite{Hosseini:2016tor,Hosseini:2016cyf} (see \cite{Hosseini:2016ume} for examples) that the topologically twisted index of three- and four-dimensional field theories, at large $N$ or high temperature, can be compactly written as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{index:theorem:original:3d:4D}
\begin{aligned}
& \log Z_{\Sigma_{\mathfrak{g}_1} \times T^d} \approx\sum_{I} \mathfrak{s}_I \frac{\partial \widetilde\mathcal{W}_d (\Delta_I) }{\partial \Delta_I} \, , \qquad
\widetilde\mathcal{W}_d (\Delta_I) \propto \Bigg\{
\begin{array}{lr}
F_{S^3} (\Delta_I), & \text{for } d = 1\\
a (\Delta_I), & \text{for } d = 2
\end{array}
\, ,
\end{aligned}
\ee
where $\widetilde\mathcal{W}_d (\Delta_I)$ is the effective twisted superpotential, evaluated on the dominant Bethe vacuum,
$F_{S^3}(\Delta_I)$ is the $S^3$ free energy of 3D $\mathcal{N} = 2$ theories, computed for example in \cite{Herzog:2010hf,Jafferis:2011zi,Fluder:2015eoa}, and $a(\Delta_I)$ is the conformal anomaly coefficient of 4D $\mathcal{N} = 1$ theories.
Here (and throughout the paper) $\approx$ denotes the equality at large $N$.%
\footnote{The relation \eqref{index:theorem:original:3d:4D} is only valid when we use a set of chemical potentials such that $\widetilde\mathcal{W}_d (\Delta_I)$ is a homogeneous function of the $\Delta_I$ (and a similar parameterization for the fluxes),
which is always possible \cite{Hosseini:2016tor,Hosseini:2016cyf}. Otherwise, \eqref{index:theorem:original:3d:4D} should be replaced by $\log Z_{\Sigma_{\mathfrak{g}_1} \times T^d} \approx( 1 - \mathfrak{g}_1 ) \mathfrak{D}^{(1)}_d \widetilde\mathcal{W}_d (\Delta_I)$,
where $\mathfrak{D}^{(1)}_d \equiv \frac{(d + 1)}{\pi} + \sum_{I} \Big( \frac{\mathfrak{s}_I}{1 - \mathfrak{g}_1} - \frac{\Delta_I}{\pi} \Big) \frac{\partial}{\partial \Delta_I}$ for $\mathfrak{g}_1 \neq 1$.}
Based on \eqref{index:theorem:original:3d:4D} it has been conjectured in \cite{Hosseini:2016tor} that
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{intro:conjecture:index:attractor}
\widetilde\mathcal{W}_d (\Delta_I) \propto \mathcal{F}_{\text{sugra}} ( X^\Lambda ) \, , \qquad
\mathcal{I}\text{-extremization} = \text{attractor mechanism} \, ,
\eea
where $\mathcal{F}_{\text{sugra}} ( X^\Lambda )$ is the prepotential of the effective $\mathcal{N} = 2$ gauged supergravity in four dimensions describing the horizon of the black hole or black string.
We refer the reader to section \ref{subsec:attractor} for details on the attractor mechanism in gauged supergravity.
\subsection{The five-dimensional index}
In this paper we take the first few steps in generalizing the above analysis to five dimensions.
We will consider the case of a generic $\mathcal{N} = 1$ gauge theory on $\mathcal{M}_4 \times S^1$, where $\mathcal{M}_4$ is a toric K\"ahler manifold and $S^1$ a circle of length $\beta$.%
\footnote{See \cite{Kim:2012tr,Kim:2013nva} for another localization computation on $\mathbb{P}^2 \times S^1$.}
One main complication compared to three and four dimensions is that, in the localization computation for five-dimensional gauge theories, there are non-perturbative contributions due to the presence of instantons.
The topologically twisted index is still given by the contour integral
\begin{equation}
\label{5Dtwisted}
Z _{\mathcal{M}_4 \times S^1}(q,\mathfrak{s}_I, y_I) =
\sum_{k=0}^\infty \sum_{\{\mathfrak{m} \} | \text{semi-stable}} \oint_\mathcal{C} q^k Z_{\text{int}}^{(k\text{-instantons})} (\mathfrak{m}, a; q, \mathfrak{s}_I , y_I) \, ,
\end{equation}
of a meromorphic form in the complex variable $a$, which parameterizes the Coulomb branch of the four-dimensional theory obtained by compactifying on $S^1$.
Here $q=e^{-8\pi^2 \beta/g_{\text{YM}}^2}$ is the instanton counting parameter with $g_{\text{YM}}$ being the gauge coupling constant,
and $\mathfrak{m}$ are a set of gauge magnetic fluxes that depend on the toric data of $\mathcal{M}_4$.
We find it useful to work first on an $\Omega$-deformed background specified by equivariant parameters $\epsilon_1$ and $\epsilon_2$ for the toric $(\mathbb{C}^*)^2$
action on $\mathcal{M}_4$, for which we write explicitly the supersymmetry background, the Lagrangian and the one-loop determinant around the classical saddle point configurations.
The non-perturbative contribution is given by contact instantons that, with the equivariant parameters turned on, are localized at the fixed point of the toric action on $\mathcal{M}_4$ and wrap $S^1$.
There are $\chi(\mathcal{M}_4)$ fixed points and each of these contributes a copy of the five-dimensional Nekrasov's partition function $Z_{\text{Nekrasov}}^{\mathbb{C}^2\times S^1}(a,\epsilon_1,\epsilon_2)$
on $\mathbb{C}^2\times S^1$, so that the partition function on $\mathcal{M}_4\times S^1$ is given by the gluing formula
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{intro:gluing}
Z_{\mathcal{M}_4 \times S^1} =
\sum_{\{\mathfrak{m} \} | \text{semi-stable}} \oint_\mathcal{C} \mathrm{d} a \prod_{l = 1}^{\chi (\mathcal{M}_4)} Z_{\text{Nekrasov}}^{\mathbb{C}^2 \times S^1} \big(a^{(l)}, \epsilon_1^{(l)}, \epsilon_2^{(l)} \big) \, ,
\ee
where $a^{(l)}$, $\epsilon_1^{(l)}$ and $\epsilon_2^{(l)}$ are determined by the toric data near each fixed point and encode the dependence on the fluxes $\mathfrak{m}$.
For brevity, we suppressed the dependence on the flavor fluxes and fugacities $\mathfrak{s}_I$ and $y_I$ that can be easily reinstated by considering them as components of a background vector multiplet.
This formula is the five-dimensional analogue of a similar four-dimensional expression that has been successfully used to evaluate equivariant Donaldson invariants \cite{Nekrasov:2003vi,Bawane:2014uka,Bershtein:2015xfa,Bershtein:2016mxz}.
In this paper we will be interested in the non-equivariant limit $\epsilon_1,\epsilon_2\rightarrow 0$. Despite the fact that $Z_{\text{Nekrasov}}^{\mathbb{C}^2\times S^1}(a,\epsilon_1,\epsilon_2)$ is singular in the non-equivariant limit, $Z_{\mathcal{M}_4 \times S^1}$
is perfectly smooth. Moreover, with an eye to holography, we will be mostly interested in the large $N$ limit and therefore we will neglect instanton contributions, since they are exponentially suppressed in this limit.
As we will see, the classical and one-loop contributions to the non-equivariant partition function still yield a non-trivial and complicated matrix model.
As for the topologically twisted index in three dimensions, we can interpret the result as the Witten index of the quantum mechanics obtained by reducing the $\mathcal{N} = 1$ gauge theory on $\mathcal{M}_4$ in the presence of background magnetic fluxes $\mathfrak{s}_I$. This index receives contributions from infinitely many topological sectors specified by the gauge magnetic fluxes $\mathfrak{m}$.
We will also initiate the study of the large $N$ limit of the topologically twisted index in five dimensions and of other related quantities, leaving a more complete analysis for the future.
We will focus on two five-dimensional $\mathcal{N} = 1$ field theories. The first is the $\mathrm{USp}(2N)$ theory with $N_f$ flavors and an antisymmetric matter field, which has a 5D ultraviolet (UV) fixed point with enhanced $E_{N_f+1}$ global symmetry \cite{Seiberg:1996bd}. The theory is
dual to AdS$_6 \times_w S^4$ in massive type IIA supergravity \cite{Brandhuber:1999np}. The second theory is $\mathcal{N} = 2$ super Yang-Mills (SYM), which we consider as the compactification of the $\mathcal{N} = (2,0)$ theory in six dimensions on a circle of radius
$R_6 = g_{\text{YM}}^2/(8 \pi^2)$ \cite{Douglas:2010iu,Lambert:2010iw,Bolognesi:2011nh}.%
\footnote{An analogous argument has been used in \cite{Kim:2012tr,Kim:2013nva,Kim:2012ava} in order to study the superconformal index of the 6D $\mathcal{N} = (2,0)$ theory.}
Using this interpretation, the index of $\mathcal{N} = 2$ SYM can be considered as the partition function of the $\mathcal{N} = (2,0)$ theory on $\mathcal{M}_4 \times T^2$.
The topologically twisted index at large $N$ for the $\mathrm{USp}(2N)$ theory should then contain information about black holes with horizons AdS$_2 \times \mathcal{M}_4$ in massive type IIA supergravity, while the index for $\mathcal{N} = 2$ SYM should contain information
about AdS$_7 \times S^4$ black strings in M-theory.
An interesting object to study in the large $N$ limit is the Seiberg-Witten (SW) prepotential $\mathcal{F}(a)$ of the four-dimensional theory obtained by compactifying on $S^1$,
which receives contributions from all the Kaluza-Klein (KK) modes on $S^1$ \cite{Nekrasov:1996cz}.
$\mathcal{F}(a)$ is expected to play a role similar to the twisted superpotential $\widetilde \mathcal{W}$ in three and four dimensions. We therefore study the distribution of its critical points in the large $N$ limit and we find that its critical value,
as a function of the chemical potentials $\Delta_\varsigma$ $(\varsigma = 1, 2)$,%
\footnote{The $\Delta_\varsigma$ parameterize the Cartan of the $\mathrm{SU}(2)$ R-symmetry and the $\mathrm{SU}(2)$ flavor symmetry of
the $\mathrm{USp}(2N)$ theory and the Cartan of the $\mathrm{SO}(5)$ R-symmetry of the $\mathcal{N} = (2,0)$ theory, respectively.
They satisfy the constraint $\sum_{\varsigma = 1}^2 \Delta_\varsigma = 2 \pi$.
Similarly, in the case of $\Sigma_{\mathfrak{g}_1}\times \Sigma_{\mathfrak{g}_2}\times S^1$ discussed below, the fluxes fulfill the constraints $\sum_{\varsigma = 1}^2 \mathfrak{s}_\varsigma = 2 (1 - \mathfrak{g}_1)$, $\sum_{\varsigma = 1}^2 \mathfrak{t}_\varsigma = 2 (1 - \mathfrak{g}_2)$.
With such a choice, all expressions in \eqref{index:theorem:F:5D}, \eqref{index:theorem:W:5D} and \eqref{index:theorem:logZ:5D} are homogeneous functions of
$\Delta_\varsigma$, $\mathfrak{s}_\varsigma$ and $\mathfrak{t}_\varsigma$. Details are given in section \ref{sec:largeN}.} is given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{index:theorem:F:5D}
\mathcal{F}(\Delta_\varsigma) \propto F_{S^5} (\Delta_\varsigma) \, ,
\ee
where $F_{S^5} (\Delta_\varsigma)$ is the free energy on $S^5$ of the corresponding $\mathcal{N}=1$ theory, in perfect analogy with \eqref{index:theorem:original:3d:4D}.
One of the reasons for analysing the critical points of $\mathcal{F}(a)$ is the expectation that they play a role similar to the Bethe vacua for the five-dimensional partition function, and, in particular, one such critical point dominates the large $N$ limit of the index.
We have no real evidence that this is the case but we will see that working under this assumption leads to interesting results.
Some motivations for this conjecture are discussed in section \ref{sec:largeN}.
We will consider the particular example of an $\mathcal{N}=1$ field theory on $\mathbb{P}^1 \times \mathbb{P}^1 \times S^1$.
With no effort, we can generalize all results to the non-toric manifold $\Sigma_{\mathfrak{g}_1}\times \Sigma_{\mathfrak{g}_2}\times S^1$,
where $\Sigma_{\mathfrak{g}_1}$ and $\Sigma_{\mathfrak{g}_2}$ are two Riemann surfaces of genus $\mathfrak{g}_1$ and $\mathfrak{g}_2$, respectively.
We denote by $\mathfrak{s}_\varsigma$ and $\mathfrak{t}_\varsigma$ $(\varsigma = 1, 2)$ the background magnetic fluxes on $\Sigma_{\mathfrak{g}_1}$ and $\Sigma_{\mathfrak{g}_2}$.
We will be able to define an effective twisted superpotential $\widetilde \mathcal{W}$ for the three-dimensional theory that we obtain by compactifying the five-dimensional $\mathcal{N}=1$ theory on $\Sigma_{\mathfrak{g}_2}$.
We refer for details to section \ref{sec:largeN}. We will find that the value of $\widetilde \mathcal{W}$, evaluated at the combined critical points of $\mathcal{F}$ and $\widetilde \mathcal{W}$, as a function of the chemical potentials $\Delta_\varsigma$ and fluxes $\mathfrak{t}_\varsigma$, satisfies
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{index:theorem:W:5D}
\begin{aligned}
\widetilde\mathcal{W} ( \mathfrak{t}_\varsigma , \Delta_\varsigma)\propto \sum_{\varsigma = 1}^{2} \mathfrak{t}_\varsigma \frac{\partial F_{S^5} (\Delta_\varsigma) }{\partial \Delta_\varsigma } \propto \Bigg\{
\begin{array}{lr}
F_{\Sigma_{\mathfrak{g}_2} \times S^3} ( \mathfrak{t}_\varsigma , \Delta_\varsigma) \, , & \text{ for } ~~ \mathrm{USp}(2N) \\
a ( \mathfrak{t}_\varsigma , \Delta_\varsigma) \, , & \text{ for } \, \mathcal{N} = (2,0)
\end{array}
\, .
\end{aligned}
\ee
Here $F_{\Sigma_{\mathfrak{g}_2} \times S^3}( \mathfrak{t}_\varsigma , \Delta_\varsigma)$ is the $S^3$ free energy of the three-dimensional $\mathcal{N} = 2$ theory obtained by compactifying
the $\mathrm{USp}(2N)$ theory on $\Sigma_{\mathfrak{g}_2}$, recently computed holographically in \cite{Bah:2018lyv},
and $a( \mathfrak{t}_\varsigma , \Delta_\varsigma)$ is the conformal anomaly coefficient of the four-dimensional $\mathcal{N} = 1$ theory obtained by compactifying the $\mathcal{N} = (2,0)$ theory on $\Sigma_{\mathfrak{g}_2}$, computed in \cite{Bah:2011vv,Bah:2012dg}.
We verified the statement for the $\mathrm{USp}(2N)$ theory only upon extremization with respect to $\Delta_\varsigma$, but we expect it to be true for all values of the chemical potentials.\footnote{This was confirmed in \cite{Crichigno:2018adf} that appeared after the completion of this work.}
We shall also consider the large $N$ limit of the topologically twisted index itself. The matrix model is too hard to compute directly even in the large $N$ limit.
The main difficulty compared to the three- and four-dimensional cases is the quadratic dependence on the gauge and background fluxes that do not allow for a simple resummation in \eqref{intro:gluing}.
The case of $\mathbb{P}^1\times \mathbb{P}^1\times S^1$ is technically simpler, since there are two sets of gauge fluxes, one for each $\mathbb{P}^1$, but still too hard to attack directly.
By resumming one set of gauge magnetic fluxes (call them $\mathfrak{m}$), we obtain a set of Bethe equations for the eigenvalues $a_i$ (these are just the Bethe vacua of the effective twisted superpotential $\widetilde \mathcal{W}$ of the compactification on $\Sigma_{\mathfrak{g}_2}$).
The result still depends on the second set of gauge magnetic fluxes (call them $\mathfrak{n}$). We expect that, in the large $N$ limit, one single distribution of eigenvalues $a_i$ and one single set of fluxes $\mathfrak{n}_i$ dominate the partition function.
At this point we shall pose the conjecture that the extremization of $\mathcal{F}(a)$ provides the missing condition for determining both $a_i$ and $\mathfrak{n}_i$. Under this conjecture we obtain
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{index:theorem:logZ:5D}
\log Z_{\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}\times S^1}\propto \sum_{\varsigma = 1}^{2} \mathfrak{s}_\varsigma \frac{\partial \widetilde \mathcal{W} (\mathfrak{t}_\varsigma , \Delta_\varsigma) }{\partial \Delta_\varsigma } \propto\sum_{\varsigma , \varrho = 1}^{2} \mathfrak{s}_\varsigma \mathfrak{t}_\varrho \frac{\partial^2 F_{S^5} (\Delta) }{\partial \Delta_\varsigma \partial \Delta_\varrho} \, ,
\ee
where we generalized the result to $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2} \times S^1$.
It is remarkable that \eqref{index:theorem:logZ:5D}, which is based on a conjecture, is completely analogous to the three- and four-dimensional result \eqref{index:theorem:original:3d:4D}.
Even more remarkably, we can compare \eqref{index:theorem:logZ:5D} with the existing results for the twisted compactification of the 6D $\mathcal{N} = (2,0)$ theory on $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$ \cite{Benini:2013cda}.
\eqref{index:theorem:logZ:5D} is expected to compute the leading behaviour, in the limit $\tau \to \mathbbm{i} 0^+$, of the elliptic genus of the two-dimensional CFT obtained by the twisted compactification.
We find that \eqref{index:theorem:logZ:5D} indeed leads to the correct Cardy behaviour
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{N=1*:SYM:Cardy:intro}
\log Z_{\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}\times S^1} \approx \frac{\mathbbm{i} \pi}{12 \tau} c_r (\mathfrak{s}_\varsigma , \mathfrak{t}_\varsigma , \Delta_\varsigma) \, ,
\ee
where $c_r (\mathfrak{s}_\varsigma , \mathfrak{t}_\varsigma , \Delta_\varsigma)$ precisely coincides with the trial central charge of the two-dimensional CFT computed in \cite{Benini:2013cda}.
Moreover, we will show in section \ref{sec:4D domain-walls} that \eqref{index:theorem:logZ:5D} is equivalent to the attractor mechanism for the corresponding black strings in AdS$_7$.
All this is in complete analogy with the four-dimensional results \eqref{Cardy:4d:intro} and \eqref{intro:conjecture:index:attractor}.
It would be very interesting to see if the conjectured result for the $\mathrm{USp}(2N)$ theory matches the entropy of magnetically charged AdS$_6 \times_w S^4$ black holes in massive type IIA supergravity, which are still to be found.
Work in this direction is in progress \cite{Hosseini:2018mIIA}.
\subsection{Overview}
The structure of this paper is as follows. In section \ref{sec:localization} we analyse the conditions of supersymmetry and the Lagrangian for a five-dimensional $\mathcal{N}=1$ gauge theory on $\mathcal{M}_4\times S^1$,
where $\mathcal{M}_4$ is a toric manifold, in a $\Omega$-background for the torus action $(\mathbb{C}^*)^2$.
We determine the classical saddle points and compute explicitly the one-loop determinants.
We finally write an expression for the (equivariant) topologically twisted index as a gluing of various copies of the K-theoretic Nekrasov's partition function,
one for each fixed point of the toric action. We then study in detail the non-equivariant limit in the sector with no instantons.
We also write explicitly the SW prepotential $\mathcal{F}(a)$ that will play an important role in the rest of the paper.
In section \ref{sec:largeN} we discuss the large $N$ limit of the topologically twisted index and of related quantities.
We first motivate the importance of finding the critical points of $\mathcal{F}(a)$.
Then we consider the partition function on $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2} \times S^1$ of two theories, $\mathcal{N} = 2$ SYM, which decompactify to the $\mathcal{N} = (2,0)$ theory in six dimensions
and the $\mathrm{USp}(2N)$ theory with $N_f$ flavors and an antisymmetric matter field, corresponding to a 5D UV fixed point.
Under some assumptions, we will derive \eqref{index:theorem:F:5D}, \eqref{index:theorem:W:5D} and \eqref{index:theorem:logZ:5D}.
Special attention will be devoted to the 6D $\mathcal{N} = (2,0)$ theory where we can compare the results with the existing holographic literature about domain walls and black strings.
For this theory, in section \ref{sec:4D domain-walls} we will be able to interpret our results as the counterpart of the attractor mechanism in four-dimensional $\mathcal{N} = 2$ gauged supergravity.
We conclude in section \ref{sec:discussion and outlook} with discussions and future problems to explore.
Some details regarding toric varieties, conventions, computation of anomaly coefficients in twisted compactifications, and polylogarithms are collected in four appendices.
\paragraph*{Note added:} while we were writing this work, we became aware of \cite{Crichigno:2018adf} which has some overlaps with the results presented here.
\section[Localization on \texorpdfstring{$\mathcal{M}_4 \times S^1$}{M(4) x S**1}]{Localization on $\mathcal{M}_4 \times S^1$}
\label{sec:localization}
In this section we evaluate the twisted indices of five-dimensional $\mathcal{N}\ge1$
theories, \textit{i.e.}\;the partition function on $\mathcal{M}_{4}\times S^{1}$,
using localization. We begin in section \ref{subsec:Geometry-of-M4} by describing
the geometry of the toric K\"ahler manifold $\mathcal{M}_{4}$. Although
the twisted theory is semi-topological, \textit{i.e.}\;does not depend on the
metric on $\mathcal{M}_{4}$, we find it useful to have a canonical
set of coordinates and a canonical metric. In section \ref{subsec:Supergravity-Background},
we describe the rigid supergravity background to which we couple the
theory in order to produce the twist and the $\Omega$-deformation.
We describe the relevant supersymmetry algebra and supersymmetric
actions in section \ref{subsec:Supersymmetry-algebra}.
The localization procedure is carried out in section \ref{subsec:Localization}. In section \ref{subsec:Nekrasov partition function} we present the relevant expression for the K-theoretic Nekrasov's partition function.
Finally, in section \ref{completePF} we present the complete partition function on $\mathcal{M}_{4}\times S^{1}$.
\subsection[Geometry of \texorpdfstring{$\mathcal{M}_{4}$}{M(4)}]{Geometry of $\mathcal{M}_{4}$}
\label{subsec:Geometry-of-M4}
We review the construction of a canonical invariant metric for a toric
K\"ahler manifold $\mathcal{M}_{4}$ in symplectic coordinates \cite{2000math......4122A,guillemin1994}.
A K\"ahler manifold $M^{2n}$ is a complex manifold of real dimension
$2n$ with an integrable almost complex structure
\begin{equation}} \newcommand{\ee}{\end{equation}
J^{2}=-\mathbbm{1}_{2n} \, .
\ee
It is also a symplectic manifold with symplectic form $\omega$,
satisfying a compatibility condition on the metric defined
by the bilinear form
\begin{equation}} \newcommand{\ee}{\end{equation}
g\equiv\omega\left(\cdot,J\cdot\right) \, ,
\ee
which states that $g$ is symmetric and positive definite.
A toric K\"ahler manifold is a K\"ahler manifold with an effective (faithful),
Hamiltonian, and holomorphic action of a real $n$-torus $\mathbb{T}^{n}$.
Given a Hamiltonian action, there exists a vector field $\tilde{v}$
for each element of the Lie algebra of $\mathbb{T}^{n}$ and a smooth
function $\mu$, the moment map, such that
\begin{equation}
\tilde{v}=\omega^{-1} \mathrm{d} \mu \, .
\label{eq:moment-map-equation}
\end{equation}
The moment map should be thought of as an equivariant map from the
Lie algebra of $\mathbb{T}^{n}$ to the space of smooth functions
on $M^{2n}$. It is defined only up to the addition of a constant.
The image of the moment map, the orbit space
\begin{equation}} \newcommand{\ee}{\end{equation}
\Delta\equiv M^{2n}/\mathbb{T}^{n} \, ,
\ee
is a convex $n$-dimensional polytope called the moment polytope. It can be written as
\begin{equation}} \newcommand{\ee}{\end{equation}
\Delta=\left\{ x\in\mathbb{R}^{n}|\left\langle x,u_{i}\right\rangle -\lambda_{i}\ge0,\; i\in\left\{ 1, \ldots, d\right\} \right\} ,
\ee
for an appropriate set of data
\begin{equation}} \newcommand{\ee}{\end{equation}
u_{i}\in\mathbb{Z}^{n} \, ,\qquad \lambda_{i}\ge0 \, .
\ee
Its vertices are located at the fixed points of the torus action and
$\Delta$ is the convex hull. The moment polytope is related to the
combinatorial description of $M^{2n}$ as a toric variety with an
associated toric fan, dual to $\Delta$, which constructed out of the vectors $n_{j}$ (see appendix \ref{sec:toric geometry}).
It will be important in the following that the number of vertices $d$ of the polytope, or equivalently the number of vectors $n_{j}$ of the fan, is equal to the
number of fixed points of the toric action. It is also equal to the
Euler characteristic of $M^{2n}$, $d=\chi(M^{2n})$.
One may describe all three structures appearing in the definition
of $M^{2n}$ explicitly using symplectic coordinates: $x^{i}$ for
$\Delta$ and $y^{i}$ for $\mathbb{T}^{n}$. Define the functions
\begin{equation}} \newcommand{\ee}{\end{equation}
l_{r}\left(x\right)\equiv\left\langle x,u_{r}\right\rangle -\lambda \, ,
\ee
and an auxiliary potential function
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
p\left(x\right)\equiv g_{p}\left(x\right)+h\left(x\right) \, ,\qquad
g_{p}\equiv\frac{1}{2}\sum_{r=1}^{d}l_{r}\left(x\right)\log l_{r}\left(x\right) \, ,\qquad
G_{ij}\left(x\right)\equiv\partial_{x^{i}}\partial_{x^{j}}p\left(x\right) \, .
\eea
The function $h\left(x\right)$ must be such that there exists a smooth,
strictly positive function $\delta\left(x\right)$ satisfying
\begin{equation}} \newcommand{\ee}{\end{equation}
\frac{1}{\det G(x)} = \delta(x)\prod_{r=1}^{d}l_{r}(x) \, .
\ee
The complex structure, symplectic (K\"ahler) form, and $\mathbb{T}^{n}$
invariant K\"ahler metric are then given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
J=\begin{pmatrix}0 & -G^{-1} \\
G & 0
\end{pmatrix} \, ,\qquad
\omega=dx^{i}\wedge dy_{i} \, ,\qquad
g=G_{ij}dx^{i}dx^{j}+\left(G^{-1}\right)^{ij}dy_{i}dy_{j} \, .
\eea
Note that $\det g=1$.
All smooth symplectic toric manifolds are simply connected \cite{fulton1993introduction}.
Compact simply connected topological four-manifolds are mostly classified
by their intersection form. Note, in particular, that
\begin{equation}} \newcommand{\ee}{\end{equation}
b_{2}^{+}=1 \, ,
\ee
for any symplectic toric four-manifold. One can check with the metric
above that%
\footnote{The orientation for which this is true is such that $\varepsilon^{x_{1}y^{1}x_{2}y^{2}}=1$.}
\begin{equation}} \newcommand{\ee}{\end{equation}
\star\omega=\omega \, .
\ee
\subsection{Nekrasov's conjecture}
\label{sec:Nekrasov's conjecture}
There is a standard way, reviewed in the next section, of putting
any four-dimensional $\mathcal{N}=2$ Lagrangian field theory on a smooth four-manifold
while preserving supersymmetry. This is done using the Witten twist
\cite{Witten:1988ze}. The resulting computations are insensitive
to, or at least piece-wise constant under, variations of the metric.
This is an example of a cohomological topological quantum field theory (TQFT),
usually called the Donaldson-Witten TQFT.
The relevant observables reside in the cohomology of the preserved supercharge.
Nekrasov has introduced a generalization of this TQFT which is valid
when the four-manifold admits a metric with an isometry \cite{Nekrasov:2003vi}.%
\footnote{In this section we restrict ourselves to describing the $\mathrm{U}(N)$ theory.}
The toric manifolds described in the previous section are prime examples of this construction.
The construction can be seen as a generalization of the computation
of the equivariant partition function for theories on $\mathbb{R}^4$,
that can be used to recover the exact effective prepotential \cite{Nekrasov:2002qd,Nekrasov:2003rj}.
The latter can be defined as \cite{Nekrasov:2002qd}
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{F}_{0}(\Lambda , a) \equiv \lim_{\epsilon_{1,2} \to 0} \epsilon_{1} \epsilon_{2} \log Z_{\text{Nekrasov}}^{\mathbb{C}^2} ( q, a, \epsilon_{1} , \epsilon_{2} ) \, , \qquad q \to \Lambda^{2h^{\lor}(G) - k (\mathfrak{R})} \, ,
\ee
where $Z_{\text{Nekrasov}}^{\mathbb{C}^2}$ is the so-called Nekrasov's partition function,
coinciding with the partition function on $\mathbb{R}^{4}$ in
the presence of the $\Omega$ deformation with parameter $\vec\epsilon = ( \epsilon_{1},\epsilon_{2})$.
$\Lambda$ is the dynamically generated scale, and $a$ represents
the vacuum expectation value for the scalar field in the vector multiplet at a specific point on the Coulomb branch.
Moreover, $h^{\lor}(G)$ is the dual Coxeter number of the gauge group $G$ and $k (\mathfrak{R})$ is the quadratic Casimir normalized such that it is $2h^{\lor}(G)$ for the adjoint representation.
It has been argued in \cite{Nekrasov:2003vi} that the analogous partition function on a compact toric manifold $\mathcal{M}_{4}$ takes the form
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{Nekrasov}
Z_{\mathcal{M}_{4}}=\sum_{\mathfrak{p}_{l} \in \mathbb{Z}^{N}} \oint_\mathcal{C} \mathrm{d} a \prod_{l = 1}^{\chi (\mathcal{M}_4)} Z_{\text{Nekrasov}}^{\mathbb{C}^2} \big(a + \epsilon_{1}^{(l)} \mathfrak{p}_{l} + \epsilon_{2}^{(l)} \mathfrak{p}_{l+1}, \epsilon^{(l)}_{1} , \epsilon^{(l)}_{2} ; q \big) .
\ee
In the equation above we have chosen to disregard insertions of operators
and the dependence on characteristic classes for non-simply connected
gauge groups, both of which are not relevant for our purposes. The
main new ingredient in this formula, in comparison to the formula
on $\mathbb{R}^{4}$, is the appearance of a sum over a set of fluxes
$\mathfrak{p}_l$. These are associated with equivariant divisors
on $\mathcal{M}_{4}$, and thus with vectors in the toric fan. The
deformation parameters $\epsilon^{(l)}_{1}$, $\epsilon^{(l)}_{2}$ are also given by the data
in the fan. Note that the modulus $a$ is now integrated over,
as should be the case on a compact space. The result presented in
\cite{Nekrasov:2003vi} is a conjecture. Specifically, the exact form
of the sum over the integers $\mathfrak{p}_{l}$ and the contour
for the integral over the modulus $a$ are not known.
It is expected that the results for the Donaldson-Witten theory are recovered in the non-equivariant limit, $\epsilon_{1,2} \to 0$.
Nekrasov conjectured that this limit is given by%
\footnote{We correct a misprint in \cite{Nekrasov:2003vi} here.}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
Z_{\mathcal{M}_{4}} = \sum_{\mathfrak{k}^{(i)} \in \mathbb{Z}^{N} } \oint_\mathcal{C} \mathrm{d} a \exp & \Bigg[\int_{\mathcal{M}_{4}}\mathcal{F}_{0} \bigg(a+\sum_{i} \mathfrak{k}^{(i)} c_{1} (L_{i}) \bigg)
+c_{1}(\mathcal{M}_{4}) \mathcal{H}_{\frac{1}{2}} \bigg (a+\sum_{i} \mathfrak{k}^{(i)} c_{1}(L_{i}) \bigg)\\
& + \chi(\mathcal{M}_{4}) \mathcal{F}_{1}(a) + \left( 3 \sigma(\mathcal{M}_{4}) + 2 \chi(\mathcal{M}_{4}) \right) \mathcal{G}_{1}(a)\Bigg].
\eea
The $ L_{i}$ are line bundles supported on two-cycles of $\mathcal{M}_{4}$,
which do not have a flat space analogue, and $ c_{1} (L_{i})$ their first Chern class. $\chi(\mathcal{M}_{4})$ is the Euler characteristic of $\mathcal{M}_{4}$ and $\sigma(\mathcal{M}_{4})$ its signature.
The additional terms in the exponential, relative to the usual effective action on $\mathbb{R}^{4}$,
come from subleading terms in $Z_{\text{Nekrasov}}^{\mathbb{C}^2}$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\log Z_{\text{Nekrasov}}^{\mathbb{C}^2} \left(a,\epsilon_{1},\epsilon_{2};q\right)=\frac{1}{\epsilon_{1}\epsilon_{2}}\mathcal{F}_{0}+\frac{\epsilon_{1}+\epsilon_{2}}{\epsilon_{1}\epsilon_{2}}\mathcal{H}_{\frac{1}{2}}+\mathcal{F}_{1}+\frac{\left(\epsilon_{1}+\epsilon_{2}\right)^{2}}{\epsilon_{1}\epsilon_{2}}\mathcal{G}_{1}+\ldots \, .
\ee
The authors of \cite{Bawane:2014uka,Bershtein:2015xfa,Bershtein:2016mxz}
have began to verify \eqref{Nekrasov} using localization. Our twisted
indices are a generalization of the partition functions on $\mathcal{M}_{4}$,
and we will follow closely the arguments used in these papers.
We will not have anything to add regarding the part of the calculation
involving the sum over fluxes. However, we will comment on the similarities
between the present setup and the calculation using localization of the
twisted indices in three and four dimensions, in which a similar contour integral arises and is given
by an explicit prescription.
\subsection[Supersymmetry on \texorpdfstring{$\mathcal{M}_4 \times S^1$}{M(4) x S**1}]{Supersymmetry on $\mathcal{M}_4 \times S^1$}
\label{subsec:Supergravity-Background}
Supersymmetric theories can sometimes be coupled to a curved background
while preserving some supersymmetry. This was originally achieved
by twisting the theory --- identifying a new euclidean rotation group
with a diagonal subgroup of rotations and R-symmetry transformations.
A subset of the supercharges become scalars under the new rotation
group, and are conserved on an arbitrary curved manifold, as long
as the coupling to the metric is implemented using this new group.
As a bonus, the energy momentum tensor turns out to be the supersymmetry
variation of a scalar supercharge $Q$. The twisted theory, where
observables are restricted to be $Q$-closed operators, then becomes
a TQFT of cohomological type \cite{Witten:1988ze}.
A more general procedure for preserving supersymmetry, initiated in
\cite{Festuccia:2011ws} and continued for four dimensions in \cite{Dumitrescu:2012ha,Klare:2012gn}, is to couple the theory to rigid supergravity
and to search for backgrounds which are fixed points of the supersymmetry
transformations. Technically, this means choosing a configuration
for the bosonic fields in the supergravity multiplet such that the
supersymmetry variation of the gravitino vanishes for some spinor.
The vanishing of the gravitino variation yields a linear differential
equation known as a generalized Killing spinor equation whose solutions
are known as generalized Killing spinors. A variation using these
spinors constitutes a rigid supersymmetry.
One can expand the scope of this construction by considering superconformal
tensor calculus instead of a specific Poincare supergravity \cite{Klare:2013dka,Klare:2012gn}. In superconformal
tensor calculus, the gravitino is part of the Weyl multiplet which
includes another fermion, the dilatino, whose supersymmetry variation
must also vanish. The resulting solutions are generalized Killing
spinors which generate an action of a subalgebra of the superconformal
algebra on the dynamical fields. In order to use this algebra to localize,
one should avoid including transformations which are not true symmetries
of the theory such as dilatations. To the best of our knowledge, this
is the most general context in which this program of preserving rigid
supersymmetry on curved backgrounds has been pursued.
A five-dimensional $\mathcal{N}=1$ theory with R-symmetry group $\mathrm{SU}(2)$
can be formulated while preserving supersymmetry on any five-manifold
as long as the holonomy group is contained in $\mathrm{SO}(4)$.
The necessary supergravity background is simply a twist. We derive
the rigid 5D $\mathcal{N}=1$ supergravity background corresponding
to the $\Omega$-background on a manifold with topology $\mathcal{M}_{4}\times S^{1}$,
where $\mathcal{M}_{4}$ is a toric K\"ahler four-manifold. Supersymmetry
is preserved using a five-dimensional uplift of the Witten twist on
$\mathcal{M}_{4}$ augmented to include the $\Omega$-deformation.
The rigid supergravity background for a twisted four-dimensional $\mathcal{N}=2$
theory, with the background corresponding to the $\Omega$-deformation,
was explicitly constructed for any toric K\"ahler manifold in \cite{Rodriguez-Gomez:2014eza}.
These backgrounds can be lifted to the 5D $\mathcal{N}=1$ theory
on $\mathcal{M}_{4}\times S^{1}$ in a straightforward manner, implicitly
described in \cite{Alday:2015lta,Pini:2015xha}. We review this below. Our spinor
and metric conventions are spelled out in appendix \ref{sec: spinor conventions}.
We consider $X\equiv\mathcal{M}_{4}\times S^{1}$ and choose coordinates
such that the $S^{1}$ is parameterized by $x_{5}\in\left[0,\beta\right)$.
The construction of a $T^{2}$ invariant K\"ahler metric $g$ for $\mathcal{M}_{4}$ was reviewed in section \textcolor{red}{\ref{subsec:Geometry-of-M4}}.
Let us define
\begin{equation}} \newcommand{\ee}{\end{equation}
\tilde{v}=\epsilon_{i}\partial_{y^{i}} \, , \qquad
x_{2}\equiv y_{1} \, ,\qquad x_{4}=y_{2} \, ,
\ee
and let
\begin{equation}} \newcommand{\ee}{\end{equation}
e_{a}^{\phantom{a}m} \, , \qquad a\in\{ 1,2,3,4\} \, , \qquad m\in\{ 1,2,3,4\} \, ,
\ee
be a vielbein for $g$. We define the metric on $X$ by augmenting $e_{a}^{\phantom{a}m}$ with
\begin{equation}} \newcommand{\ee}{\end{equation}
e_{5}^{\phantom{5}m}=\tilde{v}^{m} \, , \qquad
e_{5}^{\phantom{5}5}=1 \, .
\ee
The associated spin connection still has $\mathrm{U}(2)$ holonomy.
The Weyl multiplet of five-dimensional superconformal tensor calculus
is described, for instance, in \cite{deWit:2017cle}. Along with the
vielbein, it contains the following independent bosonic fields: an
$\mathrm{SU}(2)$ R-symmetry gauge field which we denote $A_{m}^{(\text{R})}$,
and an anti-symmetric tensor $T_{mn}$, a vector $b_{m}$, and
a scalar $D$. The remaining bosonic fields are determined in terms
of these, and of the fermions, by constraints. We will turn off $T_{mn}$
and $b_{m}$. After some renaming, the variation of the gravitino
in the remaining background can be written as
\begin{equation}} \newcommand{\ee}{\end{equation}
\delta\psi_{I m}=D_{m}\xi_{I}-\Gamma_{m}\eta_{I} \, ,
\ee
where%
\footnote{Throughout this paper, $D_m$ will denote a generic covariant derivative. The covariance is with respect to the spin, R-symmetry, gauge, and background flavor symmetry connections.
We will specify the concrete form of the derivative when appropriate.}
\begin{equation}} \newcommand{\ee}{\end{equation}
D_{m}\xi_{I}\equiv\partial_{m}\xi_{I}+\frac{1}{4} \omega_{m}^{\phantom{m}ab}\Gamma_{ab}\xi_{I} + \left(A_{m}^{(\text{R})}\right)_{I}^{\phantom{I}J}\xi_{J} \, .
\ee
We perform the twist by setting
\begin{equation}} \newcommand{\ee}{\end{equation}
A_{m}^{(\text{R})}=\frac{1}{4} \omega_{m}^{\phantom{m}ab} \sigma_{ab} \, .
\ee
One can easily check that the spinor
\begin{equation}} \newcommand{\ee}{\end{equation}
\xi= - \frac{1}{\sqrt{2}}\begin{pmatrix}\tau^2\\
0
\end{pmatrix} \, , \qquad\eta=0 \, ,
\ee
is a solution. Note that the components of both $\omega$ and $A^{(\text{R})}$
in the $x_{5}$ direction vanish in the non-equivariant limit, $\epsilon_{1,2} \to 0$.
One may verify explicitly that $\xi$ satisfies the dilatino equation with an appropriate value of $D$.
\subsection{Supersymmetry transformations and Lagrangian}
\label{subsec:Supersymmetry-algebra}
We record the supersymmetry transformations for the vector and hypermultiplets,
following the conventions of \cite{Hosomichi:2012ek,Kallen:2012va}.
For the purposes of localization it is simpler to use twisted fields
defined using the Killing spinor $\xi$. Note that $\xi$ satisfies
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& \xi_I\xi^I = 1 \, , \qquad
v^{m}v_{m}=1 \, , \qquad
v^{m}\partial_{m}\equiv\xi_I\Gamma^{m}\xi^I\partial_{m}=\epsilon_{1}\partial_{3}+\epsilon_{2}\partial_{4}+\partial_{5} \, .
\eea
For consistency of notation with \cite{Qiu:2016dyj}, we define
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{not:a:contact:form}
\kappa_{m}\equiv g_{mn}v^{n} \, .
\ee
Note that $\kappa=\mathrm{d} x^5$ is \emph{not} a contact form for $\mathcal{M}_{4}\times S^{1}$.
\subsubsection{Vector multiplet}
The five-dimensional $\mathcal{N}=1$ vector multiplet has $8+8$ off-shell components.
It comprises a connection $A_{m}$, a real scalar $\sigma$, an $\mathrm{SU}(2)$
Majorana spinor $\lambda^{\alpha}_{\phantom{\alpha}I}$, and a triplet of auxiliary
fields $D_{IJ}$ satisfying the reality condition
\begin{equation}} \newcommand{\ee}{\end{equation}
\left(D^{*}\right)^{IJ}=\varepsilon^{IK}\varepsilon^{JL}D_{KL} \, ,
\ee
all in the adjoint representation of the Lie algebra $\mathfrak{g}$.
We use the physics convention where all gauge fields are hermitian. We define the gauge covariant derivative acting on fields in the adjoint representation and the field strength as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{gauge:covariant:derivative}
D_{m}\equiv\partial_{m}-\mathbbm{i}[A_{m},\cdot] \, , \qquad F_{mn}\equiv\partial_{m}A_{n}-\partial_{n}A_{m}-\mathbbm{i}[A_{m},A_{n}] \, .
\ee
A gauge transformation with parameter $\alpha$ reads
\begin{equation}} \newcommand{\ee}{\end{equation}
G_{\alpha} = \mathbbm{i}[\alpha, \cdot ] \, , \qquad G_{\alpha}A_{m}=D_{m}\alpha \, .
\ee
In order to ensure convergence of the actions in section \ref{subsec:Supersymmetric-Lagrangians},
we will preemptively rotate both $D_{IJ}$ and $\sigma$ into the imaginary plane
\begin{equation}} \newcommand{\ee}{\end{equation}
D_{IJ}\rightarrow\mathbbm{i}D_{IJ} \, ,\qquad
\sigma\rightarrow-\mathbbm{i}\sigma \, .
\ee
The new reality conditions are such that
\begin{equation}} \newcommand{\ee}{\end{equation}
\left(D^{*}\right)^{IJ}=-\varepsilon^{IK}\varepsilon^{JL}D_{KL} \, .
\ee
The supersymmetry transformations read \cite{Hosomichi:2012ek}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& \delta A_{m}=\mathbbm{i}\xi_{I}\Gamma_{m}\lambda^{I} \, , \qquad
\delta\sigma=-\xi_{I}\lambda^{I} \, , \\
& \delta\lambda_{I}=-\frac{1}{2}\Gamma^{mn}F_{mn}\xi_{I}-\mathbbm{i}D_{m}\sigma\Gamma^{m}\xi_{I}-\mathbbm{i}D_{IJ}\xi^{J} - 2 \mathbbm{i} \tilde{\xi}_{I}\sigma \, , \\
& \delta D_{IJ}=-\xi_{I}\Gamma^{m}D_{m}\lambda_{J}-\xi_{J}\Gamma^{m}D_{m}\lambda_{I}-\left[\sigma,\xi_{I}\lambda_{J}+\xi_{J}\lambda_{I}\right]+\tilde{\xi}_{I}\lambda_{J}+\tilde{\xi}_{J}\lambda_{I} \, .
\eea
The spinor $\tilde{\xi}$ is defined as
\begin{equation}} \newcommand{\ee}{\end{equation}
\tilde{\xi}\equiv\frac{1}{5}\Gamma^{m}D_{m}\xi \, ,
\ee
and therefore vanishes in the present context.
Following \cite{Kallen:2012cs}, we define the twisted fields%
\footnote{The orientation here is the opposite of that used in \cite{Kallen:2012cs}, and corresponds with the one used in \cite{Qiu:2016dyj}.
Due to this choice, some forms which were anti-self-dual in \cite{Kallen:2012cs} are now self-dual.}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& \Psi_{m}\equiv\xi_{I}\Gamma_{m}\lambda^{I} \, , \qquad
H_{mn}\equiv2F_{mn}^{+}+\mathbbm{i}\xi^{I}\Gamma_{mn}\xi^{J}D_{IJ} \, , \\
& \chi_{mn}\equiv\xi_{I}\Gamma_{mn}\lambda^{I}+v_{n}\xi_{I}\Gamma_{m}\lambda^{I}-v_{m}\xi_{I}\Gamma_{n}\lambda^{I} \, ,
\eea
where
\begin{equation}} \newcommand{\ee}{\end{equation}
F^{+}\equiv\frac{1}{2}\left(1+i_{v}\star\right)\left(1-\kappa\wedge i_{v}\right)F \, .
\ee
The two projection operators appearing in the definition of $F^{+}$
split the two-forms on $\mathcal{M}_{4}\times S^{1}$ into vertical
and horizontal forms. The latter are further split into self-dual
and anti-self-dual forms on $\mathcal{M}_{4}$:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& F=F_{H}+F_{V}=\left(1-\kappa\wedge i_{v}\right)F+\left(\kappa\wedge i_{v}\right)F \, , \\
& F_{H}=F_{H}^{+}+F_{H}^{-}=\frac{1}{2}\left(1+i_{v}\star\right)F_{H}+\frac{1}{2}\left(1-i_{v}\star\right)F_{H} \, .
\eea
The supersymmetry algebra now takes the standard cohomological form,
up to the addition of the equivariant deformation
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{eq:vector_multiplet_susy_transformations}
& \delta A_{m}=\mathbbm{i}\Psi_{m} \, , \qquad
\delta\sigma=-v^{m}\Psi_{m} \, , \qquad
\delta\Psi_{m}=i_{v}F-\mathbbm{i}D_{m}\sigma \, , \\
& \delta\chi_{mn}=H_{mn} \, , \qquad
\delta H_{mn}=\mathbbm{i}\mathcal{L}_{v}^{A}\chi_{mn}+\mathbbm{i}\left[\sigma,\chi_{mn}\right] \, .
\eea
The square of the transformation $\delta$ contains a translation
and a gauge transformation
\begin{equation}} \newcommand{\ee}{\end{equation}
\delta^{2}=\mathbbm{i}\mathcal{L}_{v}+G_{\Phi} \, ,
\ee
where $\mathcal{L}$ is the Lie derivative on forms and
\begin{equation}} \newcommand{\ee}{\end{equation}
\Phi\equiv\sigma-\mathbbm{i}v^{m}A_{m} \, , \qquad
\mathcal{L}_{v}^{A}\equiv\mathcal{L}_{v}-\mathbbm{i}\left[v^{m}A_{m},\cdot\right].
\ee
Note that $\delta\Phi=0$.
\subsubsection{Hypermultiplet}
A hypermultiplet comprises a pair of complex scalars $q_{I}^{A}$
and a fermion $\psi^{A}$ satisfying
\begin{equation}
\left(q_{I}^{A}\right)^{*}=\Omega_{AB}\varepsilon^{IJ}q_{I}^{A} \, ,
\qquad \left(\psi^{A}\right)^{*}=\Omega_{AB}C\psi^{B} \, ,
\label{eq:hypermultiplet_reality_condition}
\end{equation}
where
\begin{equation}} \newcommand{\ee}{\end{equation}
\Omega_{AB}=\begin{pmatrix}0 & \mathbbm{1}_{N}\\
-\mathbbm{1}_{N} & 0
\end{pmatrix},
\ee
is the invariant tensor of $\mathrm{USp}(2N)$, which is a symmetry group of $N$ free hypermultiplets.
$A,B,C,\ldots$ indices are raised and lowered using $\Omega$.
The gauge group is a subgroup of $\mathrm{USp}(2N)$ whose indices we sometimes suppress.
The supersymmetry transformations read
\begin{equation}} \newcommand{\ee}{\end{equation}
\delta q_{I}=-2\mathbbm{i}\xi_{I}\psi \, , \qquad
\delta\psi=\Gamma^{m}\xi_{I}D_{m}q^{I}+\mathbbm{i}\xi_{I}\sigma q^{I} \, ,
\ee
where $D_{m}$ is covariant with respect to $A_{m}$ and $\mathrm{SU}(2)_{R}$,
and both $A_{m}$ and $\sigma$ act in the appropriate representation
\begin{equation}} \newcommand{\ee}{\end{equation}
(\sigma q)^{A}\equiv\sigma^{A}_{\phantom{A}B}q^{B} \, .
\ee
After twisting, the field $q_{I}^{A}$ becomes a spinor
\begin{equation}} \newcommand{\ee}{\end{equation}
q\equiv\xi_{I}q^{I} \, .
\ee
This spinor is actually pseudo-real, and contains only $4$ degrees
of freedom
\begin{equation}} \newcommand{\ee}{\end{equation}
\left(q^{A}\right)^{*}=\Omega_{AB}Cq^{B} \, .
\ee
Its variation includes only the part of $\psi$ given by the projection%
\begin{equation}} \newcommand{\ee}{\end{equation}
\psi_{+} \equiv\frac{1}{2}\left(\mathbbm{1}_{4}+v_{m}\Gamma^{m}\right) \psi
=\frac{1}{2}\left(\mathbbm{1}_{4}+\Gamma^{5}\right)\psi \, .
\ee
The supersymmetry transformations can be closed off-shell by introducing
a superpartner $F$ for the component\footnote{See \cite[sect.\,4.2]{Qiu:2013pta} for a more complete explanation.}
\begin{equation}} \newcommand{\ee}{\end{equation}
\psi_{-}=\frac{1}{2}\left(\mathbbm{1}_{4}-\Gamma^{5}\right)\psi \, .
\ee
The twisted supersymmetry transformations are then given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{eq:hypermultiplet_susy_transformations}
& \delta q=\mathbbm{i}\psi_{+} \, , \qquad
\delta\psi_{+}=\left(\mathcal{L}_{v}-\mathbbm{i}G_{\Phi}\right)q \, , \\
& \delta\psi_{-}=F \, , \qquad
\delta F= \left( \mathbbm{i} \mathcal{L}_{v}+G_{\Phi} \right) \psi_{-} \, .
\eea
\subsubsection[Supersymmetric actions on \texorpdfstring{$\mathcal{M}_4 \times S^1$}{M(4) x S**1}]{Supersymmetric actions on $\mathcal{M}_4 \times S^1$}
\label{subsec:Supersymmetric-Lagrangians}
The action for twisted theories is a covariantized version of the
flat space action. This is in contrast to the additional terms which
appear, for instance, in the superconformal index, on the five sphere,
and on a general contact manifold. Such actions are still supersymmetric
because the Killing spinor is covariantly constant.
The flat space Yang-Mills term is given by \cite{Hosomichi:2012ek}
\begin{equation}
S_{\text{YM}}^{\mathbb{R}^5}=\frac{1}{g_{\text{YM}}^{2}}\int \text{Tr}\left(\frac{1}{2}F^{mn}F_{mn} - D^{m}\sigma D_{m}\sigma
-\frac{1}{2}D^{IJ}D_{IJ}+\mathbbm{i}\lambda_{I}\Gamma^{m}D_{m}\lambda^{I}-\lambda_{I}\left[\sigma,\lambda^{I}\right] \right) ,
\label{eq:Yang_Mills_action}
\end{equation}
where $D_{m}$ is the covariant derivative with respect to the connection $A_{m}$, see \eqref{gauge:covariant:derivative}.
The action on $\mathcal{M}_{4}\times S^{1}$ can be written as
\begin{equation}} \newcommand{\ee}{\end{equation}
S_{\text{YM}}^{\mathcal{M}_{4}\times S^{1}}=\frac{1}{g_{\text{YM}}^{2}}\int\sqrt{g}\text{Tr}\left(\frac{1}{2}F^{mn}F_{mn}-D^{m}\sigma D_{m}\sigma
-\frac{1}{2}D^{IJ}D_{IJ}+\mathbbm{i}\lambda_{I}\Gamma^{m}D_{m}\lambda^{I}-\lambda_{I}\left[\sigma,\lambda^{I}\right]\right) ,
\ee
where $D_{m}$ is covariant with respect to $A_{m}$, the
spin connection and the $\mathrm{SU}(2)$ R-symmetry.
In order to evaluate the Euclidean path integral with action
\begin{equation}} \newcommand{\ee}{\end{equation}
\exp\left(-S_{\text{YM}}^{\mathcal{M}_{4}\times S^{1}}\right),
\ee
we must choose a contour for the bosonic fields. An appropriate contour
which ensures convergence of the integral reads
\begin{equation}} \newcommand{\ee}{\end{equation}
A_{m}^{\dagger}=A_{m} \, , \qquad
\sigma^{\dagger}=-\sigma \, , \qquad
\left(D^{*}\right)^{IJ}=-\varepsilon^{IK}\varepsilon^{JL}D_{KL} \, .
\ee
Integration of fermionic fields is an algebraic procedure and does
not require such a choice of contour. Note the change in reality conditions
for the auxiliary field $D_{IJ}$. In the rotated variables, appearing
in the supersymmetry transformations and in the rest of the paper,
\begin{equation}} \newcommand{\ee}{\end{equation}
S_{\text{YM}}^{\mathcal{M}_{4}\times S^{1}}=\frac{1}{g_{\text{YM}}^{2}}\int\sqrt{g}\text{Tr}\left(\frac{1}{2}F^{mn}F_{mn}+D^{m}\sigma D_{m}\sigma+\frac{1}{2}D^{IJ}D_{IJ}+\mathbbm{i}\lambda_{I}\Gamma^{m}D_{m}\lambda^{I}+\mathbbm{i}\lambda_{I}\left[\sigma,\lambda^{I}\right]\right) .
\label{eq:twisted_Yang_Mills_action}
\ee
The action for a hypermultiplet is similarly given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
S_{\text{\ensuremath{\mathfrak{R}}-hyper}}^{\mathcal{M}_{4}\times S^{1}} = \int \sqrt{g} \Big( & D^{m}q_{I}^{A}D_{m}q_{A}^{I}+q_{I}^{A}\sigma_{AB}\sigma^{BC}q_{C}^{I}
-2\mathbbm{i}\psi^{A}\Gamma^{m}D_{m}\psi_{A} \\
& +2\mathbbm{i}\psi^{A}\sigma_{AB}\psi^{B}-4\psi^{A}\lambda_{ABI}q^{BI}+q_{I}^{A}D_{AB}^{IJ}q^{BI} \Big) ,
\label{eq:twisted_hypermultiplet_action}
\eea
where the matrices $\sigma^{A}_{\phantom{A}B},\lambda_{I\phantom{A}B}^{\phantom{I}A}$, and
${D^{A}}_{B}$ act in the representation $\mathfrak{R}$.
This action is convergent with the contour implied by the reality condition \eqref{eq:hypermultiplet_reality_condition}.
\subsection{Localization onto the fixed points}
\label{subsec:Localization}
The actions in the previous section are invariant under the fermionic
transformation $\delta$. By a standard argument, expectation values
of $\delta$-closed observables, and in particular the partition function,
are invariant under $\delta$-exact deformations of the action
\begin{equation}} \newcommand{\ee}{\end{equation}
S_{\text{total}}=S+t \delta\mathcal{V} \, ,
\ee
provided we choose the fermionic functional $\mathcal{V}$ in such
a way that $\delta^{2}\mathcal{V}|_{\text{bosonic}}=0$, $\delta\mathcal{V}|_{\text{bosonic}}\ge0$, and
all configurations which yield a finite result when evaluated using
$S_{\text{total}}$ also yield a finite result when evaluated using $S$.
In order to localize the theory with Euclidean measure
\begin{equation}} \newcommand{\ee}{\end{equation}
\exp\left(-S_{\text{total}}\right),
\ee
we take the limit $t \to \infty$.
All configurations with $\delta\mathcal{V}|_{\text{bosonic}}\ne0$ have infinite action
in this limit and the theory localizes onto the moduli space $\delta\mathcal{V}|_{\text{bosonic}}=0$.
The semi-classical approximation around this moduli space yields the
\emph{exact} result for the functional integral.
In order to localize the five-dimensional $\mathcal{N} = 1$ twisted theories, we can add the following localizing terms
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{eq:localizing_terms}
& \delta\mathcal{V}_{\text{gauge}}\equiv\delta\int\text{Tr}\left(2\mathbbm{i}\chi\wedge\star F^{+}+\frac{1}{2}\Psi\wedge\star\left(\delta\Psi\right)^{*}\right) \, , \\
& \delta\mathcal{V}_{\text{matter}}\equiv\delta\int\sqrt{g}\left(\psi_{+}^{A}\left(\delta\psi_{+}\right)_{A}^{*}+\psi_{-}^{A}\left(\delta\psi_{-}\right)_{A}^{*}+\psi_{-}^{A}\Gamma^{m}D_{m}q_{A}\right) .
\eea
The bosonic parts of which are
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
& \delta\mathcal{V}_{\text{gauge}}|_{\text{bosonic}}=\int\text{Tr}\left(2\mathbbm{i}H\wedge\star F^{+}+\frac{1}{2}\left(i_{v}F-\mathbbm{i}D\sigma\right)\wedge\star\left(i_{v}F-\mathbbm{i}D\sigma\right)^{*}\right) , \\
& \delta\mathcal{V}_{\text{matter}}|_{\text{bosonic}}=\int\sqrt{g}\left(\left[\left(\mathcal{L}_{v}-\mathbbm{i}G_{\Phi}\right)q\right]^{A}\left(\mathcal{L}_{v}+\mathbbm{i}G_{\Phi^{*}}\right)q_{A}+F^{A}F_{A}+F^{A}\Gamma^{m}D_{m}q_{A}\right) .
\end{aligned}
\ee
The field $H$ acts as a Lagrange multiplier, setting
\begin{equation}} \newcommand{\ee}{\end{equation}
F^{+}=0 \, .
\ee
The rest of the condition $\delta\mathcal{V}_{\text{gauge}}|_{\text{bosonic}}=0$, then requires
\begin{equation}
F^{+}=0 \, ,\qquad i_{v}F=0 \, ,\qquad D\sigma=0 \, .
\label{eq:vector_multiplet_localization_equations}
\end{equation}
A similar procedure for the hypermultiplet localizing term yields
\begin{equation}
\mathcal{L}_{v}q=0 \, ,\qquad G_{\Phi}q=0 \, ,\qquad\Gamma^{i}D_{i}q=0 \, ,
\label{eq:hypermultiplet_localization_equations}
\end{equation}
where in the last term we have made explicit use of the fact that
\begin{equation}} \newcommand{\ee}{\end{equation}
F\Gamma^{5}\nabla_{5}q=0 \, ,
\ee
by summing $i \in \{1, \ldots , 4\}$.
These equations may admit solutions for certain representations of
the gauge group, which would indicate that there are moduli coming
from the hypermultiplets. However, we will consider the situation
in which the hypermultiplets are also coupled to background vector
multiplets frozen to supersymmetric configurations which effectively
give all hypermultiplets a generic mass. In this situation, there
are no solutions to \eqref{eq:hypermultiplet_localization_equations}.
In what follow, we consider only solutions of the vector multiplet
equations.
\subsubsection{Bulk solutions}
An obvious set of solutions to \eqref{eq:vector_multiplet_localization_equations}
is given by flat connections and covariantly constant $\sigma$.
The topology of our spacetime satisfies
\begin{equation}} \newcommand{\ee}{\end{equation}
\pi_{1}(\mathcal{M}_{4}\times S^{1})\simeq\mathbb{Z} \, .
\ee
Flat connections are therefore parameterized by the holonomies around
the $S^{1}$ factor, restricted only by large gauge transformations.
Using an appropriate gauge transformation, these can be brought to
the form of a constant Cartan subalgebra valued connection $A^{(0)}$:
\begin{equation}} \newcommand{\ee}{\end{equation}
A_{i}^{(0)}=0 \, ,\qquad i\in\{ 1,2,3,4\} \, , \qquad A_{5}^{(0)} \in \text{Cartan} (\mathfrak{g}) \, .
\ee
Large gauge transformations identify
\begin{equation}} \newcommand{\ee}{\end{equation}
\big(A_{5}^{(0)}\big)_{j}\sim\big(A_{5}^{(0)}\big)_{j}+\frac{2\pi}{\beta}n \, , \qquad n\in\mathbb{Z} \, ,
\ee
where $j$ is an index in the Cartan subalgebra. At generic values
of the holonomy, a covariantly constant scalar is then also constant
and Cartan valued, \textit{i.e.}\;
\begin{equation}} \newcommand{\ee}{\end{equation}
\partial_{m}\sigma^{(0)}=0 \, , \qquad \left[\sigma^{(0)},A_{5}\right]=0 \, .
\ee
We denote
\begin{equation}} \newcommand{\ee}{\end{equation}
a\equiv-\mathbbm{i}\Phi^{(0)}=-A_{5}^{(0)}-\mathbbm{i}\sigma^{(0)} \, , \qquad
a\sim a+\frac{2\pi}{\beta}\mathbb{Z} \, .
\ee
Note that the equivariant action acts as
\begin{equation}} \newcommand{\ee}{\end{equation}
\delta^{2}=\mathbbm{i} (\mathcal{L}_{v} + G_{a}) \, .
\ee
In principle, the localization calculation includes an integral over
the $\text{rank }\mathfrak{g}$ cylinders parameterized by $a$.
Later on we will find it more convenient to move to exponentiated
coordinates on this space, whereby the integration region becomes $\left(\mathbb{C}^{*}\right)^{\text{rk}(\mathfrak{g})}$.
\subsubsection{Fermionic zero modes}
The quadratic approximation of $\delta\mathcal{V}_{\text{gauge}}$
around a bulk configuration specified by $a$ allows fermionic zero
modes for both $\chi$ and $\Psi$. In the presence of such zero modes
the functional integral naively vanishes. However, following \cite{Bershtein:2016mxz,Bershtein:2015xfa,Bawane:2014uka},
we will take this as an indication that the localizing term needs
to be improved to include a fermion mass term which will soak up the
zero modes. Since the additional term is by definition $\delta$-exact,
the value of the coefficient with which it is added, as long as it
is nonzero, does not change the final result.
The $\Psi$ zero mode can be read off from the $\Psi$ kinetic term
which is proportional to
\begin{equation}} \newcommand{\ee}{\end{equation}
\int\Tr \left(\Psi\wedge\star\left(\mathcal{L}_{v}-G_{a^{*}}\right)\Psi\right) \, .
\ee
The zero mode is a constant profile for the Cartan part of $\Psi$
given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\Psi_{m}^{(0)}\propto v_{m}\propto\Psi_{5}^{(0)} \, .
\ee
It is the superpartner of the holonomy.
The zero mode for $\chi$ is also Cartan valued and can be identified
using the projection operator
\begin{equation}} \newcommand{\ee}{\end{equation}
\pi: \mathcal{M}_{4}\times S^{1}\rightarrow\mathcal{M}_{4} \, ,
\ee
with a multiple of the pullback of the K\"ahler form on $\mathcal{M}_{4}$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\chi^{(0)}\propto\pi^{*}\omega \, .
\ee
Indeed, one can check that
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{L}_{v}\chi^{(0)}=G_{a}\chi^{(0)}=0 \, .
\ee
We can construct a nowhere vanishing off-diagonal mass term by pairing
the two sets of zero modes using
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{V}^{(0)} \equiv \int \Tr \left( \sigma^{(0)}\wedge\star\chi^{(0)}\wedge\pi^{*}\omega \right) \, ,
\ee
such that
\begin{equation}
\delta\mathcal{V}^{(0)} = \int \Tr \left( i_{v}\Psi^{(0)}\wedge\star\chi^{(0)}\wedge\pi^{*}\omega+\sigma^{(0)}\wedge\star H^{(0)}\wedge\pi^{*}\omega \right) \, ,
\label{eq:fermion_zero_modes_localizing_term}
\end{equation}
where we have defined
\begin{equation}} \newcommand{\ee}{\end{equation}
H^{(0)}\equiv\delta\chi^{(0)} \, .
\ee
Note that the mass is nowhere vanishing due to the property
\begin{equation}} \newcommand{\ee}{\end{equation}
\pi^{*}\omega\wedge\star\pi^{*}\omega\ne0 \, .
\ee
We add to the localizing $\mathcal{V}$ the term $- \mathbbm{i}s\mathcal{V}^{(0)}$.
Note that the reality conditions on $H$ and $\sigma$ require $s$ to be real.
\subsubsection{Fluxes}\label{subsubsec:fluxes}
As shown in \cite{Bershtein:2015xfa}, the four-dimensional equations
defined on $\mathcal{M}_{4}$
\begin{equation}} \newcommand{\ee}{\end{equation}\label{BTequiv}
i_{\tilde{v}}F=\mathrm{d} \phi \, ,
\ee
admit Abelian solutions corresponding to equivariant line bundles
supported on
$
H^{2}\left(\mathcal{M}_{4}\right)
$.
Specifically, the flux is viewed now as a symplectic form for the
torus action represented by $\tilde{v}$, and $\phi$ represents the
moment map for the symplectic action.
The addition of $\delta\mathcal{V}^{(0)}$ to the localizing
action relaxes the constraint imposed by the Lagrange multiplier in
\eqref{eq:vector_multiplet_localization_equations} from
$
F^{+}=0
$
to
\begin{equation}} \newcommand{\ee}{\end{equation}
F^{+} = \mathfrak{j} \pi^{*}\omega \, ,
\ee
where $\mathfrak{j}$ is some element of the Cartan subalgebra.
The combined equations
\begin{equation}} \newcommand{\ee}{\end{equation}
i_{v}F=0 \, ,\qquad F^{+}= \mathfrak{j} \pi^{*}\omega \, ,
\ee
are five-dimensional versions of those analyzed in \cite{Bershtein:2015xfa},
with $A_{5}$ playing the role of $\phi$. To make the connection,
use indices $i,j,\ldots$ for $\mathcal{M}_{4}$ and write the equation
\begin{equation}} \newcommand{\ee}{\end{equation}
i_{v}F=0 \, ,
\ee
for an Abelian field strength in 4+1 notation as
\begin{equation}} \newcommand{\ee}{\end{equation}\label{equivariantcohomology}
\tilde{v}^{i}F_{ij}+\partial_{5}A_{j}-\partial_{j}A_{5}=0 \, , \qquad
\tilde{v}^{i}\partial_{i}A_{5}-\tilde{v}^{i}\partial_{5}A_{i}=0 \, .
\ee
If we set $\partial_{5}A_{j}=0$,
then the first equation is the symplectic moment map condition. The
second equation follows from the first after applying $i_{\tilde{v}}$.
The solutions above correspond to solutions of the moment map equation
\eqref{BTequiv} and define equivariant cohomology classes. The resulting equivariant line bundles
are associated to the equivariant divisors on $\mathcal{M}_{4}$.
The relationship between these divisors and the description of $\mathcal{M}_{4}$
using the toric fan is explained in appendix \ref{sec:toric geometry}. In particular, there
is an equivariant divisor $D_l$ for each vector in the fan, and therefore for each fixed point of the torus action.
The total flux is then associated with a linear combination of divisors $\sum_{l=1}^d \mathfrak{p}_l D_l$,
where $\mathfrak{p}_l$ lives in the Cartan subalgebra.
We denote the resulting field strengths $F^{(0)}$.
Note that
\begin{equation}} \newcommand{\ee}{\end{equation}
F^{(0)}=\pi^{*}F_{4}^{(0)} \, ,
\ee
for some two-form $F_{4}^{(0)}$on $\mathcal{M}_{4}$.
Notice that, due to \eqref{equivariantcohomology}, the field $a$ acquires a nonzero profile on the manifold $\mathcal{M}_4$.
Near the fixed points of the torus action, the field becomes
\begin{equation}} \newcommand{\ee}{\end{equation}\label{replacement}
a^{(l)}=a+\epsilon_{1}^{(l)}\mathfrak{p}_{l}+\epsilon_{2}^{(l)}\mathfrak{p}_{l+1} \, ,
\ee
where the identification of the parameters $\epsilon_{1}^{(l)},\epsilon_{2}^{(l)}$ is given in appendix \ref{sec:toric geometry}.
\subsubsection{Instantons}
Near the fixed circles of $v$, the complex determined in section \ref{subsec:Supersymmetry-algebra}
coincides with the one considered by Nekrasov \cite{Nekrasov:2002qd}.
We therefore conjecture, in the spirit of \cite{Pestun:2007rz,Kallen:2012cs,Hosomichi:2012ek},
that these points support point-like instantons which are accounted
for by the five-dimensional or K-theoretic version of the Nekrasov's partition function
\begin{equation}} \newcommand{\ee}{\end{equation}
Z^{\mathbb{C}^{2}\times S^{1}}_{\text{inst}} ( g_{\text{YM}} , k , a , \Delta, \epsilon_{1}, \epsilon_{2} , \beta ) \, ,
\ee
described in section \ref{subsec:Nekrasov partition function}.
The parameters appearing in this partition
function can be read off from the classical action, the toric geometry,
the metric, the fluxes, and the mass parameters. Specifically, the
identification of the parameters $\epsilon_{1},\epsilon_{2}$ and
$a$ is given in appendix \ref{sec:toric geometry} and corresponds to the values appearing in \eqref{replacement}.
The authors of \cite{Kallen:2012cs} identified a class of solutions
to the equations
\begin{equation}} \newcommand{\ee}{\end{equation}
F^{+}=0 \, ,\qquad i_{v}F=0 \, , \qquad
i_{v}\kappa=1 \, ,
\ee
on any contact five-manifolds with contact structure determined by
a one-form $\kappa$. These solutions were dubbed contact instantons.
Although the one-form $\kappa$ defined in \eqref{not:a:contact:form}
is \emph{not} a contact form, the instantons appearing in our
partition function are the same solutions.
\subsubsection{Gauge fixing}
As discussed in \cite{Pestun:2007rz}, one can add a BRST-closed term
to the action in order to gauge fix without disturbing the localization
procedure. A convenient gauge for our calculation is the background
gauge
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathrm{d}_{A^{(0)}}^{\dagger}A=0 \, ,
\ee
where $A^{(0)}$ represents the value of $A$ at a point
in moduli space. This gauge is part of the definition of the Atiyah-Hitchin-Singer
complex for the instanton moduli space \cite{Atiyah:1978wi}. We will,
in addition, gauge fix the scalar moduli such that their non-Cartan
elements vanish. The modulus $a$ will be Cartan valued
\begin{equation}} \newcommand{\ee}{\end{equation}
a^{i} \, , \qquad i \in 1, \ldots , \text{rk}(G) \, .
\ee
Doing so incurs a determinant in the matrix model which is, however,
already taken into account in the one-loop determinant described below.
An additional factor of the inverse volume of the Weyl group, $\left|\mathfrak{W}\right|^{-1}$, is also present.
\subsubsection{Integration}
Localization takes effect when the coefficient $t$ of the localizing
action is taken to be very large. The value of $s$ is up to us. Following
\cite{Bershtein:2015xfa}, we choose to take a limit
\begin{equation}} \newcommand{\ee}{\end{equation}
s \to \infty \, .
\ee
In order to keep the moduli finite, we rescale
\begin{equation}} \newcommand{\ee}{\end{equation}
\sigma^{(0)} \to \frac{1}{s}\sigma^{(0)} \, .
\ee
In taking the limit, the fermion zero modes acquire a large mass can
be trivially integrated out. In addition, $\sigma^{(0)}$
drops out of all terms except \eqref{eq:fermion_zero_modes_localizing_term}.
Following \cite{Bershtein:2015xfa}, we write the remaining integral
over the scalar moduli as
\begin{equation}} \newcommand{\ee}{\end{equation}\label{JK}
\int \mathrm{d} a \, \mathrm{d} \bar{a} \frac{\partial}{\partial\bar{a}}\int\frac{\mathrm{d} H_{0}}{H_{0}} \, e^{i\bar{a}H_{0}}\times\left(\bar{a}\text{ independent terms}\right) ,
\ee
where we have used $H_{0}$ to mean
\begin{equation}} \newcommand{\ee}{\end{equation}
\int\star H^{(0)}\wedge\pi^{*}\omega \, .
\ee
The fact that the integrand is a total derivative in $\bar{a}$ should also follow from the algebra of supersymmetry
of the zero mode supermultiplet \cite{,Benini:2015noa,Closset:2015rna}.
The integral, being a total derivative in $\bar{a}$,
reduces to a contour integral. After the integral over $H_{0}$, which takes the residue at $H_{0}=0$, we will be left with the contour integral of a meromorphic function of $a$.
Following \cite{Benini:2013nda,Hori:2014tda,Benini:2015noa,Closset:2015rna} we expect that the interplay between a proper regularization of the integrand
and the use of the zero mode $H^0$ as a regulator will lead to the determination of the correct contour of integration.
We also expect that the appropriate contour is given by some Jeffrey-Kirwan prescription \cite{JeffreyKirwan}.
In related contexts, this prescription appears in the calculation of the
instanton partition function \cite{Hori:2014tda,Hwang:2014uwa}. It
also makes an appearance in the calculation of the partition functions
of the two-dimensional $A$-model \cite{,Benini:2015noa,Closset:2015rna} and the three-
and four-dimensional topologically twisted indices \cite{Benini:2015noa}, which are lower-dimensional
analogues of the partition function on $\mathcal{M}_{4}$ and the
five-dimensional twisted index.
We postpone to future work the determination of the correct contour of integration.
\subsubsection{Classical contribution}
\label{sebsec:classical}
Classical contributions to the localization calculation come from
evaluating \eqref{eq:twisted_Yang_Mills_action} and \eqref{eq:twisted_hypermultiplet_action}
on the moduli space identified above. Since all hypermultiplet fields
vanish on the moduli space, there is no contribution from \eqref{eq:twisted_hypermultiplet_action}.
Moreover, the bulk moduli do not contribute even to \eqref{eq:twisted_Yang_Mills_action}.
This is in contrast to the contact manifold case \cite{Qiu:2016dyj}. The contribution
of instantons will be discussed when we discuss the Nekrasov's partition
function. All that is left is the contribution of the fluxes and the
auxiliary field to \eqref{eq:twisted_Yang_Mills_action}. Recall that
the evaluation of the classical action is on the configurations such that the right hand side of the fermion transformations vanish.
Specifically, this implies that $H=0$, regardless of the reality conditions of $D_{IJ}$.
In fact, $H$ appears alone on the right hand side of the transformation of $\chi$, see \eqref{eq:vector_multiplet_susy_transformations},
meaning that we are free to add an arbitrarily large quadratic term for it in the localizing action. The equation $H=0$ imposes the relation
\begin{equation}
F_{mn}^{(0)+}=-\frac{\mathbbm{i}}{2}\xi^{I}\Gamma_{mn}\xi^{J}D_{IJ}^{(0)} \, .
\label{eq:F_vs_D_relation}
\end{equation}
Similar relations involving fluxes appear in the three-dimensional computations of
the twisted indices \cite{Benini:2015noa,Benini:2016hjo}. The relevant
part of the classical action is
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
& \exp\left[-\frac{1}{g_{\text{YM}}^{2}}\int\left(F^{(0)}\wedge\star F^{(0)}+\frac{1}{2}D^{(0)IJ} \wedge \star D_{IJ}^{(0)}\right)\right] \\
& = \exp \left[ -\frac{1}{g_{\text{YM}}^{2}}\int\left(F_{V}^{(0)}\wedge\star F_{V}^{(0)}+F_{H}^{(0)+}\wedge\star F_{H}^{(0)+}
+F_{H}^{(0)-}\wedge\star F_{H}^{(0)-}+\frac{1}{2}D^{(0)IJ} \wedge \star D_{IJ}^{(0)}\right) \right] .
\end{aligned}
\ee
Due to the properties of $F^{(0)}$, we can rewrite this as
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
& \exp \left[ -\frac{\beta}{g_{\text{YM}}^{2}}\int_{\mathcal{M}_{4}} \left( F_{4}^{(0)}\wedge\star_{4}F_{4}^{(0)}+\frac{1}{2}D^{(0)IJ}D_{IJ}^{(0)}\right) \right] \\
& =\exp\left[-\frac{\beta}{g_{\text{YM}}^{2}}\int_{\mathcal{M}_{4}}\left(F_{4}^{(0)+}\wedge F_{4}^{(0)+}-F_{4}^{(0)-}\wedge F_{4}^{(0)-}+\frac{1}{2}D^{(0)IJ}D_{IJ}^{(0)}\right)\right] \\
& =\exp\left[-\frac{\beta}{g_{\text{YM}}^{2}}\int_{\mathcal{M}_{4}}\left(2F_{4}^{(0)+}\wedge F_{4}^{(0)+}-F_{4}^{(0)}\wedge F_{4}^{(0)}+\frac{1}{2}D^{(0)IJ}D_{IJ}^{(0)}\right)\right] \\
& =\exp\left(\frac{\beta}{g_{\text{YM}}^{2}}\int_{\mathcal{M}_{4}}F_{4}^{(0)}\wedge F_{4}^{(0)}\right) ,
\end{aligned}
\ee
where $F^{(0)}=\pi^{*}F_{4}^{(0)}$
and we have used \eqref{eq:F_vs_D_relation}.
Given the relationship between the field strength and the differential
representative of the first Chern class
\begin{equation}} \newcommand{\ee}{\end{equation}
c_{1}(A)=-\frac{1}{2\pi}F \, ,
\ee
the classical contribution can be written as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{Zcl:localization:computation}
\begin{aligned}
& Z_{\text{cl}}^{\mathcal{M}_4 \times S^1} (g_{\text{YM}} , \mathfrak{p} , \beta) = \exp \left[ \frac{4\pi^{2}\beta}{g_{\text{YM}}^{2}}\left(\int_{\mathcal{M}_{4}}c_{1}( \{\mathfrak{p}\} )\wedge c_{1}(\{\mathfrak{p}\})\right)\right]
= \exp\left(\frac{4\pi^{2}\beta}{g_{\text{YM}}^{2}}c (\mathfrak{p})\right) , \\
& c(\mathfrak{p} )\equiv\bigg(\sum_{l=1}^{d}\mathfrak{p}_{l}D_{l}\bigg)\cdot\bigg(\sum_{l=1}^{d}\mathfrak{p}_{l}D_{l}\bigg) .
\end{aligned}
\ee
The factor $c\left(\mathfrak{p} \right)$ can be evaluated
for any given fan and choice of $\mathfrak{p}_{l}$ using the techniques
in appendix \ref{sec:toric geometry}. We will give explicit examples in section \ref{completePF}.
\subsubsection{One-loop determinants via index theorem}
\label{subsubsec:1-loop via index theorem}
We can compute the one-loop determinant from the equivariant index theorem.
We follow the derivation in \cite{Pestun:2007rz,Gomis:2011pf}. The
fields appearing in the supersymmetry algebra \eqref{eq:vector_multiplet_susy_transformations}
and \eqref{eq:hypermultiplet_susy_transformations} can be put into
the canonical form
\begin{equation}} \newcommand{\ee}{\end{equation}
\delta\varphi_{e,o}=\hat{\varphi}_{o,e} \, , \qquad
\delta\hat{\varphi}_{o,e}=\mathcal{R}\varphi_{e,o} \, .
\ee
where $\varphi_{e},\hat{\varphi}_{e}$ are bosonic and $\varphi_{o},\hat{\varphi}_{o}$
are fermionic. The expression above is meant to represent the $\delta$-complex
linearized around a point in the moduli space. We can identify
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{R}=\mathbbm{i} ( \mathcal{L}_{v} + G_{a}) \, .
\ee
The localizing functional contains a term of the form
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{V}=\varphi_{o}D_{oe}\varphi_{e} \, .
\ee
According to \cite{Gomis:2011pf}, the result of the Gaussian integral
around a point in moduli space is given by
\begin{equation}
Z_{1\text{-loop}}= \frac{\text{det}_{\text{coker}\,D_{oe}}\mathcal{R}|_o}{\text{det}_{\text{ker}\,D_{oe}}\mathcal{R}|_e} \, .
\label{eq:one_loop_expression}
\end{equation}
Using \eqref{eq:localizing_terms} and the supersymmetry algebra, we
can identify
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
D^{\text{vector}}= (1+i_{v}\star ) (1- \kappa \wedge i_{v} ) \mathrm{d}_A \, , \qquad
D^{\text{hyper}}=\Gamma^{i}D_{i}=\pi^{*}D_{\text{Dirac}} \, .
\eea
The operator $D^{\text{vector}}$ is the projection
of the covariantized exterior derivative operator on $\mathcal{M}_{4}$, acting
on one-forms, to the self-dual two-forms. One should take into account the gauge
fixing. Together with $D^{\text{vector}}$, this forms the self-dual complex
\begin{equation}} \newcommand{\ee}{\end{equation}
D_{\text{SD}}:\Omega^{0}\xrightarrow{\mathrm{d}}\Omega^{1}\xrightarrow{\mathrm{d}_{+}}\Omega^{2+} \, ,
\ee
tensored with the adjoint representation. $D^{\text{hyper}}$ is simply the Dirac operator on $\mathcal{M}_{4}$.
The relevant complex is the Dirac complex
\begin{equation}} \newcommand{\ee}{\end{equation}
D_{\text{Dirac}}: S^{+} \to S^{-} \, ,
\ee
tensored with the representation of the gauge and flavor groups.
The bundles $S^\pm$ are the positive and negative chirality spin bundles on $\mathcal{M}_4$.
If $\mathcal{M}_4$ is not spin, these should be replaced by an appropriate bundle associated with a $\text{spin}^c$ structure.
Neither of these complexes are
elliptic. However, both are transversely elliptic with respect to
the action of $\mathcal{R}$.
The data entering \eqref{eq:one_loop_expression} can be extracted from
the computation of the $\mathcal{R}$-equivariant index for the operator $D_{oe}$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\text{ind} \, D_{oe}=\text{Tr}_{\text{ker}\,D_{oe}}e^{\mathcal{R}} - \text{Tr}_{\text{coker}\,D_{oe}}e^{\mathcal{R}} \, .
\ee
The computation of this index is described in \cite{atiyah2006elliptic}.
We will follow the exposition in \cite{Pestun:2016qko}. Let $\mathcal{E}$
be a $G$-equivariant complex of linear differential operators acting
on sections of vector bundles $E_{i}$ over $X$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{E}:\Gamma (E_{0} )\xrightarrow{D_{0}}\Gamma (E_{1} )\xrightarrow{D_{1}}\Gamma (E_{2} )\to \ldots \, , \qquad
D_{i}D_{i+1}=0 \, .
\ee
The equivariant index of the complex $\mathcal{E}$ is the virtual character of the $G$ action
on the cohomology classes $H^{k}(\mathcal{E})$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\text{ind}_{G} D (g)=\sum_{k} (-1)^{k} \Tr_{H^{k}(\mathcal{E})} g \, , \qquad g \in G \, .
\ee
If the set of fixed points of $G$ is discrete, the index can be determined
by examining the action of $G$ at those points, denoted $X^{G}$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\text{ind}_{G} D=\sum_{x\in X^{G}}\frac{\sum_{k}(-1)^{k}\text{ch}_{G}(E_{k})|_{x}}{\text{det}_{T_{x}X}(1-g^{-1})} \, .
\ee
The numerator in this expression encodes the action of $G$ on the
bundles, while the denominator is the action on the tangent space
of $X$. We refer the reader to \cite{Gomis:2011pf} for more information,
and to \cite{atiyah2006elliptic} for a complete treatment.
The index and the one-loop determinant can be computed using the equivariant
index theorem on $\mathcal{M}_{4}$. There is one fixed point at the
origin of every cone in the fan which determines $\mathcal{M}_{4}$.
The copy of $\mathbb{C}^{2}$ associated to the cone is acted upon
by the equivariant parameters, and feels the flux from equivariant
divisors associated to the two neighboring vectors, as determined
in appendix \ref{sec:toric geometry}. All that is needed to construct the character for
$\mathcal{M}_{4}$ is to add up the contributions. The subtleties
in the calculation involve the use of the right complex for the index
theorem, and the necessity of regularizing the infinite products that
it yields.
The complex identified above for the vector multiplet is the self-dual complex.
Following the discussion in \cite{Pestun:2016jze}, we use the Dolbeault complex
(the ``holomorphic projection of the vector multiplet'') and find a match to the gluing calculation in section \ref{completePF}.
The two complexes are related on a K\"ahler manifold (see \textit{e.g.}\;of \cite[sect.\,2.3.1]{Marino:1996sd}).
The relevant index for the twisted Dolbeault operator $\bar{\partial}$ on $\mathbb{C}^{2}\times S^{1}$ is given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\text{ind}_{\mathcal{R}}\pi^{*}(\bar{\partial})|_{\mathbb{C}^{2}\times S^{1}}=\sum_{\alpha\in G}\sum_{n=-\infty}^{\infty}\frac{e^{\mathbbm{i}\frac{2\pi n}{\beta}}e^{\mathbbm{i}\alpha\left(\mathfrak{p}_{1}\right)\epsilon_{1}}e^{\mathbbm{i}\alpha\left(\mathfrak{p}_{2}\right)\epsilon_{2}}e^{\mathbbm{i}\alpha\left(a\right)}}{\left(1-e^{-\mathbbm{i}\epsilon_{1}}\right)\left(1-e^{-\mathbbm{i}\epsilon_{2}}\right)}\,,
\ee
where we have incorporated the free action of the rotation on $S^{1}$
and denoted by $\mathfrak{p}_{1,2}$ the coefficients of the two divisors.
Here, $\alpha$ are the roots of the gauge group $G$ and the $\epsilon_{1,2}$ are arbitrary complex deformation parameters,
which we will take to zero at the end. The complete index reads
\begin{equation}
\text{ind}_{\mathcal{R}}\pi^{*}(\bar{\partial})|_{\mathcal{M}_{4}\times S^{1}}=
\sum_{l=1}^{d}\text{ind}_{\mathcal{R}^{(l)}}\pi^{*}(\bar{\partial})|_{\mathbb{C}^{2}\times S^{1}} \, ,
\label{eq:equivariant_index_on_M}
\end{equation}
where $d$ is the number of cones in the fan determining $\mathcal{M}_{4}$
and $\mathcal{R}^{(l)}$ signifies the use of the coefficients
and equivariant parameters relevant to that cone.
We will explicitly evaluate the one-loop determinant resulting from \eqref{eq:equivariant_index_on_M}
only in the non-equivariant limit. Define the degeneracy
\begin{equation}} \newcommand{\ee}{\end{equation}\label{indexfixedpoint}
d (\mathfrak{p} )\equiv\lim_{\epsilon_{1,2} \to 0}\sum_{l=1}^{d}\frac{e^{\mathbbm{i}\alpha(\mathfrak{p}_{l})\epsilon_{1}^{(l)}}e^{\mathbbm{i}\alpha (\mathfrak{p}_{l+1} )\epsilon_{2}^{(l)}}}
{\left(1-e^{-\mathbbm{i}\epsilon_{1}^{(l)}}\right)\left(1-e^{-\mathbbm{i}\epsilon_{2}^{(l)}}\right)} \, .
\ee
Our flux conventions are those in \cite{Pestun:2016qko}.
The limit reduces the equivariant index to the Hirzebruch-Riemann-Roch
theorem
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}\label{HRR}
d (\mathfrak{p})= \int_{\mathcal{M}_4} \text{ch} (E) \text{td} (\mathcal{M}_4) \, ,
\eea
where $E$ corresponds to $\sum_{l=1}^d \mathfrak{p}_l D_l$. We will give explicit examples of degeneracies in section \ref{completePF}.
The one-loop determinant for a vector multiplet is then given by
\begin{equation}} \newcommand{\ee}{\end{equation}
Z_{1\text{-loop}}^{\text{vector}}\left(a,\mathfrak{p},\beta\right)=\prod_{\alpha\in G}\left[\prod_{n=-\infty}^{\infty}\left(\mathbbm{i}\frac{2\pi}{\beta}n+\mathbbm{i}\alpha(a)\right)\right]^{d\left(\alpha(\mathfrak{p})\right)}\,.
\ee
The infinite product above requires regularization. A physically acceptable
regularization is given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}\label{choicereg}
& \prod_{n=-\infty}^{\infty}\left(\mathbbm{i}\frac{2\pi}{\beta}n+\mathbbm{i}\alpha(a)\right)=\left(\frac{1-x^{\alpha}}{x^{\alpha/2}}\right)\,,\\
& x^{\alpha}\equiv e^{\mathbbm{i}\beta\alpha\left(a\right)}\,,\qquad\alpha(a)=\alpha_{i}a^{i}\,.
\eea
This choice has simple transformation properties under parity and
correctly accounts for the induced Chern-Simons term when used in
the three-dimensional calculations \cite{Benini:2015noa}. See also \cite{Hori:2014tda}
for an example of its use. We thus obtain
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{1-loop:index:theorem:vector}
Z_{1\text{-loop}}^{\text{vector}}\left(a, \mathfrak{p} ,\beta\right)=\prod_{\alpha\in G}\left(\frac{1-x^{\alpha}}{x^{\alpha/2}}\right)^{d\left( \alpha(\mathfrak{p}) \right)}.
\ee
The index for a hypermultiplet is based on the twisted Dirac complex
instead of the Dolbeault complex. It also incorporates background
scalar moduli and fluxes, which we denote $\Delta$ and $\mathfrak{t}_{l}$
respectively. Given the relation on $\mathcal{M}_{4}$ between these
two complexes, one needs to change the index for the vector multiplet
by an overall flux corresponding to the square root of the canonical
bundle and take into account the opposite grading between the two
complexes \cite{Pestun:2016qko}. The canonical bundle is minus the
sum of all the equivariant divisors. In addition, there is an $\epsilon$
dependent choice of the origin of the flavor mass parameter. The choice
corresponding to the superconformal fixed point in five dimensions was discussed in \cite{Kim:2011mv,Bullimore:2014upa,Nishioka:2016guu}, following the correction to the four sphere partition function found in \cite{Okuda:2010ke}.
The authors of \cite{Bershtein:2015xfa} found a match with the results derived by Vafa and Witten in \cite{Vafa:1994tf} for the $\mathcal{N}=4$ theory with yet another choice.
If the manifold $\mathcal{M}_4$ is not spin, there may be other complications related to the choice of $\text{spin}^c$ structure.
Specifically, we have made no attempt to identify the canonical $\text{spin}^c$ structure associated with the almost complex structure of $\mathcal{M}_4$ since this choice can be shifted by background fluxes.
For notational convenience we choose a common shift of the mass parameter for all manifolds
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{shift:Delta}
\Delta \to \Delta-\frac{1}{2}(\epsilon_{1}+\epsilon_{2}) \, ,
\ee
although this may require an appropriate redefinition of the origin of the background fluxes in particular examples.
Incorporating both of these leads to
\begin{equation}} \newcommand{\ee}{\end{equation}
d^{\text{hyper}} (\mathfrak{p},\mathfrak{t} )\equiv-\lim_{\epsilon_{1,2}\to0}\sum_{l=1}^{d}\frac{e^{\mathbbm{i}\left(\rho (\mathfrak{p}_{l} )+\nu (\mathfrak{t}_{l} )-1\right)\epsilon_{1}^{(l)}}
e^{\mathbbm{i}\left(\rho(\mathfrak{p}_{l+1})+\nu(\mathfrak{t}_{l+1})-1\right)\epsilon_{2}^{(l)}}}{\left(1-e^{-\mathbbm{i}\epsilon_{1}^{(l)}}\right)\left(1-e^{-\mathbbm{i}\epsilon_{2}^{(l)}}\right)}\,,
\ee
where $\mathfrak{R}$ is the representation under the gauge group $G$, $\rho$ the corresponding weights, and
$\nu$ is the weight of the hypermultiplet under the flavor symmetry group.
The complete result can then be written as
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{1-loop:index:theorem:hyper}
& Z_{1\text{-loop}}^{\text{hyper}}\left(a,\Delta, \mathfrak{p} , \mathfrak{t} ,\beta\right)=
\prod_{\rho\in \mathfrak{R}}\left(\frac{1-x^{\rho}y^{\nu}}{x^{\rho/2}y^{\nu/2}}\right)^{d_{\text{hyper}}\left( \rho(\mathfrak{p}) , \nu(\mathfrak{t}) \right)} \, , \\
& y^{\nu}\equiv e^{\mathbbm{i}\beta\nu(\Delta)} .
\eea
\subsection{The Nekrasov's partition function}
\label{subsec:Nekrasov partition function}
In this section we collect the expressions for the K-theoretic Nekrasov's partition function:
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{K-theoretic:Nekrasov}
Z_{\text{Nekrasov}}^{\mathbb{C}^{2}\times S^{1}} (g_{\text{YM}} , k , a , \Delta , \epsilon_{1},\epsilon_{2} , \beta ) \equiv Z^{\mathbb{C}^2 \times S^1}_{\text{pert}} Z_{\text{inst}}^{\mathbb{C}^2 \times S^1} \, .
\ee
As we will discuss in the next section, the topologically twisted index on $\mathcal{M}_4 \times S^1$ can be obtained by gluing copies of the Nekrasov's partition functions in the spirit of \eqref{Nekrasov}.
\subsubsection{Perturbative contribution}
The perturbative part of the partition function on $\mathbb{C}^2\times S^1$ consists of a classical and a one-loop contribution.
\paragraph*{Classical contribution.}
The classical contribution to \eqref{K-theoretic:Nekrasov} is given by \cite{Gottsche:2006bm,Bershtein:2018srt}
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{Z:flat:space:cl}
Z_{\text{cl}}^{\mathbb{C}^{2}\times S^{1}} ( g_{\text{YM}} , k, a,\epsilon_{1},\epsilon_{2} ) =
\exp \left( \frac{4 \pi ^2 \beta}{g_{\text{YM}}^2 \epsilon_1 \epsilon_2} \Tr_{\text{F}} (a)^2 + \frac{\mathbbm{i} k \beta}{6 \epsilon_1 \epsilon_2} \Tr_{\text{F}}(a^3) \right) \, ,
\ee
where $k$ is the Chern-Simons level of $G$ and $\Tr_{\text{F}}$ is the trace in fundamental representation.%
\footnote{The generators $T_a$ are normalized as $\Tr_{\mathfrak{R}} ( T_ a T_b ) = k (\mathfrak{R}) \delta_{ab}$ with $k(\text{F}) = 1/2$ for the fundamental representation of $\mathrm{SU}(N)$.}
\paragraph*{One-loop contribution.}
We will use the perturbative part of the partition function on $\mathbb{C}^2\times S^1$ as defined in \cite{Nakajima:2005fg}. For a gauge group $G$ the contribution of a vector multiplet to the perturbative part is given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{Nakajima:pert}
Z_{\text{pert-vector}}^{\mathbb{C}^{2}\times S^{1}}\left(\epsilon_{1},\epsilon_{2} , a ; \Lambda , \beta \right)=\exp\bigg(-\sum_{\alpha\in G}\tilde{\gamma}_{\epsilon_{1},\epsilon_{2}}\left(\alpha(a) | \beta ; \Lambda\right)\bigg),
\ee
where
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\tilde{\gamma}_{\epsilon_{1},\epsilon_{2}}\left(a | \beta; \Lambda \right) & =
\frac{1}{\epsilon_{1}\epsilon_{2}}\left(\frac{\pi^{2}a}{6\beta}-\frac{\zeta(3)}{\beta^{2}}\right)+\frac{\epsilon_{1}+\epsilon_{2}}{2\epsilon_{1}\epsilon_{2}}\left(a\log\left(\beta\Lambda\right) +\frac{\pi^{2}}{6\beta}\right) \\
& + \frac{1}{2\epsilon_{1}\epsilon_{2}}\left[-\frac{\beta}{6}\left(a+\frac{1}{2}\left(\epsilon_{1}+\epsilon_{2}\right)\right)^{3}+a^{2}\log\left(\beta\Lambda\right)\right] \\
& + \frac{\epsilon_{1}^{2}+\epsilon_{2}^{2}+3\epsilon_{1}\epsilon_{2}}{12\epsilon_{1}\epsilon_{2}}\log\left(\beta\Lambda\right)
+ \sum_{n=1}^{\infty}\frac{1}{n}\frac{e^{-\beta na}}{\left(e^{\beta n\epsilon_{1}}-1\right)\left(e^{\beta n\epsilon_{2}}-1\right)} \, .
\eea
With this definition of the perturbative part of the partition function, the authors of \cite{Nakajima:2005fg} derived a formula for the partition function for the blowup
of $\mathbb{C}^2$ at a point, which is just an example of the gluing procedure we will discuss in section \ref{completePF}.
This expression is written in the conventions of \cite{Nakajima:2005fg} and also includes what we already defined as the classical contribution.
One can swap between the conventions by
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{swap:conventions}
a^{\text{there}} = -\mathbbm{i} a^{\text{here}} \, , \qquad \epsilon_{i}^{\text{there}} = - \mathbbm{i} \epsilon_{i}^{\text{here}} \, .
\ee
The one-loop contribution from a vector multiplet in our conventions is given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{vec1loop}
\begin{aligned}
& Z_{1\text{-loop, vector}}^{\mathbb{C}^{2}\times S^{1}} (a, \epsilon_1 , \epsilon_2) = Z_{\text{parity, vector}}^{\mathbb{C}^{2}\times S^{1}} (a, \epsilon_1 , \epsilon_2) \prod_{\alpha \in G} ( x^{\alpha}; p, t)_\infty \, , \\
& Z_{\text{parity, vector}}^{\mathbb{C}^{2}\times S^{1}} (a, \epsilon_1 , \epsilon_2) = \prod_{\alpha \in G} \exp \bigg[ \frac{1}{\epsilon_1 \epsilon_2} \bigg( \frac{\mathbbm{i}}{2 \beta ^2} g_3 \left( - \alpha( \beta a ) \right)
- \frac{\mathbbm{i} (\epsilon_1+\epsilon_2)}{4 \beta} g_2 \left( - \alpha( \beta a) \right) \\
& \hspace{3.cm} + \frac{\mathbbm{i} ( \epsilon_1 + \epsilon_2 )^2}{16} g_1 \left( - \alpha( \beta a ) \right)
- \frac{\mathbbm{i} \beta}{96} (\epsilon_1+\epsilon_2)^3 + \frac{\mathbbm{i} \pi}{48} \left( \epsilon_1^2 + \epsilon_2^2 \right) - \frac{\zeta (3)}{\beta^2} \bigg) \bigg] \, .
\end{aligned}
\ee
Here, we defined the double $(p,t)$-factorial as
\begin{equation}} \newcommand{\ee}{\end{equation}
( x ; p , t)_\infty = \prod_{i , j = 0}^{\infty} (1 - x p^i t^j ) \, ,
\ee
where $p = e^{- \mathbbm{i} \beta \epsilon_1}$ and $t = e^{- \mathbbm{i} \beta \epsilon_2}$.
The polynomial functions $g_s(a)$ are given in \eqref{PolyLog:inversion formulae}.
The parity contribution in \eqref{vec1loop} is related to the choice of regularization we made in \eqref{choicereg}.
It can be partially understood as an effective one-half Chern-Simons contribution \eqref{Z:flat:space:cl}.
The contribution of a hypermultiplet to the one-loop determinant instead reads
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
& Z_{1\text{-loop, hyper}}^{\mathbb{C}^{2}\times S^{1}} (a, \Delta, \epsilon_1 , \epsilon_2) = Z_{\text{parity, hyper}}^{\mathbb{C}^{2}\times S^{1}} (a, \Delta, \epsilon_1 , \epsilon_2) \prod_{\rho \in \mathfrak{R}} (x^{\rho} y^{\nu}; p, t)_{\infty}^{-1} \, , \\
& Z_{\text{parity, hyper}}^{\mathbb{C}^{2}\times S^{1}} (a, \Delta, \epsilon_1 , \epsilon_2) = \prod_{\rho \in \mathfrak{R}} \exp \bigg[ \frac{1}{\epsilon_1 \epsilon_2}
\bigg( \frac{\mathbbm{i}}{2 \beta ^2} g_3 \big( \rho( \beta a ) + \nu ( \beta \Delta ) \big) \\
& \hspace{3.cm} + \frac{\mathbbm{i} (\epsilon_1+\epsilon_2)}{4 \beta} g_2 \big( \rho( \beta a ) + \nu ( \beta \Delta ) \big)
+ \frac{\mathbbm{i} ( \epsilon_1 + \epsilon_2 )^2}{16} g_1 \big( \rho( \beta a ) + \nu ( \beta \Delta ) \big) \\
& \hspace{3.cm} + \frac{\mathbbm{i} \beta}{96} (\epsilon_1+\epsilon_2)^3
+ \frac{\mathbbm{i} \pi}{48} \left( \epsilon_1^2 + \epsilon_2^2 \right) \bigg) \bigg] \, .
\end{aligned}
\ee
Putting everything together, the perturbative part of the Nekrasov's partition function can be written as
\begin{equation}} \newcommand{\ee}{\end{equation}
Z_{\text{pert}}^{\mathbb{C}^{2}\times S^{1}} ( g_{\text{YM}} , k , a , \Delta , \epsilon_{1} , \epsilon_{2} , \beta ) =
Z_{\text{cl}}^{\mathbb{C}^{2}\times S^{1}} Z_{1\text{-loop, vector}}^{\mathbb{C}^{2}\times S^{1}} \, Z_{1\text{-loop, hyper}}^{\mathbb{C}^{2}\times S^{1}} \, .
\ee
\subsubsection{Instantons contribution}
The localization calculation for a five-dimensional gauge theory conjecturally
includes non-perturbative contributions from contact instantons \cite{Kallen:2012cs,Hosomichi:2012ek}.
With the equivariant deformation turned on, these configurations are
localized to the fixed points of the action on $\mathcal{M}_{4}$ and
wrap the $S^{1}$. Their contribution to the matrix model is given
by the five-dimensional version of the Nekrasov's instanton partition
function, which we describe below.
Nekrasov's instanton partition function, \cite{Nekrasov:2002qd,Nekrasov:2003rj},
is the equivariant volume of the instanton moduli space on $\mathbb{R}^{4}$
with respect to the action of
\begin{equation}
\mathrm{U}(1)_{a}\times \mathrm{U}(1)_{\epsilon_{1}}\times \mathrm{U}(1)_{\epsilon_{2}} \, .
\label{eq:equivariant_action}
\end{equation}
The three factors correspond to constant gauge transformations and
to rotations in two orthogonal two-planes inside $\mathbb{R}^{4}$,
respectively. The five-dimensional version of the partition function
counts instantons extended along an additional $S^{1}$ factor in
the geometry, of circumference $\beta$. The four-dimensional partition
function can be recovered by letting the size of this $S^{1}$ shrink to zero.
As with the perturbative contribution, there is an ambiguity related to the regularization
of the KK modes on the extra circle. Different looking expressions are found in \cite{Bullimore:2014upa}.
The K-theoretic instanton partition function for gauge group $\mathrm{U}(N)$ in our conventions, as derived from \cite{Tachikawa:2004ur, Nakajima:2005fg,Tachikawa:2014dja}, is given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{eq:Nekrasov_partition_function}
& Z_{\text{\text{inst}}}^{\mathbb{C}^{2}\times S^{1}}(q,k,a,\Delta,\epsilon_{1},\epsilon_{2},\beta)=
\sum_{\vec{{\bf Y}}}q^{|\vec{{\bf Y}}|}Z_{\vec{{\bf Y}},k}^{\text{CS}}\left(a,\epsilon_{1},\epsilon_{2},\beta\right)Z_{\vec{{\bf Y}}}\left(a,\Delta,\epsilon_{1},\epsilon_{2},\beta\right) , \\
& Z_{\vec{{\bf Y}}}\left(a,\Delta,\epsilon_{1},\epsilon_{2},\beta\right)=Z_{\vec{{\bf Y}}}^{\text{vector}}\left(a,\epsilon_{1},\epsilon_{2},\beta\right)\prod_{f=1}^{N_{f}}Z_{\vec{{\bf Y}}}^{\text{hyper}}\left(a,\Delta_{f},\epsilon_{1},\epsilon_{2},\beta\right) .
\eea
We have
\begin{align}
& Z_{\vec{{\bf Y}},k}^{\text{CS}} (a,\epsilon_{1},\epsilon_{2},\beta )= \prod_{i=1}^{N} \prod_{s \in\mathbf{Y}_{i}} e^{- \mathbbm{i} \beta k \phi ( a_i , s ) } \, , \nonumber \\
& Z_{\vec{{\bf Y}}}^{\text{vector}} (a,\epsilon_{1},\epsilon_{2},\beta )=\prod_{i,j=1}^{N}\left(N_{i,j}^{\vec{{\bf Y}}}(0)\right)^{-1} \, , \nonumber \\ \displaybreak[4]
& Z_{\vec{{\bf Y}}}^{\text{adj-hyper}} (a,\Delta,\epsilon_{1},\epsilon_{2},\beta )=\prod_{i,j=1}^{N}N_{i,j}^{\vec{{\bf Y}}}(\Delta) \, , \\
& Z_{\vec{{\bf Y}}}^{\text{fund-hyper}} (a,\Delta,\epsilon_{1},\epsilon_{2},\beta )=\prod_{i=1}^{N}\prod_{s \in \mathbf{Y}_{i}}
\left( 1 - e^{ \mathbbm{i} \beta \left( \phi ( a_i , s ) + \Delta + \epsilon_{1} + \epsilon_{2} \right)} \right) \, , \nonumber \\
& N_{i,j}^{\vec{{\bf Y}}}( \Delta ) \equiv \prod_{s\in Y_{j}} \left( 1 - e^{\mathbbm{i} \beta \left( E ( a_i - a_j , Y_j , Y_i , s ) + \Delta \right)} \right)
\prod_{t\in Y_{i}} \left( 1 - e^{\mathbbm{i} \beta \left( \epsilon_{1} + \epsilon_{2} - E ( a_j - a_i , Y_i , Y_j , t ) + \Delta \right)} \right) \nonumber ,
\end{align}
where for a box $s = ( i , j ) \in \mathbb{Z}_{\geq 0} \times \mathbb{Z}_{\geq 0}$ we defined the functions
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
& E ( a , Y_1 , Y_2 , s ) \equiv a - \epsilon_1 L_{Y_2} (s) + \epsilon_2 (A_{Y_1} (s) + 1 ) \, , \\
& \phi ( a , s ) \equiv a - ( i - 1 ) \epsilon_{1} - ( j - 1 ) \epsilon_{2} \, .
\end{aligned}
\ee
The rest of the symbols above are defined as follows.
\begin{itemize}
\item[--] $\vec{{\bf Y}}$ is a vector of partitions ${\bf Y}_{i}$. A partition
is a non-increasing sequence of non-negative integers which stabilizes
at zero
\begin{equation}} \newcommand{\ee}{\end{equation}
{\bf Y}_{i}=\left\{ Y_{i\,1}\ge Y_{i\,2}\ge\ldots\ge Y_{i\,n_{i}+1}=0=Y_{i\,n_{i}+2}=Y_{i\,n_{i}+3}=\ldots\right\} .
\ee
We define
\begin{equation}} \newcommand{\ee}{\end{equation}
\big|\vec{{\bf Y}} \big|\equiv \sum_{i , j = 1}^{N} Y_{i j} \, .
\ee
\item [--] For a box $s \in Y_{l}$ with coordinates $s = ( i , j )$, we define the \emph{leg length} and the \emph{arm length}
\begin{equation}} \newcommand{\ee}{\end{equation}
L_{Y_{l}}(s)\equiv Y_{lj}^{\text{T}}-i \, ,\qquad A_{Y_{l}}(s)\equiv Y_{li}-j \, ,
\ee
where $\text{T}$ stands for transpose.
\item [--] $a$ is a complex Cartan subalgebra valued scalar of framing
parameters associated with the gauge group action.
\item [--] $\Delta$ is a flavor symmetry group vector of mass parameters for the hypermultiplets.
\item [--] $q$ is a counting parameter coming from the one instanton
action
\begin{equation}} \newcommand{\ee}{\end{equation}
q=e^{-\frac{8\pi^{2}\beta}{g_{\text{YM}}^{2}}} \, .
\ee
\end{itemize}
\subsubsection{Effective Seiberg-Witten prepotential}
\label{subsubsec:SW prepotential}
In this section we provide the general form of the perturbative Seiberg-Witten (SW) prepotential for a 5D $\mathcal{N}=1$ theory with gauge group $G$, coupling constant $g_{\text{YM}}$, and Chern-Simons coupling $k$.
For a theory with hypermultiplets transforming in the representation $\mathfrak{R}_I$ of $G$, the effective prepotential
is related to the Nekrasov's partition function $Z_{\text{Nekrasov}}^{\mathbb{C}^2 \times S^1}$ as follows \cite{Nekrasov:2002qd,Nekrasov:2003rj,Nakajima:2005fg}:
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{5D:prepotential:total}
2 \pi \mathbbm{i} \mathcal{F} \equiv - \lim_{\epsilon_1 , \epsilon_2 \to 0} \epsilon_1 \epsilon_2 \log Z_{\text{Nekrasov}}^{\mathbb{C}^2 \times S^1} (g_{\text{YM}} , k , a , \Delta_{\mathfrak{R}_I} , \epsilon_1, \epsilon_2 , \beta) \, ,
\ee
whose perturbative part can be explicitly written as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{full:pert:prepotential}
\begin{aligned}
2 \pi \mathbbm{i} \mathcal{F}^{\text{pert}} (a , \Delta) & = - \frac{4 \pi^2 \beta}{g_{\text{YM}}^2} \Tr_{\text{F}} (a^2) - \frac{\mathbbm{i} k \beta}{6} \Tr_{\text{F}} (a^3)
- \frac{1}{\beta^2} \sum_{\alpha \in G} \left[ \Li_3 ( x^\alpha ) + \frac{\mathbbm{i}}{2} g_3 \left( - \alpha( \beta a ) \right) - \zeta(3) \right] \\ &
+ \frac{1}{\beta^2} \sum_{I} \sum_{\rho_I \in \mathfrak{R}_I} \left[ \Li_3 ( x^{\rho_I} y^{\nu_I} ) - \frac{\mathbbm{i}}{2} g_3 \left( \rho_I( \beta a ) + \nu_I ( \beta \Delta ) \right) - \zeta(3) \right] \, ,
\end{aligned}
\ee
where the function $g_3(a)$ is defined in \eqref{gfunctions}.
The first term in \eqref{full:pert:prepotential} comes from the classical action while the $\Li_3$ factor is the one-loop contribution of the infinite tower of KK modes on $S^1$ as discussed in \cite{Nekrasov:1996cz}.
The other polynomial terms come explicitly from the limit of the perturbative contribution in \cite{Nakajima:2005fg}.
We interpret them as effective Chern-Simons terms coming from a parity preserving regularization of the path integral, as also discussed in section \ref{subsubsec:1-loop via index theorem}.
\subsection[Partition function on \texorpdfstring{$\mathcal{M}_4 \times S^1$}{M(4) x S**1}]{Partition function on $\mathcal{M}_4 \times S^1$}
\label{completePF}
The functional integral under consideration is to be computed in an exact saddle point approximation around the moduli space which comprises the bulk moduli, the fluxes, and the instantons.
We will assume as in \cite{Bawane:2014uka,Bershtein:2015xfa,Bershtein:2016mxz} that the complete partition function is given by gluing $d$ copies of $Z_{\text{Nekrasov}}^{\mathbb{C}^2 \times S^1}$, one for each fixed point of the toric action. At each fixed point, the parameters $a$, $\Delta$, $\epsilon_1$, $\epsilon_2$ in \eqref{K-theoretic:Nekrasov} are replaced with their equivariant version
$a^{(l)}$, $\Delta^{(l)}$, $\epsilon_{1}^{(l)}$, $\epsilon_{2}^{(l)}$, whose explicit form is explained in appendix \ref{sec:toric geometry} and given, for simple cases, in examples \ref{ex1}-\ref{ex3}.
In particular, as in \cite{Bershtein:2015xfa}, the gauge magnetic fluxes are incorporated into this expression through
\begin{equation}} \newcommand{\ee}{\end{equation}
a^{(l)}=a+\epsilon_{1}^{(l)} \mathfrak{p}_{l} + \epsilon_{2}^{(l)} \mathfrak{p}_{l+1} \, .
\ee
The reason for this replacement is discussed in section \ref{subsubsec:fluxes} and it is also easy to see in the index theorem discussed in section \ref{subsubsec:1-loop via index theorem}.
Furthermore, the equivariant chemical potential is given by (see the discussion around \eqref{shift:Delta})
\begin{equation}} \newcommand{\ee}{\end{equation}
\Delta^{(l)} = \dot{\Delta} + \epsilon_{1}^{(l)} \mathfrak{t}_{l} + \epsilon_{2}^{(l)} \mathfrak{t}_{l+1} \, , \qquad \dot{\Delta} = \Delta - \big( \epsilon_1^{(l)} + \epsilon_2^{(l)} \big) \, .
\ee
As we will see this gluing is consistent with the classical and one-loop contributions determined in sections \ref{sebsec:classical} and \ref{subsubsec:1-loop via index theorem}.
The topologically twisted index of an $\mathcal{N} = 1$ theory on $\mathcal{M}_4 \times S^1$ then reads\footnote{Similar gluing formulae hold for other five-dimensional partition functions \cite{Qiu:2014oqa,Nieri:2013yra,Pasquetti:2016dyl}.}
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{completePartitionFunction}
Z_{\mathcal{M}_{4}\times S^{1}}=\sum_{\{\mathfrak{p}_{l}\}|\text{semi-stable}}\oint_{\mathcal{C}} \mathrm{d} a \prod_{l=1}^{\chi(\mathcal{M}_{4})}
Z_{\text{Nekrasov}}^{\mathbb{C}^{2}\times S^{1}} \big( a^{(l)} , \epsilon_{1}^{(l)} , \epsilon_{2}^{(l)} , \beta , q ; \Delta^{(l)} \big) \, .
\ee
Here, following \cite{Nekrasov:2003vi,Bawane:2014uka,Bershtein:2015xfa,Bershtein:2016mxz}, we restrict the sum in \eqref{completePartitionFunction} to fluxes $\mathfrak{p}_l$ corresponding to semi-stable bundles. It was argued in \cite{Bawane:2014uka,Bershtein:2015xfa,Bershtein:2016mxz} that the sum should be extended to all semi-stable equivariant bundles. These are classified by a set of fluxes $\mathfrak{p}_l$, one for each divisor,\footnote{Remember that there are $d$ divisors but only $d-2$ independent two-cycles in the cohomology of $\mathcal{M}_4$.} subject to stability conditions that have been studied by mathematicians \cite{Kool2015}. These conditions are already quite complicated for $N=2$. Summing over all semi-stable equivariant bundles, the authors of \cite{Bawane:2014uka,Bershtein:2015xfa,Bershtein:2016mxz} found perfect agreement with known Donaldson invariant results.
In this paper we have not determined either the correct contour of integration or the correct conditions to be imposed on the fluxes. We expect that the two aspects are related.
Notice that formula \eqref{completePartitionFunction} is consistent with and generalizes the blowup formula derived in \cite{Nakajima:2005fg},
which just corresponds to the case where $\mathcal{M}_4$ is the (non-compact) blowup of $\mathbb{C}^2$ at a point.%
\footnote{See \cite[(4.14)]{Nakajima:2005fg}. The blowup of $\mathbb{C}^2$ at a point can be described by a toric fan with ${\vec n}_1=(1,0)$, ${\vec n}_2=(1,1)$ and ${\vec n}_3 =(0,1)$.}
In this paper we will be interested in theories with gauge
group $\mathrm{U}(N)$ or $\mathrm{USp}(N)$. Moreover, we want
the partition function in the large $N$, non-equivariant limit $\epsilon_{1,2} \to 0$.
The appropriate large $N$ limit is defined in the next section.
We will assume that the instanton contribution to the free energy, defined by
\begin{equation}} \newcommand{\ee}{\end{equation}
F_{\mathcal{M}_{4}\times S^{1}}\equiv-\log Z_{\mathcal{M}_{4}\times S^{1}} \, ,
\ee
decays exponentially in any such limit. This is supported by the appearance
of the factor $q^{|\vec{{\bf Y}}|}$
in \eqref{eq:Nekrasov_partition_function}. In the following sections,
all partition functions are written only for the zero instanton sector.
\subsubsection{The non-equivariant limit}
The Nekrasov's partition function \eqref{K-theoretic:Nekrasov} is singular for $\epsilon_1,\epsilon_2 \to 0$ but the product in \eqref{completePartitionFunction} is perfectly smooth in this limit.
By performing explicitly the limit, we can write the classical and perturbative part of the localized partition function in the non-equivariant limit as
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}\label{finalPF}
Z_{\mathcal{M}_{4}\times S^{1}} =\frac{1}{\left|\mathfrak{W}\right|}\sum_{\{\mathfrak{p}_l\}}
\oint_{\mathcal{C}} & \prod_{i=1}^{\text{rk}(G)} \frac{\mathrm{d} x_{i}}{2\pi\mathbbm{i}x_{i}} \, e^{\frac{4 \pi^{2}\beta}{g_{\text{YM}}^{2}} \Tr_{\text{F}} c(\mathfrak{p}_l) + \frac{\mathbbm{i} k \beta}{2} \Tr_{\text{F}} \left( c(\mathfrak{p}_l ) a \right)}
\prod_{\alpha\in G} \left(\frac{1-x^{\alpha}}{x^{\alpha/2}}\right)^{d\left( \alpha(\mathfrak{p}_l) \right)} \\
& \times \prod_{I} \prod_{\rho_I \in \mathfrak{R}_I}\left(\frac{1-x^{\rho_I}y^{\nu_I}}{x^{\rho_I/2}y^{\nu_I/2}}\right)^{d_{\text{hyper}}\left(\rho_I(\mathfrak{p}_l) ,\nu_I(\mathfrak{t}_{l}) \right)} .
\eea
As one can see, the results from gluing (and taking the non-equivariant limit) precisely match the classical contributions
\eqref{Zcl:localization:computation} and the one-loop determinants evaluated using the index theorem (see \eqref{1-loop:index:theorem:vector} and \eqref{1-loop:index:theorem:hyper}).
We may also expect some simplification in the sum over fluxes compared with the equivariant formula \eqref{completePartitionFunction}. We expect that, in the non-equivariant limit, the fluxes $\mathfrak{p}_l$ should just correspond to the set of non-equivariant bundles on $\mathcal{M}_4$.
Formula \eqref{finalPF} has a natural interpretation in terms of the quantum mechanics obtained by reducing the
five-dimensional theory on $\mathcal{M}_4$. In the reduction on $\mathcal{M}_4$ in a sector with gauge and background fluxes $\mathfrak{p}_l$ and $\mathfrak{t}_l$,
we will obtain a set of zero modes whose multiplicity is given by the Hirzubruch-Riemann-Roch theorem and coincides with $d$ or $d_{\text{hyper}}$.
They will organize themselves into a set of Fermi or chiral multiplets according to the sign of $d$ and $d_{\text{hyper}}$.
\eqref{finalPF} is then precisely the sum over all sectors of gauge magnetic fluxes of the localization formula for the corresponding
quantum mechanics partition functions, as derived in \cite{Hori:2014tda}.
This is in complete analogy with the structure of the three- and four-dimensional topologically twisted indices \cite{Benini:2015noa}.
Some examples of the calculation of the degeneracy are given below. We refer to appendix \ref{sec:toric geometry} for details and notations.
\begin{example}\label{ex1}
Complex projective space, $\mathbb{P}^2$.
\begin{equation}} \newcommand{\ee}{\end{equation}
\vec{n}_{1}=(1,0) \, ,\quad\vec{n}_{2}=(0,1) \, ,\quad\vec{n}_{3}=(-1,-1) \, .
\ee
\begin{minipage}[t]{.5\textwidth}
\centering
\vspace{0pt}
\begin{tabular}{|c|c|c|c|}
\hline
\multicolumn{1}{|c}{$\mathbb{P}^{2}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \tabularnewline
\hline
$l$ & 1 & 2 & 3\tabularnewline
\hline
$\epsilon_{1}^{(l)}$ & $\epsilon_{1}$ & $\epsilon_{2}-\epsilon_{1}$ & $-\epsilon_{2}$\tabularnewline
\hline
$\epsilon_{2}^{(l)}$ & $\epsilon_{2}$ & $-\epsilon_{1}$ & $\epsilon_{1}-\epsilon_{2}$\tabularnewline
\hline
\end{tabular}
\end{minipage}\hfill
\begin{minipage}[t]{.5\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}
[scale=0.5 ]
\draw[->] (0,0) -- (2,0) node[right] {${\vec n}_1,\, D_1$};
\draw[->] (0,0) -- (0,2) node[above] {${\vec n}_2, \, D_2$};
\draw[->] (0,0) -- (-2,-2) node[ below ] {${\vec n}_3,\, D_3$};
\end{tikzpicture}
\end{minipage}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& c_{\mathbb{P}^{2}} (\mathfrak{p}_l) = \left(\mathfrak{p}_{1}+\mathfrak{p}_{2}+\mathfrak{p}_{3}\right)^2 , \\
& d_{\mathbb{P}^{2}} (\mathfrak{p}_l) = \frac{1}{2}\left(\mathfrak{p}_{1}+\mathfrak{p}_{2}+\mathfrak{p}_{3}+1\right)\left(\mathfrak{p}_{1}+\mathfrak{p}_{2}+\mathfrak{p}_{3}+2\right) , \\
& d_{\mathbb{P}^{2}}^{\text{hyper}} (\mathfrak{p}_l, \mathfrak{t}_l) = - \frac{1}{2}\left(\mathfrak{p}_{1}+\mathfrak{p}_{2}+\mathfrak{p}_{3}+\mathfrak{t}_{1}+\mathfrak{t}_{2}+\mathfrak{t}_{3}-2\right)
\left(\mathfrak{p}_{1}+\mathfrak{p}_{2}+\mathfrak{p}_{3}+\mathfrak{t}_{1}+\mathfrak{t}_{2}+\mathfrak{t}_{3}-1\right) .
\eea
\end{example}
\begin{example}\label{ex2} The product of two spheres, $\mathbb{F}_{0}\simeq\mathbb{P}^1 \times \mathbb{P}^1$.
\begin{equation}} \newcommand{\ee}{\end{equation}
\vec{n}_{1}=(1,0) \, ,\quad\vec{n}_{2}=(0,1) \, ,\quad\vec{n}_{3}=(-1,0) \, ,\quad\vec{n}_{4}=(0,-1) \, .
\ee
\begin{minipage}[t]{.5\textwidth}
\centering
\vspace{0pt}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multicolumn{1}{|c}{$\mathbb{F}_{0}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \tabularnewline
\hline
$l$ & 1 & 2 & 3 & 4\tabularnewline
\hline
$\epsilon_{1}^{(l)}$ & $\epsilon_{1}$ & $\epsilon_{2}$ & $-\epsilon_{1}$ & $-\epsilon_{2}$\tabularnewline
\hline
$\epsilon_{2}^{(l)}$ & $\epsilon_{2}$ & $-\epsilon_{1}$ & $-\epsilon_{2}$ & $\epsilon_{1}$\tabularnewline
\hline
\end{tabular}
\end{minipage}\hfill
\begin{minipage}[t]{.5\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}
[scale=0.5 ]
\draw[->] (0,0) -- (2,0) node[right] {${\vec n}_1,\, D_1$};
\draw[->] (0,0) -- (0,2) node[above] {${\vec n}_2, \, D_2$};
\draw[->] (0,0) -- (-2,0) node[ left] {${\vec n}_3,\, D_3$};
\draw[->] (0,0) -- (0,-2) node[ below] {${\vec n}_4,\, D_4$};
\end{tikzpicture}
\end{minipage}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& c_{\mathbb{F}_{0}} (\mathfrak{p}_l) = 2 \left(\mathfrak{p}_{1}+\mathfrak{p}_{3}\right)\left(\mathfrak{p}_{2}+\mathfrak{p}_{4}\right) , \\
& d_{\mathbb{F}_{0}} (\mathfrak{p}_l) =\left(\mathfrak{p}_{1}+\mathfrak{p}_{3}+1\right)\left(\mathfrak{p}_{2}+\mathfrak{p}_{4}+1\right) , \\
& d_{\mathbb{F}_{0}}^{\text{hyper}} (\mathfrak{p}_l , \mathfrak{t}_l) = - \left(\mathfrak{p}_{1}+\mathfrak{p}_{3}+\mathfrak{t}_{1}+\mathfrak{t}_{3}-1\right)\left(\mathfrak{p}_{2}+\mathfrak{p}_{4}+\mathfrak{t}_{2}+\mathfrak{t}_{4}-1\right) .
\eea
\end{example}
\begin{example}
\label{ex3} $\mathbb{F}_{1}$, the blowup of $\mathbb{P}^2$ at a point.
\begin{equation}} \newcommand{\ee}{\end{equation}
\vec{n}_{1}=(1,0) \, ,\quad\vec{n}_{2}=(0,1) \, ,\quad\vec{n}_{3}=(-1,1) \, ,\quad\vec{n}_{4}=(0,-1) \, .
\ee
\begin{minipage}[t]{.5\textwidth}
\centering
\vspace{0pt}
\begin{tabular}{|c|c|c|c|c|}
\hline
\multicolumn{1}{|c}{$\mathbb{F}_{1}$} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \multicolumn{1}{c}{} & \tabularnewline
\hline
$l$ & 1 & 2 & 3 & 4\tabularnewline
\hline
$\epsilon_{1}^{(l)}$ & $\epsilon_{1}$ & $\epsilon_{1}+\epsilon_{2}$ & $-\epsilon_{1}$ & $-\epsilon_{2}$\tabularnewline
\hline
$\epsilon_{2}^{(l)}$ & $\epsilon_{2}$ & $-\epsilon_{1}$ & $-\epsilon_{1}-\epsilon_{2}$ & $\epsilon_{1}$\tabularnewline
\hline
\end{tabular}
\end{minipage}\hfill
\begin{minipage}[t]{.5\textwidth}
\centering
\vspace{0pt}
\begin{tikzpicture}
[scale=0.5 ]
\draw[->] (0,0) -- (2,0) node[right] {${\vec n}_1,\, D_1$};
\draw[->] (0,0) -- (0,2) node[above] {${\vec n}_2, \, D_2$};
\draw[->] (0,0) -- (-2,2) node[ left] {${\vec n}_3,\, D_3$};
\draw[->] (0,0) -- (0,-2) node[ below] {${\vec n}_4,\, D_4$};
\end{tikzpicture}
\end{minipage}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& c_{\mathbb{F}_{1}}(\mathfrak{p}_l ) = \left(\mathfrak{p}_{2}+\mathfrak{p}_{4}\right)\left(2\mathfrak{p}_{1}-\mathfrak{p}_{2}+2\mathfrak{p}_{3}+\mathfrak{p}_{4}\right) , \\
& d_{\mathbb{F}_{1}}(\mathfrak{p}_l) = \frac{1}{2}\left(\mathfrak{p}_{2}+\mathfrak{p}_{4}+1\right)\left(2\mathfrak{p}_{1}-\mathfrak{p}_{2}+2\mathfrak{p}_{3}+\mathfrak{p}_{4}+2\right) , \\
& d_{\mathbb{F}_{1}}^{\text{hyper}}(\mathfrak{p}_l , \mathfrak{t}_l) = - \frac{1}{2}\left(\mathfrak{p}_{2}+\mathfrak{p}_{4}+\mathfrak{t}_{2}+\mathfrak{t}_{4}-1\right)
\left(2\mathfrak{p}_{1}-\mathfrak{p}_{2}+2\mathfrak{p}_{3}+\mathfrak{p}_{4}+2\mathfrak{t}_{1}-\mathfrak{t}_{2}+2\mathfrak{t}_{3}+\mathfrak{t}_{4}-2\right) .
\eea
\end{example}
\subsubsection{A closer look at $\protect\fakebold{\mathbb{\mathbb{P}}} ^1 \times \protect\fakebold{\mathbb{\mathbb{P}}}^1\times S^1$}
\label{subsubsec:naive P1 X P1}
We can compare the previous result with the expectations for the case $\mathbb{P}^1\times \mathbb{P}^1\times S^1$, where the computation, in principle, can be done by an explicit expansion in modes. By an obvious generalization of the results in \cite{Benini:2015noa}, we expect the following partition function
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{SYM:index:S2^2}
Z_{\mathbb{P}^1\times \mathbb{P}^1\times S^1} = \frac{1}{\left|\mathfrak{W}\right|}\sum_{\mathfrak{m} , \mathfrak{n}}
\oint_{\mathcal{C}} & \prod_{i=1}^{\text{rk}(G)} \frac{\mathrm{d} x_{i}}{2\pi\mathbbm{i}x_{i}} \, e^{\frac{8 \pi^2 \beta}{g_{\text{YM}}^2} \Tr_{\text{F}} ( \mathfrak{m} \mathfrak{n} ) + \mathbbm{i} k \beta \Tr_{\text{F}} ( \mathfrak{m} \mathfrak{n} a )}
\prod_{\alpha \in G} \bigg( \frac{1 - x^\alpha}{x^{\alpha/2}} \bigg)^{(\alpha(\mathfrak{m})+ 1) ( \alpha(\mathfrak{n})+ 1)} \\ &
\times \prod_{I} \prod_{\rho_I \in \mathfrak{R}_I} \bigg( \frac{x^{\rho_I/2} y^{\nu_I/2}}{1 - x^{\rho_I} y^{\nu_I}} \bigg)^{( \rho_I(\mathfrak{m}) + \nu_I(\mathfrak{s}) - 1 ) ( \rho_I(\mathfrak{n}) + \nu_I(\mathfrak{t}) - 1 )} \, ,
\eea
where $\mathfrak{m}/\mathfrak{n}$ (and $\mathfrak{s}/\mathfrak{t}$) are the gauge (and background) magnetic fluxes on two spheres. The degeneracies come from the Hirzebruch-Riemann-Roch theorem and the classical action comes
from an explicit computation along the lines of section \ref{sebsec:classical}.
We see that the result coincides with \eqref{finalPF} with the replacements
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathfrak{m}= \mathfrak{p}_1+\mathfrak{p}_3\, \, \qquad \mathfrak{n} = \mathfrak{p}_2+\mathfrak{p}_4\, , \qquad \mathfrak{s} = \mathfrak{t}_1 +\mathfrak{t}_3\, , \qquad \mathfrak{t} = \mathfrak{t}_2+\mathfrak{t}_4 \, .
\ee
The integral in \eqref{finalPF} indeed only depends on such combinations. We also expect that, in the non-equivariant limit, the sum over semi-stable equivariant fluxes
reduces to a sum over the two standard fluxes $\mathfrak{m}$ and $\mathfrak{n}$ on $\mathbb{P}^1\times \mathbb{P}^1$. Such set of non-equivariant fluxes was indeed used in \cite{Bawane:2014uka},
where the four-dimensional partition function on $\mathbb{P}^1\times \mathbb{P}^1$ has been studied.
Notice that, in the conventions that we are using for the background fluxes, what we will call universal twist \cite{Benini:2015bwz,Bobev:2017uzs} in section
\ref{sec:largeN} corresponds to $\mathfrak{s}=\mathfrak{t}=1$.
\section[Large \texorpdfstring{$N$}{N} limit]{Large $N$ limit}
\label{sec:largeN}
In this section, we analyze the large $N$ limit of some of the topologically twisted indices and other related quantities, finding an interesting structure.
In the large $N$ limit we may expect some simplifications. In particular, instantons are suppressed and the perturbative contribution
to the topologically twisted index of 5D $\mathcal{N} = 1$ theories discussed in section \ref{sec:localization} becomes exact.
Moreover, we may also expect that the choice of integration contour and the stability conditions on fluxes become simpler at large $N$. In particular,
we will work under the assumption that, in the large $N$ limit, the fluxes become actually independent.
In the topologically twisted index of 3D $\mathcal{N} = 2$ theories on $\Sigma_{\mathfrak{g}_1} \times S^1$ \cite{Nekrasov:2014xaa,Benini:2015noa,Benini:2016hjo,Closset:2017zgf} a distinguished role was played by the twisted superpotential $\widetilde \mathcal{W}$ of the two-dimensional theory obtained by compactification on $S^1$ (with infinitely many KK modes). In particular, the partition function can be written as a sum over the set of Bethe vacua
of the two-dimensional theory, which corresponds to the critical points of $\widetilde \mathcal{W}$ \cite{Nekrasov:2014xaa,Closset:2017zgf}. Moreover, in the large $N$ limit, one particular Bethe vacuum dominates the partition function \cite{Benini:2015eyy}. The natural quantity to consider for the 5D topologically twisted index
is the SW prepotential $\mathcal{F}(a)$ of the four-dimensional theory obtained by compactification on $S^1$. There are two reasons to expect that the critical points of $\mathcal{F}(a)$ play a distinguished role in five dimensions. First, the partition function of the topologically twisted ${\cal N}=2$ theories in four dimensions can be split into two contributions, one coming from the integration over the Coulomb plane (usually called the $u$ plane), and the other from the locus where monopoles and dyons become massless, corresponding to the critical points of
the prepotential $\mathcal{F}$ \cite{Moore:1997pc}. The integral over the $u$ plane also often reduces to boundary contributions from all the singular points in the moduli space. Secondly, in the integrable system obtained by placing the four-dimensional theory on a $\Omega$-background on $\mathbb{R}^4$ with $\epsilon_1 = \hbar$ and $\epsilon_2 = 0$, the Bethe vacua read
\cite{Nekrasov:2009rc}
\begin{equation}} \newcommand{\ee}{\end{equation}
\exp \bigg( \mathbbm{i} \frac{\partial \widetilde\mathcal{W}_\hbar (a)}{\partial a_j} \bigg) = 1 \, , \qquad j = 1, \ldots , \text{rk}(G) \, ,
\ee
where $\widetilde\mathcal{W}_\hbar (a)$ is the twisted superpotential for the two-dimensional effective theory obtained by reducing on the $\Omega$-deformed copy of $\mathbb{R}^2$. We expect that these conditions play a role in the equivariant partition function with $\epsilon_1 = \hbar$ and $\epsilon_2 = 0$. Since $\widetilde \mathcal{W}_\hbar(a)$ has the following expansion as $\hbar \to 0$,
\begin{equation}} \newcommand{\ee}{\end{equation}
\widetilde \mathcal{W}_\hbar(a) = - \frac{2 \pi}{\hbar} \mathcal{F} (a) + \ldots \, ,
\ee
we also obtain, in the limit $\hbar \rightarrow 0$, the quantization conditions (\textit{cf.}\,\cite[(3.62)]{Nekrasov:2010ka} and \cite[(4.6)]{Luo:2013nxa})
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{BAEs:general:F}
\exp \left( \frac{2 \pi \mathbbm{i}}{\hbar} a_j^{\text{D}} \right) = \exp \bigg( \frac{2 \pi \mathbbm{i}}{\hbar} \frac{\partial \mathcal{F}(a)}{\partial a_j} \bigg) = 1 \, , \qquad j = 1, \ldots , \text{rk}(G) \, .
\ee
It is possible that, in taking the non-equivariant limit of the index, the information about the pole configurations of the integrand is partially lost and we need to impose extra conditions following from \eqref{BAEs:general:F}.
For all these reasons, we may think that the critical points of $\mathcal{F}(a)$ may play a role in the evaluation of the index. In particular, in analogy with the 3D index, we may expect that, in the large $N$ limit, one particular critical point of $\mathcal{F}(a)$ dominates the partition function. We will provide some evidence of this picture by evaluating various quantities in the large $N$ limit at the critical point of $\mathcal{F}(a)$ and showing that they nicely agree with holographic predictions. Independently from these considerations, the problem of finding the distribution of critical points of $\mathcal{F}(a)$ in the large $N$ limit is interesting in itself and deserves to be studied.
We will then consider the large $N$ limit of the distribution of critical points of the functional $\mathcal{F}(a)$ focusing on two 5D theories, $\mathcal{N} = 2$ SYM, which decompactifies to the $\mathcal{N} = (2,0)$ theory in six dimensions and the $\mathcal{N} = 1$ $\mathrm{USp}(2N)$ theory with $N_f$ flavors and an antisymmetric matter field, which corresponds to a 5D UV fixed point.
In both cases we find that the value of $\mathcal{F}(a)$ at its critical points, as a function of flavor fugacities, precisely coincides, in the large $N$ limit, with the partition function of the same theory on $S^5$.
This is in parallel with what was found for the twisted superpotential of 3D $\mathcal{N} = 2$ theories in the large $N$ limit \cite{Hosseini:2016tor}, thus re-enforcing the analogy between the two quantities.
We shall then study the large $N$ limit of the topologically twisted index of 5D $\mathcal{N} = 1$ theories on $\mathbb{P}^1\times \mathbb{P}^1\times S^1$.
With no effort, we can replace $\mathbb{P}^1\times \mathbb{P}^1\times S^1$ with the more general manifold $\Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1} \times S^1$ and we will consider this more general case in the following.
The interest of this model is that we can formally dimensionally reduce on $\Sigma_{\mathfrak{g}_2}$ and obtain a three-dimensional theory. In three dimensions we can guess the form of the partition function in the large $N$ limit and use the results in \cite{Nekrasov:2014xaa,Benini:2015noa,Benini:2016hjo,Closset:2017zgf}. We expect the partition function of the three-dimensional theory to be given as a sum over topological sectors on $\Sigma_{\mathfrak{g}_2}$. We will denote the gauge/flavor magnetic fluxes on $\Sigma_{\mathfrak{g}_1}$ and $\Sigma_{\mathfrak{g}_2}$ by $\mathfrak{m} / \mathfrak{s}$ and $\mathfrak{n} / \mathfrak{t}$, respectively. We also denote by $\Delta$ a complexified chemical potential for the flavor symmetry. In each sector of gauge magnetic flux $\mathfrak{n}$ on $\Sigma_{\mathfrak{g}_2}$ we have a twisted superpotential for the compactified theory satisfying \cite{Nekrasov:2009uh}
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{WF}
\frac{\partial \widetilde \mathcal{W}(a,\mathfrak{n}, \Delta, \mathfrak{t})}{\partial \mathfrak{n}_i} = -2 \pi \frac{\partial \mathcal{F}(a,\Delta)}{\partial a_i} \, , \qquad \frac{\partial \widetilde \mathcal{W}(a,\mathfrak{n}, \Delta, \mathfrak{t})}{\partial \mathfrak{t}} = -2 \pi \frac{\partial \mathcal{F}(a,\Delta)}{\partial \Delta} \, .
\ee
One way to determine $\widetilde \mathcal{W}$ is to compare the 5D partition function for $\mathbb{P}^1\times \mathbb{P}^1\times S^1$ given in section \ref{sec:localization} with the structure of the topologically twisted index in three dimensions.
This is given by localization as a contour integral of a meromorphic quantity \cite{Benini:2015noa,Benini:2016hjo,Closset:2017zgf}
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{twisted3D}
\sum_{\mathfrak{m}\in \Gamma_\mathfrak{h}} Z^{3\text{D}}_{\text{int}} (a , \mathfrak{m} , \mathfrak{n}) = \sum_{\mathfrak{m}\in \Gamma_\mathfrak{h}} e^{\mathbbm{i} \mathfrak{m}_i \frac{\partial \widetilde \mathcal{W}(a , \mathfrak{n})}{\partial a_i}} Z^{3\text{D}}_{\text{int}} (a, \mathfrak{m}=0, \mathfrak{n}) \, ,
\ee
where $\Gamma_\mathfrak{h}$ is the lattice of gauge magnetic fluxes for the gauge group $G$. By generalizing \eqref{SYM:index:S2^2} from $\mathbb{P}^1\times \mathbb{P}^1$ to $\Sigma_{\mathfrak{g}_1} \times\Sigma_{\mathfrak{g}_2}$ as in \cite{Benini:2016hjo}, and choosing a convenient parameterization for the fluxes, we expect the integrand $Z^{3\text{D}}_{\text{int}} (a , \mathfrak{m} , \mathfrak{n})$ to be
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{SYM:index:Sigmag2xS1xSigmag10}
& \bigg( \det\limits_{i j} \frac{\partial^2 \widetilde \mathcal{W} (a , \mathfrak{n})}{\partial a_i \partial a_j} \bigg)^{\mathfrak{g}_1}
e^{\frac{8 \pi^2 \beta}{g_{\text{YM}}^2} \Tr_{\text{F}} ( \mathfrak{m} \mathfrak{n} ) + \mathbbm{i} k \beta \Tr_{\text{F}} (\mathfrak{m} \mathfrak{n} a) }
\prod_{\alpha \in G} \bigg( \frac{1 - x^\alpha}{x^{\alpha/2}} \bigg)^{(\alpha(\mathfrak{m})+ 1 - \mathfrak{g}_1 ) ( \alpha(\mathfrak{n})+ 1 - \mathfrak{g}_2 )} \\ &
\times \prod_{I} \prod_{\rho_I \in \mathfrak{R}_I} \bigg( \frac{x^{\rho_I/2} y^{\nu_I/2}}{1 - x^{\rho_I} y^{\nu_I}} \bigg)^{( \rho_I(\mathfrak{m}) + \nu_I(\mathfrak{s}) + \mathfrak{g}_1 - 1 ) ( \rho_I(\mathfrak{n}) + \nu_I(\mathfrak{t}) + \mathfrak{g}_2 - 1 )} \, .
\eea
We can therefore read off the twisted superpotential
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{full:pert:twisted superpotential}
\begin{aligned}
\widetilde \mathcal{W}^{\text{pert}} (a , \mathfrak{n} , & \Delta , \mathfrak{t}) = - \frac{8 \pi^2 \mathbbm{i} \beta}{g_{\text{YM}}^2} \Tr_{\text{F}} (\mathfrak{n} a) + \frac{k \beta}{2} \Tr_{\text{F}} (\mathfrak{n} a^2) \\ &
+ \frac{1}{\beta} \sum_{\alpha \in G} \left( \alpha (\mathfrak{n}) + 1 - \mathfrak{g}_2 \right) \left[ \Li_2 ( x^\alpha ) - \frac{1}{2} g_2 \left( - \alpha(\beta a) \right) \right] \\ &
- \frac{1}{\beta} \sum_I \sum_{\rho_I \in \mathfrak{R}_I} \left( \rho_I(\mathfrak{n}) + \nu_I (\mathfrak{t}) + \mathfrak{g}_2 - 1 \right) \left[ \Li_2 ( x^{\rho_I} y^{\nu_I} ) - \frac{1}{2} g_2 \left( \rho_I(\beta a) + \nu_I (\beta \Delta) \right) \right] \, ,
\end{aligned}
\ee
that indeed satisfies \eqref{WF}.
The topologically twisted index can then be computed as follows \cite{Nekrasov:2014xaa,Benini:2016hjo}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{Sigma_g1xM_3:Z}
& Z^{\text{pert}}_{ \Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1)} (\mathfrak{s} , \mathfrak{t} , \Delta) =
\frac{(-1)^{\text{rk}(G)}}{|\mathfrak{W}|} \sum_{\mathfrak{n} \in \Gamma_\mathfrak{h}} \sum_{a = a_{(i)}} Z^{\text{pert}} \big|_{\mathfrak{m} = 0} (a , \mathfrak{n})
\bigg( \det\limits_{i j} \frac{\partial^2 \widetilde \mathcal{W}^{\text{pert}} (a , \mathfrak{n})}{\partial a_i \partial a_j} \bigg)^{\mathfrak{g}_1 - 1} \, ,
\eea
where $a_{(i)}$ are the solutions to the Bethe ansatz equations (BAEs)
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{BAEs:M3}
\exp \bigg( \mathbbm{i} \frac{\partial \widetilde\mathcal{W}^{\text{pert}} (a , \mathfrak{n} ; \Delta , \mathfrak{t})}{\partial a_j} \bigg) = 1 \, , \qquad j = 1, \ldots , \text{rk}(G) \, .
\ee
As we will see below, these BAEs \eqref{BAEs:M3} fix the value of the gauge magnetic fluxes $\mathfrak{n}_i$ in the large $N$ limit, hence, one needs an extra input in order to fix the value of the Coulomb branch parameter $a_i$. Assuming that the right condition to be imposed in the large $N$ limit is \eqref{BAEs:general:F}, we will be able to compute the index for both $\mathcal{N} = 2$ SYM and the $\mathrm{USp}(2N)$ theory.%
\footnote{An alternative method for evaluating the partition function of the $\mathrm{USp}(2N)$ theory is discussed in appendix \ref{sec:alternative}.}
In particular, in the case of $\mathcal{N} = 2$ SYM, thought of as compactification of the 6D $\mathcal{N} = (2,0)$ theory on a circle, the index computes the elliptic genus of the 2D CFT obtained by compactifying the 6D theory on $\Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1}$. And, indeed, we find the correct Cardy behaviour of the index in the high-temperature limit. This is in parallel with what was found in \cite{Hosseini:2016cyf} for the topologically twisted index of 4D ${\cal N} =1$ SCFTs on $\Sigma_{\mathfrak{g}_1} \times T^2$.
Finally, we will also consider the large $N$ limit of the twisted superpotential \eqref{full:pert:twisted superpotential} and the distribution of its critical points.
We expect the on-shell value of $\widetilde \mathcal{W}^{\text{pert}}$ to compute some physical quantities of the intermediate compactification on $\Sigma_{\mathfrak{g}_2}$.
We find indeed that this is the case. For $\mathcal{N} = 2$ SYM, the critical value of $\widetilde \mathcal{W}^{\text{pert}}$ is precisely the trial central charge of the 4D SCFT obtained by compactifying the 6D $(2,0)$ theory on $\Sigma_{\mathfrak{g}_2}$, computed both in field theory and holographically in \cite{Bah:2011vv,Bah:2012dg}. Quite remarkably, the identification holds for an arbitrary assignment of R-charges for the trial central charge. For the $\mathrm{USp}(2N)$ theory, the critical value of $\widetilde \mathcal{W}^{\text{pert}}$, extremized with respect to $\Delta$, coincides with the $S^3$ free energy of the 3D theory, obtained by compactifying the 5D fixed point on $\Sigma_{\mathfrak{g}_2}$, recently computed holographically in \cite{Bah:2018lyv}.\footnote{The free energy on $\Sigma_{\mathfrak{g}_2} \times S^3$ as a function of $\Delta$ was explicitly computed in field theory in \cite{Crichigno:2018adf} after the completion of this work and perfectly agrees with the on-shell value of $\widetilde \mathcal{W}^{\text{pert}}$ as a function of $\Delta$.}
\subsection[\texorpdfstring{$\mathcal{N} = 2$}{N=2} super Yang-Mills on \texorpdfstring{$\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1)$}{Sigma(g2) x (Sigma(g1) x S**1)}]{$\mathcal{N}=2$ super Yang-Mills on $\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1)$}
\label{subsec:SYM}
A decoupling limit of type IIB string theory on asymptotically locally Euclidean (ALE) spaces predicts the existence of interacting 6D, $\mathcal{N} = (2,0)$ theories
labelled by an ADE Lie algebra $\mathfrak{g} = (A_{n \geq 1} , D_{n \geq 4} , E_6 , E_7 , E_8)$ \cite{Witten:1995zh}.
The $A_{N-1}$ type can also be realized as the low-energy description of the worldvolume theory of $N$ coincident M5-branes in M-theory \cite{Strominger:1995ac}.
A direct formulation of the $(2,0)$ theory has been a long standing problem. However, it has been argued in \cite{Douglas:2010iu,Lambert:2010iw,Bolognesi:2011nh} that the $(2, 0)$ theory on a circle $S^1_{(6)}$ of radius $R_6$ is equivalent to the five-dimensional $\mathcal{N} = 2$ supersymmetric Yang-Mills (SYM) theory, whose coupling constant is identified with the $S^1_{(6)}$ radius by
\begin{equation}} \newcommand{\ee}{\end{equation}
R_6 = \frac{g_{\text{YM}}^2}{8 \pi^2} \, .
\ee
We are interested in computing the partition function of the $(2,0)$ theory on $\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times T^2)$, partially topologically twisted on $\Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1}$.
Given the above relation between the $(2,0)$ theory and 5D maximally SYM, and considering the torus $T^2=S^1\times S^1_{(6)}$, this is equivalent to compute the twisted partition function of $\mathcal{N}=2$ SYM on $\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1)$. A further reduction on $S^1$ gives a four-dimensional theory. Recalling that the length of $S^1$ is $\beta$, we see that the (complexified) gauge coupling of the four-dimensional theory can be correctly identified with the modular parameter of the torus $T^2=S^1\times S^1_{(6)}$,
\begin{equation}} \newcommand{\ee}{\end{equation} \tau = \frac{4 \pi \mathbbm{i} \beta}{g_{\text{YM}}^2 } = \frac{\mathbbm{i} \beta}{2\pi R_6}\, . \ee
The twisted compactification of the $(2,0)$ theory on $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$ gives rise to an $\mathcal{N} = (0 , 2)$ SCFT in two dimensions \cite{Benini:2013cda}. The holonomy group of $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$ is $\mathrm{SO}(2)_1 \times \mathrm{SO}(2)_2$.
In order to preserve $\mathcal{N} = (0 , 2)$ supersymmetry in two dimensions we turn on a background Abelian gauge field
coupled to an $\mathrm{SO}(2)^2$ subgroup of $\mathrm{SO}(5)_R$, embedded block-diagonally.
The right-moving trial central charge $c_r (\Delta)$ and the gravitational anomaly $k = c_r - c_l$ for this class of theories were computed in \cite{Benini:2013cda} and, at large $N$, they can be rewritten as (see also appendix \ref{sec:centralcharges(2,0)})
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{cr:cl:6D(2,0):largeN}
c_r (\mathfrak{s} , \mathfrak{t}, \Delta) \approx c_l (\mathfrak{s} , \mathfrak{t}, \Delta)& \approx \frac{2 N^3 \beta^2}{(2 \pi)^2} \left[ \Delta_1 \Delta_2 ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) + ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) \right] \\ &
\approx \frac{N^3 \beta^2}{(2\pi)^2} \sum_{\varsigma , \varrho = 1}^{2} \mathfrak{t}_\varsigma \mathfrak{s}_\varrho \frac{\partial^2 (\Delta_1\Delta_2)^2}{\partial \Delta_\varsigma\partial \Delta_\varrho} \, ,
\eea
where we introduced the democratic chemical potentials and fluxes for the $\mathrm{SO}(2)^2$ subgroup of $\mathrm{SO}(5)_R$ symmetry
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{democratic:SYM}
& \Delta_1 = \Delta \, , \qquad \Delta_2 = \frac{2 \pi}{\beta} - \Delta \, , \qquad
\sum_{\varsigma = 1}^{2} \Delta_\varsigma = \frac{2 \pi}{\beta} \, , \\ &
\mathfrak{s}_1 = \mathfrak{s} \, , \qquad \mathfrak{s}_2 = 2 ( 1- \mathfrak{g}_1 ) - \mathfrak{s} \, , \qquad
\sum_{\varsigma = 1}^{2} \mathfrak{s}_\varsigma = 2 ( 1- \mathfrak{g}_1 ) \, , \\ &
\mathfrak{t}_1 = \mathfrak{t} \, , \qquad \mathfrak{t}_2 = 2 ( 1- \mathfrak{g}_2 ) - \mathfrak{t} \, , \qquad
\sum_{\varsigma = 1}^{2} \mathfrak{t}_\varsigma = 2 ( 1- \mathfrak{g}_2 ) \, .
\eea
The partition function of the $(2,0)$ theory on $\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times T^2)$ is just the elliptic genus of the two-dimensional CFT.
Thus we expect, in the high-temperature limit $\tilde \beta \rightarrow 0$, where $\tilde \beta \equiv - 2 \pi \mathbbm{i} \tau$ is a \emph{fictitious} inverse temperature,%
\footnote{The torus partition function at a given $\tau$ corresponds to a thermal ensemble while the elliptic genus is only counting extremal states. Therefore, the temperature represented by $\im \tau$ is fictitious.}
the partition function to have a Cardy behaviour
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{expect:N=1*:SYM:Cardy}
\log Z (\mathfrak{s} , \mathfrak{t} , \Delta) \approx \frac{\mathbbm{i} \pi}{12 \tau} c_r (\mathfrak{s} , \mathfrak{t} , \Delta) \, .
\ee
We will work in the 't Hooft limit
\begin{equation}} \newcommand{\ee}{\end{equation}
N \gg 1 \quad \text{ with } \quad \lambda = \frac{g_{\text{YM}}^2 N}{\beta} = \text{fixed} \, ,
\ee
for which the instanton contributions to the partition function are exponentially suppressed. The high-temperature limit of the partition function corresponds to large $\lambda$.
\subsubsection{Effective prepotential at large $N$}
\label{subsubsec:SW:SYM}
The effective SW prepotential \eqref{full:pert:prepotential} of $\mathcal{N} = 2$ SYM can be written as
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{SYM:prepotential}
\mathcal{F} (a , \Delta) & = \sum_{i = 1}^{N} \mathcal{F}^{\text{cl}} (a_i) + \sum_{i \neq j}^{N} \mathcal{F}^{1\text{-loop}} ( a_i - a_j ) - \sum_{i , j = 1}^{N} \mathcal{F}^{1\text{-loop}} ( a_i - a_j + \Delta) \\ &
= \frac{2 \pi \mathbbm{i} \beta}{g_{\text{YM}}^2} \sum_{i = 1}^{N} a_i^2 + \frac{\mathbbm{i}}{2 \pi \beta^2}
\sum_{i \neq j}^{N} \Li_3 (e^{\mathbbm{i} \beta (a_i - a_j)}) - \frac{\mathbbm{i}}{2 \pi \beta^2} \sum_{i , j = 1}^{N}\Li_3 (e^{\mathbbm{i} \beta (a_i - a_j + \Delta)}) \\ &
- \frac{1}{4 \pi \beta^2} \sum_{i \neq j}^{N} g_3 \left( \beta ( a_j - a_i ) \right) + \frac{1}{4 \pi \beta^2} \sum_{i , j = 1}^{N} g_3 \left( \beta( a_i - a_j + \Delta ) \right) \, .
\eea
The BAEs \eqref{BAEs:general:F} are then given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{BAEs:SYM}
- \frac{8 \pi^2 N}{\lambda} a_i & = \frac{\mathbbm{i}}{\beta} \sum_{j = 1}^{N} \left[ \Li_2 (e^{\mathbbm{i} \beta (a_i - a_j)}) - \Li_2 (e^{- \mathbbm{i} \beta (a_i - a_j)})
- \Li_2 (e^{\mathbbm{i} \beta (a_i - a_j + \Delta)}) + \Li_2 (e^{- \mathbbm{i} \beta (a_i - a_j - \Delta)}) \right] \\ &
+ \frac{\mathbbm{i}}{2 \beta} \sum_{j = 1}^{N} \left[ - g_2 \left( \beta ( a_j - a_i ) \right) + g_2 \left( \beta ( a_i - a_j ) \right) \right] \\ &
+ \frac{\mathbbm{i}}{2 \beta} \sum_{j = 1}^{N} \left[ g_2 \left( \beta( a_i - a_j + \Delta ) \right) - g_2 \left( \beta( a_j - a_i + \Delta ) \right) \right]
\, .
\eea
In the strong 't Hooft coupling $\lambda \gg 1$ the eigenvalues are pushed apart, \textit{i.e.}\;$| \im (a_i - a_j) | \gg 1$,
and \eqref{BAEs:SYM} can be approximated as
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{approx:BAEs:SYM}
\frac{8 \pi^2 N}{\lambda} a_k & \approx \frac{\mathbbm{i}}{2} \Delta ( 2 \pi - \beta \Delta ) \sum_{j = 1}^{N} \sign \left( \im (a_k - a_j) \right)
\, .
\eea
The $\sign$ function could be replaced by $\sign( i - j) $ if the eigenvalues $a_i$ are ordered by increasing imaginary part. We thus find the solution
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{SYM:sol:BAEs}
a_k = \frac{\mathbbm{i} \lambda}{16 \pi^2 N} \left[ \Delta ( 2 \pi - \beta \Delta ) (2 k - N - 1)
\right] \, .
\ee
It is also interesting to see what the value of the effective SW prepotential \eqref{SYM:prepotential} at the solution \eqref{SYM:sol:BAEs} is.
In the strong 't Hooft coupling $\lambda \gg 1$ we find that
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{SYM:F:approx}
\mathcal{F} (a , \Delta) & \approx \frac{2 \pi \mathbbm{i} N}{\lambda} \sum_{i = 1}^{N} a_i^2 - \frac{1}{4 \pi \beta^2}
\sum_{i , j = 1}^{N} \left[ g_3 \left( \beta ( a_j - a_i ) \right) + g_3 \left( \beta( a_i - a_j + \Delta ) \right) \right] \sign \left( \im (a_i - a_j) \right) \\ &
= \frac{2 \pi \mathbbm{i} N}{\lambda} \sum_{i = 1}^{N} a_i^2 + \frac{1}{8 \pi} \Delta ( 2 \pi - \beta \Delta ) \sum_{i , j = 1}^{N} (a_i - a_j) \sign \left( \im (a_i - a_j) \right) \\ &
= - \sum_{j = 1}^{N} \left[ \frac{2 \pi \mathbbm{i} N}{\lambda} \im(a_j) - \frac{\mathbbm{i}}{4 \pi} \Delta ( 2 \pi - \beta \Delta ) ( 2 j - N - 1) \right] \im ( a_j ) \, .
\eea
In order to get the last equality we used the relation
\begin{equation}} \newcommand{\ee}{\end{equation}
\sum_{i , j = 1}^{N} | \im ( a_ i - a_j ) | = 2 \sum_{j = 1}^{N} (2 j - 1 - N) \im ( a_j ) \, .
\ee
Plugging the solution \eqref{SYM:sol:BAEs} back into \eqref{SYM:F:approx}, we obtain
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{SYM:F:on-shell}
\mathcal{F} ( \Delta ) \approx \frac{\mathbbm{i} \beta g_{\text{YM}}^2 N^3}{384 \pi^3} \left( \Delta_1 \Delta_2 \right)^2 \, .
\ee
In order to obtain \eqref{SYM:F:on-shell} we used
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{sum:abs:sign}
\sum_{k = 1}^{N} (2 k - N - 1)^2 = \frac{1}{3} \left( N^3 - N \right) \, .
\ee
Remarkably, the BAEs \eqref{approx:BAEs:SYM} and the SW prepotential in the large $N$ limit \eqref{SYM:F:approx}, \eqref{SYM:F:on-shell} are \emph{identical}
to matrix model saddle point equations and free energy for the path integral on $S^5$ found in \cite{Minahan:2013jwa} (\textit{cf.}\,\cite[(4.13), (4.16) and (4.17)]{Minahan:2013jwa}).%
\footnote{One needs to set $\Delta_1 = \pi \left( 1 + \frac{2 \mathbbm{i}}{3} m_{\text{there}} \right)$ and $\Delta_2 = \pi \left( 1 - \frac{2 \mathbbm{i}}{3} m_{\text{there}} \right)$.}
\subsubsection{Effective twisted superpotential at large $N$}
\label{subsubsec:tildeW:SYM}
The effective twisted superpotential of the theory reads
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{Wt:SYM:pert}
\widetilde \mathcal{W} (a , \mathfrak{n} , \Delta , \mathfrak{t}) & = - \frac{8 \pi^2 \mathbbm{i} \beta}{g_{\text{YM}}^2} \sum_{i = 1}^{N} \mathfrak{n}_i a_i \\ &
+ \frac{1}{\beta} \sum_{i \neq j}^{N} ( \mathfrak{n}_i - \mathfrak{n}_j + 1 - \mathfrak{g}_2 ) \left[ \Li_2 ( e^{\mathbbm{i} \beta (a_i - a_j)} ) - \frac{1}{2} g_2 \left( - \beta (a_i - a_j) \right) \right] \\ &
- \frac{1}{\beta} \sum_{i , j = 1}^{N} \left( \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t} + \mathfrak{g}_2 - 1 \right) \left[ \Li_2 ( e^{\mathbbm{i} \beta (a_i - a_j + \Delta)} ) - \frac{1}{2} g_2 \left( \beta (a_i - a_j + \Delta) \right) \right] \, .
\eea
Let us try to find a solution to the BAEs \eqref{BAEs:M3}. For the twisted superpotential \eqref{Wt:SYM:pert}, they are given by%
\footnote{The BAEs actually read $\frac{\partial \widetilde\mathcal{W}^{\text{pert}}}{\partial a_j} = 2 \pi \mathfrak{l}_j$ where $\mathfrak{l}_j \in \mathbb{Z}$ are angular ambiguities.
Since $\mathfrak{l}_j$'s are generically of order one they are negligible in the final solution and we set them to zero in the following.}
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\label{SYM:BAEs:n:a}
& \frac{8 \pi^2 \mathbbm{i} \beta}{g_{\text{YM}}^2} \frac{\mathfrak{n}_i}{1 - \mathfrak{g}_2} = \mathbbm{i} \sum_{j (\neq i)}^{N} \left[ \left( \frac{\mathfrak{n}_i - \mathfrak{n}_j}{1 - \mathfrak{g}_2} + 1 \right) \Li_1 (e^{\mathbbm{i} \beta ( a_i - a_j )})
- \left( \frac{\mathfrak{n}_j - \mathfrak{n}_i}{1 - \mathfrak{g}_2} + 1 \right) \Li_1 (e^{- \mathbbm{i} \beta ( a_i - a_j )}) \right] \\ &
+ \frac12 \sum_{j (\neq i)}^{N} \left[ \left( \frac{\mathfrak{n}_i - \mathfrak{n}_j}{1 - \mathfrak{g}_2} + 1 \right) g_1 \left( - \beta ( a_i - a_j ) \right)
- \left( \frac{\mathfrak{n}_j - \mathfrak{n}_i}{1 - \mathfrak{g}_2} + 1 \right) g_1 \left( \beta ( a_i - a_j ) \right) \right] \\ &
- \mathbbm{i} \sum_{j = 1}^{N} \left[ \left( \frac{\mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t}}{1 - \mathfrak{g}_2} - 1 \right) \Li_1 ( e^{\mathbbm{i} \beta (a_i - a_j + \Delta)} )
- \left( \frac{\mathfrak{n}_j - \mathfrak{n}_i + \mathfrak{t}}{1 - \mathfrak{g}_2} - 1 \right) \Li_1 ( e^{- \mathbbm{i} \beta (a_i - a_j - \Delta)} ) \right] \\ &
+ \frac{1}{2} \sum_{j = 1}^{N} \left[ \left( \frac{\mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t}}{1 - \mathfrak{g}_2} - 1 \right) g_1 \left( \beta (a_i - a_j + \Delta) \right)
- \left( \frac{\mathfrak{n}_j - \mathfrak{n}_i + \mathfrak{t}}{1 - \mathfrak{g}_2} - 1 \right) g_1 \left( - \beta (a_i - a_j - \Delta) \right) \right] .
\end{aligned}
\ee
In the strong 't Hooft coupling limit, \textit{i.e.}\;$\lambda \gg 1$, \eqref{SYM:BAEs:n:a} can be approximated as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{SYM:sol:BAEs:tildeW}
\mathfrak{n}_i \approx \frac{\mathbbm{i} \lambda \beta}{16 \pi^2 N} (\Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1) \sum_{j (\neq i)}^{N} \sign \left( \im ( a_i - a_j ) \right) \, .
\ee
The $\sign$ function in \eqref{SYM:sol:BAEs:tildeW} could be replaced by $\sign( i - j) $ if the eigenvalues $a_i$ are ordered by increasing imaginary part.
We thus find that
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{SYM:ni}
\mathfrak{n}_k \approx \frac{\mathbbm{i} \lambda \beta}{16 \pi^2 N} ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) (2 k - N - 1) \, .
\ee
Note that the above result is neither \emph{real} nor \emph{integer}.
This is peculiar to the limit $\lambda \gg 1$. Since the $\mathfrak{n}_k$'s are large we treat them effectively as a continuous variable.
We also consider the above result as a complex saddle point contribution to the partition function.
The twisted superpotential \eqref{Wt:SYM:pert} at strong 't Hooft coupling limit can be approximated as
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\frac{\widetilde \mathcal{W} (a , \mathfrak{n} , \Delta , \mathfrak{t})}{1 - \mathfrak{g}_2} & \approx - \frac{8 \pi^2 \mathbbm{i} N}{\lambda (1 - \mathfrak{g}_2)} \sum_{i = 1}^{N} \mathfrak{n}_i a_i \\ &
- \frac{1}{2 \beta} \sum_{i , j = 1}^{N} \left( \frac{\mathfrak{n}_i - \mathfrak{n}_j}{1 - \mathfrak{g}_2} + 1 \right) g_2 \left( - \beta (a_i - a_j) \right) \sign \left( \im (a_i - a_j ) \right) \\ &
+ \frac{1}{2 \beta} \sum_{i , j = 1}^{N} \left( \frac{\mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t}}{1 - \mathfrak{g}_2} - 1 \right) g_2 \left( \beta (a_i - a_j + \Delta) \right) \sign \left( \im (a_i - a_j ) \right) \, .
\end{aligned}
\ee
This can be further simplified to
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{SYM:tildeW:simplified}
\begin{aligned}
\widetilde \mathcal{W} (a , \mathfrak{n} , \Delta , \mathfrak{t}) & \approx \frac{8 \pi^2 N}{\lambda} \sum_{j = 1}^{N} \im ( a_j ) \left[ \mathfrak{n}_j - \frac{\mathbbm{i} \lambda \beta}{16 \pi^2 N} ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) ( 2 j - N - 1 ) \right] \\ &
- \frac{\beta}{4} \Delta_1 \Delta_2 \sum_{i , j = 1}^{N} ( \mathfrak{n}_i - \mathfrak{n}_j ) \sign \left( \im (a_i - a_j ) \right) \, .
\end{aligned}
\ee
Plugging \eqref{SYM:ni} back into \eqref{SYM:tildeW:simplified}, all the dependence on $a_i$ goes away and we are left with
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{SYM:tildeW:on-shell:n}
\begin{aligned}
\widetilde \mathcal{W} (\Delta , \mathfrak{t}) & \approx - \frac{\beta}{4} \Delta_1 \Delta_2 \sum_{i , j = 1}^{N} ( \mathfrak{n}_i - \mathfrak{n}_j ) \sign \left( \im (a_i - a_j ) \right) \\ &
= - \frac{\mathbbm{i} \beta g_{\text{YM}}^2 N^3}{96 \pi^2} \Delta_1 \Delta_2 ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) \, .
\end{aligned}
\ee
This can be more elegantly rewritten as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{index theorem:tildeW:F}
\widetilde \mathcal{W} (\Delta , \mathfrak{t}) \approx - 2 \pi \sum_{\varsigma = 1}^{2} \mathfrak{t}_\varsigma \frac{\partial \mathcal{F}(\Delta)}{\partial \Delta_\varsigma} \, ,
\ee
where $\mathcal{F}(\Delta)$ is given in \eqref{SYM:F:on-shell}.
\paragraph*{$\widetilde \mathcal{W}(\Delta , \mathfrak{t})$ and the 4D central charge.}
Remarkably, we find the following relation between the twisted superpotential \eqref{SYM:tildeW:on-shell:n} and the conformal anomaly coefficient $a (\Delta , \mathfrak{t})$
of the four-dimensional $\mathcal{N}=1$ theory that is obtained by compactifying the 6D $\mathcal{N} = (2,0)$ theory on $\Sigma_{\mathfrak{g}_2}$ \cite{Bah:2011vv,Bah:2012dg},
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{tildeW:a central charge}
\widetilde \mathcal{W} (\Delta , \mathfrak{t}) \approx - \frac{ 8 \pi^2}{27 \beta \tau} a (\Delta , \mathfrak{t}) \, ,
\ee
where $\tau$ is the four-dimensional coupling constant.
Indeed, the central charge of the 4D theory, at large $N$, can be written as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{a4D}
a (\Delta , \mathfrak{t}) \approx -\frac{9 N^3}{128} \sum_{\varsigma=1}^2 \mathfrak{t}_\varsigma \frac{\partial (\hat \Delta_1\hat\Delta_2)^2}{\partial \hat\Delta_\varsigma} \, ,
\ee
where we used $\hat\Delta_\varsigma = \beta \Delta_\varsigma / \pi$, satisfying $\hat \Delta_1+\hat \Delta_2=2$, to parameterize a trial R-symmetry of the 4D $\mathcal{N}=1$ theory.
\eqref{a4D} can be easily derived from the results in \cite{Bah:2011vv,Bah:2012dg} (\textit{cf.}\,for example \cite[(2.22)]{Bah:2012dg}). It can be also more straightforwardly
derived as in appendix \ref{sec:centralcharges(2,0)}.
Curiously, the same relation between the twisted superpotential and the central charge in \eqref{tildeW:a central charge} was found for a class of $\mathcal{N} = 1$ gauge theories on $S^2 \times T^2$, with a partial topological twist along $S^2$ \cite{Hosseini:2016cyf}.
\subsubsection{Partition function at large $N$}
\label{subsubsec:logZ:SYM}
The topologically twisted index of 5D $\mathcal{N} = 2$ SYM on $\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1 )$ reads
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{SYM:index:Sigmag2xS1xSigmag1}
Z (y , \mathfrak{s} , \mathfrak{t}) = \frac{1}{N!} \sum_{\{\mathfrak{m} , \mathfrak{n}\} \in \mathbb{Z}^N} & \oint_{\mathcal{C}} \prod_{i = 1}^{N} \frac{\mathrm{d} x_i}{2 \pi \mathbbm{i} x_i}
e^{\frac{8 \pi^2 \beta}{g_{\text{YM}}^2} ( \mathfrak{m}_i - \mathfrak{m}_j ) ( \mathfrak{n}_i - \mathfrak{n}_j )}
\bigg( \det\limits_{i j} \frac{\partial^2 \widetilde \mathcal{W} (a , \mathfrak{n})}{\partial a_i \partial a_j} \bigg)^{\mathfrak{g}_1} \\ &
\times \prod_{i \neq j}^{N} \bigg( \frac{1 - x_i / x_j}{\sqrt{x_i / x_j}} \bigg)^{( \mathfrak{m}_i - \mathfrak{m}_j + 1 - \mathfrak{g}_1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + 1 - \mathfrak{g}_2 )} \\ &
\times \prod_{i , j = 1}^{N} \bigg( \frac{\sqrt{x_i y / x_j}}{1 - x_i y / x_j} \bigg)^{( \mathfrak{m}_i - \mathfrak{m}_j + \mathfrak{s} + \mathfrak{g}_1 - 1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t} + \mathfrak{g}_2 - 1 )} \, .
\eea
This can be evaluated using \eqref{Sigma_g1xM_3:Z}. We can write
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
Z (y , \mathfrak{s} , \mathfrak{t}) & = \frac{(-1)^{N}}{N!}
\bigg( \frac{\sqrt{y}}{1 - y} \bigg)^{N ( \mathfrak{g}_1 + \mathfrak{s} - 1 ) ( \mathfrak{g}_2 + \mathfrak{t} - 1 )}
\sum_{\mathfrak{n} \in \mathbb{Z}^N} \sum_{a = a_{(i)}} \bigg( \det\limits_{i j} \frac{\partial^2 \widetilde \mathcal{W} (a , \mathfrak{n})}{\partial a_i \partial a_j} \bigg)^{\mathfrak{g}_1 - 1} \\ &
\times \prod_{i \neq j}^{N} \bigg( \frac{1 - x_i / x_j}{\sqrt{x_i / x_j}} \bigg)^{( 1 - \mathfrak{g}_1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + 1 - \mathfrak{g}_2 )}
\bigg( \frac{\sqrt{x_i y / x_j}}{1 - x_i y / x_j} \bigg)^{( \mathfrak{s} + \mathfrak{g}_1 - 1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t} + \mathfrak{g}_2 - 1 )} \, .
\eea
We are interested in the logarithm of the partition function in the strong 't Hooft coupling limit.
The only piece which survives in this limit is given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\log Z^{(1)} & \equiv \log \prod_{i \neq j}^{N} \bigg( \frac{1 - x_i / x_j}{\sqrt{x_i / x_j}} \bigg)^{( 1 - \mathfrak{g}_1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + 1 - \mathfrak{g}_2 )}
\bigg( \frac{\sqrt{x_i y / x_j}}{1 - x_i y / x_j} \bigg)^{( \mathfrak{s} + \mathfrak{g}_1 - 1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t} + \mathfrak{g}_2 - 1 )} \\ &
= - \sum_{i \neq j}^{N} ( 1 - \mathfrak{g}_1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + 1 - \mathfrak{g}_2 ) \left[ \Li_1 ( e^{\mathbbm{i} \beta ( a_i - a_j )} ) - \frac{\mathbbm{i}}{2} g_1 \left( - \beta ( a_i - a_j ) \right) \right] \\ &
+ \sum_{i \neq j}^{N} ( \mathfrak{s} + \mathfrak{g}_1 - 1 ) ( \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t} + \mathfrak{g}_2 - 1 ) \left[ \Li_1 ( e^{\mathbbm{i} \beta ( a_i - a_j + \Delta )} ) + \frac{\mathbbm{i}}{2} g_1 \left( \beta ( a_i - a_j + \Delta ) \right) \right] \, .
\eea
In the strong 't Hooft coupling limit it can be approximated as
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{SYM:logZ:total}
\log Z^{(1)} & \approx - \frac{\mathbbm{i} \beta}{4} \sum_{i \neq j}^{N} \left[ ( a_i - a_j ) ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) + ( \mathfrak{n}_i - \mathfrak{n}_j ) ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) \right] \sign \left( \im ( a_i - a_j ) \right) \\ &
= \frac{\beta}{2} \sum_{j = 1}^{N} (2 j - 1 - N) \left[ ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) \im ( a_j ) - \mathbbm{i} ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) \mathfrak{n}_j \right] \, ,
\eea
where we assumed that the eigenvalues $a_i$ are ordered by increasing imaginary part, and used the relation
\begin{equation}} \newcommand{\ee}{\end{equation}
\sum_{i , j = 1}^{N} ( \mathfrak{n}_i - \mathfrak{n}_j ) \sign ( i - j ) = 2 \sum_{j = 1}^{N} ( 2 j - N - 1 ) \mathfrak{n}_i \, .
\ee
Plugging the solutions \eqref{SYM:sol:BAEs} and \eqref{SYM:ni} back into \eqref{SYM:logZ:total}, we find that
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{SYM:Final:logZ:index theorem}
\log Z & \approx \frac{\beta g_{\text{YM}}^2 N^3}{96 \pi^2} \left[ \Delta_1 \Delta_2 ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) + ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) \right] \\ &
= \mathbbm{i} \sum_{\varsigma = 1}^2 \mathfrak{s}_\varsigma \frac{\partial \widetilde \mathcal{W} (\mathfrak{t} , \Delta)}{\partial \Delta_\varsigma}
= - 2 \mathbbm{i} \pi \sum_{\varsigma , \varrho = 1}^2 \mathfrak{s}_\varsigma \mathfrak{t}_\varrho \frac{\partial^2 \mathcal{F} (\Delta)}{\partial \Delta_\varsigma \partial \Delta_\varrho} \\ &
= \frac{g_{\text{YM}}^2N^3}{192 \pi^2 \beta} \sum_{\varsigma , \varrho = 1}^2 \mathfrak{s}_\varsigma \mathfrak{t}_\varrho \frac{\partial^2 (\beta \Delta_1 \Delta_2)^2}{\partial \Delta_\varsigma \partial \Delta_\varrho} \, ,
\eea
where $\widetilde \mathcal{W} (\mathfrak{t} , \Delta)$ is given in \eqref{SYM:tildeW:on-shell:n} and we used \eqref{index theorem:tildeW:F} in writing the last equality.
\eqref{SYM:Final:logZ:index theorem} can be also expressed in terms of the trial right-moving central charge of the 2D $\mathcal{N} = (0,2)$ SCFT, see \eqref{cr:cl:6D(2,0):largeN}, as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{N=1*:SYM:Cardy}
\log Z (\mathfrak{s} , \mathfrak{t} , \Delta) \approx \frac{\mathbbm{i} \pi}{12 \tau} c_r (\mathfrak{s} , \mathfrak{t} , \Delta) \approx - \frac{8 \mathbbm{i} \pi^2}{27 \beta \tau} \sum_{\varsigma = 1}^2 \mathfrak{s}_\varsigma \frac{\partial a ( \mathfrak{t} , \Delta)}{\partial \Delta_\varsigma} \, .
\ee
In writing the second equality we used the relation \eqref{tildeW:a central charge}.
In \cite{Hosseini:2016cyf}, the very same Cardy behaviour \eqref{N=1*:SYM:Cardy} of the partition function in the high-temperature limit has been proved for a class of $\mathcal{N} = 1$ gauge theories on $S^2 \times T^2$, with a partial topological twist along $S^2$.
\subsubsection{Counting states and the $\mathcal{I}$-extremization principle}
\label{I-extremization principle}
The topologically twisted index of the 6D $(2,0)$ theory on $ \Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1} \times T^2$ can be interpreted as a trace over a Hilbert space of states on $\Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1} \times S^1$:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{EG}
Z (\mathfrak{s}, \mathfrak{t}, \Delta) = \Tr_{\Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1} \times S^1} (-1)^F q^{H_L} y^J \, ,
\eea
where $q=e^{2 \pi \mathbbm{i} \tau}$, $y=e^{\mathbbm{i} \beta \Delta}$ and the Hamiltonian $H_L$ on $\Sigma_{\mathfrak{g}_2} \times \Sigma_{\mathfrak{g}_1} \times S^1$ explicitly depends on the magnetic fluxes $\mathfrak{s}_\varsigma , \mathfrak{t}_\varsigma$ $(\varsigma = 1, 2)$.
The number of supersymmetric ground states $d_{\text{micro}}$ with momentum $n$ and electric charge $\mathfrak{q}$ under the Cartan subgroup of the flavor symmetry commuting with the Hamiltonian -- in the microcanonical ensemble -- is then given by
the Fourier transform of \eqref{EG} with respect to $(\tau , \Delta)$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{ch:1:micro:density}
d_{\text{micro}} (\mathfrak{s} , \mathfrak{t} ,n, \mathfrak{q}) = - \frac{\mathbbm{i} \beta}{(2 \pi)^2} \int_{\mathbbm{i} \mathbb{R}} \mathrm{d} \tilde \beta \int_{0}^{\frac{2\pi}{\beta}} \mathrm{d} \Delta \, Z (\mathfrak{s}, \mathfrak{t}, \Delta) \, e^{ \tilde \beta n - \mathbbm{i} \beta \Delta \mathfrak{q}} \, ,
\ee
where $\tilde \beta=-2 \pi \mathbbm{i} \tau$ and the corresponding integration is over the imaginary axis.
In the limit of large charges, we may use the saddle point approximation. Consider for simplicity $\mathfrak{q}=0$.
The number of supersymmetric ground states $d_{\text{micro}}$ with charges $(\mathfrak{s} ,\mathfrak{t} , n)$ can be obtained by extremizing
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{SBH}
\mathcal{I}_{\text{SCFT}} (\tilde \beta , \Delta) \equiv \log Z (\mathfrak{s}, \mathfrak{t}, \Delta) + \tilde \beta n
= \frac{N^3}{24 \tilde \beta} \sum_{\varsigma , \varrho = 1}^2 \mathfrak{s}_\varsigma \mathfrak{t}_\varrho \frac{\partial^2 (\beta \Delta_1 \Delta_2)^2}{\partial \Delta_\varsigma \partial \Delta_\varrho} + n \tilde \beta \, ,
\ee
with respect to $\Delta$ and $\tilde \beta$, \textit{i.e.}
\begin{equation}} \newcommand{\ee}{\end{equation}
\frac{\partial \mathcal{I} (\tilde \beta , \Delta)}{\partial \Delta} = 0 \, , \qquad \frac{\partial \mathcal{I} (\tilde \beta , \Delta)}{\partial \tilde \beta} = 0 \, ,
\ee
and evaluating it at its extremum
\begin{equation}} \newcommand{\ee}{\end{equation}
\log d_{\text{micro}} (\mathfrak{s} , \mathfrak{t} ,n, 0) = \mathcal{I} \big|_{\text{crit}} (\mathfrak{s} , \mathfrak{t}, n) \, .
\ee
Given \eqref{N=1*:SYM:Cardy}, we see that the extremization with respect to $\Delta$ is the $c$-extremization principle \cite{Benini:2012cz,Benini:2013cda}
and sets the trial right-moving central charge $c_r (\mathfrak{s} , \mathfrak{t} , \Delta)$ to its \emph{exact} value $c_{\text{CFT}} \approx c_r \approx c_l$ in the IR.
For the case at our disposal, \eqref{cr:cl:6D(2,0):largeN} has a critical point at
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{crit:Delta}
\bar \Delta = - \frac{2 \pi}{\beta} \frac{\mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 - \mathfrak{t}_1 \mathfrak{s}_1}{\mathfrak{s}_1 ( \mathfrak{t}_1 - 2 \mathfrak{t}_2 ) + \mathfrak{s}_2 ( \mathfrak{t}_2 - 2 \mathfrak{t}_1 )} \, ,
\ee
and its value at $\bar \Delta$ reads
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{exact:cr}
c_{\text{CFT}} (\mathfrak{s} , \mathfrak{t}) \approx - 2 N^3 \frac{\mathfrak{t}_1^2 \mathfrak{s}_2^2 + \mathfrak{t}_1 \mathfrak{t}_2 \mathfrak{s}_1 \mathfrak{s}_2 + \mathfrak{t}_2^2 \mathfrak{s}_1^2}{\mathfrak{s}_1 ( \mathfrak{t}_1 - 2 \mathfrak{t}_2) + \mathfrak{s}_2 ( \mathfrak{t}_2 - 2 \mathfrak{t}_1 )} \, .
\ee
Extremizing $\mathcal{I} (\tilde \beta , \Delta)$ with respect to $\tilde\beta$ yields
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{crit:tbeta}
\bar {\tilde \beta} (\mathfrak{s} , \mathfrak{t} , n)
= \pi \sqrt{\frac{c_{\text{CFT}} (\mathfrak{s} , \mathfrak{t})}{6 n} } \, .
\ee
Plugging back \eqref{crit:Delta} and \eqref{crit:tbeta} into $\mathcal{I} (\tilde \beta , \Delta)$, we find that
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}\label{Cardys3}
\mathcal{I}_{\text{SCFT}}\big|_{\text{crit}} (\mathfrak{s} , \mathfrak{t} , n) = 2 \pi \sqrt{ \frac{ n\, c_{\text{CFT}}(\mathfrak{s} , \mathfrak{t})}{6}} \, .
\eea
This is obviously nothing else than Cardy formula \cite{Cardy:1986ie}.
This procedure corresponds to the $\mathcal{I}$\emph{-extremization} principle used for BPS black holes (strings) \cite{Benini:2015eyy,Benini:2016rke,Hosseini:2016cyf,Hosseini:2018qsx} in the context of the AdS$_{4 (5)}$ /CFT$_{3 (4)}$ correspondence
and contains two basic pieces of information:
\begin{enumerate}
\item \label{step1} extremizing the index unambiguously determines the exact R-symmetry of the SCFT in the IR;
\item the value of the index at its extremum is the (possibly regularized) number of ground states.
\end{enumerate}
We will apply this counting to supergravity black strings and black holes in section \ref{sec:4D domain-walls}.
\subsection[\texorpdfstring{$\mathrm{USp}(2N)$}{USp(2N)} theory with matter on \texorpdfstring{$\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1)$}{Sigma(g2) x (Sigma(g1) x S**1)}]{$\mathrm{USp}(2N)$ theory with matter on $\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1)$}
\label{subsec:USp(2N)}
In this section we focus on gauge theories with a conjectured massive type IIA dual \cite{Intriligator:1997pq} (see also \cite{Brandhuber:1999np,Bergman:2012kr,Morrison:1996xf,Seiberg:1996bd}) and compute their partition function at large $N$.
The dual supergravity backgrounds have a warped AdS$_6 \times S^4$ geometry.
The stringy root of this theories is in type I' string theory as strongly coupled microscopic theories on the intersection of $N$ D4-branes and $N_f$ D8-branes and orientifold planes.
The worldvolume theory on the $N$ D4-branes plus their images is a $\mathrm{USp}(2N)$ gauge theory with $N_f$ hypermultiplets in the fundamental representation and one hypermultiplet $\mathcal{A}$ in the antisymmetric representation of $\mathrm{USp}(2N)$.
In addition to the $\mathrm{SU}(2)$ R-symmetry the theory has an $\mathrm{SU}(2)_M \times \mathrm{SO}(2 N_f) \times \mathrm{U}(1)_I$ global symmetry:
$\mathrm{SU}(2)_M$ acts on $\mathcal{A}$ as a doublet,
$\mathrm{SO}(2 N_f)$ is the flavor symmetry associated to the fundamental hypermultiplets, and $\mathrm{U}(1)_I$ is the topological symmetry associated to the conserved instanton number current $j = \ast \Tr (F \wedge F)$.
At the fixed point, the $\mathrm{SO}(2N_f) \times \mathrm{U}(1)_I$ part of the symmetry is enhanced non-perturbatively to an exceptional group $\mathrm{E}_{N_f + 1}$,
due to the instanton which becomes massless at the origin of the Coulomb branch of the fixed point theory \cite{Seiberg:1996bd}.
Finally, in the large $N$ limit the free energy of the $\mathrm{USp}(2 N)$ theory on $S^5$ reads \cite{Jafferis:2012iv}
\begin{equation}} \newcommand{\ee}{\end{equation}
F_{S^5} \approx - \frac{9 \sqrt{2} \pi N^{5/2}}{5 \sqrt{8 - N_f}} \, .
\ee
The Cartan of $\mathrm{USp}(2N)$ has $N$ elements which we denote by $u_i$, $i = 1, \ldots, N$.
We normalize the weights of the fundamental representation of $\mathrm{USp}(2N)$ to be $\pm e_i$ (so they form a basis of unit vectors for $\mathbb{R}^N$).
The antisymmetric representation thus has weights $\pm e_i \pm e_j$ with $i > j$ and $N-1$ zero weights,
and the roots are $\pm e_i \pm e_j$ with $i > j$ and $\pm 2 e_i$.
As we shall see below, at large $N$, $u_{i} = \mathcal{O}(N^{1/2})$ (see \eqref{largeN:ansatz} with $\alpha = 1/2$).
Hence, the contributions with nontrivial instanton numbers are exponentially suppressed in the large $N$ limit.
We set the length of $S^1$ to one, \textit{i.e.}\;$\beta = 1$, throughout this section.
\subsubsection{Effective prepotential at large $N$}
\label{subsubsec:SW:USp(2N)}
The effective prepotential of the theory is then given by \eqref{full:pert:prepotential}:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{prepotential:USp(2N)}
\mathcal{F} (a_i, \Delta_K) & = \sum_{i = 1}^{N} \left[ \mathcal{F}^{\text{pert}} (\pm 2 a_i) - \sum_{f = 1}^{N_f} \mathcal{F}^{\text{pert}} (\pm a_i + \Delta_f) \right] \\ &
+ \sum_{i > j}^{N} \left[ \mathcal{F}^{\text{pert}} (\pm a_i \pm a_j) - \mathcal{F}^{\text{pert}} (\pm a_i \pm a_j + \Delta_m) \right]
+ (N - 1) \mathcal{F}^{\text{pert}} (\Delta_m) \, ,
\eea
where the index $K$ labels all the matter fields in the theory.
Here, we introduced the notation $\mathcal{F} (\pm a) \equiv \mathcal{F} (a) + \mathcal{F} (- a)$.
Notice that the contribution of the last term to the large $N$ prepotential is of $\mathcal{O}(N^2)$ and thus subleading.
Let us analyze the effective prepotential \eqref{prepotential:USp(2N)} assuming that the eigenvalues grow in the large $N$ limit.
We restrict to $\im a_i > 0$ due to the Weyl reflections of the $\mathrm{USp}(2N)$ group.
We consider the following large $N$ saddle point eigenvalue distribution ansatz:
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{largeN:ansatz}
a_j = \mathbbm{i} N^{\alpha} t_j \, ,
\ee
for some number $0 < \alpha < 1$ to be determined later. At large $N$, we define the continuous function $t_j = t (j / N)$ and we introduce the density of eigenvalues
\begin{equation}} \newcommand{\ee}{\end{equation}
\rho(t) = \frac{1}{N} \frac{\mathrm{d} j}{\mathrm{d} t} \, ,
\ee
normalized so that $\int \mathrm{d} t \rho(t) = 1$. In the large $N$ limit the sums over $N$ become Riemann integrals, for example,
\begin{equation}} \newcommand{\ee}{\end{equation}
\sum_{j = 1}^{N} \to N \int \mathrm{d} t \rho (t) \, .
\ee
Consider the first line in \eqref{prepotential:USp(2N)}:
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
& \frac{\mathbbm{i}}{2 \pi} \sum_{i = 1}^{N} \bigg[ \Li_3 (e^{2 \mathbbm{i} a_i}) + \Li_3 (e^{- 2 \mathbbm{i} a_i}) - \sum_{f = 1}^{N_f} \Li_3 (e^{\mathbbm{i} (a_i + \Delta_f)})
- \sum_{f = 1}^{N_f} \Li_3 (e^{- \mathbbm{i} (a_i - \Delta_f)}) \bigg] \\ &
- \frac{1}{4 \pi} \sum_{i = 1}^{N} \bigg[ g_3 ( - 2 a_i ) + g_3 ( 2 a_i ) + \sum_{f = 1}^{N_f} g_3 ( a_i + \Delta_f ) + \sum_{f = 1}^{N_f} g_3 ( - a_i + \Delta_f ) \bigg] \, .
\end{aligned}
\ee
The second line is of $\mathcal{O} (N^{2 \alpha +1})$ and thus subleading in the large $N$ limit (as we see below).
Using the ansatz \eqref{largeN:ansatz} we may write
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\label{cont:gauge:fund}
\mathcal{F}^{(0)} \approx & \frac{\mathbbm{i} N}{2 \pi} \int \mathrm{d} t \rho(t) \bigg[ \Li_3 (e^{- 2 N^\alpha t}) + \Li_3 (e^{2 N^\alpha t})
- \sum_{f = 1}^{N_f} \Li_3 (e^{- N^{\alpha} t + \mathbbm{i} \Delta_f}) - \sum_{f = 1}^{N_f} \Li_3 (e^{N^{\alpha} t + \mathbbm{i} \Delta_f}) \bigg] \\ &
\approx - \frac{\mathbbm{i} (8 - N_f)}{12 \pi} N^{1 + 3 \alpha} \int \mathrm{d} t \rho(t) t^3 \left[ \Theta (t) - \Theta (-t) \right] + \mathcal{O} (N^{2 \alpha +1}) \\ &
= - \frac{\mathbbm{i} (8 - N_f)}{12 \pi} N^{1 + 3 \alpha} \int \mathrm{d} t \rho(t) | t |^3 + \mathcal{O} (N^{2 \alpha +1})\, ,
\end{aligned}
\ee
where $\Theta (t)$ is the Heaviside theta function.
Now, let us focus on the second line of \eqref{prepotential:USp(2N)}. Consider the following terms
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\mathcal{F}^{(1)}_{\text{hyper}} & = - \frac{\mathbbm{i}}{2 \pi} \sum_{i > j}^{N} \left[ \Li_3 (e^{\mathbbm{i} (a_i - a_j + \Delta_m)}) + \Li_3 (e^{- \mathbbm{i} (a_i - a_j - \Delta_m)}) \right] \\ &
- \frac{1}{4 \pi} \sum_{i > j}^{N} \left[ g_3 (a_i - a_j + \Delta_m) + g_3 (a_j - a_i + \Delta_m) \right] \, .
\eea
At large $N$, we obtain
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\label{F:cont:as:I}
\mathcal{F}^{(1)}_{\text{hyper}} & \approx - \frac{\mathbbm{i} N^2}{4 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left[ \Li_3 (e^{- N^{\alpha} (t - t') + \mathbbm{i} \Delta_m}) + \Li_3 (e^{ N^{\alpha} (t - t') + \mathbbm{i} \Delta_m}) \right] \\ &
- \frac{N^2}{8 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left[ (\pi - \Delta_m ) N^{2 \alpha} (t - t')^2 + 2 g_3 ( \Delta_m ) \right] \\ &
\approx \frac{N^{2}}{4 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') g_3 \left( - \mathbbm{i} N^{\alpha} (t' - t) + \Delta_m \right) \Theta ( t' - t) \\ &
+ \frac{N^{2}}{4 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') g_3 \left( - \mathbbm{i} N^{\alpha} (t - t') + \Delta_m \right) \Theta ( t - t') \\ &
- \frac{N^2}{8 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left[ (\pi - \Delta_m ) N^{2 \alpha} (t - t')^2 + 2 g_3 ( \Delta_m ) \right] \\ &
= \frac{\mathbbm{i} N^{2}}{4 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left( \frac{1}{6} N^{3 \alpha} \left| t - t' \right|^3 - g_2 ( \Delta_m ) N^{\alpha} \left| t - t' \right| \right) \, .
\end{aligned}
\ee
Next, consider
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{gauge:USp(2N):short:roots}
\mathcal{F}^{(1)}_{\text{vector}} = \frac{\mathbbm{i}}{2 \pi} \sum_{i > j}^{N} \left[ \Li_3 (e^{\mathbbm{i} (a_i - a_j)}) + \Li_3 (e^{- \mathbbm{i} (a_i - a_j)}) + \frac{\mathbbm{i}}{2} g_3 (a_j - a_i) + \frac{\mathbbm{i}}{2} g_3 (a_i - a_j) \right] \, .
\ee
Its contribution can be simply obtained by, see \eqref{Z2:gs},
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{F}^{(1)}_{\text{vector}} = - \mathcal{F}^{(1)}_{\text{hyper}} \big|_{\Delta_m = 2 \pi} \, .
\ee
We thus find
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{cont:gauge:as:I}
\mathcal{F}^{(1)}_{\text{vector}} + \mathcal{F}^{(1)}_{\text{hyper}} \approx \frac{\mathbbm{i}}{4 \pi} \left[ \frac{\pi^2}{3} - g_2 ( \Delta_m ) \right] N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left| t - t' \right| \, .
\ee
The next term that we shall consider is the following
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\begin{aligned}
\mathcal{F}^{(2)}_{\text{hyper}} & = - \frac{\mathbbm{i}}{2 \pi} \sum_{i > j}^{N} \left[ \Li_3 (e^{\mathbbm{i} (a_i + a_j + \Delta_m)}) + \Li_3 (e^{- \mathbbm{i} (a_i + a_j - \Delta_m)}) \right] \\ &
- \frac{1}{4 \pi} \sum_{i > j}^{N} \left[ g_3 (a_i + a_j + \Delta_m) + g_3 ( - a_i - a_j + \Delta_m) \right] \, .
\end{aligned}
\eea
The first term in the first line is exponentially suppressed in the large $N$ limit (since $\im a_i > 0 \, , \forall i $).
We then get
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\mathcal{F}^{(2)}_{\text{hyper}} & \approx - \frac{\mathbbm{i} N^2}{4 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \Li_3 (e^{N^{\alpha} (t + t')+ \mathbbm{i} \Delta_m}) \\ &
- \frac{N^2}{8 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left[ (\pi - \Delta_m ) N^{2 \alpha} (t + t')^2 + 2 g_3 ( \Delta_m ) \right] \\ &
\approx \frac{N^2}{4 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') g_3 (- \mathbbm{i} N^{\alpha} (t + t') + \Delta_m) \\ &
- \frac{N^2}{8 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left[ (\pi - \Delta_m ) N^{2 \alpha} (t + t')^2 + 2 g_3 ( \Delta_m ) \right] \\ &
= \frac{\mathbbm{i} N^{2}}{4 \pi} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') \left[ \frac{1}{6} N^{3 \alpha} (t + t')^3 - g_2 ( \Delta_m ) N^{\alpha} (t+ t')^2 \right] \, .
\end{aligned}
\ee
The last term that we shall consider reads
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{F}^{(2)}_{\text{vector}} = \frac{\mathbbm{i}}{2 \pi} \sum_{i > j}^{N} \left[ \Li_3 (e^{\mathbbm{i} (a_i + a_j)}) + \Li_3 (e^{- \mathbbm{i} (a_i + a_j)}) + \frac{\mathbbm{i}}{2} g_3 (- a_i - a_j) + \frac{\mathbbm{i}}{2} g_3 (a_i + a_j) \right] \, .
\ee
Its contribution can be simply obtained by, see \eqref{Z2:gs},
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{F}^{(2)}_{\text{vector}} = - \mathcal{F}^{(2)}_{\text{hyper}} \big|_{\Delta_m = 2 \pi} \, .
\ee
We thus find that
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{cont:gauge:as:II}
\mathcal{F}^{(2)}_{\text{vector}} + \mathcal{F}^{(2)}_{\text{hyper}} \approx \frac{\mathbbm{i}}{4 \pi} \left[ \frac{\pi^2}{3} - g_2 ( \Delta_m ) \right] N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') (t+ t') \, .
\ee
Putting \eqref{cont:gauge:fund}, \eqref{cont:gauge:as:I}, and \eqref{cont:gauge:as:II} together we obtain the final expression for the effective prepotential at large $N$:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{LargeN:Usp(2N):F}
\mathcal{F} \left[ \rho(t) , \Delta_m \right] & = \mathcal{F}^{(0)} + \sum_{\vartheta = 1}^{2} \left( \mathcal{F}^{(\vartheta)}_{\text{vector}} + \mathcal{F}^{(\vartheta)}_{\text{hyper}} \right) \\ &
\approx - \frac{\mathbbm{i} (8 - N_f)}{12 \pi} N^{1 + 3 \alpha} \int_{0}^{t_*} \mathrm{d} t \rho(t) | t |^3
- \mu \left( \int_{0}^{t_*} \mathrm{d} t \rho (t)- 1 \right) \\ &
+ \frac{\mathbbm{i}}{4 \pi} \left[ \frac{\pi^2}{3} - g_2 ( \Delta_m ) \right] N^{2 + \alpha} \int_{0}^{t_*} \mathrm{d} t \rho(t) \int_{0}^{t_*} \mathrm{d} t' \rho(t') \left[ | t - t' | + ( t + t' ) \right] \, ,
\eea
where we added the Lagrange multiplier $\mu$ for the normalization of $\rho(t)$.
$\alpha$ will be determined to be $1/2$ by the competition between the first and the last term in \eqref{LargeN:Usp(2N):F}, and therefore $\mathcal{F} \propto N^{5/2}$.
Remarkably, the effective prepotential \eqref{LargeN:Usp(2N):F} equals the large $N$ expression of the $S^5$ free energy computed in \cite[(3.4)]{Jafferis:2012iv} (see also \cite[(3.14)]{Chang:2017mxc}).
This is in complete analogy with the observation made in \cite{Hosseini:2016tor}.
There, the effective twisted superpotential of a three-dimensional $\mathcal{N}=2$ theory on $A$-twisted $\Sigma_{\mathfrak{g}_1} \times S^1$ was shown to be equal to the $S^3$ free energy of the same $\mathcal{N}=2$ theory, both evaluated at large $N$.
Extremizing \eqref{LargeN:Usp(2N):F} with respect to the continuous function $\rho(t)$ we find the following saddle point equation
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\frac{2 \mathbbm{i} \pi \mu}{N^{5/2}} = \frac{(8 - N_f)}{6} | t' |^3 - \left[ \frac{\pi^2}{3}-g_2(\Delta_m) \right] \int_{0}^{t_*} \rho (t) \left[ | t - t' | + ( t + t' ) \right] \, .
\eea
On the support of $\rho (t)$ the solution reads
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{LargeN:USp(2N):sol}
& \rho (t) = \frac{2 | t | }{t_{*}^2} \, , \qquad
t_* = \frac{2}{\sqrt{8 - N_f}} \left[ \frac{\pi^2}{3} -g_2 ( \Delta_m ) \right]^{1/2} \, , \\ &
\mu = \frac{4 \mathbbm{i}}{3 \pi} \frac{N^{5/2}}{\sqrt{8 - N_f}} \left[ \frac{\pi^2}{3} - g_2 ( \Delta_m ) \right]^{3/2} \, .
\eea
Evaluating \eqref{LargeN:Usp(2N):F} at \eqref{LargeN:USp(2N):sol} yields%
\footnote{$\mathcal{F} (\Delta_m) = \frac25 \mu (\Delta_m)$ due to a virial theorem for the large $N$ prepotential \eqref{LargeN:Usp(2N):F}.}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{LargeN:USp(2N):prepotential}
\mathcal{F} (\Delta_m) \approx \frac{2^{3/2} \mathbbm{i}}{15 \pi} \frac{N^{5/2}}{\sqrt{8 - N_f}} \left[ \Delta_m (2 \pi - \Delta_m ) \right]^{3/2} \, .
\eea
Finally, let us rewrite \eqref{LargeN:USp(2N):prepotential} as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{USp(2N):F:on-shell}
\mathcal{F} (\Delta) \approx \frac{2^{3/2} \mathbbm{i}}{15 \pi} \frac{N^{5/2}}{\sqrt{8 - N_f}} \left( \Delta_1 \Delta_2 \right)^{3/2} \, ,
\ee
for later use. Here we introduced the democratic chemical potentials
\begin{equation}} \newcommand{\ee}{\end{equation}
\Delta_1 = \Delta_m \, , \qquad \Delta_2 = 2 \pi - \Delta_m \, .
\ee
\subsubsection{Effective twisted superpotential at large $N$}
\label{subsubsec:tildeW:USp(2N)}
We are interested in the large $N$ limit of the effective twisted superpotential
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{Wt:USp(2N):pert}
\widetilde \mathcal{W}^{\text{pert}} (a , \mathfrak{n} , \Delta , \mathfrak{t}) & = \sum_{i = 1}^{N} \bigg[ \widetilde \mathcal{W}^{\text{pert}} (\pm 2 a_i) - \sum_{f = 1}^{N_f} \widetilde \mathcal{W}^{\text{pert}} (\pm a_i + \Delta_f) \bigg] \\ &
+ \sum_{i > j}^{N} \left[ \widetilde \mathcal{W}^{\text{pert}} (\pm a_i \pm a_j) - \widetilde \mathcal{W}^{\text{pert}} (\pm a_i \pm a_j + \Delta_m) \right]
+ (N - 1) \widetilde \mathcal{W}^{\text{pert}} (\Delta_m) \, .
\eea
We consider the following ansatz for the large $N$ saddle point eigenvalue distribution
\begin{equation}} \newcommand{\ee}{\end{equation}
a_i = \mathbbm{i} N^{\alpha} t_i
\, , \qquad \mathfrak{n}_i = \mathbbm{i} N^{\alpha} \mathfrak{N}_i
\, .
\ee
Consider the first line in \eqref{Wt:USp(2N):pert}:
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\widetilde \mathcal{W}^{(0)} & = \sum_{i = 1}^{N} ( \pm 2 \mathfrak{n}_i + 1 - \mathfrak{g}_2 ) \left[ \Li_2 (e^{\pm 2 \mathbbm{i} a_i}) - \frac{1}{2} g_2 (\mp 2 a_i) \right] \\ &
- \sum_{f = 1}^{N_f} \sum_{i = 1}^{N} ( \mathfrak{n}_i + \mathfrak{t}_f + \mathfrak{g}_2 - 1 ) \left[ \Li_2 (e^{\mathbbm{i} (a_i + \Delta_f)}) - \frac{1}{2} g_2 ( a_i + \Delta_f ) \right] \\ &
- \sum_{f = 1}^{N_f} \sum_{i = 1}^{N} ( - \mathfrak{n}_i + \tilde \mathfrak{t}_f + \mathfrak{g}_2 - 1 ) \left[ \Li_2 (e^{- \mathbbm{i} (a_i - \tilde \Delta_f)}) - \frac{1}{2} g_2 ( - a_i + \tilde \Delta_f ) \right] .
\end{aligned}
\ee
The $g_2$ terms are of $\mathcal{O} (N^{2})$ and thus subleading in the large $N$ limit. Hence,
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\label{Wt:cont:gauge:fund}
& \widetilde \mathcal{W}^{(0)} \approx \mathbbm{i} \frac{(8 - N_f)}{2} N^{1 + 3 \alpha} \int \mathrm{d} t \rho(t) \mathfrak{N} (t) t^2 \sign (t) \, .
\end{aligned}
\ee
Now, let us focus on the second line of \eqref{Wt:USp(2N):pert}. Consider the following terms
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\widetilde\mathcal{W}^{(1)}_{\text{hyper}} & = - \sum_{i > j}^{N} ( \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t}_m + \mathfrak{g}_2 - 1 ) \left[ \Li_2 (e^{\mathbbm{i} (a_i - a_j + \Delta_m)}) - \frac12 g_2 (a_i - a_j + \Delta_m) \right] \\ &
- \sum_{i > j}^{N} ( \mathfrak{n}_j - \mathfrak{n}_i + \mathfrak{t}_m + \mathfrak{g}_2 - 1 ) \left[ \Li_2 (e^{- \mathbbm{i} (a_i - a_j - \Delta_m)}) - \frac12 g_2 (a_j - a_i + \Delta_m) \right] .
\eea
At large $N$, we obtain
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\label{Wt:cont:as:I}
\widetilde\mathcal{W}^{(1)}_{\text{hyper}} & \approx - \frac{\mathbbm{i}}{4} N^{2 + 3 \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) ( t - t' )^2 \sign ( t - t' ) \\ &
+ \frac{\mathbbm{i}}{4} g_2 ( \Delta_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') \\ &
+ \frac{\mathbbm{i}}{8} ( \Delta_1 - \Delta_2 ) ( \mathfrak{t}_1 - \mathfrak{t}_2 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') | t - t' | \, ,
\end{aligned}
\ee
where we used the democratic chemical potentials and fluxes
\begin{equation}} \newcommand{\ee}{\end{equation}
\Delta_1 = \Delta_m \, , \qquad \Delta_2 = 2 \pi - \Delta_m \, , \qquad
\mathfrak{t}_1 = \mathfrak{t}_m \, , \qquad \mathfrak{t}_2 = 2 ( 1 - \mathfrak{g}_2 ) - \mathfrak{t}_m \, .
\ee
The similar contribution coming from the vector multiplet can be obtained by using
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{tW:hyper:map:gauge}
\widetilde \mathcal{W}^{\text{pert}}_{\text{vector}} = - \widetilde \mathcal{W}^{\text{pert}}_{\text{hyper}} |_{\mathfrak{t}_m = 2 ( 1 - \mathfrak{g}_2 ) , \, \Delta_m = 2 \pi} \, .
\ee
It reads
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\widetilde\mathcal{W}^{(1)}_{\text{vector}} & \approx \frac{\mathbbm{i}}{4} N^{2 + 3 \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) ( t - t' )^2 \sign ( t - t' ) \\ &
- \frac{\mathbbm{i} \pi^2}{6} N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') \\ &
- \frac{\mathbbm{i} \pi}{2} ( 1- \mathfrak{g}_2 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') | t - t' | \, .
\eea
We thus find that
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{Wt:cont:gauge:as:I}
\widetilde\mathcal{W}^{(1)}_{\text{vector}} + \widetilde\mathcal{W}^{(1)}_{\text{hyper}} & \approx - \frac{\mathbbm{i}}{4} \Delta_1 \Delta_2 N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') \\ &
- \frac{\mathbbm{i}}{4} ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') | t - t' | \, .
\eea
The next term we shall consider is given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\begin{aligned}
\widetilde\mathcal{W}^{(2)}_{\text{vector}} + \widetilde\mathcal{W}^{(2)}_{\text{hyper}} & = \sum_{i > j}^{N} ( \mathfrak{n}_i + \mathfrak{n}_j + 1 - \mathfrak{g}_2 ) \left[ \Li_2 (e^{\mathbbm{i} (a_i + a_j)}) - \frac12 g_2 ( - a_i - a_j ) \right] \\ &
+ \sum_{i > j}^{N} ( - \mathfrak{n}_i - \mathfrak{n}_j + 1 - \mathfrak{g}_2 ) \left[ \Li_2 (e^{- \mathbbm{i} (a_i + a_j)}) - \frac12 g_2 ( a_i + a_j ) \right] \\ &
- \sum_{i > j}^{N} ( \mathfrak{n}_i + \mathfrak{n}_j + \mathfrak{t}_m + \mathfrak{g}_2 - 1 ) \left[ \Li_2 (e^{\mathbbm{i} (a_i + a_j + \Delta_m)}) - \frac12 g_2 (a_i + a_j + \Delta_m) \right] \\ &
- \sum_{i > j}^{N} ( - \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t}_m + \mathfrak{g}_2 - 1 ) \left[ \Li_2 (e^{- \mathbbm{i} (a_i + a_j - \Delta_m)}) - \frac12 g_2 ( - a_i - a_j + \Delta_m) \right] .
\end{aligned}
\eea
In the large $N$ limit it can be approximated as
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{Wt:cont:gauge:as:II}
\widetilde\mathcal{W}^{(2)}_{\text{vector}} + \widetilde\mathcal{W}^{(2)}_{\text{hyper}} & \approx - \frac{\mathbbm{i}}{4} \Delta_1 \Delta_2 N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) + \mathfrak{N}(t') ) \\ &
- \frac{\mathbbm{i}}{4} ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( t + t' ) \, .
\eea
Putting \eqref{Wt:cont:gauge:fund}, \eqref{Wt:cont:gauge:as:I}, and \eqref{Wt:cont:gauge:as:II} together we obtain the final expression for the effective twisted superpotential at large $N$:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{LargeN:Usp(2N):Wt}
\widetilde \mathcal{W} \left[ \rho(t) , \mathfrak{N}(t) , \Delta_m , \mathfrak{t}_m \right] & = \widetilde \mathcal{W}^{(0)} + \sum_{\vartheta = 1}^{2} \left( \widetilde \mathcal{W}^{(\vartheta)}_{\text{vector}} + \widetilde \mathcal{W}^{(\vartheta)}_{\text{hyper}} \right) \\ &
\approx \mathbbm{i} \frac{(8 - N_f)}{2} N^{1 + 3 \alpha} \int_{0}^{t_*} \mathrm{d} t \rho(t) \mathfrak{N} (t) t^2
- \mathbbm{i} \gamma \left( \int_{0}^{t_*} \mathrm{d} t \rho (t)- 1 \right) \\ &
- \frac{\mathbbm{i}}{4} ( \Delta_1 \Delta_2 ) N^{2 + \alpha} \int_{0}^{t_*} \mathrm{d} t \rho(t) \int_{0}^{t_*} \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') \\ &
- \frac{\mathbbm{i}}{4} ( \Delta_1 \Delta_2 ) N^{2 + \alpha} \int_{0}^{t_*} \mathrm{d} t \rho(t) \int_{0}^{t_*} \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) + \mathfrak{N}(t') ) \\ &
- \frac{\mathbbm{i}}{4} ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) N^{2 + \alpha} \int_{0}^{t_*} \mathrm{d} t \rho(t) \int_{0}^{t_*} \mathrm{d} t' \rho(t') \left[ | t - t' | + ( t + t' ) \right] ,
\eea
where we added the Lagrange multiplier $\gamma$ for the normalization of $\rho(t)$.
In order to have a non-trivial saddle point we need to set $1 + 3 \alpha = 2 + \alpha$, implying that $\alpha = 1 / 2$.
The twisted superpotential thus scales as $N^{5 / 2}$.
Setting to zero the variation with respect to $\rho(t)$, we get the equation
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\frac{\gamma}{N^{5/2}} & = \frac{(8 - N_f)}{2} \mathfrak{N} (t') t'^2
- \frac12 ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) \int_{0}^{t_*} \rho(t) \left[ | t + t' | + ( t + t' ) \right] \\ &
- \frac12 ( \Delta_1 \Delta_2 ) \int_{0}^{t_*} \left[ ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign ( t - t' ) + ( \mathfrak{N} (t) + \mathfrak{N}(t') ) \right] \, .
\eea
On the support of $\rho (t)$ the solution is given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{LargeN:USp(2N):sol:Wt}
& \rho(t) = \frac{2 | t |}{t_*^2} \, , \qquad t_* = \frac{(2 \Delta_1 \Delta_2)^{1/2}}{\sqrt{8 - N_{f}}} \, , \qquad
\mathfrak{N} (t) = \frac12 \left( \frac{\mathfrak{t}_1}{\Delta_1} + \frac{\mathfrak{t}_2}{\Delta_2} \right) t \, , \\ &
\gamma = - \frac{N^{5/2}}{\sqrt{8 - N_{f}}} ( 2 \Delta_1 \Delta_2 )^{1/2} ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) \, .
\eea
Evaluating \eqref{LargeN:Usp(2N):Wt} at \eqref{LargeN:USp(2N):sol:Wt} yields%
\footnote{$\widetilde\mathcal{W} (\Delta_m , \mathfrak{t}_m) = \frac{2 i}{5} \gamma (\Delta_m , \mathfrak{t}_m)$ due to a virial theorem for the large $N$ twisted superpotential \eqref{LargeN:Usp(2N):Wt}.}
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{LargeN:USp(2N):superpotential}
\widetilde \mathcal{W} (\Delta_m , \mathfrak{t}_m) \approx - \frac{2^{3/2} \mathbbm{i} N^{5/2}}{5 \sqrt{8 - N_{f}}} ( \Delta_1 \Delta_2 )^{1/2} ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) \, .
\ee
This can be more elegantly rewritten as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{USp(2N):index theorem:tildeW:F}
\widetilde \mathcal{W} (\Delta_m , \mathfrak{t}_m) \approx - 2 \pi \sum_{\varsigma = 1}^{2} \mathfrak{t}_\varsigma \frac{\partial \mathcal{F}(\Delta)}{\partial \Delta_\varsigma} \, ,
\ee
where $\mathcal{F}(\Delta)$ is given in \eqref{USp(2N):F:on-shell}.
\paragraph*{$\widetilde \mathcal{W}$ and the free energy on $\Sigma_{\mathfrak{g}_2} \times S^3$.}
Remarkably, we find the following relation between the twisted superpotential \eqref{USp(2N):index theorem:tildeW:F} and the $S^3$ free energy
of the 3D $\mathcal{N}=2$ theory that is obtained by compactifying the 5D theory on $\Sigma_{\mathfrak{g}_2}$ with fluxes $\mathfrak{t}_\varsigma$:
\begin{equation}} \newcommand{\ee}{\end{equation}\label{relationUSP}
\widetilde \mathcal{W} (\bar\Delta_m , \mathfrak{t}) \approx \frac{\mathbbm{i} \pi}{2} \, F_{\Sigma_{\mathfrak{g}_2} \times S^3} (\mathfrak{t}) \, ,
\ee
where $\widetilde \mathcal{W} (\bar\Delta_m , \mathfrak{t})$ is evaluated at its extremum
\begin{equation}} \newcommand{\ee}{\end{equation}
\frac{\bar\Delta_m}{\pi} = \frac{5 \mathfrak{t}_1 - 3 \mathfrak{t}_2 \pm \sqrt{9 \mathfrak{t}_1^2 - 14 \mathfrak{t}_1 \mathfrak{t}_2 + 9 \mathfrak{t}_2^2}}{4 (\mathfrak{t}_1 - \mathfrak{t}_2)} \, .
\ee
Indeed, the $S^3$ free energy was very recently computed holographically in \cite{Bah:2018lyv} and it reads
\begin{equation}} \newcommand{\ee}{\end{equation}\label{FS3}
F_{\Sigma_{\mathfrak{g}_2} \times S^3} (\mathfrak{t}) \approx \frac{16 \pi}{5} \frac{ (1-\mathfrak{g}_2) N^{5/2}}{\kappa \sqrt{8 - N_f}} \frac{(z^2-\kappa^2)^{3/2} \left ( \sqrt{\kappa^2 + 8 z^2} -\kappa\right)}{\left ( 4 z^2 -\kappa^2
+\kappa \sqrt{\kappa^2 + 8 z^2}\right )^{3/2}}\, ,
\ee
where $\kappa =1$ for $\mathfrak{g}_2=0$ and $\kappa = -1$ for $\mathfrak{g}_2>1$ and the variable $z$ parameterizes the fluxes $\mathfrak{t}_\varsigma$.
In the special case of the torus, $\mathfrak{g}_2=1$, the above expression should be replaced by
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{FS32}
F_{\Sigma_{\mathfrak{g}_2} \times S^3}(\mathfrak{t}) \approx \frac{4 \sqrt{2} \pi}{5} \frac{N^{5/2}}{ \sqrt{8 - N_f}} | z | \, .
\ee
It is now a simple exercise to check that \eqref{relationUSP} holds identically for all $\mathfrak{g}_2$ upon identifying
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathfrak{t}_1 = ( 1 - \mathfrak{g}_2 ) \left( 1 + \frac{z}{\kappa} \right) \, , \qquad \mathfrak{t}_2 = ( 1 - \mathfrak{g}_2 ) \left( 1- \frac{z}{\kappa} \right) \, .
\ee
We expect that \eqref{relationUSP} holds also off-shell
\begin{equation}} \newcommand{\ee}{\end{equation}\label{relationUSP2}
\widetilde \mathcal{W} (\Delta , \mathfrak{t}) \approx \frac{\mathbbm{i} \pi}{2}\, F_{\Sigma_{\mathfrak{g}_2} \times S^3} (\Delta, \mathfrak{t}) \, ,
\ee
where $F_{\Sigma_{\mathfrak{g}_2} \times S^3} (\Delta, \mathfrak{t})$ is the free energy as a function of a trial R-symmetry.
\eqref{relationUSP} would correspond then to the statement that the $S^3$ free energy of the 3D theory is obtained by extremizing $F_{\Sigma_{\mathfrak{g}_2} \times S^3} (\Delta, \mathfrak{t})$ with respect to $\Delta$ \cite{Jafferis:2010un}.\footnote{The free energy on $\Sigma_{\mathfrak{g}_2} \times S^3$ as a function of $\Delta$ was explicitly computed in field theory in \cite{Crichigno:2018adf} after the completion of this work. The result in \cite{Crichigno:2018adf} perfectly agrees with \eqref{relationUSP2}.}
\subsubsection{Partition function at large $N$}
\label{subsubsec:logZ:USp(2N)}
The topologically twisted index of the $\mathrm{USp}(2N)$ theory with matter on $\Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1 )$ reads
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{USp(2N):index:Sigmag2xS1xSigmag1}
Z^{\text{pert}} (y , \mathfrak{s} , \mathfrak{t}) & = \frac{(-1)^{N}}{2^N N!}
\sum_{\mathfrak{n} \in \Gamma_\mathfrak{h}} \sum_{a = a_{(i)}} \bigg( \det\limits_{i j} \frac{\partial^2 \widetilde \mathcal{W} (a , \mathfrak{n})}{\partial a_i \partial a_j} \bigg)^{\mathfrak{g}_1 - 1} \\ &
\times \bigg( \frac{y_m^{1/2}}{1 - y_m} \bigg)^{(N-1) ( \mathfrak{s}_m + \mathfrak{g}_1 - 1 ) ( \mathfrak{t}_m + \mathfrak{g}_2 - 1 )}
\prod_{g = \pm 1} \prod_{i = 1}^{N} \bigg( \frac{1 - x_i^{2 g}}{x_i^{g}} \bigg)^{( 1 - \mathfrak{g}_1 ) ( 2 g \mathfrak{n}_i + 1 - \mathfrak{g}_2 )} \\ &
\times \prod_{i = 1}^{N} \prod_{f = 1}^{N_f} \bigg( \frac{x_i^{1 / 2} y_f^{1 / 2}}{1 - x_i y_f} \bigg)^{( \mathfrak{s}_f + \mathfrak{g}_1 - 1 ) ( \mathfrak{n}_i + \mathfrak{t}_f + \mathfrak{g}_2 - 1)}
\bigg( \frac{x_i^{- 1 / 2} \tilde y_f^{1 / 2}}{1 - x_i^{-1} \tilde y_f} \bigg)^{( \tilde \mathfrak{s}_f + \mathfrak{g}_1 - 1 ) ( - \mathfrak{n}_i + \tilde \mathfrak{t}_f + \mathfrak{g}_2 - 1)} \\ &
\times \prod_{g = \pm 1} \prod_{i > j}^{N} \bigg( \frac{1 - (x_i x_j )^{g}}{( x_i x_j )^{g / 2}} \bigg)^{( 1 - \mathfrak{g}_1 ) ( g \mathfrak{n}_i + g \mathfrak{n}_j + 1 - \mathfrak{g}_2 )}
\bigg( \frac{1 - (x_i / x_j )^{g}}{( x_i / x_j )^{g / 2}} \bigg)^{( 1 - \mathfrak{g}_1 ) ( g \mathfrak{n}_i - g \mathfrak{n}_j + 1 - \mathfrak{g}_2 )} \\ &
\times \prod_{g = \pm 1} \prod_{i > j}^{N} \bigg( \frac{( x_i x_j )^{g / 2} y_m^{1/2}}{1 - (x_i x_j )^{g} y_m} \bigg)^{( \mathfrak{s}_m + \mathfrak{g}_1 - 1 ) ( g \mathfrak{n}_i + g \mathfrak{n}_j + \mathfrak{t}_m + \mathfrak{g}_2 - 1 )} \\ &
\times \prod_{g = \pm 1} \prod_{i > j }^{N} \bigg( \frac{( x_i / x_j )^{g / 2} y_m^{1/2}}{1 - (x_i / x_j )^{g} y_m} \bigg)^{( \mathfrak{s}_m + \mathfrak{g}_1 - 1 ) ( g \mathfrak{n}_i - g \mathfrak{n}_j + \mathfrak{t}_m + \mathfrak{g}_2 - 1 )}
\, .
\eea
The products $\prod_{i = 1}^{N}$ are of $\mathcal{O} (N^{2})$ and thus subleading in the large $N$ limit. Then we consider the last line in \eqref{USp(2N):index:Sigmag2xS1xSigmag1}:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\log Z^{(1)}_{\text{hyper}} & = ( \mathfrak{s}_m + \mathfrak{g}_1 - 1 ) \sum_{i >j }^{N} ( \mathfrak{n}_i - \mathfrak{n}_j + \mathfrak{t}_m + \mathfrak{g}_2 - 1 ) \left[ \Li_1 (e^{\mathbbm{i} (a_i - a_j + \Delta_m)}) + \frac{\mathbbm{i}}{2} g_1 (a_i - a_j + \Delta_m) \right] \\ &
+ ( \mathfrak{s}_m + \mathfrak{g}_1 - 1 ) \sum_{i > j}^{N} ( \mathfrak{n}_j - \mathfrak{n}_i + \mathfrak{t}_m + \mathfrak{g}_2 - 1 ) \left[ \Li_1 (e^{- \mathbbm{i} (a_i - a_j - \Delta_m)}) + \frac{\mathbbm{i}}{2} g_1 (a_j - a_i + \Delta_m) \right] .
\eea
At large $N$, it can be approximated as
\begin{equation}} \newcommand{\ee}{\end{equation}
\begin{aligned}
\label{logZ:cont:as:I}
\log Z^{(1)}_{\text{hyper}} & \approx - \frac{1}{8} ( \Delta_1 - \Delta_2 ) ( \mathfrak{s}_1 - \mathfrak{s}_2 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') \\ &
- \frac{1}{8} ( \mathfrak{t}_1 - \mathfrak{t}_2 ) ( \mathfrak{s}_1 - \mathfrak{s}_2 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') | t - t' | \, ,
\end{aligned}
\ee
where we introduced the democratic fluxes
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathfrak{s}_1 = \mathfrak{s}_m \, , \qquad \mathfrak{s}_2 = 2 ( 1 - \mathfrak{g}_1 ) - \mathfrak{s}_m \, .
\ee
The similar contribution coming from the vector multiplet can be obtained by using
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{logZ:hyper:map:gauge}
\log Z^{\text{pert}}_{\text{vector}} = - \log Z^{\text{pert}}_{\text{hyper}} |_{\mathfrak{s}_m = 2 ( 1 - \mathfrak{g}_1 ) , \, \mathfrak{t}_m = 2 ( 1 - \mathfrak{g}_2 ) , \, \Delta_m = 2 \pi} \, .
\ee
It reads
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\log Z^{(1)}_{\text{vector}} & \approx \frac{\pi}{2} ( 1- \mathfrak{g}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') \\ &
+ \frac12( 1- \mathfrak{g}_2 ) ( 1- \mathfrak{g}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') | t - t' | \, .
\eea
We thus find that
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{logZ:cont:gauge:as:I}
\log Z^{(1)}_{\text{vector}} + \log Z^{(1)}_{\text{hyper}} & \approx \frac14 ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') \\ &
+ \frac14 ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') | t - t' | \, .
\eea
The terms coming from the roots $e_1 + e_2$ and $- e_1 - e_2$ are treated as in \eqref{logZ:cont:as:I}. They can be approximated at large $N$ as
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{logZ:cont:gauge:as:II}
\log Z^{(2)}_{\text{vector}} + \log Z^{(2)}_{\text{hyper}} & \approx \frac14 ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( \mathfrak{N} (t) + \mathfrak{N}(t') ) \\ &
+ \frac14 ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) N^{2 + \alpha} \int \mathrm{d} t \rho(t) \int \mathrm{d} t' \rho(t') ( t + t' ) \, .
\eea
Putting everything together we obtain the following functional for the logarithm of the partition function at large $N$:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{LargeN:Usp(2N):logZ}
\frac{\log Z}{N^{5/2}} & \approx \frac14 ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) \int_{0}^{t_*} \mathrm{d} t \rho(t) \int_{0}^{t_*} \mathrm{d} t' \rho(t') \left[ ( \mathfrak{N} (t) - \mathfrak{N}(t') ) \sign (t - t') + ( \mathfrak{N} (t) + \mathfrak{N}(t') ) \right] \\ &
+ \frac14 ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) \int_{0}^{t_*} \mathrm{d} t \rho(t) \int_{0}^{t_*} \mathrm{d} t' \rho(t') \left[ | t - t' | + ( t + t' ) \right] .
\eea
Finally we take the solution to the BAEs \eqref{LargeN:USp(2N):sol:Wt}, plug it back into \eqref{LargeN:Usp(2N):logZ} and compute the integral. We obtain
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\log Z ( \Delta_m , \mathfrak{t}_m , \mathfrak{s}_m ) & \approx \frac{\sqrt{2} N^{5/2}}{5 \sqrt{8 - N_f}}
\left[ \frac{( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 )}{( \Delta_1 \Delta_2)^{1/2}}
+ 2 ( \Delta_1 \Delta_2 )^{1/2} ( \mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1 ) \right] \\ &
= \frac{\sqrt{2} N^{5/2}}{5 \sqrt{8 - N_f}} \frac{\Delta_1 \mathfrak{s}_2 ( \Delta_1 \mathfrak{t}_2 + 3 \Delta_2 \mathfrak{t}_1 ) + \Delta_2 \mathfrak{s}_1 ( 3 \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 )}{( \Delta_1 \Delta_2 )^{1/2}} .
\eea
Remarkably, this can be rewritten as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{USp(2N):Final:logZ:index theorem}
\log Z ( \Delta_m , \mathfrak{t}_m , \mathfrak{s}_m ) \approx \mathbbm{i} \sum_{\varsigma = 1}^2 \mathfrak{s}_\varsigma \frac{\partial \widetilde \mathcal{W} (\mathfrak{t} , \Delta)}{\partial \Delta_\varsigma}
= - 2 \mathbbm{i} \pi \sum_{\varsigma , \varrho = 1}^2 \mathfrak{s}_\varsigma \mathfrak{t}_\varrho \frac{\partial^2 \mathcal{F} (\Delta)}{\partial \Delta_\varsigma \partial \Delta_\varrho} \, ,
\ee
where $ \widetilde \mathcal{W} (\mathfrak{t} , \Delta)$ and $\mathcal{F} (\Delta)$ are given in \eqref{LargeN:USp(2N):superpotential} and \eqref{USp(2N):F:on-shell}, respectively.
In analogy with \cite{Benini:2015eyy,Benini:2016rke}, we expect that the extremization of the topologically twisted index \eqref{USp(2N):Final:logZ:index theorem} reproduces the entropy of asymptotically AdS$_6$ black holes
in massive type IIA supergravity with magnetic fluxes $\mathfrak{t}_m$ and $\mathfrak{s}_m$, and horizon topology AdS$_2 \times \Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$.
Unfortunately, such black holes are still to be found. The only known example is the black hole dual to the universal twist, \textit{i.e.} \,$\mathfrak{t}_m= 1- \mathfrak{g}_2$ and $\mathfrak{s}_m=1 -\mathfrak{g}_1$,
whose entropy was computed in \cite{Bobev:2017uzs}, using gauged supergravity in six dimensions and elaborating on the results in \cite{Naka:2002jz}. The result for the entropy \cite[(4.36)]{Bobev:2017uzs} has been recently corrected by a factor of two in \cite{Suh:2018tul}. Our index \eqref{USp(2N):Final:logZ:index theorem} for the appropriate values of the chemical potentials for the universal twist, \textit{i.e.}\;$\Delta_1=\Delta_2=\pi$, predicts
\begin{equation}} \newcommand{\ee}{\end{equation} S_{\text{BH}} \approx \frac{8 \sqrt{2} \pi}{5}(1-\mathfrak{g}_1)(1-\mathfrak{g}_2) \frac{N^{5/2}}{\sqrt{8-N_f}} \approx -\frac{8}{9} (1-\mathfrak{g}_1)(1-\mathfrak{g}_2) F_{S^5} \, ,\ee
and correctly matches the result in \cite{Suh:2018tul}.%
\footnote{We thank P. Marcos Crichigno, Dharmesh Jain and Brian Willett for pointing out a numerical mistake in our computation in the first version of this paper.}
\section[4D black holes from AdS\texorpdfstring{$_7$}{(7)} black strings]{4D black holes from AdS$_7$ black strings}
\label{sec:4D domain-walls}
Our primary interest in this section is to understand the $\mathcal{I}$-extremization principle \eqref{SBH} for the topologically twisted index of the 6D $\mathcal{N} = (2,0)$ theory in terms of holography and, in particular, in terms of the attractor mechanism \cite{Ferrara:1995ih,Ferrara:1996dd} in ${\cal N}=2$ supergravity.
We shall consider the supergravity dual of two-dimensional $\mathcal{N} = (0 , 2)$ SCFTs obtained by compactifying a stack of M5-branes on $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$.
These solutions were constructed in \cite{Benini:2013cda} and can be viewed as black strings in seven dimensions, interpolating between the maximally supersymmetric AdS$_7$ vacuum at infinity
and the near-horizon AdS$_3 \times \Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$ geometry. This leads to a natural holographic interpretation of these black strings as RG flows across dimensions
--- we have a flow from the 6D $\mathcal{N} = (2,0)$ theory in the UV to a 2D $\mathcal{N} = (0,2)$ SCFT in the IR. The BPS black strings in AdS$_7$, {\it \'a la} Maldacena-Nu\~nez \cite{Maldacena:2000mw},
preserve supersymmetry due to the topological twist on the internal space $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$.
Let us briefly review these solutions. We work with a $\mathrm{U}(1)^2$ consistent truncation \cite{Liu:1999ai}
of the $\mathrm{SO}(5)$ maximal $(\mathcal{N} = 4)$ gauged supergravity in seven dimensions \cite{Pernici:1984xx}, obtained by reducing the eleven-dimensional supergravity on $S^4$ \cite{Nastase:1999cb,Nastase:1999kf}.
It contains the metric, a three-form gauge potential $S_5$, two Abelian gauge fields $A^I$ in the Cartan of $\mathrm{SO}(5)$ and two real scalars $\lambda_I$ $(I = 1 , 2)$.
The solution can then be written as%
\footnote{For notational convenience we use $\Sigma_{\sigma} \equiv \Sigma_{\mathfrak{g}_\sigma}$. The relations between the magnetic fluxes $\mathfrak{s}_I$, $\mathfrak{t}_I$ in \eqref{BB7d}
and $a_\sigma$, $b_\sigma$ in \cite[(5.26)]{Benini:2013cda} are the following: $a_1 = - \mathfrak{s}_1 / \eta_1$, $b_1 = - \mathfrak{s}_2 / \eta_1$, $a_2 = - \mathfrak{t}_1 / \eta_2 $, $b_2 = - \mathfrak{t}_2 / \eta_2 $ .}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& \mathrm{d} s_7^2 = e^{2 f(r)} (- \mathrm{d} t^2 + \mathrm{d} z^2 + \mathrm{d} r^2) + \sum_{\sigma=1}^2 e^{2 g_\sigma (r)} \mathrm{d} s^2_{\Sigma_\sigma} \, , \\
& S_5 = - \frac{1}{32 \sqrt{3} \eta_1 \eta_2} (\mathfrak{t}_1 \mathfrak{s}_2 + \mathfrak{t}_2 \mathfrak{s}_1) e^{- 4 (\lambda_1 + \lambda_2) - 2 ( g_1 + g_2 )} \mathrm{d} t \wedge \mathrm{d} z \wedge \mathrm{d} r \\
& F^I = \frac{\mathfrak{s}^I }{4 \eta_1} \text{vol} \left( \Sigma_1 \right) + \frac{\mathfrak{t}^I }{4 \eta_2} \text{vol} \left( \Sigma_2 \right) \, , \qquad
\lambda_I = \lambda_I (r) \, ,
\label{BB7d}
\eea
where $\mathrm{d} s^2_{\Sigma} = \mathrm{d} \theta^2 + f_{\kappa}^2 (\theta) \mathrm{d} \varphi^2$ defines the metric on a surface $\Sigma$ of constant scalar curvature $2\kappa$, with $\kappa = \pm 1$, and
\begin{equation}} \newcommand{\ee}{\end{equation}
f_\kappa(\theta) = \frac{1}{\sqrt{\kappa}} \sin(\sqrt{\kappa}\theta) =
\left\{\begin{array}{l@{\quad}l} \sin\theta\, & \kappa=+1\,, \\
\sinh\theta\, & \kappa=-1\,. \end{array}\right.
\ee
The volume of $\Sigma_\sigma$ is given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\text{vol}\left (\Sigma_\sigma \right )= \int f_\kappa(\theta) \mathrm{d} \theta \wedge \mathrm{d} \phi = 2 \pi \eta_\sigma \, , \qquad
\eta_\sigma =
\left\{\begin{array}{l@{\quad}l} 2 | \mathfrak{g}_\sigma - 1 | \, & \mathfrak{g} \neq 0 \, , \\
1 \, & \mathfrak{g} = 0 \, . \end{array} \right.
\ee
Moreover, $F^I = \mathrm{d} A^I$ and $f(r)$, $g_\sigma (r)$, $\lambda_I(r)$ are functions of the radial coordinate only.\
In the AdS$_3$ region the scalars are fixed in terms of the magnetic charges:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& e^{10\lambda_1} = \frac{(\mathfrak{s}_1 \mathfrak{t}_2^2 + \mathfrak{t}_1^2 \mathfrak{s}_2)(\mathfrak{s}_1^2 \mathfrak{t}_2 +\mathfrak{t}_1 \mathfrak{s}_2^2)(\mathfrak{s}_1 \mathfrak{t}_2 + \mathfrak{t}_1 \mathfrak{s}_2 -\mathfrak{s}_2 \mathfrak{t}_2)^2}{(\mathfrak{s}_1^2 \mathfrak{t}_2^2 +\mathfrak{t}_1^2 \mathfrak{s}_2^2 +\mathfrak{s}_1\mathfrak{s}_2 \mathfrak{t}_1 \mathfrak{t}_2)(\mathfrak{s}_1 \mathfrak{t}_2 + \mathfrak{t}_1 \mathfrak{s}_2 -\mathfrak{s}_1 \mathfrak{t}_1)^3} \, , \\
& e^{10\lambda_2} = \frac{(\mathfrak{s}_1 \mathfrak{t}_2^2 + \mathfrak{t}_1^2 \mathfrak{s}_2)(\mathfrak{s}_1^2 \mathfrak{t}_2 +\mathfrak{t}_1 \mathfrak{s}_2^2)(\mathfrak{s}_1 \mathfrak{t}_2 + \mathfrak{t}_1 \mathfrak{s}_2 -\mathfrak{s}_1 \mathfrak{t}_1)^2}{(\mathfrak{s}_1^2 \mathfrak{t}_2^2 +\mathfrak{t}_1^2 \mathfrak{s}_2^2 +\mathfrak{s}_1\mathfrak{s}_2 \mathfrak{t}_1 \mathfrak{t}_2)(\mathfrak{s}_1 \mathfrak{t}_2 + \mathfrak{t}_1 \mathfrak{s}_2 -\mathfrak{s}_2 \mathfrak{t}_2)^3} \, ,
\eea
and the warp factors read
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
& e^{2 g_1} = - \frac{1}{4 \eta_1}e^{-4\lambda_1-4\lambda_2}(\mathfrak{s}_1 e^{2\lambda_1} +\mathfrak{s}_2 e^{2\lambda_2}) \equiv R^2_{\Sigma_1}\, , \\ &
e^{2 g_2} = - \frac{1}{4 \eta_2} e^{-4\lambda_1-4\lambda_2}(\mathfrak{t}_1 e^{2\lambda_1} +\mathfrak{t}_2 e^{2\lambda_2}) \equiv R^2_{\Sigma_2}\, , \\
& e^f= \frac{ \mathfrak{s}_2 \mathfrak{t}_2 -\mathfrak{s}_1 \mathfrak{t}_2 -\mathfrak{t}_1 \mathfrak{s}_2}{\mathfrak{s}_1 \mathfrak{t}_1 +\mathfrak{s}_2 \mathfrak{t}_2 -2 \mathfrak{s}_1 \mathfrak{t}_2 -2 \mathfrak{t}_1 \mathfrak{s}_2} \frac{e^{2\lambda_2}}{r} \equiv \frac{R_{\text{AdS}_3}}{r} \, .
\eea
As shown in \cite{Benini:2013cda} the holographic central charge $c_{\text{sugra}} (\mathfrak{s} , \mathfrak{t})$ can be computed, {\it \'a la} Brown-Henneaux \cite{Brown1986}, as
\begin{equation}} \newcommand{\ee}{\end{equation}
c_{\text{sugra}} (\mathfrak{s} , \mathfrak{t}) = \frac{8 N^3}{\pi^2} \text{vol} \left(\Sigma_1 \times \Sigma_2 \right ) R_{\text{AdS}_3}R^2_{\Sigma_1}R^2_{\Sigma_2} \, ,
\ee
and it matches the CFT result \eqref{exact:cr}.
If we now add a momentum $n$ along the circle inside AdS$_3$ and do a compactification along this circle, we obtain a static black hole in six-dimensional gauged supergravity.
By a standard argument, the entropy of such black hole is given by the number of states of the CFT with momentum $n$, and is therefore given by the Cardy formula \eqref{Cardys3}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}\label{Cardys32}
S_{\text{BH}} (\mathfrak{s} , \mathfrak{t} , n) = \mathcal{I}_{\text{SCFT}}\big|_{\text{crit}} (\mathfrak{s} , \mathfrak{t} , n) = 2 \pi \sqrt{ \frac{ n\, c_{\text{CFT}}(\mathfrak{s} , \mathfrak{t})}{6}} \, .
\eea
As discussed in section \ref{I-extremization principle}, the entropy $S_{\text{BH}} (\mathfrak{s} , \mathfrak{t} , n)$ is the result of extremizing the functional $\mathcal{I}_{\text{SCFT}} (\tilde \beta , \Delta)$.
Both for dyonic BPS black holes in AdS$_4$ \cite{Benini:2015eyy,Benini:2016rke,Hosseini:2017fjo,Benini:2017oxt} and BPS black strings in AdS$_5$ \cite{Hosseini:2016cyf,Hosseini:2018qsx,Hristov:2018lod} the $\mathcal{I}$-extremization principle has been identified with the attractor mechanism in 4D $\mathcal{N} = 2$ gauged supergravity \cite{Cacciatori:2009iz,DallAgata:2010ejj}.
We thus expect that the $\mathcal{I}$-extremization principle \eqref{SBH} corresponds to the attractor mechanism in six-dimensional gauged supergravity.
Unfortunately, not much is known about such mechanism in six dimensions. Therefore, our strategy is to first reduce the seven-dimensional gauged supergravity on $\Sigma_{\mathfrak{g}_2}$ down to five dimensions
and then do a further reduction on the circle inside AdS$_3$ to four-dimensional $\mathcal{N} = 2$ gauged supergravity, where the attractor mechanism for static BPS black holes is well-understood. An analogous argument has been used in \cite{Amariti:2016mnz} to explain the extremization of the R-symmetry in the two-dimensional CFT.
\subsection{Attractor mechanism}
\label{subsec:attractor}
In 4D ${\cal N}=2$ gauged supergravity%
\footnote{We refer to \cite{Hristov:2014eza} and the appendices of \cite{Hosseini:2017mds} for notations and more details about gauged supergravity in five and four dimensions.}
with $n_{\text{V}}$ vector multiplets $(\Lambda = 0 , 1 , \ldots , n_{\text{V}})$ the Bekenstein-Hawking entropy of a static BPS black hole with horizon topology $\Sigma_\sigma$, with a charge vector $\mathcal{Q} = (p^\Lambda,q_\Lambda)$, can be be obtained by extremizing \cite{DallAgata:2010ejj}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{attractor:mechanism:4D}
\mathcal{I}_{\text{sugra}} (X^\Lambda) = \frac{2 \pi \mathbbm{i} \eta_\sigma}{4 G_{\text{N}}^{(4)}} \frac{ q_\Lambda X^\Lambda - p^\Lambda F_\Lambda}{g_\Lambda X^\Lambda - g^\Lambda F_\Lambda} \, ,
\eea
with respect to the symplectic sections $X^\Lambda$ such that its value at the critical point is \emph{real}. Here,
\begin{equation}} \newcommand{\ee}{\end{equation}
F_\Lambda \equiv \frac{\partial \mathcal{F}_{\text{sugra}} (X^\Lambda)}{\partial X^\Lambda} \, ,
\ee
with ${\cal F}_{\text{sugra}} (X^\Lambda)$ being the prepotential, $\mathcal{G} = (g^\Lambda , g_\Lambda)$ is
the vector of magnetic $g^\Lambda$ and electric $g_\Lambda$ Fayet-Iliopoulos (FI) parameters, and $G_{\text{N}}^{(4)}$ is the four-dimensional Newton's constant.%
\footnote{The magnetic and electric charges are defined as $\int_{\Sigma_\sigma} F^\Lambda = \text{vol}(\Sigma_\sigma) p^\Lambda$ and $\int_{\Sigma_\sigma} G_\Lambda = \text{vol}(\Sigma_\sigma) q_\Lambda$
where $G_\Lambda = 8 \pi G_\text{N}^{(4)} \delta (\mathscr{L} \mathrm{d} \text{vol}_4)/\delta F^\Lambda$. In a frame with purely electric gauging $g_\Lambda$,
the charges are quantized as $\eta_\sigma g_\Lambda p^\Lambda \in \mathbb{Z}$ and $\eta_\sigma q_\Lambda / ( 4G_\text{N}^{(4)} g_\Lambda) \in \mathbb{Z}$, not summed over $\Lambda$.}
In general, in gauged supergravity, $\mathcal{F}_{\text{sugra}} (X^\Lambda)$ is a homogeneous function of degree two, so we can equivalently define $\hat X^\Lambda \equiv X^\Lambda/(g_\Lambda X^\Lambda - g^\Lambda F_\Lambda)$ and extremize
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{attractor}
\mathcal{I}_{\text{sugra}} (\hat X^\Lambda) = - \frac{2 \pi \mathbbm{i} \eta_\sigma}{4 G_{\text{N}}^{(4)}} \left ( p^\Lambda F_\Lambda (\hat X^\Lambda) - q_\Lambda \hat X^\Lambda \right ) \, .
\eea
The critical value of \eqref{attractor} $\bar X^\Lambda$ determines the value of the physical scalars $z^i$ at the horizon,
and the entropy of the black hole is then given by evaluating the functional \eqref{attractor} at its extremum
\begin{equation}} \newcommand{\ee}{\end{equation}
S_{\text{BH}} (p^\Lambda , q_\Lambda) = \mathcal{I}_{\text{sugra}} (\bar X^\Lambda) \, .
\ee
This is the so-called \emph{attractor mechanism} \cite{Ferrara:1995ih,Ferrara:1996dd}, stating that the area of the black hole horizon is given in terms of the conserved charges and is independent of the asymptotic moduli.
In a general gauged supergravity with $n_{\text{H}}$ hypermultiplets, in addition to the gravitino and gaugino BPS equations, one needs to impose the BPS equations for the hyperino.
Altogether these equations become algebraic in the near-horizon limit and fix the horizon value of the scalars in the vector multiplets $z^I=X^I / X^0$ and the hyperscalars $q^u$, $u=1,\cdots , 4n_{\text{H}}$.
The full set of equations can be found in \cite{Klemm:2016wng}. In general, the hyperino equations at the horizon just yield a set of linear constraints on the sections $X^\Lambda$.
In simple models, this can be used to integrate out all massive fields at the horizon and write an effective theory with only massless vectors to which we can directly apply \eqref{attractor}.
This approach has been used in \cite{Hosseini:2017fjo,Benini:2017oxt} to reproduce the entropy of AdS$_4 \times S^6$ black holes in massive type IIA supergravity and, as we will see below, also works here.
\subsection{Localization meets holography}
\label{sec:localization meets holography}
In order to use the attractor mechanism \eqref{attractor}, we need to determine the matter content and the prepotential of the four-dimensional $\mathcal{N} = 2$ gauged supergravity.
This can be done in two steps, by first reducing on $\Sigma_{\mathfrak{g}_2}$, and then on a circle. Fortunately, both reductions have been worked out in the literature, respectively in \cite{Szepietowski:2012tb} and \cite{Hristov:2014eza},
and we can use the results reported there.
The consistent truncation of seven-dimensional $\mathcal{N} = 4$ $\mathrm{SO}(5)$ gauged supergravity reduced on a Riemann surface has been discussed in \cite{Szepietowski:2012tb}.
It contains two vector multiplets and a charged hypermultiplet.
The vector multiplet scalars in 5D $\mathcal{N} = 2$ supergravity are parameterized by a set of constrained \emph{real} scalars $L^I$ satisfying
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\frac{1}{6} c_{IJK} L^I L^J L^K = 1 \, ,
\eea
where the symmetric coefficients $c_{IJK}$ can be identified with the 't Hooft cubic anomaly coefficients \cite{Tachikawa:2005tq}, and for the model at hand $c_{123}=1$ is the only nonzero component.
The prepotentials and Killing vectors associated to the hypermultiplet can be found in \cite{Szepietowski:2012tb} but we will not need the explicit form of them here.
We can proceed further by adding a momentum $n$ along $S^1 \subset \text{AdS}_3$.
The near-horizon geometry of the 5D black string is then $\text{BTZ} \times \Sigma_{\mathfrak{g}_1}$, where the metric for the extremal BTZ reads \cite{Banados:1992wn}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{BTZ}
\mathrm{d} s_3^2= \frac14 \frac{ - \mathrm{d} t^2+ \mathrm{d} r^2}{r^2} + \rho \left[ \mathrm{d} z +\left( - \frac14 + \frac{1}{2 \rho r} \right) \mathrm{d} t \right]^2 \, .
\eea
Here, the parameter $\rho$ is related to the electric charge $n$.
This is \emph{locally} equivalent to AdS$_3$, since there exists locally only one constant curvature metric in three dimensions, and solves the same BPS equations; however, they are inequivalent globally.
Compactifying the 5D black string on the circle \cite{Hristov:2014eza} we obtain a static BPS black hole in four dimensions,
with magnetic charges $(\mathfrak{s}_1,\mathfrak{s}_2)$ and electric charge $n$.
It can be thought as a domain wall which interpolates between an AdS$_2 \times \Sigma_{\mathfrak{g}_1}$ near-horizon region and an asymptotic \emph{non-AdS$_4$} vacuum.
The 4D $\mathcal{N} = 2$ gauged supergravity is the STU model $(n_{\text{V}} = 3)$ coupled to a charged hypermultiplet \cite{Hristov:2014eza}. It has the prepotential
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
{\cal F}_{\text{sugra}} (X^{\Lambda}) = - \frac{1}{6} \frac{c_{IJK} X^I X^J X^K}{X^0} = - \frac{X^1 X^2 X^3}{X^0} \, .
\eea
The new vector multiplet corresponds to the isometry of the compactification circle and is associated with $\Lambda=0$. The physical scalars $z^I= X^I/X^0$ are now \emph{complex}.
Their imaginary part is proportional to the five-dimensional real scalars $L^I$ and the real part is given by the component of the gauge fields along the compactified direction.
In the near-horizon region, the hyperscalars have a nonzero expectation value and, consequently, one of the gauge fields\footnote{This vector is called $c_\mu$ in \cite{Szepietowski:2012tb}.} becomes massive. The only physical role of the hypermultiplet is indeed to Higgs one of the gauge fields leaving an effective theory with only two massless vector multiplets. We can write down the effective theory as follows.
The hyperino BPS equation can be obtained by reducing to four dimensions equation \cite[(A.22)]{Szepietowski:2012tb}. After setting the massive gauge field to zero and considering constant scalars at the horizon, we find that%
\footnote{We are using \cite[(23) and (36)]{Szepietowski:2012tb}. The relations between parameters are the following: $- 4 \eta_2 p_I = \mathfrak{t}_I$ and $L^I_{\text{here}} = X^{I}_{\text{there}}$.
Note that, in the black string of \cite{Benini:2013cda}, the 7D gauge coupling has been fixed as $m = 2$.}
\begin{equation}} \newcommand{\ee}{\end{equation}
4 \eta_2 X^3 + \mathfrak{t}_1 X^2 + \mathfrak{t}_2 X^1 =0 \, .
\ee
We see that the hyperino BPS equation only imposes a linear constraint among the sections, which can be used to write an effective prepotential
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{effective:prepotential}
\mathcal{F}_{\text{sugra}} (X^{\Lambda}) = \frac{1}{4 \eta_2} \frac{X^1 X^2 (\mathfrak{t}_1 X^2 + \mathfrak{t}_2 X^1)}{X^0}
= \frac{1}{8 \eta_2 X^0} \sum_{I = 1}^2 \mathfrak{t}_I \frac{\partial ( X^1 X^2)^2}{\partial X^I} \, ,
\eea
for the sections $X^0$, $X^1$, $X^2$ corresponding to the massless vectors at the horizon.
We can now use \eqref{attractor}. In our case, the vector of FI parameters reads \cite{Hristov:2014eza}
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{G} = ( 0 , 0 , 0 , 0 , g , g ) \, .
\ee
The charge vector of the 4D black hole is also given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{Q} = - \frac{1}{g \eta_1} \left( 0 , \mathfrak{s}^1 , \mathfrak{s}^2 , 4 g^2 G_{\text{N}}^{(4)} n , 0 , 0 \right) \, ,
\ee
so that \eqref{attractor} can be written as
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{attractor:effective:theory}
\mathcal{I}_{\text{sugra}} (\hat X^\Lambda) = \frac{\mathbbm{i} \pi}{16 g G_{\text{N}}^{(4)} \eta_2 \hat X^0}
\sum_{I , J = 1}^2 \mathfrak{s}_I \mathfrak{t}_J \frac{\partial ( \hat X^1 \hat X^2)^2}{\partial \hat X^I \partial \hat X^J} - 2 \mathbbm{i} \pi g n \hat X^0 \, .
\ee
We may fix the constant $g = 4$ in the following. Upon identifying
\begin{equation}} \newcommand{\ee}{\end{equation}
\hat X^I \equiv \frac{\beta \Delta_I}{2 \pi g} \, , \qquad \hat X^0 \equiv \frac{\mathbbm{i} \tilde \beta}{2 \pi g} \, ,
\ee
and using the relations between field theory and gravitational parameters\footnote{The reduction in \cite{Szepietowski:2012tb} and \cite{Hristov:2014eza} is done on a Riemann surface and a circle of volume $2\pi \eta_2$ and $2\pi$, respectively.}
at large $N$
\begin{equation}} \newcommand{\ee}{\end{equation}
N^3 = \frac{3 \pi^2}{16 G_{\text{N}}^{(7)}} \, , \qquad
G_{\text{N}}^{(4)} = \frac{G_{\text{N}}^{(7)}}{\text{vol}(\Sigma_2 \times S^1)} = \frac{G_{\text{N}}^{(7)}}{(2 \pi)^2 \eta_2} \, ,
\ee
we find that the attractor mechanism \eqref{attractor:effective:theory} elegantly matches the field theory result \eqref{SBH}. Explicitly, we can write
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{localization:meets:holography}
\mathcal{F}_{\text{sugra}} (\hat X^\Lambda) \propto \widetilde \mathcal{W} (\Delta , \tilde \beta) \, , \qquad \mathcal{I}_{\text{sugra}} (\hat X^\Lambda) = \mathcal{I}_{\text{SCFT}} (\Delta , \tilde \beta) \, ,
\ee
where $\widetilde \mathcal{W}$ is given in \eqref{SYM:tildeW:on-shell:n}.
We also see that the $\mathcal{I}$-extremization principle for the topologically twisted index of the 6D $\mathcal{N} = (2,0)$ theory
correctly leads to the microscopic counting for the entropy of the six-dimensional black holes obtained by compactifying on a circle the black string solutions of \cite{Benini:2013cda}.
\section{Discussion and future directions}
\label{sec:discussion and outlook}
In this paper we took the first few steps towards the derivation and evaluation of the topologically twisted index of five-dimensional $\mathcal{N}=1$ gauge theories.
There are countless aspects that we have just briefly addressed in this paper.
In particular, the structure of the index at finite $N$ needs to be studied in more details. It was argued in \cite{Bawane:2014uka,Bershtein:2015xfa,Bershtein:2016mxz} that the equivariant partition function is summed over a set of magnetic fluxes that satisfy complicated semi-stability conditions.
These conditions have been studied by mathematicians \cite{Kool2015} but they become increasingly complicated with $N$ and are almost intractable already for $N=3$ or $N=4$. A related problem is the choice of
integration contour. This is selected by supersymmetry, but it is not determined so simply in our approach. We expect, in analogy with three and four dimensions, that some sort of Jeffrey-Kirwan prescription \cite{JeffreyKirwan}
is at work. It would be very interesting if the correct determination of the contour allows to simplify the final expression for the matrix model and also the semi-stability constraints on the fluxes.
In the paper we conjectured that the Seiberg-Witten prepotential $\mathcal{F}(a)$ in five dimensions should play the role of the twisted superpotential in three and four dimensions and, in particular, its critical points should be relevant
for the evaluation of the topologically twisted index. In three and four dimensions, a large set of partition functions, not only the twisted indices,
can be written as a sum over Bethe vacua \cite{Closset:2016arn,Closset:2017zgf,Closset:2017bse,Closset:2018ghr}.
It would be interesting to see if $\mathcal{F}(a)$ plays, at least partially, a similar role in five dimensions.
For this reason, it would be interesting to evaluate explicitly the index for simple theories in five dimensions and compare the results for dual pairs \cite{Bergman:2013aca,Bergman:2015dpa}.
Also our computations at large $N$ is based on the assumption of the importance of the critical points of $\mathcal{F}(a)$.
The large $N$ results can be explicitly tested against holographic predictions.
It would be particularly interesting, from this point of view, to find a class of AdS$_6$ black holes depending non-trivially on a set of magnetic fluxes \cite{Hosseini:2018mIIA}. This would allow to test the results in section \ref{sec:largeN} and compare with possible alternatives (like, for example, the one discussed in appendix \ref{sec:alternative}).
In the large $N$ analysis we considered for simplicity two particular theories, $\mathcal{N} = 2$ SYM and the $\mathrm{USp}(2N)$ UV fixed point.
We expect that our formalism and our general results \eqref{index:theorem:F:5D}, \eqref{index:theorem:W:5D} and \eqref{index:theorem:logZ:5D} extend to other theories,
with no particular complications. We also considered just the case of $\mathbb{P}^1\times \mathbb{P}^1\times S^1$ (which can be trivially generalized to $\Sigma_{\mathfrak{g}_1}\times \Sigma_{\mathfrak{g}_2}\times S^1$).
The main reason is that, for a factorized manifold, we can perform a dimensional reduction to three dimensions and use the results for three-dimensional topologically twisted indices, which are well developed.
One main ingredient of the analysis was the twisted superpotential of such compactification, defined for each sector of the gauge magnetic flux on $\Sigma_{\mathfrak{g}_2}$.
However, let us notice that, even for a compactification on $\mathcal{M}_4\times S^1$, we can formally define a twisted superpotential. This can be defined through \eqref{WF},
by considering the theory on the equivariant background with $\epsilon_1=\hbar$ and $\epsilon_2=0$ and by gluing the corresponding Nekrasov's partition functions,
or just by analogy with \eqref{full:pert:twisted superpotential}. Once we have this twisted superpotential $\widetilde \mathcal{W}(a,\mathfrak{n})$, that depends on the Coulomb variables $a_i$ and gauge fluxes $\mathfrak{n}_i$, we can extremize both $\mathcal{F}(a)$ and $\widetilde \mathcal{W}(a,\mathfrak{n})$ with respect to $a_i$ and $\mathfrak{n}_i$. A generalization of \eqref{Sigma_g1xM_3:Z} would then give an expression for the topologically twisted index.
We do not know if this approach leads to the correct result, but we have reasons to expect that, in the large $N$ limit, the result for different $\mathcal{M}_4$ should be very similar.
In particular, one can easily see from appendix \ref{sec:centralcharges(2,0)} that the two-dimensional trial central charge of the compactification of the $\mathcal{N} = (2,0)$ theory on $\mathcal{M}_4$
depends on $\mathcal{M}_4$ only through the topological factor $p_1(\mathcal{M}_4) + 2 \chi(\mathcal{M}_4)$.
It would be very interesting to investigate further this issue.
We plan to came back to all these points in the near future.
\section*{Acknowledgements}
We would like to thank Francesco Benini, Giulio Bonelli, Kentaro Hori, Francesco Sala, Alessandro Tanzini, Masahito Yamazaki, Yutaka Yoshida and Gabi Zafrir for useful discussions and comments. We thank, in particular, P. Marcos Crichigno, Dharmesh Jain and Brian Willett for pointing out a numerical mistake in section \ref{subsec:USp(2N)} in the first version of this paper and Minwoo Suh for sharing his results \cite{Suh:2018tul} before publication. Special thanks go to Yuji Tachikawa for several illuminating conversations and bringing important references to our attention.
We also like to acknowledge the collaboration with Kiril Hristov and Achilleas Passias on related topics.
The work of SMH and IY was supported by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
AZ is partially supported by the INFN and ERC-STG grant 637844-HBQFTNCER.
IY gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University at which some of the research for this paper was performed.
\begin{appendix}
\section{Toric geometry}
\label{sec:toric geometry}
A manifold $\mathcal{M}_4$ of complex dimension two is toric if it admits a $(\mathbb{C}^*)^2$ action and $(\mathbb{C}^*)^2$ itself is dense in $\mathcal{M}_4$.
Smooth toric surfaces can be obtained by gluing together copies of $\mathbb{C}^2$ \cite{fulton1993introduction}.
In this appendix we briefly review this construction focusing on the aspects used in the paper.
A compact toric variety $\mathcal{M}_4$ of complex dimension two is described a set of $d$ integer vectors ${\vec n}_l$ in
the lattice $N=\mathbb{Z}^2$ such that the angle between any pair of adjacent vectors is less than $\pi$,
as in figure \ref{fig:toricfan}. We order the vectors such that ${\vec n}_l$ and ${\vec n}_{l+1}$ are adjacent and we also identify ${\vec n}_{d+1}={\vec n}_1$.
The variety is smooth if ${\vec n}_l$ and ${\vec n}_{l+1}$ are a basis for the lattice $N=\mathbb{Z}^2$. We will assume from now on that our varieties are smooth.
Consider also the dual lattice $M=N^*$, equipped with the natural pairing $\langle {\vec m}, {\vec n}\rangle = \sum_{i=1}^2 m_i n_i \in \mathbb{Z}$ for ${\vec m}\in M$, ${\vec n}\in N$.
Points in $N$ are associated with one-parameter subgroups of $(\mathbb{C}^*)^2$ and points in $M$ with holomorphic functions on $(\mathbb{C}^*)^2$.
In particular, we associate points ${\vec m}\in M$ with monomial functions $z_1^{m_1} z_2^{m_2}$ and define a natural $(\mathbb{C}^*)^2$ action on the variables $z_i$.
\begin{figure}
\centering
\begin{tikzpicture}
[scale=0.5 ]
\draw[->] (0,0) -- (4,0) node[right] {${\vec n}_1,\, D_1$};
\draw[->] (0,0) -- (0,4) node[above] {${\vec n}_2, \, D_2$};
\draw[->] (0,0) -- (-4,1) node[ left] {${\vec n}_3,\, D_3$};
\draw [line width=1.3, dotted] (-1,-1) arc [radius=1, start angle=200, end angle= 250];
\draw[->] (0,0) -- (2,-4) node[below ] {${\vec n}_d,\, D_d$};
\draw[->] (-1.5,3) -- (1.5,3) node[right] {${\vec m}_1$};
\draw[->] (3,-1.5) -- (3,1.5) node[right] {${\vec m}_2$};
\draw (1.7,1.5) node {$\sigma_1$};
\draw (-1.7,1.5) node {$\sigma_2$};
\draw (1.7,-1.5) node {$\sigma_d$};
\end{tikzpicture}
\caption{A toric fan for a two-dimensional complex manifold. }
\label{fig:toricfan}
\end{figure}
Each pair of adjacent vectors $({\vec n}_l,{\vec n}_{l+1})$ defines a two-dimensional cone $\sigma_l$ in the real vector space $N_{\mathbb{R}}= N \otimes_{\mathbb{Z}} \mathbb{R}$,
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\sigma_l = \{ \lambda_1 {\vec n}_l + \lambda_2 {\vec n}_{l+1} \, |\, \lambda_i\ge 0\} \, .
\eea
We can associate an affine variety $V_{\sigma_l}$ (which in the smooth case is a copy of $\mathbb{C}^2$) to each $\sigma_l$ as follows.
Consider the dual cone
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\check \sigma_l= \{ {\vec m} \in M_{\mathbb{R}} \, | \, \langle {\vec m} , {\vec u}\rangle \ge 0\, \text{ for } \, {\vec u} \in \sigma_l \} \, ,
\eea
in $M_{\mathbb{R}}= M \otimes_{\mathbb{Z}} \mathbb{R}$. The lattice of integer points in $\check\sigma_l$ are generated by the primitive integer vectors ${\vec m}_l$ normal to the faces of $\sigma_l$ and pointing inwards.%
\footnote{A vector is primitive if its components are relatively prime.} We define $V_{\sigma_l}$ as the affine variety whose set of holomorphic functions is $\{ z_1^{\mu_1} z_2^{\mu_2}\}$ for all integer vectors $(\mu_1,\mu_2)\in \check\sigma_l$.
$V_{\sigma_l}$ is just isomorphic to $\mathbb{C}^2$. For example, consider the case ${\vec n}_1=(1,0)$ and ${\vec n}_2=(0,1)$.
The dual cone is then generated by ${\vec m}_1=(1,0)$ and ${\vec m}_2=(0,1)$ and contains all the integer points in the first quadrant.
The corresponding set of functions $ z_1^{\mu_1} z_2^{\mu_2}$ with $\mu_1\ge 0, \, \mu_2\ge 0$ are precisely the holomorphic functions on $\mathbb{C}^2$.
We also associate the vector ${\vec n}_1$ with the open dense subset where $z_2\ne 0$ and the vector ${\vec n}_2$ with the open dense subset where $z_1\ne 0$.
Since ${\vec n}_l$ and ${\vec n}_{l+1}$ are a basis for $N=\mathbb{Z}^2$, the generic case can be always reduced to the previous one by a change of lattice basis.
Explicitly, we can just replace in the previous example $z_1$ and $z_2$ with local coordinates $z_1^{(l)}= z_1^{m_{l,1}} z_2^{m_{l,2}}$ and $z_2^{(l)}= z_1^{m_{l+1,1}} z_2^{m_{l+1,2}}$
that parameterize a copy of $\mathbb{C}^2$, where ${\vec m}_{l}$ and ${\vec m}_{l+1}$ are the primitive integer vectors orthogonal to ${\vec n}_{l+1}$ and ${\vec n}_l$ and pointing inwards in $\sigma_l$, respectively.
The smooth toric variety $\mathcal{M}_4$ is then constructed by gluing together the $d$ affine varieties $V_{\sigma_l}$, isomorphic to $\mathbb{C}^2$, by identifying the dense open subset associated with ${\vec n}_l$ in $V_{\sigma_{l-1}}$ and $V_{\sigma_l}$.
This is completely analogous to the construction of $\mathbb{P}_1$ as a gluing of two copies of $\mathbb{C}$.
The action of the torus $(\mathbb{C}^*)^2$ on the variables $z_1$ and $z_2$ extends naturally to a global action on $V$.
Each chart $V_{\sigma_l}$ contains a special point, the origin $(0,0)\in \mathbb{C}^2$, which is invariant under the torus action.
These are the only invariant points and each of these belongs precisely to one chart.
We then see that there exactly are $d$ fixed points under the torus action $(\mathbb{C}^*)^2$, one for each of the two-dimensional cones $\sigma_l$.
Each vector ${\vec n}_l$ determines a divisor $D_l$ in $\mathcal{M}_4$ and a corresponding line bundle.%
\footnote{In a local chart corresponding to ${\vec n}_1=(1,0)$ and ${\vec n}_2=(0,1)$, $D_1$ restricts to $z_1=0$ and $D_2$ to $z_2=0$.}
There are precisely two relations among them given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{relation}
\sum_{l=1}^d \langle {\vec m}_k , {\vec n}_l \rangle D_l =0 \, , \qquad k=1,2 \, ,
\eea
where ${\vec m}_1=(1,0)$ and ${\vec m}_2=(0,1)$ is a basis for $M$.
This means that there are $d-2$ independent two-cycles in $\mathcal{M}_4$.
The intersection number of $D_l$ and $D_{l^\prime}$, with $l\ne l^\prime$, is one if ${\vec n}_l$ and ${\vec n}_{l^\prime}$ are adjacent and otherwise is zero.
As we saw, the partition function on $\mathcal{M}_4\times S^1$ localizes at the $d$ fixed points of $(\mathbb{C}^*)^2$.
Each contributes a copy of the Nekrasov's partition function of the corresponding chart $V_{\sigma_l}$, twisted by the magnetic flux.
The total magnetic flux is given by a divisor $\sum_{l=1}^d \mathfrak{p}_l D_l$ in $\mathcal{M}_4$, but the fixed point in the chart $V_{\sigma_l}$ will only feel the contribution of the flux coming from the divisors $D_l$ and $D_{l+1}$.
In the case of the chart $\mathbb{C}^2$, specified by the vectors ${\vec n}_1=(1,0)$ and ${\vec n}_2=(0,1)$, the local variables $z_1$ and $z_2$ parameterize the tangent space around the fixed point $z_1=z_2=0$.
In this case we write a copy of the Nekrasov's partition function with equivariant parameters $\epsilon_1$ and $\epsilon_2$ and Coulomb variable $a$ given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
a+ \epsilon_1 \mathfrak{p}_1 +\epsilon_2 \mathfrak{p}_2 \, .
\eea
The replacement for a generic chart $V_{\sigma_l}$ is then easily obtained.
The equivariant parameters are replaced by the action of $(\mathbb{C}^*)^2$ on the local parameters $z_1^{(l)}$ and $z_2^{(l)}$:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\epsilon_1^{(l)}= {\vec m}_l \cdot {\vec \epsilon} \, , \qquad \epsilon_2^{(l)}= {\vec m}_{l+1} \cdot {\vec \epsilon} \, ,
\eea
where ${\vec \epsilon}=(\epsilon_1,\epsilon_2)$ and ${\vec m}_{l}$ and ${\vec m}_{l+1}$ are the primitive integer vectors orthogonal to ${\vec n}_{l+1}$ and ${\vec n}_l$, respectively, and the Coulomb parameter by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
a^{(l)} = a + \epsilon_1^{(l)} \mathfrak{p}_l + \epsilon_2^{(l)} \mathfrak{p}_{l+1}\, .
\eea
We can also use toric geometry to evaluate the contribution of the magnetic flux to the Hirzebruch-Riemann-Roch index.
This can be done be evaluating \eqref{indexfixedpoint} or using \eqref{HRR}. The two results obviously coincide, \eqref{indexfixedpoint} being the localization formula for \eqref{HRR}.
For completeness, let us also show how to evaluate \eqref{HRR} using toric geometry techniques.
In the case where $E$ is a line bundle, the index \eqref{HRR} reads
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\int_{\mathcal{M}_4} \text{ch} (E) \text{td} (\mathcal{M}_4) = \chi(\mathcal{M}_4) + \int_{\mathcal{M}_4} \left ( \text{td}_1(\mathbb{P}^2) c_1(E) + \frac{1}{2} c_1(E)^2 \right ) \, .
\eea
We need the following general information about toric variety \cite{fulton1993introduction}.
The holomorphic Euler characteristic of every smooth toric four-manifold is one and $c_1(\mathcal{M}_4)$ is associated with the divisor $\sum_{l=1}^d D_l$.
We can then evaluate the integral of products of Chern classes with the intersection of the corresponding divisors.
Since $\text{td}_1(\mathcal{M}_4) = c_1(\mathcal{M}_4)/2$ and $c_1(E) = \sum_{l=1}^d \mathfrak{p}_l D_l$ the index is given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}\label{indexHRR}
1+ \frac12 \bigg( \sum_{l=1}^d \mathfrak{p}_l D_l \bigg) \cdot \bigg( \sum_{l=1}^d (\mathfrak{p}_l +1) D_l \bigg) \, .
\eea
The intersection of divisors can be computed with the rules given above.
For example, $\mathbb{F}_1$ is the toric manifold specified by the vectors ${\vec n}_1=(1,0)$, ${\vec n}_2=(0,1)$, ${\vec n}_3=(-1,1)$ and ${\vec n}_4=(0,-1)$.
The relations \eqref{relation} gives $D_3=D_1$ and $D_4=D_1+D_2$.
By combining these relations with the fact that $D_i \cdot D_j=1$ if ${\vec n}_i$ and ${\vec n}_j$ are adjacent and $D_i \cdot D_j=0$ otherwise (for $i\ne j$),
we can determine the nonzero intersections among the independent divisors $D_1$ and $D_2$: $D_1^2=0$, $ D_1\cdot D_2=1$ and $D_2^2=-1$.
We then easily obtain the index
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\frac12 ( \mathfrak{p}_2+\mathfrak{p}_4 + 1 ) ( 2 \mathfrak{p}_1-\mathfrak{p}_2 +2 \mathfrak{p}_3 +\mathfrak{p}_4 + 2 ) \, ,
\eea
which correctly reproduces the results in example \ref{ex3}.
Notice that, by comparing \eqref{indexHRR} with \eqref{indexfixedpoint}, we can derive a general formula
for the self intersections of divisors
\begin{equation}} \newcommand{\ee}{\end{equation}
D_{l} \cdot D_{l} =
\begin{cases}
- ( n_{l+1,1} + n_{l-1,1} ) / n_{l,1} \, , & \text{if } \, n_{l,1} \ne 0 \\
- ( n_{l+1,2} + n_{l-1,2} ) / n_{l,2} \, , & \text{if } \, n_{l,2} \ne 0
\end{cases} \, .
\ee
The two expressions agree when both conditions are met.
\section{Metric and spinor conventions}
\label{sec: spinor conventions}
We work in Euclidean signature in five dimensions. We use $m,n,p,\ldots$
for spacetime and $a,b,c,\ldots$ for tangent space indices. Spacetime
indices are lowered and raised by $g_{mn}$ and its inverse $g^{mn}$.
Tangent space indices are lowered and raised by $\delta_{ab}$ and
$\delta^{ab}$. Repeated indices are summed. The two sets of indices
are related using a vielbein $e_{m}^{\phantom{m}a}$, such that
\begin{equation}} \newcommand{\ee}{\end{equation}
g_{mn}=e_{m}^{\phantom{m}a}e_{n}^{\phantom{n}a} \, .
\ee
We define the Levi-Civita connection
\begin{equation}} \newcommand{\ee}{\end{equation}
\Gamma^{m}_{\phantom{m}np}=\frac{1}{2}g^{mt}\left(\partial_{n}g_{tp}+\partial_{p}g_{nt}-\partial_{t}g_{np}\right) \, ,
\ee
and the spin connection
\begin{equation}} \newcommand{\ee}{\end{equation}
\omega_{m}^{\phantom{m}ab}=e_{n}^{\phantom{n}a}\nabla_{m}e^{n b} \, .
\label{eq:Spin_Connection}
\ee
The Riemann tensor is defined as
\begin{equation}} \newcommand{\ee}{\end{equation}
R_{mn}^{\phantom{mn}ab}(e)=\partial_{m}\omega_{n}^{\phantom{n}ab}-\partial_{n}\omega_{m}^{\phantom{m}ab}+\omega_{m}^{\phantom{m}ac}\omega_{nc}^{\phantom{nc}b}-\omega_{m}^{\phantom{m}ac}\omega_{nc}^{\phantom{nc}b} \, .
\ee
The Ricci scalar is then given by
\begin{equation}} \newcommand{\ee}{\end{equation}
R(e)=e_{a}^{n}e_{b}^{m}R_{mn}^{\phantom{mn}ab}(e) \, .
\ee
The spin group is $\text{Spin}(5)\simeq \mathrm{USp}(4)$.
A spinor is a section $\zeta^{\alpha}$ of the pseudo-real $\mathbf{4}$
representation of $\mathrm{USp}(4)$ with $\alpha \in \{1, \ldots, 4\}$.
The Clifford algebra is generated by $\Gamma^{a}$ with $a \in \{1, \ldots, 5\}$ and
\begin{equation}} \newcommand{\ee}{\end{equation}
\left\{ \Gamma^{a},\Gamma^{b}\right\} =2\delta^{ab} \, .
\ee
The $\Gamma$ matrices have index structure $\left(\Gamma^{a}\right)^{\alpha}_{\phantom{\alpha}\beta}$.
We define the Pauli matrices
\begin{equation}} \newcommand{\ee}{\end{equation}
\tau^{1}=\begin{pmatrix}0 & 1\\
1 & 0
\end{pmatrix} \, ,\qquad\tau^{2}=\begin{pmatrix}0 & -\mathbbm i\\
\mathbbm i & 0
\end{pmatrix} \, ,\qquad\tau^{3}=\begin{pmatrix}1 & 0\\
0 & -1
\end{pmatrix} \, ,
\ee
and
\begin{equation}} \newcommand{\ee}{\end{equation}
\sigma^{i}=-\bar{\sigma}^{i}=-\mathbbm{i}\tau^{i} \, , \qquad \sigma^{4}=\bar{\sigma}^{4}=\mathbbm{1}_{2} \, .
\ee
A possible choice of the $\Gamma^{a}$ is given by
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\Gamma^{i}=\begin{pmatrix}0 & \sigma^{i}\\
\bar{\sigma}^{i} & 0
\end{pmatrix} \, ,\qquad i\in\{ 1,\ldots,4\} \, ,\\
\Gamma^{5}=\Gamma^{1}\Gamma^{2}\Gamma^{3}\Gamma^{4}=\begin{pmatrix}\mathbbm{1}_{2} & 0\\
0 & -\mathbbm{1}_{2}
\end{pmatrix} \, .
\eea
We define the $\Gamma$ matrices with multiple indices using permutations
as
\begin{equation}} \newcommand{\ee}{\end{equation}
\Gamma^{a_{1}a_{2}\ldots a_{p}}\equiv\frac{1}{p!}\sum_{\sigma\in\text{perm}(p)} \sign (\sigma)\prod_{i=1}^{p}\Gamma^{a_{\sigma(i)}} \, ,
\ee
such that
\begin{equation}} \newcommand{\ee}{\end{equation}
\Gamma^{ab}=\frac{1}{2}\left(\Gamma^{a}\Gamma^{b}-\Gamma^{b}\Gamma^{a}\right).
\ee
We also define
\begin{equation}} \newcommand{\ee}{\end{equation}
\sigma^{ij}\equiv\frac{1}{2}(\sigma^{i}\bar{\sigma}^{j}-\sigma^{j}\bar{\sigma}^{i}) \, , \qquad \bar{\sigma}^{ij}\equiv\frac{1}{2}(\bar{\sigma}^{i}\sigma^{j}-\bar{\sigma}^{j}\sigma^{i}) \, .
\ee
The covariant derivatives for spinors are defined by
\begin{equation}} \newcommand{\ee}{\end{equation}
\nabla_{m}\xi=\partial_{m}\xi+\frac{1}{4}\omega_{m}^{\phantom{m}ab}\Gamma_{ab}\xi \, .
\ee
We also define a spinor Lie derivative along a Killing vector field $v$ as
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathcal{L}_{v}\xi\equiv v^{m}\nabla_{m}\xi+\frac{1}{2}\nabla_{m}v_{n}\Gamma^{mn}\xi \, .
\ee
The $\Gamma^{a}$ satisfy
\begin{equation}} \newcommand{\ee}{\end{equation}
\left(\Gamma^{a}\right)^{\dagger}=\Gamma^{a} \, ,
\ee
where $\dagger$ denotes the conjugate transpose. We define a charge conjugation matrix
\begin{equation}} \newcommand{\ee}{\end{equation}
C_{\alpha\beta}\equiv\begin{pmatrix} \sigma^{2} & 0 \\
0 & \sigma^{2}
\end{pmatrix} \, ,
\ee
satisfying
\begin{equation}} \newcommand{\ee}{\end{equation}
\left(\Gamma^{a}\right)^{\text{T}}=C\Gamma^{a}C^{-1} \, , \qquad
C^{*}=C \, , \qquad
C^{\dagger}=-C \, , \qquad
C^{*}C=-\mathbbm{1}_{4} \, ,
\ee
where $\text{T}$ denotes transposition. Spinor bilinears are defined as
\begin{equation}} \newcommand{\ee}{\end{equation}
\xi\Gamma^{a_{1}a_{2}\ldots a_{p}}\eta\equiv\xi^{\alpha}C_{\alpha\beta}\left(\Gamma^{a_{1}a_{2}\ldots a_{p}}\right)^{\beta}_{\phantom{\beta}\gamma}\eta^{\gamma} \, .
\ee
The R-symmetry group of the five-dimensional $\mathcal{N}=1$ super-algebra is $\mathrm{SU}(2)_{\text{R}}$
with invariant antisymmetric tensor $\varepsilon^{IJ}$ such that
\begin{equation}} \newcommand{\ee}{\end{equation}
\varepsilon^{12}=-\varepsilon^{21}=1 \, .
\ee
An $\mathrm{SU}(2)$ Majorana spinor is a doublet $\zeta^{\alpha}_{\phantom{\alpha}I}$
with $I\in\{ 1,2\} $ in the fundamental representation
of $\mathrm{SU}(2)_{\text{R}}$ satisfying the condition
\begin{equation}} \newcommand{\ee}{\end{equation}
{\zeta_{\alpha}^{*}}^{I}=C_{\alpha\beta}\varepsilon^{IJ}\zeta^{\beta}_{\phantom{\beta}J} \, .
\ee
If the manifold $\mathcal{M}_4$ is not spin, all spinors are valued instead in an appropriate bundle associated with the choice of a $\text{spin}^c$ structure.
This choice can be made canonically for an almost complex manifold, but doing so may require a redefinition of certain background fluxes. See \cite{Festuccia:2016gul}
for examples in the context of localization and \cite{Salamon96spingeometry} for a complete reference.
\section{Large $N$ 't Hooft anomalies of 4D/2D SCFTs from M5-branes}
\label{sec:centralcharges(2,0)}
In this appendix we provide a simple formula, at large $N$, in order to extract the 't Hooft anomaly coefficients of four-dimensional $\mathcal{N} = 1$ (or two-dimensional $\mathcal{N} = (0,2)$ field theories)
that arise from M5-branes wrapped on a Riemann surface $\Sigma_{\mathfrak{g}_2}$ (or a four-manifold $\mathcal{M}_4$).
The trial 't Hooft anomaly coefficients of this class of theories can be extracted by integrating the eight-form anomaly polynomial $I_8$ of
the 6D $\mathcal{N} = (2,0)$ theory over $\Sigma_{\mathfrak{g}_2}$ or $\mathcal{M}_4$ \cite{Alday:2009qq,Bah:2011vv,Bah:2012dg,Benini:2013cda}.
The anomaly eight-form of the 6D theory of type $\mathfrak{g} = (A_{n \geq 1} , D_{n \geq 4} , E_6 , E_7 , E_8)$ reads \cite{Harvey:1998bx,Intriligator:2000eq,Yi:2001bz}
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{total:8form}
I_8[\mathfrak{g}] = r_\mathfrak{g} I_8 [1] + d_\mathfrak{g} h_\mathfrak{g} \frac{p_2(NW)}{24} \, ,
\ee
where $I_8[1]$ is the anomaly eight-form of one M5-brane \cite{Witten:1996hc}, $NW$ is the $\mathrm{SO}(5)$ R-symmetry bundle, and $p_2(NW)$ is its second Pontryagin class.
Here we have denoted the rank, the dimension, and the Coxeter number of the Lie algebra $\mathfrak{g}$ by $r_\mathfrak{g}$, $d_\mathfrak{g}$ and $h_\mathfrak{g}$, respectively.
For the $A_{N-1}$ theory, in the large $N$ limit, the anomaly eight-form is simply given by
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{8form2}
I_8[A_{N-1}] \approx \frac{N^3}{24} e_1^2 e_2^2 \, ,
\ee
where $e_\varsigma$ $(\varsigma = 1, 2)$ are the Chern roots of $NW$.
Let us first consider the compactification of 6D theories on a Riemann surface $\Sigma_{\mathfrak{g}_2}$.
The prescription in \cite{Bah:2011vv,Bah:2012dg} for computing the anomaly coefficient $a(\hat\Delta)$ of the 4D SCFT amounts to first replace the Chern roots $e_\varsigma$ in \eqref{8form2} with
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{replacement1}
e_\varsigma \to - \frac{\mathfrak{t}_\varsigma}{2 ( 1 - \mathfrak{g}_2 )} x + \hat \Delta_\varsigma c_1(F) \, ,
\ee
implementing the topological twist along $\Sigma_{\mathfrak{g}_2}$, and then integrate the $I_8 [A_{N-1}]$ on $\Sigma_{\mathfrak{g}_2}$:
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{I8:integrated:Sigma}
I_6 = \int_{\Sigma_{\mathfrak{g}_2}} I_8 [A_{N-1}] \, .
\ee
Here $x$ is the Chern root of the tangent bundle to $\Sigma_{\mathfrak{g}_2}$, $c_1(F)$ is a flux coupled to the R-symmetry, $\mathfrak{t}_\varsigma$ are the fluxes parameterizing the twist and $\hat\Delta_\varsigma$ parameterize the trial R-symmetry.
They fulfill the following constraints
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathfrak{t}_1 + \mathfrak{t}_2 = 2 ( 1 - \mathfrak{g}_2 ) \, , \qquad \hat \Delta_1+\hat \Delta_2 = 2 \, .
\ee
On the other hand, the anomaly six-form of a 4D SCFT, at large $N$, reads
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{I6:a}
I_6 \approx \frac{16}{27} a(\hat\Delta) c_1(F)^3 \, .
\ee
The 't Hooft anomaly coefficient $a(\hat\Delta)$, at large $N$, where $c=a$, can then be read off
by expanding $I_8 [A_{N-1}] $ at first order in $x$, using \eqref{I8:integrated:Sigma} --- $\int_{\Sigma_{\mathfrak{g}_2}} x = 2 ( 1 - \mathfrak{g}_2 )$ --- and comparing the result with \eqref{I6:a}.
It is simply given by
\begin{equation}} \newcommand{\ee}{\end{equation}
a(\hat\Delta) \approx - \frac{9 N^3}{128} \sum_{\varsigma=1}^2 \mathfrak{t}_\varsigma \frac{\partial (\hat\Delta_1\hat \Delta_2)^2}{\partial \hat \Delta_\varsigma} \, .
\ee
This is \eqref{a4D} in the main text.
Consider now the compactification of 6D field theories on $\Sigma_{\mathfrak{g}_1}\times \Sigma_{\mathfrak{g}_2}$ \cite{Benini:2013cda}.
We first need to replace
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{replacemen21}
e_\varsigma \to - \frac{\mathfrak{s}_\varsigma}{2(1-\mathfrak{g}_1)} x_1 - \frac{\mathfrak{t}_\varsigma}{2(1-\mathfrak{g}_2)} x_2+ \hat \Delta_\varsigma c_1(F) \, ,
\ee
where $x_\varsigma$ are now the Chern roots of the tangent bundles to $\Sigma_{\mathfrak{g}_\varsigma}$, and $\mathfrak{t}_\varsigma$/$\mathfrak{s}_\varsigma$ the fluxes on $\Sigma_{\mathfrak{g}_2}$/$\Sigma_{\mathfrak{g}_1}$, with
\begin{equation}} \newcommand{\ee}{\end{equation}
\mathfrak{t}_1 + \mathfrak{t}_2 = 2 ( 1 - \mathfrak{g}_2 ) \, , \qquad \mathfrak{s}_1 + \mathfrak{s}_2 = 2 ( 1 - \mathfrak{g}_1) \, .
\ee
Then we integrate $I_8 [A_{N-1}]$ over $\Sigma_{\mathfrak{g}_1} \times \Sigma_{\mathfrak{g}_2}$ using $\int_{\Sigma_{\mathfrak{g}_\varsigma}} x_\varrho=2(1-\mathfrak{g}_\varsigma) \delta_{\varsigma \varrho}$.
The result should be compared with the four-form anomaly polynomial of the two-dimensional SCFT that, in the large $N$ limit, where $c_l=c_r$, reads
\begin{equation}} \newcommand{\ee}{\end{equation}
I_4 \approx \frac{c_l(\hat\Delta)}{6} c_1(F)^2 \, .
\ee
This time only the term proportional to $x_1 x_2$ in \eqref{8form2} contributes and we obtain
\begin{equation}} \newcommand{\ee}{\end{equation}
c_l(\hat\Delta) \approx \frac{N^3}{4} \sum_{\varsigma,\varrho=1}^2 \mathfrak{t}_\varsigma \mathfrak{s}_\varrho \frac{\partial^2 (\hat\Delta_1\hat \Delta_2)^2}{\partial \hat \Delta_\varsigma\partial \hat\Delta_\varrho} \, .
\ee
This is \eqref{cr:cl:6D(2,0):largeN} with the identification $\beta\Delta_\varsigma/\pi=\hat\Delta_\varsigma$.
For completeness, we also study the compactification of 6D field theories on a four-manifold $\mathcal{M}_4$ with a single flux on $\mathcal{M}_4$.
Denoting with $x_1$ and $x_2$ the Chern roots of the tangent bundle to $\mathcal{M}_4$, the topological twist can be implemented by \cite{Benini:2013cda}
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{replacement3}
e_\varsigma \rightarrow - \mathfrak{r}_\varsigma (x_1+x_2) + \hat \Delta_\varsigma c_1(F) \, ,
\ee
where $\mathfrak{r}_1+\mathfrak{r}_2=1$. The integration of $I_8[A_{N-1}]$ over $\mathcal{M}_4$ can be done by noticing that the integrals
\begin{equation}} \newcommand{\ee}{\end{equation}
p_1(\mathcal{M}_4) = 3 \, \sigma(\mathcal{M}_4) = \int_{\mathcal{M}_4} (x_1^2+x_2^2) \, ,\qquad \chi(\mathcal{M}_4) =\int_{\mathcal{M}_4} x_1 x_2 \,
\ee
give the first Pontryagin number and the Euler number of $\mathcal{M}_4$. The result is then simply
\begin{equation}} \newcommand{\ee}{\end{equation}
c_l (\hat\Delta) \approx \frac{N^3}{8} \left( p_1(\mathcal{M}_4) + 2 \chi(\mathcal{M}_4) \right) \sum_{\varsigma , \varrho = 1}^2 \mathfrak{r}_\varsigma \mathfrak{r}_\varrho \frac{\partial^2 (\hat\Delta_1\hat \Delta_2)^2}{\partial \hat \Delta_\varsigma \partial \hat \Delta_\varrho} \, .
\ee
We see that the 't Hooft anomaly coefficients of 4D (or 2D) field theories, which are obtained by wrapping M5-branes on $\Sigma_{\mathfrak{g}_2}$ (or $\mathcal{M}_4$), can always be written as (multiple) applications of operators of the form
$\sum_{\varsigma=1}^2 \mathfrak{r}_\varsigma \partial_{\hat\Delta_\varsigma}$ to the function $(\hat\Delta_1\hat \Delta_2)^2$, appearing in the eight-form anomaly polynomial through the term $p_2(NW)$.
\section{An alternative large $N$ saddle point for the $\mathrm{USp}(2N)$ theory}
\label{sec:alternative}
In this appendix, for completeness, we discuss a possible alternative method for evaluating \eqref{Sigma_g1xM_3:Z} in the saddle point approximation.
Solving \eqref{BAEs:M3} in the large $N$ limit gives a relation between $a_{(i)}$ and $\mathfrak{n}_i$.
Eliminating $a_{(i)}$, we can write \eqref{Sigma_g1xM_3:Z} as a sum over the fluxes $\mathfrak{n}_i$:
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{Sigma_g1xM_3:Z2}
& Z^{\text{pert}}_{ \Sigma_{\mathfrak{g}_2} \times ( \Sigma_{\mathfrak{g}_1} \times S^1)} (\mathfrak{s} , \mathfrak{t} , \Delta) =
\frac{(-1)^{\text{rk}(G)}}{|\mathfrak{W}|} \sum_{\mathfrak{n} \in \Gamma_\mathfrak{h}} Z^{\text{pert}} \big|_{\mathfrak{m} = 0} (a(\mathfrak{n}) , \mathfrak{n})
\bigg( \det\limits_{i j} \frac{\partial^2 \widetilde \mathcal{W}^{\text{pert}} (a(\mathfrak{n}) , \mathfrak{n})}{\partial a_i \partial a_j} \bigg)^{\mathfrak{g}_1 - 1} \, ,
\eea
where $a_{(i)}(\mathfrak{n})$ is the large $N$ solution to \eqref{BAEs:M3}.
Each term in the sum is an exponentially large function of $N$, and we can use again the saddle point approximation to find the dominant contribution to the partition function.
While for $\mathcal{N}=2$ SYM this method fails since \eqref{BAEs:M3} completely fixes the values of $\mathfrak{n}_i$, it works for the $\mathrm{USp}(2N)$ theory.
For completeness, we quote the result for the partition function evaluated with this approach
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\log Z ( \Delta_m , \mathfrak{t}_m , \mathfrak{s}_m )
= \frac{4 N^{5/2}}{5 \sqrt{8 - N_f}} \sqrt{ ( \Delta_1 \mathfrak{t}_2 + \Delta_2 \mathfrak{t}_1 ) ( \Delta_1 \mathfrak{s}_2 + \Delta_2 \mathfrak{s}_1 ) ( \mathfrak{s}_1 \mathfrak{t}_2 + \mathfrak{s}_2 \mathfrak{t}_1 )}\, .
\eea
Notice that this is different from \eqref{USp(2N):Final:logZ:index theorem}.
It coincides with \eqref{USp(2N):Final:logZ:index theorem} only in the case of the universal twist, $\mathfrak{t}_m= 1- \mathfrak{g}_2$ and $\mathfrak{s}_m=1 -\mathfrak{g}_1$.
It would be very interesting to compare the two alternative results with the entropy of asymptotically AdS$_6$ black holes in massive type IIA supergravity, with magnetic fluxes $\mathfrak{t}_m$ and $\mathfrak{s}_m$ \cite{Hosseini:2018mIIA}.
\section{Polylogarithms}
\label{sec:PolyLog}
Polylogarithms $\Li_s (z)$ are defined by
\begin{equation}} \newcommand{\ee}{\end{equation}
\Li_s (z) = \sum_{n = 1}^{\infty} \frac{z^n}{n^s} \, ,
\ee
for $|z| < 1$ and by analytic continuation outside the disk. They satisfy the following relations
\begin{equation}} \newcommand{\ee}{\end{equation}
\partial_a \Li_s (e^{\mathbbm{i} a}) = \mathbbm{i} \Li_{s-1}(e^{\mathbbm{i} a}) \, , \qquad \qquad \Li_s(e^{\mathbbm{i} a}) = \mathbbm{i} \int_{+ \mathbbm{i} \infty}^a \Li_{s-1}(e^{\mathbbm{i} a '}) \, \mathrm{d} a' \, .
\ee
For $s \geq 1$, the functions have a branch point at $z=1$ and we shall take the principal determination with a cut $[1 , + \infty)$ along the real axis.
The functions $\Li_s(e^{\mathbbm{i} a})$ are periodic under $a \to a + 2 \pi$ and have branch cut discontinuities along the vertical line $[0, - \mathbbm{i} \infty)$ and its images.
For $0< \re a < 2\pi$, polylogarithms fulfill the following inversion formula%
\footnote{The inversion formul\ae{} in the domain $- 2 \pi < \re a < 0$ are obtained by sending $a \to - a$.}
\begin{equation} \begin{aligned}} \newcommand{\eea}{\end{aligned} \end{equation}
\label{PolyLog:inversion formulae}
\Li_s (e^{\mathbbm{i} a}) + (-1)^s \Li_s (e^{- \mathbbm{i} a}) = - \frac{(2 \mathbbm{i} \pi)^s}{s!} B_s \left( \frac{a}{2 \pi} \right) \equiv \mathbbm{i}^{s - 2} g_s (a) \, ,
\eea
where $B_s (a)$ are the Bernoulli polynomials. In this paper we need, in particular,
\begin{equation}} \newcommand{\ee}{\end{equation}\label{gfunctions}
g_2 ( a ) = \frac{a^2}{2} - \pi a + \frac{\pi^2}{3} \, , \qquad g_3 ( a ) = \frac{a^3}{6} - \frac{\pi}{2} a^2 + \frac{\pi^2}{3} a \, .
\ee
One can find the formul\ae{} in the other regions by periodicity. Let us also mention that
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{Z2:gs}
g_s (2 \pi - a) = (-1)^s g_s(a) \, .
\ee
Finally, assuming $0 < \Delta < 2 \pi$, we find that
\begin{equation}} \newcommand{\ee}{\end{equation}
\label{PolyLog:asymptotic}
\Li_s (e^{t + \mathbbm{i} \Delta}) \sim \mathbbm{i}^{s-2} g_s( - \mathbbm{i} t + \Delta ) \, , \qquad \text{ as } t \to \infty \, .
\ee
\end{appendix}
\bibliographystyle{ytphys}
|
2,869,038,154,780 | arxiv | \section{\textbf{Introduction}}
Let $B(H)$ stand for the $C^{*}$-algebra of all bounded linear
operators on a complex separable Hilbert space $H$ with inner
product $\langle \cdot,\cdot\rangle$ and let $K(H)$ denote the
two-sided ideal of compact operators in $B(H)$. For $A\in B(H)$, let
$\|A\|=\sup\{\|Ax\|:\|x\|=1\}$ denote the usual operator norm of $A$
and $|A|=(A^{*}A)^{1/2}$ be the absolute value of $A$.\\
An operator $A\in B(H)$ is positive and write $A\geq0$ if $\langle
Ax,x\rangle\geq0$ for all $x\in H$. We say $A\leq B$ whenever
$B-A\geq0$.\\
We consider the wide class of unitarily invariant
norms $|||\cdot|||$. Each of these norms is defined on an ideal in
$B(H)$ and it will be implicitly understood that when we talk of
$|||T|||$, then the operator $T$ belongs to the norm ideal
associated with $|||\cdot|||$. Each unitarily invariant norm
$|||\cdot|||$ is characterized by the invariance property
$|||UTV|||=|||T|||$ for all operators $T$ in the norm ideal
associated with $|||\cdot|||$ and for all unitary operators $U$ and
$V$ in $B(H)$.
For $1\leq
p<\infty$, the Schatten $p$-norm of a compact operator $A$ is
defined by $\|A\|_{p}=(\Tr |A|^{p})^{1/p}$, where $\Tr$ is the usual
trace functional. Note that for $A\in K(H)$ we have,
$\|A\|=s_{1}(A)$, and if $A$ is a Hilbert-Schmidt operator, then
$\|A\|_{2}=(\sum_{j=1}^{\infty}s_{j}^{2}(A))^{1/2}$. These norms are
special examples of the more general class of the Schatten
$p$-norms, which are unitarily invariant \cite{bhabook}.
The direct sum $A\oplus B$ denotes the block diagonal
matrix $\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]$ defined on $H\oplus H$, see \cite{aud,zhan}. It
is easy to see that
\begin{equation}\label{eq12}
\|A\oplus B\|=\max(\|A\|,\|B\|),
\end{equation}
and
\begin{equation}\label{eq13}
\|A\oplus B\|_{p}=(\|A\|_{p}^{p}+\|B\|_{p}^{p})^{1/p}.
\end{equation}
We denote the singular values of an operator $A\in K(H)$ as
$s_{1}(A)\geq s_{2}(A)\geq \ldots$ are the eigenvalues of the
positive operator $|A|=(A^{*}A)^{1/2}$ and eigenvalues of the self-adjoint operator $A$ denote as $\lambda_{1}\geq\lambda_{2}\geq \ldots$ which repeated accordingly to
multiplicity.
There is a one-to-one correspondence between symmetric gauge
functions defined on sequences of real numbers and unitarily
invariant norms defined on norm ideals of operators. More precisely,
if $|||\cdot|||$ is unitarily invariant norm, then there exists a
unique symmetric gauge function $\Phi$ such that
$$|||A|||=\Phi(s_{1}(A),s_{2}(A),\ldots),$$
for every operator $A\in K(H)$. Let $A\in K(H)$, and if $U,V\in
B(H)$ are unitarily operators, then
$$s_{j}(UAV)=s_{j}(A),$$
for $j=1,2,\ldots$ and so unitarily invariant norms satisfies the
invariance property
$$|||UAV|||=|||A|||.$$
In this paper, we obtain some inequalities for sum and product
of operators. Some of our results generalize the previous inequalities for operators.
\section{\textbf{Some singular value inequalities for sum and product of operators}}
In this section we give inequalities for singular value of
operators. Also, some norm inequalities are obtained as an
application.\\
First we should remind the following inequalities. We apply inequalities (\ref{tao2}) and (\ref{kit2}) in our proofs.
The following inequality due to Tao \cite{tao} asserts that if $A, B, C\in K(H)$
such that $\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\geq 0$, then
\begin{equation}\label{tao2}
2s_{j}(B)\leq s_{j}\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right],
\end{equation}
for $j=1,2,\ldots$.\\
Here, we give another proof for above inequality.\\
Let $\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\geq 0$ then $\left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right]\geq 0$ and have the same singular values (see\cite[Theorem 2.1]{aud}). So, we can write
$$\left[\begin{array}{cc}
0&2B\\
2B^{*} &0\\
\end{array}\right]\leq \left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right],$$
and
$$\left[\begin{array}{cc}
0&-2B\\
-2B^{*} &0\\
\end{array}\right]\leq \left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right].$$ On the other hand, we know that for every self-adjoint compact operator $X$ we have $s_{j}(X)\leq \lambda_{j}(X\oplus -X)$, for all $j=1,2,\ldots$. By using of this fact we obtain
\begin{eqnarray*}
s_{j}\left(\left[\begin{array}{cc}
0&2B\\
2B^{*} &0\\
\end{array}\right]\right)&=&\lambda_{j}\left(\left[\begin{array}{cc}
0&2B\\
2B^{*} &0\\
\end{array}\right]\oplus \left[\begin{array}{cc}
0&-2B\\
-2B^{*} &0\\
\end{array}\right]\right)\\
&\leq&\lambda_{j}\left(\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\oplus \left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right] \right)\\
&=& s_{j}\left(\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\oplus \left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right] \right).
\end{eqnarray*}
So, we obtain $$s_{j}\left(\left[\begin{array}{cc}
0&2B\\
2B^{*} &0\\
\end{array}\right]\right)\leq s_{j}\left(\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\oplus \left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right] \right).$$
Equivalently,
$$2s_{j}(B \oplus B^{*}) \leq \left(\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\oplus \left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right] \right).$$
Since $s_{j}(B)=s_{j}(B^{*})$ and $s_{j}\left(\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right)=s_{j}\left(\left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right]\right)$, we have
$$2s_{j}(B)\leq s_{j}\left(\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right).$$
In \cite[Remark 2.2]{aud}, Audeh and Kittaneh proved that for every $A,B,C\in K(H)$ such that
$\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\geq 0$, then
\begin{equation}\label{asho}
s_{j}\left(\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right)\leq 2s_{j}(A\oplus C),
\end{equation}
for $j=1,2,\ldots$. Therefore, by inequality (\ref{tao2}) we have the following inequality
\begin{equation}\label{kit2}
s_{j}(B)\leq s_{j}(A\oplus C),
\end{equation}
for $j=1,2,\ldots$.
Since every unitarily invariant norm is a monotone function of the singular values of an
operator, we can write
\begin{equation}\label{tbo}
\left|\left|\left|\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right|\right|\right|\leq 2|||A\oplus C|||.
\end{equation}
We can obtain the reverse of inequality (\ref{tbo}) for arbitrary operators $X,Y\in B(H)$ by pointing out the following inequality holds because of norm property
$$|||X+Y|||\leq |||X|||+|||Y|||.$$
Replace $X$ and $Y$ by $X-Y$ and $X+Y$, respectively. We have
$$2|||X|||\leq |||X-Y|||+|||X+Y|||,$$
for all $X,Y\in B(H)$. \\
Let $X=\left[\begin{array}{cc}
A&0\\
0 &C\\
\end{array}\right]$ and $Y=\left[\begin{array}{cc}
0&B\\
B^{*} &0\\
\end{array}\right]$ in above inequality. So,
\begin{eqnarray*}
2\left|\left|\left|\left[\begin{array}{cc}
A&0\\
0 &C\\
\end{array}\right]\right|\right|\right|&\leq& \left|\left|\left|\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right|\right|\right|+\left|\left|\left|\left[\begin{array}{cc}
A&-B\\
-B^{*} &C\\
\end{array}\right]\right|\right|\right|\\
&=&2\left|\left|\left|\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right|\right|\right|.
\end{eqnarray*}
Hence,
$$|||
A\oplus C
|||\leq \left|\left|\left|\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right|\right|\right|,$$
for all $A,B,C\in B(H)$. $\left[\begin{array}{cc}
A&0\\
0 &C\\
\end{array}\right]$ is called a \textit{pinching} of $\left[\begin{array}{cc}
A&B\\
B^{*}&C\\
\end{array}\right]$. \\
For operator norm we have
$$\max\{\|A\|,\|C\|\}\leq \left\|\left[\begin{array}{cc}
A&B\\
B^{*} &C\\
\end{array}\right]\right\|.$$
Here we give a generalization of the inequality which has been
proved by Bhatia and Kittaneh in \cite{bha1}. They have shown that
if $A$ and $B$ are two $n\times n$ matrices, then
$$s_{j}(A+B)\leq
s_{j}\left((|A|+|B|)\oplus(|A^{*}|+|B^{*}|)\right),$$ for $1\leq
j\leq n$.\\
For giving a generalization of above inequality, we need the
following lemmas.
In the rest of this section, we always assume that $f$ and $g$ are
non-negative functions on $[0, \infty)$ which are continuous and
satisfying the relation $f(t)g(t)=t$ for all $t\in [0, \infty)$.
\\
\\
The following lemma is due to Kittaneh
\cite{kit}.
\begin{lemma}\label{function}
Let $A, B$, and $C$ be operators in $B(H)$ such that $A$ and $B$ are
positive and $BC=CA$. If $\left[\begin{array}{cc}
A&C^{*}\\
C&B\\
\end{array}\right]$ is positive in $B(H\oplus H)$, then $\left[\begin{array}{cc}
f(A)^{2}&C^{*}\\
C&g(B)^{2}\\
\end{array}\right]$ is
also positive.
\end{lemma}
Let $T$ be an operator in $B(H)$. We know that
$\left[\begin{array}{cc}
|T|&T^{*}\\
T&|T^{*}|\\
\end{array}\right]\geq0$, if $T$ is normal then we have $\left[\begin{array}{cc}
|T|&T^{*}\\
T&|T|\\
\end{array}\right]\geq0$, ( see \cite{bhabook2}).
\begin{lemma}\label{lem1}
Let $A$ be an operator in $B(H)$. Then we have
\begin{equation}\label{l1}
\left[\begin{array}{cc}
|A|^{2\alpha}&A^{*}\\
A&|A^{*}|^{2(1-\alpha)}\\
\end{array}\right]\geq0,
\end{equation}
where $0\leq\alpha\leq1$.
\end{lemma}
\begin{proof}
It is easy to check that $A|A|^{2}=|A^{*}|^{2}A$, then we have
$A|A|=|A^{*}|A$ for $A\in B(H)$. Now by making use of Lemma
\ref{function}, for $f(t)=t^{\alpha}$ and $g(t)=t^{1-\alpha}$,
$0\leq\alpha\leq1$, and positivity of $\left[\begin{array}{cc}
|A|&A^{*}\\
A&|A^{*}|\\
\end{array}\right]$, we obtain the result.
\end{proof}
\begin{theorem}\label{vd1}
Let $A$ and $B$ be two operators in $K(H)$. Then we have
$$s_{j}(A+B)\leq
s_{j}\left((|A|^{2\alpha}+|B|^{2\alpha})\oplus(|A^{*}|^{2(1-\alpha)}+|B^{*}|^{2(1-\alpha)})\right),$$
for $j=1,2,\ldots$ where $0\leq\alpha\leq1$.
\end{theorem}
\begin{proof}
Since sum of two positive operator is positive, Lemma \ref{lem1} implies that $$\left[\begin{array}{cc}
|A|^{2\alpha}+|B|^{2\alpha}&A^{*}+B^{*}\\
A+B&|A^{*}|^{2(1-\alpha)}+|B^{*}|^{2(1-\alpha)}\\
\end{array}\right]\geq0,$$
By inequality
(\ref{kit2}) we have the result.
\end{proof}
\begin{corollary}
Let $A$ and $B$ be two operators in $K(H)$. Then we have
$$s_{j}(A+B)\leq
s_{j}\left((|A|+|B|)\oplus(|A^{*}|+|B^{*}|)\right),$$ for
$j=1,2,\ldots$.
\end{corollary}
\begin{proof}
Let $\alpha=\frac{1}{2}$ in Theorem \ref{vd1}.
\end{proof}
It is easy to see that if $A$ and $B$ are normal operator in $K(H)$,
then we have
$$s_{j}(A+B)\leq
s_{j}\left((|A|+|B|)\oplus(|A|+|B|)\right),$$ for $j=1,2,\ldots$.\\
\\
\\
On the other hand, for $\alpha=1$ in Theorem \ref{vd1}, we have
\begin{eqnarray*}
s_{j}(A+B)&\leq& s_{j}(|A|^{2}+|B|^{2}\oplus 2I)\\
&=&s_{j}(|A|^{2}+|B|^{2})\cup s_{j}(2I)\\
&=&s_{j}(A^{*}A+B^{*}B)\cup s_{j}(2I),
\end{eqnarray*}
for $j=1,2,\ldots$.
\begin{theorem}\label{haj}
Let $A$,$B$ and $X$ be operators in $B(H)$ such that $X$ is compact.
Then we have the following
$$s_{j}\left(AXB^{*}\right)\leq
s_{j}\left(A^{*}f(|X|)^{2}A\oplus B^{*}g(|X^{*}|)^{2}B\right),$$ for
$j=1,2,\ldots$.
\end{theorem}
\begin{proof}
Since $\left[\begin{array}{cc}
|X|&X^{*}\\
X&|X^{*}|\\
\end{array}\right]\geq0$, by Lemma \ref{function} we have\\ $$Y=\left[\begin{array}{cc}
f(|X|)^{2}&X^{*}\\
X&g(|X^{*}|)^{2}\\
\end{array}\right]\geq0.$$ Let $Z=\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]$. Since $Y$ is positive, we have
$$Z^{*}YZ=\left[\begin{array}{cc}
A^{*}f(|X|)^{2}A&A^{*}X^{*}B\\
B^{*}XA&B^{*}g(|X^{*}|)^{2}B\\
\end{array}\right]\geq0.$$ Hence,
by inequality (\ref{kit2}), we have the desired result.
\end{proof}
\noindent In above theorem, let $X$ be a normal operator. Then we have
$$s_{j}\left(AXB^{*}\right)\leq
s_{j}\left(A^{*}f(|X|)^{2}A\oplus B^{*}g(|X|)^{2}B\right),$$ for
$j=1,2,\ldots$.
\begin{corollary}\label{131}
Let $A$, $B$ and $X$ be operators in $B(H)$ such that $X$ is
compact. Then we have
$$s_{j}\left(AXB^{*}\right)\leq
s_{j}\left(A^{*}|X|A\oplus B^{*}|X^{*}|B\right),$$ for $j=1,2,\ldots$.
\end{corollary}
\begin{proof}
Let $f(t)=t^{\frac{1}{2}}$ and $g(t)=t^{\frac{1}{2}}$ in Theorem
\ref{haj}.
\end{proof}
Here, we apply above corollary to show that singular values of $AXB^{*}$ are dominated by singular values of $\|X\|(A\oplus B)$. For our proof we need the following lemma.
\begin{lemma}\label{hz}\cite[p. 75]{bhabook}
Let $A,B\in B(H)$ such that $B$ is compact. Then
$$s_{j}(AB)\leq \|A\|s_{j}(B),$$
for $j=1,2,\ldots$.
\end{lemma}
\begin{theorem}\label{ttyo}
Let $A, B,X\in B(H)$ such that $A$ and $ B$ are arbitrary compact. Then, we have
$$s_{j}(AXB^{*})\leq \|X\|s_{j}^{2}(A\oplus B),$$
for $j=1,2,\ldots$.
\end{theorem}
\begin{proof}
From Corollary \ref{131} we have
\begin{eqnarray*}
s_{j}(AXB^{*})&\leq& s_{j}(A^{*}|X|A\oplus B^{*}|X^{*}|B)\\
&=& s_{j}\left(\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]^{*}\left[\begin{array}{cc}
|X|&0\\
0&|X^{*}|\\
\end{array}\right]\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]\right)\\
&=& s_{j}\left(\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]^{*}\left[\begin{array}{cc}
|X|^{\frac{1}{2}}&0\\
0&|X^{*}|^{\frac{1}{2}}\\
\end{array}\right]^{*}\left[\begin{array}{cc}
|X|^{\frac{1}{2}}&0\\
0&|X^{*}|^{\frac{1}{2}}\\
\end{array}\right]\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]\right)\\
&=&s_{j}\left(\left(\left[\begin{array}{cc}
|X|^{\frac{1}{2}}&0\\
0&|X^{*}|^{\frac{1}{2}}\\
\end{array}\right]\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]\right)^{*}\left(\left[\begin{array}{cc}
|X|^{\frac{1}{2}}&0\\
0&|X^{*}|^{\frac{1}{2}}\\
\end{array}\right]\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]\right)\right)\\
&=&s_{j}\left(\left|\left[\begin{array}{cc}
|X|^{\frac{1}{2}}&0\\
0&|X^{*}|^{\frac{1}{2}}\\
\end{array}\right]\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]\right|^{2}\right)\\
&=& s_{j}^{2}\left(\left[\begin{array}{cc}
|X|^{\frac{1}{2}}&0\\
0&|X^{*}|^{\frac{1}{2}}\\
\end{array}\right]\left[\begin{array}{cc}
A&0\\
0&B\\
\end{array}\right]\right)\\
&\leq& \left\|\left[\begin{array}{cc}
|X|^{\frac{1}{2}}&0\\
0&|X^{*}|^{\frac{1}{2}}\\
\end{array}\right]\right\|^{2}s_{j}^{2}(A\oplus B)\\
&=&\|X\|s_{j}^{2}(A\oplus B),
\end{eqnarray*}
for $j=1,2,\ldots$. The last inequality follows by Lemma \ref{hz}.
\end{proof}
In Theorem \ref{ttyo}, let $A$ and $B$ be positive operators in $K(H)$. Then
we have
\begin{equation}\label{maj}
s_{j}(A^{\frac{1}{2}}XB^{\frac{1}{2}})\leq \|X\|s_{j}(A\oplus B),
\end{equation}
for $j=1,2,\ldots$.
\begin{corollary}\label{gb}
Let $A$ and $B$ be two operators in $K(H)$. Then we have
\begin{equation}\label{cor1}
s_{j}(AB^{*})\leq s_{j}\left(A^{*}A\oplus B^{*}B\right),
\end{equation}
for
$j=1,2,\ldots$.
\end{corollary}
\begin{proof}
Let $X=I$ in
Corollary \ref{131}.
\end{proof}
Moreover, we can write inequality (\ref{cor1}) in the following form
\begin{eqnarray*}
s_{j}(AB^{*})&\leq& s_{j}(|A|^{2}\oplus |B|^{2})\\
&=&s_{j}^{2}(|A|\oplus|B|)=s_{j}^{2}(A\oplus B),
\end{eqnarray*}
for $j=1,2,\ldots$.\\
We should note here that inequality (\ref{cor1}) can be obtained by
Theorem 1 in \cite{bha3} and Corollary 2.2 in \cite{hir}.\\
Here, we give two results of Corollary \ref{gb}.
As the first application, let $A=\left[\begin{array}{cc}
X&Y\\
0&0\\
\end{array}\right]$ and $B=\left[\begin{array}{cc}
Y&-X\\
0&0\\
\end{array}\right]$, such that $X,Y\in K(H)$ then by easy computations we have
$$s_{j}(XY^{*}-YX^{*})\leq
s_{j}((XX^{*}+YY^{*})\oplus(XX^{*}+YY^{*})),$$ for $j=1,2,\ldots$.\\
For obtaining second application, replace $A$ and $B$ in (\ref{cor1}) by $AX^{\alpha}$ and $BX^{(1-\alpha)}$ respectively, where $X$ is a compact positive operator and $\alpha\in \mathbb{R}$. So, we have
\begin{eqnarray*}
s_{j}(AXB^{*})&\leq& s_{j}\left(X^{\alpha}A^{*}AX^{\alpha}\oplus X^{(1-\alpha)}B^{*}BX^{(1-\alpha)}\right)\\
&=& s_{j}\left(\left[\begin{array}{cc}
X^{\alpha}A^{*}AX^{\alpha}&0\\
0&X^{(1-\alpha)}B^{*}BX^{(1-\alpha)}\\
\end{array}\right] \right)\\
&=&s_{j}\left(\left[\begin{array}{cc}
X^{\alpha}A^{*}&0\\
0&X^{(1-\alpha)}B^{*}\\
\end{array}\right] \left[\begin{array}{cc}
AX^{\alpha}&0\\
0&BX^{(1-\alpha)}\\
\end{array}\right] \right)\\
&=&s_{j}\left( \left[\begin{array}{cc}
AX^{\alpha}&0\\
0&BX^{(1-\alpha)}\\
\end{array}\right]\left[\begin{array}{cc}
X^{\alpha}A^{*}&0\\
0&X^{(1-\alpha)}B^{*}\\
\end{array}\right] \right)\\
&=&s_{j}\left( \left[\begin{array}{cc}
AX^{2\alpha}A^{*}&0\\
0&BX^{2(1-\alpha)}B^{*}\\
\end{array}\right]\right)\\
&=& s_{j}(AX^{2\alpha}A^{*}\oplus BX^{2(1-\alpha)}B^{*}),
\end{eqnarray*}
for all $j=1,2,\ldots$.\\
Finally, we have
\begin{equation}\label{embr}
s_{j}(AXB^{*})\leq s_{j}(AX^{2\alpha}A^{*}\oplus BX^{2(1-\alpha)}B^{*}),
\end{equation}
for all $j=1,2,\ldots$.\\
By a similar proof of Theorem \ref{ttyo} to inequality (\ref{embr}), we obtain
$$s_{j}(AXB^{*})\leq \max\{\|X^{2\alpha}\|,\|X^{2(1-\alpha)}\|\}s_{j}^{2}(A\oplus B),$$
for all $j=1,2,\ldots$.\\
In above inequality, for positive operators $A$ and $B$ in $K(H)$ we have
$$ s_{j}(A^{\frac{1}{2}}XB^{\frac{1}{2}})\leq \max\{\|X^{2\alpha}\|,\|X^{2(1-\alpha)}\|\}s_{j}(A\oplus B)$$
for all $j=1,2,\ldots$.
\section{\textbf{Some singular value inequalities for normal operators}}\label{section4}
Here we give some results for compact normal operators. For
every operator $A$, the Cartesian decomposition is to write
$A=\Re(A)+i\Im(A)$, where $\Re(A)=\frac{A+A^{*}}{2}$ and
$\Im(A)=\frac{A-A^{*}}{2i}$. If $A$ is normal operator then $\Re(A)$
and $\Im(A)$ commute together and vice versa.
\begin{theorem}\label{normal}
Let $A_{1},A_{2},\ldots,A_{n}$ be normal operators in $K(H)$. Then
we have
\begin{eqnarray*}
\frac{1}{\sqrt{2}}s_{j}\left(\oplus_{i=1}^{n}(\Re(A_{i})+\Im(A_{i}))\right)&\leq& s_{j}(\oplus_{i=1}^{n}A_{i})\\
&\leq&
s_{j}\left(\oplus_{i=1}^{n}(|\Re(A_{i})|+|\Im(A_{i})|)\right),
\end{eqnarray*}
for
$j=1,2,\ldots$.
\end{theorem}
\begin{proof}
Let $A_{1},A_{2},\ldots,A_{n}$ be normal operators, then
$$\oplus_{i=1}^{n}A_{i}= \left(
\begin{array}{ccccc}
A_{1} & & & 0 & \\
& A_{2} & & & \\
& & & \ddots & \\
& 0 & & & A_{n}\\
\end{array}\right)$$
is normal, so we have
$$(\oplus_{i=1}^{n}\Re(A_{i}))(\oplus_{i=1}^{n}\Im(A_{i}))=(\oplus_{i=1}^{n}\Im(A_{i}))(\oplus_{i=1}^{n}\Re(A_{i}))).$$
By above equation, we obtain the following
$$\sqrt{(\oplus_{i=1}^{n}A_{i})^{*}(\oplus_{i=1}^{n}A_{i})}=\sqrt{(\oplus_{i=1}^{n}\Re(A_{i}))^{2}+(\oplus_{i=1}^{n}\Im(A_{i}))^{2}}.$$
So
\begin{eqnarray*}
s_{j}(\oplus_{i=1}^{n}A_{i})&=&s_{j}(|\oplus_{i=1}^{n}A_{i}|)\\
&=&s_{j}\left(\sqrt{(\oplus_{i=1}^{n}A_{i})^{*}(\oplus_{i=1}^{n}A_{i})}\right)\\
&=&s_{j}\left(\sqrt{(\oplus_{i=1}^{n}\Re(A_{i}))^{2}+(\oplus_{i=1}^{n}\Im(A_{i}))^{2}}\right),
\end{eqnarray*}
for $j=1,2,\ldots$.\\ By using Weyl's monotonicity principle \cite{bhabook} and the inequality\\
$$\sqrt{(\oplus_{i=1}^{n}\Re(A_{i}))^{2}+(\oplus_{i=1}^{n}\Im(A_{i}))^{2}}\leq
|\oplus_{i=1}^{n}\Re(A_{i})|+|\oplus_{i=1}^{n}\Im(A_{i})|,$$ we have
the following
$$s_{j}\left(\sqrt{(\oplus_{i=1}^{n}\Re(A_{i}))^{2}+(\oplus_{i=1}^{n}\Im(A_{i}))^{2}}\right)\leq s_{j}(|\oplus_{i=1}^{n}\Re(A_{i})|+|\oplus_{i=1}^{n}\Im(A_{i})|),$$
for $j=1,2,\ldots$. Now for proving left side inequality, we recall
the following inequality $$0\leq
(\Re(A_{i})+\Im(A_{i})^{*}(\Re(A_{i})+\Im(A_{i})\leq
2(\Re(A_{i})^{2}+\Im(A_{2})^{2}).$$ Therefore, by using the Weyl's
monotonicity principle we can write
$$s_{j}\left(\sqrt{\left((\oplus_{i=1}^{n}\Re(A_{i}))+(\oplus_{i=1}^{n}\Im(A_{i}))\right)^{*}\left((\oplus_{i=1}^{n}\Re(A_{i}))+(\oplus_{i=1}^{n}\Im(A_{i}))\right)}\right),$$
which is less than $$
\sqrt{2}s_{j}\left(\sqrt{(\oplus_{i=1}^{n}\Re(A_{i}))^{2}+(\oplus_{i=1}^{n}\Im(A_{i}))^{2}}\right).$$
for $j=1,2,\ldots$.
Therefore,
\begin{eqnarray*}
s_{j}((\oplus_{i=1}^{n}\Re(A_{i}))+(\oplus_{i=1}^{n}\Im(A_{i})))
&=&s_{j}(|(\oplus_{i=1}^{n}\Re(A_{i}))+(\oplus_{i=1}^{n}\Im(A_{i}))|)\\
&\leq&
\sqrt{2}s_{j}(\sqrt{(\oplus_{i=1}^{n}\Re(A_{i}))^{2}+(\oplus_{i=1}^{n}\Im(A_{i}))^{2}}).
\end{eqnarray*}
for $j=1,2,\ldots$.
\end{proof}
The following example shows that normal condition is necessary.
\begin{example}
Let $A=\left[\begin{array}{cc}
-1+i&1\\
i&1+2i\\
\end{array}\right]$, then a calculation shows $$s_{2}(\Re(A)+i\Im(A))\approx
1.34>s_{2}(|\Re(A)|+|\Im(A)|)\approx 1.27.$$
\end{example}
\begin{corollary}\label{coro1}
Let $A$ be a normal operator in $K(H)$. Then we have
$$(1/\sqrt{2})s_{j}(\Re(A)+\Im(A))\leq s_{j}(A)\leq s_{j}(|\Re(A)|+|\Im(A)|),$$
for $j=1,2,\ldots$.
\end{corollary}
For each complex number $x=a+ib$, we know the following inequality holds
\begin{equation}\label{vatt}
\frac{1}{\sqrt{2}}|a+b|\leq |x|\leq |a|+|b|.
\end{equation}
Now, by applying Corollary \ref{coro1}, we can obtain operator version of inequality (\ref{vatt}).\\
Here, we determine
the upper and lower bound for $A+iA^{*}$.
\begin{theorem}
Let $A_{1},A_{2},\ldots,A_{n}$ be in $K(H)$. Then
\begin{eqnarray*}
\sqrt{2}s_{j}\left(\oplus_{i=1}^{n}(\Re(A_{i})+\Im(A_{i}))\right)&\leq& s_{j}(\oplus_{i=1}^{n}(A_{i}+iA_{i}^{*}))\\
&\leq&
2s_{j}(\oplus_{i=1}^{n}(\Re(A_{i})+\Im(A_{i}))),
\end{eqnarray*}
for
$j=1,2,\ldots$.
\end{theorem}
\begin{proof}
Note that $A_{i}+iA^{*}_{i}$ is normal operator for $i=1,\ldots,n$,
so $T=\oplus_{i=1}^{n}(A_{i}+iA_{i}^{*})$ is normal. On the other
hand, we can write $T=\Re(T)+i\Im(T)$ where
$$\Re(T)=(\oplus_{i=1}^{n}(A_{i}+A_{i}^{*})+i\oplus_{i=1}^{n}(A_{i}^{*}-A_{i}))/2,$$
$$\Im(T)=(\oplus_{i=1}^{n}(A_{i}-A_{i}^{*})+i\oplus_{i=1}^{n}(A_{i}^{*}+A_{i}))/2i.$$
It is enough to compare $\Re(T)$ and $\Im(T)$ to see
$\Re(T)=\Im(T)$. So
\begin{equation}\label{mosavi}
\Re(T)+\Im(T)=\oplus_{i=1}^{n}(A_{i}+A_{i}^{*})+i\oplus_{i=1}^{n}(A_{i}^{*}-A_{i}).
\end{equation}
Now apply Theorem \ref{normal}, we have
\begin{eqnarray}
(1/\sqrt{2})s_{j}(\Re(T)+\Im(T))&\leq& s_{j}(\Re(T)+i\Im(T))\nonumber\\
&\leq&
s_{j}(|\Re(T)|+|\Im(T)|),\label{111}
\end{eqnarray}
for $j=1,2,\ldots$. Put (\ref{mosavi}),
$\Re(T)+i\Im(T)=\oplus_{i=1}^{n}(A_{i}+iA_{i}^{*})$ and $\Re(T)$ in
(\ref{111}) to obtain
\begin{equation}
(1/\sqrt{2})s_{j}(\oplus_{i=1}^{n}(A_{i}+A_{i}^{*})+i\oplus_{i=1}^{n}(A_{i}^{*}-A_{i}))\leq
s_{j}(\oplus_{i=1}^{n}(A_{i}+iA_{i}^{*})),
\end{equation}
and
\begin{eqnarray*}
s_{j}(\oplus_{i=1}^{n}(A_{i}+iA_{i}^{*})) &\leq
2s_{j}(\oplus_{i=1}^{n}(A_{i}+A_{i}^{*})/2
+i\oplus_{i=1}^{n}(A_{i}^{*}-A_{i})/2)\\
&=s_{j}(\oplus_{i=1}^{n}(A_{i}+A_{i}^{*})+i\oplus_{i=1}^{n}(A_{i}^{*}-A_{i})),
\end{eqnarray*}
for $j=1,2,\ldots$. By writing
$\Re(\oplus_{i=1}^{n}A_{i})=\oplus_{i=1}^{n}(A_{i}+A_{i}^{*})/2$ and
$\Im(\oplus_{i=1}^{n}A_{i})=\oplus_{i=1}^{n}(A_{i}-A_{i}^{*})/2i$ we
have
\begin{eqnarray*}
(1/\sqrt{2})s_{j}(2\Re(\oplus_{i=1}^{n}A_{i})+2\Im(\oplus_{i=1}^{n}A_{i}))&\leq&
s_{j}(\oplus_{i=1}^{n}(A_{i}+iA_{i}^{*}))\\
&\leq&
s_{j}(2\Re(\oplus_{i=1}^{n}A_{i})+2\Im(\oplus_{i=1}^{n}A_{i})),
\end{eqnarray*}
for
$j=1,2,\ldots$. Finally
\begin{eqnarray*}
\sqrt{2}s_{j}\left(\oplus_{i=1}^{n}(\Re(A_{i})+\Im(A_{i}))\right)&\leq& s_{j}(\oplus_{i=1}^{n}(A_{i}+iA_{i}^{*}))\\
&\leq&
2s_{j}(\oplus_{i=1}^{n}(\Re(A_{i})+\Im(A_{i}))),
\end{eqnarray*}
for
$j=1,2,\ldots$.
\end{proof}
|
2,869,038,154,781 | arxiv |
\section{Introduction}
As communication systems become more complex, \RevA{physical-layer design, i.e., devising optimal transmission and detection methods,} has become harder as well. This is true not only in wireless communication, where hardware impairments and quantization have increasingly become a limitation on the achievable performance, but also in optical communication, for which the nonlinear nature of the channel precludes the use of standard approaches. This has led to a new line of research \RevA{on physical-layer communication} where transmission and detection methods are learned from data. The general idea is to regard the transmitter and receiver as parameterized functions (e.g., neural networks) and find good parameter configurations using large-scale gradient-based optimization approaches from machine learning.
Data-driven methods have mainly focused on learning receivers assuming a given transmitter and channel, e.g., for MIMO detection \cite{Samuel2017} or decoding \cite{Nachmani2018}. These methods have led to algorithms that either perform better or exhibit lower complexity than model-based algorithms. More recently, end-to-end learning of both the transmitter and receiver has been proposed for various \RevA{physical-layer} applications including wireless \cite{OShea2017, Doerner2018}, nonlinear optical \cite{karanov2018end, li2018achievable, Jones2018}, and visible light communication\cite{Lee2018}.
In practice, gradient-based transmitter optimization is problematic since it requires a known and differentiable channel model. One approach to circumvent this limitation is to first learn a surrogate channel model, e.g., through an adversarial process, and use the surrogate model for the optimization \cite{OShea2018, Ye2018}. We follow a different approach based on stochastic transmitters, where the transmitted symbol for a fixed message is assumed to be a random variable during the training process \cite{Aoudia2018, Aoudia2018a, DeVrieze2018}. This allows for the computation of \emph{surrogate gradients} which can then be used to update the transmitter parameters. A related approach is proposed in \cite{Raj2018}.\footnote{See \cite[Sec.~III-C]{Aoudia2018a} for a discussion about the relationship between the approaches in \cite{Aoudia2018, Aoudia2018a, DeVrieze2018} and \cite{Raj2018}.}
In order to compute the surrogate gradients, the transmitter must receive a \emph{feedback signal} from the receiver. This feedback signal can either be perfect \cite{Aoudia2018, Aoudia2018a, Raj2018, DeVrieze2018} or noisy \cite{Goutay2018}. \RevC{In the latter case, it was proposed in \cite{Goutay2018} to regard the feedback transmission as a separate communication problem for which optimized transmitter and receiver pairs can again be learned. The proposed training scheme in \cite{Goutay2018} alternates between optimizing the different transmitter/receiver pairs, with the intuition that training improvements for one pair lead to better training of the other pair (and vice versa). Thus, both communication systems improve simultaneously and continuously until some predefined stopping criterion is met (see Alg.~3 in \cite{Goutay2018}). The assumed feedback link in \cite{Goutay2018} only allowed for the transmission of real numbers over an additive white Gaussian noise (AWGN) channel.} In practice, however, signals will be quantized to a finite number of bits, including the feedback signal. To the best of our knowledge, such quantization has not yet been considered in the literature. Studies on quantization have been conducted so far only in terms of the transmitter and receiver processing, for example when the corresponding learned models are implemented with finite resolution \cite{Kim2018b, Tang2018, Teng2018, Fougstedt2018ecoc, Aoudia2019}.
In this paper, we analyze the impact of quantization of the feedback signal on data-driven learning of physical-layer communication over an unknown channel. \RevC{Compared to \cite{Goutay2018}, the feedback transmission scheme is not learned. Instead, we show that due to the specific properties of the feedback signal, an adaptive scheme based on simple pre-processing steps followed by a fixed quantization strategy can lead to} \RevA{similar performance as compared to the case where unquantized feedback is used for training, even with $1$-bit quantization.} We provide a theoretical justification for the proposed approach and perform extensive simulations for both linear Gaussian and \RevC{nonlinear phase-noise channels}. \RevB{The detailed contributions in this paper are as follows:
\begin{enumerate}
\item We propose a novel quantization method for feedback signals in data-driven learning of physical-layer communication. The proposed method addresses a major shortcoming in previous work, in particular the assumption in \cite{Goutay2018} that feedback losses can be transmitted as unquantized real numbers over an AWGN channel.
\item We conduct a thorough numerical study demonstrating the effectiveness of the proposed scheme. We investigate the impact of the number of quantization bits on the performance and the training process, showing that $1$-bit quantization can provide performance similar to unquantized feedback. In addition, it is shown that the scheme is robust to noisy feedback where the quantized signal is perturbed by random bit flips.
\item We provide a theoretical justification for the effectiveness of the proposed approach in the form of Propositions 1 and 2. In particular, it is proved that feedback quantization and bit flips manifest themselves merely as a scaling of the expected gradient used for parameter training. Moreover, upper bounds on the variance of the gradient are derived in terms of the Fisher information matrix of the transmitter parameters.
\end{enumerate}
}%
\subsubsection*{Notation}
Vectors will be denoted with lower case letters in bold (e.g., $\mathbf{x}$), with $x_n$ or $[\mathbf{x}]_n$ referring to the $n$-th entry in $\mathbf{x}$; matrices will be denoted in bold capitals (e.g., $\mathbf{X}$); $\mathbb{E}(\{\mathbf{x}\}$ denotes the expectation operator; $\mathbb{V}(\mathbf{x})$ denotes the variance (the trace of the covariance matrix) of the random vector $\mathbf{x}$ (i.e., $\mathbb{V}\{\mathbf{x}\}=\mathbb{E}\{\mathbf{x}^\intercal\mathbf{x}\}-(\mathbb{E}\{\mathbf{x}\})^\intercal(\mathbb{E}\{\mathbf{x}\})$).
\begin{figure}
\centering
\includegraphics[width=0.95\columnwidth]{figures/figure1.eps}
\caption{Data-driven learning model where the discrete time index $k$ (e.g., $m_k$) is omitted for all variables. The quantization and binary feedback is shown in the lower dashed box, while the proposed pre-processor is highlighted. Note that $w=0$ for the receiver learning (Sec.~\ref{sec:receiver_learning}).}
\label{fig:model}
\end{figure}
\section{System Model}
\label{sec:model}
\newcommand{\triangleq}{\triangleq}
We wish to transmit messages \RevC{$m \in \{1, \ldots, M\}$} over an a priori unknown \RevB{static} memoryless channel which is defined by a conditional probability density function (PDF) $p(y|x)$, where $x,y \in \mathbb{C}$ and $M$ is the total number of messages.\footnote{In this paper, we restrict ourselves to two-dimensional (i.e., complex-valued) channel models, where the generalization to an arbitrary number of dimensions is straightforward.}
The communication system is implemented by representing the transmitter and receiver as two parameterized functions $f_{\ptx} : \RevC{\{1, \ldots, M\}} \to \mathbb{C}$ and $\mathbf{f}_{\rho} : \mathbb{C} \to [0,1]^M$, where \RevC{$[a,b]^M$ is the $M$--fold Cartesian product of the $[a,b]$--interval (i.e., the elements in $[a,b]^M$ are vectors of length $M$ with entries between $a$ and $b$ inclusively) and } $\tau$ and $\rho$ are sets of transmitter and receiver parameters, respectively. The transmitter maps the $k$-th message $m_k$ to a complex symbol $x_k = f_{\ptx}(m_k)$, where an average power constraint according to $\mathbb{E}\{ |x_k|^2\} \le P$ is assumed. The symbol $x_k$ is sent over the channel and the receiver maps the channel observation $y_k$ to a probability vector $\vect{q}_k = \mathbf{f}_{\rho}(y_k)$, where one may interpret the components of $\vect{q}_k$ as estimated posterior probabilities for each possible message. Finally, the receiver outputs an estimated message according to $\hat{m}_k = \arg \max_m [\vect{q}_k]_m$, where $[\vect{x}]_{m}$ returns the $m$-th component of $\vect{x}$. The setup is depicted in the top branch of the block diagram in Fig.~\ref{fig:model}, where the random perturbation $w$ in the transmitter can be ignored for now.
We further assume that there exists a feedback link from the receiver to the transmitter, which, as we will see below, facilitates transmitter learning. In general, our goal is to learn optimal transmitter and receiver mappings $f_{\ptx}$ and $\mathbf{f}_{\rho}$ using limited feedback.
\section{Data-Driven Learning}
\label{sec_theory}
\newcommand{\ExpL}{\ell}
\newcommand{\EmpL}{\ell^{\text{e}}}
In order to find good parameter configurations for $\tau$ and $\rho$, a suitable optimization criterion is required. Due to the reliance on gradient-based methods, conventional criteria such as the symbol error probability $\Pr(m_k \neq \hat{m}_k)$ cannot be used directly. Instead, it is common to minimize the expected cross-entropy loss defined by
\begin{align}
\ExpL (\tau,\rho) \triangleq - \mathbb{E}\{\log([\mathbf{f}_{\rho}(y_k)]_{m_k})\},
\end{align}
where the dependence of $\ExpL (\tau,\rho)$ on $\tau$ is implicit through the distribution of $y_k$.
A major practical hurdle is the fact that the gradient $\nabla_\tau \ExpL (\tau,\rho)$ cannot actually be evaluated because it requires a known and differentiable channel model. To solve this problem, we apply the alternating optimization approach proposed in \cite{Aoudia2018, Aoudia2018a}, which we briefly review in the following. For this approach, one alternates between optimizing first the receiver parameters $\rho$ and then the transmitter parameters $\tau$ for a certain number of iterations $N$. To that end, it is assumed that the transmitter and receiver share common knowledge about a database of training data $m_k$.
\subsection{Receiver Learning}
\label{sec:receiver_learning}
For the receiver optimization, the transmitter parameters $\tau$ are assumed to be fixed. The transmitter maps a mini-batch of uniformly random training messages $m_k$, \RevC{$k \in \{1,\ldots, B_R\}$}, to symbols satisfying the power constraint and transmits them over the channel. The receiver observes $y_1,\ldots,y_{B_R}$ and generates $B_R$ probability vectors $\mathbf{f}_{\rho}(y_1), \ldots, \mathbf{f}_{\rho}(y_{B_R})$.
The receiver then updates its parameters $\rho$ according to $\rho_{i+1} = \rho_{i} - \alpha_R \nabla_{\rho} \EmpL_R(\rho_i)$, where
\begin{align}
\EmpL_R(\rho) = -\frac{1}{B_R}\sum^{B_R}_{k=1}\log([\mathbf{f}_{\rho}(y_{k})]_{m_{k}})
\end{align}
is the empirical cross-entropy loss associated with the mini-batch and $\alpha_R$ is the learning rate. This procedure is repeated iteratively for a fixed number of iterations $N_R$.
\subsection{Transmitter Learning}
For the transmitter optimization, the receiver parameters are assumed to be fixed. The transmitter generates a mini-batch of uniformly random training messages $m_k$, \RevC{$k \in \{1,\ldots, B_T\}$}, and performs the symbol mapping as before. However, before transmitting the symbols over the channel, a small Gaussian perturbation is applied, which yields $\tilde{x}_k = x_k + w_k$, where $w_k \sim \mathcal{CN}(0,\sigma_p^2)$ and reasonable choices for $\sigma_p^2$ are discussed in Sec.~\ref{sec:simulation}. Hence, we can interpret the transmitter as stochastic, described by the PDF
\begin{align}
\label{eq:gaussian_policy}
\pi_{\tau}(\tilde{x}_k|m_k) = \frac{1}{\pi \sigma_p^2}
\exp \left(
- \frac{|\tilde{x}_k - f_{\ptx}(m_k) |^2}{\sigma_p^2}
\right).
\end{align}
Based on the received channel observations, the receiver then computes per-sample losses $l_k=-\log([\mathbf{f}_{\rho}(y_{k})]_{m_{k}}) \in \mathbb{R}$ for \RevC{$k \in \{1,\ldots,B_T\}$}, and feeds these back to the transmitter via the feedback link. The corresponding received losses are denoted by
$\hat{l}_k$, where ideal feedback corresponds to $\hat{l}_k = l_k$. Finally, the transmitter updates its parameters $\tau$ according to $\tau_{i+1} = \tau_{i} - \alpha \nabla_{\tau} \EmpL_T(\tau_i)$, where
\begin{align}
\nabla_{\tau} \EmpL_T(\tau) = \frac{1}{B_T}\sum_{k=1}^{B_T} \hat{l}_k \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k). \label{eq:PolicyGradient1}
\end{align}
This procedure is repeated iteratively for a fixed number of iterations $N_T$, after which the alternating optimization continues again with the receiver learning. The total number of gradient steps in the entire optimization is given by $N(N_T+N_R)$.
A theoretical justification for the gradient in \eqref{eq:PolicyGradient1} can be found in \cite{Aoudia2018, Aoudia2018a, DeVrieze2018}. In particular, it can be shown that the gradient of $\ExpL_T(\tau) = \mathbb{E}\left\{ l_k \right\}$ is given by
\begin{align}
\label{eq:policy_gradient}
\nabla_{\tau} \ExpL_T(\tau) = \mathbb{E}\left\{ l_k \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\right\},
\end{align}
where the expectations are over the message, transmitter, and channel distributions. Note that \eqref{eq:PolicyGradient1} is the corresponding sample average for finite mini-batches assuming $\hat{l}_k = l_k$.
\begin{rem}
As pointed out in previous work, the transmitter optimization can be regarded as a simple form of reinforcement learning. In particular, one may interpret the transmitter as an agent exploring its environment according to a stochastic exploration policy defined by \eqref{eq:gaussian_policy} and receiving (negative) rewards in the form of per-sample losses. The state is the message $m_k$ and the transmitted symbol $\tilde{x}_k$ is the corresponding action. The learning setup belongs to the class of \emph{policy gradient methods}, which rely on optimizing parameterized policies using gradient descent. We will make use of the following well-known property of policy gradient learning:\footnote{To see this, one may first apply
$\nabla_{\tau} \log \pi_{\tau} = \frac{\nabla_{\tau} \pi_{\tau} }{\pi_{\tau}}$ and then use the fact that $\int \nabla_{\tau}\pi_{\tau}(\tilde{x}|m) \text{d} \tilde{x} = 0$ since $\int \pi_{\tau}(\tilde{x}|m) \text{d} \tilde{x} = 1$.}
\begin{align}
\label{eq:exp_grad_log_policy}
\mathbb{E}\left\{ \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\right\} = 0.
\end{align}
\end{rem}
\subsection{Loss Transformation}
\label{sec:losstransform}
\RevA{The per-sample losses can be transformed through a pre-processing function $f: \mathbb{R} \to \mathbb{R}$, which is known as reward shaping in the context of reinforcement learning \cite{ng1999policy}. } Possible examples for $f$ include:
\begin{itemize}
\item Clipping: setting $f(l_k)=\min(\beta,l_k)$ is used to deal with large loss variations and stabilize training \cite{mnih2015human}.
\item Baseline: setting $f(l_k)=l_k-\beta$ is called a constant baseline \cite{sutton2018reinforcement} and is often used to reduce the variance of the Monte Carlo estimate of the stochastic gradient \cite{ng1999policy}.
\item Scaling: setting $f(l_k)=\beta l_k$ only affects the magnitude of the gradient step, but this can be compensated with methods using adaptive step sizes (including the widely used Adam optimizer \cite{Kingma2014a}). However, aggressive scaling can adversely affect the performance \cite{gu2016q,islam2017reproducibility}.
\end{itemize}
\RevA{To summarize, it has been shown that training with transformed losses, i.e., assuming $\hat{l}_k = f(l_k)$ in \eqref{eq:PolicyGradient1}, is quite robust and can even be beneficial in some cases (e.g., by reducing gradient variance through baselines). Hence, one may conclude that the training success is to a large extent determined by the relative ordering of the losses (i.e., the distinction between good actions and bad actions). In this paper, reward shaping is exploited for pre-processing before quantizing the transformed losses to a finite number of bits. }
\section{Learning with Quantized Feedback}
\label{sec:quantized}
Previous work has mostly relied on ideal feedback, where $\hat{l}_k = l_k$ \cite{Aoudia2018, Aoudia2018a, Raj2018, DeVrieze2018}. Robustness of learning with respect to additive noise according to $\hat{l}_k = l_k + n_k$, $n_k \sim \mathcal{N}(0,\sigma^2)$, was demonstrated in \cite{Goutay2018}. In this paper, we take a different view and assume that there only exists a \emph{binary feedback channel} from the receiver to the transmitter. In this case, the losses must be quantized before transmission.
\subsection{Conventional Quantization}
\subsubsection*{Optimal Quantization} Given a distribution of the losses $p(l_k)$ and $q$ bits that can be used for quantization, the mean squared quantization error is
\begin{align}
\label{eq:quantizer}
\RevC{D = \mathbb{E}\{ (l_k - Q(l_k))^2\}.}
\end{align}
With $q$ bits, there are $2^q$ possible quantization levels which can be optimized to minimize $D$, e.g., using the Lloyd-Max algorithm \cite{lloyd1982least}.
\subsubsection*{Adaptive Quantization}
In our setting, the distribution of the per-sample losses varies over time as illustrated in Fig.~\ref{fig:loss_distribution}. For non-stationary variables, adaptive quantization can be used. The source distribution can be estimated based on a finite number of previously seen values and then adapted based on the Lloyd-Max algorithm. If the source and sink adapt based on quantized values, no additional information needs to be exchanged. If adaptation is performed based on unquantized samples, the new quantization levels need to be conveyed from the source to the sink. In either case, a sufficient number of realizations are needed to accurately estimate the loss distribution and the speed of adaptation is fixed.
\begin{figure}
\centering
\includegraphics[width=0.95\columnwidth]{figures/figure2.eps}
\caption{Illustration of the non-stationary loss distribution as a function of the number of training iterations in the alternating optimization. }
\label{fig:loss_distribution}
\end{figure}
\subsubsection*{Fixed Quantization}
\label{fixed-quantization}
We aim for a strategy that does not require overhead between transmitter and receiver. A simple non-adaptive strategy is to apply a fixed quantization. Under fixed quantization, we divide up the range $[0,\bar{l}]$ into $2^q-1$ equal-size regions of size $\Delta = \bar{l}/2^q$ so that
\begin{align}
\RevC{Q(l)= \frac{\Delta}{2} + \Delta \left\lfloor \frac{l}{\Delta} \right\rfloor.}
\end{align}Here, $\bar{l}$ is the largest loss value of interest. \RevC{The corresponding thresholds are located at $m\bar{l}/2^q$, where $m \in \{1, \ldots, 2^{q}-1\}$.} Hence, the function $Q(l)$ and its inverse $Q^{-1}(l)$ are fully determined by $\bar{l}$ and the number of bits $q$.
\subsection{Proposed Quantization}
\label{sec:proposed_quantization}
Given the fact that losses can be transformed without much impact on the optimization, as described in Sec.~\ref{sec:losstransform}, we propose a novel strategy that employs adaptive pre-processing followed by a fixed quantization scheme. The proposed method operates on mini-batches of size $B_T$. In particular, the receiver (source) applies the following steps:
\begin{enumerate}
\item Clipping: we clip the losses to lie within a range $[l_{\min},l_{\max}]$. Here, $l_{\min}$ is the smallest loss in the current mini-batch, while $l_{\max}$ is chosen such that the $5\%$ largest losses in the \RevC{mini-batch} are clipped. This effectively excludes very large per-sample losses which may be regarded as outliers. We denote this operation by $f_{\text{clip}}(\cdot)$.
\item Baseline: we then shift the losses with a fixed baseline $l_{\min}$. This ensures that all losses are within the range $[0,l_{\max}-l_{\min}]$. We denote this operation by $f_{\text{bl}}(\cdot)$.
\item Scaling: we scale all the losses by $1/(l_{\max}-l_{\min})$, so that they are within the range $[0,1]$. We denote this operation by $f_{\text{sc}}(\cdot)$.
\item Fixed quantization:
finally, we use a fixed quantization with $q$ bits and send $Q(\tilde{l_k})$, where \RevC{$Q(\cdot)$ is defined in \eqref{eq:quantizer} and} $\tilde{l}_k= f(l_k) = f_{\text{sc}}(f_{\text{bl}}(f_{\text{clip}}(l_k)))$, i.e., $f \triangleq f_{\text{sc}} \circ f_{\text{bl}} \circ f_{\text{clip}}$ denotes the entire pre-processing. \RevC{For simplicity, a natural mapping of quantized losses to bit vectors $\mathbb{B}^q$ is assumed where quantization levels are mapped in ascending order to $(0,\ldots, 0,0)^\intercal$, $(0,\ldots, 0,1)^\intercal$, \ldots, $(1,\ldots, 1,1)^\intercal$. In general, one may also try to optimize the mapping of bit vectors to the quantization levels in order to improve the robustness of the feedback transmission. }
\end{enumerate}
The transmitter (sink) has no knowledge of the functions $f_{\text{clip}}(\cdot)$, $f_{\text{bl}}(\cdot)$, or $f_{\text{sc}}(\cdot)$, and interprets the losses as being in the interval $[0,1]$. It thus applies $\hat{l}_k=Q^{-1}(\tilde{l}_k) \in [0,1]$ and uses the values $\hat{l}_k$ in \eqref{eq:PolicyGradient1}. \RevA{We note that some aspects of this approach are reminiscent of the Pop-Art algorithm from \cite{van2016learning}, where shifting and scaling are used to address non-stationarity during learning. In particular, Pop-Art can be used for general supervised learning, where the goal is to fit the outcome of a parameterized function (e.g., a neural network) to given targets (e.g., labels) by minimizing a loss function. Pop-Art adaptively normalizes the targets in order to deal with large magnitude variations and also address non-stationary targets. However, Pop-Art and the proposed method are different algorithms that have been proposed in different contexts, e.g., Pop-Art does not deal with quantization issues during learning. }
\RevA{In terms of complexity overhead, the proposed method requires one sorting operation in order to identify and clip the largest losses in each mini-batch (step 1). The baseline and scaling (steps 2 and 3) can be implemented with one real addition followed by one real multiplication. Finally, the quantizer can be implemented by using a look-up table approach. At the transmitter side (sink), the method only requires the dequantization step, which again can be implemented using a look-up table. }
\subsection{Impact of Feedback Quantization}
The effect of quantization can be assessed via the Bussgang Theorem \cite{rowe1982memoryless}, which is a generalization of MMSE decomposition. If we assume $l_k \sim p(l)$ with mean $\mu_l$ and variance $\sigma^2_l$, then \begin{align}
Q(l_k)=g l_k + w_k,\label{eq:BussgangModel}
\end{align}
in which $g \in \mathbb{R}$ is the Bussgang gain and $w_k$ is a random variable, uncorrelated with $l_k$, provided we set
\begin{align}
g =\frac{\mathbb{E}\{l_k Q(l_k)\}-\mu_{l}\mathbb{E}\{Q(l_k)\}}{\sigma_{l}^{2}}. \label{eq:BussgangGain}
\end{align}
\RevC{In general, the distribution of $w_k$ may be hard (or impossible) to derive in closed form. Note that the mean of $w_k$ is $\mathbb{E}\{Q(l_k)\}- g \mu_l$ and the variance is $\mathbb{V}\{Q(l_k)\} -g^2 \sigma^2_l$. }
When the number of quantization bits $q$ increases, $Q(l_k) \to l_k$ and thus $g \to 1$.
If we replace $l_k$ with $Q(l_k)$ in \eqref{eq:policy_gradient}, denote the corresponding gradient function by $\nabla_{\tau}\ExpL_{T}^{\mathrm{q}}(\tau)$, and substitute \eqref{eq:BussgangModel}, then the following proposition holds.
\begin{prop}\label{prop:1quant}
Let $\bm{\gamma}_k = l_k \nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)$, $l_k \in [0,1]$, with $\nabla_{\tau}\ExpL_{T}(\tau)=\mathbb{E}\{ \bm{\gamma}_k\}$,
and $\bm{\gamma}^{\mathrm{q}}_k = Q(l_k) \nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)$, then
\begin{align}
\label{eq:Thmquant1}
& \mathbb{E}\{ \bm{\gamma}^{\mathrm{q}}_k\} = \nabla_{\tau}\ExpL_{T}^{\mathrm{q}}(\tau) = g\nabla_{\tau} \ExpL_T(\tau)\\
& \mathbb{V}\{ \bm{\gamma}^{\mathrm{q}}_k\} \le g^{2}\mathbb{V}\{\bm{\gamma}_{k}\}+(g\bar{w}+\bar{w}^2)\mathrm{tr}\{\mathbf{J}(\tau)\
\end{align}
where $\mathbf{J}(\tau) = \mathbb{E}\{ \nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k) \nabla^\intercal_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)\} \succeq 0$ is the Fisher information matrix of the transmitter parameters $\tau$ and $\bar{w}=\max_{l}|gl-Q(l)| = |1-1/2^{q-1}-g|$ is a measure of the maximum quantization error.
\end{prop}
\begin{IEEEproof}
See Appendix.
\end{IEEEproof}
\smallskip
Hence, the impact of quantization, under a sufficiently large mini-batch size is a scaling of the expected gradient. Note that this scaling will differ for each mini-batch. The variance is affected in two ways: a scaling with $g^2$ and an additive term that depends on the maximum quantization error and the Fisher information at $\tau$. When $q$ increases, $g \to 1$ and $\bar{w} \to 0$, so that $\mathbb{V}\{ \bm{\gamma}^{\mathrm{q}}_k\} \to \mathbb{V}\{\bm{\gamma}_{k}\}$, as expected.
In general, the value of $g$ is hard to compute in closed form, but for 1-bit quantization and a Gaussian loss distribution, \eqref{eq:BussgangGain} admits a closed-form solution.\footnote{For Gaussian losses, $\bar{w}$ in Proposition \ref{prop:1quant} is not defined. The proposition can be modified to deal with unbounded losses.} In particular,
\begin{align}
g=
\begin{cases}
1/\sqrt{8 \pi \sigma_{l}^{2}} & \mu_l={1}/{2}\\
e^{-1/(8\sigma_{l}^{2})}/\sqrt{8 \pi \sigma_{l}^{2}} & \mu_l \in \{0,1\}.
\end{cases}
\label{eq:BussgangGain2}
\end{align}
In light of the distributions from Fig.~\ref{fig:loss_distribution}, we observe that (after loss transformation) for most iterations, $\mu_l \approx 1/2$ and $\sigma^2_l$ will be moderate (around $1/(8 \pi)$), leading to $g\approx 1$. Only after many iterations $\mu_l < 1/2$ and $\sigma^2_l$ will be small, leading to $g \ll 1$. Hence, for sufficiently large batch sizes, $1$-bit quantization should not significantly affect the learning convergence rate.
\subsection{Impact of Noisy Feedback Channels}
\label{sec:impact_of_noise}
For the proposed pre-processing and quantization scheme, distortions are introduced through the function $f(\cdot)$ (in particular the clipping) and the quantizer $Q(\cdot)$. Moreover, additional impairments may be introduced when the quantized losses are transmitted over a noisy feedback channel. We will consider the case where the feedback channel is a binary symmetric channel with flip probability $p \in [0,1/2)$. Our numerical results (see Sec.~\ref{sec:noisy_feedback_channel}) indicate that the learning process is robust against such distortions, even for very high flip probabilities. In order to explain this behavior, it is instructive to first consider the case where the transmitted per-sample losses are entirely random and completely unrelated to the training data. In that case, one finds that
\begin{align}
&\mathbb{E}\{ \hat{l}_k \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\}= \mathbb{E}\{ \hat{l}_k \} \mathbb{E}\left\{ \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\right\} = 0 \nonumber
\end{align}
regardless of the loss distribution or quantization scheme.
The interpretation is that for large mini-batch sizes, random losses simply ``average out'' and the applied gradient in \eqref{eq:PolicyGradient1} is close to zero. We can exploit this behavior and make the following statement.
\begin{prop}\label{prop:1bitnoisy}
Let $\bm{\gamma}^\mathrm{e}_k = \hat{l}_k\nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)$ where the binary version of $Q({l}_k)$ has been subjected to a binary symmetric channel with flip probability $p$ to yield $\hat{l}_k$. Then, for $1$-bit and $2$-bit quantization \RevC{with a natural mapping of bit vectors to quantized losses}, we have
\begin{align}
\mathbb{E}\{ \bm{\gamma}^{\mathrm{e}}_k\} = \nabla_{\tau}\ExpL_{T}^{\mathrm{e}}(\tau) = (1-2p)\nabla_{\tau} \ExpL_T^{\mathrm{q}}(\tau).
\nonumber
\end{align}
Moreover, for $1$-bit quantization,
\begin{align}
\mathbb{V}\{ \bm{\gamma}^{\mathrm{e}}_k\} \le \mathbb{V}\{\bm{\gamma}_{k}^{\mathrm{q}}\}+4p(1-p)\Vert\nabla_{\tau}\ell_{T}^{\mathrm{q}}(\tau)\Vert^{2} +p\mathrm{tr}\{\mathbf{J}(\tau)\}.
\nonumber
\end{align}
\end{prop}
\begin{IEEEproof}
See Appendix.
\end{IEEEproof}
\smallskip
Hence, for a sufficiently large mini-batch size, the gradient is simply scaled by a factor $1-2p$. This means that even under very noisy feedback, learning should be possible.
\begin{rem}
Note that when using small mini-batches, the empirical gradients computed via \eqref{eq:PolicyGradient1} will deviate from the expected value $(1-2p)\nabla_{\tau} \ExpL_T^{\text{q}}(\tau)$: they will not be scaled exactly by $1-2p$ and they will be perturbed by the average value of $p \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)$. Hence, robustness against large $p$ can only be offered for large mini-batch sizes.
\end{rem}
\section{Numerical Results}
\label{sec:simulation}
In this section, we provide extensive numerical results to verify and illustrate the effectiveness of the proposed loss quantization scheme. In the following, the binary feedback channel is always assumed to be noiseless except for the results presented in Sec.~\ref{sec:noisy_feedback_channel}.\footnote{TensorFlow source code is available at \url{https://github.com/henkwymeersch/quantizedfeedback}.}
\subsection{Setup and Parameters}
\subsubsection{Channel Models}
We consider two memoryless channel models $p(y |x)$: the standard AWGN channel $y = x + n$, where $n\sim \mathcal{CN}(0, \sigma^2) $, and a \RevC{simplified memoryless fiber-optic channel which is defined by the recursion}
\begin{align}
\label{eq:nlpn}
x_{i+1} = x_{i} e^{\jmath L{\gamma}\mid x_{i}\mid ^{2}\slash K} + n_{i+1}, \quad 0\leq i < K,
\end{align}
where $x_0 = x$ is the channel input, $y = x_K$ is the channel output, $n_{i+1} \sim \mathcal{CN}(0, \sigma^2/K)$, $L$ is the total link length, $\sigma^2$ is the noise power, and ${\gamma} \geq 0$ is a nonlinearity parameter. Note that this channel reverts to the AWGN channel when ${\gamma} =0$. For our numerical analysis, we set $L = \RevC{5000}\,$km, ${\gamma} = 1.27\,$rad/W/km, $K = \RevC{50}$, and $\sigma^2 = -21.3\,$dBm, which are the same parameters as in \cite{li2018achievable, Aoudia2018a, Keykhosravi2019}. For both channels, we define $\text{SNR} \triangleq P/\sigma^2$. Since the noise power is assumed to be fixed, the SNR is varied by varying the signal power $P$.
The model in \eqref{eq:nlpn} assumes ideal distributed amplification across the optical link and is obtained from the nonlinear
Schr\"odinger equation by neglecting dispersive effects, see, e.g., \cite{Yousefi2011a} for more details about the derivation.
Because dispersive effects are ignored, the model does not necessarily reflect the actual channel conditions in realistic fiber-optic transmission. The main interest in this model stems from its simplicity and analytical tractability while still capturing some realistic nonlinear effects, in particular the nonlinear phase noise. The model has been studied intensively in the literature, including detection schemes \cite{Ho2005, Lau2007c, tan2011ml}, signal constellations \cite{Lau2007c, Haeger2013tcom}, capacity bounds \cite{Turitsyn2003, Yousefi2011a, keykhosravi2017tighter, Keykhosravi2019}, and most recently also in the context of machine learning \cite{li2018achievable, Aoudia2018a}.
In the following, we refer to the model as the nonlinear phase-noise channel to highlight the fact that it should not be seen as an accurate model for fiber-optic transmission.
\subsubsection{Transmitter and Receiver Networks} Following previous work, the functions $f_{\ptx}$ and $\mathbf{f}_{\rho}$ are implemented as multi-layer neural networks. A message $m$ is first mapped to a $M$--dimensional "one-hot" vector where the $m$--th element is $1$ and all other elements are $0$. Each neuron takes inputs from the previous layer and generates an output according to a learned linear mapping followed by a fixed nonlinear activation function. The final two outputs of the transmitter network are normalized to ensure
${1}/{B}\sum_{k=1}^{B} |x_k|^2=P$, $B\in \{B_T, B_R\}$, and then used as the channel input. The real and imaginary parts of the channel observation serve as the input to the receiver network. All network parameters are summarized in Table~\ref{tab:network_parameters}, where $M = 16$.
\begin{table}
\centering
\caption{Neural network parameters, where $M = 16$}
\begin{tabular}{c|ccc|ccc}
\toprule
& \multicolumn{3}{c}{transmitter $f_{\ptx}$} & \multicolumn{3}{|c}{receiver $\mathbf{f}_{\rho}$} \\ \midrule
layer & 1 & 2-3 & 4 & 1 & 2-3 & 4 \\
number of neurons & M & 30 & 2 & 2 & 50 & M \\
activation function & - & ReLU & linear & - & ReLU & softmax \\
\bottomrule
\end{tabular}
\label{tab:network_parameters}
\end{table}
\subsubsection{Training Procedure}
For the alternating optimization, we first fix the transmitter and train the receiver for $N_R = 30$ iterations with a mini-batch size of $B_R = 64$. Then, the receiver is fixed and the transmitter is trained for $N_T = 20$ iterations with $B_T = 64$. This procedure is repeated $N = 4000$ times \RevC{for the AWGN channel. For the nonlinear phase-noise channel, we found that more iterations are typically required to converge, especially at high input powers, and we consequently set $N = 6000$.} The Adam optimizer is used to perform the gradient updates, where $\alpha_T = 0.001 $ and $ \alpha_R = 0.008$. \RevC{The reason behind the unequal number of training iterations for the transmitter and receiver is that the receiver network is slightly bigger than the transmitter network and thus requires more training iterations to converge. }
\subsubsection{Transmitter Exploration Variance}
We found that the parameter $\sigma_p^2$ has to be carefully chosen to ensure successful training. In particular, choosing $\sigma_p^2$ too small will result in insufficient exploration and slow down the training process. On the other hand, if $\sigma_p^2$ is chosen too large, the resulting noise may in fact be larger than the actual channel noise, resulting in many falsely detected messages and unstable training. In our simulations, we use $\sigma_p^2 = P \cdot 10^{-3} $.
\begin{figure}
\centering
\includegraphics[width=0.95\columnwidth]{figures/figure3.eps}
\caption{Symbol error rate achieved for $M=16$. The training SNR is $15\,$dB for the AWGN channel, whereas training is done separately for each input power (i.e., SNR) for \RevC{the nonlinear phase-noise channel}. }
\label{fig:loss_distribution2}
\end{figure}
\begin{figure}%
\centering
\subfigure[]{%
\label{fig:first}%
\includegraphics[width=0.45\columnwidth]{figures/figure4_a.eps}}%
\qquad
\subfigure[]{%
\label{fig:second}%
\includegraphics[width=0.45\columnwidth]{figures/figure4_b.eps}}%
\caption{Learned decision regions for \RevC{the nonlinear phase-noise channel}, $M=16$, and $P=-3\,$dBm (a) without quantizing per-sample losses and (b) using the proposed quantization scheme and 1-bit quantization.}
\label{fig: decision_region}
\end{figure}
\subsection{Results and Discussion}
\subsubsection{Perfect vs Quantized Feedback}
We start by evaluating the impact of quantized feedback on the system performance, measured in terms of the symbol error rate (SER). For the AWGN channel, the transmitter and receiver \RevC{are} trained for a fixed $\text{SNR} = 15~\text{dB}$ (i.e., $P = - 6.3~\text{dBm}$ \RevC{such that $\text{SNR} = P/\sigma^2 = - 6.3~\text{dBm} + 21.3~\text{dBm} = 15~\text{dB}$}) and then evaluated over a range of SNRs \RevC{by changing the signal power} (similar to, e.g., \cite{Aoudia2018a}). \RevC{For the nonlinear phase-noise channel}, this approach cannot be used because optimal signal constellations and receivers are highly dependent on the transmit power.\footnote{In principle, the optimal signal constellation may also depend on the SNR for the AWGN channel.} Therefore, a separate transmitter--receiver pair is trained for each input power $P$. Fig.~\ref{fig:loss_distribution2} shows the achieved SER assuming both perfect feedback without quantization and a $1$-bit feedback signal based on the proposed method. \RevC{For both channels, the resulting communication systems with $1$-bit feedback quantization have very similar performance to the scenario where perfect feedback is used for training, indicating that the feedback quantization does not significantly affect the learning process.} As a reference, the performance of standard $16$-QAM with a maximum-likelihood (ML) detector is also shown. \RevA{The ML detector makes a decision according to
\begin{align}
\label{eq:ml}
\hat{x}_\text{ML} = \argmax\limits_{m \in \{1,\ldots,M\}} p(y|s_m),
\end{align}
where $s_1, \ldots, s_M$ are all constellation points. For the nonlinear phase-noise channel, the channel likelihood $p(y|x)$ can be derived in closed form, see \cite[p.~225]{Ho2005}. For the AWGN channel, \eqref{eq:ml} is equivalent to a standard minimum Euclidean-distance detector.} The learning approach outperforms this baseline for both channels, \RevB{which is explained by the fact that the transmitter neural network learns better modulation formats (i.e., signal constellations) compared to $16$-QAM.}
Fig.~\ref{fig: decision_region} visualizes the learned decision regions for the quantized (right) and unquantized (left) feedback schemes assuming \RevC{the nonlinear phase-noise channel with $P = -3\,$dBm}. Only slight differences are observed which can be largely attributed to the randomness of the training process.
\subsubsection{Impact of Number of Quantization Bits}
Next, \RevC{the nonlinear phase-noise channel} for a fixed input power \RevC{$P = -3~\text{dBm}$ }is considered to numerically evaluate the impact of the number of quantization bits on the performance. Fig.~\ref{fig:ser_vs_num_bits} shows the achieved SER when different schemes are used for quantizing the per-sample losses. For a fixed quantization scheme without pre-processing (see Sec.~\ref{fixed-quantization}), the performance of the trained system is highly sensitive to the number of quantization bits and the assumed quantization range $[0, \bar{l}]$. For $\bar{l}=10$ with $1$ quantization bit, the system performance deteriorates noticeably and the training outcome becomes unstable, as indicated by the error bars (which are averaged over $10$ different training runs). For the proposed quantization scheme, the performance of the trained system is (i) essentially independent on the number of bits used for quantization and (ii) virtually indistinguishable from a system trained with unquantized feedback.
\begin{figure}
\centering
\includegraphics[width=0.95\columnwidth]{figures/figure5.eps}
\caption{Impact of the number of quantization bits on the achieved performance for \RevC{the nonlinear phase-noise channel} with $M =16$, \RevC{$P=-3~\text{dBm}$}. Results are averaged over $10$ different training runs where error bars indicate the standard deviation between the runs. }
\label{fig:ser_vs_num_bits}
\end{figure}
\subsubsection{Impact on Convergence Rate}
In Fig.~\ref{fig:loss_distribution3}, we show the evolution of the empirical cross-entropy loss $\EmpL_T(\tau)$ during the alternating optimization for \RevC{the nonlinear phase-noise channel with $P=-3~\text{dBm}$}. It can be seen that quantization manifests itself primarily in terms of a slightly decreased convergence rate during training. For the scenario where per-sample losses are quantized with $5$ bits, the empirical losses $\EmpL_T(\tau)$ converged after about \RevC{$160$ iterations}, which is the same as in the case of un-quantized feedback. For $1$-bit quantization, the training converges slightly slower, after around \RevC{$200$ iterations}\RevB{, which is a minor degradation compared to the entire training time. However, the slower convergence rate implies that it is harder to deal with changes in the channel. Hence, with 1-bit quantization, the coherence time should be longer compared to with unquantized feedback.}
\begin{figure}
\centering
\includegraphics[width=0.95\columnwidth]{figures/figure6.eps}
\caption{Evolution of $\EmpL_T(\tau)$ during the alternating optimization for \RevC{the nonlinear phase-noise channel} with $M=16$, $P=-3~\text{dBm}$. Results are averaged over $15$ different training runs where the shaded area indicates one standard deviation between the runs.}
\label{fig:loss_distribution3}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.95\columnwidth]{figures/figure7.eps}
\caption{Performance on \RevC{the nonlinear phase-noise channel} with \RevC{$M=16$, $P=-3~\text{dBm}$} when transmitting quantized losses over a noisy feedback channel modeled as a binary symmetric channel with flip probability $p$. Results are average over 10 runs where the error bars indicate one standard deviation between runs.}
\label{fig:ser_vs_flip_probability}
\end{figure}
\subsubsection{Impact of Noisy Feedback}\label{sec:noisy_feedback_channel}
In order to numerically evaluate the effect of noise during the feedback transmission, we consider again \RevC{the nonlinear phase-noise channel} for a fixed input power\RevC{ $P = -3~\text{dBm}$}. Fig.~\ref{fig:ser_vs_flip_probability} shows the achieved SER when transmitting the quantized per-sample losses over a binary symmetric channel with flip probability $p$ (see Sec.~\ref{sec:impact_of_noise}). It can be seen that the proposed quantization scheme is highly robust to the channel noise. For the assumed mini-batch size $B_T = 64$, performance starts to decrease only for very high flip probabilities and remains essentially unchanged for $p<0.1$ with $1$-bit quantization and for $p<0.2$ with $2$-bit quantization. A theoretical justification for this behavior is provided in Proposition \ref{prop:1bitnoisy}, which states that the channel noise manifests itself only as a scaling of the expected gradient. Thus, one may also expect that the learning process can withstand even higher flip probabilities by simply increasing the mini-batch size. Indeed, Fig.~\ref{fig:ser_vs_flip_probability} shows that when increasing the mini-batch size from $B_T=64$ to $B_T=640$, the noise tolerance for $1$-bit quantization increases significantly and performance remains unchanged for flip probabilities as high as $p=0.3$.
Note that for $p=0.5$, the achieved SER is slightly better than $(M-1)/M \approx 0.938$ corresponding to random guessing. This is because the receiver learning is still active, even though the transmitter only performs random explorations.
\section{Conclusions}
\label{sec:conclusion}
We have proposed a novel method for data-driven learning of physical-layer communication in the presence of a binary feedback channel. Our method relies on an adaptive clipping, shifting, and scaling of losses followed by a fixed quantization at the receiver, and a fixed reconstruction method at the transmitter. We have shown that the proposed method (i) can lead to good performance even under $1$-bit feedback; (ii) does not significantly affect the convergence speed of learning; and (iii) is highly robust to noise in the feedback channel.
The proposed method can be applied beyond physical-layer communication, to reinforcement learning problems in general, and distributed multi-agent learning in particular.
\section*{Appendix}
\subsection*{Proof of Proposition \ref{prop:1quant}}
The mean of $\bm{\gamma}^{\text{q}}_k$ can be computed as
\begin{align*}
& \mathbb{E}\{ \bm{\gamma}^{\text{q}}_k\} = \nabla_{\tau}\ExpL_{T}^{\text{q}}(\tau) \nonumber \\
& =\mathbb{E}\{Q(l_k)\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)\}\\
& =g\mathbb{E}\{l_k\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)\}+\mathbb{E}\{w_k\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)\}\\
& =g\mathbb{E}\{l_k\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)\}+\mathbb{E}\{w_k\}\mathbb{E}\{\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)\}\\
& =g\mathbb{E}\{l_k\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_k|m_k)\}=g\nabla_{\tau}\ell_{T}(\tau).
\end{align*}
We have made use of the fact that $w_k$ is uncorrelated with $l_k$ and that \eqref{eq:exp_grad_log_policy} holds. The variance can similarly be bounded as follows:
\begin{align*}
& \mathbb{V}\{ \bm{\gamma}^{\text{q}}_k\} \nonumber \\
& =
\mathbb{E}\{(Q(l_{k}))^{2}\Vert\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_{k}|m_{k})\Vert^{2}\}-g^{2}\Vert\nabla_{\tau}\ell_{T}(\tau)\Vert^{2}
\\
& =g^{2}\mathbb{E}\{l_{k}^{2}\Vert\nabla\log\pi_{\tau}(x_{k}|m_{k})\Vert^{2}\}-g^{2}\Vert\nabla_{\tau}\ell_{T}(\tau)\Vert^{2}\\
& +\mathbb{E}\{w_{k}^{2}\Vert\nabla\log\pi_{\tau}(x_{k}|m_{k})\Vert^{2}\}\\
& +2\mathbb{E}\{gl_{k}w_{k}\Vert\nabla\log\pi_{\tau}(x_{k}|m_{k})\Vert^{2}\}\\
& \le g^{2}\mathbb{V}\{\bm{\gamma}_{k}\}+\bar{w}^2\text{tr}\{\mathbf{J}(\tau)\}\\
& -2g\mathbb{E}\{w_{k}l_{k}\Vert\nabla\log\pi_{\tau}(\tilde{x}_{k}|m_{k})\Vert^{2}\}\\ \nonumber
& \le g^{2}\mathbb{V}\{\bm{\gamma}_{k}\}+\bar{w}^2\text{tr}\{\mathbf{J}(\tau)\} +2g\bar{w}\text{tr}\{\mathbf{J}(\tau)\} \nonumber
\end{align*}
We have made use of $-w_{k} l_{k} =l_k (g l_k - Q(l_k)) \le \max_{l_k} |g l_k-Q(l_k)| = \bar{w}$, that $l_k \le 1$, and that $\text{tr}\{\mathbf{J}(\tau)\} = \mathbb{E}\{\Vert\nabla\log\pi_{\tau}(x_{k}|m_{k})\Vert^{2}\}$.
\subsection*{Proof of Proposition \ref{prop:1bitnoisy}}
For the proposed adaptive pre-processing and fixed $1$-bit quantization, the quantized losses $l_k$ are either $\Delta/2=1/4$ or $1-\Delta/2=3/4$. Assuming transmission over the binary symmetric channel, the gradient in \eqref{eq:policy_gradient} can be written as
\begin{align*}
\nabla_{\tau} \ExpL_T^{\text{e}}(\tau) = \mathbb{E}\{ Q(l_{k})^{1-n_k} (1-Q(l_{k}))^{n_k} \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\},
\end{align*}
where $n_k$ are independent and identically distributed Bernoulli random variables with parameter $p$. Since $n_k$ is independent of all other random variables, we can compute
\begin{align*}
\mathbb{E}[Q(l_{k})^{1-n_k} (1-Q(l_{k}))^{n_k} \,|\, Q(l_{k})] = (1-2p) Q(l_{k}) + p.
\end{align*}
Hence,
\begin{align*}
\label{eq:noisy_gradient3}
& \mathbb{E}\{ \bm{\gamma}^{\text{e}}_k\}= \nabla_{\tau} \ExpL_T^{\text{e}}(\tau)\\
& = \mathbb{E}\{ ((1-2p) Q(l_{k}) + p) \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\}\\
& = (1-2p) \mathbb{E}\{ Q(l_{k}) \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\}+ p \mathbb{E}\{ \nabla_{\tau} \log \pi_{\tau}(\tilde{x}_k|m_k)\} \nonumber \\
& = (1-2p)\nabla_{\tau} \ExpL_T^{\text{q}}(\tau),
\end{align*}
where the last step follows from \eqref{eq:exp_grad_log_policy}. For 2-bit quantization, the possible values are $\Delta/2=1/8$ (corresponding to bits 00), $3\Delta/2=3/8$ (corresponding to 01), $1-3\Delta/2=5/8$ (corresponding to 10), $1-\Delta/2=7/8$ (corresponding to 11). It then follows that when the transmitted loss is $Q(l_{k})$, the received loss is
\begin{align*}
Q(l_{k}) & \text{ with prob. }(1-p)^2\\
1-Q(l_{k}) & \text{ with prob. }p^2\\
\text{other}& \text{ with prob. }p(1-p)
\end{align*}
so that the expected received loss is $(1-2p)Q(l_{k})+p$.
The variance under 1-bit quantization can be computed as
\begin{align*}
& \mathbb{V}\{\bm{\gamma}_{k}^{\text{e}}\}\\
&=\mathbb{E}\{(\bm{\gamma}_{k}^{\text{e}})^{2}\}-(1-2p)^{2}\Vert\nabla_{\tau}\ell_{T}^{\text{q}}(\tau)\Vert^{2}\\
& = \mathbb{E}\{(Q(l_{k}))^{2(1-n_{k})}(1-Q(l_{k}))^{2n_{k}}\Vert\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_{k}|m_{k})\Vert^{2}\}\\
& -(1-2p)^{2}\Vert\nabla_{\tau}\ell_{T}^{\text{q}}(\tau)\Vert^{2}\\
& = \mathbb{E}\{Q^{2}(l_{k})\Vert\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_{k}|m_{k})\Vert^{2}\}+p\mathbb{E}\{\Vert\nabla\log\pi_{\tau}(\tilde{x}_{k}|m_{k})\Vert^{2}\}\\
& -2p\mathbb{E}\{Q(l_{k})\Vert\nabla\log\pi_{\tau}(\tilde{x}_{k}|m_{k})\Vert^{2}\}-(1-2p)^{2}\Vert\nabla_{\tau}\ell_{T}^{\text{q}}(\tau)\Vert^{2}\\
& = \mathbb{V}\{\bm{\gamma}_{k}^{\text{q}}\}+4p(1-p)\Vert\nabla_{\tau}\ell_{T}^{\text{q}}(\tau)\Vert^{2}+p\text{tr}\{\mathbf{J}(\tau)\}\\
& -2p\mathbb{E}\{Q(l_{k})\Vert\nabla_{\tau}\log\pi_{\tau}(\tilde{x}_{k}|m_{k})\Vert^{2}\}\\
& \le \mathbb{V}\{\bm{\gamma}_{k}^{\text{q}}\}+4p(1-p)\Vert\nabla_{\tau}\ell_{T}^{\text{q}}(\tau)\Vert^{2}+p \text{tr}\{\mathbf{J}(\tau)\},
\end{align*}
where the last step holds since $Q(l_k) \ge 0$.
\bibliographystyle{IEEEtran}
|
2,869,038,154,782 | arxiv | \section{Introduction}
\subsection{Motivation and main result}
We consider the $L^2$-critical generalized Korteweg-de Vries equation (gKdV)
\begin{equation} \label{gkdv}
\partial_tu+\partial_x\big(\partial_x^2u+u^5\big)=0 ,\quad (t,x) \in \mathbb R\times \mathbb R,
\end{equation}
where $u(t,x)$ is a real-valued function.
The mass $M(u)$ and the energy $E(u)$ are (formally) conserved by the flow of \eqref{gkdv} where
\begin{equation} \label{mass_energy}
M(u)=\int_{\mathbb R} u^2 \, dx \quad \text{and} \quad E(u)=\frac12 \int_{\mathbb R} (\partial_xu)^2 \, dx-\frac16 \int_{\mathbb R} u^6 \, dx .
\end{equation}
We recall the scaling invariance: if $u$ is a solution to \eqref{gkdv}, then for any $\lambda>0$
\begin{equation*}u_{\lambda}(t,x):=\lambda^{\frac12} u(\lambda^3 t, \lambda x)\end{equation*} is also a solution to \eqref{gkdv}.
Recall that the Cauchy problem for \eqref{gkdv} is locally well-posed in the energy space $H^1(\mathbb R)$ by the work of Kenig, Ponce and Vega \cite{KPV,KPV2}: for any $u_0 \in H^1(\mathbb R)$, there exists a unique (in a certain sense) \emph{maximal solution} of \eqref{gkdv} in $\mathcal{C}\big([0,T^{\star}): H^1(\mathbb R)\big)$ satisfying $u(0,\cdot)=u_0$. Moreover, we have the \emph{blow-up alternative}:
\begin{equation*} \text{if} \quad T^{\star}<+\infty, \quad \text{then} \quad \lim_{t \uparrow T^{\star}} \|\partial_xu(t)\|_{L^2}=+\infty .
\end{equation*}
For such $H^1$ solutions, the quantities $M(u)(t)$ and $E(u)(t)$ are conserved on $[0,T^{\star})$.
We recall the family of solitary wave solutions of \eqref{gkdv}. Let
$Q(x)=\big(3\sech^2(2x)\big)^{1/4}$ be the unique (up to translation) positive solution of the equation
\begin{equation} \label{eq:Q}
-Q''+Q-Q^5=0 \quad \mbox{on $\mathbb R$} .
\end{equation}
Then, the function
\begin{equation*}
u(t,x)=\lambda_0^{-\frac12}Q\big(\lambda_0^{-1}(x-\lambda_0^{-2}t-x_0)\big), \quad
\mbox{for any $(\lambda_0,x_0) \in
(0,+\infty) \times \mathbb R$} ,
\end{equation*}
is a solution of \eqref{gkdv}.
It is well-known that $E(Q)=0$ and that $Q$ is related to the following sharp Gagliardo-Nirenberg inequality (see \cite{Wei})
\begin{equation} \label{sharpGN}
\frac 13 \int_{\mathbb R} \phi^6 \le \left(\frac{\int_{\mathbb R} \phi^2}{\int_{\mathbb R} Q^2} \right)^2 \int_{\mathbb R} (\partial_x\phi)^2, \quad \forall \, \phi \in H^1(\mathbb R) .
\end{equation}
It follows from \eqref{sharpGN} and the conservation of the mass and the energy that
any initial data $u_0 \in H^1(\mathbb R)$ satisfying $\|u_0\|_{L^2} < \|Q\|_{L^2}$ generates a global in time solution of \eqref{gkdv} that is also bounded in $H^1(\mathbb R)$.
Now, we summarize available results on blow-up solutions for \eqref{gkdv} in the case of
initial data with mass equal or slightly above the threshold mass, \emph{i.e.} satisfying
\begin{equation*}
\|Q\|_{L^2}\leq \|u_0\|_{L^2}\leq (1+\delta_0) \|Q\|_{L^2} \quad \hbox{where}\quad 0<\delta_0\ll 1.
\end{equation*}
\begin{itemize}
\item At the threshold mass $\|u_0\|_{L^2}=\|Q\|_{L^2}$, there exists a unique (up to the invariances of the equation) blow-up solution $S(t)$
of the equation, which blows up in finite time (denoted by $T>0$) with the rate $\|S(t)\|_{H^1} \sim C(T-t)^{-1}$
as $t\to T$. See \cite{CoMa1,MaMeRa2}.
\item For mass slightly above the threshold, there exists a large set (including negative and zero energy solutions, and open in some topology) of blow-up solutions, with the blow-up rate $\|u(t)\|_{H^1} \sim C(T-t)^{-1}$
as $t\to T$. See \cite{MaMeRa1,Mjams} and other references therein.
\item In the neighborhood of the soliton for the same topology ($H^1$ solutions with suitable decay on the right),
there exists a $\mathcal C^1$ co-dimension one threshold manifold which separates the above stable blow-up behavior from
solutions that eventually exit the soliton neighborhood by vanishing.
Solutions on the manifold are global and locally converge to the ground state $Q$ up to the invariances of the equation.
In this class of initial data, one thus obtains the following trichotomy: stable finite time blowup, soliton behavior or exit. See~\cite{MaMeNaRa,MaMeRa1,MaMeRa2}.
\item There also exists a large class of exotic finite time blow-up solutions, close to the family of solitons, enjoying blow-up rates of the form $\|u(t)\|_{H^1} \sim C (T-t)^{-\nu}$ for any $\nu>\frac {11}{13}$.
Note that the exponent $\frac{11}{13}$ does not seem sharp
and it is an open question to determine the lowest finite time blow-up exponent for $H^1$ initial data.
Global solutions blowing up in infinite time with $\|u(t)\|_{H^1} \sim C t^{\nu}$ as $t\to \infty$,
were also constructed for any positive power $\nu>0$. See~\cite{MaMeRa3}.
Such exotic behaviors are generated by the interaction of the soliton with explicit slowly decaying tails added to the initial data.
Because of the tail, these $H^1$ solutions do not belong to the class where the trichotomy (blowup, soliton, exit) occurs.
\end{itemize}
We refer to the above mentioned articles and to the references therein for detailed results and previous references on the subject.
Recall that for the mass critical nonlinear Schr\"odinger equation (NLS), there exists a large class (stable in $H^1$) of blow-up solutions enjoying the so-called
$\log\log$ blow-up rate (see \cite{MR} and references therein), whereas (unstable) blow-up solutions with the conformal blow-up rate
$\|u(t)\|_{H^1}\sim C (T-t)^{-1}$ were also constructed by perturbation of the explicit minimal mass blow-up solution (\cite{BW,KS2,MRS}). Moreover, in the vicinity of the soliton,
it is proved in \cite{Ra} that solutions cannot have a blow-up rate
strictly between the $\log\log$ rate and the conformal rate.
It is an open question to build solutions with a blow-up rate higher than the conformal one (see however \cite{MaRa} in the case of several solitons).
The only available results concerning flattening solitons are deduced from the pseudo-conformal transformation applied to the solutions discussed above.
For the mass critical (NLS), the question of the existence of exotic behaviors is thus widely open.
The systematic study of exotic blow-up behaviors was initiated by the articles \cite{KST1,KST2} for energy critical dispersive models, followed by \cite{DK,HR,JJ,KS}.
(We also refer to \cite{GNT} for the construction of exotic solutions in other contexts.)
The article \cite{DK}, where a class of flattening bubbles is constructed for the energy critical wave equation on $\mathbb R^3$, is particularly related to our work.
More precisely, $W$ being the unique radial positive solution of $\Delta W+W^5=0$ on $\mathbb R^3$,
it is proved in \cite{DK} that for any $|\nu|\ll 1$, there exist global (for positive time) solutions of $\partial_t^2 u =\Delta u + |u|^4 u$ such that $u(t,x)\sim t^{\nu/2} W (t^{\nu} x)$ as $t\to +\infty$; the case $0<\nu\ll 1$ corresponds to blow-up in infinite time, while
$0<-\nu\ll 1$ corresponds to flattening solitons.
Such construction is especially motivated by the \emph{soliton resolution conjecture}, which
states that any global solution should decompose for large time into a certain number of decoupled solitons plus a dispersive part.
We refer to \cite{DKM} and references therein for the proof of the soliton resolution conjecture for the $3$D critical wave equation
in the radial case. It follows from \cite{DK} that some flexibility on the geometric parameters is necessary
in the statement of the conjecture.
The above mentioned works are a strong motivation for investigating exotic behaviors related to flattening solitons in the context of mass critical dispersive models.
Our main result is the existence of such solutions for the critical generalized KdV equation.
\begin{theorem}\label{th:1}
Let any $\beta \in (\frac 13 ,1)$.
For any $\delta>0$, there exist $T_\delta>0$ and $u_0\in H^1(\mathbb R)$ with
$\|u_0-Q\|_{H^1}\leq \delta$ such that the solution $u$ of \eqref{gkdv} with initial data $u_0$ is global
for $t\geq 0$ and decomposes for all $t\geq 0$ as
\begin{equation*}
u(t,x)= \frac 1{\ell^{\frac 12}(t)} Q\left( \frac{x-x(t)}{\ell(t)}\right)+w(t,x)
\end{equation*}
where the functions $\ell(t)$, $x(t)$ and $w(t,x)$ satisfy
\begin{equation}\label{def:ell}
\ell(t)\sim\left(\frac t{T_\delta}\right)^{\frac {1-\beta}2},
\quad
x(t)\sim \frac{T_\delta}{\beta} \left(\frac t{T_\delta}\right)^{\beta}\quad \mbox{as $t\to +\infty$,}
\end{equation}
and
\begin{equation}\label{def:eta}
\sup_{t\geq 0}\|w(t)\|_{H^1}\leq \delta,\quad \ \lim_{t\to +\infty} \|w(t)\|_{H^1(x>\frac 12 x(t))}=0.
\end{equation}
\end{theorem}
Theorem~\ref{th:1} states the existence of solutions arbitrarily close to the soliton $Q$ which eventually defocus in large time
with scaling $\ell(t)\sim t^{-\nu}$ where $\nu=(1-\beta)/2$ is any value in~$(0,\frac 13)$.
The values of the exponents and multiplicative constants in \eqref{def:ell} are consistent with the formal equation
$x'(t)= \ell^{-2}(t)$ relating the two geometrical parameters $x(t)$ and $\ell(t)$.
Note that by continuous dependence of the solution of \eqref{gkdv} with respect to the initial data, the constant $T_\delta$ in Theorem~\ref{th:1} satisfies $T_\delta\to \infty$ as $\delta\to 0$ .
The estimates in \eqref{def:ell} make sense only for $t\gg T_\delta$ when the flattening regime appears.
Of course, one can use the scaling invariance of the equation to generate solutions with different multiplicative constants
in \eqref{def:ell}. In the statement of Theorem~\ref{th:1}, the scaling is adjusted so that one can compare the initial data
with the soliton $Q$. We refer to Remark~\ref{rk:T0} for details.
We also notice that $w(t)$ does not converge to $0$ in $H^1(\mathbb R)$ as $t\to +\infty$;
otherwise, it would hold $E(u(t))=0$ and $\int u^2(t)=\int Q^2$ and by variational arguments, $u(t)$ would be exactly a soliton.
However, the residue $w$ is arbitrarily small in $H^1$ and converges strongly to $0$ as $t\to \infty$ in the space-time region $x>\frac 12 x(t)\gg \ell(t)$ which largely includes the soliton.
To complement Theorem~\ref{th:1},
we prove in Section~\ref{S:6.5} that the solutions do not behave as solutions of the linear Airy equation $\partial_t v+\partial_x^3 v=0$ as $t\to \infty$ (non-scattering solutions).
We claim that the restriction $\beta\in(\frac 13,1)$ in Theorem~\ref{th:1} corresponds to the full range of relevant exponents.
Indeed, the exponent $\beta=\frac 13$ is related to self-similarity, and
in the region $x<t^{1/3}$, the question of existence or non-existence of coherent nonlinear structures is of different nature. See~\cite{MP} for several results in this direction.
As mentioned above, infinite time blow-up solutions with any positive power rate were constructed in \cite{MaMeRa3}. Thus
Theorem~\ref{th:1} essentially settles the question of all possible single soliton behaviors as $t\to+\infty$.
It also sheds some light on the classification of all possible behaviors in $H^1$, while the results in \cite{MaMeNaRa,MaMeRa1,MaMeRa2} hold in a stronger topology.
\begin{remark}
We note from the proof that all initial data in Theorem~\ref{th:1} have a tail on the right of the soliton of the form $c_0 x^{-\theta}$, for $c_0>0$ and $\theta=\frac {5\beta-1}{4\beta}\in (\frac 12,1)$.
Observe that for such value of $\theta$, this tail does not belong to $L^1(\mathbb R)$.
Recall from \cite{MaMeRa3} that $\theta\in (1,\frac 54]$ corresponds
to blowup in infinite time and $\theta \in (\frac 54,\frac {29}{18})$ to exotic blowup in finite time
(for negative values of the multiplicative constant $c_0$). This means that, except the remaining question of the largest value of $\theta$ leading to exotic blowup, the influence of such tails on the soliton is now well-understood.
\end{remark}
\begin{remark}
The more general statement Theorem~\ref{th:2} given in Section~\ref{S:6} provides a large set of initial data,
related to a one-parameter condition to control the scaling instability direction (in particular responsible for blowup
in finite time).
As in the classification given by \cite{MaMeRa1}, a strong topology related to $L^2$ weighted norm is necessary to avoid destroying the tail leading to the soliton flattening.
Therefore, though the phenomenon of flattening solitons may seem exotic, it is rather robust by perturbation in weighted norms,
its only instability in such spaces being related to the scaling direction.
Moreover, it follows from formal arguments that any small perturbation in that direction should lead to blowup with the blow-up rate $C(T-t)^{-1}$ or to exit of the soliton neighborhood.
This is analogous to the situation described by the construction of the $\mathcal C^1$ threshold manifold in~\cite{MaMeNaRa}. Here, because of
weaker decay estimates on the residue, we do not address the question of the regularity of this set.
\end{remark}
\begin{remark}
Flattening solitary waves were constructed in Theorem 1.5 of \cite{Lan2} for the following double power (gKdV) equations with
saturated nonlinearities
\begin{equation*}
\partial_t u + \partial_x(\partial_x^2 u + u^5 - \gamma |u|^{q-1} u) = 0 \quad \mbox{where $q>5$ and $0<\gamma\ll 1$.}
\end{equation*}
The blow-down rate and the position of the soliton are fixed
\begin{equation*}
\ell(t)\sim c_1 t^{\frac 2{q+1}},\quad x(t)\sim c_2 t^{\frac {q-3}{q+1}} \quad \mbox{as $t\to +\infty$.}
\end{equation*}
Observe that $q>5$ corresponds to $\frac 2{q+1}\in (0,\frac 13)$, \emph{i.e.} the same range of decay rates as
in Theorem~\ref{th:1} for equation \eqref{gkdv}.
Analogous results (construction of minimal mass solutions with exotic blow-up rates) were also established for a double power nonlinear Schr\"odinger equation in \cite{LeMaRa}.
\end{remark}
\subsection*{Notation}
For $x\in \mathbb R$, we denote $x_+=\max(0,x)$.
For a given small positive constant $0<\alpha^\star \ll 1$, $\delta(\alpha^\star)$ will denote a small constant with
\begin{equation*} \delta(\alpha^\star) \to 0 \quad \text{as} \quad \alpha^\star \to 0 .\end{equation*}
We will denote by $c$ a positive constant that may change from line to line. The notation $a \lesssim b$ (respectively, $a \gtrsim b$) means that $a \le c b$ (respectively, $a \ge c b$) for some positive constant $c$.
For $1 \le p \le +\infty$, $L^p(\mathbb R)$ denote the classical Lebesgue spaces.
We define the weighted spaces $L^ 2_\textnormal{loc}=L^2(\mathbb R; e^{-\frac{|y|}{10}}dy)$ and $L^2_B(\mathbb R)=L^2(\mathbb R; e^{\frac{y}{B}}dy)$, for $B \ge 100$ to be fixed later in the proof, through the norms
\begin{equation} \label{def:L2B}
\|f\|_{L^2_\textnormal{loc}}=\left(\int_{\mathbb R} f^2(y) e^{-\frac{|y|}{10}} dy \right)^ {\frac12} \quad \text{and} \quad \|f\|_{L^2_B}=\left( \int_{\mathbb R} f^2(y)e^{\frac{y}B} dy\right)^{\frac12} .
\end{equation}
It is clear from the definition that $\|f\|_{L^2_\textnormal{loc}} \lesssim \|f\|_{L^ 2_B}$.
For $f$, $g \in L^2(\mathbb R)$ two real-valued functions, we denote the scalar product
\begin{equation*} (f,g)=\int_{\mathbb R} f(x)g(x) dx .\end{equation*}
We introduce the generator of the scaling symmetry
\begin{equation} \label{def:lambda}
\Lambda f=\frac12 f +yf' .
\end{equation}
We also define the linearized operator $\mathcal{L}$ around the ground state by
\begin{equation} \label{def:L}
\mathcal{L}f=-f''+f-5Q^4f .
\end{equation}
From now on, for simplicity of notation, we write $\int$ instead of $\int_{\mathbb R}$ and omit~$dx$ in integrals.
\subsection{Strategy of the proof}
The overall strategy of the proof, based on the construction of a suitable ansatz and energy estimates,
follows the one developed in \cite{Ma,MaMeRa1,MaMeRa2,MaMeRa3,Me,RaSz} in similar contexts.
The originality of the present work lies mainly in the
prior preparation of suitable tails and the rigorous justification of all relevant flattening regimes.
\smallskip
\noindent (i) \textit{Definition of the slowly decaying tail.}
Given $c_0 >0$, $x_0 \gg 1$ and $\frac12<\theta<1$, we introduce a smooth function $f_0$ corresponding to a slowly decaying tail on the right:
\begin{equation*}
f_0(x)=c_0x^{-\theta} \ \text{for} \ x>\frac{x_0}2, \quad f_0(x)=0, \ \text{for} \ x<\frac{x_0}4 .
\end{equation*}
In the present case, a special care has to be taken in the preparatory step of understanding the evolution of such slowly decaying tails under the (gKdV) flow. Not only the decay rate is slower than the one in \cite{MaMeRa3}
but also the control of the solution
is needed close to the larger space-time region $x\gtrsim t^{\beta}$, for $\beta>\frac 13$.
Note that the proof uses the mass criticality of the
exponent (it extends to super-critical exponents). See Section~\ref{S:2}.
\smallskip
\noindent (ii) \textit{Emergence of the flattening regime.}
For $t_0\gg 1$, we consider the rescaled time variable
\begin{equation} \label{eq:s}
\frac{ds}{dt}=\frac1{\lambda^3} \iff s(t)=s_0+\int_{t_0}^t\frac{d\tau}{\lambda^3(\tau)} \,d\tau .
\end{equation}
In the variable $s$, the equations governing the parameters $(\lambda,\sigma,b) \in (0,+\infty)\times \mathbb R^2$ write
\begin{equation} \label{eq:la:si:b}
\frac{\lambda_s}{\lambda}+b=0, \quad \sigma_s=\lambda, \quad \frac{d}{ds}\left(\frac{b}{\lambda^2}+\frac{4}{\int Q}c_0 \lambda^{-\frac 32}\sigma^{-\theta}\right)=0 ,
\end{equation}
where the term $c_0 \lambda^{-\frac 32}\sigma^{-\theta}$ comes from the tail. See computations in Lemmas~\ref{lemma:est:rF}-\ref{lemma:gh}.
We integrate these equations following the formal argument in~\cite{MaMeRa3}. First, we observe integrating the last equation in \eqref{eq:la:si:b} that
\begin{equation} \label{eq:b}
\frac{b}{\lambda^2}+\frac{4}{\int Q}c_0 \lambda^{-\frac 32}\sigma^{-\theta}=l_0 ,
\end{equation}
where $l_0$ is a constant. As in \cite{MaMeRa3}, we focus on the regime $l_0=0$, which corresponds formally to
avoid the instability by scaling.
By combining \eqref{eq:b} with the first two equations in \eqref{eq:la:si:b}, this leads to
\begin{equation*}
\lambda^{-\frac12}\lambda_s=\frac{4}{\int Q}c_0\sigma^{-\theta}\sigma_s ,
\end{equation*}
which yields after integration
\begin{equation*}
\lambda^{\frac12} -\frac{2}{\int Q}\frac{c_0}{1-\theta}\sigma^{-\theta+1} =l_1 .
\end{equation*}
Since we expect $\lambda(s)\to +\infty$ as $s\to +\infty$, we can neglect the constant $l_1$, which leads us to
\begin{equation*}
\lambda^{\frac12} =\frac{2}{\int Q}\frac{c_0}{1-\theta}\sigma^{-\theta+1} .
\end{equation*}
This imposes the conditions $\theta<1$ and $c_0>0$. Now, we use the second equation in~\eqref{eq:la:si:b} to obtain
(using the condition $\frac12<\theta$ which also ensures that the tail belongs to the space $L^2$)
\begin{equation*}
\sigma_s=\lambda=\left(\frac2{\int Q}\frac{c_0}{1-\theta}\right)^2\sigma^{2-2\theta} \implies \sigma^{2\theta-1}(s)
=(2\theta-1)\left(\frac2{\int Q}\frac{c_0}{1-\theta}\right)^2s ,
\end{equation*}
after integrating over $[s_0,s]$ and choosing $\sigma^{2\theta-1}(s_0)=(2\theta-1) \big(\frac2{\int Q}\frac{c_0}{1-\theta} \big)^2s_0$.
Hence,
\begin{equation*}
\lambda(s)=(2\theta-1)^{\frac{2(1-\theta)}{2\theta-1}}\left(\frac2{\int Q}\frac{c_0}{1-\theta}\right)^{\frac2{2\theta-1}}s^{\frac{2(1-\theta)}{2\theta-1}} .
\end{equation*}
By using the first equation \eqref{eq:la:si:b}, we also compute
\begin{equation*}
b(s)=-\frac{2(1-\theta)}{2\theta-1}s^{-1} .
\end{equation*}
To simplify constants, we choose
\begin{equation} \label{def:c0}
c_0=\frac{\int Q}{2}(1-\theta)(2\theta-1)^{-(1-\theta)}>0 ,
\end{equation}
so that
\begin{equation} \label{resol:la:si:b}
\lambda(s)=s^{\frac{2(1-\theta)}{2\theta-1}}, \quad \sigma(s)=(2\theta-1)s^{\frac1{2\theta-1}} \quad \text{and} \quad b(s)=-\frac{2(1-\theta)}{2\theta-1}s^{-1} .
\end{equation}
To come back to the original time variable, we first need to solve \eqref{eq:s}. We set
\begin{equation*}
\beta=\frac{1}{5-4\theta}\in\left(\frac 13,1\right) \iff
\theta=\frac{5\beta-1}{4\beta} .
\end{equation*}
Then,
by choosing
\begin{equation*}
t_0=\frac{2\theta-1}{5-4\theta}s_0^{\frac{5-4\theta}{2\theta-1}} \quad \text{and} \quad c_s=\left(\frac2{3\beta-1}\right)^{\frac{3\beta-1}{2}} ,
\end{equation*}
we obtain
\begin{equation*}
t=\frac{2\theta-1}{5-4\theta}s^{\frac{5-4\theta}{2\theta-1}} \iff t= \frac{3\beta-1}{2}s^{\frac2{3\beta-1}} \iff s=c_st^{\frac{3\beta-1}{2}}.
\end{equation*}
Last, we deduce from \eqref{resol:la:si:b} that
\begin{equation} \label{resol:t:la:si:b}
\lambda(t)=c_{\lambda}t^{\frac{1-\beta}{2}} \quad \text{and} \quad \sigma(t)=c_{\sigma}t^{\beta} ,
\end{equation}
for some positive constants $c_{\lambda}$ and $c_{\sigma}$
(see \eqref{def:cts}).
\smallskip
\noindent (iii) \emph{Energy estimates.}
In order to construct an exact solution of \eqref{gkdv} satisfying the formal regime~\eqref{resol:t:la:si:b}, we use a variant
of the mixed energy-virial functional first introduced for (gKdV) in \cite{MaMeRa1} (the introduction of the
virial argument in the neighborhood of the soliton for critical (gKdV) goes back to \cite{MaMejmpa}).
Considering a defocusing regime induces a simplification (see also the energy estimates in~\cite{CoMa1}) that allows us to treat the whole range
$\beta\in (\frac 13,1)$ in spite of a basic ansatz and relatively large error terms. See Section~\ref{section:energy}.
\section{Persistence properties of slowly decaying tails on the right}\label{S:2}
In this section, we present a general result concerning the persistence of a class of slowly decaying tails for the critical gKdV equation in a suitable space-time region.
Let $\theta\in (\frac 12,1]$ and define
\begin{equation}\label{def:gar}
\beta=\frac{1}{5-4\theta}\in\left(\frac 13,1\right],\quad
\theta=\frac{5\beta-1}{4\beta},\quad
\nu=\frac{1-\beta}2\in\left[0,\frac13\right).
\end{equation}
For $c_0>0$ and $x_0 \gg 1$, we consider $f_0$ any smooth nonnegative function such that
\begin{equation} \label{def:f_0}
f_0(x)= \begin{cases} c_0 x^{-\theta} & \text{for} \ x>\frac{x_0}2 \\ 0 & \text{for} \ x<\frac{x_0}4 \end{cases}
\quad \text{and} \quad \Big|f_0^{(k)}(x)\Big| \lesssim c_0|x|^{-\theta-k}, \ \forall \, k \in \mathbb N, \ \forall \, x \in \mathbb R .
\end{equation}
Note that
\begin{equation} \label{est:f_0}
\|f_0\|_{L^2} \sim c_0\left(\int_{x>x_0/4}x^{-2\theta} \, dx \right)^{\frac12} \sim c_0 x_0^{-(\theta-\frac12 )}=\delta(x_0^{-1}) .
\end{equation}
Now, for $t_0 \gg 1$ to be fixed, let $f$ be a solution of the IVP
\begin{equation} \label{eq:q_0}
\left\{ \begin{aligned} & \partial_t f+\partial_x\big(\partial_x^2 f+f^5\big)=0, \\ &f(t_0,x)=f_0(x) .\end{aligned}\right.
\end{equation}
The main result of this section states that the special asymptotic behavior of $f_0(x)$ on the right persists for $f(t,x)$ in regions of the form $x\gtrsim t^{\beta}$.
\begin{proposition} \label{decay:q_0}
Let $\theta \in \big(\frac12,1\big]$, $\beta=\frac{1}{5-4\theta}$ and $c_0>0$.
For $x_0>0$ large enough, for any $\kappa_0>0$, setting $t_0:= ({x_0}/{\kappa_0} )^{1/\beta}$,
the solution $f$ of \eqref{eq:q_0} is global, smooth and bounded in $H^1$.
Moreover, it holds
for all $t \ge t_0$ and $x>\kappa_0 t^{\beta}\ge x_0$,
\begin{align} \label{decay:q_0.1}
\forall \, k \in \mathbb N, \quad & \big|\partial_x^k f(t,x)-f_0^{(k)}(x)\big| \lesssim |x|^{-(5\theta-2+k)} ,
\\ \label{decay:q_0.2}
& \big|\partial_t f(t,x)\big|\lesssim |x|^{-(\theta+3)}.
\end{align}
\end{proposition}
The rest of this section is devoted to the proof of Proposition~\ref{decay:q_0}, which
requires preparatory monotonicity lemmas based on variants of the so-called Kato identity (see \cite{Kato,MaMejmpa,MaMe}).
This result is a substantial generalization of Lemma 2.3 in \cite{MaMeRa3}, where only the case $\theta=1$ is treated.
Our proof allows regions $x \gtrsim t^{\beta}$ for any $\beta>\frac 13$.
Complementary results are obtained in \cite{MP}, where large regions close to $x=0$ are investigated
by similar functionals.
\begin{remark} \label{rem:decay}
Without loss of generality and for simplicity of notation, we reduce ourselves to prove
estimates \eqref{decay:q_0.1} and \eqref{decay:q_0.2} for the special value $\kappa_0=2$. Indeed, consider the function $\widetilde{f}(s,y)=\lambda^{\frac12}f(\lambda^3 s, \lambda y)$. Then $\widetilde{f}$ is a solution to \eqref{eq:q_0} where $\widetilde{f}_0=\widetilde f(0)$ satisfies \eqref{def:f_0} with $\widetilde{c}_0=\lambda^{\frac12-\theta}c_0$ instead of $c_0$. Moreover, the condition $x>2t^{\beta}$ rewrites $y>2\lambda^{3\beta-1}s^{\beta}>\kappa_0 s^{\beta}$ by choosing $\lambda=(2\kappa_0)^{-\frac1{3\beta-1}}$ (recall that $\beta>\frac13$).
\end{remark}
First, note that if $x_0$ is chosen large enough, it follows directly from the Cauchy theory developed in \cite{KPV} (see Corollary 2.9) and \eqref{est:f_0} that $f \in C(\mathbb R : H^s(\mathbb R))$ for all $s \ge 0$ and
\begin{equation} \label{bound:q_0}
\sup_{t \in \mathbb R}\|f(t)\|_{H^s} \lesssim \delta(x_0^{-1}) .
\end{equation}
Moreover, by using the sharp Gagliardo-Nirenberg inequality \eqref{sharpGN} and the conservations of the mass and the energy \eqref{mass_energy}, we deduce, for $x_0$ large enough, that
\begin{equation} \label{bound:df:dx}
\sup_{t \in \mathbb R} \| \partial_xf(t)\|_{L^2} \lesssim |E(f_0)| \lesssim x_0^{-(\theta+\frac12)} .
\end{equation}
Define $q(t,x):=f(t,x)-f_0(x)$. Then, it follows from \eqref{eq:q_0} that
\begin{equation} \label{eq:q}
\left\{ \begin{aligned} &\partial_tq+\partial_x\big(\partial_x^2q+(q+f_0)^5-f_0^5\big)=F_0, \\ &q(t_0)=0 , \end{aligned}\right.
\end{equation}
where
\begin{equation*} F_0:=-\partial_x^3f_0-\partial_x(f_0^5) . \end{equation*}
For any $\bar{r} \ge 0$, we define a smooth function $\omega_{\bar{r}}$ such that
\begin{equation} \label{def:omega}
\omega_{\bar{r}}(x)= x^{\bar{r}} \ \text{for} \ x \ge 2, \quad \omega_{\bar{r}}(x)=e^{\frac{x}8} \ \text{for} \ x \le 0, \quad \omega_{\bar{r}}' > 0 \ \text{on} \ \mathbb R.
\end{equation}
Observe that
\begin{equation}\label{def:omegabis}
|\omega_{\bar{r}}'' |+|\omega_{\bar{r}}'''|\le C\omega_{\bar{r}}' \ \text{on} \ \mathbb R,
\end{equation}
for some constant $C=C(\omega_{\bar r})>0$.
\begin{lemma} \label{decay_lemma.1}
Let $0<r<2\theta+4$, $r\neq 5$ and $0<\epsilon<\frac{3\beta-1}{20}|r-5|$.
Define
\begin{equation*}
M_r(t):= \int q^2(t,x) \omega_r(\bar{x}) \, dx \quad \text{where} \quad \bar{x}=\frac{x-t^{\beta}}{t^{\nu+\epsilon}} .
\end{equation*}
Then, for $x_0>1$ large enough, and any $t \ge t_0=\left(\frac{x_0}2\right)^{\frac1{\beta}}$,
\begin{multline}\label{decay_claim.1}
M_{r}(t)+\int_{t_0}^t\int\left[s^{-3\nu-\epsilon} q^2
+s^{-1} \bar x_+ q^2
+s^{-\nu-\epsilon}(\partial_xq)^2\right]\omega_{r}'(\bar{x})\, dx ds \\
\lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-r\epsilon} & \mbox{if $r<5$.}
\end{cases}
\end{multline}
\end{lemma}
\begin{proof}
To prove \eqref{decay_claim.1}, we differentiate $M_{r}$ with respect to time, use \eqref{eq:q} and integrate by parts in the $x$ variable to obtain
\begin{align*}
M'_r &= -3t^{-\nu-\epsilon}\int (\partial_xq)^2\omega_r'(\bar{x})+t^{-3\nu-3\epsilon}\int q^2\omega_r'''(\bar{x})- \beta t^{-3\nu-\epsilon}\int q^2\omega_{{r}}'(\bar{x})\\
&\quad-(\nu+\epsilon) t^{-1} \int q^2 \, \bar{x}\omega_{{r}}'(\bar{x}) -2t^{-\nu-\epsilon} \int \left(\frac{(q+f_0)^6}{6}-(q+f_0)^5q-\frac{f_0^6}{6}\right)\omega_{{r}}'(\bar{x})
\\ &
\quad -2\int \left( (q+f_0)^5-5f_0^4q-f_0^5\right)f_0'\omega_{{r}}(\bar{x}) +2\int q F_0 \omega_{{r}}(\bar{x}) \\& =: M_1 +M_2+M_3+M_4+M_5+M_6+M_7 .
\end{align*}
By using \eqref{def:omegabis}, for $t_0$ large enough, we have
\begin{equation} \label{decay_claim1.1}
|M_2|=t^{-3\nu-3\epsilon}\left| \int q^2\omega_{{r}}'''(\bar{x})\right| \leq ct_0^{-2\epsilon} t^{-3\nu-\epsilon} \int q^2\omega_{{r}}'(\bar{x})\leq -\frac 12 M_3 ,
\end{equation}
and so
\begin{equation*}
M_1+M_2+M_3\leq
-3t^{-\nu-\epsilon}\int (\partial_xq)^2\omega_r'(\bar{x})
- \frac{\beta}2 t^{-3\nu-\epsilon}\int q^2\omega_{{r}}'(\bar{x}).
\end{equation*}
Next, we estimate $M_j$ for $j=4,\cdots,7$ separately.
For future use, observe that by the assumption $0<\epsilon<\frac{3\beta-1}{20}|r-5|$ and $0<r<6$, we also have
$0<\epsilon<\frac{3\beta-1}{4}$.
We denote
\begin{equation}\label{notation}
M_j=\int_{\bar x<-t^{1-3\nu-2\epsilon}}+
\int_{-t^{1-3\nu-2\epsilon}<\bar x<0}+\int_{\bar x>0}=:M_j^-+M_j^0+M_j^+.
\end{equation}
\smallskip
\noindent \textit{Estimate for $M_4$.} It is clear that $M_4^+(t) \le 0$. Next,
for $t_0$ large enough,
\begin{equation*}
M_4^0 \leq(\nu+\epsilon) t^{-3\nu-2\epsilon}\int q^2 \omega_{{r}}'(\bar{x})
\leq (\nu+\epsilon) t_0^{-\epsilon} t^{-3\nu-\epsilon}\int q^2 \omega_{{r}}'(\bar{x})
\leq -\frac14M_3.
\end{equation*}
Then, it follows from the definition of $\omega_{{r}}$ in \eqref{def:omega} and \eqref{bound:q_0} that,
for $t_0$ large enough,
\begin{equation*}
M_4^-
\lesssim t^{-1}\int_{\bar x<-t^{1-3\nu-2\epsilon}}q^2|\bar x|e^{\frac{\bar x}{8}}
\lesssim t^{-1} e^{-\frac1{16}(t^{1-3\nu-2\epsilon)}}\int q^2
\lesssim t^{-10} ,
\end{equation*}
since $0<\epsilon<\frac{3\beta-1}4$.
\smallskip
\noindent \textit{Estimate for $M_5$.}
Using
\begin{equation*}
\left| \frac{(q+f_0)^6}{6}-(q+f_0)^5q-\frac{f_0^6}{6}\right| \lesssim q^2 f_0^4+q^6 ,
\end{equation*}
it holds
\begin{equation*}
\big| M_5 \big| \le c t^{-\nu-\epsilon}\int q^2 f_0^4 \omega_{{r}}'(\bar{x})
+ct^{-\nu-\epsilon}\int q^6 \omega_{{r}}'(\bar{x})
=:M_{5,1}+M_{5,2} .
\end{equation*}
We observe that, for $t_0$ large enough,
\begin{equation}\label{zone}
-t^{1-3\nu-2\epsilon}=-t^{\beta-\nu-2\epsilon}<\bar x \implies t^{\beta}-t^{\beta-2\epsilon}<x
\implies \frac 12 t^\beta <x .
\end{equation}
Thus, we deduce from \eqref{def:f_0}, and then
$4\theta \beta > 2\beta>2\nu$ (since $\theta>\frac12$ and $\beta > \frac13>\nu$),
that, for $t_0$ large enough,
\begin{align}
M_{5,1}^0+M_{5,1}^+
&\leq c t^{-\nu-\epsilon} \int_{x>\frac 12 t^\beta} q^2 x^{-4\theta} \omega_{{r}}'(\bar{x})
\leq c t^{-(\nu+4\theta \beta+\epsilon)} \int q^2 \omega_{{r}}'(\bar{x}) \nonumber
\\ &\leq c t_0^{-(\nu+4\theta \beta)+3\nu} t^{-3\nu-\epsilon} \int q^2 \omega_{{r}}'(\bar{x})
\leq -\frac 18 M_3. \label{M51}
\end{align}
As before for $M_4^-$, we have for $t_0$ large enough,
\begin{equation} \label{M51bis}
M_{5,1}^- \lesssim t^{-10} .
\end{equation}
To deal with $M_{5,2}$, we follow an argument in Lemma 6 of \cite{Mjams}. We have by using the fundamental theorem of calculus
\begin{equation*}
-q^2(x,t)\sqrt{\omega_{{r}}'(\bar{x})}=2\int_x^{+\infty}q\partial_xq\sqrt{\omega_{{r}}'(\bar{x})}+\frac12t^{-\nu-\epsilon}\int_x^{+\infty}q^2 \frac{\omega_{{r}}''(\bar{x})}{\sqrt{\omega_{{r}}'(\bar{x})}} ,
\end{equation*}
and so, by Cauchy Schwarz inequality and then \eqref{def:omegabis},
\begin{align}
\left\|q^2(x,t)\sqrt{\omega_{{r}}'(\bar{x})}\right\|_{L^\infty}^2
&\lesssim \|q\|_{L^2}^2\int (\partial_xq)^2 \omega_{{r}}'(\bar{x})
+t^{-2\nu-2\epsilon}\|q\|_{L^2}^2\int q^2\frac{\big(\omega_{{r}}''(\bar{x})\big)^2}{\omega_{{r}}'(\bar{x})}\nonumber\\
&\lesssim \|q\|_{L^2}^2\left[\int (\partial_xq)^2 \omega_{{r}}'(\bar{x})
+t^{-2\nu-2\epsilon}\int q^2 \omega_{{r}}'(\bar{x})\right].
\label{est:infty}
\end{align}
Therefore, using also \eqref{bound:q_0}, for $x_0$ large enough,
\begin{align}
M_{5,2} &\lesssim t^{-\nu-\epsilon} \|q\|_{L^2}^2 \left\| q^2\sqrt{\omega_{{r}}'(\bar{x})}\right\|_{L^\infty}^2 \label{M52} \\
& \le \delta(x_0^{-1})\left[ t^{-\nu-\epsilon}\int (\partial_x q)^2 \omega_{{r}}'(\bar{x})+ t^{-3\nu-\epsilon} \int q^2 \omega_{{r}}'(\bar{x})\right]
\leq -\frac 12 M_1-\frac 1{16}M_3.\nonumber
\end{align}
\smallskip
\noindent \textit{Estimate for $M_6$.} By using interpolation, \eqref{def:f_0} and then the inequality $|x|^{-\theta}q^5 \lesssim x^{-4\theta}q^2+q^6$, we observe that
\begin{equation*}
\left| f_0'\left((q+f_0)^5-5f_0^4q-f_0^5\right)\right| \lesssim |f_0'||f_0|^3q^2+|f_0'||q|^5
\lesssim |x|^{-1} f_0^4q^2+|x|^{-1}q^6 .
\end{equation*}
It follows that
\begin{equation*}
\big| M_6 \big| \le \int_{x\geq \frac 14 x_0}q^2 f_0^4 x^{-1} \omega_r(\bar{x})
+\int_{x\geq \frac 14 x_0} x^{-1}q^6 \omega_r(\bar{x})=:M_{6,1}+M_{6,2} .
\end{equation*}
By \eqref{def:omega} and \eqref{zone}, and choosing $\epsilon>0$ such that
$0<\epsilon<\beta-\nu=\frac{3\beta-1}2$,
\begin{equation} \label{omega:est}
\omega_{{r}}(\bar{x}) x^{-1}\lesssim
\begin{cases}
\omega'_{{r}}(\bar{x}) {\bar x}{x^{-1}}
\lesssim t^{-\nu-\epsilon}\omega_{{r}}'(\bar{x}) &\mbox{for $\bar x>2$,}
\\
t^{-\beta}\omega_{{r}}'(\bar{x})\lesssim t^{-\nu-\epsilon}\omega_{{r}}'(\bar{x}) &
\mbox{for $-t^{1-3\nu-2\epsilon}<\bar x<2$,}
\end{cases}
\end{equation}
Thus, for $t_0$ and $x_0$ large enough,
\begin{equation*}
M_{6,1}^{0,+}+M_{6,2}^{0,+} \leq c( M_{5,1}^{0,+}+M_{5,2}^{0,+})
\leq c (\delta(x_0^{-1})+\delta(t_0^{-1})) (M_1+M_3)
\leq -\frac 14 M_1-\frac 1{32}M_3 .
\end{equation*}
Last, $M_{6,1}^- +M_{6,2}^- \lesssim t^{-10}$ is proved as for $M_4^-$.
\smallskip
\noindent \textit{Estimate for $M_7$.} We get from the Cauchy-Schwarz inequality that
\begin{equation*}
\big| M_7 \big| \leq 2 \left( \int q^2 \omega_{{r}}'(\bar{x})\right)^{\frac12}\left(\int F_0^2 \frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})} \right)^{\frac12}
\le -\frac 1{64} M_3 + cM_8\quad \mbox{where}\quad M_8=t^{3\nu+\epsilon}\int F_0^2 \frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})}.
\end{equation*}
First, we see from \eqref{def:f_0} that
for $x>\frac 14 x_0$,
$|F_0| \lesssim |x|^{-\theta-3}$ ($\theta>\frac12$),
and for $x<\frac 14 x_0$, $F_0=0$.
For $\bar{x} \ge 2$, it holds $ \frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})} =r^{-1} |\bar{x}|^{{r}+1}
\lesssim t^{-(\nu+\epsilon)({r}+1)}|x|^{{r}+1}$. Hence,
\begin{align*}
t^{3\nu+\epsilon}\int_{\bar x>2} F_0^2 \frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})}
&\lesssim t^{2\nu-r(\nu+\epsilon)}\int_{\bar x>2} |x|^{-2(\theta+3)}| {x}|^{{r}+1}
\\ & \lesssim t^{2\nu-r(\nu+\epsilon)}\int_{x>t^\beta} |x|^{-2\theta+{r}-5}
\\ & \lesssim t^{2\nu-r(\nu+\epsilon)}t^{-\beta(2\theta-{r}+4)}
=t^{-1+\frac{3\beta-1}2 (r-5)-r\epsilon} ,
\end{align*}
since $2\theta-r+4>0$ by assumption, and
\begin{align}
1+2\nu-r(\nu+\epsilon) -\beta(2\theta-{r}+4)
&=r(\beta-\nu)+1+2\nu-2\beta\theta-4\beta-r\epsilon\nonumber\\
&=\frac{3\beta-1}2 (r-5)-r\epsilon.\label{pourtt}
\end{align}
For $-t^{-1-3\nu-2\epsilon}< \bar x<2$, it holds $ \frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})} \lesssim 1$
and $x\geq \frac 12 t^\beta$ (from \eqref{zone}) so that
\begin{align*}
t^{3\nu+\epsilon}\int_{-t^{-1-3\nu-2\epsilon}<\bar x<2} F_0^2 \frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})} &\lesssim t^{3\nu+\epsilon} \int_{x>\frac 12t^{\beta}}x^{-2(\theta+3)}
\\
&\lesssim t^{3\nu+\epsilon}t^{-\beta(2\theta+5)}=t^{-9\beta+2+\epsilon} .
\end{align*}
Last, for $\bar x<-t^{-1-3\nu-2\epsilon}$, then $\frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})} =8e^{\frac{\bar{x}}8}$ so that as for $M_4^-$,
\begin{equation*}
t^{3\nu+\epsilon}\int_{\bar x<-t^{-1-3\nu-2\epsilon}} F_0^2 \frac{\omega_{{r}}^ 2(\bar{x})}{\omega_{{r}}'(\bar{x})}
\lesssim t^{-10} .
\end{equation*}
Gathering all those estimates, we obtain in conclusion that, for some $c>0$,
\begin{equation*}
M_r'+c\int\left[t^{-3\nu-\epsilon} q^2
+t^{-1} \bar x_+ q^2
+t^{-\nu-\epsilon}(\partial_xq)^2\right]\omega_{r}'(\bar{x})\, dx
\lesssim
t^{-1+\frac{3\beta-1}2 (r-5)-r\epsilon}.
\end{equation*}
Observe that by the assumption $0<\epsilon<\frac{3\beta-1}{20} |r-5|$,
\begin{equation} \label{assump:epsilon}
r\epsilon<6\epsilon<\frac{3}{10}(3\beta-1)|r-5|<\frac{3\beta-1}2|r-5|.
\end{equation}
Thus, integrating this estimate on $[t_0,t]$, we obtain \eqref{decay_claim.1}.
\end{proof}
We prove a similar estimate for a quantity related to the energy.
\begin{lemma}
Let $0<r<2\theta+4$, $r\neq 5$ and $0<\epsilon<\frac{3\beta-1}{20} |r-5| .$
Define
\begin{equation*}
E_{{r}}(t):=t^{2\nu+2\epsilon}\int \left[ (\partial_xq)^2-\frac13\big((q+f_0)^6-f_0^6-6qf_0^5\big) \right](t,x) \omega_{{r}+2}(\bar{x}) \, dx .
\end{equation*}
where $\bar{x}=\frac{x-t^{\beta}}{t^{\nu+\epsilon}}$.
Then, for $x_0>1$ large enough, and any $t \ge t_0=\left(\frac{x_0}2\right)^{\frac1{\beta}}$,
\begin{multline} \label{decay_claim.2}
E_{{r}}(t)+\int_{t_0}^t\int\left[s^{-\nu+\epsilon} (\partial_xq)^2
+s^{2\nu+2\epsilon-1} \bar x_+(\partial_x q)^2
+s^{\nu+\epsilon} (\partial_x^2q)^2\right]\omega_{r+2}'(\bar{x})\, dxds \\ \lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-r\epsilon} & \mbox{if $r<5$.}
\end{cases}
\end{multline}
\end{lemma}
\begin{proof}
We differentiate $E_r$ with respect to time and integrate by parts to obtain
\begin{align*}
E_r'&= -t^{\nu+\epsilon}\int \left[\partial_x^2 q+(q+f_0)^5-f_0^5\right]^2 \omega_{r+2}'(\bar x)\\
&\quad -2t^{\nu+\epsilon}\int (\partial_x^2 q)^2 \omega_{r+2}'(\bar x)
+t^{-\nu-\epsilon}\int(\partial_x q)^2 \omega_{r+2}'''(\bar x)\\
&\quad -\beta t^{-\nu+\epsilon}
\int \ (\partial_xq)^2 \omega_{r+2}'(\bar{x})
-(\nu+\epsilon) t^{2\nu+2\epsilon-1}
\int (\partial_xq)^2 \bar x \omega_{r+2}'(\bar{x})\\
&\quad +\frac\beta3 t^{-\nu+\epsilon}\int \left[(q+f_0)^6-f_0^6-6qf_0^5\right]\omega_{r+2}'(\bar{x})\\
&\quad +\frac13 (\nu+\epsilon) t^{2\nu+2\epsilon-1}
\int \left[(q+f_0)^6-f_0^6-6qf_0^5 \right] \bar x \omega_{r+2}'(\bar{x})\\
&\quad +2t^{\nu+\epsilon}\int\left[(q+f_0)^5-f_0^5\right]_x (\partial_x q) \omega_{r+2}'(\bar{x})\\
&\quad +2t^{2\nu+2\epsilon}\int (\partial_x q ) F_0' \omega_{r+2}(\bar x) -2t^{2\nu+2\epsilon}\int\left[(q+f_0)^5-f_0^5\right] F_0 \omega_{r+2}(\bar{x})\\
&\quad +2(\nu+\epsilon)t^{2\nu+2\epsilon-1}\int (\partial_xq)^2\omega_{r+2}(\bar x)\\
&\quad -\frac23(\nu+\epsilon)t^{2\nu+2\epsilon-1}\int \left[(q+f_0)^6-f_0^6-6qf_0^5 \right] \omega_{r+2}(\bar x)
\\& =: E_1 +E_2+E_3+E_4+E_5+E_6+E_7+E_8+E_9 +E_{10}+E_{11}+E_{12} .
\end{align*}
First, observe that $E_1\leq 0$, $E_2\leq 0$ and $E_4\leq 0$.
As in the proof of \eqref{decay_claim1.1}, we have for $t_0$ large, $|E_3|\leq -\frac 12 E_4$.
Next, we use the same notation as in \eqref{notation} for $E_j$, $j=5,\cdots,10$. We observe that $E_5^+\leq 0$.
Moreover, as for the estimate of $M_4$ in the proof of \eqref{decay_claim.1},
it holds $E_5^0+E_5^-\leq -\frac 14 E_4+Ct^{-10}$.
\smallskip
\noindent \emph{Estimate for $E_6$}. First, we note
\begin{equation*}
|E_6|\leq c t^{-\nu+\epsilon}\int q^2 f_0^4 \omega_{r+2}'(\bar x)
+ct^{-\nu+\epsilon}\int q^6 \omega_{r+2}'(\bar x)=:E_{6,1}+E_{6,2}.
\end{equation*}
We estimate $E_{6,1}^-\lesssim t^{-10}$ and
\begin{align*}
E_{6,1}^+
&\lesssim t^{-\nu+\epsilon} \int_{\bar x>0} q^2 x^{-4\theta} \omega_{r+2}'(\bar{x})
\lesssim t^{-\nu+\epsilon} \int_{\bar x>0} q^2 x^{-4\theta}(1+\bar x^2) \omega_{r}'(\bar{x})\\
&
\lesssim t^{-3\nu-\epsilon } \int_{x> t^\beta} q^2 x^{-4\theta+2 }\omega_{{r}}'(\bar{x})
\lesssim t^{-3\nu-(4\theta-2)\beta-\epsilon } \int q^2 \omega_{{r}}'(\bar{x})
\lesssim t^{-3\nu-\epsilon } \int q^2 \omega_{{r}}'(\bar{x});
\end{align*}
moreover, since for $\bar x<0$, $\omega_{r+2}'(\bar x)= \omega_{r}'(\bar x)$, we deduce from \eqref{zone} that
\begin{equation*}
E_{6,1}^0\lesssim t^{-\nu+\epsilon}\int_{x>\frac12t^{\beta}} x^{-4\theta}q^2\omega_{r}'(\bar x)
\lesssim t^{-\nu+\epsilon-4\beta\theta}\int q^2 \omega_{{r}}'(\bar{x})\lesssim t^{-3\nu-\epsilon } \int q^2 \omega_{{r}}'(\bar{x}),
\end{equation*}
since $\epsilon<\frac{3\beta-1}2$.
Arguing as for $M_{5,2}$ in the proof of \eqref{decay_lemma.1},
\begin{equation*}
E_{6,2} \lesssim t^{-\nu+\epsilon}\|q\|_{L^2}^4 \int (\partial_xq)^2 \omega_{r+2}'(\bar{x})
+t^{-3\nu-\epsilon}\|q\|_{L^2}^4 \int q^2\frac{\big(\omega_{r+2}''(\bar{x})\big)^2}{\omega_{r+2}'(\bar{x})} ;
\end{equation*}
and thus, as before, for $x_0$ large,
\begin{equation*}
E_{6,2} \leq-\frac 14 E_4
+t^{-3\nu-\epsilon} \int q^2 \omega_{{r}}'(\bar{x}) .
\end{equation*}
\smallskip
\noindent \emph{Estimate for $E_7$}.
We estimate $E_7^-\leq t^{-10}$ and
\begin{equation*}
|E_7^0|\lesssim
t^{-\nu} \int_{-t^{1-3\nu-2\epsilon}<\bar x<0} (q^2 f_0^4+q^6) \omega_{r+2}'(\bar{x})
\lesssim t^{-\varepsilon} (E_{6,1}+E_{6,2}).
\end{equation*}
Now, we estimate
\begin{equation*}E_7^+\lesssim t^{-1+2\nu+2\epsilon} \int_{\bar x>0} q^2 f_0^4\bar x \omega_{r+2}'(\bar{x}) +t^{-1+2\nu+2\epsilon} \int_{\bar x>0} q^6 \bar x \omega_{r+2}'(\bar{x})
=:E_{7,1}^++E_{7,2}^+.
\end{equation*}
As before, we have
\begin{align*}
E_{7,1}^+
&\lesssim t^{-1+2\nu+2\epsilon} \int_{\bar x>0} q^2 x^{-4\theta} \bar x\omega_{r+2}'(\bar{x})
\lesssim t^{-1+2\nu+2\epsilon} \int_{\bar x>0} q^2 x^{-4\theta}(1+\bar x^2)\bar x \omega_{r}'(\bar{x})\\
&
\lesssim t^{-1} \int_{x> t^\beta} q^2 x^{-4\theta+2}\bar x_+\omega_{{r}}'(\bar{x})
\lesssim t^{-1} \int q^2 \bar x_+\omega_{{r}}'(\bar{x}).
\end{align*}
Last, by arguing similarly as in \eqref{M52},
\begin{align*}
E_{7,2}^+
&\lesssim t^{-1+2\nu+2\epsilon}\|q\|_{L^2}^2\left\|q^2\sqrt{\omega_{r+2}(\bar x)}\right\|_{L^\infty(\bar x>0)}^2\\
&\lesssim t^{-1+2\nu+2\epsilon}\|q\|_{L^2}^4\left(\int_{\bar x>0}(\partial_x q)^2\omega_{r+2}(\bar x)
+t^{-2\nu-2\epsilon}\int_{\bar x>0}q^2\frac{(\omega_{r+2}')^2}{\omega_{r+2}}\right)\\
&\lesssim
t^{-1+2\nu+2\epsilon}\delta(x_0^{-1})\left(\int_{\bar x>0}(\partial_x q)^2(1+\bar x)\omega_{r+2}'(\bar x)
+t^{-2\nu-2\epsilon}\int_{\bar x>0}q^2(1+\bar x)\omega_r'(\bar x)\right)\\
&\lesssim \delta(x_0^{-1}) (E_4+E_5^+) +
t^{-1}\delta(x_0^{-1})\left( \int_{\bar x>0}q^2 \omega_r'(\bar x)+ \int_{\bar x>0}q^2\bar x\omega_r'(\bar x)\right).
\end{align*}
\smallskip
\noindent \emph{Estimate for $E_8$}.
We compute
\begin{align*}
E_8&=
10t^{\nu+\epsilon}\int (q+f_0)^4(\partial_x q)^2 \omega_{r+2}'(\bar{x})
+10t^{\nu+\epsilon}\int \left[(q+f_0)^4-f_0^4\right](\partial_x q) f_0' \omega_{r+2}'(\bar{x}) \\
& \lesssim t^{\nu+\epsilon} \int f_0^4(\partial_x q)^2 \omega_{r+2}'(\bar{x})+ t^{\nu+\epsilon} \int q^4 (\partial_x q)^2\omega_{r+2}'(\bar{x}) \\ & \quad +t^{\nu+\epsilon} \int |q||\partial_xq||f_0|^3|f_0'|\omega_{r+2}'(\bar{x})
+t^{\nu+\epsilon} \int |q|^4|\partial_xq||f_0'|\omega_{r+2}'(\bar{x}) \\
& =: E_{8,1}+E_{8,2}+E_{8,3}+E_{8,4} .
\end{align*}
First, we have as before $E_{8,1}^- \lesssim t^{-10}$. Now, arguing as for $M_{5,1}$, we get for $t_0$ large enough that
\begin{equation*}
E_{8,1}^0+E_{8,1}^+ \le ct^{\nu+\epsilon} \int_{x>\frac12t^{\beta}}(\partial_xq)^2x^{-4\theta} \omega_{r+2}'(\bar{x}) \le ct^{-(-\nu+4\theta\beta-\epsilon)}\int(\partial_xq)^2 \omega_{r+2}'(\bar{x}) \le -\frac18E_4 .
\end{equation*}
To handle $E_{8,2}$, we use a similar argument as in \eqref{M52}. Observe from the fundamental theorem of calculus that
\begin{align*}
-[q\partial_xq](x,t)\sqrt{\omega_{{r+2}}'(\bar{x})}&=\int_x^{+\infty}q\partial_x^2q\sqrt{\omega_{{r+2}}'(\bar{x})}+\int_x^{+\infty}(\partial_xq)^2\sqrt{\omega_{{r+2}}'(\bar{x})}
\\ &\quad +\frac12t^{-\nu-\epsilon}\int_x^{+\infty}q\partial_xq \frac{\omega_{{r+2}}''(\bar{x})}{\sqrt{\omega_{{r+2}}'(\bar{x})}} .
\end{align*}
Thus, by the Cauchy-Schwarz inequality,
\begin{align*}
\left\| q\partial_xq \sqrt{\omega_{{r+2}}'(\bar{x})}\right\|_{L^{\infty}} &\lesssim \|q\|_{L^2}\left(\int (\partial_x^2q)^2 \omega_{{r+2}}'(\bar{x})\right)^{\frac12} +\int (\partial_xq)^2\sqrt{\omega_{{r+2}}'(\bar{x})}\nonumber \\ &\quad +t^{-\nu-\epsilon}\|q\|_{L^2}\left(\int (\partial_xq)^2 \frac{\big(\omega_{{r+2}}''(\bar{x})\big)^2}{\omega_{{r+2}}'(\bar{x})}\right)^{\frac12} .
\end{align*}
To deal with the second term on the right-hand side of the above inequality, we integrate by parts to get
\begin{equation*}
\int (\partial_xq)^2\sqrt{\omega_{{r+2}}'(\bar{x})}=-\int q\partial_x^2q \sqrt{\omega_{{r+2}}'(\bar{x})}-\frac12t^{-\nu-\epsilon} \int q\partial_xq\frac{\omega_{{r+2}}''(\bar{x})}{\sqrt{\omega_{{r+2}}'(\bar{x})}} .
\end{equation*}
Using again the Cauchy-Schwarz inequality and then \eqref{def:omegabis} and using \eqref{bound:q_0}, we deduce
\begin{align}
\left\| q\partial_xq \sqrt{\omega_{{r+2}}'(\bar{x})}\right\|_{L^{\infty}}^2
&\lesssim \|q\|_{L^2}^2\int (\partial_x^2q)^2 \omega_{{r+2}}'(\bar{x})
+t^{-2\nu-2\epsilon}\|q\|_{L^2}^2 \int (\partial_xq)^2 \frac{\big(\omega_{{r+2}}''(\bar{x})\big)^2}{\omega_{{r+2}}'(\bar{x})} \nonumber \\
&\lesssim \|q\|_{L^2}^2\left[ \int (\partial_x^2q)^2 \omega_{{r+2}}'(\bar{x})
+t^{-2\nu-2\epsilon}\int (\partial_xq)^2 \omega_{{r+2}}'(\bar{x})\right]
. \label{Sobo:firstderiv}
\end{align}
Therefore, for $x_0$ large enough, we obtain
\begin{align*}
E_{8,2} &\lesssim t^{\nu+\epsilon} \|q\|_{L^2}^2 \left\| q\partial_xq \sqrt{\omega_{{r+2}}'(\bar{x})}\right\|_{L^\infty}^2 \\
& \lesssim \delta(x_0^{-1}) \left[t^{\nu+\epsilon} \int (\partial_x^2q)^2 \omega_{{r+2}}'(\bar{x})
+t^{-\nu-\epsilon} \int (\partial_xq)^2\omega_{{r+2}}'(\bar{x}) \right]\\
& \le -\frac14 E_2-\frac 1{16}E_4 .
\end{align*}
Next we deal with $E_{8,3}$. It is clear that $E_{8,3}^- \lesssim t^{-10}$. Moreover, since $\omega_{r+2}'(\bar{x}) \sim \omega_r'(\bar{x})$ for $\bar{x}<2$, we deduce from \eqref{zone} and the Cauchy-Schwarz inequality that
\begin{align*}
t^{\nu+\epsilon}\int_{-t^{-1-3\nu-2\epsilon}<\bar{x}<2}&|q||\partial_xq||f_0^3||f_0'| \omega'_{r+2}(\bar{x})\\&\lesssim t^{\nu+\epsilon} \int_{x>\frac12t^{\beta}} |q| |\partial_xq| |x|^{-4\theta-1}\omega'_r(\bar{x})\\
&\lesssim t^{-\beta(4\theta+1)+\epsilon}\int q^2 \omega'_r(\bar{x}) +t^{2\nu-\beta(4\theta+1)+\epsilon}\int (\partial_xq)^2 \omega'_{r+2}(\bar{x}) \\
& \le ct^{-3\nu-\epsilon}\int q^2 \omega'_r(\bar{x})-\frac1{32}E_4 ,
\end{align*}
for $t_0$ large enough, since $\beta(4\theta+1)-3\nu=\frac52(3\beta-1)>0$. Finally, we use that $\omega'_{r+2}(\bar{x})=(r+2)\bar{x}\bar{x}^{\frac{r-1}2}\bar{x}^{\frac{r+1}2}$ when $\bar{x}>2$. Thus, we get by using again the Cauchy-Schwarz inequality that
\begin{align*}
t^{\nu+\epsilon}\int_{\bar{x}>2}|q||\partial_xq||f_0^3||f_0'| \omega'_{r+2}(\bar{x})&\lesssim \int_{x>t^{\beta}} |x|^{-4\theta} |q|\bar{x}^{\frac{r-1}2} |\partial_xq|\bar{x}^{\frac{r+1}2}\\
&\lesssim t^{-\nu-4\beta\theta-\epsilon}\int q^2 \omega'_r(\bar{x}) +t^{\nu-4\beta\theta+\epsilon}\int (\partial_xq)^2 \omega'_{r+2}(\bar{x}) \\
& \le ct^{-3\nu-\epsilon}\int q^2 \omega'_r(\bar{x})-\frac1{64}E_4 ,
\end{align*}
by taking $t_0$ large enough, since $4\beta\theta-2\nu=2(3\beta-1)$.
Finally we estimate $E_{8,4}$. First, we have from Young's inequality
\begin{equation*}
E_{8,4} \lesssim
t^{\nu+\epsilon} \int q^4 (\partial_x q)^2\omega_{r+2}'(\bar{x}) + t^{\nu+\epsilon} \int q^4 (f_0')^2\omega_{r+2}'(\bar{x})
=: E_{8,2}+E_{8,5} .
\end{equation*}
As before, it is clear that $E_{8,5}^- \lesssim t^{-10}.$ By interpolation, we have $q^4|f_0'|^2 \lesssim q^2f_0^4|x|^{-2}+q^6|x|^{-2}.$ Moreover, we get arguing as in \eqref{omega:est} that $\omega'_{r+2}(\bar{x})x^{-2} \lesssim t^{-2\nu-2\epsilon} \omega'_r(\bar{x})$ for $\bar{x}>-t^{1-3\nu-2\epsilon}$. Hence, we deduce using the estimates for $M_{5,1}^0$, $M_{5,1}^+$ and $M_{5,2}$ that
\begin{equation*}
E_{8,5}^0+E_{8,5}^+ \lesssim M_{5,1}^0+M_{5,1}^++M_{5,2}
\lesssim t^{-3\nu-\epsilon} \int q^2 \omega'_r(\bar{x})+t^{-\nu-\epsilon}\int (\partial_xq)^2 \omega'_r(\bar{x}) .
\end{equation*}
\smallskip
\noindent \emph{Estimate for $E_9$}.
We have
\begin{equation*}
\big| E_9 \big| \leq 2t^{2\nu+2\epsilon} \left( \int (\partial_xq)^2 \omega_{r+2}'(\bar{x})\right)^{\frac12}\left(\int (F_0')^2 \frac{\omega_{r+2}^2(\bar{x})}{\omega_{r+2}'(\bar{x})}\right)^{\frac12}
\le -\frac 1{2^7} E_4 + c\widetilde{E}_{9}\end{equation*}
where $\widetilde{E}_{9}=t^{5\nu+3\epsilon}\int (F_0')^2 \frac{\omega_{r+2}^ 2(\bar{x})}{\omega_{r+2}'(\bar{x})}$.
From \eqref{def:f_0}, it follows that
for $x>\frac 14 x_0$,
$|F_0'| \lesssim |x|^{-\theta-4}$
and for $x<\frac 14 x_0$, $F_0'=0$.
For $\bar{x} \ge 2$, it holds $ \frac{\omega_{r+2}^ 2(\bar{x})}{\omega_{r+2}'(\bar{x})} \lesssim |\bar{x}|^{r+3}
\lesssim t^{-(\nu+\epsilon)(r+3)}|x|^{r+3}$. Hence,
\begin{align*}
t^{5\nu+3\epsilon}\int_{\bar x>2} (F_0')^2 \frac{\omega_{r+2}^ 2(\bar{x})}{\omega_{r+2}'(\bar{x})}
&\lesssim t^{2\nu-r(\nu+\epsilon)}\int_{\bar x>2} |x|^{-2(\theta+4)}| {x}|^{{r}+3}
\\ & \lesssim t^{2\nu-r(\nu+\epsilon)}\int_{x>t^\beta} |x|^{-2\theta+{r}-5}
\\ & \lesssim t^{2\nu-r(\nu+\epsilon)}t^{-\beta(2\theta-{r}+4)}
=t^{-1+\frac{3\beta-1}2 (r-5)-r\epsilon},
\end{align*}
since $2\theta-r+4>0$ by assumption, and using \eqref{pourtt}.
For $-t^{-1-3\nu-2\epsilon}< \bar x<2$, it holds $ \frac{\omega_{r+2}^ 2(\bar{x})}{\omega_{r+2}'(\bar{x})} \lesssim 1$
and $x\geq \frac 12 t^\beta$ (from \eqref{zone}) so that
\begin{align*}
t^{5\nu+3\epsilon}\int_{-t^{-1-3\nu-2\epsilon}<\bar x<2}(F_0')^2 \frac{\omega_{r+2}^ 2(\bar{x})}{\omega_{r+2}'(\bar{x})}&\lesssim t^{5\nu+3\epsilon} \int_{x>\frac 12t^{\beta}}x^{-2(\theta+4)}
\\
&\lesssim t^{5\nu+3\epsilon}t^{-\beta(2\theta+7)}=t^{-12\beta+3+\epsilon} .
\end{align*}
Last, as before,
\begin{equation*}
t^{5\nu+3\epsilon}\int_{\bar x<-t^{-1-3\nu-2\epsilon}} (F_0')^2 \frac{\omega_{r+2}^ 2(\bar{x})}{\omega_{r+2}'(\bar{x})}
\lesssim t^{-10} .
\end{equation*}
\smallskip
\noindent \emph{Estimate for $E_{10}$}. We have
\begin{equation*}
|E_{10}|\lesssim
t^{2\nu+2\epsilon}\int |q| f_0^4 |F_0| \omega_{r+2}(\bar{x}) + t^{2\nu+2\epsilon}\int |q|^5 |F_0| \omega_{r+2}(\bar{x})
=:E_{10,1}+E_{10,2}.
\end{equation*}
First,
\begin{equation*}
E_{10,1}\lesssim t^{-3\nu-\epsilon} \int q^2 \omega_{r}'(\bar{x}) +
t^{7\nu+5\epsilon} \int f_0^8F_0^2\frac{\omega_{r+2}^2(\bar{x})}{\omega_{r}'(\bar{x})} .
\end{equation*}
Then,
\begin{align*}
t^{7\nu+5\epsilon} \int_{\bar x>2} f_0^8F_0^2\frac{\omega_{r+2}^2(\bar{x})}{\omega_{r}'(\bar{x})}
&\lesssim t^{7\nu+5\epsilon} \int_{\bar x>2} x^{-10\theta-6}|\bar x|^{r+5} \\ &
\lesssim t^{2\nu-r\nu-r\epsilon} \int_{x>t^\beta} x^{-10\theta+r-1}\\
&\lesssim t^{2\nu-r(\nu+\epsilon)-\beta(10\theta-r)} \\ &\lesssim t^{2\nu-r(\nu+\epsilon)}t^{-\beta(2\theta-{r}+4)}
=t^{-1+\frac{3\beta-1}2 (r-5)-r\epsilon},
\end{align*}
for $0<\epsilon$ small enough since $\theta>\frac12$,
\begin{equation*}
t^{7\nu+5\epsilon} \int_{-t^{-1-3\nu-2\epsilon}<\bar x<2} f_0^8F_0^2\frac{\omega_{r+2}^2(\bar{x})}{\omega_{r}'(\bar{x})}
\lesssim t^{7\nu-\beta(10\theta+5)+5\epsilon},
\end{equation*}
and
\begin{equation*}
t^{7\nu+5\epsilon} \int_{\bar x<-t^{-1-3\nu-2\epsilon}} f_0^8F_0^2\frac{\omega_{r+2}^2(\bar{x})}{\omega_{r}'(\bar{x})}
\lesssim t^{-10} .
\end{equation*}
Now, we deal with $E_{10,2}$. On the one hand, we estimate as before $E_{10,2}^- \lesssim t^{-10}$. On the other hand, we deduce by using \eqref{zone} and \eqref{omega:est} that
\begin{equation*}
E_{10,2}^0+E_{10,2}^+\lesssim t^{\nu+\epsilon}
\int_{x>\frac12t^{\beta}} |q|^5 |x|^{-(\theta+2)} \omega_{r+2}'(\bar x) \lesssim t^{+\nu+\epsilon} \left\|q^2 \sqrt{\omega_{r+2}'(\bar x)}\right\|_{L^\infty}^2\int_{x>\frac12t^\beta} |q| |x|^{-(\theta+2)}.
\end{equation*}
Thus, it follows arguing as in \eqref{M52} that for $t_0$ large enough,
\begin{align*}
E_{10,2}^0+E_{10,2}^+
&\lesssim t^{\nu+\epsilon} \left[\int_{x>\frac12t^\beta} |x|^{-2(\theta+2)}\right]^{\frac 12} \|q\|_{L^2}^3
\left[\int (\partial_x q)^2 \omega_{r+2}'(\bar x)
+t^{-2\nu-2\epsilon}\int q^2 \frac{(\omega_{r+2}''(\bar x))^2}{\omega_{r+2}'(\bar x)}\right]\\
&\lesssim t^{\nu-(\theta+\frac32)\beta+\epsilon}
\left[\int (\partial_x q)^2 \omega_{r+2}'(\bar x)
+t^{-2\nu-2\epsilon}\int q^2 \omega_r'(\bar x)\right] \\
& \le-\frac1{2^8}E_4+t^{-3\nu-\epsilon} \int q^2 \omega_r'(\bar x) ,
\end{align*}
since $(\theta+\frac32)\beta=\frac{11\beta}{4}-\frac14>2\nu=1-\beta$, thanks to \eqref{def:gar}.
\smallskip
\noindent \emph{Estimate for $E_{11}$}. As before, $E_{11}^-\lesssim t^{-10}$. Moreover, observe from \eqref{def:omega}, that
\begin{equation} \label{omega:estbis}
\begin{cases}
\omega_{{r+2}}(\bar{x})=\frac{\bar{x}}{r+2} \omega_{r+2}'(\bar{x})&\mbox{for $\bar x>2$,}
\\
\omega_{{r+2}}(\bar{x}) \lesssim \omega_{r+2}'(\bar{x}) &
\mbox{for $-t^{1-3\nu-2\epsilon}<\bar x<2$}
.\end{cases}
\end{equation}
Then, it follows that for $t_0$ large enough,
\begin{align*}
E_{11}^0+E_{11}^+
&\le ct_0^{-\frac{3\beta-1}{2}+\epsilon}t^{-\nu+\epsilon} \int (\partial_x q)^2 \omega_{r+2}'(\bar x)
+\frac{2}{r+2}(\nu+\epsilon)t^{-1+2\nu+2\epsilon}\int_{\bar{x}>2}(\partial_x q)^2 \bar{x}\omega_{r+2}'(\bar x)
\\
& \le-\frac1{2^9}E_4-\frac2{r+2}E_5^+ ,
\end{align*}
since $1-3\nu=\frac{3\beta-1}2$ and $0<\epsilon<\frac{3\beta-1}4$.
\smallskip
\noindent \emph{Estimate for $E_{12}$}. On the one hand, it holds $E^{-}_{12} \lesssim t^{-10}$. On the other, we observe arguing as for $E_7$ and using \eqref{omega:estbis} that
\begin{equation*}
|E_{12}^0|+|E_{12}^+| \lesssim t^{-\epsilon}\big(E_{6,1}+E_{6,2}\big)+E_{7,1}^++E_{7,2}^+ ,
\end{equation*}
so that those terms are estimated similarly.
\smallskip
Gathering all those estimates, we obtain in conclusion, for some $c>0$,
\begin{align*}
&E_r'+c\int \left[t^{-\nu+\epsilon} (\partial_xq)^2
+t^{2\nu+2\epsilon-1} \bar x_+(\partial_x q)^2
+t^{\nu+\epsilon} (\partial_x^2q)^2\right]\omega_{r+2}'(\bar{x})\, dx
\\ & \quad \lesssim
\int\left[t^{-3\nu-\epsilon} q^2
+t^{-1} \bar x_+ q^2
+t^{-\nu-\epsilon}(\partial_xq)^2\right]\omega_{r}'(\bar{x})\, dx +t^{-1+\frac{3\beta-1}2 (r-5)-r\epsilon}.
\end{align*}
Therefore, we conclude the proof of \eqref{decay_claim.2} by using \eqref{decay_claim.1}, integrating the previous estimate over $[t_0,t]$ and using \eqref{assump:epsilon}.
\end{proof}
\begin{proof}[Proof of Proposition \ref{decay:q_0} in the case $k=0$]
First, we look for an estimate on $\int (\partial_xq)^2 \omega_{r+2}$ from the energy estimate.
Arguing as in \eqref{M51}, \eqref{M51bis} and \eqref{M52}, we get that
\begin{equation*}
t^{2\nu+2\epsilon}\int q^2f_0^4 \omega_{{r+2}}(\bar{x}) \lesssim
\int q^2 \omega_{{r}}(\bar{x}) +t^{-10}
\end{equation*}
and
\begin{equation*}
t^{2\nu+2\epsilon}\int q^6 \omega_{{r+2}}(\bar{x}) \lesssim t^{2\nu+2\epsilon}\|q\|_{L^2}^4 \int (\partial_xq)^2 \omega_{{r+2}}(\bar{x})
+\|q\|_{L^2}^4 \int q^2 \omega_{{r}}(\bar{x}) . \nonumber
\end{equation*}
Thus, it follows that for $x_0$ large enough
\begin{align*}
E_r(t) &\ge t^{2\nu+2\epsilon}\int (\partial_xq)^2 \omega_{{r+2}}(\bar{x}) -ct^{2\nu+2\epsilon}\int \big( q^6+q^2f_0^4 \big) \omega_{{r+2}}(\bar{x}) \\ &
\ge \frac12t^{2\nu+2\epsilon}\int (\partial_xq)^2 \omega_{{r+2}}(\bar{x})-cM_r(t)-ct^{-10} .
\end{align*}
Hence, we deduce by using \eqref{decay_claim.1} and \eqref{decay_claim.2} that, for $0<r<2\theta+4$,
\begin{multline}\label{decay_claim.3}
t^{2(\nu+\epsilon)}\int (\partial_xq)^2 \omega_{{r+2}}(\bar{x})+\int_{t_0}^t\int\left[s^{-\nu+\epsilon} (\partial_xq)^2
+s^{2(\nu+\epsilon)-1} \bar x_+(\partial_x q)^2
+s^{\nu+\epsilon} (\partial_x^2q)^2\right]\omega_{r+2}'(\bar{x}) \\ \lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-5\epsilon} & \mbox{if $r<5$.}
\end{cases}
\end{multline}
\smallskip
Now, we give the proof of estimate \eqref{decay:q_0.1} in the case $k=0$.
By the fundamental theorem of calculus and the properties of $\omega_r$
it holds, for any $x$,
\begin{align*}
t^{\nu+\epsilon}q^2(t,x) \omega_{{r}+1}(\bar{x}) &\lesssim t^{\nu+\epsilon}\int_x^{+\infty} |q||\partial_xq|\omega_{{r}+1}(\bar{x})+\int_x^{+\infty} q^2 \omega_{{r}+1}'(\bar{x}) \\ & \lesssim t^{2(\nu+\epsilon)}\int (\partial_xq)^2\omega_{{r}+2}(\bar{x})+\int q^2\omega_{{r}}(\bar{x}) .
\end{align*}
Hence, we obtain from estimates \eqref{decay_claim.1} and \eqref{decay_claim.3} that
\begin{equation} \label{decay_claim.4}
t^{(\nu+\epsilon)}q^2(t,x) \omega_{{r}+1}(\bar{x}) \lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-r\epsilon} & \mbox{if $r<5$}
.\end{cases}
\end{equation}
For $x>2t^{\beta}$, we have that $\bar{x}>t^{\beta-\nu}>2$, for $t \ge t_0$ large enough.
Then, we deduce from
the properties of $\omega_r$, estimate \eqref{decay_claim.4} and the identity \eqref{pourtt} that
\begin{equation} \label{decay_claim.4bis}
q^2(t,x) \left| \frac{x-t^{\beta}}{t^{\nu+\epsilon}}\right|^{{r}+1} \lesssim t^{2-\beta-\nu({r}+1)-\beta(2\theta-{r}+4)-(r+1)\epsilon} ,
\end{equation}
and thus, for such $t\geq t_0$ and $x>2t^\beta$, we have
\begin{equation*}
q^2(t,x)\lesssim x^{-(r+1)}t^{2-\beta-\beta(2\theta-r+4)}.
\end{equation*}
Taking $r$ close enough to $2\theta+4$ so that $2-\beta-\beta(2\theta-r+4)>0$,
we conclude the proof of \eqref{decay:q_0.1} in the case $k=0$ and $\kappa_0=2$ (see Remark~\ref{rem:decay})
using $t^\beta<x$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{decay:q_0} in the case $k \ge 1$] We will prove estimate \eqref{decay:q_0.1} in the case where $k \ge 1$ by an induction on $k$.
\begin{definition} Let $l \in \mathbb N$, $0<r<2\theta+4$, $r\neq 5$ and $0<\epsilon<\frac{3\beta-1}{20}|r-5|$.
We say that the induction hypothesis $\mathcal{H}_l$ holds true if
\begin{equation} \label{def:Hl}
t^{2l(\nu+\epsilon)}\int (\partial_x^lq)^2 \omega_{r+2l}(\bar{x}) \, dx
\lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-r\epsilon} & \mbox{if $r<5$}
.\end{cases}
\end{equation}
\end{definition}
First, it is clear arguing as in \eqref{decay_claim.4} that if $\mathcal{H}_l$ and $\mathcal{H}_{l-1}$ hold true for some $l \in \mathbb N$, $l \ge 1$, then
\begin{equation} \label{decay_claim.5}
t^{(2l-1)(\nu+\epsilon)}(\partial_x^{l-1}q)^2(t,x) \omega_{r+2l-1}(\bar{x}) \lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-r\epsilon} & \mbox{if $r<5$}
.\end{cases}
\end{equation}
Notice in particular that \eqref{decay_claim.5} would imply \eqref{decay:q_0.1} in the case $k=l-1$ arguing as in \eqref{decay_claim.4bis}.
\smallskip
Thus, it suffices to prove that \eqref{def:Hl} hold for any $l \in \mathbb N$ to conclude the proof of Proposition \ref{decay:q_0}. Observe from \eqref{decay_claim.1} and \eqref{decay_claim.2} that $\mathcal{H}_0$ and $\mathcal{H}_1$ hold true.
Assume that \eqref{def:Hl} holds true for $l=0,1,\cdots,k-1$.
The next lemma will prove that \eqref{def:Hl} is true for $l=k$, which will conclude the proof of Proposition \ref{decay:q_0}.
\end{proof}
\begin{lemma} \label{decay_lemma.3}
Let $k \in \mathbb N$, $k \ge 2$, $0<r<2\theta+4$, $r\neq 5$ and $0<\epsilon<\frac{3\beta-1}{20}|r-5|$. Assume moreover that \eqref{def:Hl} holds true for $l=0,1,\cdots,k-1$.
Define
\begin{equation*}
F_{r,k}(t):= t^{2k(\nu+\epsilon)}\int (\partial_x^kq)^2(t,x) \omega_{r+2k}(\bar{x}) \, dx , \quad \text{where} \quad \bar{x}=\frac{x-t^{\beta}}{t^{\nu+\epsilon}} .
\end{equation*}
Then, for $x_0>1$ large enough, and any $t \ge t_0=\left(\frac{x_0}2\right)^{\frac1{\beta}}$,
\begin{multline*}
F_{r,k}(t)+\int_{t_0}^t\int\left[s^{(2k-1)(\nu+\epsilon)-2\nu} (\partial_x^kq)^2
+s^{(2k-1)(\nu+\epsilon)}(\partial_x^{k+1}q)^2\right]\omega_{r+2k}'(\bar{x})\, dx ds \\
\lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-r\epsilon} & \mbox{if $r<5$.}
\end{cases}
\end{multline*}
\end{lemma}
\begin{proof}
We differentiate $F_{k,r}$ with respect to time and integrate by parts to obtain
\begin{align*}
F'_{r,k} &= -3t^{(2k-1)(\nu+\epsilon)}\int (\partial_x^{k+1}q)^2\omega_{r+2k}'(\bar{x})+t^{(2k-3)(\nu+\epsilon)}\int (\partial_x^{k}q)^2\omega_{r+2k}'''(\bar{x})
\\
&\quad- \beta t^{(2k-1)(\nu+\epsilon)-2\nu}\int (\partial_x^{k}q)^2\omega_{r+2k}'(\bar{x})-(\nu+\epsilon) t^{2k(\nu+\epsilon)-1} \int (\partial_x^{k}q)^2 \bar{x}\omega_{r+2k}'(\bar{x})
\\ &
\quad +2t^{2k(\nu+\epsilon)}\int \partial_x^{k}\left( (q+f_0)^5-f_0^5\right) \left((\partial_x^{k+1}q)\omega_{r+2k}(\bar{x})+t^{-(\nu+\epsilon)}(\partial_x^{k}q)\omega_{r+2k}'(\bar{x})\right)
\\ & \quad +2t^{2k(\nu+\epsilon)}\int (\partial_x^kq) F_0^{(k)} \omega_{r+2k}(\bar{x})
+2k(\nu+\epsilon)t^{2k(\nu+\epsilon)-1}\int (\partial_x^kq)^2 \omega_{r+2k}(\bar{x})
\\& =: F_1 +F_2+F_3+F_4+F_5+F_6+F_7 .
\end{align*}
First observe that $F_1 \le 0$ and $F_3 \le 0$. Moreover, arguing as in the proof of \eqref{decay_claim1.1}, we have that, for $t_0$ large enough, $|F_2|\leq -\frac 12 F_3$.
Next, we use the same notation as in \eqref{notation} for $F_j$, $j=4,\cdots,6$. We have $F_4^+\leq 0$.
Moreover, as for the estimate of $M_4$ in the proof of \eqref{decay_claim.1},
it holds $F_4^0+F_4^-\leq -\frac 14 F_3+Ct^{-10}$.
\smallskip
\noindent \emph{Estimate for $F_5$.} We will only explain how to estimate the terms
\begin{equation*}
F_{5,1}:=2t^{2k(\nu+\epsilon)}\int \partial_x^{k}(q^5) \left((\partial_x^{k+1}q)\omega_{r+2k}(\bar{x})+t^{-(\nu+\epsilon)}(\partial_x^{k}q)\omega_{r+2k}'(\bar{x})\right)
\end{equation*}
and
\begin{equation*}
F_{5,2}:=2t^{2k(\nu+\epsilon)}\int \partial_x^{k}(f_0^4q) \left((\partial_x^{k+1}q)\omega_{r+2k}(\bar{x})+t^{-(\nu+\epsilon)}(\partial_x^{k}q)\omega_{r+2k}'(\bar{x})\right)
\end{equation*}
since the other ones are estimated interpolating between these estimates.
We deal first with $F_{5,1}$. We deduce from the Cauchy-Schwarz and Young inequalities that
\begin{equation*}
\big| F_{5,1} \big| \lesssim -\frac12 F_1-\frac18 F_3+t^{(2k+1)(\nu+\epsilon)} \int \left(\partial_x^k(q^5) \right)^2 \frac{\omega^2_{r+2k}(\bar{x})}{\omega_{r+2k}'(\bar{x})}=:-\frac12 F_1-\frac18 F_3+\widetilde{F}_{5,1} .
\end{equation*}
We observe by using the Leibniz rule, the Cauchy-Schwarz inequality and the properties of $\omega$ in \eqref{def:omega} that
\begin{equation*}
\widetilde{F}_{5,1} \lesssim \!\!\!\! \sum_{\genfrac{}{}{0pt}{2}{k_1+\cdots+k_5=k}{k_1 \le \cdots \le k_5}} t^{(2k_1+\cdots 2k_5+1)(\nu+\epsilon)}\int \prod_{j=1}^5 (\partial_x^{k_j}q)^2 \omega_{r+2k_1+\cdots+2k_5+1} =:\!\!\!\!
\sum_{\genfrac{}{}{0pt}{2}{k_1+\cdots+k_5=k}{k_1 \le \cdots \le k_5}} \widetilde{F}_{5,1}(k_1,\cdots,k_5) .
\end{equation*}
When $k_5=k$, we have $k_1=\cdots=k_4=0$. It follows then applying \eqref{decay_claim.4} with $r=\frac12$ that
\begin{equation*}
\widetilde{F}_{5,1}(0,0,0,0,k) \le \Big\| t^{(\nu+\epsilon)}q^2\omega_{\frac32} \Big\|_{L^{\infty}_{x}}^4t^{(2k-3)(\nu+\epsilon)} \int (\partial_x^ kq)^2 \omega_{r+2k}'(\bar{x}) \le \delta(t_0^{-1}) F_3 .
\end{equation*}
When $k_5 \le k-1$, we consider two different cases. First if $k=2$, then $k_5=k_4=1$ and $k_1=k_2=k_3=0$, we deduce from \eqref{decay_claim.4} with $r=1$ and then \eqref{decay_claim.3} that
\begin{align*}
\widetilde{F}_{5,1}(0,0,0,1,1) &\le \Big\| t^{(\nu+\epsilon)}q^2\omega_{2} \Big\|_{L^{\infty}_{x}}^3\|\partial_xq\|_{L^{\infty}_x}^2 \left(t^{2(\nu+\epsilon)} \int (\partial_xq)^2 \omega_{r+2}(\bar{x}) \right)
\\ &\lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-5\epsilon} & \mbox{if $r<5$}.
\end{cases}
\end{align*}
In the case where $k_5 \le k-1$ and $k \ge 3$, observe that $k_4 \le k-2$. By using the induction hypothesis \eqref{def:Hl} for $l=0,1,\cdots,k-1$, we deduce that \eqref{decay_claim.5} hold for $l=1,\cdots,k-1$. Thus, it follows that
\begin{align*}
\widetilde{F}_{5,1}(k_1,k_2,k_3,k_4,k_5) &\le \prod_{j=1}^4\Big\| t^{(2k_j+1)(\nu+\epsilon)}(\partial_x^{k_j}q)^2\omega_{2k_j+\frac32} \Big\|_{L^{\infty}_{x}} \left[t^{(2k_5-3)(\nu+\epsilon)} \int (\partial_x^{k_5}q)^2 \omega_{r+2k_5}(\bar{x}) \right]
\\ &\lesssim
\begin{cases}
t^{\frac{3\beta-1}2 (r-5)-r\epsilon} & \mbox{if $r>5$}\\
t_0^{-\frac{3\beta-1}2 (5-r)-5\epsilon} & \mbox{if $r<5$}
.\end{cases}
\end{align*}
Now we deal with $F_{5,2}$. By using the Leibniz rule and integration by parts, we decompose
\begin{align*}
F_{5,2}&:=2\sum_{l=0}^{k-1}\begin{pmatrix} k \\ l \end{pmatrix}t^{2k(\nu+\epsilon)}\int \partial_x^{k-l}(f_0^4)(\partial_x^lq) (\partial_x^{k+1}q)\omega_{r+2k}(\bar{x})-t^{2k(\nu+\epsilon)}\int \partial_x(f_0^4)(\partial_x^{k}q)^2 \omega_{r+2k}(\bar{x}) \\ &
\quad +t^{(2k-1)(\nu+\epsilon)}\int f_0^4 (\partial_x^{k}q)^2\omega_{r+2k}'(\bar{x})
+2\sum_{l=0}^{k-1}\begin{pmatrix} k \\ l \end{pmatrix}t^{(2k-1)(\nu+\epsilon)}\int \partial_x^{(k-l)}(f_0^4) (\partial_x^{k}q)\omega_{r+2k}'(\bar{x}) \\ &=: F_{5,2,1}+F_{5,2,2}+F_{5,2,3}+F_{5,2,4} .
\end{align*}
First, it is clear that $ |F_{5,2,1}^-|+F_{5,2,2}^-+F_{5,2,3}^-+F_{5,2,4}^- \lesssim t^{-10}$. Next, we observe arguing as in \eqref{omega:est} that
\begin{equation} \label{omega:est.2}
\omega_{r+2k}(\bar{x})|x|^{-(k-l)-1} \lesssim t^{-((k-l)+1)(\nu+\epsilon)}\omega_{r+k+l-1}(\bar{x}), \quad \text{for} \quad \bar{x}>-t^{1-3\nu-2\epsilon} .
\end{equation}
Hence, it follows from \eqref{def:f_0}, \eqref{zone}, $\theta>\frac12$ and the Cauchy-Schwarz inequality that
\begin{align*}
F_{5,2,1}^0+F_{5,2,1}^+ &\lesssim \sum_{l=0}^{k-1}t^{(k+l-1)(\nu+\epsilon)}\int_{x>\frac12t^{\beta}} |x|^{-(4\theta-1)}|\partial_x^lq| |\partial_x^{k+1}q|\omega_{r+k+l-1}(\bar{x})
\\ & \lesssim \sum_{l=0}^{k-1} \left( t^{2l(\nu+\epsilon)} \int (\partial_x^lq)^2\omega_{r+2l}(\bar{x})\right)^{\frac12} \left( t^{(2k-2)(\nu+\epsilon)} \int (\partial_x^{(k+1)}q)^2\omega_{r+2k}'(\bar{x})\right)^{\frac12} \\
& \lesssim -\frac14F_1+\sum_{l=0}^{k-1}t^{2l(\nu+\epsilon)} \int (\partial_x^lq)^2\omega_{r+2l}(\bar{x}) .
\end{align*}
Now, observe from \eqref{def:f_0}, \eqref{zone} and \eqref{omega:est} that, for $t_0$ large enough,
\begin{align*}
F_{5,2,2}^0+F_{5,2,2}^+ +F_{5,2,3}^0+F_{5,2,3}^+&\lesssim t^{(2k-1)(\nu+\epsilon)}\int_{x>\frac12t^{\beta}} |x|^{-4\theta}(\partial_x^kq)^2\omega_{r+2k}'(\bar{x}) \\ &\lesssim t^{(2k-1)(\nu+\epsilon)-4\theta \beta}\int (\partial_x^kq)^2\omega_{r+2k}'(\bar{x})
\le -\frac1{2^4}F_3 ,
\end{align*}
since $4\theta \beta =5\beta-1=4-10\nu>2\nu$, thanks to \eqref{def:gar}.
Finally, by using an estimate similar to \eqref{omega:est.2},
\begin{align*}
F_{5,2,4}^0+F_{5,2,4}^+ &\lesssim \sum_{l=0}^{k-1}t^{(k+l-1)(\nu+\epsilon)}\int_{x>\frac12t^{\beta}} |x|^{-4\theta}|\partial_x^lq| |\partial_x^{k}q|\omega_{r+k+l}'(\bar{x})
\\ & \lesssim \sum_{l=0}^{k-1} \left( t^{2l(\nu+\epsilon)} \int (\partial_x^lq)^2\omega_{r+2l}(\bar{x})\right)^{\frac12} \left( t^{(2k-2)(\nu+\epsilon)-8\theta\beta} \int (\partial_x^{k}q)^2\omega_{r+2k}'(\bar{x})\right)^{\frac12} \\
& \lesssim -\frac1{2^5}F_3+\sum_{l=0}^{k-1}t^{2l(\nu+\epsilon)} \int (\partial_x^lq)^2\omega_{r+2l}(\bar{x}) .
\end{align*}
\smallskip
\noindent \emph{Estimate for $F_6$.} From the Cauchy-Schwarz inequality,
\begin{align*}
\big| F_6 \big| &\leq 2 t^{2k(\nu+\epsilon)}\left( \int (\partial_x^kq)^2 \omega_{{r+2k}}'(\bar{x})\right)^{\frac12}\left(\int \big(F_0^{(k)}\big)^2 \frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})} \right)^{\frac12} \\ &
\le -\frac 1{2^6} F_3 + ct^{2k(\nu+\epsilon)+3\nu+\epsilon}\int \big(F_0^{(k)}\big)^2 \frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})} .
\end{align*}
First, from \eqref{def:f_0},
for $x>\frac 14 x_0$,
$\big|F_0^{(k)}\big| \lesssim |x|^{-(\theta+k+3)}$ ($\theta>\frac12$),
and for $x<\frac 14 x_0$, $F_0=0$.
For $\bar{x} \ge 2$, it holds $ \frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})} =(r+2)^{-1} |\bar{x}|^{r+2k+1}
\lesssim t^{-(\nu+\epsilon)(r+2k+1)}|x|^{r+2k+1}$. Hence, by using \eqref{pourtt},
\begin{align*}
t^{2k(\nu+\epsilon)+3\nu+\epsilon}\int_{\bar x>2} \big(F_0^{(k)}\big)^2 \frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})}
&\lesssim t^{2\nu-r(\nu+\epsilon)}\int_{\bar x>2} |x|^{-2(\theta+k+3)}| {x}|^{r+2k+1}
\\ & \lesssim t^{2\nu-r(\nu+\epsilon)}\int_{x>t^\beta} |x|^{-2\theta+{r}-5}
\\ & \lesssim t^{2\nu-r(\nu+\epsilon)}t^{-\beta(2\theta-{r}+4)}
=t^{-1+\frac{3\beta-1}2 (r-5)-r\epsilon} ,
\end{align*}
since $2\theta-r+4>0$ by assumption.
For $-t^{-1-3\nu-2\epsilon}< \bar x<2$, it holds $ \frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})} \lesssim 1$
and $x\geq \frac 12 t^\beta$ (from \eqref{zone}) so that
\begin{align*}
t^{2k(\nu+\epsilon)+3\nu+\epsilon}\int_{-t^{-1-3\nu-2\epsilon}<\bar x<2} \big(F_0^{(k)}\big)^2 \frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})} &\lesssim t^{2k(\nu+\epsilon)+3\nu+\epsilon} \int_{x>\frac 12t^{\beta}}x^{-2(\theta+k+3)}
\\
&\lesssim t^{2k(\nu+\epsilon)+3\nu+\epsilon}t^{-\beta(2\theta+2k+5)}\\
&\lesssim t^{-\frac{k(3\beta-1)}{2}-9\beta+2+(2k+1)\epsilon} .
\end{align*}
Last, for $\bar x<-t^{-1-3\nu-2\epsilon}$, then $\frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})} =8e^{\frac{\bar{x}}8}$ so that as for $M_4^-$,
\begin{equation*}
t^{2k(\nu+\epsilon)+3\nu+\epsilon}\int_{\bar x<-t^{-1-3\nu-2\epsilon}} \big(F_0^{(k)}\big)^2 \frac{\omega_{{r+2k}}^ 2(\bar{x})}{\omega_{{r+2k}}'(\bar{x})}
\lesssim t^{-10} .
\end{equation*}
\smallskip
\noindent \emph{Estimate for $F_7$.} As before, $F_{7}^-\lesssim t^{-10}$. Moreover, observe from \eqref{def:omega}, that
\begin{equation*}
\begin{cases}
\omega_{{r+2k}}(\bar{x})=\frac{\bar{x}}{r+2k} \omega_{r+2k}'(\bar{x})&\mbox{for $\bar x>2$,}
\\
\omega_{{r+2k}}(\bar{x}) \lesssim \omega_{r+2k}'(\bar{x}) &
\mbox{for $-t^{1-3\nu-2\epsilon}<\bar x<2$}
.\end{cases}
\end{equation*}
Then, it follows that for $t_0$ large enough,
\begin{align*}
F_{7}^0+F_{7}^+
&\le ct_0^{-\frac{3\beta-1}{2}+\epsilon}t^{-\nu+\epsilon} \int (\partial_x^k q)^2 \omega_{r+2}'(\bar x)
+\frac{2k}{r+2k}(\nu+\epsilon)t^{-1+2\nu+2\epsilon}\int_{\bar{x}>2}(\partial_x^k q)^2 \bar{x}\omega_{r+2}'(\bar x)
\\
& \le-\frac1{2^7}F_3-\frac{2k}{r+2k}F_4^+ ,
\end{align*}
since $1-3\nu=\frac{3\beta-1}2$ and $0<\epsilon<\frac{3\beta-1}4$.
\smallskip
Therefore, we complete the proof of Lemma \ref{decay_lemma.3} combining all those estimates with the induction hypothesis \eqref{def:Hl} for $l=0,1,\cdots,k-1$.
\end{proof}
\section{Decomposition around the soliton} \label{section:decomp_sol}
\subsection{Linearized operator}
Here, we recall some properties of the linearized operator $\mathcal{L}$ around the soliton $Q$ defined in \eqref{def:L}.
We first introduce the function space $\mathcal{Y}$:
\begin{equation*}
\mathcal{Y}:= \Big\{ \phi \in \mathcal{C}^{\infty}(\mathbb R : \mathbb R) : \forall \, k \in \mathbb N, \, \exists \, C_k, \, r_k>0 \ \text{s.t.} \ |\phi^{(k)}(y)| \le C_k(1+|y|)^{r_k}e^{-|y|}, \ \forall \, y \in \mathbb R\Big\} \, .
\end{equation*}
\begin{lemma}[Properties of the linearized operator $\mathcal{L}$]
The self-adjoint operator $\mathcal{L}:H^2(\mathbb R) \subseteq L^2(\mathbb R) \to L^2(\mathbb R)$ defined in \eqref{def:L} satisfies the following properties.
\begin{itemize}
\item[(i)] \emph{Spectrum of $\mathcal{L}$:} the operator $\mathcal{L}$ has only one negative eigenvalue $-8$ associated to the eigenfunction $Q^3$; $\ker \mathcal{L}=\{aQ' : a \in \mathbb R\}$; and $\sigma_{ess}( \mathcal{L})=[1,+\infty)$.
\item[(ii)] \emph{Scaling:} $\mathcal{L}\Lambda Q=-2Q$ and $(Q,\Lambda Q)=0$, where $\Lambda$ is defined in \eqref{def:lambda}.
\item[(iii)] \emph{Coercivity of $\mathcal{L}$:}
for all $\phi \in H^1(\mathbb R)$,
\begin{equation*}
(\phi,Q^3)=(\phi,Q')=0 \implies (\mathcal{L}\phi,\phi) \ge \|\phi\|_{H^1}^2 .
\end{equation*}
Moreover, there exists $\nu_0>0$ such that, for all $f \in H^1(\mathbb R)$,
\begin{equation} \label{coercivity.2}
(\mathcal{L}\phi,\phi) \ge \nu_0\|\phi\|_{H^1}^2-\frac1\nu_0\Big((\phi,Q)^2+(\phi,y\Lambda Q)^2 +(\phi,\Lambda Q)^2 \Big) .
\end{equation}
\item[(iv)] \emph{Invertibility:} there exists a unique $R \in \mathcal{Y}$, even, such that
\begin{equation} \label{def:R}
\mathcal{L}R=5Q^4; \quad \text{moreover} \quad (Q,R)=-\frac34 \int Q .
\end{equation}
\item[(v)] \emph{Invertibility (bis):} there exists a unique function $P \in \mathcal{C}^{\infty}(\mathbb R) \cap L^{\infty}(\mathbb R)$ such that $P' \in \mathcal{Y}$ and
\begin{equation*}
(\mathcal{L}P)'=\Lambda Q, \ \lim_{y \to -\infty}P(y)=\frac12 \int Q, \ \lim_{y \to +\infty} P(y)=0 .
\end{equation*}
Moreover,
\begin{equation} \label{prop:P}
(P,Q)=\frac1{16}\left( \int Q\right)^2>0, \quad (P,Q')=0 .
\end{equation}
\end{itemize}
\end{lemma}
\begin{proof}
The properties (i), (ii) and (iii) are standard and we refer to Lemma 2.1 of \cite{MaMeRa1} and the references therein for their proof. Property (iv) is proved in Lemma 2.1 in \cite{MaMeRa3}, while property (v) is proved in Proposition 2.2 in \cite{MaMeRa1}.
\end{proof}
\subsection{Refined profile} We follow \cite{MaMeRa1} to define the one parameter family of approximate self similar profiles $:b\mapsto Q_b$, $|b| \ll 1$ which will provide the leading order deformation of the ground state profile $Q=Q_{b=0}$ in our construction.
More precisely, we need to localize $P$ on the left hand side. Let $\chi \in \mathcal{C}^{\infty}(\mathbb R)$ be such that
\begin{equation} \label{def:chi}
0\le \chi \le 1, \quad 0 \le (\chi'')^2 \lesssim \chi' \quad \text{on} \ \mathbb R, \quad \chi_{|_{(-\infty,-2)}} \equiv 0 \quad \text{and} \quad \chi_{|_{(-1,+\infty)}}\equiv 1 .
\end{equation}
\begin{definition}
Let $\gamma=\frac34$. The localized profile $Q_b$ is defined by
\begin{equation} \label{def:Qb}
Q_b(y)=Q(y)+bP_b(y) ,
\end{equation}
where
\begin{equation} \label{def:Pb}
P_b(y)=\chi_b(y)P(y) \quad \text{and} \quad \chi_b(y)=\chi(|b|^{\gamma}y) .
\end{equation}
\end{definition}
The properties of $Q_b$ are stated in the next lemma.
\begin{lemma}[Approximate self-similar profiles $Q_b$ , \cite{MaMeRa1}] \label{refined:profile}
There exists $b^{\star}>0$ such that for $|b|<b^{\star}$, the following properties hold.
\begin{itemize}
\item[(i)] \emph{Estimate of $Q_b$:} for all $y \in \mathbb R$,
\begin{equation} \label{est:Qb} \begin{aligned}
\big|Q_b(y) \big| &\lesssim e^{-|y|}+|b| \left(\boldsymbol{1}_{[-2,0]}(|b|^{\gamma}y)+e^{-\frac{|y|}2} \right) ; \\
\big|Q_b^{(k)}(y) \big| &\lesssim e^{-|y|}+|b|e^{-\frac{|y|}2}+ |b|^{1+k\gamma}\boldsymbol{1}_{[-2,-1]}(|b|^{\gamma}y), \quad \forall \, k \ge 1 .
\end{aligned}\end{equation}
\item[(ii)] \emph{Equation of $Q_b$:} the error term
\begin{equation*}
-\Psi_b=\big(Q_b''-Q_b+Q_b^5 \big)'+b\Lambda Q_b-2b^2\frac{\partial Q_b}{\partial b}
\end{equation*}
satisfies, for all $y \in \mathbb R$,
\begin{align}
\big|\Psi_b(y) \big| &\lesssim |b|^{1+\gamma}\boldsymbol{1}_{[-2,-1]}(|b|^{\gamma}y)+b^2 \left(e^{-\frac{|y|}2} +\boldsymbol{1}_{[-2,0]}(|b|^{\gamma}y)\right) ; \label{est:Psib} \\
\big|\Psi_b^{(k)}(y) \big| &\lesssim |b|^{1+(k+1)\gamma}\boldsymbol{1}_{[-2,-1]}(|b|^{\gamma}y)+ b^2 e^{-\frac{|y|}2}, \quad \forall \, k \ge 1 . \label{est:derivPsib}
\end{align}
Moreover,
\begin{equation}\label{est:PsibY}
\left|\big(\Psi_b,\phi\big)\right| \lesssim b^2, \quad \forall \phi \in \mathcal{Y} ,
\end{equation}
and $($recalling the definition of $L^2_B$ in \eqref{def:L2B}$)$,
\begin{equation} \label{est:PsibL2}
\big\|\Psi_b^{(k)}(y) \big\|_{L^2_B} \lesssim b^2, \ \forall \, k \ge 0 .
\end{equation}
Note that the implicit constant in \eqref{est:PsibL2} depends on the constant $B \ge 100$.
\item[(iii)] \emph{Projection of $\Psi_b$ in the direction $Q$:}
\begin{equation} \label{est:PsibQ}
\left| \big( \Psi_b,Q \big) \right| \lesssim |b|^3 .
\end{equation}
\item[(iv)] \emph{Mass and energy properties of $Q_b$:}
\begin{align}
\left| \int Q_b^2-\left(\int Q^2+2b\int PQ \right) \right| &\lesssim |b|^{2-\gamma} ; \label{est:massQb}\\
\left|E(Q_b)+b\int P Q \right| &\lesssim b^2 .\label{est:energyQb}
\end{align}
\end{itemize}
\end{lemma}
\begin{proof} The proof of Lemma \ref{refined:profile} can be found in \cite{MaMeRa1}. Actually, the properties (i), (ii) and (iv) are proved in Lemma 2.4 of \cite{MaMeRa1}. The estimate \eqref{est:PsibY} follow directly from \eqref{est:Psib} and \eqref{est:derivPsib}. Now, we explain how to prove \eqref{est:PsibL2} in the case $k=0$. It follows from \eqref{def:L2B}, \eqref{est:Psib} and the fact that $B \ge 100$ that
\begin{equation*}
\|\Psi_b\|_{L^2_B} \lesssim |b|^{1+\gamma} \left( \int_{-2|b|^{-\gamma}}^{-|b|^{-\gamma}} e^{\frac{y}B}\right)^{\frac12}+b^2\left(\int e^{\frac{y}B-\frac{|y|}2} \right)^{\frac12}+b^2\left( \int_{-2|b|^{-\gamma}}^0 e^{\frac{y}B}\right)^{\frac12} \lesssim b^2 ,
\end{equation*}
for $|b|$ small enough. The proof of \eqref{est:PsibL2} in the case $k\ge 1$ follows in a similar way by using \eqref{est:derivPsib} instead of \eqref{est:Psib}.
Note that we have added the term $-2b^2\frac{\partial Q_b}{\partial b}$ to the definition of $\Psi_b$ compared to the definition in \cite{MaMeRa1} in order to get a better estimate for the projection of $\Psi_b$ on $Q$. The property (iii) follows from the computation in the first formula of page 80 in \cite{MaMeRa1} together with \eqref{prop:P}.
\end{proof}
\begin{remark} For future reference, we also observe that
\begin{equation} \label{eq:dQb:db}
\frac{\partial Q_b}{\partial b}=\chi_b P+\gamma |b|^{\gamma}y\chi'(|b|^{\gamma}y)=P_b+\gamma y\chi_b'P .
\end{equation}
\end{remark}
\subsection{Definition of the approximate solution} \label{sec:AS}
Let any $\frac 12 < \theta <1$.
Following \eqref{def:c0}, set
\begin{equation} \label{def:c0bis}
c_0=\frac12 (1-\theta)(2\theta-1)^{-(1-\theta)} {\int Q}>0.
\end{equation}
For such $c_0$, for $x_0$ large enough and for
\begin{equation}\label{def:t0}
t_0=(2x_0)^{1/\beta},
\end{equation}
(our intention is to use Proposition~\ref{decay:q_0} with the value $\kappa_0=\frac 12$),
we consider $f(t,x)$ the solution of \eqref{eq:q_0}.
Let
\begin{equation*}v(t,x)=U(t,x)-f(t,x).\end{equation*}
Note that $U$ is solution \eqref{gkdv} if and only if $v$ satisfies
\begin{equation*}
\partial_t v +\partial_x(\partial_x^2v+(v+f)^5-f^5)=0 .
\end{equation*}
We renormalize the flow using $\mathcal C^1$ functions $\lambda(t)$ and $\sigma(t)$, defining
$V$, $F_0$ and $F$ as follows
\begin{equation*}
v(t,x)=\lambda^{-\frac 12}(t) V(t,y), \quad y=\frac{x-\sigma(t)}{\lambda(t)},
\end{equation*}
\begin{equation*}
f_0(x)=\lambda^{-\frac 12}(t) F_0\left(t,y\right),\quad f(t,x)=\lambda^{-\frac 12}(t) F\left(t,y\right).
\end{equation*}
We introduce the rescaled time variable
\begin{equation}\label{eq:sbis}
s(t)= s_0+ \int_{t_0}^t \frac {d\tau}{\lambda^3(\tau)}\quad \hbox{with} \quad t_0= \frac{2\theta-1}{5-4\theta} s_0^{\frac{5-4\theta}{2\theta-1}}
=\frac{3\beta-1}{2}s_0^{\frac2{3\beta-1}} .
\end{equation}
Note that from \eqref{def:t0} relating $t_0$ and $x_0$,
$s_0$ can be taken arbitrarily large provided $x_0$ is large.
From now, any time-dependent function can be seen as a function of $t \in \mathcal{I}$ or $s \in \mathcal{J}$, where $\mathcal{I}$ is an interval of the form $[s_0,s^{\star}]$
and $\mathcal{J}=s(\mathcal{I})$.
In view of the resolution of the ODE system in \eqref{resol:la:si:b}, we will work under the following assumptions on the parameters $(\lambda,\sigma,b)$:
\begin{equation}\label{BS:param}
\left\{
\begin{aligned}
&\left| \lambda (s) - s^{\frac{2(1-\theta)}{2\theta-1}}\right|\leq s^{\frac{2(1-\theta)}{2\theta-1}-\rho} ;\\
&\left| \sigma(s)- (2\theta-1) s^{\frac1{2\theta-1}}\right|\leq s^{\frac1{2\theta-1}-\rho} ;\\
&\left| b(s)+\frac{2(1-\theta)}{2\theta-1} s^{-1}\right|\leq s^{-1-\rho} ;
\end{aligned}
\right.
\end{equation}
where $\rho$ is a positive number satisfying
\begin{equation} \label{def:rho}
0<\rho<\min\left\{\frac1{12},\frac{1-\theta}{3(2\theta-1)} \right\} .
\end{equation}
We set
\begin{equation} \label{def:EV}
\mathcal E(V)=
V_s+\partial_y(\partial_y^2 V-V+(V+F)^5-F^5)
-\frac{\lambda_s}{\lambda}\Lambda V - \left(\frac{\sigma_s}{\lambda}-1\right) \partial_y V .
\end{equation}
Note that $u$ is solution of \eqref{gkdv} if and only if $\mathcal E(V)=0$.
We look for an approximate solution $W$ of the form
\begin{equation*}
W(s,y) = Q_{b(s)}(y) + r(s) R(y) ,
\end{equation*}
where $b(s)$ is a $\mathcal C^1$ function to be determined and where we set
\begin{equation*}
r(s)=F(s,0)=\lambda^{\frac 12}(s)f(s,\sigma(s)).
\end{equation*}
We also define (see Lemma~\ref{le:3.6})
\begin{equation} \label{def:mM}
\vec m =\begin{pmatrix} \frac{\lambda_s}{\lambda}+b \\ \frac{\sigma_s}{\lambda}-1 \end{pmatrix} \quad \text{and} \quad \vec \MM =\begin{pmatrix} \Lambda \\ \partial_y \end{pmatrix} .
\end{equation}
First, we gather some useful estimates for $r$ and $F$.
\begin{lemma} \label{lemma:est:rF}
Under the assumptions \eqref{BS:param} and for $s$ large enough, it holds
\begin{equation}\label{e:r}
\left|r-c_0\lambda^{\frac 12} \sigma^{-\theta}\right|
\lesssim \lambda^{\frac 12}\sigma^{-(5\theta-2)}\lesssim s^{-3},\quad
|r|\lesssim s^{-1} ,
\end{equation}
\begin{equation}\label{e:dr}
\left| r_s - c_0\lambda^{\frac 12}\sigma^{-\theta} \left( \frac 12 \frac{\lambda_s}{\lambda}
-\theta\frac{\sigma_s}{\sigma} \right)\right|\lesssim |\vec m|s^{-3} + s^{-4},\quad
|r_s|\lesssim |\vec m| s^{-1}+s^{-2} ,
\end{equation}
\begin{equation} \label{e:F}
\sup_{y \in \mathbb R} \left\{e^{-\frac{|y|}{10}}\big|F(s,y)\big|\right\} \lesssim s^{-1}, \quad \sup_{y \in \mathbb R} \left\{e^{-\frac{|y|}{10}}\big|\partial_yF(s,y)\big|\right\} \lesssim s^{-2} ,
\end{equation}
\begin{equation} \label{e:FLinfty}
\|F\|_{L^{\infty}(y>-2|b|^{\gamma})} \lesssim s^ {-1}, \quad \|\partial_yF\|_{L^{\infty}(y>-2|b|^{\gamma})} \lesssim s^{-2} ,
\end{equation}
\begin{equation} \label{e:r-F}
\sup_{y \in \mathbb R} \left\{e^{-\frac{3|y|}4}\big|r(s)-F(s,y)\big|\right\} \lesssim s^{-2}
\end{equation}
and
\begin{equation} \label{e2:r-F}
\left| \left(\partial_y \big( 5Q^4(r-F)\big),Q\right)-c_0\theta \left(\int Q\right) \lambda^{\frac32}\sigma^{-\theta-1} \right| \lesssim s^{-3} .
\end{equation}
\end{lemma}
\begin{proof}
Estimate \eqref{e:r} follows from \eqref{decay:q_0.1} and then \eqref{BS:param}.
Next, we compute $r_s$:
\begin{equation*}
r_s = \frac 12 \frac{\lambda_s}{\lambda} r +\lambda^{\frac 12} \sigma_s \partial_y f(s,\sigma)
+\lambda^{\frac 12}\partial_s f(s,\sigma).\end{equation*}
Note that by \eqref{e:r} and then \eqref{BS:param}
\begin{equation*}
\left| \frac 12 \frac{\lambda_s}{\lambda} r - \frac 12 c_0\frac{\lambda_s}{\lambda^{\frac 12}} \sigma^{-\theta}\right|
\lesssim \left|\frac{\lambda_s}{\lambda}\right| s^{-3}\lesssim |\vec m|s^{-3} + s^{-4}.
\end{equation*}
By \eqref{decay:q_0.1} for $k=1$, we have $|\partial_y f(s,\sigma)+c_0\theta\sigma^{-\theta-1}|\lesssim \sigma^{-5\theta+1}$
and so
\begin{equation*}
\left|\lambda^{\frac 12} \sigma_s \partial_y f(s,\sigma)+c_0\theta\lambda^{\frac 12}\sigma_s \sigma^{-\theta-1} \right|\lesssim
\lambda^{\frac 12}|\sigma_s |\sigma^{-5\theta+1}
\lesssim (|\vec m|+1)\lambda^{\frac 32} \sigma^{-5\theta+1}
\lesssim (|\vec m|+1) s^{-4}.
\end{equation*}
Last, from \eqref{decay:q_0.2} and $ds = \lambda^{-3} dt$, we have
\begin{equation*}
|\lambda^{\frac 12}\partial_s f(s,\sigma)|
=\lambda^{\frac 72} |\partial_t f(s,\sigma)|
\lesssim \lambda^{\frac 72} \sigma^{-\theta-3}\lesssim s^{-4}.
\end{equation*}
We deduce the proof of \eqref{e:dr} gathering the above estimates.
We recall that $F(s,y)=\lambda^{\frac12}(s)f(s,\lambda(s)y+\sigma(s))$ and $\partial_yF(s,y)=\lambda^{\frac32}(s)\partial_yf(s,\lambda(s)y+\sigma(s))$. Thus, splitting the integration domain into the two cases $\lambda(s)y>-\frac14\sigma(s)\iff \lambda(s)y+\sigma(s)>\frac34 \sigma(s)$ and $\lambda(s)y<-\frac14\sigma(s) \implies y<-c s$, and then using \eqref{def:f_0}, \eqref{decay:q_0.1} (with $\kappa_0=\frac12$), we deduce that, for $s$ large enough,
\begin{align*}
e^{-\frac{|y|}{10}}\big|F(s,y)\big| &\lesssim \lambda(s)^{\frac12}\left( e^{-\frac{|y|}{10}}\|f(s,\cdot)\|_{L^{\infty}(x>\frac34 \sigma(s))}+e^{-\frac{\sigma(s)}{40\lambda(s)}}\|f(s,\cdot)\|_{L^{\infty}}\right) \\
& \lesssim \lambda(s)^{\frac12} \left( \sigma(s)^{-\theta}+e^{-cs} \right),
\end{align*}
and respectively,
\begin{align*}
e^{-\frac{|y|}{10}}\big|\partial_yF(s,y)\big| &\lesssim \lambda(s)^{\frac32}\left( e^{-\frac{|y|}{10}}\|\partial_yf(s,\cdot)\|_{L^{\infty}(x>\frac34 \sigma(s))}+e^{-\frac{\sigma(s)}{40\lambda(s)}}\|\partial_yf(s,\cdot)\|_{L^{\infty}}\right)\nonumber \\
& \lesssim \lambda(s)^{\frac32} \left( \sigma(s)^{-\theta-1}+e^{-cs} \right),
\end{align*}
which, together with \eqref{BS:param}, concludes the proof of \eqref{e:F}. Note that in the case where $y>-2|b|^{-\gamma}$, we get from \eqref{BS:param} and the choice $\gamma<1$ that $\lambda(s)y+\sigma(s)>\frac34 \sigma(s)$, so that \eqref{e:FLinfty} follows from \eqref{def:f_0} and \eqref{decay:q_0.1}.
From the definition of $F$ and $r$, we have
\begin{equation*}r(s)-F(s,y)=F(s,0)-F(s,y)=\lambda^{\frac12}(s)\big(f(s,\sigma(s))-f(s,\lambda(s)y+\sigma(s)\big) .\end{equation*}
Thus, it follows applying the mean value theorem, splitting into the two cases $\lambda(s)y>-\frac14\sigma(s)$ and $\lambda(s)y<-\frac14\sigma(s)$ as above, and then using \eqref{def:f_0} and \eqref{decay:q_0.1} that, for $s$ large enough,
\begin{align*}
e^{-\frac{3|y|}4}\big|r(s)-F(s,y)\big| &\lesssim e^{-\frac{|y|}2} |y| \lambda(s)^{\frac32}\left( e^{-\frac{|y|}4}\|\partial_yf(s,\cdot)\|_{L^{\infty}(x>\frac34 \sigma(s))}+e^{-\frac{\sigma(s)}{16\lambda(s)}}\|\partial_yf(s,\cdot)\|_{L^{\infty}}\right) \\
& \lesssim e^{-\frac{|y|}4} \lambda(s)^{\frac32} \left( \sigma(s)^{-\theta-1}+e^{-cs} \right),
\end{align*}
which implies \eqref{e:r-F} by using \eqref{BS:param}.
Finally, by using the identities
\begin{equation*} \left(\partial_y\big( 5Q^4(r-F)\big),Q\right)=-5\int Q^4Q'(r-F)=-\int Q^5 \partial_yF(s,\cdot)=-\lambda^{\frac32}(s)\int Q^5\partial_yf(s,\lambda\cdot+\sigma) \end{equation*}
and $ \int Q^5=\int Q$, we deduce arguing as in \eqref{e:r-F} that
\begin{align*}
&\left| \left(\partial_y \big( 5Q^4(r-F)\big),Q\right)-c_0\theta \left(\int Q\right) \lambda^{\frac32}\sigma^{-\theta-1} \right| \\ &\lesssim
\lambda^{\frac32} \int Q^5 \left( \left| f_0'(\lambda y+\sigma)-f_0'(\sigma) \right|+\left| f_0'(\lambda y+\sigma)-\partial_y f(s,\lambda y+\sigma) \right|\right)
\lesssim \lambda^{\frac52}\left( \sigma(s)^{-\theta-2}+e^{-cs} \right).
\end{align*}
which, together with \eqref{BS:param}, concludes the proof of \eqref{e2:r-F}.
\end{proof}
In the next lemma, we derive an estimate for the mass and the energy of $W+F$.
\begin{lemma}[Mass and energy of $W+F$]
Under the assumptions \eqref{BS:param}, it holds for $s$ large enough
\begin{equation} \label{mass:W}
\left| \int (W+F)^2-\left(\int Q^2+2b\int PQ+\frac12r\int Q \right) \right| \lesssim s^{-(2-\gamma)}+x_0^{-(2\theta-1)}
\end{equation}
and
\begin{equation} \label{ener:W}
\left|\lambda^{-2}E(W+F)+c_1\left( \frac{b}{\lambda^2}+\frac{4c_0}{\int Q}\lambda^{-\frac32}\sigma^{-\theta}\right) \right| \lesssim \lambda^{-2}s^{-2}+x_0^{-(2\theta+1)} ,
\end{equation}
where $c_1=\frac1{16}\left(\int Q \right)^2$.
\end{lemma}
\begin{proof}
Observe by using the decomposition in \eqref{def:Qb} that
\begin{equation*}
\int (W+F)^2=\int Q_b^2+2r\int Q_bR+r^2\int R^2+2\int Q_bF+2r\int RF+\int F^2 ,
\end{equation*}
From the definition of $F$ and the $L^2$ conservation for \eqref{eq:q_0}, we have that
\begin{equation} \label{mass:F}
\int F^2 =\int f^2=\int f_0^2=c_0x_0^{-(2\theta-1)} .
\end{equation}
Moreover, we get from \eqref{e:FLinfty} that
\begin{equation*}
\left| \int P_b F \right| \lesssim \int_{-2|b|^{\gamma}<y<0} |F(y)|+\int_{y>0} |F(y)|e^{-\frac34{|y|}} \lesssim s^{-1+\gamma} .
\end{equation*}
Hence, it follows from \eqref{def:Qb} and \eqref{e:r-F} that
\begin{equation*}
2\int Q_bF=2r\int Q+2\int Q(F-r)+2b\int P_bF=2\int Qr +\mathcal{O}(s^{-2+\gamma}) ,
\end{equation*}
which together with \eqref{def:R}, \eqref{est:massQb}, \eqref{BS:param} and \eqref{e:r} imply \eqref{mass:W}.
Now, we compute the energy of $W$:
\begin{align*}
E(W+F)&=E(Q_b)+E(F)+\int Q_b'\partial_y(rR+F)+r\int R'\partial_yF+\frac12 r^2 \int (R')^2 \\
&\quad-\int Q_b^5(rR+F) -\frac16 \int \left( (Q_b+rR+F)^6-Q_b^6-F^6-6Q_b^5(rR+F) \right) .
\end{align*}
Moreover, it follows from the definition of $F$, the conservation of the energy for \eqref{eq:q_0} and \eqref{est:f_0} that
\begin{equation*}
E(F)=\lambda^2 E(f)=\lambda^2 E(f_0) \sim \lambda^2x_0^{-(2\theta+1)} .
\end{equation*}
Thus, we deduce then from the definition of $Q_b$ in \eqref{def:Qb}, \eqref{BS:param}, \eqref{e:r}, \eqref{e:F}, \eqref{e:FLinfty}, \eqref{e:r-F} and then \eqref{est:energyQb}, \eqref{prop:P}, \eqref{eq:Q} and \eqref{def:R} that
\begin{align*}
E(W+F)&=E(Q_b)-\int (Q''+Q^5)(rR+F)+\mathcal{O}\left(s^{-2}+\lambda^2x_0^{-(2\theta-1)}\right)\\
&=-\frac{1}{16}\left(\int Q \right)^2b-\frac14 \left(\int Q\right) r+\mathcal{O}\left(s^{-2}+\lambda^2x_0^{-(2\theta-1)}\right).
\end{align*}
This last estimate combined with \eqref{e:r} implies \eqref{ener:W}.
\end{proof}
We compute $\mathcal E(W)$ in the next lemma.
\begin{lemma} \label{le:3.6}
Under the assumptions \eqref{BS:param}, it holds
\begin{equation} \label{def:EW}
\mathcal E(W) =
- \vec m\cdot \vec \MM Q + \mathcal R,\quad
\vec m =\begin{pmatrix} \frac{\lambda_s}{\lambda}+b \\ \frac{\sigma_s}{\lambda}-1 \end{pmatrix},\quad
\vec \MM =\begin{pmatrix} \Lambda \\ \partial_y \end{pmatrix},
\end{equation}
where, for $s$ large enough,
\begin{align}
&\left|\big(\mathcal{R},\phi\big)\right| \lesssim |\vec{m}|s^{-1}+|b_s| +s^{-2}, \quad \forall \, \phi \in \mathcal{Y} , \label{est:R:Y} \\
&\|\mathcal R\|_{L^2_B}+\|\partial_y\mathcal R\|_{L^2_B} \lesssim |\vec{m}|s^{-1}+|b_s|+s^{-2} , \label{est:R:L2B}
\end{align}
where the norm $L^2_B$ is defined in \eqref{def:L2B} ,
and
\begin{equation} \label{est:RQ}
\left| (\mathcal R,Q)- c_1 \left[ b_s+2b^2-\frac{4c_0}{\int Q}\lambda^{\frac 12}\sigma^{-\theta} \left(\frac 32 \frac {\lambda_s}{\lambda}+\theta \frac{\sigma_s}{\sigma} \right)\right]
\right|\lesssim |\vec m| s^{-1}+s^{-3} ,
\end{equation}
with $c_1=\frac1{16}\left(\int Q \right)^2$.
\end{lemma}
\begin{proof}
We compute $\mathcal E(W)$ from the definition of $W$:
\begin{align*}
\mathcal E(W) &= b_s \frac{\partial Q_b}{\partial b} + r_s R
-\frac{\lambda_s}{\lambda} \Lambda W -\left(\frac {\sigma_s}{\lambda}-1\right) \partial_y W\\
&\quad + ( Q_b'' -Q_b+Q_b^5)'
+r (R''-R)' + \partial_y((Q_b+rR+F)^5-Q_b^5-F^5).
\end{align*}
Using the definition of $Q_b$ and $\Psi_b$, the definitions of $\vec m$ and $\vec \MM$ and the equation of $R$,
we rewrite the previous identity as follows
\begin{align}
\mathcal E(W) &= -\vec m \cdot \vec \MM Q-b\vec m \cdot \vec \MM P_b-r\vec m \cdot \vec \MM R
+ (b_s+2b^2) \frac{\partial Q_b}{\partial b} + r_s R +br\Lambda R-\Psi_b
\nonumber \\
&\quad + \partial_y\big((Q_b+rR+F)^5-Q_b^5-F^5-5 rQ^4 (R+1)\big) \nonumber \\ & =-\vec m \cdot \vec \MM Q + \mathcal R, \label{decomp:EW}
\end{align}
where
\begin{align*}
\mathcal R&=\mathcal R_1+\partial_y\mathcal R_2+\mathcal R_3-\Psi_b,
\\
\mathcal R_1&= r_s R + br \Lambda R
+5\partial_y\left[Q_b^4(rR+F)-Q^4 (rR+r)\right],
\\
\mathcal R_2&=(Q_b+rR+F)^5-Q_b^5-F^5-5Q_b^4(rR+F),
\end{align*}
and
\begin{equation*}
\mathcal R_3=(b_s+2b^2) \frac {\partial Q_b}{\partial b}-b\, \vec m \cdot \vec \MM P_b-r\, \vec m \cdot \vec \MM R .
\end{equation*}
\noindent \textit{Estimates for $\mathcal R_1$.}
First we deduce from Lemma \ref{lemma:est:rF} that
\begin{equation} \label{est:R1:L2}
\left| \left( \mathcal{R}_1,\phi\right) \right|+\|\mathcal R_1\|_{L^2_B}+\|\partial_y\mathcal R_1\|_{L^2_B} \lesssim |\vec{m}|s^{-1}+s^{-2}, \quad \forall \phi \in \mathcal{Y} .
\end{equation}
Now, we estimate $(\mathcal R_1,Q)$.
First, by \eqref{e:dr} and $(R,Q)=-\frac 34 \int Q$,
\begin{equation*}
\left| (r_s R,Q) +\frac 34 c_0 \left(\int Q\right) \lambda^{\frac 12}\sigma^{-\theta} \left( \frac 12 \frac{\lambda_s}{\lambda}
-\theta\frac{\sigma_s}{\sigma} \right)\right|
\lesssim |\vec m| s^{-3}+s^{-4}.
\end{equation*}
Second,
$(br\Lambda R,Q)=-br(R,\Lambda Q)=:I$.
Third, we write
\begin{align*}
5&\left(\partial_y\left[Q_b^4(rR+F)-Q^4 (rR+r)\right],Q\right)
\\&=-5 \left( Q^4(F-r),Q'\right)-5\left((Q_b^4-Q^4)(rR+F),Q'\right) \\
&=-5 \left( Q^4(F-r),Q'\right)-20rb\left(Q^3P(R+1),Q'\right)-20b\left(Q^3P(F-r),Q'\right) \\ &
\quad -5\left((Q_b^4-Q^4-4bQ^3P)(rR+F),Q'\right) \\ & =: II_1+II_2+II_3+II_4 .
\end{align*}
Moreover, by using the identity (2.52) in \cite{MaMeRa3}
\begin{equation*}-\left(R,\Lambda Q \right)-20 \left( Q^3(R+1)P,Q'\right)=0,\end{equation*}
we get that $I+II_2=0$. To deal with $II_1$, we deduce from \eqref{e2:r-F}, $|\sigma_s-\lambda| \le \lambda |\vec{m}|$ and \eqref{BS:param} that
\begin{equation*}
\left| II_1+c_0\theta \left(\int Q\right) \lambda^{\frac12}\sigma^{-\theta-1}\sigma_s \right| \lesssim |\vec{m}|s^{-2}+s^{-3} .
\end{equation*}
Next, it is clear from \eqref{BS:param} and \eqref{e:r-F} that $\big| II_3 \big| \lesssim s^{-3}$.
Finally we also claim that $\big| II_4 \big| \lesssim s^{-3}$. Indeed, a direct computation gives
\begin{equation*}
II_4=-5\left(\big(4Q^3bP(\chi_b-1)+6Q^2b^2P_b^2+4Qb^3P_b^3+b^4P_b^4\big)(rR+F),Q'\right)
\end{equation*}
which implies the claim by using the definition of $\chi_b$ in \eqref{def:Pb}, \eqref{BS:param}, \eqref{e:r} and \eqref{e:F}.
Therefore, we deduce gathering those estimates that
\begin{equation} \label{est:R1}
\left| (\mathcal R_1,Q) +\frac{c_0}4 \left(\int Q\right) \lambda^{\frac 12}\sigma^{-\theta} \left( \frac 32 \frac{\lambda_s}{\lambda}
+\theta\frac{\sigma_s}{\sigma} \right)\right| \lesssim |\vec{m}|s^{-2}+s^{-3} .
\end{equation}
\smallskip
\noindent \textit{Estimates for $\partial_y\mathcal R_2$.}
We claim that
\begin{equation} \label{est:R2}
\big|\big( \partial_y\mathcal{R}_2, \phi\big)\big|+\big\| \partial_y\mathcal{R}_2\big\|_{L^2_B}+\big\| \partial_y^2\mathcal{R}_2\big\|_{L^2_B} \lesssim s^{-2}, \ \forall \, \phi \in \mathcal{Y}, \quad \big|\big( \partial_y\mathcal{R}_2, Q\big)\big| \lesssim s^{-3} .
\end{equation}
We first develop $\mathcal{R}_2$:
\begin{align*}
\mathcal{R}_2&=10Q^3(rR+F)^2+10(Q_b^3-Q^3)(rR+F)^2+10Q_b^2(rR+F)^3\\
&\quad +5Q_b(rR+F)^4+ (rR+F)^5-F^5 .
\end{align*}
We deduce easily that $\big|\big( \partial_y\mathcal{R}_2, \phi\big)\big|+\big\| \partial_y\mathcal{R}_2\big\|_{L^2_B}+\big\| \partial_y^2\mathcal{R}_2\big\|_{L^2_B} \lesssim s^{-2}$, $\forall \, \phi \in \mathcal{Y}$, arguing as in the proof of Lemma \ref{lemma:est:rF}.
Now, we prove the second estimate in \eqref{est:R2}.
On the one hand, we get easily from the definition of $Q_b$ in \eqref{def:Qb}, \eqref{e:r} and \eqref{e:F} that
\begin{equation*}
\big|\big( \partial_y\mathcal{R}_2, Q\big)+10(Q^3(rR+F)^2,Q')\big| \lesssim s^{-3} .
\end{equation*}
On the other hand, we see by integration by parts that
\begin{align*}
10(Q^3(rR+F)^2,Q')&=-5r^2(Q^4(R+1)\partial_yR)
\\&\quad -5r(Q^4,(F-r) \partial_yR)-5r(Q^4,R\partial_yF)-5(Q^4,F\partial_yF) .
\end{align*}
Observe from the definition of $R$ in \eqref{def:R} that $5Q^4(R+1)=-\partial_y^2R+R$, so that the first term on the right-hand side of the above identity cancels out by symmetry.
Hence, it follows from \eqref{e:r}, \eqref{e:F} and \eqref{e:r-F} that
\begin{equation*}
\big|(Q^3(rR+F)^2,Q')\big|\lesssim s^{-3} ,
\end{equation*}
which yields the second estimate in \eqref{est:R2}.
\smallskip
\noindent \textit{Estimates for $\mathcal{R}_3$.} First, we deduce from \eqref{eq:dQb:db} and \eqref{BS:param} that
\begin{equation*}
|b_s+2b^2| \left| \left( \frac {\partial Q_b}{\partial b}, \phi\right) \right| \lesssim |b_s+2b^2| \lesssim |b_s|+s^{-2}, \quad \forall \phi \in \mathcal{Y} ,
\end{equation*}
\begin{equation*}
|b_s+2b^2| \left(\left\| \frac {\partial Q_b}{\partial b}\right\|_{L^2_B}+\left\| \partial_y\left(\frac {\partial Q_b}{\partial b}\right)\right\|_{L^2_B}\right) \lesssim |b_s+2b^2| \lesssim |b_s|+s^{-2} .
\end{equation*}
Arguing similarly, we get from \eqref{def:Pb}, \eqref{BS:param} and \eqref{e:r} that
\begin{equation*}
\left|b \left(\vec m \cdot \vec \MM P_b, \phi \right) \right|+\left|r \left(\vec m \cdot \vec \MM R, \phi \right) \right| \lesssim s^{-1}|\vec{m}|\quad \forall \phi \in \mathcal{Y} ,
\end{equation*}
and
\begin{equation*}
|b| \left\|\vec m \cdot \partial_y^k\vec \MM P_b\right\|_{L^2_B} +|r| \left\|\vec m \cdot \partial_y^k\vec \MM R\right\|_{L^2_B} \lesssim s^{-1}|\vec{m}|, \quad k=0,1 .
\end{equation*}
It follows combining those estimates that
\begin{equation} \label{est:R3}
(\mathcal{R}_3,\phi)+\|\mathcal{R}_3 \|_{L^2_B}+\|\partial_y\mathcal{R}_3 \|_{L^2_B} \lesssim s^{-1}|\vec{m}|+|b_s|+s^{-2}, \quad \forall \phi \in \mathcal{Y} .
\end{equation}
By using the identity \eqref{eq:dQb:db}, we get that
\begin{equation*}
\left( \frac {\partial Q_b}{\partial b},Q\right)=\left(Q,P\right)+\left((\chi_b-1)P,Q\right)+
\gamma\left(yP\chi_b',Q\right).
\end{equation*}
Thus, it follows from the definition of $\chi_b$ in \eqref{def:Pb}, the properties of $Q$, \eqref{prop:P} and \eqref{BS:param}, we deduce that
\begin{equation} \label{est:dQb:db}
\left|(b_s+2b^2)\left( \frac {\partial Q_b}{\partial b},Q\right)-\frac1{16}\left(\int Q \right)^2(b_s+2b^2)\right| \lesssim e^{-\frac{s^{\gamma}}2} .
\end{equation}
\smallskip
Therefore, we conclude the proof of \eqref{est:R:Y} gathering \eqref{est:PsibY}, \eqref{est:R1:L2}, \eqref{est:R2} and \eqref{est:R3}, the proof of \eqref{est:R:L2B} gathering \eqref{est:PsibL2}, \eqref{est:R1:L2}, \eqref{est:R2} and \eqref{est:R3}, and the proof of \eqref{est:RQ} gathering \eqref{est:PsibQ}, \eqref{BS:param}, \eqref{est:R1}, \eqref{est:R2}, \eqref{est:R3} and \eqref{est:dQb:db}.
\end{proof}
We define
\begin{equation} \label{def:gh}
g(s)=\frac{b(s)}{\lambda^2(s)}+\frac{4}{\int Q}c_0 \lambda^{-\frac 32}(s)\sigma^{-\theta}(s) \quad \text{and} \quad
h(s)=\lambda^{\frac 12}(s)-\frac{1}{\int Q} \frac{2c_0}{1-\theta} \sigma^{-\theta+1}(s) .
\end{equation}
\begin{lemma} \label{lemma:gh}
It holds
\begin{align}
\left| c_1 \lambda^{2} g_s - (\mathcal R,Q)\right|
&\lesssim |\vec m| s^{-1}+s^{-3} ; \label{est:g} \\
\left| \lambda^{-\frac12}h_s +\frac 12 \lambda^2g\right|&\lesssim |\vec m| . \label{est:h}
\end{align}
\end{lemma}
\begin{proof}
First, observe by a direct computation that
\begin{equation*}
\lambda^2g_s=b_s+2b^2-2\left(\frac{\lambda_s}{\lambda}+b\right)b-\frac{4c_0}{\int Q}\lambda^{\frac 12}\sigma^{-\theta} \left(\frac 32 \frac {\lambda_s}{\lambda}+\theta \frac{\sigma_s}{\sigma} \right)
\end{equation*}
Thus, estimate \eqref{est:g} follows from \eqref{BS:param}, \eqref{def:EW}, and \eqref{est:RQ}.
Another direct computation yields
\begin{equation*}
\lambda^{-\frac12}h_s +\frac 12 \lambda^2g=\frac12 \left( b+\frac{\lambda_s}{\lambda}\right) +\frac{2c_0}{\int Q} \left( 1-\frac{\sigma_s}{\lambda}\right)\lambda^{\frac12} \sigma^{-\theta} .
\end{equation*}
Hence, we deduce from \eqref{BS:param} and \eqref{def:EW} that
\begin{equation*}
\left| \lambda^{-\frac12}h_s +\frac 12 \lambda^2g\right| \lesssim |\vec{m}| (1+s^{-1}) ,
\end{equation*}
which implies \eqref{est:h} by choosing $s$ large enough.
\end{proof}
\subsection{Modulation and parameter estimates}
Let $U$ be a solution of \eqref{gkdv} defined on a time interval $\mathcal{I}\subset[t_0,+\infty)$ and set
\begin{equation} \label{def:v}
v(t,x)=U(t,x)-f(t,x) ,
\end{equation}
where $f$ is defined in Section~\ref{sec:AS}.
We assume that there exists $(\lambda_\sharp(t),\sigma_\sharp(t)) \in (0,+\infty)^2$ and $z \in \mathcal{C}\big(\mathcal{I} : L^2(\mathbb R)\big)$ such that
\begin{equation} \label{tube.1}
v(t,x)=\lambda_\sharp^{-\frac12}(t)(Q+r_\sharp R+ z)(t,y), \quad y=\frac{x-\sigma_\sharp(t)}{\lambda_\sharp(t)},
\quad
r_\sharp(t)=\lambda_\sharp^{\frac 12}(t) f(t,\sigma_\sharp(t)),
\end{equation}
with
\begin{equation} \label{tube.2}
\|z(t)\|_{L^2} + \lambda_\sharp (t) \sigma_\sharp^{-1}(t)+ \sigma_\sharp^{-1}(t)\le \alpha^\star ,
\end{equation}
for all $t \in \mathcal{I}$ and
where $\alpha^\star$ is small positive universal constant.
For future use, remark that \eqref{tube.2} implies (using $\frac 12<\theta<1$)
\begin{equation}\label{tube.3}
\frac{\lambda_\sharp^{\frac 32}}{\sigma_\sharp^{\theta+1}}
= \left( \frac{\lambda_\sharp}{\sigma_\sharp}\right)^{\frac 32} \sigma_\sharp^{\frac 32-\theta-1}
\lesssim (\alpha^\star)^{\frac 32}\quad\mbox{and}\quad
\frac{\lambda_\sharp^{\frac 12}}{\sigma_\sharp^{\theta}}
= \left( \frac{\lambda_\sharp}{\sigma_\sharp}\right)^{\frac 12} \sigma_\sharp^{\frac 12-\theta}
\lesssim (\alpha^\star)^{\frac 12}.
\end{equation}
We collect in the next lemma the standard preliminary estimates on this decomposition related to the choice of suitable orthogonality conditions for the remainder term.
\begin{lemma} \label{lemma:decomp}
Assume \eqref{tube.1}-\eqref{tube.2} for $\alpha^\star>0$ small enough.
Then, there exist unique continuous functions $(\lambda,\sigma,b):\mathcal{I} \to (0,\infty)\times \mathbb R^2$ such that
\begin{equation} \label{decomp:v}
\lambda^{\frac 12}(t) v(t,\lambda(t) y +\sigma(t)) = Q_{b(t)}(y) + r(t) R(y)+\varepsilon(t,y)=W(t,y)+\varepsilon(t,y) ,
\end{equation}
\begin{equation}\label{lambdacheck}
\left|\frac{\lambda(t)}{\lambda_\sharp(t)}-1\right|
+|b(t)|+\left|\frac{\sigma(t)-\sigma_\sharp(t)}{\lambda_\sharp(t)}\right|
\lesssim \|z(t)\|_{L_\textnormal{loc}^2},
\end{equation}
and where $\varepsilon$ satisfies, for all $t\in \mathcal{I}$,
\begin{equation}\label{ortho}
(\varepsilon(t),\Lambda Q)=(\varepsilon(t),y\Lambda Q)=(\varepsilon(t),Q)=0,
\end{equation}
\begin{equation}\label{1est:eps}
\|\varepsilon(t)\|_{L_\textnormal{loc}^2} \lesssim \|z(t)\|_{L_\textnormal{loc}^2},\quad
\|\varepsilon(t)\|_{L^2} \lesssim \|z(t)\|_{L^2}^{\frac 58} .
\end{equation}
\end{lemma}
\begin{proof} First, the decomposition is performed for fixed $t \in \mathcal{I}$. Let us define the map
\begin{equation*}
\Theta: ( \check\lambda,\check\sigma,\check{b},v_1 ) \in (0,+\infty)\times \mathbb R^2 \times L^2 \mapsto \left( (\check\varepsilon,y\Lambda Q),(\check\varepsilon,\Lambda Q),(\check\varepsilon, Q)\right) \in \mathbb R^3 ,
\end{equation*}
where
\begin{align*}
\check\varepsilon(y)=\check\varepsilon_{ ( \check\lambda,\check\sigma,\check{b},v_1)}(y)&
=\check\lambda^{\frac12}v_1\left(t,\check\lambda y+\check\sigma\right) -Q_{\check{b}}(y)
-\check\lambda^{\frac12}\lambda_\sharp^{\frac12}(t)f\left(t,\sigma_\sharp(t)+\lambda_\sharp(t)\check\sigma\right)R(y)
. \end{align*}
Let $v_\sharp=Q+r_\sharp R$
and $\theta_0:=(1,0,0,v_\sharp)$.
We see that $\check\varepsilon_{\theta_0}=0$, so that $\Theta(\theta_0)=0$. Moreover, it follows from explicit computations that
\begin{align*}
&\partial_{\check\lambda} \check\varepsilon_{|_{\theta_0}}=\Lambda Q+r_\sharp yR',\\
&\partial_{\check\sigma} \check\varepsilon_{|_{\theta_0}}=Q'+r_\sharp R'-\lambda_\sharp^{\frac 32} \partial_x f(t,\sigma_\sharp) R,\\
&\partial_{\check{b}} \check\varepsilon_{|_{\theta_0}}=-P .
\end{align*}
In particular, by parity properties, the identity $(Q',y\Lambda Q)=\frac12(Q,\Lambda Q)+(yQ',\Lambda Q)=\| \Lambda Q\|_{L^2}^2$,
and then (from \eqref{decay:q_0.1} and \eqref{tube.3})
\begin{equation*}
|\lambda_\sharp^{\frac 32} \partial_x f(t,\sigma_\sharp)|\lesssim \lambda_\sharp^{\frac 32} \sigma_\sharp^{-\theta-1}
\lesssim (\alpha^\star)^{\frac 32},\quad
|r_\sharp|\lesssim \lambda_\sharp^{\frac 12}\sigma_\sharp^{-\theta}
\lesssim (\alpha^\star)^{\frac 12},
\end{equation*}
we obtain
\begin{align*}
\frac{\partial \Theta}{\partial (\check\lambda,\check\sigma,\check{b})}(\theta_0)
&=
\begin{pmatrix}
(\Lambda Q,y\Lambda Q) & (Q',y\Lambda Q)+r_\sharp (R',y\Lambda Q) & -(P,y\Lambda Q) \\
\|\Lambda Q\|_{L^2}^2 +r_\sharp (yR',\Lambda Q)& (Q',\Lambda Q)-\lambda_\sharp^{\frac 32} \partial_x f(t,\sigma_\sharp)( R,\Lambda Q) & -(P,\Lambda Q)\\
(\Lambda Q,Q) +r_\sharp (yR',Q)& (Q',Q)-\lambda_\sharp^{\frac 32} \partial_x f(t,\sigma_\sharp) (R,Q) & -(P,Q)
\end{pmatrix}
\\
&=
\begin{pmatrix}
0 & \|\Lambda Q\|_{L^2}^2 & -(P,y\Lambda Q) \\
\|\Lambda Q\|_{L^2}^2 & 0 & -(P,\Lambda Q)\\
0 & 0 & -\frac1{16}\|Q\|_{L^1}^2
\end{pmatrix} + O\left((\alpha^\star)^{\frac 12}\right).
\end{align*}
Thus, the Jacobian satisfies for $\alpha^\star$ small enough
\begin{equation*}
\det \left(\frac{\partial \Theta}{\partial (\check\lambda,\check\sigma,\check{b})}(\theta_0)\right)=
\frac1{16}\|\Lambda Q\|_{L^2}^4 \|Q\|_{L^1}^2+O\left((\alpha^\star)^{\frac 12}\right) >\frac1{17}\|\Lambda Q\|_{L^2}^4 \|Q\|_{L^1}>0 .
\end{equation*}
Therefore, possibly taking a smaller constant $\alpha^\star>0$, it follows from the implicit function theorem that for
any $v_1=Q+r_\sharp R+z$ where $z$ satisfies $\|z\|_{L^2} <\alpha^\star$, there exist unique $(\check\lambda,\check\sigma,\check{b})=(\check\lambda,\check\sigma,\check{b})(v_1)$ such that $\Theta(\check\lambda,\check\sigma,\check{b},v_1)=0$, where $\check\lambda$ is close to $1$ and $\check\sigma,\check{b}$ are small.
Moreover, the map $v_1\mapsto (\check\lambda,\check\sigma,\check{b})(v_1)$ is continuous.
Now, for a function $v$ satisfying \eqref{tube.1}, we consider
\begin{equation*}
v_1(t,y)=\lambda_\sharp^{\frac 12}(t)v(t,\lambda_\sharp(t) y + \sigma_\sharp(t))=Q(y)+r_\sharp(t) R(y)+ z(t,y),
\end{equation*}
and we define
\begin{align*}
&\lambda(t)=\lambda_\sharp(t)\check\lambda(v_1(t)), \quad \sigma(t)=\lambda_\sharp(t)\check\sigma(v_1(t))+\sigma_\sharp(t), \quad b(t)=\check{b}(v_1(t)),
\\
&\varepsilon(t,y)=\check\varepsilon_{ \left( \check\lambda(v_1(t)),\check\sigma(v_1(t)),\check{b}(v_1(t)),v_1(t)\right)}(y) .
\end{align*}
In particular, $\varepsilon$ satisfies the orthogonality conditions \eqref{ortho}
and it holds
\begin{equation*}
\lambda^{\frac 12}(t) v(t,\lambda(t) y +\sigma(t))
=Q_{b(t)}(y)+\lambda^{\frac12}(t)f(t,\sigma(t))R(y)+\varepsilon(t,y) ,
\end{equation*}
which is the desired decomposition for $v$.
Now, we prove \eqref{lambdacheck} and \eqref{1est:eps}.
We omit mentioning the time dependency for simplicity.
Note from the above, the identity
\begin{equation}\label{id:eps}\begin{aligned}
\varepsilon(y)
&=\check\lambda^{\frac12} Q(\check\lambda y+\check\sigma)
- Q_{\check{b}}(y)
\\&\quad +\check\lambda^{\frac 12}\lambda_\sharp^{\frac 12}\left[f(\sigma_\sharp)R(\check\lambda y +\check\sigma)- f(\sigma_\sharp+\lambda_\sharp \check\sigma)R(y)\right]+\check\lambda^{\frac12}z(\check\lambda y+\check\sigma).
\end{aligned}\end{equation}
We will project this identity on the three orthogonality directions $y\Lambda Q$, $\Lambda Q$ and $Q$
of $\varepsilon$.
First, by direct computations, it holds
\begin{align*}
&(\check\lambda^{\frac12} Q(\check\lambda \cdot +\check\sigma)-Q, y\Lambda Q )
= \check\sigma\left( \|\Lambda Q\|_{L^2}^2 + O(|\check\lambda-1|+|\check\sigma|)\right),\\
&(\check\lambda^{\frac12} Q(\check\lambda \cdot +\check\sigma)-Q, \Lambda Q)
=(\check\lambda-1) \|\Lambda Q\|_{L^2}^2 + O(|\check\lambda-1|^2+|\check\sigma|^2),\\
&(\check\lambda^{\frac12} Q(\check\lambda \cdot +\check\sigma)-Q, Q)
=O(|\check\lambda-1|^2+|\check\sigma|^2),\quad
(P_{\check{b}},Q)=\frac {\check{b}}{16}\|Q\|_{L^1}^2+O(\check{b}^{10}).
\end{align*}
Second, by the triangle inequality and \eqref{decay:q_0.1} and \eqref{tube.3},
\begin{align*}
&\lambda_\sharp^{\frac 12}\left\|f(\sigma_\sharp)R(\check\lambda \cdot +\check\sigma)- f(\sigma_\sharp+\lambda_\sharp \check\sigma)R(\cdot)\right\|_{L^2}
\\
&\quad \lesssim \lambda_\sharp^{\frac 12} \left|f(\sigma_\sharp)- f(\sigma_\sharp+\lambda_\sharp \check\sigma) \right| \left\| R(\check\lambda \cdot +\check\sigma)\right\|_{L^2}
+\lambda_\sharp^{\frac 12} |f(\sigma_\sharp)| \left\|R(\check\lambda \cdot +\check\sigma)- R(\cdot)\right\|_{L^2}
\\&\quad \lesssim \lambda_\sharp^{\frac 32} \sigma_\sharp^{-\theta-1} |\check\sigma| +
\lambda_\sharp^{\frac 12}\sigma_\sharp^{-\theta} \left( |\check\lambda-1|+|\check\sigma|\right)
\lesssim (\alpha^\star)^{\frac 12} \left(|\check\lambda-1|+|\check\sigma|\right).
\end{align*}
Therefore, the projections yield the following estimates
\begin{equation*}
|\check\lambda-1|+|\check\sigma|\lesssim \|z\|_{L^2_\textnormal{loc}} + |\check{b}|+(\alpha^\star)^{\frac 12} \left(|\check\lambda-1|+|\check\sigma|\right),\quad
|\check{b}|\lesssim \|z\|_{L^2_\textnormal{loc}}+(\alpha^\star)^{\frac 12} \left(|\check\lambda-1|+|\check\sigma|\right).
\end{equation*}
Combining these estimates, for $\alpha^\star$ small enough, we obtain \eqref{lambdacheck}.
Then, \eqref{1est:eps} follows using the above estimates and \eqref{def:Pb} back into \eqref{id:eps}
(note in particular that from \eqref{def:Pb} and $\gamma=\frac 34$, $\|P_{\check{b}}\|_{L^2}^2 \lesssim |\check{b}|^{2-\gamma}
\lesssim \|z\|_{L^2_\textnormal{loc}}^{5/4}\lesssim \|z\|_{L^2}^{5/4}$).
\end{proof}
\begin{remark}
The $\mathcal C^1$ regularity of $t\mapsto (\lambda(t),\sigma(t),b(t))$ and the equation
\begin{equation}\label{eq:eps}
\partial_s \varepsilon = \partial_y \left[ -\partial_y^2\varepsilon+\varepsilon-\left((W+F+\varepsilon)^5-(W+F)^5\right)\right]
- \mathcal E(W) +\vec m \cdot \vec\MM \varepsilon-b\Lambda \varepsilon ,
\end{equation}
(where we have used the notation in \eqref{def:EV} and \eqref{def:mM})
follow from classical arguments and computations. We refer for example to the proof of Lemma 2.7. in \cite{CoMa2}.
\end{remark}
Next, we derive some estimates for $\varepsilon$ in $H^1$ related to the conservation of mass and energy.
\begin{lemma}[Mass and energy estimates for $\varepsilon$]\label{le:3.9}
Under the bootstrap assumptions \eqref{BS:param}, it holds
\begin{equation} \label{mass:eps}
\|\varepsilon\|_{L^2}^2 \lesssim \left| \int U_0^2-\int Q^2 \right| +s^{-1}+x_0^{-(2\theta-1)}
\end{equation}
and
\begin{equation} \label{ener:eps}
\lambda^{-2}\|\partial_y\varepsilon\|_{L^2}^2 \lesssim |E(u_0)|+|g(s)|+\lambda^{-2}s^{-2}+\lambda^{-2}\int Q\varepsilon^2+x_0^{-(2\theta+1)} ,
\end{equation}
for $s$ large enough, where $g$ is defined in Lemma \ref{lemma:gh}.
\end{lemma}
\begin{proof} By using the conservation of the $L^2$ norm for $u$, we obtain that
\begin{equation*}
\int U_0^2=\int U^2=\|W+F+\varepsilon\|_{L^2}^2=\int (W+F)^2+2\int (W+F)\varepsilon+\int \varepsilon^2 .
\end{equation*}
We observe by using the third orthogonality condition in \eqref{ortho} and then the Cauchy-Schwarz inequality, \eqref{mass:F} and \eqref{BS:param} that
\begin{align*}
\left|2\int (W+F)\varepsilon\right| &\le 2|r|\left|\int R \varepsilon\right|+2|b|\left|\int P_b \varepsilon\right|+2\left|\int F \varepsilon \right| \\
& \le \frac12 \int \varepsilon^2+\mathcal{O}(s^{-2+\gamma})+\mathcal{O}(x_0^{-(2\theta-1)}) ,
\end{align*}
which combined with \eqref{mass:W} implies \eqref{mass:eps}.
We turn to the proof of \eqref{ener:eps}. By using the conservation of the energy and the scaling properties, we get that
\begin{align*}
\lambda^2E(U_0)&=\lambda^2E(U)=E(W+F+\varepsilon)
\\ & =E(W+F)+\int \partial_y(W+F)\partial_y \varepsilon+\frac12\int (\partial_y \varepsilon)^2-\frac16 \int \big( (W+F+\varepsilon)^6-(W+F)^6\big) .
\end{align*}
Thus, it follows by using the identity $\int (-Q''-Q^5)\varepsilon=\int Q \varepsilon=0$ that
\begin{align*}
\lambda^2E(u_0)&=E(W+F)+b\int \partial_yP_b\partial_y \varepsilon+r\int \partial_yR\partial_y \varepsilon-\int( \partial_y^2 F+F^5) \varepsilon+\frac12\int (\partial_y \varepsilon)^2
\\ & \quad -\frac16 \int \big( (W+F+\varepsilon)^6-(W+F)^6-6Q^5\varepsilon-6F^5\varepsilon \big) .
\end{align*}
We get from the Cauchy-Schwarz inequality, \eqref{BS:param} and \eqref{e:r} that
\begin{equation*}
|b|\left|\int \partial_yP_b\partial_y \varepsilon\right|+|r|\left|\int \partial_yR\partial_y \varepsilon\right| \le \frac14\int (\partial_y\varepsilon)^2+\mathcal{O}(s^{-2}) .
\end{equation*}
Now, we deal with the term $\int( \partial_y^2 F+F^5) \varepsilon$. We get from the Cauchy-Schwarz inequality and \eqref{bound:df:dx} that
\begin{equation*}
\left| \int \partial_y^2 F \varepsilon \right|= \left| \int \partial_y F \partial_y \varepsilon \right| \le \frac18 \| \partial_y \varepsilon\|_{L^2}^2+c \| \partial_yF\|_{L^2}^2 \le \frac18 \| \partial_y \varepsilon\|_{L^2}^2+c\lambda^2 x_0^{-(2\theta+1)} .
\end{equation*}
Similarly, we get from H\"older's inequality, the Gagliardo-Niremberg inequality \eqref {sharpGN} and then \eqref{1est:eps} and \eqref{bound:df:dx} that
\begin{equation} \label{est:F5eps}
\left| \int F^5 \varepsilon \right| \lesssim \int \varepsilon^6+\int F^6 \lesssim \| \varepsilon \|_{L^2}^4 \| \partial_y \varepsilon\|_{L^2}^2+\| F \|_{L^2}^4 \| \partial_y F\|_{L^2}^2 \le \frac1{16}\| \partial_y \varepsilon\|_{L^2}^2+c\lambda^2 x_0^{-(2\theta+1)} .
\end{equation}
Moreover, from interpolation, the Gagliardo-Nirenberg inequality \eqref {sharpGN} and \eqref{BS:param}, \eqref{e:r}, \eqref{e:F}, it holds
\begin{align*}
&\left| \int \big( (W+F+\varepsilon)^6-(W+F)^6-6Q^5\varepsilon-6F^5\varepsilon \big)\right|
\\ & \quad \quad \quad \quad \lesssim \left| \int \big( (W+F)^5-6Q^5-6F^5\big)\varepsilon\right|+\int (W+F)^4\varepsilon^2+\int \varepsilon^6
\\& \quad \quad \quad \quad \lesssim s^{-2}+\int Q^2\varepsilon^2+\int F^4\varepsilon^2+\|\varepsilon\|_{L^2}^4\int (\partial_y\varepsilon)^2 .
\end{align*}
arguing as in \eqref{est:F5eps}, we estimate the term $\int F^4\varepsilon^2$ as follows
\begin{equation*}
\left| \int F^4 \varepsilon^2 \right| \lesssim \int \varepsilon^6+\int F^6 \le \frac1{32}\| \partial_y \varepsilon\|_{L^2}^2+c\lambda^2 x_0^{-(2\theta+1)} .
\end{equation*}
We conclude the proof of \eqref{ener:eps} by combining those estimates with \eqref{ener:W} and \eqref{1est:eps}.
\end{proof}
Next, the equation of the parameters $\lambda$, $\sigma$ and $b$ are deduced from modulation estimates.
\begin{lemma}[Modulation estimates]
Under the bootstrap assumptions \eqref{BS:param}, it holds
\begin{align}
|\vec m|&\lesssim \| \varepsilon \|_{L^2_\textnormal{loc}} +s^{-2} ,\label{mod:est:m}
\\
|b_s|&\lesssim \| \varepsilon \|_{L^2_\textnormal{loc}}^2 +s^{-2} , \label{mod:est:bs}
\\
\lambda^2|g_s|&\lesssim s^{-3}+s^{-1}\| \varepsilon \|_{L^2_\textnormal{loc}}+\| \varepsilon \|_{L^2_\textnormal{loc}}^{2} , \label{mod:est:g}
\end{align}
for $s$ large enough, where the $L^2_\textnormal{loc}$-norm is defined in \eqref{def:L2B}.
\end{lemma}
\begin{proof}
First, we differentiate the first orthogonality condition in \eqref{ortho} with respect to $s$, use the equation \eqref{eq:eps}, follow the computations in the proof of Lemma 2.7 in \cite{MaMeRa1} and use the estimate \eqref{est:R:Y} to get that
\begin{equation} \label{mod:est:m1}
\left| \left(\frac{\lambda_s}{\lambda}+b \right) +\frac{\big( \varepsilon, \mathcal{L}(\Lambda Q)'\big)}{\|\Lambda Q\|_{L^2}^2}\right| \lesssim
|\vec{m}|\big(s^{-1}+\|\epsilon\|_{L^2_\textnormal{loc}}\big)+|b_s|+s^{-2}+\|\epsilon\|_{L^2_\textnormal{loc}}^2 .
\end{equation}
Now, we derive the second orthogonality condition in \eqref{ortho} with respect to $s$. By combining similar estimates with the identity $(Q',y\Lambda Q)=\| \Lambda Q\|_{L^2}^2$, we also get that
\begin{equation} \label{mod:est:m2}
\left| \left(\frac{\sigma_s}{\lambda}-1 \right) +\frac{\big( \varepsilon, \mathcal{L}(y\Lambda Q)'\big)}{\|\Lambda Q\|_{L^2}^2}\right| \lesssim
|\vec{m}|\big(s^{-1}+\|\epsilon\|_{L^2_\textnormal{loc}}\big)+|b_s|+s^{-2}+\|\epsilon\|_{L^2_\textnormal{loc}}^2 .
\end{equation}
Next, we derive the third orthogonality condition in \eqref{ortho} with respect to $s$. It follows that
\begin{align*}
0=\big(\partial_s\varepsilon,Q \big)&=\big( \partial_y\mathcal{L}\varepsilon,Q\big)-\left(\partial_y\big((W+F+\varepsilon)^5-(W+F)^5-5Q^4\varepsilon \big), Q \right)-\big(\mathcal{E}(W),Q\big)\\
&\quad +\left( \frac{\lambda_s}{\lambda}+b\right)\big( \Lambda \varepsilon,Q\big)+
\left( \frac{\sigma_s}{\lambda}-1\right)\big( \partial_y \varepsilon,Q\big)-b\big( \Lambda \varepsilon,Q\big) .
\end{align*}
We observe the cancellations $\big( \partial_y\mathcal{L}\varepsilon,Q\big)=-(\varepsilon, \mathcal{L}(Q'))=0$ and $\big( \Lambda \varepsilon,Q\big)=-\big( \varepsilon,\Lambda Q\big)=0$. We also get by using \eqref{BS:param}, \eqref{e:r} and \eqref{e:F} that for $s$ large enough
\begin{equation*}
\left|\left(\partial_y\big((W+F+\varepsilon)^5-(W+F)^5-5Q^4\varepsilon \big), Q \right)\right| \lesssim s^{-1}\|\varepsilon\|_{L^2_\textnormal{loc}}+\|\varepsilon\|_{L^2_\textnormal{loc}}^2 .
\end{equation*}
Moreover, we have from \eqref{def:EW} that
\begin{equation*}
\big(\mathcal{E}(W),Q\big)=-\left( \frac{\lambda_s}{\lambda}+b\right)\big( \Lambda Q,Q\big)-\left( \frac{\sigma_s}{\lambda}-1\right)\big( \partial_y Q,Q\big)+(\mathcal{R},Q) .
\end{equation*}
Hence, it follows from the cancellations $(\Lambda Q,Q)=(\partial_yQ,Q)=0$ and the estimate \eqref{est:RQ} that
\begin{equation*}
\left|\big(\mathcal{E}(W),Q\big)- c_1 \left[ b_s+2b^2-4c_0\left(\int Q \right)^{-1}\lambda^{\frac 12}\sigma^{-\theta} \left(\frac 32 \frac {\lambda_s}{\lambda}+\theta \frac{\sigma_s}{\sigma} \right)\right]\right| \lesssim |\vec{m}|s^{-1}+s^{-3} .
\end{equation*}
We deduce combining those estimates and using \eqref{BS:param} the following rough estimate on $|b_s|$
\begin{equation*}
|b_s| \lesssim s^{-2}+|\vec{m}|\big(s^{-1}+\|\epsilon\|_{L^2_\textnormal{loc}}\big)+s^{-1}\|\varepsilon\|_{L^2_\textnormal{loc}}+\|\varepsilon\|_{L^2_\textnormal{loc}}^2 ,
\end{equation*}
which combined with \eqref{mod:est:m1} and \eqref{mod:est:m2} yields \eqref{mod:est:m} and \eqref{mod:est:bs} by taking $s$ large enough.
Finally, by combining the previous estimates with \eqref{est:g}, we deduce that
\begin{equation*}
\lambda^2 |g_s| \lesssim |\vec{m}| \big( s^{-1}+\|\varepsilon\|_{L^2_\textnormal{loc}}\big)+s^{-3}+s^{-1}\|\varepsilon\|_{L^2_\textnormal{loc}}+\|\varepsilon\|_{L^2_\textnormal{loc}}^2 ,
\end{equation*}
which conclude the proof of \eqref{mod:est:g} by taking $s$ large enough thanks to \eqref{mod:est:m}.
\end{proof}
\subsection{Bootstrap estimates}
Let $\psi \in\mathcal C^\infty$ be a nondecreasing function such that
\begin{equation*}
\psi(y) = \left\{ \begin{aligned} e^{2y} \quad & \mbox{for } y < -1, \\ 1 \quad & \mbox{for } y > -\frac 12, \end{aligned} \right.
\end{equation*}
For $B > 100$ large to be chosen later, we define
\begin{equation*}
\psi_B(y) = \psi\left( \frac yB \right) \quad \text{and} \quad \varphi_B(y)=e^{\frac{y}B} .
\end{equation*}
Note that, directly from the definitions of $\psi$ and $\varphi$, we have, for all $y\in\mathbb{R}$ and
\begin{equation} \label{cut:onR}
\left\{
\begin{gathered}
\psi_B(y) +\sqrt{\psi_B(y)}\lesssim \varphi_B(y), \quad |y|^4\psi_B(y) \lesssim B^4\varphi_B(y),\\
\psi_B'(y) + B^2|\psi_B'''(y)| + B^2|\varphi_B'''(y)| \lesssim \varphi_B'(y), \\
\varphi_B(y)+\psi_B(y)+\varphi_B^2(y)e^{-\frac{y}B}+\psi_B^2(y)e^{-\frac{y}B}+|y\psi_B'(y)|\lesssim B \varphi_B'(y) ,
\end{gathered}
\right.
\end{equation}
and $\frac 12 e^{\frac{2y}B } \leq \psi_B(y) \leq 3 e^{\frac{2y}B} $, for all $y<0$.
Let $0<\rho\ll 1$ and $B>100$ to be chosen later.
In addition of \eqref{BS:param}, we will work under the following bootstrap assumptions.
\begin{equation}\label{BS:eps}
\mathcal{N}_B(\varepsilon):= \left(\int \varepsilon^ 2 \varphi_B+\int (\partial_y\varepsilon)^ 2 \psi_B\right)^{\frac12} \le |s|^{-\frac 54}
\end{equation}
In particular, from the definition of the $L^2_\textnormal{loc}$-norms in \eqref{def:L2B} and $B>100$, it holds
\begin{equation} \label{est:L2loc}
\|\varepsilon\|_{L^2_\textnormal{loc}}+\|\partial_y\varepsilon\|_{L^2_\textnormal{loc}} \lesssim \mathcal{N}_B(\varepsilon) \quad \text{and} \quad \|\varepsilon\|_{L^2_\textnormal{loc}}^2 \lesssim B \int \varepsilon^2 \varphi_B' .
\end{equation}
For future reference, we state here some consequences of the bootstrap assumptions.
\begin{lemma} Under the bootstrap assumptions \eqref{BS:param} and \eqref{BS:eps}, it holds
\begin{gather} \label{BS:m}
|\vec{m}|=\left| \frac{\lambda_s}{\lambda}+b\right|+\left| \frac{\sigma_s}{\lambda}-1\right| \lesssim \|\varepsilon \|_{L^2_\textnormal{loc}} +s^{-2} \lesssim s^{-\frac54} ;
\\ \label{BS:bs}
|b_s| \lesssim \| \varepsilon \|_{L^2_\textnormal{loc}}^2 +s^{-2} \lesssim s^{-2} ;
\\ \label{BS:gs}
\lambda^2|g_s| \lesssim s^{-3}+s^{-1}\| \varepsilon \|_{L^2_\textnormal{loc}}+\| \varepsilon \|_{L^2_\textnormal{loc}}^{2} \lesssim s^{-\frac94} ;
\\\label{BS:hs}
\left| \lambda^{-\frac12}h_s +\frac 12 \lambda^2g\right| \lesssim s^{-\frac54} .
\end{gather}
\end{lemma}
\begin{proof}
Estimates \eqref{BS:m}-\eqref{BS:hs} follow from \eqref{BS:param}, \eqref{est:h}, \eqref{mod:est:m}, \eqref{mod:est:bs}, \eqref{mod:est:g} and \eqref{BS:eps}.
\end{proof}
\section{Energy estimates}\label{section:energy}
We work with the notation introduced in Section \ref{section:decomp_sol}. In particular, we assume that $\varepsilon$ satisfies \eqref{decomp:v}-\eqref{eq:eps} and that $(\lambda,\sigma,b,\varepsilon)$ satisfy \eqref{BS:param} and \eqref{BS:eps} on $\mathcal{J}=[s_0,s^{\star}]$ for some $s^{\star} \ge s_0$.
We define the mixed energy-virial functional
\begin{equation} \label{def:F}
\mathcal F = \int \left[(\partial_y\varepsilon)^2 \psi_B + \varepsilon^2 \varphi_B
- \frac 13 \left( (W +F+ \varepsilon)^6 - (W+F)^6 - 6(W+F)^5\varepsilon \right) \psi_B\right].
\end{equation}
Set
\begin{equation} \label{def:kappa}
\kappa=\frac{2(2\theta-1)}{1-\theta} \quad \text{so that} \quad \lambda^{\kappa} \sim s^{4} .
\end{equation}
\begin{proposition}
There exists $\alpha^\star>0$, $\mu_0>0$ and $B_0>100$ such that, for all $B \ge B_0$ and for all $s_0$ large enough (possibly depending on $B$), the following hold on $[s_0,s^\star]$.
\begin{enumerate}
\item Time derivative of the energy functional.
\begin{equation}\label{dsF}
\lambda^{-\kappa} \big(\lambda^{\kappa}\mathcal F\big)_s + \mu_0 \int \left[(\partial_y \varepsilon)^2+\varepsilon^2\right] \varphi_B'
\lesssim s^{-4}.
\end{equation}
\item Coercivity of $\mathcal F$.
\begin{equation}\label{coF}
\mathcal{N}_B(\varepsilon)^2
\lesssim \mathcal F +s^{-100} .
\end{equation}
\end{enumerate}
\end{proposition}
\begin{proof}
Let
\begin{equation} \label{def:GBeps}
G_B(\varepsilon)=
-\partial_y(\Psi_B \partial_y\varepsilon) + \varepsilon \varphi_B-\psi_B\left((W+F+\varepsilon)^5-(W+F)^5\right).
\end{equation}
We compute using \eqref{eq:eps},
\begin{align*}
\lambda^{-\kappa} \big(\lambda^{\kappa}\mathcal F\big)_s
& = 2 \int \left( \varepsilon_s-\frac{\lambda_s}{\lambda} \Lambda \varepsilon\right) G_B(\varepsilon)
+ \frac{\lambda_s}{\lambda} \left( 2\int \Lambda \varepsilon G_B(\varepsilon) +\kappa\mathcal F\right)\\
& \quad - 2\int (W+F)_s \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right]\psi_B \\
&= 2\int \partial_y \left[ -\partial_y^2\varepsilon+\varepsilon-\left((W+F+\varepsilon)^5-(W+F)^5\right)\right]
G_B(\varepsilon)
\\&\quad - 2 \int \mathcal E(W)G_B(\varepsilon) + \left(\frac{\sigma_s}{\lambda}-1\right) \int \partial_y\varepsilon G_B(\varepsilon)
+ \frac{\lambda_s}{\lambda} \left( 2\int \Lambda \varepsilon G_B(\varepsilon) +\kappa \mathcal F\right)\\
& \quad - 2\int (W+F)_s \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right]\psi_B\\
&=:\mathrm{f}_1+\mathrm{f}_2+\mathrm{f}_3+\mathrm{f}_4+\mathrm{f}_5.\end{align*}
\noindent \emph{Estimate of $\mathrm{f}_1$.} We claim, by choosing $\alpha^\star$ small enough, $B$ large enough and then $s$ large enough (possibly depending on $B$), that
\begin{equation} \label{est:f1}
\mathrm{f}_1+2\mu_0\int \varphi_B'\varepsilon^2 \lesssim s^{-100} ,
\end{equation}
where $\mu_0$ is a small positive constant which will be fixed below.
To prove \eqref{est:f1}, we compute following Step 3 of Proposition 3.1 in \cite{MaMeRa1},
\begin{align*}
\mathrm{f}_1&= -\int \left[3\psi_B'(\partial_y^2 \varepsilon)^2+
(3\varphi_B'+\psi_B'-\psi_B''')(\partial_y \varepsilon)^2 + (\varphi_B'-\varphi_B''')\varepsilon^2\right]\\
&\quad -\frac13\int \left[(W+F+\varepsilon)^6-(W+F)^6 - 6(W+F+\varepsilon)^5\varepsilon\right]
(\varphi_B'-\psi_B')\\
& \quad + 2\int \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right] \partial_y(W+F) (\psi_B-\varphi_B)\\
& \quad + 10 \int \psi_B' \partial_y \varepsilon \left[\partial_y(W+F) \left((W+F+\varepsilon)^4-(W+F)^4\right)
+(W+F+\varepsilon)^4 \partial_y \varepsilon\right]
\\
&\quad -\int \psi_B' \left[\left((-\partial_y^2\varepsilon+\varepsilon-\left((W+F+\varepsilon)^5-(W+F)^5\right)\right)^2-\left(-\partial_y^2\varepsilon+\varepsilon\right)^2\right]
\\ &=\mathrm{f}_1^<+\mathrm{f}_1^>,
\end{align*}
where $\mathrm{f}_1^{<,>}$ correspond respectively to integration on the three regions
$y<-\frac B2$ and $y>-\frac B2$.
\smallskip
\noindent \emph{Estimate of $\mathrm{f}_1^<$.}
In the region $y<-\frac B2$, we use the properties \eqref{cut:onR} and take $B$ large enough to deduce
\begin{align*}
\mathrm{f}_1^<&+3\int_{y<-\frac{B}2} \psi_B'(\partial_y^2\varepsilon)^2+\frac12\int_{y<-\frac{B}2} \varphi_B'\big((\partial_y\varepsilon)^2+\varepsilon^2 \big) \\
& \lesssim \int_{y<-\frac{B}2}\varphi_B' \left( \big(W^4+F^4\big)\varepsilon^2+\varepsilon^6\right)
+B\int_{y<-\frac{B}2}\varphi_B' |\partial_y(W+F)|\left(|W+F|^3|\varepsilon|^2+|\varepsilon|^5 \right)
\\ &\quad +\int_{y<-\frac{B}2} \psi_B'|\partial_y \varepsilon| \left[ |\partial_y(W+F)|\left(|W+F|^3|\varepsilon|+\epsilon^4 \right)+|\partial_y\varepsilon|\left(|W+F|^4+\epsilon^4 \right)\right]
\\ & \quad +\int_{y<-\frac{B}2} \psi_B' \left(-2(-\partial_y^2\varepsilon+\varepsilon) +(W+F+\varepsilon)^5-(W+F)^5\right)\left((W+F+\varepsilon)^5-(W+F)^5\right)
\\ &=:\mathrm{f}_{1,1}^<+\mathrm{f}_{1,2}^<+\mathrm{f}_{1,3}^<+\mathrm{f}_{1,4}^< .
\end{align*}
Observe from \eqref{est:Qb} and \eqref{e:r} that
\begin{equation*}
\int_{y<-\frac{B}2} \varphi_B'W^4\varepsilon^2 \lesssim \big(s^{-4}+e^{-\frac{B}2}\big) \int_{y<-\frac{B}2} \varphi_B'\varepsilon^2 .
\end{equation*}
To deal with $\int F^4\varepsilon^2$, recall from the definition of $F$ that $F(s,y)=\lambda^{\frac12}(s)f(s,\lambda(s)y+\sigma(s))$. By splitting the integration domain into the two cases $\lambda(s)y>-\frac14\sigma(s) \iff \lambda(s)y+\sigma(s)>\frac34 \sigma(s)$ and $\lambda(s)y<-\frac14\sigma(s)$ (from \eqref{BS:param}) and using \eqref{def:f_0}, \eqref{decay:q_0.1} and \eqref{BS:param} we get that,
\begin{equation} \label{est:Feps}
\int_{y<-\frac{B}2} \varphi_B'|F|^j\varepsilon^2 \lesssim s^{-j} \int_{y<-\frac{B}2} \varphi_B'\varepsilon^2+B^{-1}\lambda(s)^{\frac{j}2}e^{-\frac{c}{B}s}\int \varepsilon^2 \
\end{equation}
and
\begin{equation} \label{est:Feps_deriv}
\int_{y<-\frac{B}2} \varphi_B'|\partial_yF|^j\varepsilon^2 \lesssim s^{-2j}\int_{y<-\frac{B}2} \varphi_B'\varepsilon^2+B^{-1}\lambda(s)^{\frac{3j}2} e^{-\frac{c}{B}s}\int \varepsilon^2 ,
\end{equation}
for all $j \in \mathbb N$, $j \ge 1$.
To control, the purely nonlinear term in $\mathrm{f}_{1,1}$, we recall the following version of the Sobolev embedding (see Lemma 6 of \cite{Mjams} and also \eqref{est:infty}):
\begin{align*}
\left\|\varepsilon^2\sqrt{\varphi_B'}\right\|_{L^\infty(y<-\frac{B}2)}^2
&\lesssim \|\varepsilon\|_{L^2}^2\left(\int_{y<-\frac{B}2} (\partial_x\varepsilon)^2 \varphi_B'
+\int_{y<-\frac{B}2} \varepsilon^2\frac{\big(\varphi_B''\big)^2}{\varphi_B'}\right)\\ &\lesssim \delta(\alpha^\star)\int_{y<-\frac{B}2} \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right).
\end{align*}
Thus, it follows that
\begin{equation*}
\int_{y<-\frac{B}2} \varphi_B'\varepsilon^6 \le \left\|\varepsilon^2\sqrt{\varphi_B'}\right\|_{L^\infty(y<-\frac{B}2)}^2 \left(\int \varepsilon^2 \right)\lesssim \delta(\alpha^\star)\int_{y<-\frac{B}2} \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right).
\end{equation*}
Note also for future reference that the same proof yields
\begin{equation} \label{est:eps6}
\int \varphi_B'\varepsilon^6 \le \left\|\varepsilon^2\sqrt{\varphi_B'}\right\|_{L^\infty}^2 \left(\int \varepsilon^2 \right)\lesssim \delta(\alpha^\star)\int \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right).
\end{equation}
Hence, we deduce gathering those estimates and choosing $s$ and $B$ large enough and $\alpha^\star$ small enough that
\begin{equation*}
\mathrm{f}_{1,1}^< \le cs^{-100}+\frac18\int_{y<-\frac{B}2} \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right).
\end{equation*}
Observe from Young's inequality that
\begin{equation*}
\mathrm{f}_{1,2}^< \lesssim \int_{y<-\frac{B}2} \varphi_B' |\partial_y(W+F)| \left(|W+F)^3+|\partial_y(W+F)|^3 \right)\varepsilon^2+\int_{y<-\frac{B}2} \varphi_B' \varepsilon^6 ,
\end{equation*}
so that it follows arguing as for $\mathrm{f}_{1,1}^<$ that, for $s$ and $B$ large enough and $\alpha^\star$ small enough,
\begin{equation*}
\mathrm{f}_{1,2}^< \le cs^{-100}+\frac1{2^4}\int_{y<-\frac{B}2} \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right).
\end{equation*}
By using again Young's inequality, we have
\begin{equation*}
\mathrm{f}_{1,3}^< \lesssim \int_{y<-\frac{B}2} \psi_B' \left(|W+F|^4+|\partial_y(W+F)|^4 \right) (\varepsilon^2+(\partial_y \varepsilon)^2)+\int_{y<-\frac{B}2} \psi_B'\varepsilon^6+\int_{y<-\frac{B}2} \psi_B' \varepsilon^4(\partial_y\varepsilon)^2
\end{equation*}
The first two terms on the right-hand side of the above inequality are estimated as before. For the third one, we deduce arguing as in \eqref{Sobo:firstderiv} that
\begin{equation*}
\int_{y<-\frac{B}2} \psi_B' \varepsilon^4(\partial_y\varepsilon)^2 \lesssim \left\| \varepsilon \partial_y\varepsilon \sqrt{\psi_B'}\right\|_{L^{\infty}(y<-\frac{B}2)}^2 \left(\int \varepsilon^2 \right) \lesssim \delta(\alpha^\star)\int_{y<-\frac{B}2} \psi_B' \left((\partial_y\varepsilon)^2+(\partial_y^2\varepsilon)^2 \right).
\end{equation*}
Thus it follows by taking $s$ and $B$ large enough and $\alpha^\star$ small enough that
\begin{equation*}
\mathrm{f}_{1,3}^< \le cs^{-100}+\frac1{2^5}\int_{y<-\frac{B}2} \varphi_B'(\partial_y \varepsilon)^2+\frac1{2}\int_{y<-\frac{B}2} \psi_B'(\partial_y^2 \varepsilon)^2 .
\end{equation*}
Finally, we get from Young's inequality and \eqref{cut:onR} that
\begin{equation*}
\mathrm{f}_{1,4}^< \le \frac1{2^7}\int_{y<-\frac{B}2} \varphi_B'\varepsilon^2+\frac1{8}\int_{y<-\frac{B}2} \psi_B'(\partial_y^2 \varepsilon)^2+c\int_{y<-\frac{B}2} \psi_B' |W+F|^8 \varepsilon^2+c\int_{y<-\frac{B}2} \psi_B'\varepsilon^{10} .
\end{equation*}
Since $\psi_B'\sim (\varphi_B')^2$ in the region $y<-\frac{B}2$, we have
\begin{equation*}
\int_{y<-\frac{B}2} \varphi_B'\varepsilon^{10} \le \left\|\varepsilon^2\sqrt{\varphi_B'}\right\|_{L^\infty(y<-\frac{B}2)}^4 \left(\int \varepsilon^2 \right)\lesssim \delta(\alpha^\star)\int_{y<-\frac{B}2} \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right).
\end{equation*}
Hence, we deduce that
\begin{equation*}
\mathrm{f}_{1,4}^< \le cs^{-100}+\frac1{2^6}\int_{y<-\frac{B}2} \varphi_B'(\partial_y \varepsilon)^2+\frac1{4}\int_{y<-\frac{B}2} \psi_B'(\partial_y^2 \varepsilon)^2 .
\end{equation*}
Therefore, we conclude gathering all those estimates that
\begin{equation} \label{est:f1<}
\mathrm{f}_1^<+\int_{y<-\frac{B}2} \psi_B'(\partial_y^2\varepsilon)^2+\frac14\int_{y<-\frac{B}2} \varphi_B'\big((\partial_y\varepsilon)^2+\varepsilon^2 \big) \lesssim s^{-100} .
\end{equation}
\noindent \emph{Estimate of $\mathrm{f}_1^>$.}
In the region $y>-\frac B2$, one has $\varphi_B(y)=e^{\frac{y}B}$ and $\psi_B(y)=1$. Thus,
\begin{align}
\mathrm{f}_1^>&=
-\frac 1B \int_{y>-\frac{B}2} \left[ 3(\partial_y \varepsilon)^2 + (1-\frac1{B^2})\varepsilon^2 \right] e^{\frac{y}B}
\nonumber \\ & \quad -\frac{1}{3B}\int_{y>-\frac{B}2} \left[(W+F+\varepsilon)^6-(W+F)^6- 6(W+F+\varepsilon)^5\varepsilon\right]e^{\frac{y}B}
\nonumber \\
& \quad - 2\int_{y>-\frac{B}2} \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right] \partial_y(W+F) (e^{\frac{y}B}-1) \nonumber \\
&=-\frac 1B\int_{y>-\frac{B}2} \left[3(\partial_y \varepsilon)^2 + \varepsilon^2 -5Q^4\varepsilon^2+20y Q'Q^3\varepsilon^2\right]e^{\frac{y}B}
+\mathcal R_{\rm Vir}(\varepsilon) ,\label{decomp:f1>}
\end{align}
where
\begin{equation*}
\mathcal R_{\rm Vir}=\mathcal R_{\rm Vir,1}+ \mathcal R_{\rm Vir,2}+ \mathcal R_{\rm Vir,3}+ \mathcal R_{\rm Vir,4}+\mathcal R_{\rm Vir,5}
\end{equation*}
and
\begin{align*}
\mathcal R_{\rm Vir,1}&=\frac1{B^3} \int_{y>-\frac{B}2} \varepsilon^2 e^{\frac{y}B} ; \\
\mathcal R_{\rm Vir,2}&=- \frac1{3B}\int_{y>-\frac{B}2} \left[ (W+F+\varepsilon)^6-(W+F)^6 - 6 (W+F+\varepsilon)^5\varepsilon+15Q^4\varepsilon^2 \right]e^{\frac{y}B} ;\\
\mathcal R_{\rm Vir,3}&= -2\int_{y>-\frac{B}2} \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon-10(W+F)^3 \varepsilon^2\right] \\ & \quad \quad \quad \quad \quad \quad \times \partial_y(W+F)(e^{\frac{y}B}-1) ; \\
\mathcal R_{\rm Vir,4}&=- 20 \int_{y>-\frac{B}2} \left[(W+F)^3\partial_y(W+F)-Q^3Q'\right](e^{\frac{y}B}-1)\varepsilon^2 ; \\
\mathcal R_{\rm Vir,5}&=- 20 \int_{y>-\frac{B}2}Q^3Q'\left( e^{\frac{y}B}-1-\frac{y}B \right) \varepsilon^2 .
\end{align*}
To handle the first term on the right-hand side of \eqref{decomp:f1>}, we rely on the following coercivity property of the virial quadratic form (under the orthogonality conditions \eqref{ortho} ) proved in Lemma 3.5 in \cite{CoMa1} and which is a variant of Lemma 3.4 in \cite{MaMeRa1} based on Proposition 4 in \cite{MaMejmpa}.
\begin{lemma}[Localized virial estimate]
There exist $B_{0}>100$ and $\mu_1>0$ such that
for all $B\geq B_{0}$,
\begin{equation} \label{est:viriel}
\int_{y>-\frac{B}2} \left[ 3 (\partial_y\varepsilon)^2 + \varepsilon^2 - 5 Q^4 \varepsilon^2 + 20 y Q' Q^3 \varepsilon^2\right] \geq \mu_1 \int_{y>-\frac{B}2} \left[(\partial_y\varepsilon)^2 + \varepsilon^2 \right] e^{\frac{y}B}
- \frac 1 B \int \varepsilon^2 e^{-\frac{|y|}{2}} .
\end{equation}
\end{lemma}
Now, we turn our attention to $ \mathcal R_{\rm Vir}$. We begin by explaining how to control $\left|\mathcal R_{\rm Vir,1}\right|$ and $\left|\mathcal R_{\rm Vir,5}\right|$. We rely on the calculus inequality
\begin{equation} \label{est:expo}
\left| e^{\frac{y}B}-1-\frac{y}B \right| \lesssim \frac{|y|^2}{B^2}e^{\frac{|y|}B} .
\end{equation}
It follows that
\begin{equation*}
B\left|\mathcal R_{\rm Vir,1}\right| +B\left|\mathcal R_{\rm Vir,5} \right| \lesssim \frac1{B} \int_{y>-\frac{B}2}\varepsilon^2 e^{\frac{y}B} .
\end{equation*}
Hence, $\left|\mathcal R_{\rm Vir,1}\right|$ and $\left|\mathcal R_{\rm Vir,5}\right|$ will be controlled by using the contribution coming from the first term on the right-hand side of \eqref{est:viriel} and by taking $B$ large enough.
To estimate $\mathcal R_{\rm Vir,2}$, we write
\begin{align*}
&(W+F+\varepsilon)^6-(W+F)^6 - 6 (W+F+\varepsilon)^5\varepsilon +15 Q^4 \varepsilon^2
\\&\quad = \left[(W+F+\varepsilon)^6-(W+F)^6 - 6 (W+F)^5\varepsilon-15 (W+F)^4 \varepsilon^2\right] \\
&\qquad-6\left[(W+F+\varepsilon)^5-(W+F)^5 -5(W+F)^4\varepsilon\right] \varepsilon-15\left[(W+F)^4-Q^4\right]\varepsilon^2
\end{align*}
so that
\begin{align*}
&\left| \left[(W+F+\varepsilon)^6-(W+F)^6 - 6 (W+F+\varepsilon)^5\varepsilon\right] +15 Q^4 \varepsilon^2\right|
\\& \qquad \lesssim |W+F|^3|\varepsilon|^3+|\varepsilon|^6+|W+F-Q|(|W|+|F|+Q)^3\varepsilon^2.
\end{align*}
Hence, we get
\begin{align*}
B\left|\mathcal R_{\rm Vir,2} \right| &\lesssim \|\varepsilon \|_{L^{\infty}(y>-\frac{B}2)} \int_{y>-\frac{B}2} \left(1+|F|^3\right)\varepsilon^2e^{\frac{y}B} + \|\varepsilon \|_{L^{\infty}(y>-\frac{B}2)}^4 \int_{y>-\frac{B}2} \varepsilon^2 e^{\frac{y}B} \\
& \quad +\int_{y>-\frac{B}2}\left(|Q_b-Q|+|rR|+|F|\right) \left(1+|F|^3\right)\varepsilon^2e^{\frac{y}B} .
\end{align*}
On the one hand, observe from the Sobolev embedding and the bootstrap assumption \eqref{BS:eps} that
\begin{equation} \label{est:eps_>}
\|\varepsilon \|_{L^{\infty}(y>-\frac{B}2)} \lesssim \mathcal{N}_B(\varepsilon) \lesssim |s|^{-\frac54} .
\end{equation}
On the other hand, recall that $F(s,y)=\lambda^{\frac12}(s)f(s,\lambda(s)y+\sigma(s))$. For $s>2B$, we have $\lambda(s)y+\sigma(s)>\frac34 \sigma(s)$ in the region $y>-\frac{B}2$. Hence, it follows from \eqref{def:f_0}, \eqref{decay:q_0.1} and \eqref{BS:param} that
\begin{equation} \label{est:F>}
\|F\|_{L^{\infty}(y>-\frac{B}2)} \lesssim s^{-1} \quad \text{and} \quad \|\partial_yF\|_{L^{\infty}(y>-\frac{B}2)} \lesssim s^{-2} .
\end{equation}
Thus, we deduce by using \eqref{def:Qb}, \eqref{def:Pb}, \eqref{e:r}, \eqref{est:eps_>} and \eqref{est:F>} and by taking $|s|$ large enough that
\begin{equation*}
B\left|\mathcal R_{\rm Vir,2} \right| \le \frac{\mu_1}4 \int_{y>-\frac{B}2}\varepsilon^2 e^{\frac{y}B} .
\end{equation*}
To deal with $\mathcal R_{\rm Vir,3}$, we observe
\begin{align*}
\left|\left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4 \varepsilon-10(W+F)^3\varepsilon^2\right] \partial_y(W+F)(e^{\frac{y}B}-1)\right|\\
\lesssim \left(|W+F|^2|\varepsilon|^3+|\varepsilon|^5 \right) \left(|\partial_yW|+|\partial_yF|\right)e^{\frac{y}B} ,
\end{align*}
so that
\begin{align*}
\left|\mathcal R_{\rm Vir,3} \right| &\lesssim \|\varepsilon \|_{L^{\infty}(y>-\frac{B}2)} \int_{y>-\frac{B}2} \left(1+|F|^2\right)\left(1+|\partial_yF|\right)\varepsilon^2e^{\frac{y}B} \\ & \quad + \|\varepsilon \|_{L^{\infty}(y>-\frac{B}2)}^3 \int_{y>-\frac{B}2} \left(1+|\partial_yF|\right)\varepsilon^2 e^{\frac{y}B} .
\end{align*}
Hence, we deduce from \eqref{est:eps_>} and \eqref{est:F>} by taking $|s|$ large enough (possibly depending on $B$) that
\begin{equation*}
B\left|\mathcal R_{\rm Vir,3} \right| \le \frac{\mu_1}8 \int_{y>-\frac{B}2}\varepsilon^2 e^{\frac{y}B} .
\end{equation*}
To deal with $\mathcal R_{\rm Vir,4}$, we write
\begin{equation*}
(W+F)^3\partial_y(W+F)-Q^3Q' =\left( (W+F)^3-Q^3\right) \partial_y(W+F)+Q^3\partial_y(W-Q+F) ,
\end{equation*}
so that
\begin{align*}
\left|(W+F)^3\partial_y(W+F)-Q^3Q'\right| &\lesssim \left( |Q_b-Q|+|F|+|r||R|\right)\left(|W|+|F|+Q\right)^2\left( |\partial_yW|+|\partial_yF|\right)\\ & \quad +Q^3\left(|\partial_y(Q_b-Q)|+|r||R'|+|\partial_yF|\right),
\end{align*}
Thus, we deduce by using \eqref{def:Qb}, \eqref{def:Pb}, \eqref{e:r} and \eqref{est:F>} and by taking $|s|$ large enough (depending possibly on $B$) that
\begin{equation*}
B\left|\mathcal R_{\rm Vir,4} \right| \le \frac{\mu_1}{2^4} \int_{y>-\frac{B}2}\varepsilon^2 e^{\frac{y}B} .
\end{equation*}
Then, we deduce gathering all those estimates that
\begin{equation} \label{est:f1>}
\mathrm{f}_1^>+\frac{\mu_1}{2B}\int_{y>-\frac{B}2} e^{\frac{y}B}\varepsilon^2 \lesssim \frac1{B^2} \int \varepsilon^2 e^{-\frac{|y|}{2}} .
\end{equation}
The proof of \eqref{est:f1} follows by combining \eqref{est:f1<}, \eqref{est:f1>} and choosing $\mu_0=2^{-5}\min\{1,\mu_1 \}$.
\smallskip
\noindent \emph{Estimate of $\mathrm{f}_2$.} We claim that
\begin{equation} \label{est:f2}
\big| \mathrm{f}_{2}\big| \le \frac{\mu_0}2 \int \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right) +cB^2s^ {-4} .
\end{equation}
By using the decomposition in \eqref{decomp:EW}, we have
\begin{equation*}
\mathrm{f}_2=2\int (\vec{m} \cdot \vec{\MM} Q ) G_B(\varepsilon)-2\int \mathcal{R} \, G_B(\varepsilon)=: \mathrm{f}_{2,1}+\mathrm{f}_{2,2} .
\end{equation*}
We first deal with $\mathrm{f}_{2,1}$. By using the definition of $G_B(\varepsilon)$ in $\eqref{def:GBeps}$ and integration by parts, we compute
\begin{align*}
\int \Lambda Q \, G_B(\varepsilon)&=\int \psi_B \mathcal{L}(\Lambda Q)\varepsilon -\int \psi_B'\partial_y(\Lambda Q) \varepsilon +\int (\varphi_B-\psi_B)\Lambda Q \, \varepsilon\\ & \quad -\int \Lambda Q \psi_B \left[ (W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right] \\ &
\quad -5\int \Lambda Q \psi_B \left[ (W+F)^4-Q^4\right]\varepsilon .
\end{align*}
and (also using $\mathcal{L}(\partial_yQ)=0$)
\begin{align*}
\int \partial_y Q \, G_B(\varepsilon)&= -\int \psi_B'\partial_y^2 Q \varepsilon +\int (\varphi_B-\psi_B)\partial_y Q \, \varepsilon-5\int \partial_y Q \psi_B \left[ (W+F)^4-Q^4\right]\varepsilon\\ & \quad -\int \partial_yQ \psi_B \left[ (W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right]
\end{align*}
We estimate each of these terms separately. By using the identity $\mathcal{L}\Lambda Q=-2Q$, the second and third orthogonality identities in \eqref{ortho}, the localisation properties of $\psi_B$, $\psi_B'$ and $\varphi_B$, \eqref{est:expo} and the decay properties of $Q$ and $\Lambda Q$, it follows that
\begin{align*}
&\left|\int \psi_B \mathcal{L}(\Lambda Q)\varepsilon \right|=2\left| \int (\psi_B-1) Q \varepsilon\right| \lesssim \int_{y<-\frac{B}2} e^{-\frac{|y|}2} |\varepsilon| \lesssim e^{-\frac{B}4} \|\varepsilon\|_{L^2_\textnormal{loc}} ; \\
&\left|\int \psi_B'\partial_y^2 Q \varepsilon \right|+\left|\int \psi_B'\partial_y(\Lambda Q) \varepsilon \right| \lesssim \int_{y<-\frac{B}2} e^{-\frac{|y|}2} |\varepsilon| \lesssim e^{-\frac{B}4} \|\varepsilon\|_{L^2_\textnormal{loc}} ;
\end{align*}
and
\begin{align*}
&\left|\int (\varphi_B-\psi_B)\Lambda Q \, \varepsilon \right|=2\left| \int(\varphi_B-\psi_B-\frac{y}B)\Lambda Q \, \varepsilon\right| \\ & \quad \quad \quad \lesssim \int_{y<-\frac{B}2} e^{-\frac{|y|}2}|\varepsilon|+\frac1{B^2}\int_{y>-\frac{B}2}e^{-\frac{|y|}2}|\varepsilon| \lesssim \left(e^{-\frac{B}4}+\frac1{B^2}\right) \|\varepsilon\|_{L^2_\textnormal{loc}} ; \\
&\left|\int (\varphi_B-\psi_B)\partial_y Q \varepsilon \right|=\left|\int (\varphi_B-\psi_B-\frac{y}B)\partial_y Q \varepsilon \right|
\lesssim \int_{|y|>\frac{B}2} e^{-\frac{|y|}2}|\varepsilon| \\ & \quad \quad \quad \lesssim \int_{y<-\frac{B}2} e^{-\frac{|y|}2}|\varepsilon|+\frac1{B^2}\int_{y>-\frac{B}2}e^{-\frac{|y|}2}|\varepsilon| \lesssim \left(e^{-\frac{B}4}+\frac1{B^2}\right) \|\varepsilon\|_{L^2_\textnormal{loc}} ;
\end{align*}
where in the last line, we have also used the orthogonality condition (from \eqref{ortho})
\begin{equation*}
\int y\partial_yQ \varepsilon=\int \big( \Lambda Q-\frac12Q\big) \varepsilon=0 ,
\end{equation*}
Moreover, it follows from \eqref{def:Qb}, \eqref{def:Pb}, \eqref{e:r} and \eqref{e:F} that
\begin{align*}
\int \psi_B \left( |\Lambda Q|+\right. & \left. |\partial_yQ| \right) \left| (W+F)^4-Q^4\right| |\varepsilon|
\\ &\lesssim \int e^ {-\frac{3|y|}4}|W+F-Q| \left(|W+F|^3+Q^3 \right) |\varepsilon|
\lesssim s^ {-1}\|\varepsilon\|_{L^ 2_\textnormal{loc}} .
\end{align*}
To deal with the nonlinear term, we recall the Sobolev bound
\begin{equation} \label{Sob:eps_varphi}
\|\varepsilon^2 \sqrt{\psi_B}\|_{L^{\infty}}^2 \lesssim \left(\int \varepsilon^2 \right)\int \psi_B\, (\varepsilon^2+(\partial_y\varepsilon)^2)\lesssim \delta(\alpha^\star) \mathcal{N}_B(\varepsilon)^2 .
\end{equation}
Hence, we deduce from \eqref{e:r}, \eqref{e:F}, \eqref{cut:onR} and \eqref{BS:eps} that
\begin{align*}
\int \psi_B \left( |\Lambda Q|+|\partial_yQ| \right) &\left| (W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right| \\ &
\lesssim \int \psi_B e^ {-\frac{3|y|}4}\left( |W+F|^3 \varepsilon^2+|\varepsilon|^5 \right)
\\ & \lesssim B\int \varphi_B'\varepsilon^2+\|\varepsilon^ 2\sqrt{\psi_B}\|_{L^{\infty}}^2 \int e^{-\frac{3|y|}4}|\varepsilon|
\lesssim B\int \varphi_B'\varepsilon^2+s^{-\frac52} \|\varepsilon\|_{L^2_\textnormal{loc}} .
\end{align*}
Therefore, we deduce combining those estimates with \eqref{est:L2loc} and \eqref{BS:m}, and choosing $s$ and $B>100$ large enough that
\begin{equation} \label{est:f21}
\big| \mathrm{f}_{2,1}\big| \le \frac{\mu_0}4 \int \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right) +cs^ {-4} .
\end{equation}
Now, we turn to $\mathrm{f}_{2,2}$. We compute from the definition of $G_B(\varepsilon)$ in \eqref{def:GBeps}
\begin{equation*}
\mathrm{f}_{2,2}=2\int \psi_B\partial_y\mathcal{R}\, \partial_y \varepsilon+2\int \varphi_B \mathcal{R} \, \varepsilon
-2\int \psi_B \mathcal{R} \left[(W+F+\varepsilon)^5-(W+F)^5 \right] .
\end{equation*}
By the Cauchy-Schwarz inequality and the properties of $\varphi_B$ and $\psi_B$ in \eqref{cut:onR}, it holds
\begin{align*}
&\left|\int \varphi_B \mathcal{R} \varepsilon \right| \le\left(\int \varphi_B^2e^{-\frac{y}B} \varepsilon^2 \right)^{\frac12}
\left(\int e^{\frac{y}B} \mathcal{R}^2 \right)^{\frac12} \le \frac{\mu_0}{2^{4}} \int \varepsilon^2 \varphi_B' +c B^2\| \mathcal{R}\|_{L^2_B}^2 ; \\
&\left|\int \psi_B \partial_y\mathcal{R} \, \partial_y\varepsilon \right| \le\left(\int \psi_B^2e^{-\frac{y}B} (\partial_y\varepsilon)^2 \right)^{\frac12}
\left(\int e^{\frac{y}B} (\partial_y\mathcal{R})^2 \right)^{\frac12} \le \frac{\mu_0}{2^{5}} \int (\partial_y\varepsilon)^2 \varphi_B' +c B^2\| \partial_y\mathcal{R}\|_{L^2_B}^2 .
\end{align*}
To treat the nonlinear term, we observe that
\begin{equation*}
\left|\int \psi_B \mathcal{R} \left[(W+F+\varepsilon)^5-(W+F)^5 \right] \right| \lesssim \int \psi_B |\mathcal{R}|\left(|W|^4+|F|^4 \right)|\varepsilon|
+ \int \psi_B |\mathcal{R}||\varepsilon|^5 .
\end{equation*}
On the one hand, we deduce from \eqref{est:F>}, and then \eqref{cut:onR} and \eqref{est:Feps} that
\begin{align*}
\int \psi_B |\mathcal{R}|\left(|W|^4+|F|^4 \right)|\varepsilon| &\le \left(\int \psi_B^2e^{-\frac{y}B}(1+|F|^8\mathbf{1}_{\{y<-\frac{B}2\}}) \varepsilon^2 \right)^{\frac12}
\left(\int e^{\frac{y}B} \mathcal{R}^2 \right)^{\frac12}
\\ & \lesssim \frac{\mu_0}{2^{6}} \int \varepsilon^2 \varphi_B' +c B^2\| \mathcal{R}\|_{L^2_B}^2 .
\end{align*}
On the other hand, \eqref{cut:onR}, the Sobolev bound \eqref{Sob:eps_varphi}, and then \eqref{BS:eps}, \eqref{est:eps6} yield
\begin{align*}
\int \psi_B |\mathcal{R}||\varepsilon|^5 &\lesssim \left\|\varepsilon^2\sqrt{\psi_B}\right\|_{L^\infty}
\left(\int \varphi_B^2e^{-\frac{y}B} \varepsilon^6 \right)^{\frac12}
\left(\int e^{\frac{y}B} \mathcal{R}^2 \right)^{\frac12} \lesssim s^{-\frac54} \left( \int \varepsilon^2 \varphi_B' +c B^2\| \mathcal{R}\|_{L^2_B}^2\right).
\end{align*}
Then, we deduce combining those estimates with \eqref{est:R:L2B}, \eqref{est:L2loc}, \eqref{BS:m} and \eqref{BS:bs} that
\begin{equation} \label{est:f22}
\big| \mathrm{f}_{2,2}\big| \le \frac{\mu_0}{8} \int \varepsilon^2 \varphi_B'+c B^2s^{-4} .
\end{equation}
Finally, we conclude the proof of \eqref{est:f2} gathering \eqref{est:f21} and \eqref{est:f22}.
\smallskip
\noindent \emph{Estimate of $\mathrm{f}_3$.} We claim that
\begin{equation} \label{est:f3}
\left| \mathrm{f}_{3}\right| \le \frac{\mu_0}{4} \int \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right) +cs^ {-100} .
\end{equation}
From the definition of $G_B(\varepsilon)$ in \eqref{def:GBeps}, we have
\begin{align*}
\mathrm{f}_{3}&= \left(\frac{\sigma_s}{\lambda}-1\right) \int \partial_y\varepsilon \left[ -\partial_y(\psi_B\partial_y\varepsilon)+\varphi_B\varepsilon-\psi_B \left((W+F+\varepsilon)^5-(W+F)^5 \right) \right] \\ &
=: \mathrm{f}_{3,1}+\mathrm{f}_{3,2}+\mathrm{f}_{3,3} .
\end{align*}
By using the identities
\begin{equation*}
\int \partial_y\varepsilon\, \partial_y(\psi_B\partial_y\varepsilon)=\frac12 \int \psi_B' (\partial_y \varepsilon)^2 \quad \text{and} \quad \int \partial_y\varepsilon \varphi_B \varepsilon=-\frac12 \int \varphi_B' \varepsilon^2 ,
\end{equation*}
we deduce from \eqref{BS:m} and \eqref{cut:onR} that
\begin{equation*}
\left| \mathrm{f}_{3,1}\right|+\left| \mathrm{f}_{3,2}\right| \lesssim s^{-\frac54}\int \varphi_B' \left(\varepsilon^2+(\partial_y\varepsilon)^2 \right).
\end{equation*}
To deal with $\mathrm{f}_{3,3}$, we compute
\begin{align*}
\int \partial_y\varepsilon\psi_B &\left((W+F+\varepsilon)^5-(W+F)^5 \right)\\&=-\frac16\int \psi_B' \left((W+F+\varepsilon)^6-(W+F)^6-6(W+F)^5\varepsilon \right) \\
& \quad -\int \psi_B \partial_y(W+F) \left((W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon \right),
\end{align*}
so that it follows from \eqref{BS:m}, \eqref{cut:onR}, and then \eqref{est:Feps}, \eqref{est:Feps_deriv}, \eqref{est:eps6}, \eqref{est:F>}, that
\begin{equation*}
\left| \mathrm{f}_{3,2}\right| \lesssim Bs^{-\frac54} \int \varphi_B' \left[\left(|W+F|^4+|\partial_y(W+F)|^4\right)\varepsilon^2+\varepsilon^6 \right]
\lesssim Bs^{-\frac54}\int \varphi_B'\left((\partial_y \varepsilon)^2+ \varepsilon^2 \right)+s^{-100} .
\end{equation*}
Therefore, we deduce the proof of \eqref{est:f3} gathering those estimates and taking $s$ large enough (possibly depending on $B$).
\smallskip
\noindent \emph{Estimate of $\mathrm{f}_4$.} We claim that
\begin{equation} \label{est:f4}
\mathrm{f}_{4} \le \frac{\mu_0}{8} \int \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right) +cs^ {-100} .
\end{equation}
Recall from the definition of $\mathcal{F}$ in \eqref{def:F} that
\begin{align*}
\mathrm{f}_4&=2\frac{\lambda_s}{\lambda}\int \Lambda \varepsilon \left[ -\partial_y(\psi_B\partial_y \varepsilon)+\varphi_B \varepsilon-\psi_B \left( (W+F+\varepsilon)^5-(W+F)^5\right)\right] \\
& \quad +\kappa\frac{\lambda_s}{\lambda}\int \left[\psi_B (\partial_y \varepsilon)^2+\varphi_B \varepsilon^2 -\frac13\psi_B\left( (W+F+\varepsilon)^6-(W+F)^6-6(W+F)^5\varepsilon\right)\right] .
\end{align*}
We compute integrating by parts (see also page 97 in \cite{MaMeRa1})
\begin{align*}
& \int (\Lambda \varepsilon) \partial_y(\psi_B\partial_y\varepsilon)=-\int \psi_B(\partial_y\varepsilon)^2+\frac12\int y\psi_B'(\partial_y\varepsilon)^2 ; \\
& \int (\Lambda \varepsilon) \varepsilon \varphi_B=-\frac12 \int y \varphi_B' \varepsilon^2 ;\\
& \int (\Lambda \varepsilon) \psi_B \left( (W+F+\varepsilon)^5-(W+F)^5\right)\\ & \quad \quad =\frac16 \int (2\psi_B-y\psi_B')\left( (W+F+\varepsilon)^6-(W+F)^6-6(W+F)^5\varepsilon \right)\\ &\quad \quad \quad -\int \psi_B \Lambda (W+F) \left( (W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon \right).
\end{align*}
Hence, we deduce gathering those identities that
\begin{align*}
\mathrm{f}_4&=\frac{\lambda_s}{\lambda}\left((2+\kappa)\int \psi_B (\partial_y \varepsilon)^2 -\int y\psi_B'(\partial_y\varepsilon)^2 +\kappa\int \varphi_B \varepsilon^2-\int y\varphi_B' \varepsilon^2\right)\\ & \quad -\frac13 \frac{\lambda_s}{\lambda}\int \left( (k+2)\psi_B-y\psi_B'\right)\left( (W+F+\varepsilon)^6-(W+F)^6-6(W+F)^5\varepsilon \right) \\ & \quad +2 \frac{\lambda_s}{\lambda}\int \psi_B \Lambda (W+F) \left( (W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon \right)\\
&=:\mathrm{f}_{4,1}+\mathrm{f}_{4,2}+\mathrm{f}_{4,3} .
\end{align*}
We will control each of these terms separately. Observe that $\frac{\lambda_s}{\lambda} >0$ since we are in a defocusing regime (see \eqref{BS:param} and \eqref{BS:m}). Thus
\begin{equation*}
\frac{\lambda_s}{\lambda}\left( -\int_{y>0} y\psi_B'(\partial_y\varepsilon)^2 -\int_{y>0} y\varphi_B' \varepsilon^2\right) \le 0 .
\end{equation*}
Moreover, we get by using \eqref{BS:param}, \eqref{BS:m} and \eqref{cut:onR}
\begin{equation*}
\frac{\lambda_s}{\lambda} \int_{y<0}|y\psi_B'|(\partial_y\varepsilon)^2 \lesssim s^{-1}\int \varphi_B'(\partial_y\varepsilon)^2
\end{equation*}
and, by using H\"older and Young inequalities,
\begin{align*}
\frac{\lambda_s}{\lambda} \int_{y<0}|y\varphi_B'|\varepsilon^2 &\lesssim s^{-1}\left(\int |y|^{100}e^{\frac{y}B} \varepsilon^2\right)^{\frac1{100}}\left(\int_{y<0} e^{\frac{y}B} \varepsilon^2\right)^{\frac{99}{100}}
\le \frac{\mu_0}{2^4} \int \varphi_B'\varepsilon^2+s^{-100}\|\varepsilon\|_{L^2}^2 .
\end{align*}
On the other hand, the terms $(2+\kappa)\int \psi_B (\partial_y \varepsilon)^2$ and $\kappa\int \varphi_B \varepsilon^2$ are positive as well as their product with $\frac{\lambda_s}{\lambda}$. However, we can estimate them as above. It follows from \eqref{BS:param} and \eqref{BS:m} that
\begin{equation*}
\left| \frac{\lambda_s}{\lambda} \right|\left( \int \psi_B (\partial_y \varepsilon)^2+\int \varphi_B \varepsilon^2\right) \lesssim s^{-1}B \int \varphi_B'\left((\partial_y \varepsilon)^2+ \varepsilon^2 \right).
\end{equation*}
Now, we deal with the nonlinear terms. By using \eqref{BS:param}, \eqref{BS:m} and \eqref{cut:onR}, and then \eqref{est:Feps}, \eqref{est:eps6}, \eqref{est:F>}, we get that
\begin{equation*}
\left|\mathrm{f}_{4,2}\right| \lesssim Bs^{-1} \int \varphi_B' \left(|W+F|^4\varepsilon^2+\varepsilon^6 \right)
\lesssim Bs^{-1}\int \varphi_B'\left((\partial_y \varepsilon)^2+ \varepsilon^2 \right)+s^{-100} .
\end{equation*}
By definition $\Lambda (W+F)=\frac12 (W+F)+y\partial_y(W+F)$. Moreover, we use that, for $k=0$ or $1$,
\begin{align*}
&\left| \partial_y^k(W+F)\right| \left| (W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon \right| \\
& \lesssim \left| \partial_y^k(W+F)\right| \left(|W+F|^3\varepsilon^2+\varepsilon^5 \right)
\lesssim \left(\left| \partial_y^k(W+F)\right||W+F|^3+ \left| \partial_y^k(W+F)\right|^4\right) \varepsilon^2+\varepsilon^6 .
\end{align*}
Thus, we deduce from \eqref{BS:param}, \eqref{BS:m} and \eqref{cut:onR}, and then \eqref{est:Feps}, \eqref{est:Feps_deriv}, \eqref{est:eps6}, \eqref{est:F>}, that
\begin{equation*}
\left|\mathrm{f}_{4,2}\right| \lesssim Bs^{-1} \int \varphi_B' \left[\left(|W+F|^4+|\partial_y(W+F)|^4\right)\varepsilon^2+\varepsilon^6 \right]
\lesssim Bs^{-1}\int \varphi_B'\left((\partial_y \varepsilon)^2+ \varepsilon^2 \right)+s^{-100} .
\end{equation*}
Therefore, we conclude the proof of \eqref{est:f4} gathering these estimates.
\smallskip
\noindent \emph{Estimate of $\mathrm{f}_5$.} We claim that
\begin{equation} \label{est:f5}
\left| \mathrm{f}_{5}\right| \le \frac{\mu_0}{2^4} \int \varphi_B'\left(\varepsilon^2+(\partial_y \varepsilon)^2\right) +cs^ {-100} .
\end{equation}
We decompose, from the definition of $\mathrm{f}_{5}$,
\begin{align*}
\mathrm{f}_{5}&= - 2b_s\int \frac{\partial Q_b}{\partial b} \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right]\psi_B \\
&\quad - 2\int F_s \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right]\psi_B \\
&=: \mathrm{f}_{5,1}+ \mathrm{f}_{5,2} ,
\end{align*}
and estimate these two terms separately.
To deal with $\mathrm{f}_{5,1}$, we use that
\begin{align*}
&\left| \frac{\partial Q_b}{\partial b} \left[(W+F+\varepsilon)^5-(W+F)^5-5(W+F)^4\varepsilon\right] \right|\\
&\quad \quad \lesssim \left|\frac{\partial Q_b}{\partial b} \right|\left( |W+F|^3\varepsilon^2+|\varepsilon|^5\right)
\lesssim \left(|W|^4+|F|^4+ \left|\frac{\partial Q_b}{\partial b} \right|^4\right)\varepsilon^2+\varepsilon^6 .
\end{align*}
Moreover, we observe by using \eqref{eq:dQb:db} and \eqref{cut:onR} that
\begin{equation*}
\left|\frac{\partial Q_b}{\partial b}(y) \right|^4\psi_B(y) \lesssim B^4\varphi_B(y)\lesssim B^5\varphi_B'(y) .
\end{equation*}
Then, it follows from \eqref{BS:bs}, \eqref{est:Feps}, \eqref{est:eps6} and \eqref{est:F>} that
\begin{equation*}
\left| \mathrm{f}_{5,1}\right| \lesssim B^5s^{-2}\int \varphi_B'\left( 1+|W|^4+|F|^4\right) \varepsilon^2+Bs^{-2}\int \varphi_B' \varepsilon^6
\lesssim B^5s^{-2}\int \varphi_B' \left( \varepsilon^2+(\partial_y\varepsilon)^2\right).
\end{equation*}
To handle $\mathrm{f}_{5,2}$, we first observe, arguing as above,
\begin{equation*}
\left| \mathrm{f}_{5,2}\right| \lesssim B \int \psi_B |F_s|\left( |W|^3+|F|^3+|F_s|^3\right) \varepsilon^2+\int \varphi_B' \varepsilon^6 .
\end{equation*}
The second term on the right-hand side of the above inequality will be dealt by using \eqref{est:eps6} and taking $\alpha^\star$ small enough. Recalling that $F(s,y)=\lambda^{\frac12}(s)f(s,\lambda(s)y+\sigma(s))$, we compute
\begin{equation*}
F_s(s,y)=\frac12\frac{\lambda_s}{\lambda} \lambda^{\frac12} f(s,\lambda(s)y+\sigma(s))+\left[\frac{\lambda_s}{\lambda}y+\frac{\sigma_s}{\lambda} \right]\lambda^{\frac32}\partial_y f(s,\lambda(s)y+\sigma(s))+\lambda^{\frac72}f_t(s,\lambda(s)y+\sigma(s)) .
\end{equation*}
Now, we argue as in the proof of \eqref{est:Feps} and split the integration domain into the two regions $\lambda(s)y>-\frac14\sigma(s) \iff \lambda(s)y+\sigma(s)>\frac34 \sigma(s)$ and $\lambda(s)y<-\frac14\sigma(s)$ (from \eqref{BS:param}). Thus, we deduce from \eqref{def:f_0}, \eqref{decay:q_0.1}, \eqref{decay:q_0.2}, \eqref{BS:param} and \eqref{BS:m} that
\begin{equation*}
\left| \mathrm{f}_{5,2}\right| \le cB^2s^{-2} \int \varepsilon^2 \varphi_B'+\frac{\mu_0}{2^5}\int \varphi_B' \left(\varepsilon^2+(\partial_y\varepsilon)^2 \right)+cs^{-100} .
\end{equation*}
Therefore, we conclude the proof of \eqref{est:f4} gathering those estimates and taking $s$ large enough (possibly depending on $B$).
\smallskip
Finally, we conclude the proof of \eqref{dsF} gathering \eqref{est:f1}, \eqref{est:f2}, \eqref{est:f3}, \eqref{est:f4} and \eqref{est:f5}.
\smallskip
Now, we turn to the proof of \eqref{coF}. We decompose $\mathcal{F}$ as follows:
\begin{align*}
\mathcal F &= \int \left[(\partial_y\varepsilon)^2 \psi_B + \varepsilon^2 \varphi_B-5Q^4\varepsilon^2 \psi_B \right] \\
& \quad - \frac 13\int \left[ (W +F+ \varepsilon)^6 - (W+F)^6 - 6(W+F)^5\varepsilon-15Q^4\varepsilon^2 \right] \psi_B \\
&=:\mathcal F_1+\mathcal F_2 .
\end{align*}
To bound by below $\mathcal F_1$, we rely on the coercivity of the linearized energy \eqref{coercivity.2} with the choice of the orthogonality conditions \eqref{ortho} and standard localisation arguments. Proceeding for instance as in the Appendix A of \cite{MaMe} or as in the proof of Lemma 3.5. in \cite{CoMa1}, we deduce that there exists $\tilde{\nu}_0>0$ such that, for $B$ large enough,
\begin{equation*}
\mathcal F_1 \ge \tilde{\nu}_0 \, \mathcal{N}_{B}(\varepsilon)^2 .
\end{equation*}
To estimate $\mathcal F_2$, we compute
\begin{align*}
&(W+F+\varepsilon)^6-(W+F)^6 - 6 (W+F)^5\varepsilon -15 Q^4 \varepsilon^2
\\&= \left[(W+F+\varepsilon)^6-(W+F)^6 - 6 (W+F)^5\varepsilon-15 (W+F)^4 \varepsilon^2\right] -15\left[(W+F)^4-Q^4\right]\varepsilon^2,
\end{align*}
so that
\begin{align*}
\left| \mathcal F_2\right| \lesssim \int \psi_B \left[\left(|W-Q|^4+|F|^4\right)\varepsilon^2+Q^3|\varepsilon|^3+\varepsilon^6+|W+F-Q|(|W|+|F|+Q)^3\varepsilon^2\right] .
\end{align*}
We will control each term on the right-hand side separately. First observe from \eqref{def:Qb} and \eqref{e:r} and arguing as for \eqref{est:Feps} (but without the restriction $y<-\frac{B}2$) that
\begin{equation*}
\int \psi_B\left(|W-Q|^4+|F|^4\right)\varepsilon^2 \lesssim s^{-4} \mathcal{N}_B(\varepsilon)^2+s^{-100}
\end{equation*}
and
\begin{equation*}
\int \psi_B|W+F-Q|(|W|+|F|+Q)^3\varepsilon^2 \lesssim s^{-1} \mathcal{N}_B(\varepsilon)^2+s^{-100} .
\end{equation*}
Moreover, we deduce from \eqref{Sob:eps_varphi} that
\begin{equation*}
\int \psi_B Q^3 |\varepsilon|^3 \lesssim \left\| \varepsilon^2\sqrt{\psi_B} \right\|_{L^{\infty}} \left(\int Q^3|\varepsilon|\right)\lesssim \delta(\alpha^\star) \mathcal{N}_B(\varepsilon)^2
\end{equation*}
and
\begin{equation*}
\int \psi_B \varepsilon^6 \lesssim \left\| \varepsilon^2\sqrt{\psi_B} \right\|^2_{L^{\infty}} \left(\int \varepsilon^2 \right) \lesssim \delta(\alpha^\star)\mathcal{N}_B(\varepsilon)^2 .
\end{equation*}
Therefore, we conclude the proof of \eqref{coF} gathering those estimates.
\end{proof}
\section{Construction of flattening solitons}
\subsection{End of the construction in rescaled variables}
In this subsection, we still work with the notation introduced in Sections~\ref{section:decomp_sol} and~\ref{section:energy}.
We prove that for well adjusted initial data,
the decomposition of the solution introduced in \eqref{decomp:v} and the bootstrap estimates \eqref{BS:param} and \eqref{BS:eps} hold true in the whole time interval $[s_0,+\infty)$ for $s_0$ large enough (equivalently, $x_0$ large enough). The result is summarized in the next proposition.
\begin{proposition} \label{prop:BS}
For $s_0>0$ large enough, let
$\sigma_0$ and $\lambda_0$ be such that
\begin{equation} \label{def:sigma_lamba:ini}
\left| \lambda_0-s_0^{\frac{2(1-\theta)}{2\theta-1}} \right| \leq s_0^{\frac{2(1-\theta)}{2\theta-1}-2\rho}
\quad \text{and} \quad
\left|\sigma_0 - (2\theta-1)s_0^{\frac1{2\theta-1}}\right|\leq s_0^{\frac1{2\theta-1}-2\rho}
\end{equation}
where $\rho$ satisfies \eqref{def:rho}.
Let $\varepsilon_0 \in H^1(\mathbb R)$ be such that
\begin{equation} \label{def:eps:ini}
\|\varepsilon_0\|_{H^1}^2+\int\varepsilon_0^2e^{\frac{y}{10}} \le s_0^{-10}
\quad \text{and} \quad
(\varepsilon_0,\Lambda Q)=(\varepsilon_0,y\Lambda Q)=(\varepsilon_0,Q)=0 .
\end{equation}
Then there exists $b_0 \in \mathbb R$ satisfying
\begin{equation} \label{b:ini}
b_0 \in \mathcal{D}_0:=\left[-\frac{2(1-\theta)}{2\theta-1}s_0^{-1}-s_0^{-1-3\rho},-\frac{2(1-\theta)}{2\theta-1}s_0^{-1}+s_0^{-1-3\rho}\right]
\end{equation}
such that the solution of \eqref{gkdv} evolving from
\begin{equation*}
U_0(x):=\lambda_0^{-\frac12}\left(Q_{b_0}+\lambda_0^{\frac12}f(s_0,\sigma_0)R+\varepsilon_0 \right)\left(\frac{x-\sigma_0}{\lambda_0} \right)+f(s_0,x)
\end{equation*}
has a decomposition $\big(\lambda(s),\sigma(s),b(s),\varepsilon(s)\big)$ as in Lemma \ref{lemma:decomp} satisfying the bootstrap conditions~\eqref{BS:param}, \eqref{BS:eps} and
\begin{align}
|h(s)| &\le s^{\frac{1-\theta}{2\theta-1}-2\rho}\label{BS:h} ,
\\
|g(s)| &\le s^{-\frac{3-2\theta}{2\theta-1}-3\rho} ,\label{BS:g}
\end{align}
on $[s_0,+\infty)$, where $h$ and $g$ are defined in Lemma \ref{lemma:gh}.
\end{proposition}
\begin{proof} We argue by contradiction and assume that for all $b_0$ satisfying \eqref{b:ini},
\begin{equation*}
s^{\star}=s^{\star}(b_0):=\sup \big\{ s \ge s_0 : \eqref{BS:param}, \eqref{BS:eps}, \eqref{BS:h}, \eqref{BS:g} \ \text{hold} \ [s_0,s] \big\}< +\infty .
\end{equation*}
We will first show that we can strictly improve \eqref{BS:param}, \eqref{BS:eps} and \eqref{BS:h} on $[s_0,s^{\star}]$, and then find a contraction for \eqref{BS:g} by using a topological argument (see similar argument in \cite{CoMaMe}).
\smallskip
\noindent \textit{Closing \eqref{BS:eps}.} We deduce integrating \eqref{dsF} on $[s_0,s]$ with $s \le s^{\star}$ and using \eqref{def:kappa}, \eqref{coF} and \eqref{def:eps:ini} that
\begin{equation*}
\mathcal{N}_B(\varepsilon)^2(s) \lesssim \mathcal{F}(s) +s^{-100}\lesssim s^{-3}+\left(\lambda_0^{\kappa}\mathcal{F}(s_0)-s_0\right)s^{-4} \le \frac14 s^{-\frac52} ,
\end{equation*}
if $s_0$ is chosen large enough, which strictly improves \eqref{BS:eps}.
\smallskip
\noindent \textit{Closing \eqref{BS:param}.} First, by using \eqref{def:gh}, and then \eqref{BS:param} and \eqref{BS:h}, we see that
\begin{equation} \label{BS:est:lambda_sigma}
\left|\lambda(s)-\left(\frac{2}{\int Q}\frac{c_0}{1-\theta}\right)^2 \sigma^{2-2\theta}(s) \right|
\lesssim \left(\lambda^{\frac12}(s)+\sigma^{1-\theta}(s)\right)|h(s)| \lesssim s^{\frac{2(1-\theta)}{2\theta-1}-2\rho} .
\end{equation}
Thus, we deduce from \eqref{BS:param} and \eqref{BS:m} that
\begin{equation*}
\left|\sigma_s(s)-\left(\frac{2}{\int Q}\frac{c_0}{1-\theta}\right)^2 \sigma^{2-2\theta}(s) \right|
\lesssim \lambda(s) \left| \frac{\sigma_s}{\lambda}+1\right|+\lambda^{\frac12}(s) |h(s)| \lesssim s^{\frac{2(1-\theta)}{2\theta-1}-2\rho} ,
\end{equation*}
since $2\rho<\frac54$, which yields, by using \eqref{BS:param},
\begin{equation} \label{BS:est:sigma_s}
\left|\left(\sigma^{2\theta-1}\right)_s(s)-(2\theta-1)\left(\frac{2}{\int Q}\frac{c_0}{1-\theta}\right)^2 \right|
\lesssim \sigma^{2\theta-2}(s)s^{\frac{2(1-\theta)}{2\theta-1}-2\rho} \lesssim s^{-2\rho} .
\end{equation}
Observe from the definition of $c_0$ in \eqref{def:c0bis} that
\begin{equation*}
(2\theta-1)\left(\frac{2}{\int Q}\frac{c_0}{1-\theta}\right)^2=(2\theta-1)^{2\theta-1} .
\end{equation*}
This implies integrating \eqref{BS:est:sigma_s} over $[s_0,s]$, for $s \le s^{\star}$, and using the condition on $\sigma_0$ in \eqref{def:sigma_lamba:ini} that
\begin{equation*}
\left|\sigma^{2\theta-1}(s)-(2\theta-1)^{2\theta-1}s \right|
\lesssim s^{1-2\rho} \iff \left|\left(\frac{\sigma(s)}{(2\theta-1)s^{\frac1{2\theta-1}}}\right)^{2\theta-1}-1 \right|
\lesssim s^{-2\rho} .
\end{equation*}
Hence, we deduce by applying the mean value theorem to the function $r(\tau)=\tau^{\frac1{2\theta-1}}$ that
\begin{equation} \label{BS:est:sigma}
\left|\frac{\sigma(s)}{(2\theta-1)s^{\frac1{2\theta-1}}}-1 \right|
\lesssim s^{-2\rho} \iff \left|\sigma(s)-(2\theta-1)s^{\frac1{2\theta-1}} \right|
\lesssim s^{\frac1{2\theta-1}-2\rho} ,
\end{equation}
which, for $s_0$ large enough, strictly improves the estimate for $\sigma$ in \eqref{BS:param}.
Next, we get inserting \eqref{BS:est:sigma} in \eqref{BS:est:lambda_sigma} that
\begin{equation} \label{BS:est:lambda}
\left|\frac{\lambda(s)}{s^{\frac{2(1-\theta)}{2\theta-1}}}-1 \right|
\lesssim s^{-2\rho} \iff \left|\lambda(s)-s^{\frac{2(\theta-1)}{2\theta-1}} \right|
\lesssim s^{\frac{2(1-\theta)}{2\theta-1}-2\rho} ,
\end{equation}
for all $s \in [s_0,s^{\star}]$, which, for $s_0$ large enough, strictly improves the estimate for $\lambda$ in \eqref{BS:param}.
Finally, we deduce from \eqref{BS:g}, \eqref{BS:est:sigma} and \eqref{BS:est:lambda} that
\begin{align*}
\left|\frac{2\theta-1}{2(1-\theta)}sb(s)+1 \right| &=\frac{2\theta-1}{2(1-\theta)} \lambda^2(s)s\left|\frac{b(s)}{\lambda^2(s)}+\frac{2(1-\theta)}{2\theta-1}s^{-1}\lambda^{-2}(s) \right|\\
&\lesssim \lambda^2(s)s\left( |g(s)|+\left|\frac{4c_0}{\int Q}\lambda^{-\frac32}(s)\sigma^{-\theta}(s)-\frac{2(1-\theta)}{2\theta-1}s^{-1}\lambda^{-2}(s)\right|\right)\lesssim s^{-2\rho} .
\end{align*}
Thus,
\begin{equation*}
\left|b(s)+\frac{2(1-\theta)}{2\theta-1}s^{-1} \right| \lesssim s^{-1-2\rho}
\end{equation*}
for all $s \in [s_0,s^{\star}]$, which, for $s_0$ large enough, strictly improves the estimate for $b$ in \eqref{BS:param}.
\smallskip
\noindent \textit{Closing \eqref{BS:h}.} By using \eqref{BS:hs} and \eqref{BS:g}, we get for any $s \in [s_0,s^{\star}]$,
\begin{equation} \label{BS:est:hs}
\left| h_s(s) \right| \lesssim \lambda^{\frac52}(s)|g(s)|+\lambda^{\frac12}(s)s^{-\frac54}\lesssim s^{-1+\frac{1-\theta}{2\theta-1}-3\rho} ,
\end{equation}
for $s_0$ large enough, since $3\rho < \frac54$. Moreover, observe by the definition of $c_0$ in \eqref{def:c0bis}, the definition of $h$ in \eqref{def:gh} and the choice of $\sigma_0$ and $\lambda_0$ in \eqref{def:sigma_lamba:ini} that $h(s_0)=0$. Hence, it follows integrating \eqref{BS:est:hs} over $[s_0,s^{\star}]$ and using the condition $3\rho <\frac{1-\theta}{2\theta-1}$ that
\begin{equation*}
|h(s)| \lesssim s^{\frac{1-\theta}{2\theta-1}-3\rho} ,
\end{equation*}
for all $s \in [s_0,s^{\star}]$, which, for $s_0$ large enough, strictly improves the estimate for $h$ in \eqref{BS:h}.
\smallbreak
\noindent \textit{Contradiction through a topological argument.} To simplify the notation, for any $b_0$ satisfying \eqref{b:ini}, we introduce
\begin{equation}\label{def:mu}
\mu : b_0 \in \mathcal{D}_0 \mapsto \mu_0=\mu(b_0) :=\left(b_0+\frac{2(1-\theta)}{2\theta-1}s_0^{-1}\right) s_0^{1+3\rho} \in [-1,1] . \end{equation}
Then, by using the definition of $c_0$ in \eqref{def:c0bis}, the definition of $g$ in \eqref{def:gh} and the choice of $\lambda_0$ and $\sigma_0$ in \eqref{def:sigma_lamba:ini}, we compute
\begin{equation*}
g(s_0)=\mu_0s_0^{-\frac{3-2\theta}{2\theta-1}-3\rho} \in \left[-s_0^{-\frac{3-2\theta}{2\theta-1}-3\rho},s_0^{-\frac{3-2\theta}{2\theta-1}-3\rho}\right] .
\end{equation*}
We have assumed that for all $b_0 \in \mathcal{D}_0$, $s^{\star}=s^{\star}(b_0)<+\infty$. Since we have strictly improved \eqref{BS:param}, \eqref{BS:eps} and \eqref{BS:h}, then \eqref{BS:g} must be saturated in $s^{\star}$, which means that
\begin{equation*}
|g(s^{\star})|=\left(s^{\star}\right)^{-\frac{3-2\theta}{2\theta-1}-3\rho} .
\end{equation*}
Define the function
\begin{equation*}
\Phi : \mu_0 \in [-1,1] \mapsto g(s^{\star})\left(s^{\star}\right)^{\frac{3-2\theta}{2\theta-1}+3\rho} \in \left\{-1,1\right\} ,
\end{equation*}
where $s^{\star}=s^{\star}(b_0)$ and $b_0$ is given by the correspondence \eqref{def:mu}. Since \eqref{BS:g} is saturated in $s^{\star}$, it is clear that for $\mu_0=-1$, respectively $\mu_0=1$, then $s^{\star}=s_0$ and $\Phi(-1)=-1$, respectively $\Phi(1)=1$. Now, we will prove that $\Phi$ is a continuous function, which will lead to a contradiction and conclude the proof of Proposition \ref{prop:BS}.
We set
\begin{equation*}
G(s)=\left(g(s)s^{\frac{3-2\theta}{2\theta-1}+3\rho} \right)^2 .
\end{equation*}
It is clear that $G(s^{\star})=1$. Moreover we claim the following transversality property for $G$: let $s_1 \in [s_0,s^{\star}]$ such that $G(s_1)=1$; then there exists $c_0>$ such that
\begin{equation} \label{transversality}
G_s(s_1) \ge c_0\left(s_1\right)^{-1} .
\end{equation}
Indeed, we compute
\begin{equation*}
G_s(s)=2\left(\frac{3-2\theta}{2\theta-1}+3\rho\right)G(s)s^{-1}+2g(s)g_s(s)s^{2\left(\frac{3-2\theta}{2\theta-1}+3\rho\right)} ,
\end{equation*}
which yields \eqref{transversality} by choosing $s_0$ large enough and using \eqref{BS:gs} and $G(s_1)=1$, since $3\rho \le \frac14$.
Finally, it remains to show that $:\mu_0 \mapsto s^{\star}$ is continuous, which will then imply easily that $\Phi$ is continuous. Assume first that $\mu_0 \in (-1,1)$, so that $s^{\star}>s_0$, and let $0<\epsilon<s^{\star}-s_0$. Then, the transversality condition \eqref{transversality} and a continuity argument imply that there exists $\delta>0$ such that for $\epsilon$ small enough\footnote{Observe by continuity that the decomposition \eqref{decomp:v} holds on some time interval after $s^{\star}$ so that $g$ is still well defined on $[s^{\star},s^{\star}+\varepsilon_0]$, for $\varepsilon_0>0$ small enough.}
\begin{equation*}
G(s^{\star}+\epsilon) > 1+\delta \quad \text{and} \quad G(s)<1-\delta, \ \text{for all} \ s \in [s_0,s^{\star}-\epsilon] .
\end{equation*}
Now, by continuity of the flow associated to \eqref{gkdv}, there exists $\eta>0$ such that for all $\tilde{\mu}_0 \in (-1,1)$ such that $|\mu_0-\tilde{\mu}_0|<\eta$, the corresponding function $\tilde{G}$ satisfies $ | \tilde{G}(s)-G(s) | < \delta/2$ on $[s_0,s^{\star}+\epsilon]$. Then, denoting $\tilde s^{\star}=s^{\star}(\tilde{\mu}_0)$, we deduce that
\begin{align*}
\tilde{G}(s)<1-\frac{\delta}2, \ \forall s \in [s_0,s^{\star}-\epsilon] & \implies \tilde s^{\star} \ge s^{\star}-\epsilon ;\\
\tilde{G}(s^{\star}+\epsilon)>1+\frac{\delta}2 & \implies \tilde s^{\star} \le s^{\star}+\epsilon .
\end{align*}
This proves the continuity of the map $\mu_0 \mapsto s^{\star}$ at any $\mu_0 \in (-1,1)$. In the case where $\mu_0=-1$ or $\mu_0=1$, then $s^{\star}=s_0$, $G(s_0)=1$ and $G_s(s_0)>0$ (from \eqref{transversality}). Then, we conclude by using a similar argument that $:\mu_0 \mapsto s^{\star}$ is also continuous at $\mu_0 =-1$ and $\mu_0=1$.
This concludes the proof of Proposition \ref{prop:BS}.
\end{proof}
\subsection{Main result}\label{S:6}
We are now in a position to state the main result of this paper, in its full generality.
Define the constants
\begin{equation}\label{def:cts}
c_\lambda=\left(\frac {3\beta-1}2\right)^{-\frac {1-\beta}2},\quad
c_\sigma=\frac 1\beta \left(\frac {3\beta-1}2\right)^{1-\beta}.
\end{equation}
\begin{theorem}\label{th:2}
There exists $\rho_1>0$ such that for any $x_0$ large enough, the following holds.
Let $t_0=(2x_0)^{\frac 1\beta}$ and let $\sigma_0$ and $\lambda_0$ be such that
\begin{equation*}
\left| \lambda_0-c_\lambda t_0^{\frac {1-\beta}2} \right| \leq t_0^{\frac {1-\beta}2(1-\rho_1)}
\quad \text{and} \quad
\left|\sigma_0 - c_\sigma t_0^{\beta}\right|\leq t_0^{\beta(1-\rho_1)}
.
\end{equation*}
Let $\varepsilon_0 \in H^1(\mathbb R)$ be such that
\begin{equation*}
\|\varepsilon_0\|_{H^1}^2+\int\varepsilon_0^2e^{\frac{y}{10}} \leq t_0^{-\frac {3\beta-1}{20}-\rho_1}
\quad \text{and} \quad
(\varepsilon_0,\Lambda Q)=(\varepsilon_0,y\Lambda Q)=(\varepsilon_0,Q)=0 .
\end{equation*}
Then there exists $b_0=b_0(\lambda_0,\sigma_0,\varepsilon_0)$ with
$|b_0|\lesssim t_0^{-\frac 12(3\beta-1)}$
such that the solution $U(t)$ of \eqref{gkdv} corresponding to the following initial data at $t=t_0$
\begin{equation*}
U(t_0,x)
= \frac 1{\lambda_0^{\frac 12}} \left(Q_{b_0}+\lambda_0^{\frac12}f(t_0,\sigma_0)R+\varepsilon_0\right)
\left(\frac{x-\sigma_0}{\lambda_0}\right)
+ f(t_0,x)
\end{equation*}
decomposes as
\begin{equation}\label{for:U}
U(t,x)=A(t,x)+ \eta(t,x),\quad
A(t,x)=\frac 1{\lambda^{\frac 12}(t)} Q\left( \frac{x-\sigma(t)}{\lambda(t)}\right),
\end{equation}
where the functions $\lambda(t)$, $\sigma(t)$ and $\eta(t)$ satisfy
\begin{align}
&\left|\lambda-c_\lambda t^{\frac {1-\beta}{2}}\right|
\lesssim t^{ \frac {1-\beta}{2}(1-\rho_1)},\quad
\left|\sigma-c_\sigma t^{\beta}\right|
\lesssim t^{\beta(1-\rho_1)},\nonumber\\
&\|\eta\|_{L^2}\lesssim t_0^{-\frac14(3\beta-1)},\quad
\|\partial_x\eta\|_{L^2}\lesssim t_0^{-\frac{1+\beta}4},\nonumber\\
&\|\eta\|_{L^2(x>\frac12\sigma)}+\|\partial_x \eta\|_{L^2(x>\frac12\sigma)}\lesssim t^{-\frac14(3\beta-1)} .
\label{est:eta:right_ext}
\end{align}
\end{theorem}
To prove Theorem~\ref{th:2} from Proposition~\ref{prop:BS}, it is sufficient to return to the
original variables $(t,x)$ (see \S\ref{S.5.3}) and to prove the additional estimate \eqref{est:eta:right_ext} which improves the region where
the residue $\eta$ converges strongly to $0$ (see \S\ref{S.5.4}).
\subsection{Returning to original variables}\label{S.5.3}
In the context of Proposition~\ref{prop:BS} and Theorem~\ref{th:2}, we prove in this subsection the following set of estimates:
\begin{align}
& \left|\lambda-c_\lambda t^{\frac {1-\beta}{2}}\right|
\lesssim t^{ \frac {1-\beta}{2}(1-\rho_1)},
\quad \left|\frac{\lambda_t}{\lambda} - \frac {1-\beta}{2} t^{-1}\right| \lesssim t^{- 1-\frac{1-\beta}2 \rho_1},
\label{sur:lat}\\
& \left|\sigma-c_\sigma t^{\beta}\right|
\lesssim t^{\beta(1-\rho_1)},
\quad \left|\frac{\sigma_t}{\lambda} - \frac 1{\lambda^3}\right| \lesssim t^{- 1-\frac{3\beta-1}8},\label{sur:sit}
\end{align}
and
\begin{align}
& \|\lambda^{\frac 1{20}}A ^\frac1{10}\eta\|_{L^2}\lesssim t^{-\frac12(3\beta-1)},\quad
\|\lambda^{\frac 1{20}}A ^\frac1{10}\partial_x \eta\|_{L^2}\lesssim t^{-\beta},\quad
\|\lambda^{\frac 1{20}}A ^\frac1{10}\eta\|_{L^\infty}\lesssim t^{-\frac14(5\beta-1)},\label{eq:eta:loc}\\
& \|\eta\|_{L^2(x>\sigma)}\lesssim t^{-\frac12(3\beta-1)},\quad
\|\partial_x \eta\|_{L^2(x>\sigma)}\lesssim t^{-\beta},
\label{eq:eta:right}\\
& \|\eta\|_{L^2}\lesssim t_0^{-\frac14(3\beta-1)},\quad
\|\partial_x\eta\|_{L^2}\lesssim t_0^{-\frac{1+\beta}4},\label{eq:eta}\end{align}
where $\rho_1$ is a small positive number.
\begin{proof}[Proof of \eqref{sur:lat}-\eqref{eq:eta}]
First, we relate $t$ and $s$ from \eqref{eq:sbis}. We claim that for any $t\geq t_0$, $s\geq s_0$,
\begin{equation}\label{sur:t:s}
\left|t-\frac {2\theta-1}{5-4\theta} s^{\frac {5-4\theta}{2\theta-1}}\right|\lesssim \left(1-\left(\frac {s_0}s\right)^{\frac {5-4\theta}{2\theta-1}-\rho}\right) s^{\frac {5-4\theta}{2\theta-1}-\rho}.
\end{equation}
Indeed, from \eqref{BS:param} and $dt=\lambda^3(s) ds$, one has
($\rho$ is defined in \eqref{def:rho})
\begin{align*}
t-t_0=\int_{s_0}^s \lambda^3(s) ds
&\geq \int_{s_0}^s (s')^{\frac{6(1-\theta)}{2\theta-1}} ds'
-c \int_{s_0}^s (s')^{\frac{6(1-\theta)}{2\theta-1}-\rho} ds'\\
&\geq \frac{2\theta-1}{5-4\theta}\left( s^{\frac {5-4\theta}{2\theta-1}}-s_0^{\frac {5-4\theta}{2\theta-1}}\right)
-c \left( s^{\frac {5-4\theta}{2\theta-1}-\rho}-s_0^{\frac {5-4\theta}{2\theta-1}-\rho}\right),
\end{align*}
and thus using~\eqref{eq:sbis},
\begin{equation*}
t-\frac{2\theta-1}{5-4\theta} s^{\frac {5-4\theta}{2\theta-1}}
\geq -c \left( 1- \frac {s_0}{s}\right) s^{\frac {5-4\theta}{2\theta-1}-\rho}.
\end{equation*}
This proves the lower bound in \eqref{sur:t:s}.
The corresponding upper bound is proved similarly.
Note that
\begin{equation}\label{t:s:beta}
\frac {2\theta-1}{5-4\theta}=\frac{3\beta-1}{2} \quad\mbox{so that}\quad
\left|t-\frac{3\beta-1}{2} s^{\frac2{3\beta-1}}\right|\lesssim \left(1-\left(\frac {s_0}s\right)^{\frac {5-4\theta}{2\theta-1}-\rho}\right) s^{\frac2{3\beta-1}-\rho}.
\end{equation}
Observing that
\begin{equation*}
c_\lambda = \left( \frac {5-4\theta}{2\theta-1}\right)^{\frac {2(1-\theta)}{5-4\theta}},
\end{equation*}
it follows from \eqref{BS:param} and \eqref{sur:t:s} that
\begin{equation*}
\left|\lambda(t)-c_\lambda t^{\frac {2(1-\theta)}{5-4\theta}}\right|
\lesssim t^{\frac {2(1-\theta)}{5-4\theta}(1-\rho_1)}\end{equation*}
holds with $0<\rho_1:=\frac {2\theta-1}{2(1-\theta)} \rho<\frac 16$.
Moreover, we check that \eqref{BS:param} and \eqref{BS:m} imply
\begin{equation*}
\left| \frac {\lambda_t (t)}{\lambda(t)} - \frac {2(1-\theta)}{5-4\theta} \frac 1{t} \right|
\lesssim t^{-(1+\frac{2(1-\theta)}{5-4\theta}\rho_1)}.
\end{equation*}
Using $\beta=\frac 1{5-4\theta}$, one finds \eqref{sur:lat}.
Observing that
\begin{equation*}
c_\sigma = (5-4\theta)^{\frac 1{5-4\theta}} (2\theta-1)^{ \frac{4(1-\theta)}{5-4\theta}},
\end{equation*}
it follows from \eqref{BS:param} and \eqref{sur:t:s} that
\begin{equation*}
\left|\sigma(t)-c_\sigma t^{\frac {1}{5-4\theta}}\right|
\lesssim t^{\frac {1}{5-4\theta}(1-\rho_1)}.
\end{equation*}
Moreover, by \eqref{BS:param} and \eqref{BS:m}, it holds
\begin{equation*}
\left|\frac{\sigma_t}{\lambda}-\frac{1}{\lambda^3}\right|=
\left|\frac 1{\lambda^3} \left( \frac{\sigma_s}{\lambda}-1\right)\right|
\lesssim t^{-\frac{19-14\theta}{4(5-4\theta)}}.
\end{equation*}
Thus, \eqref{sur:sit} holds recalling that $\beta=\frac1{5-4\theta}$.
Last, we control $\eta$. Note that from \eqref{def:v} and \eqref{decomp:v}
\begin{equation*}
\lambda^{\frac 12} \eta\left(t,\lambda(t) y +\sigma\right)=
F(t,y) + b(t) P_{b(t)}(y) + r(t) R(y)+ \varepsilon(t, y).
\end{equation*}
Thus, the following estimates hold
\begin{align*}
\|\eta(t)\|_{L^2}&\lesssim \|f(t)\|_{L^2} + |b(t)| \|P_{b(t)}\|_{L^2}
+|r(t)| + \|\varepsilon(t)\|_{L^2},\\
\|\partial_x \eta(t)\|_{L^2}&\lesssim \|\partial_x f(t)\|_{L^2}+\lambda^{-1}(t) \left[
|b(t)| \|\partial_y P_{b(t)}\|_{L^2}
+|r(t)| + \|\partial_y \varepsilon(t)\|_{L^2}\right],
\\
\lambda^{\frac 1{20}}\|A ^\frac1{10}(t)\eta(t)\|_{L^2}&\lesssim \|Q^\frac1{10} F(t)\|_{L^2} + |b(t)| \|Q^\frac1{10}P_{b(t)}\|_{L^2}
+|r(t)| + \|Q^\frac1{10}\varepsilon(t)\|_{L^2},\\
\lambda^{\frac 1{20}}\|A ^\frac1{10}(t)\partial_x \eta(t)\|_{L^2}&\lesssim \lambda^{-1}(t) \left[ \|Q^\frac1{10}\partial_y F(t)\|_{L^2}+
|b(t)| \|Q^\frac1{10}\partial_y P_{b(t)}\|_{L^2}
\right.\\&\quad \left.+|r(t)| + \|Q^\frac1{10}\partial_y \varepsilon(t)\|_{L^2}\right].
\end{align*}
From \eqref{est:f_0}, \eqref{bound:df:dx} and $x_0= (2t_0)^{\beta}$,
\begin{align*}
&\sup_{t\in \mathbb R}\|f(t)\|_{L^2}=\|f_0\|_{L^2}\lesssim x_0^{-\frac 12 (2\theta-1)}
\lesssim t_0^{-\frac{3\beta-1}4},\\
&\sup_{t\in \mathbb R}\|\partial_x f(t)\|_{L^2}\lesssim x_0^{-\frac 12 (2\theta+1)}
\lesssim t_0^{-\frac{7\beta-1}4}.
\end{align*}
From \eqref{e:F} and \eqref{t:s:beta},
\begin{equation*}
\|Q^\frac1{10} F(t)\|_{L^2}\lesssim t^{-\frac{3\beta-1}2},
\quad
\|Q^\frac1{10}\partial_x F(t)\|_{L^2}\lesssim t^{-(3\beta-1)}.
\end{equation*}
From \eqref{def:Pb}, \eqref{BS:param} and \eqref{t:s:beta}, it holds
\begin{align*}
& \|b(t)P_{b(t)}\|_{L^2}
\lesssim |b(t)|^{\frac 58}\lesssim t^{-\frac 5{16}(3\beta-1)},
\\&
\|b(t)\partial_y P_{b(t)}\|_{L^2}+|b(t)| \|Q^\frac1{10}P_{b(t)}\|_{L^2}+ |b(t)| \|Q^\frac1{10}\partial_y P_{b(t)}\|_{L^2}\lesssim |b(t)|\lesssim t^{-\frac {3\beta-1}2}.
\end{align*}
From \eqref{e:r} and \eqref{t:s:beta}, it holds $|r(t)|\lesssim t^{-\frac 12(3\beta-1)}$.
Last, from \eqref{BS:eps},
\begin{equation*}
\|Q^\frac1{10}\varepsilon(t)\|_{L^2}+\|Q^\frac1{10}\partial_y \varepsilon(t)\|_{L^2}\lesssim
t^{-\frac 5{8}(3\beta-1)}.
\end{equation*}
This implies the first two estimates in \eqref{eq:eta:loc}. For the last estimate in \eqref{eq:eta:loc}, we use
\begin{align*}
\|\lambda^{\frac 1{20}}A ^\frac1{10}\eta\|_{L^\infty}^2&
\lesssim\|\lambda^{\frac 1{20}}A ^\frac1{10}\eta\|_{L^2}\|\lambda^{\frac 1{20}}\partial_x[A ^\frac1{10}\eta]\|_{L^2}\\
&\lesssim\|\lambda^{\frac 1{20}}A ^\frac1{10}\eta\|_{L^2}\left(\|\lambda^{\frac 1{20}}A ^\frac1{10}\partial_x\eta\|_{L^2}
+\|\lambda^{\frac 1{20}}\partial_x[A ^\frac1{10}]\eta\|_{L^2}\right)\\
&\lesssim\|\lambda^{\frac 1{20}}A ^\frac1{10}\eta\|_{L^2}\left(\|\lambda^{\frac 1{20}}A ^\frac1{10}\partial_x\eta\|_{L^2}
+\lambda^{-1}\|\lambda^{\frac 1{20}}A ^\frac1{10}\eta\|_{L^2}\right)
\lesssim t^{-\frac12(5\beta-1)}.
\end{align*}
Similarly, \eqref{eq:eta:right} follows from \eqref{e:FLinfty}, the properties of $P_b$ and \eqref{BS:eps}.
Now, we estimate $\|\varepsilon\|_{L^2}$ and $\|\partial_y \varepsilon\|_{L^2}$.
For this, the local estimates~\eqref{BS:eps} involved in the bootstrap are not sufficient, and we have to use global mass and energy estimates from Lemma~\ref{le:3.9}. First, we compute $\int u_0^2$. Using the computations of the proof of Lemma~\ref{le:3.9}
and~\eqref{mass:W},
\begin{equation*}
\int u_0^2 -\int Q^2
= 2 b_0 \int PQ + \frac 12 r_0 \int Q +\mathcal O(\|\varepsilon_0\|_{L^2}^2)
+\mathcal O(s_0^{-\frac 54}) + \mathcal O(x_0^{-(2\theta-1)}).
\end{equation*}
By the above estimates taken at $t=t_0$ and $\|\varepsilon_0\|_{L^2}\lesssim s_0^{-10} \lesssim t_0^{-5(3\beta-1)}$
(see \eqref{def:eps:ini})
we find
\begin{equation*}
\left|\int u_0^2-\int Q^2\right|
\lesssim t_0^{-\frac 12 (3\beta-1)}.
\end{equation*}
Thus, by \eqref{mass:eps}, we find, for all $t\geq t_0$,
$\|\varepsilon\|_{L^2} \lesssim t_0^{-\frac 14 (3\beta-1)}$
and the above estimates imply $\|\eta\|_{L^2} \lesssim t_0^{-\frac14 (3\beta-1)}$.
Second, we compute $E(u_0)$. Using the computations of the proof of Lemma~\ref{le:3.9}
and \eqref{ener:W},
\begin{equation*}
E(u_0) = \mathcal O(\lambda^{-2}(t_0) \|\varepsilon_0\|_{H^1}^2) + \mathcal O(\lambda^{-2}(t_0) s_0^{-2})
+\mathcal O(x_0^{-(2\theta+1)})+\mathcal O(|g(s_0)|).
\end{equation*}
Note that by \eqref{BS:g} and \eqref{t:s:beta}, it holds $|g(s_0)|\lesssim t_0^{-\frac 12 (1+\beta)}$.
We deduce from \eqref{def:eps:ini} and the previous estimates that
$|E(u_0)|\lesssim t_0^{-\frac 12 (1+\beta)}$.
Thus, by \eqref{ener:eps}, we find, for all $t\geq t_0$,
$\lambda^{-1} \|\partial_y\varepsilon\|_{L^2} \lesssim t_0^{-\frac 14 (1+\beta)}$, and the above estimates imply
$\|\partial_x \eta\|_{L^2} \lesssim t_0^{-\frac 14 (1+\beta)}$.
\end{proof}
\subsection{Additional monotonicity argument.}\label{S.5.4}
To complete the proof of Theorem~\ref{th:2}, we prove \eqref{est:eta:right_ext} by extending the local estimates \eqref{eq:eta:right} for $\eta$ on the right of the soliton to the larger region $x>\frac12\sigma$.
Write the equation for $\eta$ as follows
\begin{equation}\label{equ:eta}
\partial_t \eta+\partial_x^3 \eta
=-\partial_xN_1 + N_2
\end{equation}
where
\begin{align*}
N_1&=(A +\eta)^5-A ^5,\\
N_2&=\frac{\lambda_t}{\lambda}\frac 1{\lambda^{\frac 12}} \Lambda Q\left( \frac{\cdot-\sigma}{\lambda}\right)
+\left(\frac{\sigma_t}\lambda-\frac 1{\lambda^3}\right) \frac 1{\lambda^{\frac 12}} Q'\left( \frac{\cdot-\sigma}{\lambda}\right).
\end{align*}
Fix $t>t_0$ and for any $\tau\in [t_0,t]$, let
\begin{align*}
J(\tau)&=\int \eta^2(\tau,x)\xi(\tau,x) dx , \\
K(\tau)&=\int \left[ (\partial_x\eta)^2-\frac13\left((A +\eta)^6-A ^6-6A ^5\eta\right)\right](\tau,x)\xi(\tau,x) dx ,
\end{align*}
where $ \xi(\tau,x)=\chi\left(\frac{4x-\sigma(t)}{\sigma(\tau)}-2\right)$ and $\chi$ is defined in \eqref{def:chi}.
\begin{lemma}
For $t_0$ large enough and for all $\tau\in [t_0,t]$, it holds
\begin{equation}\label{on:Js}
\frac{dJ}{d\tau}(\tau)\lesssim \tau^{-\frac 12(3\beta+1)},
\end{equation}
and
\begin{equation}\label{on:Ks}
\frac{dK}{d\tau}(\tau)\lesssim \tau^{-\frac 12(3\beta+1)}.
\end{equation}
\end{lemma}
\begin{proof}
We begin with the proof of \eqref{on:Js}. Using \eqref{equ:eta}, we observe after integration by parts
\begin{align*}
\frac{dJ}{d\tau} &= 2\int (\partial_\tau\eta)\eta\xi+\int \eta^2\partial_\tau\xi
\\&=-3 \int (\partial_x \eta)^2 \partial_x \xi+\int \eta^2\partial_\tau\xi +\int \eta^2 \partial_x^3 \xi
-2 \int (\partial_xN_1) \eta \xi+2\int N_2 \eta\xi \\
& =:J_1+J_2+J_3+J_4+J_5 .
\end{align*}
We compute
\begin{align*}
\partial_x \xi(\tau,x) &= \frac 4{\sigma(\tau)} \chi'\left(\frac{4x-\sigma(t)}{\sigma(\tau)}-2\right) \geq 0,\\
\partial_x^3 \xi(\tau,x) &=\left(\frac 4{\sigma(\tau)}\right)^3 \chi'''\left(\frac{4x-\sigma(t)}{\sigma(\tau)}-2\right),\\
\partial_\tau \xi(\tau,x) &=-\frac{\sigma_t(\tau)}{\sigma(\tau)}\left(\frac{4x-\sigma(t)}{\sigma(\tau)}\right) \chi'\left(\frac{4x-\sigma(t)}{\sigma(\tau)}-2\right).
\end{align*}
Note that by \eqref{sur:sit},
\begin{equation} \label{est:xi3}
0 \le \partial_x\xi \lesssim \tau^{-\beta}, \quad \text{and} \quad |\partial_x^3 \xi|\lesssim \tau^{-3\beta}\lesssim\tau^{-\frac {3\beta+1}2} .
\end{equation}
Observe that $J_1 \le 0$. Since $\sigma_t\geq 0$ (see \eqref{sur:sit}), $\chi'\geq 0$ on $\mathbb{R}$ and $\chi'(x)=0$ for $x<-2$, we also have $\partial_\tau \xi\leq 0$, so that $J_2 \le 0$. Moreover, by using \eqref{eq:eta} and \eqref{est:xi3}, we have that
$\left|J_3\right| \lesssim \tau^{-\frac 12 (3\beta+1)}$.
On the other hand, more integration by parts yield
\begin{align*}
J_4
& = 2 \int \left( (A +\eta)^5-A ^5\right) (\partial_x\eta\, \xi+\eta \, \partial_x\xi)\\
& = - \frac 13 \int \left[ (A +\eta)^6-A ^6-6A ^5\eta\right]\partial_x \xi
+2 \int \left( (A +\eta)^5-A ^5\right) \eta \partial_x\xi\\
&\quad -2\int \left[ (A +\eta)^5-A ^5-5A ^4\eta\right] (\partial_x A ) \xi\\
&=\int \left(5A ^4\eta^2+\frac {40}3 A ^3\eta^3+15A ^2\eta^4+8A \eta^5+\frac 53 \eta^6\right)\partial_x \xi\\
&\quad -2\int \left[ (A +\eta)^5-A ^5-5A ^4\eta\right] (\partial_x A ) \xi ,
\end{align*}
so that
\begin{equation*}
\left|J_4\right|\lesssim \int \eta^6 \partial_x \xi +\int \eta^2 A ^4\partial_x \xi+\int \eta^2 |A |^3|\partial_xA |+\int |\eta|^5 |\partial_xA |.
\end{equation*}
For the first term on the right-hand side of the above estimate, we argue as in \eqref{est:infty} to deduce
\begin{equation*} \left\|\eta^2\sqrt{\partial_x\xi}\right\|_{L^\infty}^2
\lesssim \|\eta\|_{L^2}^2\int (\partial_x\eta)^2 \partial_x\xi
+\|\eta\|_{L^2}^2\int \eta^2\frac{\big(\partial_x^2\xi\big)^2}{\partial_x\xi} ,
\end{equation*}
in the support of $\partial_x\xi$. Thus, by using in this region
\begin{equation}\label{A-VOIR}
\frac{ (\partial_x^2\xi )^2}{\partial_x\xi} \lesssim \sigma^{-3} \frac{(\chi'')^2}{\chi'} \lesssim \sigma^{-3},
\end{equation}
we deduce from \eqref{eq:eta}, \eqref{est:xi3} that
\begin{equation} \label{full:nonlin:eta}
\int \eta^6 \partial_x \xi\lesssim
\left(\int \eta^2\right)^{2} \left[ \int (\partial_x \eta)^2 \partial_x \xi + \sigma^{-3} \int \eta^2\right] \le -\frac12J_1+c\tau^{-\frac {3\beta+1}2} ,
\end{equation}
by taking $t_0$ large enough.
For the second term, using \eqref{eq:eta:loc} and \eqref{est:xi3}, we have
\begin{equation*}
\int \eta^2 A ^4\partial_x \xi
\lesssim \sigma^{-1}\lambda^{-2} \|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\eta\|_{L^2}^2\lesssim
\tau^{-3\beta} \lesssim\tau^{-\frac {3\beta+1}2}.
\end{equation*}
Next, using \eqref{eq:eta:loc}
\begin{equation*}
\int \eta^2 |A |^3|\partial_xA |
\lesssim \lambda^{-3} \|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\eta\|_{L^2}^2
\lesssim \tau^{-\frac {3\beta+1}2},
\end{equation*}
and
\begin{equation*}
\int |\eta|^5 |\partial_xA |
\lesssim \lambda^{-\frac 32} \|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\eta\|_{L^\infty}^3\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\eta\|_{L^2}^2
\lesssim \tau^{1-6\beta}\lesssim \tau^{-\frac {3\beta+1}2}.
\end{equation*}
Gathering all these estimates, we obtain that
\begin{equation*} J_4 \le -\frac12 J_1+c\tau^{-\frac {3\beta+1}2} .\end{equation*}
Last, we observe by \eqref{sur:lat}-\eqref{sur:sit} that $|N_2(\tau,x)|\lesssim \tau^{-1} \lambda^{-\frac14}A ^{\frac12}(\tau,x)$, and so
\begin{equation*}
\left|J_5\right|\lesssim
\tau^{-1} \lambda^{-\frac14}\int A ^{\frac12} |\eta|\lesssim \tau^{-1}\lambda^{-\frac12}\|\lambda^{\frac 1{20}}A ^{\frac 1{10}} \eta\|_{L^2}
\lesssim \tau^{-\frac {3\beta+1}2}.
\end{equation*}
Collecting these estimates and taking $t_0$ large enough, we have proved \eqref{on:Js}.
\smallskip
Now, we turn to the proof of \eqref{on:Ks}. We compute using \eqref{equ:eta} and integrating by parts
\begin{align*}
\frac{dK}{d\tau} &= \int \left[(\partial_x\eta)^2-\frac13\left((A +\eta)^6-A ^6-6A ^5\eta \right) \right]\partial_{\tau}\xi
+2\int \partial_x\eta \partial_{x\tau}^2\eta \xi\\ & \quad
-2\int \left[\left((A +\eta)^5-A ^5\right)\partial_{\tau}(A +\eta)-5A ^4\partial_{\tau}A \eta \right] \xi
\\
&=-2 \int \left[\partial_x^2\eta+(A +\eta)^5-A ^5 \right]^2 \partial_x \xi-2\int (\partial_x^2\eta)^2 \partial_x\xi+\int (\partial_x\eta)^2\partial_x^3\xi\\
& \quad +2\int \partial_xN_1 \partial_x \eta\partial_x\xi +2\int \partial_xN_2 \partial_x\eta \xi-2\int N_2 N_1 \xi\\
& \quad -2\int \left[(A +\eta)^5-A ^5-5A ^4\eta \right] \partial_{\tau}A \xi \\
&\quad + \int \left[(\partial_x\eta)^2-\frac13\left((A +\eta)^6-A ^6-6A ^5\eta \right) \right]\partial_{\tau}\xi \\
& =:K_1+K_2+K_3+K_4+K_5+K_6+K_7+K_8 .
\end{align*}
Observe, since $\partial_x \xi \ge 0$, that $K_1 \le 0$ and $K_2 \le 0$. Moreover, it follows from \eqref{eq:eta} and \eqref{est:xi3} that $K_3 \lesssim \tau^{-3\beta}$. Moreover, we compute
\begin{equation*}
K_4 = 10\int (A +\eta)^4(\partial_x\eta)^2 \partial_x\xi+10\int \left[(A +\eta)^4-A ^4 \right]\partial_xA \partial_x\eta\partial_x\xi
\end{equation*}
so that
\begin{align*}
K_4& \lesssim \int A ^4(\partial_x\eta)^2 \partial_x\xi+\int \eta^4(\partial_x\eta)^2 \partial_x\xi+\int \left|A ^3\partial_xA \eta\partial_x\eta\right|\partial_x\xi
+\int \eta^4(\partial_xA )^2 \partial_x\xi \\
&=:K_{4,1}+K_{4,2}+K_{4,3}+K_{4,4} .
\end{align*}
By using \eqref{sur:lat}, \eqref{sur:sit}, \eqref{eq:eta:loc} and \eqref{est:xi3}, we deduce that
\begin{align*}
|K_{4,1}| &\lesssim \sigma^{-1}\lambda^{-2}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2}^2 \lesssim \tau^{-1-2\beta} \\
|K_{4,3}| &\lesssim \sigma^{-1}\lambda^{-3}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\eta\|_{L^2}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2} \lesssim \tau^{-1-2\beta} \\
|K_{4,4}| &\lesssim \sigma^{-1}\lambda^{-3}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\eta\|_{L^{\infty}}^2\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\eta\|_{L^2}^2\lesssim \tau^{-\frac{7\beta+1}2} \lesssim \tau^{-1-2\beta} ,
\end{align*}
since $\beta>\frac13$. To handle the purely nonlinear term $K_{4,2}$, we use, arguing as in \eqref{Sobo:firstderiv}, the improved Sobolev estimate
\begin{equation*} \left\|\eta\partial_x\eta\sqrt{\partial_x\xi}\right\|_{L^\infty}^2
\lesssim \|\eta\|_{L^2}^2\int (\partial_x^2\eta)^2 \partial_x\xi
+\|\eta\|_{L^2}^2\int (\partial_x\eta)^2\frac{\big(\partial_x^2\xi\big)^2}{\partial_x\xi} ,
\end{equation*}
in the support of $\partial_x\xi$.
Thus, using \eqref{A-VOIR}, we deduce from \eqref{eq:eta}, \eqref{est:xi3} that
\begin{equation*}
K_{4,2}\lesssim
\left(\int \eta^2\right)^{2} \left[ \int (\partial_x^2 \eta)^2 \partial_x \xi + \sigma^{-3} \int \eta^2\right] \le -\frac12K_2+c\tau^{-3\beta} ,
\end{equation*}
by taking $t_0$ large enough.
Observe from \eqref{sur:lat}-\eqref{sur:sit} that $|\partial_xN_2(\tau,x)|\lesssim \tau^{-1}\lambda^{-\frac54} A ^{\frac12}(\tau,x)$. Thus, it follows using \eqref{sur:lat} and \eqref{eq:eta:loc} that
\begin{equation*}
|K_5| \lesssim \tau^{-1}\lambda^{-\frac32}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2} \lesssim \tau^{-\frac{7+\beta}4} \lesssim \tau^{-\frac{3\beta+1}2},
\end{equation*}
since $\beta \le 1$. To estimate $K_6$, we observe
\begin{equation*}
|K_6| \lesssim \int A ^4|\eta| |N_2|+\int |\eta|^5 |N_2| =:K_{6,1}+K_{6,2} ,
\end{equation*}
so that it follows from $|N_2(\tau,x)|\lesssim \tau^{-1} \lambda^{-\frac14}A ^{\frac12}(\tau,x)$ and \eqref{sur:lat}-\eqref{eq:eta:loc},
\begin{align*}
|K_{6,1}| &\lesssim \tau^{-1}\lambda^{-\frac52}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2} \lesssim \tau^{-\frac{7+\beta}4} \lesssim \tau^{-\frac{3\beta+1}2} , \\
|K_{6,2}| &\lesssim \tau^{-1}\lambda^{-\frac12}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^{\infty}}^3\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2}^2 \lesssim \tau^{-\frac{1+17\beta}4} \lesssim \tau^{-1-2\beta} ,
\end{align*}
since $\beta>\frac13$. To control the contribution $K_7$, we observe from \eqref{sur:lat}-\eqref{sur:sit}
\begin{equation*}
\partial_tA =-\frac{\lambda_t}{\lambda} \lambda^{-\frac12} \Lambda Q\left(\frac{\cdot-\sigma}{\lambda}\right)-\frac{\sigma_t}{\lambda}\lambda^{-\frac12}Q'\left(\frac{\cdot-\sigma}{\lambda}\right) \quad \text{so that} \quad |\partial_tA (\tau,x)| \lesssim \lambda^{-3}A (\tau,x) .
\end{equation*}
Thus,
\begin{equation*}
|K_7| \lesssim \lambda^{-3} \int A ^4\eta^2+ \lambda^{-3}\int |A ||\eta|^5=: K_{7,1} +K_{7,2} .
\end{equation*}
Hence, it follows from \eqref{sur:lat}-\eqref{eq:eta:loc} that
\begin{align*}
|K_{7,1}| &\lesssim \lambda^{-5}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2}^2 \lesssim \tau^{-\frac{3+\beta}2} \lesssim \tau^{-\frac{3\beta+1}2} , \\
|K_{7,2}| &\lesssim \lambda^{-3}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^{\infty}}^3\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2}^2 \lesssim \tau^{-\frac{21\beta-1}4} \lesssim \tau^{-\frac{3\beta+1}2} ,
\end{align*}
since $\frac13<\beta\le 1$. Finally, we deal with $K_8$ by writing
\begin{equation*}
K_8 \le \int (\partial_x\eta)^2\partial_{\tau}\xi +c\int A ^4\eta^2\left|\partial_{\tau}\xi\right|+c\int \eta^6 \left|\partial_{\tau}\xi\right| =: K_{8,1}+K_{8,2}+K_{8,3} .
\end{equation*}
Observe that $K_{8,1} \le 0$, since $\partial_{\tau}\xi \le 0$. Moreover, we have $0 \le \frac{4x-\sigma(t)}{\sigma(\tau)} \le 1$ on the support of $\partial_{\tau}\xi$, so that $|\partial_{\tau}\xi(\tau,x)| \lesssim \tau^{-1}$ (from \eqref{sur:lat}-\eqref{sur:sit}). Then, it follows from \eqref{sur:lat}, \eqref{eq:eta:loc} that
\begin{equation*}
|K_{8,2}| \lesssim \tau^{-1}\lambda^{-2}\|\lambda^{\frac 1{20}} A ^{\frac 1{10}}\partial_x\eta\|_{L^2}^2 \lesssim \tau^{-1-2\beta} .
\end{equation*}
To deal with the fully nonlinear term $K_{8,3}$, we get arguing as in \eqref{full:nonlin:eta} and using \eqref{sur:lat}, \eqref{eq:eta} and
\begin{equation}\label{A-VOIR2}
\frac{\big(\partial_{x\tau}^2\xi\big)^2}{|\partial_{\tau}\xi|} \lesssim \left| \frac{\sigma_t}{\sigma} \right| \sigma^{-2} \left( \bar{x}^{-1}\chi'(\bar{x}-2)+\bar{x}\frac{(\chi'')^2}{\chi'} \right) \lesssim \tau^{-1}\sigma^{-2}
\end{equation}
on the support of $\partial_{\tau}\xi$ ($0 \le \bar{x}:=\frac{4x-\sigma(t)}{\sigma(\tau)} \le 1$),
that
\begin{equation*}
K_{8,3} \lesssim
\left(\int \eta^2\right)^{2} \left[ \int (\partial_x \eta)^2 \left|\partial_{\tau} \xi\right| + \tau^{-1}\sigma^{-2} \int \eta^2\right] \le -\frac12K_{8,1}+c\tau^{-1-2\beta} ,
\end{equation*}
for $t_0$ large enough.
We complete the proof of \eqref{on:Ks} by combining all those estimates.
\end{proof}
\begin{proof}[Proof of \eqref{est:eta:right_ext}]
We define $\underline t$ such that $\sigma(\underline t)=\frac 14 \sigma(t)$.
Note that $\underline t\gtrsim t$ from \eqref{sur:sit}.
Next, we integrate \eqref{on:Js} on $[\underline t,t]$ and we also use $\chi=0$ on $(-\infty,-2]$ and $\chi=1$ on $[-1,+\infty)$.
We obtain
\begin{equation*}
\int_{x\geq \frac{\sigma(t)}2} \eta^2(t,x) dx
\leq J(t)\leq J(\underline t)+c t^{-\frac {3\beta-1}2}
\lesssim \int_{x\geq \sigma(\underline t)} \eta^2(\underline t,x) dx + t^{-\frac {3\beta-1}2}
\lesssim t^{-\frac {3\beta-1}2} ,
\end{equation*}
where we used \eqref{eq:eta:right} in the last step. This implies the first estimate in \eqref{est:eta:right_ext}. Arguing similarly for $K$, we deduce that
\begin{equation*}
\int_{x\geq \frac{\sigma(t)}2} \left[ (\partial_x\eta)^2-\frac13\left((A +\eta)^6-A ^6-6A ^5\eta\right)\right](t,x) dx
\leq K(t)\leq K(\underline t)+c t^{-\frac {3\beta-1}2}
\lesssim t^{-\frac {3\beta-1}2} .\end{equation*}
Moreover, we deduce by using the Sobolev embedding, \eqref{eq:eta:right} and \eqref{eq:eta} that
\begin{align*}
\int_{x\geq \frac{\sigma(t)}2} \left((A +\eta)^6-A ^6-6A ^5\eta\right)
&\lesssim \int A ^4\eta^2+\left(\int \eta^2\right)^2\int_{x\geq \frac{\sigma(t)}2}(\partial_x\eta)^2 \\
&\le \frac12\int_{x\geq \frac{\sigma(t)}2}(\partial_x\eta)^2+ct^{-(3\beta-1)} ,
\end{align*}
by choosing $t_0$ large enough. Therefore, we complete the proof of the second estimate in~\eqref{est:eta:right_ext} by combining the last two estimates.
\end{proof}
\subsection{Proof of Theorem~\ref{th:1}}
The statement of Theorem~\ref{th:1} in the Introduction corresponds to a simplification of Theorem~\ref{th:2}
and to a further rescaling and translation to consider initial data at $t=0$ and close to the soliton $Q$.
Take $x_0$, $t_0$, $\lambda_0$, $\sigma_0$, $b_0$, $U_0$ and a solution $U(t)$ of \eqref{gkdv}
as in Theorem~\ref{th:2}.
Define the following rescaled version of $U_0$
\begin{equation*}
u_0(x)=
\lambda_0^{\frac 12} U_0( \lambda_0 x+ \sigma_0)=
Q_{ b_0}(x)+ \lambda_0^{\frac 12}f(t_0, \sigma_0) R(x)
+ \lambda_0^{\frac 12} f (t_0, \lambda_0 x+ \sigma_0)
\end{equation*}
and consider $u(t)$ the solution of \eqref{gkdv} with initial data $u(0)= u_0$, so that
\begin{equation*}
u(t,x)=\lambda_0^{\frac 12} U(\lambda_0^3 t+t_0,\lambda_0 x+ \sigma_0).
\end{equation*}
Let (see \eqref{eq:sbis})
\begin{equation*}
T_0= t_0 \lambda_0^{-3}= c_\lambda^{-3} t_0^{\frac 12 (3\beta -1)}
=\frac{3\beta-1}2 s_0.
\end{equation*}
The decomposition \eqref{for:U} rewrites
\begin{equation*}
u(t,x) =\frac 1{\ell^{\frac 12}(t)} Q\left( \frac{x-x(t)}{\ell(t)}\right)+ w(t,x)
\end{equation*}
where
\begin{equation*}
\ell(t)=\frac{\lambda(\lambda_0^3 t+t_0)}{\lambda_0},\quad
x(t)=\frac{\sigma(\lambda_0^3 t+t_0)-\sigma_0}{\lambda_0},
\quad
w(t,x)=\lambda_0^{\frac 12} \eta(\lambda_0^3 t+t_0,\lambda_0 x+ \sigma_0).
\end{equation*}
First, as a consequence of \eqref{sur:sit}, we have
\begin{equation*}
\left|x(t)-c_\sigma \frac{(\lambda_0^3 t+t_0)^{\beta}-t_0^\beta} {\lambda_0} \right|
\lesssim \frac{(\lambda_0^3 t+t_0)^{\beta(1-\rho_1)}}{\lambda_0}
.
\end{equation*}
Since
\begin{equation*}
c_\sigma t_0^\beta \lambda_0^{-1}
=\beta^{-1} c_\lambda^{-3} t_0^{\frac 12 (3\beta-1)}
=\beta^{-1} T_0,
\end{equation*}
we obtain, for $\epsilon=\beta\rho_1>0$,
\begin{equation}\label{on:sig}
\left|x(t)-\frac {T_0 }\beta \left[{\left(\frac t{T_0}+1\right)^{\beta}-1}\right] \right|
\lesssim T_0 \left(\frac t{T_0}+1\right)^{\beta} T_0^{-3\epsilon\frac {1-\beta}{3\beta-1}}(t+T_0)^{-\epsilon}
.
\end{equation}
Similarly, by \eqref{sur:lat}, we have
\begin{equation*}
\left|\ell(t)- \left(\frac{t}{T_0}+1\right)^{\frac {1-\beta}2}\right|
\lesssim t_0^{-\rho_1\frac {1-\beta}{2}} \left(\frac{t}{T_0}+1\right)^{\frac {1-\beta}{2}(1-\rho_1)}
\lesssim T_0^{-\rho_1\frac {1-\beta}{3\beta-1}}\left(\frac{t}{T_0}+1\right)^{\frac {1-\beta}{2}(1-\rho_1)},
\end{equation*}
and so, possibly choosing a smaller $\epsilon>0$,
\begin{equation}\label{on:lam}
\left| \ell(t) - \left(\frac{t}{T_0}+1\right)^{\frac {1-\beta}2}\right|
\lesssim (t+T_0 )^{-\epsilon}.
\end{equation}
This justifies \eqref{def:ell} for $T_\delta=T_0$.
Moreover, it follows from \eqref{eq:eta} that
\begin{align*}
\|w\|_{L^2}&\lesssim \|\eta\|_{L^2}\lesssim t_0^{-\frac14(3\beta-1)}\lesssim T_0^{-\frac 12},\\
\|\partial_x w\|_{L^2} &\lesssim
\lambda_0 \|\partial_x\eta\|_{L^2}\lesssim \lambda_0 t_0^{-\frac{1+\beta}4}\lesssim T_0^{-\frac 12}.
\end{align*}
Therefore, for arbitrary small $\delta>0$, it is enough to choose $x_0=x_0(\delta)$ large enough and take $T_{\delta}=T_0$, so that
in particular $T_{\delta}^{-1/2}\ll \delta$, which implies the first estimate in \eqref{def:eta}.
Finally, by the definition of $w(t,x)$ and then \eqref{est:eta:right_ext}, one has
\begin{equation*}
\int_{x\geq \frac12x(t)} w^2(t,x) dx
=\int_{y\geq \frac 12\sigma(\lambda_0^3 t+t_0)} \eta^2(\lambda_0^3 t+t_0, y) dy
\lesssim (\lambda_0^3t+t_0)^{-\frac {3\beta-1}{2}}
\lesssim T_0 \left(\frac {t}{T_0}+1\right)^{-\frac {3\beta-1}2}
\end{equation*}
and similarly,
\begin{multline*}
\int_{x\geq \frac12x(t)} (\partial_x w)^2(t,x) dx
=\lambda_0^2 \int_{y\geq \frac 12\sigma(\lambda_0^3 t+t_0)} (\partial_x\eta)^2(\lambda_0^3 t+t_0, y) dy\\
\lesssim \lambda_0^2 (\lambda_0^3t+t_0)^{-\frac {3\beta-1}{2}}
\lesssim \lambda_0^2 T_0 \left(\frac {t}{T_0}+1\right)^{-\frac {3\beta-1}2}.
\end{multline*}
These estimates complete the proof of \eqref{def:eta}.
\begin{remark}\label{rk:T0}
Estimates \eqref{on:sig}-\eqref{on:lam}
for $T_\delta=T_0\gg 1$ describe the behavior of the parameters both for large times and for intermediate times.
Indeed, by continuous dependence of the solution with respect to the initial data, since
$\tilde u(t,x)=Q(x-t)$ is a solution, it is clear that $T_\delta\to +\infty$ as $\delta\to 0$, and that for
$0<t\ll T_\delta$, the solution $u(t)$ behaves like $\tilde u(t)$.\end{remark}
\subsection{Non-scattering solutions}\label{S:6.5}
We prove that the solution $U$ constructed in Theorem~\ref{th:2} does not behave in $L^2$ as $t\to +\infty$ like a solution of the linear Airy equation.
For the sake of contradiction, assume that
there exists $v_0\in L^2$ such that
defining $v(t,x)$ the solution of
\begin{equation*}
\partial_t v + \partial_x^3 v= 0,\quad v(0)=v_0,
\end{equation*}
it holds
\begin{equation}\label{scat}
\lim_{t\to+\infty}\|U(t)-v(t)\|_{L^2} =0.
\end{equation}
We perform a monotonicity argument on $v$, similar to the one in \S\ref{S.5.4}.
Let $t\geq 0$.
For the same function $ \xi(\tau,x)=\chi\left(\frac{4x-\sigma(t)}{\sigma(\tau)}-2\right)$, define
\begin{equation*}
L(t)=\int v^2(\tau,x)\xi(\tau,x) dx
\end{equation*}
Then it follows from simple computations and \eqref{est:xi3} that
\begin{equation*}
\frac {dL}{d\tau }
=-3 \int (\partial_x v)^2 \partial_x \xi+\int v^2\partial_\tau\xi +\int v^2 \partial_x^3 \xi \lesssim \tau^{-3\beta}
\int v_0^2.
\end{equation*}
Let $ t_0>0$ be such that $\sigma(t_0)=\frac 1{8} \sigma(t)$ and $t_0\gtrsim t$.
Integrating on $[ t_0,t]$, and using the properties of $\chi$, we have
\begin{equation*}
\int_{x\geq \frac{\sigma(t)}2} v^2(t,x) dx
\leq L(t)\leq L( t_0)+c t^{- 3\beta+1}
\lesssim \int_{x\geq 2\sigma(t_0)} v^2(t_0,x) dx + t^{- 3\beta+1}.
\end{equation*}
As $t\to +\infty$, by \eqref{scat} and then \eqref{eq:eta:right}, \eqref{sur:lat}, \eqref{sur:sit},
it holds
\begin{align*}
\int_{x\geq 2\sigma(t_0)} v^2(t_0,x) dx
&\lesssim \int_{x\geq 2\sigma(t_0)} U^2(t_0,x) dx +o(1)
\lesssim \int_{x\geq 2\sigma(t_0)} A ^2(t_0,x) dx +o(1)\\
&\lesssim \int_{y\geq \frac{\sigma(t_0)}{\lambda(t_0)}} Q^2(y)dy + o(1)
\lesssim e^{-ct^{\frac{3\beta-1}2}}+o(1)=o(1).
\end{align*}
Thus,
\begin{equation*}
\lim_{t\to +\infty} \int_{x\geq \frac{\sigma(t)}2} v^2(t,x) dx =0,
\end{equation*}
but this is contradictory with \eqref{scat} since from \eqref{for:U} and \eqref{est:eta:right_ext}
\begin{equation*}
\lim_{t\to +\infty} \int_{x\geq \frac{\sigma(t)}2} U^2(t,x) dx =\int Q^2.
\end{equation*}
|
2,869,038,154,783 | arxiv | \section{Introduction}
In the interstellar medium, the hot molecular cores exhibit the rich chemical complexity and nursery of simple and complex molecular gases ({\color{blue}Herbst \& van Dishoeck 2009}). The hot molecular core regions are chemically rich which has a big influence on how the interstellar medium evolves ({\color{blue}Tan et al. 2014; van Dishoeck \& Black et al. 1998}). Several complex organic molecules were detected towards the many hot molecular cores in past ({\color{blue}Herbst \& van Dishoeck 2009}). The complex saturated organic molecules like \ce{CH3OCHO}, CH$_{3}$OCH$_{3}$, CH$_{3}$CH$_{2}$OH, CH$_{2}$CHCN, and \ce{C2H5CN} have detected in many major hot molecular core regions like Orion-KL, Sagittarius B2, and W51 after several years of observations {\color{blue}(Cummins et al. 1986; Turner 1989; Turner \& Steimle 1985)}. These saturated molecules were first studied towards the hot molecular cores using a large beam single-dish radio telescope where the location and emission lines were not well constrained. The emission lines of complex molecular species usually arise at $\le$ 0.2 pc diameter compact hot molecular core regions as found by different interferometric arrays.
\begin{table*}{}
\scriptsize
\caption{LTE fitted line parameters of the observed molecular lines towards IRAS 18566+040
}
\begin{adjustbox}{width=1\textwidth}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Species&Frequency&Transition&$E_{u}$$^{\color{blue}a}$&$A_{ij}$$^{\color{blue}b}$&($N$)$^{\color{blue}c}$ & $T_{ex}$$^{\color{blue}d}$&Optical depth &FWHM&V$_{LSR}$ &Remark\\
& [GHz]&[${\rm J^{'}_{K_a^{'}K_c^{'}}}$--${\rm J^{''}_{K_a^{''}K_c^{''}}}$] &[K]&[s$^{-1}$] & [cm$^{-2}$] &[K]&[$\tau$] &[km s$^{-1}$]&[km s$^{-1}$]&\\
\hline
C$_{2}$H$_{5}$CN&98.6102&11(3,9)--10(3,8)&38.39&7.32$\times$10$^{-5}$&(3.50$\pm$0.48)$\times$10$^{15}$&150$\pm$2.1&2.58$\times$10$^{-2}$&4.50$\pm$0.18&83.50&Non blend\\
C$_{2}$H$_{5}$CN&98.7011&11(3,8)--10(3,7)&38.40&7.35$\times$10$^{-5}$&(3.50$\pm$0.52)$\times$10$^{15}$&150$\pm$2.7&2.070$\times$10$^{-2}$&4.50$\pm$0.15&83.50&Non blend\\
C$_{2}$H$_{5}$CN&99.6815&11(2,9)--10(2,8)&33.00&7.90$\times$10$^{-5}$&(3.50$\pm$0.58)$\times$10$^{15}$&150$\pm$3.1&2.263$\times$10$^{-2}$&4.50$\pm$0.09&83.50&Non blend\\
\hline
CH$_{3}$OCHO&98.6069&J = 8(3,6)--7(3,5), F = 1--1&27.26&1.20$\times$10$^{-5}$&(1.55$\pm$0.31)$\times$10$^{16}$&150$\pm$1.5&6.660$\times$10$^{-3}$&4.50$\pm$0.25&85.0&Non blend\\
CH$_{3}$OCHO&98.6112&J = 8(3,6)--7(3,5), F = 0--0&27.24&1.20$\times$10$^{-5}$&(1.55$\pm$0.27)$\times$10$^{16}$&150$\pm$1.3&6.67$\times$10$^{-3}$&4.50$\pm$0.27&85.0&Non blend\\
CH$_{3}$OCHO&98.6826&J = 8(4,5)--7(4,4), F = 0--0&31.89&1.05$\times$10$^{-5}$&(1.55$\pm$0.22)$\times$10$^{16}$&150$\pm$2.5&5.660$\times$10$^{-3}$&4.50$\pm$0.35&85.0&Non blend\\
CH$_{3}$OCHO&98.7120&J = 8(4,5)--7(4,4), F = 1--1&31.90&1.02$\times$10$^{-5}$&(1.55$\pm$0.38)$\times$10$^{16}$&150$\pm$2.6&5.470$\times$10$^{-3}$&4.50$\pm$0.21&85.0&Non blend\\
CH$_{3}$OCHO&98.7479&J = 8(4,4)--7(4,3), F = 2--2&31.91&1.02$\times$10$^{-5}$&(1.55$\pm$0.33)$\times$10$^{16}$&150$\pm$2.1&5.478$\times$10$^{-3}$&4.50$\pm$0.29&85.0&Non blend\\
CH$_{3}$OCHO&98.7923&J = 8(4,4)--7(4,3), F = 0--0&31.89&1.05$\times$10$^{-5}$&(1.55$\pm$0.42)$\times$10$^{16}$&150$\pm$1.8&5.668$\times$10$^{-3}$&4.50$\pm$0.32&85.0&Non blend\\
CH$_{3}$OCHO&100.2946&J = 8(3,5)--7(3,4), F = 2--2&27.41&1.26$\times$10$^{-5}$&(1.55$\pm$0.29)$\times$10$^{16}$&150$\pm$1.6&6.7850$\times$10$^{-3}$&4.50$\pm$0.34&85.0&Non blend\\
CH$_{3}$OCHO&100.3082&J = 8(3,5)--7(3,4), F = 0--0&27.40&1.26$\times$10$^{-5}$&(1.55$\pm$0.36)$\times$10$^{16}$&150$\pm$1.9&6.791$\times$10$^{-3}$&4.50$\pm$0.29&85.0&Non blend\\
\hline
CH$_{2}$CO&100.0945&5(1,5)--4(1,4)&27.48&1.02$\times$10$^{-5}$&(2.10$\pm$0.8)$\times$10$^{15}$&150$\pm$2.7&1.59$\times$10$^{-2}$&4.50$\pm$0.18&84.0&Non blend\\
\hline
HC$_{3}$N&100.0763&11--10&28.32&7.74$\times$10$^{-5}$&(2.10$\pm$0.6)$\times$10$^{15}$&150$\pm$2.65&4.19$\times$10$^{-1}$&4.50$\pm$0.28&84.0&Non blend\\
\hline
SO&99.2998&2(3)--1(2)&9.23&1.15$\times$10$^{-5}$&(1.80$\pm$0.75)$\times$10$^{16}$&100$\pm$1.35&3.77$\times$10$^{-1}$&4.50$\pm$0.12&84.0&Non blend\\
SO&100.0296&5(4)--4(4)&38.58&1.10$\times$10$^{-6}$&(1.80$\pm$0.72)$\times$10$^{16}$&100$\pm$1.26&3.43$\times$10$^{-2}$&4.50$\pm$0.39&84.0&Non blend\\
\hline
H$_{2}$$^{34}$CS&99.7740&3(1,3)--2(1,2)&22.76&1.20$\times$10$^{-5}$&(5.20$\pm$0.28)$\times$10$^{14}$&150$\pm$2.59&4.89$\times$10$^{-3}$&4.50$\pm$0.32&85.5&Non blend\\
\hline
HC$^{13}$CCN&99.6518&11(10)--10(9)&28.70&7.57$\times$10$^{-5}$&(1.10$\pm$0.39)$\times$10$^{14}$&150$\pm$2.19&6.59$\times$10$^{-3}$&4.50$\pm$0.26&84.0&Non blend\\
\hline
HCC$^{13}$CN&99.6614&11(10)--10(9)&28.70&7.57$\times$10$^{-5}$&(1.10$\pm$0.45)$\times$10$^{14}$&150$\pm$1.97&6.56$\times$10$^{-3}$&4.50$\pm$0.59&84.0&Non blend\\
\hline
CH$_{3}$CHO&98.8633&J = 5(1,4)--4(1,3), F=2--2&16.59&2.99$\times$10$^{-5}$&(2.40$\pm$0.95)$\times$10$^{15}$&150$\pm$2.78&5.911$\times$10$^{-3}$&4.50$\pm$0.16&84.0&Non blend\\
CH$_{3}$CHO&98.9009&J = 5(1,4)--4(1,3), F=0--0&16.51&2.99$\times$10$^{-5}$&(2.40$\pm$0.89)$\times$10$^{15}$&150$\pm$2.69&8.629$\times$10$^{-3}$&4.50$\pm$0.46&84.0&Non blend\\
\hline
\end{tabular}
\end{adjustbox}
$^{\color{blue}a}$$E_{u}$ denoted the upper energy of the detected molecular emission lines.\\
$^{\color{blue}b}$$A_{ij}$ denoted the Einstain coefficient of detected molecular emission lines.\\
$^{\color{blue}c}$$N$ denoted column density of the detected molecular emission lines.\\
$^{\color{blue}d}$It is denoted excitation temperature which is define as $T_{ex} = \frac{h\nu / k}{\ln \left( \frac{n_l \, g_u}{n_u \, g_l} \right)}$ where $g_l$ and $g_u$ represented the statistical weights but $n_l$ and $n_u$ represented the number of particles in upper and lower state.\\
\label{tab:LTE}
\end{table*}
The emission lines of complex organic molecules mainly arise from the core of the hot molecular cloud and these molecular cores also contain high-velocity \ce{H2O} masers. These physical properties were also observed in some common hot molecular cores like Orion-KL, Sgr B2, G34.3+0.2, and W51 {\color{blue}(Wright et al. 1996; Miao et al. 1995; Mehringer \& Snyder 1996; Zhang et al. 1998)}. A huge amount of dust particles were observed around the hot molecular cores which were observed only in millimeter-wavelength interferometric observations. In the hot molecular core, grain-surface chemistry plays an important role in the formation of the complex organic molecules {\color{blue}(Caselli et al. 1993)}.
The hot molecular core candidate IRAS 18566+0408 was located at a 6.7 kpc distance ({\color{blue}Araya et al. 2004; Molinari et al. 1996}). The maser water (H$_{2}$O) and methanol (\ce{CH3OH}) at frequency 22 GHz and 6.7 GHz were strongly evident in the hot molecular core IRAS 18566+0408 ({\color{blue}Beuther et al. 2002}). The weak radio continuum emission in centimeter wavelength was also detected from the dust around the IRAS 18566+0408 {\color{blue}(Araya et al. 2005)}. The hot molecular core region IRAS 18566+0408 was classified as a massive disk candidate {\color{blue}(Zhang 2005)}. The maser H$_{2}$CO at 6 cm wavelength was found from the IRAS 158566+0408 {\color{blue}(Araya et al. 2005)}. Earlier, the maser H$_{2}$CO at wavelength 6 cm were also found toward five regions in the Milky way, molecular clouds, and toward Orion BN/KL {\color{blue}(Araya et al. 2002; Araya et al. 2006)}. The radio continuum emission of IRAS 18566+0408 was resolved with the VLA in the 1.3 and 6 cm wavelengths and showed four components which were consistent with an ionized jet {\color{blue}(Hofner et al. 2017)}. The emission lines of NH$_{3}$ with transition J = 1,1 and J = 2,2 were detected by the single-dish radio telescope towards IRAS 18566+0408 {\color{blue}(Miralles et al. 1994; Molinari et al. 1996; Sridharan et al. 2002)} and later VLA imaged these species using interferometric technique {\color{blue}(Zhang et al. 2007)}. Recently, CH$_{3}$CN, CH$_{3}$OH, OCS, $^{12}$CO, $^{13}$CO, and SO were detected towards IRAS 18566+0408 using Submillimeter Array (SMA) {\color{blue}(Silva et al. 2017)}.
\begin{figure*}
\centering
\scriptsize
\includegraphics[width=0.94\textwidth]{eth.pdf}
\caption{ALMA detection of unblended rotational emission lines of C$_{2}$H$_{5}$CN between the frequency range of 85.64--100.42 GHz with their different transitions towards the hot molecular core IRAS 18566+0408. The properties of detected emission lines of \ce{C2H5CN} and their spectral fitting parameters were shown in Tab.~\ref{tab:LTE}. The continuum emission has been completely subtracted from the emission spectrum. In the emission spectrum, the black colour line showed the observed transitions of C$_{2}$H$_{5}$CN, while the red colour line showed the synthetic spectra obtained from the best fitting of LTE model. The systematic velocity (V$_{\text LSR}$) of the line was $\sim$83.50 km s$^{-1}$. The best fitted $\chi^2$ value for the emission lines of \ce{C2H5CN} was $\sim$0.92.}
\label{fig:eth}
\end{figure*}
The interstaller complex organic nitrile molecule ethyl cyanide (\ce{C2H5CN}) was also known as propanenitrile or propionitrile. Earlier, this complex species has been observed mainly in hot molecular clouds in the frequency range 40--950 GHz {\color{blue}(Johnson et al. 1977; Schilke et al. 2001; White et al. 2003)}. The Berkeley-Illinois-Maryland Association (BIMA) array detected the partially blended emission lines of C$_{2}$H$_{5}$CN in Sgr B2 (N-LMH) with estimated column density 6$\times$10$^{16}$ cm$^{-2}$ {\color{blue}(Miao et al. 1995; Mehringer \& Snyder 1996)}. Earlier, {\color{blue}Miao \& Snyder (1997)} created first full synthesis imaging of \ce{C2H5CN} from Sgr B2 using BIMA Array and NRAO 12 m telescope and they estimated the column density of \ce{C2H5CN} was $\sim$9.6$\times$10$^{17}$ cm$^{-2}$. The column densities of C$_{2}$H$_{5}$CN in Orion-KL and G34.3+0.2 were 3$\times$10$^{16}$ and 3$\times$10$^{15}$ cm$^{-2}$ respectively {\color{blue}(Wright et al. 1996; Mehringer \& Snyder 1996)}. Earier, three $^{13}$C isotopologues of ethyl cyanide, $^{13}$CH$_{3}$CH$_{2}$CN, CH$_{3}$$^{13}$CH$_{2}$CN, and CH$_{3}$CH$_{2}$$^{13}$CN were observed from the Orion hot molecular cloud in the frequency range 80--40 GHz and 160--360 GHz {\color{blue}(Demyk et al. 2007)}. Recently, the emission lines of C$_{2}$H$_{5}$CN were detected in the atmosphere of Saturn largest moon Titan between the frequency 222--241 GHz using ALMA with vertical column density (1--5)$\times$10$^{15}$ cm$^{-2}$ {\color{blue}(Cordiner et al. 2014)}. We reported the first detection of the C$_{2}$H$_{5}$CN towards the hot molecular core region IRAS 18566+0408 using ALMA.
Alongside ethyl cyanide (C$_{2}$H$_{5}$CN), the complex organic molecule methyl formate (CH$_{3}$OCHO) was one of the most abundant organic molecular species which was specially found in both high mass and low mass star formation regions ({\color{blue}Cazaux et al. 2003; Brown et al. 1975}). In the interstellar medium, methyl formate (CH$_{3}$OCHO) was one of the complex organic molecule which was known as an example of ester. In the hot molecular clouds, methyl formate (\ce{CH3OCH3}) was one of the known molecule which was also responsible for the formation of multiple rotational transitions of other ester types compounds. Methyl formate was first detected towards the Sgr B2(N) ({\color{blue}Brown et al. 1975}). Earlier, many chemical formation models in the interstellar medium indicated the complex molecule methyl formate was formed after the evaporation of the methanol (\ce{CH3OH}) from grain mantle in the hot molecular cores \citep{miller91}. \citet{gar08} presented the formation mechanism of methyl formate in gas-grain interaction but how to produce methyl formate in the interstellar medium using gas-phase reactions is not well understood. The methyl formate plays a major role in the formation of biopolymers. In G31.41+0.31 hot molecular cloud, the column density of CH$_{3}$OCHO was 3.4$\times$10$^{18}$ cm$^{-2}$ ({\color{blue}Isokoski, Bottinelli \& van Dishoeck. 2013}). Earlier, CH$_{3}$OCHO was also detected in the low mass protostar IRAS 16293--2422 ({\color{blue}Cazaux et al. 2003}). The emission lines of methyl formate were also observed in a solar-type star-forming region L 1157-B1, where the molecular outflow interacts with dense clouds \citep{ari08}. Later, \citet{left17} claimed that the solar-type star-forming region L 1157-B1 was the factory of complex organic molecules. Earlier, \citet{sak06} detected the emission lines of \ce{CH3OCHO} from NGC 1333 IRAS 4B, and these molecules can be used as a tracer in the hot molecular core regions. Therefore, \ce{CH3OCHO} was an important molecule in the grain surfaces of hot corinos and hot molecular cores. We detected \ce{CH3OCHO} first time in the hot molecular core candidate IRAS 18566+0808.
In this article, we presented the first interferometric detections of \ce{C2H5CN} and \ce{CH3OCHO} between the frequency range of 85.64--100.4 GHz in the hot molecular core region IRAS 18566+0408 using ALMA band 3 observation.
Additionally, we also detected the emission lines of simple organic molecules SO, \ce{CH2CO}, \ce{HC3N}, H$_{2}$$^{34}$CS, HC$^{13}$CCN, HCC$^{13}$CN, and CH$_{3}$CHO. In Sect.~\ref{obs}, we discussed the observations and data reductions. The result of the detection of ethyl cyanide, methyl formate, and other detected simple molecules was shown in Sect.~\ref{res}. The discussion and conclusion were presented in Sect.~\ref{con} and \ref{conclusion}.
\begin{figure*}
\centering
\scriptsize
\includegraphics[width=0.96\textwidth]{CH3OCHO.pdf}
\caption{ALMA detection of unblended rotational emission lines of CH$_{3}$OCHO between the frequency range of 85.64--100.42 GHz with their different transitions towards the hot molecular core IRAS 18566+0408. The properties of detected emission lines of \ce{CH3OCHO} and their spectral fitting parameters were shown in Tab.~\ref{tab:LTE}. The continuum emission has been completely subtracted from the emission spectrum. In the emission spectrum, the black colour line showed the observed transitions of CH$_{3}$OCHO, while the blue colour line showed the synthetic spectra obtained from the best fitting of the LTE model. The systematic velocity ($V_{\text LSR}$) of the line was $\sim$85.0 km s$^{-1}$. The best fitted $\chi^2$ value for the emission lines of \ce{CH3OCHO} was $\sim$0.869.}
\label{fig:meth}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.96\textwidth]{oth.pdf}
\caption{ALMA detection of rotational emission lines of CH$_{2}$CO, HC$_{3}$N, SO, H$_{2}$$^{34}$C, HC$^{13}$CCN, HCC$^{13}$CN, and CH$_{3}$CHO between the frequency range of 85.64--100.42 GHz with their different transitions towards the hot molecular core IRAS 18566+0408. The properties of detected molecules and their spectral fitting parameters were shown in Tab.~\ref{tab:LTE}. The continuum emission has been completely subtracted. In the emission spectrum, the black colour line showed the observed transitions of detected molecules, while the red colour line showed the synthetic spectra obtained from the best fitting of LTE model.}
\label{fig:othr}
\end{figure*}
\begin{figure}
\centering
\scriptsize
\includegraphics[width=0.5\textwidth]{ethyl.pdf}
\caption{Integrated emission map of C$_{2}$H$_{5}$CN in the hot molecular core region IRAS 18566+0408 at frequency 98.6102 GHz which was obtained after combining all the calibrated visibility data. The corresponding synthesized beam of the image was 2.383$^{\prime\prime}\times$1.613$^{\prime\prime}$. The cyan color circles indicated the synthesized beam of the integrated emission map. The contour levels started at 0.5 mJy beam$^{-1}$ (3$\sigma$) increasing by a factor of $\surd$2. }
\label{fig:map}
\end{figure}
\section{Observations and data reduction}
\label{obs}
The interferometric millimeter-wavelength observation of hot molecular core or high mass star-forming region IRAS 18566+0408 was performed with Atacama Large Millimeter/Submillimeter Array (ALMA)\footnote{\href{https://almascience.nao.ac.jp/asax/}{https://almascience.nao.ac.jp/asax/}} using the band 3 (frequency range 84--116 GHz) observation. The observed phase center of IRAS 18566+0408 was $\alpha_{J2000}$: 18:59:10.000 and $\delta_{J2000}$: +04:12:16.000. During the observation, XX, YY, and XY-type signal correlators were used via the integration time 1360.800 s. The observation of the complex molecular spectral lines towards IRAS 18566+0408 was performed with band 3 having eight spectral bands covering the sky frequencies of 85.64--85.70 GHz, 86.29--86.35 GHz, 86.71--86.77 GHz, 86.80--86.86 GHz, 88.59--88.64 GHz, 89.14--89.20 GHz, 97.93--97.99 GHz, and 98.54--100.42 GHz. The corresponding spectral resolution of the interferometric data was 122.07 kHz, 121.15 kHz, 122.01 kHz, 1128.91 kHz respectively. The observation was done on 24-March-2016. During the observation, a total of thirty-six antennas were used to study the complex molecular lines from IRAS 18566+0408. The solar planet Neptune was taken as flux calibrator, J1924--2914 was taken as bandpass calibrator, and J1830+0619 was taken as phase calibrator. The systematic velocity ($V_{LSR}$) of IRAS 18566+0408 was known to be $\sim$84.5 km s$^{-1}$ ({\color{blue}Silva et al. 2017}).
\begin{figure}
\centering
\scriptsize
\includegraphics[width=0.5\textwidth]{meth.pdf}
\caption{Integrated emission map of CH$_{3}$OCHO in the hot molecular core region IRAS 18566+0408 at frequency 98.7120 GHz which was obtained after combining all the calibrated visibility data. The corresponding synthesized beam of the image was 2.391$^{\prime\prime}\times$1.610$^{\prime\prime}$. The cyan color circles indicated the synthesized beam of the resultant integrated emission map. The contour levels started at 0.5 mJy beam$^{-1}$ (3$\sigma$) increasing by a factor of $\surd$2.}
\label{fig:map1}
\end{figure}
We used the Common Astronomy Software Application ({\tt CASA 5.4.1})\footnote{\href{https://casaguides.nrao.edu/}{https://casaguides.nrao.edu/}} for initial data reduction and spectral imaging with the standard data reduction pipeline delivered by the ALMA observatory {\color{blue}(McMullin et al. 2007)}. The continuum flux density of the flux calibrator Neptune for each baseline was scaled and matched with Butler-JPL-Horizons 2012 flux calibrator model with 5\% accuracy using task {\tt SETJY} {\color{blue}(Butler 2012)}. Initially, we calibrated the bandpass and flux by flagging the bad data using {CASA} pipeline with task {\tt hifa\_flagdata} and {\tt hifa\_bandpassflag}. After the initial data reduction, we split the target data using task {\tt MSTRANSFORM} with rest frequency in each spectral windows. We used the task {\tt UVCONTSUB} for the continuum subtraction procedure from the UV plane in each raw data. After the continuum subtraction, we made the spectral image of IRAS 18566+0408 using task {\tt TCLEAN} with the rest frequency of each spectral windows.
\begin{figure*}
\includegraphics[width=0.5\textwidth]{1.png}\includegraphics[width=0.5\textwidth]{2.png}
\caption{Rotational diagram of \ce{C2H5CN} and \ce{CH3OCHO}. The black squares indicated the original statistical data points and the error bar indicated the red lines. The best-fitted column density and rotational temperature were shown on the upper right side of the image. The blue line indicated the best fitted single power-law over statistical data points.}
\label{fig:rotd}
\end{figure*}
\section{Results}
\label{res}
\subsection{Analysis of molecular emission lines towards the hot molecular core IRAS 18566+0408}
For the identification of the molecular lines towards the hot molecular core candidate IRAS 18566+0408, we used CASSIS (developed by IRAP-UPS/CNRS) with Cologne Database for
Molecular Spectroscopy (CDMS)\footnote{\href{https://cdms.astro.uni-koeln.de/cgi-bin/cdmssearch}{https://cdms.astro.uni-koeln.de/cgi-bin/cdmssearch}} \citep{mu05} and Jet Propulsion Laboratory (JPL)\footnote{\href{https://spec.jpl.nasa.gov/}{https://spec.jpl.nasa.gov/}} \citep{pic98} spectroscopic molecular databases. We detected three strong unblended rotational emission lines of C$_{2}$H$_{5}$CN and eight strong rotational emission lines of CH$_{3}$OCHO between the frequency range of 85.64--100.42 GHz in the hot molecular core region IRAS 18566+0408. The observed emission spectra of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO was extracted from ALMA spectral data cubes to make a $8.9^{''}$ diameter circular region centred at RA (J2000) = (18$^{h}$59$^{m}$09$^{s}$.92), Dec (J2000) = (4$^\circ$12$^{\prime}$15$^{\prime\prime}$.58). In Fig.~\ref{fig:eth} and \ref{fig:meth}, rotational molecular emission lines of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO were shown. The spectral line parameters like upper energy ($E_u$), quantum numbers ({${\rm J^{'}_{K_a^{'}K_c^{'}}}$--${\rm J^{''}_{K_a^{''}K_c^{''}}}$}), Full-Width Half Maximum (FWHM) and LSR velocity ($V_{LSR}$) of each transition were estimated after fitting a single Gaussian model over the observed transition of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO were shown in Tab.~\ref{tab:LTE}. We also created the integrated emission map of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO at frequency 98.6102 GHz and 98.7120 GHz which were shown in Fig.~\ref{fig:map} and \ref{fig:map1}. The integrated emission maps clearly indicated that the emission lines of complex organic molecules \ce{C2H5CN} and \ce{CH3OCHO} were mainly arose from the warm inner region of the IRAS 18566+0408.
Except for \ce{C2H5CN} and \ce{CH3OCHO}, we also observed the evidence of some simple organic molecules in IRAS 18566+0408. We have observed molecular emission lines of simple organic molecules SO, \ce{CH2CO}, \ce{HC3N}, H$_{2}^{34}$CS, HC$^{13}$CCN, HCC$^{13}$CN, and CH$_{3}$CHO. The molecular spectrum of other detected molecules was shown in Fig.~\ref{fig:othr} and corresponding spectral properties were shown in Tab.~\ref{tab:LTE}. Between the observable frequency range, we detected the hyperfine transition lines only for CH$_{3}$CHO and SO but other molecules like \ce{CH2CO}, \ce{HC3N}, H$_{2}^{34}$CS, HC$^{13}$CCN, and HCC$^{13}$CN detected with single transition line. Earlier, {\color{blue}Silva et al. (2017)} detected the single transition emission line of SO with transition J = 6(5)--5(4) towards IRAS 18566+0408 using Submillimeter Array (SMA) and he postulated that SO is coming from an outflow of gases. We also detected the dual transition emission lines of SO with transitions J = 2(3)--1(2) and 5(4)--4(4) using ALMA but we have not detected any signature of the outflow of SO towards IRAS 18566+0408. The emission lines of \ce{CH2CO}, \ce{HC3N}, H$_{2}^{34}$CS, HC$^{13}$CCN, HCC$^{13}$CN, and CH$_{3}$CHO detected the first time towards IRAS 18566+0408.
\subsection{Spatial distribution of ethyl cyanide and methyl formate towards IRAS 18566+0408}
\label{source}
We calculated the source size or emitting diameter ($\theta_{S}$) of \ce{C2H5CN} and \ce{CH3OCHO} by fitting the 2D Gaussian of the integrated emission map of \ce{C2H5CN} and \ce{CH3OCHO} which was shown in Fig.~\ref{fig:eth} and \ref{fig:meth}. The source size is one of the important parameter during the fitting of the LTE model using CASSIS. The deconvolved beam size of the emitting region was calculated by the following equation\\
\begin{equation}
\theta_{S}=\sqrt{\theta^2_{50}-\theta^2_{beam}}
\end{equation}
where $\theta^2_{50} = 2\sqrt{A/\pi}$ was the diameter of the circle whose area ($A$) was enclosing $50\%$ line peak and $\theta_{beam}$ was the half-power width of the synthesized beam \citep{riv17, mondal21}. We observed that the emission lines of \ce{C5H5CN} and \ce{CH3OCHO} have a peak at the position of the continuum. The estimated emitting diameter or source size ($\theta_{S}$) of the complex molecule \ce{C2H5CN} and \ce{CH3OCHO} was $\sim$1.20$^{\prime\prime}$. Our result clearly showed that the emitting diameters of the different detected transitions of \ce{C2H5CN} and \ce{CH3OCHO} were less than the beam size of the emission map. This implied that the detected transitions of \ce{C2H5CN} and \ce{CH3OCHO} were not well spatially resolved.
\subsection{Calculation of column density ($N$) and rotational temperature ($T_{rot}$) using rotational diagram analysis}
\label{rotd}
We used the rotational diagram method to obtain the column density ($N$) in cm$^{-2}$ and rotational temperature ($T_{rot}$) in K of detected emission lines of \ce{C2H5CN} and \ce{CH3OCHO} in hot molecular core IRAS 18566+0408. We used the rotational diagram method because we assumed that the observed species were optically thin and they satisfied the Local Tharmal Eqilibrium (LTE) condition. The column density of optically thin lines can be written as \citep{gold99},
\begin{equation}
{N_u^{thin}}=\frac{3{g_u}k_B\int{T_{mb}dV}}{8\pi^{3}\nu S\mu^{2}}
\end{equation}
where, $k_B$ is the Boltzmann constant, $\rm{\int T_{mb}dV}$ is the integrated intensity, $\mu$ is the electric dipole moment, $g_u$ is the degeneracy of the upper state, $\nu$ is the rest frequency, and the strength of the transition lines were indicated by $S$. Under the LTE conditions, the total column density of detected species can be written as,
\begin{equation}
\frac{N_u^{thin}}{g_u} = \frac{N_{total}}{Z(T_{rot})}\exp(-E_u/k_BT_{rot})
\end{equation}
where ${Z(T_{rot})}$ is the partition function at extracted rotational temperature, $T_{rot}$ is the rotational temperature, and $E_u$ is the upper energy of the observed molecular lines. The Equation 3 can be rearranged as,
\begin{equation}
ln\left(\frac{N_u^{thin}}{g_u}\right) = ln(N)-ln(Z)-\left(\frac{E_u}{k_BT_{rot}}\right)
\end{equation}
Equation 4 demonstrated the linear relationship between calculated column density and upper energy of the detected complex molecules. Using equation 4, we can estimate the rotational temperature and column density of the detected molecular species. During the rotational diagram analysis, we extracted the line parameters like FWHM, upper energy ($E_u$), line intensity, and integrated intensity ($\int T_{mb}dV$) using a single Gaussian fitting over the originally observed transition of \ce{C2H5CN} and \ce{CH3OCHO}. After the rotational diagram analysis, we found the column density of ethyl cyanide was $N$(C$_{2}$H$_{5}$CN) = (3.50$\pm$0.68)$\times$10$^{15}$ cm$^{-2}$ with rotational temperature $T_{rot}$ = 150$\pm$2.5 K. For methyl formate, the column density was $N$(CH$_{3}$OCHO) = (1.55$\pm$0.31)$\times$10$^{16}$ cm$^{-2}$ with rotational temperature $T_{rot}$ = 150$\pm$2.8 K. All detected transitions of \ce{C2H5CN} and \ce{CH3OCHO} were shown in Tab.~\ref{tab:LTE} and the computed rotational diagram was shown in Fig.~\ref{fig:rotd}. In the rotational diagram, the vertical red color error bars were the absolute uncertainty of $ln(N_{u}$/g$_{u}$) and it was created from the error of the observed $\int T_{mb}dV$ which was measured using the fitting of single Gaussian model over observed transitions of \ce{C2H5CN} and \ce{CH3OCHO}.
\subsection{Fitting of LTE model using Markov Chain Monte Carlo (MCMC) algorithm}
After the identification of emission lines of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO using the CDMS and JPL molecular databases, we used the Markov Chain Monte Carlo (MCMC) method in the CASSIS python interface for fitting the Local Thermodynamic Equilibrium (LTE) model over the original transitions of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO to calculate the column density ($N$) in cm$^{-2}$, excitation temperature ($T_{ex}$) in K, optical depth ($\tau$) and FWHM in km s$^{-1}$ of the resultant emission spectrum. We applied the LTE method because we assumed that the observed transitions of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO were optically thin and they are populated under all LTE conditions. After the fitting of the LTE model over the observed molecular emission spectrum, the best fit reduced $\chi^2$ value was calculated between the observed and simulated data, and the following equation was used to decrease the $\chi^2$ value
\begin{equation}
{ \chi^2_{red}=\frac{1}{ \sum_{i}^{N_{spec}} N_i} \sum_{i=1}^{N_{spec}} \sum_{j=1}^{N_i} \frac{(I_{obs,ij}-I_{model,ij})^2}{rms^2_i+(cal_i\times I_{obs,ij})^2}}
\end{equation}
where, ${I_{model,ij}}$ and ${ I_{obs,ij}}$ mean the intensity of the LTE model and original spectra in the channel $j$ of transition $i$ respectively. The calibration error is denoted by cal$_i$ and the rms of the spectrum $i$ is denoted by rms$_i$.
After the fitting of the LTE model over the original transition of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO, we estimated the resultant optical depth ($\tau$) of all emission lines were less than 1 that means the observed emission spectrum of \ce{C2H5CN} and \ce{CH3OCHO} were optically thin. We have obtained column density ($N$) in cm$^{-2}$ and excitation temperatures ($T_{ex}$) in K of \ce{C2H5CN} and \ce{CH3OCHO} by the MCMC calculations (Tab.~\ref{tab:LTE}) which were consistent with the column density ($N$) in cm$^{-2}$ and rotational temperatures ($T_{rot}$) in K of these molecules obtained by the rotational diagram method which was described in the Section \ref{rotd}. After the spectral analysis using rotational diagram and LTE fitting using MCMC method, we found that the estimated column density and gas temperature are same for both of the methods. During fitting LTE model, we used the source size
1.20$^{\prime\prime}$ (detail calculation presented in Sec.~\ref{source}). The resultant physical parameters were shown in Tab.~\ref{tab:LTE} and the resultant emission spectrum of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO with best-fitting LTE syntetic spectra using MCMC algorithm was shown in Fig.~\ref{fig:eth} and \ref{fig:meth}. The fractional abundance of ethyl cyanide and methyl formate with respect to \ce{H2} was f(C$_{2}$H$_{5}$CN) = 1.66$\times$10$^{-9}$ and f(CH$_{3}$OCHO) = 7.38$\times$10$^{-9}$ where the column density of hydrogen in IRAS 18566+0408 was $N$(H$_{2}$) = 2.1$\times$10$^{24}$ cm$^{-2}$ {\color{blue}(Hofner et al. 2017)}.
We also used the LTE model with MCMC method to estimate the physical properties of the other observed emission lines of SO, \ce{CH2CO}, \ce{HC3N}, H$_{2}$$^{34}$CS, HC$^{13}$CCN, HCC$^{13}$CN, and CH$_{3}$CHO in IRAS 18566+0408. The emission lines of SO and \ce{CH3CHO} were detected with multiple transitions, and the emission lines of \ce{CH2CO}, \ce{HC3N}, H$_{2}$$^{34}$CS, HC$^{13}$CCN, and HCC$^{13}$CN were detected with the single transition. The resultant physical parameters of the detected other molecular lines were shown in Tab.~\ref{tab:LTE} and the LTE fitting emission spectrum was shown in Fig.~\ref{fig:othr}.
\begin{table}{}
\centering
\scriptsize
\caption{Comparison of the observed molecular column density of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO in IRAS 18566+0408 with other molecular cores.}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Source&Species&Column density&Remark\\
& & [cm$^{-2}$] & \\
\hline
IRAS 18566+0408&C$_{2}$H$_{5}$CN&3.50$\times$10$^{15}$&Our work \\
G31.41+0.31 &C$_{2}$H$_{5}$CN&4.01$\times$10$^{15}$&{\color{blue}{Calcutt et al. 2014}} \\
Sgr B2 (N) &C$_{2}$H$_{5}$CN&1.20$\times$10$^{18}$&{\color{blue}{Belloche et al. 2009}} \\
Orion KL &C$_{2}$H$_{5}$CN& 2.4$\times$10$^{17}$&{\color{blue}{Friedel \& Looney 2008}} \\
\hline
IRAS 18566+0408&CH$_{3}$OCHO&1.55$\times$10$^{16}$&Our work\\
G31.41+0.31 &CH$_{3}$OCHO&6.50$\times$10$^{18}$ &{\color{blue}{Gorai et al. 2021}}\\
Sgr B2 (N) &CH$_{3}$OCHO&1.20$\times$10$^{18}$ &{\color{blue}{Belloche et al. 2016}}\\
Orion KL &CH$_{3}$OCHO&1.04$\times$10$^{17}$ &{\color{blue}{Favre et al. 2011}}\\
\hline
\end{tabular}
\label{tab:abun}
\end{table}
\section{Discussion}
\label{con}
\subsection{Comparision of ethyl cyanide (\ce{C2H5CN}) and methyl formate (\ce{CH3OCHO}) column density with IRAS 18566+0408 and other hot molecular cores}
Earlier, the complex molecule C$_{2}$H$_{5}$CN and CH$_{3}$OCHO were detected from several hot molecular clouds and we detected, for the first time, the emission lines of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO towards IRAS 18566+0408. In the interstellar medium, the column density of ethyl cyanide (\ce{C2H5CN}) towards the hot molecular cores were found in the order of 10$^{15}$--10$^{18}$ cm$^{-2}$ {\color{blue}(Mehringer \& Snyder 1996; Miao \& Snyder 1997)} and our calculated column density of ethyl cyanide towards the hot molecular core IRAS 18566+0408 was also consistent within the range. Earlier, the first blended and torsional emission lines of C$_{2}$H$_{5}$CN was found from Sgr B2(N-LMH) {\color{blue}(Mehringer et al. 2004)}. The emission line of \ce{C2H5CN} was also detected from low mass protostar IRAS 16293--2422 with column density 9$\times10^{14}$ cm$^{-2}$ {\color{blue}(Cazaux et al. 2003)}. {\color{blue}Cordiner et al. (2014)} also detected the emission lines of \ce{C2H5CN} in the atmosphere of Saturn largest moon Titan with vertical column density (1--5)$\times$10$^{15}$ cm$^{-2}$. The observed column density of ethyl cyanide towards Titan was very similar to the column density of ethyl cyanide towards the hot molecular core IRAS 18566+0408.
After the calculation of the column density of C$_{2}$H$_{5}$CN and CH$_{3}$OCHO towards IRAS 18566+0408, we compared these values with other molecular cores G31.41+0.31, Sgr B2 (N), and Orion KL. The Fig.~\ref{fig:bar} showed a bar diagram which compared the column density of \ce{C2H5CN} and \ce{CH3OCHO} between IRAS 18566+0408, G31.41+0.31, Sgr B2 (N), and Orion KL and corresponding values were shown in Tab.~\ref{tab:abun}. The bar diagram indicated that the column density of \ce{C2H5CN} in IRAS 18566+0408 was approximately similar with G31.41+0.31, $\sim$20 times lower than Orion KL, and $\sim$30 times lower than Sgr B2 (N). For the case of \ce{CH3OCHO}, column density in IRAS 18566+0408 was $\sim$10 times lower than Orion KL, and $\sim$20 times lower than G31.41+0.31 and Sgr B2 (N). The column density of \ce{C2H5CN} and \ce{CH3OCHO} was small with respect to G31.41+0.31, Sgr B2 (N), and Orion KL which implied that the production rate of complex organic molecules in IRAS 18566+0408 was small with respect to G31.41+0.31, Sgr B2 (N), and Orion KL.
\begin{figure}
\centering
\includegraphics[width=0.5\textwidth]{bar.jpeg}
\caption{Bar diagram of the comparison of column density of \ce{C2H5CN} and \ce{CH3COCH3} between the hot molecular cores IRAS 18566+0408, G31.41+0.31, Sgr B2 (N), and Orion KL. These column abundances were given in the Tab.~\ref{tab:abun} with their references.}
\label{fig:bar}
\end{figure}
\begin{table*}{}
\centering
\caption{Destruction pathways of \ce{C2H5CN} and \ce{CH3OCHO} with different types of chemical rections towards the hot molecular cores.}
\begin{adjustbox}{width=1\textwidth}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Reaction&Types&$\alpha$&$\beta$&$\gamma$\\
\hline
\ce{C2H5CN}+CRPHOT$\longrightarrow$\ce{CH2CHCNH}$^{+}$+H+e$^{-}$&Cosmic ray-induced photoreaction&1.30$\times$10$^{-17}$&0.00&1122.50\\
\hline
\ce{C2H5CN}+CRPHOT$\longrightarrow$CN+\ce{C2H5}&Cosmic ray-induced photoreaction&1.30$\times$10$^{-17}$&0.00&2388.00\\
\hline
\ce{CH4}$^{+}$+\ce{C2H5CN}$\longrightarrow$\ce{C2H5CNH}$^{+}$+CH3& Ion-neutral&4.0$\times$10$^{-9}$&--0.50&0.00\\
\hline
\ce{C2H5CN}+OH$\longrightarrow$\ce{C2H5}+HNCO&Neutral-neutral&8.81$\times$10$^{-14}$&6.00&500.00\\
\hline
\ce{C2H5CN}+OH$\longrightarrow$\ce{C2H5}+HOCN&Neutral-neutral&1.41$\times$10$^{-14}$&3.37&0.00\\
\hline
\ce{C2H5CN}+$h\nu$$\longrightarrow$\ce{CH2CHCNH}$^{+}$+H+e$^{-}$&Photoprocess&6.20$\times$10$^{-10}$&0.00&3.10\\
\hline
\ce{C2H5CN}+$h\nu$$\longrightarrow$CN+\ce{C2H5}&Photoprocess&3.40$\times$10$^{-9}$&0.00&2.00\\
\hline
C$^{+}$+\ce{CH3OCHO}$\longrightarrow$\ce{COOCH4}$^{+}$+C&Charge exchange&3.00$\times$10$^{-9}$&--0.50&0.00\\
\hline
H$^{+}$+\ce{CH3OCHO}$\longrightarrow$\ce{COOCH4}$^{+}$+H&Charge exchange&3.00$\times$10$^{-9}$&--0.50&0.00\\
\hline
\ce{CH3OCHO}+CRPHOT$\longrightarrow$\ce{CO2}+\ce{CH4}&Cosmic ray-induced photoreaction&1.30$\times$10$^{-17}$&0.00&1000.00\\
\hline
\ce{CH3OCHO}+CRPHOT$\longrightarrow$\ce{COOCH4}+e$^{-}$&Cosmic ray-induced photoreaction&1.30$\times$10$^{-17}$&0.00&500.00\\
\hline
\ce{H3}$^{+}$+\ce{CH3OCHO}$\longrightarrow$\ce{H5C2O2}$^{+}$+\ce{H2}&Ion-neutral&3.00$\times$10$^{-9}$&--0.50&0.00\\
\hline
\ce{H3}O$^{+}$+\ce{CH3OCHO}$\longrightarrow$\ce{H5C2O2}$^{+}$+\ce{H2O}&Ion-neutral&3.00$\times$10$^{-9}$&--0.50&0.00\\
\hline
HCO$^{+}$+\ce{CH3OCHO}$\longrightarrow$\ce{H5C2O2}$^{+}$+CO&Ion-neutral&2.90$\times$10$^{-9}$&--0.50&0.00\\
\hline
He$^{+}$+\ce{CH3OCHO}$\longrightarrow$\ce{HCO2}$^{+}$+\ce{CH3}+He&Ion-neutral&2.90$\times$10$^{-9}$&--0.50&0.00\\
\hline
\ce{CH3OCHO}+$h\nu$$\longrightarrow$\ce{H2CO}+\ce{H2CO}&Photoprocess&1.38$\times$10$^{-9}$&0.00&1.70\\
\hline
\end{tabular}
\end{adjustbox}
\label{tab:reactions}
\end{table*}
\subsection{Possible formation mechanism of ethyl cyanide (\ce{C2H5CN}) towards the hot molecular cores}
In the surface chemical route, the complex nitrile molecule ethyl cyanide (\ce{C2H5CN}) will produce via the reaction of C$_{2}$H$_{5}$ and CN towards hot molecular cores (C$_{2}$H$_{5}$ + CN $\longrightarrow$ C$_{2}$H$_{5}$CN) {\color{blue}(Belloche et al. 2009)} but unfortunately the observation of C$_{2}$H$_{5}$ by radio techniques were very difficult because it is a nonpolar molecule. In the grain surface of the hot molecular core, the complex nitrile species \ce{C2H5CN} will produce with the help of interstellar methyl cyanide (\ce{CH3CN}) and protonated \ce{CH3}. The possible chemical reaction is\\
\ce{CH3}$^{+}$+\ce{CH3CN}$\longrightarrow$\ce{C2H5CNH}$^{+}$+h$\nu$~~~~~~~~~~~~~(1)\\
\ce{C2H5CNH}$^{+}$+e$^{-}\longrightarrow$\ce{C2H5CN}+H~~~~~~~~~~~~~~~~~(2)\\
The above chemical reaction was taken from UMIST 2012 astrochemistry reaction network \citep{mce13}. In reaction 1, the methyl cyanide (\ce{CH3CN}) reacts with protonated methane (\ce{CH3}$^{+}$) to create \ce{C2H5CNH}$^{+}$ via radiative association reaction and corresponding reaction coefficients are $\alpha$ = 9.0$\times$10$^{-11}$, $\beta$ = --0.50, and $\gamma$ = 0.00 which was calculated using the Arrhenius equation. The complex molecule \ce{C2H5CN} produces after the dissociative recombination of \ce{C2H5CNH}$^{+}$ in the interstellar grains with reaction coefficients $\alpha$ = 6.32$\times$10$^{-7}$, $\beta$ = --0.76, and $\gamma$ = 0.00 which shown in reaction 2. Recently, the emission lines of CH$_{3}$CN was detected from IRAS 18566+0408 using SMA with column density 7.3$\times10^{16}$ cm$^{-2}$ {\color{blue}(Silva et al. 2017)}. This possible formation mechanism of \ce{C2H5CN} towards IRAS 18566+0408 indicated that the \ce{CH3CN} molecule acted as a precursor of \ce{C2H5CN}. In the hot molecular cores, the \ce{C2H5CN} molecule will destroy via cosmic ray-induced photoreaction, ion-neutral, neutral-neutral, and photoprocess chemical reactions which provided in Tab.~\ref{tab:reactions} \citep{mce13}.
\subsection{Possible formation mechanism of methyl formate (\ce{CH3OCHO}) towards the hot molecular cores}
In the interstellar medium, the methyl formate (\ce{CH3OCHO}) was first detected towards the hot molecular core region Sgr B2 \citep{bro75}. The emission lines of \ce{CH3OCHO} have been also detected in hot corinos and outflow regions \citep{bot07, ari08}. In the hot molecular core IRAS 18566+0408, the complex organic molecule methyl formate (\ce{CH3OCHO}) can be created on the surface of dust grains by the reaction between \ce{CH3O} and HCO, and this mechanism was the same in the case of all hot molecular cores {\color{blue}(Garrod et al. 2008; Gorai et al. 2021)}. The chemical simulation between \ce{CH3O} and HCO shows that these radicals are mobile around 30--40 K {\color{blue}(Gorai et al. 2021)}. According to the simulation of Fig.~1 in \citet{gar13}, it is clear that the gas phase methyl formate (\ce{CH3OCHO}) in the hot molecular core region mainly arose from the ice phase. The increase of UV photodissociation of methanol (\ce{CH3OH}) in the hot molecular cores leads to the formation of \ce{CH2O}, \ce{CH3O}, and \ce{CH3} around 40 K temperature {\color{blue}(Gorai et al. 2021)}. At temperature $T\sim$40 K in gas phase, the \ce{H2CO} and protonated \ce{CH3OH} react and create \ce{H5C2O2}$^{+}$ {\color{blue}(Gorai et al. 2021)}. The methyl formate (\ce{CH3OCHO}) would be created in the hot molecular core via the electron recombination of \ce{H5C2O2}$^{+}$
(\ce{H5C2O2}$^{+}$+e$^{-}\longrightarrow$\ce{CH3OCHO}+H) \citep{bon19}. Using the UMIST 2012 astrochemistry chemical network \citep{mce13}, we found the reaction coefficients of the given reaction \ce{H5C2O2}$^{+}$+ e$^{-}\longrightarrow$\ce{CH3OCHO}+H was $\alpha$ = 1.50$\times$10$^{-7}$, $\beta$ = --0.50, and $\gamma$ = 0.00. The destruction pathways of complex molecule \ce{CH3OCHO} via charge exchange, cosmic ray-induced photoreaction, ion-neutral, and photoprocess chemical reactions towards the hot molecular core regions was presented in Tab.~\ref{tab:reactions} which was adopted from UMIST 2012 astrochemistry reaction network \citep{mce13} .
Earlier, \citet{bal15} proposed that the dimethyl ether (\ce{CH3OCH3}) was the precursor of \ce{CH3OCHO} in a cold environment with gas-phase reaction. The single transition line of dimethyl ether (\ce{CH3OCH3}) with transition J=17(2,15)--16(3,14) was detected towards IRAS 18566+0408 using SMA {\color{blue}(Silva et al. 2017)} but detection of a single transition line of complex molecule is not strong conclusive evidence of presence of the gas in the hot molecular core. The complex molecule \ce{CH3OCHO} can be produced through both gas-phase reaction and grain surface reaction {\color{blue}(Gorai et al. 2021)}. Recently, \citet{ishi21} proposed a formation mechanism of \ce{CH3OCHO} in the hot molecular core region between the reaction of \ce{CH3OH} and OH radicals on ice dust which presented reaction 3, 4, and 5.\\
\ce{CH3OH} + OH$\longrightarrow$\ce{CH3O} + \ce{H2O}~~~~~~~~~~~~~~~~~~~(3)\\
\ce{CH3O} + \ce{CH2OH}$\longrightarrow$\ce{CH3OCH2OH}~~~~~~~~~~~~~(4) \\
\ce{CH3OCH2OH} + UV$\longrightarrow$\ce{CH3OCHO}~~~~~~~~~~~~(5)\\
Recently, \citet{ahm20} proposed another pathway to the formation of methyl formate in hot molecular cores which works both in gas-phase and grain surface reaction with involving formaldehyde (\ce{H2CO})\\
\ce{H2CO} + \ce{H2CO}$\longrightarrow$\ce{CH3OCHO}~~~~~~~~~~~~~~~~~~~~~(6)
After the detection of \ce{C2H5CN} and \ce{CH3OCHO} towards IRAS 18566+0408, we expect more complex molecular species would be observable from this hot molecular core region with other bands of ALMA. Since \ce{C2H5CN} and \ce{CH3OCHO} were detected successfully in IRAS 18566+0408, we recommend a search of molecular transition lines of methanol (\ce{CH3OH}) and dimethyl ether (\ce{CH3OCH3}) towards the hot molecular core region. We propose to study transition lines of complex molecule dimethyl ether in IRAS 18566+0408 because it was known as the precursor of \ce{CH3OCHO} in a cold environment \citep{bon19}. Methyl cyanide (\ce{CH3CN}), which is the precursor of ethyl cyanide (\ce{C2H5CN}), was earlier detected towards IRAS 18566+0408 hot molecular core. The detection of the C$_{2}$H$_{5}$CN and CH$_{3}$OCHO with the multiple numbers of transitions encourages more detailed studies of other molecular species towards IRAS 18566+0408 to understand the chemical complexity.
\section{Conclusion}
\label{conclusion}
We reported the first detection of the complex organic molecules ethyl cyanide (\ce{C2H5CN}) and methyl formate (\ce{CH3OCHO}) towards the hot molecular core region IRAS 18566+0408. Our main findings in this work are summarized below\\
1. We successfully detected a total of three unblended strong rotational emission lines of ethyl cyanide (\ce{C2H5CN}) and eight unblended rotational lines of methyl formate (\ce{CH3OCHO}) towards IRAS 18566+0408 using ALMA. Additionally, we also detected the rotational emission lines of SO, \ce{CH2CO}, \ce{HC3N}, H$_{2}^{34}$CS, HC$^{13}$CCN, HCC$^{13}$CN, and CH$_{3}$CHO towards IRAS 18566+0408.\\
2. Using the rotational diagram method, the statistical column density of ethyl cyanide (\ce{C2H5CN}) was $N$(C$_{2}$H$_{5}$CN) = (3.50$\pm$0.68)$\times$10$^{15}$ cm$^{-2}$ with rotational temperature $T_{rot}$ = 150$\pm$2.5 K and the statistical column density of methyl formate (\ce{CH3OCHO}) was $N$(CH$_{3}$OCHO) = (1.55$\pm$0.31)$\times$10$^{16}$ cm$^{-2}$ with rotational temperature $T_{rot}$ = 150$\pm$2.8 K. The estimated fractional abundance of ethyl cyanide and methyl formate in the high mass star-forming region IRAS 18566+0408 relative to H$_{2}$ was 1.66$\times$10$^{-9}$ and 7.38$\times$10$^{-9}$ where the column density of hydrogen was 2.1$\times$10$^{24}$ cm$^{-2}$.\\
3. We compared the calculated column density of ethyl cyanide (\ce{C2H5CN}) and methyl formate (\ce{CH3OCHO}) with other known hot molecular cores G31.41+0.31, Sgr B2 (N), and Orion KL. After the comparison, We found that the column density
of \ce{C2H5CN} and \ce{CH3OCHO} were small with respect to G31.41+0.31, Sgr B2 (N), and Orion KL. \\
4. We discussed the possible formation and destruction pathways of \ce{C2H5CN} and \ce{CH3OCHO} towards IRAS 18566+0408 using UMIST 2012 astrochemistry chemical network. We showed that the complex molecule methyl formate (\ce{CH3CN}) may be the possible precursor of the ethyl cyanide (\ce{C2H5CN}). In the case of methyl formate (\ce{CH3OCHO}), we presented different possible formation mechanisms towards IRAS 18566+0408 and claimed that dimethyl ether (\ce{CH3OCH3}), methanol (\ce{CH3OH}), and formaldehyde (\ce{H2CO}) acted as a precursor of methyl formate (\ce{CH3OCHO}). \\
5. After the successful detection of \ce{C2H5CN} and \ce{CH3OCHO} towards IRAS 18566+0408, a broader study was needed to search other molecular lines in the other frequency band of ALMA to understand the chemical complexityin this hot molecular core.
\section*{Acknowledgments}{This paper makes use of the following ALMA data: ADS /JAO.ALMA\#2015.1.00369.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in co-operation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. }\\\\
Software: CASA (v5.1.0) {\color{blue}(McMullin et al. 2007)}, CASSIS (Cassis Team At CESR/IRAP 2014)
\section*{Data availability}{The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request. The raw ALMA data are publicly
available at \href{https://almascience.nao.ac.jp/asax/}{https://almascience.nao.ac.jp/asax/} (project id: 2015.1.00369.S).}
\section*{Conflicts of interest}{The authors declare no conflict of interest.}
|
2,869,038,154,784 | arxiv | \section{Introduction}
The investigation of optical properties of atoms coupled to
dissipative environments with a structured density of modes
has been a topic of active research over the years \cite{klepner1,lmg,lzm,klepner2,kurizki,prata1,ref8,poizat}. Of particular interest is the discussion of atoms (impurities) embedded in two or three-dimensional periodic dielectric structures, known as photonic crystals \cite{yablotonovitch,ho,jw1,jw4,moss,nabiev,jq2,kofman,jw5,jw6,tarhan,zhu,paspalakis,zhu2,ref9,ref7,lai,jw3,dalton,huang,hsieh,cheng,yang2,yang1}, since they allow control
over the electromagnetic density of modes and the spatial modulation of narrow-linewidth (high-{\it Q}) modes, in both microwave and optical regimes \cite{ref9}. When these structures are used to create one or several forbidden frequency bands they allow control or complete suppression of spontaneous emission, as well as absorption from those embedded impurities \cite{jw1,moss,kofman,jw5,jw6,zhu,paspalakis,ref9,ref7,lai,jw3,dalton,huang,hsieh,cheng,yang2,yang1}. It was particularly relevant the early observation that a two-level atom embedded in a PBG \cite{jw1,jw4,jw5,jw6} could retain some population in the upper level, even when the transition frequency was in the transmitting band, being the final state a dressed state of the atom with a localized field
mode, which lies in the forbidden band. More recently the attention has been shifted to quantum dots embedded in photonic crystals where each individual quantum dot can be seen as an ``artificial atom'' \cite{vucko,john,nano}.
The important feature in any of those situations above is that the ``atom'' placed in such a structure
interacts with the field modes in the propagating frequency band and in the
forbidden photonic band gap (PBG) as well, giving rise to many interesting
coherent phenomena such as the possibility of controlling non-markovian decay \cite{kurizki,dalton}, localization of superradiance \cite{jw5}, quantum interference effects in spontaneous emission \cite{zhu,huang}, transparency to a probe field \cite{paspalakis}, and squeezing in the in-phase quadrature spectra \cite{lai}.
The majority of contributions regarding
radiative properties consider only spontaneous emission of two, three, four and five level atoms embedded in a PBG structure \cite{zhu,zhu2,zhu3,ding,yang2,yang1}, with only a few exceptions treating absorptive and dispersive properties \cite{paspalakis,tajalli}. As an important example of this last case, the absorption and dispersion properties of a $\Lambda$-type atom decaying spontaneously near the edge of a PBG was studied \cite{paspalakis}. It was
pointed out, within an isotropic PBG model, that the atom can become transparent to a probe laser
field, even when other dissipative channels are present, suggesting that many
surprising effects in the absorption and dispersion of atoms embedded in such structures can appear. Most of those effects were considered inside model systems composed by three or more levels \cite{ref7,kurizki,paspalakis, ref8, ref9, zhu, zhu3,ding,yang2,yang1}, while they were not proved to be strictly necessary.
Pursuing this line we revisit the problem of transparency of an atom placed near an isotropic band edge \cite{paspalakis}, but consider the minimal situation of transitions between two-levels only. We show that for it to be transparent to a weak driven field, the two-level atom must be coupled to a reservoir constituted of two parts - a flat and a non-flat density of modes representing a PBG structure. Transparency is therefore an inner property of the
reservoir engineering. As a side result of this approach we consider the related inverse problem considered in
\cite{nabiev,PhysRevLett.107.167404,pinkse} on the possibility to obtain information about the band edge profile
from two-level temporal decay in such structure. Here we show that is also possible to reconstruct
the band edge characteristics directly from the experimentally measured susceptibility.
This paper is organized as follows. In Section (\ref{s2}), we present the model considered and its stationary solution. In Section (\ref{s3}), the linear susceptibility is evaluated, and two models of isotropic band gap structures are analyzed. In Section (\ref{s4}), is discussed how to reconstruct the band edge characteristics from the experimentally measured susceptibility. Finally, in Section (\ref{s5}) we conclude the paper.
\section{Model}{\label{s2}}
The system considered here is a two-level
atom with excited and ground state $\left|1\right\rangle$ and
$\left|0\right\rangle$, respectively and with
transition frequency $\omega_0 $. The atom is probed by a weak electric field
with frequency $\omega $ detuned from $\omega_0 $ by $\delta=\omega - \omega_0$. The decay of the excited state
is due to a coupling with vacuum modes described by a collection
of harmonic oscillators with frequencies $\omega_m $. In the rotating wave
approximation and in the interaction picture the Hamiltonian of the system is
given by
\begin{equation}
\label{1}
H=\left(\Omega e^{i\delta t}\left|0\right\rangle\left\langle 1\right| +H.c.\right)
+\sum_{m} \left(g_m e^{i(\omega_{m}-\omega_0)t}b^{\dagger}_m\left|0\right\rangle\left\langle 1\right| +H.c.\right),
\end{equation}
where $\Omega=-\mu_{10} E_o$ is the Rabi frequency, $\mu_{10}$ is the atomic electric dipole moment, and $E_o$ is electric field
amplitude. The $g_m$ represents the coupling between
the atom and the vacuum modes and $b^{\dagger}_m$ and $b_{m}$
are the creation and annihilation operators for excitations in the reservoir,
with $m={\lambda,{\bf k}}$ indicating a photon state with polarization $\lambda$ and momentum ${\bf k}$. For sake of simplicity we assume $\Omega$ and $g_m$ as real. In the period of time $t$ the state of total system, {\it atom} + {\it reservoir modes}, can be written
as a superposition
given by
\begin{equation}
\label{2}
\left|\psi (t)\right\rangle = a_0(t)\left|0,\{0\}\right\rangle+a_1 (t)e^{-i\delta t}\left|1,\{0\}\right\rangle + \sum_{m} \alpha_m (t)\left|0,\{m\}\right\rangle.
\end{equation}
The coefficients $a_0(t)$ and $a_1(t)$ are the probability amplitudes to find
the atom in the ground and excited state and the photon reservoir in the vacuum state, respectively, while the coefficient $\alpha_{m} (t)$ gives the probability amplitude to find the atom in the ground state and a single photon in the state $m$ of the vacuum modes. Substituting Eq. (\ref{2}) into the Schr\"odinger equation containing the hamiltonian (\ref{1}) and projecting into each state at the right-hand-side of Eq. (\ref{2}) gives the following equations of motion for the time dependent coefficients $a_{l}$ and $\alpha_{m}$,
\begin{equation}
\label{a0}
i\dot{a}_0(t)=\Omega a_1(t),
\end{equation}
\begin{equation}
\label{a2}
i\dot{a}_1(t)=\Omega a_0 (t) -\delta a_1(t)+\sum_{m}g_{m}e^{-i(\omega_m-\omega_0-\delta)t}\alpha_{m}(t),
\end{equation}
\begin{equation}
\label{a3}
i\dot{\alpha}_{m} (t) = g_{m}e^{i(\omega_m-\omega_0-\delta)t}\, a_1(t).
\end{equation}
Integrating Eq. (\ref{a3}) and eliminating the vacuum amplitude in the equations for $a_0(t)$
and $a_1 (t)$ it follows that
\begin{equation}
\label{3}
i\dot{a}_0(t)=\Omega a_1(t),
\end{equation}
\begin{equation}
\label{3b}
i\dot{a}_1(t)=\Omega a_0 (t) -\delta a_1(t)-i\int_{0}^{t}K(t-t')
a_1 (t')dt',
\end{equation}
where the kernel, $ K(t-t')$, is given by
\begin{equation}
\label{5}
K(t-t')=\sum_m g_m^2e^{-i(\omega_m-\omega_0-\delta)(t-t')}.
\end{equation}
All the information about the reservoir is contained in the kernel above,
which will be dependent on the frequency distribution of the vacuum modes.
As we are interested in effects strongly dependent on the reservoir modes
distribution we keep the integro-differential equation (\ref{3b}) without any approximation.
The procedure to solve Eqs. (\ref{3}) and (\ref{3b}) is straightforward by Laplace transform, which for the initial conditions $a_{0}(0)=1$ and $a_{1}(0)=0$ results in
\begin{equation}
\label{lp1}
V_{0}(s)=\frac{s-i\delta+G(s)}{s\left[s-i\delta+G(s)\right]+\Omega^2},
\end{equation}
\begin{equation}
\label{lp2}
V_{1}(s)=\frac{-i\Omega}{s\left[s-i\delta+G(s)\right]+\Omega^2},
\end{equation}
where $V_0(s)$, $V_1(s)$ and $G(s)$ are the Laplace transforms of $a_0(t)$, $a_1(t)$ and
$K(t-t')$, respectively.
We must only assume two hypothesis to proceed with our calculations. The first
one is about the reservoir - its memory function must allow the
atomic system to reach a steady state, since we are interested in equilibrium
properties at this regime. Secondly, we are interested in the linear atomic response to the external field, so the coupling between field and atom is considered to be weak in such a way that $a_{0}(t)\approx1$
for all times. This means that we are considering a perturbative solution for $a_{1}(t)$, which is linear in the driven field amplitude. This approximation does not affects considerably the results we are discussing, since the only effect of considering a first order term is to neglect power broadening and saturation effects due the laser field intensity \cite{boyd}. In first order in $\Omega$, Eq. (\ref{lp2}) simplifies to
\begin{equation}
\label{6}
V_{1}(s)= \frac{\Omega}{s[is+\delta+iG(s)]},
\end{equation}
and the steady state solution is obtained by the limit
procedure \cite{barnett}
\begin{equation}
\label{7}
a_{1}(t \rightarrow \infty)=\lim_{s \rightarrow 0}sV_{1}(s)=\frac{\Omega}{\delta+iG(0)},
\end{equation}
where we assumed that the memory function is in such a way that the
limit $s\rightarrow 0$ for $G(s)$ shall exists.
\section{Susceptibility}{\label{s3}}
The induced polarization due the applied external field is given given by
\begin{equation}
P(t)=\int_{0}^{\infty} \tilde{\chi} (t-t^{\prime}) E (t^{\prime})\, dt^{\prime},
\end{equation}
where $\tilde{\chi }(t)$ is the complex susceptibility, whose imaginary and real part are related to the atom absorption and dispersion of energy from the laser field, respectively \cite{boyd}. For a harmonic field $E(t)=E_{o}e^{-i\omega t}+h.c.$ the polarization becomes
\begin{equation}
P(t)= 2 Re \left(e^{-i\omega t} \chi (\omega) E_o\right),
\label{su1}
\end{equation}
where $\chi (\omega)$ is the fourier transform of $\tilde{\chi} (t)$. On the other hand, the atom polarization is obtained as an average of the atomic dipole moment,
\begin{equation}
P(t)=\left\langle \mu (t)\right\rangle=2 Re\left(\mu_{01}\:a_{0}(t)a_1^{*}(t)\right)\approx 2 Re\left(\mu_{01}a_1^{*}(t)\right).
\label{su2}
\end{equation}
From Eqs. (\ref{su1}) and (\ref{su2}), and considering $N$ atoms by unit of volume, we obtain for the susceptibility at the stationary state, the following expression
\begin{equation}
\label{8}
\chi(\delta)=- N |{\bf \mu}_{01} |^2 \frac{1}{\delta-iG^{*}(0)},
\end{equation}
which contains the information about the environment through the function $G(0)$.
Here, we assume that the atomic decay, i.e., the emission rate into the vacuum modes
can be divided into two parts \cite{lmg,nabiev,kurizki,prata1,ref8} -
the response due to the flat modes of vacuum plus a response due to structured
modes. Following this assumption, we can write the kernel as
\begin{equation}
\label{9}
K(t-t')=\frac{\gamma}{2}\delta(t-t') + \tilde{K}(t-t').
\end{equation}
The first term in Eq. (\ref{9}) corresponds to a Markovian
evolution, due to the coupling to
a flat density of modes where $\gamma=4\omega_{0}^3|\mu_{01}|^2/3c^3$
is the Wigner-Weisskopf decay rate \cite{ww}.
The second term is the non-Markovian counterpart of the
evolution, corresponding to the coupling with the structured density of
modes imposed by the photonic crystal. The linear susceptibility now writes as
\begin{equation}
\label{10}
\chi(\delta)\sim - \frac{1}{\delta-i\frac{\gamma}{2}-i\tilde{G}^{*}(0) },
\end{equation}
where $\tilde{G}(0)$ is the limit $t\rightarrow \infty$ for the Laplace
transform of the non-flat part of the kernel, $\tilde K(t-t')$.
As a first example, we consider that the non-flat part is due to the coupling
to an isotropic photonic band gap where the effective mass dispersion relation is
$\omega_{{\bf k}}=\omega_g+A(|{\bf k}|-|{\bf k_0}|)^2$, with
$A\approx \omega_g/|{\bf k_0}|^2$ \cite{jw1,jw4,ref9}. In this case the kernel is given by
\begin{equation}
\label{11}
\tilde{K}(t-t')=\frac{\beta^{3/2}e^{-i[\pi /4 +(\delta_g-\delta)(t-t')]}}
{\sqrt{\pi (t-t')}},\,\,\,\, t>t',
\end{equation}
with $\beta^{3/2}=2\omega_{o}^{7/2}|\mu_{01}|^2/3c^3$ and
$\delta_g=\omega_g-\omega_o$. This model corresponds to a density
of modes given by
$\rho (\omega')=\theta(\omega '-\omega_g)/\pi \sqrt{\omega '-\omega_g}$, where $\theta(\omega '-\omega_g)$ is the Heaviside
function. The non-flat kernel then gives us
\begin{equation}
\label{12}
\tilde{G}(s)=\frac{\beta^{3/2}e^{-i\pi /4}}{\sqrt{s+i(\delta_g-\delta)}},
\end{equation}
and the linear susceptibility reads
\begin{equation}
\label{sus1}
\chi(\delta)\sim
-\frac{\sqrt{\delta_g-\delta}}{\left(\delta-i\gamma/2\right)\sqrt{\delta_g-\delta}+\beta^{3/2}}.
\end{equation}
From Eq. (\ref{sus1}) it can be seen that the susceptibility is zero at $\delta=\delta_g$ (or $\omega=\omega_g$), and the atom is transparent to the probe laser field. To illustrate, in Fig. \ref{f1} we plot $Re(\chi)$ and $-Im (\chi)$, which corresponds to dispersion and absorption, respectively, as function of $\delta$, setting $\delta_{g}=2$, $\beta =1$, and $\gamma =1$. Besides transparency (zero absorption) at $\delta=\delta_{g}=2$, it is also observed a strong deviation from the typical two-level absorption-dispersion curves in the Markovian approximation. Thus, the same transparency phenomenon of a three-level system in a $\Lambda $ configuration \cite{paspalakis} in the presence of a band edge is also possible for a two-level atom.
\begin{figure}[tdq]
\includegraphics[scale=1.4]{fig1
\caption{ Absorption and dispersion as a function of the detuning for the isotropic effective mass model. Solid line: Absorption; Dashed line: Dispersion. We set $\delta_{g}=2$, $\beta =1$, and $\gamma =1$. The parameters are dimensionless. }
\label{f1}
\end{figure}
In addition, is worthwhile to consider two extreme cases: (\textit{i}) when the flat part of the density of modes vanish, and (\textit{ii}) when the step-like function describing the band-gap changes in a smooth fashion, as in a real photonic crystal. In the first case, for $\delta < \delta_{g}$, except by a delta function absorption spike at the shifted atomic resonance frequency, the susceptibility given by Eq. (\ref{sus1}) has no imaginary part, and therefore, there is no absorption at these frequencies. For the second case we consider the one band model with
a smooth density $\rho (\omega')=\sqrt{\omega '-\omega_g}\theta(\omega '-\omega_g)/\pi (\omega '-\omega_g +\epsilon)$, where $\epsilon $ is a smooth parameter to avoid the singularity at $\omega '=\omega_g$ \cite{kurizki,ref8}. The kernel for this density is
\begin{equation}
\label{smo1}
\tilde{G}(s)=\frac{\beta^{3/2}}{ i\sqrt{\epsilon}+e^{i\pi /4}\sqrt{s+i(\delta_g-\delta)}},
\end{equation}
and the susceptibility,
\begin{equation}
\label{sus3}
\chi(\delta)\sim
-\frac{\sqrt{\delta_g-\delta}+\sqrt{\epsilon }}{\left(\delta-i\gamma/2\right)(\sqrt{\delta_g-\delta}+\sqrt{\epsilon })+\beta^{3/2}}.
\end{equation}
In this case the susceptibility is null at $\delta=\delta_g $ in the strict case of $\epsilon =0$. Otherwise, the absorption and the dispertion curves present a dip at $\delta=\delta_g $ dependent on the value of $\epsilon$. In Fig. \ref{f3} we plot $Re(\chi)$ and $-Im (\chi)$ as function of $\delta$ around $\delta=\delta_g $ considering several values of $\epsilon $. The dip in the line shapes approach zero as the values of $\epsilon $ goes to zero.
\begin{figure}[tdq]
\includegraphics[scale=1.4]{fig2
\caption{ Absorption and dispersion as a function of the detuning around $\delta =\delta_g $ for the smoothed one-band isotropic model. Solid line: $\epsilon = 1$; Dashed line: $\epsilon = 0.1$; Dotted line: $\epsilon = 0.01$. We set $\delta_{g}=2$, $\beta =1$, and $\gamma =1$. The parameters are dimensionless. }
\label{f3}
\end{figure}
As a second model for the reservoir we assume that the memory kernel has a non-markovian term due a two-band isotropic effective mass model for the photonic band gap structure. This model corresponds to a density of modes given by \cite{zhu3}
\begin{equation}
\rho (\omega ')=\frac{\theta(\omega_a-\omega ' )}{2\pi \sqrt{\omega_a-\omega ' }}+
\frac{\theta(\omega ' -\omega_b)}{2\pi \sqrt{\omega ' -\omega_b}},
\end{equation}
and resulting in
\begin{equation}
\label{12b}
\tilde{G}(s)=\frac{\beta^{3/2}e^{i\pi /4}}{2\sqrt{s+i(\delta_a-\delta)}}+\frac{\beta^{3/2}e^{-i\pi /4}}{2\sqrt{s+i(\delta_b-\delta)}},
\end{equation}
where $\delta_a=\omega_a-\omega_o$, and $\delta_b=\omega_b-\omega_o$. Now the susceptibility is given by the following expression
\begin{equation}
\label{sus2}
\chi(\delta)\sim
-\frac{\sqrt{(\delta_a-\delta )(\delta_b-\delta )}}{\left(\delta-i\gamma/2\right)\sqrt{(\delta_a-\delta )(\delta_b-\delta )}
+\frac{\beta^{3/2}}{2}\left(-i\sqrt(\delta_b-\delta )+\sqrt{(\delta_a-\delta)}\right)}.
\end{equation}
In this case the susceptibility is null at $\delta=\delta_a$ or $\delta=\delta_b$, and the atom becomes transparent at
two different frequencies. The behavior of dispersion-absorption curves is illustrated in Fig. \ref{f2} where we plot $Re(\chi)$ and $-Im (\chi)$ as function of $\delta$ and setting $\delta_{a}=1$ and $\delta_{b}=2$, $\beta =1$, and $\gamma =1$. Transparency at two different frequencies is observed.
\begin{figure}[tdq]
\includegraphics[scale=1.4]{fig3
\caption{ Absorption and dispersion as a function of the detuning for the two-band isotropic effective mass model. Solid line: Absorption; Dashed line: Dispersion. We set $\delta_{a}=1$ and $\delta_{b}=2$, $\beta =1$, and $\gamma =1$. The parameters are dimensionless. }
\label{f2}
\end{figure}
\section{Band-edge profile reconstruction}{\label{s4}}
Since we have developed all the necessary ingredients for understanding the effects of the structured reservoir on the atomic transparency to the probe, we would like to discuss on a rather important related problem, which is the inverse problem of determining the characteristics of the band gap and its profile from
experimental data. As pointed out by Nabiev \cite{nabiev}, the band gap profile by can be determined from the experimental data on the temporal behavior of the atomic spontaneous decay. Indeed, this can also be done by a stationary measure through the susceptibility function. Under the conventional continuum limit for the reservoir mode distribution, and for long times, the Laplace transform for the non-flat contribution can be written as
\begin{eqnarray}
\label{13}
\lim_{s\rightarrow 0}\tilde{G}(s)&=&\lim_{s\rightarrow 0}\int_{0}^{\infty}d\omega^{\prime}\;\Gamma (\omega^{\prime})
\int_{0}^{\infty}d\tau \:e^{-i(\omega^{\prime}-\omega -is)\tau} \nonumber \\
&=&
\pi \Gamma (\omega)-i\:\mathcal{P}\int_{0}^{\infty}
d\omega^{\prime}
\frac{\Gamma (\omega^{\prime})}{\left(\omega^{\prime}-\omega\right)},
\end{eqnarray}
where $\Gamma(\omega^{\prime})=g^2(\omega^{\prime})\rho(\omega^{\prime})$
represents the product of the coupling with the density of states for the
non-flat sector, and we have used the identity \cite{barnett}
\begin{equation}
\int_{0}^{\infty}d\tau e^{-i(\omega^{\prime}-\omega -is)\tau}=\pi \delta(\omega^{\prime}-\omega -is)-i\mathcal{P}\frac{1}{\omega^{\prime}-\omega -is},
\end{equation}
and $\mathcal{P}$ means principal value.
For the band edge profile reconstruction it must be noticed that the first term in Eq. (\ref{13}) is exactly taken in
the external field frequency. Thus, when the external field frequency is varied, the reservoir frequency is probed,
{\it i.e.}, with the variation of the probe field frequency, in fact the band gap frequency distribution is scanned. If one assumes the general expression (\ref{8}) and inverts $\tilde{G}(0)$ as
a function of the measured susceptibility, namely
\begin{equation}
\label{10b}
\tilde{G}(0) = - \frac{\gamma}{2}+i\left(\delta+\frac{N \left|\mu_{01}\right|^2}{\chi^{*}(\omega)}\right),
\end{equation}
and the band gap profile can be reconstructed.
Remark that the present approach can be applied as well to a broad range of distinct situations, such as in the recent findings on the coupling of atoms trapped in the near field of nanoscale photonic crystal cavities \cite{Tiecke2014}. In this situation the present approach would be useful for probing the cavity density of modes through a susceptibility measurement.
\section{Conclusion}{\label{s5}
In conclusion, we saw that it is possible to obtain transparency
to a laser probe field on a two-level atom if it is
considered the atomic coupling with a reservoir constituted by flat and
non-flat densities of modes existing in a PBG. We have considered two isotropic band-gap models and analized the linear response to a weak optical field through the susceptibility function. We have also discussed the
possibility of band edge profile reconstruction via the susceptibility function
knowledge. This can be an alternative to the method of reconstruction of the band edge profile involving the measurement of temporal quantities as suggested by
\cite{nabiev} and implemented with single quantum
dots embedded in a photonic crystal \cite{PhysRevLett.107.167404} (See also Ref. \cite{pinkse} for another method of reconstruction). In contrast the band
edge profile reconstruction here described can be realized in a steady state
situation. Those findings are particularly relevant for the emerging field of quantum nanophotonics \cite{nano}, as well as in the investigation of nonlinear features with actual atoms trapped in the near field of nanoscale photonic crystal cavities \cite{Tiecke2014}. The measured susceptibility can therefore be a valuable tool for band edge profile reconstruction in those systems.
This work was partially supported by CNPq and FAPESP through the Instituto Nacional de Ci\^encia e Tecnologia em Informa\c c\~ao Qu\^antica (INCT-IQ) and through the Research Center in Optics and Photonics (CePOF).
\section*{References}
\bibliographystyle{elsarticle-num}
|
2,869,038,154,785 | arxiv | \section{Introduction}
Different mathematicians mean different things when they say
``$L$-function.'' Some mean an element of the Selberg class and others
might mean a Dirichlet series with Euler product and require that it
be associated
to an automorphic form. For some people an $L$-function has to be
entire, for others it can have poles on the edge of the critical
strip, for yet
others it can even have poles in other locations.
In this paper we show how one can attach adjectives to $L$-functions (and which
adjectives one should attach, as determined by one's goals)
in such a way that the resulting classes
of $L$-functions provide a detailed framework to understanding $L$-functions.
This framework can be used to clarify
the distinctions between various classes, and also to unify by
showing connections between them. In
Section~\ref{sec:types}, we define the following sets of $L$-functions
(see that Section for definitions):
\begin{itemize}
\item Tempered balanced analytic primitive entire $L$-functions. These $L$-functions are
defined axiomatically, with precise restrictions on their
functional equation and Euler product.
\item $\mathbb Q$-automorphic $L$-functions. These $L$-functions are
associated to tempered balanced unitary cuspidal automorphic representations of $\GL(n,\mathbb A_\mathbb Q)$.
\end{itemize}
The above sets are believed to contain all primitive $L$-functions that
are expected to satisfy analogues of the Riemann Hypothesis,
and conjecturally the two sets are essentially equal.
Within each of the above sets are distinguished subsets which,
conjecturally, contain all $L$-functions arising from arithmetic
objects.
\begin{itemize}
\item $L$-functions of algebraic type (analytic and $\mathbb Q$-automorphic). These are characterized
by conditions on the $\Gamma$-factors in the functional equation.
\item $L$-functions of arithmetic type (analytic and $\mathbb Q$-automorphic). These are characterized
by conditions on the coefficients in the Dirichlet series.
\end{itemize}
Conjecturally all four of these sets of $L$-functions are equal and arise from the following arithmetic objects:
\begin{itemize}
\item Pure motives.
\item Geometric Galois representations.
\end{itemize}
Associated to each such arithmetic object is an $L$-function. Conjecturally
those sets of $L$-functions are equal, and coincide with the
four subsets of $L$-functions mentioned previously.
The conjectured relationships between these sets of $L$-functions is
shown in Figure~\ref{fig:diagram1}.
In Figure~\ref{fig:diagram2} in Section~\ref{sec:connections} we discuss in more detail the conjectured relationships between the
sets of $L$-functions described above. Precise descriptions of
each of these sets are given in Sections~\ref{sec:types} and~\ref{sec:aa}.
\usetikzlibrary{arrows,
calc,chains,
decorations.pathreplacing,decorations.pathmorphing,
fit,
positioning}
\begin{figure}
\tikzset{
line/.style={draw, thick},
}
\begin{tikzpicture}[node distance = 4mm, start chain = A going below, font = \sffamily, > = stealth', PC/.style = {
rectangle, rounded corners,
draw=black, very thick,
text width=8em,
minimum height=1.5em,
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\node[PC] {Tempered $\mathbb{Q}$-automorphic \mbox{$L$-functions} of algebraic type};
\node[PC] {Primitive analytic $L$-functions of algebraic type};
\node[blank,right=of A-1,xshift=-.375cm] {$\subsetneq$};
\node[blank,right=of A-2,xshift=-.375cm] {$\subsetneq$};
\node[PC,right=of A-3,xshift=-.375cm] {Tempered $\mathbb{Q}$-automorphic $L$-functions};
\node[PC,right=of A-4,xshift=-.375cm] {Primitive analytic $L$-functions};
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\node[PC,left=of A-1,yshift=-12mm] {$L$-functions of geometric Galois representations};
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\end{tikzpicture}
\caption{Conjectured relationships between the sets of $L$-functions considered in this paper.}\label{fig:diagram1}
\end{figure}
\section{Two views of \texorpdfstring{$L$}{}-functions}\label{sec:types}
\subsection{Analytic \texorpdfstring{$L$}{}-functions}\label{sec:axioms}
\label{sec:defn_analytic}
The first set of $L$-functions in our discussion is defined
axiomatically.
Throughout the axioms, $s=\sigma+ it$ is a complex variable with $\sigma$
and $t$ real.
A \term{tempered balanced analytic $L$-function} is a function $L(s)$ which satisfies the five axioms below. In this paper we will also refer to `tempered analytic $L$-functions' and
`analytic $L$-functions,' which are obtained by relaxing some of these axioms.
See Section~\ref{sec:axioms_comments} for a discussion.
\begin{description}
\setlength{\itemindent}{-\leftmargin}
\setlength{\listparindent}{\parindent}
\item[Axiom $1$ (Analytic properties)]
$L(s)$ is given by a Dirichlet series
\index{Dirichlet series}
\[
\label{eqn:DS}\tag{Ax1.1}
L(s)=\sum_{n=1}^\infty \frac{a_n}{n^s},
\]
where $a_n\in \mathbb C$.
{\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\smallskip
\item \emph{Convergence:} $L(s)$ converges absolutely for $\sigma>1$.
\smallskip
\item \emph{Analytic continuation:} $L(s)$ continues
to a meromorphic function having only finitely many poles,
with all poles
lying on the $\sigma=1$ line.
\end{enumerate}
}
\medskip
\item[Axiom $2$ (Functional equation)]
\index{functional equation!axioms}
There is a positive integer $N$ \linebreak called the \term{conductor} of the \hbox{$L$-function},
\index{conductor}
a positive integer $d$ called the \term{degree} of the \hbox{$L$-function},
\index{degree}
a pair of non-negative integers $(d_1, d_2)$ called the \term{signature} of the \hbox{$L$-function},
\index{signature}
where $d=d_1+2d_2$,
and complex numbers $\{\mu_j\}_{j=1}^{d_1}$ and $\{\nu_k\}_{k=1}^{d_2}$
called the \term{spectral parameters} of the \hbox{$L$-function},
\index{spectral parameters}
such that the \term{completed \hbox{$L$-function}}
\index{completed \hbox{$L$-function}}\index{$L$-function@\hbox{$L$-function}!completed}
\[
\label{eqn:Lambda}\tag{Ax2.1}
\Lambda(s) =\mathstrut N^{s/2}
\prod_{j=1}^{d_1} \Gamma_\mathbb R(s+ \mu_j)
\prod_{k=1}^{d_2} \Gamma_\mathbb C(s+ \nu_k)
\cdot L(s)
\]
has the following properties:
{\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\smallskip
\item \emph{Bounded in vertical strips:}
Away from the poles of the \hbox{$L$-function},
$\Lambda(s)$ is bounded in vertical strips $\sigma_1 \leq \sigma \leq \sigma_2$.
\smallskip
\item \emph{Functional equation:} There exists $\varepsilon\in \mathbb C$,
\index{functional equation!general}
called the \term{sign} of the functional equation, such that
\index{sign!of the functional equation}
\[
\label{eqn:FE}\tag{Ax2.2}
\Lambda(s)
=\mathstrut \varepsilon \overline{\Lambda}(1-s).
\]
\end{enumerate}
}
\medskip
\item[Axiom $3$ (Euler product)] There is a product formula
\index{Euler product}
\[
\label{eqn:EP}\tag{Ax3.1}
L(s)= \prod_{p \, {\rm prime}} F_p(p^{-s})^{-1},
\]
absolutely convergent for $\sigma > 1$.
{
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\smallskip
\item \emph{Polynomial:} $F_p$ is a polynomial with $F_p(0)=1$.
\item \emph{Degree:} Let $d_p$ be the degree of $F_p$. If $p\nmid N$ then $d_p=d$,
and if $p\mid N$ then $d_p<d$.
\smallskip
\end{enumerate}
}
\medskip
\item[Axiom $4$ (Temperedness)] The spectral parameters and Satake parameters satisfy precise bounds.
{
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\smallskip
\item \emph{Selberg bound:} \label{axiom:preciseselbergeigenvalue}
For every $j$ we have
$\Re(\mu_j) \in \{0,1\}$ and
$\Re(\nu_k) \in \{\frac12,1,\frac32,2,...\}$.
\item \emph{Ramanujan bound:} Write $F_p$ in factored form as
\[
\label{eqn:ram-bound}\tag{Ax4.1}
F_p(z) = (1-\alpha_{1,p} z)\cdots (1-\alpha_{d_p,p} z)
\]
with $\alpha_{j,p} \not = 0$. If $p\nmid N$ then $|\alpha_{j,p}| = 1$ for all $j$. If $p\mid N$ then $|\alpha_{j,p}|= p^{-m_j/2}$ for some $m_j\in \{0,1,2,...\}$, and $\sum m_j \le d-d_p$.
\end{enumerate}
}
\medskip
\item[Axiom $5$ (Central character)] There exists a Dirichlet character $\chi$ mod~$N$,
\index{central character!of an \hbox{$L$-function}}
called the \term{central character} of the \hbox{$L$-function}.
{
\renewcommand{\labelenumi}{\alph{enumi})}
\begin{enumerate}
\smallskip
\item \emph{Highest degree term:} For every prime $p$,
\[
\label{eqn:Fpchi}\tag{Ax5.1}
F_p(z)=1-a_p z + \cdots + (-1)^d\chi(p) z^d .
\]
\item \emph{Balanced:} We have
$\mathrm{Im} \left(\sum \mu_j + \sum(2\nu_k+1)\right) =0$.
\item \emph{Parity:} The spectral parameters determine the
parity of the central character:
\[
\label{eqn:parityaxiom}\tag{Ax5.2}
\chi(-1) = (-1)^{\sum \mu_j + \sum(2\nu_k+1)}.
\]
\end{enumerate}
}
\end{description}
\subsubsection{Comments on the terminology}
\label{sec:axioms_comments}
The term \term{balanced} is described by Axiom 5b).
The summand ``$+1$'' can obviously be omitted from the condition,
but we include it for uniformity with Axiom~5c).
Note that Axiom~5c) would be problematic if it were not
assumed that the exponent on $-1$ was an integer.
If we omit the modifier `balanced' when describing an $L$-function,
then we mean a function of the form $L(s+i y)$ where
$L(s)$ is balanced and $y\in \mathbb R$.
If $L(s)$ is a (not necessarily balanced) $L$-function, then it is
straightforward to check that there exists exactly one $y_0\in \mathbb R$
such that $L(s+i y_0)$ is balanced.
The term \term{tempered} refers to both the Selberg bound (Axiom 4a) and
the Ramanujan bound (Axiom 4b). Neither bound has been proven for
most automorphic $L$-functions, but if those axioms fail
for an automorphic $L$-function, they must fail
in a specific way arising from the fact that the underlying representation
is unitary. A precise description of the possibilities is given by
the \term{unitary pairing condition}, described in the appendix.
In the functional equation \eqref{eqn:FE}, the function $\overline{\Lambda}$ is the
Schwartz reflection of $\Lambda$, defined for arbitrary analytic
functions $f$ by $\overline{f}(z) = \overline{f(\overline{z})}$.
The tuple $(\varepsilon,N,\{\mu_1,\ldots,\mu_{d_1}\},\{\nu_k,\ldots,\nu_{d_2}\})$
is the \term{functional equation data} of the $L$-function. The sign $\varepsilon$ of the functional equation has absolute value 1: to see this, apply the
functional equation twice to get $\Lambda(s) = \varepsilon\overline{\varepsilon}\Lambda(s)$.
In the Euler product, the polynomials $F_p$ are known as the
\term{local factors}, and the reciprocal roots $\alpha_{j,p}$ are
called the \term{Satake parameters} at $p$.
If $p\mid N$ then we say $p$ is a \term{bad} prime,
and if $p\nmid N$ then $p$ is \term{good}.
It follows straight from the definition that if $L_1(s)$ and $L_2(s)$ are
analytic $L$-functions then so is $L_1(s) L_2(s)$.
And if both $L_1$ and $L_2$ are balanced, or tempered, then so is their product.
If the analytic $L$-function $L(s)$ cannot be written non-trivially as
$L(s)=L_1(s) L_2(s)$, then we say
that $L$ is \term{primitive}. Here `non-trivially' refers to the fact
that the constant function 1 is a degree 0 $L$-function.
It follows from the Selberg orthogonality
conjecture~\cite{sel, cg}
that $L$-functions factor uniquely into
primitive $L$-functions:
\begin{thm}\cite{cg}\label{thm:factorization} Assume the Selberg orthogonality
conjecture. If $L(s)$ is an analytic $L$-function then
\begin{equation}
L(s) = L_1(s) \cdots L_n(s)
\end{equation}
where each $L_j$ is a nontrivial primitive analytic $L$-function.
The representation is unique except for the order of the
factors.
\end{thm}
If `tempered' is included as a condition in Theorem~\ref{thm:factorization}, then each
$L_j$ is tempered. If `balanced' is included as a condition, then
the conclusion can be written as
\begin{equation}
L(s) = L_1(s+ i y_1) \cdots L_n(s+i y_n)
\end{equation}
where each $L_j$ is balanced, $y_j\in \mathbb R$, and
$\sum d_j y_j = 0$, where $d_j$ is the degree of $L_j$.
Examples such as $\zeta(s+5i)\zeta(s-5i)$ and $L(s+6i,\chi)L(s-3i, E)$
show that a non-primitive balanced $L$-function cannot necessarily be written as
a product
of primitive balanced $L$-functions.
The motivation for the Central Character axioms comes from
the $\mathbb Q$-automorphic $L$-functions which we describe in the next section.
See the discussion preceding equations~\eqref{eqn:Fpchi2} and~\eqref{eqn:parityaxiom2}: note, in particular, that discussion explains why Axioms~5a) and~5b) are equivalent
if the $L$-function is $\mathbb Q$-automorphic.
\subsection{\texorpdfstring{$\mathbb Q$}{}-automorphic \texorpdfstring{$L$}{}-functions}
For a number field $F$, let $\mathbb A_F$ denote the ring of adeles of $F$. In this section we consider $L$-functions of cuspidal automorphic representations $\pi$ of the group $\GL(n,\mathbb A_\mathbb Q)$. For such $\pi$ we will always use the same conventions as in \cite{Cogdell04}; in particular, we assume $\pi$ to be irreducible and unitary. Then $\pi$ admits a unitary central character $\omega_\pi$, which is a character of
\[
\mathbb Q^\times \backslash \mathbb A_\mathbb Q^\times = \mathbb R_{>0} \times \prod_{p< \infty} \mathbb Z_p^\times.
\]
There exists a unique real number $y$ and a character $\chi$ of
$\mathbb Q^\times\backslash\mathbb A_\mathbb Q^\times$ of finite order such that
$\omega_\pi=|\cdot|^{iy}\chi$. The character $\chi$ corresponds to
a Dirichlet character, also denoted by $\chi$:
\begin{equation}\label{ccexpleq2}
\omega_\pi(x)=x^{iy}\qquad\text{for }x\in\mathbb R_{>0}.
\end{equation}
We say $\pi$ is \term{balanced} if the restriction of $\omega_\pi$ to $\mathbb R_{>0}$ is trivial,
that is, if $y=0$.
Evidently, this is equivalent to $\omega_\pi$ being of finite order. In this case $\omega_\pi$ corresponds to a Dirichlet character $\chi$. The correspondence is such that if $\omega_\pi$ factors as $\prod_{p\leq\infty}\omega_{\pi,p}$, then
\begin{equation}\label{eqn:omegap}
\omega_{\pi,p}(p)=\chi(p)
\end{equation}
and
\begin{equation}\label{eqn:omegainfty}
\omega_{\pi,\infty}(-1)=\chi(-1).
\end{equation}
\begin{defn}\label{def:automorphic}
Let $\pi=\otimes_{p\leq\infty}\pi_p$ be a cuspidal automorphic representation of $\GL(d,\mathbb A_\mathbb Q)$. Let
\[
L(s,\pi)= \prod_{p < \infty} L(s,\pi_p)
\]
be the finite part of the Langlands $L$-function associated to $\pi$ with respect to the standard representation of the dual group $\GL(d,\mathbb C)$. We call $L(s,\pi)$ a \term{$\mathbb Q$-automorphic $L$-function}.
\end{defn}
We do not consider automorphic representations for groups other then $\GL(n)$ or for number fields other than $\mathbb Q$. This is not a serious restriction, since more general automorphic representations can always be transferred to $\GL(n)$ over $\mathbb Q$, at least conjecturally; see \cite{Arthur} for some recent and deep results. This transfer may not be cuspidal, however, so our definition \emph{will} exclude some more general automorphic $L$-functions. The following examples will illustrate why restricting to $\GL(n)$ over $\mathbb Q$ is not harmful, and in fact desirable for our purposes.
First consider Hilbert modular cuspforms on a real quadratic number field $F$. As explained in \cite{RaghuramTanabe2011}, such modular forms correspond to cuspidal automorphic representations $\pi$ of $\GL(2,\mathbb A_F)$. The $L$-function $L(s,\pi)$ is a degree $2$ $L$-function \emph{over $F$}. We may consider the automorphic induction $\mathcal{AI}_{F/\mathbb Q}(\pi)$, which is an automorphic representation of $\GL(4,\mathbb A_\mathbb Q)$. It has the same $L$-function $L(s,\pi)$, but now we consider it as a degree $4$ $L$-function \emph{over $\mathbb Q$}. If $\mathcal{AI}_{F/\mathbb Q}(\pi)$ is cuspidal, then $L(s,\pi)$ is included in our definition of automorphic $L$-function. Assume that $\mathcal{AI}_{F/\mathbb Q}(\pi)$ is not cuspidal. Then, by \textsection~3.6 of \cite{ArthurClozel1989}, $\pi$ is Galois-invariant, and therefore in the image of the base change map from $\GL(2,\mathbb A_\mathbb Q)$ to $\GL(2,\mathbb A_F)$. It follows that there exists a cuspidal, automorphic representation $\pi_\mathbb Q$ of $\GL(2,\mathbb A_\mathbb Q)$ such that $L(s,\pi)=L(s,\pi_\mathbb Q)L(s,\pi_\mathbb Q\otimes\chi_{F/\mathbb Q})$, where $\chi_{F/\mathbb Q}$ is the quadratic character corresponding to the extension $F/\mathbb Q$. We will later compare the class of $L$-functions according to Definition \ref{def:automorphic} with the class of \emph{primitive} analytic $L$-functions. It is therefore advantageous to exclude a non-primitive example like $L(s,\pi)$ from our definition of automorphic $L$-function.
As our second example, consider Siegel modular cuspforms $F$ of degree $2$. Such $F$ correspond to cuspidal automorphic representations $\pi$ of ${\rm GSp}(4,\mathbb A_\mathbb Q)$, which, at least conjecturally, can be transferred to $\GL(4,\mathbb A_\mathbb Q)$. For example, if $F$ has full level and is not a Saito-Kurokawa lifting, then the transfer is established in \cite{PitaleSahaSchmidt2014}. This transfer is again cuspidal, and thus the spin (degree $4$) $L$-function $L(s,\pi)$ is included in Definition \ref{def:automorphic}. Assume however that $F$ is a Saito-Kurokawa lifting. Then the transfer to $\GL(4,\mathbb A_\mathbb Q)$ still exists, but is no longer cuspidal. Hence, in this case, $L(s,\pi)$ is not included in our definition of automorphic $L$-function. This is desirable, since $L(s,\pi)$
is of the form $\zeta(s-\frac12)\zeta(s+\frac12)L(s, f)$, which is neither primitive nor satisfies the Ramanujan condition.
Thus, it should be excluded from our comparison with the class of primitive tempered analytic $L$-functions.
We see from these examples that $L$-functions of cuspidal automorphic representations are in general
not primitive, in the sense that they may factor as products of $L$-functions of smaller degrees.
The following lemma shows that this cannot happen for $\mathbb Q$-automorphic $L$-functions.
\begin{lem}\label{Qautoprimitivelemma}
Let $\pi$ be a cuspidal automorphic representation of \linebreak$\GL(d,\mathbb A_\mathbb Q)$. Then the $\mathbb Q$-automorphic $L$-function $L(s,\pi)$ is primitive, i.e., if
\begin{equation}\label{Qautoprimitivelemmaeq1}
L(s,\pi)=\prod_{j=1}^mL(s,\pi_j)
\end{equation}
with cuspidal automorphic representations $\pi_j$ of $\GL(d_j,\mathbb A_\mathbb Q)$ ($d_j>0$) for $1\leq j\leq
m$, then $m=1$.
\end{lem}
\begin{proof}
We consider partial $L$-functions and twist \eqref{Qautoprimitivelemmaeq1} by the contragredient
$\pi_1^\vee$ of $\pi_1$:
\begin{equation}\label{Qautoprimitivelemmaeq2}
L_S(s,\pi\times\pi_1^\vee)=\prod_{j=1}^mL_S(s,\pi_j\times\pi_1^\vee).
\end{equation}
By the Corollary to Theorem 2.4 of \cite{Cogdell04}, $L_S(s,\pi_1\times\pi_1^\vee)$ has a pole at
$s=1$. By Theorem 5.2 of \cite{Shahidi1981}, $L_S(s,\pi_j\times\pi_1^\vee)$ has no zeros on ${\rm
Re}(s)=1$ for any $j$. It follows that the right hand side of \eqref{Qautoprimitivelemmaeq2} has a
pole at $s=1$. Hence so does the left hand side, which again by Theorem 2.4 of \cite{Cogdell04}
implies that $\pi=\pi_1$. In other words, we must have $m=1$.
\end{proof}
\section{\texorpdfstring{$\mathbb Q$}{}-automorphic \texorpdfstring{$L$}{}-functions and the axioms}\label{sec:Q-auto}
Now that we have defined $\mathbb Q$-automorphic $L$-functions and have identified a collection of axioms for analytic $L$-functions,
we begin to show that $\mathbb Q$-automorphic $L$-functions satisfy the axioms.
\begin{thm}\label{thm:niceness}
Let $L(s,\pi)$ be a $\mathbb Q$-automorphic $L$-function.
There is a positive integer $N$, a pair of non-negative integers $(d_1, d_2)$ so that $d_1+2d_2=d$,
and complex numbers $\{\mu_j\}$ and $\{\nu_j\}$ such that
\begin{equation}\label{thm:nicenesseq0}
\Lambda(s,\pi) =\mathstrut N^{s/2}
\prod_{j=1}^{d_1} \Gamma_\mathbb R(s+ \mu_j)
\prod_{j=1}^{d_2} \Gamma_\mathbb C(s+ \nu_j)
\cdot L(s,\pi)
\end{equation}
has the following properties:
\begin{enumerate}
\item \label{item:ent} $\Lambda(s,\pi)$ is entire.
\item \label{item:bdd} $\Lambda(s,\pi)$ is bounded in vertical strips $\sigma_1 \leq \sigma \leq \sigma_2$.
\item\label{item:FE} There exists $\varepsilon\in \mathbb C$ such that
\begin{equation*}
\Lambda(s,\pi)=\varepsilon \overline{\Lambda}(1-s,\pi).
\end{equation*}
\item \label{item:bal} If $\pi$ is balanced, then $L(s,\pi)$ is balanced.
\end{enumerate}
\end{thm}
In other words, $\mathbb Q$-automorphic $L$-functions satisfy Axioms 1) and 2) as described in Section~\ref{sec:axioms}.
\begin{proof} For items \eqref{item:ent}, \eqref{item:bdd} and \eqref{item:FE}, see \cite[Theorems 2.3 and 2.4]{Cogdell04}. Note that the functional equation in \cite{Cogdell04} is written as $\Lambda(s,\pi)=\varepsilon\Lambda(1-s,\tilde\pi)$, where $\tilde\pi$ is the contragredient representation; one can show that $\Lambda(1-s,\tilde\pi)=\bar\Lambda(1-s,\pi)$ for unitary $\pi$ (recall that $\bar f(s)=\overline{f(\bar s)}$ for a function of a complex variable).
Item \eqref{item:bal} follows by considering the local Langlands parameter at the archimedean place, keeping in mind that the determinant of this parameter corresponds to the central character of $\pi_\infty$.
\end{proof}
The integer $N$ appearing in the functional equation equals $\prod_p p^{a(\pi_p)}$, where $a(\pi_p)$ is the exponent of the conductor of the local representation $\pi_p$. Let $M=\prod_p p^{a(\omega_{\pi,p})}$, where $a(\omega_{\pi,p})$ is the exponent of the conductor of $\omega_{\pi,p}$, the central character of $\pi_p$. By reduction to the supercuspidal case and using the existence of the distinguished vector exhibited in \cite{JacquetPSShalika1981}, one can easily prove that $a(\omega_{\pi,p})\leq a(\pi_p)$. Consequently, $M|N$. We may therefore consider $\chi$, which originally is a Dirichlet character mod $M$, as a Dirichlet character mod $N$. This character is the character required by Axiom 5).
By definition, $L(s,\pi)$ is an Euler product
\begin{equation}\label{eqn:EP2}
L(s,\pi)= \prod_{p \, {\rm prime}} F_p(p^{-s})^{-1},
\end{equation}
where each $F_p$ is a polynomial of degree at most $d$ (as required by Axiom 3b)), with $F_p(0)=1$.
Considering Langlands parameters at the non-archimedean places, it
follows
from \eqref{eqn:omegap} that
\begin{equation}\label{eqn:Fpchi2}
F_p(z)=1+ \cdots + (-1)^d\chi(p) z^d;
\end{equation}
as required by Axioms 3a) and 5a).
Considering Langlands parameters at the archimedean place, it follows
that
\begin{equation}\label{ccexpleq1}
\omega_\pi(x)=x^{\mathrm{Im}\left(\sum \mu_j + \sum(2\nu_k+1)\right)}\qquad\text{for }x\in\mathbb R_{>0},
\end{equation}
and so, from \eqref{eqn:omegainfty}, it follows that
\begin{equation}\label{eqn:parityaxiom2}
\chi(-1) = (-1)^{\sum \mu_j + \sum(2\nu_j+1)},
\end{equation}
showing that $\mathbb Q$-automorphic $L$-functions satisfy Axioms 5b) and 5c).
Conjecturally, each local component $\pi_p$ of a cuspidal, automorphic representation $\pi$ as in Definition \ref{def:automorphic} is \emph{tempered}; see \cite{Sarnak05}, or Conjecture 1.6 of \cite{Clozel1990}. The following lemma lists some consequences of temperedness for the spectral parameters and Satake parameters.
\begin{lem}\label{lem:tempered}
Assume that $\pi=\otimes\pi_p$ is a cuspidal automorphic representation of
$\GL(d,\mathbb A_\mathbb Q)$. Let $\mu_j,\nu_j$ be as in Theorem \ref{thm:niceness}. Let the polynomial $F_p(z)$ in \eqref{eqn:Fpchi2} be factored as
\begin{equation}\label{lem:temperedeq1}
F_p(z) = (1-\alpha_{1,p} z)\cdots (1-\alpha_{d_p,p} z)
\end{equation}
with $0\leq d_p\leq d$ and $\alpha_{j,p}\in\mathbb C$.
\begin{enumerate}
\item Assume that $\pi_\infty$ is tempered. Then for every $j$ we have $\Re(\mu_j) \in \{0,1\}$ and $\Re(\nu_j) \in \{\frac12,1,\frac32,2,...\}$.
\item Assume that $\pi_p$ is tempered for $p<\infty$ with $p\nmid N$. Then $|\alpha_{j,p}|=1$ for all $j\in\{1,\ldots,d\}$.
\item Assume that $\pi_p$ is tempered for $p<\infty$ with $p|N$. Then $|\alpha_{j,p}|= p^{-m_j/2}$ for some $m_j\in \{0,1,2,...\}$, and $\sum m_j \le d-d_p$.
\end{enumerate}
\end{lem}
This Lemma implies that $\mathbb Q$-automorphic $L$-functions satisfy Axiom~4):
the first item in the statement is about Axiom~4a) and the second two are about Axiom~4b).
\begin{proof}
(1) follows from the fact that a representation of $\GL(d,\mathbb R)$ is tempered if and only if its Langlands parameter has bounded image. For (2), see \cite{Sarnak05} (also, (2) is a special case of (3)). For (3), we consider the local parameter of $\pi_p$, which is an admissible homomorphism $\varphi:W'(\bar\mathbb Q_p/\mathbb Q_p)\to\GL(d,\mathbb C)$. Here, $W'(\bar\mathbb Q_p/\mathbb Q_p)$ is the Weil-Deligne group; see \cite{Rorlich1994} for precise definitions, and \cite{Kudla1994} for properties of the local Langlands correspondence. Let ${\rm sp}(n)$ be the $n$-dimensional indecomposable representation of $W'(\bar\mathbb Q_p/\mathbb Q_p)$ defined in \S5 of \cite{Rorlich1994}. We can write
\begin{equation}\label{lem:temperedeq2}
\varphi=\bigoplus_{j=1}^t\rho_j\otimes{\rm sp}(n_j),
\end{equation}
with uniquely determined irreducible representations $\rho_j$ of the Weil group $W(\bar\mathbb Q_p/\mathbb Q_p)$, and uniquely determined positive integers $n_i$. Evidently, $d=\sum_{j=1}^t\dim(\rho_j)n_j$. The Euler factor $L(s,\pi_p)$ is equal to the L-factor of $\varphi$, as defined in \S8 of \cite{Rorlich1994}. By the Proposition in \S8 of \cite{Rorlich1994},
\begin{equation}\label{lem:temperedeq3}
L(s,\pi)=\prod_{j=1}^tL(s+n_j-1,\rho_j).
\end{equation}
Assume that $\rho_1,\ldots,\rho_r$ are unramified characters of $W(\bar\mathbb Q_p/\mathbb Q_p)$, and that $\rho_{r+1},\ldots,\rho_t$ are either ramified characters or of dimension greater than $1$. Then
\begin{equation}\label{lem:temperedeq4}
L(s,\pi)=\prod_{j=1}^rL(s+n_j-1,\rho_j)=\prod_{j=1}^r\frac1{1-\rho_j(p)p^{-s-n_j+1}},
\end{equation}
where we identified $\rho_1,\ldots\rho_r$ with characters of $\mathbb Q_p^\times$. Comparison with \eqref{lem:temperedeq1} shows that $d_p=r$ and, after an appropriate permutation,
\begin{equation}\label{lem:temperedeq5}
\alpha_{j,p}=\rho_j(p)p^{-n_j+1}\qquad\text{for }j=1,\ldots,r.
\end{equation}
Now $\pi_p$ is tempered if and only if the representations $|\cdot|_p^{(n_i-1)/2}\otimes\rho_i$ are bounded for $i=1,\ldots,t$; see \textsection~2.2 of \cite{Kudla1994}. In particular, assuming $\pi_p$ is tempered, we have $|\rho_j(p)|=p^{(n_j-1)/2}$, and thus $|\alpha_{j,p}|=p^{-(n_j-1)/2}$. Setting $m_j=n_j-1$, we have $|\alpha_{j,p}|=p^{-m_j/2}$ and
$$
\sum_{j=1}^{d_p}m_j=\sum_{j=1}^{d_p}n_j-d_p\leq d-d_p.
$$
This concludes the proof.
\end{proof}
\begin{prop}\label{prop:automorphicanalytic}
Assume that $\pi=\otimes\pi_p$ is a cuspidal automorphic representation of $\GL(d,\mathbb A_\mathbb Q)$ such that each local component $\pi_p$ is tempered. Then $L(s,\pi)$ is a tempered analytic $L$-function in the sense of Section~\ref{sec:defn_analytic}.
If $\pi$ is balanced, then $L(s,\pi)$ is balanced.
\end{prop}
\begin{proof}
In the balanced case, this follows from Theorem \ref{thm:niceness}, equations \eqref{eqn:Fpchi2} and \eqref{eqn:parityaxiom2}, and Lemma \ref{lem:tempered}. Assume $\pi$ is not balanced. Then there exists $y\in\mathbb R$ such that $|\cdot|^{iy}\otimes\pi$ is balanced. Hence $L(s,|\cdot|^{iy}\otimes\pi)=L(s+iy,\pi)$ is a balanced tempered analytic $L$-function. Consequently, $L(s,\pi)$ is a tempered analytic $L$-function.
\end{proof}
\section{Algebraic and arithmetic \texorpdfstring{$L$}{}-functions}\label{sec:aa}
The most widely studied $L$-functions are those arising from arithmetic
objects such as elliptic and higher-genus curves, holomorphic modular forms,
number fields, Artin representations,
Galois representations, and motives. We give two characterizations of
such $L$-functions: one in terms of their Dirichlet coefficients, and the
other in terms of their spectral parameters.
\subsection{Analytic \texorpdfstring{$L$}{}-functions of algebraic type}\label{sec:analytic-of-alg-type}
Building on ideas of Stark and Hejhal, Booker, Str\"ombergsson,
and Venkatesh~\cite{bsv}, carried out computations that support the
conjecture that if $\lambda>\frac14$ is the Laplacian eigenvalue of a Maass
form, then $\lambda$ is transcendental. Since the $\Gamma$-shifts
in the associated $L$-function have imaginary part $\pm \sqrt{\lambda - \frac14}$,
one expects that the imaginary part of any
$\Gamma$-shift in a primitive balanced analytic $L$-function is either 0 or
transcendental. This motivates the following definition.
\begin{defn}\label{defn:Lalgebraictype} Suppose $L(s)$ is an analytic $L$-function with
spectral parameters $\{\mu_j\}$ and $\{\nu_k\}$. We say that $L(s)$
is \term{of algebraic type} if either
every $\mu_j$ and $\nu_k$ is in $\mathbb Z$, or
every $\mu_j$ and $\nu_k$ is in $\frac12 + \mathbb Z$.
The integer $w_{alg} = 2 \max\{0, \nu_1,\ldots,\nu_{d_2}\}$ is
called the \term{algebraic weight} of the $L$-function.
\end{defn}
The second option: $\mu_j$ and $\nu_k$ are in $\frac12 + \mathbb Z$,
implies that there are no $\mu_j$,
because if $L$ is tempered then $\mu_j\in \{0,1\}$,
and in general (see the Appendix on the unitary pairing condition),
$\mu_j \in (-\frac12,\frac12) \cup (\frac12,\frac32)$.
In Definition~1.6 of \cite{Clozel1990}, Clozel defined the notion of \emph{algebraic}
automorphic representation of $\GL(n)$ over a number field. We use the term
\emph{of algebraic type} because our notion applies to the $L$-function, not
its underlying representation. See Section~\ref{sec:Q-analytic-automorphic}
for more details.
The term \emph{algebraic weight} was chosen because if $M$ is a motive
of weight $w$,
then by Serre's recipe \cite{Serre1969} $L(s,M)$ will have algebraic weight~$w$.
\subsection{Analytic \texorpdfstring{$L$}{}-functions of arithmetic type}
\label{sec:defn_arithmetic}
The computations of Booker, Str\"ombergsson,
and Venkatesh~\cite{bsv}
also support the conjecture that in general the Fourier coefficients of
Maass forms with Laplacian eigenvalue $\lambda > \frac14$ are
transcendental and algebraically independent except for the constraints
imposed by the Hecke relations. Thus we have a complement to the previous
definition, involving a condition on the Dirichlet coefficients.
\begin{defn}\label{defn:Larithmetictype} Suppose $L(s) = \sum a_n n^{-s}$ is an analytic
$L$-function. We say that $L(s)$ is \term{of arithmetic type}
if there exists $w_{ar} \in \mathbb Z$
and a number field $F$ such that
$a_n n^{{w_{ar}}/2}\in \mathcal{O}_F$ for all $n$.
The smallest such $F$ is called the
\term{field of coefficients}, and the smallest such $w_{ar}$
is called the \term{arithmetic weight} of the $L$-function.
\end{defn}
An analytic $L$-function with algebraic coefficients is not
necessarily of arithmetic type, as shown by the example
$L(s) = L(s,\chi)L(s,E)$, where $\chi$ is a
primitive Dirichlet character and $E/\mathbb Q$ is an elliptic curve.
As indicated in Figure~\ref{fig:diagram2}, it is conjectured that
such examples must be non-primitive.
As we will explain in Section~\ref{sec:connections}, by combining existing
conjectures one obtains the conjecture that a primitive balanced analytic $L$-function
is of algebraic type if and only if it is of arithmetic type.
Furthermore, we have the \term{Hodge conjecture}: $w_{alg} = w_{ar}$.
\subsection{\texorpdfstring{$\mathbb Q$}{}-automorphic \texorpdfstring{$L$}{}-functions of algebraic type}\label{sec:Q-analytic-automorphic}
Let $\mathbb A_F$ be the ring of adeles of a number field $F$. In \cite{Clozel1990}, Clozel considered isobaric automorphic representations of $\GL(n,\mathbb A_F)$. He called such a representation \term{algebraic} if the local Langlands parameters at all archime\-dean places satisfy certain integrality conditions. More generally, for a connected, reductive $F$-group $G$ and an automorphic representation of $G(\mathbb A_F)$, Buzzard and Gee in \cite{BuzzardGee2014} defined the notions of \term{$C$-algebraic} and \term{$L$-algebraic}. If $G=\GL(n)$ and $\pi$ is isobaric, then
$$
\pi\text{ is algebraic }\;\Longleftrightarrow\;\pi\text{ is $C$-algebraic }\;\Longleftrightarrow\;\pi|\cdot|^{\frac{n-1}2}\text{ is $L$-algebraic}.
$$
In the case of a tempered automorphic representation $\pi\cong\otimes\pi_p$ of $\GL(d,\mathbb A_\mathbb Q)$, the notions of $C$-algebraic and $L$-algebraic can easily be expressed in terms of the archime\-dean Euler factor. Recall that this factor is of the general form
\begin{equation}\label{archLgeneraleq}
L(s,\pi_\infty)=\prod_{j=1}^{d_1}\Gamma_\mathbb R(s+\mu_j)\prod_{k=1}^{d_2}\Gamma_\mathbb C(s+\nu_k)
\end{equation}
with complex numbers $\mu_j$ and $\nu_k$, and $d=d_1+2d_2$.
\begin{lem}\label{LCalglemma}
Let $\pi\cong\otimes\pi_p$ be an automorphic representation of\linebreak $\GL(d,\mathbb A_\mathbb Q)$ such that $\pi_\infty$ is tempered. Let $L(s,\pi_\infty)$ be as in \eqref{archLgeneraleq}.
\begin{enumerate}
\item Assume that $d$ is even. Then
\begin{itemize}
\item $\pi$ is $C$-algebraic if and only if $d_1=0$ and $\nu_k\in\frac12+\mathbb Z_{\geq0}$ for $k=1,\ldots,d/2$.
\item $\pi$ is $L$-algebraic if and only if $\mu_j\in\{0,1\}$ for $j=1,\ldots,d_1$ and $\nu_k\in\mathbb Z_{>0}$ for $k=1,\ldots,d_2$.
\end{itemize}
\item Assume that $d$ is odd. Then $\pi$ is $C$-algebraic if and only if $\pi$ is $L$-algebraic if and only if $\mu_j\in\{0,1\}$ for $j=1,\ldots,d_1$ and $\nu_k\in\mathbb Z_{>0}$ for $k=1,\ldots,d_2$.
\end{enumerate}
\end{lem}
\begin{proof}
This follows in a straightforward manner from Definitions~5.7 and~5.9 of \cite{BuzzardGee2014}, and the well-known recipe of attaching $\Gamma$-factors to representations of the real Weil group.
\end{proof}
\begin{rmk} Suppose $L(s,\pi)$ is a $\mathbb Q$-automorphic $L$-function, coming from a unitary cuspidal automorphic representation $\pi=\otimes_{p\leq\infty}\pi_p$ of $\GL(d,\mathbb A_\mathbb Q)$, as in Definition \ref{def:automorphic}. In particular, as shown in the course of Section~\ref{sec:Q-auto}, $L(s,\pi)$ is an analytic $L$-function.
Thus, we can say that $L(s,\pi)$
is \term{of algebraic type} if it satisfies the conditions of
Definition~\ref{defn:Lalgebraictype}.
\end{rmk}
Therefore by the lemma we have:
\begin{rmk}\label{rem:Q1cupQ2}
By the Ramanujan conjecture, all local components $\pi_p$ of the cuspidal automorphic representation $\pi$ are tempered. Assuming this is the case, we see that $L(s,\pi)$ is of algebraic type if and only if $\pi$ is either $C$-algebraic or $L$-algebraic.
\end{rmk}
\subsection{\texorpdfstring{$\mathbb Q$}{}-automorphic \texorpdfstring{$L$}{}-functions of arithmetic type}\label{ssec:arithmetic}
Let $G$ be a connected, reductive group over the number field $F$, and let $\pi\cong\otimes\pi_v$ be an automorphic representation of $G(\mathbb A_F)$. Buzzard and Gee \cite{BuzzardGee2014} define the notions of $\pi$ being \term{$C$-arithmetic} and \term{$L$-arithmetic} in terms of the Satake parameters of $\pi_v$ at almost all places. For $G=\GL(n)$ it is true that
$$
\pi\text{ is $C$-arithmetic }\;\Longleftrightarrow\;\pi|\cdot|^{\frac{n-1}2}\text{ is $L$-arithmetic}.
$$
The conditions can easily be reformulated in terms of $L$-functions:
\begin{lem}\label{LCarithlemma}
Let $\pi\cong\otimes\pi_p$ be an automorphic representation of \linebreak$\GL(d,\mathbb A_\mathbb Q)$. Let $S$ be a finite set of primes such that $\pi_p$ is unramified for primes $p\notin S$. Let
\begin{equation}\label{LCarithlemmaeq1}
L(s,\pi_p)=\frac1{(1-\alpha_{p,1}p^{-s})\cdot\ldots\cdots(1-\alpha_{p,d}p^{-s})}
\end{equation}
be the Euler factor for $p\notin S$.
\begin{enumerate}
\item Assume that $d$ is even. Then
\begin{itemize}
\item $\pi$ is $C$-arithmetic if and only if there exists a number field $E$ such that $\alpha_{p,1}\sqrt{p},\ldots,$ $\alpha_{p,d}\sqrt{p}\in E$ for almost all $p\notin S$.
\item $\pi$ is $L$-arithmetic if and only if there exists a number field $E$ such that $\alpha_{p,1},\ldots,$ $\alpha_{p,d}\in E$ for almost all $p\notin S$.
\end{itemize}
\item Assume that $d$ is odd. Then $\pi$ is $C$-arithmetic if and only if $\pi$ is $L$-arithmetic if and only if there exists a number field $E$ such that $\alpha_{p,1},\ldots,\alpha_{p,d}\in E$ for almost all $p\notin S$.
\end{enumerate}
\end{lem}
The following is equivalent to Definition~\ref{defn:Larithmetictype}, but we include this formulation because
it is stated in terms of the parameters which are more natural for
$\mathbb Q$-automorphic $L$-functions.
\begin{defn}\label{def:automorphic-arith}
Let $L(s,\pi)$ be a $\mathbb Q$-automorphic $L$-function, coming from a cuspidal automorphic representation $\pi=\otimes_{p\leq\infty}\pi_p$ of $\GL(d,\mathbb A_\mathbb Q)$, as in Definition \ref{def:automorphic}. Let
\begin{equation}\label{def:automorphic-aritheq1}
L(s,\pi_p)=\frac1{(1-\alpha_{p,1}p^{-s})\cdot\ldots\cdots(1-\alpha_{p,d}p^{-s})}
\end{equation}
be the Euler factor at a prime $p$. We say that $L(s,\pi)$ is \term{of arithmetic type} if there exists a number field $E$ such that either
\begin{equation}\label{def:automorphic-aritheq2}
\alpha_{p,1},\ldots,\alpha_{p,d}\in E\qquad\text{for almost all good primes $p$}
\end{equation}
or
\begin{equation}\label{def:automorphic-aritheq3}
\alpha_{p,1}\sqrt{p},\ldots,\alpha_{p,d}\sqrt{p}\in E\qquad\text{for almost all good primes $p$}.
\end{equation}
\end{defn}
\begin{rmk}
Let $\pi$ be as in Definition \ref{def:automorphic-arith} and suppose $d$ is odd.
Then \eqref{def:automorphic-aritheq3} cannot occur.
Indeed, if \eqref{def:automorphic-aritheq3} would hold, then we would also have
\begin{equation}\label{def:automorphic-aritheq4}
\alpha_{p,1}\cdot\ldots\cdot\alpha_{p,d}\sqrt{p}\in E\qquad\text{for almost all good primes $p$}.
\end{equation}
But the numbers $\alpha_{p,1}\cdot\ldots\cdot\alpha_{p,d}$ are the Satake parameters of the central character $\omega_\pi$ of $\pi$, which, up to a unitary twist, corresponds to a Dirichlet character $\chi$; see \eqref{eqn:omegap}. Hence we would have
\begin{equation}\label{def:automorphic-aritheq5}
\chi(p)p^{it}\sqrt{p} = \chi(p)p^{it + \frac12}\in E\qquad\text{for almost all good primes $p$}
\end{equation}
for some real number $t$, so in particular
\begin{equation}\label{def:automorphic-aritheq6}
p^{it + \frac12}\in E\qquad\text{for infinitely many $p$. }
\end{equation}
This is impossible for $t\in \mathbb R$ because of the following consequence of
the Six Exponentials Theorem~\cite{MR972013}.
\end{rmk}
\begin{lem} If $\alpha\in \mathbb C$ and $E/\mathbb Q$ is a number field with
$p^\alpha \in E$ for infinitely many primes~$p$, then $\alpha\in \mathbb Z$.
\end{lem}
\begin{proof} Suppose $\alpha \not\in \mathbb Z$. We cannot have $\alpha\in \mathbb Q$
because $E$ is a finite extension of $\mathbb Q$.
Thus, $\{1, \alpha\}$ is linearly independent over the rationals.
(We also cannot have $\alpha$ algebraic, because by the Gelfond-Schneider
theorem~$p^\alpha$ would be transcedental. This does not seem to be
needed in the proof.)
Suppose $p_1$, $p_2$, and $p_3$ are distinct primes with $p_j^\alpha \in E$.
Since $\{\log p_1, \log p_2, \log p_3\}$ is linearly independent over the
rationals we have a contradiction because by the Six Exponentials Theorem~\cite{MR972013},
one of $p_1$, $p_2$, $p_3$, $p_1^\alpha$, $p_2^\alpha$, and $p_3^\alpha$ must
be transcendental.
\end{proof}
As a consequence of Lemma \ref{LCarithlemma} and the above remark,
we see that $L(s,\pi)$ is of arithmetic type if and only if $\pi$ is either $C$-arithmetic or $L$-arithmetic (see Section~\ref{sec:connections}).
So the conjectures of Clozel and Buzzard-Gee yield:
\begin{conj}\label{conj:Q1=Q2}
A $\mathbb Q$-automorphic $L$-function is of algebraic type if and only if it is of arithmetic type.
\end{conj}
A more general conjecture for automorphic representations $\pi$ of \linebreak $G(\mathbb A_F)$ was made
by Buzzard and Gee in \cite{BuzzardGee2014}.
Namely, $\pi$ is $L$-algebraic if and only if it is $L$-arithmetic, and $\pi$ is $C$-algebraic if and only if it is $C$-arithmetic. The surprising fact about these conjectures is that a condition purely in terms of archimedean $L$-parameters is conjecturally equivalent to a condition purely in terms of non-archimedean $L$-parameters.
For isobaric automorphic representations of $\GL(n,\mathbb A_F)$, the conjecture that $C$-algebraic implies $C$-arithmetic is also a consequence of the more general conjectures made in \cite{Clozel1990, MR3642468}; see \textsection~8.1 of \cite{BuzzardGee2014}.
\subsection{\texorpdfstring{$L$}{}-functions of motives}\label{ssec:motives}
Let $F$ be a number field. The category of motives over $F$ has been constructed by Grothendieck in the 1960s. The first article about motives seems to be \cite{Demazure1969}. More contemporary and useful surveys are \cite{Kleiman1994}, \cite{Scholl1994} and \cite{Milne2013}. We will not recall here the construction of the category of motives. What is important for us is that given a \emph{pure motive $M$ of weight $w$}, there is an $L$-function $L(M,s)$ attached to it, which, after completing it to a function $\Lambda(M,s)$ using appropriate $\Gamma$-factors, conjecturally satisfies a functional equation $\Lambda(M,s)=\pm\bar\Lambda(M,1+w-s)$. We write $\Lambda(M,s)$ and not $\Lambda(s,M)$, because the functional equation relates $s$ and $1+w-s$, and not $s$ and $1-s$.
We briefly recall the shape of $\Lambda(M,s)$, using \cite{Serre1969} as our reference. We assume that the ground field is $\mathbb Q$ for simplicity. For each non-archimedean place $p$ the characteristic polynomial $P_{w,p}$ of the action of Frobenius on the inertia-fixed points of the $w$-th \'{e}tale cohomology group (at least conjecturally) has coefficients in $\mathbb Z$. We set
\begin{equation}\label{motLeq3}
L_p(M,s)=\frac1{P_{w,p}(p^{-s})},
\end{equation}
and $L(M,s)=\prod_{p<\infty}L_p(M,s)$. There exists a positive integer $d$, called the \term{rank} of $M$, such that for almost all places the polynomial $P_{w,p}$ will have degree $d$. For all ``good'' places $p$, if we factor
\begin{equation}\label{motLeq4}
P_{w,p}(T)=\prod_{j=1}^{b_w}(1-\alpha_{j,p}T),\qquad\alpha_{j,p}\in\mathbb C^\times,
\end{equation}
then it is conjectured that
\begin{equation}\label{motLeq5}
|\alpha_{j,p}|=p^{w/2}
\end{equation}
(see $C_7$ in \textsection~2.3 of \cite{Serre1969}). At the archimedean place $\infty$ we have
\begin{equation}\label{motLeq1}
L_\infty(M,s)=\Gamma_\mathbb R\Big(s-\frac w2\Big)^{h^{w/2,+}}\Gamma_\mathbb R\Big(s-\frac w2+1\Big)^{h^{w/2,-}}\prod_{\substack{p+q=w\\p<q}}\Gamma_\mathbb C(s-p)^{h^{p,q}},
\end{equation}
with the first two factors only appearing if $w$ is even. Here, the $h^{p,q}$ are the dimensions of the spaces $H^{p,q}$ in the Hodge decomposition of the Betti realization of $M$. If $w$ is even, then there is a space $H^{p,p}$ ($p=w/2$), on which complex conjugation acts as an involution; the number $h^{p,\pm}$ is the dimension of the $\pm1$-eigenspace.
Conjecturally, there exists a positive integer $N$ such that
\begin{equation}\label{motLeq2a}
\Lambda(M,s)=N^{(s-w/2)/2}L_\infty(M,s)L(M,s)
\end{equation}
extends to an entire function on all of $\mathbb C$, bounded in vertical strips, and satisfies the functional equation $\Lambda(M,s)=\pm\bar\Lambda(M,1+w-s)$.
Replacing $s$ by $s+w/2$, we obtain the \emph{analytically normalized} functions $L(s,M):=L(M,s+w/2)$ and $\Lambda(s,M):=\Lambda(M,s+w/2)$. The functional equation becomes $\Lambda(s,M)=\pm\bar\Lambda(1-s,M)$. The factor \eqref{motLeq1} turns into
\begin{equation}\label{motLeq2}
L_\infty(M,s+w/2)=\Gamma_\mathbb R(s)^{h^{w/2,+}}\Gamma_\mathbb R(s+1)^{h^{w/2,-}}\prod_{\substack{p+q=w\\p<q}}\Gamma_\mathbb C\Big(s+\frac{q-p}2\Big)^{h^{p,q}}.
\end{equation}
By \eqref{motLeq5}, the roots of the denominator polynomials of $L(s,M)$ will have absolute value $1$.
As described in \textsection~4.3 of \cite{Clozel1990}:
\begin{conj}\label{motautconj}
There exists a one-to-one correspondence between irreducible, pure motives $M$ over $\mathbb Q$ of rank $d$, and $C$-algebraic cuspidal automorphic representations $\pi$ of $\GL(d,\mathbb A_\mathbb Q)$, such that $L(s,M)=L(s,\pi)$.
\end{conj}
The conjecture implies that the class of analytically normalized $L$-functions arising from irreducible, pure motives is the same as the class of $\mathbb Q$-automorphic $L$-functions of algebraic type.
The conjecture that $L(s,M)$ should satisfy the required analytic properties
shared by the class of analytic $L$-functions as defined in Section~\ref{sec:defn_analytic} is known as the \term{Hasse-Weil conjecture}.
\begin{rmk} Given a $\Gamma$-factor in the analytic normalization \eqref{motLeq2},
it is not possible to determine the weight of the underlying motive.
Indeed, the motives which (conjecturally) are attached to the $L$-function
form an equivalence class, where the members
are Tate twists of each other.
It is natural to choose a twist so that non-vanishing Hodge numbers
of that motive are among
$(h^{w,0},...,h^{0,w})$, with
$h^{w,0}=h^{0,w}>0$.
The weight, $w$, of that motive will equal the algebraic weight of the
$L$-function, which explains our choice of terminology.
\end{rmk}
\subsection{\texorpdfstring{$L$}{}-functions of Galois representations}\label{ssec:Galois}
Let ${\rm Gal}(\bar\mathbb Q/\mathbb Q)$ be the absolute Galois group of $\mathbb Q$. Let $L$ be a finite extension of $\mathbb Q_\ell$ for some prime $\ell$. A continuous homomorphism $\rho:{\rm Gal}(\bar\mathbb Q/\mathbb Q)\to\GL(d,L)$ will be referred to as a \term{Galois representation}. See \cite{Serre1968} for basic facts. In \cite{FontaineMazur1995} the class of \term{geometric} Galois representations was defined. As summarized in Taylor \cite{Taylor}, Galois representations arising from motives are geometric. Conversely, the Fontaine-Mazur conjecture asserts that any geometric Galois representation is motivic. Hence, the class of (analytically normalized) $L$-functions arising from geometric Galois representations should be the same as the class of (analytically normalized) $L$-functions attached to Galois representations. Assuming Conjecture \ref{motautconj}, this is also the same as the class of $\mathbb Q$-automorphic $L$-functions of algebraic type.
This explains the triangle in the upper left corner of Figure~\ref{fig:diagram2}.
Conjecture 5.16 of \cite{BuzzardGee2014}, for $G=\GL(d)$ over $\mathbb Q$, makes the statement that a geometric Galois representation is attached to an $L$-algebraic automorphic representation $\pi$ of $\GL(d,\mathbb A_\mathbb Q)$. For recent progress on this conjecture, see \cite{Shin2011} and \cite{HarrisLanTaylorThorne2016}. That every Galois representation arises from an automorphic representation is known as the Modularity Conjecture.
\section{Connections}\label{sec:connections}
The equalities between sets of $L$-functions in Figure~\ref{fig:diagram1} are a consequence of the 12 relations shown in Figure~\ref{fig:diagram2}, where an arrow means ``can be viewed as a natural subset of.'' Most of those arrows are at least partially conjectural. More detailed explanations of these arrows can be found in Tables~\ref{tbl:table1} and \ref{tbl:table2}.
\begin{table}
\def1.5{1.5}
\begin{tabular}{lll}
Connection & Label & Justification\\\hline
$Q \subset A$ & J-PS-S & \begin{minipage}[t]{.5\columnwidth}A result generally due to Cogdell--Piatetski-Shapiro and Jacquet--Piatetski-Shapiro--Shalika \cite{JacquetPSShalika1981,Cogdell04} as formulated in Proposition~\ref{prop:automorphicanalytic}.\end{minipage}\\
$A\subset Q$ & S & \begin{minipage}[t]{.5\columnwidth}Selberg \cite{sel} identified a class of axioms for $L$-functions and implicitly conjectured that this class is contained in the class associated to automorphic representations as defined by Langlands \cite{Langlands1980}. $A$ and $Q$ are, respectively, subsets of these classes with the same formal properties.\end{minipage}\\
$Q_*\subset Q$ & & \begin{minipage}[t]{.5\columnwidth}Restriction to a subset.\end{minipage}\\
$Q_1\cup Q_2 = Q_*$ & & \begin{minipage}[t]{.5\columnwidth} Remark~\ref{rem:Q1cupQ2}.\end{minipage}\\
$Q_1=Q_2$ & B-G & \begin{minipage}[t]{.5\columnwidth}A conjecture due to Buzzard--Gee \cite{BuzzardGee2014} as formulated in Conjecture~\ref{conj:Q1=Q2}.\end{minipage}\\
$A_*\subset A$ & & \begin{minipage}[t]{.5\columnwidth}Restriction to a subset.\end{minipage}\\
$Q_1 = A_1$ & & \begin{minipage}[t]{.5\columnwidth}Formal if we assume $A=Q$.\end{minipage}\\
$Q_2 \subset A_2$ & & \begin{minipage}[t]{.5\columnwidth} A result of Clozel \cite{Clozel1990} implies that the (suitably) re-scaled coefficients are integers and the inclusion otherwise is formal.\end{minipage}\\
$A_2\subset Q_2$ &&\begin{minipage}[t]{.5\columnwidth}Formal.\end{minipage}\\
$A_1=A_2$ & & \begin{minipage}[t]{.5\columnwidth} Piecing together previous connections.\end{minipage}\\
\begin{minipage}[t]{.3\columnwidth}
$A_*=A_1=A_2\\ \phantom{A_*}=Q_1=Q_2=Q_*$\end{minipage}&&\begin{minipage}[t]{.5\columnwidth}Piecing together previous connections.\end{minipage}\\
&&\\
\end{tabular}
\caption{Explanations of the arrows in Figure~\ref{fig:diagram2} between analytic $L$-functions and $\mathbb Q$-automorphic $L$-functions. These explanations and these arrows correspond to Section~\ref{sec:Q-auto} and Sections~\ref{sec:analytic-of-alg-type}--\ref{ssec:arithmetic}. If the justification for a connection is ``formal'', this means that it is an immediate corollary of the axioms or properties satisfied by the sets of $L$-functions being connected.}
\label{tbl:table1}
\end{table}
\begin{table}
\def1.5{1.5}
\begin{tabular}{lll}
Connection & Label & Justification\\\hline
$M\subset A_*$ & H-W & \begin{minipage}[t]{.5\columnwidth}In our notation this is a restatement of the Hasse--Weil conjecture.\end{minipage}\\
$M=Q_*$ & C & \begin{minipage}[t]{.5\columnwidth}Conjecture due to Clozel\cite{Clozel1990, MR3642468}.\end{minipage}\\
$M\subset G$ & T & \begin{minipage}[t]{.5\columnwidth}Taylor \cite[pp, 77, 79-80]{Taylor} distills the work of many people and describes how to attach a Galois representation to a motive.\end{minipage}\\
$G\subset M$ & F-M & \begin{minipage}[t]{.5\columnwidth}Fontaine-Mazur Conjecture \cite{FontaineMazur1995}.\end{minipage}\\
$G\subset Q_*$ & Modularity & \begin{minipage}[t]{.5\columnwidth}The general conjecture that asserts to a Galois representation one can attach an automorphic representation so that the two $L$-functions agree.\end{minipage}\\
$Q_*\subset G$ & B-G & \begin{minipage}[t]{.5\columnwidth}Conjecture due to Buzzard--Gee \cite{BuzzardGee2014}.\end{minipage}\\
$M=A_*=Q_*=G\!$ &&\begin{minipage}[t]{.5\columnwidth}Piecing together previous connections.\end{minipage}
\end{tabular}
\caption{Explanations of arrows in Figure~\ref{fig:diagram2} between various sources of $L$-functions as described in Sections~\ref{ssec:motives} and \ref{ssec:Galois}.}\label{tbl:table2}
\end{table}
\begin{landscape}
\usetikzlibrary{arrows,
calc,chains,
decorations.pathreplacing,decorations.pathmorphing,
fit,
positioning,}
\vspace{5in}
\begin{figure}\centering
\scalebox{.65}{
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rectangle, rounded corners,
draw=black, very thick,
text width=15em,
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align=center,
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\node[PC] {$\mathbb{Q}$-automorphic $L$-functions of algebraic type};
\node[PC] {$\mathbb{Q}$-automorphic $L$-functions of arithmetic type};
\node[blank] {};
\node[PC] {Analytic $L$-functions of algebraic type};
\node[PC] {Analytic $L$-functions of arithmetic type};
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fit=(A-1) (A-2)] {};
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\p2 = (A-6.south),
\n1 = {veclen(\y2-\y1,\x2-\x1)} in
node[PC, right=of A-6.east,xshift=10mm,minimum height=\n1] {$\mathbb{Q}$-automorphic:\\ $L$-functions of tempered balanced unitary cuspidal automorphic representations of $\textrm{GL}(d,\mathbb{A}_{\mathbb{Q}})$};
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\n1 = {veclen(\y2-\y1,\x2-\x1)} in
node[PC, right=of A-7.east,xshift=10mm,minimum height=\n1] {Analytic:\\ tempered balanced primitive entire analytic $L$-functions };
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\n1 = {veclen(\y2-\y1,\x2-\x1)} in
node[PC, left=of A-6.west,xshift=-15mm,minimum height=\n1] {$L(s,\rho)$ for $\rho$ an irreducible geometric Galois representation};
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node[PC, left=of A-7.west,xshift=-15mm,minimum height=\n1] {$L(s,M)$ for $M$ an irreducible pure motive};
\node[above left] at (A-1.south east) {\tiny$\mathbf{Q}_1$};
\node[above left] at (A-2.south east) {\tiny$\mathbf{Q}_2$};
\node[above left] at (A-6.south east) {\tiny$\mathbf{Q}_\star$};
\node[above left] at (A-4.south east) {\tiny$\mathbf{A}_1$};
\node[above left] at (A-5.south east) {\tiny$\mathbf{A}_2$};
\node[above left] at (A-7.south east) {\tiny$\mathbf{A}_\star$};
\node[above left] at (A-8.south east) {\tiny$\mathbf{Q}$};
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\node[above left] at (A-10.south east) {\tiny$\mathbf{G}$};
\node[above left] at (A-11.south east) {\tiny$\mathbf{M}$};
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(A-10) edge node[xshift=-5mm] {F-M} (A-11) ;
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}
\caption{More detailed conjectured relationships between the sets of $L$-functions considered in this paper.
The relationships are labeled with an attribution to give an indication of the source of the conjecture.}\label{fig:diagram2}
\end{figure}
\end{landscape}
\section{Appendix: non-tempered \texorpdfstring{$L$}{}-functions}
If an $L$-function fails to be tempered, the failure could either
occur in the $\Gamma$-factors or in the local factors of the
Euler product. The failure must occur in a specific form,
which we call the \term{unitary pairing condition}.
Our motivation is that the unitary pairing condition holds
for the factors arising from generic unitary local representations.
\subsection{The unitary pairing condition at infinity}
In the definition we use the following notation: if $x\in\mathbb R$ and
$\xi\in\mathbb C$ then $(x,\xi)^*=(x,-\overline{\xi})$.
Also, we introduce a parameter $\theta < \frac12$ which
measures how far the $L$-function is from being tempered at
infinity.
\begin{defn}
The multisets $\{\mu_j\}$ and $\{\nu_j\}$ meet the
\term{unitary pairing condition at infinity} if it is possible to
write $\mu_j=\delta_j+\alpha_j$ and $\nu_j=\eta_j+\beta_j$,
where $\delta_j\in \{0,1\}$ and $\eta_j\in\{\frac12,1,\frac32,\ldots\}$,
with $|\Re(\alpha_j)|, |\Re(\beta_j)| <\theta$, such that the multisets
$\{(\delta_j,\alpha_j)\}$ and $\{(\eta_j,\beta_j)\}$ are closed under the operation
$S\to {S}^*$.
\end{defn}
For example, the following $\Gamma$-factor satisfies the unitary
pairing condition:
\begin{align}
&
\Gamma_\mathbb R(s-0.2)
\Gamma_\mathbb R(s+0.2)
\Gamma_\mathbb R(s)^3
\Gamma_\mathbb R(s+0.9)
\Gamma_\mathbb R(s+1.1)\cr
&\phantom{xxxxx}\times\Gamma_\mathbb C(s+0.7)
\Gamma_\mathbb C(s+1.3)^2
\Gamma_\mathbb C(s+1.7)
\Gamma_\mathbb C(s+7),
\end{align}
as does this one
\begin{align}
&
\Gamma_\mathbb R(s-0.2+3i)
\Gamma_\mathbb R(s+0.2+3i)
\Gamma_\mathbb R(s+1)
\Gamma_\mathbb R(s+1-8i) \cr
&\phantom{xxxxx}\times
\Gamma_\mathbb C(s+0.7)
\Gamma_\mathbb C(s+1.3)
\Gamma_\mathbb C(s+1.3-7i)
\Gamma_\mathbb C(s+1.7-7i).
\end{align}
\subsection{The unitary pairing condition at \texorpdfstring{$p$}{}}
Just as in the archimedean case, we introduce a parameter $\theta<\frac12$
which provides a weak version of the Ramanujan bound:
$|\alpha_{j,p}|\le p^\theta$.
At a good prime the unitary pairing condition is easy to state.
\begin{defn} Suppose $p$ is a good prime.
The multiset $\{\alpha_1,\ldots,\alpha_d\}$
meets the \term{unitary pairing condition at}~$p$
with
partial Ramanujan bound~$\theta<\frac12$,
if $|\alpha_j|\le p^\theta$ and the multiset is
closed under the operation $x\to 1/\overline{x}$. Equivalently,
the polynomial $F(z)=\prod_j (1-\alpha_j z)$
has all its roots in $|z|\ge p^{-\theta}$ and satisfies the
self-reciprocal condition
\begin{equation}\label{eqn:self-reciprocal}
F(z) = \xi z^d \overline{F}(z^{-1}),
\end{equation}
where $\xi=(-1)^d \prod_j \alpha_j$.
\end{defn}
The term \emph{self-reciprocal} refers to the fact that,
up to multiplication by a constant,
the coefficients of the polynomial are the same if read in either order.
If $|\alpha_j|=1$ then the unitary pairing condition at $p$ says nothing,
because $\alpha_j=1/\overline{\alpha_j}$.
But those Satake parameters which are not on the unit circle occur in pairs:
if $\alpha_j=r e^{i\theta}$ with $r\not=1$, then $r^{-1}e^{i\theta}$ is also a
Satake parameter. Those two
points are located symmetrically with respect to the unit circle.
The general case of the unitary pairing condition at $p$, which includes
the good prime version above, closely follows the archimedean case.
Specifically, the $\Gamma_\mathbb R$ factors are like the good primes, and the
$\Gamma_\mathbb C$ factors are similar to the bad primes.
Recall the notation $(x,\xi)^*=(x,-\overline{\xi})$.
\begin{defn}
The multiset $\{\alpha_1,\ldots,\alpha_M\}$ meets the
\term{unitary pairing condition at}~$p$ of degree $d$
and partial Ramanujan bound~$\theta<\frac12$,
if it is possible to
write $\alpha_j=p^{-\eta_j-\beta_j}$
where $\eta_j\in\{0,\frac12,1,\frac32,\ldots\}$,
with $\sum 2\eta_j \le d-M$ and
$|\Re(\beta_j)| \le \theta$, such that the multiset
$S=\{(\eta_j,\beta_j)\}$ is closed under the operation
$S\to {S}^*$.
\end{defn}
|
2,869,038,154,786 | arxiv | \section{Introduction}
\setcounter{equation}0
\hspace{8mm} The very early stages of the Universe must be described with physics
beyond our current models. Around the Planck time, energy and sizes
involved would require a quantum gravity
treatment in order to account accurately for the physics
at such scale.
String Theory appears as the most promising
candidate for solving the first stages evolution.
Until now, one does not dispose of a complete string theory, valid at the
very beginning of the Universe neither
the possibility
of extracting so many phenomenological consequences from it.
Otherwise, effective and
selfconsistent string theories have been developed in the cosmological
context in the last years
(\cite{tsy}-\cite{lib3}).
These approaches
can be considered valid at the early stages inmediately after the Planck
epoch and should be linked with
the current stages, whose physics laws must be expected as the very low
energy limits of the more complete laws in the early universe.
Matters raise in this process. The Brans-Dicke
frame, emerging naturally in the low energy effective string theories,
includes both the General Relativity as well as
the low energy effective string action as different particular cases.
The former
one takes place when the Brans-Dicke parameter $\omega_{BD}=\infty$
while the last one requires $\omega_{BD}=-1$ (\cite{gdil}). Because
of being extracted from different gravity theories,
the effective string equations
are not equivalent to the Einstein Equations. Since current observational
data show agreement with General Relativity predictions, whatever another
fundamental
theory must recover it among their lowest energy limits, or at least must
give results compatible with those extracted in Einstein frameworks.
The great difficulties to incorporate string theory in a realistic
cosmological framework are not so much expected at this level, but
in the description of an early Universe evolution (string phase and
inflation) compatible with the observational evolution information.
The scope of this paper is to present a minimal model for the Universe
evolution completely extracted from selfconsistent string
cosmology (\cite{dvs94}).
In the following, we recall the selfconsistent effective treatments
in string theory and
the cosmological backgrounds arising from them. With these backgrounds,
we construct a minimal model which can be linked
with minimal observational Universe information.
We analyse the properties of this model
and confront it with General Relativity results.
Although its simplicity, interesting conclusions are found about its
capabilities as a predictive cosmological description.
The predicted current energy density is found compatible
with current observational results, since we have $\Omega \sim 1$
and in anycase $\Omega \geq \frac{4}{9}$. The energy density-dilaton
coupled term at the beginnning of radiation dominated stage is found
compatible with the order of magnitude typical of GUT scales
$\rho e^{\phi} \sim 10^{90} {\mathrm{erg \; cm^{-3}}}$.
On the other hand, by defining the corresponding critical energy density, the
energy density around the exit of inflation gives $\left. \Omega \right|_{inf}
= 1$.
This result agrees with the General Relativity statement for which
$k=0 \rightarrow \Omega=1$, but it is extracted in a Non-Einstenian
context (the low energy string effective equations). No use
of observational information neither further evolution Universe
properties are needed in order to find this agreement, only
the inflationary evolution law for the scale factor, dilaton and
density energy.
Our String Driven Model
is different from previously discussed scenarii in String
Cosmology (\cite{gv93}). Until now, no complete
description of the scale factor evolution from inmediately post-Planckian
age until current time had been extracted in String Cosmology.
The described inflationary stage, as here presented and interpretated,
is also a new feature among the solutions given by effective
string theory.
The String Driven Model does not add new problems to the yet still open
questions, but it provides a description closer and more naturally
related to the observational Universe properties. The three stages of
evolution, inflation, radiation dominated and matter dominated are
completely driven by the evolution of the string equation of state itself.
The results extracted are fully predictive without free
parameters.
This paper is organized as follows: In Sections 2 and 3 we
construct the Minimal String Driven Model. In Section 4 we discuss its
main features: enough inflation, the evolution of the Hubble
factor and the energy density predictions. We also discuss
its main properties, differences and similarities with other
string cosmology scenarii. In Section 5 we present our Conclusions.
\section{Minimal String Driven Model}
\setcounter{equation}0
{\hspace{8mm}} The String Driven Cosmological Background is a minimal
model of the Universe evolution totally extracted from effective
String Theory.
We find the cosmological backgrounds from selfconsistent
solutions of the
effective string equations.
Physical meaning of the model is preserved by
linking it with a minimal but well stablished information about
the evolution of the observational Universe.
Two ways allowing extraction of cosmological
backgrounds from string theory have
been used. The first one is
the low energy effective string equations plus the string action matter.
Solutions of these equations are
an inflationary inverse power evolution for the scale factor, as well
as a radiation dominated behaviour. On the other hand, selfconsistent
Einstein equations plus string matter, with a classical gas of strings as
sources, give us again a radiation dominated behaviour and a
matter dominated description.
From both procedures,
we obtain the evolution laws for an
inflationary stage, a radiation dominated stage and a matter
dominated stage. These behaviours are asymptotic regimes
not including strictly the transitions among stages. By
modelizing the transitions in an enoughly continuous way, we construct a
step-by-step minimal model of evolution.
In this whole and next sections, unless opposite indication, the
metric is defined in lenght units. Thus, the (0,0) component
is always time coordinate ${\mathcal{T}}$ multiplied by constant $c$,
$t = c {\mathcal{T}}$.
Derivatives are taken with respect to this coordinate $t$.
\subsection{The Low Energy Effective String Equations}
{\hspace{8mm}} We work with the low energy effective string action
(that means, to the lowest order in expansion of powers of $\alpha'$),
which in the Brans-Dicke or string frame can be written as
(\cite{tsy},\cite{dvs94},\cite{gv93}):
\begin{equation} \label{action}
S = - \frac{c^3}{16 \pi G_D} \int d^{d+1} x \sqrt{\mid g \mid} e^{- \phi}
\left(R + \partial_{\mu} \phi \partial^{\mu} \phi - \frac{H^2}{12} + V \right)
+ S_M
\end{equation}
where $S_M$ is the corresponding action for the matter sources, $H=dB$ is the
antisymmetric tensor field strength and $V$ is a constant vanishing for some
critical dimension. The dilaton field $\phi$ depends explicitly only
upon time coordinate and its potential will be considered vanishing. $D$ is
the total spacetime dimension.
We consider a spatially flat background and we write the metric
in synchronous frame ($g_{00} =1$, $g_{0i}=0=g_{0a}$) as:
\begin{equation}\label{metrica}
g_{ \mu \nu}={\mathrm diag}(1, -a^2(t) \; {{\delta}_i}_j)
\end{equation}
here $\mu, \nu = 0 \ldots (D-1)$ and $i, j = 1 \ldots (D-1)$.
The string matter is included as a classical source which stress
energy tensor in the perfect fluid approximation takes the form:
\begin{equation} \label{source}
{T_{\mu}}^{\nu} = {\mathrm diag}(\rho(t), -P(t) {{\delta}_i}^j)
\end{equation}
where $\rho$ and $P$ are the energy density and pressure for the
matter sources respectively.
The low energy effective string equations are obtained by extremizing
the variation of the effective action $S$ (\ref{action})
with respect to the metric $g_{\mu \nu}$, the dilaton field
$\phi$ and the antisymmetric field $H_{\mu \nu \alpha}$ and
taking into account
the metric (\ref{metrica}) and matter sources
(\ref{source}).
We will consider that the antisymmetric tensor $H_{\mu \alpha
\beta}$ as well as
the potential $B$ vanish.
By defining $H = {{\dot{a}}\over{a}}$ and the shifted
expressions for
the dilaton $\bar{\phi}=\phi-\ln{\sqrt{\mid g \mid}}$, matter energy
density $\bar{\rho} =\rho a^d$ and pressure $\bar{p}=P a^d$, we obtain
the low energy effective equations $L.E.E.$ (\cite{dvs94},\cite{gv93}):
\begin{eqnarray} \label{leeeq}
& & {\dot{\bar{\phi}}}^{\:2} - 2 {\ddot{\bar{\phi}}} + d H^2 = 0 \\
& & {\dot{\bar{\phi}}}^{\:2} - d H^2 = \frac{16 \pi G_D}{c^4} \;
{\bar{\rho}}
\; e^{\bar{\phi}} \nonumber \\
& & 2 ({\dot{H} - H {\dot{\bar{\phi}}}}) = \frac{16 \pi G_D}{c^4} \;
{\bar{p}
\; e^{\bar{\phi}}} \nonumber
\end{eqnarray}
The shifted expressions have the property to be invariants under the
transformations related to the scale factor duality simmetry ($a \rightarrow
a^{-1}$) and time reflection ($t \rightarrow -t$). Following this, if
($a, \phi$) is a solution of the effective equations, the dual
expression ($\hat{a}, \hat{\phi}$) obtained as:
\begin{eqnarray}
\hat{a_i} = {a_i}^{-1} \label{dual}
\ \ \ \ \ \ \ & , & \ \ \ \ \ \ \
\hat{\phi} = \phi - 2 \ln a_i
\end{eqnarray}
is also a solution of the same system of equations.
\subsubsection{String Driven Inflationary Stage Solution}
{\hspace{8mm}} The inflationary stage appears as a new selfconsistent
solution of the low energy effective
equations (\ref{leeeq}) sustained by a gas of stretched or unstable string
sources as developed in \cite{dvs94} and also in \cite{gv93}.
This kind of string behaviour is characterized by a
negative pressure and positive energy density, both growing in absolute value
with the scale factor (\cite{dvs94},\cite{lib1},\cite{gsv2},\cite{otro}).
Strings in curved backgrounds satisfy the
perfect fluid equation of state $P = (\gamma - 1) \rho$ where $\gamma$
is different for each one of the generic three different string behaviours
in curved spacetimes (\cite{dvs94}).
For the unstable (stretched) string
behaviour, it holds $\gamma_u = \frac{D-2}{D-1}$ (\cite{lib1}).
Thus, the equation of state
for these string sources in the metric (\ref{metrica}) is given
by (\cite{dvs94}):
\begin{equation} \label{eqssd}
P = - \frac{1}{d} \ \rho
\end{equation}
We find the following selfconsistent solution for the set of effective
equations (\ref{leeeq})
with the matter sources eq.(\ref{eqssd}):
\begin{eqnarray} \label{sdrin}
a(t) & = & A_I ({t_I - t})^{-Q} \ \ \ \ \ \ \ \ 0 < t < t_b < t_I
\ \ \ \ \ \ \ \
Q = \frac{2}{d+1} \\
\phi(t) & = & \phi_I + 2d \ln a(t) \nonumber \\
\rho(t) & = & \rho_I {(a(t))}^{(1-d)} \nonumber \\
P(t) & = & -{1\over{d}} \; \rho(t) \ = \ -\frac{\rho_I}{d}
{(a(t))}^{(1-d)} \nonumber
\end{eqnarray}
Notice that here $t$ is the cosmic time coordinate, running on positive
values such
that the parameter $t_I$ is greater than the end of the string
driven inflationary regime at time $t_b$,
$d$ is the number of expanding spatial dimensions; $\rho_I$, $\phi_I$ are
integration constants and $A_I$, $t_I$ parameters to be fixed by
the further evolution of scale factor.
Although the time dependence obeys a power function,
this String Driven solution is an inflationary inverse power law
proper to string cosmology.
This solution describes an inflationary stage with accelerated expansion
of scale factor since $H>0$, $\dot{H} > 0$ and can be considered
superinflationary, since $\ddot{a}(t)$ increases with time. However,
notice the negative power of time and the decreasing character of
the interval $(t_I-t)$. Notice also that the string energy density
$\rho(t)$ and the pressure $P(t)$ have a decreasing behaviour
when the scale factor grows.
\subsubsection{String Driven Radiation Dominated Stage Solution}
{\hspace{8mm}} This stage is obtained by following the same procedure
above described, but by considering now a gas of strings with dual
to unstable behaviour. Dual strings propagate in curved spacetimes
obeying a typical radiation type equation of state (\cite{dvs94},\cite{gv93}).
\begin{equation}
P = \frac{1}{d} \ \rho
\end{equation}
This string behaviour and the dilaton ``frozen'' at constant value $(\phi =
\mathrm{constant})$
gives us the scale factor for the radiation dominated stage:
\begin{eqnarray} \label{sdrad}
a(t) & = & A_{II}\: t^R \ \ \ \ \ \ \ \ \ \ \ R = \frac{2}{d+1} \\
\phi(t) & = & \phi_{II} \nonumber \\
\rho(t) & = & \rho_{II} {(a(t))}^{-(1+d)} \nonumber \\
P(t) & = & \frac{1}{d} \; \rho(t) = \frac{\rho_{II}}{d}{(a(t))}^{-(1+d)}
\nonumber
\end{eqnarray}
here $\phi_{II}$, $\rho_{II}$ are integration constants, and $A_{II}$ a
parameter to be fixed by the evolution of the scale factor.
\subsection{Selfsustained String Universes in General Relativity}
{\hspace{8mm}} As shown in ref.\cite{dvs94}, \cite{lib1} and \cite{lib3},
string solutions in curved spacetimes are
selfconsistent solutions of General Relativity equations,
in particular in a spatially flat, homogeneus and isotropic background:
\begin{equation}
ds^2 = dt^2 - {a(t)}^2 dx^2
\end{equation}
where the Einstein equations take the form:
\begin{eqnarray} \label{eins}
{1\over{2}}d(d-1)H^2 = \rho \ \ \ \ \ \ \ , \ \ \ \ \ \ \
(d-1)\dot{H} + P + \rho = 0
\end{eqnarray}
As before, the matter source is described by a gas of non interacting
classical strings (neglecting splitting
and coalescing interactions). This gas
obeys an equation of state including the three different possible behaviours
of strings in curved spacetimes: unstable, dual to unstable and stable.
Let be $\mathcal{U}$, $\mathcal{D}$ and $\mathcal{S}$ the densities for
strings with unstable, dual to unstable and stable behaviours respectively.
Taking into account the properties of each behaviour (\cite{dvs94}), the
density energy and the pressure of the string gas are described by:
\begin{equation} \label{gasro}
\rho = \frac{1}{{(a(t))}^d} \left({\mathcal{U}} a(t) + {{\mathcal{D}}\over
a(t)} + \mathcal{S}\right) \ \ \ \ \ , \ \ \ \ \
P = \frac{1}{d} \; \frac{1}{{(a(t))}^d} \left( {{\mathcal{D}}\over a(t)} -
{\mathcal{U}} a(t) \right) \nonumber
\end{equation}
Equations (\ref{gasro}) are qualitatively correct for every
$t$ and become exact in the asymptotic cases, leading to obtain the radiation
dominated behaviour of the scale factor, as well as
the matter dominated behaviour.
In the limit $a(t)\rightarrow 0$ and $t \rightarrow 0$, the dual to unstable
behaviour dominates in the equations (\ref{gasro}) and
gives us:
\begin{eqnarray} \label{rdrp}
\rho(t) & \sim & {\mathcal{D}} \; {(a(t))}^{-(d+1)} \ \ \ \ \ , \ \ \ \ \
P(t) \sim \frac{1}{d} \; {\mathcal{D}} \; {(a(t))}^{-(d+1)}
\end{eqnarray}
This behaviour is characterized by positive string density energy and
pressure, both growing when the scale factor approaches to $0$. Dual to
unstable strings behave in similar way to massless particles, i.e. radiation.
Solving selfconsistently the Einstein equations (\ref{eins}) with
sources following
eqs.(\ref{rdrp}), the scale factor solution takes the form:
\begin{equation}
a(t) \sim {\left({{2 \mathcal{D}}\over{d(d-1)}}\right)}^{1\over{d+1}}
{\left({{d+1}\over{2}}\right)}^{2\over{d+1}} (t - t_{II})^R \ \ \ \ \ , \ \
R = \frac{2}{d+1}
\end{equation}
this describes the evolution of a Friedmann-Robertson-Walker radiation
dominated Universe, the time parameter $t_{II}$ will be fixed by
further evolution of the scale factor.
On the other hand, studying the opposite limit $a(t)\rightarrow
\infty $, $t \rightarrow \infty$ and taking into account the behaviour of
the unstable density $\mathcal{U}$ which vanishes for
$a(t)\rightarrow \infty$ \cite{dvs94},
the stable behaviour becomes dominant and the
equation of state reduces to:
\begin{eqnarray} \label{mdrp}
\rho & \sim & {\mathcal{S}} \; {(a(t))}^{-d}
\ \ \ \ \ \ \ \ , \ \ \ \ \ \ \ \ P \: = \: 0
\end{eqnarray}
The stable behaviour gives a constant value for the string energy, that is,
the energy density evolves as the inverse volume decreasing with growing
scale factor, while the pressure vanishes. Thus, stable strings behave as
cold matter. Again, from solving eqs.(\ref{eins}) with
eqs.(\ref{mdrp}), the solution of a matter dominated stage emerges:
\begin{equation} \label{rgmat}
a(t) \sim {\left({d\over{(d-1)}}{{\mathcal{S}}\over{2}}\right)}^{1\over{d}}
(t-t_{III})^M \ \ \ , \ \ \ \ M = \frac{2}{d}
\end{equation}
We construct in the next sections a model with an inflationary stage
described by the String Driven solution (see eq.(\ref{sdrin})), followed by
a radiation
dominated stage (see eq.(\ref{sdrad})) and a matter dominated stage
(see eq.(\ref{rgmat})). We will consider the dilaton field remain
practically constant and vanishing from the exit of inflation until the
current time, as suggested
in the String Driven Radiation Dominated Solution.
It must be noticed that the same solution for the radiation
dominated stage emerges from the treatment with dilaton field and
without it (general relativity plus string equation of state), allowing
us to describe qualitatively the evolution of the universe by means of
these scale factor asymptotic behaviours.
\section{Scale Factor Transitions}
\setcounter{equation}0
{\hspace{8mm}} Taking the simplest option, we consider
the ``real'' scale factor evolution
minimally described as:
\begin{eqnarray} \label{real}
a_I(t) & = & A_I {(t_I - t)}^{-Q} \ \ \ \ t \in (t_i, t_r) \\
a_{II}(t) & = & A_{II} \: t^R \ \ \ \ \ \ \ \ \ t \in (t_r, t_m) \nonumber \\
a_{III}(t) & = & A_{III} \: t^M \ \ \ \ \ \ \ \ \ t \in (t_m, t_0) \nonumber
\end{eqnarray}
with transitions at least not excessively long at the beginning of radiation
dominated stage $t_r$ and of matter dominated stage $t_m$. We also define
a beginning of inflation at $t_i$, and $t_0$ is the current time.
It would be reasonable do not
have instantaneous and
continuous transitions at $t_r$ and $t_m$ for the stages extracted in the
above section, since the detail of such transitions is not provided by
the effective treatments here used. One can suspects the
existence of very brief intermediate stages at least at the end of the
inflationary stage ($t \in (t_b, t_r)$), as we will discuss in the next
section, and also at the end of radiation dominated
stage ($t \sim t_m$). The dynamics of these transitions is unknown and
not easy to modelize, it introduces in anycase free parameters (\cite{p1}).
In order to construct an evolution model for the scale factor, it is
compatible with the current level of knowledge of the theory to suppose
the transitions very brief.
We will merge our lack of knowledge on the real transitions
by means of descriptive temporal variables (\cite{p1}) in function of
which the modelized
transitions at $\bar{t_1}$ and $\bar{t_2}$ are instantaneous and continuous.
We link this descriptive scale factor with the minimal information
about the evolution of the observational Universe;
we consider the standard values for
cosmological times: the radiation-matter transition held about
${\mathcal{T}}_m \sim 10^{12}$ s, the beginning of radiation stage at
${\mathcal{T}}_r \sim 10^{-32}$ s and the current age of the Universe
${\mathcal{T}}_0 \sim {H_0}^{-1} \sim 10^{17}$ s (The exact numerical
value of ${\mathcal{T}}_0$ turns out not crucial here). We impose also
to our description satisfy the same
scale factor expansion (or scale factor ratii) reached in each one of
the three stages considered in the real model (\ref{real}).
It is also convenient to fix the temporal variable of the third (and current)
stage $t$ with our physical time (multiplied by $c$).
Explicit computations can be found in \cite{p1} and led finally to the
following scale factor in cosmic time-type variables:
\begin{eqnarray} \label{Descr}
\bar{\bar{a_I}}(\bar{\bar{t}}) & = & \bar{\bar{A_{I}}}
{(\bar{\bar{t_I}}-\bar{\bar{t}})}^{-Q} \ \ \ \ \ \ \
{\bar{\bar{t_i}}} < {\bar{\bar{t}}} < {\bar{t_1}} \\
\bar{a_{II}}(\bar{t}) & = & \bar{A_{II}} {(\bar{t}-\bar{t_{II}})}^R
\ \ \ \ \ \ \ \ \bar{t_1} < \bar{t} < \bar{t_2}\nonumber \\
a_{III}(t) & = & A_{III} {(t)}^M
\ \ \ \ \ \ \ \ \ \ \ \bar{t_2} < t < {t_0} \nonumber
\end{eqnarray}
with continuous transitions at $\bar{t_1}$ and $\bar{t_2}$ both the
scale factor and first derivatives with respect to the
variables $\bar{\bar{t}}$, $\bar{t}$ and $t$.
The parameters $\bar{\bar{t_I}}$, $\bar{\bar{A_I}}$, $\bar{t_{II}}$,
$A_{III}$ can be written in function of $\bar{A_{II}}$ and
transition times using the matching conditions.
In terms of standard observational values, the transitions $\bar{t_1}$,
$\bar{t_2}$
and the beginning of the inflationary stage description $\bar{\bar{t_i}}$
are expressed as:
\begin{eqnarray}
\bar{t_1} & = & {R \over{M}} t_r + \left(1 - {R \over{M}}\right) t_m
\label{T1} \ \ \ \ \ , \ \ \ \ \ \bar{t_2} = t_m \\
\bar{\bar{t_i}} & = &
\left({R \over{M}} - {Q\over{M}}{{t_r-t_i}\over{t_I-t_r}} \right)t_r
+ \left(1 - {R \over{M}}\right) t_m \nonumber
\end{eqnarray}
The parameters of the scale factor (\ref{Descr})
can be written also in terms of the observational values
$t_r$, $t_m$ and the global scale factor $\bar{A_{II}}$:
\begin{eqnarray} \label{desc}
\bar{\bar{t_I}} & = & {t_r} {\left({R\over{M}}+{Q\over{M}}\right)} + {t_m}
{\left({1 - {R \over{M}}}\right)} \ \ \ \ , \ \ \ \
\bar{t_{II}} = \left({1-{R\over{M}}}\right) t_m \\
\bar{\bar{A_I}} & = & \bar{A_{II}} {\left({Q \over{M}}\right)}^Q
{\left({R \over{M}}\right)}^R {t_r}^{R+Q} \nonumber \ \ \ \ \ \ , \ \ \ \ \ \
A_{III} = \bar{A_{II}} {\left({R \over{M}}\right)}^{R} {t_m}^{R-M}
\end{eqnarray}
The time variable $\bar{\bar{t}}$ of inflationary stage
and ${\bar{t}}$ of radiation stage are not a priori exactly equal to
the physical time coordinate at rest frame (multiplied by $c$), but
transformations (dilatation plus translation) of it. The low energy
effective action equations from where the scale factor, dilaton and
energy density have been extracted, allow these transformations.
With this treatment of the cosmological scale factor, we will attain
computations free of "by hand" added parameters and with full predictability
as can be seen in the next sections.
The last point is to make an approach for the dilaton field. This is
considered practically constant from the beginning of the radiation
dominated stage until the current time.
The dilaton field increases during the inflationary string driven
stage. Its value can be supposed
coincident with the value at exit inflation time in a sudden but
not continuous transition, since its temporal derivatives
can not match this asymptotic behaviour.
${\phi}_{II} \: = \: {\phi} (t_r) \: \equiv \: \phi_1 $.
Remember the expression for the dilaton in inflation dominated
stage,
\begin{equation} \label{dili}
{\phi}_{II} \: = \: {\phi}_I + 2 d \ln a(t_r)
\end{equation}
The scale factor can be written also in terms of the conformal time variable
$d\eta = \frac{dt}{a(t)}$ defined for each stage. Detailed computation
can be found elsewhere (\cite{p1}, \cite{p3}) and are not needed here.
\section{Properties of the String Driven Model}
\setcounter{equation}0
\subsection{Enough Inflation}
\hspace{8mm} The loose ends in standard FRW cosmology are the flatness
problem or why the current spatial curvature $k$ is so little and the
corresponding $\Omega$
so close to 1; the horizon problem and causality related considerations
and the large scale homogeneity of Universe and
related perturbation considerations.
The inflationary paradigm with an exponential expansion period
answers these questions provided the inflationary expansion reaches an enough
number of $e$-folders, usually considered among $80-100$.
Any other cosmological theory predicting
an inflationary stage should have to answer these questions.
As the minimum requirement in this sense, a compatibility with enough
amount of inflation must be required.
We discuss now if
enough inflation could be provided
by the String Driven inflationary stage.
In the comoving coordinates, enough inflation can be simply produced
if the additive parameter $t_I$ is found sufficiently near and
above the inflation-radiation dominated transition time $t_r$.
In this way, by increasing the time, the
scale factor inflates by approaching a pole type singularity. But the
singularity is not reached neither passed throw, because the
exit of inflation stage happens before.
Notice also that both the scale factor and dilaton
increase with time, as well as the spacetime curvature
since it is proportional to $H$. As a consequence, the
dynamics of the stage could lead to break the conditions
giving rise to the low energy effective regime. Let be
$t_b$ the instant when this breaking
in the low energy effective regime happens and, as consequence,
the string driven inflationary evolution ends.
What kind of evolution follows is an open question, a more
complete treatment
would be neccesary in order to describe the
scale factor in a higher order regime, where string
effects would play a strong role. By consistency with
the time scales, it is reasonable to suppose a very short
interval between the end of the string driven inflation and
the beginning of the standard radiation dominated
cosmology. One can suspects the
existence of a very brief intermediate stage at the end of the
inflationary stage at nearly constant curvature ($t \in (t_b, t_r)$).
Due to its briefness and supposed bounded curvature controlled
by higher-order string corrections, not so great
expansion would appear in the stage not governed by the $L.E.E.$.
Let us impose the String Driven inflationary stage to
reach an arbitrary number $f$ of e-folds between the beginning
at $t_i$ and the comeback to $L.E.E.$ at $t_r$:
\begin{equation} \label{e.i}
\frac{a_I(t_r)}{a_I(t_i)} \sim e^f
\end{equation}
This condition gives us the relation:
\begin{displaymath}
1+\frac{t_r-t_i}{t_I-t_r} \sim e^{\frac{f}{Q}}
\end{displaymath}
By defining the parameter $\epsilon$ related with the value $t_I$ through
$t_I=t_r(1+\epsilon)$ and $\delta$ relating the inflation duration
and its exit, the enough inflation condition eq.(\ref{e.i}) translates as:
\begin{displaymath}
1+\frac{\delta}{\epsilon} \sim e^{\frac{f}{Q}}
\end{displaymath}
that, in good approximation, gives us
$\epsilon \sim \delta \; e^{-\frac{f}{Q}}$.
The dilaton field increases during the inflationary stage,
however its effect in breaking the $L.E.E.$ regime is not given by
the exit inflation conditions but by the beginning inflation ones.
We have modelized a time $t_i$ being the beginning of an
inflationary stage close to Planck time $t_P \sim 10^{-43} {\mathrm{s}}$.
In this inflationary stage, from eqs.(\ref{sdrin}) the dilaton
field is $\phi(t) = \phi_I + 2d \ln a(t)$. Therefore, the dilaton
increases during the inflationary stage between $t_i$ and $t_r$
a ratio:
\begin{displaymath}
\frac{\phi(t_r)}{\phi(t_i)} = \frac{\phi_I + 2d \ln a(t_r)}
{\phi_I + 2d \ln a(t_i)}
\end{displaymath}
Thus, if an enough amount of inflation tooks place,
for instance \mbox{$a(t_r) = e^f a(t_i)$}, the ratio in the dilaton field
is given by the expression:
\begin{eqnarray} \label{radil}
\frac{\phi(t_r)}{\phi(t_i)} & = & 1 \; + \; \frac{f}{\log_{10}e}
\frac{2d }{( \phi_I + 2d \ln a(t_i))}
\end{eqnarray}
where we notice the lineal dependence with the number
of e-folds $f$ and the inverse dependence of scale factor at
the beginning of inflation stage $a(t_i)$.
No need
to place this singularity on the big bang (around $t \rightarrow 0$) is found
from the effective equations. Still more: the beginning $t_i$ of the String
Driven inflationary stage is practically irrelevant.
The usual inflationary paradigm
considers between 60-100 e-folds enough to
solve the problem of large scale homogeneity. This quantity is true for
standard (exponential De Sitter) inflation, but we can consider it
enoughly restrictive. In anycase, we realize that String Driven inflation
can, in principle, provide whatever amount of inflation, since a
finite value of $f$ gives a finite relation among $t_I$ and $t_r$ too.
One would expect the value of $t_I$ given
from theory and predicting the right amount of inflation. In our limited
minimal string driven model, we prove the compatibility of our
inflationary description with the usual standard inflation values.
In the figure[\ref{mix}] we show the scale factor evolution
for a De Sitter stage($DS$), an usual power law inflationary stage ($PL$) and
the String Driven inflationary stage ($SD$).
\begin{figure}[h]
\centering
\psfrag{a}{$10^{-34}$}
\psfrag{b}{$10^{-33}$}
\psfrag{c}{$10^{-32}$}
\psfrag{e}[r]{$a({\mathcal{T}}_r)$}
\psfrag{f}[r]{$10^{-35}$}
\psfrag{m}[r]{$\log a({\mathcal{T}})$}
\psfrag{w}{${\mathcal{T}}$ [{\it seg}]}
\psfrag{s}[r]{{\bf SD}}
\psfrag{d}[l]{ {\bf DS}}
\psfrag{p}[r]{{\bf PL}}
\includegraphics[width=100mm]{mix2.ps}
\caption{\label{mix} Comparison among different inflationary evolution laws}
\vspace{5pt}
{\parbox{140mm}{\footnotesize Representation of the inflation amount
reached by a De Sitter $(DS)$, an usual Power Law $(PL)$ and a String
Driven $(SD)$ inflationary stages. The three scale factors have been
normalized at ${\mathcal{T}}_r$. For enough inflation $(t_I
\rightarrow t_r)$, the $(SD)$ behaviour becomes very much sharper.}}
\end{figure}
We consider now the duration of inflation. In the descriptive coordinate
$\bar{\bar{t}}$, it means $\left. \Delta \bar{\bar{t}}\right|_{inf} =
\bar{t_1}-\bar{\bar{t_i}}$.
From eqs.(\ref{T1}) is seen:
\begin{displaymath}
\left. \Delta \bar{\bar{t}} \right|_{inf} = \frac{Q}{M}\frac{t_r}{t_I-t_r}
\left. \Delta t \right|_{inf}
\end{displaymath}
where $\left. \Delta t \right|_{inf} = t_r -t_i$ and remember
$Q$ and $M$ are positive values and $t_I > t_r$. Coherently, we
obtain a positive value for the duration of inflation in descriptive
variables.
Notice that as long as $t_I \rightarrow t_r$, this
duration experiments a greater
dilatation. In fact, if we have in general $t_I = t_r(1+\epsilon)$, then:
\begin{displaymath}
\left. \Delta \bar{\bar{t}} \right|_{inf} = \frac{Q}{M}\frac{1}{\epsilon}
\left. \Delta t \right|_{inf}
\end{displaymath}
Since enough inflation requires $\epsilon << 1$, the
duration in descriptive variables increases so much in order
to reproduce the expansive behaviour of the scale factor. For instance,
for an expansion of order $\sim e^f$, the last expression becames:
\begin{displaymath}
\left. \Delta \bar{\bar{t}} \right|_{inf} = \frac{Q}{M} e^{\frac{f}{Q}}
\left. \Delta t \right|_{inf}
\end{displaymath}
Such enormous dilatation in the descriptive variables is
reached by means of transformation of the transition times. From
eqs.(\ref{T1}) we see that $\bar{t_1} \sim t_m$ and always
$\bar{t_1}>0$, meanwhile
$\bar{\bar{t_i}}$ can decrease to large negative values, enlarging
in this way the duration of the stage in the descriptive variables.
By using $t_I=t_r(1+\epsilon)$ and $t_r-t_i=\delta t_r$,
the condition $\bar{\bar{t_i}}<0$ holds when:
\begin{equation} \label{tineg}
\frac{\delta}{\epsilon} \geq \frac{1}{Q} \left(R+(M-R)\frac{t_m}{t_r}\right)
\end{equation}
With the standard transition
times given in sections above, the r.h.s. of
eq.(\ref{tineg}) is $\sim 10^{43}$. In the l.h.s. we
find $\delta \rightarrow 1$ and $\epsilon \rightarrow 0$. For
beginning of inflation around the Planck time $(t_i \rightarrow t_P)$ and
enough inflation $(f \sim 80)$, we have
$\frac{\delta}{\epsilon} \sim 10^{87}$ and equation (\ref{tineg})
is satisfied. This feature is needed
in order to attain the required amount of inflation in the descriptive
variables, because it requires a great dilatation of the inflation duration
$\left. \Delta \bar{\bar{t}} \right|_{inf} \sim 10^{87}
\left. \Delta t \right|_{inf}$.
The temporal descriptive coordinate $\bar{\bar{t}}$ runs on both
positive and negative values, but the proper cosmic
time runs always on positive values.
The matter dominated stage description presents the
same duration that the real one since it runs on proper cosmic time.
The duration of the radiation dominated stage in descriptive
variables suffers a slight contraction with respect to duration
in proper time $\left. \Delta \bar{t} \right|_{rad} =\frac{R}{M}
\left. \Delta t \right|_{rad}$. For our model in three spatial
dimensions, this value is $\frac{R}{M}=\frac{1}{3}$
\subsection{Evolution of the Hubble Factor}
\hspace{8mm} The Hubble factor in the model constructed with the String Driven
inflationary stage has the following behaviours in each stage:
\begin{eqnarray}
\left. H(t) \right|_{inf} & = & c Q{\left(t_I-t\right)}^{-1} \\
\left. H(t) \right|_{rad} & = & c R{\left(t-t_{II}\right)}^{-1} \nonumber \\
\left. H(t) \right|_{mat} & = & c M{\left(t\right)}^{-1} \nonumber
\end{eqnarray}
In the figure [\ref{hub}] we can observe the evolution of Hubble
factor in the minimal String Driven cosmological model. Notice
the growth of $H(t)$ around the exit of inflation. Since the Hubble
factor is proportional to spacetime curvature, the region around
$t_r$ is a highly curved regime, where the $L.E.E.$ regime should
transitorily break down. Notice also the nearly constant curvature
at the beginning of the inflation stage
\begin{figure}[h]
\centering
\psfrag{t}{${\mathcal{T}}$[{\it seg}] }
\psfrag{H}{$H=c \frac{\dot{a}}{a} \; [{seg^{-1}}]$}
\psfrag{a}{$t_i$}
\psfrag{b}{${\mathcal{T}}_r$}
\psfrag{c}{${\mathcal{T}}_m$}
\psfrag{d}{${\mathcal{T}}_0$}
\psfrag{e}[r]{$H({\mathcal{T}}_0)$}
\psfrag{f}[ru]{$H({\mathcal{T}}_m)$}
\psfrag{g}[r]{$H({\mathcal{T}}_i)$}
\psfrag{h}[r]{$H_{inf}({\mathcal{T}}_r)$}
\includegraphics[width=100mm]{hub.ps}
\caption{\label{hub} Representation of Hubble factor in String Driven
Cosmology}
\vspace{5pt}
{\parbox{140mm}{\footnotesize Representation of the Hubble factor
evolution in the minimal String Driven cos\-mo\-lo\-gi\-cal model. During the
inflationary stage, the Hubble factor remains almost constant until
the neighbourhood of the transition to radiation dominated stage. Just
before it, the Hubble factor has an explosive but bounded growth.
During the radiation dominated and matter dominated stages, the Hubble
factor decreases with a nearly constant slope.}}
\end{figure}
\subsection{Energy Density Predictions}
\hspace{8mm} As can be easily seen, from eq.(\ref{leeeq}) we have:
\begin{displaymath}
{\dot{\bar{\phi}}}^{\:2} - 2 {\ddot{\bar{\phi}}} + d H^2 = 0
\ \ \ \ \ \ , \ \ \ \ \
{\dot{\bar{\phi}}}^{\:2} - d H^2 = \frac{16 \pi G_D}{c^4} \; {\bar{\rho}}
\; e^{\bar{\phi}}
\end{displaymath}
Both equations give:
\begin{displaymath}
\ddot{\bar{\phi}} - d H^2 = \frac{8 \pi G_D}{c^4} \; {\bar{\rho}}
\; e^{\bar{\phi}}
\end{displaymath}
By substituting the shifted expressions for the dilaton
$\bar{\phi}=\phi-\ln{\sqrt{\mid g \mid}}$ and matter energy
density $\bar{\rho} =\rho a^d$, the above equation yields:
\begin{equation} \label{nume}
\ddot{\phi}-d\left(\dot{H}+{H}^2\right) \; = \; \frac{8 \pi G_D}{c^4}
\rho \;e^{\phi}
\end{equation}
Eq.(\ref{nume}) can be considered as the generalization of the
Einstein equation in the framework of low energy effective string action.
This equation will allow us extract some predictions on the
energy density evolution in our minimal model.
\subsubsection{Energy Density at the Exit of Inflation}
\hspace{8mm} By introducing the String Driven Solution for $a(t)$, $\phi(t)$,
$\rho(t)$ (eqs.(\ref{sdrin})) in eq.(\ref{nume}), we obtain a relation
for the integration constants
$\rho_I$ and $\phi_I$:
\begin{equation} \label{rdct}
\rho_I e^{\phi_I} = {{c^4}\over{8 \pi G_D}} {{2 d (d-1)}\over
{{(d+1)}^2}} {{A_I}^{-(1+d)}}
\end{equation}
Now, is easy to find the relation between $\rho_I$, $e^{\phi_I}$
and the values of the energy density $\rho_1$ and dilaton field
$\phi_1$ at the end of inflation stage. We can compute these values
either with the real scale factor $a_I(t)$ or with the description
$\bar{\bar{a_I}}(\bar{\bar{t}})$. By defining
$\rho_1 = \rho(a_I(t_r)) = \rho(\bar{\bar{a_I}}(\bar{t_1}))$ and
$\phi_1 = \phi(a_I(t_r)) = \phi(\bar{\bar{a_I}}(\bar{t_1}))$
and making use of eqs.(\ref{sdrin}), we can write:
\begin{equation} \label{rdin}
\rho_1 e^{\phi_1} = \rho_I e^{\phi_I} {A_I}^{1+d}
{(t_I-t_r)}^{-Q(1+d)}
\end{equation}
Now, with the information about the evolution of the scale factor and the
descriptions in each stage, it is possible to relate this expression
with observational cosmological parameters. In fact, we must understand
eq.(\ref{rdin}) as one obtained in the description of inflationary
stage:
\begin{displaymath}
t_I - t_r \rightarrow \bar{\bar{t_I}}-\bar{{t_1}} =
\frac{Q}{M} t_r
\end{displaymath}
For the String Driven solution, the exponents have the values $Q={2
\over{d+1}}$, $R={2\over{d+1}}$, $M={2\over{d}}$. With these
expressions we obtain the density-dilaton coupling at the end of
inflation:
\begin{equation} \label{sdri}
\rho_1 e^{\phi_1} = {{c^4}\over{8 \pi G_D}} {{2(d-1)}\over{d}} {t_r}^{-2}
\end{equation}
With $t_r = c {\mathcal{T}}_r$ where
${\mathcal{T}}_r \sim 10^{-32} s$, this expression gives the numerical
value:
\begin{equation}
\rho_1 e^{\phi_1} = 7.1 \; \; 10^{90} \mathrm{erg} \; {\mathrm{cm}}^{-3}
\end{equation}
It must be noticed that the same result can be achieved from the
Radiation Dominated Stage, due to the continuity of the scale
factor, of the density energy and the dilaton field at the transition
time $t_r$.
It must be noticed also an interesting property of this energy
density-dilaton coupled term. We can extend the General Relativity
treatment and define $\left. \Omega \right|_{inf}$
as this coupled quantity in critical energy density units, where the critical
energy density $\rho_c(t)$ for our spatially flat metric is
$ \rho_c (t) = \frac{3 c^2}{8 \pi G} {H(t)}^2 $. We
compute the corresponding $\rho_c$ at the moment of the exit
of inflation:
\begin{equation}
\left. \rho_c \right|_{\bar{t_1}} = \frac{3 c^2}{8 \pi G} H(\bar{t_1}) =
\frac{3 c^4}{8 \pi G} M^2 {t_r}^{-2}
\end{equation}
With this, it is easily seen:
\begin{equation}
\left. \Omega \right|_{inf}
= \frac{\rho_1 e^{\phi_1}}{\left. \rho_c \right|_{\bar{t_1}}}
= \frac{2}{3} \frac{d-1}{d} \frac{1}{M^2}
\end{equation}
where $M$ is given by eq.(\ref{rgmat}). It gives for our model in the
three dimensional case
$\left. \Omega \right|_{inf}=1$.
Here we have used the descriptive variables in order to link
the scale factor transitions with
observational transition times. But the conclusion
$\left.\Omega \right|_{inf}=1$ is independent
from this choice. In fact, we have computed it again in the
proper cosmic time of the inflationary stage and taking
the corresponding values at the beginning of the radiation
dominated stage $t_r$. From eq.(\ref{rdin})
and $H(t) = Q {\left(t_I-t \right)}^{-1}$, we have:
\begin{equation}
\left. \Omega \right|_{inf}
= \frac{1}{3Q}\left(\frac{2}{Q}-1 -dQ\right)
\end{equation}
$Q$ is given by the String Driven solution eq.(\ref{sdrin}) and with it,
we recover $\left. \Omega \right|_{inf} = 1$. In fact, the same result could
be achieved also by evaluating $\left. \Omega \right|_{inf}$
exactly at the end of String Driven inflationary
stage, whatever this time may be.
That means, we have a
prediction {\bf non-dependent} of whenever the exit of inflation happens.
In the proper cosmic time coordinates, $(t_I-t_r)$ is a very
little value. But independently from this, the coupled term
energy density-dilaton at the exit of inflation gives the value one in the
corresponding critical energy density units. The value
of the critical energy density is computed following the Einstein
Equations for spatially flat metrics, but solely
the low energy effective string equations give the expression
for the coupled term eq.(\ref{nume}) and the String Driven solution
itself. In this last point, no use of further evolution, linking
with observational results neither standard cosmology have
been made. This enable us to affirm that in the $L.E.E.$
treatment, the relation between spatial curvature and
energy density holds as in General Relativity, at least
for the spatially flat case ($k=0 \rightarrow \Omega=1$).
This result means to recover a General Relativity prescription within
a Non-Einstenian framework.
\subsubsection{Predicted Current Values of Energy Density and Omega.}
\hspace{8mm} Thus, we can obtain the corresponding current value
of $\rho_0 e^{\phi_0}$
in units of critical energy density or contribution to $\Omega$.
To proceed, we remember that the evolution of the density energy in
the matter dominated stage follows
$\rho \sim {a(t)}^{-3} \sim t^{-2}$
then, at the beginning of matter dominated stage we would have
\begin{displaymath}
\rho_m = {\left({{{\mathcal{T}}_m}\over{{\mathcal{T}}_0}}\right)}^{-2}
\rho_0
\end{displaymath}
On the other hand, in the radiation dominated stage, the density
behaviour is: $\rho \sim {a(t)}^{-4} \sim t^{-2}$
which gives for the energy density
\begin{displaymath}
\rho_1 = {\left({{{\mathcal{T}}_r}\over{{\mathcal{T}}_m}}\right)}^{-2}
\rho_m
\end{displaymath}
\hspace{8mm} That is,
\begin{displaymath}
\rho_0 = {\left({{{\mathcal{T}}_r}\over{{\mathcal{T}}_0}}\right)}^2 \rho_1
\end{displaymath}
Considering that the dilaton field has been remained almost constant
since the end of inflation $\phi_0 \sim \phi_r$, we have
\begin{equation} \label{ome}
\Omega = {{\rho_0 e^{\phi_0}}\over{\rho_c}} = {\left({{{\mathcal{T}}_r}
\over{{\mathcal{T}}_0}}\right)} e^{\phi_1} {{\rho_1}\over{\rho_c}}
\end{equation}
where the current critical energy density is expressed in terms of current
Hubble factor $H_0 = H({\mathcal{T}}_0)$ as:
\begin{equation} \label{rcri}
\rho_c = {{3 c^2}\over{8 \pi G}} {H_0}^2
\end{equation}
From eq.(\ref{ome}) and with eqs.(\ref{sdri}) and (\ref{rcri}),
we obtain:
\begin{equation} \label{omeli}
\Omega = {{2 (d-1)}\over{3d}} {{{{\mathcal{T}}_0}^{-2}}\over{{H_0}^2}}
\end{equation}
As like as $H_0 \sim {{\mathcal{T}}_0}^{-1}$, we have finally
\begin{equation}
\Omega = {{2(d-1)}\over{3d}}
\end{equation}
In our three-dimensional expanding Universe, it gives $\Omega={4\over{9}}$.
In the last, we have taken ${\mathcal{T}}_0 \leq {H_0}^{-1}$
following the usual computation. In General Relativity framework,
it holds:
\begin{equation} \label{grho}
{{\mathcal{T}}_0}=\frac{2}{3}{H_0}^{-1}
\end{equation}
if the deceleration
parameter $q_0 = -\ddot{a}(t_0)\frac{a(t_0)}{{\dot{a}(t_0)}^2}> \frac{1}{2}$.
For our model, the deceleration parameter is found:
\begin{displaymath}
q_0 = \frac{1-M}{M}
\end{displaymath}
that for standard matter dominated behaviour
gives exactly $q_0=\frac{1}{2}$. For this and as well as observations
give $q_0 \sim 1$, we compute the value of $\Omega$ in this
framework. From eq.(\ref{omeli}) and eq.(\ref{grho}) we obtain:
\begin{displaymath}
\Omega = \frac{2}{3}\frac{d-1}{d}{\left(\frac{2}{3}\right)}^{-2}
\end{displaymath}
which in the three dimensional case gives exactly:
\begin{displaymath}
\Omega = 1
\end{displaymath}
We have obtained that a metric spatially flat $k=0$ leds to a
critical energy density $\Omega = 1$. This result is so known in
General Relativity, but here it has been extracted in a no General
Relativity Framework, since low energy effective equations are
not a priori equivalent to General Relativity equations. Looking
from the point of view of the Brans-Dicke metric-dilaton coupling,
General Relativity is obtained as the limit of the Brans-Dicke
parameter $\omega \rightarrow \infty$, while the low
energy effective string action (see eq.(\ref{action})) is obtained
for $\omega = -1$.
Notice that $\Omega=1$ is obtained as a result of applying
coherently General Relativity statements in the matter dominated
stage, but the initial eq.(\ref{sdri}) comes for the
inflationary stage, described in the low energy effective string
framework. Thus, a result from this string treatment is
compatible with, and leads to similar predictions that,
standard cosmology.
Since in anycase ${\mathcal{T}}_0 \leq {H_0}^{-1}$,
eq.(\ref{sdri}) can be seen as giving a lower limit for
$\Omega$:
\begin{equation}
\Omega \geq \frac{2}{3}\frac{(d-1)}{d}
\end{equation}
This is the prediction for the current energy density from
String Driven Cosmology and this is not
in disagreement with the current observational limits for
$\Omega$.
\subsection{Discussion of the Model}
\subsubsection{String Driven Cosmology is Selfconsistent}
\hspace{8mm} Matter dominated stages are not an usual result in string cosmology
backgrounds. The standard time of matter dominated beginning
${\mathcal{T}}_m \sim 10^{12}s$ is supposed too late in order
to account for string effects. But this is coherent with
the fact that matter dominated stage is described
selfconsistently by effective string treatments.
The General Relativity equations plus string sources supposes one
of the weakest regimes for describing string-metric coupling. In
this treatment, metric evolves following classical General Relativity
equations and the string effect is accounted uniquely by considering them as
the classical matter source (\cite{dvs94}). Backreaction happens since
the equation
of state for string behaviours have been obtained by studying their
propagation in curved backgrounds. The result is selfconsistent because
Einstein equations returns the correct curved background
when aymptotic behaviours
have been taken.
Thus, it is a good point that the current stage appears
selfconsistently as the asymptotic limit for large scale factor.
Similarly, the radiation dominated stage
is obtained from both effective treatments considered.
This is coherent with having such stage previous to
the current stage (where string effects must be not visible)
and successive to a stage extracted under considerations of stronger
string effects.
The intermediate behaviour between radiation and matter dominated
stages is not known. Because this and current knowledge,
this effective treatment is not able to describe suitably the
radiation dominated-matter dominated transition. As it is said
elesewhere \cite{p1}, a sudden but continuous and smooth transition among
both stages is not possible without an intermediate behaviour.
Other comment must be dedicated to the inflation-radiation
dominated transition. In the String Driven Model, this transition
requires a brief temporaly
exit of low energy effective regime for comeback within it at
the beginning of radiation dominated. This could be understood
as the conditions neccesary to modify the leading
behaviour from unstable strings to dual strings.
Further knowledge about the evolution of strings in curved
backgrounds is necessary. It has been reported that
multistring solutions (strings propagating in packets)
are present in curved backgrounds and
show different and evolving behaviours
(\cite{multi1}-\cite{multi3}).
Research in this sense could aid to overcome the
transition here considered, asymptotically rounded by
low energy effective treatments.
\subsubsection{String Driven Inflation realizes a Big-Bang}
\hspace{8mm} During the inflationary stage, the scale factor suffers
a great expansion, enough for solving
the cosmological puzzles. The almost amount of expansion
is reached around the exit time. In fact, the beginning of this
stage is characterized for a very, very slow evolution for
the scale factor and dilaton. This evolution increases
speed in approaching the exit inflation, since it is
approaching also the pole singularity in scale factor.
From a phenomenological point of view, the inflationary scale
factor describes a very little and very calm Universe emerging from the
Planck scale. The evolution of this Universe is very slow
at the beginning, but the string coupling with the metric and the
string equation of state drive this evolution, leading to a each time
more increasing and fast dynamics. In approaching the exit of
inflation, the scale factor and the spacetime
curvature increase. The metric "explodes" around
the exit of inflation. The last part of this explosion is the transition
to radiation dominated stage in a process that breaks transitorily
the effective treatment of strings. This transition is supposed
brief, the transition to standard cosmology happens at the
beginning of radiation dominated stage.
At priori, this model seems privilege an unknown parameter $t_I$, playing
the role usually assigned to singularity at $t=0$ in string cosmology.
But this is not unnatural, since in order to reach an enough amount
of inflation, this parameter is found to be very close to the
standard radiation
dominated beginning $t_r$ and so, to the beginning of standard cosmology.
On the other hand, this value appears related to the GUT scale,
which is consistent with
the freezing of the dilaton
evolution and change in the equation of state in our effective
string treatment.
In this way, the Universe starts from a classical, weak coupling
and small curvature regime. Driven by the strings, it evolves
towards a quantum regime at strong coupling and
curvature.
The argument above mentioned, where a brief
transition exits and comebacks among stages
in a low energy effective treatment,
is not an exceptional feature in String Cosmology. Pre-Big
Bang models \cite{gv93} deal with two branches
described in low energy effective treatments. Both branches
are string duality related, but the former one runs on negative
time values. The intermediate region of high curvature
is supposed containing the singularity at $t=0$ and
consequently, the Big Bang. Our String Driven Cosmology
presents also this intermediate point of high curvature,
but it is found around the inflation-radiation dominated
transition, near but not on Planck scale.
In our model, negative proper times are never considered and the
instant $t=0$ remains before of the described inflationary stage
(is not coherent include it in the effective description, since before Planck
time a fully stringy regime is expected whose effects
can not be considered merely with the effective equations).
There are not predicted singularities, neither at $t=0$ nor $t=t_I$, at
the level of the minimal model here studied.
Another great difference with the ``Pre-Big Bang scenario'' is
the predicted dynamics of universe. The Pre-Big Bang
scenario includes a Dilaton Driven phase running on
negative times. Not such
feature is found
in our Universe description. Although the curvature does not
obey a monotonic regime, time runs always on positive values and
the scale factor always expands
in our String Driven Cosmology.
The Pre-Big Bang scenario assumes
around $t \sim 0$ a ``String Phase'' with high almost constant curvature.
Our minimal String Driven Cosmology does not
assume such a phase, but a state of
high curvature is approached (and reached) at the end of
inflation stage. The low energy effective regime $(L.E.E.)$
breaks down around the inflation exit, both due to increasing curvature
(the scale factor approaches the pole singularity) and to the increasing
dilaton. The exit of inflation and beginning
of radiation dominated stage must be described with a more complete
treatment for high curvature regimes.
From eq.(\ref{radil}), the growth reached by the dilaton field
during the inflationary stage would not be so large, at least
while the low energy effective treatment holds.
The exact amount depends mainly on the initial inflationary
conditions, the parameter $\phi_I$ being constrained
by the effective equations (\ref{leeeq}). In anycase, when
the scale factor approaches its singularity and the $e$-folds
number $f$ increases very quickly with time, the same is not
true for the dilaton ratio. Comparatively, the dilaton ratio increases in
a much slower way $\frac{\phi(t_r)}{\phi(t_i)} \sim f$ than the scale
factor $\frac{a(t_r)}{a(t_i)} \sim e^f$. As a consequence,
corrections due to the high curvature regime are needed
much earlier than the corresponding to dilaton growth.
\section{Conclusions}
\setcounter{equation}0
\hspace{8mm} We have considered a minimal model for the evolution of the
scale factor
totally extracted from string cosmology.
The earliest stages (an inflationary power type expansion and a radiation
dominated stage) are obtained from the low energy effective equations,
while the radiation dominated
stage and matter dominated stage
are selfconsistent solutions of the Einstein equations with
a gas of strings as matter sources.
Such solutions suggest the low energy
effective action is asymptotically valid at earliest stages, around
and immediately after Planck time $t_P \sim 10^{-43} {\mathrm{s}}$, when
the spacetime and string dynamics would be strongly coupled.
The radiation dominated stage is extracted
from both treatments, coherently with being it an intermediate
stage among the two regimes: inflation and matter dominated stage.
On the other hand, since the stable string
behaviour
describes cold matter behaviour, current matter dominated stage
can be extracted also in the string matter treatment.
Notice that in
string theory, the equation of state of the string matter is derived
from the string dynamics itself and not given at hand from outside
as in pure General Relativity.
No detail on the transitions dynamics can be extracted in this
framework, too naive for accounting such effects. The
inflation-radiation dominated transition implies a transitory breaking of the
low energy effective regime. The radiation dominated-matter dominated stage
can not be modelized in sudden, continuous and smooth way.
Further study is required on the evolution of the equation of state for
the gas of
strings. The three string behaviours
in curved backgrounds are present in this gas and each cosmological
stage is selfconsistently driven by them.
In this way, the transition
between inflation and radiation dominated stage is related with the
evolution from unstable to dual string behaviour, while the
radiation dominated-matter dominated transition could be driven by passing
from the dual to stable behaviour.
Phenomenological information extracted from this String Driven
model is compatible with observational Universe information.
An amount of inflation, usually considered as enough for solving
the cosmological puzzles, can be obtained in the inflationary stage.
Energy ranges at exit of inflation are found coherents with GUT scales.
The inflationary stage gives a value for the energy
density-dilaton coupled term equivalent to the corresponding
critical energy density, computed as in General Relativity. That
means, we have $\left. \Omega \right|_{inf} = 1 $, whenever the
end of inflationary stage be computed. Also, the
contribution to current energy density is computed $\Omega \geq
\frac{4}{9}$, and taking account the validity of General
Relativity in the current matter dominated stage,
we find this contribution be exactly $\Omega = 1$.
Our main conclusion is to have proved that string cosmology,
althought being effective, is able to produce a
suitable minimal model for Universe evolution.
It is possible to place each effective context in a time-energy
scale range. Energy ranges are
found and General Relativity conclusions are obtained
too in a string theory context coherently. We have extracted the
General Relativity statement about spatial curvature and
energy density, at least for the spatially flat case
($k=0 \rightarrow \Omega=1$) in a totally Non-Einstenian
framework, like the low energy effective string action giving
rise to the inflationary String Driven stage.
In their validity range, no need of extra stages
is found. Only the interval around the transitions and the
very beginning epoch, probably the Planck epoch, will require
more accurate treatments that hitohere
considered.
Since the behaviours above extracted are asymptotic results, it is
not possible to give the detail of the transitions among the different
stages.
The connection among asymptotic low energy effective regimes through
a very brief stage (requiring a more complete description of
string dynamics) enables us to suppose this stage containing the evolution
in the equation of state from unstable strings domain to dual strings one.
From the point of view of the scale factor evolution, this brief transition
could be modelized as nearly instantaneous, provided curvature and
scale factor expansion have attainted nearly their maximun values.
Similar comment can be made in the radiation dominated-matter
dominated transition. It must be driven by the subsequent evolution of
the gas of strings from dual to unstable domain to string stable behaviour
domain.
Again, a brief intermediate stage could take place
among both asymptotic behaviours.
But this is an open
question in the framework of string cosmology both for inflation-radiation
dominated as well as radiation dominated-matter dominated transition.
{\bf ACKNOWLEDGMENTS}
M.P.I. wants to thank Marina Ram\'{o}n Medrano and H\'{e}ctor
J. de Vega for very helpful discussions.
|
2,869,038,154,787 | arxiv | \section{Introduction}
\noindent A \textit{mobile facility} (MF) is a facility capable of moving from one place to another, providing real-time service to customers in the vicinity of its location when it is stationary \citep{halper2011mobile}. In this paper, we study a mobile facility \textcolor{black}{fleet sizing, }routing, and scheduling problem (MFRSP) with stochastic demand. Specifically, in this problem, we aim to find the number of MFs (i.e., fleet size) to use in a given service region over a specified planning horizon and the route and schedule for each MF in the fleet. The demand level of each customer in each time period is random. The probability distribution of the demand is unknown, and only partial information about the demand (e.g., mean and range) may be available. The objective is to find the MF fleet size, routing, and scheduling decisions that minimize the sum of the fixed cost of establishing the MF fleet, the cost of assigning demand to the MFs (e.g., transportation cost), and the cost of unsatisfied demand (i.e., shortage cost).
The concept of MF routing and scheduling is very different than conventional static facility location (FL) and conventional vehicle routing (VR) problems. In static FL problems, we usually consider opening facilities at fixed locations. Conventional VR problems aims at handling the movement of items between facilities (e.g., depots) and customers. \textcolor{black}{A mobile facility is a \textit{facility-like vehicle} that functions as a traditional facility when it is stationary, except that it can move from one place to another if necessary \citep{lei2014multicut}}. Thus, the most evident advantage of MFs over fixed facilities is their flexibility in moving to accommodate the change in the demand over time and location \citep{halper2011mobile, lei2014multicut, lei2016two}.
MFs are used in many applications ranging from cellular services, healthcare services, to humanitarian relief logistics. For example, light trucks with portable cellular stations can provide cellular service in areas where existing cellular network of base stations temporarily fails \citep{halper2011mobile}. Mobile clinics (i.e., customized MFs fitted with medical equipment and staffed by health professionals) can travel to rural and urban areas to provide various (prevention, testing, diagnostic) health services. Mobile clinics also offer alternative healthcare (service) delivery options when a disaster, conflict, or other events cause stationary healthcare facilities to close or stop operations \citep{blackwell2007use, brown2014mobile, du2007mobile, gibson2011households, oriol2009calculating, song2013mobile}. For example, mobile clinics played a significant role in providing drive-through COVID-19 testing sites or triage locations during the COVID-19 pandemic. In 2019, the mobile health clinic market was valued at nearly 2 billion USD and is expected to increase to $\sim$12 billion USD by 2028 \citep{MFMarket}. In humanitarian relief logistics, MFs give relief organizations the ability to provide aid to populations dispersed in remote and dense areas. These examples motivate the need for computationally efficient optimization tools to support decision-making in all areas of the MF industry.
MF operators often seek a \textcolor{black}{strategic} and tactical plan, including the size of the MF fleet (\textcolor{black}{strategic}) and a routing plan for each MF in the fleet (tactical and operational) that minimize their fixed operating costs and maximize demand satisfaction. Determining the fleet size, in particular, is very critical as it is a major fixed investment for starting any MF-based business. The fleet sizing problem depends on the MF operational performance, which depends on the routing and scheduling decisions. The allocation of the demand to the MFs is also very important for the entire system performance \citep{lei2016two}. For example, during the COVID-19 pandemic, Latino Connection, a community health leader, has established Pennsylvania's first COVID-19 Mobile Response Unit, CATE (i.e., Community-Accessible Testing \& Education). The goal of CATE is to provide affordable and accessible COVID-19 education, testing, and vaccinations to low-income, vulnerable communities across Pennsylvania to ensure the ability to stay safe, informed, and healthy \citep{CATEMF}. \textcolor{black}{During COVID-19, CATE published an online schedule consisting of the mobile unit stops and the schedule at each stop.} A model that optimizes CATE's fleet size and schedules considering demand uncertainty could help improve CATE's operational performance and achieve better access to health services.
Unfortunately, the MFRSP is a challenging optimization problem for two primary reasons. First, customers' demand is random and hard to predict in advance, especially with limited data during the planning process. Second, even in a perfect world in which we know with certainty the amount of demand in each period, the deterministic MFRSP is challenging because it is similar to the classical FL problem \citep{halper2011mobile, lei2014multicut}. \textcolor{black}{Thus, the incorporation of demand variability increases the overall complexity of the MFRSP. However, ignoring demand uncertainty may lead to sub-optimal decisions and, consequently, the inability to meet customer demand (i.e., shortage).} Failure to meet customer demand may lead to adverse outcomes, especially in healthcare, as it impacts population health. It also impacts customers' satisfaction and thus the reputation of the service providers and may increase their operational cost (due to, e.g., outsourcing the excess demand to other providers).
To model uncertainty, \cite{lei2014multicut} assumed that the probability distribution of the demand is known and accordingly proposed the first a \textit{priori} two-stage stochastic optimization model (SP) for a closely related MFRSP. Although attractive, the applicability of the SP approach is limited to the case in which we know the distribution of the demand or we have sufficient data to model it. \textcolor{black}{In practice, however, one might not have access to a sufficient amount of high-quality data to estimate the demand distribution accurately.} This is especially true in application domains where the use of mobile facilities to deliver services is relatively new (e.g., mobile COVID-19 testing clinics). Moreover, it is challenging for MF companies to obtain data from other companies (competitors) due to privacy issues. Finally, various studies show that different distributions can typically explain raw data of uncertain parameters, indicating distributional ambiguity \citep{esfahani2018data, vilkkumaa2021causes}.
Suppose we model uncertainty using a data sample from a potentially biased distribution or an assumed distribution (as in SP). In this case, the resulting nominal decision problem evaluates the cost only at this training sample, and thus the resulting decisions may be overfitted (optimistically biased). Accordingly, SP solutions may demonstrate disappointing out-of-sample performance (`\textit{black swans}') under the true distribution (or unseen data). In other words, solutions of SP decision problems often display an optimistic in-sample risk, which cannot be realized in out-of-sample settings. This phenomenon is known as the \textit{Optimizers' Curse} (i.e., an attempt to optimize based on imperfect estimates of distributions leads to biased decisions with disappointing performance) and is reminiscent of the overfitting effect in statistics \citep{smith2006optimizer}.
\textcolor{black}{Alternatively, one can construct an ambiguity set of all distributions that possess certain partial information about the demand. Then, using this ambiguity set, one can formulate a distributionally robust optimization (DRO) problem to minimize a risk measure (e.g., expectation or conditional value-at-risk (CVaR)) of the operational cost over all distributions residing within the ambiguity set. In particular, in the DRO approach, the optimization is based on the worst-case distribution within the ambiguity set, which effectively means that the distribution of the demand is a decision variable.}
DRO \textcolor{black}{has received substantial attention recently} in various application domains due to the following striking benefits. First, as pointed out by \cite{esfahani2018data}, DRO models are more ``\textit{honest}'' than their SP counterparts as they acknowledge the presence of distributional uncertainty. \textcolor{black}{Therefore, DRO solutions often faithfully anticipate the possibility of black swan (i.e., out-of-sample disappointment). Moreover, depending on the ambiguity set used, DRO often guarantees an out-of-sample cost that falls below the worst-case optimal cost}. Second, DRO alleviates the unrealistic assumption of the decision-maker's complete knowledge of distributions. Third, several studies have proposed DRO models for real-world problems that are more computationally tractable than their SP counterparts, see, e.g., \cite{basciftci2019distributionally, luo2018distributionally, saif2020data, ShehadehSanci, shehadeh2020distributionallyTucker, tsang2021distributionally, wang2020distributionally, wang2021two,wu2015approximation}. In this paper, we propose tractable DRO approaches for the MFRSP.
The ambiguity set is a key ingredient of DRO models that must (1) capture the true distribution with a high degree of certainty, and (2) be computationally manageable (i.e., allow for a tractable DRO model or solution method). There are several methods to construct the ambiguity set. Most applied DRO literature employs moment-based ambiguity \citep{delage2010distributionally, zhang2018ambiguous}, consisting of all distributions sharing particular moments (e.g., mean-support ambiguity). The main advantage of the mean-support ambiguity set, for example, is that it incorporates intuitive statistics that a decision-maker may easily approximate and change. Moreover, various techniques have been developed to derive tractable moment-based DRO models. However, asymptotic properties of the moment-based DRO model cannot often be guaranteed because the moment information represents descriptive statistics.
Recent DRO approaches define the ambiguity set by choosing a distance metric (e.g., $\phi$--divergence \citep{jiang2016data}, Wasserstein distance \citep{esfahani2018data, gao2016distributionally}) to describe the deviation from a reference (often empirical) distribution. The main advantage of Wasserstein ambiguity, for example, is that it enable decision-makers to incorporate possibly small-size data in the ambiguity set and optimization, enjoys asymptotic properties, and often offers a strong out-of-sample performance guarantee \citep{esfahani2018data,mevissen2013data}. Recent results indicate that Wasserstein's ambiguity centered around a given empirical distribution contains the unknown true distribution with a high probability and is richer than other divergence-based ambiguity sets (in particular, they contain discrete and continuous distributions as compared to, e.g., $\phi$-divergence ball centered at the empirical distribution which does not contain any continuous distribution, and Kullback-Leibler divergence ball, which must be absolutely continuous with respect to the nominal distribution).
Despite the potential advantages, there are no moment-based, Wasserstein-based, or any other DRO approaches for the specific MFRSP that we study in this paper (see Section~\ref{Sec:LitRev}). This inspires this paper's central question: \textit{what are the computational and operational performance values of employing DRO to address demand uncertainty and ambiguity compared to the classical SP approach for the MFRSP}. To answer this question, we design and analyze two DRO models based on the demand's mean, support, and mean absolute deviation ambiguity and Wasserstein ambiguity and compare the performance of these models with the classical SP approach.
\subsection{\textcolor{black}{Contributions}}
\noindent In this paper, we present two distributionally robust MF fleet sizing, routing, and scheduling (DMFRS) models for the MFRSP, as well as methodologies for solving these models. We summarize our main contributions as follows.
\begin{enumerate}
\item \textbf{Uncertainty Modeling and Optimization Models.} We propose the first two-stage DRO models for the MFRSP. These models aim to find the optimal (1) number of MFs to use within a planning horizon, (2) a routing plan and a schedule for the selected MFs, i.e., the node that each MF is located at in each time period, (3) assignment of MFs to customers. Decisions (1)-(2) are planning (first-stage) decisions, which cannot be changed in the short run. Conversely, the assignments of the demand are decided based on the demand realization, and thus are second-stage decisions. The objective is to minimize the fixed cost (i.e., cost of establishing the MF fleet and traveling inconvenience cost) plus the maximum of a risk measure (expectation or mean CVaR) of the operational cost (i.e., transportation and unsatisfied demand costs) over all possible distributions of the demand defined by an ambiguity set. In the first model (MAD-DRO), we use an ambiguity set based on the demand's mean, support, and mean absolute deviation (MAD). In the second model (W-DRO), we use an ambiguity set that incorporates all distributions within a 1-Wasserstein distance from a reference distribution. To the best of our knowledge, and according to our literature review in Section~\ref{Sec:LitRev}, our paper is the first to address the distributional ambiguity of the demand in the MFRSP using DRO.
\item \textbf{Solution Methods.} We derive equivalent solvable reformulations of the proposed mini-max nonlinear DRO models. We propose a computationally efficient decomposition-based algorithm to solve the reformulations. In addition, we derive valid lower bound inequalities that efficiently strengthen the master problem in the decomposition algorithm, thus improving convergence.
\item \textbf{Symmetry-Breaking Constraints.} We derive two families of new symmetry breaking constraints, which break symmetries in the solution space of the first-stage routing and scheduling decisions and thus improve the solvability of the proposed models. These constraints are independent of the method of modeling uncertainty. Hence, they are valid for any (deterministic and stochastic) formulation that employ the first-stage decisions of the MFRSP. Our paper is the first to attempt to break the symmetry in the solution space of these planning decisions of the MFRSP.
\item \textbf{Computational Insights.} We conduct extensive computational experiments comparing the proposed DRO models and a classical SP model empirically and theoretically, demonstrating where significant performance improvements can be gained. Specifically, our results show (1) how the DRO approaches have superior operational performance in terms of satisfying customers demand as compared to the SP approach; (2) the MAD-DRO model is more computationally efficient than the W-DRO model; (3) the MAD-DRO model yield more conservative decisions than the W-DRO model, which often have a higher fixed cost but significantly lower operational cost; (4) how mobile facilities can move from one location to another to accommodate the change in demand over time and location; (5) efficiency of the proposed symmetry breaking constraints and lower bound inequalities; (6) the trade-off between cost, number of MFs, MF capacity, and operational performance; and (7) the trade-off between the risk-neutral and risk-averse approaches. Most importantly, our results show the value of modeling uncertainty and distributional ambiguity.
\end{enumerate}
\subsection{Structure of the paper}
\color{black}
The remainder of the paper is structured as follows. In Section~\ref{Sec:LitRev}, we review the relevant literature. Section~\ref{sec3:DMFRSformulation} details our problem setting. In Section~\ref{sec:SP}, we present our SP. In Section~\ref{sec:DRO_Models}, we present and analyze our proposed DRO models. In Section~\ref{sec:solutionmethods}, we present our decomposition algorithm and strategies to improve convergence. In Section~\ref{sec:computational}, we present our numerical experiments and corresponding insights. Finally, we draw conclusions and discuss future directions in Section~\ref{Sec:Conclusion}.
\color{black}
\section{Relevant Literature}\label{Sec:LitRev}
\noindent \textcolor{black}{In this section, we review recent literature that is most relevant to our work, mainly studies that propose stochastic optimization approaches for closely related problems to the MFRSP.} There is limited literature on MF as compared to stationary facilities. However, as pointed out by \cite{lei2014multicut}, the MFRSP share some features with several well-studied problems, including Dynamic Facility Location Problem (DFLP), Vehicle Routing Problem (VRP), and the Covering Tour Problem (CTP). First, let us briefly discuss the similarities and differences between the MFRSP and these problems. Given that we consider making decisions over a planning period, then the MFRSP is somewhat similar to DFLP, which seeks to locate/re-locate facilities over a planning horizon. To mitigate the impact of demand fluctuation along the planning period, decision-makers may open new facilities and close or relocate existing facilities at a relocation cost (\cite{albareda2009multi, antunes2009location, contreras2011dynamic, drezner1991facility,jena2015dynamic, jena2017lagrangian, van1982dual}). Most DFLPs assume that the relocation time is relatively short as compared to the planning horizon. In contrast, the MFRSP takes into account the relocation time of MFs. In addition, each MF needs to follow a specific route during the entire planning horizon, which is not a requirement in DFLP.
In CTP, one seeks to select a subset of nodes to visit that can cover other nodes within a particular coverage (\cite{current1985maximum, flores2017multi, gendreau1997covering, hachicha2000heuristics, tricoire2012bi}). In contrast to the MFRSP, CTP does not consider the variations of demand over time and assumes that the amount of demand to be met by vehicles is not related to the length of time the MF is spending at the stop. The VRP is one of the most extensively studied problem in operations research. The VRP also has numerous applications and variants \citep{subramanyam2020robust}. Both the MFRSP and the VRP consider the routing decisions of vehicles. However, the MFRSP is different than the VRP in the following ways \citep{lei2014multicut}. First, in the MFRSP, we can meet customer demand by a nearby MF (e.g., cellular stations). In the VRP, vehicles visit customers to meet their demand. Second, the amount of demand that an MF can serve at each location depends on the duration of the MF stay, which is a decision variable. In contrast, VRPs often assume a fixed service time. Finally, most VRPs require that each customer has to be visited exactly once in each route. In contrast, in the MFRSP, some customers may not be visited, and some may be visited multiple times.
Next, we review studies that proposed stochastic optimization approaches to problems similar to the MFRSP. \cite{halper2011mobile} introduced the concept of MF and proposed a continuous-time formulation to model the maximum covering mobile facility routing problem under deterministic settings. To solve their model, \cite{halper2011mobile} proposed several computationally effective heuristics. \cite{lei2014multicut} and \cite{lei2016two} are two closely related (and only) papers that proposed stochastic optimization approaches for MF routing and scheduling.
\cite{lei2014multicut} assumed that the distribution of the demand is known and accordingly proposed the first a \textit{priori} two-stage SP for MFRSP. \cite{lei2014multicut}'s SP seeks optimal first-stage routing and scheduling decisions to minimize the total expected system-wide cost, where the expectation is taken with respect to the known distribution of the demand. A priori optimization has a managerial advantage since it guarantees the regularity of service, which is beneficial for both customer and service provider. That is, a prior plan allow the customers to know when and where to receive service and enable MF service providers to be familiar with routes and better manage their time schedule during the day. The applicability of the SP approach is limited to the case in which the distribution of the demand is fully known, or we have sufficient data to model it.
Robust optimization (RO) and distributionally robust optimization (DRO) are alternative techniques to model, analyze and optimize decisions under uncertainty and ambiguity (where the underlying distributions are unknown). RO assumes that the uncertain parameters can take any value from a pre-specified uncertainty set of possible outcomes with some structure \citep{bertsimas2004price, ben2015deriving, soyster1973convex}. In RO, optimization is based on the worst-case scenario within the uncertainty set.
\textcolor{black}{Notably, \cite{lei2016two} are the first to motivate the importance of handling demand uncertainty using RO. They argue that RO is useful because it only requires moderate information about the uncertain demand rather than a detailed description of the probability distribution or a large data set. Specifically, \cite{lei2016two} proposed the first two-stage RO approach for MF feet sizing and routing problem with demand uncertainty. \cite{lei2016two}'s model aims to find the fleet size and routing decisions that minimize the fixed cost of establishing the MF fleet (first-stage) and a penalty cost for the unmet demand (second-stage).} Optimization in \cite{lei2016two}'s RO model is based on the worst-case scenario of the demand occurring within a polyhedral uncertainty set. By focusing the optimization on the worst-case scenario, RO may lead to overly conservative and suboptimal decisions for other more-likely scenarios \citep{chen2019robust, delage2018value}.
DRO models the uncertain parameters as random variables whose underlying probability distribution can be any distribution within a pre-defined ambiguity set. The ambiguity set is a family of all possible distributions characterized by some known properties of random parameters \citep{esfahani2018data}. In DRO, optimization is based on the worst-case distribution within this set. DRO is an attractive approach to model uncertainty with ambiguous distributions because: (1) it alleviates the unrealistic assumption of the decision-makers' complete knowledge of the distribution governing the uncertain parameters, (2) it is usually more computationally tractable than its SP and RO counterparts \citep{delage2018value, rahimian2019distributionally}, and (3) one can use minimal distributional information or a small sample to construct the ambiguity set and then build DRO models. \cite{rahimian2019distributionally} provide a comprehensive survey of the DRO literature.
\color{black}
The computational tractability of DRO models depends on the ambiguity sets. These sets are often based on moment information \citep{delage2010distributionally, mehrotra2014models,zhang2018ambiguous} or statistical measures such as the Wasserstein distance \citep{esfahani2018data}. To derive tractable DRO models for the MFRSP, we construct two ambiguity sets of demand, one based on 1-Wasserstein distance and one using the demand's support, mean, and mean absolute deviation (MAD). As mentioned in the introduction, we use the Wasserstein ambiguity because it is richer than other divergence-based ambiguity sets. We use the MAD as a dispersion measure instead of the variance because it allows tractable reformulation and better captures outliers and small deviations. In addition, the MAD exists for some distributions while the second moment does not \citep{ben1985approximation, postek2018robust}. We refer to \cite{postek2018robust} and references therein for rigorous discussions on properties of MAD.
\color{black}
\textcolor{black}{Next, we discuss some relevant results on the mean-support-MAD ambiguity set (henceforth denoted as MAD ambiguity). \cite{ben1972more} derived tight upper and lower bounds on the expectation of a general convex function of a random variable under MAD ambiguity. In particular, when the random variable is one-dimensional, \cite{ben1972more} show that the worst-case distribution under MAD ambiguity is a three-point distribution on the mean, support, and MAD. Recently, \cite{postek2018robust} used the results of \cite{ben1972more} to treat ambiguous expected feasibility constraints to obtain exact reformulations for both functions that are convex and concave in the components of the random variable under MAD ambiguity. These reformulations require independence of the random variables and involve an exponential number of terms. However, for the special case of linearly aggregated random variables, \cite{postek2018robust} derived polynomial-sized upper bounds on the worst-case expectations of convex functions. Finally, under the assumption of independent random variables, they derived tractable approximations of ambiguous chance constraints under mean, support, and MAD information.}
\color{black}
A reviewer of this paper brought our attention to the results in a working paper by \cite{longsupermodularity} on supermodularity in two-stage DRO problems. Specifically, \cite{longsupermodularity} identified a tractable class of two-stage DRO problems based on the scenario-based ambiguity set proposed by \cite{chen2019robust}. They showed that any two-stage DRO problem with mean, support, and upper bounds of MAD has a computationally tractable reformulation whenever the second-stage cost function is supermodular in the random parameter. Furthermore, they proposed an algorithm to compute the worst-case distribution for this reformulation. They argued that using the computed worst-case distribution in the reformulation can make the two-stage DRO problem tractable. In addition, they provided a necessary and sufficient condition to check whether any given two-stage optimization problem has the property of supermodularity. \textcolor{black}{In Appendix~\ref{Appx:Points_MAD}, we show that even if our recourse is supermodular in demand realization, the number of support points in the worst-case distribution of the demand is large, which renders our two-stage MAD-DRO model computationally challenging to solve using \cite{longsupermodularity}'s approach. In contrast, we can efficiently solve an equivalent reformulation of our MAD-DRO model using our proposed decomposition algorithm.}
\color{black}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth, height=60mm]{LitTable2}
\caption{Comparison between \cite{lei2014multicut}, \cite{lei2016two}, and our paper. }\label{Fig:Compare}
\end{figure}
Despite the potential advantages, there are no DRO approaches for the specific MFRSP that we study in this paper. Therefore, our paper is the first to propose and analyze DRO approaches for the MFRSP. In Figure~\ref{Fig:Compare}, we provide a comparison between \cite{lei2014multicut}, \cite{lei2016two}, and our approach based on the assumption made on uncertainty distribution, proposed stochastic optimization approach, decision variables, objectives, and addressing symmetry. We note that our paper and these papers share the common goal of deriving generic optimization models that can be used in any application of MF where one needs to determine the same sets of decisions under the same criteria/objective considered in each paper.
We make the following observations from Figure~\ref{Fig:Compare}. In contrast to \cite{lei2016two}, we additionally incorporate the MF traveling inconvenience cost in the first-stage objective and the random transportation cost in the second-stage objective. In contrast to \cite{lei2016two} and \cite{lei2014multicut}, we model both uncertainty and distributional ambiguity and optimize the system performance over all demand distributions residing within the ambiguity sets. Our master and sub-problems and lower bound inequalities have a different structure than those of \cite{lei2016two} due to the differences in the decision variables and objectives. We also propose two families of symmetry-breaking constraints, which break symmetries in the solution space of the routing and scheduling decisions. These constraints can improve the solvability of any formulation that uses the same routing and scheduling decisions of the MFRSP. \cite{lei2014multicut} and \cite{lei2016two} did not address the issue of symmetry in the MFRSP. Finally, to model decision makers' risk-averse attitudes, we propose both risk-neutral (expectation) and risk-averse (mean-CVaR-based) DRO models for the MFRSP. \cite{lei2014multicut} and \cite{lei2016two} models are risk-neutral.
Finally, it is worth mentioning that our work uses similar reformulation techniques in recent DRO static FL literature (see, e.g., \cite{basciftci2019distributionally, luo2018distributionally, saif2020data, ShehadehSanci, shehadeh2020distributionallyTucker, tsang2021distributionally, wang2020distributionally,wang2021two, wu2015approximation} and references therein).
\section{\textcolor{black}{Problem Setting}}\label{sec3:DMFRSformulation}
\noindent As in \cite{lei2014multicut}, we consider a fleet of $M$ mobile facilities and define the MFRSP on a directed network $G(V, E)$ with node set $V:=\lbrace v_1, \ldots, v_n\rbrace$ and edge set $E:=\lbrace e_1, \ldots, e_m \rbrace$. The sets $I \subseteq V$ and $J \subseteq V$ are the set of all customer points and the subset of nodes where MFs can be located, respectively. The distance matrix $D=(d_{i,j})$ is defined on $E$ and satisfies the triangle inequality, where $d_{i,j}$ is a deterministic and time-invafriant distance between any pair of nodes $i$ and $j$. For simplicity in modeling, we consider a planning horizon of $T$ identical time periods, and we assume that the length of each period $t\in T$ is sufficiently short such that, without loss of generality, all input parameters are the same from one time period to another (this is the same assumption made in \cite{lei2014multicut} and \cite{lei2016two}). The demand, $W_{i,t}$, of each customer $i$ in each time period $t$ is random. The probability distribution of the demand is unknown, and only a possibly small data on the demand may be available. We assume that we know the mean $\pmb{\mu}:=[\mu_{1,1}, \ldots, \mu_{|I|, |T|}]^\top$ and range [$\pmb{\underline{W}}$, $\pmb{\overline{W}}$] of $\pmb{W}$. Mathematically, we make the following assumption on the support of $\pmb{W}$.
\noindent \textbf{Assumption 1.} The support set $\mathcal{S}$ of $\pmb{W}$ in \eqref{support} is nonempty, convex, and compact.
\begin{align} \label{support}
& \mathcal{S}:=\left\{ \pmb{W} \geq 0: \begin{array}{l} \underline{W}_{i,t} \leq W_{i,t} \leq \overline{W}_{i,t}, \ \forall i \in I, \ \forall t \in T \end{array} \right\}.
\end{align}
\noindent We consider the following basic features as in \cite{lei2014multicut}: (1) each MF has all the necessary service equipment and can move from one place to another; (2) all MFs are homogeneous, providing the same service, and traveling at the same speed; (3) we explicitly account for the travel time of the MF in the model, and service time are only incurred when the MF is not in motion; (4) the travel time $t_{j,j^\prime}$ from location $j$ to $j^\prime$ is an integer multiplier of a single time period (\cite{lei2014multicut, lei2016two}; and (5) the amount of demand to be served is proportional to the duration of the service time at the location serving the demand.
We consider a cost $f$ for using an MF, which represents the expenses associated with purchasing or renting an MF, staffing cost, equipment, etc. Each MF has a capacity limit $C$, which represents the amount of demand that an MF can serve in a single time unit. Due to the random fluctuations of the demand and the limited capacity of each MF, there is a possibility that the MF fleet will fail to satisfy customers' demand fully. To minimize shortage, we consider a penalty cost $\gamma$ for each unit of unmet demand. This penalty cost can represent the opportunity cost for the loss of demand or expense for outsourcing the excess demand to other companies \citep{basciftci2019distributionally, lei2016two}. Thus, maximizing demand satisfaction is an important objective that we incorporate in our model \citep{lei2014multicut}.
Given that an MF cannot provide service when in motion, it is not desirable to keep it moving for a long time to avoid losing potential benefits. On the other hand, it is not desirable to keep the MF stationary all the time because this may lead to losing the potential benefits of making a strategic move to locations with higher demand. Thus, the trade-off of the problem includes the decision to move or keep the MF stationery. Accordingly, we consider a traveling inconvenience cost $\alpha$ to discourage unnecessary moving in cases where moving would neither improve nor degrade the total performance. As in \cite{lei2014multicut}, we assume that $\alpha$ is much lower than other costs such that its impact over the major trade-off is negligible.
We assume that the quality of service a customer receives from a mobile facility is inversely proportional to the distance between the two to account for the ``access cost'' (this assumption is common in practice and in the literature, see, e.g., \cite{ahmadi2017survey, reilly1931law, drezner2014review, lei2016two, berman2003locating,lei2014multicut}). Accordingly, we consider a demand assignment cost that is linearly proportional to the distance between the customer point and the location of an MF, i.e., $\beta d_{i,j}$, where $\beta\geq 0$ represents the assignment cost factor per demand unit and per distance unit. Table~\ref{table:notation} summarizes these notation.
Given a set of MFs, $M$, $T$, $I$, and $J$, our models aim to find: (1) the number of MFs to use within $T$; (2) a routing plan and a schedule for the selected MFs, i.e., the node that each MF is located at in each time period; and (3) assignment of MFs to customers. Decisions (1)--(2) are first-stage decisions that we make before realizing the demand. The assignment decisions (3) represent the recourse (second-stage) actions in response to the first-stage decisions \textcolor{black}{and demand realizations} (i.e., you cannot assign demand to the MFs before realizing the demand). \textcolor{black}{The objective is to minimize the fixed cost (i.e., cost of establishing the MF fleet and traveling inconvenience cost) plus a risk measure (expectation or mean CVaR) of the operational cost (i.e., transportation and unsatisfied demand costs).} We refer to \cite{lei2014multicut} for an excellent visual representation of MF operations and some of the basic features mentioned above.
\noindent \textbf{\textit{Additional Notation}}: For $a,b \in \mathbb{Z}$, we define $[a]:=\lbrace1,2,\ldots, a \rbrace$ as the set of positive integer running indices to $a$. Similarly, we define $[a,b]_\mathbb{Z}:=\lbrace c \in \mathbb{Z}: a \leq c \leq b \rbrace$ as the set of positive running indices from $a$ to $b$. The abbreviations ``w.l.o.g.'' and ``w.l.o.o.'' respectively represent ``without loss of generality'' and ``without loss of optimality.''
\section{\textbf{Stochastic Programming Model}}\label{sec:SP}
\noindent \textcolor{black}{In this section, we present a two-stage SP formulation of the MFRSP that assumes that the probability distribution of the demand is known. A complete listing of the parameters and decision variables of the model can be found in Table~\ref{table:notation}.}
\color{black}
First, let us introduce the variables and constraints defining the first-stage of this SP model. For each $m \in M$, we define a binary decision variable $y_m$ that equals 1 if MF $m$ is used, and is 0 otherwise. For all $j \in J$, $m \in M$, and $t \in T$, we define a binary decision variable $x_{j,m}^t$ that equals 1 if MF $m$ stays at location $j$ at period $t$, and is 0 otherwise. The feasible region $\mathcal{X}$ of variables $\pmb{x}$ and $\pmb{y}$ is defined in \eqref{eq:RegionX}.
\color{black}
\begin{align}
\mathcal{X}&=\left\{ (\pmb{x}, \pmb{y}) : \begin{array}{l} x_{j,m}^t+x_{j^\prime,m}^{t^\prime} \leq y_m, \ \ \forall t, m, j, \ j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace \\
x_{j,m}^t \in \lbrace 0, 1\rbrace, \ y_m \in \lbrace 0, 1\rbrace, \ \forall j, m , t \end{array} \right\}. \label{eq:RegionX}
\end{align}
As defined in \cite{lei2014multicut}, region $\mathcal{X}$ represents: (1) the requirement that an MF can only be in service when it is stationary; (2) MF $m$ at location $j$ in period $t$ can only be available at location $j^\prime\neq j$ after a certain period of time depending on the time it takes to travel from location $j$ to $j^\prime$, $t_{j,j^\prime}$ (i.e., $x_{j,m}^{t^\prime}=0$ for all $j^\prime \neq j$ and $t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace$); and (3) MF $m$ has to be in an active condition before providing service. We refer the reader to Appendix~\ref{Appx:FirstStageDec} for a detailed derivation of region $\mathcal{X}$.
\color{black}
Now, let us introduce the variables defining our second-stage problem. For all $(i,\ j, \ m, \ t)$, we define a nonnegative continuous variable $z_{i,j,m}^t$ to represent the amount of node $i$ demand served by MF $m$ located at $j$ in period $t$. For each $t \in T$, we define a nonnegative continuous variable $u_{i,t}$ to represent the amount of unmet demand of node \textit{i} in period $t$. Finally, we define a random vector $\pmb{W}:=[W_{1,1}. \ldots, W_{|I|,|T|}]^\top$. Our SP model can now be stated as follows:
\color{black}
\begin{table}[t]
\small
\center
\renewcommand{\arraystretch}{0.6}
\caption{Notation.}
\begin{tabular}{ll}
\hline
\multicolumn{2}{l}{\textbf{Indices}} \\
$m$& index of MF, $m=1,\ldots, M$\\
$i$ & index of customer location, $i=1,\ldots,I$\\
$j$ & index of MF location, $j=1,...,J$\\
\multicolumn{2}{l}{\textbf{Parameters and sets}} \\
$T$& planning horizon\\
$M$ & number, or set, of MFs \\
$J$& number, or, set of locations \\
$f$ & fixed operating cost \\
$d_{i,j}$ & distance between any pair of nodes $i$ and $j$ \\
$t_{j,j'}$ & travel time from $j$ to $j'$\\
$C$ & the amount of demand that can be served by an MF in a single time unit, i.e., MF capacity \\
$W_{i,t}$& demand at customer site $i$ for each period $t$\\
$\underline{W}_{i,t}/\overline{W}_{i,t}$ & lower/upper bound of demand at customer location $i$ for each period $t$ \\
$\gamma$ & penalty cost for each unit of unmet demand\\
\multicolumn{2}{l}{\textbf{First-stage decision variables } } \\
$y_{m}$ & $\left\{\begin{array}{ll}
1, & \mbox{if MF \textit{m} is permitted to use,} \\
0, & \mbox{otherwise.}
\end{array}\right.$ \\
$x_{jm}^t$ & $\left\{\begin{array}{ll}
1, & \mbox{if MF \textit{m} stays at location \textit{j} at period \textit{t},} \\
0, & \mbox{otherwise.}
\end{array}\right.$ \\
\multicolumn{2}{l}{\textbf{Second-stage decision variables } } \\
$z_{i,j,m}^t$ & amount of demand of node $i$ being served by MF $m$ located at $j$ in period $t$ \\
$u_{i,t}$ & total amount of unmet demand of node \textit{i} in period $t$ \\
\hline
\end{tabular}\label{table:notation}
\end{table}
\begin{subequations}\label{DMFRS_SP}
\begin{align}
(\text{SP}) \quad Z^*=& \min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} \left\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \textcolor{black}{\varrho
\big(Q(\pmb{x}, \pmb{W})\big)} \ \right\rbrace, \label{ObjSP}
\end{align}
\end{subequations}
where for a feasible $(\pmb{x}, \pmb{y}) \in \mathcal{X}$ and a realization of $\pmb{W}$
\allowdisplaybreaks
\begin{subequations}\label{2ndstage}
\begin{align}
Q(\pmb{x}, {\pmb{W}} ):= & \min_{\pmb{z, u}} \Big (\sum_{j \in J} \sum_{i \in I} \sum_{m \in M} \sum_{t \in T} \beta d_{i,j} z_{i,j,m}^t+ \gamma \sum_{t \in T} \sum_{i \in I} u_{i,t}\Big)\label{2ndstageObj}\\
& \ \ \text{s.t.} \ \ \ \ \sum_{j \in J} \sum_{m \in M} z_{i,j,m}^t+u_{i,t}= W_{i,t}, \qquad \forall i \in I, \ t \in T, \label{2ndstage_const1}\\
&\quad \ \quad \quad \sum_{i \in I} z_{i,j,m}^t \leq C x_{j,m}^t, \qquad \forall j \in J, \ m \in M, \ t \in T, \label{2ndstage_const2}\\
&\quad \ \quad \quad u_t \geq 0, \ z_{i,j,m}^t \geq 0, \qquad \forall i \in I, \ j \in J, m \in M, \ t \in T. \label{2ndstage_const3}
\end{align}
\end{subequations}
\noindent \textcolor{black}{Formulation \eqref{DMFRS_SP} aims to find first-stage decisions $(\pmb{x},\pmb{y}) \in \mathcal{X}$ that minimize the sum of (1) the fixed cost of establishing the MF fleet (first term); (2) the traveling inconvenience cost\footnote{Minimizing the traveling inconvenience cost is equivalent to maximizing the profit of keeping the MF stationary whenever possible. Parameter $\alpha$ is the profit weight factor as detailed in \cite{lei2014multicut})} (second term); and (3) a risk measure $\varrho (\cdot) $ of the random second-stage function $Q(\pmb{x}, {\pmb{W}} )$ (third term). A risk-neutral decision-maker may opt to set $\varrho(\cdot)=\mathbb{E}(\cdot)$, whereas a risk-averse decision-maker might set $\varrho(\cdot)$ as the CVaR or mean-CVaR. Classically, the MFRSP literature
employs $\varrho(\cdot)=\mathbb{E}(\cdot)$, which might be more intuitive for MF providers. For brevity, we relegate further discussion of the mean-CVaR-based SP model to Appendix~\ref{Appex:SP_CVAR}.}
\section{\textcolor{black}{Distributionally Robust Optimization (DRO) Models}}\label{sec:DRO_Models}
\noindent In this section, we present our proposed DRO models for the MFRSP that do not assume that the probability distribution of the demand is known. In Sections~\ref{sec:MAD-DRO} and \ref{sec:WDMFRS_model}, we respectively present and analyze the risk-neutral MAD-DRO and W-DRO models. For brevity, we relegate the formulations and discussions of the risk-averse mean-CVaR-based DRO models to Appendix~\ref{Sec:meanCVAR}.
\subsection{\textbf{The DRO Model with MAD Ambiguity (MAD-DRO)}}\label{sec:MAD-DRO}
\noindent \textcolor{black}{In this section, we present our proposed MAD-DRO model, which is based on an ambiguity set that incorporates the demand's mean ($\pmb{\mu}$), MAD ($\pmb{\eta}$), and support ($\mathcal{S}$)}. As mentioned earlier, we use the MAD as a dispersion or variability measure because it allows us to derive a computationally attractive reformulation \citep{postek2018robust, wang2019distributionally, wang2020distributionally}.
\textcolor{black}{First, let us introduce some additional sets and notation defining our MAD ambiguity set and MAD-DRO model}. We define $\mathbb{E}_\mathbb{P}$ as the expectation under distribution $\mathbb{P}$. We let $\mu_{i,t}=\mathbb{E}_\mathbb{P}[W_{i,t}]$ and $\eta_{i,t}=\mathbb{E}_{\mathbb{P}}(|W_{i,t}-\mu_{i,t}|)$ respectively represent the mean value and MAD of $W_{i,t}$, for all $i \in I$ and $t \in T$. \textcolor{black}{Using this notation and the support $\mathcal{S}$ defined in \eqref{support}, we construct the following MAD ambiguity set}:
\begin{align}\label{eq:ambiguityMAD}
\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta}) := \left\{ \mathbb{P}\in \mathcal{P}(\mathcal{S}) \middle|\: \begin{array}{l} \mathbb{P} (\pmb{W} \in \mathcal{S})=1,\\ \mathbb{E_P}[W_{i,t}] = \mu_{i,t}, \ \forall i \in I, \ t \in T, \\
\mathbb{E}_{\mathbb{P}}(|W_{i,t}-\mu_{i,t}|) \leq \eta_{i,t}, \ \forall i \in I, \ t \in T.
\end{array} \right\},
\end{align}
\noindent where $\mathcal{P}(S)$ in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ represents the set of distributions supported on $\mathcal{S}$ with mean $\pmb \mu$ and dispersion measure $ \leq \pmb \eta$. Using $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ defined in \eqref{eq:ambiguityMAD}, we formulate our MAD-DRO model as
\begin{align}
(\text{MAD-DRO}) & \min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} \left\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+\ \Bigg[\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P}[Q(\pmb{x},{\pmb{W}})] \Bigg] \ \right\rbrace. \label{MAD-DMFRS}
\end{align}
\noindent The MAD-DRO formulation in \eqref{MAD-DMFRS} seeks first-stage decisions ($\pmb{x},\pmb{y})$ that minimize the first-stage cost and the worst-case expectation of the second-stage (recourse) cost, \textcolor{black}{where the expectation is taken over all distributions
residing in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$}. Note that we do not incorporate higher moments (e.g., co-variance) in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ for the following primary reasons. First, the mean and range are intuitive statistics that a decision-maker may approximate and change in the model (e.g., the mean may be estimated from limited data or approximated by subject matter experts, and the range may represent the error margin in the estimates). Second, it is not straightforward for decision-makers to approximate or accurately estimate the correlation between uncertain parameters. Third, mathematically speaking, various studies have demonstrated that incorporating higher moments in the ambiguity set often undermines the computational tractability of DRO models and, therefore, their applicability in practice. Indeed, as we will show next, using $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ allows us to derive a tractable equivalent reformulation of the MAD-DRO model and an efficient solution method to solve the reformulation (see Sections~\ref{sec3:reform}, \ref{sec:Alg}, and \ref{sec5:CPU}).
\textcolor{black}{Finally, note that parameters $W_{i,t}$, $\mu_{i,t}$, $\underline{W}_{i,t}$, $\overline{W}_{i,t}$, $\eta_{i,t}$ are all indexed by time period $t$ and location $i$. Thus, if in any application, there is a relationship (e.g., correlation) between the demand of a subset of locations in a subset of periods, one can easily adjust $\mathcal{S}$, $\pmb{\mu}$, and $\pmb{\eta}$ in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ to reflect this relationship. For example, if urban cities have higher demand, then we can adjust the mean and range of the demand of these cities to reflect such a relationship. Similarly, if there is a correlation between the time period $t$ and the demand, then we can define the mean and the support based on this correlation. For example, the morning service hours may have lower demand on Monday. In this case, we can adjust $\pmb{\mu}$, $\mathcal{S}$, and $\pmb{\eta}$ to reflect this relationship. Similarly, if the demand in a subset of periods and locations is correlated, we can adjust $\mathcal{S}$, $\pmb{\mu}$, and $\pmb{\eta}$ in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ to reflect this relationship}. Nevertheless, we acknowledge that not incorporating higher moments and complex relations may be a limitation of our work and thus is worth future investigation.
\subsubsection{\textbf{Reformulation of the MAD-DRO model}}\label{sec3:reform}
\textcolor{white}{ }
\textcolor{black}{Recall that $Q(\pmb{x},{\pmb{W}})$ is defined by a minimization problem; hence, in \eqref{MAD-DMFRS}, we have an inner max-min problem. As such, it is not straightforward to solve formulation \eqref{MAD-DMFRS} in its presented form}. In this section, we derive an equivalent formulation of \eqref{MAD-DMFRS} that is solvable. \textcolor{black}{First, in Proposition~\ref{Prop1:DualMinMax}, we present an equivalent reformulation of the inner problem in \eqref{MAD-DMFRS} (see Appendix \ref{Appx:ProofofProp1} for a proof).}
\begin{proposition}\label{Prop1:DualMinMax}
For any fixed $(\pmb{x}, \pmb{y}) \in \mathcal{X}$, problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P}[Q(\pmb{x},{\pmb{W}})]$ in \eqref{MAD-DMFRS} is equivalent to
\begin{align} \label{eq:FinalDualInnerMax-1}
& \min_{\pmb{\rho}, \pmb{\psi} \geq 0} \ \left \lbrace \sum \limits_{t \in T} \sum \limits_{i \in I} ( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+\max \limits_{\pmb{W}\in \mathcal{S}} \Big\lbrace Q(\pmb{x},{\pmb{W}}) + \sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})\Big \rbrace \right \rbrace.
\end{align}
\end{proposition}
\noindent \textcolor{black}{Again, the problem in \eqref{eq:FinalDualInnerMax-1} involves an inner max-min problem that is not straightforward to solve in its presented form. However, we next derive an equivalent reformulation of the inner problem in \eqref{eq:FinalDualInnerMax-1} that is solvable.}
First, we observe that $Q(\pmb{x},{\pmb{W}})$ is a feasible linear program (LP) for a given first-stage solution $(\pmb{x}, \pmb{y}) \in \mathcal{X}$ and a realization of ${\pmb{W}}$. The dual of $Q(\pmb{x}, {\pmb{W}})$ is as follows.
\begin{subequations}\label{DualOfQ}
\begin{align}
Q(\pmb{x},{\pmb{W}})=& \max \limits_{\pmb{\lambda, v}} \ \sum_{t \in T} \sum_{i \in I} \lambda_{i,t} W_{i,t}+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t \label{Obj:Qdual}\\
& \ \text{s.t. } \ \lambda_{i,t}+ v_{j,m}^t \leq \beta d_{i,j}, \ \qquad \forall i \in I, j \in J, m \in M, t \in T, \label{Const1:QudalX}\\
& \qquad \ \ \lambda_{i,t} \leq \gamma, \qquad \qquad \qquad \ \ \forall i \in I, \ t \in T, \label{Const2:QdualU}\\
& \qquad \ \ v_{j,m}^t \leq 0, \qquad \qquad \qquad \ \ \forall j \in J, \ t \in T, \label{Const3:Qdualv}
\end{align}
\end{subequations}
\noindent where $\pmb{\lambda}$ and $\pmb{v}$ are the dual variables associated with constraints \eqref{2ndstage_const1} and \eqref{2ndstage_const2}, respectively. It is easy to see that w.l.o.o, $\lambda_{i,t} \geq 0$ for all $i \in I$ due to constraints \eqref{Const2:QdualU} and the objective of maximizing $W_{i,t}\geq 0$ times $\lambda_{i,t}$. Additionally, $ v_{j,m}^t \leq \min\{ \min_i\{\beta d_{i,j}-\lambda_{i,t}\},0 \}$ by constraints \eqref{Const1:QudalX} and \eqref{Const3:Qdualv}. Given the objective of maximizing a nonnegative term $Cx_{j,m}^t $ multiplied by $v_{j,m}^t$, $ v_{j,m}^t = \min\{ \min_i\{\beta d_{i,j}-\lambda_{i,t}\},0 \}$ in the optimal solution. Given that $\pmb{\beta}$, $\pmb d$, and $\pmb \lambda$ are finite, $\pmb{v} $ is finite. It follows that problem \eqref{DualOfQ} is a feasible and bounded LP. Note that $W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}]$ and ($\underline{W}_{i,t}, \overline{W}_{i,t})\geq 0$ by definition, in view of dual formulation \eqref{DualOfQ}, we can rewrite the inner maximization problem $\max \{ \cdot \} $ in \eqref{eq:FinalDualInnerMax-1} as
\begin{subequations}\label{DualQAndOuter}
\begin{align}
\max \limits_{\pmb{\lambda, v, W, k}} & \Bigg\{ \sum_{t \in T} \sum_{i \in I} \lambda_{i,t} W_{i,t}+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Bigg \} \\
\ \text{s.t. } &\eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \ \ W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}], \ \forall i \in I, \ \forall t \in T, \\
& k_{i,t} \geq W_{i,t}-\mu_{i,t}, \ k_{i,t} \geq \mu_{i,t}-W_{i,t}, \ \forall i \in I, \ \forall t \in T . \label{Const1:DualQAndOuter}
\end{align}
\end{subequations}
Note that the objective function in \eqref{DualQAndOuter} contains the interaction term $\lambda_{i,t} W_{i,t}$. To linearize formulation \eqref{DualQAndOuter}, we define $\pi_{i,t}=\lambda_{i,t} W_{i,t}$ for all $i \in I$ and $t \in T$. Also, we introduce the following McCormick inequalities for variables $\pi_{i,t}$:
\begin{subequations}
\begin{align}
&\pi_{i,t} \geq \lambda_{i,t} \underline{W}_{i,t}, \qquad \qquad \qquad \ \ \ \pi_{i,t} \leq \lambda_{i,t} \overline{W}_{i,t}, \ \ \forall i \in I, \ \forall t \in T, \label{McCormick1}\\
& \pi_{i,t} \geq \gamma W_{i,t}+ \overline{W}_{i,t} (\lambda_{i,t}- \gamma ), \qquad \pi_{i,t} \leq \gamma W_{i,t}+\underline{W}_{i,t}( \lambda_{i,t}-\gamma), \ \forall i \in I, \ \forall t \in T. \label{McCormick2}
\end{align}
\end{subequations}
Accordingly, for a fixed $(\pmb{x} \in \mathcal{X}, \pmb{\rho}, \pmb{\psi})$, problem \eqref{DualQAndOuter} is equivalent to the following mixed-integer linear program (MILP):
\begin{subequations}\label{eq:FinalDualInnerMax-2}
\begin{align}
\max \limits_{\pmb{\lambda, v, W, \pi, k} }& \ \Bigg\{ \sum_{t \in T} \sum_{i \in I} \pi_{i,t}+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Bigg\}\\
\text{s.t. } & \ (\pmb{\lambda, v}) \in \{\eqref{Const1:QudalX}-\eqref{Const3:Qdualv}\}, \ \pmb{\pi} \in \{ \eqref{McCormick1}-\eqref{McCormick2} \}, \label{Const1_finalinner} \\
& \ W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}], \ k_{i,t} \geq W_{i,t}-\mu_{i,t}, \ k_{i,t} \geq \mu_{i,t}-W_{i,t}, \ \forall i \in I, \ \forall t \in T. \label{Const2_finalinner}
\end{align}
\end{subequations}
Combining the inner problem in the form of \eqref{eq:FinalDualInnerMax-2} with the outer minimization problems in \eqref{eq:FinalDualInnerMax-1} and \eqref{MAD-DMFRS}, we derive the following equivalent reformulation of the MAD-DRO model in \eqref{MAD-DMFRS}:
\begin{subequations}\label{FinalDR}
\begin{align}
&\min_{\pmb{x,y,\rho, \psi, \delta}} \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big[\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big] + \delta \Bigg \} \\
& \ \ \text{s. t.} \ \ (\pmb{x}, \pmb{y}) \in \mathcal{X}, \ \psi \geq 0, \\
& \qquad \ \ \delta \geq \textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}, \label{const1:FinalDR}
\end{align}
\end{subequations}
where $\textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}=\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}): \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner} \Big\} $.
\begin{proposition}\label{Prop2:Convexh} \textcolor{black}{For any fixed values of variables $(\pmb{x}, \pmb{y}) \in \mathcal{X}$, $\pmb \rho,$ and $\pmb \psi$, $h(\pmb{x}, \pmb \rho, \pmb \psi)< +\infty$. Furthermore, function $(\pmb{x}, \pmb \rho, \pmb \psi) \mapsto\textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}$ is a convex piecewise linear function in $\pmb{x}$, $\pmb\rho$, and $\pmb \psi$ with a finite number of pieces (see Appendix \ref{Appx:Prop2} for a detailed proof).}
\end{proposition}
\vspace{1mm}
\subsection{\textit{\textbf{The DRO Model with 1-Wasserstein Ambiguity (W-DRO)}}}\label{sec:WDMFRS_model}
\noindent In this section, we consider the case that $\mathbb{P}$ may be observed via a possibly small finite set $\{\hat{\pmb{W}}^1, \ldots, \hat{\pmb{W}}^N\}$ of $N$ i.i.d. samples, which may come from the limited historical realizations of the demand or a reference empirical distribution. Accordingly, we construct an ambiguity set based on 1-Wasserstein distance, which often admits tractable reformulation in most real-life applications (see, e.g., \cite{Daniel2020, hanasusanto2018conic, jiang2019data, tsang2021distributionally, saif2020data}).
\textcolor{black}{First, let us define the 1-Wasserstein distance.} Suppose that random vectors $\pmb{\pmb{W}}_1$ and $\pmb{\pmb{W}}_2$ follow $\F_1$ and $\F_2$, respectively, where probability distributions $\F_1$ and $\F_2$ are defined over the common support $\mathcal{S}$. The 1-Wasserstein distance \textcolor{black}{$\text{dist}(\F_1, \F_2)$ between $\F_1$ and $\F_2$ is the minimum transportation cost of moving from $\F_1$ to $\F_2$, where the cost of moving masses $\pmb{W}_1$ to $\pmb{W}_2$ is the norm $||\pmb{W}_1-\pmb{W}_2||$}. \textcolor{black}{ Mathematically,}
\begin{equation}\label{W_distance}
\textcolor{black}{\text{dist}}(\mathbb{F}_1, \mathbb{F}_2):=\inf_{\Pi \in \mathcal{P}(\mathbb{F}_1, \mathbb{F}_2)} \Bigg \{\int_{\mathcal{S}} ||\pmb{W}_1-\pmb{W}_2|| \ \Pi(\text{d}\pmb{W}_1, \text{d}\pmb{W}_2) \Bigg | \begin{array}{ll} & \Pi \text{ is a joint distribution of }\pmb{W}_1 \text{ and } \pmb{W}_2\\
& \text{with marginals } \mathbb{F}_1 \text{ and } \mathbb{F}_2, \text{ respectively}
\end{array} \Bigg \},
\end{equation}
\noindent where $\mathcal{P}(\mathbb{F}_1, \mathbb{F}_2) $ is the set of all joint distributions of $(\pmb{W}_1, \pmb{W}_2)$ supported on $\mathcal{S}$ with marginals ($\mathbb{F}_1$, $\mathbb{F}_2$). Accordingly, we construct the following $1$-Wasserstein ambiguity set:
\begin{align}\label{W-ambiguity}
\mathcal{F} (\hat{\mathbb{P}}^N, \epsilon)= \left\{ \mathbb{P} \in \mathcal{P}(\mathcal{S}) \middle|\begin{array}{l} \textcolor{black}{\text{dist}}(\mathbb{P}, \hat{\mathbb{P}}^N) \leq \epsilon\end{array} \right\},
\end{align}
\noindent where $ \mathcal{P}(\mathcal{S})$ is the set of all distributions supported on $\mathcal{S}$, $\hat{\mathbb{P}}^N=\frac{1}{N} \sum_{n=1}^N \delta_{\hat \pmb{W}^n}$ is the empirical distribution of $\pmb{W}$ based on $N$ i.i.d samples, and $\epsilon >0$ is the radius of the ambiguity set. The set $\mathcal{F} (\hat{\mathbb{P}}^N, \epsilon)$ represents a Wasserstein ball of radius $\epsilon$ centered at the empirical distribution $\hat{\mathbb{P}}^N$. \textcolor{black}{Using the ambiguity set $\mathcal{F}(\hat{\mathbb{P}}^N, \epsilon)$ defined in \eqref{W-ambiguity}, we formulate our W-DRO model as}
\begin{align}
(\text{W-DRO}) \ \hat{Z}(N, \epsilon)&= \min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} \Bigg\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+\Bigg [\sup_{\mathbb{P} \in \mathcal{F} (\hat{\mathbb{P}}^N, \epsilon) }\mathbb{E}_{\mathbb{P}} [Q(\pmb{x}, \pmb{W}) ] \Bigg] \Bigg \}. \label{W-DMFRS}
\end{align}
\noindent Formulation \eqref{W-DMFRS} finds first-stage decisions ($\pmb{x},\pmb{y}) \in \mathcal{X}$ that minimize the first-stage cost and the maximum expectation of the second-stage cost over all distributions residing in $\mathcal{F} (\hat{\mathbb{P}}^N, \epsilon)$.
The W-DRO model in \eqref{W-DMFRS} can be used to model uncertainty in general and distributional ambiguity when there is a possibly small finite data sample on uncertainty. As detailed in \cite{esfahani2018data} and discussed earlier, if we have a small sample and we optimize using this sample, then the optimizer's curse cannot be avoided. To mitigate the optimizer's curse (estimation error), we robustify the nominal decision problem (the MFRSP optimization problem) against all distributions $\mathbb{P}$ under which the estimated distribution $\hat{\mathbb{P}}^N$ based on the $N$ data points has a small estimation error (i.e., with $ \textcolor{black}{\text{dist}}(\mathbb{P}, \hat{\mathbb{P}}^N) \leq \epsilon$). Therefore, in some sense one can think of Wasserstein ball as the set of all distributions under which our estimation error is below $\epsilon$, where $\epsilon$ is the maximum estimation error against which we seek protection. When $\epsilon=0$, the ambiguity set contains the empirical distribution and the W-DRO problem in \eqref{W-DMFRS} reduces to the SP problem. A larger radius $\epsilon$ indicates that we seek more robust solutions (see Appendix \ref{WDRO_Radius}).
In the next section, we show that using $\ell_1$-norm instead of the \textcolor{black}{$\ell_p$}-norm ($1<p<\infty$) in our Wasserstein ambiguity set allows us to derive a linear and tractable reformulation of the W-DRO model in \eqref{W-DMFRS}. Note that $\ell_1$-norm (i.e., the sum of the magnitudes of the vectors in space) is the most intuitive and natural way to measure the distance between vectors. In contrast, the \textcolor{black}{$\infty$-norm}-based Wasserstein ball is an extreme case. That is, the $\infty$-norm gives the largest magnitude among each element of a vector. Thus, from the perspective of the Wasserstein DRO framework, the \textcolor{black}{$\infty$-norm}-based distance metric only picks the most influential value to determine the closeness between data points \citep{chen2018robust}, which, in our case, may not be reasonable since every demand point plays a role. Deriving and comparing DRO models with different Wasserstein sets is out of the scope of this paper but is worth future investigation in more comprehensive MF optimization problems.
\subsubsection{\textbf{\textit{Reformulation of the W-DRO model}}}\label{sec:Wreformualtion}
\textcolor{white}{ }
\noindent In this section, we derive an equivalent solvable reformulation of the W-DRO model in \eqref{W-DMFRS}. First, in Proposition~\ref{Prop1} we present an equivalent dual formulation of the inner maximization problem $\sup\ [\cdot ]$ in \eqref{W-DMFRS} (see Appendix~\ref{Proof_Prop1} for a detailed proof).
\begin{proposition}\label{Prop1}
For for a fixed $(\pmb{x}, \pmb{y}) \in \mathcal{X}$, problem $\sup \limits_{\mathbb{P}\in \mathcal{F} (\hat{\mathbb{P}}^N, \epsilon) }\mathbb{E}_{\mathbb{P}} [Q(\pmb{x},\pmb{W}) ]$ in \eqref{W-DMFRS} is equivalent to
\begin{align}
& \inf_{\rho \geq 0} \Bigg\{ \epsilon \rho + \Big[ \frac{1}{N} \sum_{n=1}^N \sup_{\pmb{W} \in \mathcal{S}} \big \{ Q(\pmb{x}, \pmb{W})-\rho || \pmb{W} -\pmb{W}^n || \big \} \Big] \Bigg\}. \label{DualOfInner}
\end{align}
\end{proposition}
\noindent Formulation \eqref{DualOfInner} is potentially challenging to solve because it require solving $N$ non-convex optimization problems. Fortunately, given that the support of $\pmb{W}$ is rectangular and finite (see Assumption 1) and $Q(\pmb{x},\pmb{W})$ is feasible and bounded for every $\pmb{x}$ and $\pmb{W} $, we next recast these inner problems as LPs for each $\rho \geq 0$ and $\pmb{x} \in \mathcal{X}$. First, using the dual formulation of $Q(\pmb{x}, \pmb{W})$ in \eqref{DualOfQ}, we rewrite the inner problem $\sup \{ \cdot \} $ in \eqref{DualOfInner} for each $n$ as
\begin{subequations}\label{InnerInnerW}
\begin{align}
& \max \limits_{\pmb{\lambda}, \pmb{v}, \pmb{W}} \Big \{ \sum_{t \in T} \sum_{i \in I} \lambda_{i,t} W_{i,t}-\rho |W_{i,t}-W_{i,t}^n|+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t \Big \}\\
& \text{s.t. } \ \ (\pmb{\lambda}, \pmb{v}) \in \{ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv} \}, \pmb{W} \in [\pmb{\underline{W}}, \pmb{\overline{W}} ].
\end{align}
\end{subequations}
Second, using the same techniques in Section~\ref{sec3:reform}, we define an epigraphical random variable $\eta_{i,t}^n$ for the term $ |W_{i,t}-W_{i,t}^n|$. Then, using variables $\pmb{\eta}$, $\pi_{i,t}=\lambda_{i,t} W_{i,t}$, and inequalities \eqref{McCormick1}-\eqref{McCormick2} for variables $\pi_{i,t}$, we derive the following equivalent reformulation of \eqref{InnerInnerW} (for each $n \in [N]$):
\begin{subequations}\label{InnerInnerW2}
\begin{align}
\max \limits_{\pmb{\lambda}, \pmb{v}, \pmb{W}, \pmb{\pi}, \pmb{\eta}} &\big \{ \sum_{t \in T} \sum_{i \in I} \pi_{i,t}-\rho \eta_{i,t}^n+ \sum_{t \in T}\sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t \big \} \label{Obj:InnerInnerW2}\\
\text{s.t. } & \ (\pmb{\lambda, v}) \in \{ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv} \}, \label{C1:InnerInnerW2} \\
& \ \ \pi_{i,t} \in \{ \eqref{McCormick1}- \eqref{McCormick1}\}, \ \ W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}] , && \forall i \in I , \forall t \in T, \label{C2:InnerInnerW2} \\
& \ \ \eta_{i,t}^n \geq W_{i,t}-W_{i,t}^n, \ \eta_{i,t}^n \geq W_{i,t}^n- W_{i,t}, && \forall i \in I , \forall t \in T. \label{C_InnerInner}
\end{align}
\end{subequations}
Third, combining the inner problem in the form of \eqref{InnerInnerW2} with the outer minimization problems in \eqref{DualOfInner} and \eqref{W-DMFRS}, we derive the following equivalent reformulation of the W-DRO model in \eqref{W-DMFRS}
\begin{align}
\hat{Z}(N, \epsilon) &= \min_{ (\pmb{x}, \pmb{y}) \in \mathcal{X}, \ \rho \geq 0 } \Bigg\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \epsilon \rho \nonumber\\
& \quad \ \ \ \ +\frac{1}{N} \sum_{n=1}^N \max \limits_{\pmb{\lambda}, \pmb{v}, \pmb{W}, \pmb{\pi}, \pmb{\eta}} \Big \{ \sum_{t \in T} \sum_{i \in I} \pi_{i,t}- \rho \eta_{i,t}^n+ \sum_{t \in T}\sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t: \eqref{C1:InnerInnerW2}-\eqref{C_InnerInner} \Big \} \Bigg\}.\label{Final_W_DMFRS}
\end{align}
\noindent Using the same techniques in the proof of Proposition~\ref{Prop2:Convexh}, one can easily verify that function $[\max \limits_{\pmb{\lambda, v, \pi, W, \eta}} \big \{ \sum\limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}-\rho \eta_{i,t}^n+ \sum \limits_{t \in T}\sum\limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m}^t v_{j,m}^t \big\}]<\infty $ and is a convex piecewise linear function in $\pmb{x} \in \mathcal{X}$ and $\rho$.
\section{Solution Method}\label{sec:solutionmethods}
\noindent In this section, we present a decomposition-based algorithm to solve the MAD-DRO formulation in \eqref{FinalDR}, and strategies to improve the solvability of the formulation. The algorithmic steps for solving the W-DRO in \eqref{Final_W_DMFRS} are similar. In Section~\ref{sec:Alg}, we present our decomposition algorithm. In Section~\ref{sec:Improv}, we derive valid lower bound inequalities to strengthen the master problem in the decomposition algorithm. In Section~\ref{sec:symm}, we derive two families of symmetry breaking constraints that improve the solvability of the proposed models.
\subsection{\textbf{Decomposition Algorithm}}\label{sec:Alg}
\noindent Proposition~\ref{Prop2:Convexh} suggests that constraint \eqref{const1:FinalDR} describes the epigraph of a convex and piecewise linear function of decision variables in formulation \eqref{FinalDR}. Therefore, given the two-stage characteristics of MAD-DRO in \eqref{FinalDR}, it is natural to attempt to solve problem \eqref{FinalDR} via a separation-based decomposition algorithm. Algorithm~\ref{Alg1:CAG} presents our proposed decomposition algorithm, and the algorithm for the W-DRO model in \eqref{Final_W_DMFRS} has the same steps. Algorithm~\ref{Alg1:CAG} is finite because we identify a new piece of the function $\max \limits_{\pmb{\lambda, v, W, \pi, k}} \ \big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \big\}$ each time when the set $ \lbrace L (\pmb{x}, \delta)\geq 0 \rbrace$ is augmented in step 4, and the function has a finite number of pieces according to Proposition~\ref{Prop2:Convexh}. Note that this algorithm is based on the same theory and art of cutting plane-based decomposition algorithms employed in various other papers using decomposition to solve problems with similar structure. Nevertheless, we customized Algorithm~\ref{Alg1:CAG} to solve our proposed DRO models. In addition, in the following subsections, we derive problem-specific valid inequalities to strengthen the master problem, thus improving convergence.
\begin{algorithm}[t!]
\small
\renewcommand{\arraystretch}{0.3}
\caption{Decomposition algorithm.}
\label{Alg1:CAG}
\noindent \textbf{1. Input.} Feasible region $\mathcal{X}$; support $\mathcal{S}$; set of cuts $ \lbrace L (\pmb{x}, \delta)\geq 0 \rbrace=\emptyset $; $LB=-\infty$ and $UB=\infty.$
\vspace{2mm}
\noindent \textbf{2. Master Problem.} Solve the following master problem
\begin{subequations}\label{Master}
\begin{align}
Z=&\min_{\pmb{x}, \pmb{y}, \pmb \rho, \pmb \psi} \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big[\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big] + \delta \Bigg \} \label{MasterObj} \\
& \ \ \text{s. t.} \qquad (\pmb{x},\pmb{y} ) \in \mathcal{X}, \ \ \pmb{\psi} \geq 0, \ \ \{ L (\pmb{x}, \delta)\geq 0 \},
\end{align}
\end{subequations}
and record the optimal solution $(\pmb{x}^ *, \pmb{\rho}^*, \pmb{\psi}^*, \delta^*)$ and optimal value $Z^*$. Set $LB=Z^*$.
\noindent \textbf{3. Sub-problem.}
\begin{itemize}
\item[3.1.] With $(\pmb{x}, \pmb{\rho}, \pmb{\psi})$ fixed to $(\pmb{x}^*, \pmb{\rho}^*, \pmb{\psi}^*)$, solve the following problem
\begin{subequations}\label{MILPSep}
\begin{align}
h(\pmb{x}, \pmb \rho, \pmb \psi)= &\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big\}\\
& \ \ \ \text{s.t. } \ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner},
\end{align}
\end{subequations}
$\qquad \ $ and record optimal solution $(\pmb{\pi^*, \lambda^*, W^*, v^*, k^*})$ and $h(\pmb{x}, \pmb \rho, \pmb \psi)^*$.
\item[3.2.] Set $UB=\min \{ UB, \ h(\pmb{x}, \pmb \rho, \pmb \psi)^*+ (LB-\delta^*) \}$.
\end{itemize}
\noindent \textbf{4. if} $\delta^* \geq \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^{t*} v_{j,m}^{t*} +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}^*+k_{i,t}^*\psi_{i,t}^*) $ \textbf{then}
$\qquad \ \ $ stop and return $\pmb{x}^*$ and $\pmb{y}^*$ as the optimal solution to problem \eqref{Master} (i.e., MAD-DRO in \eqref{FinalDR}).
\noindent $\ \ $ \textbf{else} add the cut $\delta \geq \sum \limits_{t \in T} \sum \limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum \limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m} v_{j,m}^{t*}+\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}+k_{i,t}^*\psi_{i,t} )$ to the set of
$\qquad \quad$ cuts $ \{L (\pmb{x}, \delta) \geq 0 \}$ and go to step 2.
\noindent $\ \ $ \textbf{end if}
\end{algorithm}
\subsection{\textbf{Multiple Optimality Cuts and Lower Bound Inequalities}}\label{sec:Improv}
\noindent In this section, we aim to incorporate more second-stage information into the first-stage without adding optimality cuts into the master problem by exploiting the structural properties of the recourse problem. We first observe that once the first-stage solutions and the demand are known, the second-stage problem can be decomposed into independent sub-problems with respect to time periods. Accordingly, we can construct cuts for each sub-problem in step 4. Let $\delta_t$ represent the optimality cut for each period $t$, we replace $\delta$ in \eqref{MasterObj} with $\sum_{t} \delta_t$ and add constraints
\begin{align}
\delta_t\geq \sum \limits_{i \in I} \pi_{i,t}^*+ \sum \limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m} v_{j,m}^{t*}+ \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}+k_{i,t}^*\psi_{i,t} ), \qquad \forall t \in T.\label{Cut1}
\end{align}
\noindent The original single cut is the summation of multiple cuts of the form, i.e., $\delta=\sum_{t \in T} \delta_t$. Hence, in each iteration, we incorporate more or at least an equal amount of information into the master problem using \eqref{Cut1} as compared with the original single cut approach. In this manner, the optimality cuts become more specific, which may result in better lower bounds and, therefore, a faster convergence. In Proposition \ref{Prop3:LowerB1}, we further identify valid lower bound inequalities for each time period to tighten the master problem (see Appendix~\ref{Appx:Prop3} for a proof).
\begin{proposition}\label{Prop3:LowerB1}
Inequalities \eqref{Vineq1} are valid lower bound inequalities on the recourse of the MFRSP.
\begin{align}
\sum \limits_{i \in I} \min \lbrace \gamma, \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \rbrace \underline{W}_{i,t}, \qquad \forall t \in T. \label{Vineq1}
\end{align}
\end{proposition}
It follows that $\sum \limits_{i \in I} \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}+\delta_t\geq \sum \limits_{i \in I} \min \lbrace \gamma, \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \rbrace \underline{W}_{i,t} $, for all $t \in T$, are valid.
\subsection{\textbf{Symmetry-Breaking Constraints}}\label{sec:symm}
\noindent Suppose there are three homogeneous MFs. As such, solutions $y=[1,1,0]^\top$, $y=[0,1,1]^\top$, and $y=[1, 0,1 ]^\top$ are equivalent (i.e., yield the same objective) in the sense that they all permit 2 out of 3 MFs to be used in the planning period. To avoid \textcolor{black}{wasting time} exploring such equivalent solutions, we assume that MFs are numbered sequentially and add constraints \eqref{Sym1} to the first-stage.
\begin{align}\label{Sym1}
y_{m+1} \leq y_m, \qquad \forall m <M.
\end{align}
\noindent \textcolor{black}{Constraints \eqref{Sym1} enforce} arbitrary ordering or scheduling of MFs. Second, recall that in the first period, $t=1$, we decide the initial locations of the MFs. Therefore, it doesn't matter which MF is assigned to location $j$. For example, suppose that we have three candidate locations, and MFs 1 and 2 are active. Then, feasible solutions $(x_{1,1}^1=1$, $x_{3,2}^1=1)$ and $(x_{1,2}^1=1$, $x_{3,1}^1=1)$ yield the same objective. To avoid exploring such equivalent solutions, we define a dummy location $J+1$ and \textcolor{black}{add constraints \eqref{Ineq:Symm1}--\eqref{Ineq:Symm2} to the first-stage.}
\begin{subequations}\label{Sym2}
\begin{align}
& x_{j,m}^1 - \sum \limits_{j^\prime=j}^{J+1}x_{j^\prime,m+1}^1\leq 0, && \forall m <M, \forall j \in J, \label{Ineq:Symm1}\\
& x_{J+1, m}^1= 1- \sum \limits_{j=1}^Jx_{j,m}^1, && \forall m \in M.\label{Ineq:Symm2}
\end{align}
\end{subequations}
\noindent Constraints \eqref{Ineq:Symm1}--\eqref{Ineq:Symm2} are valid for any formulation that uses the same sets of first-stage routing and scheduling decisions and constraints. We derived constraints \eqref{Sym1}--\eqref{Sym2} based on similar symmetry breaking principles in \cite{ostrowski2011orbital} and \cite{shehadeh2019analysis}. Although breaking symmetry is very important and standard in integer programming problems, our paper is the first to attempt to break the symmetry in the solution space of the first-stage planning decisions of the MFRSP. In Section~\ref{sec5:symmetry}, we demonstrate the computational advantages that could be gained by incorporating these inequalities.
\section{Computational Experiments}\label{sec:computational}
\noindent In this section, we conduct extensive computational experiments comparing the proposed DRO models and a sample average approximation (SAA) of the SP model computationally and operationally, demonstrating where significant performance improvements could be gained. The sample average model solves model \eqref{DMFRS_SP} with $\mathbb{P}$ replaced by an empirical distribution based on $N$ samples of $\pmb{W}$ (see Appendix~\ref{Appx:SAA} for the formulation). In Section~\ref{sec5.1:instancegen}, we describe the set of problem instances and discuss other experimental setups. In Section~\ref{sec5:CPU}, we compare solution time of the proposed models. In Section~\ref{sec5:symmetry}, we demonstrate efficiency of the proposed lower bound inequalities and symmetry breaking constraints. \textcolor{black}{We compare optimal solutions of the proposed models and their out-of-sample performance in Sections~\ref{sec:Optimal Solutions} and \ref{sec5:OutSample}, respectively}. We analyze the sensitivity of the DRO expectation models to different parameter settings in Section~\ref{sec5:sensitivity}. \textcolor{black}{We close by comparing the risk-neutral and risk-averse models under critical parameter settings in Section~\ref{sec:CVaRExp}.}
\subsection{\textbf{Experimental Setup}}\label{sec5.1:instancegen}
\noindent We constructed 10 MFRSP instances, in part based on the same parameters settings and assumptions made in \cite{lei2014multicut} and \cite{lei2016two}. We summarize these instances in Table~\ref{table:DMFRSInstances}. Each of the 10 instances is characterized by the number of customers locations $I$, number of candidate locations $J$, and the number of periods $T$. Instances 1--4 are from \cite{lei2014multicut} and Instances 5--10 are from \cite{lei2016two}. These benchmark instances represent a wide range of potential service regions in terms of problem size as a function of the number of demand nodes/locations and time periods. For example, if we account for the scale of the problem in the sense of a static facility location problem, instance 10 consists of $30 \times 20= 600$ customers, which is relatively large for many practical applications. In addition, we constructed a service region based on 20 selected nodes (see \textcolor{black}{Figure~\ref{LehighMap}}) in Lehigh County of Pennsylvania (USA). Then, as detailed below, we constructed two instances (denoted as Lehigh 1 and Lehigh 2) based on this region.
\begin{table}[t]
\center
\footnotesize
\renewcommand{\arraystretch}{0.3}
\caption{MFRSP instances. Notation: $I$ is \# of customers, $J$ is \# of locations, and $T$ is \# of periods.}
\begin{tabular}{cccccccccc}
\hline
\textbf{Inst} & \textbf{$\pmb{I}$} & \textbf{$\pmb{J}$} & \textbf{$\pmb{T}$} \\
\hline
1& 10 & 10 & 10& \\
2 & 10 & 10 & 20& \\
3 & 15 & 15 & 10& \\
4 &15 & 15 & 20& \\
5 & 20 & 20 & 10\\
6 &20 & 20 & 20\\
7& 25 & 25& 10\\
8& 25 & 25& 20\\
9& 30 & 30& 10\\
10& 30 & 30& 20\\
\hline
\end{tabular}
\label{table:DMFRSInstances}
\end{table}
For each instance in Table~\ref{table:DMFRSInstances}, we generated a total of $I$ vertices as uniformly distributed random numbers on a 100 by 100 plane and computed the distance between each pair of nodes in Euclidean sense as in \cite{lei2014multicut}. For Lehigh county instances, we first extracted the latitude and longitude of each node and used Bing Maps Developer API to compute the travel time in minutes between each pair of nodes.
We followed the same procedures in the DRO scheduling and facility location literature to generate random parameters as follows. \textcolor{black}{For instances 1--10, we generated $\mu_{i,t}$ from a uniform distribution $U[\underline{W}, \overline{W}]=[20, 60]$, and set the standard deviation $\sigma_{i,t}=0.5\mu_{i,t}$, for all $i \in I$ and $t \in T$.} For Lehigh county instances, we used the population estimate for each node based on the most updated information posted on the 2010 US Census Bureau (see Appendix~\ref{AppexLehigh}) to construct the following two demand structures. In Lehigh 1, we generated the demand's mean as follows: if the population $\geq$10,000, we set $\mu_{i,t}=40$ (i.e., the mean of $U[20,60]$); if the population $\in$ [5,000, 10,000), we set the mean to $\mu=30$; if the population $\in$ [1,000, 5,000), we set $\mu_{i,t}=20$; and if the population $<1,000$, we set $\mu_{i,t}=15$. In Lehigh 2, we use the population percentage (weight) at each node to generate the demand's mean as $\mu_{i,t}=\min(60, \texttt{population\%} \times 1000)$. To a certain extent, these structures reflect what may be observed in real life, i.e., locations with more population may potentially create greater demand. We refer to Appendix~\ref{AppexLehigh} for the details related to Lehigh 1 and Lehigh 2.
\begin{figure}
\begin{center}
\includegraphics[scale=0.8]{MAP2.png}
\caption{Map of 20 cities in Lehigh County. We created this map using the \texttt{geoscatter} function (MATLAB).}\label{LehighMap}
\end{center}
\end{figure}
For each instance, we sample $N$ realizations $W_{i,t}^n, \ldots, W_{I,T}^n$, for $n=1, \ldots, N$, by following lognormal (LogN) distributions with the generated $\mu_{i,t}$ and $\sigma_{i,t}$. We round each parameter to the nearest integer. We solve the SAA and W-DRO models using the $N$ sample and the MAD-DRO model with the corresponding mean, MAD, and range. The Wasserstein ball's radius $\epsilon$ in the W-DRO model is an input parameter, where different values of $\epsilon$ may result in a different robust solution $\pmb{x} (\epsilon, N)$ with a very different out-of-sample performance $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$. To estimate $\epsilon^{\mbox{\tiny opt}}$ that minimizes $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$, we employed a widely used cross-validation method (see Appendix~\ref{WDRO_Radius}).
We assume that all cost parameters are calculated in terms of present monetary value. Specifically, as in \cite{lei2014multicut}, for each instance, we randomly generate (1) the fixed cost from a uniform distribution $U[a,b]$ with $a = 1000$ and $b = 1500$; (2) the assignment cost factor per unit distance per unit demand $\beta$ from $U[0.0001a,0.0001b]$; and (3) the penalty cost per unit demand $\gamma$ form $U[0.01a, 0.01b]$. Finally, we set the traveling inconvenience cost factor to $0.0001a$, and unless stated otherwise, we use a capacity parameter $C=100$. We implemented all models and the decomposition algorithm using AMPL2016 Programming language calling CPLEX V12.6.2 as a solver with default settings. We run all experiments on a MacBook Pro with Apple M1 Max Chip, 32GB of memory, and 10-core CPUs. Finally, we imposed a solver time limit of 1 hour.
\subsection{\textbf{CPU Time}}\label{sec5:CPU}
\noindent In this section, we analyze solution times of the proposed DRO models. \textcolor{black}{We consider two ranges of the demand: $\pmb{W} \in [20, 60]$ (base-case) and $\pmb{W} \in [50, 100]$ (higher demand variability and volume). We also consider two MF capacities: $C=60$ (relatively small capacity) and $C=100$ (relatively large capacity).} We focus on solving problem instances with a small sample size, which is often seen in most real-world applications (especially in healthcare) and is the primary motivation for using DRO. Specifically, we use $N=10$ and $N=50$ as a sample size for the W-DRO model. For each of the 10 instances in Table~\ref{table:DMFRSInstances}, demand range, $C$, and $N$, we generated and solved 5 instances using each model as described in Section~\ref{sec5.1:instancegen}.
\color{black}
Let us first analyze solution times of the risk-neutral DRO models. Tables~\ref{table:MAD_CPU_2060}--\ref{table:MAD_CPU_50100} present the computational details (i.e., CPU time in seconds and the number of iterations in the decomposition algorithm before it converges to the optimum) of solving the MAD-DRO model. Tables~\ref{table:WASS_CPU_2060_N10}--\ref{table:WASS_CPU_50100_N10} and Tables~\ref{table:WASS_CPU_2060_N50}--\ref{table:WASS_CPU_50100_N50} present the computational details of solving the W-DRO model with $N=10$ and $N=50$, respectively. We observe the following from these tables. First, the computational effort (i.e., solution time per iteration) increases with the size of instance ($I \times J \times T$). Second, solution times are shorter under tight capacity ($C=60$) than under large capacity ($C=100$). In addition, the decomposition algorithm takes fewer iterations to converge to the optimum under tight capacity. Intuitively, when $C=100$, each facility can satisfy more demand than $C=60$. Thus, there are more feasible choices for the MF fleet size and schedule when $C=100$. That said, the search space when $C=100$ is larger, potentially leading to a longer computational time.
Third, using the MAD-DRO model, we were able to solve all instances within the time limit. The average solution time for Instances 1--7 using the MAD-DRO model ranges from 2 to 914 seconds. The average solution time of larger instances (Instances 8--10) ranges from 241 to 1,901 seconds. In contrast, using the W-DRO model, we were able to solve Instances 1--8 with $N=10$ and Instances 1--7 with $N=50$. The average solution time of the W-DRO model with $N=10$ and $N=50$ ranges from 5 to 715 and from 14 to 1,725 seconds, respectively. Note that solution times of the W-DRO model increase with the sample size. This makes sense because the number of variables and constraints of the W-DRO model increases as $N$ increases. In addition, solution times of the W-DRO model with $N=50$ are generally longer than the MAD-DRO model. This also makes sense because the MAD-DRO model is a smaller deterministic model (i.e., it has fewer variables and constraints).
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-DRO model ($ \pmb{W} \in [20, 60]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 1 & 2 & 4 & 3 & 5 & 12 & & 5 & 6 & 8 & 18 & 23 & 30 \\
2 & 10 & 20 & & 3 & 7 & 13 & 21 & 30 & 51 & & 2 & 3 & 4 & 5 & 9 & 15 \\
3 & 15 & 10 & & 2 & 5 & 10 & 3 & 7 & 16 & & 15 & 24 & 37 & 27 & 41 & 62 \\
4 & 15 & 20 & & 4 & 7 & 12 & 6 & 11 & 22 & & 18 & 25 & 43 & 34 & 45 & 72 \\
5 & 20 & 10 & & 6 & 10 & 19 & 6 & 11 & 22 & & 19 & 26 & 45 & 34 & 45 & 72 \\
6 & 20 & 20 & & 9 & 41 & 65 & 6 & 28 & 45 & & 91 & 308 & 640 & 53 & 68 & 80 \\
7 & 25 & 10 & & 23 & 169 & 578 & 8 & 25 & 44 & & 503 & 703 & 1004 & 110 & 138 & 190 \\
8 & 25 & 20 & & 57 & 241 & 352 & 5 & 33 & 53 & & 573 & 872 & 1272 & 147 & 196 & 245 \\
9 & 30 & 10 & & 99 & 416 & 696 & 16 & 61 & 80 & & 185 & 1860 & 1871 & 113 & 147 & 161 \\
10 & 30 & 20 & & 463 & 905 & 1785 & 7 & 38 & 91 & & 1098 & 1592 & 1887 & 6 & 125 & 191 \\
\hline
\end{tabular}\label{table:MAD_CPU_2060}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-DRO model ($ \pmb{W} \in [50, 100]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 1 & 2 & 2 & 1 & 2 & 4 & & 10 & 11 & 13 & 20 & 22 & 24 \\
2 & 10 & 20 & & 1 & 2 & 3 & 2 & 4 & 9 & & 5 & 8 & 13 & 15 & 26 & 41 \\
3 & 15 & 10 & & 4 & 9 & 17 & 2 & 2 & 3 & & 29 & 47 & 65 & 28 & 34 & 45 \\
4 & 15 & 20 & & 3 & 9 & 23 & 2 & 5 & 13 & & 43 & 54 & 70 & 44 & 54 & 63 \\
5 & 20 & 10 & & 27 & 68 & 177 & 3 & 5 & 14 & & 215 & 292 & 392 & 70 & 88 & 124 \\
6 & 20 & 20 & & 6 & 14 & 38 & 3 & 5 & 14 & & 292 & 322 & 347 & 74 & 94 & 107 \\
7 & 25 & 10 & & 20 & 385 & 796 & 2 & 9 & 29 & & 553 & 686 & 914 & 175 & 205 & 258 \\
8 & 25 & 20 & & 63 & 334 & 835 & 3 & 15 & 36 & & 122 & 622 & 1635 & 3 & 14 & 36 \\
9& 30 & 10 & & 25 & 579 & 1692 & 3 & 39 & 143 & & 1851 & 1860 & 1871 & 113 & 147 & 161 \\
10& 30 & 20 & & 81 & 300 & 1240 & 3 & 10 & 41 & & 1821 & 1901 & 1937 & 123 & 147 & 156 \\
\hline
\end{tabular}\label{table:MAD_CPU_50100}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [20, 60]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 7 & 9 & 18 & 7 & 9 & 11 & & 15 & 20 & 22 & 14 & 18 & 20 \\
2 & 10 & 20 & & 8 & 10 & 12 & 7 & 8 & 10 & & 18 & 24 & 28 & 15 & 18 & 20 \\
3 & 15 & 10 & & 35 & 39 & 45 & 16 & 18 & 20 & & 12 & 15 & 18 & 6 & 8 & 9 \\
4 & 15 & 20 & & 30 & 43 & 62 & 5 & 6 & 78 & & 115 & 134 & 147 & 23 & 25 & 29 \\
5 & 20 & 10 & & 41 & 48 & 64 & 6 & 8 & 9 & & 99 & 448 & 682 & 21 & 28 & 33 \\
6 & 20 & 20 & & 49 & 139 & 177 & 6 & 8 & 10 & & 142 & 207 & 273 & 24 & 35 & 44 \\
7 & 25 & 10 & & 81 & 86 & 97 & 8 & 9 & 10 & & 296 & 410 & 506 & 32 & 41 & 50 \\
8 & 25 & 20 & & 100 & 678 & 1820 & 7 & 9 & 11 & & 534 & 715 & 1068 & 47 & 51 & 56 \\
\hline
\end{tabular}\label{table:WASS_CPU_2060_N10}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [50, 100]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 4 & 5 & 7 & 3 & 3 & 4 & & 7 & 9 & 10 & 6 & 7 & 8 \\
2 & 10 & 20 & & 9 & 12 & 17 & 3 & 3 & 4 & & 15 & 16 & 18 & 6 & 6 & 7 \\
3 & 15 & 10 & & 16 & 32 & 55 & 3 & 3 & 3 & & 35 & 48 & 61 & 6 & 7 & 9 \\
4 & 15 & 20 & & 30 & 34 & 43 & 3 & 3 & 3 & & 71 & 81 & 85 & 8 & 9 & 10 \\
5 & 20 & 10 & & 29 & 42 & 76 & 3 & 3 & 3 & & 75 & 79 & 81 & 9 & 10 & 11 \\
6 & 20 & 20 & & 82 & 92 & 108 & 3 & 3 & 3 & & 110 & 182 & 229 & 7 & 11 & 13 \\
7 & 25 & 10 & & 108 & 123 & 135 & 9 & 10 & 11 & & 151 & 192 & 243 & 9 & 10 & 12 \\
\hline
\end{tabular}\label{table:WASS_CPU_50100_N10}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [20, 60]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 18 & 28 & 35 & 6 & 10 & 12 & & 53 & 60 & 71 & 17 & 19 & 21 \\
2 & 10 & 20 & & 30 & 35 & 42 & 8 & 9 & 11 & & 89 & 104 & 114 & 18 & 20 & 21 \\
3 & 15 & 10 & & 28 & 32 & 81 & 6 & 6 & 7 & & 104 & 124 & 139 & 21 & 23 & 25 \\
4 & 15 & 20 & & 40 & 43 & 48 & 6 & 6 & 7 & & 184 & 214 & 231 & 21 & 23 & 25 \\
5 & 20 & 10 & & 33 & 57 & 95 & 4 & 6 & 7 & & 271 & 341 & 463 & 29 & 32 & 36 \\
6 & 20 & 20 & & 89 & 306 & 723 & 6 & 8 & 10 & & 455 & 1461 & 2141 & 22 & 29 & 35 \\
7 & 25 & 10 & & 109 & 127 & 169 & 6 & 7 & 8 & & 1006 & 1725 & 2095 & 37 & 40 & 42 \\
\hline
\end{tabular}\label{table:WASS_CPU_2060_N50}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [50, 100]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 12 & 14 & 17 & 3 & 3 & 4 & & 30 & 32 & 33 & 7 & 7 & 8 \\
2 & 10 & 20 & & 17 & 22 & 25 & 3 & 3 & 4 & & 44 & 57 & 63 & 7 & 9 & 9 \\
3 & 15 & 10 & & 65 & 81 & 92 & 6 & 6 & 7 & & 71 & 105 & 145 & 6 & 6 & 7 \\
4 & 15 & 20 & & 42 & 53 & 87 & 3 & 3 & 7 & & 48 & 65 & 84 & 4 & 5 & 7 \\
5 & 20 & 10 & & 50 & 60 & 66 & 3 & 3 & 3 & & 111 & 144 & 223 & 7 & 7 & 7 \\
6 & 20 & 20 & & 580 & 1174 & 2024 & 3 & 3 & 3 & & 319 & 1009 & 3000 & 8 & 10&11 \\
7 & 25 & 10 & & 106 & 113 & 129 & 3 & 3 & 3 & & 370 & 402 & 464 & 9 & 9 & 10 \\
\hline
\end{tabular}\label{table:WASS_CPU_50100_N50}
\end{table}
\color{black}
\textcolor{black}{Let us now analyze solution times of the risk-averse models. We use MAD-CVaR (W-CVaR) to denote the mean-CVaR-based DRO model with MAD (1-Wasserstein) ambiguity. Since we observe similar computational performance with different values of $\Theta \in [0, 1)$, we present results with $\Theta=0$. Tables \ref{table:MADCVAR_2060}--\ref{table:MADCVAR__50100} and Tables \ref{table:Wass_CVAR_2060_N10}--\ref{table:Wass_CVAR_50100_N50} in Appendix~\ref{Appx:CPU2} present the computational details of solving the MAD-CVaR model and W-CVaR model, respectively. Using the MAD-CVaR model, we were able to solve Instances 1--8 with $\pmb{W} \in [20,60]$ and Instances 1--7 with $\pmb{W} \in [50,100]$. In addition, solution times of MAD-CVaR model are longer than the risk-neutral MAD-DRO model. In contrast, using the W-CVaR model, we were able to solve Instances 1--5. Solution times of these instances are generally longer than the risk-neutral model, especially when $N=50$. It is not surprising that the CVaR models are more computationally challenging to solve than the risk-neutral models because the former models are larger (have more variables and constraints). In particular, the master problem of the CVaR models in the decomposition algorithm is larger than the expectation models (see Algorithm~\ref{Alg2:Decomp2} in Appendix~\ref{Sec:meanCVAR}).}
Finally, it is worth mentioning that using an \textit{enhanced multicut L-shaped} (E-LS) method to solve their SP model, \cite{lei2014multicut} were able to solve Instance 1--4. The average solution time of Instance 4 using E-LS is 3000 seconds obtained at 5\% optimality gap. The CVaR-based SP model is more challenging to solve than the risk-neutral SP model.
\subsection{\textbf{Efficiency of Inequalities \eqref{Vineq1}--\eqref{Sym2} }}\label{sec5:symmetry}
\noindent In this section, we study the efficiency of symmetry breaking constraints~\eqref{Sym1}--\eqref{Sym2} and lower bounding inequalities \eqref{Vineq1}. Given the challenges of solving large instances without \eqref{Vineq1}--\eqref{Sym2}, we use Instance 1 with $\pmb{W} \in [20, 60]$ and $C=100$ in this experiment.
First, we separately solve the proposed models with and without symmetry-breaking (SB) constraints \eqref{Sym1}--\eqref{Sym2}. First, we observe that without these SB constraints, solution times of Instance 1 using (W-DRO, MAD-DRO, SP) significantly increase from (20, 6, 70) to (1,765, 1,003, 3,600) seconds. Instances 3--10 terminated with a large gap after one hour without these SB constraints. Second, as shown in Figure~\ref{FigSB}, both the lower bound and gap (i.e., the relative difference between the upper and lower bounds on the objective value) converge faster when we include constraints \eqref{Sym1}--\eqref{Sym2} in the master problem. Moreover, constraints \eqref{Sym1}--\eqref{Sym2} lead to a stronger bound in each iteration. These results demonstrate the importance of breaking the symmetry in the first-stage decisions and the effectiveness of our SB constraints.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_LB_NoSymm.jpg}
\caption{LB values, W-DRO}
\label{FigSBa}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_LB_NoSymm.jpg}
\caption{LB value, MAD-DRO}
\label{FigSBb}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_GAP_NoSymm.jpg}
\caption{Gap values, W-DRO}
\label{FigSBGAPa}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_GAP_NoSymm.jpg}
\caption{Gap values, W-DRO}
\label{FigSBGAPb}
\end{subfigure}%
\caption{Comparisons of lower bound and gap values with and without SB constraints \eqref{Sym1}-\eqref{Sym2}.}\label{FigSB}
\end{figure}
Next, we analyze the impact of including the valid lower bounding (LB) inequalities \eqref{Vineq1} in the master problem of decomposition algorithm. We first observe that the algorithm takes a very large number of iterations and a longer time until convergence without these LB inequalities. Therefore, in Figure~\ref{FigVI}, we present the LB and gap values with and without inequalities \eqref{Vineq1} from the first 25 iterations. It is obvious that both the lower bound and gap values converge faster when we introduce inequalities \eqref{Vineq1} into the master problem. Moreover, because of the better bonding effect, the algorithm converges to the optimal solution in fewer iterations and shorter solution times.
For example, the algorithm takes 10 seconds and 9 iterations to solve the MAD-DRO instance with these inequalities and terminates with a 33\% gap after an hour without these inequalities. The results in this section demonstrate the importance and efficiency of inequalities \eqref{Vineq1}--\eqref{Sym2}.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_LB_NoBound.jpg}
\caption{LB values, W-DRO}
\label{FigVIa}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_LB_NoBound.jpg}
\caption{LB values, MAD-DRO}
\label{FigVIb}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_Gap_NoBound.jpg}
\caption{Gap values, W-DRO}
\label{FigVIaGap}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_GAP_NoBound.jpg}
\caption{Gap values, MAD-DRO}
\label{FigVIbGap}
\end{subfigure}%
\caption{Comparisons of lower bound and gap values with and without valid LB inequalities \eqref{Vineq1}.}\label{FigVI}
\end{figure}
\subsection{\textbf{Analysis of Optimal Solutions}}\label{sec:Optimal Solutions}
\textcolor{black}{In this section, we compare the optimal solutions of the SP, MAD-DRO, and W-DRO models.} Given that the SP model can only solve small instances to optimality, for a fair comparison and brevity, we use Instance 3 ($I=15$, $J=15$, and $T=10$) as an example of an average-sized instance. In addition, we present results for Lehigh 1 and Lehigh 2. Table~\ref{table:Optimal} presents the number of MFs (i.e., fleet size) for each instance.
\begin{table}[t!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.3}
\caption{Optimal number of MFs.}
\begin{tabular}{lccccccccc}
\hline
\multicolumn{4}{c}{Instance 3 (\textcolor{black}{$\pmb{W}\in [20, 60]$)}} \\
\hline
\textbf{Model} & \textbf{$N=10$} & \textbf{$N=50$} & \textbf{$N=100$} \\
\hline
SP& 7 & 8 & 8 \\
W-DRO & 9 & 8 & 8\\
MAD-DRO & 10 & 10 & 10 \\
\\
\hline
\multicolumn{4}{c}{\textcolor{black}{Instance 3 ($\pmb{W}\in [50, 100]$)}} \\
\hline
SP& 12 & 12 & 12 \\
W-DRO &13 & 13 & 13\\
MAD-DRO & 15 & 15& 15 \\
\hline
\multicolumn{4}{c}{Lehigh 1} \\
\hline
\textbf{Model} & \textbf{$N=10$} & \textbf{$N=50$} & \textbf{$N=100$} \\
\hline
SP& 6 & 7 & 7 \\
W-DRO & 9 & 8 & 8\\
MAD-DRO & 10 & 10 & 10 \\
\\
\hline
\multicolumn{4}{c}{Lehigh 2} \\
\hline
\textbf{Model} & \textbf{$N=10$} & \textbf{$N=50$} & \textbf{$N=100$} \\
\hline
SP& 5 & 5 & 5 \\
W-DRO & 6 & 6 & 6\\
MAD-DRO & 7 & 7 & 7 \\
\hline
\end{tabular}
\label{table:Optimal}
\end{table}
We observe the following from Table~\ref{table:Optimal}. First, the MAD-DRO model always activates (schedules) a higher number of MFs than the SP model, and a larger number of MFs than the W-DRO model. By scheduling more MFs, the MAD-DRO model tends to conservatively mitigate the ambiguity of the demand (reflected by lower shortage and transportation costs reported later in Section~\ref{sec5:OutSample}). Second, the W-DRO model schedules a larger number of MFs than the SP model, and the difference is significant when the sample size is small ($N=10$). This makes sense as a small sample does not have sufficient distributional information. Thus, in this case, the W-DRO model makes conservative decisions to hedge against ambiguity. As $N$ increases (i.e., more information becomes available), the W-DRO model often makes less conservative decisions. Consider Instance 3 with $\pmb{W} \in [20, 60]$, for example. The W-DRO model schedules 9 and 8 MFs when $N=10$ and $N=50$, respectively. \textcolor{black}{Third, we observe that all models scheduled more MFs when we increased the demand's range from $\pmb{W}\in [20, 60]$ to $\pmb{W}\in [50, 100]$ to hedge against the increase in the demand's volume and variability.}
\color{black}
Let us now analyze the optimal locations of the MFs. For illustrative purposes and brevity, we use Lehigh 2 in this analysis. Recall that the SP and DRO models yield different fleet sizes and thus different routing decisions. Therefore, to facilitate the analysis, we first fixed the fleet size to 4 in the three models. Second, to demonstrate how MFs can move to accommodate the change in the demand over time and location, we consider two periods (two days) with the following demand structure. In period 1, we kept the demand structure as described in Section~\ref{sec5.1:instancegen}. In the second period, we swapped the average demand of the following nodes: Allentown and Alburtis, Bethlehem and Cetronia, Emmaus and Trexlertown, Ancient Oaks and Laurys Station, Catasauqua and New Tripoli, and Wescosvill and Slatedale. That is, in the second period, we decreased the demand of the 6 nodes with the highest demand (Allentown--Wescosvill) to that of the nodes that generate the lowest demand (Alburtis--Slatedale) and increased the demand of (Alburtis--Slatedale) to that of (Allentown--Wescosvill). We refer to Table~\ref{table:period} in Appendix~\ref{AppexLehigh} for a summery of the average demand of each node in period 1 and period 2. Figure~\ref{base_swap} illustrates the MFs' locations in period 1 (Figure~\ref{period1}) and period 2 (Figure~\ref{period2}). We provide a summary of these results in Table~\ref{table:OptimalLocations} in Appendix~\ref{AppexLehigh}.
We observe the following about the initial locations in period 1 (Figure~\ref{period1}). First, all models scheduled MF \#1 and \#2 at Allentown and Bethlehem, respectively. This makes sense because, by construction, these nodes generate greater demand than the remaining nodes in period 1. Second, we do not see any MF at or near any of the nodes in the top left of the map (New Tripoli, Slatedale, Slatington, Laury Station, Schnecksville). This makes sense because these nodes generate significantly lower demand than the remaining nodes in period 1. MF \#3 and MF\#4 are scheduled at nodes that generate higher demand or near nodes that generate higher demand than the nodes in the top left of the map. For example, the DRO models scheduled MF \#3 at Emmaus (which has the greatest demand after Allentown and Bethlehem in period 1). The MAD-DRO model scheduled MF \#4 at Wescoville, while the W-DRO model scheduled this MF at Breinigsville. The SP model scheduled MF \#3 and MF \#4 at Dorneyville and Ancient Oaks. Note that Ancient Oaks generates the highest demand after Allentown, Bethlehem, and Emmaus in period 1. Moreover, Dorneyville, Breinigsville, and Wescoville are closer to demand nodes that generate higher demand in period 1 than the remaining nodes on the top left of the map (see Table~\ref{table:period} in Appendix~\ref{AppexLehigh}).
We make the following observations from the results in period 2 presented in Figure~\ref{period2}. First, it is clear that all MFs moved from their initial locations to other locations in period 2 to accommodate the change in the demand. Second, all models scheduled one MF at Alburtis, where the average demand increased from 9 to 60 (average demand of Allentown in period 1). This makes sense because, in period 2, Alburtis generates the highest demand. Third, the DRO models scheduled one MF at Trexlertown and one at New Tripoli, where the average demand increased from 8 and 3 to 43 (average demand of Emmaus in period 1) and 23 (average demand of Catasauqua in period 1), respectively. The SP and W-DRO models scheduled one MF at Cetronia, where the demand increased from 8 to 60.
\begin{figure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=\textwidth]{Base}
\caption{Initial locations of the MFs in period 1.}\label{period1}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\center
\includegraphics[width=\textwidth]{Swap1}
\caption{Locations of the MFs in period 2.}\label{period2}
\end{subfigure}
\caption{Optimal MF locations in period 1 and period 2 (Lehigh 2 instance). Color code: the red square is MF, and the black circle is a demand node/city.}\label{base_swap}
\end{figure}
\color{black}
\subsection{\textbf{Analysis of Solutions Quality}}\label{sec5:OutSample}
\noindent In this section, we compare the operational performance of the optimal solutions to Instance 3, Lehigh 1, and Lehigh 2 via out-of-sample simulation. First, we fix the optimal first-stage decisions yielded by each model in the second-stage of the SP. Then, we solve the second-stage problem in \eqref{2ndstage} with the fixed first-stage decisions and the following sets of $N'=10,000$ out-of-sample data of $W_{i,t}^n$, for all $i \in I, t \in T,$ and $n \in [N']$, to compute the corresponding out-of-sample second-stage cost.
\begin{enumerate}\itemsep0em
\item[Set 1.] \textit{Perfect distributional information}. We use the same settings and distribution (LogN) that we use for generating the $N$ data in the optimization to generate $N^\prime$ data. This is to simulate the performance when the true distribution is the same as the one used in the optimization.
\item[Set 2.] \textit{Misspecified distributional information}. We follow the same out-of-sample simulation procedure described in \cite{wang2020distributionally} and employed in \cite{shehadeh2020distributionallyTucker} to generate the $N^\prime=10,000$ data. Specifically, we perturb the distribution of the demand by a parameter $\Delta$ and use a parameterized uniform distribution $U$[$(1-\Delta)\underline{W}, (1+\Delta)\overline{W}$ ] for which a higher value of $\Delta$ corresponds to a higher variation level. We apply $\Delta \in \{ 0, 0.25, 0.5\}$ with $\Delta=0$ indicating that we only vary the demand distribution from LogN to Uniform. This is to simulate the performance when the true distribution is different from the one we used in the optimization. In addition, we generate $N'$ correlated data points with 0.2 and 0.6 correlation coefficients.
\end{enumerate}
\color{black}
For brevity, we next discuss simulation results for the solutions obtained with $N=10$. We observe similar results for solutions obtained with $N=50$ (see Appendix~\ref{Appex:AdditionalOut} for these results). In Figures~\ref{Fig3_UniN10_Inst3}, \ref{Fig3_UniN10_Inst3_Range2}, and \ref{Fig3_N10_Lehigh1}, we present the normalized histograms of out-of-sample total costs (TC) and second-stage costs (2nd) for Instance 3 (with $\pmb{W} \in [20,60]$, $N=10$), Instance 3 (with $\pmb{W} \in [50, 100]$, $N=10$), and Lehigh 1 ($N=10$). We obtained similar results for Lehigh 2 (see Appendix~\ref{Appex:AdditionalOut}). We computed TC as TC$=$first-stage cost+out-of-sample second-stage cost.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_LogN_Wass2.jpg}
\caption{TC (Set 1, LogN)}\label{Inst3_LogNTC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_LogN_Wass2.jpg}
\caption{2nd (Set 1, LogN)}\label{Inst3_LogN2nd}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Uni0}
\caption{TC (Set 2, $\Delta=0$)}\label{Inst3_Uni0_TC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}\label{Inst3_Uni0_2nd}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)} \label{Inst3_Uni25_2nd}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Instance 3 ($\pmb{W} \in [20, 60]$, $\pmb{N=10}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_UniN10_Inst3}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_Uni0.jpg}
\caption{TC (Set 1, LogN)}\label{Inst3_LogNTC_Range2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_Uni0.jpg}
\caption{2nd (Set 1, LogN)}\label{Inst3_LogN2nd_Range2}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_LogN.jpg}
\caption{TC (Set 2, $\Delta=0$)}\label{Inst3_Uni0_TC_Range2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_LogN.jpg}
\caption{2nd (Set 2, $\Delta=0$)}\label{Inst3_Uni0_2nd_Range2}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)} \label{Inst3_Uni50_2nd_Range2}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Instance 3 ($\pmb{W} \in [50, 100]$, $\pmb{N=10}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$). }\label{Fig3_UniN10_Inst3_Range2}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_LogN.jpg}
\caption{TC (Set 1, LogN)}\label{Lehigh1_LogNTC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_LogN.jpg}
\caption{2nd (Set 1, LogN)}\label{Lehigh1_LogN2nd}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_Uni0.jpg}
\caption{TC (Set 2, $\Delta=0$)}\label{Lehigh1_Uni0TC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_Uni25.jpg}
\caption{TC (Set 2, $\Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_Uni50.jpg}
\caption{TC (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)}\label{Lehigh1_Uni50TC}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 1 ($\pmb{N=10}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_N10_Lehigh1}
\end{figure}
Let us first analyze simulation results under Set 1 (i.e., perfect distributional information case) presented in Figures~\ref{Inst3_LogNTC}--\ref{Inst3_LogN2nd}, \ref{Inst3_LogNTC_Range2}--\ref{Inst3_LogN2nd_Range2}, and \ref{Lehigh1_LogNTC}--\ref{Lehigh1_LogN2nd}. The MAD-DRO model yields a higher TC on average and at upper quantiles than the W-DRO and SP models because it schedules more MFs and thus yields a higher fixed cost (i.e., cost of establishing the MF fleet). The W-DRO model yields a slightly higher TC than the SP model because it schedules more MFs (and thus yeild a higher fixed cost). However, the DRO models yield significantly lower second-stage (transportation and unmet demand) costs on average and at all quantiles than the SP model. In addition, the MAD-DRO model yields a lower second-stage cost than the W-DRO model on average and at all quantile, especially for Lehigh 1. Note that a lower second-stage cost indicates a better operational performance (i.e., lower shortage and transportation costs) and thus has a significant practical impact. These results suggest that there are benefits to using the DRO models even when we have perfect distributional information.
We observe the following from simulation results under Set 2 (i.e., misspecified distributional information case) presented in Figures~\ref{Inst3_Uni0_TC}--\ref{Inst3_Uni25_2nd}, \ref{Inst3_Uni0_TC_Range2}--\ref{Inst3_Uni50_2nd_Range2}, and \ref{Lehigh1_Uni0TC}--\ref{Lehigh1_Uni50TC}. It is clear from these figures that the DRO models consistently outperform the SP model under all levels of variation ($\Delta$) and across the criteria of mean and all quantiles of the the total and second-stage costs. Interestingly, the DRO models yield substantially lower TC and 2nd than the SP model for Lehigh 1 and Instance 3 (with $\pmb{W} \in [50, 100]$), which have higher demand volume and variations. In addition, the MAD-DRO model yields lower second-stage costs for Instance 3 and substantially lower total and second-stage costs for Lehigh 1. Finally, the MAD-DRO solutions appear to be more stable with a significantly smaller standard deviation (i.e., variations) in the total and second-stage costs than the other considered models. The superior performance of the DRO models reflects the value of modeling uncertainty and distributional ambiguity of the demand.
\color{black}
\color{black}
Next, we investigate the value of distributional robustness from the perspective of out-of-sample disappointment, which measures the extent to which the out-of-sample cost exceeds the model's optimal value \citep{Van-Parys_et_al:2021, wang2020distributionally}. We define OPT and TC as the model's optimal value and the out-of-sample objective value, respectively. That is, OPT and TC can be considered as the estimated and actual costs of implementing the model's optimal solutions in practice, respectively. Using this notation, we define the out-of-sample disappointment as in \cite{wang2020distributionally} as follows.
\begin{align}
\max \left \{ \frac{\text{TC}-\text{OPT}}{\text{OPT}}, 0 \right \} \times 100\%.
\end{align}
A disappointment of zero indicates that the model's optimal value is equal to or larger than the out-of-sample (actual) cost (i.e., TC$\leq$OPT). This, in turn, indicates that the model is more conservative and avoids underestimating costs. In contrast, a larger disappointment implies a higher level of over-optimism because, in this case, the actual cost (TC) of implementing the optimal solution of a model is larger than the estimated cost (OPT).
Figure~\ref{Fig6_Dissappt_Lehigh1} presents the histograms of the out-of-sample disappointments of the DRO and SP models for Instance 3 ($\pmb{W} \in[50, 100]$, $N=10$) and Lehigh 1 ($N=10$) with $\Delta=0$ and $\Delta=0.25$. Notably, the DRO models yield substantially smaller out-of-sample disappointments on average and at all quantiles. However, the average and upper quantiles of the disappointments of the MAD-DRO model is smaller than the W-DRO model when $\Delta=0.25$, especially for Lehigh 1. In addition, it is clear that the average and upper quantiles of the disappointments of the SP model are relatively very large (e.g., exceeding 100\% for Lehigh 1). Finally, we observe that the out-of-sample disappointment of the MAD-DRO model is more stable than the W-DRO and SP models with a smaller standard deviation. We remark that these observations are consistent for the other considered instances, and the results with $\Delta=0.5$ are similar to those with $\Delta=0.25$. This demonstrates that the DRO model provides a more robust estimate of the actual cost that we will incur in practice.
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni0_N10_Inst3_Range2.jpg}
\caption{Instance 3, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni25_N10_Inst3_Range2.jpg}
\caption{Instance 3, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni0_N10_Lehigh1_tr2.jpg}
\caption{Lehigh 1, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni25_N10_Lehigh1_tr2.jpg}
\caption{Lehigh 1, $\Delta=0.25$}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample disappointments under Set 2 with $\Delta=0$ and $\Delta=0.25$ for Instance 3 ($\pmb{W} \in[50, 100]$, $N=10$) and Lehigh 1 ($N=10$).}\label{Fig6_Dissappt_Lehigh1}
\end{figure}
\color{black}
The results in this section demonstrate that the DRO approaches are effective in an environment where the distribution is hard to estimate (ambiguous), quickly changes, or when there is a small data set on demand variability. Moreover, these results emphasize the value of modeling uncertainty and distributional ambiguity.
\subsection{\textbf{Sensitivity Analysis}}\label{sec5:sensitivity}
\noindent In this section, we study the sensitivity of DRO models to different parameter settings. Given that we observe similar results for all of the constructed instances, for presentation brevity and illustrative purposes, we present results for Instance 1 ($I$, $J$, $T$)$=$(10, 10, 10) and Instance 5 ($I$, $J$, $T$)$=$(20, 20, 10) as examples of small and relatively large instances.
First, we analyze the optimal number of active MFs as a function of the fixed cost, $f$, MF capacity, $C$, and range of demand. We fix all parameters as described in Section~\ref{sec5.1:instancegen} and solve the W-DRO and MAD-DRO models with $C\in\{ 50, \ 100, \ 150, \ 200\}$ and $f \in \{1,500 \ (\text{low}), \ 6,000 \ (\text{average}), \ 10,000 \ (\text{high})\} $ under the base range $\pmb{W} \in [20, 60]$ and $\pmb{W} \in [50, 100]$ (a higher volume of the demand). Figures \ref{Fig6:MF_vs_C_inst1} and \ref{Fig9:MF_vs_C_inst5} present the optimal number of active MFs and the associated total cost (under Set 1) for Instance 1 and Instance 5 under $\pmb{W} \in [20, 60]$, respectively. Figures \ref{Fig7:MF_vs_C_Range2_inst1}--\ref{Fig10:MF_vs_C_Range2_inst5} in Appendix~\ref{Appx:sensitivity} present the results under $\pmb{W} \in [50, 100]$.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_1500.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig6a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_1500.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig6b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_6000.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig6c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_6000.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig6d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_10000.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig6e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_10000.jpg}
\caption{Total cost, $f=10,000$}
\label{Fig6f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 1}\label{Fig6:MF_vs_C_inst1}
\end{figure}
We observe the following from these figures. First, the optimal number of scheduled MFs decreases as $C$ increases irrespective of $f$. This makes sense because, with a higher capacity, each MF can serve a larger amount of demand in each period. Second, both models schedule more MFs under $\pmb{W} \in [50, 100]$, i.e., a higher volume of the demand. For example, consider Instance 5. When $f=1,500$ and $C=100$ the (W-DRO, MAD-DRO) models schedule (10, 13) and (18, 19) MFs under $\pmb{W} \in [20, 60]$ and $\pmb{W} \in [50, 100]$, respectively. Third, the MAD-DRO model always schedules a higher number of MFs, especially when $C$ is tight. As such, the MAD-DRO model often has a slightly higher total cost (due to the higher fixed cost of establishing a larger fleet) and better second-stage cost, i.e., better operational performance (see Figures \ref{Fig8:MF_vs_C_inst1}-\ref{Fig12:MF_vs_C_range2_inst5}). For example, consider Instance 1. When $f=6,000$ and $C=50$, the W-DRO and MAD-DRO models schedule 8 and 10 MFs, respectively. The associated (total, second-stage) costs of these solutions are respectively (74,495, 26,495) and (83,004, 23,004). Finally, both models schedule fewer MFs as $f$ increases.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_1500_Inst5.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig9a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_1500.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig9b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_6000_Inst5.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig9c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_6000_Inst5.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig9d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_10000_Inst5.jpg}
\caption{Number of MFs, $f=10,000$}
\label{Fig9e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_10000_Inst5.jpg}
\caption{\textcolor{black}{Total cost, $f=10,000$}}
\label{Fig9f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 5}\label{Fig9:MF_vs_C_inst5}
\end{figure}
Second, we fix $M=20$ and solve the models with unmet demand penalty $\gamma \in \{0.10 \gamma_o, \ 0.25 \gamma_o,$ $0.35\gamma_0, \ 0.50\gamma_o\}$ (where $\gamma_o$ is the base case penalty in Section~\ref{sec5.1:instancegen}) and $f \in \{1,500, \ 6,000, \ 10,000\}$. Figure~\ref{Fig12:MF_vs_gamma} presents the number of MFs as a function of $\gamma$ and $f$. Figure~\ref{Fig13:2nd_vs_gamma} presents the associated second-stage cost. It is not surprising that as $\gamma$ increases (i.e., satisfying customer demand becomes more important), the number of scheduled MFs increases. Note that by scheduling a larger number of MFs, we could satisfy a larger amount of demand and reduce the second-stage cost (see Figure~\ref{Fig13:2nd_vs_gamma}). However, for fixed $\gamma$, the MAD-DRO model schedules more MFs, and thus yields a lower unmet demand cost (because the MAD-DRO solutions satisfy a larger amount of demand). For example, consider Instance 1 with $f=1,500$. When $\gamma$ decreases from $0.5\gamma_o$ to $0.1\gamma_o$ the optimal number of scheduled MFs of (W-DRO, MAD-DRO) decreases from (6, 6) to (1, 3) and average unmet demand cost increases from (9, 0) to (16,117, 10,827).
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_F_MF_Inst1_WDRO.jpg}
\caption{W-DRO, Instance 1}
\label{Fig12a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_F_MF_Inst1_MAD.jpg}
\caption{MAD-DRO, Instance 1}
\label{Fig12b}%
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_F_MF_Inst5_WDRO.jpg}
\caption{W-DRO, Instance 5}
\label{Fig12a2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_F_MF_Inst5_MAD.jpg}
\caption{MAD-DRO, Instance 5}
\label{Fig12b2}
\end{subfigure}%
\caption{Comparison of the results for different values of $\gamma$}\label{Fig12:MF_vs_gamma}
\end{figure}
Our experiments in this section provide an example of how decision-makers can use our DRO approaches to generate MFRSP solutions under different parameter settings. Practitioners can use these results to decide whether to adopt the MAD-DRO model (which provides a better operational and computational performance) or the W-DRO model (which provides \textcolor{black}{a lower one-time fixed cost} for establishing the MF fleet).
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma10_Inst1.jpg}
\caption{$\gamma=0.1\gamma_o$ Instance 1}
\label{Fig13a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma10_Inst5.jpg}
\caption{$\gamma=0.1\gamma_o$, Instance 5}
\label{Fig13b}%
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma25_Inst1.jpg}
\caption{$\gamma=0.25\gamma_o$, Instance 1}
\label{Fig13a2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma25_Inst5.jpg}
\caption{$\gamma=0.25\gamma_o$,Instance 5}
\label{Fig13b2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma35_Inst1.jpg}
\caption{$\gamma=0.35\gamma_o$, Instance 1}
\label{Fig13a3}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma35_Inst5.jpg}
\caption{$\gamma=0.35\gamma_o$,Instance 5}
\label{Fig13b3}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma50_Inst1.jpg}
\caption{$\gamma=0.50\gamma_o$, Instance 1}
\label{Fig13a3}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma50_Inst5.jpg}
\caption{$\gamma=0.50\gamma_o$,Instance 5}
\label{Fig13b3}
\end{subfigure
\caption{Comparison of the second-stage cost for different values of $\gamma$.}\label{Fig13:2nd_vs_gamma}
\end{figure}
\subsection{Analysis of the risk-averse solutions}\label{sec:CVaRExp}
\noindent In this section, we analyze the optimal solutions of the mean-CVaR-based models under some critical problem parameters. Specifically, we solve the models with $f\in$\{1,500, 6,000, 10,000\}, $\gamma \in \{\gamma_o, \ 0.5 \gamma_o, \ 0.25\gamma_o, \ 0.1 \gamma_o \} $ (where $\gamma_o$ is the base case penalty in Section~\ref{sec5.1:instancegen}), $\kappa=0.95$ (a typical value of $\kappa$; we observe similar results under $\kappa=0.99, \ 0.975$), and $\Theta \in \{0, 0.2, 0.5, 1\}$ (where a smaller $\Theta$ indicate that we are more risk-averse). \textcolor{black}{We use MAD-CVaR (W-CVaR) to denote the mean-CVaR-based DRO model with MAD (1-Wasserstein) ambiguity, and SP-CVaR to denote the mean-CVaR-based SP model.} Because the SP-CVaR model cannot solve large instances (even with small $N$), for a fair comparison and brevity, we present results for Instance 1. We keep all other parameters as described in Section~\ref{sec5.1:instancegen}.
Table~\ref{tableCVaROptimal} presents the optimal number of scheduled MFs (i.e., fleet size) for different $f$, $\gamma$, and $\Theta$. We make the following observations from this table. First, all models schedule fewer MFs as $f$ increases and $\gamma$ decreases irrespective of $\Theta$ (risk-aversion coefficient). This is consistent with our results in Section~\ref{sec5:sensitivity} for the risk neutral models. Second, the MAD-CVaR model often schedules a higher number of MFs than the W-CVaR model, and the latter model schedule the same or larger number of MFs than the SP-CVaR. Second, when $f=$\{6,000, 10,000\} (i.e., average and high fixed cost) and $\gamma=0.1\gamma_o$ (very low unmet demand penalty) all models schedule one MF.
\textcolor{black}{Third, the MAD-CVaR model schedules the same number of MFs under all values of $\Theta$ when ($f$, $\gamma$)$=$ (1,500, $\gamma_o$) (and similarly under (6,000, 0.25$\gamma_o$) and (10,000, $\gamma_o$)). Similarly, the W-CVaR model schedules the same number of MFs under all values of $\Theta$ when ($f$, $\gamma$)= (6,000, 0.25$\gamma_o$), and (10,000, 0.5$\gamma_o$). These results indicate that our proposed DRO expectation models with MAD and 1-Wasserstein ambiguity are risk-averse under these settings because they yield the same optimal solutions under all values of the risk-aversion coefficient $\Theta$.}
\begin{table}[t!]
\center
\caption{Comparison of the optimal number of MFs yielded by each model under different values of $\Theta$, $f$, and $\gamma$. Notation: MAD-CVaR and W-CVaR are respectively the distributionally robust mean-CVaR models with MAD and Wasserstein ambiguity, and SP-CVaR is the SP model based on the mean-CVaR criterion.}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
\multicolumn{11}{c}{\textbf{$\pmb{f=1,500}$}}\\
\hline
& \multicolumn{4}{c}{$\gamma_o$ } && & \multicolumn{4}{c}{$0.5\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 8 & 8 & 8 & 8 & & MAD-CVaR & 8 & 8 & 8 & 7 \\
W-CVaR & 8 & 8 & 8 & 6 & & W-CVaR & 7 & 7 & 6 & 6 \\
SP-CVaR & 6 & 6 & 6 & 5 & & SP-CVaR & 6 & 6 & 6 & 5 \\
\\
\hline
& \multicolumn{4}{c}{$0.25\gamma_o$ } && & \multicolumn{4}{c}{$0.1\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 6 & 6 & 6 & 5 & & MAD-CVaR & 3 & 3 & 3 & 1 \\
W-CVaR & \textcolor{red}{6} & 5 & 5 & 5 & & W-CVaR & 2 & 2 & 2 & 1 \\
SP-CVaR & 4 & 4 & 4 & 4 & & SP-CVaR & 2 & 2 & 2 & 1 \\
\hline
\multicolumn{11}{c}{\textbf{$\pmb{f=6,000}$}}\\
\hline
& \multicolumn{4}{c}{$\gamma_o$ } && & \multicolumn{4}{c}{$0.5\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & \textcolor{black}{7} & \textcolor{black}{7} & 7 & 6 & & MAD-CVaR & 7 & 7 & 6 & 5 \\
W-CVaR & 6 & 6 & 5 & 5 & & W-CVaR & 6 & 5 & 4 & 4 \\
SP-CVaR & 5 & 5 & 5 & 5 & & SP-CVaR & 5 & 5 & 4 & 4 \\
\hline
& \multicolumn{4}{c}{$0.25\gamma_o$ } && & \multicolumn{4}{c}{$0.1\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1\\
\hline
MAD-CVaR & 3 & 3 & 3 & 3 & & MAD-CVaR & 1 & 1 & 1 & 1 \\
W-CVaR & 2 & 2 & 2 & 2 & & W-CVaR & 1 & 1 & 1 & 1 \\
SP-CVaR & 2 & 2 & 2 & 2 & & SP-CVaR & 1 & 1 & 1 & 1 \\
\hline
\multicolumn{11}{c}{\textbf{$\pmb{f=10,000}$}}\\
\hline
& \multicolumn{4}{c}{$\gamma_o$ } && & \multicolumn{4}{c}{$0.5\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 6 & 6 & 6 & 6 & & MAD-CVaR & 6 & 5 & 5 & 4 \\
W-CVaR & 6 & 6 & 6 & 5 & & W-CVaR & 4 & 4 & 4 & 4 \\
SP-CVaR & 5 & 4 & 4 & 4 & & SP-CVaR & 4 & 4 & 4 & 3 \\
\hline
& \multicolumn{4}{c}{$0.25\gamma_o$ } && & \multicolumn{4}{c}{$0.1\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 1 & 1 & 1 & 1 & & MAD-CVaR & 1 & 1 & 1 & 1 \\
W-CVaR & 1 & 1 & 1 & 1 & & W-CVaR & 1 & 1 & 1 & 1 \\
SP-CVaR & 1 & 1 & 1 & 1 & & SP-CVaR & 1 & 1 & 1 & 1 \\
\hline
\end{tabular}\label{tableCVaROptimal}
\end{table}
Fourth, all models schedule more MFs under a smaller $\Theta$, especially when $\gamma=(0.25\gamma_o, 0.1\gamma_o)$ with $f=1,500$ (i.e., a low cost and lower unmet demand penalty), $\gamma=(0.5\gamma_o,0.25\gamma_o)$ with $f=6,000$, and $\gamma=(\gamma_o, 0.5\gamma_o)$ with $f=10,000$. In particular, more MFs are scheduled when $\Theta=0$ (risk-averse models with CVaR criterion) as compared to $\Theta=1$ (risk-neutral models). This makes sense because a risk-averse decision-maker may schedule more MFs to avoid high operational cost and, in particular, excessive shortages.
\color{black}
Next, we compare the out-of-sample operational performance (i.e., second-stage cost) and disappointment of the risk-neutral (i.e., expectation models) and risk-averse models (i.e., mean-CVaR-based models with $\Theta=0$, or equivalently, risk-averse models with CVaR criterion). We use MAD-E (W-E) to denote the risk-neutral DRO model with MAD (1-Wasserstein) ambiguity presented in Section~\ref{sec:MAD-DRO} (Section~\ref{sec:WDMFRS_model}). In addition, we use SP-E to denote the risk-neutral SP model. In figures~\ref{CVaR_gamma}, \ref{CVaR_50gamma}, and \ref{CVaR_25gamma}, we present histograms of the out-of-sample second-stage costs and disappointments under Set 2 with $f=$1,500 and $\gamma=\gamma_o$, $\gamma=0.5\gamma_o$, and $\gamma=0.25\gamma_o$, respectively. We obtained similar observations for the other considered values of $f$.
Let us first compare the performance of the risk-neutral and risk-averse SP models. Notably, the SP-E solutions have the worst performance with significantly higher second-stage costs and larger positive disappointments under all values of $\gamma$ and $\Delta$ than the other considered models. In contrast, the SP-CVaR solutions yield smaller second-stage costs and disappointments than the SP-E model. This makes sense because the SP-CVaR model schedules a larger numbers of MFs. In addition, when $\gamma=\gamma_o$ and $0.5\gamma_o$, the SP-CVaR model yields the same second-stage costs as the W-E model because both models schedule 6 MFs under these settings (this is why we do not see histograms for the second-stage cost of the SP-CVaR model). However, the W-E model controls the disappointments in a smaller range, while the SP-CVaR model yield larger positive disappointments than the W-E model and the other DRO models.
Let us now compare the performance of the risk-neutral and risk-averse DRO models. First, when $\gamma= \gamma_o$ (i.e., the largest unmet demand penalty), the MAD-CVaR, MAD-E, and W-CVaR models have the same and best performance because they schedule a larger fleet of 8 MFs than the other considered models. Second, when $\gamma=$0.5$\gamma_o$, the MAD-CVaR model has the lowest second-stage costs and zero disappointments under all values of $\Delta$. This makes sense because the MAD-CVaR model schedules a larger number of MFs than the other considered models when $\gamma=$0.5$\gamma_o$. Third, when $\gamma=0.25\gamma_o$, the MAD-CVaR and W-CVaR model yield the lowest second-stage costs and disappointments because they schedule a larger fleet (6 MFs) than the other considered models. However, the W-CVaR model yields slightly higher disappointments than the MAD-CVaR when $\Delta=0.5$.
Fourth, the second-stage costs and disappointments of the W-CVaR model are smaller than those of the W-E model because the W-CVaR model schedules a larger number of MFs. Fifth, the W-CVaR and MAD-E models yield the same second-stage costs when $\gamma=0.5 \gamma_o$ because they schedule 7 MFs (this is why we can only see black histograms for the MAD-E model). Finally, the MAD-E model have lower second-stage costs and disappointments than the W-E model under all values of $\gamma$ and $\Delta$, which is consistent with our results in Section~\ref{sec5:OutSample}.
Our results in this section demonstrate that the distributionally robust CVaR models tend to hedge against uncertainty, ambiguity, and risk by scheduling more MFs. Our results also indicate that the proposed DRO expectation models may be risk-averse under some parameter settings (e.g., high unmet demand penalty and low cost).
\color{black}
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_2nd_Uni0_f1500.jpg}
\caption{2nd, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_Dissappt_Uni0_f1500.jpg}
\caption{Disappointment, $\Delta=0$}
\end{subfigure}%
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_2nd_Uni25_f1500.jpg}
\caption{2nd, $ \Delta=0.25$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_Dissappt_Uni25_f1500.jpg}
\caption{Disappointment, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_2nd_Uni50_f1500.jpg}
\caption{2nd, $ \Delta=0.5$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_Dissappt_Uni50_f1500.jpg}
\caption{Disappointment, $\Delta=0.5$}
\end{subfigure}%
\caption{Normalized histograms of second-stage cost (2nd) and out-of-sample disappointments under Set 2 with $\gamma=\gamma_o$, $f=1,500$, and $\pmb{\Delta \in \{0, 0.25, 0.5\}}$.}\label{CVaR_gamma}
\end{figure}
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_2nd_Uni0_f1500.jpg}
\caption{2nd, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_Dissappt_Uni0_f1500.jpg}
\caption{Disappointment, $\Delta=0$}
\end{subfigure}%
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_2nd_Uni25_f1500.jpg}
\caption{2nd, $ \Delta=0.25$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_Dissappt_Uni25_f1500.jpg}
\caption{Disappointment, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_2nd_Uni50_f1500.jpg}
\caption{2nd, $ \Delta=0.5$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_Dissappt_Uni50_f1500.jpg}
\caption{Disappointment, $\Delta=0.5$}
\end{subfigure}%
\caption{Normalized histograms of second-stage cost (2nd) and out-of-sample disappointments under Set 2 with $\gamma=0.5\gamma_o$, $f=1,500$, and $\pmb{\Delta \in \{0, 0.25, 0.5\}}$.}\label{CVaR_50gamma}
\end{figure}
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_2nd_Uni0_f1500.jpg}
\caption{2nd, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_Dissappt_Uni0_f1500.jpg}
\caption{Disappointment, $\Delta=0$}
\end{subfigure}%
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_2nd_Uni25_f1500.jpg}
\caption{2nd, $ \Delta=0.25$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_Dissappt_Uni25_f1500.jpg}
\caption{Disappointment, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_2nd_Uni50_f1500.jpg}
\caption{2nd, $ \Delta=0.5$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_Dissappt_Uni50_f1500.jpg}
\caption{Disappointment, $\Delta=0.5$}
\end{subfigure}%
\caption{Normalized histograms of second-stage cost (2nd) and out-of-sample disappointments under Set 2 with $\gamma=0.25\gamma_o$, $f=1,500$, and $\pmb{\Delta \in \{0, 0.25, 0.5\}}$.}\label{CVaR_25gamma}
\end{figure}
\section{\textcolor{black}{Conclusion}}\label{Sec:Conclusion}
\color{black}
\noindent In this paper, we propose two DRO models for the MFRSP. Specifically, given a set of MFs, a planning horizon, and a service region, our models aim to find the number of MFs to use within the planning horizon and a route and schedule for each MF in the fleet. The objective is to minimize the fixed cost of establishing the MF fleet plus a risk measure (expectation or mean-CVaR) of the operational cost over all demand distributions defined by an ambiguity set. In the first model (MAD-DRO), we use an ambiguity set based on the demand's mean, support, and mean absolute deviation. In the second model (W-DRO), we use an ambiguity set that incorporates all distributions within a 1-Wasserstein distance from a reference distribution. To solve the proposed DRO models, we propose a decomposition-based algorithm. We also derive lower bound inequalities and two families of symmetry breaking constraints to improve the solvability of the proposed models.
\color{black}
Our computational results demonstrate (1) how the DRO approaches have superior operational performance in terms of satisfying customers demand as compared to the SP approach, (2) the MAD-DRO model is more computationally efficient than the W-DRO model, (3) the MAD-DRO model yield more conservative decisions than the W-DRO model, which often have a higher fixed cost but significantly lower operational cost, (4) how mobile facilities can move from one location to another to accommodate the change in demand over time and location, (5) efficiency of the proposed symmetry breaking constraints and lower bound inequalities, (6) the trade-off between cost, number of MFs, MF capacity, and operational performance, and (7) the trade-off between the risk-neutral and risk-averse approaches. Most importantly, our results show the value of modeling uncertainty and distributional ambiguity.
Note that we have used benchmark instances from the literature in our computational experiments, which may be a limitation of our results. However, in the sensitivity analysis section, we tested the proposed approaches under different parameters settings, demonstrating how decision-makers can use our approaches to generate MFRSP solutions under different parameter settings relevant to their specific application. Moreover, these benchmark instances represent a wide range of potential service regions, which we can efficiently solve. If we account for the scale of the problem in the sense of static facility location problem, we have demonstrated that we can solve instances of $30 \times 20= 600$ customers (instance 10), which are relatively large for many practical applications.
We suggest the following areas for future research. First, we aim to extend our models to optimize the capacity and size of the MF fleet. Second, we want to extend our approach by incorporating multi-modal probability distributions and more complex relationships between random parameters (e.g., correlation). Third, we aim to extend our approach to more comprehensive MF planning models, which consider all relevant organizational and technical constraints and various sources of uncertainties (e.g., travel time) with a particular focus on real-life healthcare settings. Although conceptually and theoretically advanced, stochastic optimization approaches such as SP and DRO are not intuitive or transparent to decision-makers who often do not have optimization expertise. Thus, future efforts should also focus on closing the gap between theory and practice.
\vspace{2mm}
\ACKNOWLEDGMENT{%
We want to thank all colleagues who have contributed significantly to the related literature. We are grateful to the anonymous reviewers for their insightful comments and suggestions that allowed us to improve the paper. Special thanks to Mr. Man Yiu Tsang (a Ph.D. student at the Department of Industrial and Systems Engineering, Lehigh University) for helping with Figure~\ref{LehighMap} and proofreading the paper. Dr. Karmel S. Shehadeh dedicates her effort in this paper to every little dreamer in the whole world who has a dream so big and so exciting. Believe in your dreams and do whatever it takes to achieve them--the best is yet to come for you.}
\vspace{1mm}
\newpage
\begin{APPENDICES}
\section{Derivation of feasible region $\mathcal{X}$ in \eqref{eq:RegionX}} \label{Appx:FirstStageDec}
\noindent In this Appendix, we provide additional details on the derivation of the constraints defining the feasible region $\mathcal{X}$ of variables ($\pmb{x},\pmb{y}$). As described in \cite{lei2014multicut}, we can enforce the requirement that an MF can only be in service when it is stationary using the following constraints:
\begin{align}\label{eq1}
& x_{j,m}^t+x_{j',m}^t \leq 1, && \forall t, m, j, \ j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace.
\end{align}
\noindent If $x_{j,m}^t=1$ (i.e., MF $m$ is stationary at some location $j$ in period $t$), it can only be available at location $j'\neq j$ after a certain period, depending on the time it takes to travel from location $j$ to location $j'$. It follows by \eqref{eq1} that $ x_{j',m}^{t'}=0$ for all $j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace$. As pointed out by \cite{lei2014multicut}, this indicates that \textit{an earlier decision of deploying an MF at one candidate location would directly affect future decisions both temporally and spatially. In fact, this correlation is a major source of complexity for optimizing the MFRSP.}
Since the MF has to be in an active condition before providing
service, we have to include the following constraints:
\begin{align}\label{eq2}
x_{j,m}^t \leq y_m, && \forall j,m,t.
\end{align}
It is straightforward to verify that constraint sets \eqref{eq1} and \eqref{eq2} can be combined into the following compact form:
\begin{align}
\mathcal{X}&=\left\{ (\pmb{x}, \pmb{y}) : \begin{array}{l} x_{j,m}^t+x_{j^\prime,m}^{t^\prime} \leq y_m, \ \ \forall t, m, j, \ j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace \\
x_{j,m}^t \in \lbrace 0, 1\rbrace, \ y_m \in \lbrace 0, 1\rbrace, \ \forall j, m , t \end{array} \right\} .
\end{align}
\section{Proof of Proposition 1 }\label{Appx:ProofofProp1}
\noindent \textit{Proof}. For a fixed $x \in \mathcal{X}$, we can explicitly write the inner problem $\sup [ \cdot ]$ in \eqref{MAD-DMFRS} as the following functional linear optimization problem.
\begin{subequations}
\begin{align}
\max_{\mathbb{P}\geq 0} & \int_{\mathcal{S}} Q(\pmb{x},\pmb{W}) \ d \mathbb{P} \\
\ \text{s.t.} & \ \int_{\mathcal{S}} W_{i,t} \ d\mathbb{P}= \mu_{i,t} \quad \quad \forall i \in I, \ t \in T, \label{ConInner:W}\\
& \int_{\mathcal{S}} |W_{i,t}-\mu_{i,t}| \ d\mathbb{P}\leq \eta_{i,t} \quad \quad \forall i \in I, \ t \in T, \label{ConInner:K}\\
& \ \int_{\mathcal{S}} d\mathbb{P}= 1. \label{ConInner:Distribution}
\end{align} \label{InnerMax2}
\end{subequations}
Letting $\rho_{i,t}, \psi_{i,t}$ and $\theta$ be the dual variables associated with constraints \eqref{ConInner:W}, \eqref{ConInner:K}, \eqref{ConInner:Distribution}, respectively, we present problem \eqref{InnerMax2} (problem (9) in the main manuscript) in its dual form:
\begin{subequations}
\begin{align}
& \min_{\pmb{\rho, \theta, \psi \geq 0}} \ \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+ \theta \label{DualInner:Obj} \\
& \ \ \text{s.t.} \ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})+ \theta \geq Q(\pmb{x},\pmb{W}) && \forall \pmb{W}\in \mathcal{S}, \label{DualInner:PrimalVariabl}
\end{align} \label{DualInnerMax}
\end{subequations}
where $\pmb{\rho}$ and $\theta$ are unrestricted in sign, $\pmb \psi \geq 0$, and constraint \eqref{DualInner:PrimalVariabl} is associated with the primal variable $\mathbb{P}$. Note that for fixed ($\pmb{\rho, \ \psi,} \ \theta$), constraint \eqref{DualInner:PrimalVariabl} is equivalent to
$$\theta \geq \max \limits_{\pmb{W} \in \mathcal{S} } \Big \lbrace Q(\pmb{x},\pmb{W}) + \sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big\rbrace. $$
Since we are minimizing $\theta$ in \eqref{DualInnerMax}, the dual formulation of \eqref{InnerMax2} is equivalent to:
\begin{align*}
& \min_{\pmb{\rho}, \pmb{\psi} \geq 0} \ \left \lbrace \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+\max \limits_{\pmb{W} \in \mathcal{S}} \Big\lbrace Q(\pmb{x},\pmb{W}) + \sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})\Big \rbrace \right \rbrace .
\end{align*}
\section{Proof of Proposition 2}\label{Appx:Prop2}
\noindent First, note that the feasible region $\Omega=\{\eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner}\}$ and $\mathcal{S}$ are both independent of $\pmb{x}$, $\pmb \rho$, and $\pmb \psi$ and bounded. In addition, the MFRSP has a complete recourse (i.e., the recourse problem is feasible for any feasible $(\pmb{x},\pmb{y}) \in \mathcal{X}$). Therefore, $\max\limits_{\pmb{\lambda, v, W, \pi, k} }\Big[ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big]< +\infty$. Second, for any fixed $\pmb{\pi, v, W, k}$, $\Big [\sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big] $ is a linear function of $\pmb{x}$, $\pmb \rho$, and $\pmb \psi$. It follows that $\max\limits_{\pmb{\lambda, v, W, \pi, k} }\Big[ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big]$ is the maximum of linear functions of $\pmb{x}$, $\pmb \rho$, and $\pmb \psi$, and hence convex and piecewise linear. Finally, it is easy to see that each linear piece of this function is associated with one distinct extreme point of $\Omega$ and $\mathcal{S}$. Given that each of these polyhedra has a finite number of extreme points, the number of pieces of this function is finite. This completes the proof.
\section{Number of Points in The Worst-Case Distribution of MAD-DRO}\label{Appx:Points_MAD}
\color{black}
The results of \cite{longsupermodularity} indicate that if the second-stage optimal value is supermodular in the realization of uncertainties under the MAD ambiguity set, the worst-case is a distribution supported on ($2n+1$) points, where $n$ is the dimension of the random vector. In this Appendix, we show that even if our recourse is supermodular in demand realization, the number of points in the worst-case distribution of the demand is large, which renders our two-stage MAD-DRO model computationally challenging to solve using \cite{longsupermodularity}' approach
Recall that the demand is indexed by $i$ and $t$, i.e., $W_{i,t}$, for all $i \in I$ and $t \in T$. Thus, the dimension of our random vector is $|I | |T|$. Accordingly, assuming that the second-stage optimal value is supermodular, then the resullts of \cite{longsupermodularity} suggest that the worst-case distribution in MAD-DRO has ($2|I| |T|$+1) points or scenarios. Note that $|I|$ and $|T|$ and thus $2|I||T|$ is large for most instances of our problem (See Example 1--2 below). Since our computational results and prior literature indicate that solving the scenario-based model using a small set of scenarios is challenging, solving a reformulation of MAD-DRO using the ($2|I||T|$+1) points is expected to be computationally challenging.
\noindent \textbf{Example 1.} Instance 6 ($I=20$ and $T=20$). The worst-case distribution has 801 points.
\noindent \textbf{Example 3.} Instance 10 ($I=30$ and $T=20$). The worst-case distribution has 1201 points.
\color{black}
\newpage
\section{Proof of Proposition~\ref{Prop1}}\label{Proof_Prop1}
Recall that $\hat{\mathbb{P}}^N=\frac{1}{N}\sum_{n=1}^n \delta_{\hat{\pmb{W}}^n}$. The definition of Wasserstein distance indicates that there exist a joint distribution $\Pi$ of $(\pmb{W}, \hat{\pmb{W}}$) such that $\mathbb{E}_{\Pi} [||\pmb{W}-\hat{\pmb{W}}||] \leq \epsilon$. In other words, for any $\mathbb{P}\in\mathcal{P}(\mathcal{S})$, we can rewrite any joint distribution $\Pi\in\mathcal{P}(\mathbb{P},\hat{\mathbb{P}}^N)$ by the conditional distribution of $\pmb{W}$ given $\hat{\pmb{W}}=\hat{\pmb{W}}^n$ for $n=1,\dots,N$, denoted as $\F^n$. That is, $\Pi=\frac{1}{N}\sum_{n=1}^N \F^n \times \delta_{\hat{\xi}_n}$. Notice that if we find one joint distribution $\Pi\in\mathcal{P}(\mathbb{P},\hat{\mathbb{P}}^N)$ such that $\int ||\pmb{W}-\hat{\pmb{W}}|| \ d\Pi \leq \epsilon$, then $\text{dist}(\mathbb{P},\hat{\mathbb{P}}^N)\leq\epsilon$. Hence, we can drop the infimum operator in Wasserstein distance and arrive at the following equivalent problem.
\begin{subequations}\label{AppxInner}
\begin{align}
& \sup_{\F^n\in\mathcal{P}(\mathcal{S}), n\in[N]} \frac{1}{N}\sum_{n=1}^N\int_\mathcal{S} Q(\pmb{x},\pmb{W}) \ d\F^n\\
& \text{s.t.} \ \ \ \ \ \ \ \ \frac{1}{N} \sum_{n=1}^N \int_\mathcal{S} ||\pmb{W}-\hat{\pmb{W}}^n|| \ d\F^n \leq \epsilon.
\end{align}
\end{subequations}
Using a standard strong duality argument and letting $\rho\geq 0$ be the dual multiplier, we can reformulate problem \eqref{AppxInner} by its dual, i.e.
\begin{align}
&\quad\,\inf_{\rho\geq 0} \sup_{\F^n\in\mathcal{P}(\mathcal{S}),n\in[N]}\left\{\frac{1}{N}\sum_{n=1}^N\int_\mathcal{S} Q(\pmb{x},\pmb{W}) d\F^n+\rho\left[\epsilon-\frac{1}{N} \sum_{n=1}^N \int_\mathcal{S} ||\pmb{W}-\hat{\pmb{W}}^n||\ d\F^n \right] \right\}\nonumber\\
&=\inf_{\rho\geq 0} \Big\{ \epsilon \rho+ \frac{1}{N}\sum_{n=1}^N \sup_{\F^n\in\mathcal{P}(\mathcal{S})}\int_\mathcal{S} \left[Q(\pmb{x},\pmb{W})-\rho||\pmb{W}-\hat{\pmb{W}}^n||\right] d\F^n\Big\} \nonumber\\
&=\inf_{\rho\geq 0} \Big\{ \epsilon \rho + \frac{1}{N} \sum_{n=1}^N \sup_{\pmb{W} \in \mathcal{S}} \{ Q(\pmb{x},\pmb{W})-\rho || \pmb{W} -\hat{\pmb{W}}^n || \} \Big\}.
\end{align}
\newpage
\section{DRO with mean-CVaR as a Risk Measure}\label{Sec:meanCVAR}
\noindent Both the MAD-DRO and W-DRO model presented in Section \ref{sec:DRO_Models} assume that the decision-maker is risk-neutral (i.e., adopt the expected value of the recourse as a risk measure). In some applications of the MFRSP, however, decision-makers might be risk-averse. Therefore, as one of our reviewers suggested, in this section, we present a distributionally robust risk-averse model for the MFRSP.
To model the decision maker's risk aversion, most studies adopt the CVaR, i.e., set $\varrho(\cdot)=\CVAR_\kappa(\cdot)$, where $\kappa\in(0,1)$. CVaR is the conditional expectation of $Q(\cdot)$ above the value-at-risk VaR (informally, VaR is the $\kappa$ quantile of the distribution of $Q(\cdot)$, see \cite{pacc2014robust, rockafellar2002conditional, sarykalin2008value, van2015distributionally}). CVaR is a popular coherent risk measure widely used to avoid solutions influenced by a bad scenario with a low probability. However, as pointed out by \cite{wang2021two} and a reviwer of this paper, neither expected value nor CVaR can capture the variability of uncertainty in a comprehensive manner. Alternatively, we consider minimizing the mean-CVaR, which balances the cost on average and avoids high-risk levels. As pointed out by \cite{lim2011conditional}, \cite{wang2021two}, and our reviewer, the traditional CVaR criterion is sensitive to the misspecification of the underlying loss distribution and lacks robustness. Therefore, we propose a distributionally robust mean-CVaR model to remedy such fragility, reflecting both risk-averse and ambiguity-averse attitudes. For brevity, we use the MAD ambiguity set to formulate and analyze this model because similar formulation and reformulation steps can be used to derive a solvable mean-CVaR-based model based on the 1-Wasserstein ambiguity.
Let us now introduce our distributionally robust mean-CVaR-based model with MAD-ambiguity (MAD-CVaR). First, following \cite{rockafellar2000optimization, rockafellar2002conditional}, and \cite{ van2015distributionally}, we formally define CVaR as
\begin{align}
\CVAR_\kappa (Q(\pmb{x},\pmb{W}))= \inf \limits_{\zeta \in \mathbb{R}} \Big \{ \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Big\}, \label{CVAR_Def_main}
\end{align}
\noindent where $[c]^+:=\max \{c,0\}$ for $c \in \mathbb{R}$. Parameter $\kappa$ measures a wide range of risk preferences, where $\kappa=0$ corresponds to the risk-neutral formulation. In contrast, when $\kappa \rightarrow 1$, the decision-makers become more risk-averse. Using \eqref{CVAR_Def_main}, we formulate the following MAD-CVaR model (see, e.g., \cite{wang2021two} for a recent application in facility location):
\begin{align}
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Bigg[ \Theta \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})] + (1-\Theta) \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W})) \Bigg] \Bigg \}, \label{CVaR-DMFRS}
\end{align}
\noindent where $0 \leq \Theta \leq 1$ is the risk-aversion coefficient, which represents a trade-off between the risk-neutral (i.e., $\mathbb{E}[\cdot]$) and risk-averse (i.e., $\CVAR(\cdot)$) objectives. A larger $\Theta$ implies less aversion to risk, and vice verse. In extreme cases, when $\Theta =1$, the decision maker is risk-neutral and \eqref{CVaR-DMFRS} reduces to the MAD-DRO expectation model in \eqref{MAD-DMFRS}. When $\Theta =0$, the decision maker is risk and ambiguity averse.
Next, we derive a solvable reformulation of \eqref{CVaR-DMFRS}. Let us first consider the inner maximization problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W})) $ in \eqref{CVaR-DMFRS}. It is easy to verify that
\begin{subequations}\label{SupCVaR}
\begin{align}
\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W}))&= \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \inf \limits_{\zeta \in \mathbb{R}} \Bigg \{ \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Bigg\} \label{CVAR_alter1} \\
&= \inf \limits_{\zeta \in \mathbb{R}} \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \Bigg \{ \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Bigg\} \label{CVAR_alter2} \\
&= \inf \limits_{\zeta \in \mathbb{R}} \Bigg \{ \zeta+ \frac{1}{1-\kappa} \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Bigg\}. \label{CVAR_alter3} \
\end{align}
\end{subequations}
\noindent Interchanging the order of $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})}$ and $\inf \limits_{\zeta \in \mathbb{R}} $ follows from Sion's minimax theorem \citep{sion1958general} because $\zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+$ is convex in $\zeta$ and concave in $\mathbb{P}$. Next, we apply the same techniques in Section~\ref{sec3:reform} and Appendix~\ref{Appx:ProofofProp1} to reformulate the inner maximization problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+$ in \eqref{CVAR_alter3} as a minimization problem and combine it with the outer minimization problem to obtain
\begin{subequations}
\begin{align}
\inf \limits _{\zeta, \pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{\zeta+ \frac{1}{1-\kappa} \Bigg [\sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+ \theta \Bigg] \Bigg\} \label{CVAR_Obj} \\
\text{s.t.} & \ \ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})+ \theta \geq Q(\pmb{x},\pmb{W})-\zeta, && \forall \pmb{W}\in \mathcal{S} , \label{CVAR_C1} \\
& \ \ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})+ \theta \geq 0, && \forall \pmb{W}\in \mathcal{S}, \label{CVAR_C2}
\end{align}
\end{subequations}
\noindent where \eqref{CVAR_C1} and \eqref{CVAR_C2} follows from the definition of $[\cdot]^+$. Accordingly, problem \eqref{SupCVaR} (equivalently last term in formulation in \eqref{CVaR-DMFRS}) is equivalent to
\begin{subequations}\label{CVARmodel_objective}
\begin{align}
\inf \limits_{\zeta, \pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{\zeta+ \frac{1}{1-\kappa} \Bigg [\sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+ \theta \Bigg] \Bigg\} \label{CVAR_Obj2} \\
\text{s.t.} & \ \ \zeta \geq \max_{\pmb{W} \in \mathcal{S}} \Big \{ Q(\pmb{x},\pmb{W})- \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big\}-\theta, \label{CVAR_equiv2}\\
& \ \ \min_{\pmb{W} \in \mathcal{S}} \Big \{ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big \} + \theta \geq 0. \label{CVAR_equiv3}
\end{align}
\end{subequations}
\textcolor{black}{Since we are minimizing $\zeta$ in \eqref{CVAR_Obj2}, we can equivalently re-write \eqref{CVAR_Obj2} as}
\color{black}
\begin{align}
\inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \Bigg \{ \max_{\pmb{W} \in \mathcal{S}} \Big \{ Q(\pmb{x},\pmb{W})- \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big\}+ \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\} \nonumber \\
\equiv \inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\}, \label{CVAR_Obj3}
\end{align}
\noindent where \eqref{CVAR_Obj3} follows from the derivation of $h(\pmb{x}, \pmb \rho, \pmb \psi) \equiv \max\limits_{\pmb{W} \in \mathcal{S}} \big \{ Q(\pmb{x},\pmb{W})- \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \big \}$ in Section~\ref{sec3:reform}. Accordingly, problem \eqref{CVARmodel_objective} is equivalent to
\begin{subequations}\label{CVARmodel_objective4}
\begin{align}
\inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\} \label{CVAR_Obj4} \\
\text{s.t.} & \ \ \min_{\pmb{W} \in \mathcal{S}} \Big \{ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big \} + \theta \geq 0. \label{CVAR_equiv4}
\end{align}
\end{subequations}
Next, derive equivalent linear constraints of the embedded minimization problem in constraint \eqref{CVAR_equiv4}. For fixed $\pmb \rho$ and $\pmb \psi$, we re-write constraint \eqref{CVAR_equiv4} as
\begin{subequations}\label{CVAR_equiv2_reform}
\begin{align}
\theta + & \min_{\pmb{W}} \ \ \Big \{ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ k_{i,t} \psi_{i,t}) \Big \} \geq 0, \label{CVAR_equiv2_reform_C1}\\
& \text{s.t.} \ \ \ W_{i,t} \leq \overline{W}_{i,t}, && \forall i, t, \label{CVAR_equiv2_reform_C2}\\
& \ \ \ \ \ \ \ W_{i,t} \geq \underline{W}_{i,t}, && \forall i, t, \label{CVAR_equiv2_reform_C22}\\
& \ \ \ \ \ \ \ k_{i,t} \geq W_{i,t}-\mu_{i,t}, && \forall i, t, \label{CVAR_equiv2_reform_C3}\\
& \ \ \ \ \ \ \ k_{i,t} \geq \mu_{i,t}-W_{i,t}, && \ \forall i, t.\label{CVAR_equiv2_reform_C4}
\end{align}
\end{subequations}
\noindent Letting $a_{i,t}$, $b_{i,t}$, $g_{i,t}$, and $o_{i,t}$ be the dual variables associated with constraints \eqref{CVAR_equiv2_reform_C2}, \eqref{CVAR_equiv2_reform_C22}, \eqref{CVAR_equiv2_reform_C3}, and \eqref{CVAR_equiv2_reform_C4}, respectively, we present the linear program in \eqref{CVAR_equiv2_reform_C1}-\eqref{CVAR_equiv2_reform_C4} in its dual form as
\begin{subequations}
\begin{align}
& \theta + \sum_{i \in I} \sum_{t \in T} \Big [ \overline{W}_{i,t}a_{i,t}+ \underline{W}_{i,t} b_{i,t} -\mu_{i,t}g_{i,t}+\mu_{i,t}o_{i,t} \Big] \geq 0, \label{dual_equiv2_C1} \\
&a_{i,t}+b_{i,t}-g_{i,t}+o_{i,t} \leq \rho_{i,t}, && \forall i, t , \label{dual_equiv2_C2} \\
&g_{i,t}+o_{i,t} \leq \psi_{i,t}, && \forall i, t, \label{dual_equiv2_C3} \\
& (b_{i,t}, g_{i,t}, \ o_{i,t}) \geq 0, \ a_{i,t} \leq 0, && \forall i, t. \label{dual_equiv2_C4}
\end{align}
\end{subequations}
\noindent Replacing \eqref{CVAR_equiv4} in \eqref{CVARmodel_objective4} with \eqref{dual_equiv2_C1}--\eqref{dual_equiv2_C4}, we derive the following equivalent reformulation of problem \eqref{CVARmodel_objective4} (equivalently, problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W}))$ in \eqref{CVaR-DMFRS}):
\begin{subequations}\label{CVARmodel_obj5}
\begin{align}
\inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\} \label{CVAR_Obj5} \\
\text{s.t.} & \ \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}.
\end{align}
\end{subequations}
\noindent Combining the inner problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W}))$ in the form of \eqref{CVARmodel_obj5} and problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})]$ in the form of \eqref{FinalDR} with the outer minimization problem in \eqref{CVaR-DMFRS}, we derive the following equivalent reformulation of the MAD-CVaR model in \eqref{CVaR-DMFRS}.
\begin{subequations}\label{Final_CVaR}
\begin{align}
\min & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \Theta \Bigg [ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big(\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big) + h(\pmb{x}, \pmb \rho, \pmb \psi) \Bigg] \nonumber \\
& \ \ + (1-\Theta) \Bigg [ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg] \Bigg \} \label{CVaR-DMFRS3} \\
\text{s.t.} & \ \ \ (\pmb{x}, \pmb{y})\in \mathcal{X}, \ \pmb \psi \geq 0, \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}.
\end{align}
\end{subequations}
\noindent It is easy to verify that problem \eqref{Final_CVaR} is equivalent to
\begin{subequations}\label{Final_CVaR2}
\begin{align}
\min & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \Theta \Bigg [ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big(\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big) \Bigg] \nonumber \\
& \ \ + \delta + (1-\Theta) \Bigg [ \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg] \Bigg \} \label{CVaR-DMFRS4_1} \\
\text{s.t.} & \ \ \ (\pmb{x}, \pmb{y})\in \mathcal{X}, \ \pmb \psi \geq 0, \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}, \\
& \ \ \ \delta \geq h(\pmb{x}, \pmb \rho, \pmb \psi), \label{CVaR-DMFRS4_2}
\end{align}
\end{subequations}
where $\textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}=\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}): \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner} \Big\} $ from Section~\ref{sec3:reform}. Finally, we observe that the right-hand side (RHS) of constraints \eqref{CVaR-DMFRS4_2} is equivalent to the RHS of constraints \eqref{const1:FinalDR} in the equivalent reformulation of the risk-neutral MAD-DRO model in \eqref{FinalDR}. Therefore, we can easily adapt Algorithm 1 to solve \eqref{Final_CVaR2} (see Algorithm~\ref{Alg2:Decomp2}).
\begin{algorithm}[t!]
\color{black}
\small
\renewcommand{\arraystretch}{0.3}
\caption{Decomposition algorithm for the MAD-CVaR Model.}
\label{Alg2:Decomp2}
\noindent \textbf{1. Input.} Feasible region $\mathcal{X}$; support $\mathcal{S}$; set of cuts $ \lbrace L (\pmb{x}, \delta)\geq 0 \rbrace=\emptyset $; $LB=-\infty$ and $UB=\infty.$
\vspace{2mm}
\noindent \textbf{2. Master Problem.} Solve the following master problem
\begin{subequations}\label{MastCVAR}
\begin{align}
\min \quad & \Bigg\{\sum_{m \in M} f y_{m}-\sum_{t \in T} \sum_{j \in J} \sum_{m \in M} \alpha x_{j, m}^{t}+\Theta\left[\sum_{t \in T} \sum_{i \in I}\left(\mu_{i, t} \rho_{i, t}+\eta_{i, t} \psi_{i, t}\right)\right]\nonumber \\
& \left.+(1-\Theta)\left[\left(\frac{\kappa}{1-\kappa}\right) \theta+\frac{1}{1-\kappa} \sum_{t \in T} \sum_{i \in I}\left(\mu_{i, t} \rho_{i, t}+\eta_{i, t} \psi_{i, t}\right)\right] +\delta \right\} \\
\text{s.t.} & \qquad (\pmb{x},\pmb{y} ) \in \mathcal{X}, \ \ \pmb{\psi} \geq 0, \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}, \ \{ L (\pmb{x}, \delta)\geq 0 \},
\end{align}%
\end{subequations}
$\ \ \ $ and record an optimal solution $(\pmb{x}^ *, \pmb{\rho}^*, \pmb{\psi}^*, \delta^*)$ and optimal value $Z^*$. Set $LB=Z^*$.
\noindent \textbf{3. Sub-problem.}
\begin{itemize}
\item[3.1.] With $(\pmb{x}, \pmb{\rho}, \pmb{\psi})$ fixed to $(\pmb{x}^*, \pmb{\rho}^*, \pmb{\psi}^*)$, solve the following problem
\begin{subequations}\label{MILPSep2}
\begin{align}
h(\pmb{x}, \pmb \rho, \pmb \psi)= &\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big\}\\
& \ \ \ \text{s.t. } \ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner},
\end{align}
\end{subequations}
$\qquad \ $ and record optimal solution $(\pmb{\pi^*, \lambda^*, W^*, v^*, k^*})$ and $h(\pmb{x}, \pmb \rho, \pmb \psi)^*$.
\item[3.2.] Set $UB=\min \{ UB, \ h(\pmb{x}, \pmb \rho, \pmb \psi)^*+ (LB-\delta^*) \}$.
\end{itemize}
\noindent \textbf{4. if} $\delta^* \geq \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^{t*} v_{j,m}^{t*} +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}^*+k_{i,t}^*\psi_{i,t}^*) $ \textbf{then}
$\qquad \ \ $ stop and return $\pmb{x}^*$ and $\pmb{y}^*$ as the optimal solution to problem \eqref{MastCVAR} (equivalently, \eqref{Final_CVaR2}).
\noindent $\ \ $ \textbf{else} add the cut $\delta \geq \sum \limits_{t \in T} \sum \limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum \limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m} v_{j,m}^{t*}+\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}+k_{i,t}^*\psi_{i,t} )$ to the set of
$\qquad \quad$ cuts $ \{L (\pmb{x}, \delta) \geq 0 \}$ and go to step 2.
\noindent $\ \ $ \textbf{end if}
\end{algorithm}
\color{black}
\newpage
\section{Two-stage SP Model with Mean-CVaR Objective}\label{Appex:SP_CVAR}
\noindent The mean-CVaR-based SP model (denoted as SP-CVaR) is as follow:
\begin{align}
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Theta \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W}))] + (1-\Theta) \CVAR (Q(\pmb{x},\pmb{W})) \Bigg \} \nonumber\\
\equiv &\nonumber \\
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}, \zeta \in \mathbb{R}^+} &\Big \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Theta \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W}))]+ (1-\Theta) \Big ( \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Big) \Bigg\}. \label{CVaR-SP}
\end{align}
\noindent The sample average deterministic equivalent of \eqref{CVaR-SP} based on $N$ scenarios, $\pmb{W}^1,\ldots, \pmb{W}^N$, is as follows:
\begin{subequations}
\begin{align}
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}, \zeta \in \mathbb{R}^+} & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Theta \sum_{n=1}^N \frac{1}{N} Q(\pmb{x},\pmb{W}^n) + (1-\Theta) \Big ( \zeta+ \frac{1}{1-\kappa} \sum_{n=1}^N \frac{1}{N} \Phi^n \Big) \Bigg\} \\
\text{s.t. } & \ \Phi^n \geq Q(\pmb{x},\pmb{W}^n) -\zeta, \qquad \forall n \in [N],\\
& \ \Phi^n \geq 0, \qquad \forall n \in [N],
\end{align}
\end{subequations}
\noindent where for each $n \in [N]$, $Q(\pmb{x}, \pmb{W}^n)$ is the recourse problem defined in \eqref{2ndstage}, and $\Theta \in [0, 1]$/
\section{Proof of Proposition~\ref{Prop3:LowerB1}}\label{Appx:Prop3}
\noindent Recall from the definition of the support set that the lowest demand of each customer $i$ in period $t$ equals to the integer parameter $\underline{W}_{i,t}$. Now, if we treat the MFs as uncapacitated facilities, then we can fully satisfy $\underline{W}_{i,t}$ at the lowest assignment cost from the nearest location $j \in J^\prime$, where $J^\prime:= \lbrace j: x_{j,m}^t=1 \rbrace$. Note that $J^\prime \subseteq J$. Thus, the lowest assignment cost must be at least equal to or larger than $\sum_{i \in I} \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \underline{W}_{i,t} $. If $\gamma < \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace$, then the recourse must be at least equal to or larger than $ \sum_{i \in I} \gamma \underline{W}_{i,t}$. Accordingly, $\sum \limits_{i \in I} \min \lbrace \gamma, \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \rbrace \underline{W}_{i,t}$ is a valid lower bound on recourse $Q(\pmb{x}, \pmb{W})$ for each $t \in T$.
\section{Sample Average Approximation}\label{Appx:SAA}
\begin{subequations}\label{SPModel}
\begin{align}
& \min \Big [ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \frac{1}{N} \sum_{n=1}^N\Big( \sum_{j \in J} \sum_{i \in I} \sum_{m \in M} \sum_{t \in T} \beta d_{i,j} z_{i,j,m}^{t,n}+ \gamma \sum_{t \in T} \sum_{i \in I} u_{i,t}^n \Big) \Big]\label{ObjSP}\\
& \ \text{s.t.} \ \ (\pmb{x}, \pmb{y}) \in \mathcal{X}, \\
&\quad \ \quad \ \ \sum_{j \in J} \sum_{m \in M} z_{i,j,m}^{t,n}+u_{i,t}^n= W_{i,t}^n, \qquad \forall i \in I, \ t \in T , \ n \in [N], \\
&\quad \ \quad \quad \sum_{i \in I} z_{i,j,m}^{t,n}\leq C x_{j,m}^t \qquad \forall j \in J, \ m \in M, \ t \in T, \ n \in [N], \\
&\quad \ \quad \quad u_{i,t}^n \geq 0, \ z_{i,j,m}^{t,n} \geq 0, \qquad \forall i \in I, \ j \in J, m \in M, \ t \in [T, \ n \in [N].
\end{align}
\end{subequations}
\newpage
\section{\textcolor{black}{Details of Lehigh County Instances}}\label{AppexLehigh}
\begin{table}[h!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.7}
\caption{A subset of 20 nodes in Lehigh County and their population based on the 2010 census of Lehigh County (from \cite{wikiLehighCounty}).}
\begin{tabular}{lll}
\hline
\textbf{City/Town/etc.} & Population \\
\hline
Allentown & 118,032 \\
Bethlehem & 74,982 \\
Emmaus & 11,211 \\
Ancient Oaks & 6,661 \\
Catasauqua & 6,436 \\
Wescosville & 5,872 \\
Fountain Hill & 4,597 \\
Dorneyville & 4,406 \\
Slatington & 4,232 \\
Breinigsville & 4,138 \\
Coplay & 3,192 \\
Macungie & 3,074 \\
Schnecksville & 2,935 \\
Coopersburg & 2,386 \\
Alburtis & 2,361 \\
Cetronia & 2,115 \\
Trexlertown & 1,988 \\
Laurys Station & 1,243 \\
New Tripoli & 898 \\
Slatedale & 455 \\
\hline
\end{tabular}
\label{table:LehighInst}
\end{table}
\begin{table}[h!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.7}
\caption{Average demand of each node in Lehigh 1 and Lehigh 2. }
\begin{tabular}{lll}
\hline
\textbf{Node} & \textbf{Lehigh 1 }& \textbf{Lehigh 2} \\
\hline
Allentown & 40 & 60\\
Bethlehem & 40 & 60 \\
Emmaus & 40 & 43 \\
Ancient Oaks & 30 & 25 \\
Catasauqua & 30 &25 \\
Wescosville & 30 & 22 \\
Fountain Hill & 20 & 18 \\
Dorneyville & 20 &17 \\
Slatington & 20 & 16\\
Breinigsville & 20 & 16 \\
Coplay & 20 & 12\\
Macungie & 20 & 12 \\
Schnecksville & 20 & 11 \\
Coopersburg & 20 &9 \\
Alburtis & 20 &9 \\
Cetronia & 20 & 8 \\
Trexlertown & 20 & 8\\
Laurys Station &20 & 5\\
New Tripoli & 15 &3 \\
Slatedale & 15 &3 \\
\hline
\end{tabular}
\label{table:AvgDemand_LehighInst}
\end{table}
\begin{table}[h!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.7}
\caption{Average demand of each node in period 1 and period 2.}
\begin{tabular}{lll}
\hline
\textbf{Node} & \textbf{Period 1}& \textbf{Period 2} \\
\hline
Allentown & 60 & 9 \\
Bethlehem & 60 & 8 \\
Emmaus & 43 & 8 \\
AncientOaks & 25 & 5 \\
Catasauqua & 25 & 3 \\
Wescosville & 22 & 3 \\
FountainHill & 18 & 18 \\
Dorneyville & 17 & 17 \\
Slatington & 16 & 16 \\
Breinigsville & 16 & 16 \\
Coplay & 12 & 12 \\
Macungie & 12 & 12 \\
Schnecksville & 11 & 11 \\
Coopersburg & 9 & 9 \\
Alburtis & 9 & 60 \\
Cetronia & 8 & 60 \\
Trexlertown & 8 & 43 \\
LaurysStation & 5 & 25 \\
NewTripoli & 3 & 25 \\
Slatedale & 3 & 22 \\
\hline
\end{tabular}
\label{table:period}
\end{table}
\begin{table}[t!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.3}
\caption{Optimal MFs locations in period 1 and period 2.}
\begin{tabular}{llllllllllllll}
\hline
\textbf{MF} & \multicolumn{3}{c}{\textbf{MAD-DRO}} & \multicolumn{3}{c}{\textbf{W-DRO}} & &\multicolumn{3}{c}{\textbf{SP}} \\ \cline{2-3} \cline{5-6} \cline{8-9}
& Period 1 & Period 2 & & Period 1 & Period 2 & & Period 1 & Period 2 \\ \cline{1-9}
\hline
1 & Allentown & New Tripoli & & Allentown & Cetronia & & Allentown & Fountain Hill \\
2 & Bethlehem & Schnecksville & & Bethlehem & New Tripoli & & Bethlehem & Slatedale \\
3 & Emmaus & Trexlertown & & Emmaus & Ancient Oaks & & Dorneyville & Cetronia \\
4 & Wescosville & Alburtis & & Breinigsville & Trexlertown & & Ancient Oaks & Alburtis \\ \cline{1-9}
\hline
\end{tabular}
\label{table:OptimalLocations}
\end{table}
\newpage
\section{Calibrating the Wasserstein Radius in the W-DRO Model}\label{WDRO_Radius}
The Wasserstein ball’s radius $\epsilon$ in the W-DRO model is an input parameter, where a larger $\epsilon$ implies that we seek more distributionally robust solutions. For each training data, different values of $\epsilon$ may result in robust solutions $\pmb{x} (\epsilon, N)$ with very different out-of-sample performance $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$. On the one hand, the radius should not be too small. \textcolor{black}{Otherwise,} the problem may behave like sample average approximation and hence, losing the purpose of robustification. In particular, if we set the radius to zero, the ambiguity set shrinks to a singleton that contains only the nominal distribution, in which case the DRO problem reduces to an ambiguity-free SP \citep{esfahani2018data}. But, on the other hand, the radius should not be too large to avoid conservative solutions, which is one of the major criticism faced by traditional RO methods. Given that the true distribution $\mathbb{P}$ is possibly unknown, it is impossible to compute $\epsilon$ that minimizes $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$. Thus, as detailed in \cite{esfahani2018data}, the best we can hope for is to approximate $\epsilon^{\mbox{\tiny opt}}$ using the training data set.
As pointed out by \cite{esfahani2018data} and \cite{gao2020finite}, practically, the radius is often selected via cross-validation. We employ the following widely used cross-validation method to estimate $\epsilon^{\mbox{\tiny opt}}$ as in \cite{jiang2019data} and \cite{esfahani2018data}. First, for each $N\in \{10, \ 50, \ 100\} $ and each $\epsilon \in \{ 0.01, \ 0.02,\ldots, \ 0.09, \ 0.1, \ldots, \ 0.9,1, \ldots, \ 10 \}$ (i.e., a log-scale interval as in \cite{esfahani2018data}, \cite{jian2017integer}, \cite{tsang2021distributionally}), we randomly partition the data into a training ($N'$) and testing set ($N''$). Using the training set, we solve W-DRO to obtain the optimal first-stage solution $\pmb{x} (\epsilon,N)$ for each $\epsilon$ and $N$. Then, we use the \textcolor{black}{testing} data to evaluate these solutions by computing $\hat{\mathbb{E}}_{\mathbb{P}_{N''}}[Q(\pmb{x} (\epsilon, N),\pmb{W})]$ (where $\mathbb{P}_{N''}$ is the empirical distribution based on the testing data $N''$) via sample average approximation. That is, we solve the second-stage with $\pmb{x}$ fixed to $\pmb{x} (\epsilon,N')$ and $N''$ data and compute the corresponding second-stage cost $\hat{\mathbb{E}}_{\mathbb{P}_{N''}}[Q(\pmb{x} (\epsilon, N''),\pmb{W})]$. Finally, we set $\epsilon^{\mbox{\tiny best}}_N$ to any $\epsilon$ that minimizes $\hat{\mathbb{E}}_{\mathbb{P}_{N''}}[Q(\pmb{x} (\epsilon, N''),\pmb{W})]$. We repeat this procedure 30 times for each $N$ and set $\epsilon$ to the average of the $\epsilon^{\mbox{\tiny best}}_N$ across these 30 replications.
We found that $\epsilon^{\mbox{\tiny best}}_N$ equals (7, 5, 2) when $N$=(10, 50, 100) for most instances. It is expected that $\epsilon$ decreases with $N$ (see, e.g., \cite{esfahani2018data}, \cite{jiang2019data}, \cite{tsang2021distributionally}). Intuitively, a small sample does not have sufficient distributional information, and thus a larger $\epsilon$ produces distributionally robust solutions that better hedge against ambiguity. In contrast, with a larger sample, we may have more information from the data, and thus we can make a less conservative decision using a smaller $\epsilon$ value.
While Wasserstein ambiguity sets offer powerful out-of-sample performance guarantees and enable practitioners to control the model's conservativeness by choosing $\epsilon$, moment-based ambiguity sets often display better tractability properties. In fact, various studies provided evidence that DRO models with moment ambiguity sets are more tractable than the corresponding SP because the intractable high-dimensional integrals in the objective function are replaced with tractable (generalized) moment problems (see, e.g., \cite{esfahani2018data, delage2010distributionally, goh2010distributionally, wiesemann2014distributionally}). In contrast, DRO models with Wasserstein ambiguity sets tend to be more computationally challenging than some moment-based DRO model and their SP counterparts, especially when $N$ is large. In this paper, we obtained similar observations. For a detailed discussion we refer to \cite{esfahani2018data} and references therein.
\newpage
\section{Additional CPU Time Results}\label{Appx:CPU2}
\color{black}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-CVaR model ($ \pmb{W} \in [20, 60]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 1 & 3 & 6 & 4 & 12 & 25 & & 2 & 7 & 15 & 4 & 12 & 25 \\
2 & 10 & 20 & & 5 & 11 & 22 & 11 & 27 & 50 & & 18 & 21 & 29 & 35 & 44 & 61 \\
3 & 15 & 10 & & 3 & 8 & 19 & 5 & 16 & 36 & & 5 & 13 & 32 & 9 & 24 & 58 \\
4 & 15 & 20 & & 28 & 63 & 107 & 23 & 56 & 93 & & 35 & 60 & 85 & 29 & 86 & 121 \\
5 & 20 & 10 & & 13 & 25 & 37 & 12 & 23 & 33 & & 19 & 75 & 107 & 17 & 47 & 68 \\
6 & 20 & 20 & & 207 & 407 & 617 & 59 & 106 & 165 & & 433 & 969 & 1576 & 103 & 198 & 303 \\
7 & 25 & 10 & & 111 & 242 & 494 & 25 & 43 & 67 & & 147 & 454 & 794 & 42 & 75 & 125 \\
8 & 25 & 20 & & 2083 & 2944 & 3903 & 11 & 94 & 169 & & 2979 & 3274 & 3600 & 164 & 237 & 316 \\
\hline
\end{tabular}\label{table:MADCVAR_2060}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-CVaR model ($ \pmb{W} \in [50, 100]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 2 & 6 & 10 & 5 & 17 & 30 & & 2 & 5 & 8 & 4 & 14 & 26 \\
2 & 10 & 20 & & 14 & 22 & 29 & 22 & 35 & 46 & & 18 & 23 & 29 & 19 & 31 & 44 \\
3 & 15 & 10 & & 18 & 31 & 57 & 9 & 21 & 49 & & 16 & 34 & 45 & 8 & 22 & 47 \\
4 & 15 & 20 & & 67 & 106 & 156 & 23 & 68 & 139 & & 64 & 132 & 255 & 27 & 62 & 106 \\
5 & 20 & 10 & & 52 & 78 & 103 & 18 & 33 & 51 & & 38 & 67 & 117 & 16 & 28 & 56 \\
6 & 20 & 20 & & 549 & 715 & 935 & 96 & 124 & 172 & & 757 & 1459 & 2802 & 91 & 124 & 184 \\
7 & 25 & 10 & & 584 & 685 & 760 & 28 & 48 & 83 & & 743 & 2277 & 2552 & 11 & 42 & 90 \\
\hline
\end{tabular}\label{table:MADCVAR__50100}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [20, 60]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 7 & 19 & 17 & 5 & 5 & 6 & & 16 & 20 & 24 & 12 & 15 & 19 \\
2 & 10 & 20 & & 12 & 13 & 16 & 6 & 6 & 8 & & 37 & 44 & 57 & 17 & 18 & 22 \\
3 & 15 & 10 & & 27 & 44 & 70 & 4 & 5 & 5 & & 22 & 28 & 38 & 10 & 12 & 13 \\
4 & 15 & 20 & & 9 & 12 & 17 & 4 & 5 & 7 & & 15 & 33 & 60 & 6 & 11 & 17 \\
5 & 20 & 10 & & 34 & 35 & 39 & 4 & 5 & 6 & & 72 & 88 & 98 & 14 & 16 & 20 \\
6 & 20 & 20 & & 20 & 31 & 46 & 4 & 6 & 9 & & 240 & 574 & 827 & 6 & 13 & 21 \\
\hline
\end{tabular}\label{table:Wass_CVAR_2060_N10}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [50, 100]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 13 & 25 & 53 & 3 & 4 & 4 & & 11 & 16 & 25 & 5 & 6 & 7 \\
2 & 10 & 20 & & 6 & 8 & 14 & 3 & 3 & 4 & & 10 & 12 & 16 & 5 & 6 & 8 \\
3 & 15 & 10 & & 16 & 26 & 57 & 3 & 3 & 3 & & 29 & 45 & 75 & 5 & 5 & 6 \\
4 & 15 & 20 & & 38 & 45 & 61 & 3 & 3 & 3 & & 63 & 82 & 136 & 5 & 6 & 7 \\
5 & 20 & 10 & & 45 & 49 & 55 & 3 & 3 & 3 & & 55 & 70 & 79 & 7 & 7 & 8 \\
\hline
\end{tabular}\label{table:Wass_CVAR_50100_N10}
\end{table}
\newpage
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [20, 60]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 28 & 35 & 40 & 5 & 6 & 7 & & 104 & 117 & 129 & 15 & 19 & 22\\
2 & 10 & 20 & & 33 & 42 & 51 & 6 & 7 & 9 & & 84 & 112 & 149 & 15 & 19 & 24\\
3 & 15 & 10 & & 28 & 34 & 43 & 4 & 5 & 5 & & 109 & 136 & 148 & 14 & 17 & 19\\
4 & 15 & 20 & & 34 & 59 & 102 & 4 & 5 & 7 & & 189 & 1471 & 3730 & 17 & 20 & 23\\
5 & 20 & 10 & & 68 & 88 & 125 & 4 & 6 & 7 & & 455 & 1,084 & 1,556 & 24 & 30 & 34\\
\hline
\end{tabular}\label{table:Wass_CVAR_2060_N50}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [50, 100]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 24 & 25 & 26 & 4 & 4 & 4 & & 37 & 46 & 51 & 6 & 7 & 8 \\
2 & 10 & 20 & & 29 & 30 & 32 & 4 & 4 & 4 & & 52 & 73 & 124 & 7 & 8 & 9 \\
3 & 15 & 10 & & 33 & 50 & 75 & 3 & 3 & 3 & & 90 & 100 & 111 & 6 & 6 & 7 \\
4 & 15 & 20 & & 50 & 59 & 68 & 3 & 3 & 3 & & 130 & 310 & 743 & 5 & 6 & 9 \\
5 & 20 & 10 & & 107 & 130 & 170 & 3 & 3 & 3 & & 118 & 182 & 321 & 5 & 6 & 8 \\
\hline
\end{tabular}\label{table:Wass_CVAR_50100_N50}
\end{table}
\textcolor{white}{sabjhasvghavwejfce}
\clearpage
\newpage
\section{Additional Out-of-Sample Results}\label{Appex:AdditionalOut}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_LogN.jpg}
\caption{2nd (Set 1, LogN)}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_Uni0}
\caption{TC (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample for Instance 3 ($\pmb{W} \in [20, 60]$, $\pmb{N=50}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_UniN50_Inst3}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_Uni0.jpg}
\caption{TC (Set 1, LogN)}\label{Inst3_LogNTC_Range2N50}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_Uni0.jpg}
\caption{2nd (Set 1, LogN)}\label{Inst3_LogN2nd_Range2N50}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_LogN.jpg}
\caption{TC (Set 2, $\Delta=0$)}\label{Inst3_Uni0_TC_Range2N50}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_LogN.jpg}
\caption{2nd (Set 2, $\Delta=0$)}\label{Inst3_Uni0_2nd_Range2N50}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)} \label{Inst3_Uni25_2nd_Range2N50}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Instance 3 ($\pmb{W} \in [50, 100]$, $\pmb{N=50}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$). }\label{Fig3_UniN50_Inst3_Range2N50}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_LogN.jpg}
\caption{2nd (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_Uni0.jpg}
\caption{TC (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}
\end{subfigure}%
%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_Uni25.jpg}
\caption{TC (Set 2, $\Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_Uni50.jpg}
\caption{TC (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 1 ($\pmb{N=50}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$). }\label{Fig3_N50_Lehigh1}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_LogN.jpg}
\caption{2nd (Set 1, LogN))}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_Uni0.jpg}
\caption{TC (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_Uni0.jpg}
\caption{2nd (Set 2, $ \Delta=0$)}
\end{subfigure}%
%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_Uni50.jpg}
\caption{TC (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_Uni50.jpg}
\caption{2nd (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 2 ($N=10$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_Uni10_Lehigh2}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_LogN.jpg}
\caption{2nd (Set 1, LogN)}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_Uni0.jpg}
\caption{TC (Set 2, $ \Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_Uni0.jpg}
\caption{2nd (Set 2, $ \Delta=0$)}
\end{subfigure}%
%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_Uni25.jpg}
\caption{TC (Set 2, $\Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_Uni50.jpg}
\caption{2nd (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 2 ($N=50$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_Uni50_Lehigh2}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Corr20.jpg}
\caption{TC (cor=0.2)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Corr20.jpg}
\caption{2nd (cor=0.2)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Corr60.jpg}
\caption{TC (cor=0.6) }
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Corr60.jpg}
\caption{2nd (cor=0.6)}
\end{subfigure}%
\caption{Out-of-sample performance under correlated data for Instance 3. Notation: cor is correlation coefficient}\label{Fig_outcorr}
\end{figure}
\clearpage
\newpage
\section{Additional Sensitivity Results}\label{Appx:sensitivity}
\begin{figure}[h!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_1500.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig7a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_1500.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig7b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_6000.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig7c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_TC_6000.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig7d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_10000.jpg}
\caption{Number of MFs, $f=10,000$}
\label{Fig7e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_TC_10000.jpg}
\caption{Total cost, $f=10,000$}
\label{Fig7f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [50, 100] $. Instance 1}\label{Fig7:MF_vs_C_Range2_inst1}
\end{figure}
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_1500_Inst5.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig10a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_1500_Inst5.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig10b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_6000_Inst5.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig10c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_6000_Inst5.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig10d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_10000_Inst5.jpg}
\caption{Number of MFs, $f=10,000$}
\label{Fig10e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_10000_Inst5.jpg}
\caption{Total cost, $f=10,000$}
\label{Fig10f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [50, 100] $. Instance 5}\label{Fig10:MF_vs_C_Range2_inst5}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_2nd_1500.jpg}
\caption{$f=1,500$}
\label{Fig6a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_2nd_6000.jpg}
\caption{$f=6,000$}
\label{Fig6b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_2nd_10000.jpg}
\caption{$f=10,000$}
\label{Fig6c}
\end{subfigure}%
\caption{Comparison of second-stage cost for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 1}\label{Fig8:MF_vs_C_inst1}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_1500_Inst5_2.jpg}
\caption{$f=1,500$}
\label{Fig11a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_F_2nd_6000_Inst5.jpg}
\caption{$f=6,000$}
\label{Fig11b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_10000_Inst5.jpg}
\caption{$f=10,000$}
\label{Fig11c}
\end{subfigure}%
\caption{Comparison of second-stage cost for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 5}\label{Fig11:MF_vs_C_inst5}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_Range2_1500_Inst5.jpg}
\caption{$f=1,500$}
\label{Fig12a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_F_2nd_Range2_6000_Inst5.jpg}
\caption{$f=6,000$}
\label{Fig12b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_Range2_10000_Inst5.jpg}
\caption{$f=10,000$}
\label{Fig12c}
\end{subfigure}%
\caption{Comparison of second-stage cost for different values of $C$ and $f$ under $\pmb{W} \in [50, 100]$. Instance 5}\label{Fig12:MF_vs_C_range2_inst5}
\end{figure}
\end{APPENDICES}
\clearpage
\newpage
\bibliographystyle{informs2014trsc}
\section{Introduction}
\noindent A \textit{mobile facility} (MF) is a facility capable of moving from one place to another, providing real-time service to customers in the vicinity of its location when it is stationary \citep{halper2011mobile}. In this paper, we study a mobile facility \textcolor{black}{fleet sizing, }routing, and scheduling problem (MFRSP) with stochastic demand. Specifically, in this problem, we aim to find the number of MFs (i.e., fleet size) to use in a given service region over a specified planning horizon and the route and schedule for each MF in the fleet. The demand level of each customer in each time period is random. The probability distribution of the demand is unknown, and only partial information about the demand (e.g., mean and range) may be available. The objective is to find the MF fleet size, routing, and scheduling decisions that minimize the sum of the fixed cost of establishing the MF fleet, the cost of assigning demand to the MFs (e.g., transportation cost), and the cost of unsatisfied demand (i.e., shortage cost).
The concept of MF routing and scheduling is very different than conventional static facility location (FL) and conventional vehicle routing (VR) problems. In static FL problems, we usually consider opening facilities at fixed locations. Conventional VR problems aims at handling the movement of items between facilities (e.g., depots) and customers. \textcolor{black}{A mobile facility is a \textit{facility-like vehicle} that functions as a traditional facility when it is stationary, except that it can move from one place to another if necessary \citep{lei2014multicut}}. Thus, the most evident advantage of MFs over fixed facilities is their flexibility in moving to accommodate the change in the demand over time and location \citep{halper2011mobile, lei2014multicut, lei2016two}.
MFs are used in many applications ranging from cellular services, healthcare services, to humanitarian relief logistics. For example, light trucks with portable cellular stations can provide cellular service in areas where existing cellular network of base stations temporarily fails \citep{halper2011mobile}. Mobile clinics (i.e., customized MFs fitted with medical equipment and staffed by health professionals) can travel to rural and urban areas to provide various (prevention, testing, diagnostic) health services. Mobile clinics also offer alternative healthcare (service) delivery options when a disaster, conflict, or other events cause stationary healthcare facilities to close or stop operations \citep{blackwell2007use, brown2014mobile, du2007mobile, gibson2011households, oriol2009calculating, song2013mobile}. For example, mobile clinics played a significant role in providing drive-through COVID-19 testing sites or triage locations during the COVID-19 pandemic. In 2019, the mobile health clinic market was valued at nearly 2 billion USD and is expected to increase to $\sim$12 billion USD by 2028 \citep{MFMarket}. In humanitarian relief logistics, MFs give relief organizations the ability to provide aid to populations dispersed in remote and dense areas. These examples motivate the need for computationally efficient optimization tools to support decision-making in all areas of the MF industry.
MF operators often seek a \textcolor{black}{strategic} and tactical plan, including the size of the MF fleet (\textcolor{black}{strategic}) and a routing plan for each MF in the fleet (tactical and operational) that minimize their fixed operating costs and maximize demand satisfaction. Determining the fleet size, in particular, is very critical as it is a major fixed investment for starting any MF-based business. The fleet sizing problem depends on the MF operational performance, which depends on the routing and scheduling decisions. The allocation of the demand to the MFs is also very important for the entire system performance \citep{lei2016two}. For example, during the COVID-19 pandemic, Latino Connection, a community health leader, has established Pennsylvania's first COVID-19 Mobile Response Unit, CATE (i.e., Community-Accessible Testing \& Education). The goal of CATE is to provide affordable and accessible COVID-19 education, testing, and vaccinations to low-income, vulnerable communities across Pennsylvania to ensure the ability to stay safe, informed, and healthy \citep{CATEMF}. \textcolor{black}{During COVID-19, CATE published an online schedule consisting of the mobile unit stops and the schedule at each stop.} A model that optimizes CATE's fleet size and schedules considering demand uncertainty could help improve CATE's operational performance and achieve better access to health services.
Unfortunately, the MFRSP is a challenging optimization problem for two primary reasons. First, customers' demand is random and hard to predict in advance, especially with limited data during the planning process. Second, even in a perfect world in which we know with certainty the amount of demand in each period, the deterministic MFRSP is challenging because it is similar to the classical FL problem \citep{halper2011mobile, lei2014multicut}. \textcolor{black}{Thus, the incorporation of demand variability increases the overall complexity of the MFRSP. However, ignoring demand uncertainty may lead to sub-optimal decisions and, consequently, the inability to meet customer demand (i.e., shortage).} Failure to meet customer demand may lead to adverse outcomes, especially in healthcare, as it impacts population health. It also impacts customers' satisfaction and thus the reputation of the service providers and may increase their operational cost (due to, e.g., outsourcing the excess demand to other providers).
To model uncertainty, \cite{lei2014multicut} assumed that the probability distribution of the demand is known and accordingly proposed the first a \textit{priori} two-stage stochastic optimization model (SP) for a closely related MFRSP. Although attractive, the applicability of the SP approach is limited to the case in which we know the distribution of the demand or we have sufficient data to model it. \textcolor{black}{In practice, however, one might not have access to a sufficient amount of high-quality data to estimate the demand distribution accurately.} This is especially true in application domains where the use of mobile facilities to deliver services is relatively new (e.g., mobile COVID-19 testing clinics). Moreover, it is challenging for MF companies to obtain data from other companies (competitors) due to privacy issues. Finally, various studies show that different distributions can typically explain raw data of uncertain parameters, indicating distributional ambiguity \citep{esfahani2018data, vilkkumaa2021causes}.
Suppose we model uncertainty using a data sample from a potentially biased distribution or an assumed distribution (as in SP). In this case, the resulting nominal decision problem evaluates the cost only at this training sample, and thus the resulting decisions may be overfitted (optimistically biased). Accordingly, SP solutions may demonstrate disappointing out-of-sample performance (`\textit{black swans}') under the true distribution (or unseen data). In other words, solutions of SP decision problems often display an optimistic in-sample risk, which cannot be realized in out-of-sample settings. This phenomenon is known as the \textit{Optimizers' Curse} (i.e., an attempt to optimize based on imperfect estimates of distributions leads to biased decisions with disappointing performance) and is reminiscent of the overfitting effect in statistics \citep{smith2006optimizer}.
\textcolor{black}{Alternatively, one can construct an ambiguity set of all distributions that possess certain partial information about the demand. Then, using this ambiguity set, one can formulate a distributionally robust optimization (DRO) problem to minimize a risk measure (e.g., expectation or conditional value-at-risk (CVaR)) of the operational cost over all distributions residing within the ambiguity set. In particular, in the DRO approach, the optimization is based on the worst-case distribution within the ambiguity set, which effectively means that the distribution of the demand is a decision variable.}
DRO \textcolor{black}{has received substantial attention recently} in various application domains due to the following striking benefits. First, as pointed out by \cite{esfahani2018data}, DRO models are more ``\textit{honest}'' than their SP counterparts as they acknowledge the presence of distributional uncertainty. \textcolor{black}{Therefore, DRO solutions often faithfully anticipate the possibility of black swan (i.e., out-of-sample disappointment). Moreover, depending on the ambiguity set used, DRO often guarantees an out-of-sample cost that falls below the worst-case optimal cost}. Second, DRO alleviates the unrealistic assumption of the decision-maker's complete knowledge of distributions. Third, several studies have proposed DRO models for real-world problems that are more computationally tractable than their SP counterparts, see, e.g., \cite{basciftci2019distributionally, luo2018distributionally, saif2020data, ShehadehSanci, shehadeh2020distributionallyTucker, tsang2021distributionally, wang2020distributionally, wang2021two,wu2015approximation}. In this paper, we propose tractable DRO approaches for the MFRSP.
The ambiguity set is a key ingredient of DRO models that must (1) capture the true distribution with a high degree of certainty, and (2) be computationally manageable (i.e., allow for a tractable DRO model or solution method). There are several methods to construct the ambiguity set. Most applied DRO literature employs moment-based ambiguity \citep{delage2010distributionally, zhang2018ambiguous}, consisting of all distributions sharing particular moments (e.g., mean-support ambiguity). The main advantage of the mean-support ambiguity set, for example, is that it incorporates intuitive statistics that a decision-maker may easily approximate and change. Moreover, various techniques have been developed to derive tractable moment-based DRO models. However, asymptotic properties of the moment-based DRO model cannot often be guaranteed because the moment information represents descriptive statistics.
Recent DRO approaches define the ambiguity set by choosing a distance metric (e.g., $\phi$--divergence \citep{jiang2016data}, Wasserstein distance \citep{esfahani2018data, gao2016distributionally}) to describe the deviation from a reference (often empirical) distribution. The main advantage of Wasserstein ambiguity, for example, is that it enable decision-makers to incorporate possibly small-size data in the ambiguity set and optimization, enjoys asymptotic properties, and often offers a strong out-of-sample performance guarantee \citep{esfahani2018data,mevissen2013data}. Recent results indicate that Wasserstein's ambiguity centered around a given empirical distribution contains the unknown true distribution with a high probability and is richer than other divergence-based ambiguity sets (in particular, they contain discrete and continuous distributions as compared to, e.g., $\phi$-divergence ball centered at the empirical distribution which does not contain any continuous distribution, and Kullback-Leibler divergence ball, which must be absolutely continuous with respect to the nominal distribution).
Despite the potential advantages, there are no moment-based, Wasserstein-based, or any other DRO approaches for the specific MFRSP that we study in this paper (see Section~\ref{Sec:LitRev}). This inspires this paper's central question: \textit{what are the computational and operational performance values of employing DRO to address demand uncertainty and ambiguity compared to the classical SP approach for the MFRSP}. To answer this question, we design and analyze two DRO models based on the demand's mean, support, and mean absolute deviation ambiguity and Wasserstein ambiguity and compare the performance of these models with the classical SP approach.
\subsection{\textcolor{black}{Contributions}}
\noindent In this paper, we present two distributionally robust MF fleet sizing, routing, and scheduling (DMFRS) models for the MFRSP, as well as methodologies for solving these models. We summarize our main contributions as follows.
\begin{enumerate}
\item \textbf{Uncertainty Modeling and Optimization Models.} We propose the first two-stage DRO models for the MFRSP. These models aim to find the optimal (1) number of MFs to use within a planning horizon, (2) a routing plan and a schedule for the selected MFs, i.e., the node that each MF is located at in each time period, (3) assignment of MFs to customers. Decisions (1)-(2) are planning (first-stage) decisions, which cannot be changed in the short run. Conversely, the assignments of the demand are decided based on the demand realization, and thus are second-stage decisions. The objective is to minimize the fixed cost (i.e., cost of establishing the MF fleet and traveling inconvenience cost) plus the maximum of a risk measure (expectation or mean CVaR) of the operational cost (i.e., transportation and unsatisfied demand costs) over all possible distributions of the demand defined by an ambiguity set. In the first model (MAD-DRO), we use an ambiguity set based on the demand's mean, support, and mean absolute deviation (MAD). In the second model (W-DRO), we use an ambiguity set that incorporates all distributions within a 1-Wasserstein distance from a reference distribution. To the best of our knowledge, and according to our literature review in Section~\ref{Sec:LitRev}, our paper is the first to address the distributional ambiguity of the demand in the MFRSP using DRO.
\item \textbf{Solution Methods.} We derive equivalent solvable reformulations of the proposed mini-max nonlinear DRO models. We propose a computationally efficient decomposition-based algorithm to solve the reformulations. In addition, we derive valid lower bound inequalities that efficiently strengthen the master problem in the decomposition algorithm, thus improving convergence.
\item \textbf{Symmetry-Breaking Constraints.} We derive two families of new symmetry breaking constraints, which break symmetries in the solution space of the first-stage routing and scheduling decisions and thus improve the solvability of the proposed models. These constraints are independent of the method of modeling uncertainty. Hence, they are valid for any (deterministic and stochastic) formulation that employ the first-stage decisions of the MFRSP. Our paper is the first to attempt to break the symmetry in the solution space of these planning decisions of the MFRSP.
\item \textbf{Computational Insights.} We conduct extensive computational experiments comparing the proposed DRO models and a classical SP model empirically and theoretically, demonstrating where significant performance improvements can be gained. Specifically, our results show (1) how the DRO approaches have superior operational performance in terms of satisfying customers demand as compared to the SP approach; (2) the MAD-DRO model is more computationally efficient than the W-DRO model; (3) the MAD-DRO model yield more conservative decisions than the W-DRO model, which often have a higher fixed cost but significantly lower operational cost; (4) how mobile facilities can move from one location to another to accommodate the change in demand over time and location; (5) efficiency of the proposed symmetry breaking constraints and lower bound inequalities; (6) the trade-off between cost, number of MFs, MF capacity, and operational performance; and (7) the trade-off between the risk-neutral and risk-averse approaches. Most importantly, our results show the value of modeling uncertainty and distributional ambiguity.
\end{enumerate}
\subsection{Structure of the paper}
\color{black}
The remainder of the paper is structured as follows. In Section~\ref{Sec:LitRev}, we review the relevant literature. Section~\ref{sec3:DMFRSformulation} details our problem setting. In Section~\ref{sec:SP}, we present our SP. In Section~\ref{sec:DRO_Models}, we present and analyze our proposed DRO models. In Section~\ref{sec:solutionmethods}, we present our decomposition algorithm and strategies to improve convergence. In Section~\ref{sec:computational}, we present our numerical experiments and corresponding insights. Finally, we draw conclusions and discuss future directions in Section~\ref{Sec:Conclusion}.
\color{black}
\section{Relevant Literature}\label{Sec:LitRev}
\noindent \textcolor{black}{In this section, we review recent literature that is most relevant to our work, mainly studies that propose stochastic optimization approaches for closely related problems to the MFRSP.} There is limited literature on MF as compared to stationary facilities. However, as pointed out by \cite{lei2014multicut}, the MFRSP share some features with several well-studied problems, including Dynamic Facility Location Problem (DFLP), Vehicle Routing Problem (VRP), and the Covering Tour Problem (CTP). First, let us briefly discuss the similarities and differences between the MFRSP and these problems. Given that we consider making decisions over a planning period, then the MFRSP is somewhat similar to DFLP, which seeks to locate/re-locate facilities over a planning horizon. To mitigate the impact of demand fluctuation along the planning period, decision-makers may open new facilities and close or relocate existing facilities at a relocation cost (\cite{albareda2009multi, antunes2009location, contreras2011dynamic, drezner1991facility,jena2015dynamic, jena2017lagrangian, van1982dual}). Most DFLPs assume that the relocation time is relatively short as compared to the planning horizon. In contrast, the MFRSP takes into account the relocation time of MFs. In addition, each MF needs to follow a specific route during the entire planning horizon, which is not a requirement in DFLP.
In CTP, one seeks to select a subset of nodes to visit that can cover other nodes within a particular coverage (\cite{current1985maximum, flores2017multi, gendreau1997covering, hachicha2000heuristics, tricoire2012bi}). In contrast to the MFRSP, CTP does not consider the variations of demand over time and assumes that the amount of demand to be met by vehicles is not related to the length of time the MF is spending at the stop. The VRP is one of the most extensively studied problem in operations research. The VRP also has numerous applications and variants \citep{subramanyam2020robust}. Both the MFRSP and the VRP consider the routing decisions of vehicles. However, the MFRSP is different than the VRP in the following ways \citep{lei2014multicut}. First, in the MFRSP, we can meet customer demand by a nearby MF (e.g., cellular stations). In the VRP, vehicles visit customers to meet their demand. Second, the amount of demand that an MF can serve at each location depends on the duration of the MF stay, which is a decision variable. In contrast, VRPs often assume a fixed service time. Finally, most VRPs require that each customer has to be visited exactly once in each route. In contrast, in the MFRSP, some customers may not be visited, and some may be visited multiple times.
Next, we review studies that proposed stochastic optimization approaches to problems similar to the MFRSP. \cite{halper2011mobile} introduced the concept of MF and proposed a continuous-time formulation to model the maximum covering mobile facility routing problem under deterministic settings. To solve their model, \cite{halper2011mobile} proposed several computationally effective heuristics. \cite{lei2014multicut} and \cite{lei2016two} are two closely related (and only) papers that proposed stochastic optimization approaches for MF routing and scheduling.
\cite{lei2014multicut} assumed that the distribution of the demand is known and accordingly proposed the first a \textit{priori} two-stage SP for MFRSP. \cite{lei2014multicut}'s SP seeks optimal first-stage routing and scheduling decisions to minimize the total expected system-wide cost, where the expectation is taken with respect to the known distribution of the demand. A priori optimization has a managerial advantage since it guarantees the regularity of service, which is beneficial for both customer and service provider. That is, a prior plan allow the customers to know when and where to receive service and enable MF service providers to be familiar with routes and better manage their time schedule during the day. The applicability of the SP approach is limited to the case in which the distribution of the demand is fully known, or we have sufficient data to model it.
Robust optimization (RO) and distributionally robust optimization (DRO) are alternative techniques to model, analyze and optimize decisions under uncertainty and ambiguity (where the underlying distributions are unknown). RO assumes that the uncertain parameters can take any value from a pre-specified uncertainty set of possible outcomes with some structure \citep{bertsimas2004price, ben2015deriving, soyster1973convex}. In RO, optimization is based on the worst-case scenario within the uncertainty set.
\textcolor{black}{Notably, \cite{lei2016two} are the first to motivate the importance of handling demand uncertainty using RO. They argue that RO is useful because it only requires moderate information about the uncertain demand rather than a detailed description of the probability distribution or a large data set. Specifically, \cite{lei2016two} proposed the first two-stage RO approach for MF feet sizing and routing problem with demand uncertainty. \cite{lei2016two}'s model aims to find the fleet size and routing decisions that minimize the fixed cost of establishing the MF fleet (first-stage) and a penalty cost for the unmet demand (second-stage).} Optimization in \cite{lei2016two}'s RO model is based on the worst-case scenario of the demand occurring within a polyhedral uncertainty set. By focusing the optimization on the worst-case scenario, RO may lead to overly conservative and suboptimal decisions for other more-likely scenarios \citep{chen2019robust, delage2018value}.
DRO models the uncertain parameters as random variables whose underlying probability distribution can be any distribution within a pre-defined ambiguity set. The ambiguity set is a family of all possible distributions characterized by some known properties of random parameters \citep{esfahani2018data}. In DRO, optimization is based on the worst-case distribution within this set. DRO is an attractive approach to model uncertainty with ambiguous distributions because: (1) it alleviates the unrealistic assumption of the decision-makers' complete knowledge of the distribution governing the uncertain parameters, (2) it is usually more computationally tractable than its SP and RO counterparts \citep{delage2018value, rahimian2019distributionally}, and (3) one can use minimal distributional information or a small sample to construct the ambiguity set and then build DRO models. \cite{rahimian2019distributionally} provide a comprehensive survey of the DRO literature.
\color{black}
The computational tractability of DRO models depends on the ambiguity sets. These sets are often based on moment information \citep{delage2010distributionally, mehrotra2014models,zhang2018ambiguous} or statistical measures such as the Wasserstein distance \citep{esfahani2018data}. To derive tractable DRO models for the MFRSP, we construct two ambiguity sets of demand, one based on 1-Wasserstein distance and one using the demand's support, mean, and mean absolute deviation (MAD). As mentioned in the introduction, we use the Wasserstein ambiguity because it is richer than other divergence-based ambiguity sets. We use the MAD as a dispersion measure instead of the variance because it allows tractable reformulation and better captures outliers and small deviations. In addition, the MAD exists for some distributions while the second moment does not \citep{ben1985approximation, postek2018robust}. We refer to \cite{postek2018robust} and references therein for rigorous discussions on properties of MAD.
\color{black}
\textcolor{black}{Next, we discuss some relevant results on the mean-support-MAD ambiguity set (henceforth denoted as MAD ambiguity). \cite{ben1972more} derived tight upper and lower bounds on the expectation of a general convex function of a random variable under MAD ambiguity. In particular, when the random variable is one-dimensional, \cite{ben1972more} show that the worst-case distribution under MAD ambiguity is a three-point distribution on the mean, support, and MAD. Recently, \cite{postek2018robust} used the results of \cite{ben1972more} to treat ambiguous expected feasibility constraints to obtain exact reformulations for both functions that are convex and concave in the components of the random variable under MAD ambiguity. These reformulations require independence of the random variables and involve an exponential number of terms. However, for the special case of linearly aggregated random variables, \cite{postek2018robust} derived polynomial-sized upper bounds on the worst-case expectations of convex functions. Finally, under the assumption of independent random variables, they derived tractable approximations of ambiguous chance constraints under mean, support, and MAD information.}
\color{black}
A reviewer of this paper brought our attention to the results in a working paper by \cite{longsupermodularity} on supermodularity in two-stage DRO problems. Specifically, \cite{longsupermodularity} identified a tractable class of two-stage DRO problems based on the scenario-based ambiguity set proposed by \cite{chen2019robust}. They showed that any two-stage DRO problem with mean, support, and upper bounds of MAD has a computationally tractable reformulation whenever the second-stage cost function is supermodular in the random parameter. Furthermore, they proposed an algorithm to compute the worst-case distribution for this reformulation. They argued that using the computed worst-case distribution in the reformulation can make the two-stage DRO problem tractable. In addition, they provided a necessary and sufficient condition to check whether any given two-stage optimization problem has the property of supermodularity. \textcolor{black}{In Appendix~\ref{Appx:Points_MAD}, we show that even if our recourse is supermodular in demand realization, the number of support points in the worst-case distribution of the demand is large, which renders our two-stage MAD-DRO model computationally challenging to solve using \cite{longsupermodularity}'s approach. In contrast, we can efficiently solve an equivalent reformulation of our MAD-DRO model using our proposed decomposition algorithm.}
\color{black}
\begin{figure}[t!]
\centering
\includegraphics[width=\textwidth, height=60mm]{LitTable2}
\caption{Comparison between \cite{lei2014multicut}, \cite{lei2016two}, and our paper. }\label{Fig:Compare}
\end{figure}
Despite the potential advantages, there are no DRO approaches for the specific MFRSP that we study in this paper. Therefore, our paper is the first to propose and analyze DRO approaches for the MFRSP. In Figure~\ref{Fig:Compare}, we provide a comparison between \cite{lei2014multicut}, \cite{lei2016two}, and our approach based on the assumption made on uncertainty distribution, proposed stochastic optimization approach, decision variables, objectives, and addressing symmetry. We note that our paper and these papers share the common goal of deriving generic optimization models that can be used in any application of MF where one needs to determine the same sets of decisions under the same criteria/objective considered in each paper.
We make the following observations from Figure~\ref{Fig:Compare}. In contrast to \cite{lei2016two}, we additionally incorporate the MF traveling inconvenience cost in the first-stage objective and the random transportation cost in the second-stage objective. In contrast to \cite{lei2016two} and \cite{lei2014multicut}, we model both uncertainty and distributional ambiguity and optimize the system performance over all demand distributions residing within the ambiguity sets. Our master and sub-problems and lower bound inequalities have a different structure than those of \cite{lei2016two} due to the differences in the decision variables and objectives. We also propose two families of symmetry-breaking constraints, which break symmetries in the solution space of the routing and scheduling decisions. These constraints can improve the solvability of any formulation that uses the same routing and scheduling decisions of the MFRSP. \cite{lei2014multicut} and \cite{lei2016two} did not address the issue of symmetry in the MFRSP. Finally, to model decision makers' risk-averse attitudes, we propose both risk-neutral (expectation) and risk-averse (mean-CVaR-based) DRO models for the MFRSP. \cite{lei2014multicut} and \cite{lei2016two} models are risk-neutral.
Finally, it is worth mentioning that our work uses similar reformulation techniques in recent DRO static FL literature (see, e.g., \cite{basciftci2019distributionally, luo2018distributionally, saif2020data, ShehadehSanci, shehadeh2020distributionallyTucker, tsang2021distributionally, wang2020distributionally,wang2021two, wu2015approximation} and references therein).
\section{\textcolor{black}{Problem Setting}}\label{sec3:DMFRSformulation}
\noindent As in \cite{lei2014multicut}, we consider a fleet of $M$ mobile facilities and define the MFRSP on a directed network $G(V, E)$ with node set $V:=\lbrace v_1, \ldots, v_n\rbrace$ and edge set $E:=\lbrace e_1, \ldots, e_m \rbrace$. The sets $I \subseteq V$ and $J \subseteq V$ are the set of all customer points and the subset of nodes where MFs can be located, respectively. The distance matrix $D=(d_{i,j})$ is defined on $E$ and satisfies the triangle inequality, where $d_{i,j}$ is a deterministic and time-invafriant distance between any pair of nodes $i$ and $j$. For simplicity in modeling, we consider a planning horizon of $T$ identical time periods, and we assume that the length of each period $t\in T$ is sufficiently short such that, without loss of generality, all input parameters are the same from one time period to another (this is the same assumption made in \cite{lei2014multicut} and \cite{lei2016two}). The demand, $W_{i,t}$, of each customer $i$ in each time period $t$ is random. The probability distribution of the demand is unknown, and only a possibly small data on the demand may be available. We assume that we know the mean $\pmb{\mu}:=[\mu_{1,1}, \ldots, \mu_{|I|, |T|}]^\top$ and range [$\pmb{\underline{W}}$, $\pmb{\overline{W}}$] of $\pmb{W}$. Mathematically, we make the following assumption on the support of $\pmb{W}$.
\noindent \textbf{Assumption 1.} The support set $\mathcal{S}$ of $\pmb{W}$ in \eqref{support} is nonempty, convex, and compact.
\begin{align} \label{support}
& \mathcal{S}:=\left\{ \pmb{W} \geq 0: \begin{array}{l} \underline{W}_{i,t} \leq W_{i,t} \leq \overline{W}_{i,t}, \ \forall i \in I, \ \forall t \in T \end{array} \right\}.
\end{align}
\noindent We consider the following basic features as in \cite{lei2014multicut}: (1) each MF has all the necessary service equipment and can move from one place to another; (2) all MFs are homogeneous, providing the same service, and traveling at the same speed; (3) we explicitly account for the travel time of the MF in the model, and service time are only incurred when the MF is not in motion; (4) the travel time $t_{j,j^\prime}$ from location $j$ to $j^\prime$ is an integer multiplier of a single time period (\cite{lei2014multicut, lei2016two}; and (5) the amount of demand to be served is proportional to the duration of the service time at the location serving the demand.
We consider a cost $f$ for using an MF, which represents the expenses associated with purchasing or renting an MF, staffing cost, equipment, etc. Each MF has a capacity limit $C$, which represents the amount of demand that an MF can serve in a single time unit. Due to the random fluctuations of the demand and the limited capacity of each MF, there is a possibility that the MF fleet will fail to satisfy customers' demand fully. To minimize shortage, we consider a penalty cost $\gamma$ for each unit of unmet demand. This penalty cost can represent the opportunity cost for the loss of demand or expense for outsourcing the excess demand to other companies \citep{basciftci2019distributionally, lei2016two}. Thus, maximizing demand satisfaction is an important objective that we incorporate in our model \citep{lei2014multicut}.
Given that an MF cannot provide service when in motion, it is not desirable to keep it moving for a long time to avoid losing potential benefits. On the other hand, it is not desirable to keep the MF stationary all the time because this may lead to losing the potential benefits of making a strategic move to locations with higher demand. Thus, the trade-off of the problem includes the decision to move or keep the MF stationery. Accordingly, we consider a traveling inconvenience cost $\alpha$ to discourage unnecessary moving in cases where moving would neither improve nor degrade the total performance. As in \cite{lei2014multicut}, we assume that $\alpha$ is much lower than other costs such that its impact over the major trade-off is negligible.
We assume that the quality of service a customer receives from a mobile facility is inversely proportional to the distance between the two to account for the ``access cost'' (this assumption is common in practice and in the literature, see, e.g., \cite{ahmadi2017survey, reilly1931law, drezner2014review, lei2016two, berman2003locating,lei2014multicut}). Accordingly, we consider a demand assignment cost that is linearly proportional to the distance between the customer point and the location of an MF, i.e., $\beta d_{i,j}$, where $\beta\geq 0$ represents the assignment cost factor per demand unit and per distance unit. Table~\ref{table:notation} summarizes these notation.
Given a set of MFs, $M$, $T$, $I$, and $J$, our models aim to find: (1) the number of MFs to use within $T$; (2) a routing plan and a schedule for the selected MFs, i.e., the node that each MF is located at in each time period; and (3) assignment of MFs to customers. Decisions (1)--(2) are first-stage decisions that we make before realizing the demand. The assignment decisions (3) represent the recourse (second-stage) actions in response to the first-stage decisions \textcolor{black}{and demand realizations} (i.e., you cannot assign demand to the MFs before realizing the demand). \textcolor{black}{The objective is to minimize the fixed cost (i.e., cost of establishing the MF fleet and traveling inconvenience cost) plus a risk measure (expectation or mean CVaR) of the operational cost (i.e., transportation and unsatisfied demand costs).} We refer to \cite{lei2014multicut} for an excellent visual representation of MF operations and some of the basic features mentioned above.
\noindent \textbf{\textit{Additional Notation}}: For $a,b \in \mathbb{Z}$, we define $[a]:=\lbrace1,2,\ldots, a \rbrace$ as the set of positive integer running indices to $a$. Similarly, we define $[a,b]_\mathbb{Z}:=\lbrace c \in \mathbb{Z}: a \leq c \leq b \rbrace$ as the set of positive running indices from $a$ to $b$. The abbreviations ``w.l.o.g.'' and ``w.l.o.o.'' respectively represent ``without loss of generality'' and ``without loss of optimality.''
\section{\textbf{Stochastic Programming Model}}\label{sec:SP}
\noindent \textcolor{black}{In this section, we present a two-stage SP formulation of the MFRSP that assumes that the probability distribution of the demand is known. A complete listing of the parameters and decision variables of the model can be found in Table~\ref{table:notation}.}
\color{black}
First, let us introduce the variables and constraints defining the first-stage of this SP model. For each $m \in M$, we define a binary decision variable $y_m$ that equals 1 if MF $m$ is used, and is 0 otherwise. For all $j \in J$, $m \in M$, and $t \in T$, we define a binary decision variable $x_{j,m}^t$ that equals 1 if MF $m$ stays at location $j$ at period $t$, and is 0 otherwise. The feasible region $\mathcal{X}$ of variables $\pmb{x}$ and $\pmb{y}$ is defined in \eqref{eq:RegionX}.
\color{black}
\begin{align}
\mathcal{X}&=\left\{ (\pmb{x}, \pmb{y}) : \begin{array}{l} x_{j,m}^t+x_{j^\prime,m}^{t^\prime} \leq y_m, \ \ \forall t, m, j, \ j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace \\
x_{j,m}^t \in \lbrace 0, 1\rbrace, \ y_m \in \lbrace 0, 1\rbrace, \ \forall j, m , t \end{array} \right\}. \label{eq:RegionX}
\end{align}
As defined in \cite{lei2014multicut}, region $\mathcal{X}$ represents: (1) the requirement that an MF can only be in service when it is stationary; (2) MF $m$ at location $j$ in period $t$ can only be available at location $j^\prime\neq j$ after a certain period of time depending on the time it takes to travel from location $j$ to $j^\prime$, $t_{j,j^\prime}$ (i.e., $x_{j,m}^{t^\prime}=0$ for all $j^\prime \neq j$ and $t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace$); and (3) MF $m$ has to be in an active condition before providing service. We refer the reader to Appendix~\ref{Appx:FirstStageDec} for a detailed derivation of region $\mathcal{X}$.
\color{black}
Now, let us introduce the variables defining our second-stage problem. For all $(i,\ j, \ m, \ t)$, we define a nonnegative continuous variable $z_{i,j,m}^t$ to represent the amount of node $i$ demand served by MF $m$ located at $j$ in period $t$. For each $t \in T$, we define a nonnegative continuous variable $u_{i,t}$ to represent the amount of unmet demand of node \textit{i} in period $t$. Finally, we define a random vector $\pmb{W}:=[W_{1,1}. \ldots, W_{|I|,|T|}]^\top$. Our SP model can now be stated as follows:
\color{black}
\begin{table}[t]
\small
\center
\renewcommand{\arraystretch}{0.6}
\caption{Notation.}
\begin{tabular}{ll}
\hline
\multicolumn{2}{l}{\textbf{Indices}} \\
$m$& index of MF, $m=1,\ldots, M$\\
$i$ & index of customer location, $i=1,\ldots,I$\\
$j$ & index of MF location, $j=1,...,J$\\
\multicolumn{2}{l}{\textbf{Parameters and sets}} \\
$T$& planning horizon\\
$M$ & number, or set, of MFs \\
$J$& number, or, set of locations \\
$f$ & fixed operating cost \\
$d_{i,j}$ & distance between any pair of nodes $i$ and $j$ \\
$t_{j,j'}$ & travel time from $j$ to $j'$\\
$C$ & the amount of demand that can be served by an MF in a single time unit, i.e., MF capacity \\
$W_{i,t}$& demand at customer site $i$ for each period $t$\\
$\underline{W}_{i,t}/\overline{W}_{i,t}$ & lower/upper bound of demand at customer location $i$ for each period $t$ \\
$\gamma$ & penalty cost for each unit of unmet demand\\
\multicolumn{2}{l}{\textbf{First-stage decision variables } } \\
$y_{m}$ & $\left\{\begin{array}{ll}
1, & \mbox{if MF \textit{m} is permitted to use,} \\
0, & \mbox{otherwise.}
\end{array}\right.$ \\
$x_{jm}^t$ & $\left\{\begin{array}{ll}
1, & \mbox{if MF \textit{m} stays at location \textit{j} at period \textit{t},} \\
0, & \mbox{otherwise.}
\end{array}\right.$ \\
\multicolumn{2}{l}{\textbf{Second-stage decision variables } } \\
$z_{i,j,m}^t$ & amount of demand of node $i$ being served by MF $m$ located at $j$ in period $t$ \\
$u_{i,t}$ & total amount of unmet demand of node \textit{i} in period $t$ \\
\hline
\end{tabular}\label{table:notation}
\end{table}
\begin{subequations}\label{DMFRS_SP}
\begin{align}
(\text{SP}) \quad Z^*=& \min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} \left\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \textcolor{black}{\varrho
\big(Q(\pmb{x}, \pmb{W})\big)} \ \right\rbrace, \label{ObjSP}
\end{align}
\end{subequations}
where for a feasible $(\pmb{x}, \pmb{y}) \in \mathcal{X}$ and a realization of $\pmb{W}$
\allowdisplaybreaks
\begin{subequations}\label{2ndstage}
\begin{align}
Q(\pmb{x}, {\pmb{W}} ):= & \min_{\pmb{z, u}} \Big (\sum_{j \in J} \sum_{i \in I} \sum_{m \in M} \sum_{t \in T} \beta d_{i,j} z_{i,j,m}^t+ \gamma \sum_{t \in T} \sum_{i \in I} u_{i,t}\Big)\label{2ndstageObj}\\
& \ \ \text{s.t.} \ \ \ \ \sum_{j \in J} \sum_{m \in M} z_{i,j,m}^t+u_{i,t}= W_{i,t}, \qquad \forall i \in I, \ t \in T, \label{2ndstage_const1}\\
&\quad \ \quad \quad \sum_{i \in I} z_{i,j,m}^t \leq C x_{j,m}^t, \qquad \forall j \in J, \ m \in M, \ t \in T, \label{2ndstage_const2}\\
&\quad \ \quad \quad u_t \geq 0, \ z_{i,j,m}^t \geq 0, \qquad \forall i \in I, \ j \in J, m \in M, \ t \in T. \label{2ndstage_const3}
\end{align}
\end{subequations}
\noindent \textcolor{black}{Formulation \eqref{DMFRS_SP} aims to find first-stage decisions $(\pmb{x},\pmb{y}) \in \mathcal{X}$ that minimize the sum of (1) the fixed cost of establishing the MF fleet (first term); (2) the traveling inconvenience cost\footnote{Minimizing the traveling inconvenience cost is equivalent to maximizing the profit of keeping the MF stationary whenever possible. Parameter $\alpha$ is the profit weight factor as detailed in \cite{lei2014multicut})} (second term); and (3) a risk measure $\varrho (\cdot) $ of the random second-stage function $Q(\pmb{x}, {\pmb{W}} )$ (third term). A risk-neutral decision-maker may opt to set $\varrho(\cdot)=\mathbb{E}(\cdot)$, whereas a risk-averse decision-maker might set $\varrho(\cdot)$ as the CVaR or mean-CVaR. Classically, the MFRSP literature
employs $\varrho(\cdot)=\mathbb{E}(\cdot)$, which might be more intuitive for MF providers. For brevity, we relegate further discussion of the mean-CVaR-based SP model to Appendix~\ref{Appex:SP_CVAR}.}
\section{\textcolor{black}{Distributionally Robust Optimization (DRO) Models}}\label{sec:DRO_Models}
\noindent In this section, we present our proposed DRO models for the MFRSP that do not assume that the probability distribution of the demand is known. In Sections~\ref{sec:MAD-DRO} and \ref{sec:WDMFRS_model}, we respectively present and analyze the risk-neutral MAD-DRO and W-DRO models. For brevity, we relegate the formulations and discussions of the risk-averse mean-CVaR-based DRO models to Appendix~\ref{Sec:meanCVAR}.
\subsection{\textbf{The DRO Model with MAD Ambiguity (MAD-DRO)}}\label{sec:MAD-DRO}
\noindent \textcolor{black}{In this section, we present our proposed MAD-DRO model, which is based on an ambiguity set that incorporates the demand's mean ($\pmb{\mu}$), MAD ($\pmb{\eta}$), and support ($\mathcal{S}$)}. As mentioned earlier, we use the MAD as a dispersion or variability measure because it allows us to derive a computationally attractive reformulation \citep{postek2018robust, wang2019distributionally, wang2020distributionally}.
\textcolor{black}{First, let us introduce some additional sets and notation defining our MAD ambiguity set and MAD-DRO model}. We define $\mathbb{E}_\mathbb{P}$ as the expectation under distribution $\mathbb{P}$. We let $\mu_{i,t}=\mathbb{E}_\mathbb{P}[W_{i,t}]$ and $\eta_{i,t}=\mathbb{E}_{\mathbb{P}}(|W_{i,t}-\mu_{i,t}|)$ respectively represent the mean value and MAD of $W_{i,t}$, for all $i \in I$ and $t \in T$. \textcolor{black}{Using this notation and the support $\mathcal{S}$ defined in \eqref{support}, we construct the following MAD ambiguity set}:
\begin{align}\label{eq:ambiguityMAD}
\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta}) := \left\{ \mathbb{P}\in \mathcal{P}(\mathcal{S}) \middle|\: \begin{array}{l} \mathbb{P} (\pmb{W} \in \mathcal{S})=1,\\ \mathbb{E_P}[W_{i,t}] = \mu_{i,t}, \ \forall i \in I, \ t \in T, \\
\mathbb{E}_{\mathbb{P}}(|W_{i,t}-\mu_{i,t}|) \leq \eta_{i,t}, \ \forall i \in I, \ t \in T.
\end{array} \right\},
\end{align}
\noindent where $\mathcal{P}(S)$ in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ represents the set of distributions supported on $\mathcal{S}$ with mean $\pmb \mu$ and dispersion measure $ \leq \pmb \eta$. Using $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ defined in \eqref{eq:ambiguityMAD}, we formulate our MAD-DRO model as
\begin{align}
(\text{MAD-DRO}) & \min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} \left\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+\ \Bigg[\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P}[Q(\pmb{x},{\pmb{W}})] \Bigg] \ \right\rbrace. \label{MAD-DMFRS}
\end{align}
\noindent The MAD-DRO formulation in \eqref{MAD-DMFRS} seeks first-stage decisions ($\pmb{x},\pmb{y})$ that minimize the first-stage cost and the worst-case expectation of the second-stage (recourse) cost, \textcolor{black}{where the expectation is taken over all distributions
residing in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$}. Note that we do not incorporate higher moments (e.g., co-variance) in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ for the following primary reasons. First, the mean and range are intuitive statistics that a decision-maker may approximate and change in the model (e.g., the mean may be estimated from limited data or approximated by subject matter experts, and the range may represent the error margin in the estimates). Second, it is not straightforward for decision-makers to approximate or accurately estimate the correlation between uncertain parameters. Third, mathematically speaking, various studies have demonstrated that incorporating higher moments in the ambiguity set often undermines the computational tractability of DRO models and, therefore, their applicability in practice. Indeed, as we will show next, using $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ allows us to derive a tractable equivalent reformulation of the MAD-DRO model and an efficient solution method to solve the reformulation (see Sections~\ref{sec3:reform}, \ref{sec:Alg}, and \ref{sec5:CPU}).
\textcolor{black}{Finally, note that parameters $W_{i,t}$, $\mu_{i,t}$, $\underline{W}_{i,t}$, $\overline{W}_{i,t}$, $\eta_{i,t}$ are all indexed by time period $t$ and location $i$. Thus, if in any application, there is a relationship (e.g., correlation) between the demand of a subset of locations in a subset of periods, one can easily adjust $\mathcal{S}$, $\pmb{\mu}$, and $\pmb{\eta}$ in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ to reflect this relationship. For example, if urban cities have higher demand, then we can adjust the mean and range of the demand of these cities to reflect such a relationship. Similarly, if there is a correlation between the time period $t$ and the demand, then we can define the mean and the support based on this correlation. For example, the morning service hours may have lower demand on Monday. In this case, we can adjust $\pmb{\mu}$, $\mathcal{S}$, and $\pmb{\eta}$ to reflect this relationship. Similarly, if the demand in a subset of periods and locations is correlated, we can adjust $\mathcal{S}$, $\pmb{\mu}$, and $\pmb{\eta}$ in $\mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})$ to reflect this relationship}. Nevertheless, we acknowledge that not incorporating higher moments and complex relations may be a limitation of our work and thus is worth future investigation.
\subsubsection{\textbf{Reformulation of the MAD-DRO model}}\label{sec3:reform}
\textcolor{white}{ }
\textcolor{black}{Recall that $Q(\pmb{x},{\pmb{W}})$ is defined by a minimization problem; hence, in \eqref{MAD-DMFRS}, we have an inner max-min problem. As such, it is not straightforward to solve formulation \eqref{MAD-DMFRS} in its presented form}. In this section, we derive an equivalent formulation of \eqref{MAD-DMFRS} that is solvable. \textcolor{black}{First, in Proposition~\ref{Prop1:DualMinMax}, we present an equivalent reformulation of the inner problem in \eqref{MAD-DMFRS} (see Appendix \ref{Appx:ProofofProp1} for a proof).}
\begin{proposition}\label{Prop1:DualMinMax}
For any fixed $(\pmb{x}, \pmb{y}) \in \mathcal{X}$, problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P}[Q(\pmb{x},{\pmb{W}})]$ in \eqref{MAD-DMFRS} is equivalent to
\begin{align} \label{eq:FinalDualInnerMax-1}
& \min_{\pmb{\rho}, \pmb{\psi} \geq 0} \ \left \lbrace \sum \limits_{t \in T} \sum \limits_{i \in I} ( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+\max \limits_{\pmb{W}\in \mathcal{S}} \Big\lbrace Q(\pmb{x},{\pmb{W}}) + \sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})\Big \rbrace \right \rbrace.
\end{align}
\end{proposition}
\noindent \textcolor{black}{Again, the problem in \eqref{eq:FinalDualInnerMax-1} involves an inner max-min problem that is not straightforward to solve in its presented form. However, we next derive an equivalent reformulation of the inner problem in \eqref{eq:FinalDualInnerMax-1} that is solvable.}
First, we observe that $Q(\pmb{x},{\pmb{W}})$ is a feasible linear program (LP) for a given first-stage solution $(\pmb{x}, \pmb{y}) \in \mathcal{X}$ and a realization of ${\pmb{W}}$. The dual of $Q(\pmb{x}, {\pmb{W}})$ is as follows.
\begin{subequations}\label{DualOfQ}
\begin{align}
Q(\pmb{x},{\pmb{W}})=& \max \limits_{\pmb{\lambda, v}} \ \sum_{t \in T} \sum_{i \in I} \lambda_{i,t} W_{i,t}+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t \label{Obj:Qdual}\\
& \ \text{s.t. } \ \lambda_{i,t}+ v_{j,m}^t \leq \beta d_{i,j}, \ \qquad \forall i \in I, j \in J, m \in M, t \in T, \label{Const1:QudalX}\\
& \qquad \ \ \lambda_{i,t} \leq \gamma, \qquad \qquad \qquad \ \ \forall i \in I, \ t \in T, \label{Const2:QdualU}\\
& \qquad \ \ v_{j,m}^t \leq 0, \qquad \qquad \qquad \ \ \forall j \in J, \ t \in T, \label{Const3:Qdualv}
\end{align}
\end{subequations}
\noindent where $\pmb{\lambda}$ and $\pmb{v}$ are the dual variables associated with constraints \eqref{2ndstage_const1} and \eqref{2ndstage_const2}, respectively. It is easy to see that w.l.o.o, $\lambda_{i,t} \geq 0$ for all $i \in I$ due to constraints \eqref{Const2:QdualU} and the objective of maximizing $W_{i,t}\geq 0$ times $\lambda_{i,t}$. Additionally, $ v_{j,m}^t \leq \min\{ \min_i\{\beta d_{i,j}-\lambda_{i,t}\},0 \}$ by constraints \eqref{Const1:QudalX} and \eqref{Const3:Qdualv}. Given the objective of maximizing a nonnegative term $Cx_{j,m}^t $ multiplied by $v_{j,m}^t$, $ v_{j,m}^t = \min\{ \min_i\{\beta d_{i,j}-\lambda_{i,t}\},0 \}$ in the optimal solution. Given that $\pmb{\beta}$, $\pmb d$, and $\pmb \lambda$ are finite, $\pmb{v} $ is finite. It follows that problem \eqref{DualOfQ} is a feasible and bounded LP. Note that $W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}]$ and ($\underline{W}_{i,t}, \overline{W}_{i,t})\geq 0$ by definition, in view of dual formulation \eqref{DualOfQ}, we can rewrite the inner maximization problem $\max \{ \cdot \} $ in \eqref{eq:FinalDualInnerMax-1} as
\begin{subequations}\label{DualQAndOuter}
\begin{align}
\max \limits_{\pmb{\lambda, v, W, k}} & \Bigg\{ \sum_{t \in T} \sum_{i \in I} \lambda_{i,t} W_{i,t}+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Bigg \} \\
\ \text{s.t. } &\eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \ \ W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}], \ \forall i \in I, \ \forall t \in T, \\
& k_{i,t} \geq W_{i,t}-\mu_{i,t}, \ k_{i,t} \geq \mu_{i,t}-W_{i,t}, \ \forall i \in I, \ \forall t \in T . \label{Const1:DualQAndOuter}
\end{align}
\end{subequations}
Note that the objective function in \eqref{DualQAndOuter} contains the interaction term $\lambda_{i,t} W_{i,t}$. To linearize formulation \eqref{DualQAndOuter}, we define $\pi_{i,t}=\lambda_{i,t} W_{i,t}$ for all $i \in I$ and $t \in T$. Also, we introduce the following McCormick inequalities for variables $\pi_{i,t}$:
\begin{subequations}
\begin{align}
&\pi_{i,t} \geq \lambda_{i,t} \underline{W}_{i,t}, \qquad \qquad \qquad \ \ \ \pi_{i,t} \leq \lambda_{i,t} \overline{W}_{i,t}, \ \ \forall i \in I, \ \forall t \in T, \label{McCormick1}\\
& \pi_{i,t} \geq \gamma W_{i,t}+ \overline{W}_{i,t} (\lambda_{i,t}- \gamma ), \qquad \pi_{i,t} \leq \gamma W_{i,t}+\underline{W}_{i,t}( \lambda_{i,t}-\gamma), \ \forall i \in I, \ \forall t \in T. \label{McCormick2}
\end{align}
\end{subequations}
Accordingly, for a fixed $(\pmb{x} \in \mathcal{X}, \pmb{\rho}, \pmb{\psi})$, problem \eqref{DualQAndOuter} is equivalent to the following mixed-integer linear program (MILP):
\begin{subequations}\label{eq:FinalDualInnerMax-2}
\begin{align}
\max \limits_{\pmb{\lambda, v, W, \pi, k} }& \ \Bigg\{ \sum_{t \in T} \sum_{i \in I} \pi_{i,t}+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Bigg\}\\
\text{s.t. } & \ (\pmb{\lambda, v}) \in \{\eqref{Const1:QudalX}-\eqref{Const3:Qdualv}\}, \ \pmb{\pi} \in \{ \eqref{McCormick1}-\eqref{McCormick2} \}, \label{Const1_finalinner} \\
& \ W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}], \ k_{i,t} \geq W_{i,t}-\mu_{i,t}, \ k_{i,t} \geq \mu_{i,t}-W_{i,t}, \ \forall i \in I, \ \forall t \in T. \label{Const2_finalinner}
\end{align}
\end{subequations}
Combining the inner problem in the form of \eqref{eq:FinalDualInnerMax-2} with the outer minimization problems in \eqref{eq:FinalDualInnerMax-1} and \eqref{MAD-DMFRS}, we derive the following equivalent reformulation of the MAD-DRO model in \eqref{MAD-DMFRS}:
\begin{subequations}\label{FinalDR}
\begin{align}
&\min_{\pmb{x,y,\rho, \psi, \delta}} \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big[\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big] + \delta \Bigg \} \\
& \ \ \text{s. t.} \ \ (\pmb{x}, \pmb{y}) \in \mathcal{X}, \ \psi \geq 0, \\
& \qquad \ \ \delta \geq \textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}, \label{const1:FinalDR}
\end{align}
\end{subequations}
where $\textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}=\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}): \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner} \Big\} $.
\begin{proposition}\label{Prop2:Convexh} \textcolor{black}{For any fixed values of variables $(\pmb{x}, \pmb{y}) \in \mathcal{X}$, $\pmb \rho,$ and $\pmb \psi$, $h(\pmb{x}, \pmb \rho, \pmb \psi)< +\infty$. Furthermore, function $(\pmb{x}, \pmb \rho, \pmb \psi) \mapsto\textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}$ is a convex piecewise linear function in $\pmb{x}$, $\pmb\rho$, and $\pmb \psi$ with a finite number of pieces (see Appendix \ref{Appx:Prop2} for a detailed proof).}
\end{proposition}
\vspace{1mm}
\subsection{\textit{\textbf{The DRO Model with 1-Wasserstein Ambiguity (W-DRO)}}}\label{sec:WDMFRS_model}
\noindent In this section, we consider the case that $\mathbb{P}$ may be observed via a possibly small finite set $\{\hat{\pmb{W}}^1, \ldots, \hat{\pmb{W}}^N\}$ of $N$ i.i.d. samples, which may come from the limited historical realizations of the demand or a reference empirical distribution. Accordingly, we construct an ambiguity set based on 1-Wasserstein distance, which often admits tractable reformulation in most real-life applications (see, e.g., \cite{Daniel2020, hanasusanto2018conic, jiang2019data, tsang2021distributionally, saif2020data}).
\textcolor{black}{First, let us define the 1-Wasserstein distance.} Suppose that random vectors $\pmb{\pmb{W}}_1$ and $\pmb{\pmb{W}}_2$ follow $\F_1$ and $\F_2$, respectively, where probability distributions $\F_1$ and $\F_2$ are defined over the common support $\mathcal{S}$. The 1-Wasserstein distance \textcolor{black}{$\text{dist}(\F_1, \F_2)$ between $\F_1$ and $\F_2$ is the minimum transportation cost of moving from $\F_1$ to $\F_2$, where the cost of moving masses $\pmb{W}_1$ to $\pmb{W}_2$ is the norm $||\pmb{W}_1-\pmb{W}_2||$}. \textcolor{black}{ Mathematically,}
\begin{equation}\label{W_distance}
\textcolor{black}{\text{dist}}(\mathbb{F}_1, \mathbb{F}_2):=\inf_{\Pi \in \mathcal{P}(\mathbb{F}_1, \mathbb{F}_2)} \Bigg \{\int_{\mathcal{S}} ||\pmb{W}_1-\pmb{W}_2|| \ \Pi(\text{d}\pmb{W}_1, \text{d}\pmb{W}_2) \Bigg | \begin{array}{ll} & \Pi \text{ is a joint distribution of }\pmb{W}_1 \text{ and } \pmb{W}_2\\
& \text{with marginals } \mathbb{F}_1 \text{ and } \mathbb{F}_2, \text{ respectively}
\end{array} \Bigg \},
\end{equation}
\noindent where $\mathcal{P}(\mathbb{F}_1, \mathbb{F}_2) $ is the set of all joint distributions of $(\pmb{W}_1, \pmb{W}_2)$ supported on $\mathcal{S}$ with marginals ($\mathbb{F}_1$, $\mathbb{F}_2$). Accordingly, we construct the following $1$-Wasserstein ambiguity set:
\begin{align}\label{W-ambiguity}
\mathcal{F} (\hat{\mathbb{P}}^N, \epsilon)= \left\{ \mathbb{P} \in \mathcal{P}(\mathcal{S}) \middle|\begin{array}{l} \textcolor{black}{\text{dist}}(\mathbb{P}, \hat{\mathbb{P}}^N) \leq \epsilon\end{array} \right\},
\end{align}
\noindent where $ \mathcal{P}(\mathcal{S})$ is the set of all distributions supported on $\mathcal{S}$, $\hat{\mathbb{P}}^N=\frac{1}{N} \sum_{n=1}^N \delta_{\hat \pmb{W}^n}$ is the empirical distribution of $\pmb{W}$ based on $N$ i.i.d samples, and $\epsilon >0$ is the radius of the ambiguity set. The set $\mathcal{F} (\hat{\mathbb{P}}^N, \epsilon)$ represents a Wasserstein ball of radius $\epsilon$ centered at the empirical distribution $\hat{\mathbb{P}}^N$. \textcolor{black}{Using the ambiguity set $\mathcal{F}(\hat{\mathbb{P}}^N, \epsilon)$ defined in \eqref{W-ambiguity}, we formulate our W-DRO model as}
\begin{align}
(\text{W-DRO}) \ \hat{Z}(N, \epsilon)&= \min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} \Bigg\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+\Bigg [\sup_{\mathbb{P} \in \mathcal{F} (\hat{\mathbb{P}}^N, \epsilon) }\mathbb{E}_{\mathbb{P}} [Q(\pmb{x}, \pmb{W}) ] \Bigg] \Bigg \}. \label{W-DMFRS}
\end{align}
\noindent Formulation \eqref{W-DMFRS} finds first-stage decisions ($\pmb{x},\pmb{y}) \in \mathcal{X}$ that minimize the first-stage cost and the maximum expectation of the second-stage cost over all distributions residing in $\mathcal{F} (\hat{\mathbb{P}}^N, \epsilon)$.
The W-DRO model in \eqref{W-DMFRS} can be used to model uncertainty in general and distributional ambiguity when there is a possibly small finite data sample on uncertainty. As detailed in \cite{esfahani2018data} and discussed earlier, if we have a small sample and we optimize using this sample, then the optimizer's curse cannot be avoided. To mitigate the optimizer's curse (estimation error), we robustify the nominal decision problem (the MFRSP optimization problem) against all distributions $\mathbb{P}$ under which the estimated distribution $\hat{\mathbb{P}}^N$ based on the $N$ data points has a small estimation error (i.e., with $ \textcolor{black}{\text{dist}}(\mathbb{P}, \hat{\mathbb{P}}^N) \leq \epsilon$). Therefore, in some sense one can think of Wasserstein ball as the set of all distributions under which our estimation error is below $\epsilon$, where $\epsilon$ is the maximum estimation error against which we seek protection. When $\epsilon=0$, the ambiguity set contains the empirical distribution and the W-DRO problem in \eqref{W-DMFRS} reduces to the SP problem. A larger radius $\epsilon$ indicates that we seek more robust solutions (see Appendix \ref{WDRO_Radius}).
In the next section, we show that using $\ell_1$-norm instead of the \textcolor{black}{$\ell_p$}-norm ($1<p<\infty$) in our Wasserstein ambiguity set allows us to derive a linear and tractable reformulation of the W-DRO model in \eqref{W-DMFRS}. Note that $\ell_1$-norm (i.e., the sum of the magnitudes of the vectors in space) is the most intuitive and natural way to measure the distance between vectors. In contrast, the \textcolor{black}{$\infty$-norm}-based Wasserstein ball is an extreme case. That is, the $\infty$-norm gives the largest magnitude among each element of a vector. Thus, from the perspective of the Wasserstein DRO framework, the \textcolor{black}{$\infty$-norm}-based distance metric only picks the most influential value to determine the closeness between data points \citep{chen2018robust}, which, in our case, may not be reasonable since every demand point plays a role. Deriving and comparing DRO models with different Wasserstein sets is out of the scope of this paper but is worth future investigation in more comprehensive MF optimization problems.
\subsubsection{\textbf{\textit{Reformulation of the W-DRO model}}}\label{sec:Wreformualtion}
\textcolor{white}{ }
\noindent In this section, we derive an equivalent solvable reformulation of the W-DRO model in \eqref{W-DMFRS}. First, in Proposition~\ref{Prop1} we present an equivalent dual formulation of the inner maximization problem $\sup\ [\cdot ]$ in \eqref{W-DMFRS} (see Appendix~\ref{Proof_Prop1} for a detailed proof).
\begin{proposition}\label{Prop1}
For for a fixed $(\pmb{x}, \pmb{y}) \in \mathcal{X}$, problem $\sup \limits_{\mathbb{P}\in \mathcal{F} (\hat{\mathbb{P}}^N, \epsilon) }\mathbb{E}_{\mathbb{P}} [Q(\pmb{x},\pmb{W}) ]$ in \eqref{W-DMFRS} is equivalent to
\begin{align}
& \inf_{\rho \geq 0} \Bigg\{ \epsilon \rho + \Big[ \frac{1}{N} \sum_{n=1}^N \sup_{\pmb{W} \in \mathcal{S}} \big \{ Q(\pmb{x}, \pmb{W})-\rho || \pmb{W} -\pmb{W}^n || \big \} \Big] \Bigg\}. \label{DualOfInner}
\end{align}
\end{proposition}
\noindent Formulation \eqref{DualOfInner} is potentially challenging to solve because it require solving $N$ non-convex optimization problems. Fortunately, given that the support of $\pmb{W}$ is rectangular and finite (see Assumption 1) and $Q(\pmb{x},\pmb{W})$ is feasible and bounded for every $\pmb{x}$ and $\pmb{W} $, we next recast these inner problems as LPs for each $\rho \geq 0$ and $\pmb{x} \in \mathcal{X}$. First, using the dual formulation of $Q(\pmb{x}, \pmb{W})$ in \eqref{DualOfQ}, we rewrite the inner problem $\sup \{ \cdot \} $ in \eqref{DualOfInner} for each $n$ as
\begin{subequations}\label{InnerInnerW}
\begin{align}
& \max \limits_{\pmb{\lambda}, \pmb{v}, \pmb{W}} \Big \{ \sum_{t \in T} \sum_{i \in I} \lambda_{i,t} W_{i,t}-\rho |W_{i,t}-W_{i,t}^n|+ \sum_{t \in T} \sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t \Big \}\\
& \text{s.t. } \ \ (\pmb{\lambda}, \pmb{v}) \in \{ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv} \}, \pmb{W} \in [\pmb{\underline{W}}, \pmb{\overline{W}} ].
\end{align}
\end{subequations}
Second, using the same techniques in Section~\ref{sec3:reform}, we define an epigraphical random variable $\eta_{i,t}^n$ for the term $ |W_{i,t}-W_{i,t}^n|$. Then, using variables $\pmb{\eta}$, $\pi_{i,t}=\lambda_{i,t} W_{i,t}$, and inequalities \eqref{McCormick1}-\eqref{McCormick2} for variables $\pi_{i,t}$, we derive the following equivalent reformulation of \eqref{InnerInnerW} (for each $n \in [N]$):
\begin{subequations}\label{InnerInnerW2}
\begin{align}
\max \limits_{\pmb{\lambda}, \pmb{v}, \pmb{W}, \pmb{\pi}, \pmb{\eta}} &\big \{ \sum_{t \in T} \sum_{i \in I} \pi_{i,t}-\rho \eta_{i,t}^n+ \sum_{t \in T}\sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t \big \} \label{Obj:InnerInnerW2}\\
\text{s.t. } & \ (\pmb{\lambda, v}) \in \{ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv} \}, \label{C1:InnerInnerW2} \\
& \ \ \pi_{i,t} \in \{ \eqref{McCormick1}- \eqref{McCormick1}\}, \ \ W_{i,t} \in [\underline{W}_{i,t}, \overline{W}_{i,t}] , && \forall i \in I , \forall t \in T, \label{C2:InnerInnerW2} \\
& \ \ \eta_{i,t}^n \geq W_{i,t}-W_{i,t}^n, \ \eta_{i,t}^n \geq W_{i,t}^n- W_{i,t}, && \forall i \in I , \forall t \in T. \label{C_InnerInner}
\end{align}
\end{subequations}
Third, combining the inner problem in the form of \eqref{InnerInnerW2} with the outer minimization problems in \eqref{DualOfInner} and \eqref{W-DMFRS}, we derive the following equivalent reformulation of the W-DRO model in \eqref{W-DMFRS}
\begin{align}
\hat{Z}(N, \epsilon) &= \min_{ (\pmb{x}, \pmb{y}) \in \mathcal{X}, \ \rho \geq 0 } \Bigg\lbrace \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \epsilon \rho \nonumber\\
& \quad \ \ \ \ +\frac{1}{N} \sum_{n=1}^N \max \limits_{\pmb{\lambda}, \pmb{v}, \pmb{W}, \pmb{\pi}, \pmb{\eta}} \Big \{ \sum_{t \in T} \sum_{i \in I} \pi_{i,t}- \rho \eta_{i,t}^n+ \sum_{t \in T}\sum_{j \in J} \sum_{m \in M} Cx_{j,m}^t v_{j,m}^t: \eqref{C1:InnerInnerW2}-\eqref{C_InnerInner} \Big \} \Bigg\}.\label{Final_W_DMFRS}
\end{align}
\noindent Using the same techniques in the proof of Proposition~\ref{Prop2:Convexh}, one can easily verify that function $[\max \limits_{\pmb{\lambda, v, \pi, W, \eta}} \big \{ \sum\limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}-\rho \eta_{i,t}^n+ \sum \limits_{t \in T}\sum\limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m}^t v_{j,m}^t \big\}]<\infty $ and is a convex piecewise linear function in $\pmb{x} \in \mathcal{X}$ and $\rho$.
\section{Solution Method}\label{sec:solutionmethods}
\noindent In this section, we present a decomposition-based algorithm to solve the MAD-DRO formulation in \eqref{FinalDR}, and strategies to improve the solvability of the formulation. The algorithmic steps for solving the W-DRO in \eqref{Final_W_DMFRS} are similar. In Section~\ref{sec:Alg}, we present our decomposition algorithm. In Section~\ref{sec:Improv}, we derive valid lower bound inequalities to strengthen the master problem in the decomposition algorithm. In Section~\ref{sec:symm}, we derive two families of symmetry breaking constraints that improve the solvability of the proposed models.
\subsection{\textbf{Decomposition Algorithm}}\label{sec:Alg}
\noindent Proposition~\ref{Prop2:Convexh} suggests that constraint \eqref{const1:FinalDR} describes the epigraph of a convex and piecewise linear function of decision variables in formulation \eqref{FinalDR}. Therefore, given the two-stage characteristics of MAD-DRO in \eqref{FinalDR}, it is natural to attempt to solve problem \eqref{FinalDR} via a separation-based decomposition algorithm. Algorithm~\ref{Alg1:CAG} presents our proposed decomposition algorithm, and the algorithm for the W-DRO model in \eqref{Final_W_DMFRS} has the same steps. Algorithm~\ref{Alg1:CAG} is finite because we identify a new piece of the function $\max \limits_{\pmb{\lambda, v, W, \pi, k}} \ \big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \big\}$ each time when the set $ \lbrace L (\pmb{x}, \delta)\geq 0 \rbrace$ is augmented in step 4, and the function has a finite number of pieces according to Proposition~\ref{Prop2:Convexh}. Note that this algorithm is based on the same theory and art of cutting plane-based decomposition algorithms employed in various other papers using decomposition to solve problems with similar structure. Nevertheless, we customized Algorithm~\ref{Alg1:CAG} to solve our proposed DRO models. In addition, in the following subsections, we derive problem-specific valid inequalities to strengthen the master problem, thus improving convergence.
\begin{algorithm}[t!]
\small
\renewcommand{\arraystretch}{0.3}
\caption{Decomposition algorithm.}
\label{Alg1:CAG}
\noindent \textbf{1. Input.} Feasible region $\mathcal{X}$; support $\mathcal{S}$; set of cuts $ \lbrace L (\pmb{x}, \delta)\geq 0 \rbrace=\emptyset $; $LB=-\infty$ and $UB=\infty.$
\vspace{2mm}
\noindent \textbf{2. Master Problem.} Solve the following master problem
\begin{subequations}\label{Master}
\begin{align}
Z=&\min_{\pmb{x}, \pmb{y}, \pmb \rho, \pmb \psi} \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big[\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big] + \delta \Bigg \} \label{MasterObj} \\
& \ \ \text{s. t.} \qquad (\pmb{x},\pmb{y} ) \in \mathcal{X}, \ \ \pmb{\psi} \geq 0, \ \ \{ L (\pmb{x}, \delta)\geq 0 \},
\end{align}
\end{subequations}
and record the optimal solution $(\pmb{x}^ *, \pmb{\rho}^*, \pmb{\psi}^*, \delta^*)$ and optimal value $Z^*$. Set $LB=Z^*$.
\noindent \textbf{3. Sub-problem.}
\begin{itemize}
\item[3.1.] With $(\pmb{x}, \pmb{\rho}, \pmb{\psi})$ fixed to $(\pmb{x}^*, \pmb{\rho}^*, \pmb{\psi}^*)$, solve the following problem
\begin{subequations}\label{MILPSep}
\begin{align}
h(\pmb{x}, \pmb \rho, \pmb \psi)= &\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big\}\\
& \ \ \ \text{s.t. } \ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner},
\end{align}
\end{subequations}
$\qquad \ $ and record optimal solution $(\pmb{\pi^*, \lambda^*, W^*, v^*, k^*})$ and $h(\pmb{x}, \pmb \rho, \pmb \psi)^*$.
\item[3.2.] Set $UB=\min \{ UB, \ h(\pmb{x}, \pmb \rho, \pmb \psi)^*+ (LB-\delta^*) \}$.
\end{itemize}
\noindent \textbf{4. if} $\delta^* \geq \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^{t*} v_{j,m}^{t*} +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}^*+k_{i,t}^*\psi_{i,t}^*) $ \textbf{then}
$\qquad \ \ $ stop and return $\pmb{x}^*$ and $\pmb{y}^*$ as the optimal solution to problem \eqref{Master} (i.e., MAD-DRO in \eqref{FinalDR}).
\noindent $\ \ $ \textbf{else} add the cut $\delta \geq \sum \limits_{t \in T} \sum \limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum \limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m} v_{j,m}^{t*}+\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}+k_{i,t}^*\psi_{i,t} )$ to the set of
$\qquad \quad$ cuts $ \{L (\pmb{x}, \delta) \geq 0 \}$ and go to step 2.
\noindent $\ \ $ \textbf{end if}
\end{algorithm}
\subsection{\textbf{Multiple Optimality Cuts and Lower Bound Inequalities}}\label{sec:Improv}
\noindent In this section, we aim to incorporate more second-stage information into the first-stage without adding optimality cuts into the master problem by exploiting the structural properties of the recourse problem. We first observe that once the first-stage solutions and the demand are known, the second-stage problem can be decomposed into independent sub-problems with respect to time periods. Accordingly, we can construct cuts for each sub-problem in step 4. Let $\delta_t$ represent the optimality cut for each period $t$, we replace $\delta$ in \eqref{MasterObj} with $\sum_{t} \delta_t$ and add constraints
\begin{align}
\delta_t\geq \sum \limits_{i \in I} \pi_{i,t}^*+ \sum \limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m} v_{j,m}^{t*}+ \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}+k_{i,t}^*\psi_{i,t} ), \qquad \forall t \in T.\label{Cut1}
\end{align}
\noindent The original single cut is the summation of multiple cuts of the form, i.e., $\delta=\sum_{t \in T} \delta_t$. Hence, in each iteration, we incorporate more or at least an equal amount of information into the master problem using \eqref{Cut1} as compared with the original single cut approach. In this manner, the optimality cuts become more specific, which may result in better lower bounds and, therefore, a faster convergence. In Proposition \ref{Prop3:LowerB1}, we further identify valid lower bound inequalities for each time period to tighten the master problem (see Appendix~\ref{Appx:Prop3} for a proof).
\begin{proposition}\label{Prop3:LowerB1}
Inequalities \eqref{Vineq1} are valid lower bound inequalities on the recourse of the MFRSP.
\begin{align}
\sum \limits_{i \in I} \min \lbrace \gamma, \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \rbrace \underline{W}_{i,t}, \qquad \forall t \in T. \label{Vineq1}
\end{align}
\end{proposition}
It follows that $\sum \limits_{i \in I} \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}+\delta_t\geq \sum \limits_{i \in I} \min \lbrace \gamma, \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \rbrace \underline{W}_{i,t} $, for all $t \in T$, are valid.
\subsection{\textbf{Symmetry-Breaking Constraints}}\label{sec:symm}
\noindent Suppose there are three homogeneous MFs. As such, solutions $y=[1,1,0]^\top$, $y=[0,1,1]^\top$, and $y=[1, 0,1 ]^\top$ are equivalent (i.e., yield the same objective) in the sense that they all permit 2 out of 3 MFs to be used in the planning period. To avoid \textcolor{black}{wasting time} exploring such equivalent solutions, we assume that MFs are numbered sequentially and add constraints \eqref{Sym1} to the first-stage.
\begin{align}\label{Sym1}
y_{m+1} \leq y_m, \qquad \forall m <M.
\end{align}
\noindent \textcolor{black}{Constraints \eqref{Sym1} enforce} arbitrary ordering or scheduling of MFs. Second, recall that in the first period, $t=1$, we decide the initial locations of the MFs. Therefore, it doesn't matter which MF is assigned to location $j$. For example, suppose that we have three candidate locations, and MFs 1 and 2 are active. Then, feasible solutions $(x_{1,1}^1=1$, $x_{3,2}^1=1)$ and $(x_{1,2}^1=1$, $x_{3,1}^1=1)$ yield the same objective. To avoid exploring such equivalent solutions, we define a dummy location $J+1$ and \textcolor{black}{add constraints \eqref{Ineq:Symm1}--\eqref{Ineq:Symm2} to the first-stage.}
\begin{subequations}\label{Sym2}
\begin{align}
& x_{j,m}^1 - \sum \limits_{j^\prime=j}^{J+1}x_{j^\prime,m+1}^1\leq 0, && \forall m <M, \forall j \in J, \label{Ineq:Symm1}\\
& x_{J+1, m}^1= 1- \sum \limits_{j=1}^Jx_{j,m}^1, && \forall m \in M.\label{Ineq:Symm2}
\end{align}
\end{subequations}
\noindent Constraints \eqref{Ineq:Symm1}--\eqref{Ineq:Symm2} are valid for any formulation that uses the same sets of first-stage routing and scheduling decisions and constraints. We derived constraints \eqref{Sym1}--\eqref{Sym2} based on similar symmetry breaking principles in \cite{ostrowski2011orbital} and \cite{shehadeh2019analysis}. Although breaking symmetry is very important and standard in integer programming problems, our paper is the first to attempt to break the symmetry in the solution space of the first-stage planning decisions of the MFRSP. In Section~\ref{sec5:symmetry}, we demonstrate the computational advantages that could be gained by incorporating these inequalities.
\section{Computational Experiments}\label{sec:computational}
\noindent In this section, we conduct extensive computational experiments comparing the proposed DRO models and a sample average approximation (SAA) of the SP model computationally and operationally, demonstrating where significant performance improvements could be gained. The sample average model solves model \eqref{DMFRS_SP} with $\mathbb{P}$ replaced by an empirical distribution based on $N$ samples of $\pmb{W}$ (see Appendix~\ref{Appx:SAA} for the formulation). In Section~\ref{sec5.1:instancegen}, we describe the set of problem instances and discuss other experimental setups. In Section~\ref{sec5:CPU}, we compare solution time of the proposed models. In Section~\ref{sec5:symmetry}, we demonstrate efficiency of the proposed lower bound inequalities and symmetry breaking constraints. \textcolor{black}{We compare optimal solutions of the proposed models and their out-of-sample performance in Sections~\ref{sec:Optimal Solutions} and \ref{sec5:OutSample}, respectively}. We analyze the sensitivity of the DRO expectation models to different parameter settings in Section~\ref{sec5:sensitivity}. \textcolor{black}{We close by comparing the risk-neutral and risk-averse models under critical parameter settings in Section~\ref{sec:CVaRExp}.}
\subsection{\textbf{Experimental Setup}}\label{sec5.1:instancegen}
\noindent We constructed 10 MFRSP instances, in part based on the same parameters settings and assumptions made in \cite{lei2014multicut} and \cite{lei2016two}. We summarize these instances in Table~\ref{table:DMFRSInstances}. Each of the 10 instances is characterized by the number of customers locations $I$, number of candidate locations $J$, and the number of periods $T$. Instances 1--4 are from \cite{lei2014multicut} and Instances 5--10 are from \cite{lei2016two}. These benchmark instances represent a wide range of potential service regions in terms of problem size as a function of the number of demand nodes/locations and time periods. For example, if we account for the scale of the problem in the sense of a static facility location problem, instance 10 consists of $30 \times 20= 600$ customers, which is relatively large for many practical applications. In addition, we constructed a service region based on 20 selected nodes (see \textcolor{black}{Figure~\ref{LehighMap}}) in Lehigh County of Pennsylvania (USA). Then, as detailed below, we constructed two instances (denoted as Lehigh 1 and Lehigh 2) based on this region.
\begin{table}[t]
\center
\footnotesize
\renewcommand{\arraystretch}{0.3}
\caption{MFRSP instances. Notation: $I$ is \# of customers, $J$ is \# of locations, and $T$ is \# of periods.}
\begin{tabular}{cccccccccc}
\hline
\textbf{Inst} & \textbf{$\pmb{I}$} & \textbf{$\pmb{J}$} & \textbf{$\pmb{T}$} \\
\hline
1& 10 & 10 & 10& \\
2 & 10 & 10 & 20& \\
3 & 15 & 15 & 10& \\
4 &15 & 15 & 20& \\
5 & 20 & 20 & 10\\
6 &20 & 20 & 20\\
7& 25 & 25& 10\\
8& 25 & 25& 20\\
9& 30 & 30& 10\\
10& 30 & 30& 20\\
\hline
\end{tabular}
\label{table:DMFRSInstances}
\end{table}
For each instance in Table~\ref{table:DMFRSInstances}, we generated a total of $I$ vertices as uniformly distributed random numbers on a 100 by 100 plane and computed the distance between each pair of nodes in Euclidean sense as in \cite{lei2014multicut}. For Lehigh county instances, we first extracted the latitude and longitude of each node and used Bing Maps Developer API to compute the travel time in minutes between each pair of nodes.
We followed the same procedures in the DRO scheduling and facility location literature to generate random parameters as follows. \textcolor{black}{For instances 1--10, we generated $\mu_{i,t}$ from a uniform distribution $U[\underline{W}, \overline{W}]=[20, 60]$, and set the standard deviation $\sigma_{i,t}=0.5\mu_{i,t}$, for all $i \in I$ and $t \in T$.} For Lehigh county instances, we used the population estimate for each node based on the most updated information posted on the 2010 US Census Bureau (see Appendix~\ref{AppexLehigh}) to construct the following two demand structures. In Lehigh 1, we generated the demand's mean as follows: if the population $\geq$10,000, we set $\mu_{i,t}=40$ (i.e., the mean of $U[20,60]$); if the population $\in$ [5,000, 10,000), we set the mean to $\mu=30$; if the population $\in$ [1,000, 5,000), we set $\mu_{i,t}=20$; and if the population $<1,000$, we set $\mu_{i,t}=15$. In Lehigh 2, we use the population percentage (weight) at each node to generate the demand's mean as $\mu_{i,t}=\min(60, \texttt{population\%} \times 1000)$. To a certain extent, these structures reflect what may be observed in real life, i.e., locations with more population may potentially create greater demand. We refer to Appendix~\ref{AppexLehigh} for the details related to Lehigh 1 and Lehigh 2.
\begin{figure}
\begin{center}
\includegraphics[scale=0.8]{MAP2.png}
\caption{Map of 20 cities in Lehigh County. We created this map using the \texttt{geoscatter} function (MATLAB).}\label{LehighMap}
\end{center}
\end{figure}
For each instance, we sample $N$ realizations $W_{i,t}^n, \ldots, W_{I,T}^n$, for $n=1, \ldots, N$, by following lognormal (LogN) distributions with the generated $\mu_{i,t}$ and $\sigma_{i,t}$. We round each parameter to the nearest integer. We solve the SAA and W-DRO models using the $N$ sample and the MAD-DRO model with the corresponding mean, MAD, and range. The Wasserstein ball's radius $\epsilon$ in the W-DRO model is an input parameter, where different values of $\epsilon$ may result in a different robust solution $\pmb{x} (\epsilon, N)$ with a very different out-of-sample performance $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$. To estimate $\epsilon^{\mbox{\tiny opt}}$ that minimizes $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$, we employed a widely used cross-validation method (see Appendix~\ref{WDRO_Radius}).
We assume that all cost parameters are calculated in terms of present monetary value. Specifically, as in \cite{lei2014multicut}, for each instance, we randomly generate (1) the fixed cost from a uniform distribution $U[a,b]$ with $a = 1000$ and $b = 1500$; (2) the assignment cost factor per unit distance per unit demand $\beta$ from $U[0.0001a,0.0001b]$; and (3) the penalty cost per unit demand $\gamma$ form $U[0.01a, 0.01b]$. Finally, we set the traveling inconvenience cost factor to $0.0001a$, and unless stated otherwise, we use a capacity parameter $C=100$. We implemented all models and the decomposition algorithm using AMPL2016 Programming language calling CPLEX V12.6.2 as a solver with default settings. We run all experiments on a MacBook Pro with Apple M1 Max Chip, 32GB of memory, and 10-core CPUs. Finally, we imposed a solver time limit of 1 hour.
\subsection{\textbf{CPU Time}}\label{sec5:CPU}
\noindent In this section, we analyze solution times of the proposed DRO models. \textcolor{black}{We consider two ranges of the demand: $\pmb{W} \in [20, 60]$ (base-case) and $\pmb{W} \in [50, 100]$ (higher demand variability and volume). We also consider two MF capacities: $C=60$ (relatively small capacity) and $C=100$ (relatively large capacity).} We focus on solving problem instances with a small sample size, which is often seen in most real-world applications (especially in healthcare) and is the primary motivation for using DRO. Specifically, we use $N=10$ and $N=50$ as a sample size for the W-DRO model. For each of the 10 instances in Table~\ref{table:DMFRSInstances}, demand range, $C$, and $N$, we generated and solved 5 instances using each model as described in Section~\ref{sec5.1:instancegen}.
\color{black}
Let us first analyze solution times of the risk-neutral DRO models. Tables~\ref{table:MAD_CPU_2060}--\ref{table:MAD_CPU_50100} present the computational details (i.e., CPU time in seconds and the number of iterations in the decomposition algorithm before it converges to the optimum) of solving the MAD-DRO model. Tables~\ref{table:WASS_CPU_2060_N10}--\ref{table:WASS_CPU_50100_N10} and Tables~\ref{table:WASS_CPU_2060_N50}--\ref{table:WASS_CPU_50100_N50} present the computational details of solving the W-DRO model with $N=10$ and $N=50$, respectively. We observe the following from these tables. First, the computational effort (i.e., solution time per iteration) increases with the size of instance ($I \times J \times T$). Second, solution times are shorter under tight capacity ($C=60$) than under large capacity ($C=100$). In addition, the decomposition algorithm takes fewer iterations to converge to the optimum under tight capacity. Intuitively, when $C=100$, each facility can satisfy more demand than $C=60$. Thus, there are more feasible choices for the MF fleet size and schedule when $C=100$. That said, the search space when $C=100$ is larger, potentially leading to a longer computational time.
Third, using the MAD-DRO model, we were able to solve all instances within the time limit. The average solution time for Instances 1--7 using the MAD-DRO model ranges from 2 to 914 seconds. The average solution time of larger instances (Instances 8--10) ranges from 241 to 1,901 seconds. In contrast, using the W-DRO model, we were able to solve Instances 1--8 with $N=10$ and Instances 1--7 with $N=50$. The average solution time of the W-DRO model with $N=10$ and $N=50$ ranges from 5 to 715 and from 14 to 1,725 seconds, respectively. Note that solution times of the W-DRO model increase with the sample size. This makes sense because the number of variables and constraints of the W-DRO model increases as $N$ increases. In addition, solution times of the W-DRO model with $N=50$ are generally longer than the MAD-DRO model. This also makes sense because the MAD-DRO model is a smaller deterministic model (i.e., it has fewer variables and constraints).
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-DRO model ($ \pmb{W} \in [20, 60]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 1 & 2 & 4 & 3 & 5 & 12 & & 5 & 6 & 8 & 18 & 23 & 30 \\
2 & 10 & 20 & & 3 & 7 & 13 & 21 & 30 & 51 & & 2 & 3 & 4 & 5 & 9 & 15 \\
3 & 15 & 10 & & 2 & 5 & 10 & 3 & 7 & 16 & & 15 & 24 & 37 & 27 & 41 & 62 \\
4 & 15 & 20 & & 4 & 7 & 12 & 6 & 11 & 22 & & 18 & 25 & 43 & 34 & 45 & 72 \\
5 & 20 & 10 & & 6 & 10 & 19 & 6 & 11 & 22 & & 19 & 26 & 45 & 34 & 45 & 72 \\
6 & 20 & 20 & & 9 & 41 & 65 & 6 & 28 & 45 & & 91 & 308 & 640 & 53 & 68 & 80 \\
7 & 25 & 10 & & 23 & 169 & 578 & 8 & 25 & 44 & & 503 & 703 & 1004 & 110 & 138 & 190 \\
8 & 25 & 20 & & 57 & 241 & 352 & 5 & 33 & 53 & & 573 & 872 & 1272 & 147 & 196 & 245 \\
9 & 30 & 10 & & 99 & 416 & 696 & 16 & 61 & 80 & & 185 & 1860 & 1871 & 113 & 147 & 161 \\
10 & 30 & 20 & & 463 & 905 & 1785 & 7 & 38 & 91 & & 1098 & 1592 & 1887 & 6 & 125 & 191 \\
\hline
\end{tabular}\label{table:MAD_CPU_2060}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-DRO model ($ \pmb{W} \in [50, 100]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 1 & 2 & 2 & 1 & 2 & 4 & & 10 & 11 & 13 & 20 & 22 & 24 \\
2 & 10 & 20 & & 1 & 2 & 3 & 2 & 4 & 9 & & 5 & 8 & 13 & 15 & 26 & 41 \\
3 & 15 & 10 & & 4 & 9 & 17 & 2 & 2 & 3 & & 29 & 47 & 65 & 28 & 34 & 45 \\
4 & 15 & 20 & & 3 & 9 & 23 & 2 & 5 & 13 & & 43 & 54 & 70 & 44 & 54 & 63 \\
5 & 20 & 10 & & 27 & 68 & 177 & 3 & 5 & 14 & & 215 & 292 & 392 & 70 & 88 & 124 \\
6 & 20 & 20 & & 6 & 14 & 38 & 3 & 5 & 14 & & 292 & 322 & 347 & 74 & 94 & 107 \\
7 & 25 & 10 & & 20 & 385 & 796 & 2 & 9 & 29 & & 553 & 686 & 914 & 175 & 205 & 258 \\
8 & 25 & 20 & & 63 & 334 & 835 & 3 & 15 & 36 & & 122 & 622 & 1635 & 3 & 14 & 36 \\
9& 30 & 10 & & 25 & 579 & 1692 & 3 & 39 & 143 & & 1851 & 1860 & 1871 & 113 & 147 & 161 \\
10& 30 & 20 & & 81 & 300 & 1240 & 3 & 10 & 41 & & 1821 & 1901 & 1937 & 123 & 147 & 156 \\
\hline
\end{tabular}\label{table:MAD_CPU_50100}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [20, 60]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 7 & 9 & 18 & 7 & 9 & 11 & & 15 & 20 & 22 & 14 & 18 & 20 \\
2 & 10 & 20 & & 8 & 10 & 12 & 7 & 8 & 10 & & 18 & 24 & 28 & 15 & 18 & 20 \\
3 & 15 & 10 & & 35 & 39 & 45 & 16 & 18 & 20 & & 12 & 15 & 18 & 6 & 8 & 9 \\
4 & 15 & 20 & & 30 & 43 & 62 & 5 & 6 & 78 & & 115 & 134 & 147 & 23 & 25 & 29 \\
5 & 20 & 10 & & 41 & 48 & 64 & 6 & 8 & 9 & & 99 & 448 & 682 & 21 & 28 & 33 \\
6 & 20 & 20 & & 49 & 139 & 177 & 6 & 8 & 10 & & 142 & 207 & 273 & 24 & 35 & 44 \\
7 & 25 & 10 & & 81 & 86 & 97 & 8 & 9 & 10 & & 296 & 410 & 506 & 32 & 41 & 50 \\
8 & 25 & 20 & & 100 & 678 & 1820 & 7 & 9 & 11 & & 534 & 715 & 1068 & 47 & 51 & 56 \\
\hline
\end{tabular}\label{table:WASS_CPU_2060_N10}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [50, 100]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 4 & 5 & 7 & 3 & 3 & 4 & & 7 & 9 & 10 & 6 & 7 & 8 \\
2 & 10 & 20 & & 9 & 12 & 17 & 3 & 3 & 4 & & 15 & 16 & 18 & 6 & 6 & 7 \\
3 & 15 & 10 & & 16 & 32 & 55 & 3 & 3 & 3 & & 35 & 48 & 61 & 6 & 7 & 9 \\
4 & 15 & 20 & & 30 & 34 & 43 & 3 & 3 & 3 & & 71 & 81 & 85 & 8 & 9 & 10 \\
5 & 20 & 10 & & 29 & 42 & 76 & 3 & 3 & 3 & & 75 & 79 & 81 & 9 & 10 & 11 \\
6 & 20 & 20 & & 82 & 92 & 108 & 3 & 3 & 3 & & 110 & 182 & 229 & 7 & 11 & 13 \\
7 & 25 & 10 & & 108 & 123 & 135 & 9 & 10 & 11 & & 151 & 192 & 243 & 9 & 10 & 12 \\
\hline
\end{tabular}\label{table:WASS_CPU_50100_N10}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [20, 60]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 18 & 28 & 35 & 6 & 10 & 12 & & 53 & 60 & 71 & 17 & 19 & 21 \\
2 & 10 & 20 & & 30 & 35 & 42 & 8 & 9 & 11 & & 89 & 104 & 114 & 18 & 20 & 21 \\
3 & 15 & 10 & & 28 & 32 & 81 & 6 & 6 & 7 & & 104 & 124 & 139 & 21 & 23 & 25 \\
4 & 15 & 20 & & 40 & 43 & 48 & 6 & 6 & 7 & & 184 & 214 & 231 & 21 & 23 & 25 \\
5 & 20 & 10 & & 33 & 57 & 95 & 4 & 6 & 7 & & 271 & 341 & 463 & 29 & 32 & 36 \\
6 & 20 & 20 & & 89 & 306 & 723 & 6 & 8 & 10 & & 455 & 1461 & 2141 & 22 & 29 & 35 \\
7 & 25 & 10 & & 109 & 127 & 169 & 6 & 7 & 8 & & 1006 & 1725 & 2095 & 37 & 40 & 42 \\
\hline
\end{tabular}\label{table:WASS_CPU_2060_N50}
\end{table}
\begin{table}[t!]
\color{black}
\center
\small
\caption{Computational details of solving the W-DRO model ($ \pmb{W} \in [50, 100]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 12 & 14 & 17 & 3 & 3 & 4 & & 30 & 32 & 33 & 7 & 7 & 8 \\
2 & 10 & 20 & & 17 & 22 & 25 & 3 & 3 & 4 & & 44 & 57 & 63 & 7 & 9 & 9 \\
3 & 15 & 10 & & 65 & 81 & 92 & 6 & 6 & 7 & & 71 & 105 & 145 & 6 & 6 & 7 \\
4 & 15 & 20 & & 42 & 53 & 87 & 3 & 3 & 7 & & 48 & 65 & 84 & 4 & 5 & 7 \\
5 & 20 & 10 & & 50 & 60 & 66 & 3 & 3 & 3 & & 111 & 144 & 223 & 7 & 7 & 7 \\
6 & 20 & 20 & & 580 & 1174 & 2024 & 3 & 3 & 3 & & 319 & 1009 & 3000 & 8 & 10&11 \\
7 & 25 & 10 & & 106 & 113 & 129 & 3 & 3 & 3 & & 370 & 402 & 464 & 9 & 9 & 10 \\
\hline
\end{tabular}\label{table:WASS_CPU_50100_N50}
\end{table}
\color{black}
\textcolor{black}{Let us now analyze solution times of the risk-averse models. We use MAD-CVaR (W-CVaR) to denote the mean-CVaR-based DRO model with MAD (1-Wasserstein) ambiguity. Since we observe similar computational performance with different values of $\Theta \in [0, 1)$, we present results with $\Theta=0$. Tables \ref{table:MADCVAR_2060}--\ref{table:MADCVAR__50100} and Tables \ref{table:Wass_CVAR_2060_N10}--\ref{table:Wass_CVAR_50100_N50} in Appendix~\ref{Appx:CPU2} present the computational details of solving the MAD-CVaR model and W-CVaR model, respectively. Using the MAD-CVaR model, we were able to solve Instances 1--8 with $\pmb{W} \in [20,60]$ and Instances 1--7 with $\pmb{W} \in [50,100]$. In addition, solution times of MAD-CVaR model are longer than the risk-neutral MAD-DRO model. In contrast, using the W-CVaR model, we were able to solve Instances 1--5. Solution times of these instances are generally longer than the risk-neutral model, especially when $N=50$. It is not surprising that the CVaR models are more computationally challenging to solve than the risk-neutral models because the former models are larger (have more variables and constraints). In particular, the master problem of the CVaR models in the decomposition algorithm is larger than the expectation models (see Algorithm~\ref{Alg2:Decomp2} in Appendix~\ref{Sec:meanCVAR}).}
Finally, it is worth mentioning that using an \textit{enhanced multicut L-shaped} (E-LS) method to solve their SP model, \cite{lei2014multicut} were able to solve Instance 1--4. The average solution time of Instance 4 using E-LS is 3000 seconds obtained at 5\% optimality gap. The CVaR-based SP model is more challenging to solve than the risk-neutral SP model.
\subsection{\textbf{Efficiency of Inequalities \eqref{Vineq1}--\eqref{Sym2} }}\label{sec5:symmetry}
\noindent In this section, we study the efficiency of symmetry breaking constraints~\eqref{Sym1}--\eqref{Sym2} and lower bounding inequalities \eqref{Vineq1}. Given the challenges of solving large instances without \eqref{Vineq1}--\eqref{Sym2}, we use Instance 1 with $\pmb{W} \in [20, 60]$ and $C=100$ in this experiment.
First, we separately solve the proposed models with and without symmetry-breaking (SB) constraints \eqref{Sym1}--\eqref{Sym2}. First, we observe that without these SB constraints, solution times of Instance 1 using (W-DRO, MAD-DRO, SP) significantly increase from (20, 6, 70) to (1,765, 1,003, 3,600) seconds. Instances 3--10 terminated with a large gap after one hour without these SB constraints. Second, as shown in Figure~\ref{FigSB}, both the lower bound and gap (i.e., the relative difference between the upper and lower bounds on the objective value) converge faster when we include constraints \eqref{Sym1}--\eqref{Sym2} in the master problem. Moreover, constraints \eqref{Sym1}--\eqref{Sym2} lead to a stronger bound in each iteration. These results demonstrate the importance of breaking the symmetry in the first-stage decisions and the effectiveness of our SB constraints.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_LB_NoSymm.jpg}
\caption{LB values, W-DRO}
\label{FigSBa}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_LB_NoSymm.jpg}
\caption{LB value, MAD-DRO}
\label{FigSBb}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_GAP_NoSymm.jpg}
\caption{Gap values, W-DRO}
\label{FigSBGAPa}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_GAP_NoSymm.jpg}
\caption{Gap values, W-DRO}
\label{FigSBGAPb}
\end{subfigure}%
\caption{Comparisons of lower bound and gap values with and without SB constraints \eqref{Sym1}-\eqref{Sym2}.}\label{FigSB}
\end{figure}
Next, we analyze the impact of including the valid lower bounding (LB) inequalities \eqref{Vineq1} in the master problem of decomposition algorithm. We first observe that the algorithm takes a very large number of iterations and a longer time until convergence without these LB inequalities. Therefore, in Figure~\ref{FigVI}, we present the LB and gap values with and without inequalities \eqref{Vineq1} from the first 25 iterations. It is obvious that both the lower bound and gap values converge faster when we introduce inequalities \eqref{Vineq1} into the master problem. Moreover, because of the better bonding effect, the algorithm converges to the optimal solution in fewer iterations and shorter solution times.
For example, the algorithm takes 10 seconds and 9 iterations to solve the MAD-DRO instance with these inequalities and terminates with a 33\% gap after an hour without these inequalities. The results in this section demonstrate the importance and efficiency of inequalities \eqref{Vineq1}--\eqref{Sym2}.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_LB_NoBound.jpg}
\caption{LB values, W-DRO}
\label{FigVIa}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_LB_NoBound.jpg}
\caption{LB values, MAD-DRO}
\label{FigVIb}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst1_WDRO_Gap_NoBound.jpg}
\caption{Gap values, W-DRO}
\label{FigVIaGap}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{Inst1_MAD_GAP_NoBound.jpg}
\caption{Gap values, MAD-DRO}
\label{FigVIbGap}
\end{subfigure}%
\caption{Comparisons of lower bound and gap values with and without valid LB inequalities \eqref{Vineq1}.}\label{FigVI}
\end{figure}
\subsection{\textbf{Analysis of Optimal Solutions}}\label{sec:Optimal Solutions}
\textcolor{black}{In this section, we compare the optimal solutions of the SP, MAD-DRO, and W-DRO models.} Given that the SP model can only solve small instances to optimality, for a fair comparison and brevity, we use Instance 3 ($I=15$, $J=15$, and $T=10$) as an example of an average-sized instance. In addition, we present results for Lehigh 1 and Lehigh 2. Table~\ref{table:Optimal} presents the number of MFs (i.e., fleet size) for each instance.
\begin{table}[t!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.3}
\caption{Optimal number of MFs.}
\begin{tabular}{lccccccccc}
\hline
\multicolumn{4}{c}{Instance 3 (\textcolor{black}{$\pmb{W}\in [20, 60]$)}} \\
\hline
\textbf{Model} & \textbf{$N=10$} & \textbf{$N=50$} & \textbf{$N=100$} \\
\hline
SP& 7 & 8 & 8 \\
W-DRO & 9 & 8 & 8\\
MAD-DRO & 10 & 10 & 10 \\
\\
\hline
\multicolumn{4}{c}{\textcolor{black}{Instance 3 ($\pmb{W}\in [50, 100]$)}} \\
\hline
SP& 12 & 12 & 12 \\
W-DRO &13 & 13 & 13\\
MAD-DRO & 15 & 15& 15 \\
\hline
\multicolumn{4}{c}{Lehigh 1} \\
\hline
\textbf{Model} & \textbf{$N=10$} & \textbf{$N=50$} & \textbf{$N=100$} \\
\hline
SP& 6 & 7 & 7 \\
W-DRO & 9 & 8 & 8\\
MAD-DRO & 10 & 10 & 10 \\
\\
\hline
\multicolumn{4}{c}{Lehigh 2} \\
\hline
\textbf{Model} & \textbf{$N=10$} & \textbf{$N=50$} & \textbf{$N=100$} \\
\hline
SP& 5 & 5 & 5 \\
W-DRO & 6 & 6 & 6\\
MAD-DRO & 7 & 7 & 7 \\
\hline
\end{tabular}
\label{table:Optimal}
\end{table}
We observe the following from Table~\ref{table:Optimal}. First, the MAD-DRO model always activates (schedules) a higher number of MFs than the SP model, and a larger number of MFs than the W-DRO model. By scheduling more MFs, the MAD-DRO model tends to conservatively mitigate the ambiguity of the demand (reflected by lower shortage and transportation costs reported later in Section~\ref{sec5:OutSample}). Second, the W-DRO model schedules a larger number of MFs than the SP model, and the difference is significant when the sample size is small ($N=10$). This makes sense as a small sample does not have sufficient distributional information. Thus, in this case, the W-DRO model makes conservative decisions to hedge against ambiguity. As $N$ increases (i.e., more information becomes available), the W-DRO model often makes less conservative decisions. Consider Instance 3 with $\pmb{W} \in [20, 60]$, for example. The W-DRO model schedules 9 and 8 MFs when $N=10$ and $N=50$, respectively. \textcolor{black}{Third, we observe that all models scheduled more MFs when we increased the demand's range from $\pmb{W}\in [20, 60]$ to $\pmb{W}\in [50, 100]$ to hedge against the increase in the demand's volume and variability.}
\color{black}
Let us now analyze the optimal locations of the MFs. For illustrative purposes and brevity, we use Lehigh 2 in this analysis. Recall that the SP and DRO models yield different fleet sizes and thus different routing decisions. Therefore, to facilitate the analysis, we first fixed the fleet size to 4 in the three models. Second, to demonstrate how MFs can move to accommodate the change in the demand over time and location, we consider two periods (two days) with the following demand structure. In period 1, we kept the demand structure as described in Section~\ref{sec5.1:instancegen}. In the second period, we swapped the average demand of the following nodes: Allentown and Alburtis, Bethlehem and Cetronia, Emmaus and Trexlertown, Ancient Oaks and Laurys Station, Catasauqua and New Tripoli, and Wescosvill and Slatedale. That is, in the second period, we decreased the demand of the 6 nodes with the highest demand (Allentown--Wescosvill) to that of the nodes that generate the lowest demand (Alburtis--Slatedale) and increased the demand of (Alburtis--Slatedale) to that of (Allentown--Wescosvill). We refer to Table~\ref{table:period} in Appendix~\ref{AppexLehigh} for a summery of the average demand of each node in period 1 and period 2. Figure~\ref{base_swap} illustrates the MFs' locations in period 1 (Figure~\ref{period1}) and period 2 (Figure~\ref{period2}). We provide a summary of these results in Table~\ref{table:OptimalLocations} in Appendix~\ref{AppexLehigh}.
We observe the following about the initial locations in period 1 (Figure~\ref{period1}). First, all models scheduled MF \#1 and \#2 at Allentown and Bethlehem, respectively. This makes sense because, by construction, these nodes generate greater demand than the remaining nodes in period 1. Second, we do not see any MF at or near any of the nodes in the top left of the map (New Tripoli, Slatedale, Slatington, Laury Station, Schnecksville). This makes sense because these nodes generate significantly lower demand than the remaining nodes in period 1. MF \#3 and MF\#4 are scheduled at nodes that generate higher demand or near nodes that generate higher demand than the nodes in the top left of the map. For example, the DRO models scheduled MF \#3 at Emmaus (which has the greatest demand after Allentown and Bethlehem in period 1). The MAD-DRO model scheduled MF \#4 at Wescoville, while the W-DRO model scheduled this MF at Breinigsville. The SP model scheduled MF \#3 and MF \#4 at Dorneyville and Ancient Oaks. Note that Ancient Oaks generates the highest demand after Allentown, Bethlehem, and Emmaus in period 1. Moreover, Dorneyville, Breinigsville, and Wescoville are closer to demand nodes that generate higher demand in period 1 than the remaining nodes on the top left of the map (see Table~\ref{table:period} in Appendix~\ref{AppexLehigh}).
We make the following observations from the results in period 2 presented in Figure~\ref{period2}. First, it is clear that all MFs moved from their initial locations to other locations in period 2 to accommodate the change in the demand. Second, all models scheduled one MF at Alburtis, where the average demand increased from 9 to 60 (average demand of Allentown in period 1). This makes sense because, in period 2, Alburtis generates the highest demand. Third, the DRO models scheduled one MF at Trexlertown and one at New Tripoli, where the average demand increased from 8 and 3 to 43 (average demand of Emmaus in period 1) and 23 (average demand of Catasauqua in period 1), respectively. The SP and W-DRO models scheduled one MF at Cetronia, where the demand increased from 8 to 60.
\begin{figure}
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=\textwidth]{Base}
\caption{Initial locations of the MFs in period 1.}\label{period1}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\center
\includegraphics[width=\textwidth]{Swap1}
\caption{Locations of the MFs in period 2.}\label{period2}
\end{subfigure}
\caption{Optimal MF locations in period 1 and period 2 (Lehigh 2 instance). Color code: the red square is MF, and the black circle is a demand node/city.}\label{base_swap}
\end{figure}
\color{black}
\subsection{\textbf{Analysis of Solutions Quality}}\label{sec5:OutSample}
\noindent In this section, we compare the operational performance of the optimal solutions to Instance 3, Lehigh 1, and Lehigh 2 via out-of-sample simulation. First, we fix the optimal first-stage decisions yielded by each model in the second-stage of the SP. Then, we solve the second-stage problem in \eqref{2ndstage} with the fixed first-stage decisions and the following sets of $N'=10,000$ out-of-sample data of $W_{i,t}^n$, for all $i \in I, t \in T,$ and $n \in [N']$, to compute the corresponding out-of-sample second-stage cost.
\begin{enumerate}\itemsep0em
\item[Set 1.] \textit{Perfect distributional information}. We use the same settings and distribution (LogN) that we use for generating the $N$ data in the optimization to generate $N^\prime$ data. This is to simulate the performance when the true distribution is the same as the one used in the optimization.
\item[Set 2.] \textit{Misspecified distributional information}. We follow the same out-of-sample simulation procedure described in \cite{wang2020distributionally} and employed in \cite{shehadeh2020distributionallyTucker} to generate the $N^\prime=10,000$ data. Specifically, we perturb the distribution of the demand by a parameter $\Delta$ and use a parameterized uniform distribution $U$[$(1-\Delta)\underline{W}, (1+\Delta)\overline{W}$ ] for which a higher value of $\Delta$ corresponds to a higher variation level. We apply $\Delta \in \{ 0, 0.25, 0.5\}$ with $\Delta=0$ indicating that we only vary the demand distribution from LogN to Uniform. This is to simulate the performance when the true distribution is different from the one we used in the optimization. In addition, we generate $N'$ correlated data points with 0.2 and 0.6 correlation coefficients.
\end{enumerate}
\color{black}
For brevity, we next discuss simulation results for the solutions obtained with $N=10$. We observe similar results for solutions obtained with $N=50$ (see Appendix~\ref{Appex:AdditionalOut} for these results). In Figures~\ref{Fig3_UniN10_Inst3}, \ref{Fig3_UniN10_Inst3_Range2}, and \ref{Fig3_N10_Lehigh1}, we present the normalized histograms of out-of-sample total costs (TC) and second-stage costs (2nd) for Instance 3 (with $\pmb{W} \in [20,60]$, $N=10$), Instance 3 (with $\pmb{W} \in [50, 100]$, $N=10$), and Lehigh 1 ($N=10$). We obtained similar results for Lehigh 2 (see Appendix~\ref{Appex:AdditionalOut}). We computed TC as TC$=$first-stage cost+out-of-sample second-stage cost.
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_LogN_Wass2.jpg}
\caption{TC (Set 1, LogN)}\label{Inst3_LogNTC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_LogN_Wass2.jpg}
\caption{2nd (Set 1, LogN)}\label{Inst3_LogN2nd}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Uni0}
\caption{TC (Set 2, $\Delta=0$)}\label{Inst3_Uni0_TC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}\label{Inst3_Uni0_2nd}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)} \label{Inst3_Uni25_2nd}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Instance 3 ($\pmb{W} \in [20, 60]$, $\pmb{N=10}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_UniN10_Inst3}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_Uni0.jpg}
\caption{TC (Set 1, LogN)}\label{Inst3_LogNTC_Range2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_Uni0.jpg}
\caption{2nd (Set 1, LogN)}\label{Inst3_LogN2nd_Range2}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_LogN.jpg}
\caption{TC (Set 2, $\Delta=0$)}\label{Inst3_Uni0_TC_Range2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_LogN.jpg}
\caption{2nd (Set 2, $\Delta=0$)}\label{Inst3_Uni0_2nd_Range2}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N10_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N10_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)} \label{Inst3_Uni50_2nd_Range2}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Instance 3 ($\pmb{W} \in [50, 100]$, $\pmb{N=10}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$). }\label{Fig3_UniN10_Inst3_Range2}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_LogN.jpg}
\caption{TC (Set 1, LogN)}\label{Lehigh1_LogNTC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_LogN.jpg}
\caption{2nd (Set 1, LogN)}\label{Lehigh1_LogN2nd}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_Uni0.jpg}
\caption{TC (Set 2, $\Delta=0$)}\label{Lehigh1_Uni0TC}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_Uni25.jpg}
\caption{TC (Set 2, $\Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N10_Uni50.jpg}
\caption{TC (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N10_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)}\label{Lehigh1_Uni50TC}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 1 ($\pmb{N=10}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_N10_Lehigh1}
\end{figure}
Let us first analyze simulation results under Set 1 (i.e., perfect distributional information case) presented in Figures~\ref{Inst3_LogNTC}--\ref{Inst3_LogN2nd}, \ref{Inst3_LogNTC_Range2}--\ref{Inst3_LogN2nd_Range2}, and \ref{Lehigh1_LogNTC}--\ref{Lehigh1_LogN2nd}. The MAD-DRO model yields a higher TC on average and at upper quantiles than the W-DRO and SP models because it schedules more MFs and thus yields a higher fixed cost (i.e., cost of establishing the MF fleet). The W-DRO model yields a slightly higher TC than the SP model because it schedules more MFs (and thus yeild a higher fixed cost). However, the DRO models yield significantly lower second-stage (transportation and unmet demand) costs on average and at all quantiles than the SP model. In addition, the MAD-DRO model yields a lower second-stage cost than the W-DRO model on average and at all quantile, especially for Lehigh 1. Note that a lower second-stage cost indicates a better operational performance (i.e., lower shortage and transportation costs) and thus has a significant practical impact. These results suggest that there are benefits to using the DRO models even when we have perfect distributional information.
We observe the following from simulation results under Set 2 (i.e., misspecified distributional information case) presented in Figures~\ref{Inst3_Uni0_TC}--\ref{Inst3_Uni25_2nd}, \ref{Inst3_Uni0_TC_Range2}--\ref{Inst3_Uni50_2nd_Range2}, and \ref{Lehigh1_Uni0TC}--\ref{Lehigh1_Uni50TC}. It is clear from these figures that the DRO models consistently outperform the SP model under all levels of variation ($\Delta$) and across the criteria of mean and all quantiles of the the total and second-stage costs. Interestingly, the DRO models yield substantially lower TC and 2nd than the SP model for Lehigh 1 and Instance 3 (with $\pmb{W} \in [50, 100]$), which have higher demand volume and variations. In addition, the MAD-DRO model yields lower second-stage costs for Instance 3 and substantially lower total and second-stage costs for Lehigh 1. Finally, the MAD-DRO solutions appear to be more stable with a significantly smaller standard deviation (i.e., variations) in the total and second-stage costs than the other considered models. The superior performance of the DRO models reflects the value of modeling uncertainty and distributional ambiguity of the demand.
\color{black}
\color{black}
Next, we investigate the value of distributional robustness from the perspective of out-of-sample disappointment, which measures the extent to which the out-of-sample cost exceeds the model's optimal value \citep{Van-Parys_et_al:2021, wang2020distributionally}. We define OPT and TC as the model's optimal value and the out-of-sample objective value, respectively. That is, OPT and TC can be considered as the estimated and actual costs of implementing the model's optimal solutions in practice, respectively. Using this notation, we define the out-of-sample disappointment as in \cite{wang2020distributionally} as follows.
\begin{align}
\max \left \{ \frac{\text{TC}-\text{OPT}}{\text{OPT}}, 0 \right \} \times 100\%.
\end{align}
A disappointment of zero indicates that the model's optimal value is equal to or larger than the out-of-sample (actual) cost (i.e., TC$\leq$OPT). This, in turn, indicates that the model is more conservative and avoids underestimating costs. In contrast, a larger disappointment implies a higher level of over-optimism because, in this case, the actual cost (TC) of implementing the optimal solution of a model is larger than the estimated cost (OPT).
Figure~\ref{Fig6_Dissappt_Lehigh1} presents the histograms of the out-of-sample disappointments of the DRO and SP models for Instance 3 ($\pmb{W} \in[50, 100]$, $N=10$) and Lehigh 1 ($N=10$) with $\Delta=0$ and $\Delta=0.25$. Notably, the DRO models yield substantially smaller out-of-sample disappointments on average and at all quantiles. However, the average and upper quantiles of the disappointments of the MAD-DRO model is smaller than the W-DRO model when $\Delta=0.25$, especially for Lehigh 1. In addition, it is clear that the average and upper quantiles of the disappointments of the SP model are relatively very large (e.g., exceeding 100\% for Lehigh 1). Finally, we observe that the out-of-sample disappointment of the MAD-DRO model is more stable than the W-DRO and SP models with a smaller standard deviation. We remark that these observations are consistent for the other considered instances, and the results with $\Delta=0.5$ are similar to those with $\Delta=0.25$. This demonstrates that the DRO model provides a more robust estimate of the actual cost that we will incur in practice.
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni0_N10_Inst3_Range2.jpg}
\caption{Instance 3, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni25_N10_Inst3_Range2.jpg}
\caption{Instance 3, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni0_N10_Lehigh1_tr2.jpg}
\caption{Lehigh 1, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Disappt_Uni25_N10_Lehigh1_tr2.jpg}
\caption{Lehigh 1, $\Delta=0.25$}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample disappointments under Set 2 with $\Delta=0$ and $\Delta=0.25$ for Instance 3 ($\pmb{W} \in[50, 100]$, $N=10$) and Lehigh 1 ($N=10$).}\label{Fig6_Dissappt_Lehigh1}
\end{figure}
\color{black}
The results in this section demonstrate that the DRO approaches are effective in an environment where the distribution is hard to estimate (ambiguous), quickly changes, or when there is a small data set on demand variability. Moreover, these results emphasize the value of modeling uncertainty and distributional ambiguity.
\subsection{\textbf{Sensitivity Analysis}}\label{sec5:sensitivity}
\noindent In this section, we study the sensitivity of DRO models to different parameter settings. Given that we observe similar results for all of the constructed instances, for presentation brevity and illustrative purposes, we present results for Instance 1 ($I$, $J$, $T$)$=$(10, 10, 10) and Instance 5 ($I$, $J$, $T$)$=$(20, 20, 10) as examples of small and relatively large instances.
First, we analyze the optimal number of active MFs as a function of the fixed cost, $f$, MF capacity, $C$, and range of demand. We fix all parameters as described in Section~\ref{sec5.1:instancegen} and solve the W-DRO and MAD-DRO models with $C\in\{ 50, \ 100, \ 150, \ 200\}$ and $f \in \{1,500 \ (\text{low}), \ 6,000 \ (\text{average}), \ 10,000 \ (\text{high})\} $ under the base range $\pmb{W} \in [20, 60]$ and $\pmb{W} \in [50, 100]$ (a higher volume of the demand). Figures \ref{Fig6:MF_vs_C_inst1} and \ref{Fig9:MF_vs_C_inst5} present the optimal number of active MFs and the associated total cost (under Set 1) for Instance 1 and Instance 5 under $\pmb{W} \in [20, 60]$, respectively. Figures \ref{Fig7:MF_vs_C_Range2_inst1}--\ref{Fig10:MF_vs_C_Range2_inst5} in Appendix~\ref{Appx:sensitivity} present the results under $\pmb{W} \in [50, 100]$.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_1500.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig6a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_1500.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig6b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_6000.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig6c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_6000.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig6d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_10000.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig6e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_10000.jpg}
\caption{Total cost, $f=10,000$}
\label{Fig6f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 1}\label{Fig6:MF_vs_C_inst1}
\end{figure}
We observe the following from these figures. First, the optimal number of scheduled MFs decreases as $C$ increases irrespective of $f$. This makes sense because, with a higher capacity, each MF can serve a larger amount of demand in each period. Second, both models schedule more MFs under $\pmb{W} \in [50, 100]$, i.e., a higher volume of the demand. For example, consider Instance 5. When $f=1,500$ and $C=100$ the (W-DRO, MAD-DRO) models schedule (10, 13) and (18, 19) MFs under $\pmb{W} \in [20, 60]$ and $\pmb{W} \in [50, 100]$, respectively. Third, the MAD-DRO model always schedules a higher number of MFs, especially when $C$ is tight. As such, the MAD-DRO model often has a slightly higher total cost (due to the higher fixed cost of establishing a larger fleet) and better second-stage cost, i.e., better operational performance (see Figures \ref{Fig8:MF_vs_C_inst1}-\ref{Fig12:MF_vs_C_range2_inst5}). For example, consider Instance 1. When $f=6,000$ and $C=50$, the W-DRO and MAD-DRO models schedule 8 and 10 MFs, respectively. The associated (total, second-stage) costs of these solutions are respectively (74,495, 26,495) and (83,004, 23,004). Finally, both models schedule fewer MFs as $f$ increases.
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_1500_Inst5.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig9a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_1500.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig9b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_6000_Inst5.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig9c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_6000_Inst5.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig9d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_10000_Inst5.jpg}
\caption{Number of MFs, $f=10,000$}
\label{Fig9e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_TC_10000_Inst5.jpg}
\caption{\textcolor{black}{Total cost, $f=10,000$}}
\label{Fig9f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 5}\label{Fig9:MF_vs_C_inst5}
\end{figure}
Second, we fix $M=20$ and solve the models with unmet demand penalty $\gamma \in \{0.10 \gamma_o, \ 0.25 \gamma_o,$ $0.35\gamma_0, \ 0.50\gamma_o\}$ (where $\gamma_o$ is the base case penalty in Section~\ref{sec5.1:instancegen}) and $f \in \{1,500, \ 6,000, \ 10,000\}$. Figure~\ref{Fig12:MF_vs_gamma} presents the number of MFs as a function of $\gamma$ and $f$. Figure~\ref{Fig13:2nd_vs_gamma} presents the associated second-stage cost. It is not surprising that as $\gamma$ increases (i.e., satisfying customer demand becomes more important), the number of scheduled MFs increases. Note that by scheduling a larger number of MFs, we could satisfy a larger amount of demand and reduce the second-stage cost (see Figure~\ref{Fig13:2nd_vs_gamma}). However, for fixed $\gamma$, the MAD-DRO model schedules more MFs, and thus yields a lower unmet demand cost (because the MAD-DRO solutions satisfy a larger amount of demand). For example, consider Instance 1 with $f=1,500$. When $\gamma$ decreases from $0.5\gamma_o$ to $0.1\gamma_o$ the optimal number of scheduled MFs of (W-DRO, MAD-DRO) decreases from (6, 6) to (1, 3) and average unmet demand cost increases from (9, 0) to (16,117, 10,827).
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_F_MF_Inst1_WDRO.jpg}
\caption{W-DRO, Instance 1}
\label{Fig12a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_F_MF_Inst1_MAD.jpg}
\caption{MAD-DRO, Instance 1}
\label{Fig12b}%
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_F_MF_Inst5_WDRO.jpg}
\caption{W-DRO, Instance 5}
\label{Fig12a2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_F_MF_Inst5_MAD.jpg}
\caption{MAD-DRO, Instance 5}
\label{Fig12b2}
\end{subfigure}%
\caption{Comparison of the results for different values of $\gamma$}\label{Fig12:MF_vs_gamma}
\end{figure}
Our experiments in this section provide an example of how decision-makers can use our DRO approaches to generate MFRSP solutions under different parameter settings. Practitioners can use these results to decide whether to adopt the MAD-DRO model (which provides a better operational and computational performance) or the W-DRO model (which provides \textcolor{black}{a lower one-time fixed cost} for establishing the MF fleet).
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma10_Inst1.jpg}
\caption{$\gamma=0.1\gamma_o$ Instance 1}
\label{Fig13a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma10_Inst5.jpg}
\caption{$\gamma=0.1\gamma_o$, Instance 5}
\label{Fig13b}%
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma25_Inst1.jpg}
\caption{$\gamma=0.25\gamma_o$, Instance 1}
\label{Fig13a2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma25_Inst5.jpg}
\caption{$\gamma=0.25\gamma_o$,Instance 5}
\label{Fig13b2}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma35_Inst1.jpg}
\caption{$\gamma=0.35\gamma_o$, Instance 1}
\label{Fig13a3}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma35_Inst5.jpg}
\caption{$\gamma=0.35\gamma_o$,Instance 5}
\label{Fig13b3}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma50_Inst1.jpg}
\caption{$\gamma=0.50\gamma_o$, Instance 1}
\label{Fig13a3}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{unmet_MF_F_2nd_gamma50_Inst5.jpg}
\caption{$\gamma=0.50\gamma_o$,Instance 5}
\label{Fig13b3}
\end{subfigure
\caption{Comparison of the second-stage cost for different values of $\gamma$.}\label{Fig13:2nd_vs_gamma}
\end{figure}
\subsection{Analysis of the risk-averse solutions}\label{sec:CVaRExp}
\noindent In this section, we analyze the optimal solutions of the mean-CVaR-based models under some critical problem parameters. Specifically, we solve the models with $f\in$\{1,500, 6,000, 10,000\}, $\gamma \in \{\gamma_o, \ 0.5 \gamma_o, \ 0.25\gamma_o, \ 0.1 \gamma_o \} $ (where $\gamma_o$ is the base case penalty in Section~\ref{sec5.1:instancegen}), $\kappa=0.95$ (a typical value of $\kappa$; we observe similar results under $\kappa=0.99, \ 0.975$), and $\Theta \in \{0, 0.2, 0.5, 1\}$ (where a smaller $\Theta$ indicate that we are more risk-averse). \textcolor{black}{We use MAD-CVaR (W-CVaR) to denote the mean-CVaR-based DRO model with MAD (1-Wasserstein) ambiguity, and SP-CVaR to denote the mean-CVaR-based SP model.} Because the SP-CVaR model cannot solve large instances (even with small $N$), for a fair comparison and brevity, we present results for Instance 1. We keep all other parameters as described in Section~\ref{sec5.1:instancegen}.
Table~\ref{tableCVaROptimal} presents the optimal number of scheduled MFs (i.e., fleet size) for different $f$, $\gamma$, and $\Theta$. We make the following observations from this table. First, all models schedule fewer MFs as $f$ increases and $\gamma$ decreases irrespective of $\Theta$ (risk-aversion coefficient). This is consistent with our results in Section~\ref{sec5:sensitivity} for the risk neutral models. Second, the MAD-CVaR model often schedules a higher number of MFs than the W-CVaR model, and the latter model schedule the same or larger number of MFs than the SP-CVaR. Second, when $f=$\{6,000, 10,000\} (i.e., average and high fixed cost) and $\gamma=0.1\gamma_o$ (very low unmet demand penalty) all models schedule one MF.
\textcolor{black}{Third, the MAD-CVaR model schedules the same number of MFs under all values of $\Theta$ when ($f$, $\gamma$)$=$ (1,500, $\gamma_o$) (and similarly under (6,000, 0.25$\gamma_o$) and (10,000, $\gamma_o$)). Similarly, the W-CVaR model schedules the same number of MFs under all values of $\Theta$ when ($f$, $\gamma$)= (6,000, 0.25$\gamma_o$), and (10,000, 0.5$\gamma_o$). These results indicate that our proposed DRO expectation models with MAD and 1-Wasserstein ambiguity are risk-averse under these settings because they yield the same optimal solutions under all values of the risk-aversion coefficient $\Theta$.}
\begin{table}[t!]
\center
\caption{Comparison of the optimal number of MFs yielded by each model under different values of $\Theta$, $f$, and $\gamma$. Notation: MAD-CVaR and W-CVaR are respectively the distributionally robust mean-CVaR models with MAD and Wasserstein ambiguity, and SP-CVaR is the SP model based on the mean-CVaR criterion.}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
\multicolumn{11}{c}{\textbf{$\pmb{f=1,500}$}}\\
\hline
& \multicolumn{4}{c}{$\gamma_o$ } && & \multicolumn{4}{c}{$0.5\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 8 & 8 & 8 & 8 & & MAD-CVaR & 8 & 8 & 8 & 7 \\
W-CVaR & 8 & 8 & 8 & 6 & & W-CVaR & 7 & 7 & 6 & 6 \\
SP-CVaR & 6 & 6 & 6 & 5 & & SP-CVaR & 6 & 6 & 6 & 5 \\
\\
\hline
& \multicolumn{4}{c}{$0.25\gamma_o$ } && & \multicolumn{4}{c}{$0.1\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 6 & 6 & 6 & 5 & & MAD-CVaR & 3 & 3 & 3 & 1 \\
W-CVaR & \textcolor{red}{6} & 5 & 5 & 5 & & W-CVaR & 2 & 2 & 2 & 1 \\
SP-CVaR & 4 & 4 & 4 & 4 & & SP-CVaR & 2 & 2 & 2 & 1 \\
\hline
\multicolumn{11}{c}{\textbf{$\pmb{f=6,000}$}}\\
\hline
& \multicolumn{4}{c}{$\gamma_o$ } && & \multicolumn{4}{c}{$0.5\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & \textcolor{black}{7} & \textcolor{black}{7} & 7 & 6 & & MAD-CVaR & 7 & 7 & 6 & 5 \\
W-CVaR & 6 & 6 & 5 & 5 & & W-CVaR & 6 & 5 & 4 & 4 \\
SP-CVaR & 5 & 5 & 5 & 5 & & SP-CVaR & 5 & 5 & 4 & 4 \\
\hline
& \multicolumn{4}{c}{$0.25\gamma_o$ } && & \multicolumn{4}{c}{$0.1\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1\\
\hline
MAD-CVaR & 3 & 3 & 3 & 3 & & MAD-CVaR & 1 & 1 & 1 & 1 \\
W-CVaR & 2 & 2 & 2 & 2 & & W-CVaR & 1 & 1 & 1 & 1 \\
SP-CVaR & 2 & 2 & 2 & 2 & & SP-CVaR & 1 & 1 & 1 & 1 \\
\hline
\multicolumn{11}{c}{\textbf{$\pmb{f=10,000}$}}\\
\hline
& \multicolumn{4}{c}{$\gamma_o$ } && & \multicolumn{4}{c}{$0.5\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 6 & 6 & 6 & 6 & & MAD-CVaR & 6 & 5 & 5 & 4 \\
W-CVaR & 6 & 6 & 6 & 5 & & W-CVaR & 4 & 4 & 4 & 4 \\
SP-CVaR & 5 & 4 & 4 & 4 & & SP-CVaR & 4 & 4 & 4 & 3 \\
\hline
& \multicolumn{4}{c}{$0.25\gamma_o$ } && & \multicolumn{4}{c}{$0.1\gamma_o$ } \\ \cline{2-5} \cline{8-11}
Model & 0 & 0.2 & 0.5 & 1 & & Model & 0 & 0.2 & 0.5 & 1 \\
\hline
MAD-CVaR & 1 & 1 & 1 & 1 & & MAD-CVaR & 1 & 1 & 1 & 1 \\
W-CVaR & 1 & 1 & 1 & 1 & & W-CVaR & 1 & 1 & 1 & 1 \\
SP-CVaR & 1 & 1 & 1 & 1 & & SP-CVaR & 1 & 1 & 1 & 1 \\
\hline
\end{tabular}\label{tableCVaROptimal}
\end{table}
Fourth, all models schedule more MFs under a smaller $\Theta$, especially when $\gamma=(0.25\gamma_o, 0.1\gamma_o)$ with $f=1,500$ (i.e., a low cost and lower unmet demand penalty), $\gamma=(0.5\gamma_o,0.25\gamma_o)$ with $f=6,000$, and $\gamma=(\gamma_o, 0.5\gamma_o)$ with $f=10,000$. In particular, more MFs are scheduled when $\Theta=0$ (risk-averse models with CVaR criterion) as compared to $\Theta=1$ (risk-neutral models). This makes sense because a risk-averse decision-maker may schedule more MFs to avoid high operational cost and, in particular, excessive shortages.
\color{black}
Next, we compare the out-of-sample operational performance (i.e., second-stage cost) and disappointment of the risk-neutral (i.e., expectation models) and risk-averse models (i.e., mean-CVaR-based models with $\Theta=0$, or equivalently, risk-averse models with CVaR criterion). We use MAD-E (W-E) to denote the risk-neutral DRO model with MAD (1-Wasserstein) ambiguity presented in Section~\ref{sec:MAD-DRO} (Section~\ref{sec:WDMFRS_model}). In addition, we use SP-E to denote the risk-neutral SP model. In figures~\ref{CVaR_gamma}, \ref{CVaR_50gamma}, and \ref{CVaR_25gamma}, we present histograms of the out-of-sample second-stage costs and disappointments under Set 2 with $f=$1,500 and $\gamma=\gamma_o$, $\gamma=0.5\gamma_o$, and $\gamma=0.25\gamma_o$, respectively. We obtained similar observations for the other considered values of $f$.
Let us first compare the performance of the risk-neutral and risk-averse SP models. Notably, the SP-E solutions have the worst performance with significantly higher second-stage costs and larger positive disappointments under all values of $\gamma$ and $\Delta$ than the other considered models. In contrast, the SP-CVaR solutions yield smaller second-stage costs and disappointments than the SP-E model. This makes sense because the SP-CVaR model schedules a larger numbers of MFs. In addition, when $\gamma=\gamma_o$ and $0.5\gamma_o$, the SP-CVaR model yields the same second-stage costs as the W-E model because both models schedule 6 MFs under these settings (this is why we do not see histograms for the second-stage cost of the SP-CVaR model). However, the W-E model controls the disappointments in a smaller range, while the SP-CVaR model yield larger positive disappointments than the W-E model and the other DRO models.
Let us now compare the performance of the risk-neutral and risk-averse DRO models. First, when $\gamma= \gamma_o$ (i.e., the largest unmet demand penalty), the MAD-CVaR, MAD-E, and W-CVaR models have the same and best performance because they schedule a larger fleet of 8 MFs than the other considered models. Second, when $\gamma=$0.5$\gamma_o$, the MAD-CVaR model has the lowest second-stage costs and zero disappointments under all values of $\Delta$. This makes sense because the MAD-CVaR model schedules a larger number of MFs than the other considered models when $\gamma=$0.5$\gamma_o$. Third, when $\gamma=0.25\gamma_o$, the MAD-CVaR and W-CVaR model yield the lowest second-stage costs and disappointments because they schedule a larger fleet (6 MFs) than the other considered models. However, the W-CVaR model yields slightly higher disappointments than the MAD-CVaR when $\Delta=0.5$.
Fourth, the second-stage costs and disappointments of the W-CVaR model are smaller than those of the W-E model because the W-CVaR model schedules a larger number of MFs. Fifth, the W-CVaR and MAD-E models yield the same second-stage costs when $\gamma=0.5 \gamma_o$ because they schedule 7 MFs (this is why we can only see black histograms for the MAD-E model). Finally, the MAD-E model have lower second-stage costs and disappointments than the W-E model under all values of $\gamma$ and $\Delta$, which is consistent with our results in Section~\ref{sec5:OutSample}.
Our results in this section demonstrate that the distributionally robust CVaR models tend to hedge against uncertainty, ambiguity, and risk by scheduling more MFs. Our results also indicate that the proposed DRO expectation models may be risk-averse under some parameter settings (e.g., high unmet demand penalty and low cost).
\color{black}
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_2nd_Uni0_f1500.jpg}
\caption{2nd, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_Dissappt_Uni0_f1500.jpg}
\caption{Disappointment, $\Delta=0$}
\end{subfigure}%
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_2nd_Uni25_f1500.jpg}
\caption{2nd, $ \Delta=0.25$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_Dissappt_Uni25_f1500.jpg}
\caption{Disappointment, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_2nd_Uni50_f1500.jpg}
\caption{2nd, $ \Delta=0.5$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_Gama_Dissappt_Uni50_f1500.jpg}
\caption{Disappointment, $\Delta=0.5$}
\end{subfigure}%
\caption{Normalized histograms of second-stage cost (2nd) and out-of-sample disappointments under Set 2 with $\gamma=\gamma_o$, $f=1,500$, and $\pmb{\Delta \in \{0, 0.25, 0.5\}}$.}\label{CVaR_gamma}
\end{figure}
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_2nd_Uni0_f1500.jpg}
\caption{2nd, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_Dissappt_Uni0_f1500.jpg}
\caption{Disappointment, $\Delta=0$}
\end{subfigure}%
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_2nd_Uni25_f1500.jpg}
\caption{2nd, $ \Delta=0.25$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_Dissappt_Uni25_f1500.jpg}
\caption{Disappointment, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_2nd_Uni50_f1500.jpg}
\caption{2nd, $ \Delta=0.5$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_50Gama_Dissappt_Uni50_f1500.jpg}
\caption{Disappointment, $\Delta=0.5$}
\end{subfigure}%
\caption{Normalized histograms of second-stage cost (2nd) and out-of-sample disappointments under Set 2 with $\gamma=0.5\gamma_o$, $f=1,500$, and $\pmb{\Delta \in \{0, 0.25, 0.5\}}$.}\label{CVaR_50gamma}
\end{figure}
\begin{figure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_2nd_Uni0_f1500.jpg}
\caption{2nd, $\Delta=0$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_Dissappt_Uni0_f1500.jpg}
\caption{Disappointment, $\Delta=0$}
\end{subfigure}%
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_2nd_Uni25_f1500.jpg}
\caption{2nd, $ \Delta=0.25$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_Dissappt_Uni25_f1500.jpg}
\caption{Disappointment, $\Delta=0.25$}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_2nd_Uni50_f1500.jpg}
\caption{2nd, $ \Delta=0.5$}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{CVaR_25Gama_Dissappt_Uni50_f1500.jpg}
\caption{Disappointment, $\Delta=0.5$}
\end{subfigure}%
\caption{Normalized histograms of second-stage cost (2nd) and out-of-sample disappointments under Set 2 with $\gamma=0.25\gamma_o$, $f=1,500$, and $\pmb{\Delta \in \{0, 0.25, 0.5\}}$.}\label{CVaR_25gamma}
\end{figure}
\section{\textcolor{black}{Conclusion}}\label{Sec:Conclusion}
\color{black}
\noindent In this paper, we propose two DRO models for the MFRSP. Specifically, given a set of MFs, a planning horizon, and a service region, our models aim to find the number of MFs to use within the planning horizon and a route and schedule for each MF in the fleet. The objective is to minimize the fixed cost of establishing the MF fleet plus a risk measure (expectation or mean-CVaR) of the operational cost over all demand distributions defined by an ambiguity set. In the first model (MAD-DRO), we use an ambiguity set based on the demand's mean, support, and mean absolute deviation. In the second model (W-DRO), we use an ambiguity set that incorporates all distributions within a 1-Wasserstein distance from a reference distribution. To solve the proposed DRO models, we propose a decomposition-based algorithm. We also derive lower bound inequalities and two families of symmetry breaking constraints to improve the solvability of the proposed models.
\color{black}
Our computational results demonstrate (1) how the DRO approaches have superior operational performance in terms of satisfying customers demand as compared to the SP approach, (2) the MAD-DRO model is more computationally efficient than the W-DRO model, (3) the MAD-DRO model yield more conservative decisions than the W-DRO model, which often have a higher fixed cost but significantly lower operational cost, (4) how mobile facilities can move from one location to another to accommodate the change in demand over time and location, (5) efficiency of the proposed symmetry breaking constraints and lower bound inequalities, (6) the trade-off between cost, number of MFs, MF capacity, and operational performance, and (7) the trade-off between the risk-neutral and risk-averse approaches. Most importantly, our results show the value of modeling uncertainty and distributional ambiguity.
Note that we have used benchmark instances from the literature in our computational experiments, which may be a limitation of our results. However, in the sensitivity analysis section, we tested the proposed approaches under different parameters settings, demonstrating how decision-makers can use our approaches to generate MFRSP solutions under different parameter settings relevant to their specific application. Moreover, these benchmark instances represent a wide range of potential service regions, which we can efficiently solve. If we account for the scale of the problem in the sense of static facility location problem, we have demonstrated that we can solve instances of $30 \times 20= 600$ customers (instance 10), which are relatively large for many practical applications.
We suggest the following areas for future research. First, we aim to extend our models to optimize the capacity and size of the MF fleet. Second, we want to extend our approach by incorporating multi-modal probability distributions and more complex relationships between random parameters (e.g., correlation). Third, we aim to extend our approach to more comprehensive MF planning models, which consider all relevant organizational and technical constraints and various sources of uncertainties (e.g., travel time) with a particular focus on real-life healthcare settings. Although conceptually and theoretically advanced, stochastic optimization approaches such as SP and DRO are not intuitive or transparent to decision-makers who often do not have optimization expertise. Thus, future efforts should also focus on closing the gap between theory and practice.
\vspace{2mm}
\ACKNOWLEDGMENT{%
We want to thank all colleagues who have contributed significantly to the related literature. We are grateful to the anonymous reviewers for their insightful comments and suggestions that allowed us to improve the paper. Special thanks to Mr. Man Yiu Tsang (a Ph.D. student at the Department of Industrial and Systems Engineering, Lehigh University) for helping with Figure~\ref{LehighMap} and proofreading the paper. Dr. Karmel S. Shehadeh dedicates her effort in this paper to every little dreamer in the whole world who has a dream so big and so exciting. Believe in your dreams and do whatever it takes to achieve them--the best is yet to come for you.}
\vspace{1mm}
\newpage
\begin{APPENDICES}
\section{Derivation of feasible region $\mathcal{X}$ in \eqref{eq:RegionX}} \label{Appx:FirstStageDec}
\noindent In this Appendix, we provide additional details on the derivation of the constraints defining the feasible region $\mathcal{X}$ of variables ($\pmb{x},\pmb{y}$). As described in \cite{lei2014multicut}, we can enforce the requirement that an MF can only be in service when it is stationary using the following constraints:
\begin{align}\label{eq1}
& x_{j,m}^t+x_{j',m}^t \leq 1, && \forall t, m, j, \ j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace.
\end{align}
\noindent If $x_{j,m}^t=1$ (i.e., MF $m$ is stationary at some location $j$ in period $t$), it can only be available at location $j'\neq j$ after a certain period, depending on the time it takes to travel from location $j$ to location $j'$. It follows by \eqref{eq1} that $ x_{j',m}^{t'}=0$ for all $j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace$. As pointed out by \cite{lei2014multicut}, this indicates that \textit{an earlier decision of deploying an MF at one candidate location would directly affect future decisions both temporally and spatially. In fact, this correlation is a major source of complexity for optimizing the MFRSP.}
Since the MF has to be in an active condition before providing
service, we have to include the following constraints:
\begin{align}\label{eq2}
x_{j,m}^t \leq y_m, && \forall j,m,t.
\end{align}
It is straightforward to verify that constraint sets \eqref{eq1} and \eqref{eq2} can be combined into the following compact form:
\begin{align}
\mathcal{X}&=\left\{ (\pmb{x}, \pmb{y}) : \begin{array}{l} x_{j,m}^t+x_{j^\prime,m}^{t^\prime} \leq y_m, \ \ \forall t, m, j, \ j\neq j^\prime, \ t^\prime \in \lbrace t, \ldots, \min \lbrace t+t_{j,j^\prime}, T\rbrace \rbrace \\
x_{j,m}^t \in \lbrace 0, 1\rbrace, \ y_m \in \lbrace 0, 1\rbrace, \ \forall j, m , t \end{array} \right\} .
\end{align}
\section{Proof of Proposition 1 }\label{Appx:ProofofProp1}
\noindent \textit{Proof}. For a fixed $x \in \mathcal{X}$, we can explicitly write the inner problem $\sup [ \cdot ]$ in \eqref{MAD-DMFRS} as the following functional linear optimization problem.
\begin{subequations}
\begin{align}
\max_{\mathbb{P}\geq 0} & \int_{\mathcal{S}} Q(\pmb{x},\pmb{W}) \ d \mathbb{P} \\
\ \text{s.t.} & \ \int_{\mathcal{S}} W_{i,t} \ d\mathbb{P}= \mu_{i,t} \quad \quad \forall i \in I, \ t \in T, \label{ConInner:W}\\
& \int_{\mathcal{S}} |W_{i,t}-\mu_{i,t}| \ d\mathbb{P}\leq \eta_{i,t} \quad \quad \forall i \in I, \ t \in T, \label{ConInner:K}\\
& \ \int_{\mathcal{S}} d\mathbb{P}= 1. \label{ConInner:Distribution}
\end{align} \label{InnerMax2}
\end{subequations}
Letting $\rho_{i,t}, \psi_{i,t}$ and $\theta$ be the dual variables associated with constraints \eqref{ConInner:W}, \eqref{ConInner:K}, \eqref{ConInner:Distribution}, respectively, we present problem \eqref{InnerMax2} (problem (9) in the main manuscript) in its dual form:
\begin{subequations}
\begin{align}
& \min_{\pmb{\rho, \theta, \psi \geq 0}} \ \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+ \theta \label{DualInner:Obj} \\
& \ \ \text{s.t.} \ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})+ \theta \geq Q(\pmb{x},\pmb{W}) && \forall \pmb{W}\in \mathcal{S}, \label{DualInner:PrimalVariabl}
\end{align} \label{DualInnerMax}
\end{subequations}
where $\pmb{\rho}$ and $\theta$ are unrestricted in sign, $\pmb \psi \geq 0$, and constraint \eqref{DualInner:PrimalVariabl} is associated with the primal variable $\mathbb{P}$. Note that for fixed ($\pmb{\rho, \ \psi,} \ \theta$), constraint \eqref{DualInner:PrimalVariabl} is equivalent to
$$\theta \geq \max \limits_{\pmb{W} \in \mathcal{S} } \Big \lbrace Q(\pmb{x},\pmb{W}) + \sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big\rbrace. $$
Since we are minimizing $\theta$ in \eqref{DualInnerMax}, the dual formulation of \eqref{InnerMax2} is equivalent to:
\begin{align*}
& \min_{\pmb{\rho}, \pmb{\psi} \geq 0} \ \left \lbrace \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+\max \limits_{\pmb{W} \in \mathcal{S}} \Big\lbrace Q(\pmb{x},\pmb{W}) + \sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})\Big \rbrace \right \rbrace .
\end{align*}
\section{Proof of Proposition 2}\label{Appx:Prop2}
\noindent First, note that the feasible region $\Omega=\{\eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner}\}$ and $\mathcal{S}$ are both independent of $\pmb{x}$, $\pmb \rho$, and $\pmb \psi$ and bounded. In addition, the MFRSP has a complete recourse (i.e., the recourse problem is feasible for any feasible $(\pmb{x},\pmb{y}) \in \mathcal{X}$). Therefore, $\max\limits_{\pmb{\lambda, v, W, \pi, k} }\Big[ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big]< +\infty$. Second, for any fixed $\pmb{\pi, v, W, k}$, $\Big [\sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big] $ is a linear function of $\pmb{x}$, $\pmb \rho$, and $\pmb \psi$. It follows that $\max\limits_{\pmb{\lambda, v, W, \pi, k} }\Big[ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big]$ is the maximum of linear functions of $\pmb{x}$, $\pmb \rho$, and $\pmb \psi$, and hence convex and piecewise linear. Finally, it is easy to see that each linear piece of this function is associated with one distinct extreme point of $\Omega$ and $\mathcal{S}$. Given that each of these polyhedra has a finite number of extreme points, the number of pieces of this function is finite. This completes the proof.
\section{Number of Points in The Worst-Case Distribution of MAD-DRO}\label{Appx:Points_MAD}
\color{black}
The results of \cite{longsupermodularity} indicate that if the second-stage optimal value is supermodular in the realization of uncertainties under the MAD ambiguity set, the worst-case is a distribution supported on ($2n+1$) points, where $n$ is the dimension of the random vector. In this Appendix, we show that even if our recourse is supermodular in demand realization, the number of points in the worst-case distribution of the demand is large, which renders our two-stage MAD-DRO model computationally challenging to solve using \cite{longsupermodularity}' approach
Recall that the demand is indexed by $i$ and $t$, i.e., $W_{i,t}$, for all $i \in I$ and $t \in T$. Thus, the dimension of our random vector is $|I | |T|$. Accordingly, assuming that the second-stage optimal value is supermodular, then the resullts of \cite{longsupermodularity} suggest that the worst-case distribution in MAD-DRO has ($2|I| |T|$+1) points or scenarios. Note that $|I|$ and $|T|$ and thus $2|I||T|$ is large for most instances of our problem (See Example 1--2 below). Since our computational results and prior literature indicate that solving the scenario-based model using a small set of scenarios is challenging, solving a reformulation of MAD-DRO using the ($2|I||T|$+1) points is expected to be computationally challenging.
\noindent \textbf{Example 1.} Instance 6 ($I=20$ and $T=20$). The worst-case distribution has 801 points.
\noindent \textbf{Example 3.} Instance 10 ($I=30$ and $T=20$). The worst-case distribution has 1201 points.
\color{black}
\newpage
\section{Proof of Proposition~\ref{Prop1}}\label{Proof_Prop1}
Recall that $\hat{\mathbb{P}}^N=\frac{1}{N}\sum_{n=1}^n \delta_{\hat{\pmb{W}}^n}$. The definition of Wasserstein distance indicates that there exist a joint distribution $\Pi$ of $(\pmb{W}, \hat{\pmb{W}}$) such that $\mathbb{E}_{\Pi} [||\pmb{W}-\hat{\pmb{W}}||] \leq \epsilon$. In other words, for any $\mathbb{P}\in\mathcal{P}(\mathcal{S})$, we can rewrite any joint distribution $\Pi\in\mathcal{P}(\mathbb{P},\hat{\mathbb{P}}^N)$ by the conditional distribution of $\pmb{W}$ given $\hat{\pmb{W}}=\hat{\pmb{W}}^n$ for $n=1,\dots,N$, denoted as $\F^n$. That is, $\Pi=\frac{1}{N}\sum_{n=1}^N \F^n \times \delta_{\hat{\xi}_n}$. Notice that if we find one joint distribution $\Pi\in\mathcal{P}(\mathbb{P},\hat{\mathbb{P}}^N)$ such that $\int ||\pmb{W}-\hat{\pmb{W}}|| \ d\Pi \leq \epsilon$, then $\text{dist}(\mathbb{P},\hat{\mathbb{P}}^N)\leq\epsilon$. Hence, we can drop the infimum operator in Wasserstein distance and arrive at the following equivalent problem.
\begin{subequations}\label{AppxInner}
\begin{align}
& \sup_{\F^n\in\mathcal{P}(\mathcal{S}), n\in[N]} \frac{1}{N}\sum_{n=1}^N\int_\mathcal{S} Q(\pmb{x},\pmb{W}) \ d\F^n\\
& \text{s.t.} \ \ \ \ \ \ \ \ \frac{1}{N} \sum_{n=1}^N \int_\mathcal{S} ||\pmb{W}-\hat{\pmb{W}}^n|| \ d\F^n \leq \epsilon.
\end{align}
\end{subequations}
Using a standard strong duality argument and letting $\rho\geq 0$ be the dual multiplier, we can reformulate problem \eqref{AppxInner} by its dual, i.e.
\begin{align}
&\quad\,\inf_{\rho\geq 0} \sup_{\F^n\in\mathcal{P}(\mathcal{S}),n\in[N]}\left\{\frac{1}{N}\sum_{n=1}^N\int_\mathcal{S} Q(\pmb{x},\pmb{W}) d\F^n+\rho\left[\epsilon-\frac{1}{N} \sum_{n=1}^N \int_\mathcal{S} ||\pmb{W}-\hat{\pmb{W}}^n||\ d\F^n \right] \right\}\nonumber\\
&=\inf_{\rho\geq 0} \Big\{ \epsilon \rho+ \frac{1}{N}\sum_{n=1}^N \sup_{\F^n\in\mathcal{P}(\mathcal{S})}\int_\mathcal{S} \left[Q(\pmb{x},\pmb{W})-\rho||\pmb{W}-\hat{\pmb{W}}^n||\right] d\F^n\Big\} \nonumber\\
&=\inf_{\rho\geq 0} \Big\{ \epsilon \rho + \frac{1}{N} \sum_{n=1}^N \sup_{\pmb{W} \in \mathcal{S}} \{ Q(\pmb{x},\pmb{W})-\rho || \pmb{W} -\hat{\pmb{W}}^n || \} \Big\}.
\end{align}
\newpage
\section{DRO with mean-CVaR as a Risk Measure}\label{Sec:meanCVAR}
\noindent Both the MAD-DRO and W-DRO model presented in Section \ref{sec:DRO_Models} assume that the decision-maker is risk-neutral (i.e., adopt the expected value of the recourse as a risk measure). In some applications of the MFRSP, however, decision-makers might be risk-averse. Therefore, as one of our reviewers suggested, in this section, we present a distributionally robust risk-averse model for the MFRSP.
To model the decision maker's risk aversion, most studies adopt the CVaR, i.e., set $\varrho(\cdot)=\CVAR_\kappa(\cdot)$, where $\kappa\in(0,1)$. CVaR is the conditional expectation of $Q(\cdot)$ above the value-at-risk VaR (informally, VaR is the $\kappa$ quantile of the distribution of $Q(\cdot)$, see \cite{pacc2014robust, rockafellar2002conditional, sarykalin2008value, van2015distributionally}). CVaR is a popular coherent risk measure widely used to avoid solutions influenced by a bad scenario with a low probability. However, as pointed out by \cite{wang2021two} and a reviwer of this paper, neither expected value nor CVaR can capture the variability of uncertainty in a comprehensive manner. Alternatively, we consider minimizing the mean-CVaR, which balances the cost on average and avoids high-risk levels. As pointed out by \cite{lim2011conditional}, \cite{wang2021two}, and our reviewer, the traditional CVaR criterion is sensitive to the misspecification of the underlying loss distribution and lacks robustness. Therefore, we propose a distributionally robust mean-CVaR model to remedy such fragility, reflecting both risk-averse and ambiguity-averse attitudes. For brevity, we use the MAD ambiguity set to formulate and analyze this model because similar formulation and reformulation steps can be used to derive a solvable mean-CVaR-based model based on the 1-Wasserstein ambiguity.
Let us now introduce our distributionally robust mean-CVaR-based model with MAD-ambiguity (MAD-CVaR). First, following \cite{rockafellar2000optimization, rockafellar2002conditional}, and \cite{ van2015distributionally}, we formally define CVaR as
\begin{align}
\CVAR_\kappa (Q(\pmb{x},\pmb{W}))= \inf \limits_{\zeta \in \mathbb{R}} \Big \{ \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Big\}, \label{CVAR_Def_main}
\end{align}
\noindent where $[c]^+:=\max \{c,0\}$ for $c \in \mathbb{R}$. Parameter $\kappa$ measures a wide range of risk preferences, where $\kappa=0$ corresponds to the risk-neutral formulation. In contrast, when $\kappa \rightarrow 1$, the decision-makers become more risk-averse. Using \eqref{CVAR_Def_main}, we formulate the following MAD-CVaR model (see, e.g., \cite{wang2021two} for a recent application in facility location):
\begin{align}
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Bigg[ \Theta \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})] + (1-\Theta) \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W})) \Bigg] \Bigg \}, \label{CVaR-DMFRS}
\end{align}
\noindent where $0 \leq \Theta \leq 1$ is the risk-aversion coefficient, which represents a trade-off between the risk-neutral (i.e., $\mathbb{E}[\cdot]$) and risk-averse (i.e., $\CVAR(\cdot)$) objectives. A larger $\Theta$ implies less aversion to risk, and vice verse. In extreme cases, when $\Theta =1$, the decision maker is risk-neutral and \eqref{CVaR-DMFRS} reduces to the MAD-DRO expectation model in \eqref{MAD-DMFRS}. When $\Theta =0$, the decision maker is risk and ambiguity averse.
Next, we derive a solvable reformulation of \eqref{CVaR-DMFRS}. Let us first consider the inner maximization problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W})) $ in \eqref{CVaR-DMFRS}. It is easy to verify that
\begin{subequations}\label{SupCVaR}
\begin{align}
\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W}))&= \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \inf \limits_{\zeta \in \mathbb{R}} \Bigg \{ \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Bigg\} \label{CVAR_alter1} \\
&= \inf \limits_{\zeta \in \mathbb{R}} \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \Bigg \{ \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Bigg\} \label{CVAR_alter2} \\
&= \inf \limits_{\zeta \in \mathbb{R}} \Bigg \{ \zeta+ \frac{1}{1-\kappa} \sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Bigg\}. \label{CVAR_alter3} \
\end{align}
\end{subequations}
\noindent Interchanging the order of $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})}$ and $\inf \limits_{\zeta \in \mathbb{R}} $ follows from Sion's minimax theorem \citep{sion1958general} because $\zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+$ is convex in $\zeta$ and concave in $\mathbb{P}$. Next, we apply the same techniques in Section~\ref{sec3:reform} and Appendix~\ref{Appx:ProofofProp1} to reformulate the inner maximization problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+$ in \eqref{CVAR_alter3} as a minimization problem and combine it with the outer minimization problem to obtain
\begin{subequations}
\begin{align}
\inf \limits _{\zeta, \pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{\zeta+ \frac{1}{1-\kappa} \Bigg [\sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+ \theta \Bigg] \Bigg\} \label{CVAR_Obj} \\
\text{s.t.} & \ \ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})+ \theta \geq Q(\pmb{x},\pmb{W})-\zeta, && \forall \pmb{W}\in \mathcal{S} , \label{CVAR_C1} \\
& \ \ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t})+ \theta \geq 0, && \forall \pmb{W}\in \mathcal{S}, \label{CVAR_C2}
\end{align}
\end{subequations}
\noindent where \eqref{CVAR_C1} and \eqref{CVAR_C2} follows from the definition of $[\cdot]^+$. Accordingly, problem \eqref{SupCVaR} (equivalently last term in formulation in \eqref{CVaR-DMFRS}) is equivalent to
\begin{subequations}\label{CVARmodel_objective}
\begin{align}
\inf \limits_{\zeta, \pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{\zeta+ \frac{1}{1-\kappa} \Bigg [\sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t})+ \theta \Bigg] \Bigg\} \label{CVAR_Obj2} \\
\text{s.t.} & \ \ \zeta \geq \max_{\pmb{W} \in \mathcal{S}} \Big \{ Q(\pmb{x},\pmb{W})- \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big\}-\theta, \label{CVAR_equiv2}\\
& \ \ \min_{\pmb{W} \in \mathcal{S}} \Big \{ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big \} + \theta \geq 0. \label{CVAR_equiv3}
\end{align}
\end{subequations}
\textcolor{black}{Since we are minimizing $\zeta$ in \eqref{CVAR_Obj2}, we can equivalently re-write \eqref{CVAR_Obj2} as}
\color{black}
\begin{align}
\inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \Bigg \{ \max_{\pmb{W} \in \mathcal{S}} \Big \{ Q(\pmb{x},\pmb{W})- \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big\}+ \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\} \nonumber \\
\equiv \inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\}, \label{CVAR_Obj3}
\end{align}
\noindent where \eqref{CVAR_Obj3} follows from the derivation of $h(\pmb{x}, \pmb \rho, \pmb \psi) \equiv \max\limits_{\pmb{W} \in \mathcal{S}} \big \{ Q(\pmb{x},\pmb{W})- \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \big \}$ in Section~\ref{sec3:reform}. Accordingly, problem \eqref{CVARmodel_objective} is equivalent to
\begin{subequations}\label{CVARmodel_objective4}
\begin{align}
\inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\} \label{CVAR_Obj4} \\
\text{s.t.} & \ \ \min_{\pmb{W} \in \mathcal{S}} \Big \{ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ |W_{i,t}-\mu_{i,t}| \psi_{i,t}) \Big \} + \theta \geq 0. \label{CVAR_equiv4}
\end{align}
\end{subequations}
Next, derive equivalent linear constraints of the embedded minimization problem in constraint \eqref{CVAR_equiv4}. For fixed $\pmb \rho$ and $\pmb \psi$, we re-write constraint \eqref{CVAR_equiv4} as
\begin{subequations}\label{CVAR_equiv2_reform}
\begin{align}
\theta + & \min_{\pmb{W}} \ \ \Big \{ \sum \limits_{t \in T} \sum \limits_{i \in I} (W_{i,t} \rho_{i,t}+ k_{i,t} \psi_{i,t}) \Big \} \geq 0, \label{CVAR_equiv2_reform_C1}\\
& \text{s.t.} \ \ \ W_{i,t} \leq \overline{W}_{i,t}, && \forall i, t, \label{CVAR_equiv2_reform_C2}\\
& \ \ \ \ \ \ \ W_{i,t} \geq \underline{W}_{i,t}, && \forall i, t, \label{CVAR_equiv2_reform_C22}\\
& \ \ \ \ \ \ \ k_{i,t} \geq W_{i,t}-\mu_{i,t}, && \forall i, t, \label{CVAR_equiv2_reform_C3}\\
& \ \ \ \ \ \ \ k_{i,t} \geq \mu_{i,t}-W_{i,t}, && \ \forall i, t.\label{CVAR_equiv2_reform_C4}
\end{align}
\end{subequations}
\noindent Letting $a_{i,t}$, $b_{i,t}$, $g_{i,t}$, and $o_{i,t}$ be the dual variables associated with constraints \eqref{CVAR_equiv2_reform_C2}, \eqref{CVAR_equiv2_reform_C22}, \eqref{CVAR_equiv2_reform_C3}, and \eqref{CVAR_equiv2_reform_C4}, respectively, we present the linear program in \eqref{CVAR_equiv2_reform_C1}-\eqref{CVAR_equiv2_reform_C4} in its dual form as
\begin{subequations}
\begin{align}
& \theta + \sum_{i \in I} \sum_{t \in T} \Big [ \overline{W}_{i,t}a_{i,t}+ \underline{W}_{i,t} b_{i,t} -\mu_{i,t}g_{i,t}+\mu_{i,t}o_{i,t} \Big] \geq 0, \label{dual_equiv2_C1} \\
&a_{i,t}+b_{i,t}-g_{i,t}+o_{i,t} \leq \rho_{i,t}, && \forall i, t , \label{dual_equiv2_C2} \\
&g_{i,t}+o_{i,t} \leq \psi_{i,t}, && \forall i, t, \label{dual_equiv2_C3} \\
& (b_{i,t}, g_{i,t}, \ o_{i,t}) \geq 0, \ a_{i,t} \leq 0, && \forall i, t. \label{dual_equiv2_C4}
\end{align}
\end{subequations}
\noindent Replacing \eqref{CVAR_equiv4} in \eqref{CVARmodel_objective4} with \eqref{dual_equiv2_C1}--\eqref{dual_equiv2_C4}, we derive the following equivalent reformulation of problem \eqref{CVARmodel_objective4} (equivalently, problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W}))$ in \eqref{CVaR-DMFRS}):
\begin{subequations}\label{CVARmodel_obj5}
\begin{align}
\inf \limits_{\pmb{\rho}, \theta, \pmb{\psi} \geq 0} & \ \ \Bigg \{ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg\} \label{CVAR_Obj5} \\
\text{s.t.} & \ \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}.
\end{align}
\end{subequations}
\noindent Combining the inner problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \CVAR (Q(\pmb{x},\pmb{W}))$ in the form of \eqref{CVARmodel_obj5} and problem $\sup \limits_{\mathbb{P} \in \mathcal{F}(\mathcal{S}, \pmb{\mu}, \pmb{\eta})} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})]$ in the form of \eqref{FinalDR} with the outer minimization problem in \eqref{CVaR-DMFRS}, we derive the following equivalent reformulation of the MAD-CVaR model in \eqref{CVaR-DMFRS}.
\begin{subequations}\label{Final_CVaR}
\begin{align}
\min & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \Theta \Bigg [ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big(\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big) + h(\pmb{x}, \pmb \rho, \pmb \psi) \Bigg] \nonumber \\
& \ \ + (1-\Theta) \Bigg [ h(\pmb{x}, \pmb \rho, \pmb \psi) + \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg] \Bigg \} \label{CVaR-DMFRS3} \\
\text{s.t.} & \ \ \ (\pmb{x}, \pmb{y})\in \mathcal{X}, \ \pmb \psi \geq 0, \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}.
\end{align}
\end{subequations}
\noindent It is easy to verify that problem \eqref{Final_CVaR} is equivalent to
\begin{subequations}\label{Final_CVaR2}
\begin{align}
\min & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \Theta \Bigg [ \sum \limits_{t \in T} \sum \limits_{i \in I} \Big(\mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}\Big) \Bigg] \nonumber \\
& \ \ + \delta + (1-\Theta) \Bigg [ \big (\frac{\kappa}{1-\kappa} \big) \theta + \frac{1}{1-\kappa} \sum \limits_{t \in T} \sum \limits_{i \in I}( \mu_{i,t} \rho_{i,t}+\eta_{i,t} \psi_{i,t}) \Bigg] \Bigg \} \label{CVaR-DMFRS4_1} \\
\text{s.t.} & \ \ \ (\pmb{x}, \pmb{y})\in \mathcal{X}, \ \pmb \psi \geq 0, \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}, \\
& \ \ \ \delta \geq h(\pmb{x}, \pmb \rho, \pmb \psi), \label{CVaR-DMFRS4_2}
\end{align}
\end{subequations}
where $\textcolor{black}{h(\pmb{x}, \pmb \rho, \pmb \psi)}=\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}): \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner} \Big\} $ from Section~\ref{sec3:reform}. Finally, we observe that the right-hand side (RHS) of constraints \eqref{CVaR-DMFRS4_2} is equivalent to the RHS of constraints \eqref{const1:FinalDR} in the equivalent reformulation of the risk-neutral MAD-DRO model in \eqref{FinalDR}. Therefore, we can easily adapt Algorithm 1 to solve \eqref{Final_CVaR2} (see Algorithm~\ref{Alg2:Decomp2}).
\begin{algorithm}[t!]
\color{black}
\small
\renewcommand{\arraystretch}{0.3}
\caption{Decomposition algorithm for the MAD-CVaR Model.}
\label{Alg2:Decomp2}
\noindent \textbf{1. Input.} Feasible region $\mathcal{X}$; support $\mathcal{S}$; set of cuts $ \lbrace L (\pmb{x}, \delta)\geq 0 \rbrace=\emptyset $; $LB=-\infty$ and $UB=\infty.$
\vspace{2mm}
\noindent \textbf{2. Master Problem.} Solve the following master problem
\begin{subequations}\label{MastCVAR}
\begin{align}
\min \quad & \Bigg\{\sum_{m \in M} f y_{m}-\sum_{t \in T} \sum_{j \in J} \sum_{m \in M} \alpha x_{j, m}^{t}+\Theta\left[\sum_{t \in T} \sum_{i \in I}\left(\mu_{i, t} \rho_{i, t}+\eta_{i, t} \psi_{i, t}\right)\right]\nonumber \\
& \left.+(1-\Theta)\left[\left(\frac{\kappa}{1-\kappa}\right) \theta+\frac{1}{1-\kappa} \sum_{t \in T} \sum_{i \in I}\left(\mu_{i, t} \rho_{i, t}+\eta_{i, t} \psi_{i, t}\right)\right] +\delta \right\} \\
\text{s.t.} & \qquad (\pmb{x},\pmb{y} ) \in \mathcal{X}, \ \ \pmb{\psi} \geq 0, \ \eqref{dual_equiv2_C1}-\eqref{dual_equiv2_C3}, \ \{ L (\pmb{x}, \delta)\geq 0 \},
\end{align}%
\end{subequations}
$\ \ \ $ and record an optimal solution $(\pmb{x}^ *, \pmb{\rho}^*, \pmb{\psi}^*, \delta^*)$ and optimal value $Z^*$. Set $LB=Z^*$.
\noindent \textbf{3. Sub-problem.}
\begin{itemize}
\item[3.1.] With $(\pmb{x}, \pmb{\rho}, \pmb{\psi})$ fixed to $(\pmb{x}^*, \pmb{\rho}^*, \pmb{\psi}^*)$, solve the following problem
\begin{subequations}\label{MILPSep2}
\begin{align}
h(\pmb{x}, \pmb \rho, \pmb \psi)= &\max \limits_{\pmb{\lambda, v, W, \pi, k} } \ \Big\{ \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^t v_{j,m}^t +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t} \rho_{i,t}+k_{i,t}\psi_{i,t}) \Big\}\\
& \ \ \ \text{s.t. } \ \eqref{Const1:QudalX}-\eqref{Const3:Qdualv}, \eqref{McCormick1}-\eqref{McCormick2}, \eqref{Const2_finalinner},
\end{align}
\end{subequations}
$\qquad \ $ and record optimal solution $(\pmb{\pi^*, \lambda^*, W^*, v^*, k^*})$ and $h(\pmb{x}, \pmb \rho, \pmb \psi)^*$.
\item[3.2.] Set $UB=\min \{ UB, \ h(\pmb{x}, \pmb \rho, \pmb \psi)^*+ (LB-\delta^*) \}$.
\end{itemize}
\noindent \textbf{4. if} $\delta^* \geq \sum \limits_{t \in T} \sum\limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum\limits_{j \in J} \sum \limits_{m \in M} Cx_{j,m}^{t*} v_{j,m}^{t*} +\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}^*+k_{i,t}^*\psi_{i,t}^*) $ \textbf{then}
$\qquad \ \ $ stop and return $\pmb{x}^*$ and $\pmb{y}^*$ as the optimal solution to problem \eqref{MastCVAR} (equivalently, \eqref{Final_CVaR2}).
\noindent $\ \ $ \textbf{else} add the cut $\delta \geq \sum \limits_{t \in T} \sum \limits_{i \in I} \pi_{i,t}^*+ \sum\limits_{t \in T} \sum \limits_{j \in J} \sum\limits_{m \in M} Cx_{j,m} v_{j,m}^{t*}+\sum \limits_{t \in T} \sum \limits_{i \in I} -(W_{i,t}^* \rho_{i,t}+k_{i,t}^*\psi_{i,t} )$ to the set of
$\qquad \quad$ cuts $ \{L (\pmb{x}, \delta) \geq 0 \}$ and go to step 2.
\noindent $\ \ $ \textbf{end if}
\end{algorithm}
\color{black}
\newpage
\section{Two-stage SP Model with Mean-CVaR Objective}\label{Appex:SP_CVAR}
\noindent The mean-CVaR-based SP model (denoted as SP-CVaR) is as follow:
\begin{align}
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}} & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Theta \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W}))] + (1-\Theta) \CVAR (Q(\pmb{x},\pmb{W})) \Bigg \} \nonumber\\
\equiv &\nonumber \\
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}, \zeta \in \mathbb{R}^+} &\Big \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Theta \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W}))]+ (1-\Theta) \Big ( \zeta+ \frac{1}{1-\kappa} \mathbb{E}_\mathbb{P} [Q(\pmb{x},\pmb{W})-\zeta ]^+ \Big) \Bigg\}. \label{CVaR-SP}
\end{align}
\noindent The sample average deterministic equivalent of \eqref{CVaR-SP} based on $N$ scenarios, $\pmb{W}^1,\ldots, \pmb{W}^N$, is as follows:
\begin{subequations}
\begin{align}
\min \limits_{(\pmb{x}, \pmb{y})\in \mathcal{X}, \zeta \in \mathbb{R}^+} & \Bigg \{ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t + \Theta \sum_{n=1}^N \frac{1}{N} Q(\pmb{x},\pmb{W}^n) + (1-\Theta) \Big ( \zeta+ \frac{1}{1-\kappa} \sum_{n=1}^N \frac{1}{N} \Phi^n \Big) \Bigg\} \\
\text{s.t. } & \ \Phi^n \geq Q(\pmb{x},\pmb{W}^n) -\zeta, \qquad \forall n \in [N],\\
& \ \Phi^n \geq 0, \qquad \forall n \in [N],
\end{align}
\end{subequations}
\noindent where for each $n \in [N]$, $Q(\pmb{x}, \pmb{W}^n)$ is the recourse problem defined in \eqref{2ndstage}, and $\Theta \in [0, 1]$/
\section{Proof of Proposition~\ref{Prop3:LowerB1}}\label{Appx:Prop3}
\noindent Recall from the definition of the support set that the lowest demand of each customer $i$ in period $t$ equals to the integer parameter $\underline{W}_{i,t}$. Now, if we treat the MFs as uncapacitated facilities, then we can fully satisfy $\underline{W}_{i,t}$ at the lowest assignment cost from the nearest location $j \in J^\prime$, where $J^\prime:= \lbrace j: x_{j,m}^t=1 \rbrace$. Note that $J^\prime \subseteq J$. Thus, the lowest assignment cost must be at least equal to or larger than $\sum_{i \in I} \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \underline{W}_{i,t} $. If $\gamma < \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace$, then the recourse must be at least equal to or larger than $ \sum_{i \in I} \gamma \underline{W}_{i,t}$. Accordingly, $\sum \limits_{i \in I} \min \lbrace \gamma, \min \limits_{j \in J} \lbrace \beta d_{i,j} \rbrace \rbrace \underline{W}_{i,t}$ is a valid lower bound on recourse $Q(\pmb{x}, \pmb{W})$ for each $t \in T$.
\section{Sample Average Approximation}\label{Appx:SAA}
\begin{subequations}\label{SPModel}
\begin{align}
& \min \Big [ \sum_{m \in M} f y_{m} -\sum_{t \in T} \sum_{j \in J} \sum_{m \in M } \alpha x_{j,m}^t+ \frac{1}{N} \sum_{n=1}^N\Big( \sum_{j \in J} \sum_{i \in I} \sum_{m \in M} \sum_{t \in T} \beta d_{i,j} z_{i,j,m}^{t,n}+ \gamma \sum_{t \in T} \sum_{i \in I} u_{i,t}^n \Big) \Big]\label{ObjSP}\\
& \ \text{s.t.} \ \ (\pmb{x}, \pmb{y}) \in \mathcal{X}, \\
&\quad \ \quad \ \ \sum_{j \in J} \sum_{m \in M} z_{i,j,m}^{t,n}+u_{i,t}^n= W_{i,t}^n, \qquad \forall i \in I, \ t \in T , \ n \in [N], \\
&\quad \ \quad \quad \sum_{i \in I} z_{i,j,m}^{t,n}\leq C x_{j,m}^t \qquad \forall j \in J, \ m \in M, \ t \in T, \ n \in [N], \\
&\quad \ \quad \quad u_{i,t}^n \geq 0, \ z_{i,j,m}^{t,n} \geq 0, \qquad \forall i \in I, \ j \in J, m \in M, \ t \in [T, \ n \in [N].
\end{align}
\end{subequations}
\newpage
\section{\textcolor{black}{Details of Lehigh County Instances}}\label{AppexLehigh}
\begin{table}[h!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.7}
\caption{A subset of 20 nodes in Lehigh County and their population based on the 2010 census of Lehigh County (from \cite{wikiLehighCounty}).}
\begin{tabular}{lll}
\hline
\textbf{City/Town/etc.} & Population \\
\hline
Allentown & 118,032 \\
Bethlehem & 74,982 \\
Emmaus & 11,211 \\
Ancient Oaks & 6,661 \\
Catasauqua & 6,436 \\
Wescosville & 5,872 \\
Fountain Hill & 4,597 \\
Dorneyville & 4,406 \\
Slatington & 4,232 \\
Breinigsville & 4,138 \\
Coplay & 3,192 \\
Macungie & 3,074 \\
Schnecksville & 2,935 \\
Coopersburg & 2,386 \\
Alburtis & 2,361 \\
Cetronia & 2,115 \\
Trexlertown & 1,988 \\
Laurys Station & 1,243 \\
New Tripoli & 898 \\
Slatedale & 455 \\
\hline
\end{tabular}
\label{table:LehighInst}
\end{table}
\begin{table}[h!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.7}
\caption{Average demand of each node in Lehigh 1 and Lehigh 2. }
\begin{tabular}{lll}
\hline
\textbf{Node} & \textbf{Lehigh 1 }& \textbf{Lehigh 2} \\
\hline
Allentown & 40 & 60\\
Bethlehem & 40 & 60 \\
Emmaus & 40 & 43 \\
Ancient Oaks & 30 & 25 \\
Catasauqua & 30 &25 \\
Wescosville & 30 & 22 \\
Fountain Hill & 20 & 18 \\
Dorneyville & 20 &17 \\
Slatington & 20 & 16\\
Breinigsville & 20 & 16 \\
Coplay & 20 & 12\\
Macungie & 20 & 12 \\
Schnecksville & 20 & 11 \\
Coopersburg & 20 &9 \\
Alburtis & 20 &9 \\
Cetronia & 20 & 8 \\
Trexlertown & 20 & 8\\
Laurys Station &20 & 5\\
New Tripoli & 15 &3 \\
Slatedale & 15 &3 \\
\hline
\end{tabular}
\label{table:AvgDemand_LehighInst}
\end{table}
\begin{table}[h!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.7}
\caption{Average demand of each node in period 1 and period 2.}
\begin{tabular}{lll}
\hline
\textbf{Node} & \textbf{Period 1}& \textbf{Period 2} \\
\hline
Allentown & 60 & 9 \\
Bethlehem & 60 & 8 \\
Emmaus & 43 & 8 \\
AncientOaks & 25 & 5 \\
Catasauqua & 25 & 3 \\
Wescosville & 22 & 3 \\
FountainHill & 18 & 18 \\
Dorneyville & 17 & 17 \\
Slatington & 16 & 16 \\
Breinigsville & 16 & 16 \\
Coplay & 12 & 12 \\
Macungie & 12 & 12 \\
Schnecksville & 11 & 11 \\
Coopersburg & 9 & 9 \\
Alburtis & 9 & 60 \\
Cetronia & 8 & 60 \\
Trexlertown & 8 & 43 \\
LaurysStation & 5 & 25 \\
NewTripoli & 3 & 25 \\
Slatedale & 3 & 22 \\
\hline
\end{tabular}
\label{table:period}
\end{table}
\begin{table}[t!]
\center
\footnotesize
\renewcommand{\arraystretch}{0.3}
\caption{Optimal MFs locations in period 1 and period 2.}
\begin{tabular}{llllllllllllll}
\hline
\textbf{MF} & \multicolumn{3}{c}{\textbf{MAD-DRO}} & \multicolumn{3}{c}{\textbf{W-DRO}} & &\multicolumn{3}{c}{\textbf{SP}} \\ \cline{2-3} \cline{5-6} \cline{8-9}
& Period 1 & Period 2 & & Period 1 & Period 2 & & Period 1 & Period 2 \\ \cline{1-9}
\hline
1 & Allentown & New Tripoli & & Allentown & Cetronia & & Allentown & Fountain Hill \\
2 & Bethlehem & Schnecksville & & Bethlehem & New Tripoli & & Bethlehem & Slatedale \\
3 & Emmaus & Trexlertown & & Emmaus & Ancient Oaks & & Dorneyville & Cetronia \\
4 & Wescosville & Alburtis & & Breinigsville & Trexlertown & & Ancient Oaks & Alburtis \\ \cline{1-9}
\hline
\end{tabular}
\label{table:OptimalLocations}
\end{table}
\newpage
\section{Calibrating the Wasserstein Radius in the W-DRO Model}\label{WDRO_Radius}
The Wasserstein ball’s radius $\epsilon$ in the W-DRO model is an input parameter, where a larger $\epsilon$ implies that we seek more distributionally robust solutions. For each training data, different values of $\epsilon$ may result in robust solutions $\pmb{x} (\epsilon, N)$ with very different out-of-sample performance $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$. On the one hand, the radius should not be too small. \textcolor{black}{Otherwise,} the problem may behave like sample average approximation and hence, losing the purpose of robustification. In particular, if we set the radius to zero, the ambiguity set shrinks to a singleton that contains only the nominal distribution, in which case the DRO problem reduces to an ambiguity-free SP \citep{esfahani2018data}. But, on the other hand, the radius should not be too large to avoid conservative solutions, which is one of the major criticism faced by traditional RO methods. Given that the true distribution $\mathbb{P}$ is possibly unknown, it is impossible to compute $\epsilon$ that minimizes $\hat{\mathbb{E}} [Q(\pmb{x} (\epsilon, N),\pmb{W})]$. Thus, as detailed in \cite{esfahani2018data}, the best we can hope for is to approximate $\epsilon^{\mbox{\tiny opt}}$ using the training data set.
As pointed out by \cite{esfahani2018data} and \cite{gao2020finite}, practically, the radius is often selected via cross-validation. We employ the following widely used cross-validation method to estimate $\epsilon^{\mbox{\tiny opt}}$ as in \cite{jiang2019data} and \cite{esfahani2018data}. First, for each $N\in \{10, \ 50, \ 100\} $ and each $\epsilon \in \{ 0.01, \ 0.02,\ldots, \ 0.09, \ 0.1, \ldots, \ 0.9,1, \ldots, \ 10 \}$ (i.e., a log-scale interval as in \cite{esfahani2018data}, \cite{jian2017integer}, \cite{tsang2021distributionally}), we randomly partition the data into a training ($N'$) and testing set ($N''$). Using the training set, we solve W-DRO to obtain the optimal first-stage solution $\pmb{x} (\epsilon,N)$ for each $\epsilon$ and $N$. Then, we use the \textcolor{black}{testing} data to evaluate these solutions by computing $\hat{\mathbb{E}}_{\mathbb{P}_{N''}}[Q(\pmb{x} (\epsilon, N),\pmb{W})]$ (where $\mathbb{P}_{N''}$ is the empirical distribution based on the testing data $N''$) via sample average approximation. That is, we solve the second-stage with $\pmb{x}$ fixed to $\pmb{x} (\epsilon,N')$ and $N''$ data and compute the corresponding second-stage cost $\hat{\mathbb{E}}_{\mathbb{P}_{N''}}[Q(\pmb{x} (\epsilon, N''),\pmb{W})]$. Finally, we set $\epsilon^{\mbox{\tiny best}}_N$ to any $\epsilon$ that minimizes $\hat{\mathbb{E}}_{\mathbb{P}_{N''}}[Q(\pmb{x} (\epsilon, N''),\pmb{W})]$. We repeat this procedure 30 times for each $N$ and set $\epsilon$ to the average of the $\epsilon^{\mbox{\tiny best}}_N$ across these 30 replications.
We found that $\epsilon^{\mbox{\tiny best}}_N$ equals (7, 5, 2) when $N$=(10, 50, 100) for most instances. It is expected that $\epsilon$ decreases with $N$ (see, e.g., \cite{esfahani2018data}, \cite{jiang2019data}, \cite{tsang2021distributionally}). Intuitively, a small sample does not have sufficient distributional information, and thus a larger $\epsilon$ produces distributionally robust solutions that better hedge against ambiguity. In contrast, with a larger sample, we may have more information from the data, and thus we can make a less conservative decision using a smaller $\epsilon$ value.
While Wasserstein ambiguity sets offer powerful out-of-sample performance guarantees and enable practitioners to control the model's conservativeness by choosing $\epsilon$, moment-based ambiguity sets often display better tractability properties. In fact, various studies provided evidence that DRO models with moment ambiguity sets are more tractable than the corresponding SP because the intractable high-dimensional integrals in the objective function are replaced with tractable (generalized) moment problems (see, e.g., \cite{esfahani2018data, delage2010distributionally, goh2010distributionally, wiesemann2014distributionally}). In contrast, DRO models with Wasserstein ambiguity sets tend to be more computationally challenging than some moment-based DRO model and their SP counterparts, especially when $N$ is large. In this paper, we obtained similar observations. For a detailed discussion we refer to \cite{esfahani2018data} and references therein.
\newpage
\section{Additional CPU Time Results}\label{Appx:CPU2}
\color{black}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-CVaR model ($ \pmb{W} \in [20, 60]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 1 & 3 & 6 & 4 & 12 & 25 & & 2 & 7 & 15 & 4 & 12 & 25 \\
2 & 10 & 20 & & 5 & 11 & 22 & 11 & 27 & 50 & & 18 & 21 & 29 & 35 & 44 & 61 \\
3 & 15 & 10 & & 3 & 8 & 19 & 5 & 16 & 36 & & 5 & 13 & 32 & 9 & 24 & 58 \\
4 & 15 & 20 & & 28 & 63 & 107 & 23 & 56 & 93 & & 35 & 60 & 85 & 29 & 86 & 121 \\
5 & 20 & 10 & & 13 & 25 & 37 & 12 & 23 & 33 & & 19 & 75 & 107 & 17 & 47 & 68 \\
6 & 20 & 20 & & 207 & 407 & 617 & 59 & 106 & 165 & & 433 & 969 & 1576 & 103 & 198 & 303 \\
7 & 25 & 10 & & 111 & 242 & 494 & 25 & 43 & 67 & & 147 & 454 & 794 & 42 & 75 & 125 \\
8 & 25 & 20 & & 2083 & 2944 & 3903 & 11 & 94 & 169 & & 2979 & 3274 & 3600 & 164 & 237 & 316 \\
\hline
\end{tabular}\label{table:MADCVAR_2060}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the MAD-CVaR model ($ \pmb{W} \in [50, 100]$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 2 & 6 & 10 & 5 & 17 & 30 & & 2 & 5 & 8 & 4 & 14 & 26 \\
2 & 10 & 20 & & 14 & 22 & 29 & 22 & 35 & 46 & & 18 & 23 & 29 & 19 & 31 & 44 \\
3 & 15 & 10 & & 18 & 31 & 57 & 9 & 21 & 49 & & 16 & 34 & 45 & 8 & 22 & 47 \\
4 & 15 & 20 & & 67 & 106 & 156 & 23 & 68 & 139 & & 64 & 132 & 255 & 27 & 62 & 106 \\
5 & 20 & 10 & & 52 & 78 & 103 & 18 & 33 & 51 & & 38 & 67 & 117 & 16 & 28 & 56 \\
6 & 20 & 20 & & 549 & 715 & 935 & 96 & 124 & 172 & & 757 & 1459 & 2802 & 91 & 124 & 184 \\
7 & 25 & 10 & & 584 & 685 & 760 & 28 & 48 & 83 & & 743 & 2277 & 2552 & 11 & 42 & 90 \\
\hline
\end{tabular}\label{table:MADCVAR__50100}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [20, 60]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 7 & 19 & 17 & 5 & 5 & 6 & & 16 & 20 & 24 & 12 & 15 & 19 \\
2 & 10 & 20 & & 12 & 13 & 16 & 6 & 6 & 8 & & 37 & 44 & 57 & 17 & 18 & 22 \\
3 & 15 & 10 & & 27 & 44 & 70 & 4 & 5 & 5 & & 22 & 28 & 38 & 10 & 12 & 13 \\
4 & 15 & 20 & & 9 & 12 & 17 & 4 & 5 & 7 & & 15 & 33 & 60 & 6 & 11 & 17 \\
5 & 20 & 10 & & 34 & 35 & 39 & 4 & 5 & 6 & & 72 & 88 & 98 & 14 & 16 & 20 \\
6 & 20 & 20 & & 20 & 31 & 46 & 4 & 6 & 9 & & 240 & 574 & 827 & 6 & 13 & 21 \\
\hline
\end{tabular}\label{table:Wass_CVAR_2060_N10}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [50, 100]$, $N=10$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 13 & 25 & 53 & 3 & 4 & 4 & & 11 & 16 & 25 & 5 & 6 & 7 \\
2 & 10 & 20 & & 6 & 8 & 14 & 3 & 3 & 4 & & 10 & 12 & 16 & 5 & 6 & 8 \\
3 & 15 & 10 & & 16 & 26 & 57 & 3 & 3 & 3 & & 29 & 45 & 75 & 5 & 5 & 6 \\
4 & 15 & 20 & & 38 & 45 & 61 & 3 & 3 & 3 & & 63 & 82 & 136 & 5 & 6 & 7 \\
5 & 20 & 10 & & 45 & 49 & 55 & 3 & 3 & 3 & & 55 & 70 & 79 & 7 & 7 & 8 \\
\hline
\end{tabular}\label{table:Wass_CVAR_50100_N10}
\end{table}
\newpage
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [20, 60]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 28 & 35 & 40 & 5 & 6 & 7 & & 104 & 117 & 129 & 15 & 19 & 22\\
2 & 10 & 20 & & 33 & 42 & 51 & 6 & 7 & 9 & & 84 & 112 & 149 & 15 & 19 & 24\\
3 & 15 & 10 & & 28 & 34 & 43 & 4 & 5 & 5 & & 109 & 136 & 148 & 14 & 17 & 19\\
4 & 15 & 20 & & 34 & 59 & 102 & 4 & 5 & 7 & & 189 & 1471 & 3730 & 17 & 20 & 23\\
5 & 20 & 10 & & 68 & 88 & 125 & 4 & 6 & 7 & & 455 & 1,084 & 1,556 & 24 & 30 & 34\\
\hline
\end{tabular}\label{table:Wass_CVAR_2060_N50}
\end{table}
\begin{table}[h!]
\color{black}
\center
\small
\caption{Computational details of solving the W-CVaR model ($ \pmb{W} \in [50, 100]$, $N=50$).}
\renewcommand{\arraystretch}{0.6}
\begin{tabular}{lllllllllllllllllllllllllllllllllll}
\hline
& & & \multicolumn{6}{c}{$C=60$ } && \multicolumn{6}{c}{$C=100$ } \\ \cline{5-10} \cline{12-17}
Inst & $I$ & $T$ & & \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} && \multicolumn{3}{c}{CPU time} & \multicolumn{3}{c}{iteration} \\ \cline{5-10} \cline{12-17}
& & & & Min & Avg & Max & Min & Avg & Max & & Min & Avg & Max & Min & Avg & Max\\
\hline
1 & 10 & 10 & & 24 & 25 & 26 & 4 & 4 & 4 & & 37 & 46 & 51 & 6 & 7 & 8 \\
2 & 10 & 20 & & 29 & 30 & 32 & 4 & 4 & 4 & & 52 & 73 & 124 & 7 & 8 & 9 \\
3 & 15 & 10 & & 33 & 50 & 75 & 3 & 3 & 3 & & 90 & 100 & 111 & 6 & 6 & 7 \\
4 & 15 & 20 & & 50 & 59 & 68 & 3 & 3 & 3 & & 130 & 310 & 743 & 5 & 6 & 9 \\
5 & 20 & 10 & & 107 & 130 & 170 & 3 & 3 & 3 & & 118 & 182 & 321 & 5 & 6 & 8 \\
\hline
\end{tabular}\label{table:Wass_CVAR_50100_N50}
\end{table}
\textcolor{white}{sabjhasvghavwejfce}
\clearpage
\newpage
\section{Additional Out-of-Sample Results}\label{Appex:AdditionalOut}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_LogN.jpg}
\caption{2nd (Set 1, LogN)}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_Uni0}
\caption{TC (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N50_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N50_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample for Instance 3 ($\pmb{W} \in [20, 60]$, $\pmb{N=50}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_UniN50_Inst3}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_Uni0.jpg}
\caption{TC (Set 1, LogN)}\label{Inst3_LogNTC_Range2N50}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_Uni0.jpg}
\caption{2nd (Set 1, LogN)}\label{Inst3_LogN2nd_Range2N50}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_LogN.jpg}
\caption{TC (Set 2, $\Delta=0$)}\label{Inst3_Uni0_TC_Range2N50}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_LogN.jpg}
\caption{2nd (Set 2, $\Delta=0$)}\label{Inst3_Uni0_2nd_Range2N50}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_TC_N50_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_Range2_2nd_N50_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)} \label{Inst3_Uni25_2nd_Range2N50}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Instance 3 ($\pmb{W} \in [50, 100]$, $\pmb{N=50}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$). }\label{Fig3_UniN50_Inst3_Range2N50}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_LogN.jpg}
\caption{2nd (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_Uni0.jpg}
\caption{TC (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_Uni0.jpg}
\caption{2nd (Set 2, $\Delta=0$)}
\end{subfigure}%
%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_Uni25.jpg}
\caption{TC (Set 2, $\Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_TC_N50_Uni50.jpg}
\caption{TC (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh1_2nd_N50_Uni50.jpg}
\caption{2nd (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 1 ($\pmb{N=50}$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$). }\label{Fig3_N50_Lehigh1}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_LogN.jpg}
\caption{2nd (Set 1, LogN))}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_Uni0.jpg}
\caption{TC (Set 2, $\Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_Uni0.jpg}
\caption{2nd (Set 2, $ \Delta=0$)}
\end{subfigure}%
%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_Uni25.jpg}
\caption{TC (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N10_MF6_Uni50.jpg}
\caption{TC (Set 2, $ \Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N10_MF6_Uni50.jpg}
\caption{2nd (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 2 ($N=10$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_Uni10_Lehigh2}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_LogN.jpg}
\caption{TC (Set 1, LogN)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_LogN.jpg}
\caption{2nd (Set 1, LogN)}
\end{subfigure
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_Uni0.jpg}
\caption{TC (Set 2, $ \Delta=0$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_Uni0.jpg}
\caption{2nd (Set 2, $ \Delta=0$)}
\end{subfigure}%
%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_Uni25.jpg}
\caption{TC (Set 2, $\Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_Uni25.jpg}
\caption{2nd (Set 2, $ \Delta=0.25$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_TC_N50_MF6_Uni50.jpg}
\caption{TC (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Lehigh2_2nd_N50_MF6_Uni50.jpg}
\caption{2nd (Set 2, $\Delta=0.5$)}
\end{subfigure}%
\caption{Normalized histograms of out-of-sample TC and 2nd for Lehigh 2 ($N=50$) under Set 1 (LogN) and Set 2 (with $\pmb{\Delta \in \{0, 0.25, 0.5\}}$).}\label{Fig3_Uni50_Lehigh2}
\end{figure}
\begin{figure}[h!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Corr20.jpg}
\caption{TC (cor=0.2)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Corr20.jpg}
\caption{2nd (cor=0.2)}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_TC_N10_Corr60.jpg}
\caption{TC (cor=0.6) }
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{Inst3_2nd_N10_Corr60.jpg}
\caption{2nd (cor=0.6)}
\end{subfigure}%
\caption{Out-of-sample performance under correlated data for Instance 3. Notation: cor is correlation coefficient}\label{Fig_outcorr}
\end{figure}
\clearpage
\newpage
\section{Additional Sensitivity Results}\label{Appx:sensitivity}
\begin{figure}[h!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_1500.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig7a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_1500.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig7b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_6000.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig7c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_TC_6000.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig7d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_10000.jpg}
\caption{Number of MFs, $f=10,000$}
\label{Fig7e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Rang2_TC_10000.jpg}
\caption{Total cost, $f=10,000$}
\label{Fig7f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [50, 100] $. Instance 1}\label{Fig7:MF_vs_C_Range2_inst1}
\end{figure}
\begin{figure}[t!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_1500_Inst5.jpg}
\caption{Number of MFs, $f=1,500$}
\label{Fig10a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_1500_Inst5.jpg}
\caption{Total cost, $f=1,500$}
\label{Fig10b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_6000_Inst5.jpg}
\caption{Number of MFs, $f=6,000$}
\label{Fig10c}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_6000_Inst5.jpg}
\caption{Total cost, $f=6,000$}
\label{Fig10d}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_Range2_10000_Inst5.jpg}
\caption{Number of MFs, $f=10,000$}
\label{Fig10e}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_Range2_TC_10000_Inst5.jpg}
\caption{Total cost, $f=10,000$}
\label{Fig10f}
\end{subfigure}%
\caption{Comparison of the results for different values of $C$ and $f$ under $\pmb{W} \in [50, 100] $. Instance 5}\label{Fig10:MF_vs_C_Range2_inst5}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_2nd_1500.jpg}
\caption{$f=1,500$}
\label{Fig6a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_Vs_F_2nd_6000.jpg}
\caption{$f=6,000$}
\label{Fig6b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_Vs_F_2nd_10000.jpg}
\caption{$f=10,000$}
\label{Fig6c}
\end{subfigure}%
\caption{Comparison of second-stage cost for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 1}\label{Fig8:MF_vs_C_inst1}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_1500_Inst5_2.jpg}
\caption{$f=1,500$}
\label{Fig11a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_F_2nd_6000_Inst5.jpg}
\caption{$f=6,000$}
\label{Fig11b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_10000_Inst5.jpg}
\caption{$f=10,000$}
\label{Fig11c}
\end{subfigure}%
\caption{Comparison of second-stage cost for different values of $C$ and $f$ under $\pmb{W} \in [20, 60]$. Instance 5}\label{Fig11:MF_vs_C_inst5}
\end{figure}
\begin{figure}[t!]
\centering
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_Range2_1500_Inst5.jpg}
\caption{$f=1,500$}
\label{Fig12a}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{C_F_2nd_Range2_6000_Inst5.jpg}
\caption{$f=6,000$}
\label{Fig12b}
\end{subfigure}%
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{C_F_2nd_Range2_10000_Inst5.jpg}
\caption{$f=10,000$}
\label{Fig12c}
\end{subfigure}%
\caption{Comparison of second-stage cost for different values of $C$ and $f$ under $\pmb{W} \in [50, 100]$. Instance 5}\label{Fig12:MF_vs_C_range2_inst5}
\end{figure}
\end{APPENDICES}
\clearpage
\newpage
\bibliographystyle{informs2014trsc}
|
2,869,038,154,788 | arxiv | \section{\label{sec:1}Introduction and Motivation}
Optical systems combining lossy and active elements provide a platform to
implement analogues of non-hermitian $\mathcal{PT}$-symmetric quantum
systems \cite{bender,ptreview1,ptreview2}, which allows to realize unique
optical switching effects
\cite{christodoulidesgroup1,christodoulidesgroup2,christodoulidesgroup3,
ptexperiments1,ptexperiments2,pteffects1,pteffects2a,pteffects4,pteffects5,
hsqoptics,longhilaserabsorber,variousstonepapers1,variousstonepapers2,yoo,schomerusyh}.
The wave scattering and dynamics in these systems provide an aspect which
also permeates mesoscopic phenomena in microcavity lasers \cite{vahala},
coherent electronic transport \cite{beenakkerreview,datta},
superconductivity \cite{beenakkerreview,altlandzirnbauer}, and
quantum-chaotic dynamics \cite{haake}. The main goal here is to relate
how tools established in these disciplines can be used to approach the
exciting spectral and dynamical features in $\mathcal{PT}$-symmetric
optics. Scattering theory \cite{beenakkerreview,datta,fyodorovsommers},
in particular, is ideally suited to deal with the complications of
complex wave dynamics in multichannel situations, which appear when
one goes beyond one-dimensional situations, and also easily accounts for
any leakage if a system is geometrically open, as is often required by
the nature of the desired optical effects, or because of the design of
practical devices.
We thus describe in detail how this approach can be applied to optical
realizations of $\mathcal{PT}$-symmetric systems with a wide range of
geometries \cite{hsqoptics,longhilaserabsorber,variousstonepapers1,
variousstonepapers2,yoo,hsrmt,birchallweyl,birchallrmt}. (For
$\mathcal{PT}$-symmetric scattering in effectively one dimension see,
\emph{e.g.}, Refs.~\cite{cannata,berry,jones}.) To formulate the approach
under these conditions it matters whether a geometric symmetry inverts or
preserves the handedness of the coordinate system, especially in the
presence of magneto-optical effects (vector potentials, which generally
cannot be gauged away beyond one dimension). An analogous bifurcation
arises in the specification of the time-reversal operation. Moreover, it
is important to note that the fields defining a device geometry are
external, a distinction from particle-physics settings which is reflected
in the subtleties of Onsager's reciprocal relations \cite{onsager}. Thus
the formulation of $\mathcal{PT}$ symmetry itself requires some care. In
particular, an alternative appears, termed $\mathcal{PTT}'$ symmetry
\cite{hsrmt}, which results in the same spectral constraints but imposes
different physical symmetry conditions. The scattering approach also
allows to restate the symmetry requirements in terms of generalized
conservation laws \cite{variousstonepapers2}, which we here extend to
multiple channels. Furthermore, the approach leads to effective models of
complex wave dynamics which relate the spectral features to universal
mesoscopic time and energy scales \cite{hsrmt,birchallweyl,birchallrmt}.
We start this exposition with a brief recapitulation of the analogy
between optics and non-hermitian quantum mechanics (\S\ref{sec:2}), and
discuss variants of geometric and time-reversal operations which can be
used to set up $\mathcal{PT}$ and $\mathcal{PTT}'$ symmetry
(\S\ref{sec:3} and \S\ref{sec:4}). We then describe how scattering theory
can be applied to study the spectral features in these settings, first
generally (\S\ref{sec:5}-\S\ref{sec:7}) and then for a coupled-resonator
geometry (\S\ref{sec:8}). This leads to effective models (\S\ref{sec:9})
which can be formulated in the energy domain (via Hamiltonians) and in
the time domain (via time evolution operators), and capture the relevant
energy and time scales (\S\ref{sec:10}). The concluding \S\ref{sec:11}
also describes possible generalizations of these models.
\section{\label{sec:2}Optical analogues of non-hermitian quantum mechanics}
The $\mathcal{PT}$-symmetric optical systems mentioned in the
introduction cover a large range of designs, including coupled optical
fibres, photonic crystals and coupled resonators, as depicted in figure
\ref{fig1}. Some of the elements are absorbing while others are
amplifying, and the absorption and amplification rates, geometry, and
other material properties are carefully matched to result in a symmetric
arrangement.
\begin{figure}[b]
\includegraphics[width=\columnwidth]{fig1.eps}
\caption{\label{fig1}(Online version in colour.)
Common designs of $\mathcal{PT}$-symmetric optical systems with absorbing
(light green) and amplifying (dark red) elements. (a) Two optical fibers.
(b) Localized modes in a photonic crystals. (c) Coupled-resonator
geometry. }
\end{figure}
We concentrate on two common designs, effectively two-dimensional systems
(planar resonators or photonic crystals, with coordinates $x$, $y$) as
well as three-dimensional pillar-like systems (such as arrangements of
aligned optical fibers with a fixed cross section) where the geometry
does not depend on the third coordinate $z$, and assume that the relevant
effects of wave scattering, gain and loss can be subsumed in a refractive
index $n(\mathbf{r})$, where ${\rm Im}\, n(\mathbf{r})>0$ signifies
absorption (loss) while ${\rm Im}\, n(\mathbf{r})<0$ signifies
amplification (gain). Later on, we also include an external vector
potential $\mathbf{A}(\mathbf{r})$ representing magneto-optical effects.
These assumptions allow to separate transverse magnetic (TM) and electric
(TE) modes, with the magnetic or electric field, respectively, confined
to the $xy$-plane (thus transverse to $z$). The electromagnetic field can
then be described by a scalar wave function $\psi({\bf r})$, which
represents the $z$-component of the electric or magnetic field,
respectively. One now arrives at two principal situations, which both
serve as an optical analogue of non-hermitian quantum mechanics.
\subsection{Helmholtz equation}
On the one hand, one can focus on the propagation in the $xy$-plane,
resulting in an effectively two-dimensional system which is described by
a Helmholtz equation
\begin{align}\label{eq:helmholtz}
&\mathcal{L}(\omega)\psi({\bf r})=0,
\nonumber \\
&\mathcal{L}(\omega)=
\left\{\begin{array}{cc}
\nabla^2+\frac{\omega^2n^2({\bf r})}{c^2} & \mbox{(TM)} \\
\nabla \frac{1}{n^{2}({\bf r})} \nabla +\frac{\omega^2}{c^2}& \mbox{(TE)} \end{array}\right.
,\quad\nabla=\left(\begin{array}{c} \partial_x\\ \partial_y\end{array}\right)
.
\end{align}
The Helmholtz equation is analogous to a stationary Schr{\"o}dinger
equation, thus, an eigenvalue problem for the frequencies $\omega$, which
can become complex due to the loss and gain encoded in $n$, or because of
leakage at the boundaries of the system in the propagation plane. This
then describes quasistationary states which decay (${\rm Im}\,\omega<0$)
or grow (${\rm Im}\,\omega>0$) over time.
\subsection{Paraxial equation}
On the other hand, one can focus on the propagation into the
$z$-direction perpendicular to this plane, which is then often described
by a paraxial equation
\begin{align}
&2i\kappa\partial_z \psi({\bf r}) +\mathcal{L}(\kappa)\psi({\bf r})=0,
\nonumber \\ &
{\mathcal{L}}(\kappa)=
\left\{\begin{array}{cc}
\nabla^2+\frac{\omega^2n^2({\bf r})}{c^2} -\kappa^2& \mbox{(TM)} \\
n^2({\bf r})\nabla \frac{1}{n^{2}({\bf r})} \nabla +\frac{\omega^2n^2({\bf r})}{c^2} -\kappa^2& \mbox{(TE)} \end{array}\right. .
\label{eq:paraxial}
\end{align}
Here $\psi({\bf r})$ now is an envelope wave function, obtained after
separating out a term $\exp(i\kappa z)$. This equation remains valid if
$n$ varies only slowly with $z$, $|\partial_z n/n|\ll|\kappa|$.
The paraxial equation is an analogue of a time-dependent Schr{\"o}dinger
equation, thus, a dynamical equation where an initial condition is
propagated forwards---here, not in time, but along the spatial direction
with coordinate $z$. It is then natural to probe the system at an initial
and a final cross-section, while the physical frequency $\omega$ is now
real.
As we assume that $n$ is $z$-independent, the paraxial equation is
associated with the eigenvalue problem $\mathcal{L}(\kappa)\psi({\bf
r})=0$ for the propagation constant $\kappa$. This eigenvalue problem
shares all relevant mathematical features with the problem
\eqref{eq:helmholtz} for $\omega$ (in particular, the eigenvalues again
can be complex because of loss, gain, and leakage in the system); so do
variants that rely on wave localization in the cores of optical fibers or
Wannier-like modes in a photonic crystal, with the gradient replaced by
hopping terms. For definiteness, we shall employ notations adapted to the
eigenvalue problem for $\omega$.
\section{\label{sec:3}Symmetries of the wave equation}
The investigation of symmetries in higher-dimensional systems of
arbitrary geometry is a central aspect of mesoscopic physics. For
hermitian systems, a complete classification requires to take care of
magnetic fields, internal degrees of freedom such as spin, pairing
potentials and particle-hole symmetries
\cite{beenakkerreview,datta,altlandzirnbauer,haake}. Non-hermitian
systems provide an even wider setting, with a mathematical
classification, \emph{e.g.}, provided in Ref. \cite{magnea}. In order to
identify and specify the role of symmetries for their optical
realizations we denote the operator in the wave equation by
$\mathcal{L}(\omega; n(\mathbf{r}))$, which explicitly takes care of the
functional dependence on the refractive index $n$.
\subsection{$\mathcal{PT}$ symmetry}
In $\mathcal{PT}$-symmetric quantum mechanics
\cite{bender,ptreview1,ptreview2}, the parity operator $\mathcal{P}$
generally stands for a unitary transformation which squares to the
identity. In optical systems, this operation is usually realized
geometrically via an isometric involution, still denoted as
$\mathcal{P}$, which inverts one, two, or three coordinates. To specify
the consequences for the wave equation we promote ${\cal P}$ to a
superoperator (we employ this notation as we will also encounter a
transformation $\mathcal{T}'$ which does not correspond to an ordinary
unitary or antiunitary operator). Then
\begin{equation}\label{eq:p}
\mathcal{P}[\mathcal{L}(\omega; n(\mathbf{r}))]
=\mathcal{L}(\omega; n(\mathcal{P}\mathbf{r})),
\end{equation}
which should be read as a rule how to write the wave equation in the
transformed coordinate system. The transformed equation is then solved by
${\cal P}\psi({\bf r})\equiv\psi({\cal P}{\bf r})$.
Conventional time reversal is implemented by complex conjugation in the
position representation, which constitutes an antiunitary operation. We
then have
\begin{equation}
\mathcal{T}[\mathcal{L}(\omega; n(\mathbf{r}))]
=\mathcal{L}(\omega^*; n^*(\mathbf{r})),
\end{equation}
which delivers a wave equation solved by ${\cal T}\psi({\bf r})\equiv\psi^*({\bf r})$.
The wave equation now displays $\mathcal{PT}$ symmetry if the refractive
index obeys
$n({\bf r})=n^*({\cal P}{\bf r})$.
In this situation ${\cal P}$ is an involution that interchanges
amplifying and absorbing regions with matching amplification and
absorption rates, and
\begin{equation}
\mathcal{PT}[\mathcal{L}(\omega; n(\mathbf{r}))]
=\mathcal{L}(\omega^*; n^*(\mathcal{P}\mathbf{r}))
=\mathcal{L}(\omega^*; n(\mathbf{r})),
\end{equation}
which constraints the spectral properties of $\mathcal{L}$ if the
boundary conditions also respect the symmetry (this is typically
\emph{not} the case if the system is open). The eigenvalues $\omega_n$
then obey
\begin{equation}\label{eq:constraint}
\omega_n=\omega^*_{\bar n}
\end{equation}
and thus either real ($n=\bar n$) if $\psi^*_n(\mathcal{P}\mathbf{r})$
linearly depends on $\psi_n(\mathbf{r})$, or occur in complex-conjugate
pairs ($n\neq\bar n$) if that is not the case.
\subsection{$\mathcal{T'}$ symmetry}
It is important to distinguish the conventional antiunitary time-reversal
operation ${\cal T}$ from another operation that is also often termed
time-reversal \cite{beenakkerreview,datta,altlandzirnbauer,haake}. If
$\mathcal{L}({\bf r})$ were hermitian ($\omega$ and $n$ both real with
appropriate boundary conditions), then the action of ${\cal T}$ could not
be distinguished from the action of a superoperator ${\cal T}'$ acting in
the position representation as
\begin{equation}
{\cal T}'[\mathcal{L}(\omega; n(\mathbf{r}))]=\mathcal{L}^T(\omega; n(\mathbf{r})).
\end{equation}
Thus, ${\cal T}'$ transforms the right eigenvalue problem ${\cal
L}\psi=0$ into the left eigenvalue problem ${\cal L}^T \bar\psi=0$. This
delivers the same spectrum, and while the right and left eigenfunctions
$\psi$ and $\bar\psi$ for a given eigenvalue generally differ they are
related by biorthogonality constraints. In the presently assumed absence
of magneto-optical effects, ${\cal T}'$ is indeed an exact symmetry,
$\mathcal{L}^T(\omega; n(\mathbf{r})) =\mathcal{L}(\omega;
n(\mathbf{r}))$ and thus $\psi=\bar\psi$. This holds even if $n$ is
complex, thus, even if ${\cal T}$ symmetry is broken. In the
non-hermitian case, therefore, these two operations are distinct and must
be treated separately.
\section{\label{sec:4}Magneto-optical effects}
In mesoscopic systems with complex wave dynamics, the alternative
time-reversal operation ${\cal T}'$ governs a multitude of effects
ranging from coherent backscattering and wave localization to minigaps in
mesoscopic superconductors
\cite{beenakkerreview,datta,altlandzirnbauer,melsen}. To clarify the role
of this operation we now consider magneto-optical effects, which are
described by a (possibly complex) external vector potential
$\mathbf{A}(\mathbf{r})$ that enters the wave equation through terms of
the generic form $(\nabla+i\mathbf{A}(\mathbf{r}))^2$. For this purpose
we denote the operator in the wave equation as $\mathcal{L}(\omega;
n(\mathbf{r}), \mathbf{A}(\mathbf{r}))$. As before, we continue to focus
on effectively two-dimensional systems; thus, $A_z=0$, see
\eqref{eq:helmholtz} and \eqref{eq:paraxial}.
\subsection{$\mathcal{PT}$ symmetry}
The involution $\mathcal{P}$ (interpreted as a coordinate transformation
of the wave equation) transforms the external vector potential according
to
$(\nabla+i\mathbf{A}(\mathbf{r}))^2\to
({\cal P}\nabla+i\mathbf{A}({\cal P}\mathbf{r}))^2
=(\nabla+i[{\cal P}\mathbf{A}]({\cal P}\mathbf{r}))^2
$,
as ${\cal P}$ is an isometry.
Thus\begin{equation}
\mathcal{P}[\mathcal{L}(\omega; n(\mathbf{r}), \mathbf{A}(\mathbf{r}))]
=\mathcal{L}(\omega; n(\mathcal{P}\mathbf{r}),
[\mathcal{P}\mathbf{A}](\mathcal{P}\mathbf{r}) ).
\end{equation}
This analysis reveals an important feature of parity when applied to
external fields, which sets them apart from internal fields that are
often discussed in the setting of particle physics, and then are subject
to additional explicit transformation rules. Here we are interested in
symmetries of a wave function $\psi({\bf r})$ that only accounts for the
internal system dynamics, not for external components such as the motion
of electrons in the inductors or the magnetic moments in the permanent
magnets generating the field. This distinction is fundamental for
Onsager's reciprocal relations, which break down in the presence of
external magnetic fields \cite{onsager}. The resulting magnetic field
depends on whether $\mathcal{P}$ preserves or inverts the handedness of
the coordinate system, as
$\nabla\times\mathcal{P}\mathbf{A}(\mathcal{P}\mathbf{r})={\rm
det}(\mathcal{P})\mathcal{P}[\nabla \times \mathbf{A}(\mathbf{r})]$.
The time-reversal operator $\mathcal{T}$ transforms
$(\nabla+i\mathbf{A}(\mathbf{r}))^2\to (\nabla-i\mathbf{A}^*(\mathbf{r}))^2
$,
so
\begin{equation}
\mathcal{T}[\mathcal{L}(\omega; n(\mathbf{r}), \mathbf{A}(\mathbf{r}))]
=\mathcal{L}(\omega^*; n^*(\mathbf{r}), -\mathbf{A}^*(\mathbf{r})).
\end{equation}
$\mathcal{PT}$ symmetry thus requires
\begin{equation}\label{eq:ptna}
n({\bf r})=n^*({\cal P}{\bf r}),\quad
\mathbf{A}({\bf r})=-\mathcal{P}\mathbf{A}^*({\cal P}{\bf r}).
\end{equation}
\emph{E.g.}, when ${\cal P}$ is a reflection $x\to-x$ this holds for a
homogeneous magnetic field that points in the $z$ direction [see figure
\ref{fig2}(a)].
\begin{figure}
\includegraphics[width=\columnwidth]{fig2.eps}
\caption{\label{fig2}(Online version in colour.)
Illustration of magneto-optical effects (associated with an effective magnetic field
$\mathbf{B}$) for (a) $\mathcal{PT}$ symmetry and (b) $\mathcal{PTT}'$
symmetry, where $\mathcal{P}$ is a reflection \cite{hsrmt}. }
\end{figure}
\subsection{$\mathcal{PTT}'$ symmetry}
We now identify a modified symmetry, which eventually yields the same
spectral constraints as obtained for $\mathcal{PT}$ symmetry, but imposes
a different condition on the magnetic field \cite{hsrmt}. This variant
follows from the inspection of the $\mathcal{T'}$ operation. In the
position representation $\partial_x^T=-\partial_x$,
$\partial_y^T=-\partial_y$ (recall that we focus on the effects in the 2D
cross-sectional plane), and thus
\begin{equation}
\mathcal{T'}:\quad (\nabla+i\mathbf{A}(\mathbf{r}))^2\to
[(\nabla+i\mathbf{A}(\mathbf{r}))^2]^T
=(\nabla-i\mathbf{A}(\mathbf{r}))^2
.
\end{equation}
This effectively inverts the vector potential, \begin{equation}
{\cal T}'[\mathcal{L}(\omega; n(\mathbf{r}), \mathbf{A}(\mathbf{r}))]=
\mathcal{L}(\omega; n(\mathbf{r}), -\mathbf{A}(\mathbf{r})),
\end{equation}
which confirms that ${\cal T}'$ symmetry is broken when $\mathbf{A}$ is
finite. Consider now
\begin{equation}
\mathcal{PTT'}[\mathcal{L}(\omega; n({\bf r}),\mathbf{A}({\bf r})]
=\mathcal{L}(\omega^*; n^*({\cal P}{\bf r}), \mathcal{P}\mathbf{A}^*({\cal P}{\bf r})).
\end{equation}
This turns into a symmetry if
\begin{equation}\label{eq:pttna}
n({\bf r})=n^*({\cal P}{\bf r}),\quad
\mathbf{A}({\bf r})=\mathcal{P}\mathbf{A}^*({\cal P}{\bf r}).
\end{equation}
\emph{E.g.}, when ${\cal P}$ is a reflection $x\to-x$ this holds for an
antisymmetric magnetic field which (in the plane of the resonator) points in the $z$ direction [see figure
\ref{fig2}(b)].
\section{\label{sec:5}Scattering formalism}
The optical systems considered here are naturally open, with leakage
occurring, \emph{e.g.}, at the fiber tips and waveguide entries, or
because the confinement in the cross-section relies on partial internal
reflection at refractive index steps or semitransparent mirrors. These
systems can thus be probed via scattering
\cite{beenakkerreview,datta,fyodorovsommers}, which delivers
comprehensive insight into their spectral and dynamical properties and
illuminates the consequences of the symmetries discussed above, as well
as the role of multiple scattering addressed later on.
These properties are encoded in the scattering matrix $S(\omega)$, which
relates the amplitudes $a_{\mathrm{in},n}$, $a_{\mathrm{out},n}$ in
incoming and outgoing scattering states $\chi_{\mathrm{in},n}$,
$\chi_{\mathrm{out},n}$,
\begin{equation}\label{eq:sdef}
\mathbf{a}_{\mathrm{out}}=S(\omega) \mathbf{a}_{\mathrm{in}}.
\end{equation}
This relation is generally obtained by the solution of the wave equation
under appropriate conditions at the boundary $\partial\Omega$ of the
scattering region $\Omega$ (outside of which we set $n=1$, ${\bf A}=0$).
The scattering states are assumed to be flux orthonormalised,
\begin{equation}\label{eq:flux}
\int_{\partial \Omega} d\mathbf{S}\cdot
[\chi_{\sigma,n}^*\nabla \chi_{\sigma',m}- \chi_{\sigma',m}\nabla\chi_{\sigma,n}^*]
=2i\sigma\delta_{nm}\delta_{\sigma\sigma'},
\end{equation}
where $\sigma=1$ for outgoing states and $\sigma=-1$ for incoming
states. Two popular choices are states with fixed angular momentum in
free space, and transversely quantized modes in a fixed-width waveguide
geometry.
We now describe the adaptation of this formalism to $\mathcal{PT}$ and
$\mathcal{PTT}'$-symmetric situations
\cite{hsqoptics,longhilaserabsorber,variousstonepapers1,
variousstonepapers2,yoo,hsrmt,birchallweyl}.
\subsection{${\cal PT}$ symmetry }
The symmetries of a scattering problem are exposed when one conveniently
groups the scattering states. To inspect ${\cal P}$ we call half of the
incoming states `incoming from the left', and the other half `incoming
from the right'. This does not need to be taken literally; all what
matters is that the two groups are converted into each other by ${\cal
P}$. The same can be done for the outgoing states. The scattering matrix
then decomposes into blocks,
\begin{equation}
S=\left(
\begin{array}{cc}
r & t' \\
t & r' \\
\end{array}
\right),
\end{equation}
where $r$ describes reflection of left incoming states into left outgoing
states, $r'$ describes the analogous reflection on the right, while $t$
and $t'$ describe transmission from the left to the right and vice
versa. The parity $\mathcal{P}$ interchanges the amplitudes of the left
and right states, which can be brought about by a $\sigma_x$ Pauli
matrix,
\begin{equation}
\mathcal{P}[S]=\sigma_x S \sigma_x=
\left(\begin{array}{cc}
r' & t \\
t' & r \\
\end{array}
\right).
\end{equation}
For consideration of the $\mathcal{T}$ operation, we group incoming and
outgoing states into time-reversed pairs. When $\mathcal{T}$ acts on a
wave function the amplitudes become conjugated, while the frequency in
the wave equation changes to $\omega^*$. Furthermore, incoming states are
converted into outgoing states, so that the relation \eqref{eq:sdef} must
be inverted. Thus,
\begin{equation}
\mathcal{T}[S(\omega)]=\{S^{-1}(\omega^*)\}^*.
\end{equation}
In combination, we have \cite{hsqoptics,variousstonepapers1,yoo}
\begin{equation}\label{eq:pts}
\mathcal{PT}[S(\omega)]=\sigma_x\{S^{-1}(\omega^*)\}^*\sigma_x.
\end{equation}
A system with $\mathcal{PT}$ symmetry is then characterized by the
invariance
\begin{equation}
\sigma_x\{S^{-1}(\omega^*)\}^*\sigma_x=S(\omega),
\end{equation}
which results in the constraint $\sigma_xS^*(\omega^*)\sigma_xS(\omega)=\openone$, or
\begin{align}\label{eq:ptcond}
t^{\prime*}(\omega^*)r(\omega)+r^*(\omega^*)t(\omega)&=0,\nonumber \\
r^{\prime*}(\omega^*)t'(\omega)+t^*(\omega^*)r'(\omega)&=0,\nonumber \\
r^*(\omega^*)r'(\omega)+t^{\prime*}(\omega^*)t'(\omega)&=\openone,\nonumber \\
r^{\prime*}(\omega^*)r(\omega)+t^{*}(\omega^*)t(\omega)&=\openone.
\end{align}
\subsection{${\cal PTT}'$ symmetry }
In hermitian problems the scattering matrix is unitary if $\omega$ is
real, and the ${\cal T}$ operation is equivalent to the operation
\cite{beenakkerreview,datta}
\begin{equation}
\mathcal{T}'[S(\omega)]=S^T(\omega),
\end{equation}
which corresponds to the operation identified above by inspection of the
wave equation. In non-hermitian settings, this delivers the solution of
the scattering problem associated with the transposed wave equation
$\mathcal{L}^T\bar\psi(\mathbf{r})=0$,
\begin{equation}
\label{eq:st}
\bar{\mathbf{a}}_{\mathrm{out}}=S^T(\omega)\bar{\mathbf{a}}_{\mathrm{in}}.
\end{equation}
In ordinary optics, where $\textbf{A}=0$, $\mathcal{T}'$ remains an exact
symmetry even when $n$ and $\omega$ are complex. We then find the
important constraint $S(\omega)=S^T(\omega)$, thus
\begin{equation}\label{eq:sst}
r=r^T,\quad r'=r'^T,\quad t'=t^T.
\end{equation}
The combined operation
\begin{equation}\label{eq:ptts}
\mathcal{PTT}'[S(\omega)]=\sigma_x\{S^{-1}(\omega^*)\}^\dagger\sigma_x
\end{equation}
turns into a symmetry if \cite{hsrmt}
\begin{align}\label{eq:pttcond}
t^{\dagger}(\omega^*)r(\omega)+r^\dagger(\omega^*)t(\omega)&=0,\nonumber \\
r^{\prime\dagger}(\omega^*)t'(\omega)+t^{\prime\dagger}(\omega^*)r'(\omega)&=0,
\nonumber \\
r^\dagger(\omega^*)r'(\omega)+t^{\dagger}(\omega^*)t'(\omega)&=\openone,\nonumber \\
r^{\prime\dagger}(\omega^*)r(\omega)+t^{\prime\dagger}(\omega^*)t(\omega)&=\openone.
\end{align}
This is realized for systems obeying the symmetry requirements
\eqref{eq:ptna}.
\section{\label{sec:6}Generalized flux conservation}
For hermitian systems ($\omega$, $n$ and $\mathbf{A}$ all real), the
unitarity $S^\dagger S=\openone$ of the scattering matrix ensures the
conservation of the probability flux, $\mathbf{a}_{\mathrm{in}}^\dagger
\mathbf{a}_{\mathrm{in}}=\mathbf{a}_{\mathrm{out}}^\dagger
\mathbf{a}_{\mathrm{out}}$. In Ref. \cite{variousstonepapers2} analogous
conservation laws were established for one-dimensional non-hermitian
systems with $\mathcal{PT}$ symmetry, and it was found that these laws
are automatically guaranteed by the symmetry conditions
\eqref{eq:ptcond}. These flux conditions only apply at real $\omega$, but
provide a useful alternative perspective on the role of symmetry. Here we
extend these conditions to higher-dimensional (multichannel) systems,
include magneto-optical effects, and also allow for $\mathcal{PTT}'$
symmetry (these considerations are original).
To illustrate the general strategy we consider the Helmholtz equation for
TM polarization, with ${\bf A}=0$ and $\omega$ real. Thus
$\psi(\mathbf{r})$ and $\psi^*(\mathcal{P}\mathbf{r})$ both solve the
same wave equation $\mathcal{L}(\omega)\psi(\mathbf{r})
=\mathcal{L}(\omega)\psi^*(\mathcal{P}\mathbf{r})=0$, with
$\mathcal{L}(\omega)$ specified in equation \eqref{eq:helmholtz}. Now take the
following volume integral over the region $\Omega$,
\begin{align}
0&=\int_\Omega d\mathbf{r}\,
[\psi^*(\mathcal{P}\mathbf{r})\mathcal{L}(\omega)\psi(\mathbf{r})
-\psi(\mathbf{r})\mathcal{L}(\omega)\psi^*(\mathcal{P}\mathbf{r})]
\nonumber \\ &=
\int_\Omega d\mathbf{r}\,
[\psi^*(\mathcal{P}\mathbf{r})\nabla^2 \psi(\mathbf{r})
-\psi(\mathbf{r})\nabla^2 \psi^*(\mathcal{P}\mathbf{r})]
\nonumber \\ &=
\int_\Omega d\mathbf{r}\,\nabla
[\psi^*(\mathcal{P}\mathbf{r})\nabla \psi(\mathbf{r})
-\psi(\mathbf{r})\nabla \psi^*(\mathcal{P}\mathbf{r})]
\nonumber \\ &=
\int_{\partial\Omega}d\mathbf{S}\cdot
[\psi^*(\mathcal{P}\mathbf{r})\nabla \psi(\mathbf{r})
-\psi(\mathbf{r})\nabla \psi^*(\mathcal{P}\mathbf{r})],
\end{align}
under application of Stoke's theorem. The resulting surface integral can
be evaluated with equation \eqref{eq:flux}. This delivers the generalized
flux-conservation law
\begin{equation}
\mathbf{a}_{\mathrm{in}}^\dagger\sigma_x \mathbf{a}_{\mathrm{in}}=\mathbf{a}_{\mathrm{out}}^\dagger\sigma_x \mathbf{a}_{\mathrm{out}}=\mathbf{a}_{\mathrm{in}}^\dagger S^\dagger(\omega)\sigma_x S(\omega)\mathbf{a}_{\mathrm{in}},
\end{equation}
which together with $S(\omega)=S^T(\omega)$ (as $\mathbf{A}=0$) amounts
to the constraints \eqref{eq:ptcond}, specialized to the case where
$\omega$ is real.
The same constraints follow for TE polarization (as $n=1$ outside
$\Omega$, so that the refractive index does not feature in the surface
integral). Next, we include a finite vector potential $\textbf{A}$. For
$\mathcal{PTT}'$ symmetry, one can follow the steps given above, which
directly delivers the constraints \eqref{eq:pttcond} (again specialized
to real $\omega$). The case of $\mathcal{PT}$ symmetry with finite vector
potential is more intricate, as one then needs to invoke the transposed
wave equation
$\mathcal{L}^T(\omega)\bar\psi(\mathbf{r})=\mathcal{L}^T(\omega)\bar\psi^*(\mathcal{P}\mathbf{r})=0$.
One then integrates
\begin{align}
0&=\int_\Omega d\mathbf{r}\,
[\bar\psi^*(\mathcal{P}\mathbf{r})\mathcal{L}(\omega)\psi(\mathbf{r})
-\psi(\mathbf{r})\mathcal{L}^T(\omega)\bar\psi^*(\mathcal{P}\mathbf{r})]
\nonumber \\ &=
\int_{\partial\Omega}d\mathbf{S}\cdot
[\bar\psi^*(\mathcal{P}\mathbf{r})\nabla \psi(\mathbf{r})
-\psi(\mathbf{r})\nabla \bar\psi^*(\mathcal{P}\mathbf{r})],
\end{align}
which results in the generalized flux-conservation law
$\bar{\mathbf{a}}_{\mathrm{in}}^\dagger \sigma_x \mathbf{a}_{\mathrm{in}}
=\bar{\mathbf{a}}_{\mathrm{out}}^\dagger\sigma_x \mathbf{a}_{\mathrm{out}}
$.
In combination with equation \eqref{eq:st}, this condition now requires
$\sigma_x=S^*(\omega)\sigma_x S(\omega)$, which is again automatically
fulfilled if the scattering matrix obeys the symmetry constraints
\eqref{eq:ptcond}.
In all these cases, the generalized flux-conservation relations at real
$\omega$ thus does not impose any extra conditions beyond the symmetry
requirements of the scattering matrix.
\section{\label{sec:7}Scattering quantization conditions}
It is now interesting to ask how the spectral properties of
$\mathcal{PT}$ or $\mathcal{PTT}'$-symmetric systems emerge within the
scattering approach. We contrast the case of open systems (where the
symmetry operation relates quasibound states with perfectly absorbed
states, thus, connects states of a different physical nature) with closed
systems (where the symmetry operation relates ordinary bound states,
thus, connects states of the same nature, whose spectral properties then
are constrained).
\subsection{General quantization conditions for quasibound and perfectly absorbed states}
Quasibound states fulfill the wave equation with purely outgoing boundary
conditions, thus, $\mathbf{a}_{\mathrm{in}}=0$ but
$\mathbf{a}_{\mathrm{out}}$ finite. In view of equation \eqref{eq:sdef},
this requires
\begin{equation}\label{eq:quant}
||S(\omega)||=\infty,
\end{equation}
and so $\omega$ has to coincide with a pole of the scattering matrix.
This results in a quantization of the admitted frequencies $\omega_n$,
which in general are complex.
Quasibound states describe systems which generate all radiation
internally, and in particular, lasers
\cite{hsqoptics,longhilaserabsorber,variousstonepapers1,variousstonepapers2,yoo}.
In a passive system the poles are confined to the lower half of the
complex plane, with the decay enforced by the leakage. In an active
system, however, gain may compensate the losses and result in a
stationary state, which signifies lasing. The threshold is attained when
the first pole reaches the real axis.
Recent works have turned the attention to perfectly absorbed states, for
which the role of incoming and outgoing states is inverted
\cite{longhilaserabsorber,stonecpa1}. These boundary conditions translate
to
$||S(\omega)||=0$,
which results in quantized frequencies $\widetilde{\omega}_n$ coinciding
with the zeros of the scattering matrix.
For a passive system, where
$\mathcal{TT}'[\mathcal{L}(\omega)]=\mathcal{L}(\omega^*)$, a quasibound
state $\psi_n(\mathbf{r})$ can be converted into a perfectly absorbed
state $\widetilde\psi_n(\mathbf{r})=\bar\psi_n^*(\mathbf{r})$ by the
$\mathcal{TT}'$ operation [for exact $\mathcal{T}'$ symmetry one simply
has $\widetilde\psi_n(\mathbf{r})=\psi_n^*(\mathbf{r})$]. This state then
fulfills the wave equation at $\widetilde{\omega}_n=\omega_n^*$, which
confines the zeros to the upper half of the complex plane.
For non-hermitian systems the relation between the poles and zeros is in
general broken. With $\mathcal{PT}$ or $\mathcal{PTT}'$ invariance,
however, the states $\widetilde\psi_n(\mathbf{r})=\psi_n^*({\cal
P}\mathbf{r})$ or $\widetilde\psi_n(\mathbf{r})=\bar\psi^*_n({\cal
P}\mathbf{r})$ are paired by the respective symmetry. Moreover, the
frequencies $\omega_n=\widetilde{\omega}_n^*$ then are no longer
constraint to the lower half of the complex plane. At the lasing
condition $\omega_n=0$, the lasing mode is then degenerate with a
perfectly absorbed mode, which leads to the concept of a
$\mathcal{PT}$-symmetric laser-absorber \cite{longhilaserabsorber}.
\subsection{Closed resonators and spectral constraints}
By taking the appropriate limit of the expressions for open systems, the
scattering approach to mode quantization can be used to study closed
systems, for which $\mathcal{PT}$ symmetry was originally defined. The
quasibound states then turn into normal bound states, and one recovers
the spectral constraints \eqref{eq:constraint}. These considerations also
apply to $\mathcal{PTT}'$ symmetry. They can also be based on the
perfectly absorbed states, which become degenerate with the quasibound
states.
When one reduces the leakage from a non-hermitian $\mathcal{PT}$ or
$\mathcal{PTT}'$-symmetric optical system, it will ultimately start to
lase. If some frequencies of the closed system are complex then lasing is
attained at a finite leakage
\cite{longhilaserabsorber,variousstonepapers1,variousstonepapers2,yoo};
otherwise one approaches at-threshold lasing \cite{hsqoptics}.
\section{\label{sec:8}
Coupled-resonator geometry}
The economy of the scattering approach is full exposed when one applies
it to multichannel systems capable of displaying complex wave dynamics.
Here we review this for a broad class of systems in which an absorbing
resonator is coupled symmetrically to an amplifying resonator, as shown
in figure \ref{fig3} \cite{hsqoptics,yoo,hsrmt,birchallweyl}.
\begin{figure}
\includegraphics[width=\columnwidth]{fig3.eps}
\caption{\label{fig3}(Online version in colour.)
Scattering amplitudes and their relation by reflection and transmission
blocks of the scattering matrix, for the coupled-resonator geometry
considered in \S\ref{sec:8}--\S\ref{sec:10}. In (b), a finite
transparency $T$ of the interface is taken into account. See also
\cite{hsqoptics,yoo,hsrmt}. }
\end{figure}
\subsection{Scattering matrix}
Denoting the scattering matrices of the two resonators as
\begin{equation}
S_L=\left(
\begin{array}{cc}
r_L & t_L' \\
t_L & r_L' \\
\end{array}
\right), \quad S_R=\left(
\begin{array}{cc}
r_R & t_R' \\
t_R & r_R' \\
\end{array}
\right),
\end{equation}
the scattering matrix $S=S_L\circ S_R$ of the composed system is given by
\begin{align}
&\left( \begin{array}{cc}r_L&t_L'\\t_L&r_L'\end{array} \right)\circ
\left( \begin{array}{cc}r_R&t_R'\\t_R&r_R'\end{array} \right)
\nonumber \\ &=
\left( \begin{array}{cc}r_L+t_L'\frac{1}{1-r_Rr_L'}r_Rt_L&t_L'\frac{1}{1-r_Rr_L'}t_R'
\\ t_R\frac{1}{1-r_L'r_R}t_L&r_R'+t_R\frac{1}{1-r_L'r_R}r_L't_R'\end{array} \right).
\label{eq:comprule}
\end{align}
$\mathcal{PT}$ symmetry follows if the scattering matrices are related by
$S_R=\mathcal{PT}[S_L]$, with this operation specified in equation
\eqref{eq:pts}, while $\mathcal{PTT}'$ symmetry is realized if the
scattering matrices are related by $S_R=\mathcal{PTT'}[S_L]$, as
specified in equation \eqref{eq:ptts}.
This construction can be amended to include an interface of finite
transparency $T$, with the scattering matrix, \emph{e.g.}, specified by
\begin{equation}\label{eq:stun}
S_T=
\left(\begin{array}{cc}-\sqrt{1-T} & i\sqrt{T}
\\ i\sqrt{T} & -\sqrt{1-T} \\ \end{array}\right).
\end{equation}
The total scattering matrix of the system is then given by $S=S_L\circ
S_T\circ S_R$.
\subsection{Quantization conditions and spectral constraints}
With equation \eqref{eq:comprule} the scattering quantization condition
\eqref{eq:quant} becomes
\begin{equation}
{\rm det}\,[r_L'(\omega) r_R(\omega) -\openone]=0.
\end{equation}
For a $\mathcal{PT}$-symmetric system obeying equation \eqref{eq:ptcond}
this condition takes the form
\begin{equation}\label{eq:ptquant}
{\rm det}\,(r_L'(\omega)-
[r_L'(\omega^*)-t_L(\omega^*)r_L^{-1}(\omega^*)t_L'(\omega^*)]^*)=0,
\end{equation}
while for a $\mathcal{PTT}$-symmetric system obeying equation
\eqref{eq:pttcond} this gives
\begin{equation}\label{eq:ptquantptt}
{\rm det}\,(r_L'(\omega)-
[r_L'(\omega^*)-t_L(\omega^*)r_L^{-1}(\omega^*)t_L'(\omega^*)]^\dagger)=0.
\end{equation}
For a closed system the scattering matrices reduce to the reflection
blocks $S_L(\omega)=r_L'(\omega)$ and $S_R(\omega)=r_R(\omega)$, while
the transmission vanishes. In the case of $\mathcal{PT}$ symmetry, with
$r_R(\omega)= \{[r_L'(\omega^*)]^{-1}\}^*$, equation \eqref{eq:ptquant}
assumes the form
\begin{equation}\label{eq:ptquant2}
{\rm det}\,(r_L'(\omega)-[r_L'(\omega^*)]^*)=0,
\end{equation}
which entails the constraints \eqref{eq:constraint}. On the real
frequency axis, this reduces to the condition $ {\rm det}\,{\rm
Im}\,r_L'(\omega)=0. $ Analogous observations also hold for closed
$\mathcal{PTT}'$-symmetric resonators, for which the quantization
condition \eqref{eq:ptquantptt} becomes
\begin{equation}\label{eq:ptquant2ptt}
{\rm det}\,(r_L'(\omega)-[r_L'(\omega^*)]^\dagger)=0.
\end{equation}
Under inclusion of a semitransparent interface, similar conditions can be
derived from the general expression
\begin{equation}\label{eq:ptquantst} {\rm
det}\,\left[S_T\left(\begin{array}{cc}r_L' & 0 \\0 & r_R \\
\end{array}\right)-\openone\right]=0.
\end{equation}
These scattering quantization conditions can all be interpreted as
conditions for constructive interference upon return to the interface
between the two parts of the resonator. The analogous conditions for
perfectly absorbed states follow from the replacement $\omega\to
\omega^*$.
\section{\label{sec:9}Effective Models}
The coupled-resonator geometry allows to make contact to well-studied
standard descriptions of multiple scattering
\cite{beenakkerreview,fyodorovsommers,fyodsomeffs,andreev,schomerusjacquod}.
This leads to effective models of complex wave dynamics in systems with
$\mathcal{PT}$ and $\mathcal{PTT}'$ symmetry, which we first formulate in
the energy domain \cite{hsrmt}, and then in the time domain
\cite{birchallweyl}.
\subsection{Effective Hamiltonians}
We first employ the Hamiltonian approach to multiple scattering
\cite{beenakkerreview,fyodorovsommers}. The hermitian part of the
dynamics in the absorbing resonator is captured by a hermitian $M\times
M$-dimensional matrix $H$. We assume $M\gg 1$ and denote the level
spacing in the energy range of interest as $\Delta$. Loss with absorption
rate $\mu$ is modeled by adding a non-hermitian term $-i\mu\openone$,
while gain with a matching rate is obtained by inverting the sign of
$\mu$. We also specify an $N\times M$-dimensional coupling matrix $V$
between the $M$ internal modes and $N$ scattering states. This matrix can
be chosen to satisfy $V^TV={\rm diag}{(v_m)}$, where $N$ finite entries
$v_m=\Delta M/\pi$ describe the open channels, while $M-N$ entries
$v_m=0$ describe the closed channels. The $N\times N$-dimensional
scattering matrix of the absorbing resonator is then given by
\begin{equation}
S_L(\omega) = 1-2i V (\omega-i\mu-H+i V^TV)^{-1}V^T.
\end{equation}
Thus, the leakage from the system effectively adds an additional
non-hermitian term $-i V^TV$ to the Hamiltonian. For $\mathcal{PT}$
symmetry the scattering matrix of the amplifying resonator follows as
\cite{hsrmt}
\begin{equation}
S_R(\omega)=[S_L^{-1}(\omega^*)]^*= 1-2i V (\omega+i\mu-H^*+i V^TV)^{-1}V^T.
\end{equation}
We also include a semitransparent interface, with scattering matrix
\eqref{eq:stun}. For a closed system, the scattering quantization
condition \eqref{eq:ptquantst} can then be rearranged into an eigenvalue
problem ${\rm det}\,(\omega-{\cal H})=0$ with effective Hamiltonian
\cite{hsrmt}
\begin{equation}
\label{eq:h}
{\cal H}=\left(\begin{array}{cc}H-i\mu & \Gamma \\ \Gamma & H^*+i\mu \\ \end{array}\right).
\end{equation}
Here $\Gamma={\rm diag}\,(\gamma_m)$ is a real positive semi-definite
coupling matrix related to $V^TV$, but with the $N$ non-vanishing entries
$\gamma_m=[\sqrt{T}/(1+\sqrt{1-T})]\Delta M/\pi\equiv \gamma$ modified to
account for the finite transparency $T$ of the interface.
The effective Hamiltonian obeys the relation
$\mathcal{PT}[\mathcal{H}]=\sigma_x \mathcal{H}^* \sigma_x =\mathcal{H}$.
In a $\mathcal{PT}$-symmetric basis the secular equation ${\rm
det}\,(\omega-{\cal H})=0$ then takes the form of a polynomial with real
coefficients, which guarantees that the eigenvalues are either real or
occur in complex-conjugate pairs, as required by equation
\eqref{eq:constraint}.
Analogous considerations apply to $\mathcal{PTT}'$-symmetric systems
\cite{hsrmt,birchallrmt}. The scattering matrix of the right resonator
then changes to
\begin{equation}
S_R(\omega)=[S_L^{-1}(\omega^*)]^\dagger= 1-2i V (\omega+i\mu-H+i V^TV)^{-1}V^T,
\end{equation}
and the effective Hamiltonian of the closed system takes the form
\begin{equation}
\label{eq:hptt}
{\cal H}=\left(\begin{array}{cc}H-i\mu & \Gamma \\ \Gamma & H+i\mu \\ \end{array}\right).
\end{equation}
We then have
$\mathcal{PTT}'[\mathcal{H}]=\sigma_x \mathcal{H}^\dagger \sigma_x =\mathcal{H}$,
which again guarantees the required spectral constraints.
While Hamiltonians with these symmetries could simply be stipulated, the
derivation of the specific manifestations \eqref{eq:h} and
\eqref{eq:hptt} reveals further constraints dictated by the
coupled-resonator geometry. In particular, the coupling is only physical
if the matrix $\Gamma$ is positive semidefinite. These Hamiltonians bear
a striking resemblance to models of mesoscopic superconductivity
\cite{beenakkerreview,melsen}.
\subsection{Quantum maps}
\begin{figure}
\includegraphics[width=\columnwidth]{fig4.eps}
\caption{\label{fig4}(Online version in colour.)
Interpretation of the $\mathcal{PT}$-symmetric quantum map
\eqref{eq:map}, which translates the resonator dynamics in (a) to a
stroboscopic evolution in two coupled Hilbert spaces (b). For the
$\mathcal{PTT}'$-symmetric map \eqref{eq:mapptt}, $F^T$ is replaced by
$F$. See also \cite{birchallweyl}. }
\end{figure}
Alternatively, one can set up effective descriptions in the time domain
\cite{birchallweyl}. For $\mathcal{PT}$ symmetry this sets out by writing
the scattering matrices as \cite{fyodsomeffs,andreev,schomerusjacquod}
\begin{subequations}
\begin{align}
S_L(\omega)&=WF[\exp(-i\omega\tau+\mu)-QF]^{-1}W^T,\\
S_R(\omega)&=WF^T[\exp(-i\omega\tau-\mu)-QF^T]^{-1}W^T,
\label{eq:smat}
\end{align}
\end{subequations}
where $F$ is an $M\times M$-dimensional unitary matrix which can be
thought to describe the stroboscopic time evolution between successive
scattering events off the resonator walls. The transfer across the
interface between the two resonators is again described by an $N\times
M$-dimensional coupling matrix $W$. However, the combination $P=W^TW$ now
projects the wave function onto the interface, such that ${\rm
rank}\,P=N$ is the number of channels connecting the resonators, while
$Q=\openone_M-P$ is the complementary projector onto the resonator wall
(${\rm rank}\,Q=M-N$). The stated frequency dependence corresponds to
stroboscopic scattering with fixed rate $\tau^{-1}$; the variable
$\omega$ thus plays the role of a quasienergy. The parameter $\mu\geq 0$
again determines the absorption and amplification rate. For the passive
system ($\mu=0$), the scattering matrices are unitary, which corresponds
to the hermitian limit of the problem.
With these specifications, and including a semitransparent interface
parameterized by $\alpha\equiv\sqrt{\sqrt{R}+i\sqrt{T}}$, the
quantization condition (\ref{eq:ptquantst}) is equivalent to the
eigenvalue problem
\begin{equation}
{\cal F}\psi_n=\lambda_n\psi_n,\quad \lambda_n=\exp(-i\omega_n\tau)
\end{equation}
for the quantum map
\begin{align}\label{eq:map}
{\cal F}&=
\sqrt{C}
\left(\begin{array}{cc} e^{-\mu}F & 0 \\ 0 & e^{\mu}F^T \\ \end{array} \right)
\sqrt{C},
\nonumber \\
\quad \sqrt{C}&=
\left(\begin{array}{cc} \mathrm{Re}\,\alpha\,P+Q & -i\mathrm{Im}\,\alpha\,P\\
-i\mathrm{Im}\,\alpha\,P &\mathrm{Re}\,\alpha\,P+Q \\ \end{array} \right)
\end{align}
where we symmetrized the coupling
\begin{equation}
C=\sqrt{C}\sqrt{C}=\left(\begin{array}{cc} \sqrt{R}P+Q &
-iP\sqrt{T} \\ -iP\sqrt{T} & \sqrt{R}P+Q \\ \end{array} \right).
\end{equation}
The $2M\times 2M$ dimensional matrix ${\cal F}$ can be interpreted as a
stroboscopic time evolution operator acting on $2M$-dimensional vectors
$\psi=\binom{\psi_L}{\psi_R}$, where $\psi_L$ and $\psi_R$ give the wave
amplitude in the absorbing and amplifying subsystem, respectively. This
is illustrated in figure \ref{fig4}. The $\mathcal{PT}$ symmetry of the
quantum map manifests itself in the relation
${\cal F}=\sigma_x[{\cal F}^{-1}]^*\sigma_x$,
which parallels the symmetry \eqref{eq:pts} for the scattering matrix.
Analogously, $\mathcal{PTT}'$ symmetry results in the quantum map
\begin{equation}
{\cal F}=
\sqrt{C}
\left(\begin{array}{cc} e^{-\mu}F & 0 \\ 0 & e^{\mu}F \\ \end{array} \right)
\sqrt{C}.
\label{eq:mapptt}
\end{equation}
This obeys the symmetry
${\cal F}=\sigma_x[{\cal F}^{-1}]^\dagger\sigma_x$,
which parallels equation \eqref{eq:ptts}.
The spectral properties associated with these symmetries are now embodied
in the secular equation $s(\lambda)=\rm{det}\,({\cal F} -\lambda)=0$,
which exhibits the mathematical property of \emph{self-inversiveness}
\cite{selfinverse2}:
\begin{equation}
s(1/\lambda^*)=[\lambda^{-2M}s(\lambda)]^*s(0),
\end{equation}
where $s(0)=\mathrm{det}\,\mathcal{F}= (\mathrm{det}\,F)^2$. For each
eigenvalue $\lambda_n$, we are thus guaranteed to find the eigenvalue
$\lambda_{\bar n}=[\lambda_n^{-1}]^*=\exp(-i\omega_n^*\tau)$, which
recovers the constraint \eqref{eq:constraint}.
\section{\label{sec:10}Mesoscopic energy and times scales}
In mesoscopic physics, a large range of spectral, thermodynamic and
transport phenomena are fully characterized by a few universal time and
energy scales \cite{beenakkerreview,datta,altlandthouless}. We now have
all the tools to make contact to these concepts and provide a
phenomenological description of the effects of multiple scattering on the
spectral features of the considered coupled resonators. These effects
generally set in when a large number $M\gg1$ of internal modes in an
energy range $M\Delta$ is mixed by scattering with a characteristic time
scale $\tau=\hbar/M\Delta$ that is much less than the dwell time in the
amplifying or absorbing regions, $\tau\ll t_{\mathrm{dwell}}$. The
coupling strength between these regions is characterized by the
associated Thouless energy $E_T=\hbar/t_{\mathrm{dwell}}\approx NT
\Delta$, which can be small or large compared to the level spacing
$\Delta$, depending on the value of the dimensionless conductance $g=NT$
of the interface. Non-hermiticity adds the new scale $\mu$, whose
interplay with the other scales determines the transition from real to
complex eigenvalues.
In the mesoscopic regime of $M\gg N \gg 1$ this transition can be
expected to be universal, with the features only depending on $M/N \gg
1$, $M\gg 1$, and $T$, which we consider fixed by the geometry, as well
as the variable $\mu$. The transition can then be investigated by
combining the effective models set up in the previous section with
random-matrix theory \cite{beenakkerreview,haake,mehta}, thus, ensembles
of Hamiltonians $H$ (usually composed with random Gaussian matrix
elements) or time-evolution operators $F$ (distributed according to a
Haar measure) which are only constrained by the symmetries of the
problem. Closer inspection identifies two natural scenarios
\cite{hsrmt,birchallrmt}, described in the following two subsections,
and a semiclassical source of corrections to random-matrix theory, discussed thereafter \cite{birchallweyl}. The interplay of the various time and energy scales is illustrated in
figure \ref{fig5}.
\subsection{Systems with coupling-driving level crossings}
In ordinary optics with exact $\mathcal{T}'$ symmetry, systems with
$\mathcal{PT}$ symmetry also display $\mathcal{PTT}'$ symmetry. The
internal Hamiltonian $H=H^*=H^T$ is then real and symmetric, and the
effective Hamiltonians \eqref{eq:h} and \eqref{eq:hptt} coincide.
(Analogously, $F=F^T$, so that the quantum maps \eqref{eq:map} and
\eqref{eq:mapptt} coincide as well.) Switching to a parity-invariant
basis one then finds
\begin{equation}\label{eq:hpar}
{\cal H}_{\cal P}=\left(\begin{array}{cc}H+\Gamma & i\mu \\ i\mu & H-\Gamma \\ \end{array}\right),
\end{equation}
which reveals the emerging $\mathcal{T}$ symmetry in the hermitian limit.
The same structure arises for systems with magneto-optical effects which
only display $\mathcal{PTT}'$. For both cases, at $\mu=0$ the system
decouples into two independent sectors. For $T=0$ the level sequences of
these sectors coincide. For very weak coupling ($T\ll 1/N\equiv T_c$,
thus $g\ll 1$) the degeneracy is slightly lifted, and one can apply
almost-degenerate perturbation theory to identify the scale $\mu\approx
N\sqrt{T}\Delta/2\pi$ at which typical eigenvalues turn complex. As $T$
exceeds $1/N\equiv T_c$ ($g>1$), however, the levels of the original
sequences cross, and when one then increases $\mu$ the bifurcations to
complex eigenvalues occur between levels that were originally
non-degenerate. In this regime, a macroscopic fraction of complex
eigenvalues appears on a \emph{coupling-independent} scale
$\mu\approx\mu_0\equiv\sqrt{N}\Delta/2\pi$.
\subsection{Systems with coupling-driving avoided crossings}
For systems with $\mathcal{PT}$ symmetry but broken $\mathcal{T}'$
symmetry, the parity basis does not partially diagonalize the effective
Hamiltonian. At $\mu=T=0$, one still starts from two degenerate level
sequences, but these interact as $T$ is increased, and instead of level
crossings one observes level repulsion. In this case the crossover to a
complex spectrum appears at $\mu\approx \sqrt{NT}\Delta/2\pi$ and thus is
always coupling dependent.
\begin{figure}
\includegraphics[width=\columnwidth]{fig5.eps}
\caption{\label{fig5}(Online version in colour.)
Sketch of the energy and time scales governing the spectral features of
$\mathcal{PT}$-symmetric resonators with separate
$\mathcal{T}'$ invariance (no magneto-optical effects), as hermiticity is
broken with absorption and amplification rate $\mu$. This universal
picture holds under the assumption of complex wave dynamics ($M\gg
N,NT\gg 1$), where $M=\hbar/\tau\Delta$ is the number of internal modes
mixed by multiple scattering with scattering time $\tau$ and $N$ is the
number of channels coupling the resonators, with transparency $T$. Here
$\Delta$ is the mean level spacing, $t_{\rm H}=\hbar/\Delta$ the
Heisenberg time, and $E_T=\hbar/t_{\rm dwell}=NT\Delta$ the Thouless
energy associated with the dwell time $t_{\rm dwell}$ in each resonator.
The critical rate at which eigenvalues turn complex is
$\mu_0=\sqrt{N}\Delta/2\pi$. When the Ehrenfest time $t_{\rm
Ehr}=\lambda^{-1}\ln M$ (with Lyapunov exponent $\lambda$) becomes
comparable to the dwell time $t_{\rm dwell}$ quantum-to
classical-correspondence sets in and suppresses multiple scattering, which reduces the
number of strongly amplified states. See also \cite{hsrmt,birchallweyl}.
}
\end{figure}
\subsection{Growth and decay rates in the semiclassical limit}
In the semiclassical limit of large $M$ at fixed $M/N$ the results above
imply that the crossover to a complex spectrum appears at a vanishingly
small rate $\mu\ll E_T$. Keeping $\mu/E_T$ in this limit fixed and
finite, the real phase is thus completely destroyed, and the focus turns
to the typical decay and growth rates encoded in the imaginary parts
${\rm Im}\, \omega_n$ of the complex eigenvalues. Numerical sampling of
the random-matrix ensembles suggests that the distribution $P({\rm Im}\,
\omega_n/\mu;\mu/E_T)$ attains a stationary limit for $M\gg 1$
\cite{hsrmt,birchallweyl,birchallrmt}. At $\mu\gtrsim E_T$, one then
finds a finite fraction of strongly amplified states with ${\rm Im}\,
\omega_n\approx \mu$, thus, a large number of candidate lasing states.
However, the fraction of these states is reduced when one takes dynamical
effects into account \cite{birchallweyl}. One then finds that the
strongly amplified states are supported by the classical repeller in the
amplifying parts of the system, whose fractal dimension $d_H$ is more and
more resolved as the phase space resolution $h\propto 1/M$ increases in
the semiclassical limit. This phenomenon can be characterized by the
Ehrenfest time $t_{\rm Ehr}=\lambda^{-1}\ln M$, where $\lambda$ the
Lyapunov exponent in the classical system. If $t_{\rm Ehr}>t_{\rm
dwell}$, the wave dynamics become quasi-deterministic, and multiple
scattering is reduced. The fraction of strongly amplified states then
follows a fractal Weyl law (a power law in $h$ with non-integer
exponent), a phenomenon which was previously observed for passive
quantum systems with finite leakage through ballistic openings
\cite{fractalweyl1,fractalweyl3}.
\section{\label{sec:11}Summary and outlook}
In summary, the scattering approach proves useful to describe general
features of non-hermitian optical systems with $\mathcal{PT}$ and
$\mathcal{PTT}'$ symmetry. In particular, the approach fully accounts for
complications that arise beyond one dimension (a choice of non-equivalent
geometric and time-reversal symmetries, the possibility of
magneto-optical effects with a finite external vector potential, and
multiple scattering), as well as additional leakage which turns the
symmetry of bound states into a relation between quasibound and perfectly
absorbed states. These effects can be captured in effective model
Hamiltonians and time evolution operators, which help to identify the
mesoscopic energy and time scales that govern the spectral features of a
broad class of systems. The construction of these models exposes physical
constraints that go beyond the mere symmetry requirements.
The models formulated here can be adapted to include, \emph{e.g.},
inhomogeneities in the gain, dissipative magneto-optical effects, leakage, or other symmetry-breaking effects \cite{birchallrmt}. Furthermore,
the construction of symmetric resonators from two subsystems can be
extended to include more elements, such as they occur in periodic or
disordered arrays. This bridges to models of higher-dimensional diffusive
or Anderson-localized dynamics \cite{west}. The models can also be
generalized to include symplectic time-reversal symmetries (with
$\mathcal{T}^2=-1$), chiral symmetries, or particle-hole
symmetries (as the $\mathcal{CT}$ symmetry obeyed by mesoscopic superconductors), for which the
consequences on the scattering matrix and effective Hamiltonians are
well known in the hermitian case \cite{beenakkerreview,altlandzirnbauer,melsen}. Some of these
effects have optical analogues; \emph{e.g.}, phase-conjugating mirrors
induce effects similar to Andreev reflection in mesoscopic
superconductivity but result in a non-hermitian Hamiltonian \cite{beenakkerphase}. Finally we note that the scattering approach also
offers a wide range of analytical and numerical methods which allow to
efficiently study individual systems \cite{datta}.
I thank Christopher Birchall for fruitful collaboration on Refs.\
\cite{birchallweyl,birchallrmt}, which form the basis of some of the
material reviewed here, as well as Uwe G{\"u}nther for fruitful
discussions concerning the role of symmetries.
|
2,869,038,154,789 | arxiv | \section{Introduction}
\label{sec:intro}
The use of the graph Laplacian to diffuse information over networks is well-established in classical and contemporary work ranging from opinion dynamics \cite{Taylor68} to distributed consensus \cite{DeGroot1974} and controls \cite{Preciado10}, synchronization \cite{Barahona2002,singer_angular_2011-1}, flocking \cite{Jadbabaie15}, and much more. In the past decade, Laplacians that are adapted to handle vector-valued data, such as graph connection Laplacians \cite{singer_vector_2012,BandeiraSS13} or matrix-weighted Laplacians \cite{Tuna16}, have been revolutionary in data science and, most notably, graph signal processing \cite{shuman_emerging_2013-1,ribeiro19GSP}.
While the ultimate form of a generalized Laplacian is as yet not present in applications, there are hints of a broader theory finding its way from algebraic geometry and topology to data science. The classical Laplacian from calculus class and the graph Laplacian are two extreme examples of a \define{Hodge Laplacian} from algebraic geometry. These are operators which act diffusively on data structures called \define{sheaves} \cite{bredon_sheaf_1997,kashiwara_sheaves_1990}: see \S\ref{sec:sheaves} for brief details.
The present work is motivated by extending recent work on distributed consensus and data fusion from the setting of vector-valued data to that of data valued in more general partially-ordered sets and, specifically, lattices (in the algebraic as opposed to geometric sense): see \S\ref{sec:lattices}. The fidelity with which lattices (including Boolean algebras) can model logical structures makes them appealing for representing distributed systems with complex logical behaviors, such as preference posets \cite{JanisMST15} and (via powerset lattices) discrete signal processing \cite{puschel2020discrete}.
\paragraph*{Related work}
The consensus literature is vast and includes Laplacian-based protocols \cite{olfati2007consensus} and (asynchronous) gossip-based protocols \cite{Kempe2003}. Several works consider consensus on general functions \cite{cortes2008distributed}, including max-consensus \cite{iutzeler2012analysis,nejad2009max}. Branch-and-bound style algorithms for reaching certain notions of lattice-valued consensus (important to the distributed computing literature) have been extensively studied \cite{zheng2021lattice,faleiro2012generalized}. Sheaves have shown promise in multi-agent systems, particularly from the perspective of concurrency \cite{goguen1992sheaf} and data fusion \cite{robinson2017sheaves}.
The Bayesian approach to modeling knowledge/belief propagation via graphical models \cite{koller2009probabilistic} is standard but fundamentally different than the approach here. Modal logics, particularly temporal logics, have seen numerous applications in model-checking \cite{baier2008principles} and control systems \cite{kantaros2019temporal,rodionova2021stl,riess2021temporal,kantaros2020reactive}. There are several use-cases of multimodal Kripke logic in the analysis of distributed systems \cite{fagin2004reasoning}.
The authors previously defined a synchronous [Tarski] Laplacian and proved a Hodge-style fixed-point convergence result \cite{ghrist2020cellular}, which is extended here to the asynchronous setting. The main algorithm in this paper (Algorithm \ref{alg:local}) generalizes certain existing algorithms in distributed systems, including that of computing solutions of two-sided max-plus linear systems \cite{cuninghame2003equation}; our algorithm goes far beyond this in computing sections of (nearly) arbitrary lattice-valued network sheaves.
\paragraph*{Outline}
Background material (\S\ref{sec:bg}) and problem spcifications (\S\ref{sec:problem}) are followed by details of a novel Laplacian for asynchronous communication on networks of lattices (\S\ref{sec:async}). It is here that the main results on stability and convergence are proved. The subsequent section (\S\ref{sec:semantics}) detail applications to multimodal logics, by using Kripke semantics, leading to a dual pair of {\em semantic} and {\em syntactic} Laplacians for diffusing knowledge and beliefs. This work ends (\S\ref{sec:examples}) with some simulation in the context of target tracking.
\section{Background}
\label{sec:bg}
\subsection{Lattices}
\label{sec:lattices}
{}
\begin{definition}
A \define{lattice} is a set $\lattice{Q}$ with two associative and commutative binary operations $\vee$ (join) and $\wedge$ (meet) such that the following additional axioms are satisfied:
\begin{itemize}
\item[] {\em Idempotence:} for all $x \in \lattice{Q}$, $x \wedge x = x, \quad x \vee x = x$.
\item[] {\em Identity:} there exist elements $\bot, \top \in \lattice{Q}$ such that $\bot$ is the identity of $\vee$ and $\top$ is the identity of $\wedge$.
\item[] {\em Absorption:} for all $x, y \in \lattice{Q}$,
\begin{align*}
x \vee (x \wedge y) &=& x &=& x \wedge (x \vee y).
\end{align*}
\end{itemize}
\end{definition}
Equivalently, lattices can be viewed as (partially) ordered sets $(\lattice{Q}, \preceq)$ with $x \preceq y \Leftrightarrow x \wedge y = x$ or, equivalently, $x \succeq y \Leftrightarrow x \vee y = x$. It will be useful to think of lattices as both partially-ordered sets and algebraic structures: the $\preceq$ notation will be crucial in proofs of all main results.
\begin{example}
Suppose $S$ is a set. The powerset $2^S$ is a (Boolean) lattice $(2^S, \cap, \cup, \emptyset, S)$. The truth values $\vec{2} = \{0,1\}$ with {\em AND}
and {\em OR} is a lattice. Other important lattices are embedded in $2^S$ such as lattices representing ontologies \cite{wille1982restructuring}, partitions \cite{davey2002introduction}, and information-theoretic content \cite{shannon1953lattice}. The extended real line $\bar{\mathbb{R}} = \mathbb{R} \cup \{- \infty, \infty \}$ is a lattice with min and max. Cartesian products of lattices are lattices with the obvious meet and join operations; the lattice $\bar{\mathbb{R}}^n$ is important in discrete event systems \cite{cuninghame1991minimax}, $\vec{2}^n$ in logic gates.
\end{example}
In order to work with systems of lattices, we will work with \define{lattice maps} $\varphi:\lattice{P}\to\lattice{Q}$. Such a map is \define{order preserving} if $x\preceq y \Rightarrow \varphi(x)\preceq\varphi(y)$ and is \define{join preserving} if $\varphi(x\vee y)=\varphi(x)\vee\varphi(y)$ and $\varphi(\bot_\lattice{P})=\bot_\lattice{Q}$. Join preserving maps are automatically order preserving. A dual definition of \define{meet preserving} maps holds with similar consequences.
\subsection{Network Sheaves}
\label{sec:sheaves}
Suppose $\graph{G} = (\graph{V}, \graph{E})$ is an undirected graph (possibly with loops). An edge between (not necessarily distinct) nodes $i$ and $j$ in $\graph{V}$ is denoted by an unordered concatenated pair of indices $ij=ji \in \graph{E}$. The set $\graph{N}_i = \{j: ij \in \graph{E} \}$ are the \define{neighbors} of $i$.
Given such a fixed network $\graph{G}$, we define a data structure over $\graph{G}$ taking values in lattices. Such a structure is an example of a \define{cellular sheaf} \cite{shepard1985cellular,curry2014sheaves} (though the details of sheaf theory are not needed here).
\begin{definition}
A \define{lattice-valued network sheaf} $\sheaf{F}$ over $\graph{G}$ is a data structure that assigns:
\begin{enumerate}
\item A lattice $(\sheaf{F}(i), \wedge_i, \vee_j)$ to every node $i \in \graph{V}$.
\item A lattice $(\sheaf{F}(ij), \wedge_{ij}, \vee_{ij})$ to every edge $ij \in \graph{E}$.
\item Structure-preserving maps
\begin{equation}
\begin{tikzcd}
\sheaf{F}(i) \arrow[r,"\sheaf{F}_{i \fc ij}"] & \sheaf{F}(ij) & \sheaf{F}(j) \arrow[l,"\sheaf{F}_{j \fc ij}", swap]
\end{tikzcd}
\end{equation}
for every $ij \in \graph{E}$.
\end{enumerate}
\end{definition}
In most applications we imagine, the maps from node data to edge data will be join-preserving. Thinking of a sheaf as a distributed system, the state of the system is given by an \define{assignment} of vertex data: a tuple $\mathbf{x} \in \prod_{i \in \graph{V}} \sheaf{F}(i)$ of choices of data $x_i\in\sheaf{F}(i)$ for each $i$. The data over the edges and the maps $\sheaf{F}_{i \fc ij}$ are used to compare the compatibility of vertex data in an assignment. The following definition is crucial.
\begin{definition}
The \define{sections} of $\sheaf{F}$ are assignments $\mathbf{x}$ that are compatible: for every $ij \in \graph{E}$,
\begin{equation}
\sheaf{F}_{i \fc ij}(x_i) = \sheaf{F}_{j \fc ij}(x_j).
\end{equation}
\end{definition}
The set of sections of $\sheaf{F}$ is denoted $\Gamma(\sheaf{F})$. These are the ``globally compatible'' states. In the simplest example of a \define{constant sheaf} (which assigns a fixed lattice to each vertex and edge, with identity maps between vertex and edge data), sections are precisely assignments of an identical element to each vertex (and edge): consensus over the network.
\begin{example}
The true utility of a sheaf lies in heterogeneity. For example, assign to each vertex $i\in V$ a finite set $S_i$ and to each edge $ij\in E$ a finite set $S_{ij}$ and set-maps $S_i\rightarrow S_{ij} \leftarrow S_j$. This induces several interesting sheaves of lattices. The \define{powerset sheaf} assigns the powersets (lattices of all subsets with union and intersection) to vertices and edges, with induced maps. The \define{partition sheaf} assigns the partition lattices (partitions of set elements with partition union and refinement) to vertex and edge sets, again with induced maps. More examples from formal concept analysis are less well-known \cite{wille1982restructuring}, but very general.
\end{example}
\section{Problem Formulation}
\label{sec:problem}
An abstract formulation is sufficient, but, for concreteness, consider a scenario in which a collection of geographically dispersed agents collect, process, and communicate data based on local sensing. Proximity gives rise to a communications network, modeled as an undirected graph $\graph{G}$. The data are lattice-valued, but each agent $i\in V$ works within its personalized lattice $\sheaf{F}(i)$. In order for two proximate agents $i$ and $j$ to communicate their individualized data (residing in $\sheaf{F}(i)$ and $\sheaf{F}(j)$ respectively), they must ``fuse'' their observiations into a common lattice $\sheaf{F}(ij)$ by means of structure-preserving lattice maps. Together, the system forms a network sheaf of lattices over the network $\graph{G}$.
The problem envisioned is distributed consensus by means of asynchronous communication and updates. By consensus, we do not mean that everyone agrees on a particular fixed lattice value; rather, each node agrees upon choices of {\em local} data that, when translated and compared to all neighbors' data over communication edges, agree. In the context of a sheaf of lattices, this is precisely the condition of an assignment being a {\em section}. The synchronous version of this problem -- everyone communicates simultaneously with neighbors and updates immediately -- is solvable via the Tarski Laplacian as per \cite{ghrist2020cellular}. The asynchronous problem is our focus here. Select sensors broadcast their data to neighboring nodes according to some \define{firing sequence}.
Denote by $\tau: \{0,1,2,\dots \} \to 2^\graph{V}$ the firing sequence of selected nodes which broadcast as a function of (ordered, discrete) time. At $t \in \{0,1,2,\dots \}$, active nodes $i\in \tau_t$ broadcast to each agent $j \in \graph{N}_{i}$.\footnote{If desired, one may choose a selection of egdes incident to the node and broadcast only to those neighbors. This results in more bookkeeping and a refined version of liveness, below, but does not substantially change the results or proofs.} No other nodes broadcast.
This notion of a firing sequence suffices to cover asynchronous updating. The regularization of time to $\{0,1,2,\dots \}$ is a convenience and does not impact results. The firing sequence is, in practice, not known {\em a priori}. This is of no consequence since our results will hold independent of the choice of firing sequence. The following assumptions will hold throughout.
\begin{assumption}[no delay]\label{ass:nodelay}
Within a single time instance $t$, nodes may broadcast (if firing), receive data (always), and compute (always).
\end{assumption}
\begin{assumption}[liveness]\label{ass:liveness}
For all $i \in \graph{V}$ and for every $t \in \{0,1,2,\dots \}$ there is a $t' \geq t$ such that $i \in \tau_{t'}$. As such, agents neither die nor are removed from the system.
\end{assumption}
\begin{assumption}[crosstalk]\label{ass:crosstalk}
Suppose $i$ is an agent and $j, j' \in \graph{N}_t \cap \tau_t$ are active neighbors. Then, $j$ and $j'$ can simultaneously broadcast to $i$ without resulting in a fault.
\end{assumption}
Under these assumptions, we wish to solve the following distributed asynchronous constrained agreement problem. Assume (1) a network $\graph{G}=(\graph{V},\graph{E})$; (2) a sheaf $\sheaf{F}$ of lattices over $\graph{G}$; (3) an firing sequence $\tau:\{0,1,2,\dots \}\to 2^{\graph{V}}$ of broadcasts; and (4) an initial condition $\vec{x}[0]$, being an assignment of an element $x_v\in\sheaf{F}(v)$ to each agent $v\in\graph{V}$. Using only local communication subordinate to the firing sequence $\tau$, evolve the initial condition $\vec{x}[0]$ to a section $\vec{x}\in\Gamma(\sheaf{F})$.
This problem has elements of consensus (because of the local agreement implied in a section) as well as data fusion (due to the lattice maps merging data from vertex lattices to edge lattices).
\section{An Asynchronous Laplacian}
\label{sec:async}
Our method for solving this asynchronous constrained agreement problem is to define an asynchronous harmonic flow on the sheaf by localizing the Tarski Laplacian of \cite{ghrist2020cellular}.
\subsection{The Tarski Laplacian}
\label{sec:tarski}
\noindent
Throughout, $\sheaf{F}$ is a lattice-valued sheaf over a network $\graph{G}$. Our first step towards a Laplacian involves preliminaries on {\em residuals} \cite{blyth2014residuation}, also known as {\em Galois connections} \cite{ore1944galois}. These are a type of {\em adjoint} or lattice-theoretic analogue of the familiar {\em Moore-Penrose pseudoinverse} in linear algebra.
\begin{definition}
Given a join-preserving lattice map $\varphi:\lattice{P}\to\lattice{Q}$, its \define{residual} is the map $\varphi^{+}\colon\lattice{Q}\to\lattice{P}$ given by
\[
\varphi^{+}(p)=\bigvee \{q~\vert~\varphi(q) \preceq p\}.
\]
\end{definition}
Like an adjoint, it reverses the direction of the map, resembling a pseudoinverse more closely in some cases. The following two lemmas have straightforward proofs via definitions.
\begin{lemma}
Suppose $\varphi:\lattice{P}\to\lattice{Q}$ is join-preserving and injective. Then $\varphi^{+}\circ\varphi = id$.
\end{lemma}
\begin{lemma}
\label{lem:residuated}
For $\varphi:\lattice{P}\to\lattice{Q}$ join-preserving, the following identities hold:
\begin{enumerate}
\item for all $p \in \lattice{P}$, $\varphi^{+}\circ\varphi(p) \succeq p$; and
\item for all $p \in \lattice{P}$ and $q \in \lattice{Q}$,
\begin{equation}
\label{eq:swap}
\varphi(p) \preceq q \, \Leftrightarrow \, p\preceq \varphi^{+}(q) .
\end{equation}
\end{enumerate}
\end{lemma}
A lattice-theoretic analogue of the graph Laplacian -- the \define{Tarski Laplacian} -- was introduced in \cite{ghrist2020cellular}. For $\sheaf{F}$ a network sheaf of lattices over $\graph{G}$ and $\vec{x}$ an assignment of vertex data, the Tarski Laplacian $L$ acts as:
\begin{equation}
\label{eq:Tarski}
\left(L\vec{x}\right)_i
=
\bigwedge_{j \in \graph{N}_{i}} \sheaf{F}^{+}_{i\fc ij} \sheaf{F}_{j \fc ij}(x_j).
\end{equation}
\noindent
The key construct is to localize this operator subordinate to a firing sequence $\tau$.
\begin{definition}
\label{def:async-Tarski}
The \define{asynchronous Tarski Laplacian} is the map
\[L: \{0,1,2,\dots \} \times \prod_{i \in \graph{V}} \sheaf{F}(i) \to \prod_{i \in \graph{V}} \sheaf{F}(i) \]
which acts on an assignment $\vec{x}$ as
\begin{equation}
\label{eq:async-laplacian}
\left(L_t\vec{x}\right)_i = \bigwedge_{j \in \graph{N}_{i} \cap \tau_t} \sheaf{F}^{+}_{i \fc ij}\sheaf{F}_{j \fc ij}(x_j).
\end{equation}
\end{definition}
This is a restriction of the Tarski Laplacian (\ref{eq:Tarski}) in that at time $t$, only immediate neighbors to broadcasting nodes are updated; all other nodes are unchanged. In the extreme of a firing sequence where all nodes broadcast at all times, the full Tarski Laplacian ensues.
\subsection{Heat flow and harmonic states}
\label{sec:heat}
\noindent
The rationale for the {\em Laplacian} moniker lies in the efficacy of $L$ as a diffusion operator on states. Given an initial state $\vec{x}[0]$, define the \define{heat flow} to be the discrete-time dynamical system:
\begin{align}
\label{eq:heatflow}
\vec{x}[t+1] & = \left(\id \wedge L_t \right)\vec{x}[t] .
\end{align}
This is the analogue of iterating the \define{random-walk} or \define{Perron} operator $I- \epsilon L$ (where $I$ is the identity matrix, $L$ the graph Laplacian matrix, $\epsilon > 0$) on scalar-valued data on a graph\cite{olfati2007consensus}. The principal result of this work is a type of {\em Hodge theorem}: the \define{harmonic} states (the equilibria of $\id\wedge L$ is a proxy for the kernel of the Laplacian) are exactly the globally consistent solutions to the sheaf.
\begin{theorem}\label{thm:main}
For any lattice-valued network sheaf $\sheaf{F}$ with join-preserving structure maps and any firing sequence $\tau$ satisfying liveness, the sections of $\sheaf{F}$, $\Gamma(\sheaf{F})$, are precisely the time-independent solutions to the heat flow (\ref{eq:heatflow}).
\end{theorem}
\begin{proof}
Suppose first that $\vec{x}[t]\in\Gamma(\sheaf{F})$ is a section. Then, for all $ij \in \graph{E}$,
\[
\sheaf{F}_{i \fc ij}(x_i[t]) = \sheaf{F}_{j \fc ij}(x_j[t]).
\]
Hence, by Lemma \ref{lem:residuated},
\begin{align*}
\left( L_t \vec{x}[t]\right)_i & = \bigwedge_{j \in \graph{N}_{i} \cap \tau_t} \sheaf{F}^{+}_{i\fc j}\sheaf{F}_{j \fc ij}(x_j[t]) \\
& = \bigwedge_{j \in \graph{N}_{i} \cap \tau_t} \sheaf{F}^{+}_{i\fc j} \sheaf{F}_{i \fc ij}(x_i[t]) \\
& \succeq x_i[t] .
\end{align*}
Then, $x_i[t+1] = \left(L_t\vec{x}[t]\right)_i \wedge x_i[t] = x_i[t]$. Hence, $x_i[t+1]=x_i[t]$ for all $i \in \graph{V}$. Inducting in $t$, sections are time-independent.
Conversely, suppose $\vec{x}[t]=\vec{x}[0]$ is a time-independent solution. Then, by (\ref{eq:heatflow}),
\begin{equation}
\left(L_t\vec{x}[0]\right)_i \succeq x_i[0]
\end{equation}
for all $i \in \graph{V}$ and all $t\in\{0,1,2,\dots \}$.
Thus, for all $i \in \graph{V}$ and $j \in \graph{N}_{i} \cap \tau_t$,
\begin{align*}
\sheaf{F}^{+}_{i\fc j}\sheaf{F}_{j \fc ij}(x_j[0]) & \\
\succeq
\bigwedge_{j \in \graph{N}_{i} \cap \tau_{t}} & \sheaf{F}^{+}_{i\fc j}\sheaf{F}_{j \fc ij}(x_j[0]) \\
& \succeq x_i[0] ,
\end{align*}
using the definition of the asynchronous Tarski Laplacian.
Applying Lemma~\ref{lem:residuated} yields:
\begin{align}
\label{eq:succeq}
\sheaf{F}_{j \fc ij}(x_j[0]) & \succeq \sheaf{F}_{i \fc ij}(x_i[0]).
\end{align}
Suppose $\eqref{eq:succeq}$ holds for a particular $i \in \graph{V}$ and a $j \in \graph{N}_{i} \cap \tau_{t}$.
By liveness, there exist $t'\geq t$ such that $i \in \tau_{t'}$ so that $i \in \graph{N}_{j} \cap \tau_t$. In particular, there is a smallest $t'>t$ with
\begin{align}
\label{eq:preceq}
\sheaf{F}_{j \fc ij}(x_j[t']) & \preceq \sheaf{F}_{i \fc ij}(x_i[t']).
\end{align}
By hypothesis, $x_{i}[0] = x_i[t']$ for all $i \in \graph{V}$. Hence, equations \eqref{eq:succeq} and \eqref{eq:preceq} imply $\sheaf{F}_{i \fc ij}(x_i[t]) = \sheaf{F}_{j \fc ij}(x_j[t])$ for all $t$. Therefore, $\vec{x}[t]$ is a section.
\end{proof}
\subsection{Convergence}
\label{sec:convergence}
\noindent The proof of Theorem \ref{thm:main} immediately suggests an iterative protocol (Algorithm \ref{alg:local}) for an agent $i \in \graph{V}$ to converge to a harmonic state by firing a finite number of times. To guarantee finite convergence, we require an assumption to avoid an agent flowing down a chain indefinitely.
\begin{assumption}[chains]\label{ass:dcc}
The lattice $\sheaf{F}(i)$ satisfies the following descending chain condition (DCC) \cite{davey2002introduction}: no strictly descending sequence $a_1 \succ a_2 \succ \cdots \succ a_n \succ \cdots$ exists in $\sheaf{F}(i)$.
\end{assumption}
\begin{remark}
All finite lattices satisfy DCC as well as lattices of subspaces of a finite-dimensional vector space; the lattice $\bar{\mathbb{R}}^n$ does not satisfy DCC.
\end{remark}
\begin{algorithm}{}
\SetKwFor{ForTo}{to}{do}{end}
\caption{$\mathtt{localHeatFlow}$}\label{alg:local}
\KwIn{local assignment $x_i \in \sheaf{F}(i)$}
\KwOut{local assignment $x_i^\ast \in \sheaf{F}(i)$}
$t \leftarrow 0$ \;
$x_i[0] \leftarrow x_i$ \;
$x_{ij}[0] \leftarrow \sheaf{F}_{i \fc ij}(x_i[0])$ \;
\Repeat{$x_i[t+1]=x_i[t]$}{
\For{$j \in \mathtt{listen}\left( \graph{N}_i\right)$}
{ $x_{ij}[t] \leftarrow \mathtt{received}(j)$ \;
$x_i[t] \leftarrow x_i[t] \wedge \sheaf{F}^{+}_{i\fc j}\left( x_{ij}[t] \right)$ \;}
\ForTo{$j \in \graph{N}_i$}{
$x_{ij}[t] \leftarrow \sheaf{F}_{i \fc ij}(x_i[t])$ \;
$\mathtt{broadcast}(x_{ij}[t]) \to j$ \;
}
$t \leftarrow t+1 $ \;}
$x_i^{\ast} \leftarrow x_i[t]$ \;
\end{algorithm}
\noindent Assumption \ref{ass:dcc} implies $\mathtt{localHeatFlow}$ terminates in finitely many iterations, since
$x_i[t+1] \preceq x_i[t]$.
Global heat flow consists of each $i$ executing the function $\mathtt{localHeatFlow}$ concurrently. Synchrony is not required, nor must the duration of each iteration be uniform.
As for complexity, suppose each $i$ has oracle access to binary meets, structure-preserving maps, and their residuals. Let $h \left( \lattice{Q} \right)$ denote the length of a maximal chain in $\lattice{Q}$; let $N$ denote the number of agents in the system; and let $H = \max_{i} h\left( \sheaf{F}(i) \right)$. For $d(i)$ denote the degree of vertex $i$ and $D = \max_i d(i)$, one clearly has convergence of any initial assignment to a section in time $O(NDH)$.
\section{Semantics}
\label{sec:semantics}
Examples of the Tarski Laplacian and heat flow are especially well-suited to distributed multimodal logic, where: vertices of a network correspond to agents; edges are communications between agents; assignments are sets of states associated to each vertex; and sections are assignments that express a consensus of knowledge across the network. For this we require the basic theory of Kripke semantics \cite{fagin2004reasoning}.
\subsection{Kripke semantics}
\label{sec:kripke}
\begin{definition}
A \define{Kripke frame} $\mathbf{F} = \left(S,\relation{K}_1, \relation{K}_2, \dots, \relation{K}_n \right)$ consists of a set of \define{states}, $S$, and a sequence of binary relations $\relation{K}_i \subseteq S \times S$ used to encode modal operators. A \define{Kripke model} over a set $\Phi$ of atomic propositions consists of the data $\mathcal{M} = \left(\mathbf{F}, \pi \right)$, where $\pi: S \to 2^{\Phi}$ validates whether or not a state $s \in S$ satisfies an atomic proposition $p \in \Phi$.
\end{definition}
Suppose $\phi$ is a formula; then one writes $(\mathcal{M}, s) \models \phi$ if $s$ {\em satisfies} the formula $\phi$ in the model $\mathcal{M}$.
The semantics of a Kripke model is defined inductively using the symbols $\mathtt{true}$, $\wedge$, and $\neg$ in their typical usage:
\begin{enumerate}
\item $(\kripke{M},s) \models \mathtt{true}$ for all $s \in S$.
\item For $p \in \Phi$ atomic, $(\kripke{M},s) \models p$ if and only if $p \in \pi(s)$.
\item For $\phi$ an formula, $(\kripke{M}, s) \models \neg \phi$ if and only if $(\kripke{M}, s) \not \models \phi$.
\item For $\phi, \psi$ formulae, $(\kripke{M}, s) \models \phi \wedge \psi$ if and only if $(\kripke{M}, s) \models \phi$ and $(\kripke{M}, s) \models \psi$.
\end{enumerate}
There are additional (dual) modal operators on formulae, $K_i$ and $P_i$ (typically corresponding to knowledge and possibility), based on the binary relations $\relation{K}_i$. One writes $(\kripke{M}, s) \models {K}_i \phi$ if and only if $(\kripke{M}, t) \models \phi$ for all $t \in S$ such that $(s,t) \in \relation{K}_i$.
The dual operators $P_i$ are defined via $P_i \phi = \neg K_i \neg \phi$. Standard operations in propositional logic such as $\phi \to \psi$ are derived in the usual way \cite{mendelson2009introduction}. By abuse of notation, we write $\models \phi$ if $(\kripke{M},s) \models \phi$ for all $s \in S$.
Depending on a number of axioms placed on the modal operators $K_i$ and $P_i$, one has rich interpretations for $K_i$ and $P_i$. For instance, the Knowledge Axiom \cite{fagin2004reasoning}
\begin{align}\label{eq:knowledge-axiom}
\models & K_i \phi \to \phi
\end{align}
and the Introspection Axiom \cite{fagin2004reasoning}
\begin{align}
\models & K_i \phi \to K_i K_i \phi
\end{align}
together suggest the interpretation of $K_i \phi$ and $P_i \phi$: \textit{agent $i$ knows $\phi$} and \textit{agent considers $\phi$ possible}, respectively. On the other hand, if \eqref{eq:knowledge-axiom} not hold, but the Consistency Axiom \cite{fagin2004reasoning} does hold,
\begin{align}
\models & \neg K_i (\mathtt{false}),
\end{align}
$K_i \phi$ and $P_i \phi$ could be interpreted as: \textit{agent $i$ believes $\phi$} and \textit{agent $i$ does not disbelieve (i.e., is undecided about) $\phi$}.\footnote{One cannot believe a logical paradox $\phi \wedge \neg \phi$, as believing something does not make it true.}
The set of all finite formulae inductively obtained by a model $\kripke{M}$ is called the \define{language} denoted $\kripke{L}(\kripke{M})$ (omitting the $\kripke{M}$ where understood). For $\phi \in \kripke{L}$, the \define{intent} of $\phi$ is the subset
\[\phi^{\kripke{M}} = \{s \in S~\vert~(\kripke{M},s) \models \phi \}.\]
Formulae $\phi, \psi \in \kripke{L}$ are \define{semantically equivalent} if $\phi^\kripke{M} = \psi^\kripke{M}$. Semantic equivalence is a \textit{bona fide} equivalence relation, written $\phi \sim \psi$, with $\kripke{L}_{\sim}(\kripke{M})$ denoting the set of equivalence classes in $\kripke{L}$ up to semantic equivalence.
\subsection{Semantic diffusion}
\label{sec:semanticdiffusion}
\noindent Our goal is to adapt the technology of sheaves of lattices and Laplacians to Kripke semantics over a network of agents. One simple approach is to use powerset lattices $2^S$ of states $S$ of a frame. Let $\graph{G}=(\graph{V},\graph{E})$ be a network and $\mathbf{F}$ a frame. Define the \define{semantic sheaf} $\sheaf{I}$ over $\graph{G}$ so that the data over each vertex and edge is precisely $2^S$. The following is crucial to define the structure-preserving maps:
\begin{definition}\label{thm:exists-forall}
For $\mathbf{F} = \left(S, \relation{K}_1, \dots, \relation{K}_n \right)$ a frame and for each $i \in \{0,1,\dots,n\}$, there is residual pair
\begin{equation}
\begin{tikzcd}
2^S \arrow[r,"\relation{K}_i^{\exists}",bend left=30] \arrow[r,"\bot", phantom] & \arrow[l, "\relation{K}_i^{\forall}", bend left = 30] 2^S
\end{tikzcd}
\end{equation}
given by the formulae
\begin{align*}
\relation{K}_i^{\exists}(\sigma) &= \{t \in S~\vert~\exists s \in \sigma,~(s,t) \in \relation{K}_i \}, \\
\relation{K}_i^{\forall}(\sigma) &= \{s \in S~\vert~\forall t \in S, (s,t) \in \relation{K}_i \Rightarrow t \in \sigma \}.
\end{align*}
\end{definition}
\begin{lemma}
Each map $\relation{K}_i^{\exists}: 2^S \to 2^S$ is join-preserving and $\left(\relation{K}_i^{\exists}\right)^{+}=\relation{K}_i^{\forall}$.
\end{lemma}
\begin{proof}
First,
\begin{align*}
\relation{K}_i^{\exists}(\sigma \cup \sigma') &= \{t \in S~\vert~\exists~s~\text{or}~s' \in \sigma,~(s,t)~\text{or}~(s',t) \in \relation{K}_i \}.
\end{align*}
Second,
\begin{align*}
\left(\relation{K}_i^{\exists}\right)^{+}(\sigma) & = \bigcup \{\alpha \in 2^S~\vert~\relation{K}_i^{\exists}(\alpha) \subseteq \sigma \} \\
& = \{s \in S~\vert~\forall t \in S, (s,t) \in \relation{K}_i \Rightarrow t \in \sigma \} \\
& = \relation{K}_i^{\forall}(\sigma) .
\end{align*}
\end{proof}
Such a sheaf of powerset lattices has an asynchronous Tarski Laplacian and a corresponding heat flow. The following definitions are straight translations from \S\ref{sec:async}.
\begin{definition}\label{def:semantic-laplacian}
Suppose $\mathbf{F} = (S, \relation{K}_1, \dots, \relation{K}_n)$ is a frame and $\sheaf{I}$ a sheaf of powersets $2^S$ over a network $\graph{G} = (\graph{V},\graph{E})$ where $\graph{V} = \{1,2, \dots, n\}$. Let $\tau: \{0,1,2,\dots \} \to \graph{V}$ be a firing sequence. The (asynchronous) \textit{semantic Laplacian} is the operator acting on $\boldsymbol \sigma \in \prod_{i \in \graph{V}} 2^S$ via:
\begin{equation}
\label{eq:semlap}
\left( L_t \boldsymbol\sigma \right)_i
=
\bigcap_{j \in \graph{N}_{i} \cap \tau_t}
\relation{K}_i^{\forall}\relation{K}_j^{\exists}(\sigma_j) .
\end{equation}
The associated \define{heat flow} is the dynamical system
\begin{equation}
\label{eq:positive-flow}
{\boldsymbol \sigma}[t+1] = (\id \wedge L_t) {\boldsymbol \sigma}[t]
\end{equation}
\end{definition}
\noindent Our main result follows directly from Theorem \ref{thm:main}, interpreted in the language of this section.
\begin{theorem}
Suppose $\graph{G} = (\graph{V}, \graph{E})$ is a network with firing sequence $\tau$ satisfying liveness. Let $\kripke{M} = (S, \relation{K}_1, \dots, \relation{K}_N, \Phi, \pi)$ be a Kripke model. Then, the sections of $\sheaf{I}$ are exactly time-independent solutions to the heat flow \eqref{eq:heatflow}.
\end{theorem}
These sections are interpretable as {\em possibility consensus} assignments of the model $\kripke{M}$ on $\graph{G}$. The assignment of formulae $(\phi_i)_{i\in\graph{V}}$ satisfies, for each edge $ij\in\graph{E}$, $P_i \phi_i \sim P_j \phi_j$. This does not mean that the formulae are in consensus as identical formulae; rather, they are semantically equivalent in the language $\kripke{L}$.
\begin{proposition}\label{prop:consensus}
Suppose $\graph{G}$ is connected. Sections port to edge-assignments $(\psi_{ij})_{ij \in \graph{E}}$ along the diagonal of $\kripke{L}_{\sim}^N$: consensus along edges.
\end{proposition}
\begin{proof}
$(\phi)_{i \in \graph{V}}$ being a section, $P_i \phi_i \sim P_j \phi_j$ for every edge $ij$. Connectedness implies any two nodes $i, i' \in \graph{V}$ are connected by a path. Transitivity of the semantic equivalence relation $\sim$ implies that $P_{i} \phi_{i} \sim P_{i'} \phi_{i'}$.
\end{proof}
\subsection{Syntactic diffusion}
\label{sec:syntacticdiffusion}
\noindent The following lemma together with Proposition \ref{prop:consensus} motivates why local residuations $\relation{K}_i^\exists \dashv \relation{K}_i^\forall$ induce a flow of knowledge.
\begin{lemma}\label{thm:galois-semantics}
Suppose $\mathcal{M} = (\mathbf{F}, \pi)$ is a model and $\phi \in \kripke{L}(\kripke{M})$. Then, the following hold
\begin{align}
\relation{K}_i^{\forall}\left( \phi^{\kripke{M}}\right) &= \left( {K}_i \phi \right)^{\kripke{M}}, \label{eq:galois-semantics-1}\\
\relation{K}_i^{\exists}\left( \phi^{\kripke{M}} \right) &= \left( P_i \phi \right)^{\kripke{M}}. \label{eq:galois-semantics-2}
\end{align}
\end{lemma}
\begin{proof}
Writing
\[
\relation{K}_i^{\forall}\left( \phi^{\kripke{M}}\right) =
\{s \in S~\vert~\forall t \in S, (s,t) \in \relation{K}_i \Rightarrow (\kripke{M},t) \models \phi \},
\]
we prove \eqref{eq:galois-semantics-1}. For \eqref{eq:galois-semantics-2}, evaluating $\relation{K}_i^{\exists}(\phi^\kripke{M})$ yields
\begin{align*}
\{t \in S~\vert~\exists s~\text{such that}~(\kripke{M},s) \models \phi~\text{with}~(s,t) \in \relation{K}_i \}.
\end{align*}
On the other hand, evaluating $\left(P_i \phi \right)^\kripke{M}$ yields
\begin{align*}
\left( \neg K_i \neg \phi \right)^{\kripke{M}} & & \\
= & \left( \left( K_i \neg \phi \right)^{\kripke{M}} \right)^c & \\
= & \{s~\vert~ \forall~t~\text{such that}~(s,t) \in \relation{K}_i,~(\kripke{M},t) \not \models \phi \}^{c} & .
\end{align*}
\end{proof}
\noindent The following simple observation together with Lemma \ref{thm:galois-semantics} allows us to freely go back and forth between syntax and semantics, opening the way for syntactic diffusion dynamics.
\begin{lemma}\label{lem:contravariant}
Suppose $\{\phi\}_{i}$ is a finite set of formula in $\kripke{L}(\kripke{M})$. Then, $\left(\bigvee_{i \in I} \phi_i\right)^{\kripke{M}} = \bigcup_{i} \phi_i^{\kripke{M}}$ and $\left(\bigwedge_{i} \phi_i\right)^{\kripke{M}} = \bigcap_{i} \phi_i^{\kripke{M}}$.
\end{lemma}
\begin{proof}
Exercise.
\end{proof}
\begin{definition}
\label{def:syntactic-laplacian}
Suppose $\boldsymbol \phi = (\phi_i)_{i \in \graph{V}}$ is a tuple of formulae in $\kripke{M}$.
The (asynchronous) \define{syntactic Laplacian} acts on assignments as:
\begin{equation}\label{synlap}
\left( L_t \boldsymbol\phi\right)_i = \bigwedge_{j \in \graph{N}_{i} \cap \tau_t} K_i P_j \phi_j .
\end{equation}
\end{definition}
This is a straight translation of the semantic Tarski Laplacian, using Lemmas \ref{thm:galois-semantics} and \ref{lem:contravariant}.
\subsection{Dual Laplacians}
\label{sec:dual}
\noindent The classical meet-join duality in lattices pushes through to all other structures built therefrom. In particular, the Tarski (and thus semantic and syntactic) Laplacians come in dual variants, implicating how syntactic and semantic consensus is interpreted.
\begin{definition}
\label{def:dual}
The \define{dual} Tarski, semantic, and syntactic Laplacians are given by, respectively:
\begin{equation}
\label{eq:dual-tarski}
\left(L^*\vec{x}\right)_i
=
\bigvee_{j \in \graph{N}_{i}} \sheaf{F}^{+}_{i\fc j} \sheaf{F}_{j \fc ij}(x_j)
\end{equation}
\begin{equation}
\label{eq:dual-semantic}
\left(L^* \boldsymbol\sigma\right)_i
=
\bigcup_{j \in \graph{N}_{i}} {\relation{K}}_{i}^{\exists}{\relation{K}}_j^{\forall}(\sigma_j)
\end{equation}
\begin{equation}
\label{eq:dual-syntactic}
\left(L^* \boldsymbol\phi \right)_i
=
\bigvee_{j \in \graph{N}_{i}} P_i K_j \phi_j. \\
\end{equation}
The corresponding asynchronous dual Laplacians $L^*_t$ intersect neighboring vertices with those in a firing sequence $\tau$.
\end{definition}
The \define{dual heat flow} iterates the operator $\id\vee L^*_t$. Convergence results to sections remain. These sections are interpretable as {\em knowledge consensus} assignments of the model $\kripke{M}$ on $\graph{G}$. The assignment of formulae $(\phi_i)_{i\in\graph{V}}$ satisfies, for each edge $ij\in\graph{E}$, $K_i \phi_i \sim K_j \phi_j$.
\section{Examples}
\label{sec:examples}
In this section, we supply an example of a semantic sheaf modeling knowledge gain; we also provide a numerical experiment demonstrating the correctness of asynchronous heat flow \eqref{eq:heatflow} as well as convergence behavior.
The \define{Lyapunov function} acts on assignments in \eqref{eq:heatflow} via
\begin{equation}
V(\vec{x}) = \sum_{ij \in \graph{E}} d\left(\sheaf{F}_{i \fc ij}(x_i),\sheaf{F}_{j \fc ij}(x_j)\right); \label{eq:lyp}
\end{equation}
each $d$ is a distance function on $\sheaf{F}(ij)$. If $d$ is a metric, then $V(\vec x) \geq 0$ and $V(\vec x) = 0$ if and only if $\vec x$ is a section. Our notion of a Lyapunov funtion on a sheaf is reminiscent of the already-studied \define{consistency radius} \cite{robinson2020assignments}.
Below, $d$ is taken to be the well-known \define{Jaccard distance}
\[
d_J(\sigma, \sigma') = \frac{\vert \sigma \cup \sigma' \vert - \vert \sigma \cap \sigma' \vert}{\vert \sigma \cup \sigma' \vert}.
\]
\subsection{Threat detection}
\label{sec:targets}
\noindent In this example, we model information gain by four sensors ($N=4$) with two-way communication links $\graph{E} = \{(1,2),(2,3),(3,4),(1,4)\}$. All four sensors are tasked with detecting malicious targets $A$ and $B$. Each sensor $i$ has a local status $s_i \in S_i=\{\hat{0},\hat{1}\}$, so that $S = \prod_{i=1}^{i=4} S_i$. The atomic propositions are $\Phi = \{p_A, p_B\}$ which are interpreted as {\em threat A is present}, or {\em threat B is present}. A global state $s = (s_1,s_2,s_3,s_4)$ may be thought of as a signature for the presence or absence of a threat via any given $\pi: S \to 2^{\{p_A, p_B\}}$.
In the context of knowledge, a natural choice of Kripke relation $\relation{K}_i$ is the equivalence relation on $S$ given by $s \sim_i t$ if and only if $s_i = t_i$ \cite{fagin2004reasoning}. This reflects the intuition that $i$ can distinguish system-wide states $s$ and $t$ insomuch as $i$ can distinguish local statuses with its own instruments.\footnote{In general, the number of sensors/number of links could be large; each sensor could have any number of possibly unique statuses; any number of threats is possible; Kripke relations can be arbitrary relations on $S_1 \times \cdots \times S_N$.} All this data is encoded in a fame $\mathbf{F} = (S_1 \times \cdots \times S_4, \sim_1, \cdots, \sim_4)$ and an associated semantic sheaf $\sheaf{I}^\mathbf{F}$.
Various scenarios of knowledge asymmetry may be encoded in this language. For instance, suppose one sensor knows that threat A is present, and the remaining nodes know threat B is not present. Syntactically,
\begin{align*}
\boldsymbol\phi [0] &=& \left( K_1 p_A, K_2\neg p_B, K_3 \neg p_B, K_4 \neg p_B \right).
\end{align*}
The first iteration $\boldsymbol\phi[1]$ of syntactic heat flow from \eqref{synlap} is computed
\begin{align*}
L(\boldsymbol\phi[0]) \wedge \boldsymbol\phi[0] &=& \\
\left(K_1P_2\neg K_2 p_B \wedge K_1P_4\neg K_4 p_B \wedge K_1 p_A, \right. \\
K_2P_1K_1p_A \wedge K_2P_3\neg K_3 p_B \wedge K_2 \neg p_B, \\
K_3 P_2 K_2 \neg p_B \wedge K_3 P_4 K_4 \neg p_B \wedge K_3 \neg p_B, \\
\left. K_4P_1K_1p_A \wedge K_4P_3\neg K_3 p_B \wedge K_4 \neg p_B \right).
\end{align*}
\noindent (Incomplete) knowledge accrues as the heat flow evolves. For instance, sensors 2 and 4 know at $t=1$ that sensor 1 thinks it is possible that it knows $p_A$. It is an interesting open question whether popular notions of distributed knowledge \cite{fagin2004reasoning} are reflected in these semantic/syntactic heat flows.
\subsection{Numerical experiments}
\label{sec:experiments}
\noindent We generate random graphs modeling sensor networks with $N = 40$ nodes with proximity radius $r = 0.05$. In each graph, positions $y_i, i \in \{1, \dots, N\}$ are sampled uniformly from the domain $D =[0,1]^2$ and $\graph{G}$ has edge set $\graph{E} = \{ij~\vert~\|y_i - y_j\|_2 < r \}$. We set $S = \{1,2,\dots,M\}$ to be the set of (abstract) states of the system; we choose $M = 10$. For each agent $i$ we generate a random Kripke relation $\relation{K}_i$ by independently selecting $(s,t) \in \relation{K}_i \subseteq S \times S$ with probability $p = 0.1$ if $s \neq t$ and with probability $q=0.9$ if $s=t$. We may interpret each distinct $\relation{K}_i$ as an internal belief system.
\begin{figure}[h]
\includegraphics[width=0.5\textwidth]{fig1.png}
\caption{Shown are the Lyapunov energies \eqref{eq:lyp} of a heat flow over sensor network ($N=40$, $r = 0.05$) with semantic sheaf $\sheaf{I}^{\mathbf{F}}$ ($M=10$; randomized $\relation{K}_i, i \in \{0,1,\dots,N\}$).}
\label{fig:1}
\end{figure}
\noindent For a random initial assignment $\boldsymbol \sigma [0]$, we simulate the heat flow dynamics using the dual asynchronous semantic Laplacian \eqref{eq:dual-semantic} for the frame $\mathbf{F} = (S, \relation{K}_1, \dots, \relation{K}_{N})$. In our experiment, we compare the Lyapunov functions for a single heat flow with fixed initial $\boldsymbol \sigma[0]$ for three different firing sequences: a fully-synchronous sequence, an asynchronous random sequence (one node fires at a time), a sequence in which the probability that a node fires is inversely proportional to its degree, and a sequence where teams of $k=8$ nodes alternate firing simultaneously (Figure \ref{fig:1}).
\bibliographystyle{IEEEtr}
|
2,869,038,154,790 | arxiv | \section{Introduction}
After the most recent findings by BES~\cite{bes} and Belle~\cite{belle}, and after years of experimental research on $X,Z$ exotic spectroscopy, we can rely on a set of well established states which present very similar features and pose a clear problem about their interpretation.
The most studied is the $X(3872)$, a $1^{++}$ state with apparently {\it no} close in mass charged partners.
The $X(3872)$ happens to be exactly at the $D^0 \bar D^{*0}$ threshold -- a very loosely bound state according to several authors.
Most recently, very close to the $D^\pm \bar D^{*0}$ threshold, but {\it sensibly above it} by about $20$~MeV, a $1^{+-}$ charged state has been discovered: the $Z_c^\pm(3900)$~\cite{bes,belle}.
Some hints of an almost degenerate $Z_c^0$ neutral partner seem encouraging~\cite{cleo}. In spite of the positive binding energy, the $Z_c$ is described in several papers as a hadron molecule, a relative of the $X(3872)$~\cite{han2}.
The issue with molecules is wether nuclear forces between color singlets are enough binding to explain the large production cross sections at hadron colliders~\cite{bigna}. On the other hand such Van der Waals-like forces will not be flavor blind and could give rise to SU(3) and even SU(2) incomplete multiplets (as is the case of the $X$).
Yet, are there alternative explanations of the $X(3872)$ neutrality not relying on the complications of inter-hadron forces?
Sligthy above $Z_c$, close to the $D^*\bar D^*$ threshold, another resonance, the $Z^\prime_c(4025)$ has been found, likely carrying $1^{+-}$ quantum numbers~\cite{bes2}. This state is {\it above threshold} as well, by about 5~MeV.
It is still unclear if there are indeed two almost degenerate states observed in two different decay modes: $D^*\bar D^* $ and $h_c\pi$, with similar widths. However, the latter decay should imply $P=-$, if in $S$-wave.
We shall assume here that there is only one $Z_c^\prime(1^{+-})$ state, decaying into $h_c \pi$ in $P$-wave.
Similarly, there are two more $1^{+-}$ states, very close to $B\bar B^*$ and $B^*\bar B^* $ thresholds: $Z_b(10610)$ and $Z_b^\prime(10650)$~\cite{zb}. These look very much like the partners in the $b$-sector of $Z_c$ and $Z_c^\prime$, but closer to their `related' thresholds.
\section{An alternative interpretation of the\newline incomplete $X$ multiplet}
One of the most striking facts to deal with is that $X$
has no charged partners, whereas $Z_c$s or $Z_b$s, are observed in three charged states.
The $Z_c$ particle, its mass and quantum numbers, were predicted with surprising accuracy in the diquark-antidiquark model~\cite{noi} together with a lower partner (which for the moment cannot be excluded). Unfortunately the tetraquark model, in its diquark-antidiquark incarnation, also predicts $G=-$ charged partners of the $X$ and an hyperfine splitting between two neutral $X$ particles almost degenerate in mass. The experimental exclusion of charged partners is rather compelling, but not yet decisive~\cite{maianier}.
In several papers all of these states are described in terms of hadron molecules -- even if their binding energy is positive. Why should such resonances, having the same nature, occur in different charge configurations? Is this due to the (uncontrolled) complexity of nuclear forces binding them or can we formulate alternative explanations?
Here we consider the hypothesis that the $X$, as well as $Z$ resonances, are indeed compact tetraquark states of the kind $c\bar c q \bar q$, although we will not stick to the diquark-antidiquark description. The four quarks are free to rearrange in two ways: $i)$ a closed charm configuration in which the $J/\psi$, or any heavier charmonium, is surrounded by light quak matter -- what Voloshin calls an {\it hadrocharmonium}~\cite{volo}, $ii)$ an open charm configuration of two barely interacting $D$ mesons~\cite{mols,braaten}. These two alternatives are superimposed and the compact tetraquark (as long as we can consider it as `stable') is described by
$|\Psi\rangle=|\Psi_P\rangle+|\Psi_Q\rangle$,
$P$ and $Q$ being the labels of the Hilbert subspaces of open and closed charm states respectively.
\begin{figure}[t]
\centering
\includegraphics[width=0.4\textwidth]{feshbach-picture-2}
\caption{\small $P$ and $Q$ are the open (shallow) and closed channels respectively. (1) Charged $X(3872)$ partners suppression, (2) $X$ case (3), $Z_{c,b}$ case.
}
\label{fig:mol}
\end{figure}
We assume that the hadrocharmonium system admits bound states giving rise to a discrete spectrum of levels. A resonance occurs if one of such levels falls close
to some open-charm threshold. An interaction between these two channels is understood.
The level we name as `$J/\psi \,\rho$' has $1^{++}$ quantum numbers: it consists of a $c\bar c$ pair with the quantum numbers of the $J/\psi$ and a light component with the quantum numbers of the $\rho$, held together by hadronic Van der Waals-like forces. The masses of the two components do not have to be {\it a priori} equal to the masses of the `constituent' hadrons, here $J/\psi$ and $\rho$. On the other hand, note that $J/\psi \,\rho^0$ would have a mass of 3872~MeV if the masses of the two constituents were just summed.
If the closest hadrocharmonium level in the $Q$ subspace happens to be above the onset of the continuum spectrum of levels in the $P$ subspace, the coupling between channels gives rise to an attractive interaction and favours the formation of a metastable (Feshbach) resonance at the hadrocharmonium level. The effect is enhanced the smaller the difference in energy $E-E_{\rm{th}}$, $E$ being the bound state level and $E_{\rm{th}}$ the open-charm threshold energy~\cite{bec}.
The Feshbach phenomenon is therefore the formation of a resonance in the `scattering' between different internal tetraquark states.
Because of the interaction between channels $P$ and $Q$ we expect level repulsion and a consequent increase of the detuning parameter $\nu=E-E_\text{th}$, represented in Fig.~1. If the unperturbed detuning between the $J/\psi \,\rho^\pm$ level and the $D^\pm\bar D^{*}$ threshold is small and negative, as one might guess using naively the central values of the masses of the constituent mesons
($M(\rho^\pm)-M(\rho^0)\approx 0.7\pm 0.8$~MeV, $M(D^\pm)-M(D^0)\approx 4.8\pm 0.1$~MeV), the interaction between channels will tend to make it more negative. On the other hand, as stated above, this would give rise to a repulsive interaction (the Fermi scattering length $a$ gets a positive correction) likely destabilizing the system and favoring its fall-apart decay. In this case, a bound state solution could be found if the open channel had discrete levels. Indeed such solution would predominantly be in the open channel with a radius $\sim |\nu|^{-1}\approx 40$~fm, much larger than in the original closed channel. We might infer that
no charged partners of the $X$ have been observed because of these reasons.
The coupling between the $P$ and $Q$ subspaces, described by some $H_{QP}$ Hamiltonian term, appears in the expression of the scattering length
\begin{equation*}
a\simeq a_P+C\sum_n\frac{|\langle \Psi_n|H_{QP}|\Psi_{{\rm th}}\rangle|^2 }{E_{\rm{th}}-E_n} \simeq a_{NR} - C\frac{|\langle \Psi_{\rm res}|H_{QP}|\Psi_{{\rm th}}\rangle|^2 }{\nu}
\end{equation*}
where $C$ is some (positive) constant and $a_P$ is the scattering length (at zero energy) when coupling between open and closed channels is neglected.
Including higher order terms, leading to $H_{QQ}$ interaction terms, the energy of bound states gets shifted (real part of the correction)
and a width is introduced (imaginary part) {\it if} decay into open channel is possible (into levels higher than the threshold one): a metastable state is formed. The last term in the scattering amplitude will then be $\propto 1/(\nu +i\Gamma/2)$. For $\Gamma \ll |\nu|$ the scattering amplitude has a steep $1/\nu$ behavior otherwise, for $\Gamma/2 \gtrsim |\nu|$, the divergence is smoothed. Consider also that the actual detuning $\nu$ has been shifted from the unperturbed one (being larger in absolute value).
In the case of the $X$, given that the unperturbed detuning is $\nu\sim 0$ (assuming the mass $J/\psi \,\rho^0$ to be $3872$~MeV), the second order correction to the energy level should be proportional to $\lim_{\epsilon\to 0} \fint_{0}^{2\epsilon} d\nu |f(\nu)|^2/(\nu-\epsilon)=0$~\footnote{The integral is in the sense of the Principal Value. The function $f(\nu)$ has to be some regular function in $\nu$. Consider that the less prominent term corresponding to $J/\psi \;\omega$ in the closed channel could also be taken into account although the most simple approximation is that of reabsorb all the non-resonant closed channel states (including the continuum) in $a_{NR}$.}. This should leave the detuning small, $\nu\to 0^+$,
enhancing the Feshbach resonance phenomenon and making the $X$ a metastable state with very long lifetime~\footnote{A Feshbach mechanism between the open charm spectrum and the $\chi_{c1}(2P)$ charmonium level was first proposed in~\cite{braaten} to explain the narrowness of the $X(3872)$. More precise estimates for the mass of the $\chi_{c1}(2P)$ made this hypothesis unlikely.} -- being essentially driven by the instability of $D^*$, $\Gamma_{D^*}\sim 100$~keV. In this sense the $X$ is a particularly fine-tuned state and we do not expect to have its analog in the beauty sector.
\section{$Z_c^{(\prime)}$ and $Z_b^{(\prime)}$}
The same line of reasoning could apply to the charged states. The quantum numbers of the $Z_c$ are $1^{+-}$, thus we could choose a hidden charm combination $\psi(3770)\pi^{0,\pm}$ which falls about $10$ MeV above the $Z_c$ mass.
This could be the reason why this resonance, differently from the $X(3872)$, is observed in all charged states. Moreover, for the reasons discussed above, we expect it to be broader than the $X$. In fact $Z_c$ has a total width of $\approx 50$~MeV to be compared with $\Gamma_X\lesssim 1$~MeV.
As for the $Z_c^\prime$ we might consider a $h_c(2P)\pi$ hadrocharmonium level. The $h_c(2P)$ has not been observed yet but its mass, according to a rough order of magnitude estimate, might be found with the $h_c(1P)+h_b(2P)-h_b(1P)$ mass formula (locating $h_c(2P)\pi$ at 4025~MeV, just a few MeVs above the threshold value $\bar D^* D^{*\pm}$ at 4017~MeV).
As a {\it caveat} we notice that the $h_c(2P)\pi$ hadrocharmonium level would have negative parity if no unit of orbital angular momentum is introduced. The decay $Z_c^\prime \to h_c(1P)\pi$, if confirmed to be $P$-wave, would support this hypothesis~\footnote{On the other hand, orbital angular momentum could have the effect of rising the mass of the state.}.
As we do not yet know the hadrocharmonium spectrum, one might wonder how the scheme here described might be preliminarily tested. As a first step we apply the Feshbach formalism to estimate the width of the states.
According to the Fermi golden rule, the width of a metastable Feshbach resonance, with $\nu>0$, has to be proportional to the density of accessible levels $dn/dE$ thus leading to $\Gamma\simeq A\sqrt{\nu}$:
when $\nu\to 0^+$ the two channels will be strongly hybridized and the resulting metastable state be very narrow~\footnote{The width formula is, for $\nu>0$, given by $\Gamma(\nu)\sim \sqrt{\nu_c \nu}$ where the critical value of the detuning $\nu_c$ is a function of the details of the potentials involved, including the hybridizing one. Narrow width approximation would require $\Gamma(\nu)\lesssim\nu$. We have to relax it to $\Gamma(\nu)<C\nu$ where $C$ is some positive constant; in other words we need $\nu>\nu_c/C^2$. In our computation (see the text) we estimate a critical value $\nu_c\sim 100$~MeV, thus $C\sim 2$ is sufficient when considering the $Z_c^{(\prime)}$, but we would need $C\sim 10$ for the $Z_b^{(\prime)}$.
If $\nu<0$ and the open channel admits discrete levels, one can indeed find a bound state in the open channel.
We assume to be in the conditions where the open channel has no discrete levels, {\it i.e.} the pion forces are not enough binding~\cite{rux}.
}.
Hence, the detuning for $Z_c$ is $\nu \approx 20$~MeV and $\Gamma_{\rm exp}\simeq 46\pm 22 $~MeV. This would lead to $A\simeq 10\pm 5~{\rm MeV}^{1/2}$.
As for the $Z_c^\prime$, the detuning is roughly estimated to be $\nu \approx 9$~MeV, which would then lead to $\Gamma\approx 29\pm 16$~MeV, using $A$ computed above.
The measured width is $\Gamma_{\rm exp}\simeq 24.8\pm 9.5$~MeV (at a mass of 4026~MeV in the decay channel $D^*\bar D^*$), sensibly smaller than the measured width of $Z_c$ and in good agreement with our simple estimate. This is also compatible with the $Z_c^\prime \to h_c \pi$ channel, having a mass of 4023~MeV but a width of $\Gamma_{\rm exp}\simeq 7.9\pm 3.7$~MeV.
Under the hypothesis that the quantum numbers of $Z_b$ and $Z_b^\prime$ are $1^{+-}$, we might conjecture that the two closed charm states are $\chi_{b0}(1P)\rho$ and $\chi_{b1}(1P)\rho$, with a unit of orbital angular momentum to give the overall positive parity. Both combinations would be only slightly higher than the related $B \bar B^*$ and $B^* \bar B^*$ thresholds (the effect of the orbital motion on the mass of these hadrobottomonium levels should be estimated).
In the case of $Z_b$ and $Z_b^\prime$ the detunings are roughly $\nu=2.7$~MeV and $\nu^\prime=1.8$~MeV respectively. The measured decay widths reported by Belle are $\Gamma_{\rm exp}=18.4\pm 2.4$~MeV and $\Gamma^\prime_{\rm exp}=11.5\pm 2.2$~MeV for the lighter state $Z_b(10610)$ and the heavier $Z_b^\prime(10650)$ respectively, {\it i.e.} $(\Gamma/\Gamma^\prime)_{\rm exp}\approx 1.6\pm 0.4$, to be compared with $\Gamma/\Gamma^\prime=\sqrt{\nu/\nu^\prime}=1.2$.
We observe that in the beauty sector $A\approx 10\pm 3~{\rm MeV}^{1/2}$, as for the charm.
More states, together with their widths, might be predicted in the charm and beauty sector along the same lines. There is however an important selection rule:
the lifetime $\tau_D$ of the open charm molecular constituents must be confronted with the characteristic time $\tau_F$ of the Feshbach resonance. If $\tau_D \gg \tau_F$ our formulation of Feschbach mechanism can neglect the instability of the constituents. If \mbox{$\tau_D \sim \tau_F$} the decay of the constituents might happen during the lifetime of the molecule, thus challenging the Feshbach mechanism. We could consider only the stablest open flavor mesons $D$,$B$ and $D^*$,$B^*$, reproducing the observed $X$ and $Z_{c,b}^{\left(\prime\right)}$. If we weaken this requirement, for example in charm sector with an $S$-wave hadrocharmonium, the first state we predict appears at the thresholds $D_1^0 \bar D^0$ and $h_c \,\omega$, thus having quantum numbers $\left(I^G\right)\,J^{PC}=\left(0^+\right)\,1^{-+}$, a mass of $\approx 4310$~MeV and a width of $\approx 40$~MeV.
\section{Conclusions}
We confront with the problem of the absence of charged partners of $X(3872)$
and with the recent findings on the $Z_c$s and $Z_b$s resonances. We attempt a description of all these states as Feshbach resonances arising in the scattering between internal states of a tetraquark configuration. We distinguish between a closed subspace of hadrocharmonium levels and an open subspace of open-charm thresholds. There could be only a few hadrocharmonium-like states almost matching open charm thresholds. The most perfect match is that realized by the $D^0 \bar D^{*0}$ threshold with $J/\psi \,\rho^0$, which we refer to as the neutral $X$.
The $J/\psi \,\rho^\pm$ might fail to match $D^\pm \bar D^{*0}$, being below it, and a metastable state might be forbidden. Few more matchings can be found corresponding to $Z_c$s and $Z_b$s. We relate a parameter of the Feshbach resonances, the detuning, to the observed width of the states, reporting some interesting agreement with data.
\section*{Acknowledgements} ADP whish to thank Ben Grinstein for his hospitality at UCSD and for intersting discussions.
|
2,869,038,154,791 | arxiv | \section{Introduction}
In this paper we consider the generic unconstrained minimization problem
\begin{equation}\label{eq:mainP}
\compactify \min_{x\in \R^n} f(x),
\end{equation}
where $f: \R^n \rightarrow \R$ is a smooth objective function and bounded from below. One of the most fundamental methods for solving \eqref{eq:mainP} is {\em gradient descent (GD)}, on which many state-of-the-art methods are based. Given current iterate $x_k\in \R^n$, the update rule of GD is
\begin{eqnarray}\label{GD}
\compactify x_{k+1} = x_k - \alpha_k \nabla f(x_k),
\end{eqnarray}
where $\alpha_k>0$ is a stepsize. The efficiency of GD depends on further properties of $f$. Assuming $f$ is $L$--smooth and $\mu$--strongly convex, for instance, the iteration complexity of GD is ${\cal O}(\kappa{\rm log}(1/\epsilon))$, where $\kappa=L/\mu$ and $\epsilon$ is the target error tolerance. However, it is known that GD is not the ``optimal'' gradient type method: it can be {\em accelerated}.
The idea of accelerating converging optimization algorithms can track its history back to 1964 when Polyak proposed his ``heavy ball'' method \citep{polyak}.
In 1983, Nesterov proposed his accelerated version for general convex optimization problems. Comparing with Polyak's method, Nesterov's method gives acceleration for
general convex and smooth problems and the iteration complexity improves
to ${{\cal O}(1/\sqrt{\epsilon})}$ \citep{nesterov}. In 2009, Beck and Teboulle proposed fast iterative shrinkage thresholding algorithm (FISTA) \citep{fista} that uses Nesterov's momentum coefficient and accelerates {\em proximal} type algorithms to solve a more complex class of objective functions that combine a smooth, convex loss function (not necessarily differentiable) and a strongly convex, smooth penalty function (also see \citep{nesterov2007, Nesterov2013}). To develop further insights into Nesterov's method, Su et al.\ \citep{ode_boyd} examined a continuous time 2nd-order ODE which at its limit reduces to Nesterov's accelerated gradient method. In addition, Lin et al. \citep{lin_katalyst} introduced a generic approach known as {\em catalyst} that minimizes a convex objective function via an accelerated proximal point algorithm and gains acceleration in Nesterov's sense. \citep{BubeckLS15} proposed a geometric alternative to gradient descent that is inspired by ellipsoid method and produces acceleration with complexity $\compactify{{\cal O}(1/\sqrt{\epsilon})}$. Recently, \citep{Zhu2017LinearCA} used a linear {\em coupling} of gradient descent and mirror descent and claimed to attend acceleration in Nesterov's sense as well.
In contrast, the {\em sequence acceleration} techniques accelerate a sequence independently from the iterative method that produces this sequence.
In other words, these techniques take a sequence $\{x_k\}$
and produce an accelerated sequence based on the linear combination of $x_k{\rm s}$ such that the new accelerated sequence converges faster than the original.
In the same spirit, recently, Scieur et al.~\citep{rna_16,rna_18} proposed an acceleration technique called regularized nonlinear acceleration~(RNA).
Scieur et al.'s idea is based on Aitken's $\Delta^2$-algorithm \citep{aitken} and Wynn's $\epsilon$-algorithm \citep{wynn} (or recursive formulation of generalized Shanks transform \citep{ shanks, wynn, brezinski}).
To achieve acceleration,
Scieur et al. considered a technique known as minimum polynomial approximation and they assumed a {\em linear} model for the iterates near the optimum. They also proposed a {\em regularized} variant of their method to stabilize it numerically.
The intuition behind the regularized nonlinear acceleration of Scieur et al. is very natural. To minimize $f$ as in \eqref{eq:mainP}, they considered the sequence of iterates $\{x_k\}_{k\geq 0}$ is generated by a fixed-point map. If $x^\star$ is a minimizer of $f$, $\nabla f(x^\star)=0$, and hence through extrapolation one can find:
\begin{eqnarray}\label{eq:rna0}
\compactify c^\star \approx \arg\min_{c}
\left\{
\left\|
\nabla f\left(\sum_{k = 0}^K c_k x_k \right)
\right\|
\;:\; c\in\R^{K+1}, \;\sum_{k = 0}^K c_k =1
\right\},
\end{eqnarray}
such that the next (accelerated) point can be generated as a linear combination of $K+1$ previous iterates:
$
x = \sum_{i=0}^K c_i^\star x_i.
$
We review RNA in detail in Section \ref{sec:rna}.
{\em Notation.} We denote the $\ell_2$-norm of a vector $x$ by $\| x \|$ and define $\| x \|_M$ by $\|x\|_M\stackrel{\text{def}}{=} \sqrt{x^\top M x}.$
\subsection{Contributions}
We highlight our main contributions in this paper as follows:
\textbf{Direct nonlinear acceleration~(DNA).} Inspired by Anderson's acceleration technique \citep{Anderson} (see Appendix for a brief description of Anderson's acceleration) and the work of Scieur et al. \citep{rna_16}, we propose an extrapolation technique that accelerates a converging iterative algorithm. However, in contrast to \citep{rna_16}, we find the extrapolation coefficients $c^\star$ by directly minimizing the function at the linear combination of $K+1$ iterates $\{x_k\}_{k\geq 0}^K$ with respect to $c\in\R^{K+1}$. In particular, for a given sequence of iterates $\{x_k\}_{k\ge 0}^K$ we propose to approximately solve:
\begin{eqnarray}\label{mainDNA}
\boxed{\compactify \min_{c\in\R^{K+1}} f\left(\sum_{k = 0}^K c_k x_k \right)+\lambda g(c),}
\end{eqnarray}
where
$\lambda>0$ is a balancing parameter and
$g$ is a penalty function. As our approach tries to minimize the functional value directly, we call it as \emph{direct nonlinear acceleration} (DNA). We also note that our formulation shares some similarities with \citep{riseth,zhang2018globally}. However, unlike \citep{riseth}, we do not require line search and check a decrease condition at each step of our algorithm. On the other hand, Zhang et al.\ \citep{zhang2018globally} do not consider a direct acceleration scheme as they deal with a fixed-point problem.
\textbf{Regularization.}
We propose several versions of DNA by varying the penalty function $g(c)$. This helps us to deal with the numerical instability in solving a linear system as well as to control errors in gradient approximation. In our first version, we let $g(c) = 1_S(c)$, where $S\stackrel{\text{def}}{=} \{c\;:\; \sum_i c_i=1\}$ and $1_{S}(c)=0$ if $c\in S$, while $1_{S}(c)=+\infty$ otherwise. Later, we propose two regularized {\it constraint-free} versions to find a better minimum of the function $f$ by expanding the search space of extrapolating coefficients to $\R^{K+1}$ rather than restricting them over the space $S$. To this end, the first {constraint-free version}
adds a quadratic regularization $\compactify{ g(c) = \left\|\sum_{i=0}^K c_ix_i-y\right\|^2}$ to the objective function, where $y$ is a reference point and $g(c)$
controls how far we want the linear combination $\sum_ic_ix_i$ to deviate from $y$. In the second {constraint-free version}, we add the regularization directly on $c$. We add a quadratic term of the form $g(c) =\compactify{ \|c-e\|^2}$ to the objective function, where $e$ is a reference point to $c$ and $g(c)$ controls how far we want $c$ to deviate from $e$. In contrast, the {\em regularized} version of RNA only considers a ridge regularization $\|c\|^2$ for numerical stability. Trivially, we note that by setting $e=0$, we recover the regularization proposed in RNA. We argue that by using a different penalty function $g(c)$ as regularizer our DNA is more robust than RNA.
\textbf{Quantification between RNA and DNA in minimizing quadratic functions by using GD iterates.}
If $g(c)=0$ or $g(c)=1_{\sum_ic_i=1}$, in terms of the functional value, we always obtain a better accelerated point than
RNA. Moreover, the acceleration obtained by DNA can be {\em theoretically} directly implied from the existing results of Scieur et al.\citep{rna_16}.
If $g(c)=0$, we show by a simple example on quadratic functions that DNA outperforms RNA by an arbitrary large margin.
If $g(c)=1_{\sum_ic_i=1}$, we also quantify the functional values obtained from both RNA and DNA for quadratic functions and provide a bound on how DNA outperforms RNA in this setup.
\textbf{Numerical results.} Our empirical results show that for smooth and strongly convex functions, minimizing the functional value converges faster than RNA. In practice, our acceleration techniques are robust and outperform that of Scieur et al.~\citep{rna_16} by large margins in almost all experiments on both synthetic and real datasets. To further push the robustness of our methods, we test them on nonconvex problems as well. As a proof of concept, we trained a simple neural network classifier on MNIST dataset \citep{mnist} via GD and accelerate the GD iterates via the online scheme in \citep{rna_16} for both RNA and DNA. Next, we train ResNet18 network \citep{resnet18} on CIFAR10 dataset \citep{cifar10} by SGD and accelerate the SGD iterates via the online scheme in \citep{rna_16} for both RNA and DNA. In both cases, DNA outperform RNA in lowering the generalization errors of the networks.
\section{Regularized Nonlinear Acceleration}\label{sec:rna}
In RNA, one solves \eqref{eq:rna0} by assuming that the gradient can be approximated by linearizing it in the neighborhood of $\{x_k\}_{k=0}^K.$
Thus, by assuming $\sum_{k = 0}^K c_k =1$, the relation
$
\compactify{ \left\|\nabla f\left(\sum_{k = 0}^K c_k x_k \right)\right\|\approx \left\| \sum_{k = 0}^K c_k\nabla f\left(x_k \right)\right\|}
$
holds. Hence, one can approximately solve \eqref{eq:rna0} via:
\begin{eqnarray}\label{eq:rna}
\compactify c^\star=\arg\min_{c} \left\{ \left\| \sum_{k = 0}^K c_k \nabla f\left(x_k \right)\right\| =\left\| \sum_{k = 0}^K c_k \tilde{R}_k\right\|\;:\; c\in\R^{K+1}, \; \sum_{k = 0}^K c_k =1 \right\},
\end{eqnarray}
where $\tilde{R}_k$ is the $k^{\rm th}$ column of the matrix $\tilde{R}$, which holds $\nabla f\left(x_k \right).$~Moreover \eqref{eq:rna} does not need an explicit access to the gradient and it can be seen as an approximated minimal polynomial extrapolation (AMPE) as in \citep{ampe,rna_16,rna_18}.
In this context, we should note that Scieur et al. indicated that the summability condition $c^\top\mathbbm{1}=1$ is not restrictive, where $\mathbbm{1}$ is a vector of all 1s.
If the sequence $\{x_k\}$ is generated via GD (as in \eqref{GD}), then $\tilde{R} =\left[\nicefrac{(x_0 - x_{1})}{\alpha_0},\ldots,\nicefrac{(x_K - x_{K+1})}{\alpha_K} \right]$.
Also, if $\tilde{R}^\top \tilde{R}$ is nonsingular, then the minimizer of \eqref{eq:rna}
is explicitly given as:
$\compactify{c^\star = \frac{(\tilde{R}^\top \tilde{R})^{-1}\mathbbm{1}}{\mathbbm{1}^\top(\tilde{R}^\top \tilde{R})^{-1}\mathbbm{1}}}$. If $\tilde{R}^\top \tilde{R}$ is singular then $c$ is not necessarily unique. Any $c$ of the form $\frac{z}{z^\top\mathbbm{1}}$, where $z$ is a solution of $\tilde{R}^\top \tilde{R} z = \mathbbm{1}$, is a solution
of (\ref{eq:rna}). To deal with the numerical instabilities and the case when the matrix
$\tilde{R}^\top \tilde{R}$ is singular, Scieur et al. proposed to add a regularizer of the form
$\lambda \|c\|^2$ to their problem, where $\lambda>0$. As a result, $c^\star$ is unique and given as
$\compactify{c^\star = \frac{(\tilde{R}^\top \tilde{R}+\lambda I)^{-1}\mathbbm{1}}{\mathbbm{1}^\top(\tilde{R}^\top \tilde{R}+\lambda I)^{-1}\mathbbm{1}}}.$
The numerical procedure of RNA is given in Alg~\ref{alg:RNA}.
For further details about RNA we refer the readers to \citep{rna_16,rna_18}. Scieur et al. also explained several acceleration schemes to use with Algorithm~\ref{alg:RNA}.
\begin{algorithm}
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwInOut{Init}{Initialize}
\SetKwInOut{Compute}{Compute}
\Input{Sequence of iterates $x_0,\ldots,x_{K+1}$; sequence of step sizes $\alpha_0,\ldots,\alpha_{K}$; $\mathbbm{1}\in\R^{K+1}$: a vector of all 1s; and $\lambda >0$.}
\nl Set $\tilde{R} =\left[\frac{x_0 - x_{1}}{\alpha_0},\ldots,\frac{x_K - x_{K+1}}{\alpha_K} \right] $\;
\nl Solve the linear system: $\left(\tilde{R}^\top \tilde{R} + \lambda I \right)z = \mathbbm{1}$\;
\nl Set $c = \frac{z}{z^\top\mathbbm{1}}\in\mathbb{R}^{K+1}$\;
\Output{$x = \sum_{k = 0}^K c_k x_k$.}
\caption{RNA} \label{alg:RNA}
\end{algorithm}
\section{ Direct Nonlinear Acceleration}\label{sec:dna}
Instead of minimizing the norm of the gradient, we propose to minimize the objective function $f$ directly to obtain the coefficients $\{c_k\}$.
We set $g(c)=0$ in \eqref{mainDNA} and we propose to solve the unconstrained minimization problem
\begin{eqnarray}\label{eq:fkrna}
\compactify \boxed{ \min_{c\in\R^{K+1}} f\left(Xc \right),}
\end{eqnarray}
where $X = [x_0, \ldots,x_K]$. We call problem \eqref{eq:fkrna} as direct nonlinear acceleration (DNA) without any constraint.
If $f$ is quadratic, then we have the following lemma:
\begin{lemma}\label{lemma:dna1}
Let the objective function $f$ be quadratic and let $\{x_k\}$ be the iterates produced by \eqref{GD} to minimize $f$.
Then
$c$ is a solution of the linear system $X^\top R z = - X^\top \nabla f(0),$
where $R \in\R^{n\times (K+1)}$ is a matrix such that its $i^{\rm th}$ column is
$R_i = \frac{x_i-x_{i+1}}{\alpha_i}-\nabla f(0)$ and $X = [x_0, \ldots,x_K]$.
\end{lemma}
If $f$ is non-quadratic then we can {\em approximately} solve problem (\ref{eq:fkrna})
by approximating its gradient by a linear model.
In fact, we use the following approximation $\nabla f(x)\approx A(x-y_x)+\nabla f(y_x),$ where we assume that $x$ is close to $y_x$ and $A$ is an approximation of the Hessian. Therefore, by setting $x=Xc$ and $y_x=y$ in the above, we have
$
\nabla f(Xc) \approx A(Xc-y)+\nabla f(y)=\sum_i c_iAx_i - A y +\nabla f(y),
$
where $y$ is a referent point that is assumed to be in the neighborhood of $Xc$. For instance, one may choose $y$ to be $x_K$. Let $x_{i-1}$ be a referent point for $x_i$, that is, assume that $\nabla f(x_i) \approx A(x_{i}-x_{i-1})+\nabla f(x_{i-1})$. Then one can show that
$Ax_i = \nabla f(x_i)-\nabla f(0)$. As a result, we have
\begin{eqnarray}\label{approximate_grad}
\compactify \nabla f(Xc)& \approx &\compactify \sum_i c_i (\nabla f(x_i)-\nabla f(0))- A y +\nabla f(y)\nonumber\\
&=&\compactify \sum_i c_i \left({\tfrac{x_i-x_{i+1}}{\alpha_i}}-\nabla f(0)\right)- A y +\nabla f(y)\nonumber\\
&=&\compactify \sum_i c_iR_i- A y +\nabla f(y) \quad = \quad Rc - A y +\nabla f(y) \quad \approx \quad Rc +\nabla f(0).
\end{eqnarray}
Therefore, from the first optimality condition and by using
(\ref{approximate_grad}), we conclude that the solutions of (\ref{eq:fkrna})
can be approximated by the solutions of the linear system $X^\top R z = - X^\top \nabla f(0)$.
We describe the numerical procedure in Alg~1 in the Appendix.
\begin{figure}
\centering
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/LS_online2016_1-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/LS_online2016_2-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/LS_online2016_3-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/RR_online2016_1-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/RR_online2016_2-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/RR_online2016_3-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/LR_online2016_1-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/LR_online2016_2-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Synthetic2/LR_online2016_3-eps-converted-to.pdf}
\end{minipage}
\caption{\small{Acceleration on synthetic data by using online acceleration scheme in \citep{rna_16}. First and second row represent quadratic, strong convex objective function as Least Squares and Ridge Regression, respectively. The last row represents non-quadratic but strong convex objective function as Logistic Regression. For all plots we use $k=3$. For RNA, we have $\lambda=10^{-8}$; for DNA, we set $\lambda=10^{-8}$, except for the last LR plot where for DNA-2, we set $\lambda=10$.} }\label{fig:synthetic_ol16}
\end{figure}
\begin{algorithm}
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwInOut{Init}{Initialize}
\SetKwInOut{Compute}{Compute}
\Input{Sequence of iterates $x_0,\ldots,x_{K+1}$ and sequence of step sizes $\alpha_0,\ldots,\alpha_{K}$\;}
\nl Set $R =\left[\frac{x_0 - x_{1}}{\alpha_0} -\nabla f(0),\ldots,\frac{x_K - x_{K+1}}{\alpha_K} -\nabla f(0) \right]$ and $X = [x_0, \ldots,x_K]$\;
\nl Set $c$ as a solution of the linear system $X^\top R z = - X^\top \nabla f(0)$\;
\Output{$x = \sum_{k = 0}^K c_k x_k$.}
\caption{DNA } \label{alg:fkRNA}
\end{algorithm}
\paragraph{Comments on the convergence of DNA.}
Let $H$ be the Hessian of $f$, where we assume that $f$ is quadratic.
Also let $\lambda_{\rm max}(H)$ be the maximum eigenvalue of $H$
\begin{lemma}\label{lemma:krna}
Let $c_D$ and $c_R$ be the extrapolation coefficients produced by DNA and RNA, respectively. Then
$
\compactify{\|Xc_D-x^\star\|_H^2\le \|Xc_R-x^\star\|_H^2\le \lambda_{\rm max}(H)\|Xc_R-x^\star\|^2.}
$
\end{lemma}
\begin{theorem}\label{thm:dna_conv}
Denote $\xi = \nicefrac{(\sqrt{L}-\sqrt{\mu})}{(\sqrt{L}+\sqrt{{\mu}})}.$ It is given in \citep{rna_16} that for the coefficients $c_R$ produced by Alg~\ref{alg:RNA}:
$
\compactify{\|Xc_R-x^\star\|^2\le \kappa(H)\tfrac{2\xi^k}{1+\xi^{2k}}\|x_0-x^\star\|^2. }
$
Further, for Alg~\ref{alg:kRNA} we have
$
\compactify{\|Xc_D-x^\star\|_H^2\le \lambda_{\rm max}(H)\kappa(H)\tfrac{2\xi^k}{1+\xi^{2k}}\|x_0-x^\star\|^2.}
$
\end{theorem}
\begin{remark}
The convergence rate for DNA for quadratic functions in Theorem \ref{thm:dna_conv} is the same as that for Krylov subspace methods (for example, conjugate gradient algorithm) up to a multiplicative scalar.
\end{remark}
However, numerically DNA is unstable like RNA without regularization. In fact, the matrix $X^\top R$ can be very ill-conditioned and can lead to large errors in computing $c^\star$. Moreover, we accumulate errors in approximating the gradient via linearization as our approximation of the gradient is valid only in the neighborhood of the iterates $x_0,\ldots,x_K$. To solve these problems, we propose three regularized versions of DNA by using three different regularizers in the form of $g(c)$ and show that they work well in practice. But one can explore different forms of $g(c)$ as regularizer.~We explain them in the following sections.
\begin{algorithm}
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwInOut{Init}{Initialize}
\SetKwInOut{Compute}{Compute}
\Input{Sequence of iterates $x_0,\ldots,x_{K+1}$; sequence of step sizes $\alpha_0,\ldots,\alpha_{K}$; and $\mathbbm{1}\in\R^{K+1},$ a vector of all 1s\;}
\nl Set $\tilde{R} =\left[\frac{x_0 - x_{1}}{\alpha_0},\ldots,\frac{x_K - x_{K+1}}{\alpha_K} \right]$ and $X = [x_0, \ldots,x_K]$\;
\nl Solve the linear system for $z\in\R^{K+1}$: $X^\top\tilde{R} z = \mathbbm{1}$\;
\nl Set $c = \frac{z}{z^\top\mathbbm{1}}\in\mathbb{R}^{K+1}$\;
\Output{$x = \sum_{k = 0}^K c_k x_k$.}
\caption
\kRNA -1} \label{alg:kRNA}
\end{algorithm}
\vspace{-05pt}
\subsection{DNA-1}
\vspace{-05pt}
This regularized version of \kRNA is directly influenced by Scieur et al. \citep{rna_16}.
Here, we generate the extrapolated point $x$ as a linear combination of the set of $K+1$ iterates such that, $x=\sum_k c_kx_k$. Additionally, as in \citep{rna_16,rna_18}, we assume the sum of the coefficients $c_k$ to be equal to 1.
Therefore, for $c\in\R^{K+1}$ with sum of its elements equal to 1, we set $g(c) = 1_{\sum_i c_i=1}$ in \eqref{mainDNA} and consider the following constrained problem:
\begin{eqnarray}\label{eq:krna}
\compactify {{\boxed{\min_{c \in\R^{K+1}}\ f(Xc)+\lambda 1_{S}(c)=\min_{c} \left\{ f\left(Xc \right) \;:\; c\in\R^{K+1}, \; \compactify \sum_{k = 0}^K c_k =1 \right\},}}}
\end{eqnarray}
where $X = [x_0, \ldots,x_K]$. We call this version of DNA as DNA-1.
\begin{lemma}\label{lemma:ckrna_lemma}
If the objective function $f$ is quadratic and $X^\top \tilde{R}$ is nonsingular then
$c = (X^\top \tilde{R})^{-1}\mathbbm{1} / \delta, $ where $\delta=\mathbbm{1}^\top (X^\top \tilde{R} )^{-1}\mathbbm{1},
$
and $\mathbbm{1}$ is the vector of dimension $K+1$ with all the components equal to 1 and $ \compactify{\tilde{R}=\left[\nicefrac{(x_0 - x_{1})}{\alpha_0},\ldots,\nicefrac{(x_K - x_{K+1})}{\alpha_K} \right]}$.
\end{lemma}
\begin{figure*}
\centering
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_glass-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_letterscale-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_pendigits-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_satimage-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_segment-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_shuttlescale-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_svmguide4-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LS_online2016_vehicle-eps-converted-to.pdf}
\end{minipage}
\caption{\small{Acceleration on {\tt LIBSVM} dataset by using online acceleration scheme in \citep{rna_16} on Least Squares problems. For all datasets, we use $k=3$. For RNA and DNA, we set $\lambda=10^{-8}$.}}\label{fig:real_ls_ol16}
\end{figure*}
\begin{figure*}
\centering
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_glass-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_pendigits-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_segment-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_shuttlescale-eps-converted-to.pdf}
\end{minipage}
\\
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_svmguide4-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_vehicle-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_vowel-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/RR_online2016_wine-eps-converted-to.pdf}
\end{minipage}
\caption{\small{Acceleration on {\tt LIBSVM} dataset by using online acceleration scheme in \citep{rna_16} on Ridge Regression problems. For all datasets, we use $k=3$. For RNA and DNA, we set $\lambda=10^{-8}$.}}\label{fig:realdata_rr_ol16}
\end{figure*}
Similar to RNA, if $X^\top \tilde{R}$ is singular then $c$ is not necessarily unique. Any $c$ of the form $\frac{z}{z^\top\mathbbm{1}}$, where $z$ is a solution of $X^\top\tilde{R}z = \mathbbm{1}$, is a solution
of (\ref{eq:krna}). DNA-1 is described in Alg~\ref{alg:kRNA}.
\paragraph{Comparison with RNA on simple quadratic functions.}
Denote the functional value obtained by DNA, DNA-1 (Alg~\ref{alg:kRNA}) and RNA (Alg~\ref{alg:RNA}) at an extrapolated point as $f_{D}$, $f_{D1}$ and $f_R$, respectively.
\begin{proposition}\label{fn_value}
Let $A\in\R^{n\times n}$ be symmetric and positive definite and $f(x)=\frac{1}{2}x^\top Ax$ be a quadratic
objective function. Let $X=[x_0\;\;x_1\;\cdots x_k]$ be a matrix generated by stacking $k$ iterates of GD to minimize $f$.
Then the functional value of DNA, DNA-1 and RNA at the accelerated point are: $f_D = 0$,
$
\compactify{f_{\rm D1}=\frac{1}{2 \mathbbm{1}^\top (X^\top A X)^{-1} \mathbbm{1}}},
$
and
$
\compactify{ f_{\rm R}=\frac{\mathbbm{1}^\top (X^\top A^2 X)^{-1}X^\top A X (X^\top A^2 X)^{-1}\mathbbm{1}}{2(\mathbbm{1}^\top (X^\top A^2X)^{-1} \mathbbm{1})^2}},
$
respectively.
\end{proposition}
We conclude that for this simple objective function, DNA reaches the optimal solution after the first acceleration. Moreover, one can choose the matrix $A$ such that $f_R$ is arbitrary large, and this example shows that DNA may outperform RNA by a large margin.~The comparison between DNA-1 and RNA on the previous example is given in the following lemma and theorem.
\begin{figure*}
\centering
\begin{minipage}{0.26\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LR_online2016_fourclass-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.26\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LR_online2016_germannumer-eps-converted-to.pdf}
\end{minipage}
\begin{minipage}{0.26\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/Realdata/LR_online2016_splice-eps-converted-to.pdf}
\end{minipage}
\caption{\small{Acceleration on {\tt LIBSVM} dataset by using online acceleration scheme in \citep{rna_16} on Logistic Regression problems. For all datasets, we use $k=3$. For RNA and DNA, we set $\lambda=10^{-8}$.}}\label{fig:realdata_lr_ol16}
\end{figure*}
\begin{lemma}\label{lemma:ratio}
We assume that the matrix $\tilde{R}$
has full column rank.
With the notations used in Proposition \ref{fn_value}, we have $\compactify{\nicefrac{f_{{\rm R}}}{f_{{\rm D1}}}=\|z\|_{A^{-1}}^2\|y\|^2\|z\|^{-4},}$ where $\compactify{z\stackrel{\text{def}}{=} (\tilde{R}^\dagger)^\top \mathbbm{1} = ((AX)^\dagger)^\top \mathbbm{1}}$ and $\compactify{y\stackrel{\text{def}}{=}((A^{1/2}X)^\dagger)^\top\mathbbm{1}}$.
We have $y^\top A^{-1/2} z = z^\top z$; then, by using Cauchy-Schwarz inequality, we conclude that $\compactify{{\|z\|_{A^{-1}}\|y\|_2} \ge y^\top A^{-1/2} z= {\|z\|_2^2}}$
whence $\nicefrac{f_{\rm R}}{f_{\rm D1}} \ge 1$
\end{lemma}
Note that the ratio $\compactify{\frac{f_{\rm R}}{f_{\rm D1}} \ge 1}$ can be directly concluded from the definition of ${f_{\rm R}}$ and ${f_{\rm D1}}$. The main goal of the previous lemma is to exactly quantify the ratio between these two quantities. The following theorem gives more insight.
\begin{theorem}\label{lemma:UB}
We have $ \compactify{\frac{f_{{\rm R}}}{f_{{\rm D1}}}\le U_R \stackrel{\text{def}}{=} \|z\|_{A^{-1}}^2\|z\|_A^2 \|z\|^{-4},}$ and $U_R \in \left[\nicefrac{1}{2}+ \nicefrac{\kappa(A)}{2},\kappa(A)\right],$ where
$\kappa(A)$ is the condition number of $A$.
\end{theorem}
\begin{algorithm}
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwInOut{Init}{Initialize}
\SetKwInOut{Compute}{Compute}
\Input{Sequence of iterates $x_0,\ldots,x_{K+1}$; sequence of step sizes $\alpha_0,\ldots,\alpha_{K}$; regularizer $\lambda>0$; and reference vector $y\in\R^{k+1}$\;}
\nl Set $R =\left[\frac{x_0 - x_{1}}{\alpha_0} -\nabla f(0),\ldots,\frac{x_K - x_{K+1}}{\alpha_K} -\nabla f(0) \right]$ and $X = [x_0, \ldots,x_K]$\;
\nl Set $c$ as a solution of the linear system
$(X^\top R+\lambda X^\top X) z= \lambda X^\top y-X^\top \nabla f(0)$\;
\Output{$x = \sum_{k = 0}^K c_k x_k$.}
\caption{ DNA-2 } \label{alg:DNA-2}
\end{algorithm}
The above theorem tells us, for a simple quadratic function, the ratio of the objective function values of DNA-1 and RNA may attain an order of $\kappa(A)$, but it never exceeds it. The theoretical quantification of the acceleration obtained by DNA and its different versions compared to RNA in more general problems is left for future work.
Although DNA-1 can be seen as a regularized version of DNA, we still need to remedy the fact that the linearization of the gradient is not a {\em good} approximation in the entire space, and that the matrix $X^\top\tilde{R}$ may be singular. To this end, we impose some regularization such that the new extrapolated point {\em stays} near to some reference point. We propose two different ways in the following two sections.
\subsection{DNA-2}
We set $g(c)=\|Xc-y\|^2$ in \eqref{mainDNA} and consider a regularized version of problem \eqref{eq:fkrna}:
\begin{eqnarray}\label{eq:fkrnareg-}
\compactify \min_{\substack{c\in\R^{K+1}}} f\left(Xc \right)+\frac{\lambda}{2}\|Xc-y\|^2,
\end{eqnarray}
where $\lambda>0$ is a balancing parameter and $y$ is a reference point (a point supposed to be in the neighborhood of $Xc$).
By taking the derivative of the objective in \eqref{eq:fkrnareg-} with respect to $c$ and setting it to 0, we find
$
\compactify{ X^\top\nabla f(Xc)+\lambda X^\top(Xc-y)=0},
$
which after using the approximation \eqref{approximate_grad} becomes
$
\compactify{ X^\top(Rc+\nabla f(0) )+\lambda X^\top(Xc-y)=0.}
$
Finally, $c^\star$ is given as a solution to the linear system
\begin{eqnarray}\label{solution2}
\compactify (X^\top R+\lambda X^\top X) c=\lambda X^\top y-X^\top \nabla f(0).
\end{eqnarray}
In general, $X^\top R$ is not necessarily symmetric. To justify the regularization further, one might symmetrize $X^\top R$ by its transpose. In our experiments, we obtained good performance without this.
\begin{figure*}
\centering
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/NN/MNIST_GD_with_markers.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/NN/MNIST_SGD_with_markers.pdf}
\caption{}
\end{subfigure}
\begin{subfigure}{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{ICML_plots/NN/CIFAR_SGD_with_markers.pdf}
\caption{}
\end{subfigure}
\caption{\small{Accelerating neural-network training. (a) A 2-layer neural network with GD optimizer and fixed stepsize $0.0001$. (b) A 2-layer neural network with SGD optimizer and fixed stepsize $0.0001$. For both, we use $k=5$. (c) ResNet18 on CIFAR10 dataset with SGD optimizer and fixed stepsize $0.0001$. We use $k=6$. Note that these are not the best stepsize setting for the networks. Codebase {\tt Pytorch}.
}}\label{fig:neu_net}
\end{figure*}
We call this method DNA-2 (see Alg~\ref{alg:DNA-2}).
Note that $X^\top R+\lambda X^\top X$ can be singular, especially near the optimal solution. To remedy this, we propose either to add another regularization to the problem (\ref{eq:fkrnareg-}), or to consider a direct regularization on $c$ instead of $Xc$. We explain this next.
\subsection{DNA-3}
We set $g(c)=\|c-e\|^2$ in \eqref{mainDNA} and consider a regularized version of \eqref{eq:fkrna} as
\begin{eqnarray}\label{eq:fkrnareg}
\compactify \min_{\substack{c\in\R^{K+1}}} f\left(Xc \right)+\frac{\lambda}{2}\|c-e\|^2,
\end{eqnarray}
where $\lambda>0$ and $e$ is a reference point for $c$. By taking the derivative with respect to $c$ and setting it to 0, we find
$
\compactify{ X^\top\nabla f(Xc)+\lambda (c-e)=0},
$
which after using the approximation \eqref{approximate_grad} becomes
$
\compactify{X^\top(Rc+\nabla f(0))+\lambda X^\top(c-e)=0.}
$
Therefore, $c^\star$ is given as a solution to the linear system:
$
\compactify{(X^\top R+\lambda I)c=\lambda e-X^\top \nabla f(0).}
$
We call this method DNA-3, and describe it in Alg~\ref{alg:DNA-3}.
\begin{algorithm}
\SetAlgoLined
\SetKwInOut{Input}{Input}
\SetKwInOut{Output}{Output}
\SetKwInOut{Init}{Initialize}
\SetKwInOut{Compute}{Compute}
\Input{Sequence of iterates $x_0,\ldots,x_{K+1}$; sequence of step sizes $\alpha_0,\ldots,\alpha_{K}$; regularizer $\lambda>0$; and $e\in\R^{k+1}$\;}
\nl Set $R =\left[\frac{x_0 - x_{1}}{\alpha_0} -\nabla f(0),\ldots,\frac{x_K - x_{K+1}}{\alpha_K} -\nabla f(0) \right]$ and $X = [x_0, \ldots,x_K]$\;
\nl Set $c$ as a solution of the linear system
$(X^\top R+\lambda I) z = \lambda e-X^\top \nabla f(0)$\;
\Output{$x = \sum_{k = 0}^K c_k x_k$.}
\caption{ DNA-3 } \label{alg:DNA-3}
\end{algorithm}
\section{Numerical illustration}
We evaluate our techniques and compare against RNA and GD
by using both synthetic data as well as real-world datasets. Overall, we find that DNA outperforms RNA in most settings by large margins.
{\em Experimental setup.}
Our experimental setup comprises of 3 typical problems, least squares, ridge regression, and logistic regression, for which the optimal solution $x^\star$ is either known or can be evaluated using a numerical solver. We apply the online acceleration scheme in \citep{rna_16} and compare 3 versions of DNA against RNA and GD.
Our results show the difference between the functional values at the extrapolated point and at the optimal solution on a logarithmic scale (the lower the better), as the iterations progress.
The primary objective of our simulations is to show the effectiveness of DNA and its different versions to accelerate a converging, deterministic optimization algorithm. Therefore, we do not report any computation time of the algorithms and we do not claim these implementations are optimized. Note that the computation bottleneck of all algorithms (including RNA) is solving the linear system to calculate $c$, and because the dimensionality of the linear systems is the same in RNA and DNA, the extra cost is the same in both approaches. In our experiments,
we consider a fixed stepsize $\alpha_k=1/L$ for GD, where $L$ is the Lipschitz constant of $\nabla f$.
We note that for DNA-1 and 2 we need to use the stepsize explicitly to construct $R$ as defined in Lem~\ref{lemma:dna1}.
{\em Least Squares.}
We consider a least squares regression problem of the form
\begin{eqnarray}\label{ls}
\compactify \min_x f(x)\stackrel{\text{def}}{=} \frac{1}{2}\|Ax-y\|^2
\end{eqnarray}
where $A\in\mathbb{R}^{m\times n}$ with $m>n$ is the data matrix, $y\in\mathbb{R}^m$ is the response vector. For $m>n$ and ${\rm rank}(A)=n$, the objective function $f$ in \eqref{ls} is strongly convex. The optimal solution $x^\star$ to \eqref{ls} is given by
$
\compactify{x^\star=\argmin_x f(x)=(A^\top A)^{-1}A^\top y.}
$
For least squares we only consider the overdetermined systems, that is, $m>n$.
{\em Ridge Regression.}
The classic ridge regression problem is of the form:
\begin{eqnarray}\label{ridge}
\compactify \min_x f(x)\stackrel{\text{def}}{=} \frac{1}{2}\|Ax-y\|^2+\frac{1}{2n}\|x\|^2,
\end{eqnarray}
where $A\in\mathbb{R}^{m\times n}$ is the data matrix, $y\in\mathbb{R}^n$ is the response vector.~The optimal solution $x^\star$ to \eqref{ridge} is given by
$
x^\star=\argmin_x f(x)=(A^\top A+\tfrac{1}{2n} I)^{-1}A^\top y.
$
{\em Logistic Regression.}
In logistic regression with $\ell_2$ regularization, the objective function $f(x)$ is the summation of $n$ loss function of the form:
\begin{eqnarray}\label{lr}
\compactify f_i(x) = {\rm log}(1 +{\rm exp}(-y_i\langle A(:,i), x \rangle)+\frac{1}{2m}\|x\|^2.
\end{eqnarray}
We use the MATLAB function {\tt fminunc} to numerically obtain the minimizer of $f$ in this case.
{\em Synthetic Data.}
To compare the performance of different methods under different acceleration schemes, we are interested in the case where matrix $A$ has a known singular value distribution and we consider the cases where $A$ has varying condition numbers. We note that the condition number of $A$ is defined as $\kappa(A)\stackrel{\text{def}}{=}\nicefrac{\lambda_{\max}(A)}{\lambda_{\min}(A)}$, where $\lambda$ is the eigenvalue of $A$. We first generate a random matrix and let $U\Sigma V^\top$ be its SVD. Next we create a vector $S\in\R^{\min\{m,n\}}$ with entries $s_i\in\R^+$ arranged in an nonincreasing order such that $s_1$ is maximum and $s_{\min\{m,n\}}$ is minimum. Finally, we form the test matrix $A$ as $A=U{\rm diag}(s_1\;\;s_2\;\cdots s_{\min\{m,n\}})V^\top$ such that $A$ will have a higher condition number if $\nicefrac{s_1}{s_{\min\{m,n\}}}$ is large and smaller condition number if $\nicefrac{s_1}{s_{\min\{m,n\}}}$ is small. We create the vector $y$ as a random vector.
{\em Real Data.} We use 15 different real-world datasets from the {\tt LIBSVM} repository \citep{libsvm}.
We set apart the datasets with $y$-labels as $\{-1,1\}$ for Logistic regression and used the remaining 12 multi-label datasets for least squares and ridge regression problems. We use the matrix $A$ in its crude form, that is, without any scaling/normalizing or centralizing its rows or columns.
{\em Acceleration Results.} We use GD as our baseline algorithm and the online acceleration scheme as explained in \citep{rna_16} for both RNA and DNAs to accelerate the GD iterates.
For synthetic data (see Fig~\ref{fig:synthetic_ol16}), we see that for smaller condition numbers and for quadratic objective functions, DNA-1 and RNA has almost similar performance and DNA-2 and 3 show faster decrease of $f(x_k)-f(x^\star)$, but all of them are very competitive. As the condition number of the problems becomes huge, DNA-2 and 3 outperform RNA by large margins. However, for logistic regression problems we see performance gains for all versions of DNA compared to RNA. Though for huge condition numbers, for logistic regression problems, the performance of DNA-2 depends on the hyperparameter $\lambda$. We argue with experimental evidence as in Fig~\ref{fig:synthetic_ol16} that for huge condition numbers the sensitivity of the performance of DNA-2 is problem specific. We use an additional regularizer $\epsilon \|c\|^2$, where $\epsilon\approx 10^{-14}$ to find a stable solution to \eqref{solution2} of DNA-2.~Next, on real-world datasets in Figs~ \ref{fig:real_ls_ol16} and \ref{fig:realdata_rr_ol16}, we see that all versions of DNA outperform RNA, except in a few cases, where RNA and DNA-1 have almost similar performance. We indicate the oscillating nature of DNA-2 in some plots is due to its problem-specific sensitivity to the regularizer. In Fig~\ref{fig:realdata_lr_ol16}, we find for logistic regression problems on real datasets, DNA outperfoms RNA. We owe the success of DNA on non-quadratic problems to its adaptive gradient approximation. We note that the performance of all algorithms on the offline scheme of \citep{rna_16} are similar to online scheme of \citep{rna_16}. However, on the second online scheme used in \citep{rna_18}, all the algorithms perform extremely poorly. Therefore, we do not report the results in this paper.
{\bf Application to the non-convex world: Accelerating neural network training.}
Modern deep learning requires optimization algorithms to work in a nonconvex setup. Although this is not the main goal of this paper, nevertheless, we implement our acceleration techniques for training neural networks and obtain surprisingly promising results.
We only use DNA-1 for experiments in this section. Tuning the hyperparameter $\lambda$ for the other versions of DNAs requires more time, and we leave this for future research. The {\tt Pytorch} implementation of RNA is based on \citep{rna_18}. Finally, see Fig~7
in Appendix for more results.
{\em MNIST Classification.}
First, we trained a simple two-layer neural network classifier on MNIST dataset \citep{mnist} via GD and accelerate the GD iterates via the online scheme in \citep{rna_16} for both RNA and DNA-1. The two-layer neural network is wildely adopted in most tutorials that use MNIST dataset \footnote{https://github.com/pytorch/examples/blob/master/mnist/main.py}. In Fig~\ref{fig:neu_net} (a), DNA-1 gains acceleration by using GD iterates with a window size $k=5$. However, RNA fails to accelerate the GD iterates. This motivated us to train the same network on MNIST dataset classification \citep{mnist} via SGD as baseline algorithm and accelerate the SGD iterates via the online scheme in \citep{rna_16} for both RNA and DNA-1 (as in Fig~\ref{fig:neu_net} (b)). Again, with window size $k=5$, DNA-1 achieves better acceleration than RNA.
{\em ResNet18 on CIFAR10.}
Finally, we train the ResNet18 network \citep{resnet18} on CIFAR10 dataset \citep{cifar10} by SGD. Each epoch of SGD consists of multiple iterations and each iteration applies to $128$ training samples. The size of the training set is $5\times 10^4$ and the size of validation set is $10^4$. Each sample is a $32\times 32$ resolution color image and they are categorized into 10 classes. We accelerate the SGD iterates via the online scheme in \citep{rna_16} for both RNA and DNA-1. Again DNA-1 outperforms RNA in lowering the generalization error of the network (see Fig~\ref{fig:neu_net} (c)).
\bibliographystyle{plainnat}
{
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2,869,038,154,792 | arxiv | \section{Introduction}
Coupled cluster (CC) calculations~\cite{cizek1966} are considered to be one of the most reliable computational methods in quantum chemistry and thus often used as benchmarks~\cite{bartlett2007}. Their application, however, is strictly limited to single-reference systems, where the CC ansatz of
$ \Psi_{\rm CC} = e^T \Phi_{\rm HF} $
contains a sufficient number of electron configurations to accurately describe the electronic structure, even if the excitation operator $T$ is truncated. Upon the increase of the multi-reference character of the system, that is, upon the decrease of the weight of the Hartree-Fock (HF) orbital~\cite{hartree1935} derived Slater determinant $\Phi_{\rm HF}$ in the ansatz, the reliability of the commonly applied truncated CC methods - such as CCSD, CCSDT or the gold standard CCSD(T)~\cite{raghavachari1989} - becomes increasingly questionable. From a practical point of view, this is a serious problem because several important homogeneous and enzyme catalysts relevant for various applications contain transition metal centers with strong multi-reference character.
To tackle the latter issue, an alternative CC ansatz was proposed based on Kohn-Sham Density Functional Theory (KS-DFT).
The KS-CC formalism, $\Psi_{\rm KS\scalebox{0.75}[1.0]{$-$} CC} = e^T \Phi_{\rm KS}$, differs practically from the original coupled cluster methodology in the reference state $\Phi_{\rm KS} $.
This Slater determinant is constructed from KS-DFT derived auxiliary Kohn-Sham one-electron orbitals~\cite{tong1966} whose physical interpretation is considered somewhat vague~\cite{stowasser1999}.
Nevertheless, during the last two decades, a considerable number of KS-CC studies appeared in the literature \cite{harvey2003,beran2003,olah2009,chen2010,radon2011,vasiliu2015,fang2016,fang2017,feldt2019,bertels2021} which repeatedly argued that the use of KS reference determinants is not only permitted by the CC equations, but it might even lead to improved results due to the decrease of the multi-reference character in some aspects.
To the best of our knowledge, KS-CC calculations were proposed for the first time by Harvey and Aschi in 2003~\cite{harvey2003}.
In their work on the interconversion of iron carbonyl complexes (Fe(CO)$_4$ + CO $\rightleftharpoons$ Fe(CO)$_5$, in various spin states), they initially utilized the usual HF-CCSD(T) methodology and calculated the $T_1$ diagnostic for the studied systems as a test of multi-reference character.
This measure is defined as
\begin{equation}
T_1 = \frac{\| t_1 \|}{\sqrt{n_{\rm corr}}}\,,
\label{t1}
\end{equation}
where $\| t_1 \|$ is the Euclidean norm of the vector of $t_1$ amplitudes, i.e., the coefficients of singly excited determinants in the CC ansatz, and $n_{\rm corr}$ denotes the number of correlated electrons~\cite{lee1989}.
Empirically, $T_1 < 0.02$ is recommended for reliable CCSD(T) results but 0.04-0.06 was obtained for iron carbonyls.
To gain more reliable energies at the same computational cost, the authors also performed CCSD(T) calculations using BP86 Kohn-Sham orbitals~\cite{perdew1986,becke1988}, which resulted in improved, but borderline (0.02-0.025) $T_1$ diagnostics.
In addition, considerable deviations of up to 4 kcal/mol were observed between the respective HF-CCSD(T) and KS-CCSD(T) net reaction energies and the authors suggested the KS referenced results to be more accurate.
Unfortunately, the latter hypothesis could not be tested against experimental energies which are only approximately determined.
Later, the same group also applied KS-CCSD(T) methodology for Heme model iron complexes, in the case of which HF-CCSD(T) calculations turned out to be practically unfeasible due to bad convergence behavior~\cite{olah2009} whereas the use of B3LYP Kohn-Sham orbitals~\cite{lee1988,stephens1994} eliminated the convergence issues.
Furthermore, in the few cases where both KS and HF referenced results are available, it was observed that KS shows decreased singles and doubles CC amplitudes; the electronic energy, however, only negligibly ($< 1$ kcal/mol) altered compared to HF counterpart.
In 2019, as their most recent publication related to the topic, the group extensively studied the effect of different settings in CCSD(T) calculations, including the choice of the reference orbital type~\cite{feldt2019}.
They selected a C-H activation reaction on a non-heme type iron complex, i.e., [Fe(NH$_3$)$_5$O]$^{2+}$ + CH$_4$ $\rightarrow$ [Fe(NH$_3$)$^5$O-H-CH$_3$]$^{2+}$ in various spin states, as model system and found significant, up to 4 kcal/mol deviations between HF-CCSD(T) and KS-CCSD(T) energetics.
Interestingly, the choice of the KS functional (B3LYP, BP86, TPSS~\cite{tao2003} and M06~\cite{zhao2006,zhao2008} were tested) appeared to be practically insignificant, as the different KS-CCSD(T) energies varied within a mere 1 kcal/mol.
Again, the authors suggested the KS reference to be more probably reliable without external benchmarking.
In the meantime, relying on the above works, Shaik~\cite{chen2010} and Pierloot~\cite{radon2011} also published B3LYP referenced coupled cluster studies on various iron complexes.
These papers already assumed that the KS-CCSD(T) approach is equally reliable to HF-CCSD(T), thus, the traditional HF referenced calculations were not even performed.
Moving beyond iron, the Dixon group investigated the compounds of numerous $d$ and $f$ transition metals using both HF-CCSD(T) and KS-CCSD(T).
At first~\cite{vasiliu2015}, the group explored the energy profile of water addition to metal dioxides, i.e., MO$_2$ + H$_2$O $\rightarrow$ M(OH)$_2$O), and found that the change of reference orbitals (Hartree-Fock or PW91 Kohn-Sham~\cite{perdew1992}) results in deviations of up to 5 kcal/mol in the energy level of intermediates and transition states.
Since the $T_1$ diagnostic was considerably smaller with KS reference, the ambiguity between the methods was interpreted as the improvement of CCSD(T) energetics by the inclusion of KS orbitals.
Unfortunately, experimental data cannot be used for the assessment of accuracy in this case, as they are only available for metal oxide surfaces, rather than for separate dioxide molecules.
In their subsequent works~\cite{fang2016,fang2017}, novel computational protocols were developed for the prediction of the heat of formation of transition metal hydrides, oxides, sulfides and chlorides, which treats the electronic energy at CCSD(T) level of theory.
Again, it was observed that the use of PW91 or B3LYP orbitals instead of the usual HF orbitals decreases the $T_1$ diagnostic and also alters the energy (or the derived enthalpy) by up to several kcal/mol.
In this case, although experimental dissociation energies and enthalpies are abundantly available in the literature, most of these come with error bars exceeding the orbital choice related deviation.
The Dixon group still managed to find one example where the B3LYP-KS reference appears to be unequivocally superior to HF, i.e., the enthalpy of the reaction of UCl$_6$ + 3F$_2$ $\rightarrow$ UF$_6$ + 3Cl$_2$, experimentally measured to be 278.0$\pm$1.2 kcal/mol at room temperature, was predicted to be 285.9 kcal/mol and 277.8 kcal/mol by HF-CCSD(T) and KS-CCSD(T) based protocols, respectively~\cite{fang2016}.
Even though KS-CCSD(T) clearly produces better numerical agreement, assessing the quality of the computational models for the single benchmark is still challenging.
In fact, not only an incidental cancellation of numerical errors cannot be ruled out, but thermochemical correction factors of the temperature-dependent enthalpy might also be a source of error.
Altogether, in the above pioneering KS-CC papers, the practicality of the KS based approach is typically argued on the basis of favorable $T_1$ diagnostics.
This is however a questionable argument as the usefulness of $T_1$ as a measure of the quality of CC energy is disputed~\cite{liakos2011}.
In contrast, the Head-Gordon group, for example, compared the performance of HF-CC and KS-CC calculations based on the reproduction of experimental vibrational frequencies \cite{beran2003,bertels2021}.
On a database of diatomic molecules, KS-CCSD(T) was found to outperform HF-CCSD(T) in accuracy by nearly a factor of 5 regardless of the density functional, among which BLYP~\cite{becke1988,lee1988}, B97M-rV~\cite{mardirossian2017}, B97~\cite{becke1997}, $\omega$B97X-V~\cite{mardirossian2014} and $\omega$B97M-V~\cite{mardirossian2016} were tested.
This result was attributed to the smooth change of KS orbitals with nuclear displacement contrasted to the rapid change of HF orbitals.
Thus, the less efficient frequency calculations with HF-CCSD(T) do not necessarily indicate less accurate electronic energies at the optimal ground state nuclear configuration, which is of primary interest in typical coupled cluster studies.
In view of all these results, in order to validate the suspected favorable performance of the truncated KS-CC approaches with unquestionable mathematical rigor high-level numerical reference quantities are needed.
As the truncated CC methods approximate the full configuration interaction (FCI) wave function, the upper limit of their accuracy is the FCI energy for the applied basis set which is invariant to the chosen reference orbitals, see Appendix for details.
Accordingly, the main goal of the present work is to compare the truncated KS-CC and HF-CC results to calculations of essentially FCI quality benchmark results, which - in contrast to the ambiguous experimental data - allows an unequivocal evaluation of accuracy.
As such an analysis is unfeasible for transition metal complexes using today’s classical computers, we selected diatomic systems of both small and large multi-reference character for this purpose.
Nevertheless, we will demonstrate that it is possible to generalize the results to experimentally more relevant larger molecules
The rest of the paper is structured as follows.
In Sect.~\ref{sect:methods}, the computational details are given and we present the selection of diatomic model systems in Sect.~\ref{sect:model}.
We discuss the effect of orbital choice on the difference between FCI and truncated CC energies in Sect.~\ref{sect:energy}.
As an additional evaluation of the quality of the truncated CC results, we also investigate the orbital dependence of similarity between FCI and CC electron densities in Sect.~\ref{sect:density}.
Furthermore, to compare the behavior of HF-CC and KS-CC from the practical points of view, we examine the reference dependence of the wave-function diagnostics in Sect.~\ref{sect:diagnostics}, the expected error of the routinely applied approximations in Sect.~\ref{sect:approximation}, and the propagation of molecular energy errors on concrete examples of chemical reactions in Sect.~\ref{sect:reaction}.
The analysis of larger electronic system is found in Sect.~\ref{sect:transition}.
The conclusion, \ref{sect:conclusion}, is followed by the appendix on the FCI invariance of reference.
\begin{table*}[!t]
\begin{tabular*}{0.99\textwidth}{@{\extracolsep{\fill} } lccccccccc
\hline
\hline
Molecule & BH & HF & CF & BF& NO & OF & BN & C$_2$ & B$_2$\\ \hline
TAE$_{\rm corr}$\% (HF-CCSD) & 97.8 & 95.0 & 89.5 & 89.8 & 89.9 & 88.2 & 83.3 & 84.5 & 75.7 \\
TAE$_{\rm corr}$\%(HF-CCSDT) & 99.8 & 99.7 & 99.5 & 99.3 & 99.0 & 99.1 & 98.0 & 97.8 & 97.0 \\
TAE$_{\rm corr}$\%(HF-CCSD(T)) & 99.8 & 100.1 & 99.8 & 100.1 & 99.6 & 98.5 & 100.3 & 99.6 & 96.9 \\
\hline
\end{tabular*}
\caption{ Diatomic model systems selected for the present study. As measures of multi-reference character, the correlation contribution to CCSD/CBS, CCSDT/CBS and CCSD(T)/CBS atomization energies are provided in terms of the correlation contribution to CCSDTQP6/CBS total atomization energy, according to the W4-17 database~\cite{karton2017}. Consult the Supplementary Material for the details of the calculations.}
\label{tab:mr}
\end{table*}
\section{Computational Details}
\label{sect:methods}
All quantum chemical calculations were carried out using the MRCC program (version 2020-02-22)~\cite{kallay2020}, which was linked to the LIBXC density functional library (version 4.3.4.)~\cite{lehtola2018}. Sample input files are found in the Supplementary Material (SM).
Optimized geometry and multiplicity of the studied molecules was taken from previous theoretical works~\cite{karton2017,feldt2019}.
\subsection{Electronic energies }
For all selected model systems, coupled cluster calculations were performed using STO-3G~\cite{stewart1970}, 6-31G~\cite{hehre1972}, cc-pVDZ and cc-pVTZ~\cite{dunning1989} basis sets, at CCS, CCSD, CCSD(T), CCSDT, CCSDTQ and CCSDTQP level of theory.
Calculations containing even higher order excitations would require unreasonably large computational resources, even for diatomic systems; nevertheless, as pointed out in the next sections, CCSDTQP produces converged energies of practically FCI quality.
The self-consistent field (SCF) calculations generating the reference orbitals were performed with restricted (RHF/RKS) formalism in the case of singlet molecules and with unrestricted (UHF/UKS) formalism in the case of higher multiplicities.
Since we aim to draw general conclusions on the effect of Kohn-Sham orbitals on CC accuracy, we selected various types of density functionals for KS-CC calculations.
Our conception was to pick one functional from each rung of the Jacobs's ladder of KS-DFT methods~\cite{perdew2001}.
Following this logic of hierarchy, we chose the Hartree~\cite{hartree1928}, LDA~\cite{kohn1965}, BP86~\cite{perdew1986,becke1988}, M06-L~\cite{zhao2006,zhao2008} and B3LYP~\cite{lee1988,stephens1994} formulas as representative examples for the 0th, 1st, 2nd, 3rd and 4th rung, respectively.
Double hybrid functionals were omitted from the present study as their SCF procedure, hence the resulting set of orbitals, does not differ from that of conventional hybrid functionals.
However, we added a range-separated hybrid, HSE06~\cite{krukau2006,heyd2003}, as the sixth functional, which although belongs to the 4th rung, shows distinct features from regular hybrids in many aspects.
Note that CC calculations over range-separated hybrid KS orbitals require a slight modification in the MRCC source code which is described in the SM.
The accuracy of the electronic energy can be easily evaluated by comparing it to the (near-FCI) reference value.
Still, it must be kept in mind that CC methods are non-variational, which means that a more accurate energy does not necessarily indicate a more accurate electronic wave function.
Thus, to better establish the role of reference molecular orbitals, we also analyzed their effect on the three-dimensional function of electron density.
\subsection{Electron density}
The unrelaxed electron density~\cite{jensen2006} of a $\Psi_{\rm CC}$ or $\Psi_{\rm KS-CC}$ many-body wave function, expanded in the basis of HF or KS molecular molecular orbitals $\{\varphi(r)\}$, reads
\begin{equation}
\rho (r) = \sum_{\sigma i j} P^\sigma_{ij} \varphi_{i \sigma} (r) \varphi_{j \sigma} (r)
\label{eq:dens}
\end{equation}
with $P$ one-electron reduced density matrix (1RDM) where $i$, $j$ and $\sigma$ denote the spatial orbital and the spin index, respectively.
MRCC provides both the 1RDM matrix and the MO information required for the computation of density according to Eq.~\eqref{eq:dens}, i.e., 1RDM is obtained from the CCDENSITIES output in FCIDUMP format~\cite{knowles1989}, while orbital data are saved as a MOLDEN file~\cite{molden}.
The density was mapped to a finite grid by Multiwfn~\cite{multiwfn}, whose relevant input, i.e., the data of the corresponding natural orbitals~\cite{jensen2006}, was generated by our hand-coded routine.
In order to ensure sufficient numerical accuracy, the density was calculated for a grid which was set fine enough to keep the deviation of the integrated electronic charge from the theoretical value below 0.1\%.
\section{Results and Discussion}
\subsection{Selection of model systems}
\label{sect:model}
As the first step of our investigation, we selected a set of diatomic molecules with versatile degrees of multi-reference character.
The selection was based on the W4-17 dataset where the atomization energy is computed for 200 molecular systems at various HF-CC levels up to sextuple excitations, using complete basis set (CBS) extrapolation~\cite{karton2017}.
In contrast to $T_1$ or other wave-function diagnostics, the accuracy of total atomization energies (TAE) are investigated as an unambiguous indicator of multi-reference effects.
In Ref.~\onlinecite{karton2017}, the commonly applied CCSD, CCSD(T) or CCSDT level TAE results were compared to the TAE for CCSDTQP6 of FCI quality.
The reliability of truncated CC methods can be quantified by the proportion of correlation effects taken into account by the calculation.
The percentage of correlation energy (${\rm TAE_{corr}}\%$) recovered by a particular truncated method $X$ is given by
\begin{equation}
{\rm TAE_{corr}}\% (X) = 100 \frac{ {\rm \Delta TAE( HF\scalebox{0.75}[1.0]{$-$} } X)}{{\rm \Delta TAE( HF\scalebox{0.75}[1.0]{$-$} CC\scalebox{0.75}[1.0]{$-$} SDTQP6})}\,.
\label{TAEcorr}
\end{equation}
Here, we introduced the relative TAE for method HF-CC-$X$ which measures the TAE difference obtained for HF-CC-$X$ and for HF-SCF level, i.e.,
\begin{equation}
{\rm \Delta TAE( HF\scalebox{0.75}[1.0]{$-$}}X)= {\rm TAE( HF\scalebox{0.75}[1.0]{$-$}}X)-{\rm TAE( HF\scalebox{0.75}[1.0]{$-$} SCF})\,.
\end{equation}
Eq.~\eqref{TAEcorr} gives 100\% for methods of FCI quality and 0\% for plain Hartree-Fock SCF.
The typically applied CC methods are expected to give a value between 0\% and 100\%, depending on the CC level and the degree of multi-reference.
Nevertheless, as the implemented coupled cluster equations are not based on the variational principle, values over 100\%, i.e. the overestimation of correlation effects, are also not excluded.
We note that the advantageous feature of TAE$_{\rm corr}$\% is that it quantifies the multi-reference character in a system size independent manner, allowing simple comparisons among different molecules.
As summarized in Tab.~\ref{tab:mr}, nine diatomic systems were chosen for the study (BH, HF, BF, CF, NO, C$_2$, OF, B$_2$, BN), which are all sufficiently small for near-FCI calculations with basis sets of reasonable size.
According to the TAE$_{\rm corr}$\% covered by lower CC levels, the multi-reference character gradually increases from BH to B$_2$, as ordered from left to right in Tab.~\ref{tab:mr}. The order is primarily based on TAE$_{\rm corr}$\% for CCSDT, but it is mostly consistent with the relation of CCSD and CCSD(T) data.
The rationale for the reversed order of NO and OF regarding TAE$_{\rm corr}$\%(CCSDT) is the significantly lower TAE$_{\rm corr}$\%(CCSD) and TAE$_{\rm corr}$\%(CCSD(T)) percentages of the latter.
\subsection{Convergence of HF-CC and KS-CC absolute electronic energies to the FCI limit}
\label{sect:energy}
We began our investigations by computing the electronic energies of the nine molecules in Tab.~\ref{tab:mr} at different HF-CC and KS-CC levels.
In theory, HF-CC and KS-CC converge to the same energy limit, i.e., to the FCI limit for a given molecule and basis set, when gradually increasing the maximal number of excitations, in spite of the fact that considerably different energies might result from HF-CC and KS-CC at the commonly used truncated levels, such as the popular CCSD(T).
A mathematical proof for the invariance of the FCI limit with respect to the choice of reference orbitals is provided in the Appendix.
Importantly, once this universal energy limit is determined, the evaluation of the effect of reference orbitals becomes straightforward as the reference enabling the fastest convergence to FCI, i.e., the reference producing the closest energy to FCI at a given CC level, can be considered the most beneficial.
The main issue with the latter approach is that the routinely applied frozen core and density fitting approximations need to be turned off, otherwise the FCI limit is not independent of the reference in practice, which makes any rigorous comparison between HF-CC and KS-CC absolute electronic energies meaningless.
All-electron coupled cluster calculations without density fitting are, however, very expensive, even for diatomic systems.
Results of FCI quality can only be obtained with relatively small basis sets: we utilized STO-3G, 6-31G and cc-pVDZ at this stage of the research.
In the case of all three bases, we managed to perform calculations up to CCSDTQP level, and the deviation between HF-CCSDTQP and KS-CCSDTQP electronic energies was found to be below 0.05 kcal/mol for all molecules.
Taken this negligibly low value together with the expected negligible contribution of sextuple excitations to the electronic energy (according to the W4-17 dataset, $< 0.1$ kcal/mol change in $TAE$ was observed, even for pathologically multi-reference systems,), it can be stated that CCSDTQP results essentially represent the FCI limit.
The resulting electronic energies are summarized in the Supplementary Material (SM).
As the size of the basis set was found to have little influence on the relation of HF-CC and KS-CC results, we only present the evaluation of cc-pVDZ calculations in the following whereas the discussion of STO-3G and 6-31G results are found in the SM.
The effect of reference orbitals on the convergence of CC/cc-pVDZ energies to the FCI limit is illustrated in Fig.~\ref{fig:energy}.
Each of the seven chosen reference orbital sets (generated using the habitual Hartree-Fock method and the six Kohn-Sham density functionals listed in the Computational Details) is assigned to a color, as depicted by the legend on the very top.
Going from the top black frame to the bottom one of Fig.~\ref{fig:energy}, we present the performance of the increasingly accurate and expensive non-perturbative coupled cluster methods of CCSD, CCSDT, CCSDTQ and CCSDTQP.
In each frame, the horizontal axis shows the diatomic model systems in the order of increasing multi-reference character (see also Tab.~\ref{tab:mr}).
On the top vertical axes, the percentage of recovered correlation energy is depicted, which is defined - analogously to Eq.~\ref{TAEcorr}, in a molecule size independent manner - as
\begin{equation}
E_{corr} \% (X) = 100 \frac{\Delta E(X)}{\Delta E({\rm HF\scalebox{0.75}[1.0]{$-$} CCSDTQP})}
\end{equation}
using $\Delta E(X)=E(X)-E({\rm HF\scalebox{0.75}[1.0]{$-$} SCF})$,
\begin{figure*}[!t]
\includegraphics[width=0.85\textwidth]{./energy_error.png}
\caption{
Effect of the choice of reference orbitals on the accuracy of CC/cc-pVDZ electronic energies, measured against the FCI energy limit (computed at HF-CCSDTQP/cc-pVDZ level in practice).
Results for C$_2$ based on KS-LDA and KS-B686 reference are not presented due to convergence issue.
\label{fig:energy}}
\end{figure*}
where E stands for the electronic energy and $X$ is the computational method to be characterized.
Note that this expression gives 100\% for $X$ = HF-CCSDTQP, and should negligibly deviate from 100\% in the case of any KS-CCSDTQP energy, as any of these are practically equal to the FCI limit.
For sake of brevity, we introduce notation $E_{\rm corr}= \Delta E({\rm HF\scalebox{0.75}[1.0]{$-$} CCSDTQP})$.
It is notable that all electronic energies, including KS-CC results, were analyzed in terms of the dimensionless $E_{\rm corr} \% (X)$ using the HF referenced electronic energies.
In this way, $E(X)$ is the only quantity in the measure which varies with the reference, accordingly it is ensured that a higher $E_{\rm corr} \%$ percentage indicates higher accuracy, i. e. smaller deviation from the near-FCI energy.
The convergence to the FCI energy limit upon increasing the CC level can be observed on the change of the scale of the top vertical axes in Fig.~\ref{fig:energy}. At CCSD level (top frame of Fig.~\ref{fig:energy}) 89-99\% of the correlation energy is recovered, depending on the molecule and the reference orbitals.
The inclusion of higher and higher excitations to the CC ansatz brings the depicted values closer and closer to the FCI limit of 100\%, i.e., CCSDT: 98.7-100.0\%; CCSDTQ: 99.82-100.01\%.
At CCSDTQP level, all values reach 100.0\% with a negligible variation of $\pm$0.02\% in $E_{\rm corr} \%$, regardless of the multi-reference character of the molecule and the choice of reference orbitals.
The bottom vertical axes represent the alteration in the recovered correlation energy percentage upon changing the usual HF reference to KS orbitals, which we defined as
\begin{eqnarray}
\Delta E_{\rm corr} \% (({\rm KS\scalebox{0.75}[1.0]{$-$} CC}X) &=& |100-E_{\rm corr} \% ({\rm HF\scalebox{0.75}[1.0]{$-$} CC}X)| \\
&&- |100-E_{\rm corr}\% ({\rm KS\scalebox{0.75}[1.0]{$-$} CC}X)|\,.\nonumber
\label{eq:deltaE}
\end{eqnarray}
\begin{figure*}[!ht]
\includegraphics[width=0.85\textwidth]{./ccsd_pert_energy.png}
\caption{
Effect of the choice of reference orbitals on the accuracy of CCSD(T)/cc-pVDZ electronic energies, measured against the FCI energy limit (computed at HF-CCSDTQP/cc-pVDZ level in practice).
Results for C$_2$ based on KS-LDA and KS-B686 reference as well as for CF, NO, OF based on Hartree reference are not presented due to convergence issue.
\label{fig:energy_pert}}
\end{figure*}
A positive $\Delta E_{\rm corr} \%$ indicates the improvement effect of the given KS reference, i.e., $E_{\rm corr}\%({\rm KS\scalebox{0.75}[1.0]{$-$} CC}X)$ approximate more accurately the FCI limit of 100\%, while a negative value means deterioration relative to the HF-CC$X$ result.
We note that the absolute values in Eq.~\eqref{eq:deltaE} are introduced due to the non-variational nature of the applied coupled cluster implementation, which might lead not only to the underestimation, but also to the overestimation of correlation energy ($E_{\rm corr} \%>100\%$) at truncated CC levels.
Benchmarks are summarized in Fig.~\ref{fig:energy} which clearly shows that the change of the habitual HF reference (black bars) to KS (colored bars) yields clear deterioration relative to the FCI limit ($\Delta E_{\rm corr} \%<0$) for the commonly used functionals.
In contrast to this, the most basic Hartree formula (pink bars) provides an opposite trend for some of the studied molecules and it might facilitate the convergence to FCI ($\Delta E_{\rm corr} \%>0$).
The interesting tendency might be interpreted as the consequence of the intimate relation of the Hartree and the Hartree-Fock functionals which is also witnessed at low orders in the M{\o}ller-Plesset perturbation theory which is closely connected to the CC expansion~\cite{jensen2006}.
In particular, at CCSD level (top black frame of Fig.~\ref{fig:energy}), the use of any of LDA, BP86, M06-L, B3LYP and HSE06 references consistently decreases the accuracy of the electronic energy.
In the following figures molecules are arranged according to their multi-reference character given in Tab.~\ref{tab:mr}.
The deviation from HF-CCSD in the unfavorable direction ranges from a minimal change of 0.0-0.1\% in terms of correlation energy (BH, HF) to up to 0.5\% in moderately multi-reference cases, and up to 1\% in the case of the four highly multi-reference systems.
Apart from the least correlated BH and HF molecules, the KS-CCSD energies - with the exception of those based on Hartree orbitals - are typically more similar to one another than to the HF-CCSD results, i.e., the difference between the highest and lowest $\Delta E_{\rm corr} \%$ values is generally below 0.2\% and even the highest variations, observed at exceptionally high multi-reference character, BN, B$_2$, do not exceed 0.35\%.
Note that results for C$_2$ based on KS-LDA and KS-B686 reference are not presented due to convergence issue.
Even though this result is in complete agreement with previous findings~\cite{feldt2019}, it is still a remarkable outcome considering the great variety of density functionals from local density approximation to hybrid formulas.
We also note that the order of functionals in KS-CC accuracy somewhat reflects the Jacob's ladder of DFT, i.e., LDA from the 1st rung (blue color) and HSE06 from the 4th rung (red color) appear to be the least and most favorable functionals, respectively.
Nevertheless, even the most complex functional produces a less beneficial reference than the HF method.
In the unique, deviant case of the Hartree functional (pink color), for most of the investigated systems, Hartree reference provides lower KS-CCSD energy than the HF-CCSD, i.e., 5 molecules out of 9 give positive $\Delta E_{\rm corr} \%$, among which the values of 1.9\% (C$_2$), 1.3\% (B$_2$) and 0.9\% (NO) are the most significant.
On the other hand, the improvement is not systematic as Hartree orbitals were identified to be even the least accurate in the case of BF and three other negative $\Delta E_{\rm corr} \%$ values were also found.
Although the inclusion of triple excitations (CCSDT/cc-pVDZ frame of Fig.~\ref{fig:energy}) considerably decreases the variation of electronic energies as expected upon approaching the FCI limit, the above described trends are sustained.
KS-CCSDT results, which remain practically independent of the functional choice except for Hartree, underperform HF-CCSDT by $0.1-0.3\%$ of the correlation energy, going from low to high multi-reference character.
In particular, for BH and HF, $\Delta E_{\rm corr}\%$ decreases practically to zero, indicating that CCSDT is of near-FCI quality due to the low extent of correlation.
The beneficial effect of the Hartree functional also appears at CCSDT level for the previously highlighted molecules of C$_2$, B$_2$ and NO ($\Delta E_{\rm corr}\% = +0.06-0.24\%$).
Furthermore, the electronic energy of BH and HF is also improved by Hartree at some extent ($\Delta E_{\rm corr}\% \leq +0.05\%$).
Taking another step further to CCSDTQ level, the first four molecules of BH, HF, CF and BF are found to reach the convergence of electronic energy, regardless of the reference choice.
The deviation of KS-CCSDTQ from HF-CCSDTQ is barely detectable ($|\Delta E_{\rm corr}\% |\leq 0.01\%$) with the sole exception of the Hartree reference in the case of BN.
For NO and the four highest multi-reference cases, the previously noted trends apply, i.e., HF-CCSDTQ provides the closest approximation of the FCI limit, slightly outperforming KS-CCSDTQ by 0.01-0.1\% of $E_{\rm corr}$, with the size of deviation depending on the degree of multi-reference character.
The effect of orbital choice within the Kohn-Sham framework has negligible influence as only the Hartree functional stands out with some improvements up to $\Delta E_{\rm corr}\% =+0.05\%$ (C$_2$, B$_2$, NO) or, on the contrary, with exceptional deterioration for BN.
Altogether, it can be concluded that none of the standard KS references is suitable for taking into account the correlation energy part neglected by HF-CC.
What is more, KS orbitals actually found to slightly decelerate the convergence to the FCI limit.
A systematic deterioration of electronic energy was observed compared to HF-CC, which increases with degree of multi-reference character and amounts up to 1\%, 0.3\% and 0.1\% of $E_{\rm corr}\%$ at CCSD, CCSDT and CCSDTQ level, respectively.
It should not escape our attention that, given that these proportions also apply to larger molecules such as transition metal complexes, the reference dependent deviation might easily reach multiple kcal/mol units for practically relevant chemical reactions.
As the next step, we extended the previously established analyses for the most commonly applied CCSD(T) method presented in Fig.~\ref{fig:energy_pert} where the recovered correlation energy percentages are quite similar to those at CCSDT level.
$\Delta E_{\rm corr}\%$ values, however, range down to -1\%, which better recalls the CCSD results.
The behavior of high multi-reference systems closely follows the patterns of Fig.~\ref{fig:energy}, i.e., the change from HF to any KS reference results in the consistent deterioration of the electronic energy, the extent of which has little dependence on the density functional.
The case of low multi-reference character is slightly more complex, as certain systems show distinct CCSD(T) results, i.e., positive $\Delta E_{\rm corr}\%$ values appear with non-Hartree references, ranging up to +0.1\% (for BH and BF).
Still, the practical significance of this magnitude of improvement is debatable, as it seems to be rather incidental and it is peculiar to two of the least correlated systems with the lowest absolute correlation energy.
As for the Hartree functional based perturbative results, the calculations are found to be rather unstable, i.e., convergence issues for CF, NO, OF, C$_2$, even the converged energies deviate significantly from the HF-CCSD(T) value.
In summary, according our numerical exploration, the use of typical KS references in CCSD(T) generally provides energy comparable to the HF counterpart.
\subsection{Convergence of HF-CC and KS-CC electron densities to the FCI limit}
\label{sect:density}
\begin{figure*}[!t]
\includegraphics[width=0.85\textwidth]{./density.png}
\caption{
Effect of the choice of reference orbitals on the accuracy of CC/cc-pVDZ electron densities.
A value of 0 in the relative error corresponds to the full similarity to the FCI density (calculated at HF-CCSDTQP/cc-pVDZ in practice), whereas the value of 1 represents the size of error in HF/cc-pVDZ density, see Eq.~\ref{eq:rho_norm}-\ref{eq:rho_d} in the main text for detailed mathematical definitions.
Results for C$_2$ based on KS-LDA and KS-B686 reference are not presented due to convergence issue.
\label{fig:density}}
\end{figure*}
\begin{figure*}[!t]
\includegraphics[width=0.85\textwidth]{./Tdiag.png}
\caption{
Effect of the choice of reference orbitals on the coupled cluster wave-function diagnostics at CCSD(T)/cc-pVDZ level. (a) $T_1$ diagnostic; (b) largest double amplitude, max$(|t_2|)$. Results for C$_2$ based on KS-LDA and KS-B686 reference are not presented due to convergence issue.
\label{fig:diagnostic}}
\end{figure*}
It is notable that the applied CC approach is not a variational approach hence lower CC energy does not necessarily indicate a more accurate description of the problem. Therefore, in the following, we attempt to assess the accuracy of the computations in an alternative way, i.e., we benchmark the precision of the CC calculations in terms of the associated electron density instead of the numerically demanding direct comparison of the corresponding many-body wave functions.
The application of this analysis is also motivated by getting physical insight on the effect of using various correlation-exchange functionals as we not only monitor densities obtained for the coupled cluster excitation levels but for Kohn-Sham theory as well.
In practice, densities to be compared are evaluated on the same finite grid. In general, the similarity of two vectors, such as electron distributions $\rho$ and $\rho'$, can be quantified by various measures, e.g., cosine similarity, relative entropies, Minkowski distances.
In this paper, we apply the standard Euclidean distance, i.e., the norm of the difference of the vectors, defined as
\begin{equation}
\| \rho - \rho' \| = \sqrt{\sum ( \rho_i - \rho_i' )^2}
\label{eq:rho_norm}
\end{equation}
where index $i$ runs over the grid.
Similarly to the energy based analysis presented in the \ref{sect:energy}, HF-CCSDTQP level density result serves as reference of practically FCI quality in the following CC density benchmarks.
Accordingly,
\begin{equation}
D(X) = \| \rho(X) - \rho({\rm HF\scalebox{0.75}[1.0]{$-$} CCSDTQP}) \|
\label{eq:rho_D}
\end{equation}
measures the deviation of density obtained for $X$ computational model from the reference result. The relative error of method $X$ is defined as
\begin{equation}
d(X) = \frac{D(X)}{D({\rm HF\scalebox{0.75}[1.0]{$-$} SCF})}
\label{eq:rho_d}
\end{equation}
which measures the error in units of $D$(HF-SCF).
By definition, $d(X)$ is exactly 1 and 0 for $X$=HF-SCF and $X$=HF-CCSDTQP, respectively.
For general $X$ cases, where the corresponding wave function recovers a significant portion of the correlation effects, $d(X)$ tends towards 0.
The numerical data of $d(X)$ is presented in Fig.~\ref{fig:density} for various excitation levels.
Comparing SCF level error measures, we find that the high-level Kohn-Sham theories provide typically the most accurate single-determinant approximation, i.e., the corresponding errors are a small fraction of the HF-SCF error.
Interestingly, molecules of intermediate multi-reference character are found to be represented the most faithfully by the high-level KS-SCF theories.
Also in agreement with the expectations, the error of the solution Hartree-SCF, which neglects any exchange and correlation effects, is outstandingly large, i.e., it is typically multiple of $D$(HF-SCF).
For increasing CC excitation levels, the tendencies of electron-density errors are generally in line with the conclusions of the correlation-energy analysis detailed in Sect.~\ref{sect:energy}.
Most importantly, the lowest deviations are typically found for CC results of Hartree and Hartree-Fock references, i.e. the addition of computationally rather cheap CCSD excitations already compensate their limitations found at SCF level of theory.
Similarly to the observations found for CC energies, the deviation of electron-density error is quite even for the various KS functionals compared to tendencies found for the aforementioned correlation-free references.
In line with expectations corroborated by the findings for molecular energy, low-level CC theory also shows an overall tendency to provide more accurate electron density for single-reference molecules.
Also remarkable that low-level CC theory applied on high-level functionals (B3LYP and HS06) can be even counterproductive for the moderately correlated problems, i.e, their extraordinarily accurate SCF electron density is slightly deteriorated at KS-CCSD level of theory.
For CCSDTQP level of theory, the error of the wave functions is already found to be practically zero confirming the convergence of the high-level CC calculations to the exact solution.
\subsection{The relation of $T_1$ and max($|t_2|$) diagnostics to the accuracy of electronic energy}
\label{sect:diagnostics}
The $T_1$ diagnostic of Eq.~\eqref{t1} and the $\max(|t_2|$) value, i.e., the largest absolute value among doubles amplitudes, are commonly applied in the literature to check the reliability of coupled cluster calculations.
Thus, we were naturally interested in how the reference choice affects these wave-function diagnostics and whether the accuracy trends described in the previous sections (especially the outstanding performance of HF-CC) are reflected in $T_1$ and $\max(|t_2|)$.
Fig.~\ref{fig:diagnostic}a shows the $T_1$ diagnostic for the diatomic model systems at CCSD(T)/cc-pVDZ level of theory.
The magnitude of singles CC amplitudes has remarkable reference dependence, i.e, in most cases, $T_1$ value for HF based result are drastically larger relative to KS counterparts.
We find that all types of non-Hartree density functionals typically decrease the $T_1$ diagnostic except for a minor number of increases with LDA and BP86 functionals in the case of the least multi-reference BH, HF and BF systems.
Furthermore, with the exception of BN and B$_2$, the decreasing order of $T_1$ values from LDA to HSE06 fairly corresponds to the Jacob's ladder hierarchy.
The most conspicuous example is probably OF, where the HF-based $T_1$ diagnostic of 0.04 is already reduced to 0.026 when using a simple LDA functional, and is decreased to as low as 0.007 with HSE06.
These findings are in line with the previous KS-CC studies reporting decreased $T_1$ values relative to HF-CC~\cite{harvey2003,vasiliu2015,fang2016}.
Furthermore, we also observe an overall trend of increased $T_1$ values for the more correlated problems in the case of the common references.
In contrast to this trend, in the case of CC solutions based on the Hartree reference, which completely neglects quantum exchange and correlation effects, the diagnostic produces exceedingly large value, i.e., 0.11-0.15, regardless of the particular molecule.
The tendencies found for $T_1$ might be interpreted as the footprint of orbital relaxation effect induced by single excitations, i.e., larger $T_1$ is detected for the less accurate single-determinant approximations of Hartree-Fock and -in particular- Hartree theories.
This argument is corroborated by our systematic numerical investigation of energy and density, see Sect.~\ref{sect:energy} and \ref{sect:density}, which also shows that lower $T_1$ values do not indicate more accurate approximation of the molecule.
The reference dependence of the other widely used wave-function diagnostic, $\max(|t_2|)$, is plotted in Fig.~\ref{fig:diagnostic}b.
In contrast to $T_1$ diagnostic, $\max(|t_2|)$ is found to be practically independent of the reference, with a variation below 0.01 in most cases.
Even the Hartree functional, which otherwise always shows distinctive behavior, fits the rest of the data points, though it slightly stands out at high multi-reference character.
It is also notable that $\max(|t_2|)$ systematically increases for problems of strongly correlated character, see the outstandingly $\max(|t_2|)$ values of BN, C$_2$ and B$_2$.
To conclude on the use of CC-amplitude based diagnostics, according to our numerical exploration, $\max(|t_2|)$ - rather than $T_1$ - could be applied to estimate the degree of multi-reference character, hence the reliability of lower-level truncated CC calculations.
Nevertheless, we found no obvious relation between the reference dependent deviations in the diagnostic value and the relative accuracy of HF-CC and different KS-CC methods reflected by energy and electron density.
\begin{figure*}[!t]
\includegraphics[width=0.85\textwidth]{./frozencore.png}
\caption{
Effect of the choice of reference orbitals on the size of error in absolute electronic energy deriving from frozen core and density fitting approximations, computed at CCSDTQP/cc-pVDZ level. The error is given in terms of percentage of correlation energy. Results for C$_2$ based on KS-LDA and KS-B686 reference are not presented due to convergence issue.
\label{fig:frozencore}}
\end{figure*}
\subsection{Effects of frozen core and density fitting approximations }
\label{sect:approximation}
Until this point, we discussed coupled cluster calculations without any additional approximations, in order to focus purely on the effects of reference choice and exclude any other factors.
In practice, however, frozen core approximation (FCA) and density fitting (DF) are commonly applied to speed up calculations, which introduce additional sources of error in the electronic energy.
In FCA the core orbitals are kept doubly occupied in the CC calculations, thus the number of active orbitals can be significantly reduced for heavy elements by the negligible loss of correlation effects of the low-lying core electrons.
DF is used to provide computationally cheap approximation for numerical integrals, correspondingly, we found that the error introduced by DF is detectable but rather marginal compared to the chemical accuracy scale.
To investigate the magnitude and the reference dependence of these typical error sources, we repeated the energy calculations described in \ref{sect:energy} applying both FCA and DF.
The obtained CC electronic energies are found in the Supplementary Material.
As already discussed in \ref{sect:energy}, the use of FCA and DF breaks down the theoretically established reference invariance of the FCI energy; thus, we firstly examined how the reference affects the approximated CCSDTQP (practically FCI) level energy.
In Fig.~\ref{fig:frozencore}, we plotted the deviation of these energies from those computed at non-approximated CCSDTQP level, expressing it in terms of the percentage of the exact correlation energy, $E_{\rm corr}$.
Fig.~\ref{fig:frozencore} clearly shows that the Hartree-Fock reference comes with the smallest error, which amounts approximately 1\% of the correlation energy for all molecules.
As for Kohn-Sham referenced methods, the error of approximations consistently follows the order of M06-L $<$ HSE06 $<$ B3LYP $<$ BP86 $<$ LDA $<$ Hartree.
The first four functionals in the latter order, which produce errors between 1\% and 2\% of $E_{\rm corr}$, show very similar behavior, in the sense that their approximated FCI limits are closer to each other than to that of HF-CC.
In the case of LDA, the use of frozen core and density fitting causes 2\% and 4\% error.
Most notably, due to the complete neglect of exchange processes of core electrons, 10-30\% portion of $E_{\rm corr}$ is lost with Hartree references which also implies that the application of Hartree orbitals is severely limited for small electronic systems.
Of course, the magnitude of the error is a function of the CC level, but we find that the truncation of the CC ansatz leaves the results shown in Fig.~\ref{fig:frozencore} practically unchanged, see SM for the error of approximations with CCSD, CCSD(T) and CCSDT.
That is, the same amount of electronic energy (as given in Fig.~\ref{fig:frozencore}) is neglected by frozen core and density fitting at all CC levels, implying that the slower convergence of KS-CC methods to the FCI limit, as established in \ref{sect:energy}, also holds true for approximated CC.
This tendency also reveals that the contribution of the cores is rather marginal in the overall correlation effects validating the applicability of the FCA.
Altogether, our results indicate that the HF-CC methods not only converge faster to its FCI, but also have more accurate FCI limit if the common approximations are taken into account.
The consistent performance of the HF reference is rather natural as these orbitals are optimized for describing exchange effects which can be also captured in the FCA at SCF level of theory.
\subsection{Implications on the accuracy of reaction energies: error propagation}
\label{sect:reaction}
\begin{figure*}[!t]
\includegraphics[width=0.85\textwidth]{./reaction_error.png}
\caption{
Effect of the choice of reference orbitals on the error in CCSD(T)/cc-pVDZ reaction energies (kcal/mol), determined against non-approximated CCSDTQP/cc-pVDZ (practically FCI/cc-pVDZ) data.
Two hypothetical chemical reactions among the studied diatomic model systems ((a) 2CF + 2BH $\rightarrow$ C$_2$ + B$_2$ + 2HF; (b) BN + OF + BH $\rightarrow$ B$_2$ + HF + NO) are presented as representative examples.
The left-side (dark colored) bar and the right-side (light colored) bar belonging to each reference orbital set show the error of the non-approximated calculation and that of the approximated calculation (with frozen core (FC) and density fitting (DF)), respectively.
Tables below discuss the origin of reaction energy errors at the level of molecular CC electronic energies: provided are the standard deviation of individual molecular errors, i.e., differences between molecular CCSD(T) and FCI electronic energies in kcal/mol, and the average of the percentages of recovered molecular correlation energy.
Missing data is not presented due to convergence issue of molecular calculations.
Graph (a) shows the generally expected behavior, i.e., larger deviation of molecular errors for typical KS-CC yields larger reaction energy error.
Graph (b) illustrates the case of an accidental error cancellation, i.e., smaller reaction energy errors in spite of the slightly larger variety of molecular errors for KS-CC.
\label{fig:reaction}}
\end{figure*}
As we have seen in the previous sections, the use of Hartree-Fock reference orbitals in truncated CC calculations is expected to lead to the closest approximation of the FCI limit of the molecular electronic energy $E$.
In quantum chemical studies, however, we are typically interested in the accuracy of reaction energies, which are calculated as the difference of the algebraic sum of the individual $E$ energy of the product molecules and the reactants.
Although it could be argued at first sight that the substitution of the most reliable HF-CC energies necessarily provides the most accurate results, this seemingly reasonable statement is not strictly true.
Note that the variation – rather than the magnitude – of molecular errors determines the accuracy of the reaction energy.
As KS-CC energies are spread in a wider range compared to HF-CC results, the chance for less favorable reaction energy error for HF-CC is expected to be relatively low.
Nevertheless, it is possible that a fortuitous cancellation of molecular errors results in the outperformance of KS-CC compared to HF-CC.
To demonstrate the above reasoning on the example of the popular CCSD(T) method, we presented the accuracy of CCSD(T)/cc-pVDZ reaction energies of two hypothetical reactions among the diatomic model systems.
Fig.~\ref{fig:reaction}a shows the reference dependence of CCSD(T) error in kcal/mol for the reaction of 2CF + 2BH $\rightarrow$ C$_2$ + B$_2$ + 2HF, which follows the expected pattern, i.e., KS-CC produces approximately 1 kcal/mol higher errors than HF-CC, regardless of the functional applied.
Frozen core and density fitting approximations (see light-colored bars for each reference) slightly enhance the error with all references but have negligible influence on the relation of accuracies.
As shown by the table below the bar graph, the enhanced uncertainty of KS-CC can be traced back to the larger variation of molecular CCSD(T) errors.
Namely, upon replacing the HF orbitals to KS reference, the standard deviation of errors increases from 0.65 to 0.8 and from 1.1 to 1.3-1.4 in the case of all-electron and frozen-core calculations, respectively.
We also note that the standard deviation correlates with the overall accuracy of molecular electronic energies (represented by the average of percentages of recovered molecular correlation energy $E_{\rm corr}\%$ in the table): as suggested above, the closer the energies are located the FCI limit (100\%), the smaller the deviation is.
The only ambiguous result in Fig.~\ref{fig:reaction}a is the small error of the Hartree functional with FC and DF approximations ($\sim$0.5 kcal/mol).
As this method is unequivocally inaccurate, i.e., only 78.8\% of $E_{\rm corr}$ is taken into account by CCSD(T), and its performance also varies drastically among molecules (standard deviation: 9.8 kcal/mol), it is clear that a case of fortunate error cancellation was found.
While the majority of chemical reactions involving the nine diatomic species shows similar behavior to Fig.~\ref{fig:reaction}a, consult SM for further examples, some exceptions are also identified which understood by fortuitous error cancellation.
Fig.~\ref{fig:reaction}b shows the reaction energy of BN + OF + BH $\rightarrow$ B$_2$ + HF + NO, where most KS-CCs reaction energy errors halves the HF-CC counterpart.
The low KS-CC reaction energy errors are unexpected regarding both the decreased average $E_{\rm corr}\%$ values and the slightly enhanced standard deviations of molecular errors compared to HF-CC.
We also note the outstandingly high reaction energy error (4.4 kcal/mol) of the Hartree functional, which is in this case fully consistent with the enormous variation of individual errors.
The high consistency of KS-CC results is to be investigated in a subsequent work.
To summarize the above analysis, higher precision in calculating the molecular electronic energies does not guarantee superior performance in predicting reaction energies.
Nevertheless, the tested reactions confirm the expectations in general, i.e., we find that the reaction energy is tendentiously more accurate for HF-CC relative to the analogous KS-CC results as the consequence of its lower variation of errors in molecular energies.
It is also notable that incidental outperformance of KS-CC owing to error cancellations can be observed in some cases.
\begin{figure}[!t]
\includegraphics[width=0.48\textwidth]{./transition.png}
\caption{
Analysis of the triplet and quintet states of the helium model of [FeHe$_5$O]$^{2+}$:
a) Energy gap computed at truncated CC models with FCA+DF.
b) Error analysis of the gap.
c) Wave-function diagnostics using $T_1$ and max($|t_2|$) measures.
\label{fig:transition}}
\end{figure}
\subsection{Can the results be generalized to larger systems? – Example of a practically relevant transition metal complex}
\label{sect:transition}
Given our results obtained for diatomic model systems, it is naturally of interest whether similar trends are also valid for larger molecules including transition metal (TM) complexes.
Thus, we searched for a practically relevant complex, which is however small enough to perform calculations of closely FCI quality.
After deliberation and performing extensive tests on the limits of computational performance, we selected the helium model of Harvey~\cite{feldt2019}.
This tetragonal bipyramidal complex, structured as [FeHe$_5$O]$^{2+}$, is constructed in a way to resemble the spin density of C-H bond activating iron-oxo catalysts.
Herein, we examine the effect of reference orbital choice on the triplet-quintet gap of [FeHe$_5$O]$^{2+}$.
Considering basis sets of reasonable size, such as cc-pVDZ, all-electron CCSDTQP calculations, used as near-FCI reference for diatomic molecules, would be unfeasible even for such a small TM complex.
Nevertheless, it is possible to calculate the electronic energy of triplet and quintet states at FCA-CCSDT(Q)/cc-pVDZ level with frozen core and density fitting approximations.
Furthermore, corresponding to findings of Sect.~\ref{sect:approximation}, we attempted to approximate all-electron CCSDT(Q)/cc-pVDZ level theory
as
\begin{equation}
E{\rm(CCSDT(Q)} \approx E{\rm(FCA\scalebox{0.75}[1.0]{$-$} CCSDT(Q)} + \Delta E_{\rm FCA}\,,
\label{eq:approx}
\end{equation}
i.e., the error of FCA and DF can be already estimated at low CC levels, such as $\Delta E_{\rm FCA}=E$(FCA-CCCSD(T))$-E$(CCSD(T)).
Finding that the approximated energy of Eq.~\ref{eq:approx} is nearly independent of the orbital choice for the investigated TM model, the numerical result can be considered of near-FCI quality.
More specifically, the difference between HF and B3LYP referenced triplet-quintet gaps obtained in this manner ($E_{\rm gap}{\rm (HF)}= 58.8$ kcal/mol and $E_{\rm gap}{\rm (KS\scalebox{0.75}[1.0]{$-$} B3LYP)}= 57.3$ kcal/mol, respectively) is below 1.5 kcal/mol, which approaches the requirement of chemical accuracy and also indicates close convergence to the FCI limit.
We note that only the popular B3LYP functional was tested here among the KS-CC methods due to the significant computational cost. Additionally, the computed gaps deviate from the values presented in ref.~\onlinecite{feldt2019} due to the application of different basis sets.
Fig.~\ref{fig:transition}a visualizes the effect of orbital choice on the triplet-quintet gap at truncated levels of CCSD, CCSD(T), CCSDT and CCSDT(Q) with FCA+DF approximation.
Though the exact value of the FCI limit is uncertain (due to the 1.5 kcal/mol difference of the two aforementioned estimations shown as red dashed lines), the performance of HF-CC seems superior to B3LYP-CC according to the following analysis.
In Fig.~\ref{fig:transition}b the error of the computational models, i.e., black bars for HF-CC and orange bars for B3LYP-CC, is estimated relative to both $E_{\rm gap}{\rm (HF)}$ and $E_{\rm gap}{\rm (KS\scalebox{0.75}[1.0]{$-$} B3LYP)}$ reference values in left and right bars, respectively.
It is notable that lower errors are obtained for HF-CC approach regardless of FCI estimation.
The deterioration of the results by the B3LYP reference is attributed partly to the slower convergence of B3LYP-CC electronic energies to the FCI limit and partly to the larger frozen-core error of B3LYP-CC (see SM for details), which is in complete agreement with our observations on smaller diatomic systems.
Moreover, as summarized in Fig.~\ref{fig:transition}c, the reference dependence of wave-function diagnostics follows the trends described in Sect.~\ref{sect:diagnostics}, i.e., the $T_1$ diagnostic for B3LYP-CCSD(T) is considerably decreased compared to HF-CCSD(T), while max($|t_2|$) remains essentially constant.
Altogether, these findings suggest that the conclusions drawn for the nine diatomic molecules are generally applicable for KS-CC.
\section{Conclusion}
\label{sect:conclusion}
The present computational study investigated the effect of orbital choice on the accuracy of coupled cluster calculations compared to reference calculations of FCI quality for the first time to the best of our knowledge.
According to our numerical exploration, the following points should be taken into consideration in future CC studies involving Kohn-Sham reference orbitals.
\begin{enumerate}
\item KS-CC calculations, regardless of the exchange-correlation density functional used for the generation of reference orbitals, produce less accurate molecular electronic energies and densities (relative to the FCI result) than the habitual HF-CC approach with equal CC level and basis set.
Although HF-CC and KS-CC all-electron methods converge to the same FCI limit upon increasing the CC level, this convergence occurs somewhat slower with KS common references.
What is more, the error of the routinely applied frozen core and density fitting approximations is larger for KS-CC, which further enhances the difference from HF-CC in accuracy, in favor of the latter methodology.
\item The $T_1$ diagnostic of KS-CC wave functions of typical functionals is considerably lower than that of the corresponding HF-CC wave functions in line with literature findings.
Nevertheless, this seemingly favorable decrease in $T_1$ is neither corroborated by the improvement of correlation energy, nor by the accuracy of the corresponding density.
On the other hand, the largest $t_2$ amplitudes seem as a reference-free indicator of multi-reference character.
\item In spite of the higher errors in individual molecular electronic energies, KS-CC might still outperform HF-CC in the accuracy of reaction energies corroborating previous observations as a result of cancellation of errors.
However, the better performance of HF-CC in absolute energy indicate that HF-CC is expected to provide more reliable energy differences in general.
\end{enumerate}
Although the application of KS-CC might seem unfavorable considering the above points, we emphasize that this methodology is a very useful alternative of HF-CC in case convergence issues encountered in HF-SCF or HF-CC calculations and also because our calculated KS-CC reference energies does not indicate significant differences in absolute energy.
Additionally, at sufficiently high CC level, determined by the multi-reference character of the studied molecule, the difference between HF-CC and KS-CC results disappears in the calculations and both can be reliably applied.
Furthermore, the relatively small variation of errors using truncated KS-CC with different references of the Jacob's ladder suggest that already computationally cheap functionals, e.g., BP86, might be considered competitive.
Finally, our results also imply that alternative post-HF methods approaching the FCI limit become less sensitive to the underlying orbital set. Frozen core approximation introduces some additional error in KS-based molecular energies but it is expected to cause a diminishing effect for light elements in estimating vertical electronic excitations.
\section*{Acknowledgements}
Discussions with \'Ad\'am Gali and P\'eter R. Nagy as well as support from the National Research, Development and Innovation Office (NKFIH FK-20-135496) are greatly acknowledged.
P.T. and G.B. thank the support from the Wigner Student Scholarship of the Wigner Research Centre for Physics and the Bolyai Research Scholarship of the Hungarian Academy of Sciences, respectively.
We acknowledge KIF\"U for awarding us access to computational resources based in Hungary.
Z.B., T.S., and G.B. would like to thank the University of Alabama and the Office of Information Technology for providing high performance computing resources and support that have contributed to these research results.
This work was made possible in part by a grant of high-performance computing resources and technical support from the Alabama Supercomputer Authority.
Dedicated to the loving memory of professors G\'eza Tichy, J\'anos Pipek and Gyula Radnay.
\section*{Appendix}
In the following we give an illustrative mathematical derivation for the invariance of the FCI solution with respect to the choice of reference orbitals.
Let us consider two different sets of reference orbitals, e.g., a set of Hartree-Fock and a set of Kohn-Sham molecular orbitals which will be referred to as $\{\varphi'\}$ and $\{\varphi''\}$ in the following, respectively.
Considering that analogous formula are derived for both orbital sets, for sake of compactness, double notation is introduced in the following derivations, e.g., $\varphi'^{(\prime)}$ denotes both $\varphi'$ and $\varphi''$ simultaneously.
The basis consisting of $m$ basis functions $\{\chi\}$ is used to expand the molecular orbitals as
\begin{equation}
\varphi_i'^{(\prime)} = \sum_{j=1}^m c_{ij}'^{(\prime)} \chi_j\,.
\label{eq:MO}
\end{equation}
Note that the HF and KS procedure creates $m$ molecular orbitals (MOs) from the $m$ atomic basis functions.
However, constructing a Slater determinant $\Phi'^{(\prime)}$, only $n$ MOs, corresponding to the number of electrons, are selected out of $m$ MOs.
The $k$th Slater determinant of the wave function ansatz, containing the $k_1$th, $k_2$th, $\dots$, $k_n$th orbitals reads as
\begin{equation}
\Phi_k'^{(\prime)} = \hat{A} \left(\prod_{i=1}^n\varphi_{k_i}'^{(\prime)} (i)\right)\,,~
\label{eq:SD}
\end{equation}
where $\hat{A}$ is the antisymmetrizing operator and the parentheses indicate that each one-electron orbital contains the spatial coordinates of a single electron as variable.
There are $\binom{m}{n}$ possible ways of orbital selection, i.e, the order among orbitals does not make a difference because $\hat{A}$ permutes them anyway, thus FCI ansatz $\Psi'^{(\prime)}$ is expanded in the basis of $d=\binom{m}{n}$ physically relevant Slater determinants,
\begin{equation}
\Psi'^{(\prime)} = \sum_{k=1}^{d} C_k'^{(\prime)} \Phi_k'^{(\prime)}\,.
\label{eq:FCI}
\end{equation}
The Slater determinants in Eq.~\ref{eq:FCI} are expressed in terms of the basis functions using Eq.~\ref{eq:MO} and Eq.~\ref{eq:SD} as
\begin{equation}
\Psi'^{(\prime)} = \sum_{k=1}^{d} C_k'^{(\prime)} \hat{A}
\left(\prod_{i=1}^n \left( \sum_{j=1}^m c_{k_ij}'^{(\prime)} \chi_j (i) \right)
\right)\,.
\label{eq:FCI-MO1}
\end{equation}
By expanding the products of Eq.~\ref{eq:FCI-MO1}, the Slater determinants can be rewritten as a linear combination of $n$-factored products of basis functions, i.e.,
\begin{equation}
\Psi'^{(\prime)} = \sum_{k=1}^{d} C_k'^{(\prime)} \sum_S w_{Sk}'^{(\prime)} \prod_{i=1}^n\chi_{S_i} (i)\,,
\end{equation}
where $S$ denotes a selection scenario.
By interchanging the order of summations, the coefficients of basis product $\prod_{i=1}^n\chi_{S_i} (i)$ corresponding to selection $S$ can be merged into a single variable, i.e.,
\begin{equation}
W_{S}'^{(\prime)} = \sum_{k=1}^{d} C_k'^{(\prime)} w_{Sk}'^{(\prime)}\,.
\label{eq:W}
\end{equation}
Finally a simple expression for the many-body wave function is written as
\begin{equation}
\Psi'^{(\prime)} = \sum_S W_{S}'^{(\prime)}\prod_{i=1}^n\chi_{S_i} (i) \,.
\label{eq:FCI2}
\end{equation}
In order to prove that the FCI wave functions $\Psi'$ and $\Psi''$ are identical (and hence so are their corresponding electronic energies), it must be shown that $W'$ and $W''$ are equal for all $S$ selections.
As a first step, we show that the number of the linearly independent $S$ scenarios is exactly $\binom{m}{n}$ due to the antisymmetrization.
In particular, if any two (or more) $\chi$ functions within of selection $S$ are identical, the corresponding basis product drops out in the course of antisymmetrization, as the same expression is obtained with opposite sign upon variable exchange.
Furthermore, $S$ selections that only differ in the order of $\chi$ functions necessarily have the same weight up to a sign as the antisymmetrization process interconverts them.
FCI defined in Eq.~\eqref{eq:FCI} optimizes $d$ variational parameters explicitly in order to minimize energy.
According to Eq.~\eqref{eq:W}, the reference-free expansion of FCI of Eq.~\eqref{eq:FCI2} implicitly varies parameters $W'^{(\prime)}$ of dimension $\binom{m}{n}$ which implies the invariance of reference.
As for truncated CI and CC methods or frozen core approximation, the many-body solution is no longer independent of the reference molecular orbital set.
The breaking of the invariance is owed to the fact that the dimension of $W'^{(\prime)}$ equals $\binom{m}{n}$ independently of the applied computational method whereas the dimension of $C'^{(\prime)}$ decreases for truncated approaches, i.e., the summation in Eq.~\eqref{eq:W} runs on a restricted configuration space of dimension $d<\binom{m}{n}$.
This means that $W'^{(\prime)}$ values become inevitably interdependent, which also prevents them from reaching their optimal values.
\bibliographystyle{achemso}
|
2,869,038,154,793 | arxiv | \section{Introduction}
Stability analyis of vector fields around an equilibrium point is a subject of interest in nonlinear dynamical systems.
For the nonlinear system if the eigenvalues of the linearized Jacobian has a mixture of positive, negative and zero real parts the analysis is not so straightforward and we have to go to the centre manifold theory to properly analyse the dynamics. The eigenspaces of the eigenvectors corresponding to positive, negative and zero real parts are considered. We assign a 3-D coordinate system for representing the eigenspaces of these three particular types of eigenvectors. But in case of centre manifold theory we consider only a 2-D coordinate system for representing the eigenspaces of the eigenvectors of negative and zero real parts of eigenvalues. They are termed as the stable and centre directions along the $Y$ and $X$ axis, respectively so that it boils down only to the analysis along the centre direction to determine stability. In case the spectrum of the linearized Jacobian contains purely imaginary eigenvalues, a slowly decaying oscillatory amplitude can appear. An investigation with application of centre manifold theory for limit cycle behaviour\cite{liu2000application} are also made in the context of nonlinear aeroelasticity. Limit cycle stability reversal of a two and three degrees of freedom
airfoils has been investigated in \cite{dessi2002limit,dessi2004limit}. Limitations of centre manifold theory for limit cycle calculation has been shown in \cite{grzkedzinski2005limitation}. \par
In our work we primarily focus on application of centre manifold theory for a $3-D$ system with general second order nonlinear terms. We perform the stability analysis by reducing the state equations on a $2-D$ centre manifold. The centre manifold is found to possess two fixed points one stable and another unstable. The system moves towards the stable fixed point starting from the unstable fixed point in asymptotic time. Along the course of discussion we also state and prove a theorem along with a conjecture made in the theorem
Then we present an example of a system which describes the kinetics of the protein molecules\cite{sanchez2010protein,manning2004structural} in an assembly. Stability analysis is done for it in the light of centre manifold theory. \par
The paper is organised as follows. In the prelude we introduce the centre manifold theory, its various ramifications and technical details like eigenbasis and similarity transformations, block diagonal representation of the system, seek for the leading order term through centre manifold equation. All these are discussed at length to adapt the mathematical concepts for applied dynamical systems\cite{strogatz2018nonlinear,thompson2002nonlinear} with a few degrees of freedom. Then we discuss about a general 3-D system with second order nonlinearities for the stability analysis. Then we state and prove a theorem for the condition of stability and in our example of the $2-D$ model through centre manifold theory we have shown the verification of the theorem in the light of obtained results. Finally we conclude with a short and modest concluding paragraph.
\section{Centre Manifold Theory}
Consider a $n\times n$ nonlinear system represented by the following vector equation.
\begin{equation}
\begin{split}
\dot{\vec{z}}=\vec{f}(\vec{z})
\end{split}
\end{equation}
where $f(\vec{z})$ is comprised of linear and nonlinear terms of the individual variables.\\
To analyse the sysem using centre manifold theory we have to transform the equations into the following form\cite{wiggins2003introduction}
\begin{equation}\label{Decouple}
\begin{split}
&\dot{\vec{x}}=A\vec{x}+\vec{f}(\vec{x},\vec{y})\\&
\dot{\vec{y}}=B\vec{y}+\vec{g}(\vec{x},\vec{y})
\end{split}
\end{equation}
where $\vec{x}$ is a $c$ dimensional vector and $A$ is a $c\times c$ constant matrix having eigenvalues of zero real parts. Similarly $\vec{y}$ is a $s$ dimensional vector and $B$ is a $s\times s$ constant matrix having eigenvalues of negative real part. We have $c+s=n$. Here $\vec{f}(\vec{x},\vec{y})$ and $\vec{g}(\vec{x},\vec{y})$ are the nonlinear parts of the system. The advantage of such a transformation is twofold. On one hand we decouple the centre direction from the stable direction and also separate the linear part from the nonlinear part. But if the system cannot be written in this form then we have to go for a change of basis transformation. We have to transform the system in its eigenbasis. Transforming the system in its eigenbasis may lead to interprete the system in its simplest form and brings the system in the form as represented by the equation (\ref{Decouple}). For converting to eigenbasis we follow the following procedure.
\subsection{Eigenbasis analysis}
For converting to eigenbasis we have to first compute the linearized Jacobian($J$) around the equilibrium point. Then we have to compute the eigenvalues of the Jacobian. Let there be $c$ eigenvalues with zero real parts and $s$ eigenvalues with negative real parts. The eigenvectors of the eigenvalues of the zero real part span the centre eigenspace represented by the axis $X$. The eigenvectors of the eigenvalues of negative real part span the stable eigenspace represented by $Y$ axis. Let the eigenvectors of the zero real part eigenvalues be $n_{c_{1}}$ \dots $n_{c_{c}}$. Let the eigenvectors of negative real part eigenvalues be $n_{s_{1}}$ \dots $n_{s_{s}}$. Let the vector field in the eigenbasis be represented by $U$. We do the following transformation.
\begin{equation}
\begin{split}
\vec{X}=P\vec{U}
\end{split}
\end{equation}
where $\vec{X}$ is the original vector field and $P$ is a $n\times n$ matrix which is constructed as follows.
\begin{equation}
\begin{split}
P=\begin{bmatrix}
n_{c_{1}} \dots n_{c_{c}} \; n_{s_{1}} \dots n_{s_{s}}
\end{bmatrix}
\end{split}
\end{equation}
If $\dot{\vec{X}}=E\vec{X}+\vec{f}(\vec{x},\vec{y})$, where $\vec{f}$ represents the nonlinear part and thereby the original system of equations on transformation of $\vec{X}=P\vec{U}$ the equation takes the form,
\begin{equation}\label{Transformed vector eqn}
\begin{split}
& P\dot{\vec{U}}=EP\vec{U}+\vec{f}(\vec{x},\vec{y}) \\ &
\implies \dot{\vec{U}}=P^{-1}EP\vec{U}+P^{-1}\vec{f}(\vec{x},\vec{y}).
\end{split}
\end{equation}
The transformation of $E \rightarrow P^{-1}EP$ is known as similarity transformation. Now it can be checked the matrix $P^{-1}EP$ is in block diagonal form and in case of real eigenvalues it will be a diagonal matrix with all the eigenvalues(both zero and negative) aligned along the diagonal. The system can be reduced to the form as given by equation (\ref{Decouple}) with centre directions decoupled from the stable directions. If it is a two dimensional system with $n=2$ we should have $c=1$ and $s=1$. Then the matrix $A$ in equation (\ref{Decouple}) will be just the eigenvalue of the centre direction that is $0$ and matrix $B$ will just be the eigenvalue of the stable direction that is a negative number.
\subsection{Centre Manifold analysis}
Once the system is in the form as given by equation (\ref{Decouple}) we can carry forward our analysis from that point.\par
Consider the decoupled directions $\vec{x}$ and $\vec{y}$. We write,
\begin{equation}
\begin{split}
\vec{y}=\vec{h}(\vec{x})
\end{split}
\end{equation}
where $\vec{h}$ represent the expressions we use to represent $\vec{y}$ in terms of $\vec{x}$. We call the above equation is the centre manifold equation. We consider $\vec{h}(0)=0$ and $D\vec{h}(0)=0$ where $0$ is the equilibrium point.The first condition implies that $y=0$ at the equilibrium point and the second condition implies that the centre manifold is tangent to the centre eigenspace at the point $0$. We expand $\vec{h}(x)$ in Taylor series expansion. Then we can write,
\begin{equation}
\vec{y}=\vec{h}(\vec{x})=a_{0}\vec{x^2}+a_{1}\vec{x^3}+\mathcal{O}(4)
\end{equation}
Then,
\begin{equation}
\dot{\vec{y}}=D\vec{h}(\vec{x})\dot{\vec{x}}=(2a_{0}\vec{x}+3a_{1}\vec{x^2}+\mathcal{O}(3))\dot{\vec{x}}
\end{equation}
where $D$ stands for the jacobian of the vector $\vec{h}$. For $2$ dimensional system $D\vec{h}$ will be just the derivative of function $h$ with respect to $x$. The above equation can be written as,
\begin{equation}\label{N(x)=0}
\begin{split}
&\mathcal{N}(\vec{x})=(2a_{0}\vec{x}+3a_{1}\vec{x^2}+\mathcal{O}(3))\dot{\vec{x}}-\dot{\vec{y}}=0 \\ &
\implies \mathcal{N}(\vec{x})=(2a_{0}\vec{x}+3a_{1}\vec{x^2}+\mathcal{O}(3))(A\vec{x}+\vec{f}(\vec{x},\vec{y}))-(B\vec{y}+g(\vec{x},\vec{y}))=0 \\ &
\implies \mathcal{N}(\vec{x})=(2a_{0}\vec{x}+3a_{1}\vec{x^2}+\mathcal{O}(3))(A\vec{x}+\vec{f}(\vec{x},\vec{h}(\vec{x})))-(B\vec{h}(\vec{x}))+g(\vec{x},\vec{h}(\vec{x})))=0
\end{split}
\end{equation}
Now $\mathcal{N}(\vec{x})$ is a system of equations containing only $\vec{x}$. For a two dimensional system $\mathcal{N}$ will be just a expression in a single variable $x$. We can equate the coefficients of various powers of $x$ to zero and solve for $a_{0}$, $a_{1}$ etc. We will always consider the leading order term. If $a_{0}=0$ then we go to the next higher order term and solve for $a_{1}$ and so on. Then the centre manifold will be given by the equation,
\begin{equation}
\vec{y}=\vec{h}(\vec{x})=a_{0}\vec{x^2}+a_{1}\vec{x^3}+\mathcal{O}(4)
\end{equation}
upto leading order. Once we obtain $\vec{y}$ in terms of $\vec{x}$ that is the centre manifold equation we no longer consider the $\vec{y}$ direction. We consider only the $\vec{x}$ direction and write,
\begin{equation}
\dot{\vec{x}}=A\vec{x}+\vec{f}(\vec{x},\vec{h}(\vec{x}))
\end{equation}
The above equation is called the reduced system of equations. We can use it to analyse stability around the equilibrium point.
\subsection{Centre Manifold Analysis of a general 3-D system of second order nonlinearities}
Let the linearized Jacobian around the equilibrium point yields eigenvalues $i\lambda_{1},-i\lambda_{1},-\lambda_{2}$.
We consider $\lambda_{1} > 0$ and $\lambda_{2} > 0$. The real parts of first two eigenvalues is $0$, so there is a centre manifold. The third eigenvalue is real negative, so there is a stable manifold. Let on eigenbasis transformation the system equations take the form
\begin{equation}
\begin{split}
\begin{bmatrix}
\dot{x} \\ \dot{y} \\ \dot{z}
\end{bmatrix}=\begin{bmatrix}
0 & \lambda_{1} & 0 \\
-\lambda_{1} & 0 & 0 \\
0 & 0 & -\lambda_{2}
\end{bmatrix}\begin{bmatrix}
x \\ y \\z
\end{bmatrix}+\begin{bmatrix}
f_{1}(x,y,z) \\ f_{2}(x,y,z) \\ f_{3}(x,y,z)
\end{bmatrix}
\end{split}
\end{equation}
where $f_{1},f_{2},f_{3}$ are the nonlinear parts containing second order nonlinearities in $x,y,z$. We can assume some second order polynomial structure in $x,y,z$ for $f_{1},f_{2},f_{3}$. Let,
\begin{equation}
\begin{split}
& f_{1}(x,y,z)=c_{0}x^2+c_{1}y^2+c_{2}z^2+c_{3}xy+c_{4}yz+c_{5}zx \\ &
f_{2}(x,y,z)=d_{0}x^2+d_{1}y^2+d_{2}z^2+d_{3}xy+d_{4}yz+d_{5}zx \\ &
f_{3}(x,y,z)=e_{0}x^2+e_{1}y^2+e_{2}z^2+e_{3}xy+e_{4}yz+e_{5}zx
\end{split}
\end{equation}
Therefore the system equations are given by,
\begin{equation}
\begin{split}
& \dot{x}=\lambda_{1}y+c_{0}x^2+c_{1}y^2+c_{2}z^2+c_{3}xy+c_{4}yz+c_{5}zx \\ &
\dot{y}=-\lambda_{1}x+d_{0}x^2+d_{1}y^2+d_{2}z^2+d_{3}xy+d_{4}yz+d_{5}zx \\ &
\dot{z}= -\lambda_{2}z+e_{0}x^2+e_{1}y^2+e_{2}z^2+e_{3}xy+e_{4}yz+e_{5}zx
\end{split}
\end{equation}
The equilibrium point is the origin$(0,0,0)$. If the equilibrium point is a point other than the origin say $(x_{0},y_{0},z_{0})$ then we would have made the transformation $x \rightarrow x-x_{0}, y \rightarrow y-y_{0}, z \rightarrow z-z_{0}$ and have brought the equilibrium position to the origin. \\ The centre manifold is two dimensional and the stable manifold is one dimensional. Let the centre manifold is given by the following taylor series expansion.
\begin{equation}
z=h(x,y)=a_{0}x^2+a_{1}xy+a_{2}y^2+\mathcal{O}(3)
\end{equation}
We take upto $\mathcal{O}(2)$. So
\begin{equation}
z=h(x,y)=a_{0}x^2+a_{1}xy+a_{2}y^2
\end{equation}
Therefore,
\begin{equation}
\begin{split}
& \dot{z} =\frac{\partial h}{\partial x}\dot{x}+\frac{\partial h}{\partial y}\dot{y} \\ &
\implies \dot{z}-(2a_{0}x+a_{1}y)\dot{x}-(2a_{2}y+a_{1}x)\dot{y}=0
\end{split}
\end{equation}
Substituting the values of $\dot{x},\dot{y},\dot{z}$ from the system equations and replacing $z$ by $(a_{0}x^2+a_{1}xy+a_{2}y^2)$ in the equations we have,
\begin{equation}
\begin{split}
& -\lambda_{2}(a_{0}x^2+a_{1}xy+a_{2}y^2)+e_{0}x^2+e_{1}y^2+e_{2}(a_{0}x^2+a_{1}xy+a_{2}y^2)^2+e_{3}xy\\ & +e_{4}y(a_{0}x^2+a_{1}xy+a_{2}y^2)+e_{5}x(a_{0}x^2+a_{1}xy+a_{2}y^2)\\ &-(2a_{0}x+a_{1}y)[\lambda_{1}y+c_{0}x^2+c_{1}y^2+c_{2}(a_{0}x^2+a_{1}xy+a_{2}y^2)^2+c_{3}xy \\ &+c_{4}y(a_{0}x^2+a_{1}xy+a_{2}y^2)+c_{5}x(a_{0}x^2+a_{1}xy+a_{2}y^2)]\\ &-(2a_{2}y+a_{1}x)[-\lambda_{1}x+d_{0}x^2+d_{1}y^2+d_{2}(a_{0}x^2+a_{1}xy+a_{2}y^2)^2+d_{3}xy \\ & +d_{4}y(a_{0}x^2+a_{1}xy+a_{2}y^2)+d_{5}x(a_{0}x^2+a_{1}xy+a_{2}y^2)]=0
\end{split}
\end{equation}
Considering only $\mathcal{O}(2)$ and equating the coefficients of $x^2,xy,y^2$ to $0$ respectively we have,
\begin{equation}
\begin{split}
&-\lambda_{2}a_{0}+e_{0}+a_{1}\lambda_{1}=0 \\ &
-\lambda_{2}a_{1}+e_{3}-2a_{0}\lambda_{1}+2a_{2}\lambda_{1}=0 \\ &
-\lambda_{2}a_{2}+e_{1}-a_{1}\lambda_{1}=0
\end{split}
\end{equation}
Solving for $a_{0},a_{1},a_{2}$ we have,
\begin{equation}
\begin{split}
& a_{0}=\frac{\lambda_{2}(e_{0}\lambda_{2}+e_{3}\lambda_{1})+4e_{0}\frac{\lambda_{1}^2}{\lambda_{2}}+2\lambda_{1}^2(e_{1}-e_{0})}{\lambda_{2}^3+4\lambda_{1}^2} \\ &
a_{1}=\frac{\lambda_{2}^2e_{3}+2\lambda_{1}\lambda_{2}(e_{1}-e_{0})}{\lambda_{2}^3+4\lambda_{1}^2} \\ &
a_{2}=\frac{\lambda_{2}(e_{1}\lambda_{2}-e_{3}\lambda_{1})+4e_{1}\frac{\lambda_{1}^2}{\lambda_{2}}+2\lambda_{1}^2(e_{0}-e_{1})}{\lambda_{2}^3+4\lambda_{1}^2}
\end{split}
\end{equation}
Therefore the centre manifold equation is given by,
\begin{equation}
\begin{split}
& z=\frac{\lambda_{2}(e_{0}\lambda_{2}+e_{3}\lambda_{1})+4e_{0}\frac{\lambda_{1}^2}{\lambda_{2}}+2\lambda_{1}^2(e_{1}-e_{0})}{\lambda_{2}^3+4\lambda_{1}^2}x^2+\frac{\lambda_{2}^2e_{3}+2\lambda_{1}\lambda_{2}(e_{1}-e_{0})}{\lambda_{2}^3+4\lambda_{1}^2}xy
\\ & +\frac{\lambda_{2}(e_{1}\lambda_{2}-e_{3}\lambda_{1})+4e_{1}\frac{\lambda_{1}^2}{\lambda_{2}}+2\lambda_{1}^2(e_{0}-e_{1})}{\lambda_{2}^3+4\lambda_{1}^2}y^2
\end{split}
\end{equation}
The reduced equations on the centre manifold on considering terms of $\mathcal{O}(2)$ and neglecting higher order terms is given by,
\begin{equation}\label{reduced eqn x y}
\begin{split}
& \dot{x}=\lambda_{1}y+c_{0}x^2+c_{1}y^2+c_{3}xy \\ &
\dot{y}=-\lambda_{1}x+d_{0}x^2+d_{1}y^2+d_{3}xy
\end{split}
\end{equation}
Now we seek to find if there is any oscillations on this reduced phase space.
For this we write,
\begin{equation}\label{r theta coord}
\begin{split}
&x=r\cos\theta \\ &
y=r\sin\theta
\end{split}
\end{equation}
on transformation to $r,\theta$ coordinate. Then
\begin{equation}
r^2=x^2+y^2
\end{equation}
Differentiating both sides with respect to time $t$ we have,
\begin{equation}
\begin{split}
& r\dot{r}=x\dot{x}+y\dot{y} \\ &
\implies \dot{r}=\frac{x\dot{x}+y\dot{y}}{r}
\end{split}
\end{equation}
On puting the values of $\dot{x}$, $\dot{y}$ and simplifying we have,
\begin{equation}
\begin{split}
\dot{r}=\frac{x(c_{0}x^2+c_{1}y^2+c_{3}xy)+y(d_{0}x^2+d_{1}y^2+d_{3}xy)}{r}
\end{split}
\end{equation}
At this point for the sake of simplification we assume values for $c_{0},c_{1},c_{3},d_{0},d_{1},d_{3}$.\\ We consider $c_{0}=c_{1}=c_{3}=d_{0}=d_{1}=d_{3}=1$. \\ Then the above equation takes the form,
\begin{equation}\label{r dot}
\begin{split}
\dot{r}& =\frac{x(x^2+y^2+xy)+y(x^2+y^2+xy)}{r} \\ &
\implies \dot{r}=r^2(\cos\theta+\sin\theta)(1+\cos\theta\sin\theta)
\end{split}
\end{equation}
on puting $x=r\cos\theta,y=r\sin\theta$.
Now for fixed point prediction we should solve for $r$ from $\dot{r}=0$. Therefore we have to solve for $r$ from the equation,
\begin{equation}
r^2(\cos\theta+\sin\theta)(1+\cos\theta\sin\theta)=0
\end{equation}
Now equating each of the factors to $0$ we have,
\begin{equation}
\begin{split}
&r=0 \\ & \cos\theta=-\sin\theta \\ &
\cos\theta\sin\theta=-1
\end{split}
\end{equation}
The solution $r=0$ implies a fixed point at the origin which is a sink as we will see. The third solution is discarded because $\cos\theta\sin\theta > -1$ for any $\theta$. From the second solution we have,
\begin{equation}
\begin{split}
& \cos\theta=-\sin\theta \\ &
\implies \theta=n\pi+\frac{3\pi}{4}
\end{split}
\end{equation}
Puting in equation (\ref{r theta coord}),
\begin{equation}
\begin{split}
& x=-\frac{r}{\sqrt{2}} \\ &
y=\frac{r}{\sqrt{2}}
\end{split}
\end{equation}
for $\theta=2n\pi+\frac{3\pi}{4}$ and
\begin{equation}
\begin{split}
& x=\frac{r}{\sqrt{2}} \\ &
y=-\frac{r}{\sqrt{2}}
\end{split}
\end{equation}
for $\theta=(2n+1)\pi+\frac{3\pi}{4}$. \\
Puting the the first solution of $x,y$ in the first equation of (\ref{reduced eqn x y}) and recalling we have considered $c_{0}=c_{1}=c_{3}=1$ we have,
\begin{equation}
\begin{split}
&-\frac{\dot{r}}{\sqrt{2}}=\lambda_{1}\frac{r}{\sqrt{2}}+\frac{r^2}{2}+\frac{r^2}{2}-\frac{r^2}{2} \\ &
\implies -\dot{r}=r(\lambda_{1}+\frac{r}{\sqrt{2}})
\end{split}
\end{equation}
Now the above equation is a equation only in $r$. For solving the amplitude $r$ we equate $\dot{r}=0$. Thus we have,
\begin{equation}
\begin{split}
&r(\lambda_{1}+\frac{r}{\sqrt{2}})=0 \\ &
\implies r=0,-\sqrt{2}\lambda_{1}
\end{split}
\end{equation}
Now $r=0$ implies a sink at the origin since $\dot{r} < 0$ for $r > 0$ in the neighbourhood of origin. $r=-\sqrt{2}\lambda_{1}$ is discarded because amplitude cannot be negative. \\
Puting the second solution of $x,y$ in the first equation of (\ref{reduced eqn x y}) we have,
\begin{equation}\label{r dot 2nd}
\begin{split}
\frac{\dot{r}}{\sqrt{2}}& =-\lambda_{1}\frac{r}{\sqrt{2}}+\frac{r^2}{2}+\frac{r^2}{2}-\frac{r^2}{2} \\ &
\implies \dot{r} =r(\frac{r}{\sqrt{2}}-\lambda_{1})
\end{split}
\end{equation}
For solving for amplitude we equate $\dot{r}=0$. Thus we have,
\begin{equation}
\begin{split}
& r(\frac{r}{\sqrt{2}}-\lambda_{1})=0 \\
&\implies r=0,\sqrt{2}\lambda_{1}
\end{split}
\end{equation}
$r=0$ gives a sink at the origin since $\dot{r} < 0$ for $r > 0$ in the neighbourhood of origin. $r=\sqrt{2}\lambda_{1}$ gives a source because $\dot{r}<0$ for $r<\sqrt{2}\lambda_{1}$ and $\dot{r}>0$ for $r>\sqrt{2}\lambda_{1}$ in the neighbourhood of $r=\sqrt{2}\lambda_{1}$. We want to show as $t \rightarrow \infty$ the system approaches $r=0$ starting from $r=\sqrt{2}\lambda_{1}$.\\
For this we make the following coordinate transformation. We write $r^{\prime}=r-\sqrt{2}\lambda_{1}$. Then $r=r^{\prime}+\sqrt{2}\lambda_{1}$. We have to do this transformation because $r=\sqrt{2}\lambda_{1}$ is the starting point of the system. Puting the value of $r$ in terms of $r^{\prime}$ in equation(\ref{r dot 2nd}) we have,
\begin{equation}
\begin{split}
& \dot{r^{\prime}}=(r^{\prime}+\sqrt{2}\lambda_{1})\frac{r^{\prime}}{\sqrt{2}} \\ &
\implies \sqrt{2}\frac{dr^{\prime}}{dt}=(r^{\prime}+\sqrt{2}\lambda_{1})r^{\prime} \\ &
\implies \frac{\sqrt{2}\lambda_{1}}{(r^{\prime}+\sqrt{2}\lambda_{1})r^{\prime}}dr^{\prime}=\lambda_{1}dt \\ &
\implies \frac{(r^{\prime}+\sqrt{2}\lambda_{1})-r^{\prime}}{(r^{\prime}+\sqrt{2}\lambda_{1})r^{\prime}}dr^{\prime}=\lambda_{1}dt \\ &
\implies \left[\frac{1}{r^{\prime}}-\frac{1}{r^{\prime}+\sqrt{2}\lambda_{1}}\right]dr^{\prime}=\lambda_{1}dt
\end{split}
\end{equation}
Integrating both sides we have,
\begin{equation}
\begin{split}
\ln\frac{r^{\prime}k}{r^{\prime}+\sqrt{2}\lambda_{1}}=\lambda_{1}t
\end{split}
\end{equation}
where $k$ is the constant of integration. Then from the above equation,
\begin{equation}
\begin{split}
& \frac{r^{\prime}+\sqrt{2}\lambda_{1}}{r^{\prime}k}=e^{-\lambda_{1}t} \\ &
\implies r^{\prime}=\frac{\sqrt{2}\lambda_{1}}{ke^{-\lambda_{1}t}-1}
\end{split}
\end{equation}
Puting $t=\infty$ in the above equation we get $r^{\prime}=-\sqrt{2}\lambda_{1}$. Since $r^{\prime}=r-\sqrt{2}\lambda_{1}$ when $r^{\prime}=-\sqrt{2}\lambda_{1}$ we have $r=0$. So the system approaches $r=0$ as $t \rightarrow \infty$.
\\
For the $\theta$ dynamics we write,
\begin{equation}
\begin{split}
\theta=\tan^{-1}\frac{y}{x}
\end{split}
\end{equation}
Differentiating the above equation with respect to time $t$ we have,
\begin{equation}
\begin{split}
&\dot{\theta}=\frac{x\dot{y}-y\dot{x}}{x^2+y^2} \\ &
\implies \dot{\theta}=\frac{x(-\lambda_{1}x+x^2+y^2+xy)-y(\lambda_{1}y+x^2+y^2+xy)}{x^2+y^2}
\end{split}
\end{equation}
on puting the values of $\dot{x},\dot{y}$ and recalling $c_{0}=c_{1}=c_{3}=d_{0}=d_{1}=d_{3}=1$.
On simplication it gives,
\begin{equation}
\begin{split}
\dot{\theta}& =-\lambda_{1}+x-y+\frac{xy(x-y)}{x^2+y^2} \\ &
=-\lambda_{1}+r(\cos\theta-\sin\theta)(1+\cos\theta\sin\theta)
\end{split}
\end{equation}
on puting $x=r\cos\theta,y=r\sin\theta$. Since $r$ and $\theta$ are functions of $t$, $\dot{\theta}$ is also a function of $t$. $\dot{\theta}$ is not explicitly solvable in terms of $t$ since $r$ and $\theta$ are not explicitly solvable in terms of $t$. Still we can do some analysis and prediction. If $\dot{\theta}(0) \neq 0$ due to the nonzero angular velocity the system will exhibit a periodic spiral motion with approach towards the point $r=0$. For oscillation prediction we look into the equations.
\begin{equation}
\begin{split}
&x=r\cos\theta \\ &
y=r\sin\theta
\end{split}
\end{equation}
Since $r$ is approaching towards $0$ we can say the amplitude is decreasing so in the $x-t$ and $y-t$ plane the amplitude becomes small and small as $t \rightarrow \infty$ and eventually becomes $0$.
\\ \\
Now we proceed to prove a theorem.
\begin{theorem}
Consider the system.
\begin{equation}
\begin{split}
&\dot{\vec{x}}=A\vec{x}+\vec{f}(\vec{x},\vec{y})\\&
\dot{\vec{y}}=B\vec{y}+\vec{g}(\vec{x},\vec{y})
\end{split}
\end{equation}
If the nonlinear parts $\vec{f}(\vec{x},\vec{y})$ and $\vec{g}(\vec{x},\vec{y})$ are even that is $\vec{f}(\vec{x},\vec{y})=\vec{f}(-\vec{x},-\vec{y})$ and $\vec{g}(\vec{x},\vec{y})=\vec{g}(-\vec{x},-\vec{y})$ then $\vec{h}(\vec{x})$ is even that is $\vec{h}(\vec{x})=\vec{h}(-\vec{x})$
\begin{proof}
We have
\begin{equation}
\begin{split}
&\dot{\vec{y}}=D\vec{h}(\vec{x})\dot{\vec{x}} \\ &
\implies B\vec{y}+\vec{g}(\vec{x},\vec{y})=D\vec{h}(\vec{x})(A\vec{x}+\vec{f}(\vec{x},\vec{y})) \\ &
\implies B\vec{h}(x)+\vec{g}(\vec{x},\vec{y})=D\vec{h}(\vec{x})A\vec{x}+D\vec{h}(\vec{x})\vec{f}(\vec{x},\vec{y})
\end{split}
\end{equation}
Now, $\vec{h}(\vec{x})=a_{0}\vec{x^2}+a_{1}\vec{x^3}+\mathcal{O}(4)$. Upto leading order $\vec{h}(\vec{x})$ will be $\vec{h}(\vec{x})=a_{0}\vec{x^2}$. Then $D\vec{h}(\vec{x})=2a_{0}\vec{x}$. Then we should have
\begin{equation}\label{thm1}
\begin{split}
B\vec{h}(x)+\vec{g}(\vec{x},\vec{y})=(2a_{0}\vec{x})A\vec{x}+(2a_{0}\vec{x})\vec{f}(\vec{x},\vec{y})
\end{split}
\end{equation}
Puting $-\vec{x}$ at both sides of the equation we have,
\begin{equation}
\begin{split}
B\vec{h}(-x)+\vec{g}(-\vec{x},-\vec{y})=(-2a_{0}\vec{x})A(-\vec{x})+(-2a_{0}\vec{x})\vec{f}(-\vec{x},-\vec{y})
\end{split}
\end{equation}
Since $\vec{f}(\vec{x},\vec{y})$ and $\vec{g}(\vec{x},\vec{y})$ are even we should have,
\begin{equation}\label{thm2}
\begin{split}
B\vec{h}(-x)+\vec{g}(\vec{x},\vec{y})=(2a_{0}\vec{x})A\vec{x}+(-2a_{0}\vec{x})\vec{f}(\vec{x},\vec{y})
\end{split}
\end{equation}
Now $\vec{f}(\vec{x},\vec{y})$ being nonlinear is at least of $\mathcal{O}(2)$. Therefore $(2a_{0}\vec{x})\vec{f}(\vec{x},\vec{y})$ is at least of $\mathcal{O}(3)$ which we do not want to consider. Therefore the last terms in equations (\ref{thm1}) and (\ref{thm2}) go to $0$. Therefore from equations (\ref{thm1}) and (\ref{thm2}) we have.
\begin{equation}
\begin{split}
& B\vec{h}(x)+\vec{g}(\vec{x},\vec{y})=(2a_{0}\vec{x})A\vec{x} \\ &
B\vec{h}(-x)+\vec{g}(\vec{x},\vec{y})=(2a_{0}\vec{x})A\vec{x}
\end{split}
\end{equation}
Subtracting one from the other we have
\begin{equation}
\begin{split}
& B(\vec{h}(\vec{x})-\vec{h}(-\vec{x}))=0 \\ &
\implies \vec{h}(\vec{x})=\vec{h}(-\vec{x})
\end{split}
\end{equation}
If in $\vec{h}(\vec{x})$, $a_{0}=0$ we go to the next leading order term and consider $\vec{h}(\vec{x})=a_{1}\vec{x^3}$. Then we have $D\vec{h}(\vec{x})=3a_{1}\vec{x^2}$. Therefore in the equation $B\vec{h}(\vec{x})+\vec{g}(\vec{x},\vec{y})=D\vec{h}(\vec{x})A\vec{x}+D\vec{h}(\vec{x})\vec{f}(\vec{x},\vec{y})$ we put $\vec{h}(\vec{x})=a_{1}\vec{x^3}$. Therefore we have,
\begin{equation}\label{thm3}
\begin{split}
B(a_{1}\vec{x^3})+\vec{g}(\vec{x},\vec{y})=(3a_{1}\vec{x^2})A\vec{x}+(3a_{1}\vec{x^2})\vec{f}(\vec{x},\vec{y})
\end{split}
\end{equation}
Puting $\vec{x}=-\vec{x}$ in the above equation we have,
\begin{equation}\label{thm4}
\begin{split}
& Ba_{1}(-\vec{x^3})+\vec{g}(-\vec{x},-\vec{y})=(3a_{1}\vec{x^2})A(-\vec{x})+(3a_{1}\vec{x^2})\vec{f}(-\vec{x},-\vec{y}) \\ &
\implies -B(a_{1}\vec{x^3})+\vec{g}(\vec{x},\vec{y})=-(3a_{1}\vec{x^2})A\vec{x}+(3a_{1}\vec{x^2})\vec{f}(\vec{x},\vec{y})
\end{split}
\end{equation}
Now $\vec{f}(\vec{x},\vec{y})$ being nonlinear is at least of $\mathcal{O}(2)$. So $(3a_{1}\vec{x^2})\vec{f}(\vec{x},\vec{y})$ is at least of $\mathcal{O}(4)$ which we do not want to consider in equations (\ref{thm3}) and (\ref{thm4}). Therefore from equations (\ref{thm3}) and (\ref{thm4}) we have,
\begin{equation}
\begin{split}
& B(a_{1}\vec{x^3})+\vec{g}(\vec{x},\vec{y})=(3a_{1}\vec{x^2})A\vec{x} \\ &
-B(a_{1}\vec{x^3})+\vec{g}(\vec{x},\vec{y})=-(3a_{1}\vec{x^2})A\vec{x}
\end{split}
\end{equation}
Adding these two equations we have,
\begin{equation}
\begin{split}
& 2\vec{g}(\vec{x},\vec{y})=0 \\ &
\implies \vec{g}(\vec{x},\vec{y})=0
\end{split}
\end{equation}
which is absurd since $\vec{g}(\vec{x},\vec{y})$ is a defined function in the problem. So we can say the leading order term in $\vec{h}(\vec{x})$ cannot be $\vec{x^3}$ if coefficient of $\vec{x^2}$ is $0$. So we go to the next leading order term that is $\vec{x^4}$. Considering $\vec{h}(\vec{x})=a_{2}\vec{x^4}$ and following the procedure as shown for the case when the leading order term is of $\mathcal{O}(2)$, we can show $\vec{h}(\vec{x})=\vec{h}(-\vec{x})$. We can make the conjecture that the leading order term in $\vec{h}(\vec{x})$ will be always of even power of $\vec{x}$ if the nonlinear parts are even functions of $\vec{x}$ and $\vec{y}$.
\end{proof}
\end{theorem}
In the example that we consider to illustrate the centre manifold problem we show that our above theorem and conjecture are true.
\section{Kinetic Stability analysis of protein assembly around a fixed point}
We pick up the nonlinear equations as given in the article\cite{tsuruyama2017kinetic}. The Nonlinear equations describe the kinetics of protein molecules. We write down the equation as follows.
\begin{equation}\label{master eqn}
\begin{split}
& \dot{x}=-(D_{1}-aX_{e}-k)x+(-bX_{e}+p)z+ax^2-bxz \\ &
\dot{z}=(k-cY_{e})x-pz+cx^2+cxz
\end{split}
\end{equation}
The above two equations form the master equations for the analysis that follows. Following the path as shown in the paper\cite{tsuruyama2017kinetic} we consider sufficiently small values of $D_{1},k,Y_{e},p$. They can be set to $0$. Then the equations (\ref{master eqn}) can be written down as follows.
\begin{equation}
\begin{split}
& \dot{x}=aX_{e}x-bX_{e}z+ax^2-bxz \\ &
\dot{z}=cx^2+cxz
\end{split}
\end{equation}
The equilibrium point is $(0,0)$. The linearized Jacobian$(J)$ around the equilibrium point $(0,0)$ is givven by
\begin{equation}
J=\begin{bmatrix}
aX_{e} & -bX_{e} \\ 0 & 0
\end{bmatrix}
\end{equation}
The eigenvalues of $J$ are $\lambda=0,aX_{e}$. So we see one of the eigenvalues is 0 which indicates that it has a centre eigenspace and permits us to do the centre manifold anaysis. The other eigenspace is a stable eigenspace corresponding to negative eigenvalue. This prompts us to consider $aX_{e} < 0$ which implies $a<0$ because $X_{e}$ cannot be negative. The eigenvector correponding to $0$ eigenvalue is $\begin{bmatrix}
b \\ a \end{bmatrix}$ and the eigenvector corresponding to $aX_{e}$ eigenvalue is $\begin{bmatrix} 1 \\ 0 \end{bmatrix}$. As the Jacobian $J$ is not block diagonal we have to resort to eigenbasis transformation. As mentioned in the theory we consider a transformation $\vec{X}=P\vec{U}$ where $\vec{X}$ is the original vector and $\vec{U}$ is the vector in the changed basis. In our case the system being two dimensional $\vec{X}$ and $\vec{U}$ are two dimensional vectors and $P$ is a $2 \times 2$ matrix formed by the eigenvectors of the Jacobian. We place the eigenvector of the centre direction as the first vector and that of the stable direction as the second vector. So,
\begin{equation}
\begin{split}
P=\begin{bmatrix}
b & 1 \\ a & 0
\end{bmatrix}
\end{split}
\end{equation}
The inverse is given by
\begin{equation}
\begin{split}
P^{-1}=\begin{bmatrix}
0 & \frac{1}{a} \\ 1 & -\frac{b}{a}
\end{bmatrix}
\end{split}
\end{equation}
Following equation (\ref{Transformed vector eqn}) we have
\begin{equation}
\dot{\vec{U}}=P^{-1}AP\vec{U}+P^{-1}\vec{f}(x,y)
\end{equation}
Now as mentioned in the theory $P^{-1}AP$ will be a block diagonal matrix which in our case will be just a diagonal matrix with the eigenvalues along the diagonal. We make the first diagonal entry as the eigenvalue of the centre direction and the second diagonal entry as the eigenvalue of the stable direction. Therefore we have,
\begin{equation}
P^{-1}AP=\begin{bmatrix} 0 & 0 \\ 0 & aX_{e} \end{bmatrix}
\end{equation}
Therefore,
\begin{equation}\label{transformed basis eqn}
\begin{split}
\dot{\vec{U}}=\begin{bmatrix} \dot{u} \\ \dot{v} \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 0 & aX_{e} \end{bmatrix}\begin{bmatrix} u \\ v \end{bmatrix}+\begin{bmatrix} 0 & \frac{1}{a} \\ 1 & -\frac{b}{a} \end{bmatrix}\begin{bmatrix} ax^2-bxz \\ cx^2+cxz \end{bmatrix}
\end{split}
\end{equation}
Now the nonlinear part in the above equation is still in terms of $x,z$ which we would like to convert in terms of $u,v$ using the equation $\vec{X}=P\vec{U}$. Therefore we have
\begin{equation}
\begin{split}
\begin{bmatrix} x \\ z \end{bmatrix}=\begin{bmatrix} b & 1 \\ a & 0 \end{bmatrix}\begin{bmatrix} u \\ v \end{bmatrix}
\end{split}
\end{equation}
Therefore we have,
\begin{equation}
\begin{split}
& x=bu+v \\ &
z=au
\end{split}
\end{equation}
Puting in equation (\ref{transformed basis eqn}) we have
\begin{equation}
\begin{bmatrix} \dot{u} \\ \dot{v} \end{bmatrix}=\begin{bmatrix} 0 & 0 \\ 0 & aX_{e} \end{bmatrix}\begin{bmatrix} u \\ v \end{bmatrix}+\begin{bmatrix} 0 & \frac{1}{a} \\ 1 & -\frac{b}{a} \end{bmatrix}\begin{bmatrix} a(bu+v)^2-b(bu+v)au \\ c(bu+v)^2+c(bu+v)au \end{bmatrix}
\end{equation}
Therefore we have the system of equations as,
\begin{equation}\label{eqns in u v}
\begin{split}
& \dot{u}=\frac{1}{a}\left[c(bu+v)^2+c(bu+v)au\right] \\ &
\dot{v}=aX_{e}v+a(bu+v)^2-b(bu+v)au-\frac{b}{a}\left[c(bu+v)^2+c(bu+v)au\right]
\end{split}
\end{equation}
From equation (\ref{N(x)=0}) we have
\begin{equation}
(2a_{0}x+3a_{1}x^2+\mathcal{O}(3))(Ax+f(x,h(x)))-(Bh(x)+g(x,h(x)))=0
\end{equation}
But here we replace $x$ by $u$ because we are working in the eigenbasis.\\
Therefore we have,
\begin{equation}\label{pre centre manifold}
(2a_{0}u+3a_{1}u^2+\mathcal{O}(3))(Au+f(u,h(u)))-(Bh(u)+g(u,h(u)))=0
\end{equation}
Now we recall,
\begin{equation}
h(u)=a_{0}u^2+a_{1}u^3+\mathcal{O}(4)
\end{equation}
We consider upto $\mathcal{O}(2)$.
Therefore,
\begin{equation}
\begin{split}
& Au+f(u,h(u))=\frac{1}{a}\left[c(bu+a_{0}u^2)+c(bu+a_{0}u^2)au\right] \\ &
Bh(u)+g(u,h(u))=aX_{e}(a_{0}u^2)+a(bu+a_{0}u^2)^2-b(bu+a_{0}u^2)au-\frac{b}{a}\left[c(bu+a_{0}u^2)^2+c(bu+a_{0}u^2)au\right]
\end{split}
\end{equation}
As we consider upto $\mathcal{O}(2)$ we search for the coefficient of $u^2$ that is $a_{0}$. If $a_{0}=0$, then we go to the next higher order term that is $u^3$. Puting in equation (\ref{pre centre manifold}) we have,
\begin{equation}
\begin{split}
& (2a_{0}u+3a_{1}u^2)\frac{1}{a}\left[c(bu+a_{0}u^2)+c(bu+a_{0}u^2)au\right] \\ &-\left[aX_{e}(a_{0}u^2)+a(bu+a_{0}u^2)^2-b(bu+a_{0}u^2)au-\frac{b}{a}\left[c(bu+a_{0}u^2)^2+c(bu+a_{0}u^2)au\right]\right]=0
\end{split}
\end{equation}
The coefficient of $u^2$ in the above expression is $aX_{e}a_{0}-\frac{bc}{a}b^2-b^2c$. \\
Equating the coefficient of $u^2$ to $0$ we have,
\begin{equation}
\begin{split}
& aX_{e}a_{0}-b^2c(1+\frac{b}{a})=0 \\ &
\implies a_{0}=\frac{b^2c}{aX_{e}}(1+\frac{b}{a})
\end{split}
\end{equation}
\subsection{Equation of Centre manifold and reduced system}
Equation of centre manifold is given by
\begin{equation}
\begin{split}
v=h(u)=a_{0}u^2+\mathcal{O}(3)
\end{split}
\end{equation}
Considering upto $\mathcal{O}(2)$ which is the leading order and substituting the value of $a_{0}$ we have,
\begin{equation}
v=\frac{b^2c}{aX_{e}}(1+\frac{b}{a})u^2
\end{equation}
The above equations is the equation of \textit{Centre Manifold}. From the first equation of the set (\ref{eqns in u v}) we have
\begin{equation}
\dot{u}=\frac{1}{a}\left[c(bu+v)^2+c(bu+v)au\right]
\end{equation}
Puting the value of $v$ as obtained in the centre manifold equation, in the above equation we have,
\begin{equation}
\dot{u}=\frac{1}{a}\left[c\left[bu+\frac{b^2c}{aX_{e}}(1+\frac{b}{a})u^2\right]^2+c\left[bu+\frac{b^2c}{aX_{e}}(1+\frac{b}{a})u^2\right]au\right]
\end{equation}
Considering upto $\mathcal{O}(2)$ of the power of $u$ in the above equation we have,
\begin{equation}\label{reduced system}
\dot{u}=cb(1+\frac{b}{a})u^2
\end{equation}
This gives the reduced system of equations.
\subsubsection{Stability Analysis of Equilibrium Point}
After we obtain the reduced system we no longer consider the equation of $\dot{v}$. We know the stable direction that is the $v$ direction will converge exponentially fast towards the equilibrium. So for analysis of stability of equilibrium point we can only consider equation (\ref{reduced system}).
We have,
\begin{equation}
\dot{u}=cb(1+\frac{b}{a})u^2
\end{equation}
Now consider the neigbourhood of equilibrium point $(0,0)$ along the $u$ direction. Two cases may arise. \\ \\
\textit{Case 1}:
$cb(1+\frac{b}{a}) > 0$. In this case if $u > 0$ near the origin that is the equilibrium point, we have $cb(1+\frac{b}{a})u^2 > 0$. If $u < 0$ near the origin we have $cb(1+\frac{b}{a})u^2 > 0$. In both case we have $\dot{u} > 0$ that is the velocity vector of the $u$ direction is along the positive $u$ axis. So the origin that is the equilibrium point is unstable \\ \\
\textit{Case 2}:
$cb(1+\frac{b}{a}) < 0$. In this case if $u > 0$ near the origin that is the equilibrium point, we have $cb(1+\frac{b}{a})u^2 < 0$. If $u < 0$ near the origin we have $cb(1+\frac{b}{a})u^2 < 0$. In both case we have $\dot{u} < 0$ that is the velocity vector of the $u$ direction is along the negative $u$ axis. So the origin that is the equilibrium point is unstable \\ \\
So we see in either of cases \textit{Case 1} and \textit{Case 2}, the origin that is the equilibrium point is unstable.
\subsubsection{Verification of \textit{Theorem 1}}
We have the system in the eigenbasis as,
\begin{equation}
\begin{split}
& \dot{u}=\frac{1}{a}\left[c(bu+v)^2+c(bu+v)au\right] \\ &
\dot{v}=aX_{e}v+a(bu+v)^2-b(bu+v)au-\frac{b}{a}\left[c(bu+v)^2+c(bu+v)au\right]
\end{split}
\end{equation}
Denote,
\begin{equation}
\begin{split}
& f(u,v)=\frac{1}{a}\left[c(bu+v)^2+c(bu+v)au\right] \\ &
g(u,v)=a(bu+v)^2-b(bu+v)au-\frac{b}{a}\left[c(bu+v)^2+c(bu+v)au\right]
\end{split}
\end{equation}
These are the nonlinear parts of the system. \\
In the above equations puting $u=-u$ and $v=-v$ we see that
\begin{equation}
\begin{split}
& f(u,v)=f(-u,-v) \\ &
g(u,v)=g(-u,-v)
\end{split}
\end{equation}
that is they are even. On the other hand the centre manifold equation is,
\begin{equation}
\begin{split}
h(u)=\frac{b^2c}{aX_{e}}(1+\frac{b}{a})u^2
\end{split}
\end{equation}
which is even and has leading order term as a even power of $u$. This verifies our \textit{Theorem 1} and the conjecture made in \textit{Theorem 1}.
\section{Conclusion}
Through our work we have discussed the centre manifold theory with application to a general $3-D$ nonlinear system with second order nonlinearities. We did the stability analysis of the system after reducing the state equations on a $2-D$ centre manifold. The reduced phase space was found to possess a stable and unstable fixed point. The system approaches the stable fixed point which is the origin from the unstable fixed point in asymptotic time with a spiral motion towards the origin. In the $x-t$ and $y-t$ plane it was shown to exhibit a oscillatory behaviour with amplitude decaying with time. Then we cast the example of protein molecules which is a $2-D$ nonlinear system. Stability analysis is done around a fixed point which happens to be the origin in our case. The equations obtained point towards a unstable system which very much matches with the predictions of standard theory.
With reference\cite{tsuruyama2017kinetic} to the chosen example of the kinetics of protein assembly, the success of our study lies
in the fact that we have shown mathematically for the oscillation to occur
we require a minimum of three original dimensions with a reduced system of 2-
dimensions on the centre manifold for obtaining the phase plane where the
oscillations take place. If the dimension of the original system is two we can only predict the
stability around the equilibrium point as we have shown in our example.
Along the course of discussion we also state and prove a theorem along with a conjecture made in the theorem. We showed that our theorem and conjecture are true in the light of obtained results for the example that we consider. The significance of the theorem that we prove lies in the fact that we can at once do the stability analysis around the equilibrium point without delving into detailed analysis. Centre manifold analysis requires finding the centre manifold equation and further the reduced system of equations on the centre manifold. But if the nonlinear parts of the decoupled equations are either even or odd we can at once guess the parity of the centre manifold equation and subsequently the stability of the system around the equilibrium point without doing the detailed analysis.\par
As a future extension of our work we would like to mention the above analysis could be done along the line shown in our work for a standard $n-D$ system with higher order nonlinearities. Higher dimensional systems often arise in physics for example a Hamiltonian system with two degrees of freedom where the two positions and two momenta give rise to a 4-D system of O.D.E in position and momentum. Higher dimensional systems with high degrees of nonlinearities are also encountered frequently in engineering problems specifically those related to aerodynamical models. So investigating such a system is of natural interest to see if the predictions of the theory also incorporates the higher order nonlinearities of a higher dimensional system and if the theory works well in all different ranges and all different situations.
\bibliographystyle{unsrtnat}
|
2,869,038,154,794 | arxiv | \section{Introduction}
In many research areas, it is important to assess the distributional effects
of covariates on an outcome variable. Several methods have been implemented in
the literature to study this. A prolific line of research is a combination of
conditional mean and quantile regression models together with micro simulation
exercises, as in \cite{AutorKatzKearney05}, \cite{MachadoMata05}, and
\cite{Melly05} (see \citet{FortinLemieuxFirpo11} for a review). A more recent
and popular method is the recentered influence function (RIF) regression of
\cite{FirpoFortinLemieux09}, which directly estimates the effect of a change
in the covariate distribution on a functional of the unconditional
distribution of the outcome variable. The functional of interest can be the
mean, quantile, or any other aspect of the unconditional distribution.
Consider, as an example, the unconditional quantile of the outcome variable
$Y$. Let $F_{Y}$ be the unconditional distribution function of $Y,$ then the
$\tau$-quantile of $F_{Y}$ is defined by
\[
Q_{\tau}[Y]:=\arg\min\{q:\tau\leq F_{Y}(q)\}\ \ \mbox{for}\ \ \tau\in(0,1).
\]
In this paper, we seek to study how $Q_{\tau}[Y]$ changes when we induce an
infinitesimal change in a covariate $X\in\mathbb{R}$, \ allowing the presence
of other observable covariates $W$ and unobservable covariates collected in
$U.$ These covariates and the outcome variable are related via a structural or
causal function $h$ so that $Y=h(X,W,U)$. We consider a sequence of policy
experiments that change $X$ into $X_{\delta}=\mathcal{G}(X;\delta)$ for a
smooth function $\mathcal{G}(\cdot;\cdot)$. The policy experiments are indexed
by $\delta$ satisfying $\mathcal{G}(X;0)=X.$ That is, $\delta=0$ corresponds
to the \emph{status quo} policy. With this induced change in $X$, the outcome
variable becomes $Y_{\delta}=h(X_{\delta},W,U)=h(\mathcal{G}(X;\delta),W,U)$
where the distribution of $\left( X,W,U\right) $ is held constant. Our
policy experiment has a \emph{ceteris paribus} interpretation at the
population level: we change $X$ into $X_{\delta}$ while holding the stochastic
dependence among $X,W,$ and $U$ constant. Such a policy experiment is
implementable if the covariate $X$ is not a causal factor for either $W$ or
$U.$ In this case, when we intervene $X$ and change it into $X_{\delta}$, $W$
and $U$ will not change. This does not rule out the stochastic dependence
among $X,$ $W,$ and $U.$ In the meanwhile, the structural function
$h(\cdot,\cdot,\cdot)$ is also held constant. The main parameter of interest
is the marginal effect of the change on the unconditional quantile of the
outcome variable:
\[
\Pi_{\tau}:=\lim_{\delta\rightarrow0}\frac{Q_{\tau}[Y_{\delta}]-Q_{\tau}%
[Y]}{\delta}.
\]
\cite{FirpoFortinLemieux09} develop methods to study what corresponds to a
location shift $X_{\delta}=X+\delta$. This shift affects the entire
unconditional distribution of $Y=h(X,W,U)$, moving it towards a counterfactual
distribution of $Y_{\delta}=h(X+\delta,W,U)$. One of the main results in
\citet[][p.958,
eq. (6)]{FirpoFortinLemieux09} is that $\Pi_{\tau}$ can be represented as an
average derivative:
\[
\Pi_{\tau}=E\left[ \dot{\psi}_{x}\left( X,W\right) \right] ,
\]
where
\[
\dot{\psi}_{x}\left( x,w\right) =\frac{\partial E\left[ \psi\left(
Y,\tau,F_{Y}\right) |X=x,W=w\right] }{\partial x},
\]
$\psi\left( y,\tau,F_{Y}\right) =\left[ \tau-1\left\{ y\leq Q_{\tau
}[Y]\right\} \right] /f_{Y}(Q_{\tau}[Y])$ is the influence function of the
quantile functional, and $f_{Y}(Q_{\tau}[Y])$ is the unconditional density of
$Y$ evaluated at the $\tau$-quantile $Q_{\tau}[Y].$ The unconditional quantile
effect $\Pi_{\tau}$ can then be estimated by first running an unconditional
quantile regression (henceforth, UQR), which involves regressing the influence
function $\psi\left( Y_{i},\tau,F_{Y}\right) $ on the covariates
$(X_{i},W_{i})$ and then taking an average of the partial derivatives of the
regression function with respect to $X.$
The same method is applicable to other functionals of interest --- we only
need to replace $\psi\left( y,\tau,F_{Y}\right) $ by the influence function
underlying the functional we care about. This leads to the general RIF
regression of \cite{FirpoFortinLemieux09}. The potential simplicity and
flexibility that the methodology offers motivates subsequent research to expand
the use of RIF regressions. On the empirical side, after its introduction, RIF
regressions became a popular method for analyzing and identifying the
distributional effects on outcomes in terms of changes in observed
characteristics in areas such as labor economics, income and inequality,
health economics, and public policy. On the theoretical side,
\cite{Rothe2012} provides a generalization of \cite{FirpoFortinLemieux09} for
the case of location shifts, and, more recently, \cite{SasakiUraZhang20}
study the high-dimensional setting while \cite{InoueLiXu21} focus on the two-sample
problem.
This paper extends the UQR and RIF regression in several ways. First, we study
general counterfactual policy changes, of which the location shift is a
special case. Our framework allows for any smooth and invertible intervention
of the target covariates. As a complement to the existing literature that
focuses on changing the \emph{marginal distribution} of the target covariates,
we consider changing the \emph{values} of the target covariates directly. An
advantage of our approach is that the changes under consideration are directly
implementable. We note that it may not be easy to induce a desired shift in
the marginal distribution, and when possible, such a shift is often achieved
via transforming the target covariates, which is what we consider here.
Second, we provide extensive discussions of a counterfactual policy that, in
addition to the location shift, affects the scale of a covariate. For example,
we may consider $X_{\delta}=X/(1+\delta)+\delta.$ We find that in this case,
the marginal effect can be decomposed as the sum of two effects: one related
to the location shift and the other related to the scale shift. In order to
interpret the scale effect, we introduce the \textit{quantile-standard
deviation elasticity}: the percentage change in the unconditional quantiles of
the outcome variable induced by a 1\% change in the standard deviation of the
target covariate.
Third, we allow the target covariates to be endogenous, and we characterize
the asymptotic bias of the unconditional effect estimator when the endogeneity
is not appropriately accounted for. We eliminate the endogeneity bias using a
control variable/function approach. Such an approach is analogous to the
method of causal inference under the unconfoundedness assumption.
Fourth, in the Supplemental Appendix we also consider the case of simultaneous
shifts in different covariates. We focus on the case of simultaneous
location shifts in two covariates. This happens when a location shift in one
covariate induces a location shift in another covariate at the same time. For
example, $Y=h\left( X_{1},X_{2},W,U\right) $ for two scalar target
covariates $X_{1}$ and $X_{2}$, and the policy induces $X_{1\delta}%
=X_{1}+\delta$ and $X_{2\delta}=X_{2}-\delta$. Our approach can easily
accommodate this case, and we show the simultaneous effect can be obtained as
a linear combination of individual effects obtained by considering one change
at a time.
Finally, we propose consistent and asymptotically normal semiparametric
estimators of the location-scale effect and the simultaneous effect. The
estimators can be easily implemented in empirical work using either a probit
or logit specification of the conditional distribution function. We conduct an
extensive Monte Carlo study evaluating the finite sample performances of the
location-scale effect estimator and the accuracy of the normal approximation.
Simulation results show that the estimator works reasonably well under
different specifications and that the standard normal distribution provides a
good approximation to the finite sample distribution of a studentized test
statistic introduced in this paper.
As potential applications of our proposed approach, consider the following
empirical examples to motivate its use.
\begin{example}
\label{Example 1} \textbf{Effect of increasing education on wage inequality.}
In a Mincer equation, log wages are modeled as a function of certain
observable covariates such as years of education. A study of the effect of a
shift in education on wage inequality could be implemented using our proposed
framework. We can accommodate a counterfactual policy experiment where there
may be not only a general increase in the education level but also a change in
its dispersion.
\end{example}
\begin{example}
\label{Example_Smoking} \textbf{Smoking and birth weight.} Consider a tax
levied on the consumption of cigarettes. It is reasonable to think that the
consumption $X$ will be reduced to $X/(1+\delta)$, where $\delta$ is the tax
burden on the consumer. Thus, the tax induces a reduction in the level and
dispersion of cigarette consumption. We will use the proposed method to assess its effect on the
distribution of birth weights.
\end{example}
\begin{example}
\textbf{Wage controls and earnings distribution} During War World II, the
National War Labor Board imposed wage controls in the form of brackets: wages
below the bracket were allowed to rise, while wages above the bracket were not
allowed to rise. Importantly, these brackets differed across industries,
occupations, and regions. \cite{ziebarth2022} use the tools developed in this
paper to analyze the effect of a more uniform (less dispersion) distribution
of brackets on the distributions of earnings.
\end{example}
\begin{example}
\textbf{Trade integration and skill distribution} \cite{Gu2020} document the
impact of trade integration on both the mean and the standard deviation of the
skill distribution across municipalities in Denmark. Moreover, as argued by
\cite{Hanushek2008}, skills are related to income distribution. Thus, a
quantification of the impact of a scale effect in the skills distribution on
the quantiles of the income distribution appears to be relevant.
\end{example}
\begin{example}
\textbf{Days in a job training program.} \cite{SasakiUraZhang20} develop
high-dimensional UQR to analyze the effect on wages of counterfactual increase
in: $(i)$ the days of participation in a job training program; and $(ii)$ the
days actually taking classes in the same job training program. Our
simultaneous effect analysis can consider, for example, a reduction in $(i)$
with a simultaneous increase in $(ii)$. Thus, our paper can be used to study
the effect of a more concentrated job training program.
\end{example}
We illustrate the proposed method with two empirical applications. The first
one is related to Example \ref{Example 1}: the effect of changing education on
wage inequality, decomposing it into location and scale effects. Empirical
results reveal the contrasting nature of the two effects. The location effects
are seen to be positive and relatively similar across quantiles. On the other
hand, the scale effects are highly heterogeneous and monotonically decreasing
across quantiles. Hence, the scale effects can more than offset the location
effects. This shows that not accounting for both shifts may result in a biased
assessment of the policy effects on the quantiles of the outcome variable. The
second application is related to Example \ref{Example_Smoking} where we
estimate the unconditional effects of smoking during pregnancy on the birth
weight. The effects from reducing the mean and variance of the\ number of
cigarettes smoked are positive and are different for different quantiles of
the birth weight distribution.
The paper is organized as follows. Section \ref{locsca} studies the
unconditional effects of general policy interventions with the location-scale
shift as the main example. Section \ref{sec:dist_vs_cov} provides some further
discussion on the methodological contribution of this paper relative to
\cite{FirpoFortinLemieux09}. Section \ref{estimation} describes the estimator
of the location-scale effect and studies its asymptotic properties. Section
\ref{MC} reports the finite sample performance of the location-scale effect
estimator and the associated tests. Section \ref{app} presents the empirical
applications. Section \ref{conclusion} concludes. The proofs are in the
Appendix. The case of simultaneous changes and the details for a theoretical
example are given in the Supplementary Appendix.
A word on notation: we use $F_{Y|X}(y|x)$ and $f_{Y|X}(y|x)$ to denote the
cumulative distribution function and the probability density function of $Y,$
respectively, conditional on $X=x$. For a random variable $Z$, the
unconditional $\tau$-quantile is denoted by $Q_{\tau}[Z]$, \textit{i.e.},
$\Pr(Z\leq Q_{\tau}[Z])=\tau$, and its variance is denoted by $var(Z).$ For a
pair of random variables $Z_{1}$ and $Z_{2}$, the conditional quantile is
denoted by $Q_{\tau}[Z_{1}|z_{2}]$, \textit{i.e.}, $\Pr(Z_{1}\leq Q_{\tau
}[Z_{1}|z_{2}]|Z_{2}=z_{2})=\tau$. We adopt the following notational
conventions:
\[
\frac{\partial E(Z|X)}{\partial X}=\left. \frac{\partial E\left(
Z|X=x\right) }{\partial x}\right\vert _{x=X},\RIfM@\expandafter\text@\else\expandafter\mbox\fi{ }\frac{\partial
F_{Z|X}(z|X)}{\partial X}=\left. \frac{\partial F_{Z|X}(z|X=x)}{\partial
x}\right\vert _{x=X}.
\]
For a column vector $v,$ $d_{v}$ stands for the number of elements in $v.$
\section{Unconditional effects of general policy interventions}
\label{locsca}
\subsection{Introducing location-scale shifts}
We start with a general structural model $Y=h(X,W,U)$, where the function $h$
is unknown, and we only observe $(X,W)$ and $Y$. Here $X$ is univariate but
the dimension of $W$ is left unrestricted. All the unobserved causal factors
of $Y$ are collected in $U$. We are concerned with the effect on the
distribution of $Y$ of general (infinitesimal) changes in $X$, the
\textit{target} variable.
Perhaps the simplest example of a counterfactual change in $X$ is a location
shift: $X_{\delta}=X+\delta$. The popular method of UQR of
\cite{FirpoFortinLemieux09} can be used to assess the effect of such changes
in the unconditional quantiles of $Y$.\footnote{See Section
\ref{sec:dist_vs_cov} for a discussion about how this paper relates to
\cite{FirpoFortinLemieux09}.} In this paper we provide results for the general
case where $X_{\delta}=\mathcal{G}(X;\delta)$ for some (suitable) policy
function $\mathcal{G}$ chosen by the researcher or policy maker. A
counterfactual change in $X$ to $X_{\delta}=\mathcal{G}(X;\delta)$ induces a
counterfactual outcome $Y_{\delta}=h(\mathcal{G}(X;\delta),W,U)=h(X_{\delta
},W,U)$. Our parameter of interest, the \textit{marginal effect for the }%
$\tau$\textit{-quantile}, is an infinitesimal contrast of unconditional
quantiles and is defined as
\begin{equation}
\Pi_{\tau}:=\lim_{\delta\rightarrow0}\frac{Q_{\tau}[Y_{\delta}]-Q_{\tau}%
[Y]}{\delta},\label{eq:pi_tau}%
\end{equation}
whenever this limit exists.
A particular policy function that we analyze in detail is the following
\emph{location-scale shift} in $X$
\begin{equation}
X_{\delta}=\mathcal{G}(X;\delta)=\left( X-\mu\right) s(\delta)+\mu
+\ell(\delta).\label{eq_location_scale}%
\end{equation}
Here, $\mu$ is a \textit{known} policy parameter, and we refer to $\ell
(\delta)$ as the location shift and to $s(\delta)>0$ as the scale shift. In
order to take limits to find $\Pi_{\tau}$, we assume that $\ell(\delta)$ and
$s(\delta)$ are continuously differentiable functions of the scalar $\delta$.
Both $\ell(\delta)$ and $s(\delta)$ are chosen by the researcher or policy
maker subject to the restriction that $s(0)=1$ and $\ell(0)=0$. Note that this
choice of $\mathcal{G}$ nests the case $X_{\delta}=X+\delta$ by choosing
$\ell(\delta)=\delta$ and $s(\delta)\equiv1$.
A distinctive feature of $X_{\delta}$ in \eqref{eq_location_scale} is that
\[
var[X_{\delta}]=s(\delta)^{2}var[X],
\]
and so it allows for the study of counterfactual changes in the dispersion of
the target variable. To see this, suppose that $s(\delta)<1$, then,
realizations of $X$ that are above/below $\mu$ are \textquotedblleft
moved\textquotedblright\ towards $\mu$, followed by a location shift of
$\ell(\delta)$. Therefore, we have a constant location shift, given by
$\ell(\delta)$, and a relative location shift induced by the scale shift,
which tends to bunch observations near $\mu$. The result is a reduction of the
variance of $X$. If, on the other hand, $s(\delta)>1$, then the counterfactual
policy moves $X$ away from $\mu$ and consequently increases its variance.
Under some regularity assumptions spelled below, the marginal effect
$\Pi_{\tau}$ corresponding to the policy function $\mathcal{G}$ given in
\eqref{eq_location_scale} can be decomposed into the sum of two effects: one
associated with the location shift governed by $\ell(\delta)$, and the other
associated with the scale shift $s(\delta)$. The former corresponds to a
version of the estimand studied by \cite{FirpoFortinLemieux09}. The latter
effect is, to the best of our knowledge, new.
Subsection \ref{subsection_general_policy} contains a rigorous development of
our main results for a general policy function. Readers interested in the
location-scale shift only can skip subsection \ref{subsection_general_policy}
and focus on subsections \ref{subsection_location_scale_policy} and
\ref{Sec: elasticity} where we provide the specific results for the
location-scale shift, discuss their interpretations, and offer examples.
\subsection{Results for a general policy function}
\label{subsection_general_policy}
Central to our results is the counterfactual policy function $\mathcal{G}$,
which maps $X$ to $X_{\delta}$ and generates a counterfactual outcome
$Y_{\delta}$. As mentioned before, our parameter of interest $\Pi_{\tau}$
given in \eqref{eq:pi_tau}, compares the quantiles of
\begin{equation}
Y=h(X,W,U) \label{eq_model_0}%
\end{equation}
to the quantiles of
\begin{equation}
Y_{\delta}=h(X_{\delta},W,U)=h(\mathcal{G}(X;\delta),W,U). \label{eq_model_1}%
\end{equation}
An important assumption is that the distribution of $\left( X,W,U\right) $
in (\ref{eq_model_1}) is held the same as that in (\ref{eq_model_0}). To
understand the latter condition, we can consider two parallel worlds: the
worlds before and after the intervention. For each given $\delta,$ let
$\mathcal{G}^{-1}(x;\delta)$ be the inverse function of $\mathcal{G}%
(x;\delta)$ such that $\mathcal{G}(\mathcal{G}^{-1}(x;\delta);\delta)=x.$
After applying the inverse transform to the target covariate in the
post-intervention world, the distribution of $\left( \mathcal{G}%
^{-1}\mathcal{(}X^{\delta};\delta),W^{\delta},U^{\delta}\right) $ in the
post-intervention world is assumed to be the same as that of $\left(
X,W,U\right) $ in the pre-intervention world. Here, no change is induced on
$W$ and $U$ and so $\left( W^{\delta},U^{\delta}\right) $ is actually the
same as $(W,U)$ for every individual in the population.
In essence, we keep the structural function $h\left( \cdot,\cdot
,\cdot\right) $ and the distribution of $\left( X,W,U\right) $ intact
during the policy intervention. The effect under consideration is then the
policy effect due to the policy intervention only and thus has a \emph{ceteris
paribus} causal interpretation.
For notational economy, we write $x^{\delta}=\mathcal{G}^{-1}(x;\delta)$. Then
$X_{\delta}=x$ if and only if $X=x^{\delta}.$ Define the Jacobian of the
inverse transform $x\mapsto x^{\delta}:=\mathcal{G}^{-1}(x;\delta)$ as%
\[
J(x^{\delta};\delta):=\frac{\partial x^{\delta}}{\partial x}=\left[
\frac{\partial\mathcal{G}\left( x;\delta\right) }{\partial x}\right]
^{-1}\bigg|_{x=x^{\delta}}.
\]
Then, the joint probability density functions\ of the covariate vector before
and after the intervention satisfy
\[
f_{X_{\delta},W}(x,w)=J(x^{\delta};\delta)\cdot f_{X,W}(x^{\delta},w).
\]
For $\varepsilon>0$, define $\mathcal{N}_{\varepsilon}:=\left\{
\delta:\left\vert \delta\right\vert \leq\varepsilon\right\} $. We maintain
the following assumption.
\begin{assumption}
\label{Assumption:main} (i.a) For some $\varepsilon>0,$ $\mathcal{G}\left(
x;\delta\right) $ is continuously differentiable on $\mathcal{X\otimes
N}_{\varepsilon}$, where $\mathcal{X}$ is the support of $X.$
(i.b) $\mathcal{G}\left( x;\delta\right) $ is strictly increasing in $x$ for
each $\delta\in\mathcal{N}_{\varepsilon}.$
(i.c) $\mathcal{G}\left( x;0\right) =x$ for all $x\in\mathcal{X}$.
(ii) for $\delta\in\mathcal{N}_{\varepsilon}$, the conditional density of $U$
satisfies $f_{U|X_{\delta},W}(u|x,w)=f_{U|X,W}(u|x^{\delta},w)$, and the
support $\mathcal{U}$ of $U$ given $X$ and $W$ does not depend on $\left(
X,W\right).$
(iii.a) $x\mapsto f_{X,W}(x,w)$ is continuously differentiable for all
$w\in\mathcal{W}$ and
\[
\int_{\mathcal{W}}\int_{\mathcal{X}}\sup_{\delta\in\mathcal{N}_{\varepsilon}%
}\left\vert \frac{\partial\left[ J\left( x^{\delta};\delta\right)
f_{X,W}(x^{\delta},w)\right] }{\partial\delta}\right\vert dxdw<\infty
\]
where $\mathcal{W}$ is the support of $W.$
(iii.b) $x\mapsto f_{U|X,W}(u|x,w)$ is continuously differentiable for all
$\left( u,w\right) $ and
\begin{align*}
\int_{\mathcal{W}}\int_{\mathcal{X}}\int_{\mathcal{U}}\sup_{\delta
\in\mathcal{N}_{\varepsilon}}\left\vert \frac{\partial}{\partial\delta}\left[
f_{U|X,W}(u|x^{\delta},w)f_{X,W}(x^{\delta},w)\right] \right\vert dudxdw &
<\infty,\\
\int_{\mathcal{W}}\int_{\mathcal{X}}\int_{\mathcal{U}}\sup_{\delta
\in\mathcal{N}_{\varepsilon}}\left\vert \frac{\partial f_{X,W}(x^{\delta}%
,w)}{\partial\delta}\right\vert f_{U|X,W}(u|x,w)dudxdw & <\infty.
\end{align*}
(iv) $f_{X,W}(x,w)$ is equal to $0$ on the boundary of the support of $X$
given $W=w$ for all $w\in\mathcal{W}.$
(v) $f_{Y}(Q_{\tau}[Y])>0.$
\end{assumption}
\begin{remark}
Assumption \ref{Assumption:main}(i) imposes some restrictions on the policy
function $\mathcal{G}\left( x;\delta\right) .$ It is reasonable that
$\mathcal{G}\left( x;\delta\right) $ is strictly increasing in $x,$ as a
non-monotonic and non-invertible function does not seem to be practically
relevant. The strictly increasing property implies that $J\left(
x;\delta\right) >0$ for all $x\in\mathcal{X}$ and $\delta\in\mathcal{N}%
_{\varepsilon}.$ The condition that $\mathcal{G}\left( x;0\right) =x$ says
that there is no intervention when $\delta=0,$ and it implies that $J\left(
x;0\right) =1$ for all $x\in\mathcal{X}.$ Assumption \ref{Assumption:main}%
(ii) assumes that how $U$ depends on the covariate vector is maintained when
we induce a change in the covariate vector. Note that Assumption
\ref{Assumption:main}(ii) is different from $f_{U|X_{\delta},W}%
(u|x,w)=f_{U|X,W}(u|x,w)$, which in general cannot hold when $U$ depends on
$X$ and $W$. The counterfactual model in (\ref{eq_model_1}) says that we
maintain the structure of the causal system. Assumption \ref{Assumption:main}%
(ii) says that we also maintain how the unobservable depends on the
observables. As discussed above, we also implicitly assume that $\left(
\mathcal{G}^{-1}\mathcal{(}X^{\delta};\delta),W^{\delta}\right) $ has the
same distribution as $\left( X,W\right) .$ The rest of Assumption
\ref{Assumption:main} consists of regularity conditions.
\end{remark}
\begin{remark}
Assumption \ref{Assumption:main} does not assume that $U$ is independent of
$\left( X,W\right) .$ It does not assume that $U$ is conditionally
independent of $X$ given $W$ either. Assumption \ref{Assumption ID} below will
impose identification assumptions.
\end{remark}
The following theorem characterizes the effects of the policy change on the
distribution of $Y_{\delta}$ and its quantiles.
\begin{theorem}
\label{th:uqpe_scale} Let Assumption \ref{Assumption:main} hold.
(i) For each $\left( x,w\right) \in\mathcal{X}\otimes\mathcal{W},$%
\[
\lim_{\delta\rightarrow0}\frac{f_{X_{\delta},W}(x,w)-f_{X,W}(x,w)}{\delta
}=-\frac{\partial}{\partial x}\left[ \kappa\left( x\right) f_{X,W}%
(x,w)\right] ,
\]
where
\[
\kappa\left( x\right) :=\frac{\partial\mathcal{G}(x;\delta)}{\partial\delta
}\bigg|_{\delta=0}.
\]
(ii) As $\delta\rightarrow0$, we have
\begin{align*}
& \frac{F_{Y_{\delta}}(y)-F_{Y}\left( y\right) }{\delta}\\
& \rightarrow E\left[ \left( \frac{\partial F_{Y|X,W}(y|X,W)}{\partial
X}-\mathds1\left\{ h(X,W,U)\leq y\right\} \frac{\partial\ln f_{U|X,W}%
(U|X,W)}{\partial X}\right) \kappa\left( X\right) \right]
\end{align*}
uniformly in $y\in\mathcal{Y}$, the support of $Y$.
(iii) The marginal effect of the intervention $X_{\delta}=\mathcal{G}%
(X;\delta)$ on the $\tau$-quantile of the outcome variable $Y$ can be
represented by
\begin{equation}
\Pi_{\tau}=A_{\tau}-B_{\tau} \label{Pi_theorem}%
\end{equation}
where%
\begin{align*}
A_{\tau} & =E\left[ \frac{\partial E\left[ \psi\left( Y,\tau
,F_{Y}\right) |X,W\right] }{\partial X}\kappa\left( X\right) \right] ,\\
B_{\tau} & =E\left[ \psi\left( Y,\tau,F_{Y}\right) \frac{\partial\ln
f_{U|X,W}(U|X,W)}{\partial X}\kappa\left( X\right) \right] ,
\end{align*}
and
\[
\psi\left( y,\tau,F_{Y}\right) =\frac{\tau-1\left( y<Q_{\tau}[Y]\right)
}{f_{Y}(Q_{\tau}[Y])}.
\]
\end{theorem}
\begin{remark}
To understand Theorem \ref{th:uqpe_scale}(i), we can write%
\[
f_{X_{\delta},W}(x,w)-f_{X,W}(x,w)=f_{X_{\delta},W}(x,w)-f_{X,W}(x^{\delta
},w)+f_{X,W}(x^{\delta},w)-f_{X,W}(x,w).
\]
It is quite intuitive that the second term is approximately $\delta\cdot
\frac{\partial x^{\delta}}{\partial\delta}|_{\delta=0}\cdot\frac{\partial
f_{X,W}(x,w)}{\partial x}=-\delta\cdot\kappa\left( x\right) \cdot
\frac{\partial f_{X,W}(x,w)}{\partial x}$ when $\delta$ is small. Here we have
used $\kappa\left( x\right)$ also equals $-\frac{\partial x^{\delta}}{\partial\delta
}|_{\delta=0}$ (see the Appendix for a proof). The first term reflects the effect from the Jacobian of the
transformation. Indeed, $f_{X_{\delta},W}(x,w)-f_{X,W}(x^{\delta},w)=\left[
J(x^{\delta};\delta)-J(x^{\delta};0)\right] f_{X,W}(x^{\delta},w)$ as
$J(x^{\delta};0)=1.$ The first term is then approximately equal to
$\delta\cdot f_{X,W}(x,w)\cdot\frac{\partial J\left( x,\delta\right)
}{\partial\delta}\big|_{\delta=0}.$ But
\[
\frac{\partial J\left( x,\delta\right) }{\partial\delta}\bigg|_{\delta
=0}=\frac{\partial}{\partial\delta}J\left( x,\delta\right) \bigg|_{\delta
=0}=\frac{\partial}{\partial\delta}\frac{\partial x^{\delta}}{\partial
x}\bigg|_{\delta=0}=\frac{\partial}{\partial x}\frac{\partial x^{\delta}%
}{\partial\delta}\bigg|_{\delta=0}=-\frac{\partial\kappa\left( x\right)
}{\partial x}%
\]
and hence the first term is approximately $-\delta\cdot f_{X,W}(x,w)\cdot
\frac{\partial\kappa\left( x\right) }{\partial x}.$ Combining these two
approximations yields Theorem \ref{th:uqpe_scale}(i).
\end{remark}
\begin{remark}
By definition, $\kappa\left( x\right) $ measures the marginal change of
$\mathcal{G}(x;\delta)$ as we increase $\delta$ from zero infinitesimally.
Theorems \ref{th:uqpe_scale} (ii) and (iii) show that only $\kappa\left(
x\right) $ appears in the marginal effect and the Jacobian does not. This is
not surprising, as what matters for the marginal effect is the marginal change
in the policy function.
\end{remark}
\begin{remark}
Theorem \ref{th:uqpe_scale}(iii) represents the structural parameter
$\Pi_{\tau}$ in terms of statistical objects. While the first term $A_{\tau}$
is identifiable, the second term $B_{\tau}$, which involves the conditional
density of $U$ given $X$ and $W,$ is not. If we use $\hat{A}_{\tau},$ a
consistent estimator of $A_{\tau},$ as an estimator of $\Pi_{\tau},$ then the
second term $B_{\tau}$ is the asymptotic bias of $\hat{A}_{\tau}.$ This bias
is an endogeneity bias, as it is in general not equal to zero when $X$ is not
independent of $U$ (conditioning on $W$.) Similar results have been
established in \cite{yixiao2021} but only for location shifts. If we do not
have the identification condition such as what is given in Assumption
\ref{Assumption ID} below, Theorem \ref{th:uqpe_scale}(iii) allows us to use a
bound approach to bound $B_{\tau}$ and infer the range of the policy effect or
conduct a sensitivity analysis similar to that in \cite{martinez2020}.
\end{remark}
\begin{remark}
\label{Remark: more general functional}While the paper focuses on the quantile
functional, Theorem \ref{th:uqpe_scale}(iii) is formulated in a general way.
The result holds for any Hadamard differentiable functional and for the mean
functional. We only need to replace $\psi\left( y,\tau,F_{Y}\right) $ by the
influence function of the functional that we are interested in. For example,
for the mean functional, we can replace $\psi\left( y,\tau,F_{Y}\right) $ by
$y-E(Y)$, and Theorem \ref{th:uqpe_scale}(iii) remains valid.
\end{remark}
To identify $\Pi_{\tau},$ we make the following independence or conditional
independence assumption.
\begin{assumption}
\label{Assumption ID}For $\delta\in\mathcal{N}_{\varepsilon}$, the
unobservable $U$ satisfies either $f_{U|X,W}(u|x,w)=f_{U|X,W}(u|x^{\delta
},w)=f_{U}(u)$ or $f_{U|X,W}(u|x,w)=f_{U|X,W}(u|x^{\delta},w)=f_{U|W}(u|w).$
\end{assumption}
Under the above assumption, $\partial\ln f_{U|X,W}(u|x,w)/\partial x=0$ and
the second term $B_{\tau}$ in (\ref{Pi_theorem}) vanishes. In this case,
$\Pi_{\tau}=A_{\tau}$ and hence is identified. The corollary below then
follows directly from Theorem \ref{th:uqpe_scale}(iii).
\begin{corollary}
\label{Corrollary:uqpe_scale}Let Assumption \ref{Assumption:main} hold with
Assumption \ref{Assumption:main} (ii) strengthened to Assumption
\ref{Assumption ID}. Then \
\begin{equation}
\Pi_{\tau}=E\left[ \frac{\partial E\left[ \psi\left( Y,\tau,F_{Y}\right)
|X,W\right] }{\partial X}\kappa\left( X\right) \right] =\frac{1}%
{f_{Y}(Q_{\tau}[Y])}E\left[ \frac{\partial\mathcal{S}_{Y|X,W}\left( Q_{\tau
}[Y]|X,W\right) }{\partial X}\kappa\left( X\right) \right]
\label{eq_pi_tau_general}%
\end{equation}
where $\mathcal{S}_{Y|X,W}\left( \cdot|x,w\right) :=1-F_{Y|X,W}\left(
\cdot|x,w\right) $ is the\ conditional survival function.
\end{corollary}
\begin{remark}
Both conditions in Assumption \ref{Assumption ID} require that $f_{U|X,W}%
(u|x,w)=f_{U|X,W}(u|x^{\delta},w)$. This is related to the assumption in
\citet[][pp.955-957]{FirpoFortinLemieux09}, framed as \textquotedblleft
maintaining the conditional distribution of Y given X
unaffected.\textquotedblright\ In essence, \citet{FirpoFortinLemieux09}
requires\ $f_{U|X}(u|x)=f_{U|X}(u|x^{\delta}).$ When this condition fails, we
may still have $f_{U|X,W}(u|x,w)=f_{U|X,W}(u|x^{\delta},w).$ Such a condition
has also been used in \cite{Lieli2020} and \cite{Spini2021} in a context of
extrapolation to populations with different distributions of the covariates.
\end{remark}
\begin{remark}
\label{control_variable} The first condition in Assumption \ref{Assumption ID}
is satisfied if $U$ is independent of $(X,W)$.
In our view, this condition is hard to
achieve in empirical applications. The second condition in Assumption
\ref{Assumption ID}, which is commonly used to achieve identification in
applied work, is a conditional independence assumption. Such a condition is
often referred to as the unconfoundedness condition in the causal inference
literature. When $W$ consists of only causal variables entering the causal
function $h(X,W,U),$ the second condition in Assumption \ref{Assumption ID}
may not hold and $U$ may be endogenous. In this case, we can find control
variables $W_{c}$ so that
\[
f_{U|X,W,W_{c}}(u|x,w,w_{c})=f_{U|X,W,W_{c}}(u|x^{\delta},w,w_{c}%
)=f_{U|W,W_{c}}(u|w,w_{c}).
\]
After replacing $W$ by $W^{\ast}=(W,W_{c}),$ Corollary
\ref{Corrollary:uqpe_scale} continues to hold. To see this, we can write the
structural function as $h^{\ast}(X,W^{\ast},U)$, but $h^{\ast}(X,W^{\ast
},U)=h(X,W,U).$ That is, we include the control variables in the structural
function and restrict the structural function to be a constant function of the
control variables. With such a conceptual change, our proof goes through
without any change. We have, therefore, removed the endogenous bias, namely
$B_{\tau}$, by using a control variable/function approach.
\end{remark}
We note in passing that Corollary \ref{Corrollary:uqpe_scale} has the
following alternative representation:
\[
\Pi_{\tau}=\left\langle E\left[ \frac{\partial E\left[ \psi\left(
Y,\tau,F_{Y}\right) |X,W\right] }{\partial X}\bigg| X\right] ,\frac
{\partial\mathcal{G}(X;\delta)}{\partial\delta}\bigg|_{\delta=0}\right\rangle
,
\]
where $\left\langle \cdot,\cdot\right\rangle $ is the inner product defined by
$\left\langle h\left( X\right) ,g\left( X\right) \right\rangle :=E\left[
h\left( X\right) g\left( X\right) \right] $ in the space $\mathcal{L}%
_{2}\left( X\right) .$ By the Cauchy-Schwarz inequality, $|\langle h\left(
X\right) ,g\left( X\right) \rangle|\leq\left\Vert h\left( X\right)
\right\Vert \left\Vert g\left( X\right) \right\Vert $, where $\left\Vert
\cdot\right\Vert $ is the norm defined by $\left\Vert h\left( X\right)
\right\Vert :=\sqrt{\left\langle h\left( X\right) , h\left( X\right)
\right\rangle }$. Consider the class of policy functions with a unit norm,
namely $\left\Vert \frac{\partial\mathcal{G}(X;\delta)}{\partial\delta
}\bigg|_{\delta=0}\right\Vert =1.$ Then
\[
|\Pi_{\tau}|\leq\left\Vert E\left[ \frac{\partial E\left[ \psi\left(
Y,\tau,F_{Y}\right) |X,W\right] }{\partial X}\bigg|X\right] \right\Vert .
\]
Thus, if a policy function satisfies
\[
\frac{\partial\mathcal{G}(X;\delta)}{\partial\delta}\bigg|_{\delta=0}=E\left[
\frac{\partial E\left[ \psi\left( Y,\tau,F_{Y}\right) |X,W\right]
}{\partial X}\bigg|X\right] \cdot\left\Vert E\left[ \frac{\partial E\left[
\psi\left( Y,\tau,F_{Y}\right) |X,W\right] }{\partial X}\bigg|X\right]
\right\Vert ^{-1},
\]
then it achieves the highest $\Pi_{\tau}$ (in magnitude) in this class. We
leave optimal policy designs based on a cost-benefit analysis for future research.
\subsection{Results for the location-scale shift}
\label{subsection_location_scale_policy}
In this subsection we obtain a representation for $\Pi_{\tau}$ for the
particular case of the location-scale shift given in (\ref{eq_location_scale}%
):
\[
X_{\delta}=\mathcal{G}(X;\delta) = \left( X-\mu\right) s(\delta)+\mu
+\ell(\delta).
\]
The corollary below also follows directly from Theorem \ref{th:uqpe_scale}(iii).
\begin{corollary}
\label{Corrollary:uqpe_scale_2}Let Assumption \ref{Assumption:main} hold with
Assumption \ref{Assumption:main} (ii) strengthened to Assumption
\ref{Assumption ID}. Then, for the location-scale shift in
(\ref{eq_location_scale}) with $\ell(0)=0,$ $s(0)=1,$ and $s(\delta)>0,$ the
marginal effect can be decomposed as
\begin{equation}
\Pi_{\tau}=\Pi_{\tau,L}+\Pi_{\tau,S}, \label{eq_pi_tau}%
\end{equation}
where%
\begin{align*}
\Pi_{\tau,L} & =\frac{\dot{\ell}\left( 0\right) }{f_{Y}(Q_{\tau}[Y])}%
\int_{\mathcal{W}}\int_{\mathcal{X}}\frac{\partial\mathcal{S}_{Y|X,W}(Q_{\tau
}[Y]|x,w)}{\partial x}f_{X,W}(x,w)dxdw,\\
\Pi_{\tau,S} & =\frac{\dot{s}\left( 0\right) }{f_{Y}(Q_{\tau}[Y])}%
\int_{\mathcal{W}}\int_{\mathcal{X}}\frac{\partial\mathcal{S}_{Y|X,W}(Q_{\tau
}[Y]|x,w)}{\partial x}\left( x-\mu\right) f_{X,W}(x,w)dxdw,
\end{align*}
and $\mathcal{S}_{Y|X,W}\left( \cdot|x,w\right) :=1-F_{Y|X,W}\left(
\cdot|x,w\right) $ is the\ conditional survival function.
\end{corollary}
Corollary \ref{Corrollary:uqpe_scale_2} shows that the overall effect
$\Pi_{\tau}$ can be decomposed into the sum of $\Pi_{\tau,L}$ and $\Pi
_{\tau,S}.$ Here $\Pi_{\tau,L}$ is the location effect and is the estimand in
\cite{FirpoFortinLemieux09} when we set $\dot{\ell}(0)=1$ and $s\left(
\delta\right) \equiv1$. $\Pi_{\tau,S}$ is the scale effect and is present
whenever $s(\delta)$ is not identically 1 and $\dot{s}\left( 0\right) \neq
0$.\footnote{It can be seen that $\Pi_{\tau,S}$ depends on $\mu$. However, we
suppress this dependence from the notation for simplicity.}
To better understand the location and scale effects in Corollary
\ref{Corrollary:uqpe_scale_2}, consider the case that $X$ and $U$ are
independent and there is no $W.$ Then
\begin{align}
\Pi_{\tau,L} & =\frac{\dot{\ell}\left( 0\right) }{f_{Y}(Q_{\tau}[Y])}%
\int_{\mathcal{X}}\frac{\partial\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)}{\partial
x}f_{X}(x)dx,\label{Pi_L_independence_case}\\
\Pi_{\tau,S} & =\frac{\dot{s}\left( 0\right) }{f_{Y}(Q_{\tau}[Y])}%
\int_{\mathcal{X}}\frac{\partial\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)}{\partial
x}\left( x-\mu\right) f_{X}(x)dx.\nonumber
\end{align}
To sign the location effect $\Pi_{\tau,L}$, we can assess whether
$\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)$ is increasing in $x$ or not. If $\dot{\ell
}\left( 0\right) \geq0$ and $\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)$ is increasing
in $x$ on average, more precisely, $\int_{\mathcal{X}}\frac{\partial
\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)}{\partial x}f_{X}(x)dx\geq0,$ then $\Pi
_{\tau,L}\geq0.$ As an example, consider the case that $h\left( x,u\right) $
is increasing in $x$ for each $u.$ Then, $\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)$ is
increasing in $x$ for all $x\in\mathcal{X}$, and so $\Pi_{\tau,L}\geq0$ if
$\dot{\ell}\left( 0\right) \geq0.$
It is a bit more challenging to determine the sign of the scale effect
$\Pi_{\tau,S}$, which depends on, not only the function form of $\frac
{\partial\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)}{\partial x}$, but also the
distribution of $X.$ The next example provides some insight into the scale effect.
\begin{example}
\textbf{Normal Covariate.}\label{example_stein} Consider the linear model
$Y=\lambda+X\gamma+U$ where $X$ and $U$ are independent and $X\sim N(\mu
_{X},\sigma_{X}^{2})$. We can use Stein's lemma (see, for example,
\citet[][pp.124-125]{casella} and references therein) to gain some insight
into the scale effect. The lemma states that for a differentiable function $m$
such that $E[|m^{\prime}(X)|]<\infty$, $E[m(X)(X-\mu_{X})]=\sigma
^{2}E[m^{\prime}(X)]$ whenever $X\sim N(\mu_{X},\sigma_{X}^{2})$. Taking
$m(x)=\partial\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)/\partial x$ and using Stein's
lemma, we can express the scale effect for $\mu=\mu_{X}$ as
\[
\Pi_{\tau,S}=\frac{\dot{s}\left( 0\right) }{f_{Y}(Q_{\tau}[Y])}E\left[
\frac{\partial\mathcal{S}_{Y|X}(Q_{\tau}[Y]|X)}{\partial X}\left( X-\mu
_{X}\right) \right] =\frac{\dot{s}\left( 0\right) \sigma^{2}_X}%
{f_{Y}(Q_{\tau}[Y])}E\left[ \frac{\partial^{2}\mathcal{S}_{Y|X}(Q_{\tau
}[Y]|X)}{\partial X^{2}}\right] .
\]
Therefore, when $X$ is normal and $\dot{s}(0)>0,$ the scale effect is
non-negative (non-positive) if $\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)$ is a convex
(concave) function of $x$. It is interesting to see that the location effect
depends on the first order derivative of $\mathcal{S}_{Y|X}(Q_{\tau}[Y]|x)$
(see equation (\ref{Pi_L_independence_case})) while the scale effect depends
on its second-order derivative.
\end{example}
In the next example, we simplify $\Pi_{\tau,S}$ under the additional
assumption that $U$ is also normal.\footnote{See Section
\ref{Sec: Details Example Normal Location Model} in the Supplementary Appendix
for details.}
\begin{example}
\textbf{Normal Covariate and Normal Noise. }%
\label{example_normal_location_model} Consider a linear model $Y=\lambda
+X\gamma+U$ where $X$ and $U$ are independent. We have: $\Pi_{\tau,L}%
=\dot{\ell}\left( 0\right) \gamma$. In addition to the normal covariate
assumption $X\sim N(\mu_{X},\sigma^{2}_{X})$, suppose $U$ is also normal
$U\sim N(0,\sigma_{U}^{2})$. Then, for $\mu=\mu_{X},$
\[
\Pi_{\tau,S}=\dot{s}\left( 0\right) \sqrt{R_{YX}^{2}}Q_{\tau}[X_{\gamma
}^{\circ}]
\]
where $R_{YX}^{2}$ is the population R-squared defined by $R_{YX}%
^{2}:=var(\lambda+X\gamma)/var(Y)$ and $X_{\gamma}^{\circ}=\left( X-\mu
_{X}\right) \gamma.$ While the location effect is constant across quantiles, the scale
effect varies across quantiles.
\end{example}
In Example \ref{example_normal_location_model}, the scale effect, when
$\mu=\mu_{X}$, does not depend on $\mu_{X}$ or \textrm{sign}$\left(
\gamma\right) $. To understand this and obtain a more general result, we note
that $\Pi_{\tau,S}$ is proportional to the following covariance:
\begin{align}
cov\left( \frac{\partial\mathcal{S}_{Y|X}(Q_{\tau}[Y]|X)}{\partial
X},X\right) & =cov\left( f_{U}(Q_{\tau}[Y]-\delta-X\gamma),X\right)
\gamma\nonumber\\
& =cov\left( f_{U}(Q_{\tau}[U^{\circ}]-X_{\gamma}^{\circ}),X_{\gamma}%
^{\circ}\right) , \label{signing_covariance}%
\end{align}
where $U^{\circ}:=U+X_{\gamma}^{\circ}$ and we have used%
\begin{align*}
Q_{\tau}[Y] & =Q_{\tau}[U+\delta+X\gamma]=Q_{\tau}\left[ U+\left(
X-\mu_{X}\right) \gamma+\delta+\mu_{X}\gamma\right] \\
& =Q_{\tau}[U^{\circ}]+\delta+\mu_{X}\gamma.
\end{align*}
Now, if $X-\mu_{X}$ is symmetrically distributed around zero, then $X_{\gamma
}^{\circ}$ also shares this property. In this case, the covariance in
(\ref{signing_covariance}) does not depend on \textrm{sign}$\left(
\gamma\right) $ as the distributions of $X_{\gamma}^{\circ}$ and $U^{\circ}$
remain the same if we flip the sign of $\gamma.$ Also, since the distributions
of $X_{\gamma}^{\circ}$ and $U^{\circ}$ do not depend on $\mu_{X},$ the
covariance in (\ref{signing_covariance}) does not depend on $\mu_{X}.$ On the
other hand, for the denominator of the scale effect, we have%
\[
f_{Y}(Q_{\tau}[Y])=f_{Y}(Q_{\tau}[U^{\circ}]+\delta+\mu_{X}\gamma)=f_{U^{\circ}%
}\left( Q_{\tau}[U^{\circ}]\right) .
\]
If $X-\mu_{X}$ is symmetrically distributed around zero, then the distribution
of $U^{\circ}$ does not depend on $\mu_{X}$ or \textrm{sign}$\left(
\gamma\right) .$ Hence, $f_{Y}(Q_{\tau}[Y])$ does not depend on $\mu_{X}$ or
\textrm{sign}$\left( \gamma\right) .$
Since both the numerator and the denominator of $\Pi_{\tau,S}$ are invariant
to $\mu_{X}$ and \textrm{sign}$\left( \gamma\right) $, we obtain the
following proposition immediately.
\begin{proposition}
Consider the linear model $Y=\lambda+X\gamma+U$ where $X$ and $U$ are
independent. If $X-E\left[ X\right] $ is symmetrically distributed around
zero, then the scale effect computed for $\mu=E\left[ X\right] $ does not
depend on either $E\left[ X\right] $ or \textrm{sign}$\left( \gamma\right)
.$
\end{proposition}
\subsection{Interpretation of the scale effects\label{Sec: elasticity}}
Consider a situation where we only care about the scale effect, that is, we
set $\ell(\delta)\equiv0$. Then, we have $X_{\delta}=\mu+\left( X-\mu\right)
s(\delta)$. If we denote by $\sigma_{X}$ and $\sigma_{X_{\delta}}$ the
standard deviations of $X$ and $X_{\delta}$, respectively, then $\sigma
_{X_{\delta}}=\sigma_{X}s\left( \delta\right) $. To interpret $\Pi_{\tau,S}%
$, we assume $Q_{\tau}[Y_{\delta}]\neq0$ and consider the following
\textit{quantile-standard deviation elasticity}
\[
\mathcal{E}_{\tau,\delta}:=\frac{dQ_{\tau}[Y_{\delta}]}{Q_{\tau}[Y_{\delta}%
]}\left( \frac{d\sigma_{X_{\delta}}}{\sigma_{X_{\delta}}}\right) ^{-1}.
\]
By straightforward calculations, we have%
\[
\mathcal{E}_{\tau,\delta}=\frac{1}{Q_{\tau}[Y]}\frac{dQ_{\tau}[Y_{\delta}%
]}{d\delta}\left( \frac{1}{s\left( \delta\right) }\frac{ds\left(
\delta\right) }{d\delta}\right) ^{-1}.
\]
When $s(0)=1$ and $\dot{s}\left( 0\right) \neq0$, the elasticity at
$\delta=0$ is
\begin{equation}
\mathcal{E}_{\tau,\delta=0}=\frac{\Pi_{\tau,S}}{\dot{s}\left( 0\right)
Q_{\tau}[Y]}. \label{eq:elasticity_0}%
\end{equation}
Therefore, a $1\%$ increase in the standard deviation of $X$ results in a
$\Pi_{\tau,S}/\{\dot{s}\left( 0\right) Q_{\tau}[Y]\}\%$ change in the $\tau
$-quantile of $Y$.
\addtocounter{example}{-1}
\begin{example}
[Continued]Plugging $\Pi_{\tau,S}=\dot{s}\left( 0\right) \sqrt{R_{YX}^{2}%
}Q_{\tau}[X_{\gamma}^{\circ}]$, we obtain the \textit{quantile-standard
deviation elasticity} at $\delta=0$ as
\[
\mathcal{E}_{\tau,\delta=0}=\sqrt{R_{YX}^{2}}\frac{Q_{\tau}[X_{\gamma}^{\circ
}]}{Q_{\tau}[Y]}.
\]
So, $\mathcal{E}_{\tau,\delta=0}$ is positive if $Q_{\tau}[X_{\gamma}^{\circ
}]$ and $Q_{\tau}[Y]$ have the same sign. When $\alpha=0$, $\mu_{X}=0$, and
$X$ and $U$ are independent normals, we have $Q_{\tau}[X_{\gamma}^{\circ
}]/Q_{\tau}[Y]=\sqrt{R_{YX}^{2}}$ and so $\mathcal{E}_{\tau,\delta=0}%
=R_{YX}^{2}$. Interestingly, the \textit{quantile-standard deviation
elasticity} is equal to the population R-squared for all quantile levels.
\end{example}
Often times, when the outcome of interest (e.g., price and wage) is strictly
positive, we are interested in $\log Y$. In such a case, we denote the scale
effect by $\tilde{\Pi}_{\tau,S}$, and, since we set $\ell(\delta)\equiv0$ and
there is no location effect, it is given by
\[
\tilde{\Pi}_{\tau,S}:=\lim_{\delta\rightarrow0}\frac{Q_{\tau}[\log Y_{\delta
}]-Q_{\tau}[\log Y]}{\delta}.
\]
Since $\log\left( \cdot\right) $ is a strictly increasing transformation, we
have
\[
\tilde{\Pi}_{\tau,S}=\lim_{\delta\rightarrow0}\frac{\log Q_{\tau}[Y_{\delta
}]-\log Q_{\tau}[Y]}{\delta},
\]
and we can relate $\tilde{\Pi}_{\tau,S}$ to ${\Pi}_{\tau,S}$ by
\[
\tilde{\Pi}_{\tau,S}=\frac{1}{Q_{\tau}[Y]}{\Pi}_{\tau,S}.
\]
Comparing this to \eqref{eq:elasticity_0}, we obtain that the elasticity at
$\delta=0$ is
\[
\mathcal{E}_{\tau,\delta=0}=\frac{\tilde{\Pi}_{\tau,S}}{\dot{s}(0)}.
\]
This says that a $1\%$ increase in the standard deviation of $X$ results in a
$\tilde{\Pi}_{\tau,S}/\dot{s}\left( 0\right) \%$ change in the $\tau
$-quantile of $Y$. When $\dot{s}\left( 0\right) =1$ (\emph{e.g.}, $s\left(
\delta\right) =1+\delta),$ the scale effect $\tilde{\Pi}_{\tau,S}$ (based on
$\log\left( Y)\right) $ can be interpreted directly as the quantile-standard
deviation elasticity. When $\dot{s}\left( 0\right) =-1$ (\emph{e.g.},
$s\left( \delta\right) =1/(1+\delta)),$ the scale effect $\tilde{\Pi}%
_{\tau,S}$ has the same magnitude as the quantile-standard deviation
elasticity but with an opposite sign.
\section{Distribution intervention vs. value intervention}
\label{sec:dist_vs_cov}
The seminal paper by \cite{FirpoFortinLemieux09} (FFL hereafter in this
section) considers the effect of a change in the marginal distribution of $X$
from $F_{X}$ to either $(i)$ a \emph{fixed} $G_{X}$ or $(ii)$ a
\textquotedblleft variable\textquotedblright\ $G_{X,\delta}$ which depends on
$\delta$. \cite{Rothe2012} also focuses on these two cases.
In the first case, FFL considers a change from $F_{X}$ to a \emph{fixed}
$G_{X}$. Keeping $F_{Y|X}$ the same, a counterfactual distribution can be
obtained by $F_{Y}^{\ast}(y)=\int_{\mathcal{X}}F_{Y|X}(y|x)dG_{X}(x)$. For
$\delta\in\lbrack0,1]$, the convex combination $F_{Y,\delta}:=(1-\delta
)F_{Y}+\delta F_{Y}^{\ast}$ is a cdf and can be interpreted as a perturbation
of $F_{Y}$ in the direction of $F_{Y}^{\ast}-F_{Y}$. For a certain statistic
$\rho(F)$ of interest, such as a particular quantile of $Y$, we have
\begin{equation}
\frac{\partial\rho(F_{Y,\delta})}{\partial\delta}\bigg\vert_{\delta=0}%
=\int_{\mathcal{Y}}\psi_{\rho}(y,F_{Y})d(F_{Y}^{\ast}-F_{Y})(y)
\end{equation}
where $\psi_{\rho}(y,F_{Y})$ is the influence function of $\rho$ at $F_{Y}$.
See Chapter 20 in \cite{vanderVaart98} or Section 2.1 in
\cite{NeweyIchimura2022}. Since $F_{Y}^{\ast}(y)-F_{Y}(y)=\int_{\mathcal{X}%
}F_{Y|X}(y|x)d(G_{X}-F_{X})(x)$, we have
\begin{align}
\frac{\partial\rho(F_{Y,\delta})}{\partial\delta}\bigg\vert_{\delta=0} &
=\int_{\mathcal{Y}}\int_{\mathcal{X}}\psi_{\rho}(y,F_{Y})f_{Y|X}%
(y|x)dyd(G_{X}-F_{X})(x)\nonumber\\
& =\int_{\mathcal{X}}E[\psi_{\rho}(Y,F_{Y})|X=x]d(G_{X}-F_{X})(x).
\end{align}
This is essentially Theorem 1 in FFL, which provides a characterization of a
directional derivative of the functional $\rho\left( \cdot\right) $ in the
direction induced by a change in the marginal distribution of $X.$ The theorem
is silent on how the change in the marginal distribution is implemented.
The second case, covered in Corollary 1 in FFL, is closer to what we consider
here. In this case, $G_{X,\delta}$ is the distribution induced by the location
shift $X+\delta$. The counterfactual distribution is $F_{Y,\delta}^{\ast
}(y)=\int_{\mathcal{X}}F_{Y|X}(y|x)dG_{X,\delta}(x)$. The parameter of
interest is $\lim_{\delta\rightarrow0}\left[ \rho(F_{Y,\delta}^{\ast}%
)-\rho(F_{Y})\right] /\delta.$ Corollary 1 in FFL shows that
\begin{equation}
\lim_{\delta\rightarrow0}\frac{\rho(F_{Y,\delta}^{\ast})-\rho(F_{Y})}{\delta
}=\int_{\mathcal{X}}\frac{\partial E[\psi_{\rho}\left( y,F_{Y}\right)
|X=x]}{\partial x}dF_{X}(x). \label{eq:equivalence}%
\end{equation}
Our general intervention $X_{\delta}=\mathcal{G}(X;\delta)$ includes the above
location shift as a special case. To see this, we assume that $W$ is not
present and set $\mathcal{G}(X;\delta)=X+\delta$, in which case $\kappa\left(
x\right) =1,$ and it follows from Remark
\ref{Remark: more general functional} and Corollary
\ref{Corrollary:uqpe_scale} that $\Pi_{\rho}:=E\left[ \frac{\partial E\left[
\psi_{\rho}\left( Y,F_{Y}\right) |X\right] }{\partial X}\right]
=\int_{\mathcal{X}}\frac{\partial E[\psi_{\rho}(Y,F_{Y})|X=x]}{\partial
x}dF_{X}(x)$, which is identical to the right-hand side of
(\ref{eq:equivalence}). This shows that our approach is strictly more general
than the second case considered by FFL.
There is another main difference between FFL and our paper. From a broad point
of view, FFL considers the scenario where the conditional distribution of $Y$
given $X$ is fixed, and ask how the unconditional distribution of $Y$ would
change if the marginal distribution of $X$ had changed. This is largely a
predictive exercise unless the conditional distribution of $Y$ given $X$ has a
structural or causal interpretation, that is, $X$ is exogenous. In our paper,
we allow for an endogenous $X$ in the sense that $X$ and $U$ may be
correlated. This could arise, for example, when a common factor causes both
$X$ and $U$. As discussed in Remark \ref{control_variable}, $X$ and $U$ may be
dependent even after conditioning on the causal variable $W$. In such a case,
we need to find additional control variables that do not necessarily enter the
structural function $h$ such that $X$ and $U$ become conditionally independent
conditional on $W$ and these additional control variables. The endogeneity
problem is then addressed by using the control variable approach.
At the conceptual level, we consider the policy experiment where both the
structural function $h$ and the distribution of ($X,W,U$) are kept intact.
Given that $h$ is the same, we can say that the effect is causal and have a
\emph{ceteris paribus} interpretation. Given that the distribution of
($X,W,U$) is the same, the policy experiment applies to the current population
under consideration and is fully implementable. Hence the effect is what a
policy maker can achieve under the current environment and is therefore fully policy-relevant.
Furthermore, our counterfactual exercise focuses on manipulating the value of
the target covariate, while the bulk of the literature focuses more on
manipulating its marginal distribution and often uses a value intervention as
an example of how the marginal distribution may be shifted. The advantage of
using a value intervention is that the policy function $\mathcal{G}%
(\cdot;\delta)$ defines clearly how the policy can be implemented. This is in
contrast to the intervention of the marginal distribution where the policy
maker is not given a clear recipe to achieve such an intervention. In
addition, it seems to be easier to attach a cost implication to the value
intervention. A policy maker may want to trade off the cost with the policy
goal they hope to achieve. A marginal distribution intervention seems to be
more of theoretical interest unless it can be implemented empirically via a
value intervention as considered in this paper.\footnote{An important example
of value interventions is the literature on policy relevant treatment effects
where an instrumental variable is manipulated in order to shift the program
participation rate. See, for example, \cite{Heckman2005}.}
\section{Estimation and asymptotic results}
\label{estimation}
In this section, we focus on the estimation of $\Pi_{\tau}$ given in
\eqref{eq_pi_tau}. The estimator involves several preliminary steps. Firstly,
for a given quantile, we need to estimate $Q_{\tau}[Y]$. This is given by
\begin{equation}
\hat{q}_{\tau}=\arg\min_{q}\sum_{i=1}^{n}\left( \tau-\mathds{1}\left\{
Y_{i}\leq q\right\} \right) (Y_{i}-q). \label{eq_hat_q}%
\end{equation}
Next, we need to estimate the density of $Y$ evaluated at $Q_{\tau}[Y]$. This
can be estimated by
\begin{equation}
\hat{f}_{Y}\left( \hat{q}_{\tau}\right) =\frac{1}{n}\sum_{i=1}%
^{n}\mathcal{K}_{h}\left( Y_{i}-\hat{q}_{\tau}\right) \label{eq_hat_f}%
\end{equation}
where $\mathcal{K}_{h}(u)=h^{-1}\mathcal{K}(h^{-1}u)$ for a given kernel
$\mathcal{K}$ and a bandwidth $h$. For the average derivative of the
conditional cdf, we propose either a logit model as in
\cite{FirpoFortinLemieux09} or a probit model. Note that $\mathcal{S}%
_{Y|X,W}(Q_{\tau}[Y]|x,w)=1-F_{Y|X,W}(Q_{\tau}[Y]|x,w).$ We model
$\mathcal{S}_{Y|X,W}(Q_{\tau}[Y]|x,w)$ via $F_{Y|X,W}(Q_{\tau}[Y]|x,w)$ by
assuming that
\begin{equation}
F_{Y|X,W}(Q_{\tau}[Y]|x,w)=G(\phi_{\mathrm{x}}\left( x\right) ^{\prime
}\alpha_{\tau}+\phi_{\mathrm{w}}(w)^{\prime}\beta_{\tau})
\label{eq:model_probit}%
\end{equation}
where $\phi_{\mathrm{x}}\left( \cdot\right) $ and $\phi_{\mathrm{w}}\left(
\cdot\right) $ are column vectors of smooth basis functions and $G(\cdot)$ is
either the cdf of a logistic random variable (logit) or a standard normal
random variable (probit). Note that the subscripts \textquotedblleft%
$\mathrm{x}$\textquotedblright\ and \textquotedblleft$\mathrm{w}%
$\textquotedblright\ serve only to distinguish $\phi_{\mathrm{x}}\left(
\cdot\right) $ from $\phi_{\mathrm{w}}\left( \cdot\right).$ They are not
related to the arguments of these functions. For the choices of $\phi
_{\mathrm{x}}\left( \cdot\right) $ and $\phi_{\mathrm{w}}\left(
\cdot\right) ,$ we may take $\phi_{\mathrm{x}}\left( x\right) =x$ or
($x,x^{2})^{\prime}$ and $\phi_{\mathrm{w}}\left( w\right) =\left(
1,w\right) ^{\prime}.$ By default, we include the constant in the vector $w.$
Other more flexible choices are possible, but it is beyond the scope of this
paper to consider a fully nonparametric specification.
Let $Z_{i}=[\phi_{\mathrm{x}}\left( X_{i}\right) ^{\prime},\phi_{\mathrm{w}%
}(W_{i})^{\prime}]^{\prime}$ and $\theta_{\tau}=(\alpha_{\tau}^{\prime}%
,\beta_{\tau}^{\prime})^{\prime}.$ We estimate $\theta_{\tau}$ by the maximum
likelihood estimator:%
\begin{align}
\hat{\theta}_{\tau} & :=(\hat{\alpha}_{\tau},\hat{\beta}_{\tau}^{\prime
})^{\prime}=\arg\max_{\theta\in\Theta}\sum_{i=1}^{n}l_{i}(\theta;\hat{q}%
_{\tau})\nonumber\\
& =\arg\max_{\theta\in\Theta}\sum_{i=1}^{n}\bigg\{\mathds1\left\{ Y_{i}%
\leq\hat{q}_{\tau}\right\} \log\left[ G(Z_{i}^{\prime}\theta)\right]
+\mathds1\left\{ Y_{i}>\hat{q}_{\tau}\right\} \log\left[ 1-G(Z_{i}^{\prime
}\theta)\right] \bigg\}, \label{eq_likelihood}%
\end{align}
where $\Theta$ is a compact parameter space that contains $\theta_{\tau}$ as
an interior point. The estimator of $\Pi_{\tau}$ is then
\[
\hat{\Pi}_{\tau}=\hat{\Pi}_{\tau,L}+\hat{\Pi}_{\tau,S}%
\]
where
\begin{align}
\hat{\Pi}_{\tau,L} & =-\frac{\dot{\ell}(0)}{\hat{f}_{Y}\left( \hat{q}%
_{\tau}\right) }\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\hat{\theta}_{\tau
})\dot{\phi}_{\mathrm{x}}\left( X_{i}\right) ^{\prime}\hat{\alpha}_{\tau
},\label{eq:est_pi_l}\\
\hat{\Pi}_{\tau,S} & =-\frac{\dot{s}(0)}{\hat{f}_{Y}\left( \hat{q}_{\tau
}\right) }\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}^{\prime}\hat{\theta}_{\tau}%
)\dot{\phi}_{\mathrm{x}}\left( X_{i}\right) ^{\prime}\hat{\alpha}_{\tau
}\left( X_{i}-\mu\right) . \label{eq:est_pi_s}%
\end{align}
In the above, $g$ is the derivative of $G$, that is, the logistic density or
the standard normal density and $\dot{\phi}_{\mathrm{x}}\left( x\right)
=\partial\phi_{\mathrm{x}}\left( x\right) /\partial x$, which has the same
dimension as $\phi_{\mathrm{x}}\left( x\right) $. In order to establish the
asymptotic distribution of $\hat{\Pi}_{\tau}$, we need the following three
sets of assumptions, one for each preliminary estimation step.
\begin{assumption}
\label{assumption_quantile}\textbf{Quantile.} The density of $Y$ is positive,
continuous, and differentiable at $Q_{\tau}[Y]$.
\end{assumption}
\begin{assumption}
\label{assumption_logit_probit}\textbf{Logit/Probit.} For $G$ either the cdf
of a logistic or a standard normal random variable, we have
\begin{enumerate}
\item[(i)] $F_{Y|Z}(Q_{\tau}[Y]|z)=G(z^{\prime}\theta_{\tau})$ for an interior
point $\theta_{\tau}\in\Theta$ and $\hat{\theta}_{\tau}=\theta_{\tau}%
+o_{p}\left( 1\right) .$
\item[(ii)] For
\[
H_{i}\left( \theta;q\right) =\frac{\partial^{2}l_{i}\left( \theta;q\right)
}{\partial\theta\partial\theta^{\prime}},
\]
which is the Hessian of observation $i$, the following holds
\[
\sup_{(\theta,q)\in\mathcal{N}}\left\Vert \frac{1}{n}\sum_{i=1}^{n}%
H_{i}(\theta;q)-E[H_{i}(\theta;q)]\right\Vert \overset{p}{\rightarrow}0,
\]
where $\mathcal{N}$ is a neighborhood of $(\theta_{\tau}^{\prime},Q_{\tau
}[Y]^{\prime})^{\prime}$, and $H:=E[H_{i}(\theta_{\tau};Q_{\tau}[Y])]$ is
negative definite.
\item[(iii)] For the score $s_{i}$ defined by%
\[
s_{i}\left( \theta,q\right) =\frac{\partial l_{i}\left( \theta;q\right)
}{\partial\theta},
\]
the following stochastic equicontinuity assumption holds:
\[
\frac{1}{n}\sum_{i=1}^{n}\left\{ s_{i}(\theta_{\tau};\hat{q}_{\tau})-E\left[
s_{i}(\theta_{\tau};q)\right] |_{q=\hat{q}_{\tau}}\right\} =\frac{1}{n}%
\sum_{i=1}^{n}s_{i}(\theta_{\tau};Q_{\tau}[Y])+o_{p}(n^{-1/2}),
\]
and the map $q\mapsto E\left[ s_{i}(\theta_{\tau};q)\right] $ is
continuously differentiable at $Q_{\tau}[Y]$ with
\[
\frac{\partial E\left[ s_{i}(\theta_{\tau};q)\right] }{\partial
q}\bigg\vert_{q=Q_{\tau}[Y]}=:H_{Q}.
\]
\item[(iv)] For $\tilde{X}_{i}=(1,X_{i})^{\prime}$,
\begin{align*}
M_{1}\left( \theta\right) & :=E\left\{ [\dot{g}(Z_{i}^{\prime}\theta
)\dot{\phi}_{\mathrm{x}}\left( X_{i}\right) ^{\prime}\alpha]\tilde{X}%
_{i}Z_{i}^{\prime}\right\} \in\mathbb{R}^{2\times d_{Z}}\\
M_{2}\left( \theta\right) & :=E\left[ g(Z_{i}^{\prime}\theta)\tilde
{X}_{i}\dot{\phi}_{\mathrm{x}}(X_{i})^{\prime}\right] \in\mathbb{R}^{2\times
d_{\phi_{\mathrm{x}}}}%
\end{align*}
are well defined for any $\theta\in\mathcal{N}_{\theta_{\tau}}$, a
neighborhood of $\theta_{\tau}$; and the following uniform law of large numbers
holds:
\begin{align*}
& \sup_{\theta\in\mathcal{N}_{\theta_{\tau}}}\bigg\|\frac{1}{n}\sum_{i=1}%
^{n}[\dot{g}(Z_{i}^{\prime}\theta)\dot{\phi}_{\mathrm{x}}\left( X_{i}\right)
^{\prime}\alpha]\tilde{X}_{i}Z_{i}^{\prime}-M_{1}\left( \theta\right)
\bigg\|\overset{p}{\rightarrow}0,\\
& \sup_{\theta\in\mathcal{N}_{\theta_{\tau}}}\bigg\|\frac{1}{n}\sum_{i=1}%
^{n}g(Z_{i}^{\prime}\theta)\tilde{X}_{i}\dot{\phi}_{\mathrm{x}}(X_{i}%
)^{\prime}-M_{2}\left( \theta\right) \bigg\|\overset{p}{\rightarrow}0,
\end{align*}
where $\dot{g}$ is the derivative of $g$.
\end{enumerate}
\end{assumption}
\begin{assumption}
\label{assumption_density}\textbf{Density.}
\begin{enumerate}
\item[(i)] The kernel function $K(\cdot)$ satisfies (i) $\int_{-\infty
}^{\infty}K(u)du=1$, (ii) $\int_{-\infty}^{\infty}u^{2}K(u)du<\infty$, and
(iii) $K(u)=K(-u)$, and it is twice differentiable with Lipschitz continuous
second-order derivative $K^{\prime\prime}\left( u\right) $ satisfying (i)
$\int_{-\infty}^{\infty}K^{\prime\prime}(u)udu<\infty$ and $\left( ii\right)
$ there exist positive constants $C_{1}$ and $C_{2}$ such that $\left\vert
K^{\prime\prime}\left( u_{1}\right) -K^{\prime\prime}\left( u_{2}\right)
\right\vert \leq C_{2}\left\vert u_{1}-u_{2}\right\vert ^{2}$ for $\left\vert
u_{1}-u_{2}\right\vert \geq C_{1}.$
\item[(ii)] As $n\uparrow\infty$, the bandwidth satisfies: $h\downarrow0$,
$nh^{3}\uparrow\infty$, and $nh^{5}=O(1)$.
\end{enumerate}
\end{assumption}
Under Assumption \ref{assumption_quantile}, $\hat{q}_{\tau}$ given in
\eqref{eq_hat_q} is asymptotically linear with
\[
\hat{q}_{\tau}-Q_{\tau}[Y]=\frac{1}{n}\sum_{i=1}^{n}\frac{\tau
-\mathds{1}\left\{ Y_{i}\leq Q_{\tau}[Y]\right\} }{f_{Y}(Q_{\tau}[Y])}%
+o_{p}(n^{-1/2})=\frac{1}{n}\sum_{i=1}^{n}\psi(Y_{i},\tau,F_{Y})+o_{p}%
(n^{-1/2}).
\]
See, for example, \cite{serfling1980}. Assumption
\ref{assumption_logit_probit} is mostly necessary to deal with the preliminary
estimator $\hat{q}_{\tau}$ that enters the likelihood in
\eqref{eq_likelihood}. Assumption \ref{assumption_density} is taken from
\cite{yixiao2020}.
The following lemma contains the influence function for the maximum likelihood
estimator $\hat{\theta}_{\tau}$.
\begin{lemma}
\label{lemma_mle_alpha_beta} Under Assumptions \ref{assumption_quantile} and
\ref{assumption_logit_probit}, we have
\[
\hat{\theta}_{\tau}-\theta_{\tau}=-H^{-1}\frac{1}{n}\sum_{i=1}^{n}s_{i}%
(\theta_{\tau};Q_{\tau}[Y])-H^{-1}H_{Q}\frac{1}{n}\sum_{i=1}^{n}\psi
(Y_{i},\tau,F_{Y})+o_{p}(n^{-1/2}).
\]
\end{lemma}
\begin{theorem}
\label{theorem_est_pi} Under Assumptions \ref{assumption_quantile},
\ref{assumption_logit_probit}, and \ref{assumption_density}, the estimators
given in \eqref{eq:est_pi_l} and \eqref{eq:est_pi_s} satisfy
\[%
\begin{pmatrix}
\hat{\Pi}_{\tau,L}\\
\hat{\Pi}_{\tau,S}%
\end{pmatrix}
-%
\begin{pmatrix}
\Pi_{\tau,L}\\
\Pi_{\tau,S}%
\end{pmatrix}
=\frac{1}{n}\sum_{i=1}^{n}\Phi_{i,\tau}+O\left( h^{2}\right) +o_{p}%
(n^{-1/2})+o_{p}(n^{-1/2}h^{-1/2}),
\]
where
\begin{align*}
\Phi_{i,\tau} & =\frac{1}{f_{Y}\left( Q_{\tau}[Y]\right) }D_{\mu}\left\{
g(Z_{i};\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left( X_{i}\right) ^{\prime
}\alpha_{\tau}\tilde{X}_{i}-E\left[ g(Z_{i};\theta_{\tau})\dot{\phi
}_{\mathrm{x}}\left( X_{i}\right) ^{\prime}\alpha_{\tau}\tilde{X}%
_{i}\right] \right\} \\
& -\frac{1}{f_{Y}\left( Q_{\tau}[Y]\right) }D_{\mu}MH^{-1}s_{i}%
(\theta_{\tau};Q_{\tau}[Y])\\
& -\left[
\begin{pmatrix}
\Pi_{\tau,L}\\
\Pi_{\tau,S}%
\end{pmatrix}
\frac{\dot{f}_{Y}(Q_{\tau}[Y])}{f_{Y}\left( Q_{\tau}[Y]\right) }+\frac
{1}{f_{Y}\left( Q_{\tau}[Y]\right) }D_{\mu}MH^{-1}H_{Q}\right] \psi
(Y_{i},\tau,F_{Y})\\
& -%
\begin{pmatrix}
\Pi_{\tau,L}\\
\Pi_{\tau,S}%
\end{pmatrix}
\frac{1}{f_{Y}\left( Q_{\tau}[Y]\right) }\left\{ \mathcal{K}_{h}\left(
Y_{i}-Q_{\tau}[Y]\right) -E\mathcal{K}_{h}\left( Y_{i}-Q_{\tau}[Y]\right)
\right\} ,
\end{align*}
$\dot{f}_{Y}\left( \cdot\right) $ is the derivative of $f_{Y}\left(
\cdot\right) ,$
\[
D_{\mu}=%
\begin{pmatrix}
D_{L}^{\prime}\\
D_{\mu,S}^{\prime}%
\end{pmatrix}
=%
\begin{pmatrix}
-\dot{\ell}(0) & 0\\
\mu\dot{s}(0) & -\dot{s}(0)
\end{pmatrix}
,
\]%
\[
M=M_{1}\left( \theta_{\tau}\right) +%
\begin{pmatrix}
M_{2}\left( \theta_{\tau}\right) , & O
\end{pmatrix}
\in\mathbb{R}^{2\times d_{Z}},
\]
and $O\in\mathbb{R}^{2\times d_{\phi_{\mathrm{w}}}}$ is a matrix of zeros.
\end{theorem}
Theorem \ref{theorem_est_pi} establishes the contribution from each estimation
step. In particular, the last term in $n^{-1}\sum_{i=1}^{n}\Phi_{i,\tau}$ is
the contribution from estimating the density of $Y$ non-parametrically. This
term converges at a non-parametric rate, which is slower than other terms. As
a result, the asymptotic distribution of the location-scale effect estimator
is determined by the last term in $n^{-1}\sum_{i=1}^{n}\Phi_{i,\tau}$.
However, we do not recommend dropping all other terms. Instead, we write the
asymptotic normality result in the form
\begin{equation}
\left[ \frac{1}{n^{2}}\sum_{i=1}^{n}\hat{\Phi}_{i,\tau}\hat{\Phi}_{i,\tau
}^{\prime}\right] ^{-1/2}\left[
\begin{pmatrix}
\hat{\Pi}_{\tau,L}\\
\hat{\Pi}_{\tau,S}%
\end{pmatrix}
-%
\begin{pmatrix}
\Pi_{\tau,L}\\
\Pi_{\tau,S}%
\end{pmatrix}
\right] \overset{d}{\rightarrow}N(0,I_{2}) \label{asym_norm_full_form2}%
\end{equation}
as $n\uparrow\infty$, $nh^{3}\uparrow\infty$, and $nh^{5}\downarrow0$ where
$\hat{\Phi}_{i,\tau}$ is a plug-in estimator of $\Phi_{i,\tau}.$ In
particular,
\begin{align}
& \left[ n^{-2}\sum_{i=1}^{n}(l_{1}^{\prime}\hat{\Phi}_{i,\tau})^{2}\right]
^{-1/2}\left( \hat{\Pi}_{\tau,L}-\Pi_{\tau,L}\right) \overset{d}{\rightarrow
}N(0,1),\nonumber\\
& \left[ n^{-2}\sum_{i=1}^{n}(l_{2}^{\prime}\hat{\Phi}_{i,\tau})^{2}\right]
^{-1/2}\left( \hat{\Pi}_{\tau,S}-\Pi_{\tau,S}\right) \overset{d}{\rightarrow
}N(0,1), \label{asym_norm_individual_form2}%
\end{align}
as $n\uparrow\infty$, $nh^{3}\uparrow\infty$, and $nh^{5}\downarrow0$ where
$l_{1}=\left( 1,0\right) ^{\prime}$ and $l_{2}=\left( 0,1\right) ^{\prime
}$. Note that Theorem \ref{theorem_est_pi} has shown that the estimation error
in $\hat{\Pi}_{\tau,L}$ or $\hat{\Pi}_{\tau,S}$ is an average of independent
observations. The above asymptotic normality results can be proved using a
Lyapunov CLT under the following conditions:
(i) $E\left\Vert \Phi_{i,\tau}\Phi_{i,\tau}^{\prime}\right\Vert $ is finite,
$n^{-1}\sum_{i=1}^{n}\left[ \hat{\Phi}_{i,\tau}\hat{\Phi}_{i,\tau}^{\prime
}-E\Phi_{i,\tau}\Phi_{i,\tau}^{\prime}\right] $ converges in probability to
zero, and $n^{-1}\sum_{i=1}^{n}E\left[ \Phi_{i,\tau}\Phi_{i,\tau}^{\prime
}\right] $ is nonsingular for all large enough $n;$
(ii) $\int_{-\infty}^{\infty}\left\vert K(u)\right\vert ^{2+\Delta}du<\infty$
for some $\Delta>0$ and $\left\vert f_{Y}^{\prime\prime}(Q_{\tau
}[Y]\right\vert <C$ for some constant $C.$
Inferences based on our asymptotic results account for the estimation errors
from all estimation steps and are more reliable in finite samples. This is
supported by simulation evidence not reported here, but available upon
request. On the other hand, if we parametrize the density of $Y$ and estimate
it at the parametric $\sqrt{n}$-rate, then the last term in $n^{-1}\sum
_{i=1}^{n}\Phi_{i,\tau}$ will take a different form and will be of the same
order as the other terms. In this case, the location-scale effect estimator is
$\sqrt{n}$-asymptotically normal, and all the terms in Theorem
\ref{theorem_est_pi} will contribute to the asymptotic variance. With an
obvious modification of the last term in $\Phi_{i,\tau},$ the asymptotic
normality can be presented in the same way as in (\ref{asym_norm_full_form2}).
Let
\[
\Gamma_{\tau,S}=D_{\mu,S}^{\prime}E\left[ \frac{\partial F_{Y|X,W}(Q_{\tau
}[Y]|X,W)}{\partial X}\tilde{X}\right]
\]
be the numerator of $\Pi_{\tau,S}.$ Then the scale effect $\Pi_{\tau,S}$ is
zero if and only if $\Gamma_{\tau,S}=0.$ To test the null hypothesis
$H_{0}:\Pi_{\tau,S}=0,$ we can equivalently test the null hypothesis
$H_{0}:\Gamma_{\tau,S}=0.$ Unlike $\Pi_{\tau,S},$ $\Gamma_{\tau,S}$ can be
estimated at the parametric rate even if $f_{Y}\left( \cdot\right) $ is not
parametrically specified. More specifically, under Assumption
\ref{assumption_logit_probit}, we can estimate $\Gamma_{\tau,S}$ by
\[
\hat{\Gamma}_{\tau,S}:=D_{\mu,S}^{\prime}\frac{1}{n}\sum_{i=1}^{n}%
[g(Z_{i}^{\prime}\hat{\theta}_{\tau})\dot{\phi}_{\mathrm{x}}\left(
X_{i}\right) ^{\prime}\hat{\alpha}_{\tau}]\tilde{X}_{i},
\]
where $D_{\mu,S}^{\prime}=\left( \mu,-1\right) $ upon setting $\dot{s}(0)=1$
without loss of generality.
Under the assumptions of Theorem \ref{theorem_est_pi}, we can show that
\[
\hat{\Gamma}_{\tau,S}-\Gamma_{\tau,S}=D_{\mu,S}^{\prime}\frac{1}{n}\sum
_{i=1}^{n}\Phi_{i,\tau}^{\Gamma}+o_{p}\left( \frac{1}{\sqrt{n}}\right) ,
\]
where
\begin{align*}
\Phi_{i,\tau}^{\Gamma} & =g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi
}_{\mathrm{x}}\left( X_{i}\right) ^{\prime}\alpha_{\tau}\tilde{X}%
_{i}-E\left\{ [g(Z_{i}^{\prime}\theta_{\tau})\dot{\phi}_{\mathrm{x}}\left(
X_{i}\right) ^{\prime}\alpha_{\tau}]\tilde{X}_{i}\right\} \\
& -{MH}^{-1}s_{i}(\theta_{\tau};Q_{\tau}[Y])-{MH}^{-1}H_{Q}\psi(Y_{i}%
,\tau,F_{Y}).
\end{align*}
Define
\[
V_{\tau}=\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{i=1}^{n}E(D_{\mu,S}%
^{\prime}\Phi_{i,\tau}^{\Gamma})^{2}.
\]
If $D_{\mu,S}^{\prime}\Phi_{i,\tau}^{\Gamma}$ has a finite second moment and
$V_{\tau}>0,$ then a standard CLT yields $V_{\tau}^{-1/2}\sqrt{n}(\hat{\Gamma
}_{\tau,S}-\Gamma_{\tau,S}^{\mu})\overset{d}{\rightarrow}N\left( 0,1\right)
$. To test $H_{0}:\Gamma_{\tau,S}=0,$ we construct the test statistic
\[
t_{\tau,S}:=\frac{\sqrt{n}\hat{\Gamma}_{\tau,S}}{\sqrt{\hat{V}_{\tau}}}\RIfM@\expandafter\text@\else\expandafter\mbox\fi{
for }\hat{V}_{\tau}=\frac{1}{n}\sum_{i=1}^{n}(D_{\mu,S}^{\prime}\hat{\Phi
}_{i,\tau}^{\Gamma})^{2},
\]
where
\begin{align}
\hat{\Phi}_{i,\tau}^{\Gamma} & =g(Z_{i}\hat{\theta}_{\tau})\dot{\phi
}_{\mathrm{x}}\left( X_{i}\right) ^{\prime}\hat{\alpha}_{\tau}\tilde{X}%
_{i}-\frac{1}{n}\sum_{i=1}^{n}g(Z_{i}\hat{\theta}_{\tau})\dot{\phi
}_{\mathrm{x}}\left( X_{i}\right) ^{\prime}\hat{\alpha}_{\tau}\tilde{X}%
_{i}\nonumber\\
& -{\hat{M}\hat{H}}^{-1}s_{i}(\hat{\theta}_{\tau};\hat{q}_{\tau})-{\hat
{M}\hat{H}}^{-1}\hat{H}_{Q}\hat{\psi}(Y_{i},\tau,F_{Y}). \label{eq:var_t_test}%
\end{align}
In the above, $\hat{\psi}(Y_{i},\tau,F_{Y})=\left[ \tau-1\left\{ Y_{i}%
\leq\hat{q}_{\tau}\right\} \right] /\hat{f}_{Y}\left( \hat{q}_{\tau
}\right) $ and the score $s_{i}(\hat{\theta}_{\tau};\hat{q}_{\tau})$ is
obtained by evaluating the expression given in \eqref{eq_score} at
$\theta=\hat{\theta}_{\tau}$ and $q=\hat{q}_{\tau}$. ${\hat{M}},$ ${\hat{H},}$
and $\hat{H}_{Q}$ are the sample versions of ${M},$ ${H,}$ and $H_{Q},$
respectively. Details are given in the proof of the corollary below.
\begin{corollary}
\label{corollary_est_pi} Let the assumptions of Theorem \ref{theorem_est_pi}
hold. Assume that $D_{\mu,S}^{\prime}\Phi_{i,\tau}^{\Gamma}$ has a finite
second moment and $\hat{V}_{\tau}/V_{\tau}\overset{p}{\rightarrow}1$ for some
$V_{\tau}>0.$ Then, under the null hypothesis $H_{0}:\Pi_{\tau,S}=0,$
\[
t_{\tau,S}\overset{d}{\rightarrow}N(0,1).
\]
\end{corollary}
\section{Monte Carlo experiments}
\label{MC}
In this section, we use Monte Carlo simulations to evaluate the finite sample
performances of the proposed estimators and tests of location and scale
effects. We employ the same data generating process as in Example
\ref{example_normal_location_model} for which we have derived the closed-form
expressions for the location and scale effects. In particular, we let
\[
Y=\lambda+X\gamma+U,
\]
where $X\sim N(\mu_{X},\sigma_{X}^{2})$ and $U\sim N(0,\sigma_{U}^{2})$. We
set $\lambda=0,$ $\sigma_{U}^{2}=1,$ $\dot{\ell}(0)=1$ and $\dot{s}(0)=-1$.
The last derivative corresponds to, for example, $s(\delta)=(1+\delta)^{-1}$.
Then, from the results in Example \ref{example_normal_location_model}, the
true location effect is $\Pi_{\tau,L}=\gamma,$ and the true scale effect is
\[
\Pi_{\tau,S}^{\mu_{X}}=-\sqrt{R_{YX}^{2}}Q_{\tau}[X_{\gamma}^{\circ}%
]=-\sqrt{R_{YX}^{2}}\sqrt{var(X_{\gamma}^{\circ})}Q_{\tau}[\varepsilon
]=-\sqrt{R_{YX}^{2}}\cdot\sigma_{X}\cdot\left\vert \gamma\right\vert \cdot Q_{\tau
}[\varepsilon]
\]
where $\varepsilon$ is standard normal.
We consider quantiles $\tau\in\{0.10,0.25,0.50,0.75,0.90\}$ and sample sizes
$n=500$ and $n=1000$. The number of simulations is set to $10,000$ for each experiment.
We implement our estimators in \texttt{Matlab}. The unconditional quantile
estimator in equation \eqref{eq_hat_q} is easily computed as an order
statistic. The density function is estimated as a kernel density estimator as
in equation \eqref{eq_hat_f} using a standard normal kernel. For the bandwidth
choice in the kernel density estimation, we use a modified version of
Silverman's rule of thumb. More specifically, since we require $nh^{3}%
\uparrow\infty$ and $nh^{5}\downarrow0$ as $n\uparrow\infty$, we take
$h=1.06\hat{\sigma}_{Y}n^{-1/4}$, where $\hat{\sigma}_{Y}$ is the sample
standard deviation of $Y$.
\subsection{Bias, variance, and mean squared error}
In this subsection, we consider the bias, variance, and mean-squared error
(MSE) of the proposed location and scale effects estimators. For each effect
estimator, we consider either a probit or a logit specification for the
conditional cdf $F_{Y|X}(Q_{\tau}[Y]|X).$ Under our data generating process,
the probit with $F_{Y|X}(Q_{\tau}[Y]|X)=\boldsymbol{\Phi}(X\alpha_{\tau}%
+\beta_{\tau})$ for the standard normal CDF $\boldsymbol{\Phi}$ is correctly
specified while the logit with $F_{Y|X}(Q_{\tau}[Y]|X)=\left[ 1+\exp\left(
X\alpha_{\tau}+\beta_{\tau}\right) \right] ^{-1}$ is misspecified.
The bias, variance, and MSE are reported in Table \ref{table:MC} when $\mu
_{X}=0$, $\gamma=1$ and $\sigma_{X}^{2}=1$ so that the true location effect is
$1$ for any $\tau$ and the true scale effect is $-\sqrt{0.5}Q_{\tau
}[\varepsilon]\approx -0.707Q_{\tau}[\varepsilon].$ To save space, simulation results
for other values of $\gamma$ and $\sigma_{X}^{2}$ are omitted.
Table \ref{table:MC} shows that the estimator based on the probit
specification outperforms that based on the logit one. This is consistent with
the correct specification of probit. For each estimator, the bias decreases as
the sample size $n$ increases. The variance also decreases as the sample size
$n$ increase, and as a result, the MSE also becomes smaller when the sample
size grows. For our purposes, the scale effect estimator performs well. For
non-central quantiles, the difference in the scale effect estimates under the
probit and logit specifications is in general larger than the difference in
the location effect estimates. For central quantiles, the probit and logit
specifications lead to more or less the same estimates for both the scale
effect and the location effect.
\begin{center}
{\scriptsize \begin{table}[ptbh]
\caption{The biases, variances, and mean-squared errors of the location and
scale effects estimators with $\gamma=1$ and $\sigma_{X}^{2}=1$.}%
\label{table:MC}%
{\scriptsize \centering\hspace{2cm}
\begin{tabular}
[c]{l|cccccc}\hline\hline
& & {$\tau=0.1$} & {$\tau=0.25$} & {$\tau=0.50$} & {$\tau=0.75$} &
{$\tau=0.90$}\\\hline
& & & & $n=500$ & & \\\hline
Bias & {$\Pi_{L}$ (probit)} & -0.015 & 0.013 & 0.023 & 0.012 & -0.016\\
& {$\Pi_{L}$ (logit)} & -0.016 & 0.012 & 0.023 & 0.012 & -0.016\\
& {$\Pi_{S}$ (probit)} & -0.008 & 0.008 & 0.000 & -0.007 & 0.008\\
& {$\Pi_{S}$ (logit)} & 0.039 & 0.034 & 0.000 & -0.034 & -0.039\\\hline
Variance & {$\Pi_{L}$ (probit)} & 0.019 & 0.010 & 0.008 & 0.010 & 0.019\\
& {$\Pi_{L}$ (logit)} & 0.019 & 0.010 & 0.008 & 0.010 & 0.020\\
& {$\Pi_{S}$ (probit)} & 0.032 & 0.007 & 0.003 & 0.008 & 0.033\\
& {$\Pi_{S}$ (logit)} & 0.033 & 0.007 & 0.003 & 0.008 & 0.034\\\hline
MSE & {$\Pi_{L}$ (probit)} & 0.019 & 0.010 & 0.009 & 0.010 & 0.019\\
& {$\Pi_{L}$ (logit)} & 0.020 & 0.011 & 0.009 & 0.010 & 0.020\\
& {$\Pi_{S}$ (probit)} & 0.033 & 0.007 & 0.003 & 0.008 & 0.033\\
& {$\Pi_{S}$ (logit)} & 0.035 & 0.009 & 0.003 & 0.009 & 0.035\\\hline\hline
& & & & $n=1000$ & & \\\hline
Bias & {$\Pi_{L}$ (probit)} & -0.011 & 0.009 & 0.017 & 0.008 & -0.013\\
& {$\Pi_{L}$ (logit)} & -0.011 & 0.009 & 0.017 & 0.008 & -0.013\\
& {$\Pi_{S}$ (probit)} & -0.007 & 0.005 & -0.000 & -0.004 & 0.010\\
& {$\Pi_{S}$ (logit)} & 0.041 & 0.032 & -0.000 & -0.031 & -0.038\\\hline
Variance & {$\Pi_{L}$ (probit)} & 0.011 & 0.006 & 0.005 & 0.006 & 0.011\\
& {$\Pi_{L}$ (logit)} & 0.011 & 0.006 & 0.005 & 0.006 & 0.011\\
& {$\Pi_{S}$ (probit)} & 0.018 & 0.004 & 0.001 & 0.004 & 0.017\\
& {$\Pi_{S}$ (logit)} & 0.018 & 0.004 & 0.001 & 0.004 & 0.018\\\hline
MSE & {$\Pi_{L}$ (probit)} & 0.011 & 0.006 & 0.005 & 0.006 & 0.011\\
& {$\Pi_{L}$ (logit)} & 0.011 & 0.006 & 0.005 & 0.006 & 0.011\\
& {$\Pi_{S}$ (probit)} & 0.018 & 0.004 & 0.001 & 0.004 & 0.017\\
& {$\Pi_{S}$ (logit)} & 0.020 & 0.005 & 0.001 & 0.005 & 0.019\\\hline\hline
\end{tabular}
}\end{table}}
\end{center}
\subsection{Accuracy of the normal approximation}
In this subsection, we investigate the finite sample accuracy of the normal
approximation given in (\ref{asym_norm_individual_form2}). Using the same data
generating process as in the previous subsection and employing the probit
specification, we simulate the distributions of the studentized statistics
\[
\left[ n^{-2}\sum_{i=1}^{n}(l_{1}^{\prime}\hat{\Phi}_{i,\tau})^{2}\right]
^{-1/2}(\hat{\Pi}_{\tau,L}-\Pi_{\tau,L})
\]
and
\[
\left[ n^{-2}\sum_{i=1}^{n}(l_{2}^{\prime}\hat{\Phi}_{i,\tau})^{2}\right]
^{-1/2}(\hat{\Pi}_{\tau,S}-\Pi_{\tau,S}).,
\]
for the location and scale effects, respectively. We plot each distribution
and compare it with the standard normal distribution. We consider $\gamma
\in\left\{ 0.25,0.50,0.75,1\right\} $ and use the same $\tau$ values as in
the previous subsection. Simulation results for the two sample sizes $n=500$
and $n=1000$ are qualitatively similar, and we report only the case when
$n=1000$ here. Figures \ref{fig:beta_025_location}--\ref{fig:beta_075_scale}
report the (simulated) finite sample distributions when $\sigma_{X}^{2}=1$ and
$n=1000$ for some selected values of $\gamma$ and $\tau$ together with a
standard normal density that is superimposed on each figure. It is clear from
these figures that the standard normal distribution provides an accurate
approximation to the distribution of the studentized test statistic for both
the location and scale effects.
\begin{figure}[ptb]
\centering
\includegraphics[
height=3.4411in,
width=4.4529in
] {location_n1000beta25-eps-converted-to.pdf} \caption{Finite sample exact
distribution of the studentized location effect statistic when $\gamma=0.25$,
$\sigma_{X}^{2}=1$, and $n=1000.$}%
\label{fig:beta_025_location}%
\end{figure}
\begin{figure}[ptb]
\centering
\includegraphics[
height=3.4247in,
width=4.4322in
] {location_n1000beta75-eps-converted-to.pdf}\caption{Finite sample exact
distribution of the studentized location effect statistic when $\gamma=0.75$,
$\sigma_{X}^{2}=1$, and $n=1000.$}%
\end{figure}
\bigskip
\begin{figure}[ptb]
\centering
\includegraphics[
height=3.4541in,
width=4.4745in
] {scale_n1000beta25-eps-converted-to.pdf}\caption{Finite sample exact
distribution of the studentized scale effect statistic when $\gamma=0.25$,
$\sigma_{X}^{2}=1$, and $n=1000.$}%
\end{figure}
\begin{figure}[ptb]
\centering
\includegraphics[
height=3.4601in,
width=4.4815in
] {scale_n1000beta75-eps-converted-to.pdf} \caption{Finite sample exact
distribution of the studentized scale effect statistic when $\gamma=0.75$,
$\sigma_{X}^{2}=1$, and $n=1000.$}%
\label{fig:beta_075_scale}%
\end{figure}
Table \ref{table:emp_cov_location_scale} reports the empirical coverage of
95\% confidence intervals for the location and scale effects. The empirical
coverage is close to the nominal coverage in all cases. This is consistent
with Figures \ref{fig:beta_025_location}--\ref{fig:beta_075_scale}. We may
then conclude that the normal approximation can be reliably used for inference
on the location and scale effects.%
\begin{table}[ptb]
\caption
{Empirical coverage of 95\% confidence intervals for the location and scale effects when $\sigma
_{X}^2=1.$}
\centering
{\scriptsize
\begin{tabular}
[c]{l|lccccc}\hline\hline
& $\gamma$ & $\tau=0.1$ & $\tau=0.25$ & $\tau=0.50$ & $\tau=0.75$ &
$\tau=0.90$\\\hline
& \multicolumn{6}{|c}{$n=500$}\\\hline
Location & ${0.25}$ & $0.946$ & $0.950$ & $0.951$ & $0.950$ & $0.947$\\
& ${0.5}$ & $0.942$ & $0.952$ & $0.950$ & $0.953$ & $0.938$\\
& ${0.75}$ & $0.940$ & $0.954$ & $0.952$ & $0.956$ & $0.937$\\
& ${1}$ & $0.937$ & $0.957$ & $0.950$ & $0.957$ & $0.935$\\\hline
Scale & ${0.25}$ & $0.900$ & $0.921$ & $0.973$ & $0.916$ & $0.902$\\
& ${0.5}$ & $0.930$ & $0.943$ & $0.957$ & $0.939$ & $0.928$\\
& ${0.75}$ & $0.937$ & $0.950$ & $0.954$ & $0.946$ & $0.933$\\
& ${1}$ & $0.939$ & $0.952$ & $0.951$ & $0.945$ & $0.933$\\\hline\hline
& \multicolumn{6}{|c}{$n=1000$}\\\hline
Location & $0.25$ & $0.948$ & $0.951$ & $0.951$ & $0.954$ & $0.945$\\
& $0.5$ & $0.946$ & $0.950$ & $0.952$ & $0.957$ & $0.943$\\
& $0.75$ & $0.945$ & $0.952$ & $0.953$ & $0.957$ & $0.940$\\
& $1$ & $0.941$ & $0.952$ & $0.952$ & $0.958$ & $0.942$\\\hline
Scale & $0.25$ & $0.922$ & $0.939$ & $0.965$ & $0.940$ & $0.921$\\
& $0.5$ & $0.938$ & $0.949$ & $0.955$ & $0.950$ & $0.933$\\
& $0.75$ & $0.942$ & $0.951$ & $0.952$ & $0.952$ & $0.938$\\
& $1$ & $0.939$ & $0.952$ & $0.950$ & $0.953$ & $0.940$\\\hline\hline
\end{tabular}
}%
\label{table:emp_cov_location_scale}
\end{table}%
\subsection{Power of the t-test of a zero scale effect}
To investigate the power of the t-test proposed in Corollary
\ref{corollary_est_pi}, we simulate the following model:
\[
Y=\lambda+X\gamma+U,
\]
where
\[%
\begin{pmatrix}
X\\
U
\end{pmatrix}
\sim N\left(
\begin{pmatrix}
1\\
0
\end{pmatrix}
,%
\begin{pmatrix}
1 & 0\\
0 & 1
\end{pmatrix}
\right) .
\]
Here we set $\lambda=0,$ $\mu_{X}=1$ and $\dot{s}(0)=-1$. When $\gamma=0$, $X$
is excluded from the outcome equation and thus the scale effect is 0. The null
hypothesis of a zero scale effect corresponds to the case that $\gamma=0$. The
power of the test is obtained by varying $\gamma$ around 0 in a grid from
$-0.4$ to $0.4$ with an increment of $0.01$.
Figure \ref{fig:power} graphs the size-adjusted power of the t-test for
different quantile levels when $n=500$ and when $n=1000$. The power is
calculated using the probit specification, namely $F_{Y|X}(Q_{\tau
}[Y]|X)=\boldsymbol{\Phi}(X\alpha_{\tau}+\beta_{\tau})$. The size adjustment
is based on the empirical critical value such that the test rejects the null
5\% of the time. Figure \ref{fig:power} shows that the power increases as
$\gamma$ deviates more from its null value of zero, and that for a given
nonzero value of $\gamma,$ the power increases with the sample size. Results
not reported here show that the test has a quite accurate size in that the
empirical rejection probability under the null is close to 5\%, the nominal
level of the test.
\begin{figure}[ptb]
\hspace*{-1.9cm} \centering
\includegraphics[scale=0.65]{power_S10000-eps-converted-to.pdf}\caption{Size-adjusted power of
the t-test for a zero scale effect.}%
\label{fig:power}%
\end{figure}
\section{Empirical application}
\label{app}
In this section, we consider two applications: education and wages, and
smoking and birth weights.
\subsection{Education and wages}
Our first application is based on a household labor survey from
\cite{wooldridge} that can be accessed online for replication.\footnote{See
\url{http://fmwww.bc.edu/ec-p/data/wooldridge/wage1.des} and
\url{http://fmwww.bc.edu/ec-p/data/wooldridge/wage1.dta} for the data in the
Stata data file format.} The idea is to evaluate the effects of
education on the quantile of the unconditional distribution of
log wages. In this application, $Y=lwage,$ which is log hourly wage, and
$X=educ$, which is years of education is our target variable. The controls are:
$W=[exper\ tenure\ nonwhite\ female]$, where $exper$ is years of working
experience, $tenure$ is years with current employer, $nonwhite$ is a dummy
that equals 1 if the individual is non-white, and $female$ is a dummy that
equals 1 if the individual is female. We assume that Assumption
\ref{Assumption ID} holds for this choice of $W.$
While the main goal is to study the scale effect, we also present results for
the location effect. We set $\dot{\ell}(0)=1$ and $\dot{s}(0)=-1$. Note that
when $\dot{s}(0)=-1,$ the estimated effects we present below are the
unconditional scale effects when the variance of the covariate is
\emph{reduced} by a small amount. For the mean of years of education $\mu_{X}%
$, we let $\mu_{X}=12.29$ based on the Barro-Lee Data on Educational
Attainment.\footnote{The dataset is available from
\url{https://databank.worldbank.org/reports.aspx?source=Education Statistics}
We use the series \textquotedblleft Barro-Lee: Average years of total
schooling, age 25+, total\textquotedblright\ for the US between 1970-2010 and
find that the average years of schooling is 12.29.} We set $\mu=\mu_{X}=12.29$
to study the location and scale effects. In a similar fashion to the Monte
Carlo analysis, we consider $\tau\in\{0.10,0.25,0.50,0.75,0.90\}$. The sample
size for the household labor survey is $n=526$, which is comparable to $n=500$
in the simulation exercises. We compute the standard errors using the
approximation in \eqref{asym_norm_individual_form2}.
\begin{table}[ptb]
\caption{Effects of location-scale shifts in education on the unconditional
quantiles of log-wage.}%
\label{tab:wage1}%
\centering
{\scriptsize {\
\begin{tabular}
[c]{l|cccccc}\hline\hline
& & \textbf{$\tau=0.10$} & \textbf{$\tau=0.25$} & \textbf{$\tau=0.50$} &
\textbf{$\tau=0.75$} & \textbf{$\tau=0.90$}\\\hline
Location (probit) & {Estimate} & 0.039 & 0.062 & 0.101 & 0.101 & 0.118\\
& {} & (0.008) & (0.011) & (0.015) & (0.016) & (0.021)\\
& {$95\% \ CI_{L}$} & 0.025 & 0.041 & 0.072 & 0.069 & 0.076\\
& {$95\% \ CI_{U}$} & 0.054 & 0.083 & 0.129 & 0.132 & 0.160\\\hline
Location (logit) & {Estimate} & 0.038 & 0.065 & 0.103 & 0.100 & 0.120\\
& {} & (0.007) & (0.010) & (0.015) & (0.016) & (0.021)\\
& {$95\% \ CI_{L}$} & 0.024 & 0.044 & 0.074 & 0.069 & 0.080\\
& {$95\% \ CI_{U}$} & 0.053 & 0.085 & 0.131 & 0.132 & 0.160\\\hline\hline
Scale (probit) & {Estimate} & 0.045 & 0.029 & -0.025 & -0.103 & -0.203\\
& {} & (0.014) & (0.011) & (0.013) & (0.028) & (0.065)\\
& {$95\% \ CI_{L}$} & 0.018 & 0.007 & -0.051 & -0.158 & -0.330\\
& {$95\% \ CI_{U}$} & 0.071 & 0.052 & 0.001 & -0.049 & -0.077\\\hline
Scale (logit) & {Estimate} & 0.045 & 0.034 & -0.024 & -0.110 & -0.227\\
& {} & (0.014) & (0.012) & (0.014) & (0.029) & (0.066)\\
& {$95\% \ CI_{L}$} & 0.017 & 0.011 & -0.051 & -0.167 & -0.356\\
& {$95\% \ CI_{U}$} & 0.072 & 0.058 & 0.002 & -0.053 & -0.099\\\hline\hline
\end{tabular}
} }
\par
{\scriptsize Notes: standard errors are in parentheses. }\end{table}
\begin{figure}[ptb]
\centering
\includegraphics[scale=0.65]{effects_probit-eps-converted-to.pdf}\caption{Point and interval
estimates of the location and scale effects of education on the unconditional
quantiles of log-wage based on the probit specification: $\Pi_{\tau,L}$
(dashed blue) and $\Pi_{\tau,S}$ (solid red). }%
\label{fig:emp_results}%
\end{figure}
The most interesting results in Table \ref{tab:wage1} appear in the
unconditional scale effects. As discussed in Section \ref{Sec: elasticity},
the scale effects can be interpreted as percentage changes of the
unconditional quantiles. Consider the scale effect for $\tau=0.10$. Both the
probit and logit specifications suggest an effect of about .045. Then, using
the quantile-standard deviation elasticity, a $1\%$ decrease in the standard
deviation of education would produce a positive effect of $.045\%$ on the
unconditional quantile at the quantile level $\tau=0.10$. Given that the
sample standard deviation of $educ$ is $2.77$, the $1\%$ decrease is
approximately a change in the standard deviation from $2.77$ to $2.74$.
Consider now the scale effect for $\tau=0.50$. In this case, both probit and
logit specifications provide a statistically insignificant effect (at the
$5\%$ level). Confront this with the results of Example
\ref{example_normal_location_model} where in the linear model $Y=\lambda
+X\gamma+U$, the scale effect $\Pi_{0.50,S}=0$ if both $X$ and $U$ are
symmetrically distributed around 0. Thus, $\hat{\Pi}_{0.50,S}\approx0$ is
consistent with a linear model with symmetrically distributed $X$ and $U.$
Finally, consider the scale effect for $\tau=0.90$, again using both probit
and logit specifications. In this case, the effects are negative, suggesting a
$1\%$ decrease in the standard deviation would reduce the upper $\tau=0.90$
quantile by $.20\%$ (probit) and $.23\%$ (logit). Overall this analysis shows
that the scale effects are monotonically decreasing in $\tau$. This can be
seen in Figure \ref{fig:emp_results} that plots, for a finer grid of $\tau
$,\footnote{For Figure \ref{fig:emp_results} we use $\tau
=0.10,0.11,...,0.89,0.90$.} the probit estimates for both the location (dashed
blue) and scale (solid red) effects.
How can this be interpreted? The location effects suggest that the marginal
contribution of one more year of education benefits more the upper parts of
the unconditional distribution of wages. The scale effects suggest the
contrary. Reducing the overall dispersion of education would increase the
lower quantile wages, but reduce the upper ones.
\begin{table}[ptb]
\caption{Effects of location-scale shifts in education on the unconditional
quantiles of log-wage.}%
\label{tab:wage2}%
\centering
{\scriptsize {\
\begin{tabular}
[c]{l|cccccc}\hline\hline
& & \textbf{$\tau=0.10$} & \textbf{$\tau=0.25$} & \textbf{$\tau=0.50$} &
\textbf{$\tau=0.75$} & \textbf{$\tau=0.90$}\\\hline
Location (probit) & {Estimate} & 0.039 & 0.062 & 0.101 & 0.101 & 0.118\\
& {} & (0.008) & (0.011) & (0.015) & (0.016) & (0.021)\\
& {$95\% \ CI_{L}$} & 0.025 & 0.041 & 0.072 & 0.069 & 0.076\\
& {$95\% \ CI_{U}$} & 0.054 & 0.083 & 0.129 & 0.132 & 0.160\\\hline
Location (logit) & {Estimate} & 0.038 & 0.065 & 0.103 & 0.100 & 0.120\\
& {} & (0.007) & (0.010) & (0.015) & (0.016) & (0.021)\\
& {$95\% \ CI_{L}$} & 0.024 & 0.044 & 0.074 & 0.069 & 0.080\\
& {$95\% \ CI_{U}$} & 0.053 & 0.085 & 0.131 & 0.132 & 0.160\\\hline\hline
Scale (probit) & {Estimate} & 0.045 & 0.029 & -0.025 & -0.103 & -0.203\\
& {} & (0.014) & (0.011) & (0.013) & (0.028) & (0.065)\\
& {$95\% \ CI_{L}$} & 0.018 & 0.007 & -0.051 & -0.158 & -0.330\\
& {$95\% \ CI_{U}$} & 0.071 & 0.052 & 0.001 & -0.049 & -0.077\\\hline
Scale (logit) & {Estimate} & 0.045 & 0.034 & -0.024 & -0.110 & -0.227\\
& {} & (0.014) & (0.012) & (0.014) & (0.029) & (0.066)\\
& {$95\% \ CI_{L}$} & 0.017 & 0.011 & -0.051 & -0.167 & -0.356\\
& {$95\% \ CI_{U}$} & 0.072 & 0.058 & 0.002 & -0.053 & -0.099\\\hline\hline
\end{tabular}
} }
\par
{\scriptsize Notes: standard errors are in parentheses. }\end{table}
\subsection{Smoking and birth weight}
This second application considers the relationship between smoking during
pregnancy and the child's birth weight. This was previously studied by
\cite{Abrevaya2001}, \cite{KoenkerHallock01}, \cite{Rothe2010}, and
\cite{ChernozhukovFernandezVal11}. We use the natality data from the National
Vital Statistics System for the year 2018.\footnote{Available here:
\url{https://www.nber.org/research/data/vital-statistics-natality-birth-data}.}
The outcome variable is \emph{birth weight} in grams, while the target
variable is the average number of cigarettes smoked daily during pregnancy. We
focus on the sample of mothers who are smokers. The sample consists of 219,667 observations.
For this model $Y$ is \emph{birth weight} in grams and $X$ is the mother's
reported average number of cigarettes smoked per day during pregnancy. We use
the same covariates as \cite{Abrevaya2001}:\footnote{We omit the dummy of
whether the mother smoked during pregnancy because we focus on the sample of
smoking mothers.} $(i)$ a dummy for whether the mother is black; $(ii)$ a
dummy for marital status; $(iii)$ age and age squared; $(iv)$ a set of dummies
for education attainment: high school graduate, some college, and college
graduate; $(v)$ weight gain during pregnancy, $(vi)$ a set of dummies for
prenatal visit: visit during the second trimester, visit during the third
trimester, and no visit at all; and $(vii)$ a dummy for the sex of the child.
For this application, we set $\mu=0$, $l(\delta)\equiv0$, and $s(\delta
)=1/\left( 1+\delta\right) $, so that according to
\eqref{eq_location_scale}, counterfactual cigarette consumption is now
$X_{\delta}=X/(1+\delta)$, which has a smaller mean and variance than $X.$
Note, again, that $\dot s(0)=-1$. To motivate such a counterfactual policy, we
can think of a tax on the price of cigarettes, which induces the consumer to
reduce cigarette consumption from $X$ to $X/(1+\delta)$. .\footnote{Suppose that $\alpha_{x}$ is the exponent of $X$ in the
Cobb-Douglas utility function. Suppose further that the exponents are
normalized to sum to 1. Then, if $M$ is the income, and $p_{x}$ is the price
of $X$, we have that $\alpha_{x}M=p_{x}X$. Similarly, under the proposed
counterfactual tax $\alpha_{x}M=p_{x}(1+\delta)X_{\delta}$. It follows that
$X_{\delta}=X/(1+\delta)$.}
Table \ref{tab:smoking1} and Figure \ref{fig:emp_results_smoking} show the
results. The effects are positive and monotonically increasing across
quantiles. This means that the marginal impact on the \emph{birth weight} of a
tax on cigarettes is positive. The effects are stronger for upper quantiles of
the distribution of \emph{birth weight}. In order to interpret the magnitudes,
we use the quantile-standard deviation elasticity. According to
\eqref{eq:elasticity_0}, the elasticity can be calculated as $\mathcal{E}%
_{\tau,\delta=0}=-\Pi_{\tau,S}/Q_{\tau}[Y]$ as $\dot{s}\left( 0\right) =-1.$
For example, for $\tau=0.50$, $\mathcal{E}_{0.50,\delta=0}=-0.0128$. This
means that a $1\%$ decrease in the standard deviation of the consumption of
cigarettes increases the median birth weight by $0.0128\%$.
\begin{table}[ptb]
\caption{Effects of a negative location-scale shift in smoking ($X_{\delta
}=X/\left( 1+\delta)\right) $ on the unconditional quantiles of \emph{birth
weight}.}%
\label{tab:smoking1}
\centering
{\scriptsize {\
\begin{tabular}
[c]{l|cccccc}\hline\hline
& & \textbf{$\tau=0.10$} & \textbf{$\tau=0.25$} & \textbf{$\tau=0.50$} &
\textbf{$\tau=0.75$} & \textbf{$\tau=0.90$}\\\hline
Scale (probit) & {Estimate} & 26.562 & 35.412 & 39.096 & 41.316 & 46.249\\
& {} & (2.784) & (1.870) & (1.708) & (2.024) & (2.834\\
& {$95\% \ CI_{L}$} & 21.106 & 31.746 & 35.749 & 37.349 & 40.694\\
& {$95\% \ CI_{U}$} & 32.018 & 39.078 & 42.443 & 45.282 & 51.803\\\hline
Scale (logit) & {Estimate} & 25.242 & 34.412 & 40.038 & 44.232 & 50.077\\
& {} & (2.722) & (1.848) & (1.711) & (1.984) & (2.695)\\
& {$95\% \ CI_{L}$} & 19.908 & 30.790 & 36.684 & 40.343 & 44.795\\
& {$95\% \ CI_{U}$} & 30.577 & 38.034 & 43.392 & 48.122 & 55.360\\\hline\hline
\end{tabular}
} }
\par
{\scriptsize Notes: standard errors are in parentheses.}\end{table}
\begin{figure}[ptb]
\caption{Point and interval estimates of the location-scale effects of smoking
on the unconditional quantiles of \emph{birth weight} based on the probit
specification.}%
\label{fig:emp_results_smoking}
\centering
\includegraphics[scale=0.65]{smoking_scale_probit_mu_x0-eps-converted-to.pdf}
\end{figure}
\section{Conclusion}
\label{conclusion}
This paper has provided a general procedure to analyze the distributional
impact of changes in covariates on an outcome variable. The standard
unconditional quantile regression analysis focuses on a particular impact
coming from a location shift. We have
provided a framework to study the unconditional policy effects generated by a
smooth and invertible intervention of one or more target variables, allowing
them to be possibly endogeneous. We focus particularly on a location-scale shift and
show how to additively decompose the total effect into a location effect and a
scale effect. They can be analyzed and estimated separately.
Additionally, we consider the case of simultaneous changes in
different covariates. We show how this can be obtained from the usual
vector-valued unconditional quantile regressions.
\bibliographystyle{econometrica}
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2,869,038,154,795 | arxiv | \section{Introduction} It is well-known that a Riemannian manifold could be embedded isometrically in a Euclidean space, via the Nash embedding theorems. However, this is difficult technically. On the other hand, manifolds are often encountered as a differentiable submersion of a differentiable embedding. It turns out it is relatively easy to do geometry in this context, where the metric on the manifold is not necessarily induced from the Euclidean space, but defined by an operator.
This approach to computation, initiated in \cite{Edelman_1999} in the optimization literature, has been very successful, leading to applications in optimization, statistics, and computer vision. The computation of the Levi-Civita connection either uses the calculus of variation or the well-known formulas for embedded or submersed manifolds. We attempted to suggest a simplified framework in \cite{Nguyen2020a}.
We show this approach is also fruitful in studying Riemannian geometry itself in this article. In a sense, the approach could be considered dual to the local chart approach. Its main advantage is all formulas are defined and computed globally. The local formulas involving Christoffel symbols, for example, the {\it curvature formula}, have very straightforward global\slash embedded counterparts which we will explain shortly. The applicability comes from the fact that we only need a differentiable embedding instead of a Riemannian embedding, and that there are also similarly simple formulas in the submersion case. Furthermore, we will show Jacobi fields and tangent\slash horizontal bundle metrics can be expressed and computed easily in this formulation, and obtain several new results. It is interesting to note that the global formulas are very similar to the local ones and are easy to use.
For a Riemannian manifold $\mathrm{M}$ embedded (differentiably) in a Euclidean space $\mathcal{E}$, at each point $x\in\mathrm{M}$, the tangent space $\mathcal{T}_x\mathrm{M}$ is identified with a subspace of $\mathcal{E}$, thus the tangent bundle $\mathcal{T}\mathrm{M}$ is a subbundle of $\mathrm{M}\times \mathcal{E}$. We show there exists a positive-definite operator-valued function $\mathsf{g}$ from $\mathrm{M}$ into the space of linear operators $\mathfrak{L}(\mathcal{E}, \mathcal{E})$ on $\mathcal{E}$, inducing the original Riemannian metric on $\mathrm{M}$. It extends the bundle metric from $\mathcal{T}\mathrm{M}$ to $\mathrm{M}\times \mathcal{E}$. The extension is not unique given an intrinsic metric on $\mathrm{M}$. To use this approach to compute intrinsic Riemannian measures, we need to make a choice of $\mathsf{g}$, and the computational result will be independent of the choice. The operator $\mathsf{g}$ induces a {\it projection} $\Pi$ from $\mathrm{M}\times\mathcal{E}$ to the tangent bundle $\mathcal{T}\mathrm{M}$ of $\mathrm{M}$, or in the case where we have a Riemannian submersion $\mathfrak{q}:\mathrm{M} \to \mathcal{B}$, a projection $\ttH$ from $\mathrm{M}\times\mathcal{E}$ to the horizontal subbundle $\mathcal{H}\mathrm{M}\subset \mathcal{T}\mathrm{M}$ associated with this submersion. These projections (considered as operator-valued functions) are pivotal in this approach.
First, the projection $\Pi$ to the tangent bundle defines a connection on $\mathrm{M}$, defined simply as $\Pi(\rD_{\mathtt{X}} \mathtt{Y})$ for vector fields $\mathtt{X}$ and $\mathtt{Y}$, where $\mathtt{Y}$ is identified with an $\mathcal{E}$-valued function on $\mathrm{M}$, using the identification just discussed, $\rD_{\mathtt{X}}\mathtt{Y}$ denotes the directional derivative (covariant derivative using the trivial connection on the trivial bundle defined by the embedding of $\mathrm{M}$ in $\mathcal{E}$). In general, this connection is not compatible with metric, but if the metric operator $\mathsf{g}$ is constant, it is identical to the Levi-Civita connection. Otherwise, it differs from the Levi-Civita connection by a tensor $\mathring{\Gamma}$, evaluated on two tangent vectors $\xi$ and $\eta$ to $\mathrm{M}$ as
$$\mathring{\Gamma}(\xi, \eta) = \frac{1}{2}\Pi\mathsf{g}^{-1}((\rD_{\xi}\mathsf{g})\eta + (\rD_{\eta}\mathsf{g})\xi - \mathcal{X}(\xi, \eta))
$$
($\mathcal{X}$ is the index-raised term, see \cref{prop:Levi}). This is analogous to the usual formula for Christoffel symbols. Following \cite{Edelman_1999}, we define a concept of a Christoffel function $\Gamma$, that could be used to compute Levi-Civita covariant derivatives. It is a function from $\mathrm{M}$ to the space of bilinear functions from $\mathcal{E}\times\mathcal{E}$ to $\mathcal{E}$, such that $\nabla_{\mathtt{X}}\mathtt{Y} = \rD_{\mathtt{X}}\mathtt{Y} + \Gamma(\mathtt{X}, \mathtt{Y})$, where $\mathtt{X}, \mathtt{Y}$ are vector fields and $\nabla$ is the Levi-Civita covariant derivative. On tangent vectors, $\Gamma(\xi, \eta) = -(\rD_{\xi}\Pi)\eta + \mathring{\Gamma}(\xi, \eta)$ (again, $\rD_{\xi}\Pi$ is the directional derivative of the operator-valued function $\Pi$). Given a Christoffel function $\Gamma$, the curvature of $\mathrm{M}$ could be computed by the familiar formula
\begin{equation}\label{eq:cur_local}
\rR_{\xi, \eta}\phi = -(\rD_{\xi}\Gamma)(\eta, \phi)
+(\rD_{\eta}\Gamma)(\xi, \phi) - \Gamma(\xi, \Gamma(\eta, \phi)) +\Gamma(\eta, \Gamma(\xi, \phi))
\end{equation}
for three tangent vectors $\xi, \eta, \phi$ at $x\in\mathrm{M}$. Thus, for the textbook example of the sphere $\mathrm{M}=S^n\subset \mathcal{E}=\mathbb{R}^{n+1}$, $\Pi_x\omega = \omega - xx^{\mathsf{T}}\omega $ (with $x\in S^n, \omega\in \mathbb{R}^{n+1}$), where $\Pi_x$ denotes the projection to the tangent space $\mathcal{T}_x\mathrm{M}$ at $x$, $\Gamma(\xi, \eta) = -(\rD_{\xi}\Pi)\eta = x\xi^{\mathsf{T}}\eta$, and \cref{eq:cur_local} gives us the curvature. There is no need to convert to trigonometric coordinates. In this instance, \cref{eq:cur_local} is equivalent to a $(1,3)$-form of the Gauss-Codazzi equation, but it could be used for metric operators defined only on $\mathrm{M}$.
The approach also works for a Riemannian submersion, in \cref{theo:rsub}, we provide a formula similar to \cref{eq:cur_local}. It is equivalent to the $(1, 3)$ form of the O'Neil formula \cite{ONeil1966}. In both the embedded and submersed cases, the projections $\Pi$ (to the tangent bundle) and $\ttH$ (to the horizontal bundle $\mathcal{H}\mathrm{M}$) allow us to extend a tangent or a horizontal vector $\xi$ at a point $x\in \mathrm{M}$ to a vector field (or horizontal vector field) $p_{\xi}$ on $\mathrm{M}$, defined as $p_{\xi}(y)=\Pi_y\xi$ (or $\ttH_y\xi$) for $y\in \mathrm{M}$. In the embedded case, we could show that $[p_\xi, p_\eta]$ evaluated at $x$ vanishes for two tangent vectors $\xi$ and $\eta$, or equivalently $(\rD_{\xi}\Pi)_x\eta = (\rD_{\eta}\Pi)_x\xi$. This is not the case with horizontal projections, and the difference $(\rD_{\xi}\ttH)_x\eta - (\rD_{\eta}\ttH)_x\xi$ is exactly $2\mathrm{A}_{\xi}\eta$, where $\mathrm{A}$ is the O'Neil tensor. Thus, knowing $\ttH$ and its directional derivative is sufficient to compute the lift of the curvature of a submersed manifold $\mathcal{B}$ if the curvature of $\mathrm{M}$ is known. We can derive easily the {\it curvature of flag manifolds} from this approach, obtaining an alternative form of the curvature formula for {\it naturally reductive homogeneous spaces} (\cite{KobNom}, chapter 10). In general, the curvature computed by this approach could produce rather complicated expressions if the underlying fibration or symmetries of the manifold is not apparent. However, it makes available a procedural approach for all metrics.
Our next goal is to study {\it Jacobi fields}, which could be considered as curves on the tangent bundle $\mathcal{T}\mathrm{M}$ of $\mathrm{M}\subset \mathcal{E}$, obtained by taking directional derivatives of geodesics, considered as a function of both time and initial conditions, by a change in initial conditions. The initial data of the Jacobi field equation could be identified as a point on the double tangent bundle $\mathcal{T}\cT\mathrm{M}$, considered as a submanifold of $\mathcal{E}^4$. For $(x, v)\in \mathcal{T}\mathrm{M}\subset \mathcal{E}^2$, a Jacobi field $\mathfrak{J}(t)$ along a geodesic $\gamma(t)$, with initial condition $\gamma(0)=x, \dot{\gamma}(0) = v, \mathfrak{J}(0) = (x, \Delta_{\mathfrak{m}})\in\mathcal{T}\mathrm{M}\subset\mathcal{E}^2$ has initial time derivative of the form $\dot{\mathfrak{J}}(0) = (x, \Delta_{\mathfrak{m}}, v, \Delta_{\mathfrak{t}})\in \mathcal{T}\cT\mathrm{M}\subset \mathcal{E}^4$, where $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in\mathcal{E}^2$ is a tangent vector in $\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$, the tangent space of $\mathcal{T}\mathrm{M}$ at $(x, v)$. We describe $\mathcal{T}\cT\mathrm{M}$ as a submanifold of $\mathcal{E}^4$, with constraints given in terms of the projection $\Pi$ and its directional derivative. To the best of our knowledge, the horizontal lift of a Jacobi field (from a curve on the tangent bundle $\mathcal{T}\mathcal{B}$ of a base manifold $\mathcal{B}$ to a curve on the horizontal bundle $\mathcal{H}\mathrm{M}$ in a Riemannian submersion $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$) has not been studied before, and it could be described quite explicitly in our framework. We show $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}:\mathcal{H}\mathrm{M}\to\mathcal{T}\mathcal{B}$ is a differentiable submersion, describe the vertical bundle $\mathcal{V}\mathcal{H}\mathrm{M}$ of this submersion explicitly by a map $\mathrm{b}$ from the vertical bundle $\mathcal{V}\mathrm{M}$ to $\mathcal{V}\mathcal{H}\mathrm{M}$, constructed by directional derivatives of $\ttH$. We also identify a subbundle $\mathcal{Q}\mathcal{H}\mathrm{M}$ of $\mathcal{T}\mathcal{H}\mathrm{M}$, which is transversal to the vertical bundle $\mathcal{V}\mathcal{H}\mathrm{M}$, which will play the role of a horizontal bundle in the submersion $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$. We define the {\it canonical flip} $\mathfrak{j}_{\cH}$ on $\mathcal{Q}\mathcal{H}\mathrm{M}$ (again, with the help of the O'Neil tensor) which corresponds to the {\it canonical flip} on $\mathcal{T}\cT\mathcal{B}$. The initial data of a lifted Jacobi field could be identified with a point of $\mathcal{Q}\mathcal{H}\mathrm{M}$. Our main result for Jacobi fields is a horizontal lift formula in \cref{theo:jacsub}, using this canonical flip. We also obtain a formula for horizontal lifts of Jacobi fields of naturally reductive homogeneous spaces in \cref{theo:jacobi_submerse_nat}, further clarify the relationship between invariant vector fields and Jacobi fields. We also add partial results to a conjecture of Ziller characterizing symmetric spaces by zeros of Jacobi fields.
As we have a description of the double tangent bundle $\mathcal{T}\cT\mathrm{M}\subset\mathcal{E}^4$, we can also explicitly construct {\it natural metrics} \cite{Sasaki,TriMuss,KowalSeki,Abbassi2005,GudKap} on $\mathcal{T}\cT\mathrm{M}$. We describe explicitly the connection map $\mathrm{C}$, the metric operator $\mathsf{G}$, and its projection $\Pi_{\mathsf{G}}$ on $\mathcal{T}\cT\mathrm{M}$, for a family of metrics constructed based on two real-valued functions $\alpha, \beta$. The Sasaki metric \cite{Sasaki} ($\alpha=1, \beta=0$) and the Cheeger-Gromoll metric ($\alpha=\beta=(1+t)^{-1}$, constructed by Tricerri and Musso \cite{TriMuss}) are special cases of this family. Using the results in these cited works, we express the Christoffel function $\Gamma_{\mathsf{G}}$ corresponding to this metric in our framework.
Since $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}:\mathcal{H}\mathrm{M}\to\mathcal{T}\mathcal{B}$ is a differentiable submersion, naturally we wish to give $\mathcal{H}\mathrm{M}$ a metric $\mathsf{G}_{\cQ}$ so $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$ is a Riemannian submersion if $\mathcal{T}\mathcal{B}$ is equipped with the metric $\mathsf{G}$ above. The construction is similar to the embedded case, but the connection map $\mathrm{C}$ is replaced by a modified counterpart $\mathrm{C}^{\mathrm{\cQ}}$, constructed with the help of both the Christoffel function and the map $\mathrm{b}$ defining the vertical bundle $\mathcal{V}\mathcal{H}\mathrm{M}$. We provide formulas for the horizontal Christoffel function $\Gamma^{\mathcal{H}}_{\mathsf{G}_{\cQ}}$, allowing us to compute the horizontal lift of covariant derivatives on $\mathcal{T}\mathcal{B}$.
Besides the examples used to illustrate the concepts (including $\SOO(n)$, the tangent bundle of a sphere and flag manifolds), we present a detailed calculation for the Grassmann manifold $\Gr{p}{n}$, considered as the submersed image of the Stiefel manifold $\St{p}{n}$, providing explicit formulas for its Jacobi field, and the natural metric on the horizontal bundle $\mathcal{H}\St{p}{n}$ corresponding to this submersion.
\section{Related works}The formulas for the Levi-Civita connections for Riemannian embedding and submersion are classical, for example in \cite{ONeil1983}, we make the observation that $\mathsf{g}$ is only required to be defined on $\mathrm{M}$. Some results in \cref{sec:christ_func} overlap with \cite{Nguyen2020a}, but the focus of that paper is on numerical implementation and optimization, the proofs given here are also different. The foundational paper \cite{Edelman_1999} provided formulas in the style studied here for Grassmann and Stiefel manifolds, popularizing the method of Riemannian optimization from the earlier works of \cite{GaLu,Gabay1982}. A rather extensive collection of manifolds have been studied by this method, as quotients of products of Stiefel, Grassmann manifolds, Lie groups, and symmetric spaces and a few of their differential geometric measures have been implemented in computer codes, available in \cite{Manopt,Pymanopt,Geomstats}, among others. For the most part, these examples consider metrics that extend almost everywhere to the ambient space \cite{AMS_book, ONeil1983}, thus the role of the metric operator has not been emphasized, and the tensor $\mathring{\Gamma}$ has not been studied. The treatment of the Gauss-Codazzi equation as a result about subbundles could be found in \cite{taylor2011partial}. We learn about the canonical flip from \cite{Michor}. The constant-coefficient differential equation for Jacobi fields of a naturally reductive space appeared in \cite{Rauch,Chavel,Ziller}. The closed-form formula for Jacobi fields for symmetric spaces, but not homogeneous spaces, is also well-known. The formula for Jacobi fields for $\SOO(n)$ traces back to \cite{GKR}. The idea of the horizontal lift of Jacobi fields in \cref{theo:jacsub} has been noted in Section 8 of \cite{Chavel}. Natural metrics were first studied in \cite{Sasaki} and subsequently extended by many authors. We provide the construction of families of natural metrics for horizontal bundles using our framework.
\section{Notations} By $\mathbb{R}^{m\times n}$, we denote the space of real matrices of size $m\times n$. The base inner product on an inner product space $\mathcal{E}$ is denoted by $\langle,\rangle_{\mathcal{E}}$. We use $\mathfrak{L}$ to denote the space of linear operators, for example, $\mathfrak{L}(\mathcal{E}, \mathcal{E})$ is the space linear operator on $\mathcal{E}$, $\mathfrak{L}(\mathcal{E}\otimes\mathcal{E}, \mathcal{E})$ is the space of $\mathcal{E}$-valued bilinear operators on $\mathcal{E}$. The metric operator on a manifold $\mathrm{M}$ is often denoted by $\mathsf{g}$, $\Pi$ denotes the projection (under $\mathsf{g}$) from $\mathrm{M}\times\mathcal{E}$ to the tangent bundle $\mathcal{T}\mathrm{M}$ of $\mathrm{M}$, considered as a function from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$. If $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$ is a Riemannian submersion, we denote by $\mathcal{V}\mathrm{M}$ and $\mathcal{H}\mathrm{M}$ the vertical and horizontal subbundles of $\mathcal{T}\mathrm{M}$, and $\ttV$, $\ttH$, the projections from $\mathcal{E}$ to $\mathcal{V}\mathrm{M}$ and $\mathcal{H}\mathrm{M}$. For a tangent vector $\xi$, $\rD_{\xi}$ denotes the directional derivative in direction $\xi$ of a scalar, vector or operator-valued function. We use the same notation $\rD_{\mathtt{X}}$ for the derivatives by a vector field of scalar, vector, or operator-valued function. In particular, for two vector fields $\mathtt{X}$ and $\mathtt{Y}$, $\rD_{\mathtt{X}}\mathtt{Y}$ makes sense if we identify $\mathtt{Y}$ with a function from $\mathrm{M}$ to $\mathcal{E}\supset \mathrm{M}$. For a geodesic $\gamma(t)$, we denote by $\nabla_{d/dt}$ the covariant derivative in direction $\dot{\gamma}(t)$ along the curve. By $\Gamma$ and $\GammaM$ we denote the Christoffel function of $\mathrm{M}$, $\GammaH$ and $\GammaV$ denote the horizontal and vertical Christoffel functions defined in \cref{sec:christ_func}. By $\nabla^{\mathcal{H}}$ we denote $\ttH\nabla$, the horizontal component of the Levi-Civita connection. The O'Neil tensor is denoted by $\mathrm{A}$ with adjoint $\mathrm{A}^{\dagger}$, defined in \cref{subsec:Rsubmerse}. The tangent bundle projection map is typically denoted by $\pi$, $\mathrm{C}$ is the connection map, $\mathfrak{J}$ denotes the Jacobi field and $J$ denotes the tangent component of $\mathfrak{J}$ in the embedding in $\mathcal{E}^2$. By $\mathcal{Q}$ we denote the horizontal subbundle of $\mathcal{T}\mathcal{H}\mathrm{M}$, defined in \cref{sec:subm_tangent}.
We also use $\mathsf{g}_x$ to denote the valuation of $\mathsf{g}$ at $x$, this also applies for projections $\ttH_x$, vector fields $\mathtt{Y}_x$, etc. Because of this, partial derivatives will always be denoted by $\partial$ to avoid confusion. The Riemannian exponential map is denoted by $\Exp$, with $\Exp_x v$ denotes the point $\gamma(1)$ for the geodesic $\gamma(t)$ with $\gamma(0) = x, \dot{\gamma}(0) = v$. The exponential map corresponding to a one-parameter subgroup of a Lie group is denoted by $\exp(tX)$, with $X$ an element of the corresponding Lie algebra, $t\in\mathbb{R}$.
For integers $n$ and $p$, we denote the orthogonal group in $\mathbb{R}^{n\times n}$ by $\OO(n)$, $\St{p}{n}$ the Stiefel submanifold of $\mathbb{R}^{n\times p}$, $\Gr{p}{n}$ the corresponding Grassmann manifold, $\Herm{p}$ the vector space of symmetric matrices in $\mathbb{R}^{p\times p}$, and $\mathfrak{o}(p)$ the space of antisymmetric matrices. We denote the symmetrize\slash asymmetrize operators by $\mathrm{sym}(A) = \frac{1}{2}(A + A^{\ft})$ and $\mathrm{skew}(A) = \frac{1}{2}(A - A^{\ft})$. By $\|\|_{\mathsf{g}}$ we denote the norm associated with the metric operator $\mathsf{g}$, and $\|\|_{\mathsf{g}, x}$ denotes the norm at a particular point.
\section{Embedded ambient structure: metric operator, projection, and the Levi-Civita connection}\label{sec:christ_func}
\subsection{Differentiable embedding and metric operator}\label{subsec:subman}
Let $\mathcal{E}$ be an inner product space with the inner product of $\omega_1, \omega_2\in\mathcal{E}$ denoted by $\langle\omega_1, \omega_2\rangle_{\mathcal{E}}$. Assume $\mathrm{M}$ is a differential submanifold of $\mathcal{E}$. It is well-known, \cite{LeeSmooth} any (Hausdorff, $\sigma$-compact) differential manifold $\mathrm{M}$ could be embedded to an inner product space $\mathcal{E}$, for example, by the Whitney embedding theorem. We will assume $\mathrm{M}$ is equipped with a Riemannian metric $\langle\rangle_R$, not necessarily the metric induced from the embedding in $\mathcal{E}$. Let $\mathcal{T}\mathrm{M}$ be the tangent bundle of $\mathrm{M}$, so at each point $x\in\mathrm{M}$, the tangent space $\mathcal{T}_x\mathrm{M}$ is identified with a subspace of $\mathcal{E}$. We define a metric operator as follows.
\begin{definition} Let $\mathrm{M}$ be a Riemannian manifold, $\mathrm{M}\subset \mathcal{E}$ is a differentiable embedding of $\mathrm{M}$ in a Euclidean space $(\mathcal{E}, \langle\rangle_{\mathcal{E}})$. A metric operator $\mathsf{g}$ on $\mathrm{M}$ is a smooth operator-valued function $\mathsf{g}$ from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$ such that $\mathsf{g}(x)$ is positive-definite for all $x\in\mathrm{M}$ and $\langle \xi, \eta\rangle_{R, x} = \langle\xi, \mathsf{g}(x)\eta\rangle_{\mathcal{E}}$ for all $x\in\mathrm{M}, \xi, \eta\in \mathcal{T}_x\mathrm{M}$, where $\langle\rangle_{R, x}$ denotes the Riemannian metric evaluated at $x$. The triple $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ is called an embedded ambient structure (or simply ambient structure) of $\mathrm{M}$.
\end{definition}
We will also write $\mathsf{g}_x$ for $\mathsf{g}(x)$. In general, a positive-definite operator $\mathsf{g}$ on an inner product space $\mathcal{E}$ induces a new inner product $\langle\rangle_{\mathsf{g}}$, defined as $\langle\omega_1, \mathsf{g}\omega_2\rangle_{\mathcal{E}}$, and we write $\langle\rangle_{\mathsf{g}, x}$ for the inner product defined by $\mathsf{g}_x$. Recall if $\mathcal{E}_1, \langle\rangle_{\mathcal{E}_1}$ and $\mathcal{E}_2, \langle\rangle_{\mathcal{E}_2}$ are two inner product spaces, and $f:\mathcal{E}_1\to\mathcal{E}_2$ is a linear map, the adjoint $f^{\mathsf{T}}:\mathcal{E}_2\to\mathcal{E}_1$ of $f$ is the unique map satisfying $\langle \nu_2, f \omega_1\rangle_{\mathcal{E}_2} = \langle f^{\mathsf{T}}\nu_2, \omega_1\rangle_{\mathcal{E}_1}$, for $\nu_2\in\mathcal{E}_2, \omega_1\in \mathcal{E}_1$.
In \cref{prop:emb_exists}, we show metric operators always exist. The proof uses a standard result on projections stated below (which is also used in \cite{Nguyen2020a}).
\begin{lemma}\label{lem:projprop}Let $(\mathcal{E}, \langle\rangle_{\mathcal{E}})$ be an inner product space and $\mathcal{V}\subset \mathcal{E}$ be a subspace. Let $\mathsf{g}$ be a positive-definite operator on $\mathcal{E}$. Assume there exists an inner product space $(\mathcal{E}_{\mathrm{N}},\langle\rangle_{\mathcal{E}_{\mathrm{N}}})$ and a linear map $\mathrm{N}:\mathcal{E}_{\mathrm{N}}\to \mathcal{E}$ such that $\mathrm{N}(\mathcal{E}_{\mathrm{N}}) = \mathcal{V}$ and $\mathrm{N}$ is injective. Let $\mathrm{N}^{\mathsf{T}}:\mathcal{E}\to\mathcal{E}_{\mathrm{N}}$ be the adjoint of $\mathrm{N}$ under the inner products $\langle\rangle_{\mathcal{E}}$ and $\langle\rangle_{\mathcal{E}_{\mathrm{N}}}$, then $\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N}$ is invertible and the projection $\Pi_{\mathsf{g}}$ from $\mathcal{E}$ to $\mathcal{V}$ under the metric induced by $\mathsf{g}$ is given by
\begin{equation}\label{eq:Pig}
\Pi_{\mathsf{g}}\omega = \mathrm{N}(\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N})^{-1}\mathrm{N}^{\mathsf{T}}\mathsf{g}\omega
\end{equation}
for $\omega\in\mathcal{E}$. That means $\Pi_{\mathsf{g}}\omega\in \mathcal{V}$, and $\langle \Pi_{\mathsf{g}}\omega, \mathsf{g}\eta\rangle_{\mathcal{E}} = \langle \omega, \mathsf{g}\eta\rangle_{\mathcal{E}}$ for any $\eta\in\mathcal{V}$.
Moreover, $\Pi_\mathsf{g}$ is idempotent and $\mathsf{g}\Pi_\mathsf{g}$ is self-adjoint under $\langle\rangle_{\mathcal{E}}$.
\end{lemma}
\begin{proof} If $\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N}\delta = 0$ for $\delta\in \mathcal{E}_{\mathrm{N}}$, then $\langle\mathsf{g}\mathrm{N}\delta, \mathrm{N}\delta \rangle_{\mathcal{E}} = \langle \mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N}\delta, \delta\rangle_{\mathcal{E}}=0$, this means $\mathrm{N}\delta = 0$ since $\mathsf{g}$ is positive-definite, and so $\delta = 0$ since $\mathrm{N}$ is injective. Thus $\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N}$ is invertible. It is clear $\Pi_{\mathsf{g}}\omega\in \mathcal{V}$. If $\eta\in \mathcal{V}$, then $\eta = \mathrm{N}\delta$ for $\delta\in \mathcal{E}_{\mathrm{N}}$, thus
$$\langle\mathrm{N}(\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N})^{-1}\mathrm{N}^{\mathsf{T}}\mathsf{g} \omega, \mathsf{g}\eta\rangle_{\mathcal{E}} =
\langle \omega,\mathsf{g}\mathrm{N} (\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N})^{-1}\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N}\delta\rangle_{\mathcal{E}} =
\langle \omega, \mathsf{g}\mathrm{N}\delta\rangle_{\mathcal{E}}=\langle \omega, \mathsf{g}\eta\rangle_{\mathcal{E}}$$
Finally, it is clear from \cref{eq:Pig} that $\Pi_{\mathsf{g}}$ is idempotent and $\mathsf{g}\Pi_{\mathsf{g}}$ is self-adjoint.
\end{proof}
\begin{proposition}\label{prop:emb_exists}If a manifold $\mathrm{M}$ is differentiably embedded in $\mathcal{E}$, then $x\mapsto \Pi^{\mathcal{E}}_x$ is a smooth operator-valued map from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$, the space of linear operators in $\mathcal{E}$, where $\Pi^{\mathcal{E}}_x$ denotes the projection from $\mathcal{E}$ to $\mathcal{T}_x\mathrm{M}$ under the inner product in $\mathcal{E}$. Assume $\mathrm{M}$ is equipped with a Riemannian metric $\langle\rangle_R$, then there exists a smooth operator-valued function $\mathsf{g}$ from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$ such that $\mathsf{g}_x :=\mathsf{g}(x)$ is positive-definite for all $x\in\mathrm{M}$ and for two tangent vectors $\eta, \xi\in \mathcal{T}_x\mathrm{M}$ we have
\begin{equation}\langle \eta, \xi\rangle_{R, x} = \langle\eta, \mathsf{g}_x\xi\rangle_{\mathcal{E}}
\end{equation}
If $\dI_{\mathcal{E}}$ is the identity map of $\mathcal{E}$, we can take $\mathsf{g}$ to be the operator defined by
\begin{equation}\label{eq:stdg}
\mathsf{g}_x\omega = (\dI_{\mathcal{E}} - \Pi^{\mathcal{E}}_x)\omega + \mathsf{g}_{R, x}(\Pi^{\mathcal{E}}_{x}\omega)
\end{equation}
where $\mathsf{g}_{R, x}$ is the unique self-adjoint operator on $\mathcal{T}_x\mathrm{M}$ such that
$$\langle \xi, \eta\rangle_R = \langle \xi, \mathsf{g}_{R, x}\eta\rangle_{\mathcal{E}}$$
\end{proposition}
\begin{proof} As a projection is defined for any subspace, $\Pi_x^{\mathcal{E}}$ is a well-defined map for all points $x\in \mathrm{M}$. Let $m$ be the dimension of $\mathrm{M}$, $\psi_{\mathcal{U}}: \mathbb{R}^m\to \mathcal{U}\subset\mathrm{M}\subset \mathcal{E}$ be a coordinate chart for $\mathcal{U}\subset\mathrm{M}$ near $x$ considered as a map from $\mathbb{R}^m$ to $\mathcal{E}$. Then $d\psi_{\mathcal{U}}(x)$ is a map from $\mathbb{R}^m$ to $\mathcal{E}$, injective with image precisely $\mathcal{T}_x\mathrm{M}$. Applying \cref{lem:projprop} for the identity operator, $\Pi_x^{\mathcal{E}} = d\psi_{\mathcal{U}}(x)(d\psi_{\mathcal{U}}^{\mathsf{T}}(x)d\psi_{\mathcal{U}}(x))^{-1}d\psi_{\mathcal{U}}^{\mathsf{T}}(x)$. Since $\psi$ is assumed to be smooth, $\Pi_x^{\mathcal{E}}$ is smooth in $\mathcal{U}$.
To show $\mathsf{g}$ defined in \cref{eq:stdg} is self-adjoint, note that $\Pi_x^{\mathcal{E}}$, thus $\dI_{\mathcal{E}} - \Pi_x^{\mathcal{E}}$ is self-adjoint. For $\omega_1, \omega_2\in \mathcal{E}$ we have
$$\langle \omega_1,\mathsf{g}_{R, x} \Pi^{\mathcal{E}}_{x}\omega_2\rangle_{\mathcal{E}}= \langle\Pi^{\mathcal{E}}_{x} \omega_1,\mathsf{g}_{R, x} \Pi^{\mathcal{E}}_{x}\omega_2\rangle_{\mathcal{E}}
= \langle\mathsf{g}_{R, x}\Pi^{\mathcal{E}}_{x} \omega_1, \Pi^{\mathcal{E}}_{x}\omega_2\rangle_{\mathcal{E}}
$$
where the first equality is the defining property of a projection, and the second is because $\mathsf{g}_R$ is self-adjoint on $\mathcal{T}_x\mathrm{M}$. The last expression is $\langle\mathsf{g}_{R, x}\Pi^{\mathcal{E}}_{x} \omega_1, \omega_2\rangle_{\mathcal{E}}$ by property of a projection. Thus $\mathsf{g}_{R, x}\Pi^{\mathcal{E}}_{x}$ is self-adjoint, hence $\mathsf{g}$ in \cref{eq:stdg} is self-adjoint. If $\langle\omega, \mathsf{g}_x\omega\rangle_{\mathcal{E}} =0$ then $\langle\omega, (\dI_{\mathcal{E}} - \Pi^{\mathcal{E}}_x)\omega\rangle_{\mathcal{E}} + \langle \omega, \mathsf{g}_{R, x}\Pi^{\mathcal{E}}_x\omega\rangle_{\mathcal{E}} = 0$, or
$$\langle(\dI_{\mathcal{E}} - \Pi^{\mathcal{E}}_x)\omega, (\dI_{\mathcal{E}} - \Pi^{\mathcal{E}}_x)\omega\rangle_{\mathcal{E}} + \langle\Pi^{\mathcal{E}}_x \omega, \mathsf{g}_{R, x}\Pi^{\mathcal{E}}_x\omega\rangle_{\mathcal{E}}=0$$
using the fact that $(\dI_{\mathcal{E}} - \Pi^{\mathcal{E}}_x)$ is idempotent and self-adjoint on the first term and that $\Pi^{\mathcal{E}}_x$ is a $\langle\rangle_{\mathcal{E}}$-projection on the second term (because $ \mathsf{g}_{R, x}\Pi^{\mathcal{E}}_x\omega\in \mathcal{T}_x\mathrm{M}$). Both terms are nonnegative, therefore both have to be zero, hence $\omega=0$.
\end{proof}
It is easy to see $\mathsf{g}$ satisfying the proposition above is not unique: for example, for the sphere $\mathrm{M}=S^n\subset\mathcal{E}=\mathbb{R}^{n+1}$ with $\omega\in \mathcal{E}$ and $x\in S^n$, the operator $\mathsf{g}(x)\omega = \omega +\beta xx^{\mathsf{T}}\omega$ for $\beta \geq 0$ is a positive-definite operator. While each $\beta$ defines a different operator on $\mathcal{E}$, the induced metrics on $\mathcal{T}_x\mathrm{M}$ are given by the same expression $\eta^{\mathsf{T}}\eta$ for a tangent vector $\eta$.
\begin{proposition}\label{prop:Levi} Let $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ be an ambient structure of a Riemannian manifold $\mathrm{M}$ with metric operator $\mathsf{g}$. Then at each $x\in \mathrm{M}$, $\mathsf{g}_x$ defines an inner product $\langle \omega_1, \omega_2\rangle_{\mathsf{g}, x} =
\langle \omega_1, \mathsf{g}_x\omega_2\rangle_{\mathcal{E}}$ on $\mathcal{E}$ for $\omega_1, \omega_2\in \mathcal{E}$. Denote by $\Pi_{\mathsf{g}, x}$ the associated projection to $\mathcal{T}_x\mathrm{M}$. Then $\Pi_{\mathsf{g}}:x\mapsto \Pi_{\mathsf{g}, x}$ is a smooth map from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$. There exists a smooth operator-valued function $\Gamma$ from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}\otimes\mathcal{E}, \mathcal{E})$, the space of $\mathcal{E}$-valued bilinear forms on $\mathcal{E}$, such that if $\mathtt{X}$ and $\mathtt{Y}$ are vector fields on $\mathrm{M}$, then
\begin{equation}\label{eq:LvC}
(\nabla_{\mathtt{X}} \mathtt{Y})_x = (\rD_{\mathtt{X}}\mathtt{Y})_x + \Gamma(\mathtt{X}_x, \mathtt{Y}_x)\end{equation}
for all $x\in\mathrm{M}$, where $\nabla_{\mathtt{X}} \mathtt{Y}$ is the covariant derivative defined by the Levi-Civita connection of the metric induced by $\mathsf{g}$. Here, we identify $\mathtt{Y}$ with a $\mathcal{E}$-valued function and $(\rD_{\mathtt{X}}\mathtt{Y})_x$ is its directional derivative in the direction $\mathtt{X}_x$. We call such $\Gamma$ a {\bf Christoffel function} of $(\mathrm{M}, \mathsf{g}, \mathcal{E})$. There could be more than one Christoffel function given a metric operator $\mathsf{g}$, however for two tangent vector $\xi, \eta$, $\Gamma(\xi, \eta)$ is independent of the vector field extensions of $\xi$ and $\eta$, and is only dependent on the restriction of the metric operator $\mathsf{g}$ to the tangent bundle.
If $\mathcal{X}$ is a smooth function from $\mathrm{M}$ to the space $\mathfrak{L}(\mathcal{E}\times\mathcal{E}, \mathcal{E})$ of $\mathcal{E}$-bilinear forms with value in $\mathcal{E}$, such that at any point $x\in \mathrm{M}$ and any triple $\xi, \eta, \phi$ of tangent vectors to $\mathrm{M}$ at $x$,
\begin{equation}\label{eq:cX}
\langle \mathcal{X}(\xi, \eta)_x, \phi\rangle_{\mathcal{E}} = \langle(\rD_{\phi}\mathsf{g})_x\xi, \eta\rangle_{\mathcal{E}}\end{equation}
Then for two tangent vectors $\xi, \eta$ at $x\in\mathrm{M}$
\begin{equation}\label{eq:GammaForm}\Gamma(\xi, \eta)_x = -(\rD_{\xi}\Pi)\eta + \frac{1}{2} \Pi_{\mathsf{g}, x}\mathsf{g}_x^{-1}(\rD_{\xi}\mathsf{g}\eta +\rD_{\eta}\mathsf{g}\xi-\mathcal{X}(\xi, \eta))_x\end{equation}
\end{proposition}
\begin{proof} Let $m$ be the dimension of $\mathrm{M}$ and $\psi_{\mathcal{U}}:\mathbb{R}^m\to\mathcal{U}\subset \mathcal{E}$ be a coordinate function for an open subset $\mathcal{U}$ of $\mathrm{M}$, considered as a map from $\mathbb{R}^m$ to $\mathcal{E}$. Then $\Pi_{\mathsf{g}} = d\psi_{\mathcal{U}}(d\psi_{\mathcal{U}}^{\mathsf{T}}\mathsf{g} d\psi_{\mathcal{U}})^{-1}d\psi_{\mathcal{U}}^\mathsf{T}\mathsf{g}$ in $\mathcal{U}$, hence $\Pi_{\mathsf{g}}$ is smooth in $\mathcal{U}$, thus, on $\mathrm{M}$.
To construct $\Gamma$, let $\mathcal{N}\subset\mathcal{E}$ be a tubular neighborhood of $\mathrm{M}$ (see \cite{LeeRiemann}), $\mathcal{N}$ is an open subset in $\mathcal{E}$ and there is a retraction $r$, a smooth map from $\mathcal{N}$ to $\mathrm{M}$, such that $r$ is the identity map on $\mathrm{M}$. We extend $\mathsf{g}$ from $\mathrm{M}$ to $\mathcal{N}$, setting $\mathsf{g}(x) = \mathsf{g}(r(x))$ for $x\in \mathcal{N}$. This gives $\mathcal{N}$ a metric. Since $\mathcal{N}$ is an open subset of $\mathcal{E}$, the Levi-Civita connection on $\mathcal{N}$ is defined via the classical Christoffel symbols, which could be written as a bilinear form
\begin{equation}\label{eq:GammaN}\Gamma_{\mathcal{N}}(\omega_1, \omega_2)=\frac{1}{2}\mathsf{g}^{-1}(\rD_{\omega_1}\mathsf{g}\omega_2 + \rD_{\omega_2}\mathsf{g}\omega_1 - \mathcal{X}_0(\omega_1, \omega_2))\end{equation}
for $\omega_1, \omega_2\in\mathcal{E}$, with $\mathcal{X}_0(\omega_1, \omega_2)$ satisfies $
\langle\mathcal{X}_0(\omega_1, \omega_2), \omega_3\rangle_{\mathcal{E}} = \langle \omega_2,\rD_{\omega_3}\omega_1\rangle_{\mathcal{E}}$ for $\omega_3\in \mathcal{E}$. Since $\mathrm{M}$ is a Riemannian submanifold of $\mathcal{N}$ with this metric, its Levi-Civita connection for two vector fields $\mathtt{X}, \mathtt{Y}$ on $\mathrm{M}$ is given by $\Pi_{\mathsf{g}}(\rD_{\mathtt{X}}\mathtt{Y} + \Gamma_{\mathcal{N}}(\mathtt{X}, \mathtt{Y}))$ (\cite{ONeil1983}, Lemma 4.3). As $\Pi_{\mathsf{g}} \mathtt{Y} = \mathtt{Y}$, $\Pi_{\mathsf{g}}(\rD_{\mathtt{X}}\mathtt{Y}) = \rD_{\mathtt{X}}(\Pi_{\mathsf{g}} \mathtt{Y}) - (\rD_{\mathtt{X}}\Pi_{\mathsf{g}})\mathtt{Y} = \rD_{\mathtt{X}} \mathtt{Y} - (\rD_{\mathtt{X}}\Pi_{\mathsf{g}})\mathtt{Y}$, so
$$\nabla_{\mathtt{X}}\mathtt{Y} = \rD_\mathtt{X} \mathtt{Y} -(\rD_\mathtt{X}\Pi_{\mathsf{g}})\mathtt{Y} + \Pi_{\mathsf{g}}\Gamma_{\mathcal{N}}(\mathtt{X}, \mathtt{Y})$$
Hence, we can define $\Gamma(\xi, \omega)_x := -(\rD_{\xi}\Pi_{\mathsf{g}})_x\Pi_{\mathsf{g}, x}\omega + \Pi_{\mathsf{g}, x}\Gamma_{\mathcal{N}}(\xi, \omega)_x$ for $\xi\in \mathcal{T}_x\mathrm{M}, \omega\in\mathcal{E}$, then $\Gamma(\mathtt{X}, \mathtt{Y})_x = -(\rD_{\mathtt{X}}\Pi_{\mathsf{g}})_x\mathtt{Y}_x + \Pi_{\mathsf{g}, x}\Gamma_{\mathcal{N}}(\mathtt{X}, \mathtt{Y})_x$ satisfies \cref{eq:LvC}. We can extend $\Gamma$ to an operator from $\mathcal{E}\times\mathcal{E}$ to $\mathcal{E}$ by defining $\Gamma(\omega_1, \omega_2)_x = \Gamma(\Pi_{\mathsf{g}}\omega_1, \omega_2)_x$. Thus we have proved the existence of a Christoffel function $\Gamma$, and it is clearly smooth.
It is clear the condition in \cref{eq:LvC} implies $\Gamma(\mathtt{X}, \mathtt{Y})_x$ is only dependent on the value of $\mathtt{X}_x, \mathtt{Y}_x$ at $x$. Since $(\rD_{\mathtt{X}}\mathtt{Y})_x$ is not dependent on the metric, while $\nabla_{\mathtt{X}}\mathtt{Y}$ is only dependent on the restriction of the metric to the tangent bundle, $\Gamma$ evaluated on tangent vectors is only dependent on the restriction of a metric operator on the tangent bundle.
Let $\mathcal{X}(\xi, \eta)$ be a function satisfying \cref{eq:cX}. $(\Pi_{\mathsf{g}}\mathsf{g}^{-1}\mathcal{X}(\xi, \eta))_x$ is a vector in $\mathcal{T}_x\mathrm{M}$ satisfying $\langle (\Pi_{\mathsf{g}}\mathsf{g}^{-1}\mathcal{X}(\xi, \eta))_x, \mathsf{g}_x\phi \rangle_{\mathcal{E}}= \langle(\rD_{\phi}\mathsf{g})_x\xi, \eta\rangle_{\mathcal{E}}$ for all $\phi\in \mathcal{T}_x\mathrm{M}$. Such vector is unique as $\mathsf{g}_x$ is non degenerated in $\mathcal{T}_x\mathrm{M}$, so $(\Pi_{\mathsf{g}}\mathsf{g}^{-1}\mathcal{X}(\xi, \eta))_x = (\Pi_{\mathsf{g}}\mathsf{g}^{-1}\mathcal{X}_0(\xi, \eta))_x$ with $\mathcal{X}_0$ as in \cref{eq:GammaN}, so the right-hand side of \cref{eq:GammaForm} evaluate to the same value as that of $\Gamma$ constructed from the existence part when restricted to tangent vectors, this proves the last statement.
\end{proof}
This last statement allows us to work with a more convenient $\mathcal{X}$ in some cases, as we only need \cref{eq:cX} to be valid on tangent vectors.
\begin{remark}
We will use the {\it derivative of projection trick} converting $\Pi_1(\rD_{\xi}\Pi_2)\omega$ to $(\rD_{\xi}(\Pi_1\Pi_2))\omega - (\rD_{\xi}\Pi_1)\Pi_2\omega$ often in this paper, for two projections $\Pi_1$ and $\Pi_2$. It is used in the following lemma, providing standard commuting vector fields extension of tangent vectors useful in tensor calculations.
\end{remark}
\begin{lemma}Let $\xi, \eta$ be two tangent vectors to $\mathrm{M}$ at $x\in\mathrm{M}$, where $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ is an embedded ambient structure. Define vector fields $p_{\xi}, p_{\eta}$ on $\mathrm{M}$ by setting $p_{\xi}(y) = \Pi_{\mathsf{g}, y}\xi$ and $p_{\eta}(y) = \Pi_{\mathsf{g}, y}\eta$ for $y\in \mathrm{M}$. Then $p_{\xi}(x) = \xi, p_{\eta}(x) = \eta$ and
\begin{equation}\label{eq:proj_deriv}\begin{gathered}\Pi_{\mathsf{g}, x}(\rD_{\xi}p_{\eta})_x = 0\\
[p_{\xi},p_{\eta}]_x = 0\\
(\rD_{\xi}\Pi_{\mathsf{g}})_x\eta = (\rD_{\eta}\Pi_{\mathsf{g}})_x\xi
\end{gathered}
\end{equation}
\end{lemma}
\begin{proof}It is clear $p_{\xi}(x) = \xi, p_{\eta}(x) = \eta$. $p_{\eta}$ could be considered as an $\mathcal{E}$-valued function on $\mathrm{M}$, so $(\rD_{\xi}p_{\eta})_x=(\rD_{\xi}(\Pi_{\mathsf{g}}\eta))_x$ is defined and is equal to $(\rD_{\xi}\Pi_{\mathsf{g}})_x\eta$. Note that at $x$, $\Pi_{\mathsf{g}, x}(\rD_{\eta}p_{\xi})_x$ expands to
$$\Pi_{\mathsf{g}, x}(\rD_{\eta}\Pi_{\mathsf{g}})_x\xi = (\rD_{\eta}(\Pi_{\mathsf{g}}^2)_x)\xi - (\rD_{\eta}\Pi_{\mathsf{g}})_x(\Pi_{\mathsf{g}, x}\xi) =
(\rD_{\eta}\Pi_{\mathsf{g}})_x\xi - (\rD_{\eta}\Pi_{\mathsf{g}})_x\xi = 0$$
and similarly $\Pi_{\mathsf{g}, x}(\rD_{\xi}p_{\eta})_x=0$, thus $\Pi_{\mathsf{g}, x}[p_{\xi}, p_{\eta}]_x =
\Pi_{\mathsf{g}, x}(\rD_{\xi}p_{\eta})_x - \Pi_{\mathsf{g}, x}(\rD_{\eta}p_{\xi})_x= 0$. But $[p_{\xi}, p_{\eta}]_x$ is tangent to $\mathrm{M}$ at $x$, hence $[p_{\xi}, p_{\eta}]_x=\Pi_{\mathsf{g}, x}[p_{\xi}, p_{\eta}]_x =0$. This implies $(\rD_{\xi}(\Pi_{\mathsf{g}}\eta))_x = (\rD_{\eta}(\Pi_{\mathsf{g}}\xi))_x$.
\end{proof}
\begin{remark}Note that when the metric operator $\mathsf{g}$ is constant, $\Gamma(\xi, \eta) = -\rD_{\xi}\Pi_{\mathsf{g}}\eta$ for tangent vectors $\xi, \eta$.
Another way to look at \cref{eq:GammaForm} is for vector fields $\mathtt{X}$ and $\mathtt{Y}$ on $\mathrm{M}$ considered as functions from $\mathrm{M}$ to $\mathcal{E}$, $\Pi_{\mathsf{g}}\rD_{\mathtt{X}}\mathtt{Y}$ is a connection, although in general not compatible with metric. The Levi-Civita connection is another connection, thus the difference gives rise to a $(1, 2)$-tensor $\mathring{\Gamma}$ described below.
\end{remark}
\begin{proposition}Let $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ be an embedded tangent structure with a Christoffel function $\Gamma$. Then for all tangent vectors $\xi, \eta$ at $x\in \mathrm{M}$, we have
\begin{equation}\label{eq:mrGamma0}
\mathring{\Gamma}(\xi, \eta)_x := (\rD_{\xi}\Pi_{\mathsf{g}})_x\eta + \Gamma(\xi, \eta)_x = \Pi_{\mathsf{g}, x}\Gamma(\xi, \eta)_x \in \mathcal{T}_x\mathrm{M}\end{equation}
Also, $\Gamma(\xi, \eta)_x=\Gamma(\eta,\xi)_x$ and $\mathring{\Gamma}(\xi, \eta)_x= (\nabla_{p_{\xi}}p_{\eta})_x$.
\end{proposition}
\begin{proof} This follows from \cref{prop:emb_exists} and \cref{eq:GammaForm}. $\Gamma(\xi, \eta)_x = \Gamma(\eta, \xi)_x$ follows from torsion-freeness of the Levi-Civita connection.
\end{proof}
In the following examples, for an ambient matrix space $\mathcal{E}$ we always use the trace inner product, $\langle \omega, \omega\rangle_{\mathcal{E}} = \Tr\omega\omega^{\mathsf{T}}$ for $\omega\in\mathcal{E}$ as the base inner product.
\begin{example}\label{ex:son}
For the special orthogonal group $\SOO(n)$ of matrices satisfying $UU^{\mathsf{T}} = \dI_n$ in $\mathcal{E}=\mathbb{R}^{n\times n}$, with determinant $1$, define the metric $\langle\omega, \omega\rangle_{\mathsf{g}} = \frac{1}{2}\Tr(\omega \omega^{\mathsf{T}})$ for $\omega\in\mathcal{E}$, and the metric operator $\mathsf{g} = \frac{1}{2}\dI_n$. The tangent space at $U\subset\SOO(n)$ consists of matrices $\eta \in \mathcal{E}$ satisfying $U^{\mathsf{T}}\eta + \eta^{\mathsf{T}}U= 0$. Applying \cref{lem:projprop} with $\mathcal{E}_{\mathrm{N}} =\mathfrak{o}(n)$, the space of antisymmetric matrices, and $\mathrm{N}\delta = U\delta$ for $\delta\in\mathfrak{o}(n)$, then $\mathrm{N}^{\mathsf{T}}\omega = \frac{1}{2}(U^{\mathsf{T}}\omega - \omega^{\mathsf{T}}U)=\mathrm{skew}(U^{\mathsf{T}}\omega)$, where $\mathrm{skew}$ is the antisymmetrize operator. From here the projection is $\Pi\omega = \frac{1}{2}(\omega - U\omega^{\mathsf{T}}U)$ and the Christoffel function is just $-(\rD_{\xi}\Pi)\eta = \frac{1}{2}(\xi \eta^{\mathsf{T}}U + U\eta^{\mathsf{T}}\xi)$.
\end{example}
\begin{example}\label{ex:sasaki}Let $\mathrm{M}=\mathcal{T} S^{n-1}$ be the tangent bundle of the unit sphere $S^{n-1}$, consisting of pairs of $\mathbb{R}^n$-vectors $(x, v)$ with $x^{\mathsf{T}}x = 1, x^{\mathsf{T}}v = 0$. The ambient space is $\mathcal{E} = \mathbb{R}^{2n}$. Consider the Sasaki metric \cite{Sasaki}, given by the operator
\begin{equation}\mathsf{g}_{(x, v)}\begin{bmatrix} \omega_{\mathfrak{m}} \\ \omega_{\mathfrak{t}}\end{bmatrix} =\begin{bmatrix}\dI_n & vx^{\mathsf{T}} \\0 & \dI_n
\end{bmatrix}
\begin{bmatrix}\dI_n & 0 \\xv^{\mathsf{T}} & \dI_n
\end{bmatrix}\begin{bmatrix} \omega_{\mathfrak{m}} \\ \omega_{\mathfrak{t}}\end{bmatrix} = \begin{bmatrix} \dI_n +vv^{\mathsf{T}} & vx^{\mathsf{T}}\\xv^{\mathsf{T}} & \dI_n
\end{bmatrix}\begin{bmatrix} \omega_{\mathfrak{m}} \\ \omega_{\mathfrak{t}}\end{bmatrix}
\end{equation}
for $(\omega_{\mathfrak{m}}^{\mathsf{T}}, \omega_{\mathfrak{t}}^{\mathsf{T}})^{\mathsf{T}}\in \mathbb{R}^{2n}$. To avoid repeated use of the adjoint $\mathsf{T}$, we will write a vector in $\mathbb{R}^{2n}$ as $(\omega_{\mathfrak{m}}, \omega_{\mathfrak{t}})$ only, but will let the matrices operate on column vectors. Then, $\mathsf{g}^{-1}$ is given by the matrix $\mathsf{g}^{-1} =\begin{bmatrix} \dI_n & -vx^{\mathsf{T}}\\-xv^{\mathsf{T}} & \dI_n + xv^{\mathsf{T}}vx^{\mathsf{T}} \end{bmatrix}$. The tangent space of $\mathrm{M}$ consists of pairs of vectors $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ satisfying the conditions $x^{\mathfrak{t}}\Delta_{\mathfrak{m}} = 0, \Delta_{\mathfrak{m}}^{\mathsf{T}} v + x^{\mathsf{T}}\Delta_{\mathfrak{t}} = 0$. The normal space at $(x, v)$ consists of vectors $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ satisfying $\langle\mathsf{g}(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}}), (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\rangle_{\mathbb{R}^{2n}} = 0$. The constraints show that $\langle(x, 0), (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\rangle_{\mathbb{R}^{2n}} = 0$, $\langle(v, x), (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\rangle_{\mathbb{R}^{2n}} = 0$. Thus, a normal vector will have the form $\mathrm{N}(a_0, a_1)$, with $\mathrm{N}$ is the linear map from $\mathcal{E}_{\mathrm{N}} = \mathbb{R}^2$ to the normal space at $(x, v)$ such that
$$\mathsf{g}\mathrm{N}\begin{bmatrix}a_0 \\ a_1\end{bmatrix} = \begin{bmatrix}x & v\\0& x \end{bmatrix}\begin{bmatrix} a_0\\ a_1\end{bmatrix}$$
so $\mathrm{N}$ is given by the matrix $\begin{bmatrix} x & 0\\ 0 &x\end{bmatrix}$, since $v^{\mathsf{T}}x =0$. By direct computation, $\mathrm{N}^{\mathsf{T}}\mathsf{g}\mathrm{N}= \dI_2$, so the projection to the normal space is given by $\mathrm{N}(\dI_2)^{-1}\mathrm{N}^{\mathsf{T}}\mathsf{g} = \begin{bmatrix} xx^{\mathsf{T}} & 0 \\xv^{\mathsf{T}} & xx^{\mathsf{T}}\end{bmatrix}$. The projection to the tangent space is given by
\begin{equation} \Pi_{(x, v)} = \begin{bmatrix}\dI_n- xx^{\mathsf{T}} & 0 \\-xv^{\mathsf{T}} & \dI_n-xx^{\mathsf{T}}
\end{bmatrix}\end{equation}
For three tangent vectors $\xi = (\xi_{\mathfrak{m}}, \xi_{\mathfrak{t}}), \eta = (\eta_{\mathfrak{m}}, \eta_{\mathfrak{t}}), \phi=(\phi_{\mathfrak{m}}, \phi_{\mathfrak{t}})$, as $\rD_{\phi}\mathsf{g}\xi =((\phi_{\mathfrak{t}} v^{\mathsf{T}} + v\phi_{\mathfrak{t}}^{\mathsf{T}})\xi_{\mathfrak{m}} + (\phi_{\mathfrak{t}} x^{\mathsf{T}} + v\phi_{\mathfrak{m}}^{\mathsf{T}})\xi_{\mathfrak{t}}, (\phi_{\mathfrak{m}}v^{\mathsf{T}} + x\phi_{\mathfrak{t}}^{\mathsf{T}})\xi_{\mathfrak{m}} )$, we can take $\mathcal{X}(\xi, \eta) = ( \xi_{\mathfrak{t}}\eta_{\mathfrak{m}}^{\mathsf{T}}v +
\eta_{\mathfrak{t}}\xi_{\mathfrak{m}}^{\mathsf{T}}v, \eta_{\mathfrak{m}}(\xi_{\mathfrak{m}}^{\mathsf{T}}v + \xi_{\mathfrak{t}}^{\mathsf{T}}x) + \xi_{\mathfrak{m}}(\eta_{\mathfrak{m}}^{\mathsf{T}}v + \eta_{\mathfrak{t}}^{\mathsf{T}}x))$. We can compute the Christoffel function by \cref{eq:GammaForm}. As is well-known, it is easier to compute covariant derivatives on lifts of vector fields from $S^{n-1}$, we will see the general result in \cref{sec:nat_metric}.
\end{example}
An interesting example is that of a Stiefel manifold $\St{p}{n}$, consisting of matrices $Y\in \mathbb{R}^{n\times p}$ ($n >p$ are two positive integers) with $Y^{\mathsf{T}}Y = \dI_p$. In this case, we can define a metric operator of the form $\mathsf{g}\omega = \omega +(\alpha-1)YY^{\mathsf{T}}\omega$ for $\omega\in\mathbb{R}^{n\times p}$. This is a metric operator if $\alpha > 0$. In \cite{Nguyen2020a}, we derived the Levi-Civita connection and projection for this metric using the metric operator approach. This example is interesting, as when $\alpha < 1$, $\mathsf{g}$ is not positive-definite for some values of $Y\in\mathcal{E}$, thus the usual Riemannian embedding approach cannot be used on the full $\mathcal{E}$. The metric operator approach avoids this difficulty, see details in \cite{Nguyen2020a}. See also \cite{ExtCurveStiefel}, where the authors use a pseudo-Riemannian metric on $\SOO(n)\times \SOO(k)$, which induces a Riemannian metric on $\St{p}{n}$ for $\alpha > 0$. Finally, we can realize this metric as a quotient metric on $\SOO(n)/\SOO(n-p)$ with a left-invariant metric on $\SOO(n)$.
\subsection{Submersed ambient structure}
Recall (\cite{ONeil1983}, Definition 7.44) a Riemannian submersion $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$ between two manifolds $\mathrm{M}$ and $\mathcal{B}$ is a smooth, onto map, such that the differential $d\mathfrak{q}$ is onto at every point $x\in\mathrm{M}$, the fiber $\mathfrak{q}^{-1}(\mathfrak{q} x)$ is a Riemannian submanifold of $\mathrm{M}$ containing $x$, and $d\mathfrak{q}$ preserves scalar products of vectors normal to fibers. In particular, quotient space by free and proper action of a group of isometries is a Riemannian submersion. In a Riemannian submersion, the tangent bundle $\mathcal{T}\mathrm{M}$ of $\mathrm{M}$ has a decomposition $\mathcal{T}\mathrm{M} = \mathcal{V}\mathrm{M}\oplus\mathcal{H}\mathrm{M}$, where $\mathcal{V}\mathrm{M}$ is the vertical bundle, defined to be the tangent space of the submanifold $\mathfrak{q}^{-1}(\mathfrak{q} x)$ at each $x\in\mathrm{M}$, and $\mathcal{H}\mathrm{M}$ is its orthogonal complement. By the submersion assumption, each tangent vector of $\mathcal{B}$ at $\mathfrak{q} x\in \mathcal{B}$ has a unique inverse image in $\mathcal{T}_x\mathrm{M}$ that is orthogonal to $\mathcal{V}_x\mathrm{M}$, called the horizontal lift. Denoted by $\ttH$ the projection from $\mathcal{E}$ to $\mathcal{H}\mathrm{M}$, we will call it the {\it horizontal projection}. The {\it vertical projection} $\ttV$ from $\mathcal{E}$ to $\mathcal{V}\mathrm{M}$ is defined similarly, and we have $\ttH + \ttV = \Pi_{\mathsf{g}}$. Tangent vectors in $\mathcal{H}\mathrm{M}$ and $\mathcal{V}\mathrm{M}$ are called {\it horizontal and vertical vectors} respectively. By the submersion assumption, we have the following lemma
\begin{lemma}Let $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$ be a Riemannian submersion, where $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ is an embedded ambient structure with the projection $\Pi_{\mathsf{g}}$ and a Christoffel function $\GammaM$. If $v_{\mathcal{B}}$ and $w_{\mathcal{B}}$ are vector fields on $\mathcal{B}$, which lift to horizontal vector fields $v_{\mathcal{H}}$ and $w_{\mathcal{H}}$ then the horizontal lift of $\nabla^{\mathcal{B}}_{v_B}w_{\mathcal{B}}$ is $\nabla^{\mathcal{H}}_{v_{\mathcal{H}}} w_{\mathcal{H}} :=\ttH\nabla_{v_{\mathcal{H}}} w_{\mathcal{H}}$, or
\begin{equation}\label{eq:gamma_sub}
\begin{gathered}
\ttH\nabla_{v_{\mathcal{H}}} w_{\mathcal{H}}= \ttH(\rD_{v_{\mathcal{H}}}w_{\mathcal{H}} + \GammaM(v_{\mathcal{H}}, w_{\mathcal{H}})) = \rD_{v_{\mathcal{H}}}w_{\mathcal{H}} +\GammaH(v_{\mathcal{H}}, w_{\mathcal{H}})\\
\GammaH(v_{\mathcal{H}}, w_{\mathcal{H}}) := -(\rD_{v_{\mathcal{H}}}\ttH)w_{\mathcal{H}} + \ttH\GammaM(v_{\mathcal{H}}, w_{\mathcal{H}}) \end{gathered}
\end{equation}
We also have $\GammaH(v_{\mathcal{H}}, w_{\mathcal{H}})= -(\rD_{v_{\mathcal{H}}}\ttH)w_{\mathcal{H}} + \ttH\mathring{\Gamma}_{\mathrm{M}}(v_{\mathcal{H}}, w_{\mathcal{H}})$.
\end{lemma}
\begin{proof} The lift of $\nabla^{\mathcal{B}}_{v_B}w_{\mathcal{B}}$ is $\ttH\nabla_{v_{\mathcal{H}}} w_{\mathcal{H}}$ follows from \cite{ONeil1983}, Lemma 7.45. The formula for $\GammaH$ follows from the facts $\ttH\Pi = \ttH$ and $\ttH w_{\mathcal{H}} = w_{\mathcal{H}}$, so
$$\ttH\rD_{v_{\mathcal{H}}}w_{\mathcal{H}} = \ttH\rD_{v_{\mathcal{H}}}(\Pi_{\mathsf{g}} w_{\mathcal{H}}) = \rD_{v_{\mathcal{H}}}(\ttH\Pi_{\mathsf{g}} w_{\mathcal{H}}) - (\rD_{v_{\mathcal{H}}}\ttH)\Pi_{\mathsf{g}} w_{\mathcal{H}}
$$
or $\rD_{v_{\mathcal{H}}} w_{\mathcal{H}} - (\rD_{v_{\mathcal{H}}}\ttH) w_{\mathcal{H}}$. The alternative formula for $\GammaH$ follows from
$$\begin{gathered}
\ttH(\rD_{v_{\mathcal{H}}} \Pi_{\mathsf{g}})w_{\mathcal{H}} = (\rD_{v_{\mathcal{H}}}(\ttH \Pi_{\mathsf{g}}))w_{\mathcal{H}} -(\rD_{v_{\mathcal{H}}}\ttH)\Pi_{\mathsf{g}} w_{\mathcal{H}} =0
\end{gathered}$$
\end{proof}
\begin{remark}
In general, $\GammaH(\xi, \eta) \neq \GammaH(\eta, \xi)$, because $(\rD_{\xi}\ttH){\eta} \neq (\rD_{\eta}\ttH)\xi$. The difference $(\rD_{\xi}\ttH){\eta} - (\rD_{\eta}\ttH)\xi$ is a vertical tangent vector (we can prove using the derivative of projection trick that $\Pi((\rD_{\xi}\ttH){\eta} - (\rD_{\eta}\ttH)\xi) = (\rD_{\xi}\ttH){\eta} - (\rD_{\eta}\ttH)\xi$ and $\ttH((\rD_{\xi}\ttH){\eta} - (\rD_{\eta}\ttH)\xi) =0$). It is $2\mathrm{A}_{\xi}\eta$, where $\mathrm{A}_{\xi}\eta$ is the O'Neil tensor, which will be defined in \cref{subsec:Rsubmerse}.
We will define a {\it submersed ambient structure} as a tuple $(\mathrm{M}, \mathfrak{q}, \mathcal{B}, \mathsf{g}, \mathcal{E})$ where $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ is an embedded ambient structure, and $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$ is a Riemannian submersion. The results in this section apply to submersed ambient structures.
\end{remark}
Expressions for $\GammaH$ could be lengthy, and there may exist a simpler expression for $\GammaH$ that is only valid for horizontal vectors. To calculate curvatures, we will need $\GammaH$ to relate to $\GammaM$ as in \cref{eq:gamma_sub}, and we cannot use an expression that is valid only for horizontal vectors.
\begin{example} If $\sum_{j=0}^q d_i = n$, with $d_i > 0$ is an integer partition of $n$,
consider the action of $\mathrm{S}(\mathrm{O}(d_0)\times \cdots\times\mathrm{O}(d_q))$ (the group of block-diagonal orthogonal matrices of size $n\times n$ with determinant $1$) acting by right multiplication on $\SOO(n)$. In this case, $\mathrm{M} = \SOO(n)$ with the metric $\mathsf{g}=\frac{1}{2}\dI_n$ as in \cref{ex:son}, while the flag manifold is the quotient $\mathcal{B} = \SOO(n)/\mathrm{S}(\mathrm{O}(d_0)\times \cdots\times\mathrm{O}(d_q))$ under this action, with the case $q=1$ corresponding to a Grassmann manifold. Let $\mathfrak{d}$ denote the Lie algebra $\mathfrak{o}(d_0)\times \cdots\times\mathfrak{o}(d_q)$ of the stabilizer group $\mathrm{S}(\mathrm{O}(d_0)\times \cdots\times\mathrm{O}(d_q))$. It is a subalgebra of $\mathfrak{o}(n)$ consisting of block-diagonal anti-symmetric matrices. Its orthogonal complement is the subspace $\mathfrak{b}$ of antisymmetric matrices with diagonal blocks (corresponding to $\mathfrak{d}$) equal to zero.
The vertical space at $U\in \SOO(n)$ consists of matrices $U\diag(b_0,\cdots b_q)= UD$ with $U\in\SOO(n)$ and $D=\diag(b_0,\cdots b_q)\in\mathfrak{d}$ denotes the block diagonal matrix with diagonal blocks $b_i\in \mathfrak{o}(d_i)\subset \mathbb{R}^{d_i\times d_i}$, for $i\in \{0,\cdots q\}$. Let $\mathcal{E}_{\mathrm{N}} := \mathfrak{d}$, define $\mathrm{N}:\mathcal{E}_{\mathrm{N}}\to\mathcal{E} =\mathbb{R}^{n\times n}$ by setting $\mathrm{N}(b_0,\cdots, b_q) = U\diag(b_0,\cdots b_q)$, let $K_i := \diag(0_{d_0\times d_0},\cdots, \dI_{d_i},\cdots, 0_{d_q\times d_q})$ then $\mathrm{N}^{\mathsf{T}}\omega = \frac{1}{2}\sum_{i=0}^q K_i(U^{\mathsf{T}}\omega - \omega^{\mathsf{T}}U)K_i\in \mathfrak{d}$ (which is the operator symmetrizing $U^{\mathsf{T}}\omega$ then taking the diagonal part). By \cref{lem:projprop}, the projection $\ttV_U$ to the vertical space at $U$ is $\frac{1}{2}\sum_{i=0}^q UK_i(U^{\mathsf{T}}\omega - \omega^{\mathsf{T}}U)K_i$, hence the horizontal projection is
\begin{equation}
\ttH_U\omega = \frac{1}{2}(\omega - U\omega^{\mathsf{T}}U - \sum_{i=0}^q UK_i(U^{\mathsf{T}}\omega - \omega^{\mathsf{T}}U)K_i)
\end{equation}
since $\Pi\omega = \frac{1}{2}(\omega - U\omega^{\mathsf{T}}U)$ from \cref{subsec:subman}. A matrix $\eta$ is horizontal at $U$ if and only if $U^{\mathsf{T}}\eta$ is antisymmetric and $K_iU^{\mathsf{T}}\eta K_i = 0$ for $i\in \{0\cdots q\}$. It is more intuitive to translate this to the picture at the identity. For any matrix $X$ in $\mathbb{R}^{n\times n}$, let $X_{\mathfrak{d}}$ denote the block diagonal
matrix in $\mathbb{R}^{n\times n}$ with the same diagonal blocks as $X$, (corresponding to the structure of $\mathfrak{d}$), and $X_{\mathfrak{b}} = X - X_{\mathfrak{d}}$, for example when $q=2$
$$
X = \begin{bmatrix} * & * & *\\
* & * & *\\
* & * & *
\end{bmatrix}\;\;\; X_{\mathfrak{d}} = \begin{bmatrix} * & 0 & 0\\
0 & * & 0\\
0 & 0 & *
\end{bmatrix}\;\;\;
X_{\mathfrak{b}} = \begin{bmatrix} 0 & * & *\\
* & 0 & *\\
* & * & 0
\end{bmatrix}
$$
The tangent space of $\SOO(n)$ at $\dI_n$ is identified with $\mathfrak{o}(n)$, the vertical projection of an antisymmetric matrix $Z$ at $\dI_n$ is just $Z_{\mathfrak{d}}$, and the horizontal projection is $Z_{\mathfrak{b}}$. The projection $\ttH_U$ is $U\mathrm{skew}(U^{\mathsf{T}}\omega)_{\mathfrak{b}}$, where $\mathrm{skew}$ is the antisymmetrize operator, $\mathrm{skew} X = 1/2(X-X^{\mathsf{T}})$. This picture could be generalized to other homogeneous spaces, ($\mathfrak{d}$ and $\mathfrak{b}$ are written as $\mathfrak{h}$ and $\mathfrak{m}$ in \cite{KobNom}, for example. We rename the subspaces to avoid notational conflict in this paper).
We have $\mathring{\Gamma} = 0$, thus, a horizontal Christoffel function $\GammaH$ is simply $-(\rD_{\xi}\ttH)\omega$
\begin{equation}\label{eq:flag_gamma_omg}
\GammaH(\xi,\omega) = \frac{1}{2}( \xi\omega^{\mathsf{T}}U + U\omega^{\mathsf{T}}\xi+ \sum_{i=0}^q \xi K_i(U^{\mathsf{T}}\omega - \omega^{\mathsf{T}}U)K_i) + UK_i(\xi^{\mathsf{T}}\omega - \omega^{\mathsf{T}}\xi)K_i)
\end{equation}
If $\omega = \eta$ is a horizontal vector then $\eta^{\mathsf{T}}U = -U^{\mathsf{T}}\eta$ and $K_i U^{\mathsf{T}}\eta K_i = 0$, thus
\begin{equation}\label{eq:Axe}
\rD_{\xi}\ttH\eta - \rD_{\eta}\ttH\xi = -U\sum_{i=0}^q K_i (\xi^{\mathsf{T}}\eta - \eta^{\mathsf{T}}\xi) K_i = -U(\xi^{\mathsf{T}}\eta - \eta^{\mathsf{T}}\xi)_{\mathfrak{d}}
\end{equation}
Translating to the identity, \cref{eq:Axe} is $U[U^{\mathsf{T}}\xi, U^{\mathsf{T}}\eta]_{\mathfrak{d}}$ and
for $A, B\in \mathbb{R}^{n\times n}$,
\begin{equation}\label{eq:UAB}\GammaH(UA, UB) = \frac{1}{2}U(AB^{\mathsf{T}} +B^{\mathsf{T}}A + A(B-B^{\mathsf{T}})_{\mathfrak{d}} + (A^{\mathsf{T}}B - B^{\mathsf{T}}A)_{\mathfrak{d}})
\end{equation}
Denote by $\kappa_A$ the invariant vector field $X\mapsto XA$ for $X\in\SOO(n), A\in \mathfrak{o}(n)$ with $A_{\mathfrak{d}}=0$, then for $A, B\in\mathfrak{o}(n)$ with $A_{\mathfrak{d}} = B_{\mathfrak{d}} = 0$, $(\nabla_{\kappa_A} \kappa_B)_U = (\rD_{\kappa_A}\kappa_B)_U + \GammaH(UA, UB) = UAB -\frac{1}{2}U(AB+BA+[A,B]_{\mathfrak{d}}) = \frac{1}{2}(\kappa_{[A,B]_{\mathfrak{b}}})_U$, (\cite{KobNom}, theo. II.10.3.3).
\end{example}
\section{Curvature formulas}
\subsection{Curvature formula for an embedded manifold}
As explained, Christoffel functions are not unique for an ambient structure $(\mathrm{M}, \mathsf{g}, \mathcal{E})$, but any choice of Christoffel function could be used to evaluate the curvature tensor:
\begin{theorem}\label{theo:cur_embed} Let $\mathrm{M}$ be a submanifold of $\mathcal{E}$, with a metric operator $\mathsf{g}: \mathrm{M}\to\mathfrak{L}(\mathcal{E},\mathcal{E})$. Let $\Gamma:\mathrm{M}\to \mathfrak{L}(\mathcal{E}\otimes \mathcal{E},\mathcal{E})$ be a Christoffel function of $\mathsf{g}$. Then the Riemannian curvature of $\mathrm{M}$ is given by one of the following:
\begin{equation}\label{eq:rc1}
\RcM_{\xi,\eta}\phi = -(\rD_{\xi}\Gamma)(\eta, \phi) + (\rD_{\eta}\Gamma)(\xi, \phi)-\Gamma(\xi, \Gamma(\eta, \phi)) +\Gamma(\eta, \Gamma(\xi, \phi))
\end{equation}
\begin{equation}\label{eq:rc1a}
\RcM_{\xi,\eta}\phi = -(\rD_{\xi}\Gamma)(\eta, \phi) + (\rD_{\eta}\Gamma)(\xi, \phi)-\Gamma(\Gamma(\phi, \eta), \xi)) +\Gamma(\Gamma(\phi, \xi), \eta)
\end{equation}
where $\xi, \eta, \phi$ are three tangent vectors to $\mathrm{M}$ at $x\in \mathrm{M}$, $(\rD_{\xi}\Gamma)(\eta, \phi), (\rD_{\eta}\Gamma)(\xi, \phi)$ denote the directional derivatives, and all expressions are evaluated at $x$. With $\mathring{\Gamma}(\xi, \phi) = (\rD_{\xi}\Pi) \phi +\Gamma(\xi, \phi)$ as in \cref{eq:mrGamma0}, then
\begin{equation}\label{eq:rc2}
\RcM_{\xi,\eta}\phi = -(\rD_{\xi}\mathring{\Gamma})(\eta, \phi) +\
(\rD_{\eta}\mathring{\Gamma})(\xi, \phi) -\Gamma(\xi,\Gamma(\eta, \phi)) +
\Gamma(\eta,\Gamma(\xi, \phi))
\end{equation}
\end{theorem}
We use \cref{eq:rc2} when the directional derivative of $\mathring{\Gamma}$ is easier to compute than that of $\Gamma$, for example, if $\mathsf{g}$ is constant then $\mathring{\Gamma}=0$. Note that to use \cref{eq:rc2}, derivatives of $\mathring{\Gamma}$ have to be computed with $\mathring{\Gamma}$ satisfying \cref{eq:mrGamma0}, if we simplify $\mathring{\Gamma}$, we must make sure $\Gamma$ is simplified accordingly, and vice-versa.
\iffalse
A particular case is when $\mathring{\Gamma}(\eta, \phi) = \Pi\mathsf{g}^{-1}\mathrm{K}(\eta, \phi))$ for a bilinear $\mathcal{E}$-valued form $\mathrm{K}$ as in \cite{Nguyen2020a}. We can expand:
$$
\rD_{\xi}(\Pi\mathsf{g}^{-1}\mathrm{K}(\eta, \phi)) = (\rD_{\xi}\Pi)\mathsf{g}^{-1}\mathrm{K}(\eta, \phi) -\Pi\mathsf{g}^{-1}(\rD_{\xi}\mathsf{g})\mathsf{g}^{-1}\mathrm{K}(\eta, \phi) +\Pi\mathsf{g}^{-1}(\rD_{\xi}\mathrm{K})(\xi, \eta)
$$
\fi
\begin{proof} Let $\xi, \eta, \phi$ be three tangent vectors at $x\in\mathrm{M}$, identified with elements of $\mathcal{E}$. Define three vector fields on $\mathrm{M}$ by setting $p_{\xi}(y) = \Pi(y)\xi, p_{\eta}(y)= \Pi(y)\eta, p_{\phi}(y) = \Pi(y)\phi$ for $y\in\mathrm{M}$. By \cref{eq:proj_deriv}, $[p_{\xi}, p_{\eta}]_x = 0$ and
$$\begin{gathered}
(\RcM_{\xi\eta }\phi)_x = (\nabla_{[p_{\xi}, p_{\eta}]} p_{\phi} - \nabla_{p_{\xi}} \nabla_{p_{\eta}} p_{\phi} + \nabla_{p_{\eta}} \nabla_{p_{\xi}} p_{\phi})_x\\
=-\{\rD_{p_{\xi}}(\rD_{p_{\eta}}p_{\phi} +\Gamma(p_{\eta}, p_{\phi}))\}_{x} -\Gamma(p_{\xi}, (\rD_{p_{\eta}}p_{\phi} +\Gamma(p_{\eta}, p_{\phi})))_{x} +\\
\{\rD_{p_{\eta}}(\rD_{p_{\xi}}p_{\phi} +\Gamma(p_{\xi}, p_{\phi}))\}_{x} + \Gamma(p_{\eta}, (\rD_{p_{\xi}}p_{\phi} +\Gamma(p_{\xi}, p_{\phi})))_{x}\end{gathered}
$$
We have $(-\rD_{p_{\xi}}\rD_{p_{\eta}}p_{\phi} + \rD_{p_{\eta}}\rD_{p_{\xi}}p_{\phi})_{x} = (\rD_{[p_{\xi}, p_{\eta}]}p_{\phi})_{x} = 0$, thus
\begin{equation}\label{eq:rcm_alt}
\begin{gathered}(\RcM_{\xi\eta }\phi)_x = -\{\rD_{p_{\xi}}(\Gamma(p_{\eta}, p_{\phi}))\}_{x} -\Gamma(p_{\xi}, (\rD_{p_{\eta}}p_{\phi} +\Gamma(p_{\eta}, p_{\phi})))_{x} +\\
\{\rD_{p_{\eta}}(\Gamma(p_{\xi}, p_{\phi}))\}_{x} + \Gamma(p_{\eta}, (\rD_{p_{\xi}}p_{\phi} +\Gamma(p_{\xi}, p_{\phi})))_{x}\end{gathered}\end{equation}
As $\Gamma$ is linear in the $\mathcal{E}$ variables, we have
$$\rD_{p_{\xi}}(\Gamma(p_{\eta}, p_{\phi})) = (\rD_{p_{\xi}}\Gamma)(p_{\eta}, p_{\phi}) + \Gamma(\rD_{p_{\xi}}p_{\eta}, p_{\phi}) + \Gamma(p_{\eta}, \rD_{p_{\xi}}p_{\phi})
$$
Substitute the above and the similar combination for $(p_{\eta}, p_{\xi}, p_{\phi})$ to \cref{eq:rcm_alt}, the terms $-\Gamma(\rD_{p_{\xi}}p_{\eta}, p_{\phi})_x$ and $\Gamma(\rD_{p_{\eta}}p_{\xi}, p_{\phi})_x$ from the similar combination cancels as $(\rD_{p_{\xi}}p_{\eta})_x = (\rD_{p_{\eta}}p_{\xi})_x$ by \cref{eq:proj_deriv}. Thus
$$\begin{gathered}(\RcM_{\xi\eta }\phi)_x = -(\rD_{p_{\xi}}\Gamma)(p_{\eta}, p_{\phi})_x - \Gamma(p_{\eta}, \rD_{p_{\xi}}p_{\phi})_x -\Gamma(p_{\xi}, (\rD_{p_{\eta}}p_{\phi} +\Gamma(p_{\eta}, p_{\phi})))_{x}\\
+(\rD_{p_{\eta}}\Gamma)(p_{\xi}, p_{\phi})_x + \Gamma(p_{\xi}, \rD_{p_{\eta}}p_{\phi})_x
+ \Gamma(p_{\eta}, (\rD_{p_{\xi}}p_{\phi} +\Gamma(p_{\xi}, p_{\phi})))_{x}=\\
-(\rD_{p_{\xi}}\Gamma)(p_{\eta}, p_{\phi})_x -\Gamma(p_{\xi}, \Gamma(p_{\eta}, p_{\phi}))_{x} +
(\rD_{p_{\eta}}\Gamma)(p_{\xi}, p_{\phi})_x
+ \Gamma(p_{\eta}, \Gamma(p_{\xi}, p_{\phi}))_{x}
\end{gathered}
$$
which gives us \cref{eq:rc1}. In \cref{eq:rcm_alt},
$\Gamma(p_{\xi}, (\rD_{p_{\eta}}p_{\phi} +\Gamma(p_{\eta}, p_{\phi})))_{x} = \Gamma((\rD_{p_{\eta}}p_{\phi} +\Gamma(p_{\eta}, p_{\phi})), p_{\xi})_{x}$ because each variable is a tangent vector, similarly
$\Gamma(p_{\eta}, (\rD_{p_{\xi}}p_{\phi} +\Gamma(p_{\xi}, p_{\phi})))_{x} = \Gamma((\rD_{p_{\xi}}p_{\phi} +\Gamma(p_{\xi}, p_{\phi})), p_{\eta})_{x}$, and in the preceding calculation, the affected terms are
$-\Gamma((\rD_{p_{\eta}}p_{\phi} +\Gamma(p_{\eta}, p_{\phi})), p_{\xi})_{x} + \Gamma((\rD_{p_{\xi}}p_{\phi} +\Gamma(p_{\xi}, p_{\phi})), p_{\eta})_{x}$, or $-\Gamma(\Gamma(p_{\eta}, p_{\phi}), p_{\xi})_{x} + \Gamma(\Gamma(p_{\xi}, p_{\phi}), p_{\eta})_{x}$ which gives us \cref{eq:rc1a}.
From $(\rD_{\xi}\rD_{\eta}\Pi)\phi - (\rD_{\eta}\rD_{\xi}\Pi)\phi =0$, we deduce \cref{eq:rc2}.
\end{proof}
We now give a version of the Gauss-Codazzi equation for a metric operator.
\begin{proposition}\label{prop:gauss_cod} Let $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ be an embedded ambient structure and $\Pi=\Pi_{\mathsf{g}}$ the associated projection. Let $(\GammaE)_x$ be a bilinear map from $\mathcal{T}_x\mathrm{M}\times\mathcal{E}$ to $\mathcal{E}$ satisfying $\GammaE(\xi, \eta)_x = \GammaE(\eta, \xi)$ for all pair of tangent vectors $\xi, \eta$, and for all $\omega\in\mathcal{E}$
\begin{equation}\label{eq:metriccomp}
\langle \eta, (\rD_{\xi}\mathsf{g})_x \omega\rangle_{\mathcal{E}} =\langle \GammaE(\xi, \eta)_x, \mathsf{g}_x\omega\rangle_{\mathcal{E}} + \langle \eta, \mathsf{g}_x\GammaE(\xi, \omega)_x\rangle_{\mathcal{E}}
\end{equation}
and $\GammaE$ is smooth as a function from $\mathrm{M}$ to $\mathcal{L}(\mathcal{E}\otimes\mathcal{E}, \mathcal{E})$. Let $\mathtt{X}, \mathtt{Y}$ be two vector fields on $\mathrm{M}$ and $\mathtt{s}$ be a $\mathcal{E}$-valued function on $\mathrm{M}$. Define a connection on $\mathrm{M}\times \mathcal{E}$ by $\nabla^{\mathcal{E}}_{\mathtt{X}} \mathtt{s} = \rD_{\mathtt{X}} \mathtt{s} + \GammaE(\mathtt{X}, \mathtt{s})$. Then $\nabla^{\mathcal{E}}$ satisfies $\nabla^{\mathcal{E}}_{\mathtt{X}} \mathtt{Y} - \nabla^{\mathcal{E}}_{\mathtt{Y}}\mathtt{X} = [\mathtt{X}, \mathtt{Y}]$ and
\begin{equation}\label{eq:metriccomp2}
\rD_{\mathtt{X}}\langle\mathtt{Y}, \mathsf{g} \mathtt{s} \rangle_{\mathcal{E}} = \langle\nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Y}, \mathsf{g} \mathtt{s}\rangle_{\mathcal{E}} + \langle \mathtt{Y}, \mathsf{g} \nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{s}\rangle_{\mathcal{E}}
\end{equation}
Also, $\Gamma(\xi, \omega) := -(\rD_{\xi}\Pi)\omega + \Pi\GammaE(\xi, \omega)$ could be extended to a Christoffel function for $\mathsf{g}$, thus, for the Levi-Civita connection $\nabla$ of the induced metric, we have
\begin{equation}\label{eq:proj_compab}
\nabla_{\mathtt{X}}\mathtt{Y} = \rD_{\mathtt{X}}\mathtt{Y} -(\rD_{\mathtt{X}}\Pi)\mathtt{Y} + \Pi\GammaE(\mathtt{X}, \mathtt{Y})
\end{equation}
Let $\xi, \eta, \phi$ be tangent vectors on $\mathrm{M}$ at $x$. Define the second fundamental form
\begin{equation}\Two(\xi, \eta) = \GammaE(\xi, \eta) - \Gamma(\xi, \eta) = (\rD_{\xi}\Pi)\eta+(\dI_{\mathcal{E}}-\Pi)\GammaE(\xi, \eta)\end{equation}
where $\dI_{\mathcal{E}}$ denotes the identity map of $\mathcal{E}$. Then
$\Two(\xi, \eta) =\Two(\eta, \xi)$ and $\Pi\Two(\xi, \eta) = 0$. Consider $\Two_{\xi}:\eta\mapsto \Two(\xi, \eta)$ as a linear map from $\mathcal{T}_x\mathrm{M}$ to $\mathcal{E}$. Then its adjoint $\Two_{\xi}^{\dagger}$ as a map from $\mathcal{E}$ to $\mathcal{T}_x\mathrm{M}$ under the inner products induced by $\mathsf{g}$ is given by:
\begin{equation}\label{eq:Twodual}
\Two_{\xi}^{\dagger}\omega = \Two^{\dagger}(\xi, \omega)= (\rD_{\xi}\Pi)(\dI_{\mathcal{E}}-\Pi) \omega - \Pi\GammaE(\xi, (\dI_{\mathcal{E}}-\Pi)\omega) =-\Gamma(\xi, (\dI_{\mathcal{E}}-\Pi)\omega)
\end{equation}
for $\omega\in\mathcal{E}$. Define:
\begin{equation}
\RcE_{\xi, \eta}\phi = -(\rD_{\xi}\GammaE)(\eta, \phi) + (\rD_{\eta}\GammaE)(\xi, \phi) - \GammaE(\xi, \GammaE(\eta, \phi)) + \GammaE(\eta, \GammaE(\xi, \phi))
\end{equation}
Then the Gauss-Codazzi equation holds:
\begin{equation}\label{eq:gaussco31}
\RcM_{\xi \eta}\phi = \Pi\RcE_{\xi, \eta}\phi + \Two^{\dagger}(\eta,\Two(\xi, \phi)) - \Two^{\dagger}(\xi, \Two(\eta, \phi))
\end{equation}
\end{proposition}
\begin{proof}Equation \ref{eq:metriccomp2} is equivalent to
$$\langle\mathtt{Y}, (\rD_{\mathtt{X}}\mathsf{g}) \mathtt{s}\rangle_{\mathcal{E}} = \langle\GammaE(\mathtt{X}, \mathtt{Y}), \mathsf{g}\mathtt{s} \rangle_{\mathcal{E}} + \langle\mathtt{Y}, \mathsf{g}\GammaE(\mathtt{X}, \mathtt{s}) \rangle_{\mathcal{E}}
$$
which follows from \ref{eq:metriccomp}. $\nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Y} -\nabla^{\mathcal{E}}_{\mathtt{Y}}\mathtt{X} = [\mathtt{X}, \mathtt{Y}]$ follows from the expression of $\nabla^{\mathcal{E}}$ and the symmetry of $\Gamma^{\mathcal{E}}$ when evaluated on vector fields. The right-hand side of \cref{eq:proj_compab} is a vector field, since
$$\Pi(\rD_{\mathtt{X}}\mathtt{Y} -(\rD_{\mathtt{X}}\Pi)\mathtt{Y} + \Pi\GammaE(\mathtt{X}, \mathtt{Y})) = \Pi\rD_{\mathtt{X}}\mathtt{Y} + \Pi\GammaE(\mathtt{X}, \mathtt{Y}) =\rD_{\mathtt{X}}\mathtt{Y}-
(\rD_{\mathtt{X}}\Pi)\mathtt{Y} + \Pi\GammaE(\mathtt{X}, \mathtt{Y}) $$
by the derivative of projection trick, thus we have a connection, and the right-hand side could be written as $\Pi\nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Y}$. Compatibility with metric for $\nabla$ follows from compatibility of $\nabla^{\mathcal{E}}$, as for three vector fields $\mathtt{X}, \mathtt{Y}, \mathtt{Z}$
$$\langle \Pi\nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Y}, \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}} +
\langle \mathtt{Y}, \mathsf{g} \Pi\nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Z}\rangle_{\mathcal{E}}
=\langle \nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Y}, \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}} +
\langle \mathtt{Y}, \mathsf{g} \nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Z}\rangle_{\mathcal{E}} = \rD_{\mathtt{X}}\langle \mathtt{Y}, \mathsf{g} \mathtt{Z}\rangle_{\mathcal{E}}
$$
For an ambient vector $\omega$, $\langle (\dI_{\mathcal{E}} - \Pi)\omega, \mathsf{g} \mathtt{Y}\rangle_{\mathcal{E}} = 0$ hence
$$\begin{gathered}0 = \rD_{\mathtt{X}}\langle (\dI_{\mathcal{E}} - \Pi)\omega, \mathsf{g} \mathtt{Y}\rangle_{\mathcal{E}} =
\langle \nabla^{\mathcal{E}}_{\mathtt{X}}(\dI_{\mathcal{E}} - \Pi)\omega, \mathsf{g} \mathtt{Y}\rangle_{\mathcal{E}} + \langle (\dI_{\mathcal{E}} - \Pi)\omega, \mathsf{g} \nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Y}\rangle_{\mathcal{E}}=\\
\langle-(\rD_{\mathtt{X}}\Pi)\omega +\GammaE((\dI_{\mathcal{E}}-\Pi)\omega, \mathsf{g}\mathtt{Y})\rangle_{\mathcal{E}} +
\langle \omega, \mathsf{g} (\dI_{\mathcal{E}} - \Pi) \nabla^{\mathcal{E}}_{\mathtt{X}}\mathtt{Y}\rangle_{\mathcal{E}}
\end{gathered}$$
Note the last term is $\langle \omega, \mathsf{g} \Two(\mathtt{X}, \mathtt{Y})\rangle_{\mathcal{E}}$, thus
$$\Two^{\dagger}(\mathtt{X}, \omega) = -\Pi(-(\rD_{\mathtt{X}}\Pi)\omega +\GammaE(\mathtt{X}, (\dI_{\mathcal{E}}-\Pi)\omega)) = -\Gamma(\mathtt{X}, (\dI_{\mathcal{E}}-\Pi)\omega)$$
which gives us \cref{eq:Twodual}. We note if $\nu$ is normal, $\Pi\nu = 0$ then $\Two^{\dagger}(\xi, \nu) = -\Gamma(\xi,\nu)$. Expanding the right-hand side of \cref{eq:gaussco31}
$$\begin{gathered}\Pi(-\rD_{\xi}(\GammaE(\eta, \phi)) + \rD_{\eta}(\GammaE(\xi, \phi)) - \GammaE(\xi, \GammaE(\eta, \phi)) + \GammaE(\eta, \GammaE(\xi, \phi))) -\\
\Gamma(\eta, \GammaE(\xi, \phi) - \Gamma(\xi, \phi)) +\Gamma(\xi, \GammaE(\eta, \phi)-\Gamma(\eta, \phi))\end{gathered}$$
Expand the first line, using $\Pi\rD_{\xi}(\GammaE(\eta, \phi)) = \rD_{\xi}(\Pi\GammaE(\eta, \phi)) - (\rD_{\xi}\Pi)\GammaE(\eta, \phi)$ and
$\Pi\GammaE(\xi, \omega) = \Gamma(\xi, \omega) + (\rD_{\xi}\Pi)\omega$
then permute the roles of $\xi$ and $\eta$
$$\begin{gathered} -\rD_{\xi}(\Pi\GammaE(\eta, \phi)) + (\rD_{\xi}\Pi)\GammaE(\eta, \phi)
+\rD_{\eta}(\Pi\GammaE(\xi, \phi)) - (\rD_{\eta}\Pi)\GammaE(\xi, \phi)
\\- \Gamma(\xi, \GammaE(\eta, \phi))-(\rD_{\xi}\Pi) \GammaE(\eta, \phi))
+ \Gamma(\eta, \GammaE(\xi, \phi)) +(\rD_{\eta}\Pi) \GammaE(\xi, \phi))
-\\\Gamma(\eta, \GammaE(\xi, \phi) - \Gamma(\xi, \phi)) +\Gamma(\xi, \GammaE(\eta, \phi)-\Gamma(\eta, \phi))=\\
-\rD_{\xi}(\Gamma(\eta, \phi)) + \rD_{\eta}(\Gamma(\xi, \phi)) -\Gamma(\xi,\Gamma(\eta, \phi)) +
\Gamma(\eta,\Gamma(\xi, \phi))
\end{gathered}$$
which is $\RcM_{\xi\eta}\phi$. We have used $-\rD_{\xi}(\Pi\GammaE(\eta, \phi)) +\rD_{\eta}(\Pi\GammaE(\xi, \phi)) = -\rD_{\xi}(\Gamma(\eta, \phi)) -\rD_{\xi}(\rD_\eta\Pi\phi)+ \rD_{\eta}(\Gamma(\xi, \phi)) +\rD_{\eta}(\rD_{\xi}\Pi\phi)= -\rD_{\xi}(\Gamma(\eta, \phi)) + \rD_{\eta}(\Gamma(\xi, \phi))$.
\end{proof}
The equation is often given in the $(0, 4)$ form:
\begin{equation}
\langle\RcM_{\xi \eta}\phi, \mathsf{g}\zeta\rangle_{\mathcal{E}} = \langle\RcE_{\xi\eta}\phi, \mathsf{g}\zeta\rangle_{\mathcal{E}} + \langle\Two(\xi, \phi), \mathsf{g}\Two(\eta, \zeta)\rangle_{\mathcal{E}} - \langle\Two(\xi, \zeta),\mathsf{g}\Two(\eta, \phi)\rangle_{\mathcal{E}}
\end{equation}
$\GammaE$ could be constructed by extending the metric operator $\mathsf{g}$ to a region near $\mathrm{M}$ in $\mathcal{E}$ then applying the usual Christoffel formula. $\RcE_{\xi\eta}\phi$ could also be calculated as $(\nabla_{[\mathtt{X}, \mathtt{Y}]}^{\mathcal{E}}\mathtt{Z} - \nabla_{\mathtt{X}}^{\mathcal{E}}\nabla_{\mathtt{Y}}^{\mathcal{E}}\mathtt{Z} + \nabla_{\mathtt{Y}}^{\mathcal{E}}\nabla_{\mathtt{X}}^{\mathcal{E}}\mathtt{Z})_x$ for three vector fields $\mathtt{X}, \mathtt{Y}, \mathtt{Z}$ such that $\mathtt{X}_x = \xi, \mathtt{Y}_x = \eta, \mathtt{Z}_x = \phi$, and as before we can choose the vector fields to be $p_{\xi}, p_{\eta}, p_{\phi}$.
For a Riemannian embedding in $\mathcal{E}$, $\GammaE$ is zero and only the derivative of the projection needs to be evaluated. Otherwise, $\GammaE$ could be more complicated than $\Gamma$. As mentioned, the relationship between the Gauss-Codazzi equation and metric connections on subbundles is discussed in \cite{taylor2011partial}, appendix C. Results of \cref{subsec:Rsubmerse} below could also be considered from this point of view.
\begin{example}
Continue with $\mathrm{M}=\SOO(n)$, consider $U\in\SOO(n)$ and let $\xi=UA, \eta=UB, \phi=UC$ be three tangent vectors at $U$, with $A, B, C\in\mathfrak{o}(n)$, and $\Gamma$ from \cref{ex:son}
$$\begin{gathered}\Gamma(\xi, \Gamma(\eta, \phi)) = \frac{1}{4}\{\xi(\eta\phi^{\mathsf{T}}U+U\phi^{\mathsf{T}}\eta)^{\mathsf{T}} U + U(\eta\phi^{\mathsf{T}}U+U\phi^{\mathsf{T}}\eta)^{\mathsf{T}}\xi\}=\\
\frac{1}{4}\{\xi U^{\mathsf{T}}\phi\eta^{\mathsf{T}}U + \xi\eta^{\mathsf{T}}\phi + \phi\eta^{\mathsf{T}}\xi + U\eta^{\mathsf{T}}\phi U^{\mathsf{T}}\xi\}\\
= \frac{1}{4}\{UA C(-B) + UA(-B)C + UC(-B)A + U(-B)CA\}
\end{gathered}
$$
Using \cref{eq:rc2}, the curvature is $\frac{1}{4}U[[A, B], C]$, as is well-known.
\end{example}
\subsection{Curvature formulas for a Riemannian submersion}\label{subsec:Rsubmerse}
The following lemma expresses the O'Neil tensor in \cite{ONeil1966} in terms of projections and Christoffel functions. We mostly follow the original paper. In a curvature calculation, we need to evaluate the Christoffel function on ambient, or not necessarily horizontal vectors, so the expression for the Christoffel function has to be valid on the whole tangent space, we cannot use simplified formulas that are valid for horizontal vectors only.
\begin{lemma}\label{lem:subdual}
Let $\mathrm{M}$ be a Riemannian manifold and $\mathcal{H}\oplus\mathcal{V}\subset \mathcal{T}\mathrm{M}$ be two orthogonal subbundles of $\mathcal{T}\mathrm{M}$. Let $\ttH$ and $\ttV$ be the projection operators from $\mathcal{T}\mathrm{M}$ to $\mathcal{H}$ and $\mathcal{V}$, respectively. Let $\nabla$ be the Levi-Civita connection of a Riemannian metric $\langle\rangle_{R}$, and $c_1, c_2$ be two vector fields. Then $\ttV\nabla_{c_1}(\ttH c_2)$ is a $(1, 2)$ tensor. For fixed tangent vectors $\xi, \eta$
at $x\in\mathrm{M}$ and two vector fields $c_1, c_2$ such that $c_1(x)=\xi, c_2(x)=\eta$,
the map $\mathrm{A}_{\xi}:\eta\mapsto (\ttV\nabla_{c_1}(\ttH c_2))_{x}$ maps $\mathcal{T}_x\mathrm{M}$ to $\mathcal{T}_x\mathrm{M}$ for $x\in\mathrm{M}$, and induce an operator on vector fields also denoted by $\mathrm{A}_{c_1}$.
Its adjoint $\mathrm{A}^{\dagger}_{c_1}$ in Riemannian inner product is given by $-\ttH\nabla_{c_1}\ttV$, that is:
\begin{equation}\label{eq:orthadj}
\langle \ttH\nabla_{c_1}(\ttV c_3), c_2 \rangle_{R} = - \langle c_3, \ttV\nabla_{c_1}(\ttH c_2) \rangle_{R}
\end{equation}
for all vector fields $c_2, c_3$. Further, if $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ is an embedded ambient structure inducing the Riemannian metric $\langle\rangle_R$, and $\Gamma$ is a Christoffel function, with $\mathring{\Gamma}(c_1, c_2 ) = \Gamma(c_1, c_2) + (\rD_{c_1}\Pi)c_2$. Then we have:
\begin{equation}\label{eq:rAV}
\mathrm{A}_{c_1} c_2 = -(\rD_{c_1}\ttV)\ttH c_2 + \ttV\mathring{\Gamma}(c_1, \ttH c_2)
\end{equation}
\begin{equation}
\mathrm{A}^{\dagger}_{c_1} c_3 = (\rD_{c_1}\ttH)\ttV c_3 - \ttH\mathring{\Gamma}(c_1, \ttV c_3)
\end{equation}
For a subbundle $\mathcal{F}$ of $\mathcal{T}\mathrm{M}$, let $\Pi_{\mathcal{F}}$ be the projection to $\mathcal{F}$. Set
\begin{equation}\label{eq:gammaF}
\Gamma_{\mathcal{F}}(\xi, \omega) = -(\rD_{\xi}\Pi_{\mathcal{F}})\omega + \Pi_{\mathcal{F}}\mathring{\Gamma}(\xi, \omega)
\end{equation}
In this notation, $\mathrm{A}_{c_1}c_2 = \Gamma_{\mathcal{V}}(c_1, \ttH c_2)$ and $\mathrm{A}^{\dagger}_{c_1} c_2 = -\Gamma_{\mathcal{H}}(c_1, \ttV c_2)$, or $\mathrm{A}_{\xi}\eta = \Gamma_{\mathcal{V}}(\xi, \ttH \eta)$ and $\mathrm{A}^{\dagger}_{\xi} \eta = -\Gamma_{\mathcal{H}}(\xi, \ttV \eta)$ for two tangent vectors $\xi, \eta$ at $x$.
If $\mathcal{H} = \mathcal{H}\mathrm{M}$ is the horizontal bundle of a Riemannian submersion $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$ and $c_1$ and $c_2$ are two horizontal vector fields with $c_1(x) = \xi, c_2(x)= \eta$ then
\begin{equation}\label{eq:oneilLie}
2\mathrm{A}_{\xi}\eta = (\ttV[c_1, c_2])_x = (\rD_{\xi}\ttH)_x\eta - (\rD_{\eta}\ttH)_x\xi
\end{equation}
\end{lemma}
\begin{proof} We have
$$\ttV\nabla_{c_1}(\ttH (fc_2)) = (\rD_{c_1}f) \ttV \ttH c_2 + f\ttV\nabla_{c_1}(\ttH c_2) = f\ttV\nabla_{c_1}(\ttH c_2)$$
So the map $(c_1, c_2)\to \ttV\nabla_{c_1}(\ttH c_2)$ is a tensor. From compatibility with metric: $$0 = \rD_{c_1}\langle\ttH c_3, \ttV c_2 \rangle_R= \langle \nabla_{c_1}\ttH c_3, \ttV c_2 \rangle_R + \langle \ttH c_3, \nabla_{c_1}\ttV c_2 \rangle_R$$
where $\langle\ttH c_2, \ttV c_3\rangle_R=0$ by orthogonality. This implies \cref{eq:orthadj}. When the Riemannian metric on $\mathrm{M}$ is induced by a metric operator $\mathsf{g}$
$$\ttV\nabla_{c_1}(\ttH c_2) = \ttV(\rD_{c_1}(\ttH c_2) -(\rD_{c_1}\Pi)(\ttH c_2) + \mathring{\Gamma}(c_1, \ttH c_2))$$
Expand the first two terms using the derivative of projection trick
$$\begin{gathered}\ttV\rD_{c_1}(\ttH c_2) = \rD_{c_1}(\ttV\ttH c_2) - (\rD_{c_1}\ttV)\ttH c_2 = - (\rD_{c_1}\ttV)\ttH c_2 \\
\ttV(\rD_{c_1}\Pi)(\ttH c_2) = (\rD_{c_1}\ttV ) \ttH c_2 - (\rD_{c_1}\ttV)\Pi\ttH c_2 = 0
\end{gathered}$$
Hence $\ttV\nabla_{c_1}\ttH c_2 = - (\rD_{c_1}\ttV)\ttH c_2 +\ttV\mathring{\Gamma}(c_1, \ttH c_2)
$, and we can switch the role of $\ttV$ and $\ttH$ for the dual formula.
For \cref{eq:oneilLie}, the first part follows from Lemma 2 of \cite{ONeil1966}, while the last equality follows by defining $c_1(y) = \ttH_y\xi$ and $c_2(y) = \ttH_y\eta$, in this case $[c_1, c_2]_x$ is vertical by the derivative of projection trick.
\end{proof}
To calculate the curvature in the following theorem, we repeat that $\GammaH$ must be evaluated from \cref{eq:gammaF} for ambient vectors, if $\GammaH$ is only valid for horizontal vectors the calculation may not be valid.
\begin{theorem}\label{theo:rsub} Assume $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ is an embedded ambient structure and there is a bundle decomposition $\mathcal{T}\mathrm{M} = \mathcal{H}\mathrm{M}\oplus\mathcal{V}\mathrm{M}$. For $x\in\mathrm{M}$, if $\xi, \eta, \phi\in\mathcal{H}_x\mathrm{M}$ are three horizontal vectors at $x$), let $\mathrm{A}_{\xi}$ be the operator defined by $\mathrm{A}
_{\xi}\omega=\ttV\nabla_{\xi}(\ttH\omega) = \GammaV(\xi, \ttH\omega)$ for $\omega\in\mathcal{E}$ as in \cref{lem:subdual}, and $\mathrm{A}^{\dagger}_{\xi}$ be its adjoint (thus if $\omega \in \mathcal{T}_x \mathrm{M}$ then $\mathrm{A}^{\dagger}_{\xi}\omega=-\ttH\nabla_{\xi}\ttV\omega = -\GammaH(\xi, \ttV\omega)$), with all expressions evaluated at $x$. Set
\begin{equation}\label{eq:cursubmer}
\RcH_{\xi\eta}\phi := 2\mathrm{A}^{\dagger}_{\phi}\mathrm{A}_{\xi}\eta -(\rD_{\xi}\GammaH)(\eta, \phi) + (\rD_{\eta}\GammaH)(\xi, \phi)-\GammaH(\xi, \GammaH(\eta, \phi)) +
\GammaH(\eta, \GammaH(\xi, \phi))
\end{equation}
Then $(\RcH_{\xi\eta}\phi)_x$ is in $\mathcal{H}_x$. $\RcH$ satisfies the O'Neil's equations:
\begin{equation}\label{eq:ONeil13}
\RcH_{\xi\eta}\phi = \ttH \RcM_{\xi\eta}\phi + 2 \mathrm{A}^{\dagger}_{\phi} \mathrm{A}_{\xi}\eta - \mathrm{A}^{\dagger}_{\xi} \mathrm{A}_{\eta}\phi + \mathrm{A}^{\dagger}_{\eta} \mathrm{A}_{\xi}\phi
\end{equation}
where $\RcM$ is the curvature tensor of $\mathrm{M}$. Thus, if $\mathcal{H}=\mathcal{H}\mathrm{M}$ is the horizontal bundle from a Riemannian submersion $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$, $\RcH$ given by \cref{eq:cursubmer} is the horizontal lift of the Riemannian curvature tensor on $\mathcal{B}$.
Alternatively,
\begin{equation}\label{eq:cursubmer2}
\begin{gathered}
\RcH_{\xi\eta}\phi = 2\mathrm{A}^{\dagger}_{\phi}\mathrm{A}_{\xi}\eta
-(\rD_{\xi}(\ttH\mathring{\Gamma}(\eta, \phi)) +\
(\rD_{\eta}(\ttH\mathring{\Gamma}(\xi, \phi))\\
-\GammaH(\xi, \GammaH(\eta, \phi)) +
\GammaH(\eta, \GammaH(\xi, \phi))
\end{gathered}
\end{equation}
\end{theorem}
Equation \ref{eq:ONeil13} is the $(1, 3)$ form of the classical O'Neil's equation in \cite{ONeil1966}.
\begin{proof} We expand the right-hand side of \cref{eq:ONeil13}, every expression will be evaluated at $x\in\mathrm{M}$
\begin{equation}\label{eq:tmp1}
\begin{gathered}\ttH(-(\rD_{\xi}(\mathring{\Gamma}(\eta, \phi)) +
\rD_{\eta}(\mathring{\Gamma}(\xi, \phi)) -\Gamma(\xi,\Gamma(\eta, \phi)) +
\Gamma(\eta,\Gamma(\xi, \phi)) ) -\\
2\GammaH(\phi, \GammaV(\xi, \eta)) +\GammaH(\xi, \GammaV(\eta, \phi)) - \GammaH(\eta, \GammaV(\xi, \phi))
\end{gathered}
\end{equation}
We will reduce it to \cref{eq:cursubmer}. We have, with $\omega = \Gamma(\eta, \phi)$, using \cref{eq:gammaF}
$$\begin{gathered}\ttH\Gamma(\xi, \omega) = -\ttH(\rD_{\xi}\Pi_{\mathsf{g}})\omega +\ttH\mathring{\Gamma}(\xi, \omega) =
-\ttH(\rD_{\xi}\Pi_{\mathsf{g}})\omega +\GammaH(\xi, \omega) +(\rD_{\xi}\ttH)\omega =\\
-(\rD_{\xi}(\ttH\Pi_{\mathsf{g}}))\omega +(\rD_{\xi}\ttH)\Pi_{\mathsf{g}}\omega+\GammaH(\xi, \omega) +(\rD_{\xi}\ttH)\omega = (\rD_{\xi}\ttH)\Pi_{\mathsf{g}}\omega+\GammaH(\xi, \omega)
\end{gathered}
$$
Hence, $\ttH\Gamma(\xi, \Gamma(\eta, \phi)) = (\rD_{\xi}\ttH)\Pi_{\mathsf{g}}\Gamma(\eta, \phi)+\GammaH(\xi, \Gamma(\eta, \phi))$. By product rule
$$\begin{gathered}
\ttH\rD_{\xi}(\mathring{\Gamma}(\eta, \phi))= \rD_{\xi}(\ttH\mathring{\Gamma}(\eta, \phi)) - (\rD_{\xi}\ttH)\mathring{\Gamma}(\eta, \phi) \\
\end{gathered}
$$
and permuting the role of $\xi$ and $\eta$ the first line of \cref{eq:tmp1} is:
$$\begin{gathered}
-\rD_{\xi}(\ttH\mathring{\Gamma}(\eta, \phi)) + (\rD_{\xi}\ttH)\mathring{\Gamma}(\eta, \phi)
+\rD_{\eta}(\ttH\mathring{\Gamma}(\xi, \phi)) - (\rD_{\eta}\ttH)\mathring{\Gamma}(\xi, \phi)\\
-(\rD_{\xi}\ttH)\Pi_{\mathsf{g}}\Gamma(\eta, \phi)-\GammaH(\xi, \Gamma(\eta, \phi))
+(\rD_{\eta}\ttH)\Pi_{\mathsf{g}}\Gamma(\xi, \phi)+\GammaH(\eta, \Gamma(\xi, \phi))
\\=
-\rD_{\xi}(\ttH\mathring{\Gamma}(\eta, \phi)) -\GammaH(\xi, \Gamma(\eta, \phi))
+\rD_{\eta}(\ttH\mathring{\Gamma}(\xi, \phi)) +\GammaH(\eta, \Gamma(\xi, \phi))
\end{gathered}
$$
Where we have used \cref{eq:mrGamma0}. Combine with the second line of \cref{eq:tmp1}:
$$\begin{gathered}-\rD_{\xi}(\ttH\mathring{\Gamma}(\eta, \phi)) -\GammaH(\xi, \Gamma(\eta, \phi)-\GammaV(\eta, \phi)
+\rD_{\eta}(\ttH\mathring{\Gamma}(\xi, \phi)) +\\
\GammaH(\eta, \Gamma(\xi, \phi)-\GammaV(\xi, \phi)) +2\GammaH(\phi, \GammaV(\xi, \eta)
\end{gathered}
$$
which reduces to the right-hand side of \cref{eq:cursubmer}, as $\Gamma(\eta, \phi) = \GammaV(\eta, \phi) + \GammaH(\eta, \phi)$ since $\Pi_{\mathsf{g}} =\ttH + \ttV$.
When $\mathcal{H}$ is the horizontal lift of a Riemannian submersion, $\mathrm{A}_{\xi}\eta$ is antisymmetric from Lemma 2 of \cite{ONeil1966} or \cref{eq:oneilLie}, $\mathrm{A}_{\xi}\phi = -\mathrm{A}_{\phi}\xi$. From Theorem 2 of \cite{ONeil1966}, the horizontal lift of the Riemannian curvature tensor on $\mathcal{B}$ satisfies \cref{eq:ONeil13}.
\end{proof}
\begin{example}\label{ex:flag_curv}
Continuing with our example of flag manifolds, as before, let $\mathfrak{d} = \mathfrak{o}(d_0)\times \cdots\times\mathfrak{o}(d_q)$, $\mathfrak{b}$ is its orthogonal complement in $\mathfrak{o}(n)$, consisting of antisymmetric matrices with zero diagonal blocks. For any antisymmetric matrix $X$, let $X_{\mathfrak{d}}$ be the block-diagonal component of $X$ (block size determined by $\mathfrak{d}$), and $X_{\mathfrak{b}} = X - X_{\mathfrak{d}}$. Consider $U\in \SOO(n)$ and three horizontal tangent vectors $\xi = UA, \eta = UB, \phi = UC$ at $U$, with $A, B, C\in\mathfrak{b}$. Let us compute \cref{eq:ONeil13} first. From \cref{eq:Axe}, the term $2\mathrm{A}^{\dagger}_{\phi}\mathrm{A}_{\xi}\eta$ is $-\GammaH(UC, U[A, B]_{\mathfrak{d}})$. Using \cref{eq:UAB}, for an antisymmetric block diagonal matrix $D$, $\GammaH(UC, UD) = \frac{1}{2}U[C, D]_{\mathfrak{b}}$,
So $\mathrm{A}^{\dagger}_{\phi}\mathrm{A}_{\xi}\eta = (1/4)U[[A, B]_{\mathfrak{d}}, C]$ (also note $[[A, B]_\mathfrak{d}, C] = [[A, B]_{\mathfrak{d}}, C]_{\mathfrak{b}}$), and \cref{eq:ONeil13} gives the following formula for lift of the curvature of flag manifolds at $U\in\SOO(n)$
$$\RcH_{\xi, \eta}\phi = \frac{1}{4}U\{[[A, B], C]_{\mathfrak{b}} +2[[A, B]_{\mathfrak{d}}, C] - [[B, C]_{\mathfrak{d}}, A] - [[C, A]_{\mathfrak{d}}, B]\}$$
or more formally, if $\mathcal{L}_U$ is the left multiplication by $U$ and $d\mathcal{L}_U$ its differential
\begin{equation}\label{eq:flag_curv}
\RcH_{\xi, \eta}\phi = \frac{1}{4}d\mathcal{L}_U\{[[A, B], C]_{\mathfrak{b}} +2[[A, B]_{\mathfrak{d}}, C] - [[B, C]_{\mathfrak{d}}, A] - [[C, A]_{\mathfrak{d}}, B]\}
\end{equation}
This formula has a generalization to naturally reductive homogeneous spaces, which we will review shortly. Alternatively, to use \cref{eq:cursubmer2}, from \cref{eq:UAB}
$$\begin{gathered}U^{\mathsf{T}}\GammaH(\xi, \GammaH(\eta, \phi)) = -\frac{1}{4}A(BC+CB-[B,C]_{\mathfrak{d}}) - \frac{1}{4}(BC+CB-[B,C]_{\mathfrak{d}})A \\
-\frac{1}{2}\{A[B,C]_{\mathfrak{d}} +\frac{1}{4}[A, BC+CB+[B,C]_{\mathfrak{d}}]_{\mathfrak{d}}
\end{gathered}$$
and a lengthy but routine computation eventually gives us \cref{eq:flag_curv}.
\end{example}
\begin{remark}
We briefly review a few main facts about naturally reductive homogeneous spaces, used later in \cref{sec:jac}. Follow \cite{KobNom}, (where $\mathfrak{m}, \mathfrak{d}$ and $\mathfrak{b}$ are denoted by $\mathfrak{k}, \mathfrak{h}$ and $\mathfrak{m}$ respectively), we call a homogeneous space $\mathcal{B} = \mathrm{M}/\mathcal{K}$ a naturally reductive homogeneous space where $\mathrm{M}$ is a Lie group, $\mathcal{K}$ is a closed subgroup with Lie algebras $\mathfrak{m}, \mathfrak{d}$ and $\mathfrak{m} = \mathfrak{d}\oplus\mathfrak{b}$, such that $[\mathfrak{d}, \mathfrak{b}]\subset \mathfrak{b}$, and the subspace $\mathfrak{b}$ is equipped with an $\Ad(\mathcal{K})$-invariant (positive-definite) non-degenerate symmetric bilinear form $\langle\rangle_{\mathfrak{b}}$ satisfying
\begin{equation}\label{eq:natural}\langle X, [Z, Y]_{\mathfrak{b}}\rangle_{\mathfrak{b}} + \langle [Z, X]_{\mathfrak{b}}, Y\rangle_{\mathfrak{b}} = 0\text{ for } X, Y, Z\in\mathfrak{b}
\end{equation}
here, for $W\in \mathfrak{m}$, $W_{\mathfrak{d}}$ and $W_{\mathfrak{b}}$ are components of $W = W_{\mathfrak{d}} +W_{\mathfrak{b}}$ in the decomposition $\mathfrak{m} = \mathfrak{d}\oplus\mathfrak{b}$. The form $\langle\rangle_{\mathfrak{b}}$ induces an invariant Riemannian metric on $\mathcal{B}$. If $\langle\rangle_{\mathfrak{b}}$ is induced from a bi-invariant positive-definite inner product on $\mathfrak{m}$ and $\mathfrak{b}$ is orthogonal to $\mathfrak{d}$ then
\cref{eq:natural} is satisfied, in particular, flag manifolds are naturally reductive. For $A\in\mathfrak{b}$ and $U\in \mathrm{M}$, if $\mathcal{L}_U$ is the operator of left multiplication by $U$, $d\mathcal{L}_UA$ is a tangent vector at $U$, which we will sometimes denote by $UA$. Denote by $\kappa_A$ the invariant vector field $X\mapsto d\mathcal{L}_XA$ on $\mathrm{M}$. The naturally reductive assumption implies (\cite{KobNom}, theorem 3.3)
\begin{equation}
\nabla^{\mathcal{H}}_{\kappa_A}\kappa_B := \ttH\nabla_{\kappa_A}\kappa_B = \frac{1}{2}\kappa_{[A,B]_{\mathfrak{b}}}
\end{equation}
From \cite{KobNom}, proposition II.10.3.4, we have (note the opposite sign convention)
\begin{equation}\label{eq:natret} 4\RcH_{\xi, \eta}\phi = d\mathcal{L}_U\{4[[A, B]_{\mathfrak{d}}, C] - [A, [B, C]_{\mathfrak{b}}]_{\mathfrak{b}} -\
[B, [C, A]_{\mathfrak{b}}]_{\mathfrak{b}} + 2[[A, B]_{\mathfrak{b}}, C]_{\mathfrak{b}}\}
\end{equation}
We show it is equivalent to \cref{eq:flag_curv}. We have $4[[A, B]_{\mathfrak{d}}, C] + 2[[A, B]_{\mathfrak{b}}, C]_{\mathfrak{b}} = 2[[A, B]_{\mathfrak{d}}, C] + 2[[A, B], C]_{\mathfrak{b}}$, as $[[A, B]_{\mathfrak{d}}, C]\in\mathfrak{b}$. Expand $2[[A, B], C]_{\mathfrak{b}} = 2[[A, C], B]_{\mathfrak{b}} + 2[A,[B, C]]_{\mathfrak{b}}$, the curly bracket of the above becomes
$$\begin{gathered}2[[A, B]_{\mathfrak{d}}, C] + 2[[A, C], B]_{\mathfrak{b}} + 2[A,[B, C]]_{\mathfrak{b}} -[A, [B, C]_{\mathfrak{b}}]_{\mathfrak{b}} - [B, [C, A]_{\mathfrak{b}}]_{\mathfrak{b}}=\\
2[[A, B]_{\mathfrak{d}}, C] + [B, [C, A]]_{\mathfrak{b}} + [A,[B, C]]_{\mathfrak{b}} -[[B, C]_{\mathfrak{d}}, A] - [[C, A]_{\mathfrak{d}}, B]=\\
2[[A, B]_{\mathfrak{d}}, C] + [[A, B], C]]_{\mathfrak{b}} -[[B, C]_{\mathfrak{d}}, A] - [[C, A]_{\mathfrak{d}}, B]
\end{gathered}$$
using the Jacobi identity again in the last expression. Thus, \cref{eq:flag_curv} is an alternative formula for the curvature of naturally reductive homogeneous spaces.
\end{remark}
\section{Double tangent bundle}
\subsection{Tangent bundle of a tangent bundle in embedded ambient structure}
The embedding $\mathrm{M}\subset \mathcal{E}$ as differentiable manifolds allows us to identify the tangent bundle of $\mathrm{M}$ with a subspace of $\mathcal{E}^2 = \mathcal{E}\oplus\mathcal{E}$. If $\mathrm{M}$ is defined by a system of equations, we can differentiate them to derive the defining equation for $\mathcal{T}\mathrm{M}$. We have seen in \cref{ex:sasaki}, if $\mathrm{M}$ is the unit sphere with defining equation $x^{\mathsf{T}}x = 1$, then $\mathcal{T}\mathrm{M}$ is considered as a pair $(x, v)$ with $x^{\mathsf{T}}x = 1$ and $x^{\mathsf{T}}v = 0$, the second equation is linear in $v$, obtained by taking the directional derivative of $x^{\mathsf{T}}x - 1$.
In general, a tangent vector to $\mathcal{T}\mathrm{M}$ could be considered as an element in $\mathcal{E}^2$. Corresponding to the $\mathfrak{m}$anifold and $\mathfrak{t}$angent components $x$ and $v$ of $\mathcal{T}\mathrm{M}$, a tangent vector $\tilde{\Delta}$ to $\mathcal{T}\mathrm{M}$ at $(x, v)$ has two components $\Delta_{\mathfrak{m}}$ and $\Delta_{\mathfrak{t}}$, $\tilde{\Delta}= (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in \mathcal{E}^2$. In the case of the sphere, the constraints on $\Delta_{\mathfrak{m}}$ and $\Delta_{\mathfrak{t}}$ are $x^{\mathsf{T}}\Delta_{\mathfrak{m}} = 0$ and $\Delta_{\mathfrak{m}}^{\mathsf{T}}v + x^{\mathsf{T}}\Delta_{\mathfrak{t}} = 0$.
Instead of working with specific constraints, our approach will be to define the double tangent space $\mathcal{T}\cT\mathrm{M}$ via the projection operators $\Pi_{\mathsf{g}}$.
\begin{proposition}\label{prop:TTM} Let $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ be an embedded ambient structure. The tangent bundle $\mathcal{T}\mathrm{M}$ of $\mathrm{M}$ is a submanifold of $\mathcal{E}^2$ consisting of pairs $(x, v)$ with $x\in \mathrm{M}$, $v\in \mathcal{E}$ such that $\Pi_{\mathsf{g}, x}v = v$. The tangent bundle $\mathcal{T}\cT\mathrm{M}$ of $\mathcal{T}\mathrm{M}$ is a submanifold of $\mathcal{E}^4$ consisting of quadruples $(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in\mathcal{E}^4$ with $x\in \mathrm{M}$ satisfying
\begin{equation}\label{eq:TTM}
\begin{gathered}
\Pi_{\mathsf{g}, x} v = v\\
\Pi_{\mathsf{g}, x} \Delta_{\mathfrak{m}} = \Delta_{\mathfrak{m}}\\
(\rD_{\Delta_{\mathfrak{m}}}\Pi_{\mathsf{g}, x})v + \Pi_{\mathsf{g}, x}\Delta_{\mathfrak{t}} = \Delta_{\mathfrak{t}}
\end{gathered}
\end{equation}
In particular, $\Delta_{\mathfrak{m}}$ is a tangent vector at $x$. If $v = 0$ or $\Delta_{\mathfrak{m}}=0$ then $\Delta_{\mathfrak{t}}$ is also a tangent vector at $x$. If $(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in\mathcal{T}\cT\mathrm{M}$ then $(x, \Delta_{\mathfrak{m}}, v, \Delta_{\mathfrak{t}})\in\mathcal{T}\cT\mathrm{M}$.
\end{proposition}
Let $\gamma(t)$ be the geodesic associated with the metric $\mathsf{g}$ on $\mathrm{M}$. We will use the notation $\Exp$ to denote the exponential map, with $\Exp_x v = \gamma(1)$ where $\gamma$ is the geodesic with $\gamma(0) = x, \dot{\gamma}(0) = v$. If the manifold is not complete, $\gamma(1)$ may not exist, but below we look at $\Exp_x tv$, which exists if $t$ is small enough.
\begin{proof} The statement $v\in\mathcal{T}_x\mathrm{M}$ if and only if $\Pi_{\mathsf{g}, x}v = v$ is from the definition of the projection. The constraint on a tangent vector $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$ at $(x, v) \in \mathcal{T}\mathrm{M}$ follows by differentiating the constraints on $x$ and $v$. The condition $x\in\mathrm{M}$ implies $\Delta_{\mathfrak{m}}\in \mathcal{T}_x\mathrm{M}$, or $\Pi_{\mathsf{g}, x} \Delta_{\mathfrak{m}} = \Delta_{\mathfrak{m}}$. The condition $\Pi_{\mathsf{g}, x} v = v$ implies $(\rD_{\Delta_{\mathfrak{m}}}\Pi_{\mathsf{g}, x})v + \Pi_{\mathsf{g}, x}\Delta_v = \Delta_v$. Conversely, assuming the pair $(\Delta_{\mathfrak{m}}, \Delta_v)$ satisfies the conditions of \cref{eq:TTM}. Consider the curve $c(t) = (\Exp_xt\Delta_{\mathfrak{m}}, \Pi_{\Exp_xt\Delta_{\mathfrak{m}}} (v+t\Delta_v))\in\mathcal{E}^2$. It is clear that it is a curve on $\mathcal{T}\mathrm{M}$, with $c(0) = (x, v)$ and $\dot{c}(0) = (\Delta_{\mathfrak{m}}, (\rD_{\Delta_{\mathfrak{m}}}\Pi_{\mathsf{g}})_xv + \Pi_{\mathsf{g}, x}\Delta_v) = (\Delta_{\mathfrak{m}}, \Delta_v)$, thus $(\Delta_{\mathfrak{m}}, \Delta_v)$ is a tangent vector to $\mathcal{T}\mathrm{M}$. The last paragraph is clear, with the last statement follows from $(\rD_{\Delta_{\mathfrak{m}}}\Pi_{\mathsf{g}, x})v = (\rD_{v}\Pi_{\mathsf{g}, x})\Delta_{\mathfrak{m}}$.
\end{proof}
Define the map $\rU$ from $\mathcal{T}\mathrm{M}$ to $\mathcal{E}^4$ by $\rU(x, v)= (x, v, 0, v)\in\mathcal{E}^4$, then $\rU(x, v)$ satisfies \cref{eq:TTM}, so $\rU(x, v) \in\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$. It is the familiar {\it canonical vector field}. The map $\mathfrak{j}:(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto (x, \Delta_{\mathfrak{m}}, v, \Delta_{\mathfrak{t}})\in\mathcal{T}\cT\mathrm{M}$ is the {\it canonical flip}.
We will write $\pi:\mathcal{T}\mathrm{M}\to\mathrm{M}$ for the tangent bundle projection. We now introduce the {\it connection map} following \cite{Dombrowski1962,GudKap}, where it is defined via parallel transport. It is shown (Lemma 3.3 of \cite{GudKap} or section 3 of \cite{Dombrowski1962}) that it is a linear bundle map $\mathrm{C}$ from $\mathcal{T}\cT\mathrm{M}$ to $\mathcal{T}\mathrm{M}$, satisfying, for all $x\in \mathrm{M}$, $\Delta \in \mathcal{T}_{x}\mathrm{M}$
\begin{equation}\label{eq:conn_map} \mathrm{C}((dZ)_{x}\Delta) = (\nabla_{\Delta}Z)_x = (\rD_{\Delta}z)_x + \Gamma(\Delta, z(x))_{x}
\end{equation}
for all vector fields $Z: x\mapsto (x, z(x))$ defined on a geodesic $\gamma(t)$ from $x$ on $\mathrm{M}$ with $\dot{\gamma}(0) = \Delta$. The statement in \cite{GudKap} is for vector fields on $\mathrm{M}$, but the proof only requires a vector field along a curve.
\begin{lemma}\label{lem:conn}
The connection map $\mathrm{C}:\mathcal{T}\cT\mathrm{M}\to\mathcal{T}\mathrm{M}$ at $(x, v) \in \mathcal{T}\mathrm{M}$ is given by
\begin{equation} (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto \Delta_{\mathfrak{t}} + \Gamma(\Delta_{\mathfrak{m}}, v)_x\end{equation}
The map $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto (\Delta_{\mathfrak{m}}, \mathrm{C}_{(x, v)}(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}))$ is a bijection between $\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$ and $(\mathcal{T}_x\mathrm{M})^2$. Alternatively, the map $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}} - (\rD_{\Delta_{\mathfrak{m}}}\ttH)_xv)$ is also a bijection between $\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$ and $(\mathcal{T}_x\mathrm{M})^2$.
\end{lemma}
\begin{proof} Write $\Pi_x$ for $\Pi_{\mathsf{g}, x}$. The curve $c(t) = (\Exp_xt\Delta_{\mathfrak{m}}, \Pi_{\Exp_xt\Delta_{\mathfrak{m}}}(v+t\Delta_{\mathfrak{t}}))$ on $\mathcal{T}\mathrm{M}$ gives us a vector field $Z: \Exp_xt\Delta_{\mathfrak{m}}\mapsto c(t)$ along the geodesic $\Exp_xt\Delta_{\mathfrak{m}}$. At $x=\pi c(0)$, $(dZ)_x\Delta_{\mathfrak{m}} =
(\Delta_{\mathfrak{m}}, (\rD_{\Delta_{\mathfrak{m}}}\Pi)_x v + \Pi_x\Delta_{\mathfrak{t}}) = (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$, hence the left-hand side of \cref{eq:conn_map} is $\mathrm{C}(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ and the right-hand side is $\Delta_{\mathfrak{t}} + \Gamma(\Delta_{\mathfrak{m}}, v)_x$.
To show $f:(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto (\Delta_{\mathfrak{m}}, \mathrm{C}_{(x, v)}(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}))$ is injective, if $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ is such that $f(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}}) = f(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ then $\delta_{\mathfrak{m}} = \Delta_{\mathfrak{m}}$, and it is clear from the affine format of $\mathrm{C}$ that $\Delta_{\mathfrak{t}} = \delta_{\mathfrak{t}}$. To show it is onto, take $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{c}})\in \mathcal{T}_x\mathrm{M}^2$. Define $\Delta_{\mathfrak{t}} := \Delta_{\mathfrak{c}} - \Gamma(v, \Delta_{\mathfrak{m}})_x$. We can verify the tangent relation
$$(\rD_{\Delta_{\mathfrak{m}}}\Pi) v + \Pi(\Delta_{\mathfrak{c}} - \Gamma(v, \Delta_{\mathfrak{m}})_x) =
\Pi\Delta_{\mathfrak{c}} - \Gamma(v, \Delta_{\mathfrak{m}})_x = \Delta_{\mathfrak{t}}
$$
from \cref{eq:mrGamma0}, and $\Pi\Delta_{\mathfrak{c}} = \Delta_{\mathfrak{c}}$. It is clear $\mathrm{C}(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{c}} - \Gamma(v, \Delta_{\mathfrak{m}})_x) =\Delta_{\mathfrak{c}}$, so $f$ is onto. The alternate identification with $(\mathcal{T}_x\mathrm{M})^2$ is also clear, as the difference between them is $(\rD_{\Delta_{\mathfrak{m}}}\ttH)_x v + \Gamma(\Delta_{\mathfrak{m}}, v) = \mathring{\Gamma}(\Delta_{\mathfrak{m}}, v)\in \mathcal{T}_x\mathrm{M}$.
\end{proof}
The connection map appears in the initial condition for Jacobi fields and plays a pivotal role in natural metrics on tangent bundles.
\begin{example}
For our $\mathrm{M}=\SOO(n)\subset\mathcal{E}$ example, recall if $U\in\mathrm{M}$, $\eta \in \mathcal{T}_U\mathrm{M}$ if and only if $U^{\mathsf{T}}\eta + \eta^{\mathsf{T}}U= 0$, and the projection is $\Pi\omega = \frac{1}{2}(\omega - U\omega^{\mathsf{T}}U)$ for $\omega \in \mathcal{E}$. The equation $\Pi\omega=\omega$ is equivalent to $U^{\mathsf{T}}\eta + \eta^{\mathsf{T}}U= 0$. The defining equations for $\tilde{\Delta} = (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in\mathcal{T}\cT\mathrm{M}$ at $(U, \eta)\in \mathcal{T}\mathrm{M}$ are $\Pi\Delta_{\mathfrak{m}} = \Delta_{\mathfrak{m}}$ and
$$-\frac{1}{2}\Delta_{\mathfrak{m}}\eta^{\mathsf{T}}U - \frac{1}{2}U\eta^{\mathsf{T}}\Delta_{\mathfrak{m}} + \Pi\Delta_{\mathfrak{t}} = \Delta_{\mathfrak{t}}$$
The expressions are simpler if we translate to the identity, with $\eta = UA, \Delta_{\mathfrak{m}} = UB$ where $A$ and $B$ are antisymmetric matrices. We can set $\Pi\Delta_{\mathfrak{t}} = UD$, and hence $\Delta_{\mathfrak{t}} = U\{\frac{1}{2}(BA+AB) + D\}$ for an antisymmetric matrix $D$.
The connection map is $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto \Delta_{\mathfrak{t}} + \frac{1}{2}(\Delta_{\mathfrak{m}}\eta^{\mathsf{T}}U + U\eta^{\mathsf{T}}\Delta_{\mathfrak{m}}) = UD$.
\end{example}
\subsection{Tangent bundle of horizontal space in submersed ambient structures}\label{sec:subm_tangent}
We have an analogous result for submersed ambient structures. The main actors, to be introduced subsequently, have their relationship described in \cref{fig:HMB}.
\begin{figure}
\centering
\begin{tikzpicture}[node distance={17mm}, scale=.70]
\node (1){$\mathcal{V}\mathcal{H}\mathrm{M}$};
\node[right of=1] (2){$\mathcal{Q}\mathcal{H}\mathrm{M}$};
\node[right of=2] (3){$\mathcal{T}\mathcal{H}\mathrm{M}$};
\node[right of=3] (4){$\mathcal{T}\cT\mathcal{B}$};
\node[below of=3] (5){$\mathcal{H}\mathrm{M}$};
\node[below of=1] (9){$\mathcal{V}\mathrm{M}$};
\node[left of=9] (10){$\mathcal{T}\mathrm{M}$};
\node[right of=5] (6){$\mathcal{T}\mathcal{B}$};
\node[below of=5] (7){$\mathrm{M}$};
\node[below of=6] (8){$\mathcal{B}$};
\fill (1) --node[]{$\oplus$}(2);
\fill (2) --node[]{$=$}(3);
\fill (10) --node[]{$=$}(9);
\fill (9) --node[]{$\oplus$}(5);
\draw[->] (3) -- (4);
\draw[->] (9) -- node[left]{$\mathrm{b}$}(1);
\draw[->] (5) -- node[left]{$\pi_{|\mathcal{H}\mathrm{M}}$}(7);
\draw[->] (6) -- node[left]{$\pi_{\mathcal{B}}$}(8);
\draw[->] (7) -- node[above]{$\mathfrak{q}$}(8);
\draw[->] (5) -- node[above]{$d\mathfrak{q}$}(6);
\draw[->] (3) -- node[above]{$d^2\mathfrak{q}$}(4);
\draw[->] (5) to [bend right=10] node[above]{$\mathrm{h}$}(2);
\draw[->] (5) to [bend left=10] node[below]{$\mathrm{v}$}(2);
\draw[->] (2) to [bend right=30] node[below]{$\mathrm{C}^{\mathrm{\cQ}}$}(5);
\draw[->] (3) -- node[left]{$d\pi$}(5);
\draw[->] (4) -- node[left]{$d\pi_{\mathcal{B}}$}(6);
\draw[->] (10) -- node[left]{$\pi$}(7);
\end{tikzpicture}
\caption{Relationship between bundles in a Riemannian submersion}
\label{fig:HMB}
\end{figure}
The main idea is if $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$ is a Riemannian submersion, then $d\mathfrak{q}:\mathcal{H}\mathrm{M}\to\mathcal{T}\mathcal{B}$ is a differentiable submersion, with the vertical space $\mathcal{V}\mathcal{H}\mathrm{M}$ having an explicit description via the map $\mathrm{b}$ which we will explain here. When the submersion is a quotient by a right action of a group of isometries $P$, then $P$ also acts on the tangent bundle. If $\psi$ belongs to the Lie algebra of $P$, with $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ is an embedded ambient structure and $\psi$ and $\exp t\psi$ act as operators on $\mathcal{E}$ for $t\in \mathbb{R}$, the action of $\exp(t\psi)$ on the tangent bundle is given by $(x, v)\exp(t\psi) = (x\exp(t\psi), v\exp(t\psi))$ for $(x, v)\in\mathcal{T}\mathrm{M}$. The action maps the vertical space at $x$ to the vertical space at $x\exp(t\psi)$, hence the horizontal space at $x$ to that at $x\exp(t\psi)$. So if $v$ is horizontal, $(x\exp(t\psi), v\exp(t\psi))$ is a curve on $\mathcal{H}\mathrm{M}$ and differentiating, $(x\psi, v\psi)\in \mathcal{T}_{(x, v)}\mathcal{H}\mathrm{M}$. The map $\mathrm{b}$ represents the correspondence $(x, x\psi)\mapsto (x, v, x\psi, v\psi)$ (note $v\psi$, like $\Delta_{\mathfrak{t}}$ in the previous section, does not belong to $\mathcal{T}_x\mathrm{M}$). We show this correspondence extends to submersions in general and could be defined using projections, thus providing an explicit decomposition of $\mathcal{T}\mathcal{H}\mathrm{M}$ to vertical and horizontal spaces. Eventually, we will equip $\mathcal{T}\mathcal{H}\mathrm{M}$ with a metric to make $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}:\mathcal{H}\mathrm{M}\to\mathcal{T}\mathcal{B}$ a Riemannian submersion, so the bundle $\mathcal{Q}\mathcal{H}\mathrm{M}$ described below is the horizontal bundle of this submersion.
\begin{proposition}\label{prop:QHM}
Let $(\mathrm{M}, \mathfrak{q}, \mathcal{B}, \mathsf{g}, \mathcal{E})$ be a submersed ambient structure with $\mathfrak{q}:\mathrm{M}\rightarrow\mathcal{B}$ is a Riemannian submersion. Let $\mathcal{T}\mathcal{B}$ be the tangent bundle of $\mathcal{B}$ and $\mathcal{H}\mathrm{M}$ the horizontal bundle of $\mathrm{M}$ in the submersion. Then $\mathcal{H}\mathrm{M}$ could be considered as a submanifold of $\mathcal{E}^2$ consisting of pairs $(x, v)$ with $x\in\mathrm{M}$ and $\ttH v = v$. The map $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}: \mathcal{H}\mathrm{M}\to \mathcal{T}\mathcal{B}$ is a differentiable submersion, with fiber at $(b, v_b)$ the submanifold $\{(x, v_{b,x})|x\in \mathfrak{q}^{-1}b\}$ where $v_{b, x}$ denotes the horizontal lift of $v_b$ at $x$. The tangent bundle $\mathcal{T}\mathcal{H}\mathrm{M}$ of $\mathcal{H}\mathrm{M}$ is a submanifold of $\mathcal{E}^4$ consisting of quadruples $(x, v, \delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})\in\mathcal{E}^4$ with $x\in\mathrm{M}, v\in \mathcal{H}_x\mathrm{M}$ and
\begin{equation}\label{eq:submerg}
\begin{gathered}
\Pi_x \Delta_{\mathfrak{m}} = \Delta_{\mathfrak{m}}\\
(\rD_{\Delta_{\mathfrak{m}}}\ttH_x) v + \ttH_x\Delta_{\mathfrak{t}} = \Delta_{\mathfrak{t}}
\end{gathered}
\end{equation}
The bundle map $\ttQ:\mathcal{T}\mathcal{H}\mathrm{M}\to\mathcal{T}\mathcal{H}\mathrm{M}$ over $\mathcal{H}\mathrm{M}$, mapping $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in \mathcal{T}_{(x, v)}\mathcal{H}\mathrm{M}$ to $(\ttH_x\Delta_{\mathfrak{m}}, (\rD_{\ttH_x\Delta_{\mathfrak{m}}}\ttH_x) v + \ttH_x\Delta_{\mathfrak{t}})$ is idempotent, $\ttQ^2=\ttQ$, with image a subbundle $\mathcal{Q}\mathcal{H}\mathrm{M}$ of $\mathcal{T}\mathcal{H}\mathrm{M}$ with fibers over $(x, v)$ vectors $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ satisfying
\begin{equation}\label{eq:submerg_h}
\begin{gathered}
\ttH_x \delta_{\mathfrak{m}} = \delta_{\mathfrak{m}}\\
(\rD_{\delta_{\mathfrak{m}}}\ttH)_x v + \ttH_x\delta_{\mathfrak{t}} = \delta_{\mathfrak{t}}
\end{gathered}
\end{equation}
For each vertical vector $\epsilon_{\mathfrak{m}}\in \mathcal{V}_x\mathrm{M}$, there exists a unique vector $\epsilon_{\mathfrak{t}}\in\mathcal{E}$ such that $(x, v, \epsilon_{\mathfrak{m}}, \epsilon_{\mathfrak{t}})\in\mathcal{V}\mathcal{H}\mathrm{M}$, where $\mathcal{V}\mathcal{H}\mathrm{M}$ is the vertical subbundle of the tangent bundle $\mathcal{T}\mathcal{H}\mathrm{M}$ under the differentiable submersion $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$. Its fiber $\mathcal{V}_{(x, v)}\mathcal{H}\mathrm{M}$ at $(x, v) \in \mathcal{H}\mathrm{M}$ is the subspace of $\mathcal{T}_{(x, v)}\mathcal{H}\mathrm{M}$ that $d^2\mathfrak{q}_{|\mathcal{H}\mathrm{M}} := d(d\mathfrak{q}_{|\mathcal{H}\mathrm{M}})$ maps to the zero tangent vector at $\mathcal{T}_{d\mathfrak{q}(x, v)}\mathcal{T}\mathcal{B}$. We have
\begin{equation}\label{eq:rmb}
\epsilon_{\mathfrak{t}} = (\rD_{\epsilon_{\mathfrak{m}}}\ttH)_x v - (\rD_{v}\ttH)_x\epsilon_{\mathfrak{m}} = (\rD_{v}\ttV)_x\epsilon_{\mathfrak{m}} - (\rD_{\epsilon_{\mathfrak{m}}}\ttV)_x v = \GammaH(v, \epsilon_{\mathfrak{m}})-\GammaH(\epsilon_{\mathfrak{m}}, v)
\end{equation}
Thus, $\rB_v:\epsilon_{\mathfrak{m}} \mapsto \epsilon_{\mathfrak{t}}$ defines a linear map from $\mathcal{V}_x\mathrm{M}$ to $\mathcal{E}$. $(\rD_{v}\ttH)_x\epsilon_{\mathfrak{m}}$ is a horizontal tangent vector, $(\rD_{\epsilon_{\mathfrak{m}}}\ttV)_x v$ is a vertical tangent vector of $\mathrm{M}$. For each $(x, v)\in \mathcal{H}\mathrm{M}$, the map $\mathrm{b}$, mapping $\epsilon_{\mathfrak{m}}$ to $(\epsilon_{\mathfrak{m}}, \epsilon_{\mathfrak{t}})$ is a bijection between $\mathcal{V}_x\mathrm{M}$ and $\mathcal{V}_{(x, v)}\mathcal{H}\mathrm{M}$.
We have a differentiable bundle decomposition $\mathcal{T}\mathcal{H}\mathrm{M} = \mathcal{Q}\mathcal{H}\mathrm{M} \oplus \mathcal{V}\mathcal{H}\mathrm{M}$, decomposing $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in \mathcal{T}_{(x, v)}\mathcal{H}\mathrm{M}$ to
\begin{equation}
\begin{gathered}
(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})= (\ttH_x\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}} - (\rB_v\ttV_x\Delta_{\mathfrak{m}})) + (\ttV_x\Delta_{\mathfrak{m}}, \rB_v(\ttV_x\Delta_{\mathfrak{m}}))
\end{gathered}
\end{equation}
At each fiber, the bundle map $d(d\mathfrak{q})_{|\mathcal{H}\mathrm{M}})_{|\mathcal{Q}\mathcal{H}\mathrm{M}}:\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}\mapsto \mathcal{T}_{d\mathfrak{q} (x, v)}\mathcal{T}\mathcal{B}$ is a linear bijection. Both $\mathrm{b}$ and $\mathcal{Q}\mathcal{H}\mathrm{M}$ (hence $\ttQ$) are intrinsic, they are only dependent on the submersion $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$.
\end{proposition}
Note that we have not defined a metric on $\mathcal{H}\mathrm{M}$, so the decomposition $\mathcal{T}\mathcal{H}\mathrm{M} = \mathcal{Q}\mathcal{H}\mathrm{M} \oplus \mathcal{V}\mathcal{H}\mathrm{M}$ is not yet an orthogonal decomposition. We note both equations
\cref{eq:submerg} and \cref{eq:submerg} have the property that if $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ satisfies them, and $\delta_{1}$ is a horizontal vector, then $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}} +\delta_1)$ also satisfies them. We will use
$\ttH_x(\rD_{\Delta_{\mathfrak{m}}}\ttH_x) v= (\rD_{\Delta_{\mathfrak{m}}}(\ttH_x)^2)v - (\rD_{\Delta_{\mathfrak{m}}}\ttH_x)\ttH_xv = 0$ for any tangent vector $\Delta_{\mathfrak{m}}$ and horizontal vector $v$ in the following.
\begin{proof} The descriptions of $\mathcal{H}\mathcal{B}$ and $\mathcal{T}\mathcal{H}\mathcal{B}$ are similar to the tangent bundle case, the curve used to prove $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ satisfying \cref{eq:submerg} is horizontal is
$$c(t) = (\Exp_xt\Delta_{\mathfrak{m}}, \ttH_{\Exp_xt\Delta_{\mathfrak{m}}} (v+t\Delta_{\mathfrak{t}}))\in\mathcal{E}^2$$
Since $\mathfrak{q}$ is a submersion, $d\mathfrak{q}:\mathcal{T}\mathrm{M}\to\mathcal{T}\mathcal{B}$ is surjective everywhere, $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$ is also surjective. Let $(\mathcal{B}, \mathsf{g}_{\mathcal{B}}, \mathcal{E}_{\mathcal{B}})$ be an ambient space with a metric operator of $\mathcal{B}$, thus $\mathfrak{q}$ could be considered a map from $\mathrm{M}$ to $\mathcal{E}_{\mathcal{B}}$, and $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$ a map from $\mathcal{H}\mathrm{M}$ to $\mathcal{E}_{\mathcal{B}}^2$ mapping $(x, v)$ to $(\mathfrak{q}(x), d_{\mathfrak{t}}\mathfrak{q}(x, v))$ where $d_{\mathfrak{t}}(x, v)$ denotes the tangent component of $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$. For an element $(\gamma_{\mathfrak{b}}, \gamma_{\mathfrak{t}})\in\mathcal{T}_{d\mathfrak{q}(x, v)}\mathcal{T}\mathcal{B}$, let $\Delta_{\mathfrak{m}}\in \mathcal{H}_x\mathrm{M}$ be the horizontal lift of $\gamma_{\mathfrak{b}}$. Then $(\Delta_{\mathfrak{m}}, (\rD_{\Delta_{\mathfrak{m}}}\ttH)_x v)$ satisfies the last equation of \cref{eq:submerg} hence belongs to $\mathcal{T}_{(x, v)}\mathcal{H}\mathrm{M}$, the second component of its image under $d^2\mathfrak{q}_{|\mathcal{H}\mathrm{M}} := d(d\mathfrak{q}_{|\mathcal{H}\mathrm{M}})$ differs from $\gamma_{\mathfrak{t}}$ by a tangent vector in $\mathcal{T}_{\mathfrak{q}(x)}\mathcal{B}$, which lifts to a horizontal vector $\Delta_1\in\mathcal{H}_x\mathrm{M}$, and hence $(\Delta_{\mathfrak{m}}, (\rD_{\Delta_{\mathfrak{m}}}\ttH)_x v +\Delta_1)\in \mathcal{H}_{x}\mathrm{M}$ maps to $(\gamma_{\mathfrak{b}}, \gamma_{\mathfrak{t}})\in\mathcal{T}_{d\mathfrak{q}(x, v)}\mathcal{T}\mathcal{B}$. Thus, $d^2\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$ is surjective, and $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$ is a differentiable submersion.
We can verify directly $\ttQ$ is idempotent, as $\ttH_x$ is idempotent and
$$(\rD_{\ttH_x\Delta_{\mathfrak{m}}}\ttH_x) v + \ttH_x((\rD_{\ttH_x\Delta_{\mathfrak{m}}}\ttH_x) v + \ttH_x\Delta_{\mathfrak{t}}) = (\rD_{\ttH_x\Delta_{\mathfrak{m}}}\ttH_x) v + \ttH_x\Delta_{\mathfrak{t}}
$$
The description in \cref{eq:submerg_h} of $\mathcal{Q}\mathcal{H}\mathrm{M}$ is clear from idempotency.
Since $\mathfrak{q}$ maps $\mathrm{M}$ to $\mathcal{B}$, it could be considered a map from $\mathrm{M}$ to $\mathcal{E}_{\mathcal{B}}$ (an ambient space of $\mathcal{B}$). Extend $\mathfrak{q}$ to a smooth map on an open subset of $\mathcal{E}$ near $\mathrm{M}$, thus we have an extension of $d\mathfrak{q}$ to a map from $\mathrm{M}\times\mathcal{E}$ to $\mathcal{B}\times \mathcal{E}_{\mathcal{B}}$, and the second derivative $d^2\mathfrak{q}$ would map $(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in \mathcal{T}\cT\mathrm{M}$ to $(\mathfrak{q}(x), d\mathfrak{q}_{\mathfrak{t}, x} v, d\mathfrak{q}_{\mathfrak{t}, x} \Delta_{\mathfrak{m}}, d\mathfrak{q}_{\mathfrak{t}, x}\Delta_{\mathfrak{t}} + \mathsf{Hess}\mathfrak{q}(\Delta_{\mathfrak{m}}, v)$, the last component is the result of taking the directional derivative of $d_{\mathfrak{t}, x}\mathfrak{q}$, considered as a function from $\mathrm{M}\times\mathcal{E}$ to $\mathcal{E}$, in direction $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$, where $\mathsf{Hess}\mathfrak{q}(\Delta_{\mathfrak{m}}, v)$ denotes the Hessian $\mathcal{E}_{\mathcal{B}}$-valued bilinear form of $\mathfrak{q}$.
If $\epsilon_{\mathfrak{m}}\in \mathcal{V}_x\mathrm{M}$, consider the curve $e(t) = (\Exp_x tv, \ttV_{\Exp_x tv}\epsilon_{\mathfrak{m}})\in \mathcal{V}\mathrm{M}$. The velocity curve $\dot{e}(t)$ is a curve in $\mathcal{T}\mathcal{V}\mathrm{M}$, which evaluates at $t=0$ to be $\dot{e}(0) = (x, \epsilon_{\mathfrak{m}}, v, (\rD_{v} \ttV)_x\epsilon_{\mathfrak{m}})\in\mathcal{E}^4$. Via the differential of $d\mathfrak{q}$, $\dot{e}(0)$ maps to
$$d^2\mathfrak{q} \dot{e}(0) = (\mathfrak{q}{x}, 0, d\mathfrak{q}_{\mathfrak{t}, x}v, d\mathfrak{q}_{\mathfrak{t}, x}(\rD_{v} \ttV)_x\epsilon_{\mathfrak{m}} + \mathsf{Hess}\mathfrak{q}(\epsilon_{\mathfrak{m}}, v))_x$$
On the other hand, the differential $d\mathfrak{q}$ maps $e(t)$ to the curve $(\Exp_{\mathfrak{q}(x)} td\mathfrak{q}_{\mathfrak{t}}(x, v), 0)\in \mathcal{T}\mathcal{B}$, and the velocity curve of $d\mathfrak{q} e$ at $t=0$ has components $(\mathfrak{q}(x), 0, d\mathfrak{q}_{\mathfrak{t}, x}v, 0)$. This gives us the equality
$$d\mathfrak{q}_{\mathfrak{t}, x}(\rD_{v} \ttV)_x\epsilon_{\mathfrak{m}} + \mathsf{Hess}\mathfrak{q}(\epsilon_{\mathfrak{m}}, v)_x=0$$
But $\ttV_x (\rD_{\epsilon_{\mathfrak{m}}}\ttV_x) v = (\rD_{\epsilon_{\mathfrak{m}}}\ttV)_x v - (\rD_{\epsilon_{\mathfrak{m}}}\ttV)_x\ttV_x v = \rD_{\epsilon_{\mathfrak{m}}}\ttV_x v$ using the derivative of projection trick, (as $v$ is a horizontal vector, we apologize for the possible confusion), so $\rD_{\epsilon_{\mathfrak{m}}}\ttV_x v$ is vertical, hence we have
\begin{equation}\label{eq:VHess}d\mathfrak{q}_{\mathfrak{t}, x}\{(\rD_{v} \ttV)\epsilon_{\mathfrak{m}}
- (\rD_{\epsilon_{\mathfrak{m}}} \ttV)v\}
+ \mathsf{Hess}\mathfrak{q}(\epsilon_{\mathfrak{m}}, v)=0\end{equation}
But $(\rD_{v} \ttV)\epsilon_{\mathfrak{m}} - (\rD_{\epsilon_{\mathfrak{m}}} \ttV)v =
(\rD_{\epsilon_{\mathfrak{m}}} \ttH)v - (\rD_{v} \ttH)\epsilon_{\mathfrak{m}} $ as $\Pi_{\mathsf{g}} = \ttV + \ttH$, and on $\mathrm{M}$ $(\rD_{\epsilon_{\mathfrak{m}}}\Pi_{\mathsf{g}})_xv = (\rD_{v}\Pi_{\mathsf{g}})_x\epsilon_{\mathfrak{m}}$. This adjustment ensures $\hat{\epsilon} := (x, v, \epsilon_{\mathfrak{m}}, (\rD_{\epsilon_{\mathfrak{m}}} \ttH)_xv - (\rD_{v} \ttH)_x\epsilon_{\mathfrak{m}})$ is in $\mathcal{T}\mathcal{H}\mathrm{M}$ by direct verification. Thus, \cref{eq:VHess} shows $d^2\mathfrak{q}_{|\mathcal{H}\mathrm{M}}\hat{\epsilon} = 0 \in \mathcal{T}_{d\mathfrak{q}(x, v)}\mathcal{T}\mathcal{B}$, hence, it belongs to $\mathcal{V}_{x, v}\mathcal{H}\mathrm{M}$. The last equality of \cref{eq:rmb} follows by expanding $\GammaH$ using its definition. The statement that $(\rD_{v}\ttH)_x\epsilon_{\mathfrak{m}} $ is horizontal is proved by verifying $\ttH_x(\rD_{v}\ttH)_x\epsilon_{\mathfrak{m}} =
(\rD_{v}\ttH^2)_x\epsilon_{\mathfrak{m}} - (\rD_{v}\ttH)_x(\ttH_x \epsilon_{\mathfrak{m}}) = (\rD_{v}\ttH)_x\epsilon_{\mathfrak{m}}$.
If another $\epsilon_{\mathfrak{t}}$ is with the same property that $(\epsilon_{\mathfrak{m}}, \epsilon_{\mathfrak{t}})$ maps to zero, then it differs from the constructed vector by a horizontal vector, which maps to zero. From the bijectivity of horizontal projections, we have the uniqueness of $\epsilon_{\mathfrak{t}}$.
The decomposition $\mathcal{T}\mathcal{H}\mathrm{M} = \mathcal{Q}\mathcal{H}\mathrm{M} \oplus \mathcal{V}\mathcal{H}\mathrm{M}$ is now clear, the bijectivity on fibers of $\mathcal{Q}\mathcal{H}\mathrm{M}$ to $\mathcal{T}\cT\mathcal{B}$ follows from the bijectivity of horizontal projections.
With $\pi:\mathcal{T}\mathrm{M}\to\mathrm{M}$ is the tangent bundle projection, from \cref{fig:HMB}, the map $\mathrm{b}$ is intrinsic because $(\epsilon_{\mathfrak{m}}, \epsilon_{\mathfrak{t}})$ is described intrinsically as the only tangent vector on $\mathcal{T}\mathcal{H}\mathrm{M}$ such that $d\pi(\epsilon_{\mathfrak{m}}, \epsilon_{\mathfrak{t}})=\epsilon_{\mathfrak{m}}$ and $d^2\mathfrak{q}(\epsilon_{\mathfrak{m}}, \epsilon_{\mathfrak{t}})$ is a zero vector in $\mathcal{T}\cT\mathcal{B}$, while $\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$ could be described as the space of vectors in $\mathcal{T}_{(x, v)}\mathcal{H}\mathrm{M}$ mapped to $\mathcal{H}_x\mathrm{M}$ under $d\pi$.
\end{proof}
\begin{proposition}\label{prop:frjH} Let $\mathrm{A}$ be the O'Neil tensor. The map $\mathfrak{j}_{\cH}$ defined by
\begin{equation}\mathfrak{j}_{\cH}:(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto (x, \ttH_x\Delta_{\mathfrak{m}}, v, (\rD_v\ttH)_x\Delta_{\mathfrak{m}} +\ttH_x \Delta_{\mathfrak{t}})
= (x, \ttH_x\Delta_{\mathfrak{m}}, v, 2\mathrm{A}_{v}\Delta_{\mathfrak{m}} + \Delta_{\mathfrak{t}})
\end{equation}
maps $\mathcal{T}\mathcal{H}\mathrm{M}$ to $\mathcal{Q}\mathcal{H}\mathrm{M}$. Restricting to $\mathcal{Q}\mathcal{H}\mathrm{M}$, it is an involution. It corresponds to the canonical flip of $\mathcal{T}\cT\mathcal{B}$, that is, if $d^2\mathfrak{q}$ maps $(x, v, \delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})\in \mathcal{Q}\mathcal{H}\mathrm{M}$ to $(b, w, \delta_b, \delta_w)\in\mathcal{T}\cT\mathcal{B}$ then it maps $(x, \delta_{\mathfrak{m}}, v, (\rD_v\ttH)_x\delta_{\mathfrak{m}} +\ttH_x \delta_{\mathfrak{t}})$ to $(b, \delta_b, b, \delta_w)$.
\end{proposition}
\begin{proof} First, we prove $(x, \ttH_x\Delta_{\mathfrak{m}}, v, (\rD_v\ttH)\Delta_{\mathfrak{m}} + \ttH\Delta_{\mathfrak{t}})$ is in $\mathcal{Q}_{(x, \ttH_x\Delta_{\mathfrak{m}})}\mathcal{H}\mathrm{M}$, using \cref{eq:submerg}. Using the derivative of projection trick
$$(\rD_{v}\ttH)_x \ttH_x\Delta_{\mathfrak{m}} + \ttH_x\{(\rD_v\ttH)_x\Delta_{\mathfrak{m}} + \ttH_x\Delta_{\mathfrak{t}}\} =
(\rD_v\ttH^2)_x\Delta_{\mathfrak{m}} + \ttH_x\Delta_{\mathfrak{t}}$$
which verifies the last condition in \cref{eq:submerg}. By \cref{eq:oneilLie}
$$(\rD_v\ttH)_x\Delta_{\mathfrak{m}} + \ttH_x\Delta_{\mathfrak{t}} = 2\mathrm{A}_v\Delta_{\mathfrak{m}} + (\rD_{\Delta_{\mathfrak{m}}}\ttH)_x +\ttH_x\Delta_{\mathfrak{t}} = 2\mathrm{A}_v\Delta_{\mathfrak{m}} + \Delta_{\mathfrak{t}}$$
It is an involution because $\mathrm{A}_{v}\xi = -\mathrm{A}_{\xi}v$ , or
$$(\rD_{\Delta_{\mathfrak{m}}}\ttH)_x v + \ttH_x((\rD_v\ttH)_x\Delta_{\mathfrak{m}} +\ttH_x \Delta_{\mathfrak{t}}) = (\rD_{\Delta_{\mathfrak{m}}}\ttH)_x v +\ttH_x\Delta_{\mathfrak{t}} = \Delta_{\mathfrak{t}}
$$
To show it corresponds to the canonical flip of $\mathcal{T}\cT\mathcal{B}$, note, if $\mathcal{E}_{\mathcal{B}}$ is an embedded ambient space of $\mathcal{B}$ and consider $d\mathfrak{q}$ as a map to $\mathcal{E}_{\mathcal{B}}^2$, let $d\mathfrak{q}_{\mathfrak{t}, x}$ be its tangent component, $d\mathfrak{q}(x, v) = (\mathfrak{q}(b), d\mathfrak{q}_{\mathfrak{t}, x} v)$, then for $(x, v, \delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})\in\mathcal{Q}\mathcal{H}\mathrm{M}$
$$d^2\mathfrak{q}(x, v, \delta_{\mathfrak{m}}, \delta_{\mathfrak{t}}) = (\mathfrak{q}(x), d\mathfrak{q}_{\mathfrak{t}, x} v, d\mathfrak{q}_{\mathfrak{t}, x}\delta_{\mathfrak{m}}, d\mathfrak{q}_{\mathfrak{t}, x}\delta_{\mathfrak{t}} + \mathsf{Hess}\mathfrak{q}(\delta_{\mathfrak{m}}, v)) = (b, w, \delta_b, \delta_w)$$
The components of $d^2\mathfrak{q}(x, \delta_{\mathfrak{m}}, v, (\rD_v\ttH)_x\delta_{\mathfrak{m}} +\ttH_x\delta_{\mathfrak{t}})$ are $b, \delta_b, w$, and
$$\begin{gathered}
(d\mathfrak{q}_{\mathfrak{t}, x}\{(\rD_v\ttH)_x\delta_{\mathfrak{m}} +\ttH_x\delta_{\mathfrak{t}}\} + \mathsf{Hess}\mathfrak{q}(v, \delta_{\mathfrak{m}})) = \delta_w + d\mathfrak{q}_{\mathfrak{t}, x}\{(\rD_v\ttH)_x\delta_{\mathfrak{m}}+\ttH_x\delta_{\mathfrak{t}} -\delta_\mathfrak{t}\}
\end{gathered}
$$
But $(\rD_v\ttH)_x\delta_{\mathfrak{m}}+\ttH_x\delta_{\mathfrak{t}} - \delta_\mathfrak{t}= (\rD_v\ttH)_x\delta_{\mathfrak{m}}- (\rD_{\delta_{\mathfrak{m}}}\ttH)_xv=2\mathrm{A}_v\delta_{\mathfrak{m}}$ is vertical, so it maps to zero in $\mathcal{T}\mathcal{B}$.
\end{proof}
We now define the horizontal connection map $\mathrm{C}^{\mathrm{\cQ}}$.
\begin{lemma}\label{lem:connHor} If $\tilde{\delta} = (\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})\in \mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$, then
$\mathrm{C}^{\mathrm{\cQ}}_{(x, v)}\tilde{\delta} := \delta_{\mathfrak{t}} + \GammaH(\delta_{\mathfrak{m}}, v)$ is in $\mathcal{H}_x\mathrm{M}$. The map $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})\mapsto (\delta_{\mathfrak{m}}, \mathrm{C}^{\mathrm{\cQ}}_{(x, v)}(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}}))$ is a bijection between $\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$ and $(\mathcal{H}_x\mathrm{M})^2$. The map $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})\mapsto (\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}} - (\rD_{\delta_{\mathfrak{m}}}\ttH) v)$ is also a bijection between these spaces. We have the compatibility equation on $\mathcal{Q}\mathcal{H}\mathrm{M}$
\begin{equation}d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}\mathrm{C}^{\mathrm{\cQ}} = \mathrm{C} d^2\mathfrak{q}
\end{equation}
\end{lemma}
\begin{proof} We have
$$\ttH_x(\delta_{\mathfrak{t}} + \GammaH(\delta_{\mathfrak{m}}, v)) = \ttH_x\delta_{\mathfrak{t}} + \ttH_x\GammaH(\Delta_{\mathfrak{m}}, v) = \delta_{\mathfrak{t}} -(\rD_{\delta_{\mathfrak{m}}}\ttH)_x v+ \ttH_x\GammaH(\delta_{\mathfrak{m}}, v)$$
The last expression reduces to $\delta_{\mathfrak{t}} +\GammaH(\delta_{\mathfrak{m}}, v)$, thus $\delta_{\mathfrak{t}} +\GammaH(\delta_{\mathfrak{m}}, v)$ is $\mathcal{H}_x\mathrm{M}$. The next two statements are proved similar to the embedded case. Since $\mathrm{C}^{\mathrm{\cQ}}$ maps to a horizontal vector, to prove compatibility, if $d^2\mathfrak{q}(x, v, \delta_\mathfrak{m}, \delta_\mathfrak{t})$ maps to $(b, w, \delta_b, \delta_w)$, the vector field $(\Exp_b t\delta_b, \Pi^\mathcal{B}_{\Exp_b t\delta_b}(w+t\delta_w))$ along the geodesic $\Exp_b t\delta_b$ lifts to $(\Exp_w t\delta_\mathfrak{m}, \ttH_{\Exp_v t\delta_\mathfrak{m}}(v+t\delta_\mathfrak{t}))$ along the geodesic $\Exp_x t\delta_\mathfrak{m}$, from here $\mathrm{C}_{(b, w)} (\delta_b, \delta_w)$ lifts to $\mathrm{C}^{\mathrm{\cQ}}_{(x, v)}(\delta_\mathfrak{t}, \delta_\mathfrak{m})$.
\end{proof}
\begin{example}
Continuing with our example of a flag manifold, with $\mathrm{M} = \SOO(n)$ and $\mathcal{B} = \SOO(n)/\mathrm{S}(\mathrm{O}(d_0)\times \cdots\times\mathrm{O}(d_q))$, a vertical vector at $U\in\SOO(n)$ is of the form $\epsilon_{\mathfrak{m}} = U\diag(b_0, \cdots, b_q)$, and $\mathcal{H}\mathrm{M}$ consists of pairs $(U, \eta)$ where $U^{\mathsf{T}}\eta$ is antisymmetric, with zero diagonal blocks. As explained, we expect $\rB_{\eta} \epsilon_{\mathfrak{m}}$ to be $\eta \diag(b_0, \cdots, b_q)=\eta U^{\mathsf{T}}\epsilon_{\mathfrak{m}}$, as it is indeed invariant with respect to the induced action. Note by \cref{eq:UAB}, for two tangent (not necessarily horizontal) vectors to $\SOO(n)$ of form $UA$ and $UB$, with $A$ and $B$ antisymmetric matrices $(\rD_{UA}\ttH)_xUB = \frac{1}{2}(U(AB+BA-2AB_{\mathfrak{d}} + (AB-BA)_{\mathfrak{d}}))$ (we recall $X_{\mathfrak{d}}$ means taking the block diagonals of a matrix $X$).
From here, if $\eta = UA$ is horizontal ($A_{\mathfrak{d}} = 0$) and $\epsilon_{\mathfrak{m}} = UB$ with $B_{\mathfrak{d}} = B$, we have $(AB)_{\mathfrak{d}} = (BA)_{\mathfrak{d}} = 0$ and
$$\rB_{\eta}\epsilon_{\mathfrak{m}} = (\rD_{\epsilon_{\mathfrak{m}}}\ttH)_U\eta - (\rD_{\eta}\ttH)_U\epsilon_{\mathfrak{m}}= UAB =\eta U^{\mathsf{T}}\epsilon_{\mathfrak{m}}
$$
as expected, and $\mathcal{V}_{(U, \eta)}\mathcal{H}\mathrm{M}$ consists of vectors of the form $(\epsilon_{\mathfrak{m}}, \eta U^{\mathsf{T}}\epsilon_{\mathfrak{m}})$. The space $\mathcal{Q}_{(U, \eta)}\mathcal{H}\mathrm{M}$ could be identified affinely with two copies of $\mathcal{H}\mathrm{M}$, for example $\delta_{\mathfrak{m}} = UC, \delta_1 = \ttH_x\delta_{\mathfrak{t}} = UD$ with two antisymmetric matrices $C$ and $D$ such that $C_{\mathfrak{d}} = D_{\mathfrak{d}} = 0$, then $\delta_{\mathfrak{t}} = U\{\frac{1}{2}(CA+AC + [C, A]_{\mathfrak{d}}) + D\}$.
The canonical flip would map $(U, UA, UC, U\{\frac{1}{2}(CA+AC + [C, A]_{\mathfrak{d}}) + D\})$ to $(U, UC, UA, U\{\frac{1}{2}(CA+AC + [A, C]_{\mathfrak{d}}) + D\})$ as $2\mathrm{A}_{\eta}\delta_{\mathfrak{m}} = U[A, C]_{\mathfrak{d}}$.
\end{example}
\subsection{Jacobi fields}
\subsection{Jacobi fields of embedded spaces}
It is known Jacobi fields are derivatives of the exponential map, so if the exponential map is known explicitly, Jacobi fields should also be known explicitly. The following proposition assumes a simplified condition and shows how a Jacobi field and its time derivative, identified as $\mathcal{E}$-valued functions, can be evaluated in our embedded manifold setup.
\begin{proposition} Let $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ be a Riemannian manifold with metric operator $\mathsf{g}$ on an ambient space $\mathcal{E}$ with Christoffel function $\Gamma$ and $I\subset \mathbb{R}$ be an interval containing $0$. Let $\gamma: \mathcal{T}\mathrm{M}\times I\to \mathrm{M}, (x, v,t) \mapsto \Exp_x tv$ be the geodesic family with initial condition $\gamma(x, v; 0) = x, \dot{\gamma}(x, v; 0) = v$, and assume $\gamma$ is defined on $\mathcal{T}\mathrm{M}\times I$. Then $\gamma$ is a smooth map from $\mathcal{T}\mathrm{M}\times I$ to $\mathrm{M}$. We write $\gamma(t)$ for $\gamma(x, v;t)$ when $(x, v)$ is fixed and understood. Let $(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ be a tangent vector to $\mathcal{T}\mathrm{M}$ at $(x, v) \in \mathcal{T}\mathrm{M}$, and for fixed $t$, let $J(t)= J_{(x,v, \Delta_\mathfrak{m}, \Delta_\mathfrak{t})}(t)$ be the tangent component of $d\gamma$, that is $d\gamma: (x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}, t) \mapsto (\gamma(t), J(t))_{(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})}\in \mathcal{T}\mathrm{M}\subset \mathcal{E}^2$, thus, $J(t) = (\partial^{\mathcal{T}\mathrm{M}}_{\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}}\gamma)(t)_{x, v}$, the directional derivative of $\gamma$ in direction $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ at $(x, v)$. Then $\mathfrak{J}(t):=(\gamma(t), J(t))$ is a vector field along the curve $\gamma(x, v; t)$ satisfying:
\begin{equation}\begin{gathered}
J(0) = \Delta_{\mathfrak{m}}\\
\dot{J}(0) = \Delta_{\mathfrak{t}}\\
(\nabla_{d/dt})^2J(t) = \rR_{J(t), \dot{\gamma}(t)} \dot{\gamma}(t)
\end{gathered}
\end{equation}
Thus $\mathfrak{J}$ is the Jacobi field with the given initial conditions. For any $t$, $\dot{\mathfrak{J}}(t) =(\gamma(t), J(t), \dot{\gamma}(t), \dot{J}(t))$ belongs to $\mathcal{T}\cT\mathrm{M}$, in particular $\dot{\mathfrak{J}}(0) = (x, \Delta_{\mathfrak{m}}, v, \Delta_{\mathfrak{t}})$, the canonical flip of $(x, v,\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$. Alternatively, $\Delta_{\mathfrak{c}}:=\Delta_{\mathfrak{t}} + \Gamma(\Delta_{\mathfrak{m}}, v)_x= \mathrm{C}_{(x, v)}(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ is tangent to $\mathrm{M}$ and the initial condition could be written as
\begin{equation}\begin{gathered}
J(0) = \Delta_{\mathfrak{m}}\\
\nabla_{d/dt}J(0) = \Delta_{\mathfrak{c}}
\end{gathered}
\end{equation}
for two tangent vectors $\Delta_{\mathfrak{m}}$ and $\Delta_{\mathfrak{c}}$ to $\mathrm{M}$ at $x$.
\end{proposition}
Here, $\mathrm{C}$ is the connection map. The two formulations of the initial conditions are equivalent by \cref{lem:conn}. The Jacobi field $J_0(t) = \dot{\gamma}(t)$ corresponds to the initial data $\Delta_{\mathfrak{m}} = v, \Delta_{\mathfrak{t}} = - \Gamma(v, v), \Delta_{\mathfrak{c}} = 0$ (as $\ddot{\gamma} + \Gamma(\dot{\gamma}, \dot{\gamma}) = 0$). The Jacobi field $J_1(t) = t\dot{\gamma}(t)$ corresponds to the initial data $\Delta_{\mathfrak{m}} = 0, \Delta_{\mathfrak{t}} = v = \Delta_{\mathfrak{c}}$.
The theorem should still work with some modifications in the situation where $\gamma$ is not defined on the whole $\mathcal{T}\mathrm{M}\times I$, but the initial data for geodesics belong to a subset of $\mathcal{T}\mathrm{M}$, satisfying conditions as in the setup of a geodesic variation.
\begin{proof} That $\gamma$ is a smooth map when it is defined follows from the Gr{\"o}nwall inequality as is standard in the theory of differential equations.
Let $\alpha(s, t) = \gamma({\Exp_{x, s\Delta_{\mathfrak{m}}}}, \Pi_{\Exp_{x, s\Delta_{\mathfrak{m}}}}(v + s\Delta_{\mathfrak{t}});t)$. Then $\alpha(0, t)= \gamma(x, v; t)$ is a geodesic, thus $\alpha(s, t)$ is a geodesic variation. Hence $\frac{\partial}{\partial s}\alpha(s, t)|_{s=0}$ is a Jacobi field, which is $\partial^{T\mathrm{M}}_{d/ds\Exp_{x, s\Delta_{\mathfrak{m}}}(s=0),d/ds(\Pi_{\Exp_{x, s\Delta_{\mathfrak{m}}}}(v + s\Delta_{\mathfrak{t}}))(s=0)}\gamma_{|(x, v; t)}$, and simplifies to $\partial^{T\mathrm{M}}_{\Delta_{\mathfrak{m}},(\rD_{\Delta_{\mathfrak{m}}}\Pi)_xv + \Pi_x\Delta_{\mathfrak{t}}}\gamma(x, v; t) = \partial^{T\mathrm{M}}_{\Delta_{\mathfrak{m}},\Delta_{\mathfrak{t}}}\gamma(x, v; t)$. For initial conditions, first, $J(0) = \partial^{T\mathrm{M}}_{\Delta_{\mathfrak{m}},\Delta_{\mathfrak{t}}}\gamma(x, v; 0)=\Delta_{\mathfrak{m}}$, as $\gamma(x, v, 0) = x$. As $\dot{\gamma}(x, \Delta; 0)=\Delta$ for any tangent vector $\Delta$, $\dot{J}(0)= \partial^{T\mathrm{M}}_{\Delta_{\mathfrak{m}},\Delta_{\mathfrak{t}}}\dot{\gamma}(x, v; 0) =
\partial^{T\mathrm{M}}_{\Delta_{\mathfrak{m}},\Delta_{\mathfrak{t}}}((x, \Delta)\mapsto \Delta) = \Delta_{\mathfrak{t}}$. It follows $(\nabla_{\dot{\gamma}(0)})J(0) = \dot{J}(0) + \Gamma(\dot{\gamma}(x, v, 0), J(0)) = \Delta_{\mathfrak{t}} + \Gamma(v, \Delta_{\mathfrak{m}})_{x}=\Delta_{\mathfrak{c}}$.
The remaining statements about $\mathfrak{J}$ and $\dot{\mathfrak{J}}$ are just standard statements about differentials and velocities of curves on a manifold.
\end{proof}
\begin{example}
Continuing with the example $\mathrm{M} := \SOO(n)\subset \mathcal{E} := \mathbb{R}^{n\times n}$. The geodesic for this metric is $\gamma(U, \eta; t) = U\exp(tU^{\mathsf{T}}\eta)$, or $\Exp_U t\eta=U\exp(tU^{\mathsf{T}}\eta)$. We have already characterized $\mathcal{T}\cT\mathrm{M}$ previously.
The above proposition states that the Jacobi field $J(t)$ with $J(0)=\Delta_{\mathfrak{m}}, \dot{J}(0) = \Delta_{\mathfrak{t}}$ is just the directional derivative of $U\exp(U^{\mathsf{T}}\eta)$ in the tangent direction $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$, by the chain rule it is
\begin{equation}\label{eq:jac_son}
J(t) = J(U, \eta, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}; t) = \Delta_{\mathfrak{m}} \exp(tU^{\mathsf{T}}\eta) + tU \frL_{\exp}(tU^{\mathsf{T}}\eta, \Delta_{\mathfrak{m}}^{\mathsf{T}}\eta + U^{\mathsf{T}}\Delta_{\mathfrak{t}})
\end{equation}
where for two square matrices $A$ and $E$, $\frL_{\exp}(A, E)$ denotes the Fr{\'e}chet derivative of $\exp$ at $A$ in direction $E$, as reviewed in \cref{sec:frechet}. For the exponential function, it is well-known $\frL_{\exp}(A, E) = \exp A \sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!}\ad_A^n E$. However, as will be reviewed, $\frL_{\exp}(A, E)$ could be evaluated more efficiently by Pad{\'e} approximant.
If $\eta=UA, \Delta_{\mathfrak{m}} = UB, \Pi\Delta_{\mathfrak{t}} = UD$ then $\Delta_{\mathfrak{m}}^{\mathsf{T}}\eta + U^{\mathsf{T}}\Delta_{\mathfrak{t}} = \frac{1}{2}(AB-BA)+ D$
\begin{equation}\begin{gathered}
J(t) = U\exp(tA)\{\exp(-tA)B\exp(tA) + t\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!}t^{n}\ad_A^n (\frac{1}{2}[A, B]+ D)\}\\
= UB\exp(tA) + tU\frL_{\exp}(tA, \frac{1}{2}[A, B] +D)
\end{gathered}
\end{equation}
It is clear $J(t)$ is tangent to $\SOO(n)$ at $U\exp(tA)$. The expression on the first line extends to any compact Lie group with bi-invariant metric.
\end{example}
\subsection{Jacobi fields and Riemannian submersion}\label{sec:jac}
Let $(\mathrm{M}, \mathfrak{q}, \mathcal{B}, \mathsf{g}, \mathcal{E})$ be a submersed ambient structure of the submersion $\mathfrak{q}: \mathrm{M}\to\mathcal{B}$. Denote by $\nabla$ and $\nabla^{\mathcal{B}}$ the Levi-Civita connections on $\mathrm{M}$ and $\mathcal{B}$, respectively. We recall for two vector fields $\mathtt{X}$ and $\mathtt{Y}$ on $\mathcal{B}$, the horizontal lift of $\nabla^{\mathcal{B}}_{\mathtt{X}}\mathtt{Y}$ is $\ttH\nabla_{\bar{\mathtt{X}}}\bar{\mathtt{Y}}$, where $\bar{\mathtt{X}}$ and $\bar{\mathtt{Y}}$ are horizontal lifts of $\mathtt{X}$ and $\mathtt{Y}$. We will use the notation $\nabla^{\mathcal{H}}_{\bar{\mathtt{X}}}\bar{\mathtt{Y}}:=\ttH\nabla_{\bar{\mathtt{X}}}\bar{\mathtt{Y}}$.
Recall $\mathfrak{j}^{\mathcal{H}}: (x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}) = (x, \ttH_x\Delta_{\mathfrak{m}}, v, 2\mathrm{A}_v\Delta_{\mathfrak{m}} + \Delta_{\mathfrak{t}})$ from $\mathcal{T}\mathcal{H}\mathrm{M}$ to $\mathcal{Q}\mathcal{H}\mathrm{M}$ defined in \cref{prop:frjH} is the canonical flips when restricted to $\mathcal{Q}\mathcal{H}\mathrm{M}$. The construction below is independent of the embedding $\mathcal{B}\subset \mathcal{E}_{\mathcal{B}}$, it allows us to lift Jacobi fields on $\mathcal{T}\mathcal{B}$ to curves on $\mathcal{H}\mathrm{M}$. See \cite{ONeil1983,LeeRiemann,Gallier} for background materials.
\begin{theorem}\label{theo:jacsub} Let $(\mathrm{M}, \mathfrak{q}, \mathcal{B}, \mathsf{g}, \mathcal{E})$ be a submersed ambient structure of the Riemannian submersion $\mathfrak{q}:\mathrm{M}\mapsto\mathcal{B}$, with horizontal bundles $\mathcal{H}\mathrm{M}$ and horizontal projection $\ttH$. Let $\mathcal{E}_{\mathcal{B}}$ be an inner product space containing $\mathcal{B}$, so $\mathcal{T}\mathcal{B}$ and $\mathcal{T}\cT\mathcal{B}$ are considered as subspaces of $\mathcal{E}_{\mathcal{B}}^2$ and $\mathcal{E}_{\mathcal{B}}^4$. For $(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\in \mathcal{T}\mathcal{H}\mathrm{M}\subset\mathcal{E}^4$, let $\gamma(t) = \gamma(x,v, ;t) = \Exp_x tv$ be the geodesic family on $\mathrm{M}$ with initial conditions $\gamma(0) = x, \dot{\gamma}(0) = v$ and let
$J^{\mathcal{H}}(t) = \ttH_{\gamma(t)}(\partial^{\mathcal{T}\mathrm{M}}_{x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}}\gamma)(t)$, $\nu_{\mathfrak{m}} = \ttH_x\Delta_{\mathfrak{m}}$, $\nu_{\mathfrak{t}} = (\rD_v\ttH)\Delta_{\mathfrak{m}} + \ttH\Delta_{\mathfrak{t}}$ and $\RcH$ be the horizontal lift of the curvature tensor, then $\mathfrak{J}^{\mathcal{H}}(t) := (\gamma(t), J^{\mathcal{H}}(t))$ is a curve in $\mathcal{H}\mathrm{M}\subset\mathcal{E}^2$ satisfying the Jacobi field equation
\begin{equation}\label{eq:JacoH}
\begin{gathered}
J^{\mathcal{H}}(0) = \nu_{\mathfrak{m}}\\
\dot{J}^{\mathcal{H}}(0) = \nu_{\mathfrak{t}}\\
(\nabla^{\mathcal{H}}_{d/dt})^2 J^{\mathcal{H}}(t) = \rR^{\mathcal{H}}_{J^{\mathcal{H}}(t), \dot{\gamma}(t)}\dot{\gamma}(t)
\end{gathered}
\end{equation}
$\dot{\mathfrak{J}}^{\mathcal{H}}(t) = (\gamma(t), J^{\mathcal{H}}(t), \dot{\gamma}(t), \dot{J}^{\mathcal{H}}(t))$ is a curve in $\mathcal{Q}\mathcal{H}\mathrm{M}$ with
\begin{equation}\dot{\mathfrak{J}}^{\mathcal{H}}(0) = (x, \nu_{\mathfrak{m}}, v, \nu_{\mathfrak{t}}) = \mathfrak{j}^{\mathcal{H}}(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})
\end{equation}
Thus, $d\mathfrak{q}$ maps $\mathfrak{J}^{\mathcal{H}}$ to the Jacobi field $\mathfrak{J}^{\mathcal{B}}$on $\mathcal{T}\mathcal{B}$ with $\dot{\mathfrak{J}}^\mathcal{B}(0) = d^2\mathfrak{q}(\mathfrak{j}_{\cH}(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}))$. Conversely, for any Jacobi field $\mathfrak{J}^{\mathcal{B}}$ on $\mathcal{T}\mathcal{B}$ along the geodesic $\gamma^{\mathcal{B}}(t) = \Exp^{\mathcal{B}}_b tw$ on $\mathcal{B}$, with $\dot{\mathfrak{J}}^\mathcal{B}(0) = (b, \Delta_b, w, \Delta_w)\in \mathcal{T}\cT\mathcal{B}$, let $(x, \nu_{\mathfrak{m}}, v, \nu_{\mathfrak{t}})\in \mathcal{Q}\mathcal{H}\mathrm{M}$ be the unique vector in $\mathcal{Q}\mathcal{H}\mathrm{M}$ such that $d^2\mathfrak{q} (x, \nu_{\mathfrak{m}}, v, \nu_{\mathfrak{t}}) = (b, \Delta_b, v, \Delta_w)$, then
\begin{equation}J^{\mathcal{H}}(t) = \ttH_{\gamma(t)}(\partial^{\mathcal{T}\mathrm{M}}_{x, v, \nu_{\mathfrak{m}},\nu_{\mathfrak{t}} +2\mathrm{A}_{\nu_{\mathfrak{m}}}v}\gamma^{\mathcal{H}})(t)
\end{equation}
satisfies \cref{eq:JacoH} and thus $\mathfrak{J}^{\mathcal{H}}$ is the lift of the Jacobi field $\mathfrak{J}^{\mathcal{B}}$ from $\mathcal{T}\mathcal{B}$ to $\mathcal{H}\mathrm{M}$ with the given initial condition. The initial conditions could also be stated as
\begin{equation}\label{eq:submerse_Jac_C}
\begin{gathered}
J^{\mathcal{H}}(0) = \nu_{\mathfrak{m}}\\
(\ttH_x\nabla_{d/dt}J^{\mathcal{H}})(0) = \nu_{\mathfrak{t}}+\GammaH(v, \nu_{\mathfrak{m}})_x = \mathrm{C}^{\mathrm{\cQ}}_{(x, v)}(\nu_{\mathfrak{m}}, \nu_{\mathfrak{t}})=:\nu_\mathfrak{c}
\end{gathered}
\end{equation}
\end{theorem}
\begin{proof}It is clear $J^{\mathcal{H}}(0) = \ttH_x\Delta_{\mathfrak{m}}$, set $J(t) =\partial^{T\mathrm{M}}_{x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}}\gamma(t)$ then
$$\dot{J}^{\mathcal{H}}(0) = (\rD_{\dot{\gamma}(0)}\ttH)_x J(0) + \ttH_x J(0) = (\rD_v\ttH)_x\Delta_{\mathfrak{m}} + \ttH_x\Delta_t=\Delta_{\mathfrak{t}}$$
For the alternate initial condition \cref{eq:submerse_Jac_C}, if $\Gamma$ is the Christoffel function on $\mathrm{M}$
$$\begin{gathered}\ttH_x(\nabla_{\dot{\gamma}(0)}{J}^{\mathcal{H}})(0) = \ttH_x (\dot{J}^{\mathcal{H}}(0) + \Gamma(v, J^{\mathcal{H}}(0))_x) =\\ \ttH_x\{(\rD_v\ttH)_x\Delta_{\mathfrak{m}} + \ttH_x\Delta_{\mathfrak{t}} + \Gamma(v, \ttH_x\Delta_{\mathfrak{m}})_x\}\\
= (\rD_v\ttH)_x\Delta_{\mathfrak{m}} - (\rD_v\ttH)_x\ttH_x\Delta_{\mathfrak{m}}+ \ttH_x\Delta_{\mathfrak{t}} +\ttH_x\Gamma(v, \ttH_x\Delta_{\mathfrak{m}})_x
=\\ \ttH_x\Delta_{\mathfrak{t}} + (\rD_v\ttH)_x\Delta_{\mathfrak{m}} + \GammaH(v, \ttH_x\Delta_{\mathfrak{m}})_x = \nu_{\mathfrak{t}} + \GammaH(v, \nu_{\mathfrak{m}})_x=\nu_{\mathfrak{c}}
\end{gathered}
$$
For the differential equation, let $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ be a tangent vector to $\mathcal{H}\mathrm{M}$ at $(x, v)\in \mathcal{H}\mathrm{M}$ and set $d\mathfrak{q}(x, v) = (b, w)\in \mathcal{T}\mathcal{B}$, $d^2\mathfrak{q}(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}) = (b, w, \Delta_b, \Delta_w)$. We have $\gamma^{\mathcal{B}} = \mathfrak{q}\gamma$, hence, the chain rule gives:
$$(\gamma^{\mathcal{B}}(t), \partial^{\mathcal{T}\mathcal{B}}_{b, w, \Delta_{b}, \Delta_{w}}\gamma^{\mathcal{B}}(t)) = d\mathfrak{q} (\gamma(t), \partial^{\mathcal{T}\mathrm{M}}_{x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}}\gamma(t))$$
Thus, $d\mathfrak{q}$ maps the tangent vector $\partial^{\mathcal{T}\mathrm{M}}_{x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}}\gamma(t)$ at $\gamma(t)$ to $\partial^{\mathcal{T}\mathcal{B}}_{b, w, \Delta_{b}, \Delta_{w}}\gamma^{\mathcal{B}}(t)$ at $\gamma^{\mathcal{B}}(t)$, hence $J^{\mathcal{B}}(t):=\partial^{\mathcal{T}\mathcal{B}}_{b, w, \Delta_{b}, \Delta_{w}}\gamma^{\mathcal{B}}$ lifts horizontally to $J^{\mathcal{H}}(t) :=\ttH_x\partial^{\mathcal{T}\mathrm{M}}_{x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}}\gamma$. Both $(\nabla^{\mathcal{B}}_{\dot{\gamma}^{\mathcal{B}}})^2{J}^{\mathcal{B}}(t)$ and $\rR^{\mathcal{B}}_{J^{\mathcal{B}}(t), \dot{\gamma}^{\mathcal{B}}(t)}\dot{\gamma}^{\mathcal{B}}(t)$ are in $\mathcal{T}\mathcal{B}$, with lifts $(\nabla^{\mathcal{H}}_{\dot{\gamma}^{\mathcal{H}}})^2{J}^{\mathcal{H}}(t)$ and $\rR^{\mathcal{H}}_{J^{\mathcal{H}}(t), \dot{\gamma}(t)}\dot{\gamma}(t)$ respectively. By linearity of the lift, we have the differential equation \cref{eq:submerse_Jac_C}. The argument shows $\mathfrak{J}^{\mathcal{H}}$ is the lift of $\mathfrak{J}^{\mathcal{B}}$. The remaining statements on the initial condition follows from the correspondence between $\mathfrak{j}_{\cH}$ with the canonical flip on $\mathcal{T}\cT\mathcal{B}$ in \cref{prop:frjH}.
\end{proof}
\begin{example}
Continuing with flag manifolds, recall geodesics on $\SOO(n)$ are of the form $\gamma(U, \eta; t) = U\exp(tU^{\mathsf{T}}\eta)$ for $(U, \eta) \in \mathcal{T}\SOO(n)$. With the initial data given by $\eta = UA, \nu_{\mathfrak{m}} = UC$, $\nu_{\mathfrak{t}} =U(\frac{1}{2}(AC + CA + [A, C]_{\mathfrak{d}}) + E)$ (for $UE = \ttH_x\nu_{\mathfrak{t}}=\nu_{\mathfrak{c}}$ in this case), then $\delta_{\mathfrak{m}} = UC, \delta_{t} = U(\frac{1}{2}(AC + CA + [C, A]_{\mathfrak{d}}) + E)$ with $A, C, E$ are horizontal, hence $J^{\mathcal{H}}(t) = \ttH_{\gamma(t)}J(t)$ is
\begin{equation}\begin{gathered}
U\exp(tA)\{\exp(-tA)C\exp(tA) + t\exp(-tA) \frL_{\exp}(tA, \frac{1}{2}[A, C]_{\mathfrak{b}} + E)\}_{\mathfrak{b}}=\\
U\exp(tA)\{C+ \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n!}t^n(\ad_A^{n-1} (\frac{1}{2}[A, C]_{\mathfrak{b}}+ E-[A, C]) \}_{\mathfrak{b}}
\end{gathered}
\end{equation}
\end{example}
This could be generalized to naturally reductive homogeneous spaces in the next theorem. Let $\mathrm{M}$ be a Lie group with Lie algebra $\mathfrak{m}$, identified with the tangent space at the identity of $\mathrm{M}$. For $U\in \mathrm{M}$, let $\mathcal{L}_U$ be the left multiplication by $U$. From \cite{KobNom}, chapter 10, section 2, geodesics on a naturally reductive homogeneous space lifts to a one-parameter exponential family $U\exp(tA)$ for $U\in \mathrm{M}, A\in \mathfrak{b}$.
\begin{theorem}\label{theo:jacobi_submerse_nat} If $\mathcal{B}=\mathrm{M}/\mathcal{K}$ is a naturally reductive homogeneous space, with $\mathrm{M}$ and $\mathcal{K}$ are Lie groups with Lie algebras $\mathfrak{m}$ and $\mathfrak{d}$, with a decomposition $\mathfrak{m} = \mathfrak{d} \oplus \mathfrak{b}$ such that $[\mathfrak{b}, \mathfrak{d}]\subset \mathfrak{b}$ and an $\ad(\mathfrak{d})$-invariant and naturally reductive metric $\langle\rangle_{\mathfrak{b}}$ on $\mathfrak{b}$. Thus, any element $S\in\mathfrak{m}$ has a decomposition $S = S_{\mathfrak{b}} + S_{\mathfrak{d}}$. Let $\gamma(t) = U\exp(tA)$ be a horizontal geodesic on $\mathrm{M}$, the lift of a geodesic $\gamma^{\mathcal{B}}(t)$ on $\mathcal{B}$, with $U=\gamma(0)\in\mathrm{M}$ and $A\in\mathfrak{b}$. A Jacobi field $\mathfrak{J}^{\mathcal{B}}$ along $\gamma^{\mathcal{B}}$ lifts to a horizontal vector field $\mathfrak{J}^{\mathcal{H}}$ along $\gamma(t)$ on $\mathrm{M}$. Assume $\mathfrak{J}^{\mathcal{H}}(0) = d\mathcal{L}_UC$, $\ttH\nabla^{\mathcal{H}}_{d/dt}\mathfrak{J}^{\mathcal{H}}(0)=d\mathcal{L}_UE$ for $C, E\in \mathfrak{b}$, let $F(t)$ be the function from $\mathbb{R}$ to $\mathfrak{b}$ such that $\mathfrak{J}^{\mathcal{H}}(t) = d\mathcal{L}_{\gamma(t)}F(t)$, then $F(t)$ satisfies
\begin{equation}\label{eq:FJacobi}
\ddot{F}(t) + [A, \dot{F}(t)]_{\mathfrak{b}} - [A,[A,F(t)]_{\mathfrak{d}}] = 0
\end{equation}
and the lift of the Jacobi field is
\begin{equation}\label{eq:jafieldH} \mathfrak{J}^{\mathcal{H}}(t) = d\mathcal{L}_{\gamma(t)}\{C+ t\mathcal{Z}(t\ad_A) (\frac{1}{2}[A, C]_{\mathfrak{b}}+ E-[A, C]) \}_{\mathfrak{b}}
\end{equation}
with $\mathcal{Z}(x) = \frac{1-\exp(-x)}{x}=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{(n+1)!}x^n$.
\end{theorem}
The case $C=A, E=0$ corresponds to the Jacobi field $d\mathcal{L}_{\gamma(t)}A=\dot{\gamma}(t)$, the case $C=0, E= A$ corresponds to the Jacobi field $td\mathcal{L}_{\gamma(t)}A = t\dot{\gamma}(t)$. Equation \ref{eq:FJacobi} appeared in \cite{Rauch,Chavel,Ziller}. We will use the notations $\nabla_{d/dt}^{\mathcal{H}}$ for $\ttH\nabla_{d/dt}$ and $\mathcal{Z}_{x=P}=\mathcal{Z}(P)$ for an operator $P$.
\begin{proof} Let $F(t)$ be a smooth function from $\mathbb{R}$ to $\mathfrak{b}$. Then we have for $A \in \mathfrak{b}$
\begin{equation}\label{eq:nablaF}
\nabla^{\mathcal{H}}_{d/dt}d\mathcal{L}_{\gamma(t)}F(t) = d\mathcal{L}_{\gamma(t)}\{\dot{F}(t) + \frac{1}{2}[A, F(t)]_{\mathfrak{b}}\}
\end{equation}
this follows from the fact that for a fixed $t$, $d\mathcal{L}_{\gamma(s)} F(t)$ and $\dot{\gamma}(s)$ are invariant vector fields along $\gamma$ (in the variable $s$), hence, by the naturally reductive assumption $(\nabla_{d/ds}^{\mathcal{H}}d\mathcal{L}_{\gamma(s)}F(t))_{s=t} = \frac{1}{2}d\mathcal{L}_{\gamma(t)}[A, F(t)]_{\mathfrak{b}}$. On the other hand, and at $s=t$, $d\mathcal{L}_{\gamma(s)}(F(s)-F(t))$ is the zero tangent vector, thus $(\nabla^{\mathcal{H}}_{d/ds}d\mathcal{L}_{ \gamma(s)}(F(s)-F(t))_{s=t} = \lim_{s\to t}d\mathcal{L}_{ \gamma(s)}\frac{1}{s-t}(F(s)-F(t)) = d\mathcal{L}_{\gamma(t)}\dot{F}(t)$, and together we have \cref{eq:nablaF}. Repeating this, we have
\begin{equation}\label{eq:nbl2}
(\nabla^{\mathcal{H}})^2_{d/dt}\gamma(t)F(t) = d\mathcal{L}_{\gamma(t)}\{\ddot{F}(t) + [A, \dot{F}(t)]_{\mathfrak{b}} + \frac{1}{4}[A, [A, F(t)]_{\mathfrak{b}}]_{\mathfrak{b}}\}
\end{equation}
Now we use $F(t) = C + (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{b}}$ where $G := \frac{1}{2}[A, C]_{\mathfrak{b}}+ E-[A, C]$. Differentiate $1-\exp(-x) = x\mathcal{Z}(x)$ we have $\exp(-x) = \mathcal{Z}(x) + x \mathcal{Z}'(x)$ and
$$\dot{F}(t) = (\mathcal{Z}_{x=t\ad_A}G +t\ad_A\mathcal{Z}'_{x=t\ad_A}G)_{\mathfrak{b}} = (\exp(-t\ad_A)G)_{\mathfrak{b}}$$
hence $\ddot{F}(x) = -(\ad_A\exp(-t\ad_A)G)_{\mathfrak{b}}= - [A, (\exp(-t\ad_A)G)_\mathfrak{b} + (\exp(-t\ad_A)G)_\mathfrak{d}]_\mathfrak{b}$. Noting $G_{\mathfrak{d}} = -[A, C]_{\mathfrak{d}}$, $\exp(-tx) = 1-tx\mathcal{Z}(tx)$
$$\begin{gathered}\ddot{F}(t) + [A, \dot{F}(t)]_{\mathfrak{b}} = -[A, (\exp(-t\ad_A)G)_{\mathfrak{d}}]_{\mathfrak{b}} =
-[A, G_{\mathfrak{d}}-t(\ad_A\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{d}}]_{\mathfrak{b}}\\= [A[A, t\mathcal{Z}_{x=t\ad_A}G+C]_{\mathfrak{d}}]
= [A[A, (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{b}}+C]_{\mathfrak{d}}]
\end{gathered}$$
which is \cref{eq:FJacobi} since $[A, t\mathcal{Z}_{x=t\ad_A}G+C]_{\mathfrak{d}} = [A, (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{b}}+C]_{\mathfrak{d}}$, because $[\mathfrak{b}, \mathfrak{d}]\subset\mathfrak{b}$ implies $[A, (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{d}}]_{\mathfrak{d}}=0$.
From here, the inside of the curly brackets on the right-hand side of \cref{eq:nbl2} is
$$\begin{gathered}
[A[A, (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{b}}+ C]_{\mathfrak{d}}]_{\mathfrak{b}} + \frac{1}{4}[A, [A, (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{b}} +C]_{\mathfrak{b}}]_{\mathfrak{b}}=\\
\frac{3}{4}[A[A, (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{b}}+ C]_{\mathfrak{d}}]_{\mathfrak{b}} + \frac{1}{4}[A[A, (t\mathcal{Z}_{x=t\ad_A}G)_{\mathfrak{b}} +C]]_{\mathfrak{b}}
\end{gathered}
$$
which is $\frac{3}{4}[A, [A, F(t)]_{\mathfrak{d}}] +\frac{1}{4}[A[A,F(t)]]_{\mathfrak{b}}$. On the other hand, from \cref{eq:flag_curv}, which, as mentioned, is equivalent to the curvature formula for naturally reductive homogeneous space in \cref{eq:natret}
$$\RcH_{J^{\mathcal{H}}(t), \dot{\gamma}(t)}\dot{\gamma}(t) = \frac{1}{4}d\mathcal{L}_{\gamma(t)}\{[[F(t), A], A]_{\mathfrak{b}} +2[[F(t), A]_{\mathfrak{d}}, A] - [[A, F(t)]_{\mathfrak{d}}, A]\}
$$
and the result follows from anticommutativity of Lie brackets.
\end{proof}
We finish this section with the well-known connection between Killing fields and Jacobi fields. For naturally reductive homogeneous spaces, some results in the next lemma are well-known (\cite{Ziller}) but the formula for the Jacobi field helps clarify them.
We assume the setup of \cref{theo:jacobi_submerse_nat}. If $X\in\mathfrak{m}$, then the Killing field or {\it isotropic Jacobi field} associated with $X$ is $d\mathcal{L}_{\gamma(t)}(\exp(-t\ad_A)X)_{\mathfrak{b}}$, which is also a Jacobi field, if we substitute $C = X_{\mathfrak{b}}, E = -\frac{1}{2}[A, X_{\mathfrak{b}}]_{\mathfrak{b}} - [A, X_{\mathfrak{d}}]$ in \cref{eq:jafieldH} and note $\exp(-x) = 1 - x\mathcal{Z}(x)$. A Jacobi field is called isotropic if it arises from $X\in\mathfrak{m}$ in this manner. We will use the notation $\mathfrak{J}^{\mathcal{H}}_{U, C, A, E}$ to denote the (lift of the) Jacobi field with initial condition $\mathfrak{J}^{\mathcal{H}}(0) = d\mathcal{L}_U C$, $\nabla^{\mathcal{H}}_{d/dt}\mathfrak{J}^{\mathcal{H}}(0) = d\mathcal{L}_U E$ along the geodesic $\gamma(t) = \Exp_U (td\mathcal{L}_UA)= U\exp(tA)$ for $A, C, E\in\mathfrak{b}$ and $U\in\mathrm{M}$.
\begin{lemma}\label{lem:tech_zil} Let $\mathcal{B} = \mathrm{M}/\mathcal{K}$ be a connected naturally reductive homogeneous space as in \cref{theo:jacobi_submerse_nat}.\hfill\break
1) A Jacobi field $\mathfrak{J}^{\mathcal{H}}_{U, C, A, E}$ along the geodesic $\gamma(t) = \Exp_U (d\mathcal{L}_UtA)$ is isotropic if and only if there is $X=X_{\mathfrak{b}} + X_\mathfrak{d}\in \mathfrak{m}$ such that $C = X_{\mathfrak{b}}, E =-\frac{1}{2}[A, C]_{\mathfrak{b}}-[A, X_{\mathfrak{d}}]$. In particular, $\mathfrak{J}^{\mathcal{H}}_{U, 0, A, E}$ for $C\in\mathfrak{b}$ is isotropic if and only if $E=-[A, D]$ for $D\in\mathfrak{d}$. Also, $\mathfrak{J}^{\mathcal{H}}_{U, C, A, -\frac{1}{2}[A, C]_\mathfrak{b}}$ is isotropic.
\hfill\break
2) For all $A\in \mathfrak{b}$, the map $\ad_A^{\mathfrak{b}}: X\mapsto [A, X]_{\mathfrak{b}}$ from $\mathfrak{b}$ to itself is anti-self-adjoint under $\langle\rangle_{\mathfrak{b}}$, thus its eigenvalues are either zero or purely imaginary. If $\lambda\sqrt{-1}$ is a purely imaginary eigenvalue of $\ad_A^{\mathfrak{b}}$ as an operator on $\mathfrak{b}$, with eigenvector $V + \sqrt{-1}V_*$ with $V, V_*\in\mathfrak{b}$, then $[A,V] =-\lambda V_* +D_*, [A, V_*] = \lambda V + D$ for $D, D_*\in \mathfrak{d}$. If $[A,D_*] = [A, D] = 0$ then
\begin{equation}\label{eq:zjac}
(\mathcal{Z}(t\ad_A) V)_{\mathfrak{b}} = \frac{\sin t\lambda}{t\lambda} V - t\frac{1- \cos t\lambda }{(t\lambda)^2}[A, V]_\mathfrak{b}
\end{equation}
Thus, $t(\mathcal{Z}(t\ad_A) V)_{\mathfrak{b}}=0$ for $t = \frac{2k\pi}{\lambda}$, $k\in \mathbb{Z}$ and the Jacobi field $\mathfrak{J}^{\mathcal{H}}_{U, 0, A, V}$ vanishes at those points. In particular, if $[A, \mathfrak{d}] = 0$, this formula holds.
\end{lemma}
\begin{proof} For 1), we compare the values and first time-derivatives of two functions
$\{C+ t\mathcal{Z}_{x=t\ad_A} (\frac{1}{2}[A, C]_{\mathfrak{b}}+ E-[A, C]) \}_{\mathfrak{b}}$ and $(\exp(-t\ad_A)X)_{\mathfrak{b}}$ at $t= 0$ and get $C = X_{\mathfrak{b}}$ and $\frac{1}{2}[A, C]_{\mathfrak{b}}+ E-[A, C] =-[A, X]_{\mathfrak{b}}$, which gives us the relation between $X$, $C$ and $E$, and conversely, a direct substitution proves that the first function reduces to the second for $C$ and $E$ satisfying the conditions of 1. From here, when $C =0$, $E=-[A, X_\mathfrak{d}]$ and when $X_\mathfrak{d}=0$ we get $E=-\frac{1}{2}[A, C]_\mathfrak{b}$.
For 2)For $A, B, C\in \mathfrak{b}$, from the naturally reductive assumption,
$$\langle [A, B]_{\mathfrak{b}}, C\rangle_{\mathfrak{b}} + \langle B, [A, C]_{\mathfrak{b}} \rangle_{\mathfrak{b}} = 0$$
hence $\ad_A^{\mathfrak{b}}$ is anti-self-adjoint.
If $\ad_A^{\mathfrak{b}}(V + \sqrt{-1}V_*) = \sqrt{-1}\lambda(V + \sqrt{-1}V_*)$, we have $[A, V]_{\mathfrak{b}} = - \lambda V_*$ and $[A, V_*]_{\mathfrak{b}} = \lambda V$, hence $[A, V] = -\lambda V_* + D_*$, and $[A, V_*] = \lambda V + D$, where $V, V_*\in \mathfrak{b}$, for some $D_*, D\in \mathfrak{d}$. We have
$$[A, [A, V]] = -\lambda [A, V_*] + [A, D_*] = -\lambda^2 V + \lambda D$$
by assumption. By induction, $\ad_A^{2n}V = (-1)^n \lambda^{2n} V + c_nD$ for $c_n\in\mathbb{R}$, $\ad_A^{2n+1}V = (-1)^n \lambda^{2n}[A, V]$. Thus
$$(\mathcal{Z}(t\ad_A) V)_{\mathfrak{b}} =\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}t^{2n}\lambda^{2n}V
- \sum_{n=0}^{\infty} \frac{(-1)^n}{(2n+2)!}t^{2n+1}\lambda^{2n}[A, V]_\mathfrak{b}
$$
which gives us \cref{eq:zjac}.
\end{proof}
\begin{remark} In \cite{Ziller}, the author suggested that manifolds satisfying the condition
\begin{itemize}
\item[$(ZC)$] $\mathcal{B}=\mathrm{M}/\mathcal{K}$ is a naturally reductive homogeneous space and all twice vanishing Jacobi fields of $\mathcal{B}$ are isotropic.
\end{itemize}
are locally symmetric spaces. The case of 3-symmetric naturally reductive manifolds was settled in \cite{Gonzalez}, where \cref{eq:zjac} appeared. Lemma \ref{lem:tech_zil} helps to show, for example, if $\mathfrak{b}$ contains an element $A$ such that $[A, \mathfrak{d}] = 0$ and $[A, \mathfrak{b}]_{\mathfrak{b}}\neq 0$ then $(ZC)$ is not satisfied, as in that case, $\ad_A^{\mathfrak{b}}$ must have an imaginary eigenvalue with eigenvector $V+\sqrt{-1}V_*$ satisfies \cref{eq:zjac}, and $\mathfrak{J}^{\mathcal{H}}_{U, 0, A, V}$ is isotropic by $(ZC)$. Hence $V = [A, D]=0$ for some $D\in\mathfrak{d}$ by 1). In particular, the Stiefel manifold $\SOO(n)/\SOO(n-p)$ with the bi-invariant metric on $\SOO(n)$ for integers $n>p$ does not satisfy $(ZC)$. In this case, $\mathcal{K}=\SOO(n-p)\subset\mathrm{M}=\SOO(n)$, with $\mathcal{K}$ identified with the bottom right diagonal $(n-p)\times (n-p)$ block. Here, $\mathfrak{m} = \mathfrak{o}(n) = \mathfrak{d}\oplus\mathfrak{b}_0\oplus\mathfrak{b}_1$ where $\mathfrak{d}$ is formed by the bottom right $(n-p)\times(n-p)$ blocks, $\mathfrak{b}_0\subset\mathfrak{o}(n)$ formed by the top left $p\times p$ diagonal block, $\mathfrak{b}_1\subset \mathfrak{o}(n)$ is the space with those diagonal blocks vanishes and $\mathfrak{b} = \mathfrak{b}_0\oplus\mathfrak{b}_1$. Then $[\mathfrak{b}_0, \mathfrak{d}] = 0$ but $[\mathfrak{b}_0, \mathfrak{b}]_{\mathfrak{b}} \neq 0$.
\end{remark}
\section{Natural metrics on tangent bundles}\label{sec:nat_metric}
In \cite{Sasaki}, Sasaki introduced a metric on the tangent bundle of a manifold which makes the bundle projection a submersion. Later works, including \cite{Dombrowski1962,KowalSeki}, clarify our understanding of this metric. In \cite{CheegerGromoll} Cheeger and Gromoll proposed a method to construct complete, non-negative metrics on vector bundles on compact homogeneous spaces (thus satisfies the condition of the {\it soul theorem}). This method was extended in \cite{TriMuss} to give a complete metric on tangent bundles of complete manifolds, which the authors named the Cheeger-Gromoll metric. Sasaki and Cheeger-Gromoll metrics are examples of natural metrics, their Levi-Civita connections and curvatures could be computed from those of the base manifold metric \cite{GudKap}. We will start with a general setup, then focus on a metric family inspired by \cite{Abbassi2005,BLW}, which includes both the Sasaki and Cheeger-Gromoll metrics.
The following proposition defines the vertical and horizontal spaces of the fibration $\pi: \mathcal{T}\mathrm{M}\to\mathrm{M}$.
\begin{proposition} With the same notation as \cref{prop:TTM}, let $\pi:\mathcal{T}\mathrm{M}\to \mathrm{M}$, $\pi: (x, v)\mapsto x$ with $(x, v)\in \mathcal{T}\mathrm{M}\subset\mathcal{E}^2$ be the tangent bundle projection map. Its differential $d\pi:\mathcal{T}\cT\mathrm{M}\to \mathcal{T}\mathrm{M}$ maps $(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ to $(x, \Delta_{\mathfrak{m}})$, thus its kernel, the {\bf $\pi$-vertical space} at $(x, v)$ consists of elements $(x, v, 0, \Delta_{\mathfrak{t}})\in\mathcal{E}^4$, where $\Delta_{\mathfrak{t}}$ is tangent to $\mathrm{M}$ at $x$ (i.e.\ $\Pi_x\Delta_{\mathfrak{t}} = \Delta_{\mathfrak{t}}$). Let $\mathrm{C}:\mathcal{T}\cT\mathrm{M}\to \mathcal{T}\mathrm{M}$ be the connection map $(x, v, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})\mapsto (x, \Delta_{\mathfrak{t}} + \Gamma(v, \Delta_{\mathfrak{m}})_x$ in \cref{lem:conn}. The {\bf $\pi$-horizontal space} at $(x, v)$, defined as the kernel of $\mathrm{C}_{(x, v)}$ consists of vectors satisfying $\Delta_{\mathfrak{t}} + \Gamma(v, \Delta_{\mathfrak{m}})_x=0$. $\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$ decomposes to vertical and horizontal components as follows:
\begin{equation}\label{eq:TTM_decomp}
(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}) = (0, \Delta_{\mathfrak{t}} + \Gamma(\Delta_{\mathfrak{m}}, v)_{x}) + (\Delta_{\mathfrak{m}}, -\Gamma(\Delta_{\mathfrak{m}}, v)_x)
\end{equation}
\end{proposition}
\begin{proof}The statements for $\pi$ and $d\pi$ are straightforward. It is clear the first component of \cref{eq:TTM_decomp} is vertical and the second is horizontal.
\end{proof}
\begin{remark}\label{rem:rmv_rmh}
Consider $(x, v)\in \mathcal{T}\mathrm{M}$. If $\xi \in\mathcal{T}_{x}\mathrm{M}$, define the horizontal lift $\xi^{\mathrm{h}}\in \mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$ by $\xi^{\mathrm{h}}:=(\xi, -\Gamma(\xi, v)_x)$, which is in the horizontal space, the kernel of $\mathrm{C}_{(x, v)}$. Define the vertical lift $\xi^\mathrm{v}$ by $\xi^\mathrm{v}:=(0, \xi)\in\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$. Let $\mathtt{X}$ and $\mathtt{Y}$ be two vector fields on $\mathrm{M}$, we use the same notations $\mathrm{h}, \mathrm{v}$ to denote the corresponding lifts of vector fields. Applying derivative rules:
\begin{equation} (\rD_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{h}})_{(x, v)} = ((\rD_{\mathtt{X}}\mathtt{Y})_{x}, (-\rD_{\mathtt{X}}\Gamma)(\mathtt{Y})_{x}+\Gamma(\mathtt{Y}, \Gamma(\mathtt{X}, v))_{x} -\Gamma((\rD_{\mathtt{X}}\mathtt{Y}))_{x}, v)
\end{equation}
\begin{equation} \rD_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}} = (0, \rD_{\mathtt{X}}\mathtt{Y})
\end{equation}
\begin{equation} \rD_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{h}} = (0, -\Gamma(\mathtt{X}, \mathtt{Y}))
\end{equation}
\begin{equation} \rD_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{v}} = 0
\end{equation}
From here we have the Dombrowski Lie bracket relations, using our global formula for curvature
\begin{equation} [\mathtt{X}^{\mathrm{h}}, \mathtt{Y}^{\mathrm{h}}]_{(x, v)} = ([\mathtt{X}, \mathtt{Y}]_{(x, v)}, \rR_{\mathtt{X}_x, \mathtt{Y}_x}v -\Gamma([\mathtt{X}, \mathtt{Y}], v)_{x})
\end{equation}
\begin{equation} [\mathtt{X}^{\mathrm{h}}, \mathtt{Y}^{\mathrm{v}}] = (\nabla_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}}
\end{equation}
\begin{equation} [\mathtt{X}^{\mathrm{v}}, \mathtt{Y}^{\mathrm{v}}] = 0
\end{equation}
\end{remark}
We now state a purely linear algebra lemma for metric and projection, which gives us the main idea of the natural metric construction
\begin{lemma}\label{lem:two_func} Let $\mathcal{E}, \langle\rangle_{\mathcal{E}}$ be an inner product space, and $\mathcal{T}\subset\mathcal{E}$, $\mathcal{T}_2\subset\mathcal{E}^2$ be a pair of vector subspaces of $\mathcal{E}$ and $\mathcal{E}^2$. Assume $f_1:\mathcal{E}^2\to\mathcal{E}, f_2:\mathcal{E}^2\to\mathcal{E}$ are two linear maps such that $f_1(\mathcal{T}_2)=\mathcal{T}, f_2(\mathcal{T}_2) = \mathcal{T}$ and $f_1\oplus f_2=[f_1^{\mathsf{T}}, f_2^{\mathsf{T}}]^{\mathsf{T}}$ is invertible in $\mathcal{E}^2$ and restricts to a bijection between $\mathcal{T}_2$ and $\mathcal{T}^2$. Let $\mathsf{g}_1, \mathsf{g}_2$ be two positive-definite operators on $\mathcal{E}$, and $\Pi_1, \Pi_2$ be the projections of $\mathcal{E}$ to $\mathcal{T}$ with respect to the inner products defined by these operators. Then the operator
\begin{equation}\mathsf{G}:= \begin{bmatrix}f_1^{\mathsf{T}}, f_2^{\mathsf{T}}\end{bmatrix}\begin{bmatrix}\mathsf{g}_1 & 0 \\ 0 &\mathsf{g}_2\end{bmatrix}\begin{bmatrix}f_1\\ f_2\end{bmatrix}
\end{equation}
which maps $\tilde{\omega}\in\mathcal{E}^2$ to $f_1^{\mathsf{T}}\mathsf{g}_1 f_1\tilde{\omega} + f_2^{\mathsf{T}}\mathsf{g}_2f_2\tilde{\omega}$ is a positive definite operator, which defines an inner product on $\mathcal{E}^2$. The projection to $\mathcal{T}_2$ under $\mathsf{G}$ is given by
\begin{equation}\Pi_{\mathsf{G}} = \begin{bmatrix}f_1\\ f_2\end{bmatrix}^{-1}\begin{bmatrix}\Pi_1 & 0 \\ 0 &\Pi_2\end{bmatrix}\begin{bmatrix}f_1\\ f_2\end{bmatrix}
\end{equation}
or for $\tilde{\omega}\in \mathcal{E}^2$, $\Pi_{\mathsf{G}}\tilde{\omega} = (f_1\oplus f_2)^{-1}(\Pi_1 f_1 \tilde{\omega}, \Pi_2 f_2 \tilde{\omega})$. Under this metric, the kernels of $f_1$ and $f_2$ are orthogonal complement subspaces.
\end{lemma}
\begin{proof}It is clear from the bijective assumption of $(f_1\oplus f_2)$ that $\mathsf{G}$ is positive-definite, and it is clear from construction that the image of $\Pi_{\mathsf{G}}$ is in $\mathcal{T}_2$. To show $\Pi_{\mathsf{G}}$ is the projection, note for $\tilde{\omega}\in \mathcal{E}^2$ and $\tilde{\eta}\in \mathcal{T}_2$
$$\begin{gathered}\langle \Pi_{\mathsf{G}}\omega, \mathsf{G} \tilde{\eta}\rangle_{\mathcal{E}^2} = \langle\mathsf{G} \Pi_{\mathsf{G}}\omega, \tilde{\eta}\rangle_{\mathcal{E}^2} = \langle \begin{bmatrix}f_1^{\mathsf{T}}, f_2^{\mathsf{T}}\end{bmatrix}\begin{bmatrix}\mathsf{g}_1\Pi_1 & 0 \\ 0 &\mathsf{g}_2\Pi_2\end{bmatrix}\begin{bmatrix}f_1\\ f_2\end{bmatrix}\tilde{\omega}, \tilde{\eta}\rangle_{\mathcal{E}^2} = \\
\langle \mathsf{g}_1\Pi_1f_1\tilde{\omega}, f_1\tilde{\eta} \rangle_{\mathcal{E}} + \langle \mathsf{g}_2\Pi_2f_2\tilde{\omega}, f_2\tilde{\eta}\rangle_{\mathcal{E}}=
\langle f_1\tilde{\omega}, \mathsf{g}_1\Pi_1f_1\tilde{\eta} \rangle_{\mathcal{E}} + \langle f_2\tilde{\omega}, \mathsf{g}_2\Pi_2 f_2\tilde{\eta}\rangle_{\mathcal{E}}\\= \langle f_1\tilde{\omega}, \mathsf{g}_1 f_1\tilde{\eta} \rangle_{\mathcal{E}} + \langle f_2\tilde{\omega}, \mathsf{g}_2 f_2\tilde{\eta}\rangle_{\mathcal{E}}
\end{gathered}
$$
where aside from the abuse of the notation to write operator expressions in matrix notation, we use the facts that $\mathsf{G}$, $\mathsf{g}_1\Pi_1, \mathsf{g}_2\Pi_2$ are self-adjoint together with $f_i\tilde{\eta}\in \mathcal{T}$ for $i=1,2$. The last expression is $\langle \tilde{\omega}, \mathsf{G}\tilde{\eta}\rangle_{\mathcal{E}^2}$. Since $f_1\oplus f_2$ is a bijection, $f_1$ and $f_2$ are surjective onto $\mathcal{E}$. Thus, by the rank-nullity theorem, the kernels of $f_1$ and $f_2$ each has dimension $\dim\mathcal{E}$. If $\tilde{\omega}_1\in \Null(f_1), \tilde{\omega}_2\in \Null(f_2)$ then
$$\langle \tilde{\omega}_1,\mathsf{G} \tilde{\omega}_2\rangle_{\mathcal{E}^2}
=\langle f_1\tilde{\omega}_1, \mathsf{g}_1 f_1\tilde{\omega}_2 \rangle_{\mathcal{E}} + \langle f_2\tilde{\omega}_1, \mathsf{g}_2f_2\tilde{\omega}_2\rangle_{\mathcal{E}}=0
$$
Thus $\Null(f_1)$ is orthogonal to $\Null(f_2)$, therefore $\mathcal{E}^2=\Null(f_1)\oplus \Null(f_2)$
\end{proof}
If $\omega\in \mathcal{E}$, we will use the notation $\Gamma[\omega]$ to denote the operator from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$, mapping $x\in \mathrm{M}$ to the operator $\eta\mapsto \Gamma[\omega]_{x}\eta :=\Gamma(\eta, \omega)_{x}$. For each $x$, $\Gamma[\omega]_{x}$ is a linear map from $\mathcal{E}$ to itself so we can define the adjoint $\Gamma^{\mathsf{T}}[\omega]_{x}$ under $\langle\rangle_{\mathcal{E}}$, and we define the operator-valued function $\Gamma^{\mathsf{T}}[\omega]: x\to \Gamma^{\mathsf{T}}[\omega]_x$ from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$. For a vector field $\mathtt{X}$ we use the notations $\Gamma[\mathtt{X}]$ and $\Gamma[\mathtt{X}]_{|x}$ to define the operator-valued function evaluated as $\Gamma(\nu, \mathtt{X}_{x})_{x}$ for $x\in \mathrm{M}, \nu\in \mathcal{E}$. We will use block matrix notation to define operators on $\mathcal{E}^2$ to $\mathcal{E}^2$, where an operator block $\begin{bmatrix} A & B \\ C & D\end{bmatrix}$ acts on $\tilde{\Delta} = (\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ by $(A \Delta_{\mathfrak{m}} + B \Delta_{\mathfrak{t}}, C\Delta_{\mathfrak{m}} + D\Delta_{\mathfrak{t}})$ (thus we think of $\tilde{\Delta}$ as a vertical vector for the operator action but write it in horizontal form for convenience). We use the subscript $(x, v)$ to denote the value of a function at point $(x, v)\in \mathcal{E}^2$, for example, $\mathsf{G}_{(x, v)}$ or $\hat{\sfg}_{(x, v)}$ below.
\begin{theorem}\label{theo:SCGMT}Let $(\mathrm{M}, \mathsf{g}, \mathcal{E})$ be an embedded ambient structure of a manifold $\mathrm{M}$. Let $\hat{\sfg}$ be a positive-definite operator-valued function from $\mathcal{T}\mathrm{M}$ to $\mathcal{E}$. Let $\mathsf{G}$ be the operator-valued function from $\mathcal{T}\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}^2, \mathcal{E}^2)$ defined by
\begin{equation}\label{eq:sfG_metric}
\begin{gathered}\mathsf{G}_{(x, v)} = \begin{bmatrix} \dI & \Gamma^{\mathsf{T}}[v]_{x}\\
0 & \dI
\end{bmatrix}\begin{bmatrix} \mathsf{g}_x & 0 \\ 0 & \hat{\sfg}_{(x, v)}\end{bmatrix} \begin{bmatrix} \dI & 0\\
\Gamma[v]_{x} & \dI
\end{bmatrix}
\end{gathered}
\end{equation}
for $(x, v) \in \mathcal{T}\mathrm{M}$. Then $(\mathcal{T}\mathrm{M}, \mathsf{G}, \mathcal{E}^2)$ is an embedded ambient structure, and
\begin{equation}\begin{gathered}\mathsf{G}^{-1}_{(x, v)} =
\begin{bmatrix} \dI & 0\\
-\Gamma[v]_{x} & \dI
\end{bmatrix}
\begin{bmatrix}\mathsf{g}^{-1}_x & 0 \\
0 & \hat{\sfg}^{-1}_{(x, v)}
\end{bmatrix}
\begin{bmatrix} \dI & -\Gamma^{\mathsf{T}}[v]_{x}\\
0 & \dI
\end{bmatrix}
\end{gathered}
\end{equation}
The metric induced by $\mathsf{G}$ is natural, this means for $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}}) \in \mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$, the vertical component $(0, \Delta_{\mathfrak{t}} + \Gamma(\Delta_{\mathfrak{m}}, v)_x$ and horizontal component $(\Delta_{\mathfrak{m}}, -\Gamma(\Delta_{\mathfrak{m}}, v)_x$ are orthogonal and $\pi:\mathcal{T}\mathrm{M}\to\mathrm{M}$ is a Riemannian submersion. Let $\Pi_{\mathsf{g}}$ and $\Pi_{\hat{\sfg}}$ be the projections corresponding to $\mathsf{g}$ and $\hat{\sfg}$, then the projection $\Pi_{\mathsf{G}}$ from $\mathcal{E}^2$ to $\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$ corresponding to $\mathsf{G}$ is
\begin{equation}\label{eq:bundl_proj}
\Pi_{\mathsf{G},(x, v)} = \begin{gathered}
\begin{bmatrix} \dI & 0\\
-\Gamma[v]_{x} & \dI
\end{bmatrix}\begin{bmatrix} \Pi_{\mathsf{g}, x} & 0 \\ 0 & \Pi_{\hat{\sfg}, x, v}\end{bmatrix}
\begin{bmatrix} \dI & 0\\
\Gamma[v]_x & \dI
\end{bmatrix}
\end{gathered}
\end{equation}
In other words, for ${\tilde{\omega}} = (\omega_x, \omega_v)\in \mathcal{E}^2$, its projection to $\mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$ is
\begin{equation}\label{eq:bundl_proj2}
\Pi_{\mathsf{G}, (x, v)}(\omega_{\mathfrak{m}}, \omega_{\mathfrak{t}}) = (\Pi_{\mathsf{g}, x}\omega_{\mathfrak{m}},
(-\Gamma[v]_{x}\Pi_{\mathsf{g}, x} + \Pi_{\hat{\sfg}, x, v}\Gamma[v]_{x})\omega_{\mathfrak{m}} + \Pi_{\hat{\sfg}, x, v}\omega_{\mathfrak{t}})
\end{equation}
A Christoffel function $\Gamma_{\mathsf{G}}$, evaluated at $\tilde{\xi} = (\xi_x, \xi_v)$, $\tilde{\eta} = (\xi_x, \xi_v)$ in $\mathcal{E}^2$ is given by $-\rD_{\tilde{\xi}}\Pi_{\mathsf{G}}\tilde{\eta} + \mathring{\Gamma}_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta})$ with
\begin{equation}
\begin{gathered}
\mathring{\Gamma}_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta}) = \Pi_{\mathsf{G}}\mathsf{G}^{-1}\mathrm{K}_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta})\\
\mathrm{K}_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta}) = \frac{1}{2}
((\rD_{\tilde{\xi}}\mathsf{G})\tilde{\eta} + (\rD_{\tilde{\eta}}\mathsf{G})\tilde{\xi} -\mathcal{X}_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta}))
\end{gathered}
\end{equation}
where $\mathcal{X}_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta})$ satisfies $\langle \tilde{\phi}, \mathcal{X}_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta})\rangle_{\mathcal{E}^2} = \langle\tilde{\xi}, \rD_{\tilde{\phi}}\mathsf{G}\tilde{\eta}\rangle_{\mathcal{E}^2}$ for all tangent vectors $\tilde{\xi}, \tilde{\eta}, \tilde{\phi}$ of $\mathcal{T}\mathrm{M}$.
\end{theorem}
Thus, $\mathsf{G}$ is diagonal after being transformed by the combination of $\pi$ and the connection map. Direct calculations show $\Pi_{\mathsf{G}}\mathsf{G}^{-1}$, $\rD_{\tilde{\xi}}\Pi_{\mathsf{G}}$ and $\Pi_{\mathsf{G}}\mathsf{G}^{-1}\rD_{\tilde{\xi}}\mathsf{G}$ also have simpler forms after that transformation. For computational purposes, $\rD_{\tilde{\xi}}\mathsf{G}$ can be evaluated by block, and $\mathcal{X}_{\mathsf{G}}$ can be evaluated from index-raising expressions of derivatives of $\mathsf{g}, \hat{\sfg}$ and $\Gamma$.
\begin{proof} Applying \cref{lem:two_func} with $\mathcal{T} = \mathcal{T}_x\mathrm{M}$, $\mathcal{T}_2= \mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$, $f_1 = (d\pi)_{(x, v)}$ and $f_2$ the connection map $\mathrm{C}_{(x, v)}$, we get the first four equations in the theorem, the kernels of $d\pi$ and of the connection map are orthogonal, and it is clear $\pi$ is a submersion from the expression of $\mathsf{G}$. The remaining statements about the Christoffel function follows from \cref{prop:Levi}.
\end{proof}
There are choices of $\hat{\sfg}$ such that the projection $\Pi_{\hat{\sfg}}$ is the same as $\Pi_{\mathsf{g}}$. While this could be done more generally, we focus on a subfamily of the $g$-natural metrics in \cite{KowalSeki,Abbassi2005}, that contains both the Sasaki and the Cheeger-Gromoll metric.
We denote $\alpha(t) = \alpha_t, \beta(t) =\beta_t$ for scalar functions $\alpha$ and $\beta$ in the following.
\begin{proposition}\label{prop:nat_TTM2} Let $\alpha$ be a smooth, positive scalar function from $\mathbb{R}_{\geq 0}$ to $\mathbb{R}_{> 0}$ and $\beta$ is a non-negative smooth function from $\mathbb{R}_{\geq 0}$ to $\mathbb{R}_{\geq 0}$ . Define the operator-valued function $\hat{\sfg}:\mathcal{T}\mathrm{M}\mapsto \mathfrak{L}(\mathcal{E}, \mathcal{E})$ for $\omega\in\mathcal{E}$ by:
\begin{equation}\label{eq:BLW}
\hat{\sfg}_{(x, v)}\omega = \alpha_{\|v\|^2_{\mathsf{g}, x}}\mathsf{g}_x\omega + \beta_{\|v\|^2_{\mathsf{g}, x}} \langle v,\omega\rangle_{\mathsf{g}, x}\ \mathsf{g}_x v
\end{equation}
then $\Pi_{\mathsf{g}, x}$ is the projection of $\hat{\sfg}_{(x, v)}$. The inverse of $\hat{\sfg}$ is given by
\begin{equation}\label{eq:hsfginv}
\hat{\sfg}_{(x, v)}^{-1}\omega = \frac{1}{\alpha_{\|v\|^2_{\mathsf{g}, x}}}\mathsf{g}_x^{-1}\omega -\frac{\beta_{\|v\|^2_{\mathsf{g}, x}}}{\alpha_{\|v\|^2_{\mathsf{g}, x}}(\alpha_{\|v\|^2_{\mathsf{g}, x}} +\beta_{\|v\|^2_{\mathsf{g}, x}}\|v\|_{\mathsf{g}, x}^2)} \langle v, \omega\rangle_{\mathcal{E}}v
\end{equation}
In that case, if $\Gamma$ is a Christoffel function of $\mathsf{g}$, and $\nabla$ is the Levi-Civita connection associated with $\mathsf{g}$, for three vector fields $\mathtt{X}, \mathtt{Y}, \mathtt{Z}$ of $\mathrm{M}$ we have
\begin{equation}\label{eq:hsfg_compa}
\rD_{\mathtt{X}^{\mathrm{h}}}\langle \mathtt{Y}\circ\pi, \hat{\sfg}(\mathtt{Z}\circ\pi)\rangle_{\mathcal{E}} =
\langle \nabla_{\mathtt{X}}\mathtt{Y}, \hat{\sfg}\mathtt{Z}\rangle_{\mathcal{E}} + \langle \mathtt{Y}, \hat{\sfg}\nabla_{\mathtt{Y}}\mathtt{Z}\rangle_{\mathcal{E}}
\end{equation}
Recall the horizontal lift of a tangent vector $\delta$ at $x$ to $(x, v)$ is given by $\delta^{\mathrm{h}}=(\delta, -\Gamma(\delta, v)_x)$, the vertical lift is $\delta^{\mathrm{v}} = (0, \delta)$, and the connection map is $\mathrm{C}_{(x, v)}\tilde{\xi} = \xi_{\mathfrak{t}} +\Gamma(\xi_{\mathfrak{m}}, v)_x$ for a tangent vector $\tilde{\xi}\in \mathcal{T}_{(x, v)}\mathcal{T}\mathrm{M}$. A Christoffel function $\Gamma_{\mathsf{G}}$ of $\mathsf{G}$ could be evaluated at two tangent vectors $\tilde{\xi} =(\xi_{\mathfrak{m}}, \xi_{\mathfrak{t}}), \tilde{\eta} = (\eta_{\mathfrak{m}}, \eta_{\mathfrak{t}})$ at $(x, v)$ as
\begin{equation}\label{eq:GammasfGDec}
\Gamma_{\mathsf{G}}(\tilde{\xi}, \tilde{\eta}) =
\Gamma_{\mathsf{G}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, \eta_{\mathfrak{m}}^{\mathrm{h}}) +
\Gamma_{\mathsf{G}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, (\mathrm{C}\tilde{\eta})^{\mathrm{v}}) +
\Gamma_{\mathsf{G}}((\mathrm{C}\tilde{\xi})^{\mathrm{v}}, \eta_{\mathfrak{m}}^{\mathrm{h}}) +
\Gamma_{\mathsf{G}}((\mathrm{C}\tilde{\xi})^{\mathrm{v}}, (\mathrm{C}\tilde{\eta})^{\mathrm{v}})
\end{equation}
\begin{equation}\label{eq:GammasfGparts}
\begin{gathered}
\Gamma_{\mathsf{G}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, \eta_{\mathfrak{m}}^{\mathrm{h}}) = (\Gamma(\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}), -\Gamma(\Gamma(\xi_{\mathfrak{m}},\eta_{\mathfrak{m}}), v) + (\rD_{\xi_{\mathfrak{m}}}\Gamma)(\eta_{\mathfrak{m}}, v)-\\\Gamma( \eta_{\mathfrak{m}}, \Gamma(\xi_{\mathfrak{m}}, v)) + \frac{1}{2}\rR_{\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}}v)\\
\Gamma_{\mathsf{G}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, (\mathrm{C}\tilde{\eta})^{\mathrm{v}}) =
(-\frac{\alpha}{2}\rR_{v, \mathrm{C}\tilde{\eta}}\xi_{\mathfrak{m}}, \frac{\alpha}{2}\Gamma(\rR_{v, \mathrm{C}\tilde{\eta}}\xi_{\mathfrak{m}}, v) +
\Gamma(\xi_{\mathfrak{m}}, \mathrm{C}\tilde{\eta}))\\
\Gamma_{\mathsf{G}}((\mathrm{C}\tilde{\xi})^{\mathrm{v}}, \eta_{\mathfrak{m}}^{\mathrm{h}}) = (-\frac{\alpha}{2}\rR_{v,\mathrm{C}\tilde{\xi}}\eta_{\mathfrak{m}}, \frac{\alpha}{2}\Gamma(\rR_{v,\mathrm{C}\tilde{\xi}}\eta_{\mathfrak{m}}, v)
+\Gamma(\mathrm{C}\tilde{\xi}, \eta_{\mathfrak{m}}))\\
\Gamma_{\mathsf{G}}((\mathrm{C}\tilde{\xi})^{\mathrm{v}}, (\mathrm{C}\tilde{\eta})^{\mathrm{v}}) = (0,\frac{\alpha'}{\alpha}(\|v\|_{\mathsf{g}}^2) \{\langle v, \mathsf{g}\mathrm{C}\tilde{\eta}\rangle_{\mathcal{E}} \mathrm{C}\tilde{\xi} + \langle v,\mathsf{g} \tilde{\xi}\rangle_{\mathcal{E}}\mathrm{C}\tilde{\eta}\} + \rF v)
\end{gathered}
\end{equation}
where the operator-valued functions $\Gamma$, its directional derivatives, $\rR$ and $\mathsf{g}$ are evaluated at $x$, $\mathrm{C}$ is evaluated at $(x, v)$ and the functions $\alpha, \alpha', \beta, \beta'$ are evaluated at $\|v\|_{\mathsf{g}}^2$ and $\rF := (\alpha + \|v\|_{\mathsf{g}}^2\beta)^{-1}\{(\beta-\alpha')\langle\mathrm{C}\tilde{\xi}, \mathrm{C}\tilde{\eta}\rangle_{\mathsf{g}} + (\beta'-2\frac{\alpha'}{\alpha}\beta)\langle v, \mathrm{C}\tilde{\xi})\rangle_{\mathsf{g}}\langle v, \mathrm{C}\tilde{\eta})\rangle_{\mathsf{g}}\}$.
\end{proposition}
It is known the case $\alpha = 1$, $\beta=0$ corresponds to the Sasaki metric and
$\Gamma_{\mathsf{G}}((\mathrm{C}\tilde{\xi})^{\mathrm{v}}, (\mathrm{C}\tilde{\eta})^{\mathrm{v}}) = (0,0)$ in this case. The case $\alpha(t) = \beta(t) = (1 +t)^{-1}$ corresponds to the Cheeger-Gromoll metric \cite{TriMuss}. In that case, the coefficient $\rF$ in $\Gamma_{\mathsf{G}}((\mathrm{C}\tilde{\xi})^{\mathrm{v}}, (\mathrm{C}\tilde{\eta})^{\mathrm{v}})_{\mathfrak{t}}$ equals
$$\frac{2+\|v\|_{\mathsf{g}}^2}{(1+\|v\|_{\mathsf{g}}^2)^2}\langle\mathrm{C}\tilde{\xi}, \mathrm{C}\tilde{\eta}\rangle_{\mathsf{g}} + \frac{1}{(1+\|v\|_{\mathsf{g}}^2)^2}\langle v, \mathrm{C}\tilde{\xi})\rangle_{\mathsf{g}}\langle v, \mathrm{C}\tilde{\eta})\rangle_{\mathsf{g}}$$
Parametrizing $\alpha, \beta$ by additional parameters provides subfamilies of metrics, for example, those considered in \cite{BLW}. In the following, recall the canonical vector field $\rU$ on $\mathcal{T}\mathrm{M}$ is the vector field defined by $\rU(x, v) = (x, v, 0, v)$ for $(x, v)\in\mathcal{T}\mathrm{M}$.
\begin{proof} Unless stated otherwise, we will evaluate expressions at $(x, v)$, thus, avoid showing variables, if possible, to shorten expressions. Direct computation shows
$$\begin{gathered}
\langle\hat{\sfg}\omega_1,\Pi_{\mathsf{g}} \omega_2\rangle = \langle\Pi_{\mathsf{g}}^{\mathsf{T}}\hat{\sfg}\omega_1, \omega_2\rangle = \alpha_{\|v\|^2_{\mathsf{g}}}\langle\Pi^{\mathsf{T}}_{\mathsf{g}} \mathsf{g}\omega_1, \omega_2\rangle_{\mathcal{E}} + \beta_{\|v\|^2_{\mathsf{g}}} \langle v,\Pi_{\mathsf{g}}^{\mathsf{T}}\mathsf{g}\omega_1\rangle_{\mathcal{E}}\langle\mathsf{g} v, \omega_2\rangle_{\mathcal{E}}\\
= \alpha_{\|v\|^2_{\mathsf{g}}}\langle \mathsf{g}\Pi_{\mathsf{g}}\omega_1, \omega_2\rangle_{\mathcal{E}} + \beta_{\|v\|^2_{\mathsf{g}}} \langle v,\mathsf{g}\omega_1\rangle_{\mathcal{E}}\langle\mathsf{g} v, \omega_2\rangle_{\mathcal{E}}
\end{gathered}$$
where we use self-adjointness of $\mathsf{g}\Pi_{\mathsf{g}}$, and the fact that $v\in \mathcal{T}_x\mathrm{M}$. The last expression is $\langle\hat{\sfg}\Pi_{\mathsf{g}}\omega_1, \omega_2\rangle$, so $\hat{\sfg}\Pi_{\mathsf{g}}$ is self-adjoint. The formula for $\hat{\sfg}^{-1}$ is the operator form of the Sherman-Morrison matrix identity, proved by direct substitution.
We can verify \cref{eq:hsfg_compa} by a direct calculation, for vector fields $\mathtt{X}, \mathtt{Y}, \mathtt{Z}$, noting
$$(\rD_{(\mathtt{X}, -\Gamma(\mathtt{X}, \rU))}\|\rU\|_{\mathsf{g}}^2)_{(x, v)} = 2\langle v, \mathsf{g}_x \Gamma(\mathtt{X}, v)_x\rangle_{\mathcal{E}} + 2\langle v, \mathsf{g}_x(-\Gamma(\mathtt{X}_x, v)_x)\rangle_{\mathcal{E}}=0$$
Hence $(\rD_{(\mathtt{X}, -\Gamma(\mathtt{X}, \rU))}\alpha(\|\rU\|_{\mathsf{g}}^2))_{(x, v)}=(\rD_{(\mathtt{X}, -\Gamma(\mathtt{X}, \rU))}\beta(\|\rU\|_{\mathsf{g}}^2))_{(x, v)}=0$, thus
$$\begin{gathered}(\rD_{(\mathtt{X}, -\Gamma(\mathtt{X}, \rU))}\{\alpha(\|\rU\|_{\mathsf{g}}^2)\langle \mathtt{Y}, \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}} +\beta(\|\rU\|_{\mathsf{g}}^2)\langle \rU, \mathsf{g}\mathtt{Y}\rangle_{\mathcal{E}}\langle \rU, \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}}\})_{(x, v)}=\\
\alpha(\|v\|_{\mathsf{g}, x}^2)\{\langle \nabla_{\mathtt{X}}\mathtt{Y}, \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}} +
\langle \mathtt{Y}, \mathsf{g}\nabla_{\mathtt{X}}\mathtt{Z}\rangle_{\mathcal{E}}\}_x +\\
\beta(\|v\|_{\mathsf{g}, x}^2)\{\langle v, \mathsf{g}\nabla_{\mathtt{X}}\mathtt{Y}\rangle_{\mathcal{E}}\langle v,\mathsf{g} \mathtt{Z}\rangle_{\mathcal{E}}+\langle \Gamma(v, \mathtt{X}), \mathsf{g}\mathtt{Y}\rangle_{\mathcal{E}}\langle v,\mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}}
+\langle -\Gamma(v, \mathtt{X}), \mathsf{g}\mathtt{Y}\rangle_{\mathcal{E}}\langle v,\mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}}\\
+\langle v, \mathsf{g}\mathtt{Y}\rangle_{\mathcal{E}}\langle v, \mathsf{g}\nabla_{\mathtt{X}}\mathtt{Z}\rangle_{\mathcal{E}}
+\langle v, \mathsf{g}\mathtt{Y}\rangle_{\mathcal{E}}\langle \Gamma(\mathtt{X}, v), \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}}
+\langle v, \mathsf{g}\mathtt{Y}\rangle_{\mathcal{E}}\langle -\Gamma(\mathtt{X}, v), \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}} \}_x
\end{gathered}$$
which, writing $\alpha$ and $\beta$ for their values at $\|v\|_{\mathsf{g}, x}^2$, could be rearranged to
$$\{\langle \nabla_{\mathtt{X}}\mathtt{Y},\alpha\mathsf{g}\mathtt{Z} +\beta\langle v, \mathsf{g}\mathtt{Z}\rangle_{\mathcal{E}}v \rangle_{\mathcal{E}} + \langle \alpha\mathsf{g} \mathtt{Y} +\beta\langle v, \mathsf{g}\mathtt{Y}\rangle_{\mathcal{E}} , \nabla_{\mathtt{X}}\mathtt{Z}\rangle_{\mathcal{E}}\}_x
$$
which is the right-hand side of \cref{eq:hsfg_compa}.
For the Christoffel function, we will formulate a more general result in \cref{theo:NaturalSubmerse} and will provide the rest of the proof.
\end{proof}
For a Riemannian submersion, we have the following
\begin{proposition}\label{prop:projH} Let $(\mathrm{M}, \mathfrak{q}, \mathcal{B}, \mathsf{g}, \mathcal{E})$ be a submersed ambient structure of $\mathcal{B}$. By \cref{prop:QHM}, $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}\to\mathcal{T}\mathcal{B}$ is a differentiable submersion, with the vertical bundle $\mathcal{V}\mathcal{H}\mathrm{M}$ and at $(x, v)\in \mathcal{H}\mathrm{M}$, $\mathcal{V}_{(x, v)}\mathcal{H}\mathrm{M}$ consists of vectors of form $(\xi, (\rB_v\xi)_x)\in \mathcal{T}\mathcal{H}\mathrm{M}$, with $\xi\in \mathcal{V}_x\mathrm{M}$ and $\rB_v$ defined in \cref{prop:QHM}. Set $\rB(\phi, v)_x := (\rD_{\ttV\phi}\ttH) v - (\rD_v\ttH)\ttV\phi$ for $\phi\in \mathcal{T}_x\mathrm{M}$, we have $\rB(\epsilon, v)_x = \rB_v\epsilon$ for $\epsilon\in \mathcal{V}_x\mathrm{M}$ and $\rB(\xi, v)_x = 0$ if $\xi\in \mathcal{H}\mathrm{M}$. Extend $\rB$ to a smooth bilinear map from $\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}\otimes \mathcal{E}$, $\mathcal{E})$. Let $\hat{\sfg}$ be a positive-definite operator-valued function from $\mathcal{H}\mathrm{M}$ to $\mathcal{L}(\mathcal{E}, \mathcal{E})$, such that $\hat{\sfg}_{(x_1, v_1)} = \hat{\sfg}_{(x_2, v_2)}$ if $d\mathfrak{q}(x_1, v_1) = d\mathfrak{q}(x_2, v_2)$. Define
\begin{equation}\begin{gathered}
(\Gamma^{\cQ})_{(x, v)}\omega = \Gamma^{\cQ}[v]_x\omega := \GammaH(\ttH\omega, v)_x - \rB(\omega, v)_x
\end{gathered}
\end{equation}
\begin{equation}\label{eq:Hproj}
\mathsf{G}_{\mathcal{Q}, (x, v)} := \begin{bmatrix} \dI & (\Gamma^{\cQ})^{\mathsf{T}}[v]_{x}\\
0 & \dI
\end{bmatrix}\begin{bmatrix} \mathsf{g}_x & 0 \\ 0 & \hat{\sfg}_{(x, v)}\end{bmatrix} \begin{bmatrix} \dI & 0\\
\Gamma^{\cQ}[v]_{x} & \dI
\end{bmatrix}
\end{equation}
Then $\Gamma^{\cQ}$ is an operator-valued function from $\mathcal{H}\mathrm{M}$ to $\mathfrak{L}(\mathcal{E}, \mathcal{E})$ and $\mathsf{G}_{\cQ}$ is a metric operator from $\mathcal{H}\mathrm{M}$ to $\mathcal{E}^2$, defining a Riemannian metric on $\mathcal{H}\mathrm{M}$.
Let $\mathsf{g}_{\mathcal{B}}$ be the metric on $\mathcal{B}$ in the Riemannian submersion. For $(x, v)\in \mathcal{H}\mathrm{M}$, let $(b, w) = d\mathfrak{q}(x, v)
\in \mathcal{T}\mathcal{B}$, $\hat{\sfg}$ induces an inner product $\hat{\sfg}_{\mathcal{B}, (b, w)}$ on $\mathcal{T}_b\mathcal{B}$, evaluated on two tangent vectors $\xi^{\mathcal{B}}$ and $\xi^{\mathcal{B}}\in\mathcal{T}_b\mathcal{B}$ as $\langle \xi^{\mathrm{M}}, \hat{\sfg}_{(x, v)}\eta^{\mathrm{M}}\rangle_{\mathcal{E}}$ where $\xi^{\mathrm{M}}$ and $\eta^{\mathrm{M}}$ are horizontal lifts of $\xi^{\mathcal{B}}$ and $\eta^{\mathcal{B}}$. The inner products $\hat{\sfg}_{\mathcal{B}, (b, w)}$ is well-defined, independent of the lifts. With the associated metric tensors $\mathsf{g}_{\mathcal{B}}$ and $\hat{\sfg}_{\mathcal{B}}$, $\mathcal{T}\mathcal{B}$ could be equipped with the metric $\mathsf{G}_\mathcal{B}$ as defined in \cref{theo:SCGMT}. Under the metrics $\mathsf{G}_{\cQ}$ and $\mathsf{G}_{\mathcal{B}}$, the bundle projection $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}:\mathcal{H}\mathrm{M}\to\mathcal{T}\mathcal{B}$ is a Riemannian submersion.
Let $\ttH_{\mathsf{g}, x}$ and $\ttH_{\hat{\sfg}, (x, v)}$ be the projections of $\mathcal{E}$ to $\mathcal{H}_x\mathrm{M}$ under the inner products $\mathsf{g}_x$ and $\hat{\sfg}_{(x, v)}$, respectively. Under $\mathsf{G}_{\cQ}$, the projection of $\mathcal{E}^2$ to $\mathcal{Q}_x\mathcal{H}\mathrm{M}$ is
\begin{equation}\label{eq:bundl_projH}
\ttH_{\mathsf{G},(x, v)} = \begin{gathered}
\begin{bmatrix} \dI & 0\\
-\Gamma^{\cQ}[v]_{x} & \dI
\end{bmatrix}\begin{bmatrix} \ttH_{\mathsf{g}, x} & 0 \\ 0 & \ttH_{\hat{\sfg}, x}\end{bmatrix}
\begin{bmatrix} \dI & 0\\
\Gamma^{\cQ}[v]_x & \dI
\end{bmatrix}
\end{gathered}
\end{equation}
As a Riemannian metric on $\mathcal{H}\mathrm{M}$, $\mathsf{G}_{\mathcal{Q}}$ is only dependent on the values of $\mathsf{g}$ and $\hat{\sfg}$ evaluated on tangent vectors of $\mathrm{M}$.
The bundle $\mathcal{V}\mathcal{H}\mathrm{M}$ is the vertical bundle, and $\mathcal{Q}\mathcal{H}\mathrm{M}$ is the horizontal bundle of $\mathcal{H}\mathrm{M}$ under the Riemannian submersion $d\mathfrak{q}_{\mathcal{H}\mathrm{M}}$. Let $\ttQ$ be the idempotent map defining $\mathcal{Q}\mathcal{H}\mathrm{M}\subset\mathcal{T}\mathcal{H}\mathrm{M}$ in \cref{prop:QHM}, then $\ttQ$ is the projection from $\mathcal{T}\mathcal{H}\mathrm{M}$ to $\mathcal{Q}\mathcal{H}\mathrm{M}$ under $\mathsf{G}_{\cQ}$, thus $\ttQ$ is the restriction of $\ttH_{\mathsf{G}_{\cQ}}$ to $\mathcal{T}\mathcal{H}\mathrm{M}$.
\end{proposition}
\begin{proof}We will apply \cref{lem:two_func}, with $f_1(\omega_{\mathfrak{m}}, \omega_{\mathfrak{t}}) = \omega_{\mathfrak{m}}, f_2(\omega_{\mathfrak{m}}, \omega_{\mathfrak{t}}) = \Gamma^{\cQ}[v]_x\omega_{\mathfrak{m}} + \omega_{\mathfrak{t}}$, $\mathcal{T}_2 = \mathcal{Q}_x\mathcal{H}\mathrm{M}$, $\mathcal{T} = \mathcal{H}_x\mathrm{M}$. Restricting to $\mathcal{T}_2 = \mathcal{Q}_x\mathcal{H}\mathrm{M}$, $f_1 = (d\pi_{|\mathcal{H}\mathrm{M}})_{(x, v)}$ and $f_2 = \mathrm{C}^{\mathrm{\cQ}}_{(x,v)}$, as by construction $\rB(\eta, v)_x = 0$ for a horizontal vector $\eta$. This gives us the statements that $\mathsf{G}_{\cQ}$ defines a metric under \cref{eq:Hproj}.
Let us show that $\mathcal{V}_x\mathcal{H}\mathrm{M}$ and $\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$ are orthogonal. Consider $(\epsilon, \rB(\epsilon, v)_x)$ in $\mathcal{V}_x\mathcal{H}\mathrm{M}$ where $\epsilon\in\mathcal{V}_x\mathrm{M}$ is a vertical vector, and $\tilde{\eta} = (\eta_{\mathfrak{m}}, \eta_{\mathfrak{t}}) \in \mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$
$$\begin{gathered}\langle (\epsilon, \rB(\epsilon, v)_x), (\mathsf{G}_{\mathcal{Q}})_{(x, v)}\tilde{\eta}\rangle_{\mathcal{E}^2} = \langle (\epsilon, \Gamma^{\cQ}(\epsilon, v)_x +\rB(\epsilon, v)_x), (\mathsf{g}_x\eta_{\mathfrak{m}}, \hat{\sfg}_{(x, v)}(\Gamma^{\cQ}(\eta_{\mathfrak{m}}, v)_x +\eta_{\mathfrak{t}}))\rangle_{\mathcal{E}^2} \\
= \langle (\epsilon, 0), (\mathsf{g}_x\eta_{\mathfrak{m}}, \hat{\sfg}_{(x, v)}(\Gamma^{\cQ}(\eta_{\mathfrak{m}}, v)_x +\eta_{\mathfrak{t}}))\rangle_{\mathcal{E}^2} = 0
\end{gathered} $$
as we have constructed $\Gamma^{\cQ}$ such that $\Gamma^{\cQ}(\epsilon, v)_x +\rB(\epsilon, v)_x = 0$, and $\epsilon$ and $\eta_{\mathfrak{m}}$ are orthogonal by the assumption that $\eta_{\mathfrak{m}}$ is a horizontal vector. By construction, $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$ is an isometry from $\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$ to $\mathcal{T}_{(b, w)}\mathcal{T}\mathcal{B}$, hence $d\mathfrak{q}_{|\mathcal{H}\mathrm{M}}$ is a Riemannian submersion. The statements about the independence of the metric on $\mathcal{B}$ with respect to the lift follow by simple checks, based on the assumptions of $\mathsf{g}$ and $\hat{\sfg}$. The statements about $\ttQ$ and $\mathcal{Q}\mathcal{H}\mathrm{M}$ follow from the submersion property of $\mathsf{G}_{\cQ}$.
\end{proof}
We will recall the lift $\mathrm{b}$ in \cref{fig:HMB} in \cref{eq:rmb} and describe the lifts $\mathrm{h}, \mathrm{v}$.
\begin{definition}\label{def:pqb} For $(x, v)\in \mathcal{H}\mathrm{M}$, let $\eta$ be a horizontal vector at $x\in\mathrm{M}$. Define the $\pi$-horizontal lift $\eta^{\mathrm{h}}:= (\eta,-\Gamma^{\cQ}(\eta, v))\in\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$ , the $\pi$-vertical lift $\eta^{\mathrm{v}}:= (0, \eta)\in\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$. For a vertical vector $\epsilon\in\mathcal{V}_x\mathrm{M}$, recall the $\mathfrak{q}$-vertical lift $\epsilon^{\mathrm{b}} =(\epsilon, (\rB_v\epsilon)_x) = (\epsilon, \rB(\epsilon, v)_x)\in \mathcal{V}_{(x, v)}\mathcal{H}\mathrm{M}$. We define $\pi$-horizontal and $\pi$-vertical lifts of horizontal vector fields, as well as $\mathfrak{q}$-vertical lifts of vertical vector fields on $\mathrm{M}$ similarly. We have $\mathrm{h}$ is a bijection between $\mathcal{H}_x\mathrm{M}$ and the nullspace of $(\mathrm{C}^{\mathrm{\cQ}})_{|\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}}$, $\mathrm{v}$ is a bijection between $\mathcal{H}_x\mathrm{M}$ and the nullspace of $d\pi_{_{|\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}}}$, and $\mathrm{b}_{|\mathcal{V}_x\mathrm{M}}$ is a bijection between $\mathcal{V}_x\mathrm{M}$ and $\mathcal{V}_{(x, v)}\mathcal{H}\mathrm{M}$.
\end{definition}
We have the following bracket formulas for lifts of horizontal vector fields.
\begin{lemma} Let $(\mathrm{M},\mathfrak{q}, \mathcal{B}, \mathsf{g}, \mathcal{E})$ be a submersed ambient structure of the Riemannian submersion $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$ with horizontal bundle $\mathcal{H}\mathrm{M}$. For two horizontal vector fields $\mathtt{X}, \mathtt{Y}$ on $\mathrm{M}$ we have
\begin{equation}\label{eq:Dom1}[\mathtt{X}^{\mathrm{h}}, \mathtt{Y}^{\mathrm{h}}] = (\ttH[\mathtt{X}, \mathtt{Y}])^{\mathrm{h}} +
(\RcH_{\mathtt{X}, \mathtt{Y}}\rU)^{\mathrm{v}} + (\ttV[\mathtt{X}, \mathtt{Y}])^{\mathrm{b}}
\end{equation}
\begin{equation}[\mathtt{X}^{\mathrm{h}}, \mathtt{Y}^{\mathrm{v}}] = (\nabla_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}}
\end{equation}
\begin{equation}[\mathtt{X}^{\mathrm{v}}, \mathtt{Y}^{\mathrm{v}}] = [\mathtt{X}, \mathtt{Y}]^{\mathrm{v}}
\end{equation}
\end{lemma}
\begin{proof}
Similar to \cref{rem:rmv_rmh}, the first component of $\rD_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{h}}$ is $\rD_{\mathtt{X}}\mathtt{Y}$, from here the first component of $[\mathtt{X}^{\mathrm{h}}, \mathtt{Y}^{\mathrm{h}}]$ is $[\mathtt{X}, \mathtt{Y}]$. The second (tangent) component of $\rD_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{h}}$ is
$$-(\rD_{\mathtt{X}}\Gamma^{\cQ})(\mathtt{Y}, \rU) - \Gamma^{\cQ}(\rD_{\mathtt{X}}\mathtt{Y}, \rU)
+ \Gamma^{\cQ}(\mathtt{Y}, \Gamma^{\cQ}(\mathtt{X}, \rU))
$$
where $\rU$ is the canonical vector field. Since $\mathtt{X}$ is horizontal, $\Gamma^{\cQ}(\mathtt{X}, \omega) = \GammaH(\mathtt{X}, \omega)$ for $\omega\in \mathcal{E}$. Changing the role of $\mathtt{Y}$ and $\mathtt{X}$, the tangent component of $[\mathtt{X}^{\mathrm{h}}, \mathtt{Y}^{\mathrm{h}}]$ is
$$-(\rD_{\mathtt{X}}\Gamma^{\cQ})(\mathtt{Y}, \rU) -\Gamma^{\cQ}([\mathtt{X}, \mathtt{Y}], \rU) +\GammaH(\mathtt{Y}, \GammaH(\mathtt{X}, \rU)) +
(\rD_{\mathtt{Y}}\Gamma^{\cQ})(\mathtt{X}, \rU) -\GammaH(\mathtt{X}, \GammaH(\mathtt{Y}, \rU))
$$
We split the left-hand side of \cref{eq:Dom1} to
$([\mathtt{X}, \mathtt{Y}], -\Gamma^{\cQ}([\mathtt{X}, \mathtt{Y}], \rU))$ and $(0, -(\rD_{\mathtt{X}}\Gamma^{\cQ})(\mathtt{Y}, \rU) \\ +\GammaH(\mathtt{Y}, \GammaH(\mathtt{X}, \rU)) +(\rD_{\mathtt{Y}}\Gamma^{\cQ})(\mathtt{X}, \rU) -\GammaH(\mathtt{X}, \GammaH(\mathtt{Y}, \rU))$. As $[\mathtt{X}, \mathtt{Y}]$ is a vector field, using the definition of $\mathrm{h}$ and $\mathrm{b}$,
$$\begin{gathered}
([\mathtt{X}, \mathtt{Y}], -\Gamma^{\cQ}([\mathtt{X}, \mathtt{Y}], \rU)) = (\ttV[\mathtt{X}, \mathtt{Y}]+\ttH[\mathtt{X}, \mathtt{Y}], -\GammaH(\ttH[\mathtt{X}, \mathtt{Y}], \rU)+\rB([\mathtt{X}, \mathtt{Y}], \rU))\\
=(\ttH[\mathtt{X}, \mathtt{Y}])^{\mathrm{h}} + (\ttV[\mathtt{X}, \mathtt{Y}])^{\mathrm{b}}
\end{gathered}$$
as $\rB(\ttH[\mathtt{X}, \mathtt{Y}], \rU)=0$. Expand
$$(\rD_{\mathtt{X}}\Gamma^{\cQ})(\mathtt{Y}, \rU)=(\rD_{\mathtt{X}}\GammaH)(\mathtt{Y}, \rU) +\GammaH((\rD_{\mathtt{X}}\ttH)\mathtt{Y}, \rU)_x - (\rD_{\mathtt{X}}\rB)(\mathtt{Y}, \rU)$$
we need to show the remaining terms below is $\RcH_{\mathtt{X}, \mathtt{Y}}\rU$:
$$\begin{gathered}
- (\rD_{\mathtt{X}}\GammaH)(\mathtt{Y}, \rU)-\GammaH(\mathtt{X}, \GammaH(\mathtt{Y}, \rU))
+(\rD_{\mathtt{Y}}\GammaH)(\mathtt{X}, \rU) +\GammaH(\mathtt{Y}, \GammaH(\mathtt{X}, \rU))\\
-\GammaH((\rD_{\mathtt{X}}\ttH)\mathtt{Y}, \rU)_x +\
\GammaH((\rD_{\mathtt{Y}}\ttH)\mathtt{X}, \rU)_x + (\rD_{\mathtt{X}}\rB)(\mathtt{Y}, \rU) -
(\rD_{\mathtt{Y}}\rB)(\mathtt{X}, \rU)
\end{gathered}$$
In the second line, $((\rD_{\mathtt{X}}\ttH)\mathtt{Y} - (\rD_{\mathtt{Y}}\ttH)\mathtt{X})_x = \ttV_x[\mathtt{X}, \mathtt{Y}]_x$. Compared with \cref{eq:cursubmer}, we need to show
\begin{equation}\label{eq:needtoshow}-\GammaH(\ttV_x[\mathtt{X}, \mathtt{Y}]_x, v)_x -
(\rD_{\mathtt{Y}}\rB)(\mathtt{X}, \rU)_x + (\rD_{\mathtt{X}}\rB)(\mathtt{Y}, \rU)_x= -\GammaH(v, \ttV_x[\mathtt{X}, \mathtt{Y}]_x)_x\end{equation}
as the rightmost expression is $(\mathrm{A}^{\dagger}_{v} \ttV[\mathtt{X}, \mathtt{Y}])_x$. Note that for $\phi\in\mathcal{T}_{(x, v)}\mathcal{H}\mathrm{M}$, $\rB(\phi, v)_x = (\rD_{\ttV\phi}\ttH)_x v - (\rD_v\ttH)_x\ttV\phi$. Hence, set $\xi = \mathtt{X}_x, \eta = \mathtt{Y}_x$
$$\begin{gathered} - (\rD_{\mathtt{Y}}\rB)_x(\xi, v) + (\rD_{\mathtt{X}}\rB)_x(\eta, v) =\\
- \rD_\eta\{(\rD_{\ttV\xi}\ttH)v - (\rD_{v}\ttH)\ttV\xi)\} + \rD_\xi\{(\rD_{\ttV\eta}\ttH)v - (\rD_{v}\ttH)\ttV\eta)\} \\
= -(\rD_{\ttV[\mathtt{X}, \mathtt{Y}]}\ttH)_xv + ((\rD_{v}\ttH)\ttV[\mathtt{X}, \mathtt{Y}])_x
\end{gathered}
$$
Where we have used $(\rD_{\eta}\ttV)_x\xi - (\rD_{\xi}\ttV)_x\eta = (\rD_{\xi}\ttH)_x\eta - (\rD_{\eta}\ttH)_x\xi = 2\mathrm{A}_{\xi}\eta = (\ttV[\mathtt{X}, \mathtt{Y}])_x$. Substitute in \cref{eq:needtoshow} and expand $-\GammaH(\ttV_x[\mathtt{X}, \mathtt{Y}]_x, v)_x = (\rD_{(\ttV[\mathtt{X}, \mathtt{Y}])}\ttH)_x v -\ttH\mathring{\Gamma}(\ttV[\mathtt{X}, \mathtt{Y}], v)_x$ on the left-hand side, we finally get the right-hand side. The rest of the lemma is clear.
\end{proof}
In the following theorem, expressions are evaluated at one point $(x, v)\in\mathcal{H}\mathrm{M}$ under consideration and we will omit the point to keep the expressions compact. We write $\langle\rangle_{\mathsf{g}}, \langle\rangle_{\hat{\sfg}}, \langle\rangle_{\mathsf{G}_{\cQ}}$ for the inner products using the corresponding operators.
\begin{theorem}\label{theo:NaturalSubmerse}Let $\tilde{\nabla}$ be the Levi-Civita covariant derivative with respect to $\mathsf{G}_{\cQ}$, $\ttQ$ be the horizontal projection from $\mathcal{T}\mathcal{H}\mathrm{M}$ to $\mathcal{Q}\mathcal{H}\mathrm{M}$ in \cref{prop:QHM} and $\nabla^{\mathcal{H}}=\ttH\nabla$. At $(x, v)\in \mathcal{H}\mathrm{M}$, for horizontal vector fields $\mathtt{X}$ and $\mathtt{Y}$ on $\mathrm{M}$, we have
\begin{equation}\label{eq:tnablaLift}
\begin{gathered}
\ttQ \tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{h}} = (\nabla^{\mathcal{H}}_{\mathtt{X}} \mathtt{Y})^{\mathrm{h}} + \frac{1}{2}(\RcH_{\mathtt{X}, \mathtt{Y}} v)^{\mathrm{v}}\\
\ttQ\tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}} = (\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}} -\frac{\alpha}{2}(\RcH_{v, \mathtt{Y}}\mathtt{X})^{\mathrm{h}}\\
\ttQ\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{h}} = -\frac{\alpha}{2}(\RcH_{v, \mathtt{X}}\mathtt{Y})^{\mathrm{h}}\\
\ttQ\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{v}} =(\frac{\alpha'}{\alpha}(\mathsf{g}( v, \mathtt{X}) \mathtt{Y} + \mathsf{g}( v, \mathtt{Y})\mathtt{X}) +\rF v)^{\mathrm{v}}\\
\rF := \frac{(\beta-\alpha')\langle\mathtt{X}, \mathtt{Y}\rangle_{\mathsf{g}} + (\beta'-2\alpha'\beta/\alpha)\langle v, \mathtt{X}\rangle_{\mathsf{g}}\langle v, \mathtt{Y}\rangle_{\mathsf{g}}}{\alpha + \| v\|_{\mathsf{g}}^2\beta}
\end{gathered}
\end{equation}
where the scalar functions $\alpha, \alpha', \beta, \beta'$ are evaluated at $\|v\|_{\mathsf{g}}^2$. Let $\tilde{\xi}=(\xi_{\mathfrak{m}}, \xi_{\mathfrak{t}}), \tilde{\eta} =(\eta_{\mathfrak{m}}, \eta_{\mathfrak{t}})$ be two tangent vectors in $\mathcal{Q}_{(x, v)}\mathcal{H}\mathrm{M}$. A horizontal Christoffel function $\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}$ corresponding to the horizontal component $\ttQ\tilde{\nabla}$ of the Levi-Civita connection for the metric $\mathsf{G}_{\cQ}$ is given by
\begin{equation}\label{eq:GammaHsfGDec}\begin{gathered}
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}(\tilde{\xi}, \tilde{\eta}) =
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, \eta_{\mathfrak{m}}^{\mathrm{h}}) +
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, (\mathrm{C}^{\mathrm{\cQ}}\tilde{\eta})^{\mathrm{v}}) +\\
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}((\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi})^{\mathrm{v}}, (\eta_{\mathfrak{m}})^{\mathrm{h}}) +
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}((\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi})^{\mathrm{v}}, (\mathrm{C}^{\mathrm{\cQ}}\tilde{\eta})^{\mathrm{v}})
\end{gathered}
\end{equation}
\begin{equation}\label{eq:GammaHsfGparts}
\begin{gathered}
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, \eta_{\mathfrak{m}}^{\mathrm{h}}) = (\GammaH(\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}), -\Gamma^{\cQ}(\GammaH(\xi_{\mathfrak{m}},\eta_{\mathfrak{m}}), v) + (\rD_{\xi_{\mathfrak{m}}}\Gamma^{\cQ})( \eta_{\mathfrak{m}}, v)-\\\GammaH( \eta_{\mathfrak{m}}, \GammaH(\xi_{\mathfrak{m}}, v)) + \frac{1}{2}\RcH_{\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}} v)\\
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, (\mathrm{C}^{\mathrm{\cQ}}\tilde{\eta})^{\mathrm{v}}) =
(-\frac{\alpha}{2}\RcH_{ v, \mathrm{C}^{\mathrm{\cQ}}\tilde{\eta}}\xi_{\mathfrak{m}}, \frac{\alpha}{2}\GammaH(\RcH_{ v, \mathrm{C}^{\mathrm{\cQ}}\tilde{\eta}}\xi_{\mathfrak{m}}, v) +
\GammaH(\xi_{\mathfrak{m}}, \mathrm{C}^{\mathrm{\cQ}}\tilde{\eta}))\\
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}((\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi})^{\mathrm{v}}, \eta_{\mathfrak{m}}^{\mathrm{h}}) = (-\frac{\alpha}{2}\RcH_{ v,\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi}}\eta_{\mathfrak{m}}, \frac{\alpha}{2}\GammaH(\RcH_{ v,\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi}}\eta_{\mathfrak{m}}, v)
+\GammaH(\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi}, \eta_{\mathfrak{m}}))\\
\Gamma_{\mathsf{G}_{\cQ}}^{\mathcal{H}}((\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi})^{\mathrm{v}}, (\mathrm{C}^{\mathrm{\cQ}}\tilde{\eta})^{\mathrm{v}}) = (0,\frac{\alpha'}{\alpha}\{ \langle v, \mathrm{C}^{\mathrm{\cQ}}\tilde{\xi}\rangle_{\mathsf{g}}\mathrm{C}^{\mathrm{\cQ}}\tilde{\eta} + \langle v, \mathrm{C}^{\mathrm{\cQ}}\tilde{\eta}\rangle_{\mathsf{g}} \mathrm{C}^{\mathrm{\cQ}}\tilde{\xi}\} +\rF v)
\end{gathered}
\end{equation}
where $\rF$ is evaluated from \cref{eq:tnablaLift} with $\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi}, \mathrm{C}^{\mathrm{\cQ}}\tilde{\eta}$ in place of $\mathtt{X}$, $\mathtt{Y}$.
\end{theorem}
Again, the case $\alpha = 1, \beta = 0$ is the case of the Sasaki metric on $\mathcal{T}\mathcal{B}$, the case $\alpha=\beta = (1+t)^{-1}$ is that of the Cheeger-Gromoll metric.
\begin{proof}First, we have the following relations for three horizontal vector fields $\mathtt{X}, \mathtt{Y}, \mathtt{Z}$ on $\mathrm{M}$. We will not repeat the proof (identical to that of Lemma 6.2 of \cite{GudKap}, with the opposite sign convention for $\RcH$, using the Koszul formula):
\begin{equation}\label{eq:qqq}\langle \tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{h}}, \mathtt{Z}^{\mathrm{h}}\rangle_{\mathsf{G}_{\cQ}} = \langle \nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Y}, \mathtt{Z}\rangle_{\mathsf{g}}
\end{equation}
\begin{equation}\label{eq:qqp}2\langle \tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{h}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} = \langle (\RcH_{\mathtt{X}, \mathtt{Y}} v)^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}}
\end{equation}
\begin{equation}\label{eq:qpq}2\langle \tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{h}}\rangle_{\mathsf{G}_{\cQ}} = -\langle (\RcH_{\mathtt{X}, \mathtt{Z}} v)^{\mathrm{v}}, \mathtt{Y}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}}
\end{equation}
\begin{equation}\label{eq:qpp}2\langle \tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} = \rD_{\mathtt{X}^{\mathrm{h}}}\langle\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} - \langle\mathtt{Y}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Z})^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}} +\langle\mathtt{Z}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}}
\end{equation}
\begin{equation}2\label{eq:pqq}\langle\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{h}}, \mathtt{Z}^{\mathrm{h}}\rangle_{\mathsf{G}_{\cQ}} = -\langle (\RcH_{\mathtt{Y}, \mathtt{Z}}v)^{\mathrm{v}}, \mathtt{X}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}}
\end{equation}
\begin{equation}2\label{eq:pqp}\langle\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{h}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} = \rD_{\mathtt{Y}^{\mathrm{h}}}\langle \mathtt{Z}^{\mathrm{v}}, \mathtt{X}^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}} - \langle\mathtt{Z}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{Y}}\mathtt{X})^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} - \langle\mathtt{X}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{Y}}\mathtt{Z})^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}}
\end{equation}
\begin{equation}\label{eq:ppq}2\langle\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{h}}\rangle_{\mathsf{G}_{\cQ}} = -\rD_{\mathtt{Z}^{\mathrm{h}}}\langle\mathtt{X}^{\mathrm{v}}, \mathtt{Y}^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}} + \langle\mathtt{Y}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{Z}}\mathtt{X})^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} +\langle\mathtt{X}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{Z}}\mathtt{Y})^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}}
\end{equation}
\begin{equation}\label{eq:ppp}2\langle\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} = \rD_{\mathtt{X}^{\mathrm{v}}}\langle\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} +\rD_{\mathtt{Y}^{\mathrm{v}}}\langle\mathtt{Z}^{\mathrm{v}}, \mathtt{X}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} -\rD_{\mathtt{Z}^{\mathrm{v}}}\langle\mathtt{X}^{\mathrm{v}}, \mathtt{Y}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}}
\end{equation}
The first equality of \cref{eq:tnablaLift} follows from \cref{eq:qqq,eq:qqp} as $\mathsf{G}_{\cQ}$ and $\mathsf{g}$ are related by a submersion. The remaining equalities are proved in a similar way to Proposition 8.2 of \cite{GudKap} and of theorem 2 of \cite{Abbassi2005}, which we present below. From \cref{eq:qpq}, with $\alpha, \beta$ evaluated at $\| v\|^2_{\mathsf{g}}$
$$2\langle \tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{h}}\rangle_{\mathsf{G}_{\cQ}} = \langle \RcH_{\mathtt{Z}, \mathtt{X}} v, \mathtt{Y}\rangle_{\hat{\sfg}} = \alpha\langle \RcH_{\mathtt{Z}, \mathtt{X}} v, \mathtt{Y} \rangle_{\mathsf{g}} +\beta\langle\mathtt{Y}, v \rangle_{\mathsf{g}}\langle \RcH_{\mathtt{Z}, \mathtt{X}} v, v \rangle_{\mathsf{g}}
$$
which is $-\alpha\langle \RcH_{v, \mathtt{Y}}\mathtt{X}, \mathtt{Z} \rangle_{\mathsf{g}}$, using Bianchi's identities. This gives us the $\mathrm{h}$-component of $\tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}}$. From metric compatibility of $\nabla$, \cref{eq:hsfg_compa} and property of projection
$$\rD_{\mathtt{X}^{\mathrm{h}}}\langle\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} =
\langle(\nabla_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} + \langle\mathtt{Y}^{\mathrm{v}}, (\nabla_{\mathtt{X}}\mathtt{Z})^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} =
\langle(\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} + \langle\mathtt{Y}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Z})^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}}$$
From here and \cref{eq:qpp}, we get $\mathrm{v}$-component of $\tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}}$ because
$$
2\langle \tilde{\nabla}_{\mathtt{X}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} =\langle(\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} + \langle\mathtt{Y}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Z})^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}}
- \langle\mathtt{Y}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Z})^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}} +\langle\mathtt{Z}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{X}}\mathtt{Y})^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}}
$$
We will skip the calculation of $\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{h}}$ as it is similar. Expanding \cref{eq:ppq}, using $\mathsf{G}_{\cQ}$-metric compatibility then use the just proved expressions for $\tilde{\nabla}_{\mathtt{Z}^{\mathrm{h}}}\mathtt{X}^{\mathrm{v}}, \tilde{\nabla}_{\mathtt{Z}^{\mathrm{h}}}\mathtt{Y}^{\mathrm{v}}$, note that the $\mathrm{h}$ and $\mathrm{v}$ components are orthogonal
$$\begin{gathered}2\langle \tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{h}}\rangle_{\mathsf{G}_{\cQ}} = -\langle(\nabla^{\mathcal{H}}_{\mathtt{Z}}\mathtt{X})^{\mathrm{v}}, \mathtt{Y}^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}}
+\frac{\alpha}{2}\langle(\RcH_{v, \mathtt{X}}\mathtt{Z})^{\mathrm{h}}, \mathtt{Y}^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}}
-\langle\mathtt{X}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{Z}}\mathtt{Y})^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}}\\
+\frac{\alpha}{2}\langle \mathtt{X}^{\mathrm{v}}, (\RcH_{v, \mathtt{Y}}\mathtt{Z})^{\mathrm{h}} \rangle_{\mathsf{G}_{\cQ}} + \langle\mathtt{Y}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{Z}}\mathtt{X})^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} +\langle\mathtt{X}^{\mathrm{v}}, (\nabla^{\mathcal{H}}_{\mathtt{Z}}\mathtt{Y})^{\mathrm{v}} \rangle_{\mathsf{G}_{\cQ}} =0
\end{gathered}$$
Next, for any real function $f$, $\rD_{\mathtt{X}^{\mathrm{v}}} f(\|\rU\|^2_{\mathsf{g}}) = 2f'(\|\rU\|^2_{\mathsf{g}})\langle \mathtt{X}, \rU\rangle_{\mathsf{g}}$. Write $\alpha, \beta, \alpha', \beta'$ for their values at $\|v\|^2_{\mathsf{g}}$ at the horizontal tangent point $(x, v)$:
$$\begin{gathered}\rD_{\mathtt{X}^{\mathrm{v}}}\langle\mathtt{Y}^{\mathrm{v}}, \mathtt{Z}^{\mathrm{v}}\rangle_{\mathsf{G}_{\cQ}} =
\rD_{\mathtt{X}^{\mathrm{v}}}\{\alpha(\|v\|^2_{\mathsf{g}})\langle\mathtt{Y}, \mathtt{Z}\rangle_{\mathsf{g}} +
\beta(\|v\|^2_{\mathsf{g}})\langle\mathtt{Y}, v\rangle_{\mathsf{g}}\langle \mathtt{Z}, v\rangle_{\mathsf{g}}
\} =\\
\{2\alpha'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, \mathtt{Z}\rangle_{\mathsf{g}} +
2\beta'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, v\rangle_{\mathsf{g}}\langle \mathtt{Z}, v\rangle_{\mathsf{g}} +
\beta\{\langle\mathtt{Y}, \mathtt{X}\rangle_{\mathsf{g}}\langle \mathtt{Z}, v\rangle_{\mathsf{g}} +
\langle\mathtt{Y}, v\rangle_{\mathsf{g}}\langle \mathtt{Z}, \mathtt{X}\rangle_{\mathsf{g}}\}
\end{gathered}
$$
By \cref{eq:hsfginv}, $\hat{\sfg}^{-1}\mathsf{g}\omega = \frac{1}{\alpha}\omega - \frac{\beta}{\alpha(\alpha +\|v\|^2_{\mathsf{g}}\beta)}\langle v,\mathsf{g}\omega \rangle_{\mathcal{E}}v$, in particular, $\hat{\sfg}^{-1}\mathsf{g} v = \frac{1}{\alpha + \|v\|^2_{\mathsf{g}}\beta}v$. Thus, from \cref{eq:ppp}, with two permutations of the above equality, the $\mathrm{v}$-component of $\ttQ\tilde{\nabla}_{\mathtt{X}^{\mathrm{v}}}\mathtt{Y}^{\mathrm{v}}$ is
$$\begin{gathered}\frac{1}{2}\hat{\sfg}^{-1}\mathsf{g}\{
2\alpha'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\mathtt{Y} +
2\beta'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, v\rangle_{\mathsf{g}}v +
\beta\langle\mathtt{Y}, \mathtt{X}\rangle_{\mathsf{g}} v +
\beta\langle\mathtt{Y}, v\rangle_{\mathsf{g}}\mathtt{X} +\\
2\alpha'\langle\mathtt{Y}, v \rangle_{\mathsf{g}}\mathtt{X} +
2\beta'\langle\mathtt{Y}, v \rangle_{\mathsf{g}}\langle\mathtt{X}, v\rangle_{\mathsf{g}}v +
\beta\langle\mathtt{X}, \mathtt{Y}\rangle_{\mathsf{g}} v +
\beta\langle\mathtt{X}, v\rangle_{\mathsf{g}}\mathtt{Y}
-\\
2\alpha'\langle\mathtt{Y}, \mathtt{X}\rangle_{\mathsf{g}}v -
2\beta'\langle\mathtt{Y}, v\rangle_{\mathsf{g}}\langle \mathtt{X}, v\rangle_{\mathsf{g}}v -
\beta\langle \mathtt{X}, v\rangle_{\mathsf{g}}\mathtt{Y} -
\beta\langle\mathtt{Y}, v\rangle_{\mathsf{g}}\mathtt{X}\}=\\
\hat{\sfg}^{-1}\mathsf{g}\{\alpha'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\mathtt{Y} + \alpha'\langle\mathtt{Y}, v \rangle_{\mathsf{g}}\mathtt{X} + (\beta'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, v\rangle_{\mathsf{g}} +
\beta\langle\mathtt{Y}, \mathtt{X}\rangle_{\mathsf{g}}-\alpha'\langle \mathtt{Y}, \mathtt{X}\rangle_{\mathsf{g}})v
\}=\\
\hat{\sfg}^{-1}\mathsf{g}\{\alpha'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\mathtt{Y} + \alpha'\langle\mathtt{Y}, v \rangle_{\mathsf{g}}\mathtt{X}\} + (\beta'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, v\rangle_{\mathsf{g}} +
\beta\langle\mathtt{Y}, \mathtt{X}\rangle_{\mathsf{g}}-\alpha'\langle \mathtt{Y}, \mathtt{X}\rangle_{\mathsf{g}})\hat{\sfg}^{-1}\mathsf{g} v\\
=\frac{\alpha'}{\alpha}(\langle\mathtt{X}, v \rangle_{\mathsf{g}}\mathtt{Y} + \langle\mathtt{Y}, v \rangle_{\mathsf{g}}\mathtt{X}) -\frac{\beta\alpha'}{\alpha(\alpha +\|v\|^2_{\mathsf{g}}\beta)}(2\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, v\rangle_{\mathsf{g}})v\\
+\frac{1}{\alpha + \|v\|^2_{\mathsf{g}}\beta}(\beta'\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, v\rangle_{\mathsf{g}} +
(\beta-\alpha')\langle\mathtt{X}, \mathtt{Y}\rangle_{\mathsf{g}})v\\
=\frac{\alpha'}{\alpha}(\langle\mathtt{X}, v \rangle_{\mathsf{g}}\mathtt{Y} + \langle\mathtt{Y}, v \rangle_{\mathsf{g}}\mathtt{X}) +
\frac{
(\beta'-2\beta\alpha'/\alpha)\langle\mathtt{X}, v \rangle_{\mathsf{g}}\langle\mathtt{Y}, v\rangle_{\mathsf{g}}
+ (\beta-\alpha')\langle\mathtt{X}, \mathtt{Y}\rangle_{\mathsf{g}}}{\alpha +\|v\|^2_{\mathsf{g}}\beta}v
\end{gathered}
$$
This completes the proof of \cref{eq:tnablaLift}. To compute $\Gamma_{\mathsf{G}}^{\mathcal{H}}$, we have \cref{eq:GammaHsfGDec} by linearity. Fix $(x, v)\in \mathcal{H}\mathrm{M}$. Let $q_{\tilde{\xi}}, r_{\tilde{\xi}}, q_{\tilde{\eta}}, r_{\tilde{\eta}}$ be horizontal vector fields on $\mathrm{M}$ by defining $q_{\tilde{\xi}}(y) = \ttH_y\xi_{\mathfrak{m}}, r_{\tilde{\xi}}(y)= \ttH_y(\xi_{\mathfrak{t}}+\GammaH(\xi_{\mathfrak{m}}, v)_x), q_{\tilde{\eta}}(y) = \ttH_y\eta_{\mathfrak{m}}, r_{\tilde{\eta}}(y)= \ttH_y(\eta_{\mathfrak{t}}+\GammaH(\eta_{\mathfrak{m}}, v)_x)$ for $y\in\mathrm{M}$, then $\xi_{\mathfrak{m}}^{\mathrm{h}}, (\mathrm{C}^{\mathrm{\cQ}}_{(x, v)}\tilde{\xi})^{\mathrm{v}}, \eta_{\mathfrak{m}}^{\mathrm{h}}, (\mathrm{C}^{\mathrm{\cQ}}_{(x, v)}\tilde{\eta})^{\mathrm{v}}$ are $q_{\tilde{\xi}}^{\mathrm{h}}, r_{\tilde{\xi}}^{\mathrm{v}}, q_{\tilde{\eta}}^{\mathrm{h}}, r_{\tilde{\eta}}^{\mathrm{v}}$ evaluated at $(x, v)$. We have
$$\begin{gathered}(\rD_{q_{\tilde{\xi}}^{\mathrm{h}}}q_{\tilde{\eta}}^{\mathrm{h}})_{(x, v)} = \rD_{q_{\tilde{\xi}}^{\mathrm{h}}}(q_{\tilde{\eta}}, -\Gamma^{\cQ}(q_{\tilde{\eta}}, \rU))_{(x, v)} =\\
((\rD_{q_{\tilde{\xi}}}q_{\tilde{\eta}})_x, -(\rD_{\xi_{\mathfrak{m}}}\Gamma^{\cQ})_x(\eta_{\mathfrak{m}}, v) -\Gamma^{\cQ}((\rD_{q_{\tilde{\xi}}}q_{\tilde{\eta}})_x, v)_x+\Gamma^{\cQ}( \eta_{\mathfrak{m}}, \Gamma^{\cQ}(\xi_{\mathfrak{m}}, v))_x)
\end{gathered}$$
Using \cref{eq:tnablaLift}, $(\ttQ\tilde{\nabla}_{q_{\tilde{\xi}}^{\mathrm{h}}}q_{\tilde{\eta}}^{\mathrm{h}})_{(x, v)} = (\nabla^{\mathcal{H}}_{q_{\tilde{\xi}}}q_{\tilde{\eta}}, -\Gamma^{\cQ}(\nabla^{\mathcal{H}}_{q_{\tilde{\xi}}}q_{\tilde{\eta}}, v) +\frac{1}{2}\RcH_{\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}}v)_x$, thus
$$\begin{gathered}
\Gamma^{\mathcal{H}}_{\mathsf{G}_{\cQ}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, \eta_{\mathfrak{m}}^{\mathrm{h}})_{(x, v)} =
(\ttQ\tilde{\nabla}_{q_{\tilde{\xi}}^{\mathrm{h}}}q_{\tilde{\eta}}^{\mathrm{h}} - \rD_{q_{\tilde{\xi}}^{\mathrm{h}}}q_{\tilde{\eta}}^{\mathrm{h}})_{(x, v)}=
((\nabla^{\mathcal{H}}_{q_{\tilde{\xi}}}q_{\tilde{\eta}} - \rD_{q_{\tilde{\xi}}^{\mathrm{h}}}q_{\tilde{\eta}})_{(x, v)},\\
-\Gamma^{\cQ}(\nabla^{\mathcal{H}}_{q_{\tilde{\xi}}}q_{\tilde{\eta}} - \rD_{q_{\tilde{\xi}}^{\mathrm{h}}}q_{\tilde{\eta}}, v) +\frac{1}{2}\RcH_{\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}}v +
(\rD_{\xi_{\mathfrak{m}}}\Gamma^{\cQ})(\eta_{\mathfrak{m}}, v)-\GammaH(\eta_{\mathfrak{m}}, \GammaH(\xi_{\mathfrak{m}}, v)))
\\
=(\GammaH(\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}),\\
-\Gamma^{\cQ}(\GammaH(\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}), v) +
(\rD_{\xi_{\mathfrak{m}}}\Gamma^{\cQ})(\eta_{\mathfrak{m}}, v)-\GammaH(\eta_{\mathfrak{m}}, \GammaH(\xi_{\mathfrak{m}}, v)) +
\frac{1}{2}\RcH_{\xi_{\mathfrak{m}}, \eta_{\mathfrak{m}}}v)
\end{gathered}
$$
where expressions are evaluated at $(x, v)$. This gives us the first equation in \cref{eq:GammaHsfGparts}. Similarly, the expression for $\Gamma^{\mathcal{H}}_{\mathsf{G}_{\cQ}}(\xi_{\mathfrak{m}}^{\mathrm{h}}, (\mathrm{C}^{\mathrm{\cQ}}\tilde{\eta})^{\mathrm{v}})_{(x,v)}$ follows from
$$(\ttQ\nabla_{q_{\tilde{\xi}}^{\mathrm{h}}}r_{\tilde{\eta}}^{\mathrm{v}})_{(x, v)} =(-\frac{\alpha}{2}\rR_{v, \mathrm{C}^{\mathrm{\cQ}}\tilde{\eta}}\xi_{\mathfrak{m}},
\frac{\alpha}{2}\GammaH(\RcH_{v, \mathrm{C}^{\mathrm{\cQ}}\tilde{\eta}}\xi_{\mathfrak{m}}, v) + (\nabla^\mathcal{H}_{q_{\tilde{\xi}}}r_{\tilde{\eta}}))_x$$
$$(\rD_{q_{\tilde{\xi}}^{\mathrm{h}}}r_{\tilde{\eta}}^{\mathrm{v}})_{(x, v)} = (0, (\rD_{q_{\tilde{\xi}}}r_{\tilde{\eta}})_x)$$
Then, the formula for $\Gamma^{\mathcal{H}}_{\mathsf{G}_{\cQ}}((\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi})^{\mathrm{v}}, \eta_{\mathfrak{m}}^{\mathrm{h}})$ follows from
$$(\rD_{r_{\tilde{\xi}}^{\mathrm{v}}}q_{\tilde{\eta}}^{\mathrm{h}})_{(x, v)} = (0, -\GammaH(\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi}, \eta_{\mathfrak{m}}))_{(x, v)}$$
$$(\ttQ\nabla_{r_{\tilde{\xi}}^{\mathrm{v}}}q_{\tilde{\eta}}^{\mathrm{h}})_{(x, v)} = -\frac{\alpha}{2}(\RcH_{v,\mathrm{C}^{\mathrm{\cQ}}\xi_{\mathfrak{m}}}\tilde{\eta})^{\mathrm{h}}_{(x, v)}$$
and $(\rD_{r_{\tilde{\xi}}^{\mathrm{v}}}r_{\tilde{\eta}}^{\mathrm{v}})_{(x, v)}=0$ gives us the formula for $\Gamma^{\mathcal{H}}_{\mathsf{G}_{\cQ}}((\mathrm{C}^{\mathrm{\cQ}}\tilde{\xi})^{\mathrm{v}}, (\mathrm{C}^{\mathrm{\cQ}}\tilde{\eta})^{\mathrm{v}})$.
\end{proof}
\section{Application to Grassmann manifold}\label{sec:grass}
The Grassmann manifold could be considered as the simplest example of a flag manifold, that we have realized as a quotient of the orthogonal group. Following \cite{Edelman_1999}, we will construct it as a quotient of the Stiefel manifold. The horizontal projection and Levi-Civita connection are well-known and will be reviewed briefly. We will show the computation of curvature, Jacobi fields and horizontal bundle metric in this section.
Let $n > p > 0$ be two positive integers. Recall a Stiefel manifold could be considered as a submanifold of $\mathbb{R}^{n\times p}$ of matrices satisfying the equation $Y^{\mathsf{T}}Y = \dI_p$, $Y\in \mathbb{R}^{n\times p}$. We will use the embedded metric on the Stiefel manifold $\mathsf{g}\omega = \omega$ for $\omega\in\mathcal{E}=\mathbb{R}^{n\times p}$.
For an ambient vector $\omega \in \mathcal{E} = \mathbb{R}^{n\times p}$, the projection to the tangent space of Stiefel manifold at $Y$ is $\Pi\omega = \omega-\frac{1}{2}(YY^{\ft}\omega + Y\omega^{\ft}Y)$ \cite{Edelman_1999}, and $\mathring{\Gamma} = 0$.
The Grassmann manifold $\Gr{p}{n}$ could be considered as the quotient of $\St{p}{n}$ by a right action of the orthogonal group $\OO(p)$, that is under the equivalence $YQ\sim Y$ for $Y\in \St{p}{n}$ and $Q \in \OO(p)$. We will use the notation $\llbracket Y\rrbracket$ to denote the equivalent class of $Y$. Thus, in our convention, $\mathrm{M} = \St{p}{n}$ and $\mathcal{B}= \Gr{p}{n}$, and we have a submersion $\mathfrak{q}:\mathrm{M}\to\mathcal{B}$. In this submersion, the vertical space consists of vectors $Yb$, where $b = -b^{\mathsf{T}}\in \mathfrak{o}(p)$. The vertical projection is therefore $\ttV\omega = \frac{1}{2}Y(Y^{\mathsf{T}}\omega-\omega^{\mathsf{T}}Y)$ using \cref{lem:projprop}, with the constant metric $\mathsf{g}\omega = \omega$ and the map $\mathrm{N}:\mathfrak{o}(p)\to\mathbb{R}^{n\times p}=:\mathcal{E}$, $\mathrm{N}:b\mapsto Yb$ for $b\in\mathfrak{o}(p)$, and $\mathrm{N}^{\mathsf{T}}\omega = \frac{1}{2}(Y^{\mathsf{T}}\omega - \omega^{\mathsf{T}}Y)$. Hence, the projection to the horizontal space is $\ttH\omega = (\Pi - \ttV)\omega = \omega-YY^{\ft}\omega$. A horizontal vector $\eta$ satisfies $Y^{\mathsf{T}}\eta = 0$. We try to keep the formulas compact and omit explicit subscripting $Y$ for $\Pi, \ttH$ and $\ttV$.
If $\xi$ is tangent to the Stiefel manifold and $\omega \in \mathcal{E}$, $\GammaH(\xi, \omega) = -(\rD_{\xi}\ttH)\omega$
$$\GammaH(\xi, \omega) = Y\xi^{\ft}\omega +\xi Y^{\ft}\omega$$
and for horizontal vectors $\xi, \eta, \phi$, $Y^{\ft}\xi = Y^{\ft}\eta= Y^{\ft}\phi =0$, from \cref{eq:rAV}
\begin{equation}\label{eq:rA_grass}\mathrm{A}_{\xi}\eta =-(\rD_{\xi}\ttV)\eta = -\frac{1}{2}(Y\xi^{\ft}\eta +\xi Y^{\ft}\eta -\xi\eta^{\ft} Y - Y\eta^{\ft}\xi)
=-\frac{1}{2}Y(\xi^{\ft}\eta - \eta^{\ft}\xi)\end{equation}
$$\mathrm{A}^{\dagger}_{\phi}\mathrm{A}_{\xi}\eta = -\GammaH(\phi,-\frac{1}{2}Y(\xi^{\ft}\eta - \eta^{\ft}\xi))=
(Y\phi^{\ft} +\phi Y^{\ft})\{\frac{1}{2}Y(\xi^{\ft}\eta - \eta^{\ft}\xi)\}=\frac{1}{2}(\phi\xi^{\ft}\eta-\phi\eta^{\ft}\xi)
$$
$$(\rD_{\xi}\GammaH)(\eta, \phi) = \xi\eta^{\ft}\phi +\eta\xi^{\ft}\phi$$
Thus $-(\rD_{\xi}\GammaH)(\eta, \phi) + (\rD_{\eta}\GammaH)(\xi, \phi) =0$ and
$\GammaH(\xi, \GammaH(\eta, \phi)) = (Y\xi^{\ft} +\xi Y^{\ft})(Y\eta^{\ft}\phi +\eta Y^{\ft}\phi)= \xi\eta^{\ft}\phi$. We now get the classical curvature formula for $\Gr{p}{n}$:
\begin{proposition}
Let $\xi, \eta, \phi$ be three horizontal vectors at $Y\in\St{p}{n}$ as horizontal lifts of tangent vectors at $\llbracket Y\rrbracket \in \Gr{p}{n}$. The lift of the Riemannian curvature tensor of $\Gr{p}{n}$ to $\St{p}{n}$ is given by:
\begin{equation}\label{eq:r13_grass}\RcH_{\xi, \eta}\phi= -\xi\eta^{\ft}\phi + \eta\xi^{\ft}\phi +\phi\xi^{\ft}\eta-\phi\eta^{\ft}\xi\end{equation}
If $(Y_{\perp} | Y) \in \OO(n)$ and $\xi = Y_{\perp} B_1, \eta = Y_{\perp} B_2$, we have $\htK(\xi, \eta) = \langle\hcR_{\xi, \eta}\xi, \eta\rangle_{\mathsf{g}}$ is
\begin{equation}\label{eq:sec_grass_cur}
\begin{gathered}
\htK_{\mathcal{H}}(\xi, \eta) = \TrR(B_1B_1^{\ft}B_2B_2^{\ft} + B_2B_1^{\ft}B_1B_2^{\ft} - 2B_1B_2^{\ft}B_1B_2^{\ft}) \\= ||B_2^{\ft}B_1 - B_1^{\ft}B_2||_F^2 + ||B_1B_2^{\ft} - B_2B_1^{\ft}||_F^2
\end{gathered}
\end{equation}
\end{proposition}
Note that the expression for $\htK_{\mathcal{H}}$ is dependent on $Y_{\perp}\Yperp^{\mathsf{T}} = \dI_n - YY^{\mathsf{T}}$, not on the choice of $Y_{\perp}$. Without using \cref{eq:cursubmer} or \cref{eq:flag_curv}, the curvatures could be derived from the theory of symmetric spaces, where the above expression comes from the Lie bracket $[\tilde{B}_3[\tilde{B}_1\tilde{B}_2]]$ in the embedding $B_i\mapsto \tilde{B}_i=\begin{bmatrix}0 & -B_i^{\ft}\\B_i & 0\end{bmatrix}\in \mathfrak{o}(n)$.
\begin{proof}Equation (\ref{eq:r13_grass}) follows from the preceding calculation and \cref{eq:cursubmer}, which reduces to $2\mathrm{A}^{\dagger}_{\phi}\mathrm{A}_{\xi}\eta -\GammaH(\xi\GammaH(\eta, \phi)) + \GammaH(\eta, \GammaH(\xi, \phi))$, or $\phi(\xi^{\ft}\eta - \eta^{\ft}\xi) -\xi\eta^{\mathsf{T}}\phi + \eta\xi^{\mathsf{T}}\phi$.
Let $\xi=Y_{\perp}B_1$, $\eta=Y_{\perp}B_2, \phi=Y_{\perp}B_3$, $B_1, B_2, B_3\in \mathbb{R}^{(n-p)\times p}$. Then
$$Y_{\perp}\RcH_{\xi, \eta} \phi= -B_1B_2^{\ft}B_3 + B_2B_1^{\ft}B_3 + B_3B_1^{\ft}B_2-B_3B_2^{\ft}B_1$$
The sectional curvature numerator in \cref{eq:sec_grass_cur} follows from a substitution.
\end{proof}
We now describe the horizontal bundle $\mathcal{H}\St{p}{n}\subset \mathcal{T}\St{p}{n}$ of the submersion $\mathfrak{q}:\St{p}{n}\to\Gr{p}{n}$ and its structure as in \cref{sec:subm_tangent}. $\mathcal{H}\St{p}{n}$ could be identified with a submanifold of $(\mathbb{R}^{n\times p})^2$ of pairs of matrices $(Y, \eta)$ satisfying $Y^{\mathsf{T}}Y=\dI_p$, $Y^{\mathsf{T}}\eta = 0$. Its tangent bundle could be considered as a quadruple $(Y, \eta, \Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ with $\eta$ a horizontal vector, $\Delta_{\mathfrak{m}}$ a Stiefel-tangent vector, thus $Y^{\mathsf{T}}\Delta_{\mathfrak{m}} + \Delta^{\mathsf{T}}_{\mathfrak{m}}Y = 0$, and $\Delta_{\mathfrak{m}}^{\mathsf{T}} \eta + Y^{\mathsf{T}}\Delta_{\mathfrak{t}} = 0$. For $b\in\mathfrak{o}(p)$, $(\rD_{Yb}\ttH)\eta=0$ and the operator $\rB$ in \cref{prop:QHM} is
$$\rB(Yb, \eta) = (\rD_{Yb}\ttH)\eta - (\rD_{\eta}\ttH) Yb = \eta Y^{\mathsf{T}}Y b = \eta b$$
as expected. The extension $\rB(\phi, \eta) = \eta Y^{\mathsf{T}}\phi$ satisfies $\rB(\phi, \eta) = 0$ if $\phi$ is horizontal and $\rB(Yb, \eta)= \eta b$, and we will use this expression to extend $\rB$ to $\mathcal{E}^2$. The vertical bundle $\mathcal{V}\mathcal{H}\mathrm{M}$ consists of quadruples $(Y, \eta, Yb, \eta b)$, while $\mathcal{Q}\mathcal{H}\mathrm{M}$ consists of tuples $(Y, \eta, \delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ with the same relation as $(\Delta_{\mathfrak{m}}, \Delta_{\mathfrak{t}})$ but now $\delta_{\mathfrak{m}}$ is horizontal. The connection map sends ($\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ to $\delta_{\mathfrak{c}} = \delta_{\mathfrak{t}} + Y\delta_{\mathfrak{m}}^{\mathsf{T}}\eta$. A horizontal vector in $\mathcal{Q}\mathcal{H}\mathrm{M}$ is thus of the form $(Y, \eta, \delta_{\mathfrak{m}}, \delta_{\mathfrak{c}} -Y\delta_{\mathfrak{m}}^{\mathsf{T}}\eta)$ for three tangent vectors $\eta, \delta_{\mathfrak{m}}, \delta_{\mathfrak{c}}$.
In this Stiefel coordinate, from \cref{prop:frjH} and \cref{eq:rA_grass}, the canonical flip maps $(Y, \eta, \delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ to $(Y, \delta_{\mathfrak{m}}, \eta, \delta_{\mathfrak{t}} + Y(\delta_{\mathfrak{m}}^{\mathsf{T}}\eta-\eta^{\mathsf{T}}\delta_{\mathfrak{m}}))$. From \cref{theo:jacsub}
\begin{proposition} Let $\csr(z)$ and $\ssr z$ be analytic continuations of $\cos z^{1/2}$ and $z^{-1/2} \sin z^{1/2}$ to entire functions. For $Y\in\St{p}{n}$ and $\eta$ a tangent vector at $Y$, a geodesics for the Grassmann manifold $\Gr{p}{n}$ lifts to a horizontal geodesics $\gamma(t)$ on the Stiefel manifold $\St{p}{n}$, with $(\gamma(0), \dot{\gamma}(0)) = (Y, \eta)$ as:
\begin{equation}\label{eq:exp_grass}
\gamma(t) = \Exp_{Y}t\eta = Y \csr t^2\eta^{\ft}\eta + t\eta\ssr t^2 \eta^{\ft}\eta
\end{equation}
The tangent component of the horizontal lift $\mathfrak{J}^{\mathcal{H}}(t)=(\gamma(t), J^{\mathcal{H}}(t))$ of a Jacobi field on $\Gr{p}{n}$ to $\St{p}{n}$ with initial data $\dot{\mathfrak{J}}^{\mathcal{H}}(0) = (Y, \nu_{\mathfrak{m}}, \eta, \nu_{\mathfrak{t}})\in\mathcal{Q}\mathcal{H}\St{p}{n}$ is
\begin{equation}
\begin{gathered}
J^{\mathcal{H}}(t) = (\dI_n - \gamma(t)\gamma(t)^{\ft})\{
\nu_{\mathfrak{m}}\csr t^2\eta^{\ft}\eta +
t\delta_{\mathfrak{t}}\ssr(t^2\eta^{\ft}\eta)
+\\ t^2 Y \frL_{\csr}(t^2\eta^{\ft}\eta, \eta^{\ft}\delta_{\mathfrak{t}} + \delta_{\mathfrak{t}}^{\ft}\eta) + t^3\eta \frL_{\ssr}(t^2\eta^{\ft}\eta, \eta^{\ft}\delta_{\mathfrak{t}} + \delta_{\mathfrak{t}}^{\ft}\eta)\}
\end{gathered}
\end{equation}
where $\delta_{\mathfrak{t}} = \nu_{\mathfrak{t}} - Y(\nu_{\mathfrak{m}}^{\mathsf{T}}\eta - \eta^{\mathsf{T}}\nu_{\mathfrak{m}})$ and for an analytic function $f(z) = \sum_{i=0}^\infty f_i z^i$, $\frL_f(A, E)$ denotes the Fr{\'e}chet derivative $\sum_{i=1}^\infty f_i (\sum_{b+c=i-1}A^bEA^c)$.
\end{proposition}
We will review Fr{\'e}chet derivatives in \cref{sec:frechet}. As mentioned, when $f(x)=\exp(x)$, $\frL_{\exp}(A, E) = \exp(tA)((1-\exp(-x))/x)_{x=t\ad_A}E$. Fr{\'e}chet derivatives have the advantage that it is defined for all differentiable functions, and for analytic functions it has about three times the computational complexity of evaluating $f(A)$, so for any practical purpose it could be considered as a closed-form expression. We hope the expression of Jacobi fields in terms of Fr{\'e}chet derivatives is also useful theoretically. The Fr{\'e}chet derivatives $\frL_{\csr}$ and $\frL_{\ssr}$ are not available in numerical packages but are simple to implement. It is easy to verify the initial condition $J^{\mathcal{H}}(0) = (\dI_n-YY^{\mathsf{T}})\nu_{\mathfrak{m}} = \nu_{\mathfrak{m}}$, and $\dot{J}^{\mathcal{H}}(0) = -(Y\eta^{\mathsf{T}} +\eta Y^{\mathsf{T}})\nu_{\mathfrak{m}} + (\dI_n-YY^{\mathsf{T}})\delta_{\mathfrak{t}} =\delta_{\mathfrak{c}} - Y\eta^{\mathsf{T}}\nu_{\mathfrak{m}} = \delta_{\mathfrak{t}} + Y\nu_{\mathfrak{m}}^{\mathsf{T}}\eta - Y\eta^{\mathsf{T}}\nu_{\mathfrak{m}}=\nu_{\mathfrak{t}}$.
\begin{proof}The formula for $\gamma(t)$ is proved in \cite{NguyenGeodesic}, or follows from either direct substitution of \cref{eq:exp_grass} to the geodesic equation for the lift of the Grassmann, which is $\ddot{\gamma} + \gamma\dot{\gamma}^\mathsf{T}\dot{\gamma} = 0$ (a calculation similar to verification the geodesic equation of a sphere), or from the first $p$ columns of $U\exp t\hat{\eta}$ (the geodesic for $SO(n)$), where $U = (Y | Y_{\perp}^{\mathsf{T}})$, $\hat{\eta} = \begin{pmatrix} 0 & -\eta^{\mathsf{T}}\\ \eta & 0\end{pmatrix}$, breaking $\exp t\hat{\eta}$ to even and odd powers (note the horizontal geodesics are the same for the induced metric $\mathsf{g}_i\eta = \eta$ and the canonical metric $\mathsf{g}_c\eta = \eta - \frac{1}{2}YY^{\mathsf{T}}\eta$, as $Y^{\mathsf{T}}\eta = 0$).
The expression for $J^{\mathcal{H}}(t)$ is just the horizontal projection of the directional derivative of $\gamma(Y, \eta; t)$ in the direction $(\delta_{\mathfrak{m}}, \delta_{\mathfrak{t}})$ defined by the canonical flip.
\end{proof}
For the natural metric on $\mathcal{H}\St{p}{n}$ corresponding to the submersion to the Grassmann manifold, at a point $(Y, V)\in \mathcal{H}\St{p}{n}$, for $\omega\in \mathbb{R}^{n\times p}$ we set $\Gamma^{\cQ}(\omega, V)_Y = \GammaH(\ttH_Y\omega, V) - \rB(\omega, V) = Y\omega^{\mathsf{T}}V - V Y^{\mathsf{T}}\omega$ from \cref{eq:Hproj} and extend it to an operator on $\mathcal{E}^2$. This expression could be used to evaluate the metric $\mathsf{G}_{\cQ}$ in \cref{eq:Hproj}, the projection in \cref{eq:bundl_projH}. To evaluate the Christoffel function at $(Y, V)\in \mathcal{H}\St{p}{n}$ for horizontal vectors $\tilde{\xi}=(\xi_{\mathfrak{m}}, \xi_{\mathfrak{t}}), \tilde{\eta}=(\eta_{\mathfrak{m}}, \eta_{\mathfrak{t}})\in \mathcal{Q}_{(Y, V)}\mathcal{H}\St{p}{n}$, in \cref{eq:GammaHsfGparts}, with the curvature known, we can use
\begin{equation}(\rD_{\xi_{\mathfrak{m}}}\Gamma^{\cQ})( \eta_{\mathfrak{m}}, V)=\xi_{\mathfrak{m}}\eta_{\mathfrak{m}}^{\mathsf{T}}V - V \xi_{\mathfrak{m}}^{\mathsf{T}}\eta_{\mathfrak{m}}\end{equation}
\begin{equation}\Gamma^{\cQ}(\GammaH(\xi_{\mathfrak{m}},\eta_{\mathfrak{m}}), V) = - V \xi_{\mathfrak{m}}^{\mathsf{T}}\eta_{\mathfrak{m}}\end{equation}
\section{Discussion} We have demonstrated differential geometric measures of a Riemannian manifold could be computed effectively using a metric operator if the manifold is embedded in a Euclidean space, or if it is a submersed image of such manifold, and have derived several new results using this approach. We believe the approach could be effective for other types of geometries, for example, Finsler geometry or generalized complex geometry. Jacobi field and tangent bundle metrics appear in the problem of geodesic regression in computer vision \cite{MachaLeite,Niethammer2011,Fletcher2013}, thus our present work presents an approach to evaluate them for common manifolds. We hope researchers, both applied and pure mathematics will find the approach useful in their future works.
\begin{appendix}
\section{Fr{\'e}chet Derivative}\label{sec:frechet}Recall \cite{Higham,Mathias,Havel} if $f(A)=\sum_{i=0}^{\infty}f_iA^i$ is a power series with scalar coefficients and $A$ is a square matrix, then the Fr{\'e}chet derivative $\frL_f(A, E) = \lim_{h\to 0}\frac{1}{h}(f(A+hE) - f(A))$ in direction $E$ could be expressed under standard convergence condition as
$$\frL_f(A, E) = \sum_{i=0}^{\infty} f_i \sum_{a+b=i-1}A^aEA^b$$
If $\hat{A} = \begin{pmatrix} A & E \\ 0 & A\end{pmatrix}$ then $f(\hat{A}) = \begin{pmatrix} f(A) & \frL_f(A, E) \\ 0 & f(A)\end{pmatrix}$, this could be used to show $\frL_f(A, E)$ and $f(A)$ could be computed together with a computational complexity of around three times the complexity of $f(A)$. There exist routines to compute Fr{\'e}chet derivatives of the exponential function in open source or commercial packages. We have mentioned $\frL_{\exp}(A, E) = \exp A\sum_{n=0}^{\infty} \frac{(-1)^n}{(n+1)!}\ad_A^n E$. For Jacobi fields of the Grassmann manifold, we need to evaluate $\frL_{\csr}$ and $\frL_{\ssr}$, where we recall $\csr z$ and $\ssr$ are analytic continuations of $\cos z^{1/2}$ and $z^{-1/2}\sin z^{1/2}$. Based on the ideas in \cite{HighamCosine}, the evaluation for $A$ with small eigenvalues could be done by Pad{\'e} approximant, then use functional equations for $\csr$ and $\ssr$ (based on equations for {\it cosine} and {\it sine}) for $A$ with large eigenvalues. To evaluate time derivatives of Jacobi fields, the following formula is handly. While it is easy to prove, we could not find a reference.
\begin{lemma} If $f(x) = \sum_{i=0}^{\infty} f_i x^i$ is an analytic function near zero and $f'(x) = \sum_{i=0}^{\infty} (i+1)f_{i+1} x_i$, then
\begin{equation}
\frac{d}{dt}L_f(tA, tE) = AL_{f'}(tA, tE) + Ef'(tA)
\end{equation}
\end{lemma}
\begin{proof} We only need to prove this for monomials $f(x) = x^n$. This follows from
$$ \frac{d}{dt}\sum_{a+b=n-1}t^nA^aEA^b = A(t^{n-1}\sum_{a+b=n-2}nA^a E A^b) + E(n t^{n-1} A^{n-1})
$$
\end{proof}
\end{appendix}
\bibliographystyle{amsplain}
|
2,869,038,154,796 | arxiv | \section{Introduction}
Weak gravitational lensing (WL) by large scale structures (LSS) is recognized as a powerful tool for probing
the distribution of dark matter (DM) in the Universe. The details of this distribution depend both on DM particle
properties and on the cosmological growth factor, itself a function of the equation of state, and thus weak lensing
measurements can provide important constraints on cosmology (Kneib et al.
2003; Sheldon et al. 2004; Hoekstra et al. 2004, 2005; Clowe et al. 2006; Mandelbaum et al. 2006; Rozo et al. 2010;
Leauthaud et al. 2010, 2011a, 2011b; Kneib \& Natarajan 2011).
The two-dimensional (2D) WL convergence map is proportional to the density projected along each line of sight.
High signal-to-noise ratio (SNR) peaks in the convergence map generally correspond to massive clusters
(Hamana et al. 2004). It turns out that a simple Gaussian filter of width $\theta_G\approx1~\arcmin$
is close to the optimal linear filter for cluster detection, and this choice has been extensively studied in
simulations (White et al.\ 2002; Hamana et al.\ 2004; Tang \& Fan 2005; Gavazzi \& Soucail 2007). Shape noise
from intrinsic ellipticity of galaxies and projection effects of the LSS will produce spurious noise peaks, degrading
the completeness and purity of cluster detection. Such effects can also influence the WL signals of the LSS, increasing
the SNR of smaller structures above $3\sigma$.
As a cosmological probe, the peak abundance is complementary to the WL power spectrum, and
is similar to galaxy cluster abundance (Dietrich \& Hartlap 2010; Kratochvil et al. 2010; Maturi et al. 2010;
Yang et al. 2011, 2013; Marian et al. 2012; Shan et al. 2012; Bard et al. 2013; Van Waerbeke et al. 2013).
A major advantage of WL peaks and a motivation for
their use is that they avoid the issue of having to identify genuine bound clusters and measure their masses.
Peaks can be directly compared to cosmological N-body simulations without the need to make the correspondence to
observed or simulated ``galaxy clusters''. Since the abundance of WL peaks can be used as a cosmological tool, we expect
their clustering to also be valuable. With simulations, Marian et al. (2013) studied the high-order statistics of WL peaks,
including the stacked tangential-shear profiles and the peak-peak correlation function. They found that the
marginalized constraints are tightened by a factor of $\sim$2 compared to the peak abundance alone, the
least contributor to the error reduction being the correlation function.
First we present the WL convergence map of the $173$-tile CS82 field, and study the WL peak statistics.
We will analyze peak abundance, peak correlation functions
and the tangential shear profiles around peaks. We count the positive and negative peaks in the mass map, measure the
peak abundance as a function of SNR, and compare with the $\Lambda$ Cold Dark Matter ($\Lambda$CDM) cosmological
model using the analytical predictions by Fan et al. (2010). We then measure higher-order statistics of
WL peaks for the first time with real data. We investigate the correlation functions of
WL peaks. For galaxies and clusters, we expect the
correlation functions of WL peaks with different SNR to be well-fitted with power laws. Furthermore, combining with
the Constant Mass galaxies (CMASS) from the Sloan Digital Sky Survey III DR10 Baryonic Oscillation Spectroscopic Survey
(SDSS-III/DR10/BOSS, Eisenstein et al. 2011; Dawson et al. 2013) experiment, we study the cross-correlation between the
CMASS galaxies and WL peaks. We also compare our WL peak detections with catalogs of overdensities detected
via the red sequence Matched-filter Probabilistic Percolation (redMaPPer) algorithm (Rykoff et al. 2013).
We fit the tangential-shear profiles of different SNR WL peaks and ``dark clumps''
(WL peaks without any obvious optical cluster counterpart) with singular isothermal sphere (SIS) profile and
Navarro-Frenk-White (NFW, Navarro, Frenk, \& White 1996) profile plus a 2-halo term.
This paper is organized in the following way. In Section~2, we describe the CS82 data used. In Section~3,
we reconstruct the 2D lensing convergence ``mass map'', and extract a catalog of peaks. In Section~4, we study the
peak statistics with peak abundance, correlation functions and tangential-shear profiles. Section~5
summarizes and discusses the results.
Throughout this paper, we adopt a fiducial, flat $\Lambda$CDM cosmological model with $\Omega_{\rm CDM}=0.226$,
$\Omega_b=0.0455$, $\Omega_{\Lambda}=0.7285$, $\sigma_8=0.81$, $n_{\rm initial}=0.966$,
$H_0=100~h~\mathrm{km}~{\mathrm s}^{-1}~\mathrm{Mpc}^{-1}$ with $h=0.71$.
\section{CFHT/MegaCam Stripe-82 Survey and weak lensing catalog} \label{sec:CS82}
SDSS equatorial Stripe 82, which covers more than $200$ square degrees, has a high density of spectroscopic redshifts,
with $> 100,000$ redshift measurements. On-going surveys such as the SDSS-III Baryon Oscillation Spectroscopic Survey
(BOSS) and Wiggle-Z are now adding more than $> 40,000$ new spectra to this legacy.
The CFHT/MegaCam Stripe 82 Survey (CS82) is a large collaborative $i$-band survey between the Canadian/French
and Brazilian communities, which has been successfully conducted down to $i_{AB}=24.0$ in excellent seeing conditions
(between $0.4$ and $0.8$~arcsec with a median of $0.59$~arcsec) (Erben et al. 2014, in preparation). This area contains a
total of $173$ tiles ($165$ tiles CS82 and $8$ CFHT-LS Wide tiles). Each CS82 tile was obtained in four dithered
observations with an exposure time of $410$s, each resulting in a $5$-$\sigma$ limiting magnitude in a 2~arcsec
diameter aperture of about $i_{AB}=24.0$. After applying all the masks across the entire survey, the final effective
sky coverage drops from $173~{\rm deg}^2$ to $\sim$124~${\rm deg}^2$. Figure~\ref{fig:ra_area} shows a clear correlation
between effective sky coverage $S_{\rm eff}$ and RA direction of the data. On the two edges of stripe data, the mask
region is larger because of the higher stellar density.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{ra_area.eps}
\caption{The relation between RA and effective sky coverage. Each dot
corresponds to one CS82 tile. The dashed line shows a polynomial fit of
the data, which displays the impact of stellar density contamination.
\label{fig:ra_area}}
\end{center}
\end{figure}
The shapes of faint galaxies are measured with the Lensfit method (Miller et al 2007, 2013), the details of the calibration
and systematic effects are shown and discussed in Heymans et al. (2012). We use all galaxies with
magnitudes $i_{AB} <23.5$, signal-to-noise $\nu>10$, weight $w>0$ and FITCLASS$=0$, in which $w$ represents an inverse
variance weight accorded to each source galaxy by LensFit, and FITSCLASS is a star/galaxy classification provided by
Lensfit. The magnitude cut is quite conservative as the limiting magnitude of each tile is higher than $23.5$. These
criteria result in a total of $2,846,452$ source galaxies, and the average source surface number density is
$6.4$ galaxies per ${\rm arcmin}^2$.
\section{Convergence map}\label{sec:mass2d}
\begin{figure*}
\begin{center}
\includegraphics[width=1.0\columnwidth]{seeing_ng.eps}
\includegraphics[width=1.0\columnwidth]{seeing_sigame.eps}
\caption{The relation between seeing and effective galaxy number density (left panel)/intrinsic ellipticity dispersion
(right panel) of each tile. Each dot corresponds to one CS82 tile. The dashed line shows a linear fit to the data.
\label{fig:seeing_ng}}
\end{center}
\end{figure*}
The convergence field, $\kappa$, is estimated from the shear field $\gamma$ using the Kaiser \& Squires (1993)
inversion algorithm as
\begin{equation}
\kappa(\theta)=\frac{1}{\pi}\int d^2 \theta' \Re [D^*(\theta-\theta')\gamma(\theta')],
\end{equation}
where $D(\theta)=\frac{-1}{(\theta_1-i \theta_2)^2}$ is a complex convolution kernel to obtain $\kappa$ from the
shear $\gamma$. In this paper, we do the mass reconstruction per tile. The pixel scale for the binning of
$\gamma$ is $0.0586~\arcmin$.
We treat individual CS82 tiles as ``empty fields'' with average mass properties. As the tiles are degree-scale,
the net mass sheet density on this scale should be negligeable.
For the finite density of source galaxies resolved by CFHT, the scatter of their intrinsic ellipticities
means that a raw, unsmoothed convergence map $\kappa(\theta)$ will be infinitely noisy. We smooth
the convergence map by convolving it (while still in Fourier space) with a Gaussian window function,
\begin{equation}
W_G(\theta)=\frac{1}{\pi \theta_G^2} \exp \left( -\frac{\theta^2}{\theta_G^2} \right ).
\end{equation}
where $\theta_G$ is the smoothing scale. As shown by Van Waerbeke (2000), if different galaxies' intrinsic
ellipticities are uncorrelated, the statistical properties of the resulting noise field can be described by Gaussian
random field theory (Bardeen et al.\ 1986; Bond \& Efstathiou 1987) on scales where the discreteness effect of
source galaxies can be ignored. The Gaussian field is uniquely specified by the variance of the noise, which is in
turn controlled by the number of galaxies within a smoothing aperture (Kaiser \& Squires 1993; Van Waerbeke 2000)
\begin{equation}
\sigma_{\rm noise}^2=\frac{\sigma_e^2}{2} \frac{1}{2\pi \theta_G^2 n_g},
\end{equation}
where $\sigma_e$ is the rms amplitude of the intrinsic ellipticity
distribution and $n_g$ is the density of source galaxies. In Figure~\ref{fig:seeing_ng}, we show the
relation between the seeing and effective galaxy number density/intrinsic ellipticity dispersion of each tile.
We define the signal-to-noise ratio for WL detections as
\begin{equation}
\nu\equiv\frac{\kappa}{\sigma_{\rm noise}}.
\end{equation}
To define the noise level in theoretical calculations of $\nu$, we adopt a constant effective density of galaxies
equal to the mean within our survey. For our CS82 survey, we use the mean galaxy density in each
tile, which is corrected for the masked area (the area goes from $173~{\rm deg}^2$ to $124~{\rm deg}^2$) --- but do
not consider the non-uniformity of the density within each tile due to masks or galaxy clustering.
\begin{figure*}
\begin{center}
\includegraphics[width=2.0\columnwidth]{cs82_field.ps}
\caption{Reconstructed ``DM mass'' convergence map for an area of
CS82 fields, smoothed with a Gaussian filter of width $\theta_G= 1'$. The red circles (multi-circles)
denote the redMaPPer (high richness) clusters. }\label{fig:map}
\end{center}
\end{figure*}
In order to better display large-scale features, we show the reconstructed ``DM mass'' convergence map for an
area of CS82 fields in Figure~\ref{fig:map} with a smoothing scale $\theta_G=1~\arcmin$.
We also compare our WL peak detections with catalogs of overdensities detected via optical observation. The red
sequence Matched-filter Probabilistic Percolation (redMaPPer) algorithm is an efficient red sequence cluster
finder developed by Rykoff et al. (2013) based on the optimized red-sequence richness estimator.
The red circles (multi-circles) denote the redMaPPer (high richness) clusters. While there may be some correlation
between our WL peaks and the positions of the redMaPPer cluster candidates, there is generally not a clear one-to-one
correspondence between individual peaks and individual clusters, presumably due to shape noise and projection effects
in the LSS.
\section{Peak statistics}
In this section, we will analyze the peaks in the WL mass maps, defining a peak as any pixel that has a higher values
of $\kappa$ than any of the surrounding eight pixels (Jain \& Van Waerbeke 2000). Considering the data, we can find that
the number of all WL peaks in each tile is related to effective galaxy number density (See Figure~\ref{fig:ng_nptot}).
We study three kinds of peak statistics: peak abundance, peak
correlation functions and the mean tangential shear around peaks based on the mass map of the CS82 survey.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{ng_nptot.eps}
\caption{The relation between effective galaxy number density and
the number of all WL peaks of each tile. The dashed line is a linear fit to the data.
By increasing the galaxy number density, there is an increase on the detection level,
so we can detect smaller structures as shown in this plot. However, there is a big
variance in the distribution of massive structures from field to field.
\label{fig:ng_nptot}}
\end{center}
\end{figure}
\subsection{Peak abundance}\label{sec:aboud}
To assess the reliability of this map, we shall first investigate the statistical properties of local maxima and minima.
Figure~\ref{fig:count_peak} shows the distribution of peak heights, as a function of SNR.
The bimodal distribution in both the $B$-mode and $E$-mode signals is dominated by positive and negative noisy
fluctuations, but an asymmetric excess in the $E$-mode signal is apparent at $\nu>3.0$.
Local minima could correspond to voids (Jain \& Van Waerbeke 2000; Miyazaki et al.\ 2002).
But the large angular extent of voids is ill-matched to our $\theta_G=1\arcmin$ filter width, and their
density contrast can never be greater than unity, so this aspect of our data is likely just noise.
The dashed curve shows the prediction from Gaussian random field theory (van Waerbeke 2000).
The low galaxy number density of CS82 survey will introduce some Poisson noise, making even the B-mode
peak count histograms have a non-Gaussian component.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{count_peak_random.eps}
\caption{Nomalized numbers of local maxima (solid line) and minima (dashed line) in our $E$-mode (black) and $B$-mode (red)
convergence map of the CS82 field, with smoothing scales $\theta_G=1~\arcmin$.
Local maxima can still have a slightly negative peak height if they occur along the same line of sight as a negative
noise fluctuation (or a large void), and local minima can similarly have a slightly positive peak height.
The dashed curve shows the prediction from Gaussian random field theory (van Waerbeke 2000).
\label{fig:count_peak}}
\end{center}
\end{figure}
Taking into account the effects of noise on the main-cluster-peak heights and the enhancement of the
number of noise peaks near DM halos, Fan et al.\ (2010) developed an anlytical model incorporating the mass
function of DM halos to calculate the statistical abundance of WL peaks over large scales. They pointed out that
because of the mutual effects of the mass distribution of DM halos and noise, the noise peak abundance also carries
important cosmological information, especially the information related to the density profile of DM halos.
This model can allow us to use directly the peaks detected in the large-scale reconstructed convergence map from WL
observations as cosmological probes without the need to differentiate true or false peaks with follow-up observations.
We adopt the Sheth-Tormen mass function (1999) and the NFW density profile for DM halos in the calculations.
Figure~\ref{fig:count_pv} recasts the peak distribution into a cumulative density of positive maxima.
The prediction from the model by Fan et al.\ (2010) is shown as solid line. In these theoretical calculations, we model
the population of background galaxies as having an intrinsic ellipticity dispersion
$\sigma_e=0.3$, density $n_g=6.4~\rm arcmin^{-2}$ and the redshift distribution $n(z)=A\frac{z^a+z^{ab}}{z^b+c}$
with $a=0.531, b=7.810, c=0.517$ and $A=0.688$. This galaxy distribution has a median redshift
$z_m=0.75$ and a mean redshift $z=0.82$ (see details in Erben et al. 2014, in prep.).
The dotted curve shows the prediction from Gaussian random field theory (van Waerbeke 2000). The measurements, especially
the high SNR peaks, are inconsistent with a pure Gaussian noise. The model including LSS signals and shape noise are more
reasonable.
We also show different cosmological models (dashed color curves) as in Bard et al. (2013).
At large SNR, the statistical errors are large with poor constraints on cosmological parameters.
The low SNR peak distribution is also shown on a linear scale in the sub-panel of Figure~\ref{fig:count_pv}.
The low SNR peaks contain most of the power to separate between different cosmological models, as discussed in
Kratochvil, Haiman \& May (2010). As pointed out by Yang et al. (2011), the reason why noise can boost the sensitivity
to cosmological parameters, contrary to simple intuition, is because the signal for WL peaks is a non-linear function of
the noise.
These low SNR peaks are dominated by random galaxy shape noise,
but the projection of multiple (typically, $4-8$) halos along the line of sight also contribute to the signal of the low
SNR peaks, making their number counts sensitive to cosmological parameters (Yang et al. 2011). However, the analytical
model in Fan et al. (2010) only considers the effects of shape noise. In the noise-dominated case, the distribution of peak
heights will roughly follow that expected for a Gaussian random field, but will differ in detail because of the contribution
from large-scale structures (Yang et al. 2011). A model including the projection effects of LSS should be developed before
interpreting the results in terms of cosmology.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{count_pv_cosmo.eps}
\caption{Cumulative density of local maxima $N(>\nu)$ (black circles) using smoothing scale $\theta_G=1~\arcmin$.
The dashed black curve show the expected peak number from true DM halos. Error bars are simply $1\sigma$
uncertainties assuming Poisson shot noise. The black solid curve shows an analytical
prediction in a fiducial $\Lambda$CDM universe (Fan et al.\ 2010), with the influence of
shape noise. The dashed color curves show different cosmological models as in Bard et al. 2013.
The dotted curve shows the prediction from Gaussian random field theory (van Waerbeke 2000). The sub-panel shows the
low SNR peak distribution on a linear scale.
\label{fig:count_pv}}
\end{center}
\end{figure}
\subsection{Auto-correlation function}\label{sec:corr}
The auto-correlation function $w(\theta)$ is measured by comparing the
actual peak distribution to a catalog of positions distributed randomly
over the unmasked region of the survey.
We use the estimator of Lancy \& Szalay (1993, LS) to calculate
$w(\theta)$, as this has been found to be the most reliable
estimator for the two-point correlation function (Kerscher et al. 2000).
The LS estimator is given by,
\begin{eqnarray}
w(\theta)&=&\frac{DD-2DR+RR}{RR}, \\
&& =1+\left ( \frac{N_{\rm rd}}{N} \right)^2 \frac{DD}{RR} - 2\left(
\frac{N_{\rm rd}}{N} \right) \frac{DR}{RR}
\end{eqnarray}
where DD, DR and RR are pair counts in bins of $\theta \pm \delta
\theta$ of the data-data, data-random and random-random points
respectively, and $N$ and $N_{\rm rd}$ are the numbers of data and random
points in the sample.
Historically, measurements of the cluster correlation function found
results consistent with a power law over scales $\rm r< 60 h^{-1}
Mpc$ or so (Bahcall \& Soneira 1983; Nichol et al. 1992; Peacock \& West 1992;
Croft et al. 1997; Gonzalez et al. 2002),
\begin{equation}
w(\theta)=\left( \frac{\theta}{\theta_0} \right)^{-\gamma},
\end{equation}
where the correlation length $\theta_0$ depends on cluster richness
(peak richness here). Thus, we fit a power-law $w(\theta)=A_w(\theta/1')^{-\gamma}$ with different SNR peaks.
A set of random points will produce $\gamma=0$.
In Figure~\ref{fig:auto}, we show the auto-correlation functions of WL peaks with SNR $\nu>0$ and $\nu>2$. We
fit the measured correlation function for $\theta>4~\rm arcmin$. The solid lines are the fitted power law.
The auto-correlation function of peaks can be well fitted with a power law
$w_{pp}(\theta)=A_w(\theta/1')^{-\gamma}$ (see Table~\ref{tab:tab1}). For the peaks with $\nu>0$, the
exponent of the power law has a value $\sim$0.64, which is even lower than the angular correlation
function of galaxies with $\gamma \sim$0.8 (Zehavi et al. 2002). This suggests that the $\nu>0$ peaks include lots of
small structures and also noise peaks. For the peaks with $\nu>2$ which are related to more massive structures and
less noise peaks, we find $\gamma_{\nu>2}=1.32 \pm 0.01$, which is close to the measured auto-correlation
functions of SDSS clusters with $\gamma \sim$0.8-1.3 (Estrada et al. 2009; Hong et al. 2012). The number of the
higher SNR peaks is too small to be well fitted with a power-law. Note that there is a turn-around at scale of
$\sim$3~$\rm arcmin$, which depends on the size of the Gaussian smoothing scale applied to the shear data.
\begin{table}
\centering
\caption{Slope of the power-law fit to $w(\theta)$}
\begin{tabular}{ccccc}
\hline
\hline
SNR & $\gamma$ & $A_{w}$\\
\hline
$\nu>0$ & $0.64 \pm 0.01$ & $0.20 \pm 0.01$\\
$\nu>2$ & $1.32 \pm 0.01$ & $1.59 \pm 0.03$ \\
\hline
\end{tabular}
\label{tab:tab1}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[angle=0.0, width=1.0\columnwidth]{w_auto.eps}
\caption{Auto-correlation functions of WL peaks with SNR $|\nu|>0$ (red points) and $|\nu|>2$
(black points). The drop of the auto-correlation function at small scales depends on the
smoothing scale used in the analysis. \label{fig:auto}}
\end{center}
\end{figure}
\subsection{Cross-correlation function}
Because WL peaks are related to LSS in the Universe, we expect that there to be a cross-correlation between
WL peaks and biased systems, such as clusters and massive galaxies. However, the redMaPPer catalog does not contain
enough objects, $\sim$432 clusters with $0.1<z<0.6$ and richness $\lambda>20$, to estimate a correlation function.
Therefore, we use CMASS galaxies instead.
In this section, we present the cross-correlation functions between CMASS galaxies and WL peaks with
$\nu>2.0$. The CMASS sample is the SDSS-III/BOSS experiment BAO tracer (Dawson et al. 2013). The parent catalog of
CMASS selection on Stripe 82 contains $22,034$ tracers, covering $\sim$98~$\rm deg^{-2}$ of the CS82 region. As in
Comparat et al. (2013), we use the complete CMASS selection, not only the galaxies confirmed by spectroscopy, in order
to avoid fiber collision issues. The mean redshift is $0.53$ with a dispersion of $0.1$. On the same scale where
we have determined the auto-correlation functions of WL peaks, we measure the slope of the auto-correlation function for
CMASS galaxies to be $\gamma \sim$0.73$\pm$0.01. We conclude that the WL peaks with $\nu>2.0$ are more biased than
CMASS galaxies, suggesting that these peaks are related to groups or clusters as expected.
Eventually we checked the cross-correlation between CMASS galaxies and WL peaks. Figure~\ref{fig:cmass} shows that this
can also be fit with a power law $w_{cp}(\theta)=A_w(\theta/1')^{-\gamma}$. The slope of cross correlation is $0.78\pm0.01$.
\begin{figure}
\begin{center}
\includegraphics[angle=0.0, width=1.0\columnwidth]{w_cmass.eps}
\caption{Cross-correlation functions between CMASS galaxies and WL peaks with SNR $|\nu|>2$. \label{fig:cmass}}
\end{center}
\end{figure}
\subsection{Tangential shear}\label{sec:tan}
In this section, we estimate the average tangential shear profile of WL peaks. Stacking the signal from many peaks
can reduce the contribution from shape noise, uncorrelated structures along the line of sight and
substructures. We calculate the excess surface mass density $\Delta \Sigma(R)=\Sigma_{\rm crit}g_t(R)$,
where $g_t(R)$ is the tangential shear and $\Sigma_{\rm crit}$ is the critical surface density. The mean source and lens redshifts
$<z_s>$ and $<z_l>$ are {$0.75$} and $0.45$, respectively.
\subsubsection{Matched redMaPPer peaks}
Tangential shear measurements require the identification of the DM density peak, but WL peaks
are expected to be offset of the centers of the main DM halos associated with them (Yang et al. 2013).
In this paper, halo centers are assumed to contain central galaxies (CGs) which can be used as good tracers, on condition
that they can be correctly identified.
As in Shan et al. (2012), we search for matched redMaPPer clusters within a $3.0\arcmin$ radius of peaks that appear in
the WL mass map. This search radius is chosen to be larger than the smoothing scale, but smaller than the angular virial
radius of a massive cluster at $0.1<z<0.9$ (Hamana et al.\ 2004). If more than one pair exists within $3.0\arcmin$, we
adopt the closest match as the primary candidate. In total, $19$ redMaPPer clusters have no corresponding WL peaks. We show
the separation histogram of redMaPPer matched peaks in Figure~\ref{fig:matchedpeak_d}. The separation of WL peaks and
optical centers is from various systematic noise sources, such as the effect of projected LSS
(Gavazzi \& Scoucail 2007; Geller et al. 2010), smoothing of the mass maps, and shape noise in the maps (Dietrich et al. 2012).
The purity, defined as the fraction of peaks above a given detection threshold
$\nu_{\rm th}$ that are associated with an optically detected cluster (Shan et al. 2012) of our sample is much lower
($\sim$15$\%$ for $\nu>3.5$ WL peaks), which is due to the lower galaxy number density.
\begin{figure}
\begin{center}
\includegraphics[width=1.0\columnwidth]{matchedpeak_d.eps}
\caption{The separation histogram of redMaPPer matched peaks.
\label{fig:matchedpeak_d}}
\end{center}
\end{figure}
In Figure~\ref{fig:center}, we present $\Delta \Sigma$ of redMaPPer matched peaks with SNR $\nu>3$
using the WL peak positions (red points) and the matched redMaPPer cluster center positions (black points),
respectively. The WL peak positions can introduce noise on the tangential shear measurement. As in Johnston
et al. (2007), Leauthaud et al. (2010) and George et al. (2012), centroid errors will lead to a smoothing of the
lensing signal on small scales and to an underestimate of halo masses. On large scales ($R>0.7~\rm Mpc$), the
tangential shear signals are almost the same. Thus, we will use the matched redMaPPer cluster centers in this paper.
\begin{figure}
\begin{center}
\includegraphics[angle=0.0, width=1.0\columnwidth]{gglensing_peak_center.eps}
\caption{The measured excess surface mass density $\Delta \Sigma$, obtained by
stacking the tangential shear signals for the redMaPPer matched peaks with $\nu>3.0$ using the WL
peak positions (red points) and the matched redMaPPer cluster center positions (black points), respectively.
\label{fig:center}}
\end{center}
\end{figure}
In Figure~\ref{fig:peak}, we present $\Delta \Sigma$ of matched WL peaks with SNR $0<\nu<1$ (top-left panel),
$1<\nu<2$ (top-right panel), $\nu>3$ (bottom-left panel) and $\nu>4$ (bottom-right panel). From the figure,
the higher SNR peaks have higher $\Delta \Sigma$: there is about an order of magnitude difference between the
lensing signals corresponding to the $\nu>4$ and $0<\nu<1$ bins.
Low SNR peaks often include contributions from several halos or structures along the line of sight, thus
they are relatively insensitive to the inner profile of individual halos (Yang et al. 2013), but reflect instead the
``2-halo term'', or halo-halo clustering. We fit the stacked profiles with both a SIS and a NFW model plus a 2-halo term
following Covone et al. (2014). Fitting results are given in Table~\ref{tab:tab2}. Note that although the full NFW model has
two free parameters, we assume the Bullock et al. (2001) relation between concentration c and virial mass $M_{\rm vir}$ seen
in numerical simulations, leaving only a single parameter to fit.
\begin{figure}
\begin{center}
\includegraphics[angle=0.0, width=1.0\columnwidth]{gglensing_peak_tot.eps}
\caption{{Black circles show the measured excess surface mass density $\Delta \Sigma$ for $4$ samples of WL peaks
with different SNR. The solid and dashed black curves are the best-fit NFW model plus a 2-halo term
and SIS model, respectively. The blue and red curves are main galaxy cluster halo and 2-halo term, respectively.
The SIS model is more disfavored. The corresponding best-fit parameters are listed in Table~\ref{tab:tab2}. }
\label{fig:peak}}
\end{center}
\end{figure}
\begin{table*}
\centering
\caption{Density Profile Models in Figure~\ref{fig:peak}.}
\begin{tabular}{ccccc}
\hline
\hline
SNR & $M_{\rm vir}$ & $\chi^2_{\rm NFW}/\rm d.o.f$ & $\sigma_v$ & $\chi^2_{\rm SIS}/\rm d.o.f$ \\
& $10^{14} M_{\odot}/h$ & & km/s & \\
\hline
$0<\nu<1$ & $0.97^{+0.26}_{-0.25}$ & $0.47$ & $536.65^{+144.45}_{-152.34}$ & $2.38$ \\
$1<\nu<2$ & $1.25^{+0.31}_{-0.28}$ & $0.43$ & $554.53^{+135.67}_{-87.17}$ & $1.54$ \\
$\nu>3$ & $4.07^{+0.98}_{-0.83}$ & $0.76$ & $775.41^{+129.91}_{-92.79}$ & $4.54$ \\
$\nu>4$ & $6.28^{+1.36}_{-1.21}$ & $1.27$ & $928.37^{+150.94}_{-220.32}$ & $3.27$ \\
\hline
\end{tabular}
\label{tab:tab2}
\end{table*}
As expected, the mass of NFW profiles and the velocity dispersion of SIS profiles increase with SNR. The SIS model is
strongly disfavored for the high SNR peaks $\nu>3$ and $\nu>4$, and is rejected at the $3~\sigma$ level. This could be
due to the inner slope of the DM mass density in halos (Rocha et al. 2013). The NFW model plus a 2-halo term gives
more acceptable fits to the data.
\subsubsection{Dark clump peaks}
\begin{figure}
\begin{center}
\includegraphics[angle=0.0, width=1.0\columnwidth]{gglensing_dark_tot.eps}
\caption{The measured $\Delta \Sigma$ for the $\nu>3.0$ peaks, which is
obtained by stacking the distortion signals for the peaks with the matched redMaPPer
cluster center positions (black) and without any matched clusters (red), respectively. The solid and dashed
lines are best-fitted NFW and SIS models, respectively, together with a 2-halo term.
\label{fig:dark}}
\end{center}
\end{figure}
We also study the profiles of the peaks without any matched clusters (``dark clumps'' hereafter), which could include
both smaller structures and noisy peaks. The same redshift distribution with $z_l=0.45$ as the redMaPPer matched peaks
is used for ``dark clumps''.
Because of mis-centering problems, only $\Delta \Sigma$ on large scales $\rm R>0.7~\rm Mpc$
can be used. For the peaks with $\nu>3.0$, the tangential shear signals of dark clumps on large scales are lower
than the redMaPPer matched peaks (see Figure~\ref{fig:dark}). The best fits of the dark clumps are:
$M_{\rm vir}=2.40^{+1.57}_{-1.23} \times 10^{14} M_{\odot}/h$ with $\chi^2=13.80$
(NFW profile plus 2-halo term) and $\sigma_v=740.86^{+531.37}_{-518.25} \rm km/s$ with
$\chi^2=22.17$ (SIS profile). The SIS profile is even more strongly disfavored for dark clumps. It is rejected
at $22\sigma$. Comparing with the matched clumps (to the redMaPPer clusters) in Table~\ref{tab:tab2}, there is a
difference in virial mass of a factor of $2$.
\section{Conclusions}\label{sec:conc}
In this paper, we have reconstructed the WL convergence map of CS82 fields. With the peaks in the WL mass map, we
further study three kinds of WL peak statistics, the peak height distribution, the peak auto-correlation function,
and the mean tangential shear around peaks. This is the first measurement of high-order statistics of
WL peaks with real data.
The use of peak abundance as a cosmological tool have been discussed extensively in the literature (Dietrich \& Hartlap 2010;
Kratochvil et al. 2010; Yang et al. 2011, 2013; Marian et al. 2012, 2013; Shan et al. 2012; Bard et al. 2013). In our paper, we
measure the abundance of peaks as a function of SNR, and compare it with the analytical prediction in Fan et al.
(2010). The peak abundance detected in CS82 is consistent with predictions from a $\Lambda$CDM cosmological
model, once shape noise effects are properly included. If other noise effects, including projection effects and
mask effects, were included accurately in analytic models, we suggest that WL peak abundance could become
a better method to constrain cosmology than pure cluster counts, because we could use the information contained
in the large number of low SNR peaks.
The slope and amplitude of the peak auto-correlation functions depends on the SNR of WL peaks. The auto-correlation
function of different SNR WL peaks $\nu>0$ and $\nu>2$ can be well fitted with power laws with the following slopes:
$\gamma_{\nu>0}=0.64\pm 0.01$ and $\gamma_{\nu>2}=1.32\pm 0.01$. {We conclude that the WL peaks with $\nu>2.0$ are more
biased than CMASS galaxies, suggesting that these peaks are related to groups or clusters as expected.} Combining with the
CMASS galaxies, the cross-correlation with a power-law slope $\gamma \sim$0.78 between CMASS galaxies and high SNR peaks can be
found.
We also fit spherical models to the mean tangential shear profiles around peaks, including the singular isothermal sphere (SIS)
model and Navarro, Frenk \& White (NFW) model plus 2-halo term. The SIS model is strongly disfavored for the high SNR
peaks $\nu>3$ and $\nu>4$, which is rejected at $3~\sigma$. The NFW model plus 2-halo term gives more
acceptable fits to the data. We also compared the dark and matched clumps (to the redMaPPer clusters) and found that there
is a difference in virial mass of a factor of $2$, assuming the matched and unmatched peaks have the
same mass function. This could indicate that approximately half of the dark clumps are false detections, in the sense
that they do not correspond to single massive halos along the line of sight. This assumption would require better data
(such as extensive spectroscopic follow-up) to validate, however.
The high SNR peaks in the WL mass map are related to the LSS in the Universe. In an upcoming paper,
we will constrain cosmology with WL peak statistics. Future surveys, such as the Dark Energy Survey (DES),
Large Synopic Survey Telescope (LSST), Kunlun Dark Universe Survey Telescope (KDUST) and Euclid
surveys, will allow us to map WL peaks throughout much larger cosmological volumes, thus probing cosmology more sensitively.
\chapter{\flushright{\bf{Acknowledgments}}}
\flushleft{Based on observations obtained with MegaPrime/MegaCam, a joint project of CFHT and CEA/DAPNIA, at the
Canada-France-Hawaii Telescope (CFHT), which is operated by the National Research Council (NRC) of Canada,
the Institut National des Science de l'Univers of the Centre National de la Recherche Scientifique (CNRS) of
France, and the University of Hawaii. The Brazilian partnership on CFHT is managed by the Laborat\'{o}rio Nacional
de Astrof\'isica (LNA). This work made use of the CHE cluster, managed and funded by ICRA/CBPF/MCTI, with financial
support from FINEP and FAPERJ. We thank the support of the Laborat\'{o}rio Interinstitucional de e-Astronomia (LIneA).
We thank the CFHTLenS team.
The authors thank Zoltan Haiman for useful discussions. This research was supported by a Marie Curie International Incoming
Fellowship within the $7^{th}$ European Community Framework Programme. HYS acknowledges the support from Swiss National Science
Foundation (SNSF) and NSFC of China under grants 11103011. JPK acknowledges support from the ERC advanced grand LIDA. TE is
supported by the Deutsche Forschungsgemeinschaft through project ER 327/3-1 and by the
Transregional Collaborative Research Centre TR 33 - `The Dark Universe'.}
|
2,869,038,154,797 | arxiv | \section{Introduction}
A positive integer $A$ is called a {\bf congruent number} if
$A$ is the area of a right-angled triangle with three rational
sides. So, $A$ is congruent if and only if there exists a
rational Pythagorean tripel $(a,b,c)$ (\ie, $a,b,c\in\mathds{Q}$,
$a^2+b^2=c^2$, and $ab\neq 0$), such that $\frac{ab}2=A$.
The sequence of integer congruent numbers starts with
$$
5, 6, 7, 13, 14, 15, 20, 21, 22, 23, 24, 28, 29, 30, 31, 34, 37,\ldots
$$
For example, $A=7$ is a congruent number,
witnessed by the rational Pythagorean triple $$\Bigl(\frac{24}{5}\,,
\frac{35}{12}\,,\frac{337}{60}\Bigr).$$
It is well-known that $A$ is a congruent number if
and only if the cubic curve $$C_A:\ y^2=x^3-A^2 x$$
has a rational point $(x_0,y_0)$ with $y_0\neq 0$.
The cubic curve $C_A$ is called a {\bf congruent number curve}.
This correspondence between rational points on congruent number curves and
rational Pythagorean triples can be made explicit as follows:
Let
$$
C(\mathds{Q}):= \{(x,y,A)\in \mathds{Q}\times\mathds{Q}^*\times \mathds{Z}^*:y^2=x^3-A^2x\},
$$
where $\mathds{Q}^*:=\mathds{Q}\setminus\{0\}, \mathds{Z}^*:=\mathds{Z}\setminus\{0\}$, and
$$
P(\mathds{Q}):=\{(a,b,c,A)\in \mathds{Q}^3\times\mathds{Z}^*:a^2+b^2=c^2\ \textsl{and\/}\ ab=2A\}.
$$
Then, it is easy to check that
\begin{equation}\label{psi}
\begin{aligned}
\psi\ :\ \quad P(\mathds{Q})&\ \to\ C(\mathds{Q})\\
(a,b,c,A)&\ \mapsto \ \Bigl(\frac{A(b+c)}{a}\,,\,\frac{2A^2(b+c)}{a^2}\,,\,A\Bigr)
\end{aligned}
\end{equation}
is bijective and
\begin{equation}\label{psi-1}
\begin{aligned}
\psi^{-1}\ :\qquad C(\mathds{Q})&\ \to\ P(\mathds{Q})\\
(x,y,A)&\ \mapsto\ \Bigl(\frac{2x A}{y}\,,\;\frac{x^2-A^2}{y}\,,\;\frac{x^2+A^2}{y}\,,\,A\Bigr).
\end{aligned}
\end{equation}
For positive integers $A$, a triple $(a,b,c)$ of
non-zero rational numbers is called a {\bf rational Pythagorean $\boldsymbol{A}$-triple}
if $a^2+b^2=c^2$ and $A=\big{|}\frac{ab}{2}\big{|}$.
Notice that if $(a,b,c)$ is a {rational Py\-tha\-go\-re\-an $A$-triple}, then $A$
is a congruent number and $|a|,|b|,|c|$ are the
lengths of the sides of a right-angled triangle
with area $A$. Notice also that we allow $a,b,c$
to be negative.
It is convenient to consider the curve $C_A$ in the
projective plane $\mathds{R} P^2$, where the curve is given by
$$
C_A :\ y^2z = x^3-A^2xz^2.
$$
On the points of $C_A$, one can define a commutative, binary,
associative operation ``$+$'', where $\mathscr{O}$, the neutral
element of the operation, is the projective point $(0,1,0)$
at infinity. More formally, if $P$ and $Q$ are two points on $C_A$,
then let $P\# Q$ be the third intersection point of
the line through $P$ and $Q$ with the curve $C_A$.
If $P=Q$, the line through $P$ and $Q$ is replaced by the tangent in $P$.
Then
$P+Q$ is defined by stipulating
$$P+Q\;:=\;\mathscr{O}\# (P\# Q),$$
where for a point $R$ on $C_A$, $\mathscr{O}\# R$ is the point reflected across the $x$-axis.
The following figure shows the congruent number curve $C_A$ for
$A=5$, together with two points $P$ and $Q$ and their sum $P+Q$.
\begin{center}
\psset{xunit=.6cm,yunit=.4cm,algebraic=true,dimen=middle,dotstyle=o,dotsize=5pt 0,linewidth=1.6pt,arrowsize=3pt 2,arrowinset=0.25}
\begin{pspicture*}(-7.1329967371431815,-7.588280903625012)(8.234736979744335,9.360344080789211)
\psaxes[labelFontSize=\scriptstyle,xAxis=true,yAxis=true,Dx=2.,Dy=2.,ticksize=-2pt 2pt,subticks=1,linewidth=.6pt,]{->}(0,0)(-7.1329967371431815,-7.588280903625012)(8.234736979744335,9.360344080789211)
\psplotImp[linewidth=1.2pt,linecolor=blue,stepFactor=0.1](-9.0,-9.0)(9.0,10.0){1.0*y^2+25.0*x^1-1.0*x^3}
\psplot[linewidth=.6pt]{-7.1329967371431815}{8.234736979744335}{(-43.01566031988557-1.5476943995438228*x)/-7.215284285929719}
\psline[linewidth=.6pt,linestyle=dashed,dash=4pt 4pt](-4.388182328551535,-7.588280903625012)(-4.388182328551535,9.360344080789211)
\begin{small}
\psdots[dotsize=4pt 0](5.824738905621608,7.211160924647724)
\rput[bl](5.3,7.4){$Q$}
\psdots[dotsize=4pt 0](-1.3905453803081107,5.663466525103901)
\rput[bl](-1.3,6){$P$}
\psdots[dotsize=4pt 0](-4.388182328551535,5.020466785549749)
\rput[bl](-6.,5.2){$P\# Q$}
\psdots[dotsize=4pt 0](-4.388182328551535,-5.020466785549749)
\rput[bl](-6.3,-5.4){$P+Q$}
\end{small}
\end{pspicture*}
\end{center}
More formally, for two points $P=(x_0,y_0)$ and $Q=(x_1,y_1)$ on
a congruent number curve $C_A$, the point $P+Q=(x_2,y_2)$ is given by
the following formulas:
\begin{itemize}
\item If $x_0\neq x_1$, then
$$x_2=\lambda^2-x_0-x_1,\qquad y_2=\lambda(x_0-x_2)-y_0,$$
where $$\lambda:=\frac{y_1-y_0}{x_1-x_0}.$$
\item If $P=Q$, \ie, $x_0=x_1$ and $y_0=y_1$, then
\begin{equation}\label{eq:2P}
x_2=\lambda^2-2x_0,\qquad y_2=3x_0\lambda-\lambda^3-y_0,
\end{equation}
where
\begin{equation}\label{eq:lambda}
\lambda:=\frac{3x_0^2-A^2}{2y_0}.
\end{equation}
Below we shall write $2*P$ instead of $P+P$.
\item If $x_0=x_1$ and $y_0=-y_1$, then $P+Q:=\mathscr{O}$. In particular,
$(0,0)+(0,0)=(A,0)+(A,0)=(-A,0)+(-A,0)=\mathscr{O}$.
\item Finally,
we define $\mathscr{O}+P:=P$ and $P+\mathscr{O}:=P$ for any point $P$, in particular,
$\mathscr{O}+\mathscr{O}=\mathscr{O}$.
\end{itemize}
With the operation~``$+$'',
$(C_A,+)$ is an abelian group with neutral element $\mathscr{O}$.
Let $C_A(\mathds{Q})$ be the set of rational points on $C_A$ together
with $\mathscr{O}$. It is easy to see that $\bigl(C_A(\mathds{Q}),+\bigr)$.
is a subgroup of $(C_A,+)$. Moreover, it is well known that
the group $\bigl(C_A(\mathds{Q}),+\bigr)$ is finitely generated.
One can readily check that the three points $(0,0)$ and
$(\pm A,0)$ are the only points on $C_A$ of order~$2$,
and one easily finds other points of finite order on $C_A$.
But do we find also rational points of finite order on~$C_A$?
This question is answered by the following
\begin{thm}\label{thm:main}
If $A$ is a congruent number and $(x_0,y_0)$ is a rational
point on $C_A$ with $y_0\neq 0$, then the order of $(x_0,y_0)$
is infinite. In particular, if there exists one {rational Py\-tha\-go\-re\-an $A$-triple}, then
there exist infinitely many such triples.
\end{thm}
The usual proofs of {\sc Theorem}\;\ref{thm:main} are quite involved. For example,
Koblitz~\cite[Ch.\,I, \S\,9, Prop.\,17]{Koblitz} gives a proof using
Dirichlet's theorem on primes in an arithmetic progression, and in
Chahal~\cite[Thm.\,3]{Chahal}, a proof is given using the Lutz-Nagell theorem,
which states that rational points of finite order are integral.
However, both results, Dirichlet's theorem and the Lutz-Nagell theorem, are
quite deep results, and the aim of this article is to provide a simple proof
of {\sc Theorem}\;\ref{thm:main} which relies on an elementary theorem of Fermat.
\section{A Theorem of Fermat}
In~\cite{Fermat}, Fermat gives an algorithm to construct
different right-angled triangles with three rational
sides having the same area (see also Hungerb\"uhler~\cite{Noebi}).
Moreover, Fermat claims that his
algorithm yields infinitely many distinct such right-angled
triangles. However, he did not provide a proof for this
claim. In this section, we first present Fermat's algorithm
and then we show that this algorithm delivers infinitely many
pairwise distinct rational right-angled triangles of the
same area.
\begin{algo}\label{algo}
Assume that $A$ is a congruent number, and
that $(a_0,b_0,c_0)$ is a {rational Py\-tha\-go\-re\-an $A$-triple},
\ie, $A=\big{|}\frac{a_0 b_0}2\big{|}$.
Then
\begin{equation}\label{eq:fermat}
a_1:=\frac{4c_0^2a_0b_0}{2c_0(a_0^2-b_0^2)},\quad
b_1:=\frac{c_0^4-4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)},\quad
c_1:=\frac{c_0^4+4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)},
\end{equation}
is also a {rational Py\-tha\-go\-re\-an $A$-triple}. Moreover, $a_0b_0=a_1b_1$, \ie, if $(a_0,b_0,c_0,A)\in P(\mathds{Q})$,
then $(a_1,b_1,c_1,A)\in P(\mathds{Q})$.
\end{algo}
\begin{proof}
Let $m:=c_0^2$, let $n:=2a_0b_0$, and let
$$X:=2 m n,\quad Y:=m^2-n^2,\quad Z:=m^2+n^2,$$
in other words,
$$X=4c_0^2a_0b_0,\quad
Y=c_0^4-4a_0^2b_0^2,\quad
Z=c_0^4+4a_0^2b_0^2.$$
Then obviously, $X^2+Y^2=Z^2$, and since $a_0,b_0,c_0\in\mathds{Q}$,
$\bigl(|X|,|Y|,|Z|\bigr)$ is a rational Pythagorean triple, where the
area of the corresponding right-angled triangle
is $$\tilde A=\bigg{|}\frac{X Y}{2}\bigg{|}=
\big{|}2 a_0 b_0 c_0^2 (c_0^4-4 a_0^2 b_0^2)\big{|}.$$
Since $a_0^2+b_0^2=c_0^2$, we get $c_0^4=(a_0^2+b_0^2)^2=a_0^4+2a_0^2b_0^2+b_0^4$
and therefore $$c_0^4-4 a_0^2 b_0^2\;=\;a_0^4-2a_0^2b_0^2+b_0^4\;=\;(a_0^2-b_0^2)^2>0.$$
So, for $$a_1=\frac{X}{2c_0(a_0^2-b_0^2)},\quad
b_1=\frac{Y}{2c_0(a_0^2-b_0^2)},\quad
c_1=\frac{Z}{2c_0(a_0^2-b_0^2)},$$
we have $a_1^2+b_1^2=c_1^2$ and
$$\frac{a_1b_1}{2}\;=\;
\frac{XY}{2\cdot 4c_0^2(a_0^2-b_0^2)^2}\;=\;
\frac{2 a_0 b_0 c_0^2 (c_0^4-4 a_0^2 b_0^2)}{4c_0^2(a_0^2-b_0^2)^2}\;=\;
\frac{2 a_0 b_0 c_0^2 (a_0^2-b_0^2)^2}{4c_0^2(a_0^2-b_0^2)^2}\;=\;
\frac{a_0 b_0}{2}.
$$
\end{proof}
\begin{thm}\label{thm:FermatClaim}
Assume that $A$ is a congruent number,
that $(a_0,b_0,c_0)$ is a {rational Py\-tha\-go\-re\-an $A$-triple},
and for positive integers~$n$, let
$(a_n,b_n,c_n)$ be the {rational Py\-tha\-go\-re\-an $A$-triple}
we obtain by {\sc Fermat's Algorithm} from $(a_{n-1},b_{n-1},c_{n-1})$.
Then for any distinct non-negative integers $n,n'$, we have
$|c_n|\neq |c_{n'}|$.
\end{thm}
\begin{proof} Let $n$ be an arbitrary but fixed
non-negative integer. Since $A=\big{|}\frac{a_n b_n}2\big{|}$,
we have $2A=|a_nb_n|$, and consequently
\begin{equation}\label{*}
a_n^2b_n^2=4A^2
\end{equation}
Furthermore,
since $a_n^2+b_n^2=c_n^2$, we have
$$(a_n^2+b_n^2)^2=a_n^4+2a_n^2b_n^2+b_n^4=a_n^4+8A^2+b_n^4=c_n^4,$$
and consequently we get
$$c_n^4-16A^2=a_n^4-8A^2+b_n^4=a_n^4-2a_n^2b_n^2+b_n^4=(a_n^2-b_n^2)^2>0.$$
Therefore, $$\sqrt{(a_n^2-b_n^2)^2}=|a_n^2-b_n^2|=\sqrt{c_n^4-16A^2},$$
and with~(\ref{eq:fermat}) and~(\ref{*}) we finally have
$$|c_{n+1}|=\frac{c_n^4+16A^2}{2c_n\sqrt{c_n^4-16A^{2\mathstrut}}}\,.$$
Now, assume that $c_n=\frac uv$ where $u$ and $v$ are in lowest terms.
We consider the following two cases:
{\it $u$ is odd\/}: First, we write $v=2^k\cdot\tilde v$, where $k\ge 0$
and $\tilde v$ is odd. In particular, $c_n=\frac{u}{2^{k\mathstrut}\cdot\tilde v}$.
Since $c_{n+1}$ is rational, $\sqrt{c_n^4-16A^2}\in\mathds{Q}$.
So, $$\sqrt{c_n^4-16A^2}=\sqrt{\frac{u^4-16A^2v^4}{v^4}}=\frac{\tilde u}{v^2}$$
for a positive odd integer $\tilde u$. Then
$$|c_{n+1}|=\frac{\frac{u^4+16A^2v^4}{v^4}}{\frac{2u\tilde u}{v^3}}=
\frac{\bar{u}}{2u\tilde u v}=
\frac{\bar{u}}{2u\tilde u 2^{k\mathstrut}\tilde v}=\frac{\bar{u}}{2^{k+1}u\tilde u\tilde v}
=\frac{u'}{2^{k+1\mathstrut}\cdot v'}$$
where $\bar{u},u',v'$ are odd integers and $\gcd(u',v')=1$. This shows
that $$c_n=\frac{u}{2^{k\mathstrut}\cdot\tilde v}\quad\Rightarrow\quad
|c_{n+1}|=\frac{u'}{2^{k+1\mathstrut}\cdot v'}$$ where $u,\tilde v,u',v'$ are odd.
{\it $u$ is even\/}: First, we write $u=2^k\cdot\tilde u$, where $k\ge 1$
and $\tilde u$ is odd. In particular, $c_n=\frac{2^k\cdot\tilde u}{v}$, where $v$ is odd.
Similarly, $A=2^l\cdot\tilde A$, where $l\ge 0$ and $\tilde A$ is odd.
Then $$c_n^4\pm 16A^2=\frac{2^{4k}\cdot\tilde u^4\pm 2^{4+2l}\tilde A^2 v^4}{v^4},$$
where both numbers are of the form $$\frac{2^{2m}\bar{u}}{v^4}\,,$$
where $\bar{u}$ is odd and $4\le 2m\le 4k$, \ie, $2\le m\le 2k$.
Therefore, $$|c_{n+1}|=\frac{2^{2m}u_0\cdot v^3}{2\cdot 2^k\tilde u\cdot
v^4\cdot2^{m\mathstrut}u_1}=\frac{2^{m-k-1\mathstrut}\cdot u'}{v'},$$
where $u_0,u_1,u',v'$ are odd. Since $m<2k+1$, we have $m-k-1<k$, and
therefore we obtain
$$c_n=\frac{2^k\cdot\tilde u}{v}\quad\Rightarrow\quad
|c_{n+1}|=\frac{2^{k'}\cdot u'}{v'}$$ where $\tilde u,v,u',v'$ are odd and
$0\le k'<k$.
Both cases together show that whenever $c_n=2^k\cdot\frac uv$, where
$k\in\mathds{Z}$ and $u,v$ are odd, then $|c_{n+1}|=2^{k'}\cdot\frac{u'}{v'}$,
where $u',v'$ are odd and $k'<k$. So, for any distinct non-negative
integers $n$ and $n'$, $|c_n|\neq |c_{n+1}|$.
\end{proof}
The proof of {\sc Theorem}\;\ref{thm:FermatClaim} gives us the following
reformulation of {\sc Fermat's Algorithm}:
\begin{cor}\label{cor:FA}
Assume that $A$ is a congruent number, and
that $(a_0,b_0,c_0)$ is a {rational Py\-tha\-go\-re\-an $A$-triple},
\ie, $A=\big{|}\frac{a_0 b_0}2\big{|}$.
Then
$$a_1=\frac{4Ac_0}{\sqrt{c_0^4-16A^2}},\quad
b_1=\frac{\sqrt{c_0^4-16A^2}}{2c_0},\quad
c_1=\frac{c_0^4+16A^2}{2c_0\sqrt{c_0^4-16A^2}},
$$
is also a {rational Py\-tha\-go\-re\-an $A$-triple}.
\end{cor}
\begin{proof} Notice that $c_0^4-4a_0^2b_0^2=c_0^4-16A^2$
and recall that $|a_0^2-b_0^2|=\sqrt{c_0^4-16A^2}$.
\end{proof}
\section{Doubling points with Fermat's Algorithm}
Before we prove {\sc Theorem}\;\ref{thm:main} (\ie,
that congruent number curves do not
contain rational points of finite order), we first prove
that {\sc Fermat's Algorithm}\;\ref{algo} is essentially doubling points
on congruent number curves.
\begin{lem}\label{lem:doubling}
Let $A$ be a congruent number, let $(a_0,b_0,c_0)$
be a {rational Py\-tha\-go\-re\-an $A$-triple}, and let $(a_1,b_1,c_1)$ be
the {rational Py\-tha\-go\-re\-an $A$-triple} obtained by {\sc Fermat's Algorithm} from $(a_0,b_0,c_0)$.
Furthermore, let $(x_0,y_0)$ and $(x_1,y_1)$ be the rational
points on the curve $C_A$ which correspond to
$(a_0,b_0,c_0)$ and $(a_1,b_1,c_1)$, respectively.
Then we have
$
2*(x_0,y_0)=(x_1,-y_1).$$
\end{lem}
\begin{proof}
Let $(a_0,b_0,c_0)$ be a {rational Py\-tha\-go\-re\-an $A$-triple}. Then, according to~(\ref{eq:fermat}), the
{rational Py\-tha\-go\-re\-an $A$-triple} $(a_1,b_1,c_1)$ which we obtain by {\sc Fermat's Algorithm} is given
by $$a_1:=\frac{4c_0^2a_0b_0}{2c_0(a_0^2-b_0^2)},\quad
b_1:=\frac{c_0^4-4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)},\quad
c_1:=\frac{c_0^4+4a_0^2b_0^2}{2c_0(a_0^2-b_0^2)}.
$$
Now, by~(\ref{psi}), the coordinates of the rational point
$(x_1,y_1)$ on $C_A$ which corresponds to the {rational Py\-tha\-go\-re\-an $A$-triple}
$(a_1,b_1,c_1)$ are given by
\begin{align*}
x_1&
\frac{a_0b_0\cdot(b_1+c_1)}{2\cdot a_1}=\frac{a_0b_0\cdot 2c_0^4}{2\cdot 4c_0^2a_0b_0}=
\frac{c_0^2}{4}\,,\\
y_1&=\frac{2(\frac{a_0b_0}2)^2(b_1+c_1)}{a_1^2}=\frac18 (a_0^2 - b_0^2) c_0.
\end{align*}
Let still $(a_0,b_0,c_0)$ be a {rational Py\-tha\-go\-re\-an $A$-triple}. Then, again by~(\ref{psi}), the corresponding
rational point $(x_0,y_0)$ on $C_A$ is given by
$$x_0=\frac{b_0(b_0+c_0)}{2}\,,\qquad
y_0=\frac{b_0^2(b_0+c_0)}{2}\,.$$
Now, as we have seen in~(\ref{eq:2P}) and~(\ref{eq:lambda}),
the coordinates of the point $(x'_1,y'_1):=2*(x_0,y_0)$ are given by
$x'_1=\lambda^2-2x_0$, $y'_1=3x_0\lambda-\lambda^3-y_0$, where
\begin{multline*}
\lambda=\frac{3x_0^2-(\frac{a_0b_0}2)^2}{2y_0}=
\frac{\frac{3 b_0^2(b_0+c_0)^2-a_0^2b_0^2}{4}}{b_0^2(b_0+c_0)}=
\frac{3 (b_0+c_0)^2-a_0^2}{4(b_0+c_0)}=
\frac{3 (b_0+c_0)^2+(b_0^2-c_0^2)}{4(b_0+c_0)}=\\[3ex]
\frac{(3b_0^2+6b_0c_0+3c_0^2)+(b_0^2-c_0^2)}{4(b_0+c_0)}=
\frac{4b_0^2+6b_0c_0+2c_0^2}{4(b_0+c_0)}=
\frac{2b_0^2+3b_0c_0+c_0^2}{2(b_0+c_0)}=\\[3ex]
\frac{(2b_0+c_0)(b_0+c_0)}{2(b_0+c_0)}=
\frac{(2b_0+c_0)}{2}\,.
\end{multline*}
Hence,
$$x_1'=\lambda^2-2x_0=\frac{(2b_0+c_0)^2}{4}-b_0(b_0+c_0)=
\frac{(4b_0^2+4b_0c_0+c_0^2)-(4b_0^2+4b_0c_0)}{4}=\frac{c_0^2}{4}\,
$$
and
$$
y'_1=3x_0\lambda-\lambda^3-y_0=\frac18 (2 b_0^2 c_0 - c_0^3)=\frac18(b_0^2-a_0^2)c_0,
$$
\ie, $x_1=x'_1$ and $y_1=-y'_1$, as claimed.
\end{proof}
\begin{comment}
and finally we show that this implies that
for a congruent number $A$, the congruent number curve
$y^2=x^3-A^2x$ does not have rational points of finite
order other than $(0,0)$ and $(\pm A,0)$.
\end{comment}
With {\sc Lemma}\;\ref{lem:doubling}, we are now able to
prove {\sc Theorem}\;\ref{thm:main}, which states that
for a congruent number $A$, the curve
$C_A:y^2=x^3-A^2x$ does not have rational points of finite
order other than $(0,0)$ and $(\pm A,0)$.
\begin{proof}[Proof of Theorem\;\ref{thm:main}]
Assume that $A$ is a congruent number,
let $(x_0,y_0)$ be a rational point on $C_A$ which
$y_0\neq 0$, and let $(a_0,b_0,c_0)$ be the {rational Py\-tha\-go\-re\-an $A$-triple}
which corresponds to $(x_0,y_0)$ by~(\ref{psi-1}).
Furthermore, for positive integers~$n$, let
$(a_n,b_n,c_n)$ be the {rational Py\-tha\-go\-re\-an $A$-triple}
we obtain by {\sc Fermat's Algorithm} from $(a_{n-1},b_{n-1},c_{n-1})$,
and let $(x_n,y_n)$ be the
rational point on $C_A$ which corresponds to the
{rational Py\-tha\-go\-re\-an $A$-triple} $(a_n,b_n,c_n)$ by~(\ref{psi}).
By the proof of {\sc Lemma}\;\ref{lem:doubling} we know that the
$x$-coordinate of $2*(x_n,y_n)$ is equal to $\frac{c_n^2}{4}$,
and by {\sc Theorem}\;\ref{thm:FermatClaim} we have that for any distinct
non-negative integers $n,n'$, $|c_n|\neq |c_{n'}|$.
Hence, for all distinct
non-negative integers $n,n'$ we have
$$(x_n,y_n)\neq (x_{n'},y_{n'}),$$ which shows that the order
of $(x_0,y_0)$ is infinite.
\end{proof}
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|
2,869,038,154,798 | arxiv | \section{Introduction}
Reading comprehension is the task of answering questions pertaining to a given passage. An AI agent which can display such capabilities would be useful in a wide variety of commercial applications such as answering questions from financial reports of a company, troubleshooting using product manuals, answering general knowledge questions from Wikipedia documents, \textit{etc}. Given its widespread applicability, several variants of this task have been studied in the literature. For example, given a passage and a question, the answer could either (i) match some span in the passage or (ii) be generated from the passage or (iii) be one of the $n$ given candidate answers. The last variant is typically used in various high school, middle school, and competitive examinations. We refer to this as Reading Comprehension with Multiple Choice Questions (RC-MCQ). There is an increasing interest in building AI agents with deep language understanding capabilities which can perform at par with humans on such competitive tests. For example, recently \cite{DBLP:conf/emnlp/LaiXLYH17} have released a large scale dataset for RC-MCQ collected from high school and middle school English examinations in China comprising of approximately $28000$ passages and $100000$ questions. The large size of this dataset makes it possible to train and evaluate complex neural network based models and measure the scientific progress on RC-MCQ.
\begin{figure}
\fbox{\begin{minipage}{23em}\small
Passage: \color{red}{One day, I was studying at home. Suddenly, there was a loud noise}...\color{black}{A building in my neighborhood was on fire...A few people jumped out of the window... Those who were still on the second floor were just crying for help...Firefighters arrived at last. They fought the fire bravely.} \color{blue}{Water pipes were used and a ladder was put near the second-floor window.}\color{green}{ Then the people inside were taken out by the firefighters}\color{black}{...}\color{orange}{Thanks to the firefighters, the people inside were saved and the fire was put out in the end}\color{black}{, but many things, such as desk, pictures and clothes, were damaged.} \newline
\color{black}{
\textit{\textbf{Question:} How did the people who didn't jump out of the window get out of the building?}} \newline \color{black}{\textbf{Option A:}} \color{green}{They were taken out by the firefighters.}\\ \color{black}{\textbf{Option B:}} \color{blue}{They climbed down a ladder by themselves.}\\ \color{black}{\textbf{Option C:}} \color{orange}{They walked out after the fire was put out.}\\ \color{black}{\textbf{Option D:} They were taken out by doctors}\\ \textbf{Correct Option:} A
\end{minipage}}
\caption{Example of RC-MCQ from RACE dataset}
\label{fig:example}
\end{figure}
While answering such Multiple Choice Questions (MCQs) (\textit{e.g.}, Figure \ref{fig:example}), humans typically use a combination of \textit{option elimination} and \textit{option selection}. More specifically, it makes sense to first try to eliminate options which are completely irrelevant to the given question. While doing so, we may also be able to discard certain portions of the passage which are not relevant to the question (because they revolve around the option which has been eliminated, \textit{e.g.}, portions marked in blue and orange, corresponding to Option B and Option C respectively in Figure ~\ref{fig:example}). This process can then be repeated multiple times, each time eliminating an option and refining the passage (by discarding irrelevant portions). Finally, when it is no longer possible to eliminate any option, we can pick the best option from the remaining options. In contrast, the current state of the art models for RC-MCQ focus explicitly on option selection. Specifically, given a question and a passage, they first compute a question aware representation of the passage (say $d_q$). They then compute a representation for each of the $n$ options and select an option whose representation is closest to $d_q$. There is no iterative process where options get eliminated and the representation of the passage gets refined in the light of this elimination.
We propose a model which tries to mimic the human process of answering MCQs. Similar to the existing state of the art method \cite{DBLP:conf/acl/DhingraLYCS17}, we first compute a question-aware representation of the passage (which essentially tries to retain portions of the passage which are only relevant to the question). We then use an elimination gate (depending on the passage, question and option) which takes a soft decision as to whether an option needs to be eliminated or not. Next, akin to the human process described above, we would like to discard portions of the passage representation which are aligned with this eliminated option. We do this by subtracting the component of the passage representation along the option representation (similar to Gram-Schmidt orthogonalization). The amount of orthogonalization depends on the soft decision given by the elimination gate. We repeat this process multiple times, during each pass doing a soft elimination of the options and refining the passage representation. At the end of a few passes, we expect the passage representation to be orthogonal (hence dissimilar) to the irrelevant options. Finally, we use a selection module to select the option which is most similar to the refined passage representation. We refer to this model as \textit{ElimiNet}. Note that such a model will not make sense in cases where the options are highly related. For example, if the question is about life stages of a butterfly and the options are four different orderings of the words \textit{butterfly, egg, pupa, caterpillar} then it does not make sense to orthogonalize the passage representation to the incorrect option representations. However, the dataset that we focus on in this work does not contain questions which have such permuted options.
We evaluate \textit{ElimiNet} on the RACE dataset and compare it with Gated Attention Reader (GAR) \cite{DBLP:conf/acl/DhingraLYCS17}, the current state of the art model on this dataset. We show that of the $13$ question types in this dataset our model outperforms GAR on 7 question types. We also visualize the soft elimination probabilities learnt by \textit{ElimiNet} and observe that it indeed learns to iteratively refine the passage representation and push the probability mass towards the correct option. Finally, we show that an ensemble model combining \textit{ElimiNet} with \textit{GAR} gives an accuracy of $47.2\%$ which is $3.1\%$ (relative) better than the best-reported performance on this dataset
The code for our model is publicly available\footnote{\url{https://github.com/sohamparikh94/ElimiNet}}.
\section{Related Work}
Over the last few years, the availability of large scale datasets has led to an increasing interest in the task of Reading Comprehension. These datasets cover different variations of the Reading comprehension task. For example, SQuAD \cite{DBLP:conf/emnlp/RajpurkarZLL16}, TriviaQA \cite{DBLP:conf/acl/JoshiCWZ17}, NewsQA \cite{DBLP:journals/corr/TrischlerWYHSBS16}, MS MARCO \cite{ms-marco-human-generated-machine-reading-comprehension-dataset}, NarrativeQA \cite{DBLP:journals/corr/abs-1712-07040}, \textit{etc.} contain \{\textit{passage, question, answer}\} where the answer matches a span of the passage or it has to be generated. On the other hand, CNN/Daily Mail \cite{DBLP:conf/nips/HermannKGEKSB15}, Children's Book Test (CBT) \cite{DBLP:journals/corr/HillBCW15} and Who Did What (WDW) dataset \cite{DBLP:conf/emnlp/OnishiWBGM16} offer cloze-style RC where the task is to predict a missing word/entity (from the passage) in the question. Some other datasets such as MCTest \cite{DBLP:conf/emnlp/RichardsonBR13}, AI2 \cite{DBLP:conf/ijcai/KhashabiKSCER16} and RACE contain RC with multiple choice questions (RC-MCQ) where the task is to select the right answer.
The advent of these datasets and the general success of deep learning for various NLP tasks, has led to a proliferation of neural network based models for RC. For example, the models proposed in \cite{DBLP:journals/corr/XiongZS16,DBLP:journals/corr/SeoKFH16,DBLP:conf/acl/WangYWCZ17,DBLP:journals/corr/HuPQ17} address the first variant of RC requiring span prediction as in the SQuAD dataset. Similarly, the models proposed in \cite{DBLP:conf/acl/ChenBM16,DBLP:conf/acl/KadlecSBK16,DBLP:conf/acl/CuiCWWLH17,DBLP:conf/acl/DhingraLYCS17} address the second variant of RC requiring cloze-style QA. Finally, \cite{DBLP:conf/emnlp/LaiXLYH17} adapt the the models proposed in \cite{DBLP:conf/acl/ChenBM16,DBLP:conf/acl/DhingraLYCS17} for cloze-style RC and use them to address the problem of RC-MCQ. Irrespective of which of the three variants of RC they address, these models use a very similar framework. Specifically, these models contain components for (i) encoding the passage (ii) encoding the question (iii) capturing interactions between the question and the passage (iv) capturing interactions between question and the options (for MCQ) (v) making multiple passes over the passage and (vi) a decoder to predict/generate/select an answer. The differences between the models arise from the specific choice of the encoder, decoder, interaction functions and iteration mechanism. Most of the current state of the art models can be seen as special instantiations of the above framework.
The key difference between our model and existing models for RC-MCQ is that we introduce components for (soft-)eliminating irrelevant options and refining the passage representation in the light of this elimination. The passage representation thus refined over multiple (soft-)elimination rounds is then used for selecting the most relevant option. To the best of our knowledge, this is the first model which introduces the idea of option elimination for RC-MCQ.
\begin{figure}
\includegraphics[width=0.49\textwidth]{eliminet.png}
\caption{A simplistic diagram of the proposed model}
\label{fig:diagram}
\end{figure}
\section{Proposed Model}
Given a passage $D = [w_1^d, w_2^d,\ldots, w_M^d]$ of word-length $M$, a question $Q = [w_1^q, w_2^q,\ldots, w_N^q ]$ of word-length $N$ and $n$ options $Z_k = [w_1^z, w_2^z, \ldots, w_{J_k}^z]$ where $1 \leqslant k \leqslant n$ and each option is of word-length $J_k$, the task is to predict a conditional probability distribution over the options (\textit{i.e.}, to predict $P(Z_i| D, Q)$). We model this distribution using a neural network which contains modules for encoding the passage/question/options, capturing the interactions between them, eliminating options and finally selecting the correct option. We refer to these as the encoder, interaction, elimination and selection modules as shown in Figure \ref{fig:diagram}. Among these, the main contribution of our work is the introduction of a module for elimination. Specifically, we introduce a module to (i) decide whether an option can be eliminated (ii) refine the passage representation to account for eliminated/un-eliminated options and (iii) repeat this process multiple times. In the remainder of this section, we describe the various components of our model.
\paragraph{Encoder Module:} We first compute vectorial representations of the question and options. We do so by using a bidirectional recurrent neural network which contains two Gated Recurrent Units (GRU) \cite{DBLP:journals/corr/ChungGCB14}, one which reads the given string (question or option) from left to right and the other which reads the string from right to left. For example, given the question $Q = [w_1^q, w_2^q,\ldots, w_N^q ]$, each GRU unit computes a hidden representation for each time-step (word) as:
\begin{align*}
\overrightarrow{h_i^q} = \overrightarrow{GRU_q} (\overrightarrow{h_{i-1}^q} , e(w_i^q)) \\
\overleftarrow{h_i^q} = \overleftarrow{GRU_q} (\overleftarrow{h_{i-1}^q} , e(w_i^q))
\end{align*}
where $e(w_i^q) \in \mathbb{R}^d$ is the $d$-dimensional embedding of the question word $w_i^q$. The final representation of each question word is a concatenation of the forward and backward representations (\textit{i.e.}, $h_i^q = [\overleftarrow{h_i^q}, \overrightarrow{h_i^q}]$). Similarly, we compute the bi-directional representations for each word in each of the $k$ options as $h_i^{z_k} = [\overleftarrow{h_i^{z_k}}, \overrightarrow{h_i^{z_k}}]$. Just to be clear, $h_i^{z_k}$ is the representation of the $i$-th word in the $k$-th option ($z_k$). We use separate GRU cells for the question and options, with the same GRU cell being used for all the $n$ options. Note that the encoder also computes a representation of each passage word as simply the word embedding of the passage word (\textit{i.e.}, $h_i^d = e(w_i^d)$). Later on in the interaction module we use a GRU cell to compute the interactions between the passage words.
\paragraph{Interaction Module:} Once the basic question and passage word representations have been computed, the idea is to allow them to interact so that the passage words' representations can be refined in the light of the question words' representations. This is similar to how humans first independently read the passage and the question and then read the passage multiple times, trying to focus on the portions which are relevant and ignoring portions that are irrelevant (\textit{e.g.}, portion marked in red in Figure \ref{fig:example}) to the question. To achieve this, we use the same multi-hop architecture for iteratively refining passage representations as proposed in Gated Attention Reader \cite{DBLP:conf/acl/DhingraLYCS17}. At each hop $t$, we use the following set of equations to compute this refinement:
\begin{align}
\nonumber \alpha^{t}_i &= \text{softmax}(Q^Td_i^{t})
\end{align}
where, $Q \in \mathbb{R}^{N \times l}$ is a matrix whose columns are $h_1^q, h_2^q, ..., h_N^q$ as computed by the encoder. $\alpha^{t}_{i} \in \mathbb{R}^N$ such that each element $j$ of $\alpha^{t}_{i}$ essentially computes the importance of the $j$-th question word for the $i$-th passage word during hop $t$. At the 0-th hop, $d^{0}_i = h_i^d = e(w_i^d) \in \mathbb{R}^l$ is simply the embedding of the $i$-th passage word. The goal is to refine this embedding over each hop based on interactions with the question. Next, we compute,
\begin{align}
\nonumber \tilde{q}^{t}_i &= Q\alpha^{t}_i
\end{align}
where $\tilde{q}^{t}_i \in \mathbb{R}^l$ computes the importance of each dimension of the current passage word representation and is then used as a gate to scale up or scale down different dimensions of the passage word representation.
\begin{align}
\nonumber \tilde{d}^{t}_i &= d^{t}_i \odot \tilde{q}^{t}_i
\end{align}
We now allow these refined passage word representations to interact with each other using a bi-directional recurrent neural network to compute $d^{(t+1)}_i$ for the next hop.
\begin{align}
\nonumber \overrightarrow{d^{(t+1)}_{i}} &= \overrightarrow{\text{GRU}}_D^{(t+1)}(\overrightarrow{d^{(t+1)}_{i-1}},\tilde{d}^{(t)}_{i})\\
\nonumber \overleftarrow{d^{(t+1)}_{i}} &= \overleftarrow{\text{GRU}}_D^{(t+1)}(\overleftarrow{d^{(t+1)}_{i-1}},\tilde{d}^{(t)}_{i})\\
\nonumber
d^{(t+1)}_{i} &= [\overleftarrow{d^{(t+1)}_{i}}, \overrightarrow{d^{(t+1)}_{i}}]
\end{align}
The above process is repeated for $T$ hops wherein each hop takes $d^{(t)}_i, Q$ as the input and computes a refined representation $\tilde{d}^{(t+1)}_i$. After $T$ hops, we obtain a fixed-length vector representation of the passage by combining the passage word representations using a weighted sum.
\begin{align}
\nonumber m_{i} &= \text{softmax}(\tilde{d}^{(T)}_{i}W_{att}h^{q}_{N})\\
\label{doc} x &= \sum_{i=1}^{M}m_{i}\tilde{d}^{(T)}_{i}
\end{align}
where $m_{i}$ computes the importance of each passage word and $x$ is a weighted sum of the passage representations.
\paragraph{Elimination Module:} The aim of the elimination module is to refine the passage representation so that it does not focus on portions which correspond to irrelevant options. To do so we first need to decide whether an option can be eliminated or not and then ensure that the passage representation gets modified accordingly. For the first part, we introduce an \textit{elimination} gate to enable a soft-elimination.
\begin{align}
\nonumber e_i = \text{sigmoid}(W_ex + V_eh^{q} + U_eh^{z_i})
\end{align}
Note that this gate is computed separately for each option $i$. In particular, it depends on the final state of the bidirectional option GRU ($h^{z_i} = h^{z_i}_{J_i}$). It also depends on the final state of the bidirectional question GRU ($h^{q} = h^{q}_{N}$) and the refined passage representation ($x$) computed by the interaction module. $W_e, V_e, U_e$ are parameters which will be learned.
Based on the above soft-elimination, we want to now refine the passage representation. For this, we compute $x^{e}_{i}$ which is the component of the passage representation ($x$) orthogonal to the option representation ($h^{z_i}$) and $x^{r}_{i}$ which is the component of the passage representation along the option representation.
\begin{align}
\nonumber r_i &= \frac{<x,h^{z_i}>h^{z_i}}{|x|^{2}}\\
\label{Xe} x^{e}_{i} &= x - r_{i} \\
\label{Xk} x^{r}_{i} &= x - x^{e}_{i}
\end{align}
The \textit{elimination gate} then decides how much of $x^{e}_{i}$ and $x^{r}_{i}$ need to be retained.
\begin{align}
\nonumber \tilde{x}_{i} &= e_{i}\odot x^{e}_{i} + (1 - e_{i})\odot x^{r}_{i}
\end{align}
If ${e}_{i} = 1$ (eliminate, \textit{e.g.}, portions corresponding to Option D in Figure \ref{fig:example}) then the passage representation will be made orthogonal to the option representation (akin to ignoring portions of the passage relevant to the option) and ${e}_{i} = 0$ (don't eliminate, \textit{e.g.}, portions marked in green, corresponding to Option A in Figure \ref{fig:example}) then the passage representation will be aligned with the option representation (akin to focusing on portions of the passage relevant to the option).
Note that in equations \eqref{Xe} and \eqref{Xk} we completely subtract the components along or orthogonal to the option representation. We wanted to give the model some flexibility to decide how much of this component to subtract. To do this we introduce another gate, called the subtract gate,
\begin{align}
\nonumber s_{i} &= \text{sigmoid}(W_{s}x+ V_{s}h^{q} + U_{s}h^{z_{i}})
\end{align
where $W_s , V_s , U_s$ are parameters that need to be learned. We then replace the RHS of Equations \ref{Xe} and \ref{Xk} by $x - s_{i} \odot r_{i}$ and $x - s_{i} \odot x^e_{i}$ respectively. Thus the components $r_i$ and $r^{\perp}_i$ used in Equation \eqref{Xe} and \eqref{Xk} are gated using $s_i$. One could argue that $e_i$ itself could encode this information but empirically we found that separating these two functionalities (elimination and subtraction) works better.
For each of the $n$ options, we independently compute representations $\tilde{x}_{1}, \tilde{x}_{2}, ... ,\tilde{x}_{n}$. These are combined to obtain a single refined representation for the passage.
\begin{align}
\nonumber b_{i} &= v^{T}_{b}\text{tanh}(W_{b}\tilde{x}_{i} + U_{b}h^{z_{i}})\\
\nonumber \beta_{i} &= \text{softmax}(b_{i})\\
\label{tildeX} \tilde{x} &= \sum^{n}_{i=1}\beta_{i}\tilde{x}_{i}
\end{align}
Note that $\tilde{x}_{1}, \tilde{x}_{2}, ... ,\tilde{x}_{n}$ represent the $n$ option-specific passage representations and $\beta_{i}$'s give us a way of combining these option specific representations into a single passage representation. We repeat the above process for $L$ hops wherein the $m$-th hop takes $\tilde{x}^{m-1}$, $h^q$ and $h^{z_i}$ as input and returns a refined $\tilde{x}^{m}$ computed using the above set of equations.
\paragraph{Selection Module} Finally, the selection module takes the refined passage representation $\tilde{x}^L$ after $L$ elimination hops and computes its bilinear similarity with each option representation.
\begin{align}
\nonumber \text{score}(i) &= \tilde{x}^{L}W_{att}h^{z_{i}}
\end{align}
where $\tilde{x}^{L}$ and $h^{z_{i}}$ are vectors and $W_{att}$ is a matrix which needs to be learned. We select the option which gives the highest score as computed above. We train the model using the cross entropy loss by normalizing the above scores (using softmax) first to obtain a probability distribution.
\section{Experimental Setup}
In this section, we describe the dataset used for evaluation, the hyperparameters of our model, training procedure and state of the art models used for comparison.
\newline
\newline
\textbf{Dataset:} We evaluate our model on the RACE dataset which contains multiple choice questions collected from high school and middle school English examinations in China. The high school portion of the dataset (RACE-H) contains $62445$, $3451$ and $3498$ questions in train, validation, and test sets respectively. The middle school portion of the dataset (RACE-M) contains $18728$, $1021$ and $1045$ questions for train, validation, and test sets respectively.
\newline
This dataset contains a wide variety of questions of varying degrees of complexity. For example, some questions ask for the most appropriate title for the passage which requires deep language understanding capabilities to comprehend the entire passage. There are some questions which ask for the meaning of a specific term or phrase in the context of the passage. Similarly, there are some questions which ask for the key idea in the passage. Finally, there are some standard Wh-type questions. Given this wide variety of questions, we wanted to see if there are specific types of questions for which an elimination module makes more sense. To do so, with the help of in-house annotators, we categorize the questions in the test dataset into the following $13$ categories using scripts with manually defined rules: (i) $6$ Wh-question types, (ii) questions asking for the title/meaning/key idea of the passage, (iii) questions asking whether the given statement is True/False, (iv) questions asking for a quantity (e.g., how much, how many) (v) fill-in-the-blanks questions. We were able to classify $91.26\%$ of questions in the test set into these $12$ categories and the remaining $8.74\%$ of questions were labeled as miscellaneous.
The distribution of questions belonging to each of these categories in RACE-H and RACE-M are shown in Figure \ref{fig:cat_dist1}.
\begin{figure}[!t]
\centering
\begin{subfigure}{.38\textwidth}
\includegraphics[width=1\linewidth]{pie_mid}
\end{subfigure}%
\hfill
\begin{subfigure}{.38\textwidth}
\includegraphics[width=1\linewidth]{pie_high}
\end{subfigure}
\caption{Distribution of different question types in the RACE-Mid (top) and RACE-High (bottom) portions of the dataset}
\label{fig:cat_dist1}
\end{figure}
\paragraph{Training Procedures:}
We try two different ways of training the model. In the first case, we train the parameters of all the modules (encoder, interaction, elimination, and selection) together. In the second case, we first remove the elimination module and train the parameters of the remaining modules. We then fix the parameters of the encoder and interaction module and train only the elimination and selection module. The idea was to first help the model understand the document better and later focus on elimination of options (in other words, ensure that the entire learning is focused on the elimination module). Of course, we also had to learn the parameters of the selection module from scratch because it now needs to work with the refined passage representations. Empirically, we find that this pre-training step does not improve over the performance obtained by end-to-end training. Hence, we report results only for the first case (\textit{i.e.}, end-to-end training).
\newline
\newline
\textbf{Hyperparameters:}
We restrict our vocabulary to the top 50K words appearing in the passage, question, and options in the dataset. We use the same vocabulary for the passage, question, and options. We use the same train, valid, test splits as provided by the authors. We tune all our models based on the accuracy achieved on the validation set. We initialize the word embeddings with $100$ dimensional Glove embeddings \cite{DBLP:conf/emnlp/PenningtonSM14}. We experiment with both fine-tuning and not fine-tuning these word embeddings. The hidden size for BiGRU is the same across the passage, question, and option and we consider the following sizes :$\{64, 128, 256\}$. We experiment with $\{1, 2, 3\}$ hops in the interaction module and $\{1,3,6\}$ passes in the elimination module. We add dropout at the input layer to the BiGRUs and experiment with dropout values of $\{0.2, 0.3, 0.5\}$. We try both Adam and SGD as the optimizer. For Adam, we set the learning rate to $10^{-3}$ and for SGD we try learning rates of $\{0.1, 0.3, 0.5\}$. In general, we find that Adam converges much faster. We train all our models for upto 50 epochs as we do not see any benefit of training beyond 50 epochs.
\newline
\newline
\textbf{Models Compared:} We compare our results with the current state of the art model on RACE dataset, namely, Gated Attention Reader \cite{DBLP:conf/acl/DhingraLYCS17}. This model was initially proposed for cloze-style RC and is, in fact, the current state of the art model for cloze-style RC. The authors of RACE dataset adapt this model for RC-MCQ by replacing the output layer with a layer which computes the bilinear similarity between the option and passage representations.
\begin{figure}[!b]
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=1\textwidth]{one_model}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=1\linewidth]{mid_plot}
\end{subfigure}
\begin{subfigure}{0.49\textwidth}
\includegraphics[width=1\linewidth]{high_plot}
\end{subfigure}
\caption{Performance of ElimiNet and Gated Attention Reader (GAR) on different question categories in RACE-Full (top), RACE-Mid (mid) and RACE-High (bottom). The categories in which our model outperforms GAR are marked with *.}
\label{perf_comp_mid}
\end{figure}
\section{Results and Discussions}
In this section, we discuss the results of our experiments.
\if 0
\begin{figure}
\centering
\includegraphics[width=1\linewidth, height=5cm]{one_model}
\caption{Performance of ElimiNet and Gated Attention Reader (GAR) on different question categories in RACE-full. The categories in which our model outperforms GAR are marked with *.}
\label{perf_comp_full}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{high_plot}
\caption{Performance of ElimiNet and Gated Attention Reader (GAR) on different question categories in RACE-High. The categories in which our model outperforms GAR are marked with *.}
\label{perf_comp_high}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=1\linewidth]{mid_plot}
\caption{Performance of ElimiNet and Gated Attention Reader (GAR) on different question categories in RACE-Mid. The categories in which our model outperforms GAR are marked with *.}
\label{perf_comp_mid}
\end{figure}
\fi
\if 0
\begin{figure}[!bh]
\begin{subfigure}{1\textwidth}
\includegraphics[width=1\textwidth]{one_model}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=1\linewidth]{mid_plot}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=1\linewidth]{high_plot}
\end{subfigure}
\caption{Category-wise performance of End-to-End ElimiNet on questions from RACE-Mid and RACE-High test sets. Our model performs better than baseline on 9 and 6 categories out of $13$ on RACE-Mid and RACE-High respectively.}
\label{perf_comp_mid}
\end{figure}
\fi
\subsection{Performance of Individual Models}
We compare the accuracy of different models on RACE-Mid (middle school), RACE-High (high school) and full RACE test-set comprising of both RACE-Mid and RACE-High. For each dataset, we compare the accuracy for each question type. These results are summarized in Figure \ref{perf_comp_mid}. We observe that on RACE-Mid ElimiNet performs better than Gated Attention Reader (GAR) on $9$ out of $13$ categories. Similarly, on RACE-High ElimiNet performs better than GAR on $6$ out of $13$ categories. Finally, on RACE-full, ElimiNet performs better than GAR on $7$ out of $13$ categories. Note that, overall on the entire test set (combining all question types) our model gives a slight improvement over GAR. The main reason for this is that the dataset is dominated by fill in the blank style questions and our model performs worse by only $2\%$ on such questions. However, since nearly $50\%$ of the questions in the dataset are fill in the blank style questions even a small drop in the performance on these questions, offsets the gains that we get on other question types.
\subsection{Ensemble of Different Models}
Since ElimiNet and GAR perform well on different question types we believe that taking an ensemble of these models should lead to an improvement in the overall performance. For a fair comparison, we also want to see the performance when we independently take an ensemble of $n$ GAR models and $n$ ElimiNet models. We refer to these as GAR-ensemble and ElimiNet-ensemble models. Each model in the ensemble is trained using a different hyperparameter setting and we use $n =6$ (we do not see any benefit of using $n > 6$). The results of these experiments are summarized in Table \ref{table:perfcomp}. ElimiNet-ensemble performs better than GAR-ensemble and the final ensemble gives the best results. We observe the ElimiNet-ensemble performs significantly better on RACE-Mid dataset than the GAR-ensemble and gives almost the same performance on the RACE-High dataset. Overall, by taking an ensemble of the two models we get an accuracy of $47.2\%$ which is $3.1\%$ (relative) better than GAR and $1.3\%$ (relative) better than GAR-ensemble.
\subsection{Effect of Subtract Gate}
We wanted to see if the subtract gate enables the model to learn better (by performing partial orthogonalization/alignment). For this, we compared the accuracy with and without the subtract gate (we set the subtract gate to a vector of $1$s). We observed that the accuracy of our model drops from $44.33\%$ to $42.58\%$ and we outperformed the GAR model only in $3$ out of $13$ categories. This indicates that the flexibility offered by the subtract gate does help the model
\if 0
\begin{figure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=\linewidth]{one_model}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\includegraphics[width=0.5\linewidth]{mid_plot}
\end{subfigure}
\begin{subfigure}
\includegraphics[width=0.5\linewidth]{high_plot}
\end{subfigure}
\caption{Category-wise performance of End-to-End ElimiNet on questions from RACE-Mid and RACE-High test sets. Our model performs better than baseline on $9$ and $6$ categories out of $13$ on RACE-Mid and RACE-High respectively.}
\label{perf_comp_mid}
\end{figure}
\fi
\begin{table}[!tbh]
\centering
\begin{tabular}{p{3.6cm}p{1.0cm}p{1.0cm}p{1.0cm}}
\hline
\textbf{Model} & \textbf{RACE-Mid} & \textbf{RACE-High} & \textbf{RACE-Full} \\ \hline
SA Reader & 44.2 & 43.0 & 43.3 \\
GA Reader (GAR) & 43.7 & 44.2 & 44.1 \\
\textbf{ElimiNet } & \textbf{44.4} & \textbf{44.5} & \textbf{44.5} \\
\cline{1-4}
GAR Ensemble & 45.7 & 46.2 & 45.9 \\
ElimiNet Ensemble & \textbf{47.7} & 46.1 & 46.5 \\
GAR + ElimiNet (ensemble of above 2 ensembles) & 47.4 & \textbf{47.4} & \textbf{47.2} \\
\cline{1-4}
\end{tabular}
\caption{Performance of individual and ensemble models}
\label{table:perfcomp}
\end{table}
\begin{figure}[!t]
\centering
\begin{subfigure}{0.325\textwidth}
\includegraphics[width=1\linewidth]{4_probs}
\end{subfigure}%
\hfill
\begin{subfigure}{0.325\textwidth}
\includegraphics[width=1\linewidth]{30_probs}
\end{subfigure}
\caption{Change in the probability of correct option and incorrect option (initially predicted with highest score) over multiple passes of the \textit{elimination} module. The two figures correspond to two different examples from the test set.}
\label{fig:vis}
\end{figure}
\if 0
\begin{table}[!tbh]
\centering
\begin{tabular}{cccc}
\hline
\textbf{Model} & \textbf{RACE} \\ \hline
Stanford Attentive Reader & 43.3 \\
Gated Attention Reader (GAR) & 44.1 \\
\textbf{ElimiNet } & \textbf{44.5} \\
\cline{1-4}
GAR Ensemble (6 models) & 45.9 \\
ElimiNet Ensemble (6 models) & 46.5 \\
GAR + ElimiNet & \textbf{47.2} \\
(ensemble of above 2 ensembles) & & & \\
\cline{1-4}
\end{tabular}
\caption{Performance Comparison for Proposed Models}
\label{table:perfcomp}
\end{table}
\fi
\if 0
\begin{table}[!h]
\begin{minipage}{0.5\textwidth}
\begin{tabular}{|c|c|c|c|l}
\cline{1-4}
\textbf{Iterations} & \textbf{Race-M} & \textbf{Race-H} & \textbf{Race} & \\ \cline{1-4}
1 & 43.11 & 43.6 & 43.51 & \\
3 & 43.7 & 44.6 & 44.3 & \\
6 & 43.5 & 44.0 & 43.8 & \\
\textbf{9} & \textbf{44.4} & \textbf{44.5} & \textbf{44.5} & \\ \cline{1-4}
\end{tabular}
\caption{2-stage training for ElimiNet}
\end{minipage}
\begin{minipage}{0.5\textwidth}
\begin{tabular}{|c|c|c|c|l}
\cline{1-4}
\textbf{Iterations} & \textbf{Race-M} & \textbf{Race-H} & \textbf{Race} & \\ \cline{1-4}
1 & 43.25 & 43.05 & 43.11 & \\
3 & 44.36 & 43.57 & 43.8 & \\
\textbf{6} & \textbf{44.5 } & \textbf{43.68} & \textbf{43.92} & \\
9 & 42.20 & 43.45 & 43.09 & \\
\cline{1-4}
\end{tabular}
\caption{End-to-End training for ElimiNet}
\end{minipage}
\end{table}
\fi
\subsection{Visualizing Shift in Probability Scores}
If the elimination module is indeed learning to eliminate options and align/orthogonalize the passage representation w.r.t the uneliminated/eliminated options then we should see a shift in the probability scores as we do multiple passes of elimination. To visualize this, in Figure \ref{fig:vis}, we plot the probabilities of the correct option and the incorrect option with the highest probability before passing through $elimination$ module for two different test instances. We observe that as we do multiple passes of elimination, the probability mass shifts from the incorrect option (blue curve) to the correct option (green curve). This indicates that the elimination module is learning to align the passage representation with the correct option (hence, increasing its similarity) and moves it away from the incorrect option (hence, decreasing its similarity).
\section{Conclusion}
We focus on the task of Reading Comprehension with Multiple Choice Questions and propose a model which mimics how humans approach this task. Specifically, the model uses a combination of elimination and selection to arrive at the correct option. This is achieved by introducing an elimination module which takes a soft decision as to whether an option should be eliminated or not. It then modifies the passage representation to either align it with uneliminated options or orthogonalize it to eliminated options. The amount of orthogonalization or alignment is determined by two gating functions. This process is repeated multiple times to iteratively refine the passage representation. We evaluate our model on the recently released RACE dataset and show that it outperforms current state of the art models on $7$ out of $13$ question types. Finally, using an ensemble of our elimination-selection approach with a state of the art selection approach, we get an improvement of $3.1\%$ over the best reported performance on RACE dataset. As future work, instead of soft elimination we would like to use reinforcement learning techniques to learn a policy for hard elimination.
\section*{Acknowledgments}
We thank Google for supporting Preksha Nema through their Google India Ph.D. Fellowship program. We thank our anonymous reviewer for suggesting the butterfly example which is mentioned in the introduction.
\bibliographystyle{named}
|
2,869,038,154,799 | arxiv | \section{Introduction}
\label{sec_introduction}
As discussed in
\citet[][hereafter Paper~I]{per04}
and
\citet[][hereafter Paper~II]{per05},
accretion disks are commonly believed to be present in both cataclysmic
variables
(CVs)
and quasi-stellar objects and active galactic nuclei
(QSOs/AGNs).
A property that CVs and QSOs have in common is that both types of
objects sometimes present blue-shifted absorption troughs in UV
resonance lines,
giving direct observational evidence for an outflowing wind.
Both types of objects also show the existence of a persistent velocity
structure in their absorption troughs
(when present)
over significantly long time scales (Paper~I).
In order for a disk wind to account for the wide/broad resonance line
absorption structures observed in many CVs and QSOs,
it must be able to account for the steady velocity structure that is
observed.
The 2.5D time-dependent disk wind models of Pereyra and collaborators,
both for CVs and QSOs,
have steady disk wind solutions
\citep{per97a,per97b,per00,hil02,per03}.
The earlier 1D disk wind models of Murray and collaborators also find
steady disk wind solutions
\citep{mur95,mur96,chi96,mur98}.
However,
in the literature it has been argued that
line-driven disk winds are ``intrinsically unsteady''
because of the increasing gravity along the streamlines at wind base
that is characteristic of disk winds
(see Paper~I; Paper~II).
Since the steady nature of CV and QSO wind flows is an observational
constraint,
whether line-driven disk winds are steady is a significant issue.
Our objective is to study this issue through a semi-analytical method
independent of our previous 2.5D numerically intensive simulation
efforts
\citep[e.g.,][]{per97a,per00}.
In Paper~I we showed that an increase in gravity at wind base,
that is characteristic of disk winds,
does {\it not} imply an unsteady wind solution.
We also developed mathematically ``simple'' models that mimic the disk
environment and we showed that
{\it line-driven disk winds can be steady}.
In Paper~II we extended the concepts presented in Paper~I,
and discussed in detail many aspects that were mentioned in Paper~I.
The goal of this paper is to apply the concepts introduced in Paper~I
and Paper~II to more realistic models that implement the exact flux
distribution above a standard accretion disk.
We compare and find good agreement with the steady CV disk wind
solutions found through numerically intensive calculations by
\citet{per97a}
and
\citet{per00}.
In \S\ref{sec_generalcomments} we present a general discussion of the
reduction of the 2.5D models of \citet{per97a,per00} to a series of 1D
models by neglecting radial terms.
We shall see that,
although this a rough approximation high above the disk,
it is quite accurate near the disk surface.
We reduce the 2.5D model of Pereyra and collaborators
to a series of 1D equations in \S\ref{sec_reduction}.
In \S\ref{sec_results} we present the results of our series of 1D models
for different disk radius.
We show that the critical point in the inner disk regions is found near
the disk surface,
allowing for an analysis of the solutions with steady wind mass loss
rates.
A summary and conclusions are presented in \S\ref{sec_sumcon}.
\section{General Comments}
\label{sec_generalcomments}
The analysis of existence for steady line-driven wind solutions
discussed in Paper~I and Paper~II correspond to 1D models.
Thus,
to apply them to a 2.5D model,
we first reduce the 2.5D model to a series of 1D models.
To do this we consider the following:
the wind in CVs tend to be vertical
(perpendicular to the disk).
There is observational,
theoretical,
and computational evidence for this.
In CVs the P-Cygni type line profiles are observed only in low
inclination systems
(close to disk face-on)
and with high inferred disk mass accretion rates
\citep[e.g.,][]{war95}.
This orientation angle effect was one of the observational properties
of CVs that led to the early suggestion by
\citet{cor82}
that the outflows detected in CVs originate from the accretion disk.
From a theoretical standpoint,
if the wind originates from a steady disk,
then the radial forces at wind base
(disk surface)
must be at equilibrium,
thus the wind must start off vertical.
We note here that by
``vertical''
we actually mean
``non-radial,''
since the wind starts off with an azimuthal component due to disk
rotation.
We also note here that although the disk wind must in principal start
vertical at its base,
this does not necessarily imply that the wind will continue
predominantly vertical.
For example,
in the case of QSOs/AGNs,
disk wind models
\citep{mur95,hil02}
indicate that the wind tends to quickly become radial in nature.
However,
computational 2.5D models of CV disk winds have produced winds that
tend to be vertical in nature.
Theoretical resonance line profiles calculated from these models are
consistent with observed line profiles in their general form,
in the magnitudes of wind velocities implied by the absorption
components,
in the FWHM of the emission components,
and in the strong dependence with inclination angle
\citep{per97a,per00}.
Therefore,
near the wind base
(disk surface),
a CV disk wind at a given radius can be well approximated by a vertical
1D model.
High above the disk rotational forces come into play
(no longer at equilibrium with the radial component of gravity),
streamlines collide,
and the density/velocity wind structure will be 2.5D
(i.e.,
2D with an azimuthal velocity component)
\citep{per97a,per97b,per00,per03}.
Although a 1D vertical model
is not accurate high above the disk for the above reasons,
it does give accurate results near wind base.
As was shown by \citet{cas75} for early type stars,
and was discussed in Paper~II for 1D line-driven winds,
the wind mass loss rate is determined by the position of the critical
point.
As we show below,
for the CV parameters we use in this section
[taken from the models of
\citet{per00}],
at radii where there is significant contributions to the wind mass
loss,
the critical point is near the disk surface.
Therefore,
disk wind characteristics for 2.5D CV models can be addressed.
\section{Reduction to a Series of 1D Models}
\label{sec_reduction}
The equations of state, mass, and momentum of the 2.5D CV disk wind
model of
\citet{per00}
are, respectively:
\begin{equation}
P
=
\rho \, b^2
\;\;\;\; ;
\end{equation}
\begin{equation}
\label{equ_pkb2_masscons}
{\partial\rho \over \partial t}
+ {1 \over r}{\partial(r \rho V_r) \over \partial r}
+ {\partial(\rho V_z) \over \partial z}
=
0
\;\;\;\; ;
\end{equation}
\noindent
and
\clearpage
\begin{eqnarray}
\rho{\partial V_r \over \partial t}
+ \, \rho V_r{\partial V_r \over \partial r}
- \rho {{V_\phi}^2 \over r}
+ \rho V_z{\partial V_r \over \partial z}
&=&
- \, \rho{GM_{wd} \over (r^2 + z^2)}{r \over (r^2+ z^2)^{1/2}}
- {\partial P \over \partial r}
+ \rho{\kappa_e \, {\cal F}_r(r,z) \over c}
\nonumber
\\
&& \\
&& + \, \rho {\kappa_e \, S_r(r,z) \over c} \;
\times
k \left( \max \left[{ 1 \over \rho \kappa_e V_{th}}
\left|
{\partial V_z \over \partial z}
\right|\; ,\; 10^8 \right] \;
\right)^\alpha
\;\;\;\; ;
\nonumber
\end{eqnarray}
\begin{equation}
\rho {\partial V_\phi \over \partial t}
+ \rho V_r{\partial V_\phi \over \partial r}
+ \rho {V_\phi V_r \over r}
+ \rho V_z{\partial V_\phi \over \partial z}
=
0
\;\;\;\; ;
\end{equation}
\begin{eqnarray}
\label{equ_pkb2_momz}
\rho{\partial V_z \over \partial t}
+ \rho V_r{\partial V_z \over \partial r}
+ \rho V_z{\partial V_z \over \partial z}
&=&
- \, \rho{GM_{wd} \over (r^2+z^2)}{z \over (r^2+z^2)^{1/2}}
- {\partial P \over \partial z}
+ \rho{\kappa_e \, {\cal F}_z(r,z) \over c}
\nonumber
\\
&& \\
&& + \, \rho {\kappa_e \, S_z(r,z) \over c} \;
\times k \left( \max \left[{ 1 \over \rho \kappa_e V_{th}}
\left|
{\partial V_z \over \partial z}
\right|\; ,\; 10^8 \right] \;
\right)^\alpha \nonumber
\;\;\;\; ;
\end{eqnarray}
\noindent
where
$P$
is the pressure,
$\rho$
is the density,
$b$
is the isothermal sound speed,
$V_r$,
$V_\phi$,
and
$V_z$
are the corresponding velocity components in cylindrical coordinates,
$G$
is the gravitational constant,
$M_{wd}$
is the mass of the white dwarf,
$\kappa_e$
is the Thomson cross section per mass,
$c$
is the speed of light,
$V_{th}$
is the ion thermal velocity,
and
$k$
and
$\alpha$
are the CAK line force parameters.
${\cal F}_z$
and
${\cal F}_r$
are the corresponding radiation flux components given by:
\begin{equation}
\label{equ_pkb2_fz}
{\cal F}_z(r,z)
=
\int_{r_i}^\infty \int_0^{2\pi} {Q(r') \over \pi} \;
{z^2
\over
\left[\left(r^2+r'^2+z^2-2rr'\cos\phi\right)^{1/2}\right]^4 }
\; r' d\phi dr'
\;\;\;\; ,
\end{equation}
\noindent
and
\begin{equation}
\label{equ_pkb2_fr}
{\cal F}_r(r,z)
=
\int_{r_i}^\infty \int_0^{2\pi} {Q(r') \over \pi} \;
{z (r-r'\cos\phi)
\over
\left[\left(r^2+r'^2+z^2-2rr'\cos\phi\right)^{1/2}\right]^4 }
\; r' d\phi dr'
\;\;\;\; ,
\end{equation}
\noindent
$S_z$
and
$S_r$
are defined as:
\begin{equation}
\label{equ_pkb2_sz}
S_z(r,z)
\equiv
\int_{r_i}^\infty \int_0^{2\pi} {Q(r') \over \pi} \;
{z^{\alpha+2}
\over
\left[\left(r^2+r'^2+z^2-2rr'\cos\phi\right)^{1/2}\right]^{\alpha+4}}
\; r' d\phi dr'
\;\;\;\; ,
\end{equation}
\noindent
and
\begin{equation}
\label{equ_pkb2_sr}
S_r(r,z)
\equiv
\int_{r_i}^\infty \int_0^{2\pi} {Q(r') \over \pi} \;
{z^{\alpha+1}(r-r'\cos\phi)
\over
\left[\left(r^2+r'^2+z^2-2rr'\cos\phi\right)^{1/2}\right]^{\alpha+4}}
\; r' d\phi dr'
\;\;\;\; ,
\end{equation}
\noindent
where in turn
$Q(r)$
is the rate of energy per area,
at a given radius
$r$,
radiated by a standard disk
\citep{sha73},
that is:
\begin{equation}
\label{equ_pkb2_q1}
Q(r)
=
{3 \dot M_{accr} G M_{wd} \over 8 \pi r^3}
\left[1 - \left({r_i \over r} \right)^{1/2} \right]
\;\;\;\; ,
\end{equation}
\noindent
where
$r_i$
is the inner disk radius.
In CVs,
$r_i$ is approximately equal to the white dwarf radius.
We note that in equations~(\ref{equ_pkb2_fz})-(\ref{equ_pkb2_sr})
there is an implicit assumption that
$r_i \ll r_f$,
where
$r_f$
is the outer disk radius.
The total disk luminosity
${\cal L}_{disk}$
is given by
\begin{equation}
{\cal L}_{disk}
=
\int_{r_i}^{r_f} 4 \pi r Q(r) dr
\;\;\;\; ,
\end{equation}
\noindent
where
$r_f$
is the outer disk radius.
Thus
\begin{equation}
{\cal L}_{disk}
=
{\dot M_{accr} G M_{wd} \over 2 r_i}
\left\{ 1 - {3r_i \over r_f}
\left[ 1 - {2 \over 3}
\left({r_i \over r_f} \right)^{1/2}
\right]
\right\}
\;\;\;\; ;
\end{equation}
\noindent
assuming
$r_i \ll r_f$
\begin{equation}
\label{equ_pkb2_ldisk}
{\cal L}_{disk}
\approx
{\dot M_{accr} G M_{wd} \over 2 r_i}
\;\;\;\; .
\end{equation}
We define
\begin{equation}
\label{equ_pkb2_gamma}
\Gamma
\equiv
{\kappa_e \, {\cal L}_{disk} \over 4 \pi G M_{wd} \, c}
\;\;\;\; .
\end{equation}
\noindent
Considering equations~(\ref{equ_pkb2_ldisk}) and (\ref{equ_pkb2_gamma}),
equation~(\ref{equ_pkb2_q1}) can be rewritten as
\begin{equation}
\label{equ_pkb2_q2}
Q(r)
=
3 \, {c \over \kappa_e} \, {G M_{wd} \over {r_i}^2} \, \Gamma
\left( r_i \over r \right)^3
\left[1 - \left({r_i \over r} \right)^{1/2} \right]
\;\;\;\; .
\end{equation}
\noindent
Additionally,
we define two dimensionless functions
$\Upsilon(r,z)$
and
$\zeta(r,z)$:
\begin{equation}
\label{equ_pkb21_upsilon}
\Upsilon(r,z)
\equiv
\int_{r_i}^\infty \int_0^{2\pi} {1 \over \pi} \;
\left( r_i \over r' \right)^3
\left[1 - \left({r_i \over r'} \right)^{1/2} \right]
{z^2
\over
\left[\left(r^2+r'^2+z^2-2rr'\cos\phi\right)^{1/2}\right]^4}
\; r' d\phi dr'
\;\;\;\; ;
\end{equation}
\begin{equation}
\label{equ_pkb21_zeta}
\zeta(r,z)
\equiv
\int_{r_i}^\infty \int_0^{2\pi} {1 \over \pi} \;
\left( r_i \over r' \right)^3
\left[1 - \left({r_i \over r'} \right)^{1/2} \right]
{z^{\alpha+2}
\over
\left[\left(r^2+r'^2+z^2-2rr'\cos\phi\right)^{1/2}\right]^{\alpha+4}}
\; r' d\phi dr'
\;\;\;\; .
\end{equation}
\noindent
It follows that
\begin{equation}
{\cal F}_z(r,z)
=
3 \, {c \over \kappa_e} \, {G M_{wd} \over {r_i}^2} \, \Gamma \,
\Upsilon(r,z)
=
3 {{\cal L}_{disk} \over 4 \pi {r_i}^2} \Upsilon(r,z)
\;\;\;\; ;
\end{equation}
\begin{equation}
S_z(r,z)
=
3 \, {c \over \kappa_e} \, {G M_{wd} \over {r_i}^2} \, \Gamma \,
\zeta(r,z)
=
3 {{\cal L}_{disk} \over 4 \pi {r_i}^2} \zeta(r,z)
\;\;\;\; .
\end{equation}
In reducing the 2.5D model to a series of 1D models,
for simplicity,
we assume an isothermal wind.
We do not expect this approximation to significantly change the results
of this work,
as is illustrated in the intermediate models of
\citet{per00}
[see Figures~6 and 7 of
\citet{per00}].
This,
on one hand,
significantly simplifies the analysis,
and on the other hand gives a predefined temperature structure
[namely:
$b^2(x) = {\rm constant}$]
that is a requirement of the 1D models of this work
(see Paper~I; Paper~II).
Thus the equation of state we assume,
for the application to the 2.5D CV disk wind model,
is
\begin{equation}
\label{equ_pkb2_state}
{P \over \rho}
=
b^2 \hskip 28pt ({\rm constant})\ .
\end{equation}
\noindent
We also assume that the streamlines,
starting from the disk at a given radius
$r$,
are predominantly vertical.
As discussed in \S\ref{sec_generalcomments},
this assumption is justified on observational,
theoretical,
and computational grounds.
The mass conservation equation is equivalent to the 1D models of
\citet{per97b},
namely,
for a given
$r$
\begin{equation}
\label{equ_pkb21_masscons}
\dot{M}
=
\rho \, V \, 4 \pi R^2 \hskip 28pt (\rm constant)
\;\;\;\; ,
\end{equation}
\noindent
where we are determining the existence of 1D steady
(stationary)
solutions,
and where
\begin{equation}
R^2
\equiv
r^2+z^2
\;\;\;\; ,
\end{equation}
\noindent
and
$\dot{M}$
is an integration constant,
$z$
is the height above the disk,
and
$r$
is the radius where a given streamline begins.
Thus for each value of
$r$
considered,
a given 1D model is developed.
We note that in equation~(\ref{equ_pkb21_masscons}),
$\dot{M}$
is not the wind mass loss rate
(as in Paper~I and Paper~II),
but only an integration constant.
We have used the symbol
$\dot{M}$
following the notation of Paper~I and Paper~II.
However,
the wind mass loss rate
$\dot{M}_{wind}$
can be obtained by
(after first determining the values of
$\dot{M}$
for different values of
$r$)
\begin{equation}
\dot{M}_{wind}
=
2 \int_{r_i}^{r_f} { \dot{M}(r) \over 4 \pi (r^2 + z_s^2) } \,
2 \pi r \, dr
\;\;\;\; .
\end{equation}
For
$z \ll r$,
$R^2 \approx r^2 \; (= \hbox{\rm constant for a given 1D model})$,
thus for
$z \ll r$,
$\rho V = {\rm constant}$
[equation~(\ref{equ_pkb21_masscons})]
that is the integration of equation~(\ref{equ_pkb2_masscons})
assuming stationary solutions and neglecting radial velocities.
Therefore,
equation~(\ref{equ_pkb21_masscons}) is accurate near the disk surface
and it is also accurate towards infinity
($z \gg r$,
$R^2 \approx z^2$).
However,
equation~(\ref{equ_pkb21_masscons}) is not accurate at intermediate
values of
$z$
where 2.5D effects become important
\citep{per97a,per97b,per00}.
Using the accurate results near the disk surface,
as discussed in \S\ref{sec_generalcomments},
we study whether wind solutions with steady mass loss rates exist.
In reducing the 2.5D model to a series of 1D models,
we additionally neglect the effect of a maximum possible value for the
line radiation force
\citep{cas75,abb82},
that are represented in the models of
\citet{per97a} and \citet{per00}
by including a
``$\max$''
function in the line force terms
[see equation~(\ref{equ_pkb2_momz})].
On one hand,
we do not expect this approximation to significantly change the results
of this work.
In the intermediate models of
\citet{per00}
this effect was found to be most significant in the low density region
high above the inner disk region.
On the other hand,
this allows the momentum equation to take a form equivalent to
the momentum equation of Paper~II,
that is the form of the momentum equation assumed in the
1D models of this work.
Thus,
within the assumption of a predominately vertical wind,
assuming stationary solutions,
neglecting radial velocities,
and considering equation~(\ref{equ_pkb2_momz}),
the momentum equation for the 1D models
(at a given
$r$)
becomes
\newpage
\begin{eqnarray}
\rho V {d V \over d z}
& = &
- \, \rho{GM_{wd} \over R^2} \, {z \over R} \,
+ \, \rho { 3 \, GM_{wd} \over r_i^2} \, \Gamma \, \Upsilon(r,z) \,
- \, {dP \over dz} \,
\nonumber
\\
&&
\\
&& + \, \rho { 3 \, GM_{wd} \over {r_i}^2} \, \Gamma \, \zeta(r,z) \,
k \left( { 1 \over \rho \kappa_e V_{th}}
{\partial V \over \partial z} \right)^\alpha
\;\;\;\; ,
\nonumber
\end{eqnarray}
\noindent
where
$\Upsilon(r,z)$
and
$\zeta(r,z)$
are given by equations~(\ref{equ_pkb21_upsilon}) and
(\ref{equ_pkb21_zeta}),
respectively.
The series of 1D models derived from the 2.5D disk wind model,
can be represented within the 1D models of Paper~II,
through the following parameterization:
\begin{eqnarray}
B(z)
&=&
{G M_{wd} \over R^2} \, {z \over R}
- {3 G M_{wd} \over r_i^2} \Gamma \Upsilon(r,z)
\;\;\;\; ,
\\
& & \nonumber
\\
A(z)
&=&
4 \, \pi \, R^2 = 4 \, \pi \, \left(r^2 + z^2 \right)
\;\;\;\; ,
\\
& & \nonumber \\
\hbox{and \ \ \ \ \ \ \ \ } \gamma(z)
&=&
{3 \, G M_{wd} \over {r_i}^2} \, \Gamma \, \, \zeta(r,z) \,
k \left(1 \over \kappa_e V_{th}\right)^\alpha
\;\;\;\; ,
\end{eqnarray}
\noindent
where the body force $B(z)$,
the area function
$A(z)$,
and the line opacity weighted flux
$\gamma(z)$
are defined for a given $r$.
Thus,
the mass conservation equation and the momentum equation have the form
of the equations of Paper~II.
The isothermal equation of motion is therefore,
\begin{equation}
\left(1 -{b^2 \over V^2} \right) A V {dV \over dz}
=
- \, B A
+ \, \gamma A \left({A \over \dot{M}} V { dV \over dz } \right)^\alpha
+ b^2 {dA \over dz}
\;\;\;\; .
\end{equation}
Defining the characteristic distance
$r_0$,
gravitational force
$B_0$,
area $A_0$,
and line-opacity weighted flux
$\gamma_0$
(see Paper~I; Paper~II)
through
\begin{equation}
r_0
=
r
\hskip 16pt
;
\hskip 16pt
B_0
=
{G M_{wd} \over r^2}
\hskip 16pt
;
\hskip 16pt
A_0
=
4 \pi r^2
\hskip 16pt
;
\hskip 16pt
\gamma_0
=
{3 \, G M_{wd} \over r_i^2} \, \Gamma \,
k \left(1 \over \kappa_e V_{th}\right)^\alpha
\;\;\;\; ;
\end{equation}
\noindent
the equation of motion becomes
\begin{equation}
\left(1-{s \over \omega} \right) a \, {d \omega \over d x}
=
- g a
+ f a \left({a \over \dot{m}} {d\omega \over dx} \right)^\alpha
+ 4 s x
\;\;\;\; ,
\end{equation}
\noindent
where
\begin{equation}
x
=
{z \over r}
\;\;\;\; ;
\end{equation}
\begin{equation}
g
=
{x \over \left(1+x^2\right)^{3/2}}
-
{3 \Gamma \over x_i^2} \, \Upsilon
\hskip 24pt
;
\hskip 24pt
x_i
\equiv
{r_i \over r}
\;\;\;\; ;
\end{equation}
\begin{equation}
a
=
1 + x^2
\;\;\;\; ;
\end{equation}
\begin{equation}
f
=
{1 \over \alpha^\alpha (1-\alpha)^{1-\alpha}} \, \zeta
\;\;\;\; .
\end{equation}
\noindent
The parameters
$s$, $\omega$, and $\dot{m}$
(see Paper~I; Paper~II)
are given by
\begin{equation}
s
\equiv
{b^2 \over 2 W_0}
\hskip 24pt
;
\hskip 24pt
\omega
\equiv
{W \over W_0}
\hskip 24pt
;
\hskip 24pt
\dot{m}
\equiv
{\dot{M} \over \dot{M}_{CAK}}
\;\;\;\; ,
\end{equation}
\noindent
where
\begin{equation}
W_0
\equiv
B_0 r_0
\hskip 24pt
;
\hskip 24pt
W
\equiv
{V^2 \over 2}
\hskip 24pt
;
\hskip 24pt
\dot{M}_{CAK}
\equiv
\alpha (1-\alpha)^{(1 - \alpha) / \alpha}
{ \left( \gamma_0 A_0 \right)^{1 / \alpha}
\over
\left(B_0 A_0 \right)^{(1-\alpha) / \alpha}
}
\;\;\;\; .
\end{equation}
Taking
$x_0=0$
and
$q_0=0$,
the spatial variable
$q$
(see Paper~II)
\begin{equation}
\label{equ_q}
q
\equiv
\int\displaylimits_{x_0}^{x} {1 \over a(x')} \, dx' + q_0
\;\;\;\; ,
\end{equation}
\noindent
becomes
\begin{equation}
q
=
\arctan(x)
\;\;\;\; .
\end{equation}
\noindent
The equation of motion,
expressed in terms of $q$,
becomes
\begin{equation}
\left(1-{s \over \omega} \right) {d \omega \over dq}
=
h(q)
+ f a \left({1 \over \dot{m}} {d\omega \over dq} \right)^\alpha
\;\;\;\; ,
\end{equation}
\noindent
where the function
$h(q)$
is now given by
\begin{equation}
h(q)
=
- g a
+ 4 s \tan(q)
\;\;\;\; .
\end{equation}
The nozzle function $n$ for each $r$ is given by (Paper~II):
\begin{eqnarray}
\label{equ_n}
n(q)
\equiv
&&
\alpha (1-\alpha)^{(1 - \alpha) / \alpha}
{(fa)^{1 / \alpha} \over
(-h)^{(1-\alpha) / \alpha}}
\hskip 48pt
{\rm for}
\hskip 24pt
h(q)
<
0
\nonumber \\
&&
\left[
=
\alpha (1-\alpha)^{(1 - \alpha) / \alpha}
{(fa)^{1 / \alpha} \over
( g a - 4 s \tan(q) )^{(1-\alpha) / \alpha}
}
\right]
\;\;\;\; .
\end{eqnarray}
\noindent
Additionally we shall recall the definition of the $\beta$ function
as defined in Paper~I and Paper~II
\begin{equation}
\label{equ_beta}
\beta(\omega)
\equiv
1 - {s \over w}
\;\;\;\; ,
\end{equation}
that we shall apply in the following section.
Thus,
we have reduced the 2.5D models to a series of 1D models by solving the
hydrodynamic equations in the vertical direction and neglecting radial
terms.
That is,
for a given radial distance
$r$,
we have a 1D model with
$q=\arctan(x)=\arctan(z/r)$
as the independent spatial variable.
As discussed above these models are accurate near the disk surface.
Since,
as we see below,
the corresponding critical point for inner radii is near the disk
surface,
the 1D models for the inner disk regions determine that disk wind flows
with steady mass loss rates are possible.
The models can be used to estimate local mass loss rates
(by determining local density at the sonic point)
as well as velocity and density wind structure near the disk surface
in the inner region.
\section{Results}
\label{sec_results}
For the 1D wind models,
we implement the set of parameters used by
\citet{per00}
corresponding to a CV,
namely:
\begin{eqnarray}
M_{wd}
=
0.6 M_{\sun}
\hskip 14pt
;
\hskip 14pt
{\cal L}_{disk}
&=&
{\cal L}_\sun
\hskip 33pt
;
\hskip 33pt
z_s = 0.0229 R_{\sun}
\;\;\;\; ;
\nonumber
\\
\\
r_i
=
0.01 R_{\sun}
\hskip 21pt
;
\hskip 21pt
b
&=&
10 \; \hbox{km s}^{-1}
\hskip 14pt
;
\hskip 14pt
V_{th} = 2.67 \; \hbox{km s}^{-1}
\;\;\;\; .
\nonumber
\end{eqnarray}
\noindent
We use the line force parameters also used by
\citet{per00}
[and
\citet{cas75}
and
\citet{abb82}],
namely
\begin{equation}
k
=
1/3
\hskip 14pt
;
\hskip 14pt
\alpha
=
0.7 \ .
\end{equation}
As with the analysis of the isothermal CAK stellar wind
(Paper~II),
we first consider,
for a given
$r$,
the
$h$
function for the 1D vertical wind model with the flux distribution of a
standard
\citet{sha73}
disk.
As discussed in Paper~II,
$h$
must be negative at the critical point
(i.e.,
$h$ must be negative in order for there to exist values of $q$,
$\omega$,
and
$d\omega/dq$
that satisfy the critical point conditions).
Second,
we consider the nozzle function,
that as we found in Paper~II,
for the case of an isothermal wind must be a locally increasing
function at the critical point
(i.e.,
it must have a positive spatial derivative at the critical point).
Third,
for the spatial points where the first two conditions hold,
through the nozzle function
$n$,
the
$\beta$
function,
the normalized wind mass loss rate
$\dot{m}$,
and the calculation of
$d^2\omega/dq^2$
under critical point conditions,
as discussed in Paper~II,
we determine if a local solution exists by verifying whether or not
the condition
\begin{equation}
\label{equ_local4}
\beta''(\omega) \, \dot{m} \, (\omega')^2
+ \beta'(\omega) \, \dot{m} \, \omega''
- n''(q)
<
0
\;\;\;\;
\end{equation}
\noindent
holds.
Fourth,
within the set of points that fulfill the first three conditions,
we iteratively determine the position of the critical point such that
the wind reaches sound speed at the sonic height of the model
(as discussed in Paper~II).
As a consistency check,
for a given
$r$,
we superimpose a plot of the nozzle function
$n$
and the
$\beta(\omega) \, \dot{m}$
function evaluated with the corresponding velocities
$V(z)$.
As discussed in Paper~II,
in the supersonic wind regime where the
$h$
function is negative
(Region II),
for points other than the critical point,
the following conditions must hold
\begin{equation}
\beta(\omega) \, \dot{m}
<
n(q)
\hskip 24pt
({\rm for \ } \omega > s {\rm \ and \ }
h(q) < 0 {\rm \ and \ } q \not= q_c)
\;\;\;\; ,
\label{equ_app1}
\end{equation}
\noindent
and at the critical point
\begin{equation}
\beta(\omega_c) \, \dot{m}
=
n(q_c)
\;\;\;\; .
\label{equ_app2}
\end{equation}
For
$r = 2 \, r_i$,
Figure~\ref{fig_pkb21_02_h} shows that the
$h$
function is negative
from the sonic point to beyond 1000 times the sonic height.
Figure~\ref{fig_pkb21_02_n} shows that the nozzle function
$n$,
corresponding to a vertical streamline at
$r = 2 \, r_i$,
is a monotonically increasing function from the sonic point to beyond
1000 times the sonic height.
It is shown,
in Figure~\ref{fig_pkb21_02_l},
that equation~(\ref{equ_local4}) holds from the sonic height
to beyond 1000 times the sonic height,
showing that local solutions to the equation of motion exist throughout
the fore mentioned spatial region.
By
``local solution''
at a given spatial point,
we mean the integration of the equation of motion in the vicinity
of the given point,
assuming the point to be the critical point.
Figure~\ref{fig_pkb21_02_v} shows the velocity vs. height obtained
upon integrating the equation of motion with the condition of
achieving the correct sonic height of the model.
As a consistency check,
Figure~\ref{fig_pkb21_02_nb} verifies that equations~(\ref{equ_app1})
and (\ref{equ_app2}) hold for the solution obtained.
Figures~\ref{fig_pkb21_10_h}-\ref{fig_pkb21_10_nb},
Figures~\ref{fig_pkb21_20_h}-\ref{fig_pkb21_20_nb},
and Figures~\ref{fig_pkb21_50_h}-\ref{fig_pkb21_50_nb} plot the results
from an equivalent analysis to wind streamlines starting at
$r = 10 \, r_i$,
$r = 20 \, r_i$,
and $r = 50 \, r_i$ respectively.
The results show that steady line-driven disk wind solutions exist for
a standard Shakura-Sunyaev disk.
A significant result we find is that the critical point tends to be
closer to the disk surface in the inner disk regions,
and farther out from the disk surface in the outer disk regions.
Physically this is due to the length scales in these different regions.
The disk wind,
at each radii,
results from the balancing of the gravitational forces and the
radiation pressure forces.
This balance can be presented quantitatively by the nozzle function
defined in Paper~I and Paper~II.
The spatial dependence of the nozzle function in turn is crucial in
determining the critical point position.
As larger disk radii are considered,
the scale length of the corresponding nozzle function also increases,
and therefore the critical point position increases as well.
Also,
the resulting local wind mass loss rates,
for the different radii considered,
are in good agreement with the 2.5D numerically intensive CV disk wind
models of
\cite{per97a,per00};
for example for a radius of $10r_i$ the local wind mass loss rates
found in this work is
$5.0 \times 10^{-7} {\rm g} \, {\rm s}^{-1} \, {\rm cm}^{-2}$,
and for the same physical model parameters the previous numerically
intensive 2.5D calculations estimated a value of
$7.1 \times 10^{-7} {\rm g} \, {\rm s}^{-1} \, {\rm cm}^{-2}$
for the same radius.
By confirming the overall earlier results of Pereyra and collaborators,
we conclude that the likely scenario for the wind outflows in CVs is
a line-driven disk wind.
This scenario was suggested for CVs early on by \citet{cor82}.
\citet{per97a,per00} showed that the line-driven disk wind scenario
was able to account for the general forms of the \ion{C}{4}
$\lambda\lambda$1549 line profile and its general dependence on
viewing angle
[for a review on CV properties in general, see \citet{war95}].
The steady nature of line-driven disk wind that we find here may also
be relevant to QSO studies. \citet{tur84} had suggested this scenario
early-on based on observational constraints.
The first serious attempt to study this scenario was by
\citet{mur95},
who developed 1D streamline models and found that this scenario could
account for several QSO observational features,
such as the terminal wind velocities inferred from
broad absorption lines (BALs)
(when present)
and the general form of single-trough BALs
(when present).
As discussed in the Introduction,
an additional constraint on the BAL region,
is that it presents a steady velocity structure.
Our results here show that the line-driven disk wind scenario is
consistent with the observed steady velocity structure.
Thus both observational constraints and theoretical/computational
studies to date seem to indicate that the line-driven disk wind
scenario is a promising one to account for the BALs commonly
observed in QSOs.
In future work we plan to study several aspects of this scenario and
develop more realistic models.
\section{Summary and Conclusions}
\label{sec_sumcon}
In Paper~I we had shown that steady wind solutions can exist by using
``simple'' models that mimic the disk environment.
These models are more readily analyzable than the more detailed models
presented here.
In Paper~II we extended the concepts of Paper~I,
and discussed in detail aspects of the steady/unsteady wind analysis
that was presented in Paper~I.
The objective of this work is to determine,
in a manner independent of the results of previous
numerically-intensive 2.5D hydrodynamic simulations,
whether steady line-driven disk wind solutions exist under the flux
distribution of a standard disk \citep{sha73}.
Our main conclusion pertaining to the more realistic models presented
here is that a line-driven wind,
arising from a steady disk using the flux distribution of a standard
Shakura-Sunyaev disk model,
is steady.
As we had discussed in Paper~II,
when including gas pressures effects,
the spatial dependence of the nozzle function continues to play a key
role in determining the steady/unsteady nature of supersonic line-driven
wind solutions.
These results are consistent with the steady nature of the 1D
streamline disk wind models of Murray and collaborators
\citep{mur95,mur96,chi96,mur98}.
These results also confirm the results of the 2.5D steady disk wind
models of Pereyra and collaborators
\citep{per97a,per00,hil02,per03}.
Another result that we find is that the critical point of the wind tends
to be closer to the disk surface as smaller radii closer to the inner
disk region are considered.
As we develop more realistic models,
we aim towards placing stronger constraints on the accretion disk
wind scenario,
particularly as they apply to QSOs.
If the scenario proves tenable,
then we may have a well-defined route towards a better understanding
of these astronomically fundamental objects.
\acknowledgments
We wish to thank Kenneth G. Gayley and Norman W. Murray
for many useful discussions.
This work is supported by the National Science
Foundation under Grant AST-0071193,
and by the National Aeronautics and Space Administration
under Grant ATP03-0104-0144.
|
2,869,038,154,800 | arxiv | \section{Introduction}
The study of the Cosmic Microwave Background (CMB) radiation holds the key
to understanding the seeds of the structure we see around us in the universe,
and could potentially enable precision measures for most of the important
cosmological parameters.
For this reason, as well as because of its intrinsic interest, one would
like a physically transparent framework for the study of CMB anisotropies
which is as general, powerful, and flexible as possible.
Theoretically, the calculation of CMB anisotropies is ``clean'', involving as
it does only linear perturbation theory.
However the calculations can become quite complex once one allows for the
possibility of non-flat universes, non-scalar perturbations to the metric,
and polarization as well as temperature anisotropies.
Recently Hu \& White \cite{TAMM} presented a formalism for calculating CMB
anisotropies which treats all types of perturbations,
temperature and polarization anisotropies,
and hierarchy and integral solutions on an equal footing.
The formalism, named the total angular momentum method, greatly simplifies
the physical interpretation of the equations and the form of their solutions
(see e.g.~\cite{Polar}).
However it was presented in detail only for the case of flat spatial
hypersurfaces. Here we generalize the treatment for the curved spaces
of open and closed Friedman-Robertson-Walker (FRW) universes.
Aspects of this method in open (hyperbolic, negatively curved) geometries have
been introduced
in Hu \& White \cite{OpenTen} and Zaldarriaga, Seljak \& Bertschinger
\cite{ZalSelBer} for the cases of tensor temperature and scalar polarization
respectively.
The latter work also addressed methods for efficient implementation
through the line of sight integration technique \cite{LOS}.
In this paper, we complete the total angular momentum method for
arbitrary perturbation type and
FRW metric, paying particular attention to the case of open
universes because of its strong observational motivation.
As an example we use this formalism to compute the temperature and polarization
angular power spectra of both scalar and tensor modes in critical density and
open inflationary models.
We incorporated the formalism into the CMBFAST code of Seljak \& Zaldarriaga
\cite{LOS}, which has been made publically available.
The outline of the paper is as follows:
we begin by establishing our notation for fluctuations about a FRW background
cosmology in \S\ref{sec:metric}.
We then present the Boltzmann equation in our formalism in
\S\ref{sec:boltzmann}, which contains the main results.
We give some examples and discuss applications in \S\ref{sec:discussion}.
Some of the more technical parts of the derivations (the Einstein, radial
and hierarchy equations) are presented in a series of three Appendices.
\section{Metric and Stress-Energy Perturbations} \label{sec:metric}
In this section, we discuss the representation of the perturbations for
the cosmological fluids and the geometry of space-time.
We start by defining the basis in which we shall expand such perturbations
and their representation under various gauge choices.
We assume that the background is described by an FRW metric
$g_{\mu\nu} = a^2 \gamma_{\mu\nu}$
with scale factor $a(t)$ and constant comoving curvature
$K = - H_0^2(1-\Omega_{\rm tot})$ in the spatial metric $\gamma_{ij}$.
Here greek indices run from $0$ to $3$ while latin indices run over the
spatial part of the metric: $i,j=1,2,3$.
It is often convenient to represent the metric in spherical coordinates where
\begin{equation}
\gamma_{ij} dx^i dx^j = |K|^{-1} \left[ d\chi^2 + \sin_K^2\chi
( d \theta^2 + \sin^2\theta\, d\phi^2 ) \right]\,,
\end{equation}
with
\begin{equation}
\sin_K(\chi) = \cases { \sinh(\chi)\,, & $K<0\,,$ \cr
\sin(\chi)\,, & $K>0\,,$ \cr}
\end{equation}
where the flat-limit expressions are regained as $K \rightarrow 0$ from
above or below. The component corresponding to conformal time
\begin{equation}
x^0 \equiv \eta = \int {dt\over a(t)}
\end{equation}
is $\gamma_{00}=-1$.
Small perturbations $h_{\mu\nu}$ around this FRW metric
\begin{equation}
g_{\mu\nu} = a^2(\gamma_{\mu\nu} + h_{\mu\nu})\,,
\end{equation}
can be decomposed into
scalar ($m=0$, compressional),
vector ($m=\pm 1$, vortical) and tensor ($m=\pm 2$, gravitational wave)
components from their transformation properties under spatial rotations
\cite{AbbSch,TAMM}.
\subsection{Eigenmodes}
\label{sec:eigenmodes}
In linear theory, each eigenmode of the Laplacian for the perturbation
evolves independently, and so it is useful to decompose the perturbations
via the eigentensor ${\bf Q}^{(m)}$, where
\begin{equation}
\nabla^2 {\bf Q}^{(m)} \equiv \gamma^{ij} {\bf Q}_{|ij}^{(m)} =
-k^2 {\bf Q}^{(m)} ,
\end{equation}
with ``$|$'' representing covariant differentiation with respect to
the three metric $\gamma_{ij}$.
Note that the eigentensor ${\bf Q}^{(m)}$ has $|m|$ indices (suppressed
in the above). Vector and tensor modes also satisfy the auxiliary conditions
\begin{eqnarray}
Q^{(\pm 1)}_i{}^{|i} &=& 0\, , \nonumber\\
\gamma^{ij} Q^{(\pm 2)}_{ij} &=& Q^{(\pm 2)}_{ij}{}^{|i} = 0 \,,
\end{eqnarray}
which represent the divergenceless and transverse-traceless conditions
respectively, as appropriate for vorticity and gravity waves.
In flat space, these modes are particularly simple and may be expressed as
\begin{equation}
Q_{i_1 \ldots i_{m}}^{(\pm m)} \propto
(\hat{e}_1 \pm i \hat{e}_2)_{i_1} \ldots
(\hat{e}_1 \pm i \hat{e}_2)_{i_m}
\exp(i \vec{k} \cdot \vec{x})\,, \qquad (K=0, m\ge 0)\,,
\end{equation}
where the presence of $\hat{e}_i$, which forms
a local orthonormal basis with $\hat{e}_3=\hat{k}$, ensures the
divergenceless and transverse-traceless conditions.
It is also useful to construct (auxiliary) vector and tensor objects out
of the fundamental scalar and vector modes through covariant differentiation
\begin{equation}
Q_i^{(0)} = -k^{-1} Q_{|i}^{(0)}\,, \qquad Q_{ij}^{(0)}
= k^{-2} Q_{|ij}^{(0)} + {1 \over 3} \gamma_{ij} Q^{(0)} \,,
\end{equation}
\begin{equation}
Q^{(\pm 1)}_{ij} = -(2k)^{-1}( Q^{(\pm 1)}_{i|j} + Q^{(\pm 1)}_{j|i} ).
\end{equation}
The completeness properties of these eigenmodes are
discussed in detail in
\cite{AbbSch}, where it is shown that in terms of the generalized wavenumber
\begin{equation}
q = \sqrt{k^2+(|m|+1)K} \, , \qquad \nu = q/|K|\,,
\end{equation}
the spectrum is complete for
\begin{equation}
\begin{array}{rll}
\nu &\ge 0, \qquad & K<0\,, \\
& = 3,4,5\ldots, \qquad &K>0\,.
\end{array}
\end{equation}
A deceptive aspect of this labelling is that for an open universe
the characteristic scale of
the structure in a mode is $2\pi/k$ and {\it not\/} $2\pi/q$, so all
functions have structure only out to the curvature scale even as
$q \rightarrow 0$.
We often go between the variable sets $(k,\eta)$, $(q,\eta)$ and
$(\nu,\chi)$ for convenience.
\subsection{Perturbation Representation}
A general metric perturbation can be broken up into the normal modes of
scalar ($m=0$), vector ($m=\pm 1$) and tensor ($m=\pm 2)$ types,
\begin{eqnarray}
h_{00} &=& - \sum_m 2 A^{(m)} Q^{(m)} \,, \nonumber \\
h_{0i} &=& - \sum_m B^{(m)} Q_i^{(m)} \,, \nonumber \\
h_{ij} &=& \sum_m 2 H_L^{(m)} Q^{(m)} \gamma_{ij}+2 H_T^{(m)} Q_{ij}^{(m)} \,.
\end{eqnarray}
Note that scalar quantities cannot be formed from vector and tensor modes
so that $A^{(m)}=0$ and $H_L^{(m)}=0$ for $m\ne 0$; likewise vector quantities
cannot be formed from tensor modes so that $B^{(m)}=0$ for $|m| = 2$.
There remains gauge freedom associated with the coordinate
choice for the metric perturbations (see Appendix \ref{sec:gauge}).
It is typically employed to eliminate two out of four of these
quantities for scalar perturbations and one of the two for vector
perturbations. The metric is thus specified by four quantities.
Two popular choices are the {\it synchronous\/} gauge, where
\begin{eqnarray}
H_L^{(0)} = h_L, &\qquad& H_T^{(0)} = h_T , \nonumber\\
H_T^{(1)} = h_V, &\qquad& H_T^{(2)} = H,
\end{eqnarray}
and the generalized (or conformal) {\it Newtonian\/} gauge, where
\begin{eqnarray}
A^{(0)} = \Psi\,, &\qquad& B^{(1)}=V \,, \nonumber \\
H_L^{(0)} = \Phi\,, &\qquad& H_T^{(2)}=H \, .
\end{eqnarray}
Here and below, when only the $m\ge 0$ expressions are displayed, the
$m < 0$ expressions should be taken to be identical unless otherwise
specified.
The stress energy tensor can likewise be broken up into scalar, vector,
and tensor contributions. Furthermore one can separate fluid ($f$)
contributions and seed ($s$) contributions.
The latter is distinguished by the fact that the net effect can be viewed
as a perturbation to the background.
Specifically $T_{\mu\nu} = \bar T_{\mu\nu} + \delta T_{\mu\nu}$ where
$\bar T^0_{\hphantom{0}0} = -\rho_f$,
$\bar T^0_{\hphantom{0}i} = \bar T_0^{\hphantom{i}i} =0$
and $\bar T^i_{\hphantom{i}j} = p_f \delta^i_{\hphantom{i}j}$
is given by the fluid alone.
The fluctuations can be decomposed into the normal modes of
\S\ref{sec:eigenmodes} as
\begin{equation}
\begin{array}{lcl}
\delta T^0_{\hphantom{0}0} &=&
- \sum_m [\rho_f \delta_f^{(m)} + \rho_s] \, Q^{(m)}, \vphantom{\displaystyle{\dot a \over a}}\\
\delta T^0_{\hphantom{0}i} &=& \sum_m [(\rho_f + p_f) (v_f^{(m)}-B^{(m)})
+ v_s^{(m)}]
\, Q_i^{(m)} ,
\vphantom{\displaystyle{\dot a \over a}}\\
\delta T_0^{\hphantom{i}i} &=&
-\sum_m [(\rho_f + p_f)v_f^{(m)} + v_s^{(m)}] \, Q^{(m)}{}^i,
\vphantom{\displaystyle{\dot a \over a}}\\
\delta T^i_{\hphantom{i}j} &=& \sum_m [\delta p_f^{(m)} + p_s]
\delta^i_{\hphantom{i}j} Q^{(m)} + [ p_f\pi_f^{(m)} + p_s]
Q^{(m)}{}^i_{\hphantom{i}j} \vphantom{\displaystyle{\dot a \over a}} \, .
\end{array}
\label{eqn:stress}
\end{equation}
Since $\delta^{(m)}_f=\delta p^{(m)}_f= 0$ for $m \ne 0$, we hereafter
drop the superscript from these quantities.
A minimally coupled scalar field $\varphi$ with Lagrangian
\begin{equation}
{\cal L} = -{1 \over 2} \sqrt{-g} \left[ g^{\mu \nu} \partial_\mu \varphi
\partial_\nu \varphi + 2V(\varphi) \right]
\end{equation}
can be treated in the same way with the associations
\begin{equation}
\rho_\phi = p_\phi + 2{\cal V} =
{1 \over 2} a^{-2} \dot \phi^2 + {\cal V} \,,
\label{eqn:scalarfieldbac}
\end{equation}
for the background density and pressure. The fluctuations $\varphi = \phi+ \delta\phi$
are related to the fluid quantities as \cite{KodSas}
\begin{eqnarray}
\delta \rho_\phi =\delta p_\phi + 2{\cal V}_{,\phi}\delta\phi &=& a^{-2}(\dot\phi
\dot{\delta\phi}-A^{(0)} \dot\phi^2)+{\cal V}_{,\phi} \delta\phi \nonumber\,,\\
(\rho_\phi+p_\phi) (v^{(0)}_\phi- B^{(0)}) &=& a^{-2} k \dot\phi \delta\phi
\nonumber\,,\\
p_\phi \pi_\phi^{(0)} & = & 0\,.
\label{eqn:scalarfieldpert}
\end{eqnarray}
The evolution of the matter and metric perturbations follows from the
Einstein equations $G_{\mu\nu} = 8\pi G T_{\mu\nu}$ and encorporates
the continuity and Euler equations through the implied energy-momentum
conservation $T^{\mu\nu}{}_{;\nu} = 0$.
We give these relations explicitly for the scalar, vector and tensor
perturbations in both Newtonian and synchronous gauge in
Appendix \ref{sec:einstein} (see also \cite{HSW}).
These equations hold equally well for relativistic matter such as the
CMB photons and the neutrinos. However in that case they do not represent
a closed system of equations (the equation of motion of the anisotropic
stress perturbations $\pi_f^{(m)}$ is unspecified) and do not account for
the higher moments of the distribution or for momentum exchange between
different particle species.
To include these effects, we require the Boltzmann equation
which describes the evolution of the full distribution function under
collisional processes.
\section{Boltzmann Equation}
\label{sec:boltzmann}
The Boltzmann equation describes the evolution in time $(\eta)$ of the
spatial ($\vec{x}$) and angular ($\hat{n}$) distribution of the radiation
under gravity and scattering processes. In the notation of \cite{TAMM}, it
can be written implicitly as
\begin{equation}
{d \over d\eta} \vec{T}(\eta,\vec{x},\hat{n}) \equiv
{\partial \over \partial\eta} \vec{T} + n^i \vec{T}_{|i}
= \vec{C}[\vec{T}] + \vec{G}[h_{\mu\nu}] \, ,
\label{eqn:boltzmannimplicit}
\end{equation}
where $\vec{T} = (\Theta, Q+iU, Q-iU)$ encapsulates the perturbation
to the temperature $\Theta=\Delta T/T$ and the polarization
(Stokes $Q$ and $U$ parameters) in units of the temperature fluctuation.
The term $\vec{C}$ accounts for collisions, here Compton scattering of the
photons with the electrons,
while the term $\vec{G}$ accounts for gravitational redshifts.
\subsection{Metric and Scattering Sources}
The gravitational term $\vec{G}$
is easily evaluated from the Euler-Lagrange equations
for the motion of a massless particle in the background given by $g_{\mu\nu}$
\cite{AbbSch,SacWol,WSS}:
\begin{equation}
\vec{G}[h_{\mu\nu}] = \left({1 \over 2}
{n}^i {n}^j \dot h_{ij}
+ {n}^i \dot h_{0i}
+ {1 \over 2} n^i h_{00|i} \,, 0 \, , 0 \right) .
\label{eqn:gravred}
\end{equation}
Note that gravitational redshifts affect different polarization states alike.
As should be expected, the modification from the flat space case involves the
replacement of ordinary spatial derivatives with covariant ones.
The Compton scattering term $\vec{C}$ was derived in \cite{TAMM,ZalSelBer}
in the total angular momentum language.
Though the basic result has long been known \cite{Cha,BonEfs}, this
representation has the virtue of explicitly showing that complications due
to the angular and polarization dependence of Compton scattering come simply
through the quadrupole moments of the distribution. Here
\begin{eqnarray}
\vec{C}[{\vec {T}}] &=& -\dot\tau
\left[ \vec{T}(\hat{n})
- \left( \int {d\hat{n}'\over 4\pi}
\Theta' + \hat{n} \cdot \vec{v}_B \, , 0 , \, 0 \right)
\right]
+
{\dot\tau \over 10} \int d\hat{n}'
\sum_{m=-2}^2 {\bf P}^{(m)}(\hat{n},\hat{n}') \vec{T}(\hat{n}')\, ,
\label{eqn:fullcollision}
\end{eqnarray}
where the differential cross section for Compton scattering is
$\dot\tau = n_e \sigma_T a$ where $n_e$ is the free electron number density
and $\sigma_T$ is the Thomson cross section.
The bracketed term in the collision integral describes the isotropization of
the photons in the rest frame of the electrons.
The last term accounts for the angular and polarization dependence of the
scattering with
\begin{eqnarray}
{\bf P}^{(m)} =
\left(
\begin{array}{ccc}
Y_2^{m}{}'\, Y_2^m \quad &
- \sqrt{3 \over 2} \Spy{2}{2}{m}{}'\, Y_2^m
\quad & - \sqrt{3 \over 2}
\Spy{-2}{2}{m}{}'\, Y_2^m
\vphantom{\displaystyle{\dot a \over a}}\\
- \sqrt{6} Y_2^{m}{}' \Spy{2}{2}{m} \quad &
3 \Spy{2}{2}{m}{}'\Spy{2}{2}{m} \quad &
3 \Spy{-2}{2}{m}{}'\Spy{2}{2}{m}{} \quad
\vphantom{\displaystyle{\dot a \over a}}\\
- \sqrt{6} Y_2^{m}{}' \Spy{-2}{2}{m}{} \quad &
3 \Spy{2}{2}{m}{}' \Spy{-2}{2}{m} \quad &
3 \Spy{-2}{2}{m}{}' \Spy{-2}{2}{m} ,
\vphantom{\displaystyle{\dot a \over a}}\\
\end{array}
\right),
\label{eqn:scatmatrix}
\end{eqnarray}
where $Y_\ell^m {}'\equiv Y_\ell^{m*} (\hat{n}')$ and
$\Spy{s}{\ell}{m}{}'\equiv\Spy{s}{\ell}{m*}(\hat{n}')$ and the unprimed
harmonics have argument $\hat{n}$.
Here $\Spin{Y}{s}{\ell}{m}$ are the spin-weighted spherical harmonics
\cite{Spin,SelZal,KamKosSte,TAMM}.
\subsection{Normal Modes}
\label{sec:normal}
The temperature and polarization distributions are functions of the position
$\vec{x}$ and the direction of propagation of the photons $\vec{n}$.
They can be expanded in modes which account for both the local angular and
spatial variations: $\Spin{G}{s}{\ell}{m}(\vec{x},\hat{n})$, i.e.
\begin{equation}
\begin{array}{rcl}
\Theta(\eta,\vec{x},\hat{n}) &=& \displaystyle{
\int {d^3 q \over (2\pi)^3} }
\sum_{\ell}
\sum_{m=-2}^2 \Theta_\ell^{(m)} \Gm{0}{\ell}{m} \, , \\
(Q \pm i U)(\eta,\vec{x},\hat{n}) &=& \displaystyle{\int {d^3q
\over (2\pi)^3}}
\sum_{\ell} \sum_{m=-2}^2
(E_\ell^{(m)} \pm i B_\ell^{(m)}) \, \Gm{\pm 2}{\ell}{m} \,,
\end{array}
\label{eqn:decomposition}
\end{equation}
with spin $s=0$ describing the temperature fluctuation and $s=\pm 2$
describing the polarization tensor.
$E_\ell$ and $B_\ell$ are the angular moments of the electric and magnetic
polarization components.
It is apparent that the effects of the local scattering process $\vec{C}$
is most simply evaluated in a representation where the separation of the
local angular and spatial distribution is explicit \cite{TAMM}, with the
former being an expansion in $\Spy{s}{\ell}{m}$.
The subtlety lies in relating the local basis at two {\it different\/}
coordinate points, say the last scattering event and the observer.
In flat space, the representation is straightforward since the parallel
transport of the angular basis in space is trivial. The result is a product
of spin-weighted harmonics for the local angular dependence and plane waves
for the spatial dependence:
\begin{equation}
\Gm{s}{\ell}{m}(\vec{x},\hat{n}) =
(-i)^\ell \sqrt{ {4\pi \over 2\ell+1}}
[\Spy{s}{\ell}{m}(\hat{n})] \exp(i\vec{k} \cdot \vec{x})\,,
\qquad (K=0)\,.
\label{eqn:flatG}
\end{equation}
Here we seek a similar construction in an curved geometry.
We will see that this construction greatly simplifies the scalar harmonic
treatment of \cite{Wil,WhiSco,ZalSelBer} and extends it to vector and
tensor temperature \cite{OpenTen} modes as well as all polarization modes.
To generalize these modes to the curved geometry, we wish to replace the
plane wave with some spatially dependent phase factor
$\exp[i \delta(\vec{x},\vec{k})]$ related to the eigenfunctions
${\bf Q}^{(m)}$ of \S\ref{sec:eigenmodes} while keeping the same local
angular dependence (see Eq.~\ref{eqn:generalizedGA}).
By virtue of this requirement, the Compton scattering terms, which
involve only the local angular dependence, retain the same form as in
flat space.
In Appendix \ref{sec:derivation}, we derive $\Spin{G}{s}{\ell}{m}$ by
recursion from covariant contractions of the fundamental basis ${\bf Q}^{(m)}$.
The result is a recursive definition of the basis
\begin{equation}
n^i (\Spin{G}{s}{\ell}{m})_{|i}
= {q \over 2\ell +1} \left[
{\Spin{\kappa}{s}{\ell}{m}} (\Spin{G}{s}{\ell-1}{m})
- {\Spin{\kappa}{s}{\ell+1}{m}} (\Spin{G}{s}{\ell+1}{m})
\right]
- i{q m s \over \ell(\ell+1)}\ \Spin{G}{s}{\ell}{m} \, ,
\label{eqn:recursion}
\end{equation}
constructed from the lowest $\ell$-mode of Eq.~(\ref{eqn:Gl0})
with the coupling coefficient
\begin{equation}
\Spin{\kappa}{s}{\ell}{m} = \sqrt{ \left[
{(\ell^2-m^2)(\ell^2-s^2)\over\ell^2}\right]
\left[1 - {\ell^2\over q^2} K \right]}.
\label{eqn:coupling}
\end{equation}
The structure of this relation is readily apparent. The recursion relation
expresses the addition of angular momentum and is the defining equation in
the total angular momentum method.
It says the ``total'' local angular dependence at (say) the origin is the
sum of the local angular dependence at distant points (``spin'' angular
momentum) plus the angular variations induced by the spatial dependence of
the mode (``orbital'' angular momentum).
The recursion relation represents the addition of angular momentum for the
case of an infinitesimal spatial separation.
Here the leading order spatial variation is the gradient
[$n^i (\Spin{G}{s}{\ell}{m})_{|i}$]
term which has an angular structure of a dipole $Y_1^0$.
The first term on the rhs of equation~(\ref{eqn:coupling}) arises from the
Clebsch-Gordan relation that couples the orbital $Y_1^0$ with the intrinsic
$\Spin{Y}{s}{\ell}{m}$ to form $\ell\pm 1$ states,
\begin{eqnarray}
\sqrt{4 \pi \over 3} Y_1^0 (\Spin{Y}{s}{\ell}{m})
&=& { \Spin{c}{s}{\ell}{m} \over \sqrt{(2\ell+1)(2\ell-1)}}
\left(\Spin{Y}{s}{\ell-1}{m}\right)
+ {\Spin{c}{s}{\ell+1}{m} \over \sqrt{(2\ell+1)(2\ell+3)}}
\left(\Spin{Y}{s}{\ell+1}{m}\right)
- {m s \over \ell (\ell +1)} \left({}_s Y_{\ell}^m\right) \,,
\label{eqn:streamingcg}
\end{eqnarray}
where the coupling coefficient is $\Spin{c}{s}{\ell}{m} =
\sqrt{(\ell^2-m^2)(\ell^2-s^2)/\ell^2}$.
The second term on the rhs of the coupling equation (\ref{eqn:coupling})
accounts for geodesic deviation factors in the conversion of spatial
structure into orbital angular momentum.
Consider first a closed universe with radius of curvature ${\cal R}=K^{-1/2}$.
Suppressing one spatial coordinate, we can analyze the problem as geometry on
the 2-sphere with the observer situated at the pole.
Light travels on radial geodesics or great circles of fixed longitude.
A physical scale $\lambda$ at fixed latitude (given by the polar angle $\chi$)
subtends an angle $\alpha = \lambda/{\cal R}\sin\chi$.
In the small angle approximation, a Euclidean analysis would infer a distance
related by
\begin{equation}
{\cal D}(d) = {\cal R} \sin \chi =
K^{-1/2} \sin \chi \, , \qquad (K>0),
\end{equation}
called here the {\it angular diameter distance}.
For negatively curved or open universes, a similar analysis implies
\begin{equation}
{\cal D}(d) = |K|^{-1/2} \sinh \chi\, ,
\qquad (K<0).
\label{eqn:angularsize}
\end{equation}
Thus the angular scale corresponding to an eigenmode of
wavelength $\lambda$ is
\begin{equation}
\theta = {\lambda \over {\cal R} \sinh{\chi}} \,
\approx {1 \over \nu \sinh{\chi}} \, .
\end{equation}
For an infinitesimal change $\chi$, orbital angular momentum
of order $\ell$ is stimulated when
\begin{eqnarray}
\chi &\approx& {1\over\nu\theta} [1+{\cal O}(\nu^2\theta^2)]\,, \nonumber\\
\eta &\approx& {\ell\over q}[1+{\cal O}(\ell^2 K / q^2) ]\,,
\end{eqnarray}
which explains the factors of $\ell^2 K / q^2$ in the coupling term in a
curved geometry.
We shall see in \S\ref{sec:integral} that these infinitesimal additions of
angular momentum and geodesic deviation may be encorporated into a single
step by finding the integral solutions to the coupling equation
(\ref{eqn:recursion}).
\subsection{Evolution Equations}
It is now straightforward to rewrite the Boltzmann equation
(\ref{eqn:boltzmannimplicit}) as the evolution equations for the amplitudes
of the normal modes of the temperature and polarization
$\vec{T}_\ell^{(m)} = (\Theta^{(m)}_\ell, E^{(m)}_\ell, B_\ell^{(m)})$.
The gravitational sources and scattering sources of these equations
follow from Eq. (\ref{eqn:gravred}) and (\ref{eqn:fullcollision}) by noting
that the spin harmonics are orthogonal,
\begin{equation}
\int d\Omega\ (\Spin{Y}{s}{\ell}{m})(\Spin{Y}{s}{\ell'}{m'}{}^*)
= \delta_{\ell,\ell'} \delta_{m m'}.
\end{equation}
The term $n^i \vec{T}_{|i}$ is evaluated by use of the coupling relation
Eq.~(\ref{eqn:recursion}) for $n^i(\Spin{G}{s}{\ell}{m})_{|i}$.
It represents the fact that spatial gradients in the distribution become
orbital angular momentum as the radiation streams along its trajectory
$\vec{x}(\hat{n})$.
For example, a temperature variation on a distant surface surrounding the
observer appears as an anisotropy on the sky.
This process then simply reflects a projection relation that relates distant
sources to present day local anisotropies.
With these considerations, the temperature fluctuation evolves as
\begin{equation}
\dot\Theta_\ell^{(m)}
= q \Bigg[ {\Spin{\kappa}{0}{\ell}{m} \over (2\ell-1)}
\Theta_{\ell-1}^{(m)}
-{\Spin{\kappa}{0}{\ell+1}{m}
\over (2\ell+3)}
\Theta_{\ell+1}^{(m)} \Bigg]
- \dot\tau \Theta_\ell^{(m)} + S_\ell^{(m)},
\qquad (\ell \ge m),
\label{eqn:boltz}
\end{equation}
and the polarization as
\begin{eqnarray}
\dot E_\ell^{(m)} &=&
q
\Bigg[ {\Spin{\kappa}{2}{\ell}{m} \over (2\ell-1)}
E_{\ell-1}^{(m)} - {2m \over \ell (\ell + 1)} B_\ell^{(m)}
- {\Spin{\kappa}{2}{\ell+1}{m} \over (2 \ell + 3)}
E_{\ell + 1}^{(m)} \Bigg]
- \dot\tau [E_\ell^{(m)}+\sqrt{6}P^{(m)}\delta_{\ell,2}]\,,\nonumber\\
\dot B_\ell^{(m)} &=&
q \Bigg[ {\Spin{\kappa}{2}{\ell}{m} \over (2\ell-1)}
B_{\ell-1}^{(m)} + {2m \over \ell (\ell + 1)} E_\ell^{(m)}
- {\Spin{\kappa}{2}{\ell+1}{m} \over (2 \ell + 3)}
B_{\ell + 1}^{(m)} \Bigg] -\dot\tau B_\ell^{(m)}.
\label{eqn:boltzpol}
\end{eqnarray}
The temperature fluctuation sources in Newtonian gauge are
\begin{equation}
\begin{array}{lll}
S_0^{(0)} = \dot\tau \Theta_0^{(0)} - \dot\Phi \, , \qquad &
S_1^{(0)} = \dot\tau v_B^{(0)} + k\Psi \, , \qquad &
S_2^{(0)} = \dot\tau P^{(0)} \, , \vphantom{\displaystyle{\dot a \over a}}\\
\qquad &
S_1^{(1)} = \dot\tau v_B^{(1)} + \dot V \, , \qquad &
S_2^{(1)} = \dot\tau P^{(1)} \, , \vphantom{\displaystyle{\dot a \over a}}\\
\qquad &
\qquad &
S_2^{(2)} = \dot\tau P^{(2)} - \dot H \vphantom{\displaystyle{\dot a \over a}} \, ,
\end{array}
\end{equation}
and in synchronous gauge,
\begin{equation}
\begin{array}{lll}
S_0^{(0)} = \dot\tau \Theta_0^{(0)} - \dot h_L \, , \qquad &
S_1^{(0)} = \dot\tau v_B^{(0)} \, , \qquad &
S_2^{(0)} = \dot\tau P^{(0)} -{2 \over 3}\sqrt{1-3K/k^2}\ \dot h_T
\, , \vphantom{\displaystyle{\dot a \over a}}\\
\qquad &
S_1^{(1)} = \dot\tau v_B^{(1)} \, , \qquad &
S_2^{(1)} = \dot\tau P^{(1)} -{\sqrt{3} \over 3}\sqrt{1-2K/k^2}\ \dot h_V
\, , \vphantom{\displaystyle{\dot a \over a}}\\
\qquad &
\qquad &
S_2^{(2)} = \dot\tau P^{(2)} - \dot H \vphantom{\displaystyle{\dot a \over a}} \, ,
\end{array}
\label{eqn:tempsources}
\end{equation}
The $\ell=m=2$ source doesn't contain a curvature factor because we have
recursively defined the basis functions in terms of the lowest member,
which is $\ell=2$ in this case. In the above
\begin{equation}
P^{(m)} = {1 \over 10} \left[ \Theta_2^{(m)} - \sqrt{6} E_2^{(m)} \right]
\label{eqn:polsource}
\end{equation}
and note that the photon density and velocities are related to the $\ell=0,1$
moments as
\begin{equation}
\delta_\gamma = 4\Theta_0^{(0)}$\,, \qquad $v_\gamma^{(m)}=
\Theta_1^{(m)} \, ;
\end{equation}
whereas the anisotropic stresses are given by
\begin{equation}
\pi_\gamma^{(m)} Q_{ij}^{(m)} =
12 \int {d \Omega \over 4\pi}\ (n_i n_j-{1\over 3}\gamma_{ij})\Theta^{(m)},
\end{equation}
which relates them to the quadrupole moments ($\ell=2$) as
\begin{equation}
(1-3K/k^2)^{1/2} \pi_\gamma^{(0)} = {12 \over 5}\Theta_2^{(0)}, \qquad
(1-2K/k^2)^{1/2} \pi_\gamma^{(1)} = {8\sqrt{3} \over 5} \Theta_2^{(1)},
\qquad \pi_\gamma^{(2)} = {8\over 5} \Theta_2^{(2)} .
\end{equation}
The evolution of the metric and matter sources are given in
Appendices \ref{sec:scalar}---\ref{sec:tensor}.
\subsection{Integral Solutions}
\label{sec:integral}
The Boltzmann equations have formal integral solutions that are simple to
write down. The hierarchy equations for the temperature distribution
Eq.~(\ref{eqn:boltz}) merely express the projection of the various plane
wave temperature sources $S_\ell^{(m)} \Gm{0}{\ell}{m}$ on the sky today
(see Eq.~(\ref{eqn:tempsources})).
Likewise Eq.~(\ref{eqn:boltzpol}) expresses the projection of
$-\sqrt{6} P^{(m)} \dot\tau e^{-\tau} \Spin{G}{\pm 2}{\ell}{m}$.
The projection is obtained by extracting the total angular dependence of
the mode from its decomposition in spherical coordinates: i.e.~into radial
functions times spin harmonics $\Spin{Y}{s}{\ell}{m}$.
We discuss their explicit construction in Appendix \ref{sec:radial}.
The full solution immediately follows by integrating the projected source
over the radial coordinate,
\begin{eqnarray}
{\Theta_\ell^{(m)}(\eta_0,q) \over 2\ell + 1}\, & = &
\int_0^{\eta_0} d\eta \ e^{-\tau} \, \sum_{j} S_{j}^{(m)} \,
\phi_{\ell}^{(jm)} ,\nonumber \\
{E^{(m)}_\ell(\eta_0,q) \over 2\ell+1} &=&
\int_0^{\eta_0} d\eta \ \dot\tau e^{-\tau} (-\sqrt{6} P^{(m)}_{\vphantom{\ell}})
\,\epsilon_{\ell}^{(m)} ,\nonumber\\
{B^{(m)}_\ell(\eta_0,q) \over 2\ell+1} &=&
\int_0^{\eta_0} d\eta \ \dot\tau e^{-\tau} (-\sqrt{6} P^{(m)}_{\vphantom{\ell}})
\,\beta_{\ell}^{(m)} ,
\label{eqn:los}
\end{eqnarray}
where the arguments of the radial functions
($\phi_\ell,\epsilon_\ell,\beta_\ell$) are the distance to the source
$\chi = \sqrt{-K}(\eta_0-\eta)$ and the reduced wavenumber
$\nu=q/\sqrt{-K}$ (see Appendix \ref{sec:radial} for explicit forms).
The interpretation of these equations is also readily apparent from their form
and construction.
The decomposition of $\Spin{G}{s}{j}{m}$ into radial and spherical parts
encapsulates the summation of spin and orbital angular momentum as well as
the geodesic deviation factors described in \S\ref{sec:normal}.
The difference between the integral solution and the differential form
is that in the former case the coupling is performed in one step from the
source at time $\eta$ and distance $\chi(\eta)$ to the present, while in the
latter the power is steadily transferred to higher $\ell$ as the time advances.
Take the flat space case.
The intrinsic local angular momentum at the point $(\chi,\hat{n})$ is
$\Spin{Y}{s}{j}{m}$ but must be added to the orbital angular momentum
from the plane wave which can be expanded in terms of $j_\ell Y_\ell^0$.
The result is a sum of $|\ell-j|$ to $\ell+j$ angular momentum states with
weights given by Clebsch-Gordan coefficients. Alternately a state of definite
angular momentum involves a sum over the same range in the spherical Bessel
function.
These linear combinations of Bessel functions are exactly the radial functions
in Eq.~(\ref{eqn:los}) for the flat limit \cite{TAMM}.
For an open geometry, the same analysis follows save that the spherical
Bessel function must be replaced by a hyperspherical Bessel function
(also called ultra-spherical Bessel functions) in the manner described
in Appendix \ref{sec:radial}. The qualitative aspect of this modification
is clear from considering the angular diameter distance arguments of
\S\ref{sec:normal}.
The peak in the Bessel function picks out the angle which a scale
$k^{-1} \approx \sqrt{-K}\nu^{-1}$ subtends at distance
$d \approx \chi/\sqrt{-K}$.
A spherical Bessel function peaks when its argument $kd \approx \ell$
or $\lambda/d \approx \theta$ in the small angle approximation.
The hyperspherical Bessel function peaks at
$k{\cal D}=\nu\sinh\chi \approx \ell$ for $\nu \gg 1$ or
$\lambda/{\cal D} \approx \theta$ in the small angle approximation.
The main effect of spatial curvature is simply to shift features in
$\ell$-space with the angular diameter distance, i.e.~to higher $\ell$
or smaller angles in open universes.
Similar arguments hold for closed geometries \cite{WhiSco}.
By virtue of this fact the division of polarization into $E$ and $B$-modes
remains the same as that in flat space. More specifically, for a single
mode the ratio in power is given by
\begin{equation}
{\sum_\ell [\ell \beta_\ell^{(m)} ]^2
\over
\sum_\ell [\ell \epsilon_\ell^{(m)} ]^2 }
= \cases{
0, & $m= 0, $ \cr
6, & $m= \pm 1,$ \cr
8/13, & $m= \pm 2,$ \cr}
\label{eqn:ebratio}
\end{equation}
at fixed source distance $\chi$ with $\nu \sin_K \chi \gg 1$.
The integral solutions (\ref{eqn:los}) are the basis of the ``line of sight''
method \cite{LOS,SelZal} for rapid numerical calculation of CMB spectra,
which has been employed in CMBFAST.
The numerical implementation of equations (\ref{eqn:los}) requires an efficient
way of calculating the radial functions ($\phi_\ell,\epsilon_\ell,\beta_\ell$).
This is best done acting the derivatives of the hyperspherical Bessel function
in the radial equations~(\ref{eqn:phiradial})-(\ref{eqn:betaradial}) and
(\ref{eqn:phiradialaux}) on the sources through integration by parts.
The remaining integrals can be efficiently calculated with the techniques of
\cite{ZalSelBer} for generating hyperspherical Bessel functions.
The tensor CMBFAST code has now been modified to use the formalism described
in this paper and the results have been cross-checked against solutions of
the Boltzmann hierarchy equations (\ref{eqn:boltz})-(\ref{eqn:boltzpol}) with
very good agreement.
\subsection{Power Spectra}
The final step in calculating the anisotropy spectra is to integrate over
the $k$-modes. The power spectra of temperature and polarization anisotropies
today are defined as,
e.g.~$C_\ell^{\Theta\Theta}\equiv\left\langle | a_{\ell m} |^2\right\rangle$
for $\Theta = \sum a_{\ell m} Y_\ell^m$ with the average being over the
($2\ell+1$) $m$-values. In terms of the moments of the previous section,
\begin{equation}
(2\ell+1)^2 C_\ell^{X\widetilde X} = {2 \over \pi}
\int {dq \over q} \sum_{m=-2}^2
q^3 \ X_\ell^{(m)*} \widetilde X_\ell^{(m)}\, ,
\label{eqn:cldef}
\end{equation}
where $X$ takes on the values $\Theta$, $E$ and $B$ for the temperature,
electric polarization and magnetic polarization evaluated at the present.
For a closed geometry, the integral is replaced by a sum over
$q/|K|=3,4,5\ldots$
Note that there is no cross correlation $C_\ell^{\Theta B}$ or $C_\ell^{E B}$
due to parity.
We caution the reader that power spectra for the metric fluctuation sources
$P_h(q) = \left\langle h^*(q) h(q) \right\rangle$ must be defined in a similar
fashion for consistency and choices between various authors differ by factors
related to the curvature (see \cite{OpenInflation} for further discussion).
To clarify this point, the initial power spectra of the metric fluctuations
for a scale-invariant spectrum of scalar modes and minimal inflationary
gravity wave modes \cite{OpenTen} are
\begin{eqnarray}
P_{\Phi}(q) &\propto& {\displaystyle{1 \over \nu(\nu^2+1)}}\,, \nonumber\\
P_{H}(q) &\propto& {\displaystyle{(\nu^2+4)\over \nu^3(\nu^2+1)}} \tanh(\pi \nu/2)\,,
\label{eqn:initialpower}
\end{eqnarray}
where the normalization of the power spectrum comes from the underlying
theory for the generation of the perturbations. This proportionality constant
is related to the amplitude of the matter power spectrum on large scales or
the energy density in long-wavelength gravitational waves \cite{OpenInflation}.
The vector perturbations have only decaying modes and so are only present in
seeded models.
The other initial conditions follow from detailed balance of the evolution
equations and gauge transformations (see Appendix \ref{sec:einstein}).
Our conventions for the moments also differ from those in
\cite{SelZal,KamKosSte}.
They are related to those of \cite{SelZal} by\footnote{Footnote 3 of
\cite{TAMM} incorrectly gives the relation between $\Theta$ and $\Delta_T$.}
\begin{eqnarray}
(2\ell+1)\Delta_{T\ell}^{(S)} &=& \Theta_\ell^{(0)}/(2\pi)^{3/2}\,, \nonumber\\
(2\ell+1)\Delta_{T\ell}^{(T)} &=& \sqrt{2}\Theta_\ell^{(2)}/(2\pi)^{3/2}\,,
\end{eqnarray}
where the factor of $\sqrt{2}$ in the latter comes from the quadrature sum
over equal $m=2$ and $-2$ contributions.
Similar relations for $\Delta_{(E,B)\ell}^{(S,T)}$ occur but with an extra
minus sign so that $C_{C,\ell}=-C_\ell^{\Theta E}$ with the other power
spectra unchanged.
The output of CMBFAST {\it continues\/} to be $C_{C,\ell}$ with the sign
convention of \cite{SelZal}.
In the notation of \cite{KamKosSte}, the temperature power spectra agree
but for polarization $C^{EE,BB}_\ell=C^{G,C}_\ell/2$ and
$C^{\Theta E}_\ell=-C^{TG}_\ell/\sqrt{2}$.
\begin{figure*}
\begin{center}
\leavevmode
\epsfxsize=6in \epsfbox{otamm_1.ps}
\end{center}
\caption{The scalar (left) and tensor (right) angular power spectra
for anisotropies in a critical density model (thick lines) and an open
model (thin lines) with $\Omega_0=0.4$.
Solid lines are $C_\ell^{\Theta\Theta}$, dashed $C_\ell^{EE}$ and
dotted $C_\ell^{BB}$.}
\label{fig:temp}
\end{figure*}
\section{Results}
\label{sec:discussion}
We now employ the formalism developed here to calculate the scalar and tensor
temperature and polarization power spectra for two CDM models one with
critical density and one with $\Omega_0=1-\Omega_K=0.4$ with initial
conditions given by Eq.~(\ref{eqn:initialpower}).
In general, there are two classes of effects: the geometrical and dynamical
aspects of curvature.
On intermediate to small scales (large $\ell$), only geometrical aspects of
curvature affect the spectra.
Changes in the angular diameter distance to last scattering move features
in the low-$\Omega_0$ models to smaller angular scales (higher $\ell$) as
discussed in \S\ref{sec:boltzmann}.
Since the low-$\ell$ tail of the $E$-mode polarization is growing rapidly
with $\ell$, shifting the features to higher $\ell$ results in smaller
large-angle polarization in an open model for both scalar and tensor
anisotropies. The suppression is larger in the case of scalars than tensors
since the low-$\ell$ slope is steeper \cite{TAMM}.
\begin{figure*}
\begin{center}
\leavevmode
\epsfxsize=6in \epsfbox{otamm_2.ps}
\end{center}
\caption{The scalar (left) and tensor (right) temperature-polarization
cross correlation $C_\ell^{\Theta E}$ with the same parameters and notation
as Fig.~\protect\ref{fig:temp} (thick: flat; thin open). Dotted lines
represent negative correlation.}
\label{fig:cross}
\end{figure*}
The presence of curvature also affects the late-time dynamics and initial
power spectra. As is well known, the scalar temperature power spectrum
exhibits an enhancement of power at low multipoles due to the integrated
Sachs-Wolfe (ISW) effect during curvature domination.
This does not affect the polarization, assuming no reionization, as it is
generated at last scattering. However it {\it does\/} affect the
temperature-polarization cross correlation (see Fig.~\ref{fig:cross}).
In an open universe, the largest scales (lowest $\ell$) pick up unequal-time
correlations with the ISW contributions which are of opposite sign to the
ordinary Sachs-Wolfe contribution.
This reverses the sign of the correlation and formally violates the
predictions of \cite{CriCouTur}.
In practice this effect is unobservable due to the smallness of signal.
Even minimal amounts of reionization will destroy this effect.
Open universe modifications to the initial power spectrum are potentially
observable in the large angle CMB spectrum.
Unfortunately subtle differences in the temperature power spectrum can be
lost in cosmic variance.
While polarization provides extra information, in the absence of late
reionization the large-angle polarization is largely a projection of small-scale
fluctuations.
Nonetheless in our universe (where reionization occured before redshift
$z\approx 5$) the large-angle polarization is sensitive to the primordial
power spectrum at the curvature scale.
Thus if the fluctuations which gave rise to the large-scale structure and
CMB anisotropy in our universe were generated by an open inflationary scenario
based on bubble nucleation, a study of the large-angle polarization can in
principle teach us about the initial nucleation event \cite{OpenInflation}.
\vskip 0.5truecm
In summary, we have completed the formalism for calculating and interpreting
temperature and polarization anisotropies in linear theory from arbitrary
metric fluctuations in an FRW universe.
The results presented here are new for non-flat vector and tensor
(polarization) perturbations and we have calculated the scalar and tensor
temperature and polarization contributions for open inflationary spectra.
The open tensor perturbation equations have been added to CMBFAST which is now
publically available.
\vskip 0.5truecm
\noindent
{\it Acknowledgments:} We thank the Aspen Center for Physics
where a portion of this work was completed. W.H was supported
by the W.M. Keck Foundation and NSF PHY-9513835;
M.Z. by NASA Grant NAG5-2816.
|
2,869,038,154,801 | arxiv |
\section{Introduction}
Determining the physical properties and chemical abundances of the interstellar medium can help us to get a better understanding about the formation and evolution of galaxies. Photoionized regions such as planetary nebulae (PNe) and \ion{H}{ii} regions, give us information about the past and the present composition of the interstellar medium in which they were formed. The method for estimating chemical abundances in these regions is well detailed in \citet{2017Peimbert_pasp129}. However, there are two big problems in the determination of physical parameters and chemical abundances in photoionized regions. The first problem is that in most cases the temperature T(BJ) estimated using the Balmer jump of H is lower than the temperature T([\ion{O}{iii}]) determined with the emission line ratio [\ion{O}{iii}]~4363/(4959+5007) \citep{1971Peimbert_Bole6}. The second problem is that the ionic abundances estimated with optical recombination lines (ORLs) are systematically higher than the ones estimated with collisionally excited lines (CELs) for the same ion. The ratio of these two ionic abundances is defined as the abundance discrepancy factor (ADF). \cite{2001Liu_mnras327} shows that there is a positive correlation between the ADF(O$^{++}$) and the difference in temperatures T([\ion{O}{iii}])-T(BJ) for a sample of 11 PNe. \Draft{\citet{2004Tsamis_mnra353} show that there is some evidences for a cold plasma using the ORLs observations.} Some mechanisms have been proposed to explain the abundance discrepancy, such as temperature fluctuations in the gas \citep{1967Peimbert_apj150}, non-thermal electron energy distribution \citep{2012Nicholls_apj752}, density condensations (Viegas \& Clegg 1994)\nocite{1994Viegas_mnra271}, and abundance inhomogeneities \citep{1990Torres-Peimbert_aap233, 2000Liu_mnra312, 2002Pequignot_12}.
Chemically homogeneous photoionization models of PNe can not reproduce both CELs and ORLs in the cases of high ADFs \Draft{\citep{2009Bohigas_rmxa45}}. Models combining chemically inhomogeneous regions have been able to reproduce the CEL and ORL simultaneously \citep[][for the cases of NGC 6153, M 2-36, M 1-42, Abell 30, 30 Doradus, and PN SMC N87]{2002Pequignot_12, 2003IAUS..209..389T, 2003Ercolano_mnra344, 2005Tsamis_mnra364, 2006pnbm.conf..192T}.
Afterward, a 3D chemically inhomogeneous model developed by \citet{2011Yuan_mnra411} for the PN NGC~6153, with an ADF$\sim$10, successfully reproduces the observed intensities of the CELs and ORLs. In this chemically inhomogeneous model, the H-poor (or metal-rich) region that only has less than 2\% of the total mass contributes to the majority of the emission of heavy elements ORLs, while the emission of UV and optical CELs of heavy elements comes mainly from the gas with "normal" \Draft{metallicity}. If the amount of metals in the H-poor region increases, the emissivity of the ORLs will be higher. The same line emission could then be obtained with a more massive and not so metal-rich region or a less massive and more metal-rich region, leading to a degeneracy between the metallicity of the metal-rich region and its mass contribution.
Observational evidence of chemical inhomogeneities has been shown by \citet{2016Garcia-Rojas_apjl824}, where narrow-band images of PN NGC 6778 are obtained. The images show that the regions emitting the ORLs \ion{O}{ii} 4649+51 is located in the central parts of the nebula, while the CEL [\ion{O}{iii}]~5007 comes from the outer parts of the nebula that coincides with the H$\alpha$ emission. \citet{2017Pena_mnras472} measured the expansion velocity (V$_{exp}$) for different ions using ORLs and CELs, for 14 PNe, finding that in most cases the kinematics is different for ORLs and CELs (V$_{exp}^{ORLs}$ < V$_{exp}^{ CELs}$). They suggest that the ORLs are emitted by a plasma that was ejected in a different event that the plasma emitting the CELs, causing the difference in expansion velocities.
An important parameter when chemical inhomogeneities are present, is the electronic temperature of the H-poor region, since it can give us an idea of how H-poor is the gas. However, determining this temperature it is not an easy task, given that ORLs have a low dependence on temperature, and are very faint. Small uncertainties in the recombination line ratios will give a large variations in the temperature estimation.
In this work we want to investigate the ADF values (determined as an observer would do) for models composed of different metallicity regions: a gas with "normal" \Draft{metallicity} and a metal-rich gas. We study the degeneracy between the metallicity and the mass of the metal-rich gas, and the consequences of the overestimation of H$\beta$ when determining abundances with ORLs. We show that the total mass of oxygen involved in the metal-rich region is not strongly affected by the degeneracy and we determine a range of values for this parameter in the case of PN NGC 6153.
The structure of the paper is outlined here. In sect. \ref{sec:models} we describe how we make the models, its components and the hypothesis that are assumed. In sect. \ref{sec:results} we show the estimations of plasma physical conditions and chemical abundances for the outputs of the bi-abundance models. In sect.~\ref{sec:compare_mod_object} we compare the results of our models to an observed object.
In sect. \ref{sec:discussion} we discuss the obtained results and detail the limitations of the work. Finally in sect. \ref{sec:conclusions} we present the conclusions of the work.
\section{Modeling strategy}
\label{sec:models}
We use the code Cloudy \DraftThree{v.17.02} \citep{2017Ferland_rmxaa53}, via the python library pyCloudy \citep{2013Morisset_}, to compute the photoionization models for this work. This code solves the thermal equilibrium, the photoionization equilibrium and the radiative transfer for each concentric cell in a spherical symmetric simulation. \Draft{The code outputs the electron temperature and density, the ionic fractions of all the elements, as well as the line emission for each cell of the model.} \DraftTwo{The line emissions are obtained with PyNeb \citep{2012Luridiana_283} in the case of: \ion{O}{II}\ from \citet{2017Storey_mnra470}, and \ion{C}{II}\ and \ion{N}{II}\ from \citet{1991Pequignot_aap251}. We use the add\_emis\_from\_pyneb method from the CloudyModel class of the pyCloudy library, which use the electron temperature and density, and the ionic abundance of the corresponding element, to compute the emissivities of lines which are not computed by Cloudy. It calls the PyNeb corresponding RecAtom class and computes the emission of the lines for each zone of the Cloudy model. The total intensity is obtained by integrating over the volume, in the same way than it is done for the lines computed by Cloudy itself.}
To simulate a PN one needs to describe the properties of the ionizing star and its surrounding gas. We simplify the emission of the central star to a black body at a given effective temperature (Teff) and luminosity (L). The gas is described with the following properties: its morphology, its \Draft{metallicity} and its density distribution. We assume a spherical morphology with constant hydrogen density in the whole nebula. In the figure \ref{fig:squeme} the different regions of the modeled PN are schematically represented in different colors. The nebula is made of a close to solar \Draft{metallicity} gas represented in blue with different darkness (see below) and some metal-rich clumps shown in red. To model such a chemically inhomogeneous nebula with a 1D photoionization code, we actually run different models that will be combined afterward. One model (M$_1$) corresponds to the "normal" gas (\textit{N}, middle darkness blue in Fig.~\ref{fig:squeme}). Another model (M$_2$) deals with the metal-rich clumps (\textit{MR}, in red) and the gas behind these clumps (\textit{BC}, light blue). This model M$_2$ is in fact computing 2 regions at once.
It appears that in some conditions this \textit{BC}\ gas recombines at a smaller radius than the \textit{N}\ gas outer radius: this implies the existence of a fourth region: a shadowed component (\textit{S}, in dark blue of fig.~\ref{fig:squeme}) of same \Draft{metallicity} than the \textit{N}\ component, that requires its own model (M$_3$).
As a summary we present in table \ref{tab:summary_names} the names of the different components, their corresponding abbreviations, the reference of the photoionization model, and the regime of metallicity for each one of the four components. These four gas components are detailed in the following sections.
\begin{table}
\centering
\caption{Some properties of the four components used to build the PN model in this work.}
\begin{tabular}{c c c c}
\hline \hline
Name & Abbreviation & Model & Metallicity \\ \hline \hline
"normal" & \textit{N} & M$_1$ & close to solar \\
meta-rich & \textit{MR} & M$_2$ & rich \\
behind clump & \textit{BC} & M$_2$& close to solar\\
shadow & \textit{S} & M$_3$& close to solar \\
\hline \B
\end{tabular}
\label{tab:summary_names}
\end{table}
\subsection{"Normal" (\textit{N}) component}
\label{sec:normal-model}
The "normal" (\textit{N}) region is the base component (with close to solar metallicity) of our nebular models. We choose to compare our models with the PN NGC~6153 (see section \ref{sec:obs}), because it has UV, optical and IR observations \citep{1984Pottasch_apjl278, 2000Liu_mnra312} and a previous 3D bi-abundance model has been published by \citet[][here after Y11]{2011Yuan_mnra411}. The parameters for the \textit{N}\ component are based on the Bn component of the 3D bi-abundance model by Y11. From the column of the Bn component of their table 2, we take the effective temperature (92~kK), luminosity (1.3$\times 10^{37}$~erg/s) and chemical abundances \Draft{(shown in table \ref{tab:N_abunds}).}
The inner and outer radius of our \textit{N}\ component are approximated to 5 and 15~arcsec based on the observations and models presented in Y11. The distance to NGC~6153 used by Y11 is of 1.5~kpc \citep{2003Pequignot_209}, but more recent data has been published by \citet{2018Gaia_vizier1345} resulting in distance of 1.36~kpc. With such a distance the inner and outer radius of the nebula are 1.02$\times 10^{17}$~cm and 3.05$\times 10^{17}$~cm, respectively. The hydrogen density is simplified to a constant value of 3,000~cm$^{-3}$, taken as an approximation of the the density distribution of Y11. The H$\beta$\ flux from the model with the described parameters is compared to the observed value of NGC~6153, finding that the H$\beta$\ flux in the model is about twice the observed one. To lower the H$\beta$\ flux to about half in the same physical size, we add a filling factor (ff) and diminish the hydrogen density (nH). The adopted ff is 0.6 and the nH explored are: 1000, 1500, 2000, and 2500~cm$^{-3}$). We find the best fit of H$\beta$\ with an hydrogen density of 2500~cm$^{-3}$ with filling factor of 0.6. In this model the ionization stage (measured with line rations such as [\ion{S}{iii}]/[\ion{S}{ii}] y [\ion{O}{III}]/[\ion{O}{II}]) is too high, to fix this we lower the luminosity (to 1.07$\times 10^{37}$~erg/s) by a factor equal to the square of the ratio of the distances 1.36 \citep{2018Gaia_vizier1345} and 1.5~kpc \citep{2003Pequignot_209}. Since the ionization stage was still slightly higher, we refine the fit by decreasing the luminosity to 9$\times 10^{36}$~erg/s. With this parameters the total H$\beta$\ flux and ionization stage are simultaneously fitted. We include graphite and silicate dust grains, each following a -3.5 slope size distribution with ten sizes from 0.005$\mu$m to 0.25$\mu$m; the total dust to gas ratio is 4.2$\times$10$^{-3}$ by mass (about 2/3 of the canonical dust for ISM in Cloudy), see Sec.~\ref{sec:compare_mod_object} for detailed justification.
\begin{table}
\centering
\caption{Chemical abundances of the \textit{N}\ region in units of 12+log(X/H).}
\begin{tabular}{c c c}
\hline \hline
\input{abunds.tex}
\end{tabular}
\label{tab:N_abunds}
\end{table}
The gas modeled here will be referred to as the \textit{N}\ component, it has the main contribution in terms of mass in the bi-abundance model (see section \ref{sec:bi-model}). The electron temperature, density and the ionic fraction of H$^+$, He$^+$, He$^{++}$, O$^+$, O$^{+2}$ and O$^{+3}$ of the different components are shown as a function of \DraftThree{depth} in figure \ref{fig:normal_model}. In particular, solid lines are drawing the behaviour of the \textit{N}\ component generated by the model M$_1$. At \DraftThree{depth $\sim$ 6$\times$10$^{16}$~cm} we can see the small drop in electronic density (upper right panel) corresponding to the recombination of He$^{++}$.
\begin{figure}
\centering
\includegraphics[scale = 0.55]{scheme.pdf}
\caption{Schematic representation of the morphology of the 4 components of our chemically inhomogeneous PN. All blue regions are gas with close to solar \Draft{metallicity}, and the red regions represents metal-rich (\textit{MR}) clumps that in this case are about 630 times more metallic. The region of middle darkness blue is the "normal" gas (\textit{N}), the light blue region is the gas behind the clumps (\textit{BC}), and the dark blue region is a shadow (\textit{S}) ionized by the diffuse of the normal component. The fraction of solid angle for: \textit{MR}, \textit{BC}\ and \textit{S}\ is $\Omega$/4$\pi$~=~0.25.}
\label{fig:squeme}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale = 0.65]{normal_model.pdf}
\caption{Solid line is for model M$_1$ that represents the \textit{N}\ component (sec. \ref{sec:normal-model}) and dashed line is for model M$_2$ that represents: the \textit{MR}\ component with log ACF = 2.1 (sec. \ref{sec:rich-model}) and the \textit{BC}\ component (sec. \ref{sec:behind-clumps-region}). The parameter depth is the radius of the model minus the inner radius of the \textit{MR}\ component. Top left panel: electron temperature as a function of depth. Top right panel: electron density as a function of depth. Bottom left panel: ionic fraction of H$^+$, He$^+$ and He$^{++}$ in red, green and blue, respectively. Bottom right panel: ionic fraction of O$^+$, O$^{+2}$ and O$^{+3}$ in red, green and blue, respectively. }
\label{fig:normal_model}
\end{figure*}
\subsection{Metal-rich (MR) component}
\label{sec:rich-model}
The metal-rich (\textit{MR}) component can be seen as a collection of clumps in the inner parts of the nebula. Given the large amount of free parameters, we decided to fix the inner and outer radius of the clumps at 4.5\arcsec\ and 5\arcsec\ (or 9.2$\times$10$^{16}$~cm and 1$\times$10$^{17}$~cm at a distance of 1.36~kpc), respectively. We only vary the fraction of volume that is occupied by the clumps (but see the Sec.~\ref{sec:discussion} where we explore different values for the density and the inner radius of this \textit{MR}\ region). The effective temperature, the hydrogen density and the helium abundance are the same than in the \textit{N}\ component. \citet{2003IAUS..209..389T} did a 2-component model of NGC~6153 and found that a better fit is obtained when both components have similar densities. The \textit{MR}\ and \textit{N}\ regions are not supposed to be in contact and then not pressure equilibrium hypothesis is used.
The mean ionization parameter\footnote{$U(r) = Q_0 / 4\pi r^2 N_e c $, where $Q_0$ is the ionizing photon rate of the central star per second, $r$ is the distance to the central star, $N_e$ is the electron density and $c$ the light velocity. \DraftTwo{The mean value is obtained over the volume of the nebula: $<U> = \int U dV / V.$}} in this region is \DraftTwo{log~<U>} $\sim$ -1.6. The abundances we adopt for the metals are detailed bellow.
For a given element X, the abundance in the \textit{MR}\ region is defined by the following equation:
\begin{equation}
X/H_{MR} = X/H_{N} \cdot ACF(X)
\label{eq:ACF}
\end{equation}
where ACF(X) is the abundance contrast factor between the \textit{N}\ and the \textit{MR}\ components. In the following we will mainly use the 10-logarithmic value of the ACF (expressed in "dex"). The enhancement of the abundance in the \textit{MR}\ clumps is certainly not uniform (it may differ from one element to another one), but it is out of the scope of this paper to take this effect into account. Thus for a given ACF, the same factor is applied to all the metals. \DraftThree{The dust content included in the close-to-solar components (\textit{N}, \textit{BC}\ and \textit{S}) is less than the canonical D/G for the ISM. The IR emission from this regions reproduces the IRAS observations (see sec. \ref{sec:em_spec}), so we don't include dust in the \textit{MR}\ region, assuming all the dust is in the close-to-solar gas. The opposite extreme assumption that all the dust is in the \textit{MR}\ component is discussed in sec. \ref{sec:dust_mr}.}
In the figure \ref{fig:normal_model}, the \textit{MR}\ component is drawn as dashed lines and corresponds to \Draft{depth $\leq$ 9$\times$10$^{15}$~cm}. Its temperature (upper-left panel) is close to 650~K in this case (ACF = 2.1 dex). The electron density (upper-right panel) is higher than in the \textit{N}\ component, due to the contribution of the metallic ions to the free electrons soup.
\begin{figure*}
\centering
\includegraphics[scale = 0.68]{opd_IP.pdf}
\caption{Total optical depth as a function of energy. This value is measured at the outer radius of the \textit{MR}\ component for ACF(X): 0.1, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0~dex. The horizontal black dotted line indicates the optical depth equal to 1. The vertical dashed lines denote ionization potentials, for H and He: purple (magenta) represents once (twice) ionized ions, for metals: the colors green, blue and red are for once, twice and tree times ionized ions respectively.}
\label{fig:opd_IP}
\end{figure*}
\Draft{The ionization structure of a given element is determined by the balance between recombination and ionization, with the more ionized ions being closer to the ionization source. In the lower-right panel of figure \ref{fig:normal_model} we can see that at the inner part of model M$_2$, where the \textit{MR}\ component is present, the dominant ion is O$^{+2}$, while for the model M$_1$ of the \textit{N}\ component, the dominant ion at the inner part is O$^{+3}$. Although the ionization source is the same for both models, the O/H abundance is not the same, being 2.1 dex larger for the \textit{MR}\ component (in this example) making the amount of photons with energies larger than 54.9~eV not enough for the O$^{+3}$ to be the dominant ion in the \textit{MR}\ component. We can see that just before the region of the M$_2$ model where the abundance goes back to the same value than in the \textit{N}\ component (at depth $9\times 10^{15}$cm), the O$^+$ abundance begins to increase while the O$^{+2}$ and O$^{+3}$ are decreasing, because of the large absorption of photons with energies larger than 35~eV. The decrease in the metal abundances at this transition zone has an effect on the structure of the oxygen ionization, with an abrupt increase of the O$^{+2}$ ions, and a decrease of the O$^+$ ions.}
In order to confirm that the decrease of O$^{++}$ is due to the lack of 35~eV we show in fig.~\ref{fig:opd_IP} the total optical depth at the outer radius of the \textit{MR}\ component as a function of energy. We clearly see the effect of the metals dominating the changes in the optical depth at high metallicities. We can see that for ACF $\gtrsim$ 2.0 dex the optical depth starts to be larger than 1 at 35~eV.
To illustrate this behaviour in more details, we show the ionic fractions (integrated over the volume) of oxygen as a function of the ACF(O) in the top panel of fig. \ref{fig:acf_ion_frac}. When the ACF increases, we observe a decrease of the mean ionization of the \textit{MR}\ region. We fixed the size of this region, which is then matter-bounded (we consider here only the \textit{MR}\ region. There is actually close to solar metallicity gas behind this region, it will be discussed in sec.~\ref{sec:behind-clumps-region}). Increasing the metallicity increases the opacity of the gas, reducing its Str\"omgren size. In the case of the smallest metallicity, the relatively small \textit{MR}\ region only includes the inner part of what would be the \textit{MR}\ region if not matter-bounded, corresponding to the highest ionization. On the other side, when the metallicity is very high, the \textit{MR}\ region is almost radiation-bounded and shows a global lower ionization.
\begin{figure}
\includegraphics[scale = 0.43]{acf_vs_oxy_frac.pdf}
\caption{Distribution of the ionic fraction integrated over the volume of O$^+$, O$^{++}$, and O$^{+3}$ in red, green and blue dashed lines, respectively, as a function of the log~ACF(O) for the \textit{MR}\ component in the top panel, and the \textit{BC}\ component in the bottom panel.}
\label{fig:acf_ion_frac}
\end{figure}
\subsection{Behind clumps (\textit{BC}) component}
\label{sec:behind-clumps-region}
Behind the \textit{MR}\ component we found gas with the same chemical abundances \Draft{and dust} than the \textit{N}\ component but with a different ionization stage, we will call this gas "behind clump" (\textit{BC}) gas. Cloudy is able to model the \textit{BC}\ region in the same run than the \textit{MR}\ region. So for each \textit{MR}\ gas a \textit{BC}\ gas will be modeled, in the same run. \DraftThree{We use the \textit{function} command of Cloudy, to include dust only in the \textit{BC}\ component of the model.}
The model including the \textit{MR}\ and \textit{BC}\ components will be referred to as the M$_2$ model. The \textit{BC}\ component has the same hydrogen density, chemical abundances and inner radius than the normal component but a different outer radius and ionization parameter since the \textit{MR}\ gas that is at a smaller radius than the \textit{BC}\ gas is absorbing some of the ionizing radiation.
\Draft{In the bottom panel of fig. \ref{fig:acf_ion_frac} we show the \textit{BC}\ component ionic fractions of oxygen integrated over the volume as a function of the ACF(O). We can see that at log~ACF $\sim$ 2.3 the dominant ion changes from O$^{++}$ to O$^+$.}
For values of ACF smaller than 2.3~dex, there is still some photons more energetic than 35~eV entering the \textit{BC}\ region and ionizing all the O$^+$. Above 2.3~dex, no more photons ionizing O$^+$ escape the \textit{MR}\ region, and the \textit{BC}\ region is purely O$^+$. \Draft{This will have an effect in the contribution of the O$^{++}$ emission coming from the \textit{BC}\ region for ACF > 2.3~dex.}
The outer radius and the mean ionization parameter of the \textit{BC}\ gas are shown as a function of the ACF(X) in figure \ref{fig:bc_rout_logU}, for comparison the outer radius and the mean ionization parameter of the \textit{N}\ component are also shown. We notice that when the ACF(X) reaches values of 2.6~dex, the \textit{BC}\ gas starts to recombine at a smaller radius than the outer radius of the \textit{N}\ component. The size of the \textit{BC}\ regions starts to decrease, leading to an increase of the \Draft{mean} ionization parameter \Draft{<U>} (lower panel of fig.~\ref{fig:bc_rout_logU}). The space between the recombination front of the \textit{BC}\ region and the outer radius of the nebula is then filled by the shadow region (see Sec.~\ref{sec:shadow-model}).
\begin{figure}
\includegraphics[scale = 0.5]{bc_rcut_logU.pdf}
\caption{The top panel represents the outer radius as a function of the log~ACF(X) for: the \textit{MR}, \textit{BC}\ and \textit{N}\ components as a solid green line, orange dots and solid orange line, respectively. The bottom panel represents the log of the mean ionization parameter as a function of log~ACF(X) for: the \textit{MR}, \textit{BC}\ and \textit{N}\ components as solid green line, orange dots and solid orange line, respectively.}
\label{fig:bc_rout_logU}
\end{figure}
\subsection{Shadow (\textit{S}) component}
\label{sec:shadow-model}
Since the models M$_1$ and M$_2$ are combined to form a spherical morphology, when the model M$_2$ recombines at a smaller radius than M$_1$, there is a shadow after M$_2$. The ionization of the shadow is due to the diffuse radiation (mainly Lyman recombination photons) coming from the \textit{N}\ gas around it. We build a model (M$_3$) to get an approximate idea of the physical conditions and the emission of the shadow (\textit{S} ) region. We use one single model for all the shadow regions obtained with different values of ACF(X), \Draft{this model has the same abundances and dust content than the \textit{N}\ component}. We use as representative ionizing flux the nebular continuum computed by Cloudy during the M$_1$ model, at the radius corresponding to half of the nebula. The figure \ref{fig:shadow_scheme} gives a schematic representation of the shadow region in dark blue. The shadow region is what follows the recombination front of the \textit{BC}\ region. We are actually computing a model where the radial direction of the Cloudy 1D model corresponds to the tangential direction of our object (illustrated by the "$h\nu$" arrows in Fig.~\ref{fig:shadow_scheme}).
The radial size of the \textit{S}\ regions are determined by the difference in radius between the recombination front of the \textit{BC}\ region and the outer radius of the \textit{N}\ region (see fig. \ref{fig:bc_rout_logU}). On the other side, the tangential size ($a$ in Fig.~\ref{fig:shadow_scheme}) of the clumps (the depth of the Cloudy model) needs to be defined.
As we are only interested in the representative emission produced by the \textit{S}\ ionized region, we will adopt a value for this size $a$ that avoid a neutral tube inside the \textit{S}\ region. To obtain this result, we cut the Cloudy model at the half of its Str{\"o}mgren size. The electron temperature of this region is almost constant and close to 6300~K. The electron temperature of the surrounding material (\textit{N}\ region) is 30\% higher than this value; this could imply a higher density for the \textit{S}\ region in case of pressure equilibrium. We did not took this effect into account, keeping the hydrogen density at the same value in all the regions.
\begin{figure}
\includegraphics[scale = 0.28]{shadow_scheme2.pdf}
\caption{Colors represent the same regions as in figure \ref{fig:squeme}. Scheme characterizing the parameters of the \textit{S}\ region. The outer part of the \textit{S}\ region has an arch size of \textit{a}, the $h\nu$ arrows represent the direction of the diffuse ionizing photons coming from the \textit{N}\ region to ionize the \textit{S}\ region.}
\label{fig:shadow_scheme}
\end{figure}
\subsection{Bi-abundance models}
\label{sec:bi-model}
We combine the three models M$_1$, M$_2$ and M$_3$ described in the previous sections to obtain the emission of the whole nebula. The parameter we use to define the fraction of volume each model occupies is the solid angle ($\Omega$) of the \textit{MR}\ region (and the \textit{BC}\ and \textit{S}\ regions).
We actually do not perform a 3D model, not even a pseudo-3D models {\it a la} \citet{2016Gesicki_aap585}. We rather add the intensities computed by each models using a weight determined by $\Omega$.
In this combined model the intensity of a line is obtained with the following equation:
\begin{equation}
I(\lambda) = I(\lambda)_{M_1} \left(1 - \dfrac{\Omega}{4\pi} \right) + I(\lambda)_{M_2} \left(\dfrac{\Omega}{4\pi}\right) + I(\lambda)_{M_3} \left(\dfrac{\Omega}{4\pi}\right)
\label{eq:I_lambda}
\end{equation}
where $I(\lambda)_{Mi}$ is the intensity at the wavelength $\lambda$ for model M$_i$ \Draft{as computed by Cloudy for a $4\pi$ sphere.} \DraftTwo{In the case of $M_3$, the intensity $I(\lambda)_{M_3}$ already takes into account the thickness of the shadow which depends on the position of the BC recombination front.}
It is almost topologically equivalent to use a single model with ${\Omega}/{4\pi}$ or a collection of \textit{n} clumps, each one with an angular size of ${\Omega}/({4\pi n})$. This is not totally the case for the \textit{S}\ region: a large clump may lead to a neutral tube in its shadow, while a smaller clump will have a fully ionized shadow. We choose to model the shadows without neutral component, imposing an implicit upper limit of the clump sizes.
In the following sections we will explore the effect of the metallicity enhancement (ACF) and the contribution of the \textit{MR}\ (and \textit{BC}\ and \textit{S} ) component. This is achieved by considering a grid of models where ACF ranges from 0.1 to 3.0 dex by 0.1 dex steps, and ${\Omega}/{4\pi}$ ranges from 0.01 to 0.5.
\subsection{NGC~6153: a reference PN with high ADF}
\label{sec:obs}
In this section we make a brief summary of the observations taken from the literature that are used to compare with our models.
NGC~6153 is an intrinsically bright PN located in the southern sky at a distance of 1.36~kpc \citep{2018Gaia_vizier1345}, it has an angular size of 32" x 30" \citep[NIR]{2006Skrutskie_aj131}. It was observed with the ESO 1.52 m telescope by \citet{2000Liu_mnra312}. They used the B\&C spectrograph, the slit was 2 arcsec wide and 3.5 arcmin long. The spatial sampling was 1.63 arcsec per pixel. A "minor-axis" of the PN was observed by placing the slit in a PA of 122.8$^\circ$ centered in the central star. The mean spectra of the whole nebula was obtained scanning across the nebula with a long-slit oriented nort-south. A high resolution spectra (FWHM = 1.5\AA) was obtained for the spectral range $\lambda\lambda$3995-4978\AA, and a low resolution spectra (FWHM = 4.5\AA) in the spectral range $\lambda\lambda$3535-7400\AA. The observations were corrected for bias, flats, cosmic rays, they were wavelength calibrated with an He-Ar calibration lamp. The flux was calibrated using standard stars, and dereddened using the galactic extinction curve of \cite{1983Howarth_mnra203} with c(H$\beta$)=1.30 and R~=~3.1. \citet{2000Liu_mnra312} determined the chemical abundances for NGC~6153 and an ADF(O$^{++}$) of 9.2.
\section{Plasma diagnostics with the model outputs}
\label{sec:results}
The emission of the bi-abundance models (see \ref{sec:bi-model}) \Draft{are used as a simulation of a real PN} (with equation \ref{eq:I_lambda}), to determine the physical parameters and chemical abundances with the same methods than for a real object. The results are shown in the following subsections. The electronic temperature and density, and the ionic abundances are determined using the code PyNeb version 1.1.9 \citep{2013Luridiana_ascl}. The atomic data used for each ion are listed in table \ref{tab:pn_atomic_data}, if the electronic temperature is under 500~K (the case for the \textit{MR}\ component at very high ACF) an extrapolation is made to the \ion{H}{i} recombination coefficients from \citet{1995Storey_mnra272}.
\begin{table*}
\centering
\caption{Atomic data \Draft{selected from PyNeb that are} used in the determination of physical parameters and chemical abundances.}
\begin{threeparttable}
\begin{tabular}{l l l}
\hline \hline
~ & \multicolumn{2}{c}{Collisionally excited lines} \T\\
Ion & Transition probabilities & Collision strength \B \\
\hline \hline
N$^+$ & \citet{2004Froese-Fischer_Atom87} & \citet{2011Tayal_apjs195} \T \\
O$^+$ & \citet{2004Froese-Fischer_Atom87} & \citet{2009Kisielius_mnra397} \\
S$^+$ & \citet{2009Podobedova_Jour38} & \citet{2010Tayal_apjs188} \\
O$^{++}$ & \citet{2004Froese-Fischer_Atom87}$^{a}$ & \citet{2014Storey_mnras441} \\
Cl$^{++}$ & \citet{1986Kaufman_jpcrd15}$^{b}$ & \citet{1989Butler_aap208}\\
Ar$^{3+}$ & \citet{1982Mendoza_mnra198} & \citet{1997Ramsbottom_ADNDT66} \\
~ & \multicolumn{2}{c}{Recombination lines} \T\B\\
Ion & Recombination coefficients & Case \\
H$^+$ & \citet{1995Storey_mnra272}$^{c}$ & B \\
He$^+$ & \citet{1996Smits_mnra278} & B\\
He$^{++}$ & \citet{1995Storey_mnra272} & B \\
O$^{++}$ & \citet{2017Storey_mnra470} & B \\
\hline
\end{tabular}
\begin{tablenotes
\item $^{a}$ For transitions 4-2 and 4-3 we use \citet{2000Storey_mnra312}.
\item $^{b}$ For transition 4-3 we use \citet{1983Mendoza_103}.
\item $^c$ We make an extrapolation at low Te (< 500~K).
\end{tablenotes}
\end{threeparttable}
\label{tab:pn_atomic_data}
\end{table*}
\subsection{Physical parameters with CELs}
\label{sec:te-cel}
The line emissions are obtained from the bi-abundance models using eq. \ref{eq:I_lambda}, for the grid of ACF and $\Omega$/4$\pi$ explored. As we are mainly interested by the medium ionization zone (20~eV $<$ IP $<$ 45~eV), we determine the temperature and the density simultaneously using the line ratios: [\ion{O}{iii}]~4363/5007~\AA\ and [\ion{Cl}{III}]~5538/5518~\AA. The contribution of recombination to the [\ion{O}{iii}]~4363 line is naturally included in the models, no correction from this effect has been performed (but see the Sec.~\ref{sec:auroral} for a discussion of the [\ion{O}{iii}]~4363 emission). The results are shown in figure \ref{fig:high_te_ne_pc}, were the x-axis represents the changes in the ACF, the y-axis represents the changes in $\Omega$/4$\pi$ and the color represents the electronic temperature (density) in the top (bottom) panels respectively. Both parameters are found almost constant in the ACF-$\Omega$ plane, with variations of less than 5\% and 10\% for electronic density and temperature, respectively. This is due to the fact that they are determined using CEL which are mainly emitted by the close to solar components (\textit{N}, \textit{BC}\ and \textit{S}\ regions). Small variations appear in the high~ACF-high~$\Omega$ corner of the plots. In the case of the temperature, it may seems counter-intuitive that the temperature increases in the case of a higher contribution of the cold \textit{MR}\ region. What we see here is the effect of the contribution of the recombination to the [\ion{O}{iii}]~4363 emission, coming from the \textit{MR}\ region, which increases at low temperatures. The strong change in the temperature at log ACF $\sim$ 2.2 is due to the sudden vanishing of O$^{++}$ emission in the \textit{BC}\ region (see sec.~\ref{sec:behind-clumps-region}). The decrease of the temperature at constant $\Omega$ and high values of ACF is due to the increase of the contribution of the \textit{S}\ region to the emission of [\ion{O}{iii}]~5007.
\begin{figure}
\begin{center}
\includegraphics[scale = 0.55]{Te_o3_pc.pdf}
\includegraphics[scale = 0.55]{ne_cl3_pn.pdf}
\caption{Top (bottom) panel: the color represents the electronic temperature (density) estimation for the bi-abundance models (see sec. \ref{sec:bi-model}). The x-axis represents the variations in the ACF for oxygen (see eq. \ref{eq:ACF}) and goes from 0.1 to 3.0~dex. The y-axis represents the normalized solid angle of: the metal-rich clumps, the gas behind the clumps and the shadow, in the range of 0.01 to 0.50. Diagnostics are made with the sensitive line ratios: [\ion{O}{iii}]~4363/5007~\AA\ and [\ion{Cl}{III}]~5538/5518~\AA. The contribution of recombination in [\ion{O}{iii}]~4363~\AA\ is taken into account. Line intensities of the bi-abundance models are obtained using eq. \ref{eq:I_lambda}.}
\label{fig:high_te_ne_pc}
\end{center}
\end{figure}
\subsection{Electronic temperature from the Balmer Jump}
\label{sec:te-bj}
The electronic temperature can also be estimated from the Hydrogen Balmer jump normalized to a Balmer line \citep{1967Peimbert_apj150}. For the ACF-$\Omega$ grid, we estimate the Balmer jump temperature, T(BJ), using the Continuum class from PyNeb. The method requires: the flux of the continuum (in units erg$\cdot$s$^{-1}$cm$^{-3}$\AA$^{-1}$) at a wavelength previous and latter to the Balmer jump (such wavelengths can be defined by the user, we choose 3643 and 3861~\AA), a line intensity of H (in units erg$\cdot$s$^{-1}$cm$^{-3}$, we choose H11), the electronic density and the He$^+$/H$^+$ and He$^{++}$/H$^+$ abundances ratios. As an observer would do, we use for the electron density the estimation obtained from a density diagnostic, here the [\ion{Cl}{iii}]~5538/5518\AA\ line ratio, as derived in the previous section (bottom panel of fig. \ref{fig:high_te_ne_pc}). In the range of density we are expecting for a PN (lower than 10$^5$ cm$^{-3}$), the exact adopted value does not strongly affect the derived temperature.
The He$^+$/H$^+$ and He$^{++}$/H$^+$ abundances ratios are estimated consistently by iterative process with the T(BJ) using \Draft{the following lines normalized to H$\beta$}: \ion{He}{i}~4471, 5876 and 6678 for He$^+$/H$^+$, and \ion{He}{ii}~4686 for He$^{++}$/H$^+$. The resulting T(BJ) is shown in figure \ref{fig:te_bj}. We can see that as metallicities (or ACF) and $\Omega$ increase, the temperature starts to decrease, mainly because of the contribution of the \textit{MR}\ component that are cooler at higher metallicity and also because of the contribution of \textit{BC}\ and \textit{S}\ components that are at a lower temperature than the \textit{N}\ component.
\begin{figure}
\includegraphics[scale = 0.55]{Te_BJ.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}. The color represents the Balmer jump temperature determination for the bi-abundance models.}
\label{fig:te_bj}
\end{figure}
\subsection{Estimation of O$^{++}$/H$^+$ with CELs}
We estimate the ionic abundance of O$^{++}$/H$^{+}$ for the ACF-$\Omega$ map with the CEL [\ion{O}{iii}]~5007\AA\ in two ways: taking the T([\ion{O}{iii}]) for both O$^{++}$ and H$^{+}$, and taking T([\ion{O}{iii}]) for O$^{++}$ and T(BJ) for H$^{+}$. In figure \ref{fig:opp_cel} we show the determination with T(BJ) for H$^{+}$. The other determination that only uses T([\ion{O}{iii}]) differs by a factor roughly equal to T(BJ)/T([\ion{O}{iii}]) to what is shown in fig. \ref{fig:opp_cel}. From fig. \ref{fig:opp_cel} we see a 0.3~dex decrease of O$^{++}$/H$^{+}$ at high $\Omega$ and ACF larger than 2.3~dex. This is due to the abrupt decrease of O$^{++}$/O from the \textit{BC}\ region (see bottom panel of fig. \ref{fig:acf_ion_frac}). In this high ACF regime (> 2.3~dex): at constant ACF when the $\Omega$ decreases the contribution of the \textit{BC}\ region also decreases and at constant $\Omega$ when the ACF increases the outer radius of the \textit{BC}\ region decreases (see top panel of fig. \ref{fig:bc_rout_logU}). In both cases the contribution in volume of the \textit{BC}\ region to the total O$^{++}$/H$^{+}$ is smaller. Since there is less (if any) O$^{++}$ in this region (at high ACF) the total O$^{++}$/H$^{+}$ increases again.
The value of 12 + log O/H used in the close to solar components (\textit{N}, \textit{BC}\, and S) is 8.75. The value obtained from the CELs is slightly lower than this value, because of the ionic fraction O$^{++}$/O being lower than one. We also computed the "real" value of O$^{++}$/H$^{+}$ by integrating over the volume of the nebula the contribution of the \textit{N}, \textit{BC}\, and \textit{S}\ regions (no contribution from \textit{MR}\ is taken into account, as this region does not emit [\ion{O}{iii}]~5007\AA ), which is close to both empirical determinations, being slightly closer for the T(BJ) estimation.
\begin{figure}
\includegraphics[scale = 0.55]{Opp_cel_bj.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}. The color represents the O$^{++}$/H$^{+}$ ionic abundance estimation with CELs for the bi-abundance models, using the [\ion{O}{iii}] temperature for O$^{++}$ (shown in top panel of fig. \ref{fig:high_te_ne_pc}), and the Balmer jump temperature (shown in fig. \ref{fig:te_bj}) for H$^{+}$.}
\label{fig:opp_cel}
\end{figure}
\subsection{Estimation of O$^{++}$/H$^+$ with ORLs}
We estimate the ionic abundances of O$^{++}$/H$^{+}$ with ORLs for the grid of ACF-$\Omega$ explored in the bi-abundance models. Given that the dependence of \ion{O}{ii} recombination lines to the electronic temperature is small, we use the Balmer jump temperature estimated in sec. \ref{sec:te-bj}. We make an average of the abundance determined with the same 3-3 recombination lines of \ion{O}{ii} than in table 14 of \citet{2000Liu_mnra312}.
The O$^{++}$/H$^{+}$ estimation is shown in figure \ref{fig:opp_orl}. The same calculation was made with the temperature from [\ion{O}{iii}] 4363/5007~\AA, and the results were very similar. We see in fig. \ref{fig:opp_orl} changes of 1 order of magnitude from the low ACF-low $\Omega$ corner to the high ACF-high $\Omega$ opposite corner. At ACF larger than 1 dex the contribution of the \textit{MR}\ region becomes important given the low temperature of the region, favoring the recombination line emission.
The determination of O$^{++}$/H$^{+}$ from ORLs is expected to provide the actual value of the O$^{++}$/H$^{+}$ in the \textit{MR}\ region (if H$\beta$\ is dominant in this region). We see here that this is far from being the case at high ACFs, the highest value obtained for 12 + log O$^{++}$/H$^{+}$ being 9.9 (at $\Omega$/4$\pi$ = 0.5 and ACF(O) = 2.4~dex) while for the \textit{MR}\ region it reaches 10.8.
\begin{figure}
\includegraphics[scale = 0.55]{Opp_orl_apparent_bj.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}. The color represents the O$^{++}$/H$^{+}$ ionic abundance estimation with ORLs for the bi-abundance models, using the T(BJ) (shown in fig. \ref{fig:te_bj}).}
\label{fig:opp_orl}
\end{figure}
\subsection{Estimation of the ADF(O$^{++}$)}
\begin{figure}
\includegraphics[scale = 0.55]{log_ADFOpp_bj.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}. The color represents the log ADF(O$^{++}$) determined for the bi-abundance models. }
\label{fig:adf_opp}
\end{figure}
From the values of O$^{++}$/H$^{+}$ determined from CELs and ORLs (see previous sections) we can compute the ADF(O$^{++}$), which is shown in fig.~\ref{fig:adf_opp}. This can be compared to the ACF(O) running on the x-axis. In almost the whole ACF-$\Omega$ plane, the difference is important, reaching \Draft{a factor of 50} in the upper right corner. \Draft{For the \textit{MR}\ component, the ionic fraction O$^{++}$/O is always higher than 0.06 in the ACF-$\Omega$ plane, so even though we are comparing the discrepancy factor of O$^{++}$ to the contrast factor of O, this difference can not be completely attributed to the O$^{++}$/O ratio}. The value for the close to solar regions determined from the CELs being actually close to the "real" value used as input of the models, the discrepancy is coming from the ORLs determination that does not match the "real" value. The main issue of this determination is not the temperature but rather the estimation of the H$\beta$ proportion coming from the \ion{O}{ii} line emitting region (\textit{MR}\ component) as described in the next section.
We compare in figure~\ref{fig:adf_acf_opp} the ADF(O$^{++}$) to the ACF(O$^{++}$), defined by the ratio of O$^{++}$/H$^{+}$ integrated over the \textit{MR}\ region and O$^{++}$/H$^{+}$ integrated over the \textit{N} +\textit{BC} +\textit{S}\ regions.
This can be seen as the measure of the error one do when determining the ADF, the ACF being the "true" value of the ionic abundance difference. Whatever the value of $\Omega$, the log ACF(O) around 0.8 corresponds to a good determination of the ADF (white solid line in fig.~\ref{fig:adf_acf_opp}). For lower values of the ACF(O), the ADF(O$^{++}$) overestimates the value of ACF(O$^{++}$) (by up to 0.7~dex). This corresponds to situations where the contribution to \ion{O}{II}\ coming from the close to solar regions actually dominates the total emission of the \ion{O}{II}\ lines. For values of ACF(O) greater than 0.8 dex, the \ion{O}{II}\ emission mainly comes from the \textit{MR}\ region and the ADF(O$^{++}$) underestimates the true value given by the ACF(O$^{++}$) by a factor up to $\sim$100 (dark solid line in fig.~\ref{fig:adf_acf_opp}).
\begin{figure}
\includegraphics[scale = 0.55]{adf_acf_opp_BJ.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}. The color represents the difference in log of the ADF(O$^{++}$) determined for the bi-abundance models and the ACF(O$^{++}$) (difference in O$^{++}$\ of the \textit{MR}\ and the \textit{N}, \textit{BC}\ and \textit{S}\ components). White dashed (solid) is for log ADF(O$^{++}$) - log ACF(O$^{++}$) equal to 0.5 (0.0) and black dash-dotted, dotted, dashed and solid lines are for: -0.5, -1.0, -1.5, -2.0, respectively. }
\label{fig:adf_acf_opp}
\end{figure}
\subsection{Contributions of the \textit{MR}\ clumps to recombination lines.}
\label{sec:clumps_fraction}
In Figs.~\ref{fig:hb_rich} and \ref{fig:o2_rich} we draw the contribution of some recombination lines (namely for H$\beta$ and the V1 multiplet of \ion{O}{ii}) emitted by the \textit{MR}\ region relative to the total emission.
The H$\beta$ emission is mainly coming from the close to solar components, as the \textit{MR}\ contribution is never higher than 9\%. On the other side, the \ion{O}{ii} lines are well representative of the \textit{MR}\ region when this one is strongly H-poor (ACF(O) > 1.5 dex). The apparent incapacity of the \ion{O}{ii} lines to correctly predict the ACF of the nebula (see previous sections) is actually mainly due to the impossibility to only take into account the H$\beta$ emitted by the \textit{MR}\ region.
\begin{figure}
\includegraphics[scale = 0.50]{hb_rich.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}. The color represents the fraction of the H$\beta$ from the clumps to the total emission for the bi-abundance models.}
\label{fig:hb_rich}
\end{figure}
\begin{figure}
\includegraphics[scale = 0.50]{O2_rich_total.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}. The color represents the fraction of the V1 multiplet of \ion{O}{ii} from the \textit{MR}\ to the total emission for the bi-abundance models.}
\label{fig:o2_rich}
\end{figure}
\section{Models fitting a real object}
\label{sec:compare_mod_object}
\subsection{Looking for solutions for NGC~6153}
The two main observables related to the Abundance Discrepancy problem are the value of the ADF and $\Delta T$ the difference T([\ion{O}{iii}]) - T(BJ). We determine this parameters for NGC~6153 (with the same method than for our models), namely ADF(O$^{++}$)=8.2 (using T(BJ)) and $\Delta T$=3025~K \citep[with the observations presented in][]{2000Liu_mnra312}, to define an area in the ACF-$\Omega$ plane where the models fit these values. This area is shown in fig.~\ref{fig:adf_dT}, where the blue (green) color band shows where the model fits ADF(O$^{++}$) ($\Delta T$ resp.) within $\pm$15\%. Another way to perform a similar determination is by plotting the observables predicted by the models, as in fig.~\ref{fig:adf_te}. The color code corresponds to the values of ACF (right panel) and $\Omega$/4$\pi$\ (left panel). The diamond corresponds to the observations for NGC~6153. Using a box of $\pm$15\% around the observed values, we can extract the models and exhibit the values of ACF-$\Omega$ combinations that reproduce the observations. This is done in fig.~\ref{fig:sol_fam}. Two families of solutions appear around [ACF(O$^{++}$), $\Omega$/4$\pi$] = [2.1, \Draft{0.50}] and [2.8, 0.18]. The best solutions correspond to S$_1$: a lower ACF but high volume for the \textit{MR}\ region, and S$_2$: a higher ACF but smaller volume.
The characteristics of these two solutions are summarized in table~\ref{tab:phys_params}, where T$_e$, n$_e$, the ionic fractions of H$^+$, O$^+$, O$^{++}$\ and O$^{+3}$\ and the fractions of volume and mass are given for each one of the 4 regions. The electron temperature of the \textit{N}\ region is close to \Draft{9500}~K, while the \textit{BC}\ and \textit{S}\ components are at slightly lower temperature. The \textit{MR}\ region is found to be as cold as 500-600~K. The high value of n$_e$ found in the \textit{MR}\ region is due to the contribution of the metals to the free electrons. The ionic fractions indicates that the two solutions are very different in the \textit{MR}\ - \textit{BC}\ - \textit{S}\ regions: the S$_1$ solution almost does not have \textit{S}\ region, and the \textit{MR}\ and \textit{BC}\ regions are mainly O$^{++}$, while the S$_2$ solution exhibits a \textit{S}\ component and show the \textit{MR}\ and \textit{BC}\ regions well recombined to O$^+$. In both cases, the mass or volume fraction of the \textit{MR}\ region relative to the whole nebula is rather small.
One interesting result is that despite the fact that both solutions differs by a factor of $\sim$ \Draft{4} in the O/H abundance, the mass of oxygen embedded in the \textit{MR}\ region only differs by a factor \Draft{of $\sim$ 1.5}. This points out that the \textit{MR}\ oxygen mass is somewhere robust against the degeneracy of the solutions relative to ACF(O) and $\Omega$, each one acting in opposite direction. We then determine the oxygen in the metal-rich region in NGC~6153 to be $\sim$40\% of the total oxygen in the nebula, this in a volume that is less than 1\%.
These two solutions S$_1$ and S$_2$ will be used in the next sections to explore the emission of the corresponding models.
\begin{figure}
\centering
\includegraphics[scale = 0.55]{adf_dT.pdf}
\caption{Same axis as figure \ref{fig:high_te_ne_pc}.
The black dashed (solid) line represents the bi-abundance models with estimations of $\Delta T$ = T([\ion{O}{iii}]) - T(BJ) (ADF(O$^{++}$)) equal to the observed value of 3025~K (8.2) for NGC~6153 \citep[our determination using the observations from][]{2000Liu_mnra312} , the green (blue) contours are for 15\% above and bellow this value.}
\label{fig:adf_dT}
\end{figure}
\begin{figure*}
\centering
\includegraphics[scale = 0.6]{relation_adf_te.pdf}
\caption{Left panel: color represents the normalized solid angle of: the metal-rich clumps, the gas behind the clumps and the shadow, in the range of 0.01 to 0.50. Right panel: color represents the variations in the ACF for oxygen (see eq. \ref{eq:ACF}) and goes from 0.1 to 3.0~dex for the bi-abundance models. The x-axis shows the difference in temperature estimated with: [\ion{O}{iii}]~4363/5007~\AA\ and the Balmer jump. The y-axis shows the log ADF(O$^{++}$) estimated for the bi-abundance models. The red diamond represents the value for the PN NGC~6153 taken from \citet{2000Liu_mnra312} (for the minor axis in the case of Te(BJ) and for the whole nebula in the case of Te([\ion{O}{iii}]) and ADF(O$^{++}$)). The red square represents the region selected for the models that are closer to the observed value in NGC~6153.}
\label{fig:adf_te}
\end{figure*}
\begin{figure}
\centering
\includegraphics[scale = 0.55]{solutions_alla_liu.pdf}
\caption{Selected models, as being inside the red square of figure \ref{fig:adf_te}. The color represents the distance from the model to the observed value represented as a red diamond in fig \ref{fig:adf_te}. The distance ($\sqrt{(x/x_0)^2+(y/y_0)^2}$, where x = T([\ion{O}{iii}]) - T(BJ), y = ADF(O$^{++}$)) is normalized through $x_0$ and $y_0$ so the distance from the centre of the square to a corner is 1.}
\label{fig:sol_fam}
\end{figure}
\begin{table*}
\centering
\caption{Physical parameters for each one of the four components, for the two solutions S$_1$, and S$_2$ in parenthesis.}
\begin{threeparttable}
\begin{tabular}{l l l l l}
\hline \hline
& Normal & Rich clumps & Behind clumps & Shadow \T\B\\
\hline \hline \T
\input{table2.tex}
\end{tabular}
\begin{tablenotes
\item $^{*}$ Fraction of solid angle for each component.
\end{tablenotes}
\end{threeparttable}
\label{tab:phys_params}
\end{table*}
\subsection{Emitted spectra }
\label{sec:em_spec}
We use the solutions S$_1$ and S$_2$ determined in the previous section to show in fig.~\ref{fig:lines_cel_orl} the contribution to the total emission coming from each one of the 4 regions (in different colors), for a set of 28 representative emission lines.
An extensive list of emission lines with the intensity emitted by each region, normalized to H$\beta$, for both solutions, is given in table~\ref{tab:intensities}.
\Draft{For both solutions we compare in table \ref{tab:IR_emission} the infrared emission to the observed values through the 4 IRAS bands at 12, 25, 60 and 100 $\mu$m obtained for PN NGC~6153 from \citet{1988iras7H}.}
\Draft{We can see that in both solutions the IR emission from the model is very close to the observed emission. \DraftThree{For the S$_1$, the IR band fluxes are well reproduced. For the S$_2$, the model is higher than the observed fluxes at 12, 25, and 60 $\mu$m, and a good fit for 100~$\mu$m is obtained. The emission at the shorter wavelengths bands is already higher than what is observed, adding dust to the \textit{MR}\ component will increase the emission in this range.} Thus there is no evidence of the presence of dust in the \textit{MR}\ region, \DraftThree{thus we don't include dust in the \textit{MR}\ component.} This result agrees with what \citet[][]{2003Ercolano_mnra344} found for the polar knots of Abell~30, but is in contradiction with what \citet[][]{1994Borkowski_apj435} found in the case of the equatorial H-poor ring of the same object. The interpretation of this lack of dust in the metal rich region, in terms of creation and destruction of dust, is out of the scope of this paper, \DraftThree{nonetheless, the extreme opposite case where all the dust is in the \textit{MR}\ component is discussed in sec. \ref{sec:dust_mr}.} }
\begin{table}
\centering
\caption{Emission from IRAS bands for PN NGC~6153 \citep{1988iras7H} and the two solutions S$_1$ and S$_2$ in Jansky units.}
\begin{tabular}{c c c c}
\hline \hline
Wavelength & Observations & S$_1$ & S$_2$ \\ \hline \hline
12~$\mu$m & 6.9 & 7.2 & 10.3 \\
25~$\mu$m & 52.1 & 72.8 & 107.0 \\
60~$\mu$m & 120.0 & 127.7 & 173.6 \\
100~$\mu$m & 52.1 & 40.3 & 52.0 \\
\hline \B
\end{tabular}
\label{tab:IR_emission}
\end{table}
We see again that the contribution of H$\beta$ coming from the \textit{MR}\ region is very small. This is the key problem in the determination of the true value of the \textit{MR}\ region metallicity, translated in the difference between the ADF (observed) and ACF (real).
The important contribution of the \textit{MR}\ region coming from the metal recombination lines is clearly seen, as well as from the infrared lines. The contribution of the \textit{BC}\ and \textit{S}\ regions are different in the two solutions.
From the table~\ref{tab:intensities} we can see that the total intensities of some lines (latest column) change between the 2 solutions: the \ion{C}{ii} (\ion{N}{ii}) recombination lines increase by a factor of $\sim$\Draft{2.8 (1.7)}, while \Draft{the \ion{Ne}{ii} recombination lines decrease by $\sim$ 25\%}, between the S$_1$ and the S$_2$ solutions.
This is mainly due to a change in the ionization of the \textit{MR}\ region. For the same reason we see a difference in the emission of the infrared lines of [\ion{N}{ii}] and [\ion{Ne}{ii}]. The other IR lines are from higher charged ions, less affected by the ionization changes in the \textit{MR}\ regions. The \ion{O}{II}\ recombination lines are not changing, by construction of our solutions based on fitting the observed ADF(O$^{++}$).
The optical CELs are mainly unchanged between the two solutions
We use a simplistic relation in the way metals are enhanced in the \textit{MR}\ region (same ACF for all the metals). The real situation may certainly be more complex and these differences in the \ion{C}{ii} and \ion{N}{ii} emission between the two solutions \Draft{can not be used to derive properties related to the oxygen abundance like ACF(O).} \DraftTwo{Nevertheless, the \ion{C}{ii}, \ion{O}{ii}, and \ion{N}{ii} intensities predicted by both models are close to the observed values given by \citet[][]{2011Yuan_mnra411}. This could be seen as an indication that the C:N:O relative abundances used in \textit{MR}\ region are adequate.}
\begin{figure*}
\centering
\includegraphics[scale = 0.53]{lines_orl_cel.pdf}
\caption{Normalized intensity of ORLs and CELs showing the contribution of the four components of the bi-abundance models (see sec. \ref{sec:bi-model}). Top panel represents the S$_1$ solution (log ACF(O)~=~2.1, and $\Omega$/4$\pi$~=~\Draft{0.50}), and bottom panel the S$_2$ solution (log ACF(O)~=~2.8, and $\Omega$/4$\pi$~=~0.18). The two solutions are selected from the two solution families shown in figure \ref{fig:sol_fam}. }
\label{fig:lines_cel_orl}
\end{figure*}
\section{Discussion}
\label{sec:discussion}
\subsection{Helium abundance in the \textit{MR}\ component}
The material that is responsible for the observed high ADF ($>$~10) in some PNe is unique and original. It is not a common ISM component. It is very hard to observe as it is blended with "normal" PN gas in the observations. It mainly emits metal recombination lines, and some H recombination lines. The contribution from this component to the total He recombination lines is almost negligible (as well as in the case of HI lines), and barely distinguishable from the "normal" component emission. The exact He abundance in this \textit{MR}\ component is then very hard to determine. On the other side, the fact that this component is cold points to a small heating and strong cooling. We explore a variant of the solutions obtained above, where the He/H abundance is also enhanced. The results are presented in details in the Appendix~\ref{sec:App-He}. An acceptable solution can be found for He/H as high as 6.3, leading to a slight overprediction of the \ion{He}{I} line intensities compared to values reported by \citet[][]{2011Yuan_mnra411}. The total amount of oxygen in the whole nebula derived for this solution does not significantly changed compared to what is found in solutions S$_1$ and S$_2$.
This result is similar to that of \citet{2011Yuan_mnra411}, where the He abundance is less enhanced than metals in their metal rich region (He/H (MR/NR) $\simeq$ 5, while O/H (MR/NR) $\simeq$ 80, from their Tab.2.).
\DraftThree{On the other hand, in the case of the model of Abell~30 presented by \citet[][]{2003Ercolano_mnra344}, the He/H abundance is reaching a value as high as 40, while O was determined to be almost in the same amount than H. We have here lower values for O/H (between 0.07 and 0.35) and for He/H (lower than 6.3). The process leading to the H-poor components in these two object seems to be different.
Finally, if we compare with the abundances determined in some nova shells (nova event is not excluded by \citet[][]{2018Wesson_mnra480} as a proxy for a scenario to these H-poor clumps), one can see that the strong enhancement determined for the C:N:O elements is not associated to an equivalent enhancement in Helium \citep[See e.g.][]{1996Morisset_aap312}.}
\subsection{Dust in the \textit{MR}\ component}
\label{sec:dust_mr}
\DraftThree{The creation and destruction of dust in the \textit{MR}\ component is unknown, and due to the lack of spatially resolved IR observations, we cannot constrain the radial distribution of dust.
In the previous sections we have assumed that no dust is in the \textit{MR}\ component, and all the dust necessary to reproduce the IRAS observations is located in the close-to-solar components. Here we explore the opposite case where the dust is only present in the \textit{MR} component, although the reality may be an intermediate solution between both hypothesis. We generate grids of models including dust of graphite and silicate types with two sizes for each: 1~$\mu$m and 0.01$\mu$m, we search for the D/G values that have the closest fit to the IRAS observations. We define the D/G for each model to be proportional to the O/H of the \textit{MR}\ component. We follow the same procedure searching for solutions that reproduce $\Delta$T and the ACF(O$^{++}$), like in the figures \ref{fig:adf_te} and \ref{fig:sol_fam}. In these models, at some point when the ACF increases the dust becomes optically thick (ACF > 2.7~dex) and the gas behind the clumps is no longer ionized by the star, becoming a shadow ionized by the Lyman continuum radiation from the \textit{N}\ component. The solution at "lower" ACF and higher $\Omega$/4$\pi$ is lost, and only one solution with ACF = 2.8~dex and $\Omega$/4$\pi$ = 0.27, marginally reproduces simultaneously $\Delta$T and ACF(O$^{++}$). For this solution the IR emission at 12, 25, 60 and 100~$\mu$m is 10.2, 44.6, 134.8 and 71.4~Jy, close to the observed values (See Tab.~\ref{tab:IR_emission}). For this solution, the D/G by mass is 1.04, which is considerably higher than the canonical value in the ISM of 6.3$\times$10$^{-3}$. The abundances of the elements trapped in the dust (in log, by number) are: C = -1.21, O = -0.4, Mg = Si = Fe = -1.0. These values of abundances of elements in dusty phase are even higher than the corresponding abundances in the gaseous phase for Mg, Si and Fe, for O is roughly the same amount and for C is about 30\% of the gaseous phase. A more detailed modeling of the distribution of dust could be of interest as further work, but additional observations are needed to constrain the free parameters.}
\subsection{Recombination contribution to auroral lines}
\label{sec:auroral}
\DraftThree{The recombination contribution to the auroral lines for [\ion{N}{ii}]~5755 is taken from Cloudy based on \citet{1984Nussbaumer_Astr56}, for [\ion{O}{II}]~7332 from Cloudy based on \citet{2001Liu_mnra323}, for [\ion{O}{III}]~4363 from Cloudy (version 17.02) using \citet{1991Pequignot_aap251} and \citet{1984Nussbaumer_Astr56}}. We see in Fig.~\ref{fig:lines_cel_orl} that the contribution of the \textit{MR}\ component to the auroral lines [\ion{N}{ii}]~5755 and \Draft{[\ion{O}{II}]~7332} emission is not negligible. This is due to the recombination of N$^{++}$ and O$^{++}$, favored by the low temperature of the \textit{MR}\ region. In the grid we explored, the most important contributions reach 1/3 of the total auroral emission, for a model with a determined ADF(O$^{++}$) of $\sim$30. In case of more extreme ADFs, the recombination contribution may even be dominant. \Draft{In case of higher excitation nebula the recombination of [\ion{O}{III}]~4363 may become significant.} This is in total coherence with observations from \citet{2016Jones_mnra455} and \citet{2016Garcia-Rojas_apjl824} who found the spacial location of the [\ion{O}{III}]~4363 emission in coincidence with the O$^{++}$\ recombination lines and not with the [\ion{O}{III}]~5007 emission. This effect can be amplified when the observation is obtained in the direction of the metal rich region rather than for the whole nebula, which is the case in high spatial resolution observations. Without taking the effect of the recombination into account, one can determine a gradient of temperature increasing toward the central part of the nebula, where the \textit{MR}\ regions are located. On the contrary, the increase of [\ion{O}{III}]~4363 emission is actually related to a very strong decrease of the electron temperature! \DraftTwo{We will explore the contribution of the recombination to [\ion{O}{III}]~4363 in a forthcoming paper \citep{2020Gomez-Llanos_arxiv}}
\subsection{Ionization correction factor for the \textit{MR}\ region}
\label{sec:ICF-MR}
The results presented in this work are about the ADF(O$^{++}$). If one is interested in the ADF(O), an Ionization Correction Factor (ICF) needs to be applied. Even in the case of a low ionization nebula where only O$^+$\ and O$^{++}$\ are present, these ionic abundances are determined from the [\ion{O}{II}] and [\ion{O}{III}] lines that are mainly emitted by the close to solar regions (\textit{N}, \textit{BC}\ and \textit{S}). For the \textit{MR}\ region where only O$^{++}$\ is observed through the \ion{O}{II}\ lines, the ICF(O$^{++}$) needs to be used. Since the ICF(O$^{++}$) for ORLs can't be determined, the one derived for the forbidden lines has to be used. From table ~\ref{tab:phys_params}, we see that the ICF(O$^{++}$) - which is O/O$^{++}$\ - is almost the same for the \textit{N}, \textit{MR}\ and \textit{BC}\ regions in the S$_1$ solution (\Draft{between 1.1 and 1.2}). But in the case of the S$_2$ solution, the ICF(O$^{++}$) for the close to solar region is close to 1.1, while the ICF(O$^{++}$) that needs to be applied to O$^{++}$\ in the \textit{MR}\ region is 4.8.
If the density of the \textit{MR}\ region, or its radial size, is changed, the ionization of the region will change, and so will the ICF(O$^{++}$).
The \ion{He}{ii}~4686 emission is classically used to determine the ICF(O$^{++}$) taking into account the presence of O$^{+3}$\ \citep{2014Delgado-Inglada_mnra440}. But we see that in the case of the S$_1$ solution, 20\% of the intensity of this line comes from the \textit{MR}\ region and should be removed before computing the ICF.
\subsection{Effects of density and size of the \textit{MR}\ region}
\label{sec:density-MR}
We explored in the previous sections the behaviour of a grid of models. In this grid some parameters have been fixed to reasonable values. This is the case of the \textit{MR}\ region density, its distance to the central star and its radial size.
The electron density of the \textit{MR}\ region is hard to constrain by observations: recombination lines (the only ones emitted by this cold region) have quite low dependency on the density. There is no indication that a pressure equilibrium must exist between the warm and the cold regions, as they are not supposed to be in contact. We adopted then the same hydrogen density as for the \textit{N}, \textit{BC}\ and \textit{S}\ regions. Increasing for example the density of the \textit{MR}\ region would increase its optical depth, decreasing its global ionization and possibly leading to a vanishing of the \textit{BC}\ region, replaced by a pure shadow region.
The effect of changing the radial size of the \textit{MR}\ region, decreasing for example its inner radius, would lead to a very similar effect: increasing its optical depth.
In both cases, the global emission of the \textit{MR}\ region, for a given $\Omega$, will increase. To recover the same ADF and $\Delta T$, a lower ACF will be needed for a given $\Omega$.
We performed test cases to explore the effect of changing the hydrogen density and the inner radius of the \textit{MR}\ component.
In a first test, we explore the density range from 1$\times$10$^3$~cm$^{-3}$ to 4$\times 10^3$~cm$^{-3}$. At lower densities the models fitting the observed ADF(O$^{++}$) and $\Delta T$, as shown in fig.~\ref{fig:adf_te}, have higher $\Omega$ eventually reaching impossible values higher than 1. On the other side, at higher densities no solution is found that reproduces simultaneously the ADF(O$^{++}$) and $\Delta T$.
The other test varying the inner radius between 4.3\arcsec\ and 4.7\arcsec, leads to a similar behaviour. A smaller inner radius needs less $\Omega$ to fit the two observables.
These tests show that the main result obtained in the previous section is still valid when changing some of the parameters we fixed {\it a priori} in our grid of models: the ACF (the real metal enhancement of the rich region) is higher than the ADF determined from observations.
From the exploration of this grid, we determine the mass of oxygen embedded in the metal rich clumps to be between 25\% and 60\% of the total mass of the nebula.
Another limitation of our study is the reality of the \textit{BC}\ and \textit{S}\ regions. The very simplistic model presented here assumes a very sharp separation between the \textit{N}\ region and what is happening behind the \textit{MR}\ region. Some observations exhibit cometary tails \citep[see e.g.][in the Helix nebula]{2005AJ....130..172O} that are very aligned in the direction of the central star. On the other side, the image of NGC~6153 does not show clear evidences of this kind of structure.
\section{Conclusions}
\label{sec:conclusions}
In the previous sections we described how a grid of models is obtained considering two metallicity models when varying the abundance of a metal-rich (\textit{MR}) component and its volume. The ratio between the two metallicities is called the Abundance Contrast Factor (ACF). We explored how the physical parameters and chemical abundances can be determined from the observables produced by the models, especially the [\ion{O}{III}] and \ion{O}{II}\ emission lines leading to the determination of O$^{++}$/H$^+$\ by two methods. These two values are classically used to define the Abundance Discrepancy Factor (ADF(O$^{++}$)).
We show that the determination of the ADF(O$^{++}$) does not represent an good estimation of the ACF(O) given as input of the models, even taking into account that the ACF is defined for the oxygen element and the ADF only for the O$^{++}$\ ion. The difference between the ACF and the ADF can be as high as two orders of magnitude. The main limitation is coming from the fact that the ionic abundance O$^{++}$/H$^+$\ determined from the \ion{O}{II}\ region is computed using the \ion{H}{i} intensity resulting from the emission of the whole nebula, when only a small fraction is actually coming from the \textit{MR}\ region (a maximum of 8\% is found in the models we present here).
It is today almost impossible to isolate the contribution of the \textit{MR}\ region from the total \ion{H}{i} emission by observations. Very high spatial and/or spectral resolution observation may solve this important issue.
Nevertheless, we show that it is possible to determine an estimation of the mass of metals (at least oxygen in our case) embedded in the rich component, from the optical recombination lines.
\DraftTwo{The models reproduce the Infrared observations without needs of an enhanced dust content in the metal rich region.}
\section*{Acknowledgements}
\DraftTwo{We thank the anonymous referee for helping to improve the quality of this research, especially on the discussion of the presence of dust in the models, and the He/H abundance in the \textit{MR}\ region.}
We are very grateful to Grazyna Stasi\'nska and Jorge Garc\'ia-Rojas for reading and helpfully commenting the draft of this paper.
This work was supported by CONACyT-CB2015 254132, DGAPA/PAPIIT-107215, and DGAPA/PAPIIT-101220 projects. VGL acknowledges the support of CONACyT and PAEP-UNAM grants. The manuscript of this paper has been written in the Overleaf environment.
\section*{Data availability}
The models presented in the paper are available under request to the authors.
|
2,869,038,154,802 | arxiv | \section{Introduction}\label{sec:intro}
This paper presents a method to actively sample individuals in a population as a way to mitigate the spread of pandemics such as COVID-19. Sampling algorithms are commonly used in machine learning to acquire training data labels for classification ("active learning" \cite{AL-MJ, pz2006}) and in bandit algorithms \cite{bandit-wolpert} to explore complex search spaces through exploration and exploitation. The method we present in this paper builds on these ideas, but do so in the context of containing the spread of an epidemic in a population.
The literature on managing disease spread through ideas based on active sampling is primarily from the public health area. There are two reasons why this literature has considered sampling, although both these are sometimes intertwined in the context of population surveillance \cite{lee2010principles}.
The first is to estimate the actual incidence or spread of a disease, such as HIV, in a population. In the case of estimating HIV incidence , methods such as a population survey and "sentinel surveillance" \cite{magnani2005review, fylkesnes_studying_1998} have been shown to be useful. The population survey in a catchment area (i.e. where there is likely disease) is (stratified/cluster) random sampling, and is generally considered the gold standard but also known to be expensive \cite{lee2010principles}. Sentinel surveillance, in contrast, gathers data from a subset of venues (the "sentinels", e.g. clinics) where the subset with the disease is likely to visit for treatment. Recognizing that sensitive diseases like HIV may be prevalent in hard-to-reach populations who might stay under the radar due to activities that are illegal or illicit (e.g. sex workers, drug users) the literature \cite{magnani2005review, goel2010assessing} has critically studied other techniques such as snowball sampling (starting with a few seeds who have the disease and asking them to name others who might), respondent-driven sampling (similar to snowball, but longer chains with fewer referrals per seed) and time-location sampling (e.g. sampling users at nights in districts with brothels). With sensor data newer forms of sampling are emerging as well. It has been shown for instance, that to quantify disease spread in livestock that sampling the nodes (physical locations) where there is greatest movement can be effective \cite{dawson2015sampling}.
The second reason for considering sampling strategies is to mitigate the spread of disease by proactively identifying individuals who need to be quarantined or tested. One method to do this, common in the public health literature is contact-tracing \cite{PhysRevE.66.056115, eames2003contact, underwood_contact_2003}. The idea is to sample contacts of known infected individuals for presence of disease, and, along with quarantine, is the main mechanism used worldwide during the COVID-19 pandemic \cite{salathe2020covid, hellewell2020feasibility}. Traditionally contact-tracing is done through questionnaires, but newer automated contact tracing based on tracking cell phone proximities have also been proposed \cite{ferretti2020quantifying}.
The SARS-Cov-2 pathogen and its transmission have the following unique characteristics, though, that make none of these sampling methodologies sufficient by itself.
\begin{itemize}
\item The possibility of pre-symptomatic and asymptomatic spread rule out pure sentinel-based strategies that would test those who report at clinics with symptoms. For the same reason, contact tracing alone would be insufficient. This is because contact-tracing pursues only contacts of those known to have tested positive. Asymptomatic carriers may have also been transmitting the disease to their contacts, but those will not be actively sought by this strategy. Hence, random sampling is also useful.
\item The pathogen has also been shown to survive on surfaces or in air in specific locations for extended periods of time. Hence, location-based sampling may also be useful.
\item There is uncertainty about whether individuals develop long-lasting immunity. Hence sampling strategies may have to allow for re-testing the same individuals repeatedly.
\item Information about how covariates influence getting the disease, and being affected by it, are still evolving. For instance, while we know that comorbities and age affect outcomes, there is still uncertainty about whether this knowledge is complete. Moreover, how covariates might affect contracting the disease is still uncertain. In this environment, covariate-based sampling strategies alone might be insufficient. Sampling strategies that target specific \emph{individuals} to test is therefore critical.
\item The highly infectious nature of the pathogen and the extent of worldwide spread have placed constraints on testing capacities, requiring prioritizing tests if they are to be useful in mitigating the spread.
\end{itemize}
The method presented in this paper addresses all five aspects noted above. This paper presents dynamic active sampling strategies based on multi-armed bandit algorithms that can optimally combine the different sampling ideas to generate real-time lists of whom to sample. Bandit algorithms are particularly good at combining exploration with exploitation and have seen success in many scenarios where this combination is important. This is the case with SARS-Cov-2, where exploration (random sampling) has to be effectively combined with exploitation (contact-tracing, location-based sampling) in a dynamic manner based on data. One of the key aspects of our work is that we identify specific individuals to test/sample, as opposed to criteria or covariate-based sampling strategies.
Furthermore, our algorithm operates in a high uncertainty network environment. Specifically, the contacts between individuals are initially only partially observable and are revealed gradually based on individual sampling. The bandit algorithm therefore first uses the Thompson sampling strategy to trade off between expansion in unobserved nodes to identify possible new hot spots, vs. densification on the observed portion to utilize the knowledge learnt for testing. If densification is chosen, an inner level upper confidence bounding (UCB) policy is designed to balance between the individuals having higher probability of infection vs. those having greater information uncertainty. To quantify the closeness of individuals and risk of getting infected, we construct heterogeneous network embedding to encode the social interactions by a continuous latent representation. In order to model how the sampling affects, and in turn is affected by, the underlying environment, we implement an agent-based model constructed from realistic data from New York City.
Several recent Covid-19 studies have focused on the ability to predict and flatten the spread of the disease by applying different intervention policies \cite{Keskinocak2020.07.22.20160036,Keskinocak2020.04.29.20084764, topirceanu2020centralized,atkeson2020will,ferguson2020report,chang2020modelling}. Among the policies that were examined were case isolation, home quarantine, social distancing, restrictions on air travels, and school closures and re-opening. To assess the efficiency of the restrictions, most of these studies coupled S/ I/ R (stands for susceptible / infected / recovered) diffusion modeling \cite{hethcote2000mathematics,cooper2020sir}, with Agent Based simulation Model (ABM) \cite{perez2009agent} of infectious spread under different restrictions scenarios. Among these studies, an ABM of the Covid-19 spread in the city of NY under different quarantine policies was offered by \cite{hoertel2020facing}. This model serves as a basis for the ABM that we developed in this work. Also in contemporaneous work, \cite{grushka2020framework} present a framework that uses historical data to build a classifier that computes a ``risk score'', which is the basis of determining how to combine exploration and exploitation. While the idea of using bandit-based algorithms for testing is important, the challenge is in developing specific methods that address all the complexities noted previously. Our work presents a specific method that is adaptive, dynamic and comprehensive. Along with the lines of whom to sample for testing, there is a similar question of whom to vaccinate. \cite{chen2020allocation}, for example, present adaptive and dynamic covariate-based policies for vaccination. Our framework allows vaccination to be incorporated into the background dynamics in order to test the effectiveness of combinational strategies such as smart testing with vaccination.
We make the following important contributions. First, we present a novel active sampling algorithm to effectively manage pandemics such as COVID-19 (as far as we know there are no methods in the literature developed for this problem yet). We develop a general multi-armed bandit framework for this problem that (a) can leverage information of different types such as individuals and locations, (b) can handle uncertainty in both the underlying disease dynamic as well as information and (c) is built to sample based on individuals and not covariates. Second, we make several modeling contributions related to how bandit based algorithms can work in this setting. Specifically, we build a heterogeneous network to capture social relations, neighborhood similarity and community membership and show how this can drive a two-level active sampling strategy to sample individuals. The use of network embedding ideas within the heterogeneous network is novel. Third, we make important contributions to the literature on multi-armed bandits as well. Beyond classic bandit problems, there exists limited literature on active search on graph, with the objective of finding as many target nodes as possible with some given property. Most of the existing work assumes that the complete network structure and underlying process is known beforehand. There is very little work on partially observed and dynamically changing networks and underlying (disease) dynamics. We model the partially observed scenarios within this framework and show how the network can be strategically expanded over time to support the active sampling strategy.
\iffalse
since we consider a scenario where the sampling is not independent of the environment, but itself affects the underlying mechanism. While there are similarities to reinforcement learning, the state-action space is so large to render existing reinforcement learning paradigms intractable for this problem.There are two levels of exploration/ exploitation trade-offs. First, the network structure is partially observable and can only be revealed gradually based on individual sampling. Second, since we do not have full knowledge of the disease states of the entire population, the probability of infections is not known and needs to be learnt as we sample their contacts and observe their test results.
\fi
Before presenting the details, it is worth emphasizing that smart testing strategies are needed even if tests are cheap and/or vaccines are available for the following reasons.
\begin{itemize}
\item Tests are never truly free, there is an infrastructure behind the testing (generating the tests, distributing, obtaining and storing results) that is expensive.
\item In active infections that have high economic impact, policy makers may need to provide incentives to make individuals take these tests, particularly if they are asymptomatic (e.g. in the US this may require some compensation for the time an individual who tested positive might have to be in quarantine if the government mandated the test).
\item For different reasons, some individuals have higher propensity of being infected yet asymptomatic, or pose higher risk on the society when infected. A smart testing strategy can help decision makers detecting these individuals.
\item Pandemics such as COVID-19 are likely going to recur in the future, with no guarantees that tests for future infections will be cheap (particularly in the early stages, which active testing can play a critical role). The methodology presented in this paper applies to mitigating the spread of any pandemic.
\item We view vaccines as complementary to testing, and, as we show, active testing strategies remain important even with vaccinations as a combinational strategy to eradicate pandemics.
\end{itemize}
\section{Problem Formulation}\label{sec:problem}
We approach the problem from the perspective of policy makers who can enact sampling policies. Below we describe a general setting from this perspective that can apply to any disease scenario. COVID-19 specific instantiations of these are presented as brief examples here, but those are discussed in more detail in Sections \ref{sec:ABM} and \ref{sec:MAB}. As we present the setting considered here, we distinguish between information known to the policy-makers and the true state of information in the world. As we will see later in the paper, this distinction enables us to design and run carefully constructed agent-based models to evaluate sampling policies.
Since disease spread occurs in a spatio-temporal manner in a population, the setting we consider has three components. The first component, \emph{data setting}, represents knowledge about the individuals, their spatio-temporal behavior along with characteristics of the physical area. The second component, \emph{process setting}, represents knowledge about the underlying disease and its mechanism of spread. The third component, \emph{policy setting}, represents what policy makers are assumed to have access to, and accordingly what the sampling algorithm will use. These are described below.
\noindent \\ Data setting:
\begin{itemize}
\setlength\itemsep{0em}
\item a population of individuals $P$, with set of covariates $Z$ (e.g., age, gender),
\item a set of locations $L$ in the city,
\item spatial-temporal data about individuals in $P$ over $L$.
\end{itemize}
\noindent \\ Process setting:
\begin{itemize}
\setlength\itemsep{0em}
\item initial actual disease state (e.g., S/E/I/R) of each individual in $p \in P$,
\item a spatio-temporal model of contagion that represents how the disease spreads in the population,
\end{itemize}
\noindent \\ The policy makers' setting involves two components, the information setting and the policy setting. The information setting for the policy maker includes:
\begin{itemize}
\setlength\itemsep{0em}
\item complete list of individuals $P$, with partial covariates information,
\item complete list of locations $L$,
\item initial disease state of each individual in $P$ (this can be incomplete),
\item the contagion process is partially known - the policy maker (algorithm) is assumed to be aware of only the high-level factors affecting disease diffusion, such as contacts with exposed individuals or the role of locations where infected individuals have been. As information about a disease is more well-known (such as the specific impact of covariates on disease spread, or the exact probabilities based on locations) we can augment this setting to reflect this information, but don't assume this to be the case in this paper.
\end{itemize}
\noindent The policy setting includes:
\begin{itemize}
\item a quarantine policy $Q$, that determines when and how individuals will isolate in the population,
\item an objective function $O(T)$. This paper focuses on sampling strategies for bridging the gap between the actual infected population and the known infected individuals, as a step towards controlling disease spread.
\item constraints on number of tests per day ($D_{max}$).
\item Sampling an individual to test reveals the following:
\begin{itemize}
\item disease state of the individual being tested.
\item spatio-temporal data about the individual, if infected. In some cases (e.g. access to full GPS data) this will be complete information on the places visited and the actual contact network. If this is by self-reporting (as is common in most places) then this information will be incomplete.
\end{itemize}
\end{itemize}
Table \ref{tbl:notation} summarized the common notations we use through the paper, and in which section each notation first appears.
\begin{table}[]
\centering
\begin{tabular}{p{0.15\linewidth} p{0.2\linewidth} p{0.7\linewidth}}
\toprule
\textbf{Section} & \textbf{Notation} & \textbf{Description } \\
\midrule
Section 2 & $P = \{p_i\}$ & Population \\
& $Z= \{z_{ip}\}$ & Individuals’ covariates (e.g. age, gender) \\
& $L = \{l_j\}$ & City layout (list of locations, e.g. a specific school, restaurant) \\
& $T = \{t_l\}$ & Time in days \\
& $Q$ & Quarantine policy (e.g. 14-day isolation for positively tested individuals) \\
& $O(T)$ & High level objective (e.g. minimizing total deaths) \\
& $D_{max}$ & Constraints of number of daily tests \\
\midrule
Section 3.1 & $K$ & Number of arms in multi-armed bandit algorithms \\
& $x_t$ & Selected arm at time $t$ \\
& $r_t(x_t)$ & Payoff at time $t$ for selecting arm $x_t$ \\
\midrule
Section 3.2 & $\mathcal{G} = <V,E,Z,D>$ & Network of individuals (in the contact network) and locations (in the heterogenous network) connected based on proximity \\
& $V = \{v_i\}$ & Nodes in the graph \\
& $E = \{e_{ij}\}$ & Edges in the graph \\
& $D = \{d_i\}$ & Disease states (e.g. S/E/I/R ) \\
& $y_v$ & Whether the individual is infected (partial disease state) \\
& $\mathcal{G}_t$ & Network at time $t$ \\
& $t$ & Known network at time $t$ \\
& $f(v)$ & Reward of testing node $v$ \\
& $U/ U_t$ & Subset of nodes to sample from / at time $t$ \\
& $f$ & Function that maps the node to [0,1] as the probability of infection. \\
& $F(U, f)$ & Set function on subset $U$ \\
& $N$ & Neighborhood in the heterogenous network, a set of nearby locations \\
& $TYPE_v$ & Set of node types (e.g. individual, location) \\
& $f_v$ & Mapping from $v$ to $\{TYPE_v\}$ \\
& $TYPE_e$ & Set of edge types (e.g., strong contact, nearby locations) \\
& $f_e$ & Mapping from $e$ to $\{TYPE_e\}$ \\
\midrule
Section 3.3 & $x_v$ & Node embedding \\
& $DIST$ & Euclidean distance between network embedding of nodes \\
& $\sigma_t$ & Confidence interval to represent the uncertainty of estimates \\
& $N_k(v)$ & $k$-neighborhood of node $v$ \\
\midrule
Section 4 & $\tau$ & Simulation epoch in days \\
& $A = \{a_i\}$ & Agents that represent the population \\
& $AL$ & Represents which location each agent tends to visit and how often \\
\bottomrule
\end{tabular}
\caption{Frequently Used Notations}
\label{tbl:notation}
\end{table}
\section{Active Sampling Framework and Algorithm}\label{sec:MAB}
In Section \ref{sec:intro} we identified five unique characteristics of COVID-19. The solution approach presented here builds on ideas from the multi-armed bandit literature and heterogeneous network embedding literature to model these five characteristics. Specifically, the multi-armed bandit framework offers a general approach to combine exploration and exploitation (i.e. sampling neighbors of infected individuals and simultaneously exploring broadly to identify asymptomatic cases as well). Strategies for populating the arms of the bandit framework allow for re-testing individuals who may be repeatedly exposed as well, addressing another key requirement. The need to sample individuals specifically (as opposed to covariates) and leveraging locations is addressed in our work through a heterogeneous network embedding framework, which is used by the sampling algorithm. We present these details as follows in this section. First, we present a high-level overview of the multi-armed bandit framework and related literature. We then present the hetergeneous network embedding framework that is used to model individuals and locations. Finally, we present the sampling algorithm used to dynamically identify specific individuals to test.
\subsection{Background and Literature Review}
The Multi-Armed Bandit (MAB) is a generic framework to address the problem of decision making under uncertainty \cite{auer2002finite,langford2008epoch}. In this setting, the learner must choose from among a variety of actions and only observes partial feedback from the environment, without prior knowledge of which action is the best. In the classic stochastic K-armed bandit problem, at each time step $t$, the learner selects a single action/arm $x_t$ among a set of $K$ actions and observes some payoff $r_t(x_t)$. The reward of each arm is assumed to be drawn stochastically from some unknown probability distribution. The goal of the learner is maximize the cumulative payoff obtained in a sequence of $n$ allocations over time, or equivalently minimize the \emph{regret} \cite{bubeck2012regret}, which is defined as the difference between the cumulative reward obtained by always playing the optimal arm and the cumulative reward achieved by the learning policy,
$$\mathcal{R}_n = \max_{i = 1,...,K}\mathbb{E}\big{[}\sum_{t = 1}^n r_t(i) - \sum_{t = 1}^n r_t(x_t)\big{]}.$$
The fundamental exploration/exploitation dilemma is to: (1) gain as many rewards as possible in the current round, but also (2) have a high probability of correctly identifying the better arm. Many MAB algorithms with near-optimal guarantees have been proposed and applied in various domains. Out of all existing approaches to MAB, Upper Confidence Bound (UCB) policies \cite{auer2002finite, garivier2011kl} are the most popular approaches that are designed based on the principle of ``optimism in the face of uncertainty'', with arms chosen based on an upper confidence bound on the expected reward for each arm. Other approaches include Thompson sampling \cite{chapelle2011empirical, thompson1933likelihood}, expected improvement \cite{huang2006global,picheny2013benchmark} and knowledge gradient policies \cite{frazier2008knowledge}.
Beyond classic bandit problems, there exists literature on active search, with the objective of finding as many target nodes as possible with some given property. Most of the existing work assumes that the complete network structure is known before hand. For example, \cite{ma2015active} models the fully observed network using Gaussian random fields and proposes a UCB-type policy based on sigma optimality. Limited attention was paid to partially observed networks. \cite{bnaya2013social} proposed modeling the problem of social network querying and targeted crawling as an MAB problem with the goal of adaptively choosing the next profile to explore. \cite{singla2015information} deals with the case where each node’s visibility is limited to its local neighborhood and new nodes become visible and available for selection only once one of their neighbors
has been chosen. However, it is assumed that exploring a node reveals its 2-hop neighborhood which is usually not feasible in real-world social networks. \cite{soundarajan2017varepsilon} proposes $\epsilon-$WGX to solve the Active Edge Probing problem in incomplete networks, and yet in their case, a node can be queried multiple times and a single random edge adjacent to the queried
node is revealed in each query. \cite{madhawa2019multi} proposes a $k$NN-UCB policy for partially observed network based on structure graph features including degree, average neighbor degree, median neighbor
degree and average fraction of probed neighbors.
In the MAB context of active search on graphs, the question our research addresses can be posed, more generally, as:
{\it given a partially observed network with no information about
how it was observed, and a budget to query the partially observed network nodes, can we
learn to sequentially ask optimal queries?} This is a novel extension to the MAB literature.
\subsection{The Sampling Framework}\label{network_embedding}
COVID-19 is a pandemic that spreads via social contacts, directly or indirectly through locations. There are two approaches that can be taken to model this. The first approach, which we present below, is based on using contact networks only. This is close to how contacts are traced today and might therefore be immediately applicable. The second approach, which we present further below, leverages locations to build \emph{heterogeneous} networks that combine individuals and locations. This is useful when both contact information as well as information about locations are available.
Contact networks are modeled as undirected, time dependent, network $\mathcal{G}_t = <V, E, Z, D> $,
where $V$ is the node representation of the individuals $P$, with covariates information $Z$. The network is evolving over time with new contacts reported to the policy maker. For example, if the policy maker aims to make daily decisions, $\mathcal{G}_t$ represents the daily snapshot of the network on day $t$.
An edge $e_{ij} \in E$ represents a direct contact between individuals $v_i$ and $v_j$. At any time point, every node $v$ in our graph $\mathcal{G}_t $ has one disease state in $D$. We use $y_v$ to represent whether the individual is tested positive. The actual disease state can never be fully observed (e.g. the difference between susceptible, exposed and recovered cannot be identified).
The graph $\mathcal{G}_t$ is not fully observable and only partial information $\hat{\mathcal{G}}_t $ is available to the decision maker. When a node tests "positive", its \textit{known} contacts will be revealed at once, with the understanding that there might be other contacts that are latent and not readily observed.
Mathematically, this problem of active sampling can be formalized as a stochastic combinatorial optimization problem with $<V, \mathcal{U}, f>$, where $V$ is the set of nodes to be sampled from, $\mathcal{U} = \{U\subset V: |U| \leq D_{max} \}$ is a family of subsets of $V$ with up to $D_{max}$ (testing capacity) nodes, and $f$ a function that maps the node to [0,1] as the probability of infection. The objective function for all the nodes in a set $U \subset V$ is defined by $F(U, f)$. For example, if the objective is to find as many infected nodes as possible in each day, $F(U, f)$ can be defined as:
$$F(U,f) = \sum_{v \in U}f(v).$$
The goal of the sampling policy is to adaptively select a sequence of subsets $U_t$ to test, in a way that maximizes the cumulative rewards of $F(U_t,f)$ over time, recognizing that we observe, on testing, the realized reward $f(v)$ of each node $v \in U_t$ immediately, i.e. whether the tested individual is infected or not ($y_v$).
An important observation is that immediate contacts alone may be insufficient to consider for COVID-19 testing and we need a richer representation of similarity between individuals. We seek to model \emph{similarity} such that if one agent is infected, then a \emph{similar} node is also possibly infected. This can help identify potential regions at high risk of outbreaks given previous outbreak locations, the mobility of agents, and their network structure. Local structure features, like degree, number of triads, and centrality, have been used in previous network analysis on finding structurally similar nodes \cite{Henderson}. Although these features can help infer roles of each node, such as super spreaders, or periphery nodes, they do not capture the information about the neighborhood similarity, social relations, and community membership. For example, if two agents are living in the same household, COVID-19 can easily spread through contact transmission or droplet transmission. Due to large amount of pre-symptomatic and asymptomatic disease transmissions, even if there is no direct contact link between two agents, if they constantly went to the same supermarket, or live in nearby neighborhoods, the agents can end up infecting each other. This complex interplay of social relationships and locations has resulted in several infections from the same gathering (super-spreaders, hotspots) and brought up the importance of even seemingly minor occurrences such as how even a few agents with the same travel history may have caused unknown breakouts along the way.
Our goal is to quantify these above mentioned social relations by a continuous latent representation of nodes, which can then be exploited to guide the sampling policy to more effectively allocate testing kits of limited capacity. A prioneering method, DeepWalk \cite{perozzi2014deepwalk}, uses language modeling approaches to learn latent node embedding in the following two steps (please refer to Appendix A for the technical details of this approach). First it traverses the network with random walks to infer local structures by neighborhood relations, and then uses a SkipGram model \cite{mikolov2013distributed} to learn node embedding based on the produced samples. In our context, by translating the nodes in the network into a continuous space this way, the node embedding thus generated provides a method to efficiently sample individuals to test based on how ``close" they are to others.
The above discussion is restricted for representation learning for contact networks where nodes and relationships are of a singular type. However, the network embedding ideas help in ``indirectly" modeling locations (e.g. ``similar" individuals constructed as described above could have been similar because they went to the same location). Still, if location information was available to the policy maker then explicitly modeling is useful. Specifically, it is well-known that in the case of COVID-19, the transmission patterns and severity are largely depending on the type of the contacts. For example, nursing homes and long-term care facilities are especially vulnerable to the prevalence and spread of COVID-19 because they combine numerous risk factors for transmission: elderly people with underlying health conditions, congregate living, short of medical resources, frequent staff and visitors entering and leaving facilities, inadequate staffing, and infection control for an emergency. While contact networks capture whether two individuals are in close proximity in a certain period of time, it does not include the the type of the contact and/or the locations where the individuals met. In order to be able to identify the specific locations and/or gathering of the COVID breakouts, we augment the contact network with diversity of node types and multiple types of relationships between nodes.
Specifically, as illustrated in Fig. \ref{fig:network}, we represent the contact information with people (P), locations (L), neighborhood (N) as nodes. We also augment this network with edge types that reflects the number (or frequency) of contacts, or the intensity of each dyadic interaction. For example, individuals from the same household will have a much stronger interaction than the contact between random people that potentially met at the same subway station. The link between locations and neighborhoods is intended to generalize the observation of the transmissions to local neighborhood, even if two individuals are not reported in contact tracing due to unawareness.
\begin{definition}[heterogeneous network]
A heterogeneous network is a network $\mathcal{G}$ with multiple types of nodes or multiple types of edges. Formally, each node $v$ and each edge $e$ are associated with a type mapping function $f_v: v \rightarrow \text{TYPE}_v$, and $f_e: e \rightarrow \text{TYPE}_e$, where $\text{TYPE}_v$ and $\text{TYPE}_e$ are the set of node and edge types, respectively.
\end{definition}
\begin{figure}[htp!]
\centering
\includegraphics[trim=150 140 30 200, clip, width=\textwidth]{Figures/network.jpg}
\caption{An illustrative example of a heterogeneous contact network.}
\label{fig:network}
\end{figure}
Heterogeneous networks present unique challenges that cannot be handled by node embedding models that are specifically designed for homogeneous network, such as the DeepWalk method mentioned above \cite{dong2017metapath2vec, chang2015heterogeneous}. When performing the random walk, DeepWalk ignores their node types. However, it is demonstrated that such random walks on heterogeneous networks are biased to highly visible types of nodes and concentrated nodes. With this regards, Metapath2vec \cite{dong2017metapath2vec} extends DeepWalk to heterogeneous networks by incorporating meta-paths which have been shown to be effective in many data mining tasks in heterogeneous information networks \cite{sun2013pathselclus,dong2015coupledlp, fu2017hin2vec}.
A meta-path is a sequence of node types encoding key composite relations among
the involved node types. Different meta-paths express different
semantic meaning, and thus different disease transmission pathways. For example, in Fig \ref{fig:network}, a meta-path "P-L-P" represents the direct interactions of two individuals in the same place, e.g. household, workplace, school. A meta-path "P-L-N-L-P" represents two individuals residing in the same neighborhood. Then the random walk is confined on the pre-defined meta-paths. Additionally, we extend the metapath2vec by assigning different weights to edges corresponding to the contact intensity, which in turn translates to the transition probability in the biased random walk. For example, in a homogeneous network and traditional random walk, the next node of $P_3$ can be all types of nodes connected to it - $P_2, P_4, P_5, L_6, L_2, L_3$. However, under the meta-path "P-L-P", the random walk can only result in location nodes (L) - $L_2, L_3$, given that itself is an individual node (I). Meanwhile, the transition probability from $P_3$ to $L_2$ is higher since their link indicates a more intensive relationship. The meta-paths commonly requires that the starting nodes and ending nodes are of the same node type so that random walk can be performed recursively.
After the random walk sequences are generated, a modified heterogeneous SkipGram is used considering node type information during model optimization.
\subsection{The Sampling Algorithm}
In the classic stochastic multi-arm bandit problem, the learner selects one of the $K$ arms at each time step and receives a stochastic reward sampled from some unknown reward distribution. In our case, each node is one possible arm. Due to finite but large number of arms, feature-based exploration is needed to share the knowledge learnt for similar nodes. In this paper, we formulate the active sampling problem as a contextual bandit, in which before making the choice of action, the learner observes the currently known disease states, and a feature vector $x_i$ associated with each of the possible arm (a.k.a. individual node $v_i$) encoded from the contact network structure. The learner then selects an action and reveals whether the node is a search target.
As suggested by \cite{chen2020allocation, long2018spatial}, it is challenging to compute the optimal dynamic allocation using dynamic programming even for 35 age-compartment pairs and the heuristic obtained from approximate dynamic programming performs worse than simple heuristics such as single-step myopic policy. Hence full learning model at individual level is challenging, if not infeasible. In the same time, there are many unobserved positive cases affecting the disease transmission, as well as un-reported (random) contacts. With this regards, we treat the graph context drawn $i.i.d.$ from some context density.
It combines two major challenges. First, the number of possible actions grows exponentially with the cardinality constraint $D_{max}$. Second, the policy maker can only observe the partially reported portion of the network.
To deal with the partially observed network, we thus consider two levels of explorations:
\begin{itemize}
\item Given the current observed network $\hat{\mathcal{G}}_t$, the probability of infection is not known beforehand, and needs to be learnt as we gradually sample individuals and observe their test results.
\item Expansion on (randomly selected) unobserved nodes to identify new hotspot.
\end{itemize}
Hence the proposed sampling policy concerns two levels of exploration/exploitation tradeoff.
\paragraph{Outer Level: expansion vs. densification.} The outer level is modeled as a two-armed bandit, corresponding to the two choices of expansion and/ or densification. Consistent with the literature, we choose to use the Thompson sampling policy with Beta-Bernoulli distribution \cite{thompson1933likelihood, chapelle2011empirical}. The expected reward $\theta$ of the two choices are modeled with a Beta distribution. After one choice is chosen, the realization of the reward (whether an individual is tested positive) is sampled from a Bernoulli distribution. With a testing capacity constraint $D_{max}$, we repeatedly sample $D_{max}$ realizations of $\hat{\theta}$ and choose between the two choices of expansion and/or densification that has the highest sampled value. If expansion is chosen, our policy samples a node from the network that was not in $\hat{\mathcal{G}}_t$ uniformly at random. If densification is chosen, our policy use the inner level exploration/ exploitation algorithm to choose individuals to sample. We use $\bar{D}_{max}$ to represent the total number of tests that are allocated to densification. After the test result $y_v$ for each chosen individual is revealed, the posterior Beta distribution will be updated.
\iffalse
\begin{algorithm}\label{TS}
\caption{Thompson sampling}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{$\alpha$, $\beta$ for prior Beta distribution, $S_i = 0$, $F_i = 0$}
\For{$t=1$ to $T$}{
1. Draw $\hat{\theta}_i$ from $Beta(S_i + \alpha, F_i + \beta)$ for $ i = 1,2$\\
2. Draw arm $a_t=\arg\max_{i}\hat{\theta}_i$\\
~~~~Observe reward $r_t$ \\
3. Update posterior distribution\\
\If{$r_t = 1$}{$S_i = S_i +1$}
\Else{$F_i = F_i +1$}
}
\end{algorithm}
\fi
\paragraph{Inner Level: individual selection under combinatorial optimization: combinatorial $k$NN-UCB.}
On each day, we first use network embedding $x_v$ from Section \ref{network_embedding} to find the node representation based on the past 14-days contacts of each known individual. The node embedding encodes the network structure, i.e. neighbors and connection information, and quantifies the similarity between two nodes in the sense that if one is infected then the other is highly likely to be infected from potential contact chains. Our goal is to effectively allocate $\bar{D}_{max}$ lab tests to individuals to alleviate the disease spread.
The goal of the inner level bandit algorithm is to adaptively select a sequence of subsets $U_t \subset V$ to test and maximize the cumulative rewards of $F(U_t,f)$ over time. We also observe the realized reward $f(v)$ of each node $v \in U_t$, i.e. whether the tested individual is infected or not.
In this paper, to deal with the challenge that the number of possible individuals to sample is a combinatorial set given the the cardinality constraint $\bar{D}_{max}$,
we allow the policy maker to use any exact/ approximation/ randomized algorithm, termed as ORACLE, to find solutions for
$$U^{\text{OPT}} \in \arg \max_{U \in \mathcal{U}} F(U, f) = \arg \max_{U \in \mathcal{U}} \sum_{v \in U}f(v).$$ We denote the solution as $U^* = \text{ORACLE}(V, \mathcal{U}, f, \bar{D}_{max})$.
Given that the contextual information is presented in the format of a network rather than numeric feature vectors, we thus adopt non-parametric strategies and use $K$NN-UCB for structured bandits. Specifically, we define the $k-$nearest neighbor upper confidence bound as:
\begin{equation}\label{ucb}
\hat{f}(v_i) + \eta\sigma_t(v_i),
\end{equation}
where the expected reward of node $i$ is estimated with weighted $k$NN regression as
$$\hat{f}(v_i) = \frac{1}{k}\sum_{v_j \in \mathcal{N}_k(v_i)} \frac{y_j}{\text{DIST}(x_i, x_j)},$$
and $\text{DIST}(x_i, x_j)$ is the euclidean distance between network embedding of node $i$ and $j$, and $\mathcal{N}_k(v)$ is the $k$-neighborhood of node $v$. The uncertainty is chosen to be the average distance to points in the $k$-neighborhood,
$$\sigma_t(v_i) = \frac{1}{k}\sum_{v_j \in \mathcal{N}_k(v_i)}\text{DIST}(x_i, x_j).$$
If at each time step, only one arm is selected, the traditional UCB policy selects the arm with the highest upper confidence bound index, i.e. Eq. (\ref{ucb}). However, in the case of combinatorial optimization, we propose the algorithm Combinatorial $k$NN-UCB which makes use of the $\text{ORACLE}(V, \mathcal{U}, f, \bar{D}_{max})$ to provide (approximate) solutions for the offline optimization problem. The pseudocode is provided in Algorithm \ref{comKNN}. Due to the fact that the number of individuals is huge, scalable space partition method is needed to restrict the search of nearest neighbors to local partition.
\iffalse
\begin{algorithm}[htp!]\label{comKNN}
\caption{Combinatorial KNN-UCB}
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{Combinatorial structure $<V, \mathcal{U}, F>$, parameter $\alpha$, oracle ORACLE, initial known disease states $y_v$ }
\For{$t=1$ to $T$}{
Update the contact network and find the network embedding of each node $x_v$\\
Update disease states $y_v$\\
Compute the UCB for each individual (P) nodes as
\begin{eqnarray*}
\hat{f}(v_i) &=& \frac{1}{k}\sum_{v_j \in \mathcal{N}_k(v_i)} \frac{y_j}{\text{DIST}(a_i, a_j)}\\
\sigma_t(v_i) &=& \frac{1}{k}\sum_{v_j \in \mathcal{N}_k(v_i)}\text{DIST}(a_i, a_j),\\
\bar{f}_t(v_i) &=& \hat{f}(v_i) + \eta \sigma_t(v_i), \forall v_i \in I.
\end{eqnarray*}\\
Compute $U_t \leftarrow \text{ORACLE}(V, \mathcal{U}, \bar{f}_t)$\\
Test individuals in set $U_t$\\
Observe the disease state and update $y_v, \forall v \in U_t$
}
\end{algorithm}
\fi
\begin{algorithm}[htp!]\label{comKNN}
\caption{Pseudo-code for Active Sampling}
\SetAlgoLined
\SetKwInOut{Input}{input}\SetKwInOut{Output}{output}
\Input{Past 14-day heterogeneous network $\hat{\mathcal{G}}_0 = <V, E,Z, D>$, oracle ORACLE, trade-off parameter $\eta$, daily constraint $D_{max}$, reward function $f$, $\alpha$, $\beta$ for prior Beta distribution, $S_i = 0$, $F_i = 0$ for $i = 1,2$ }
\For{$t=1$ to $T$}{
Update the contact network $\hat{\mathcal{G}}_t$ and find the network embedding of each node $x_v$\\
initialize the proportion allocated to densification $\bar{D}_{max} = 0$\\
\For {$l == 1$ to $D_{max}$}{
\textbf{[Outer Level]}\\
Draw $\hat{\theta}_i$ from $Beta(S_i + \alpha, F_i + \beta)$ for $ i = 1,2$\\
Choose arm $a_t=\arg\max_{i}\hat{\theta}_i$\\
\If{$a_t$ == expansion}{Randomly sample a node $v$ that is not in $\hat{\mathcal{G}}_t$ to test, and update $y_v$}
\Else{$\bar{D}_{max} = \bar{D}_{max} + 1$}
}
\textbf{[Inner Level]}\\
Compute the UCB for each individual (P) nodes in $\mathcal{G}_t$ as
\begin{eqnarray*}
\hat{f}(v_i) &=& \frac{1}{k}\sum_{v_j \in \mathcal{N}_k(v_i)} \frac{y_j}{\text{DIST}(a_i, a_j)}\\
\sigma_t(v_i) &=& \frac{1}{k}\sum_{v_j \in \mathcal{N}_k(v_i)}\text{DIST}(a_i, a_j)\\
\bar{f}_t(v_i) &=& \hat{f}(v_i) + \eta \sigma_t(v_i)
\end{eqnarray*}\\
Compute $U_t \leftarrow \text{ORACLE}(V, \mathcal{U}, \bar{f}_t, \bar{D}_{max})$\\
Test individuals in set $U_t$\\
Observe the testing results and update $y_v$\\
Compute the number of positive cases under expansion and densification, respectively, $\bar{S}_1, \bar{S}_2$\\
Compute the number of negative cases under expansion and densification, respectively, $\bar{F}_1, \bar{F}_2$\\
Update posterior Beta distribution:
\begin{eqnarray*}
S_i &=& S_i +\bar{S}_i\\
F_i &=& F_i +\bar{F}_i, \text{ for } i = 1,2
\end{eqnarray*}
}
\end{algorithm}
\section{Agent-Based Model}\label{sec:ABM}
In the real-world setting, as long as we have daily spatio-temporal data on individuals in a population, the algorithm presented in the previous section can be used to select individuals to test. However, and somewhat counter-intuitive, real-world data alone is insufficient to fully understand the impact of different sampling policies. The "real world" represents a single run of how reality is shaped, and does not offer the advantage of running experiments to test counterfactuals. For instance, would the sampling algorithm have worked as well under a different quarantine policy, or under various compliance scenarios? Agent-based models, on the other hand, are particularly effective to answer these types of more general questions.
In this paper we couple a disease progression model \cite{hethcote2000mathematics} with Agent Based simulation Model (ABM) \cite{perez2009agent} of infectious spread under different settings. Disease progression is commonly modelled with the deterministic compartmental S/I/R \cite{hethcote2000mathematics,cooper2020sir}, which divides the population into three compartments - susceptible to the disease (S), actively infected with the disease (I), and recovered (or dead) and no longer contagious (R)) - and defines transmission rates between the compartments. To account for additional aspects of the Covid-19 transmission, we use an extended S/I/R model that includes individuals that are exposed (E) to the disease \cite{silva2020covid,prem2020effect}, and asymptomatic (A) individuals \cite{manchein2020strong,koo2020interventions} - an S/E/A/I/R model.
We constructed a data driven stochastic Agent-Based Model (ABM) of the COVID-19 epidemic in New York city. Based on \cite{hoertel2020facing}, our ABM model includes four components: (1) synthetic population with demographic characteristics and spatial information that is generated to resemble the city of New York, (2) daily interaction network between individuals (agents) in the population, (3) disease dynamics that spreads via interactions, and progresses as an S/E/A/I/R model, (4) policy makers, as agents, who influence the environment based on the sampling strategies and quarantine policies in place.
In the reminder of this section we describe components (2) - (4) of the ABM. The population generating approach in component (1) is detailed in Appendix B. In algorithm \ref{alg:ABM} we provide the pseudo code of our model.
\begin{algorithm}
\caption{Pseudo-code for the ABM}
\label{alg:ABM}
\DontPrintSemicolon
\SetAlgoLined
\textbf{Input:} \;
\Indp Agents $A = \{a_1, ..., a_n\}$; Locations $L = \{l_1, .. l_m\}$;
Agents-locations interaction $AL = \big\{\{\phi_{a_i, l_l}\}\big\}$;
Number of initial infected agents $\tilde{n}$; \;
\Indm \textbf{Output:} \;
\Indp Disease state $D(A) = \{d(a_1), ... d(a_n)\}$; meetings log; \;
\Indm \textbf{Initialize:} \;
(1) Set initial disease state $D(A)$ such that: \;
\Indp for $\tilde{n}$ agents $d(a) \leftarrow Ia$, \;
for $N-\tilde{n}$ agents $k(a) \leftarrow S$; \;
\Indm (2) Set simulation day $\tau \leftarrow 1$; \;
(3) Initialize set of agents in quarantine $Q \leftarrow \{\}$; \;
(4) Initialize agent-location log; \;
\While{some agents are still infected}{
\textit{// Draw locations per agents:} \;
\For {$a_i \in A \cap Q$}{
Draw daily locations $l_{a_i}$ according to the agent-location interactions propensity $AL$; \;
Append tuples $\big\{\{a_i, l_j \in l_{a_i}\}\big\}, \tau$ to the agent-location log;
}
\textit{// Get new infected agents:} \;
\For {$l_j \in L$}{
Get the list of agents $a_{l_j}$ that visited $l_j$ at time $tau$ from the agent-location log; \;
If one or more agents is infected ($d(a \in a_{l_j}) \in \{Ia, Is, Ic\}$, set the disease status of all \textbf{\textit{Susceptible (S)}} agents $\in a_{l_j}$ as \textbf{\textit{Infected asymptomatic (Ia)}};
}
\textit{// Progress disease state:} \;
Progress disease state $d_i$ for all agents according to Figure \ref{fig:MC}; \;
\textit{// Put critically infected in self-quarantine:} \;
\For {$d_i \in D$}{
\Switch{$d_i$}{
\Case{$Is$: }{Append $a_i$ to $Q$} \;
\Case{$R$ or $D$: }{
Remove $a_i$ from $Q$} \;
}
}
\textit{// Put agents in enforced-quarantine:} \;
\SetKwBlock{SMPL}{Interact with policy-makers (sampling policy)}{end}
\SMPL{
\textbf{Send:} $\tilde{A} \leftarrow $ set of externally sampled agents to be tested\;
\textbf{Receive:} $\doubletilde{\textit{A}} \leftarrow $ set of agents to be set in quarantine\;
\textbf{Respond:} Append $\doubletilde{\textit{A}}$ to $Q$ \;
}
$\tau \leftarrow \tau + 1$\textit{// Advance simulation time:} \;
}
\end{algorithm}
\subsection{Interaction data structure}
Our data is composed of a set of individual agents $A = \{a_1, ..., a_n\}$, a set of geographic locations $L = \{l_1, .. l_m\}$, and the propensity of the interactions between the agents and the locations: $AL = \big\{\{\phi_{a_i, l_l}\}\big\}$ (how often an agent goes to the location).
Agents in our data is characterized by age, gender, and a list of locations that the agents goes to, at a given probability. Locations can take different location types, and are spread across the neighborhoods of New York City. For this paper we set the following location types: household, workplace, school, station, and supermarket. Two agents may meet if they go to the same location at the same time. The meeting probability at a given location is a function of the location type (for example, agents that live at the same house will meet with a probability of $\phi_{household}=1$, while agents that go the the same supermarket, will meet at a much lower probability: $\phi_{supermarket} << 1$). In addition to the different location types, we add a special location type that we call a "mixing location", in which agents can meet at random. Mixing locations in our model simulates meeting at public spaces, such at parks, shops, theater, etc.
Note that our sampling algorithm works as an agent within this framework and can be used with any ABM setting - all that is required is for the ABM to simulate contacts and movements of individuals in a population. The above are used primarily for exposition to simulate one specific city (New York).
Agents in our model can potentially be in one of seven disease states, as described below. We formally define $D(A) = \{d(a_1), ... d(a_n)\}$ to be the set of disease states of all agents.
\subsection{Disease dynamic}
The disease process used in this paper follows the deterministic \textit{S/E/A/I/R} (Susceptible ($S$) - Exposed ($E$) - Asymptomatic ($A$) - Infectious ($I$) - Recovered ($R$)) compartmental model, with death ($D$) rate. We further split the infectious state into three (instead of two) disease stages: Infected asymptomatic ($Ia$), Infected symptomatic ($Is$), and Infected critical ($Ic$). Beyond being a more accurate description of COVID-19, this allows us to flexibly define different policies and different behaviours to infected individuals according to the state of their disease. Similar to \textit{S/E/A/I/R}, we assume Recovered ($R$) is an absorbing state, implying that infected individuals can either die or become immune. Figure \ref{fig:MC} presents the disease states and the transmission dynamics in this paper.
Transmission probabilities are assumed to be constant across individuals. This is due to the fact that the impact of patient's covariates on the disease dynamics is still unknown. The transmission rate between states $S$ and $E$ is $\Lambda_t$ that determines the rate at which uninfected individuals are exposed to infected individuals. Once exposed, they become infected with probability of $p_{I|E}$. Otherwise they remain susceptible with the complementary probability ($1-p_{I|E}$).
In infected individuals, the disease may escalate and become mild ($Is$), at a probability of $p_{Is|Ia}$, then critical ($Ic$), at a probability of $p_{Ic|Is}$. At any stage of the disease, infected patient can recover. Recovery takes on average $\lambda_{Ia}$ for asymptomatic agents, $\lambda_{Ia} + \lambda_{Is}$ for symptomatic agents, and $\lambda_{Ia} + \lambda_{Is}$ for critically ill agents. The latter disease state can result in death, at a rate of $p_{D|Ic}$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.9\textwidth]{Figures/MarkovChain.jpg}
\caption{Transmission dynamics of the Corona virus}
\label{fig:MC}
\end{figure}
We note that the sampling algorithm presented in this paper only uses data observed as a result of the actual disease diffusion process, and does not depend specifically on the details of how this is implemented in the ABM. In that sense, our approach will work with any extensions of the model implemented in here.
\subsection{Policy makers as agents}
At the end of each ABM simulation epoch we sample a set of agents to be tested for COVID-19. Once an agent is sampled, her disease state as well as (possibly incomplete) contact network are revealed to policy makers, whom can then decide which agents should be put into quarantine.
Other than sampled agents, we assume that critically infected agents self quarantine themselves. The quarantine duration is set for 14 days, following the current global practice. While in quarantine, an agent's disease dynamics continues, but her interactions with others stop.
\section{Implications for Policy}
The paper presented a smart-testing approach that was implemented on simulated data. In practice, our method can be used by policy makers to determine whom to test if the following information can be provided: (1) a list of all individuals in the population; (2) a partial mapping of individuals to locations that they can be associated with (e.g., household, workplaces, schools); (3) the set of known infected individuals in the population; and (4) daily-contact network for individuals - ideally, this should be provided for all individuals that were tested positive, but the method can work even with partial information. This daily contact network, for instance, can be computed easily from readily available mobile tracking data in some cases.
Our method can also be used by policy makers to examine carefully constructed scenarios in a world with smart-testing strategies. These scenarios may include different quarantine policies, intervention strategies or social behavior such as compliance or willingness to be vaccinated. The scenarios constructed can also experimentally then show the outcomes under smart-testing versus other testing strategies.
\bibliographystyle{abbrv}
|
2,869,038,154,803 | arxiv | \section{Introduction}
Space debris consists of a range of human-made objects in a variety of orbits around the Earth, representing the remnants of payloads and payload delivery systems accumulated over a number of decades. With the increasing accumulation of debris in Earth orbit, the chance of collisions between the debris increases (causing an increase in the number of debris fragments). More importantly, the chance of collisions between debris and active satellites increases, posing a risk of damage to these expensive and strategically important assets. This risk motivates the need to obtain better information on the population characteristics of debris (distribution of sizes and masses, distribution of orbits etc).
Research into space debris is considered a critical activity world-wide and is the subject of the Inter-Agency Space Debris Coordination Committee\footnote{http://www.iadc-online.org/}, previously described in detail in many volumes, including a United Nations Technical Report on Space Debris \citep{UNreport} and a report from the Committee on the Peaceful Uses of Outer Space \citep{demp12}. The American Astronomical Society maintains a Committee on Light Pollution, Radio Interference and Space Debris\footnote{http://aas.org/comms/committee-light-pollution-radio-interference-and-space-debris}, since space debris poses a risk to important space-based astrophysical observatories.
The risks posed by space debris, and the difficulties inherent in tracking observations and orbit predictions, were illustrated starkly on the 10$^{\rm th}$ of February 2009, when the defunct Russian Kosmos-2251 satellite collided with the active Iridium-33 satellite at a relative speed of over 11 km/s, destroying the Iridium satellite\footnote{Orbital Debris Quarterly News, 2009, NASA Orbital Debris Program Office, 13 (2)}. Even with the global efforts to track space debris and predict their orbits, this collision between two satellites of 900 kg and 560 kg, respectively, and of $\sim$10 m$^{2}$ debris size, was unanticipated. Further exploration of SSA capabilities is required to continue to minimise collision risks and provide better early warnings following collision breakups that multiply debris numbers and randomise orbits.
Methods for obtaining data on space debris include ground-based (radar observations and optical observations) and space-based measurements. Previously, large ground-based radio telescopes (whose primary operations are for radio astronomy) have been used in a limited fashion to track space debris. For example, the 100 m Effelsberg telescope in Germany, outfitted with a seven-beam 1.4 GHz receiver, was used for space debris tracking using reflected radiation generated by a high power transmitter \citep{ruitz05}. Recently, a new method of ground-based space debris observation has been trialed, using interferometric radio telescope arrays (also primarily operated for radio astronomy), in particular using the Allen Telescope Array \citep{wel09}. This technique utilises stray radio frequency emissions originating from the Earth that reflect off space debris and are received and imaged at the interferometric array with high angular resolution. Scenarios such as this, with a passive receiver and a non-cooperative transmitter, are described as “passive radar”, a sub-class of the bi-static radar technique (transmitter and receiver at different locations).
This paper explores the possibilities offered by this technique, using a new low frequency radio telescope that has been built in Western Australia, the Murchison Widefield Array (MWA). The MWA is fully described in \citet{tin13}. Briefly, the MWA operates over a frequency range of 80 - 300 MHz with an instantaneous bandwidth of 30.72 MHz, has a very wide field of view ($\sim$2400 deg$^{2}$ at the lower end of the band), and reasonable angular resolution ($\sim$6 arcmin at the lower end of the band).
Recent observations with the MWA by \citet{mck12} have shown that terrestrial FM transmissions (between 87.5 and 108.0 MHz) reflected by the Moon produce a significant signal strength at the MWA. \citet{mck12} estimate that the Equivalent Isotropic Power (EIP) of the Earth in the FM band is approximately 77 MW. The ensemble of space debris is illuminated by this aggregate FM signal and will reflect some portion of the signal back to Earth, where the MWA can receive the reflected signals and form images of the space debris, tracking their positions on the sky as they traverse their orbits.
It should be noted that this technique makes use of the global distribution of FM radio transmitters, but that the technique is likely to work most effectively with receiving telescopes that are well separated from the transmitters. If the receiving array is in close proximity to an FM transmitter, the emissions directly received from the transmitter will be many orders of magnitude stronger than the signals reflected from the space debris. Therefore, the technique can likely only be contemplated using telescopes like the MWA, purposefully sited at locations such as the radio-quiet Murchison Radio-astronomy Observatory (MRO), located in Murchison Shire, 700 km north of Perth in Western Australia \citep{jon08}. The MWA is a science and engineering Precursor for the much larger and more sensitive low frequency component of the Square Kilometre Array (SKA) \citep{dew09}, which will also be located at the MRO and is currently under development by the international SKA Organisation.
In this paper we estimate the conditions under which the MWA can detect space debris, in passive radar mode in the FM band. In section 2 we consider a simple calculation of the expected signal strength at the MWA for one idealised scenario. We also present comprehensive but idealised electromagnetic simulations that agree well with the simple calculation and show that space debris detection is feasible with the MWA. For normal MWA observation modes, the detection of space debris of order $\sim$0.5 m radius and larger appears feasible. If efforts are made to modify standard MWA observation modes and data processing techniques, then substantially smaller debris could be detected, down to $\sim$0.2 m in radius in low Earth orbits. In section 3 we present observational tests that show the basic technique to be feasible, but also illustrate challenges that must be overcome for future observations. In section 4 we discuss the utility of the technique, complementarity with other techniques, and suggest future directions for this work.
\section{Detectability of space debris with the MWA}
As an example that illustrates the feasibility of this technique, we
consider the MWA as the receiving element in a passive radar system in
the FM band and consider a single transmitter based in Perth, Western
Australia (specifically the transmitter for call sign 6JJJ: Australian Communications and Media Authority [ACMA]
Licence Number 1198502). The transmitter is located at
(LAT,LONG)=($-32\degr00\arcmin42\arcsec$,
$116\degr04\arcmin58\arcsec$) and transmits a mixed polarisation
signal, with an omnidirectional radiation pattern in the horizontal plane, at 99.3 MHz over the FM
emission standard 50 kHz bandwidth and with an Effective Radiated
Power (ERP) of 100 kW. The MWA, acting as receiver, is located at
(LAT,LONG)=($-26\degr42\arcmin12\arcsec$,
$116\degr40\arcmin15\arcsec$), $\sim$670 km from the
transmitter.
We consider an idealised piece of space debris (a
perfectly conducting sphere of radius $r$, in metres) at a distance
of $R_{r}$ km
from the MWA site. We
denote the distance between the transmitter and the space debris as
$R_{t}$ km. For simplicity, we model the
Radar Cross Section (RCS, denoted by $\sigma$) of the debris as the
ideal backscatter from a conducting sphere in one of two domains that
are approximations of Mie scattering \citep{str41}: 1) when the wavelength
$\lambda>2\pi r$, the RCS is described by Rayleigh scattering and
$\sigma= 9\pi r^{2}(\frac{2 \pi r}{\lambda})^{4}$; and 2) when
$\lambda<<2\pi r$, $\sigma=\pi r^{2}$, corresponding to
the geometrical scattering limit. The greatest interest is in
case 1, as the space debris size distribution is dominated by small
objects (although the example given in the introduction shows that
even the largest items of space debris pose serious unknown risks).
For case 1 we obtain Equation 1, below, following the well-known
formula for bi-static radar \citep{wills05},
\begin{equation}
S=3.5\times10^{6}\frac{P_{t} G r^{6} \nu^{4}}{R_{t}^{2} R_{r}^{2} B}
\label{eqn:flux}
\end{equation}
where S is the spectral flux density of the received signal at the MWA in astronomical units of Jansky (1 Jy = 10$^{-26}$ Wm$^{-2}$Hz$^{-1}$), $P_{t}$ is the ERP in kW, $G$ is the transmitter gain in the direction of the space debris, $B$ is the bandwidth in kHz over which the signal is transmitted, $R_{t}$, $R_{r}$ and $r$ are as described above, and $\nu$ is the center frequency of the transmitted signal in MHz.
For $P_{t}$=100 kW, $R_{r}$=1000 km, $R_{t}=$1200 km, $G$=0.5, $B$=50 kHz, $r$=0.5 and $\nu$=99.3 MHz, the spectral flux density, $S$, is approximately 4 Jy. The assumption of $G$=0.5 is based on an idealised dipole antenna transmitter radiation pattern which is omni-directional in the horizontal plane. FM transmitter antenna geometries vary significantly, with local requirements on transmission coverage dictating the directionality and thus the antenna geometry. Phased arrays can be used to compress the radiation pattern in elevation and different antenna types can produce gain in preferred azimuthal directions. Therefore, in reality, significant departures from a simple dipole is normal. Propagation losses in signal strength due to two passages through the atmosphere are assumed to be negligible. Due to the approximations described above, this estimate should be considered as order-of-magnitude only. However, the calculation indicates that the detection of objects of order 1 m in size is feasible. In a 1 second integration with the MWA over a 50 kHz bandwidth, an image can be produced with a pixel RMS of approximately 1 Jy \citep{tin13}, making the reflected signal from the space debris a several sigma detection in one second.
According to equation 1, the MWA will be sensitive to sub-metre scale space debris. However, for FM wavelengths, sub-metre scale debris are in the Rayleigh scattering regime, meaning that the RCS drops sharply for smaller debris, dramatically reducing the reflected and received power. In order to more systematically explore the detectibility of debris of different sizes and relative distances between transmitter, debris, and receiver, we have performed a series of electromagnetic simulations.
The simulations performed take the basic form described above, in
which a single FM transmitter (idealised dipole radiation pattern for the transmitter, omni-directional in the horizontal plane) at a given location was considered
(separate simulations were performed for 11 different transmitters
located in the south-west of Western Australia), along with the known
location of the MWA as receiving station. The debris were modelled as being directly above the MWA. A
range of different debris altitudes was modeled: 200 km; 400 km; 800
km; and 1600 km (as for the calculation above, propagation losses due to the atmosphere are considered negligible). The debris was assumed to be a perfectly conducting
sphere, with a range of radii: 0.2 m; 0.5 m; 1 m; and 10 m. The
simulations proceeded using the {\it XFdtd} code from Remcom
Inc\footnote{http://www.remcom.com/xf7}. {\it XFdtd} is a
general-purpose electromagnetics analysis code based on the
finite-difference, time-domain technique and can model objects of
arbitrary size, shape and composition. The $11$ (transmitters) $\times$ $4$
(radii) $\times$ $4$ (altitudes) $= 176$ runs of {\it XFdtd} were calculated
on a standard desktop computer over the course of several hours. The
simulations compute output that includes the RCS according to the full
Mie scattering solution and the spectral flux density at the location
of the MWA, for each combination of debris size, altitude and
transmitter.
Figure 1 shows a selection of results from the simulations, for the Perth-based 6JJJ transmitter and the ranges of debris size and altitudes listed above. Also shown is a single point representing the result of the simple calculation from Equation 1. The simple calculation overestimates, by a factor of approximately two, relative to the {\it XFdtd} simulation, the flux density at the MWA. This level of agreement is reasonable, given the simplifying assumptions made for Equation 1.
\begin{figure}[ht]
\plotone{Fig1.pdf}
\caption{The results of electromagnetic simulations, as described in the text. These simulations model the flux density at the MWA for space debris modelled as a perfectly conducting sphere (0.2, 0.5, 1 and 10 m radii) at altitudes of 200, 400, 800 and 1600 km above the MWA site, with a transmitter at 99.3 MHz based near Perth ($\sim$700 km from the MWA). Also shown is a single point representing the example calculation based on Equation 1, with $r=0.5$ m}
\end{figure}
Figure 1 shows that debris with $r >$ 0.5 m can plausibly be detected up to altitudes of approximately 800 km with a one second observation. The solid line in Figure 1 denotes the one sigma sensitivity calculation made for the simple example, above, based on information from \citet{tin13}, showing that a three sigma detection can be made for debris of $r =$0.5 m at 800 km. Of course, if longer observations can be utilised, the detection thresholds can be reduced. For example, a ten second observation would yield a three sigma detection of debris with $r =$ 0.5 m at 1000 km.
The solution to the detection of smaller ($\sim$10 cm) scale debris with low frequency passive radar in the past has been to use ERPs of gigawatts, as with the 217 MHz NAVSPACECOM ``Space Fence'' \citep{pet12}. An alternative approach is to build larger and more sensitive receiving antennas. The natural evolution of the MWA is the much larger and more sensitive low frequency SKA, which will have a factor of $\sim$1000 more receiving area and be far more sensitive than the MWA. The low frequency SKA may have great utility for space debris tracking later this decade.
\section{Verification of the technique}
We verified the technique outlined above with an observation using the
MWA during its commissioning phase. This is a subarray of 32/128 tiles with a maximum east-west baseline of roughly
1\,km and a maximum north-south baseline of roughly 2\,km, giving a
synthesized beam of roughly $5\arcmin \times 10\arcmin$ at a
frequency of $\sim$100 MHz. We observed an overflight of the
International Space Station (ISS) on 2012~November~26, as
shown in Figure~\ref{fig:track}. The center frequency of our
30.72\,MHz bandpass was 103.4\,MHz, and we correlated the data with a
resolution of 1\,s in time and 10\,kHz in frequency. We used two
different array pointings, as plotted in the right panel of
Figure~\ref{fig:track}. The first was from 12:20:00\,UT to
12:24:56\,UT and was pointing at an azimuth of $180\degr$ (due south)
and an elevation of $60\degr$. The second was from 12:24:56\,UT to
12:29:52\,UT at an azimuth of $90\degr$ (due east) and an elevation of
$60\degr$. As discussed in \citet{tin13} and verified by
\citet{wil12}, the MWA has an extremely broad primary beam (roughly
$35\degr$ FWHM) which was
held fixed during each observation. We observed the overflight of
the ISS between 12:23:00\,UT and 12:26:00\,UT, when it was at a
primary beam gain of $>1$\%, traversing about $112\degr$ on the sky.
During the pass, the ISS ranged in distance between approximately
850\,km (at the low elevation limit of the MWA, approximately
30$^{\circ}$) and approximately 400\,km (near zenith at the MWA) (Figure 2); its angular speed across the sky is
approximately 0.5$^{\circ}$ per second, corresponding to approximately
five synthesised beams with this antenna configuration.
\begin{figure}[ht]
\plottwo{Fig2a.pdf}{Fig2b.pdf}
\caption{Left: The track of the ISS over Western Australia on
2012~November~26, between 12:23:00 and 12:26:00 UT. The red and
green segments of the track denote the time ranges for the two
different pointed observations undertaken with the MWA, with ticks
located every 20\,s and the square filled marker denoting the changeover time for two pointings. Blue circles denote the location and power
(proportional to diameter of circle, with largest circles
representing 100 kW transmitter ERP) of FM transmitters. Green
diamonds denote the names of towns/regions associated with the
transmitters. The image is a projection representing the view of
Australia from a low-Earth orbit satellite directly over the MWA
site. Right: The track of the ISS plotted on the celestial sphere as
visible from the MWA site. The image is an orthographic projection
of the \citet{has82} 408\,MHz map, with some individual radio
sources labeled. The red and green contours are contours of primary
beam gain (at 1\%, 10\%, 25\%, 50\%, and 75\% of the peak gain) for
the two pointed observations. The blue points are the track of the
ISS, with color from dark blue to white representing the primary
beam gain from low to high and the square filled marker denoting the changeover time for two pointings. North is up and east is to the left
(note that this orientation is opposite that in the left-hand
panel).}
\label{fig:track}
\end{figure}
To image the data, we started with an observation pointed at the
bright radio galaxy 3C~444 taken before the ISS observations. This
observation was used as a calibration observation, determining complex
gains for each tile assuming a point-source model of
3C~444, which was sufficient to model the source given that its extent
($<4$\arcmin) is the same as the FWHM of the synthesised beam of the
MWA subarray at this frequency. While we changed pointings between
this observation and those of the ISS, we have previously found that the
instrumental phases are very stable across entire days, and that only
the overall amplitudes need to be adjusted for individual
observations.
Antenna-based gain solutions were obtained
using the \textsc{CASA} task \textit{bandpass} on a 2-minute
observation. Solutions were obtained separately for each 10\,kHz
channel, integrating over the full two minutes. The flux density used
for 3C~444 was 116.7\,Jy at the central frequency of 103\,MHz, with a
spectral index of $-0.88$. A S/N$>3$
was required for a successful per-channel, per-baseline solution, and
the overall computed gain solutions for each antenna were of S/N$>5$.
Calibration solutions were applied to the ISS observations using the \textsc{CASA} task
\textit{applycal} and the corrected data were written to
\textsc{uv-fits} format for imaging in \texttt{miriad} \citep{sau95}.
For the two ISS observations, we flagged bad data in individual
fine channels (10\,kHz wide) associated with the centers and edges of
our 1.28\,MHz coarse channels. Because of the extremely fast motion
of the ISS, we created images in 1\,s intervals (the integration time
used by the correlator) using \texttt{miriad}. We started by imaging the
whole primary beam ($40\degr \times 40\degr$), separating the
data into $3\times 10.24$\,MHz bandpasses. The bottom two bandpasses covered the FM band,
while the top bandpass was above the FM band. We used a
$1\arcmin$ cell size and cleaned emission from Fornax~A. Note that
\texttt{miriad} does not properly implement wide-field imaging, but our
images used a slant-orthographic projection such that the synthesized
beam is constant over the image \citep{ord10}.
The results covering the two time intervals and two bandpasses are shown in
Figure~\ref{fig:images}. The ISS is readily visible as a streak moving
through our images, but is only visible in the images that cover the
FM band as would be expected from reflected terrestrial FM emission
(as in \citealt{mck12}). We predicted the position of the ISS based
on its two-line ephemeris (TLE) using the \texttt{pyephem} package\footnote{http://rhodesmill.org/pyephem}. The predicted position agrees with the observed position to within the observational uncertainties.
\begin{figure}[ht]
\plotone{Fig3.pdf}
\caption{Images generated for the ISS overflight. The left panels
show the 1\,s interval starting at 12:25:13\,UT, while the right
panels show the interval starting 1\,s later at 12:25:14\,UT. The
top panels show the 10\,MHz covering the top end of the FM band
(98.7\,MHz--108.7\,MHz) while the bottom panels show the 10\,MHz
above that, which does not include any of the FM band. The
double-lobed radio source Fornax\,A as well as the radio galaxies
PKS~J0200$-$3053 and PKS~J0237$-$1932 are labeled (with arrows). The predicted position of the ISS at the two times is indicated; it
is readily visible as a streak in the top images, but is not visible in the bottom images.
The greyscale scales linearly with flux, and are the same for all the panels. The images have been
corrected for primary beam gain, which is why the noise appears to
increase toward the left hand side. North is up, east to the left, and
these images cover roughly $35\degr$ on a side. 10 MHz bands have been used, even though the FM signals only occupy a fraction of the 98.7 -- 108.7 MHz band, so that the radio galaxies can be detected.
}
\label{fig:images}
\end{figure}
To determine more quantitatively the nature of the emission seen in
Figure~\ref{fig:images}, we imaged each 1\,s interval and each 10\,kHz
fine channel separately (as before, this was done in \texttt{miriad}).
We did not deconvolve the images at all; most of them had no signal or
the emission from the ISS was very weak. The emission from the ISS is
not a point source (smeared due to its motion) and adding deconvolution for a non-stationary object to our already
computationally-demanding task was not feasible. In each of our $180
\times 3072$ dirty images, we located the position of the ISS based on
its ephemeris. We then measured its flux density by adding up the
image data over a rectangular region of $10\arcmin$ (comparable to our
instrumental resolution) in width and with a length appropriate for
the instantaneous speed of the ISS (up to $40\arcmin$ in a 1\,s
interval), as seen in Figure~\ref{fig:zoom}. The resulting dynamic spectra
showing flux density as functions of time and frequency are shown in
Figure~\ref{fig:dynspec}. These spectra have been corrected for the
varying primary beam gain of the MWA over its field-of-view based on
measurements of the sources PKS~2356$-$61 (first observation, assuming
a flux density of $166\,$Jy at 97.7\,MHz) and PKS~J0237$-$1932 (second
observation, assuming a flux density of 35\,Jy at 97.7\,MHz); we have
ignored the frequency-dependence of the primary beam gain for the
purposes of this calculation. Note that the emission seen in
Figure~\ref{fig:dynspec} is manifestly \textit{not} radio-frequency
interference (RFI) that is observed as a common-mode signal by all of
our tiles. It is only visible at a discrete point in our synthesized
images corresponding to the position of the ISS
(Figure~\ref{fig:zoom}). We ran the same routines used to create the
dynamic spectra in Figure~\ref{fig:dynspec} but using a position offset from the ISS
position, and we see almost no emission, with what we do
see consistent with sidelobes produced by bright emission in
individual channels, due to the ISS.
\begin{figure}[ht]
\plotone{Fig4.pdf}
\caption{A zoom in of the image of the ISS in the FM band over a 1
second integration period. The left panel shows the 1\,s interval
starting at 12:25:13\,UT, while the right panel shows the interval
starting 1\,s later at 12:25:14\,UT (same times and data as shown in \ref{fig:images}). These images cover the
98.7--108.8\,MHz bandpass and are centered on the position of the
ISS at 12:25:13\,UT. The red boxes show the region over which the
flux was summed to create the dynamic spectra in
Figure~\ref{fig:dynspec}. North is up, east to the left, and these
images cover roughly $4\degr$ on a side. The synthesised beam is shown in the bottom left hand corner of the images.}
\label{fig:zoom}
\end{figure}
The ISS has an approximate maximum projected area of $\sim$1400 m$^{2}$, is of mixed composition (thus not well approximated as perfectly conducting), has a highly complex geometry, and was at an unknown orientation relative to the transmitter(s) and receiver during the observations. The detailed transmitter antenna geometries and gain patterns are not known. Thus, it is very difficult to accurately predict the flux density we would expect from either simple calculations or electromagnetic simulations.
\begin{figure}[ht]
\epsscale{0.7}
\plotone{Fig5a.pdf}\\
\plotone{Fig5b.pdf}
\caption{Dynamic spectra over the time range 12:23:00 to 12:26:00 UT
and over the frequency ranges 88--98\,MHz (top panel) and
98--108\,MHz (bottom panel). The black regions are those with no
data, either because of the gap between the observations or the
individual channels flagged during imaging. The color scale in both
images is the same and is linearly proportional to flux density, with the white stretch indicating the strongest signals.
The specific time discussed in the text, 12:24:20\,UT, is marked
with a dashed line. Individual FM frequencies from
Table~\ref{tab:transmit} are also labeled. The noisier portions at
the beginning and end of the first observation and at the end of the
second observation are because the flux densities have been
corrected for the primary beam gain, which is low at those times.}
\label{fig:dynspec}
\end{figure}
We can, however, unambiguously identify the origin of some of the transmissions reflected off the ISS. For example, we consider the dynamic spectrum from our observations in Figure 5 at the time 12:24:20 UT. At this time there are clearly a number of strong signals detected, plus a large number of much weaker signals. The five strongest signals at this time, in terms of the integrated fluxes (Jy.MHz, measured by summing over individual transmitter bands in the dynamic spectra to a threshold defined by the point at which the derivative of the amplitude changed sign), are listed in Table 1, along with transmitters that can be identified as the origin of the FM broadcasts \citep{acma}. Each of the five identified transmitters are local to Western Australia and, as can be seen from Figure 2, are all relatively close to the ISS at this time. Each of the five transmitters have omni-directional antenna radiation patterns in the horizontal plane.
If we take the reported ERP for a typical station from
Table~\ref{tab:transmit} and use
it with Equation~\ref{eqn:flux} to estimate the received flux, we
find values of roughly $10^4\,{\rm Jy.MHz}$, assuming the full
$1400\,{\rm m}^2$ of reflecting area for the ISS. Given its varying
composition and orientation, the measured fluxes in
Table~\ref{tab:transmit} are reasonably consistent with this estimate.
\begin{table}[ht]
\begin{tabular}{ c | c c r r c c c} \hline
\# & Freq. & Flux & Call sign(s) & Location(s) & ERP(s) &$R_{t}$ & $R_{r}$ \\
& (MHz) & (Jy.MHz) & & & (kW) & (km) & (km)\\ \hline
1 &98.1&3055&6JJJ&Central Agricultural&80&537 & 489 \\
2 &96.5&1907&6NAM&Northam&10&518 & 489 \\
3 &98.9&1799&6ABCFM/&Central Agricultural/&80/10&537/498&489 \\
& & &6JJJ &Geraldton&&& \\
4 &99.7&1718&6ABCFM/&Central Agricultural/&80/10&537/498&489 \\
& & & 6PNN&Geraldton&&& \\
5 &101.3&1672&6PNN&Geraldton&10&498&489 \\ \hline
\end{tabular}
\caption{The five strongest detected signals at 12:24:20 UT and transmitters identified with those signals\label{tab:transmit}}
\end{table}
Many, but not all, of the much weaker signals detected can also be
identified. However, it is clear that some of these weaker signals
are from powerful transmitters at much larger distances, making unique
identification of the transmitter more difficult (FM broadcasts are
made on the same frequencies in different regions). Furthermore, it
is clear that transmitters local to Western Australia, and of
comparable power to those listed in Table 1, have not been detected in
our observations. This is likely to be the result of the complex
transmitter/reflector/receiver geometries mentioned above and the
rapid evolution of the geometry with time, illustrated by the rapid
evolution of the received signals identified in Table 1. The five
strongest signals at 12:24:20 UT are strongest over a 5 second period,
have a highly modulated amplitude over a further 30 second period, but
are generally very weak or not detected over the majority of our
observation. Thus, it is highly likely that other transmitters of
comparable strength would be detected at other times, when favourable
geometries prevail. At 12:24:20 UT, it appears that the geometries
were favourable for the Central Agricultural, Northam and Geraldton
transmitters, but not for transmitters of comparable power in Perth,
Bunbury and Southern Agricultural. We note that, while the strongest signals detected originated from transmitters with omni-directional transmitting antennas (see above), the transmittors of comparable power not detected (Perth, Southern Agricultural and Bunbury) also have omni-directional antennas. Thus, the most likely factor driving detection or non-detection is the transmitter/reflector/receiver geometry, rather than transmitting antenna directionality.
The modulation in power of the reflected radiation is likely at least partly due to ``glints'' from the large, complex structure, which creates a reflected radiation pattern whose finest angular angular scale is $\sim \lambda/d$, where $\lambda$ is the wavelength of the radiation and $d$ is the extent of the object. For $d\sim100$ m (full extent of the ISS) and $\lambda \sim 3$ m, the glints are $>2^{\circ}$ in angular extent. At an altitude of $\sim$500 km, the smallest glint footprint on the surface of the Earth is $\sim10$ km. With an ISS orbital speed of 8 kms$^{-1}$, the glint duration at the MWA is of order one second. This corresponds well to some of the modulation timescales seen in Figure 5, although modulation exists on longer timescales (10 seconds and longer). These longer timescales may be due to glints involving structures smaller than the full extent of the ISS (single or multiple solar panels, for example). A full analysis of the modulation structure of the signals is extremely complex and cannot be performed at a sufficiently sophisticated level to be useful at present. In the future, detailed electromagnetic simulations of complex reflector geometries may be used to gain insight into the modulations, but such an analysis is beyond the scope of the current work.
In long integrations, such as required for Epoch of Reionisation experiments \citep{bow13}, the effect of having 10 pieces of space debris in the MWA primary beam at any given time with flux densities $>$1 Jy is the equivalent of a $<$1 mJy additional confusion noise foreground component in the FM band. This contribution will have no discernable impact on the MWA's science goals for observations in the FM band.
\section{Discussion and future directions}
At metre size scales, of order 1000 pieces of debris are currently known and tracked. On average, up to approximately 10 such pieces will be present within the MWA field-of-view at any given time, allowing continuous, simultaneous detection searches and tracking opportunities.
These estimated detection rates naturally lead to two primary capabilities. The first is a blind survey for currently unknown space debris, provided by the very large instantaneous field-of-view of the MWA. The second capability is the high cadence detection and tracking of known space debris. The wide field-of-view of the MWA means that large numbers of debris can be simultaneously detected and tracked, for a substantial fraction of the time that they appear above the local horizon.
Imaging with the MWA can provide measurements of the right ascension and declination (or azimuth and elevation) of the space debris, as seen from the position of the MWA (convertable to topocentric right ascension and declination). The MWA will also be able to measure the time derivatives of right ascension and declination, giving a four vector that defines the ``optical attributable''. Measurements of the four vector at two different epochs allow an estimate of the six parameters required to describe an orbit. \citet{farn09} describe methods for orbit reconstruction using the optical attributable, taking into account the correlation problem (being able to determine that two measurements of the four vector at substantially different times can be attributed to the same piece of space debris).
The calculations and practical demonstrations presented above show that the basic technique using the MWA is feasible and worthy of further consideration as a possible addition to Australia's SSA capabilities, particularly considering that the MWA is the Precursor to the much larger and more sensitive low frequency SKA. However, the calculations and observational tests presented here also point to the challenges that will need to be met in order to develop this concept into an operational capability.
Overall, these challenges relate to the non-standard nature of the imaging problem for space debris detection, compared to the traditional approach for astronomy. In radio interferomeric imaging, a fundamental assumption is made that the structure of the object on the sky does not change over the course of the observation. This assumption is broken in spectacular fashion when imaging space debris, due to their rapid angular motion across the sky relative to background celestial sources and due to the variation in RCS caused by the rapidly changing transmitter/reflector/receiver geometry.
Another standard assumption used in astronomical interferometric imaging is that the objects being imaged lie in the far field of the array of receiving antennas; the wavefronts arriving at the array can be closely approximated as planar. It transpires that for space debris, most objects lie in the transition zone between near field and far field at low radio frequencies, for an array the size of the MWA. Signal to noise is degraded somewhat when the standard far-field assumption is adopted, as an additional smearing of the imaged objects results. It is worth noting that this will be a much larger effect for the SKA, with a substantially larger array footprint on the ground. Thus, accounting for this effect for the MWA will be an important step toward using the SKA for SSA purposes. For example, for an object at 1000 km and the MWA spatial extent of 3 km, the deviation from the plane wave assumption translates into more than a radian of phase error across the array at a frequency of 100 MHz, which produces appreciable smearing of the reconstructed signal with traditional imaging. It may be possible to use the near field nature of the problem to undertake 3-dimensional imaging of the debris, using the wavefront curvature at the array to estimate the distance to the debris.
Further significant work is required to implement modified signal
and image processing schemes for MWA data that take account of these
time-dependent and geometrical effects and sharpen the detections
under these conditions. For example, the flexible approach to
producing MWA visibility data (the data from which MWA images are
produced) allows modifications to the signal processing chain to
incorporate positional tracking of an object in motion in real-time.
Additionally, it is possible to collect data from the MWA in an even
more basic form (voltages captured from each antenna) and apply high
performance computing in an offline mode to account for objects in
motion. This would also enable measurement of the Doppler
shifts of individual signals (expected to be $\sim {\rm few}\,$kHz,
which is smaller than our current 10\,kHz resolution) that could
enable separation of different transmitters at the same frequency
and unambiguous identification of transmitters (a Doppler shift
pattern fixes a one-dimensional locus perpendicular to the path of
the reflecting object, so combining two passes allows
two-dimensional localizations). That in turn would help with
constraining the basic properties of the reflecting objects\footnote{Given the changing geometry between passes, more than two might be required before a given transmitter is identified more than once with sufficient signal-to-noise for localization. Conversely, if the Doppler measurement is sufficiently significant and the transmitter is sufficiently isolated, a single one-dimensional localization might be enough for unique identification. We intend to test these ideas with future observations.}.
Finally, modifications to image processing algorithms can be applied
in post-processing to correct for near-field effects, essentially a
limited approximation of the same algorithms that can be applied
earlier in the signal chain. The results of this future work will be
reported in subsequent papers.
Once these improvements are addressed, the remaining significant challenges revolve around searching for and detecting signals from known and unknown space debris. These challenges are highly aligned with similar technical requirements for astronomical applications, in searches for transient and variable objects of an astrophysical nature. A large amount of effort has already been expended in this area by the MWA project, as one of the four major science themes the project will address \citep{bow13,mur13}. To illustrate the challenges, our flux estimates require (for example) greater
than 4$\sigma$ significance, but that is for a single trial. This
is correct when looking to recover debris with an approximately
known ephemeris, but blind searches have large trial factors that
can require significantly revised thresholds. Given
the full $2400\,{\rm deg}^2$ field-of-view of the MWA, we must
search over $\sim 10^4$ individual positions in each image to
identify unknown debris. Combined with 3000 channels and 3600 time
samples (for an hour of observing; note that we might observe with a
shorter integration time in the future) this results in $10^{11}$
total trials. So our 4$\sigma$ threshold (which implies 1 false
signal in $10^4$ trials) would generate $10^7$ false signals, and we
would need something like an $8\sigma$ threshold to be assured of a
true signal. Given the steep dependence of flux on object size
(Equation~\ref{eqn:flux}) this means our size limit may need to be
increased significantly. However, knowledge of individual bright
transmitters (from the type of analysis presented in this paper) means we can reduce the number of trials in the frequency axis
substantially, and seeking patterns among adjacent time samples that
fit plausible ephemerides will also help mitigate this effect.
If the challenges above can be addressed and a robust debris detection and tracking capability can be established with the MWA, it could become a very useful element of a suite of SSA facilities focussed on southern and eastern hemispheric coverage centered on Australia.
In November 2012 an announcement was made via an Australia-United States Ministerial Consultation (AUSMIN) Joint Communiqu\'{e} that a US Air Force C-band radar system for space debris tracking would be relocated to Western Australia\footnote{http://foreignminister.gov.au/releases/2012/bc\_mr\_121114.html}. This facility will provide southern and eastern hemispheric coverage, allowing the tracking of space debris to a 10 cm size scale. This system can produce of order 200 object determinations per day (multiple sets of range, range-rate, azimuth and elevation per object) and will be located near Exmouth in Western Australia, only approximately 700 km from the MWA site.
Additionally, the Australian Government has recently made investments into SSA capabilities, via the Australian Space Research Program (ASRP): RMIT Univerity's ``Platform Technologies for Space, Atmosphere and Climate''; and a project through EOS Space Systems Pty Ltd, ``Automated Laser Tracking of Space Debris''. The EOS project will result in an upgrade to a laser tracking facility based at Mt Stromlo Observatory, near Canberra on Australia's east coast, capable of tracking sub-10 cm debris at distances of 1000 km\footnote{http://www.space.gov.au}.
In principle, the C-band radar system, a passive radar facility based on the MWA, and the laser tracking facility could provide a hierarchy of detection and tracking capabilities covering the southern and eastern hemispheres. The C-band radar system could potentially undertake rapid but low positional accuracy detection of debris, with a hand-off to the MWA to achieve better angular resolution and rapid initial orbit determination (via long duration tracks). The MWA could then hand off to the laser tracking facility for the most accurate orbit determination. Such a diversity of techniques and instrumentation could form a new and highly complementary set of SSA capabilities in Australia. A formal error analysis, describing the benefits of combining the data from such a diverse set of tracking assets will be the subject of work to be presented in a future paper.
Such a hierarchy of hand-off between facilities has implications for scheduling the MWA, whose primary mission is radio astronomy. Currently the MWA is funded for science operations at a 25\% observational duty cycle (approximately 2200 hours of observation per year). This duty cycle leaves approximately 6500 hours per year available to undertake SSA observations (assuming that funds to operate the MWA for SSA can be secured). Even taking a conservative approach where only night time hours are used for observation, up to approximately 2200 hours could be available for SSA. It would be trivial to schedule regular blocks of MWA time, per night or per week, to undertake SSA observations. Given the rapid sky coverage of the MWA and the short observation times effective for SSA, the entire sky could be scanned over the course of approximately one hour each night, or using multiple shorter duration observations over the course of a night. Since the MWA can be rapidly repointed on the sky, the possibility of interupting science observations for short timescale follow-up SSA observations, triggered from other facilities such as the C-band radar, could be considered.
A review of Australia's capabilities in SSA was presented by \citet{new11}, pointing out that better connections between the SSA community and latent expertise and capabilities in the Australian astronomical community could be leveraged for significant benefit. In particular the authors suggested involving new radio astronomy facilities in SSA activities, given the advances in Western Australia connected with the SKA. The development of a passive radar facility in the FM band using the MWA would be a significant step in this direction.
Immediate future steps are to repeat the ISS measurements with the full 128 tile MWA, as an improvement on the 32 tile array used for the observations reported here, as well as to target smaller known satellites as tests and to implement the improvements described above.
\acknowledgements
We acknowledge the Wajarri Yamatji people as the traditional owners of
the Observatory site. Support for the MWA comes from the
U.S. National Science Foundation (grants AST CAREER-0847753,
AST-0457585, AST-0908884, AST-1008353, and PHY-0835713), the Australian Research Council (LIEF grants LE0775621 and LE0882938), the U.S. Air Force Office of Scientic Research (grant FA9550-0510247), the Centre for All-sky Astrophysics (an Australian Research Council Centre of Excellence funded by grant CE110001020), New Zealand Ministry of Economic Development (grant MED-E1799), an IBM Shared University Research Grant (via VUW \& Curtin), the Smithsonian Astrophysical Observatory, the MIT School of Science, the Raman Research Institute, the Australian National University, the Victoria University of Wellington, the Australian Federal government via the National Collaborative Research Infrastructure Strategy, Education Investment Fund and the Australia India Strategic Research Fund and Astronomy Australia Limited, under contract to Curtin University, the iVEC Petabyte Data Store, the Initiative in Innovative Computing and NVIDIA sponsored CUDA Center for Excellence at Harvard, and the International Centre for Radio Astronomy Research, a Joint Venture of Curtin University and The University of Western Australia, funded by the Western Australian State government. The electromagnetic simulations were performed by Gary Bedrosian and Randy Ward of Remcom Inc.
{\it Facility:} \facility{MWA}.
|
2,869,038,154,804 | arxiv | \section{Introduction}
\label{sec:intro}
The spatial curvature of the Universe is one of the most fundamental physical parameters, and spatial flatness is an essential ingredient of the current concordance $\Lambda$CDM cosmological model \cite{Planck2020}.
This work is driven by simple questions; why do we believe that the existing observational datasets suggest the spatial flatness?
To what extent does this question depend on the assumptions we make in a standard cosmological scenario?
These questions are partly motivated by the Hubble tension \cite{Riess:2021aa} which offers us excellent opportunities not only to reveal new physics but also to challenge any assumptions behind a concordance flat $\Lambda$CDM model.
Many possible solutions to the Hubble tension have been proposed (see \cite{Di_Valentino_2021} for a review).
Not surprisingly, the vast majority of such studies minimally extended the concordance flat $\Lambda$CDM model (see e.g. \cite{Poulin:2018bb,Fondi:2022tt} for exceptions) so that an extended model is still based on the success of the concordance model.
Nevertheless, it is not trivial whether and how a minimal extension best explains a series of cosmological observations. \par
Let us first summarize how spatial curvature is constrained within the context of a $\Lambda$CDM model.
In Fig.~\ref{Figure:OmKcomp}, we show a summary of examples from recent measurements.
The spatial curvature is commonly parameterized by a dimensionless parameter, $\Omega_{K}\equiv -c^{2}K/H_{0}^{2}$ where $c$ is the speed of light, $H_{0}=100h\,[{\rm km/s/Mpc}]$ is the Hubble constant, and $K$ is the curvature parameter.
$\Omega_{K}>0\,(K<0)$, $\Omega_{K}=0\,(K=0)$, and $\Omega_{K}<0\,(K>0)$ correspond to an open, flat, and closed universe, respectively.
The spatial curvature cannot be determined solely by the information from the primary anisotropies of the cosmic microwave background (CMB, hereafter) due to geometric degeneracy.
As discussed in detail in \cite{Planck2020}, the CMB temperature and polarization data in Planck preferred the closed universe ($\Omega_{K}=-0.044^{+0.018}_{-0.015}$ for TT,TE,EE+lowE) at the $\sim 2\sigma$ level.
This negative curvature was driven by a smooth temperature power spectrum at $\ell\gtrsim 1000$, which essentially degenerates with the effect of the lensing amplitude, $A_{L}>1$ \cite{Planck:2015,DiValentino:2020na}.
This situation was relaxed when the Planck temperature and polarization data was combined with the lensing reconstruction ($\Omega_{K}=-0.0106\pm 0.0065$) and the Baryon Acoustic Oscillation (BAO) data ($\Omega_{K}=0.0007\pm 0.0019$) \cite{Planck2020}.
In addition, Ref.~\cite{Alam:2021eb} showed that $\Omega_{K}=0.078^{+0.086}_{-0.099}$ was obtained only from a recent compilation of the BAO measurements from the Sloan Digital Sky Survey, combined with a prior on the baryon density, $\Omega_{\rm b}h^{2}$ (and hence on the sound horizon scale) from Big Bang Nucleosynthesis and CMB monopole temperature from the COBE/FIRAS data.
Adding the Planck temperature and polarization data (without marginalizing over $A_{L}$) to the compiled BAO data gave $\Omega_{K}=-0.0001\pm 0.0018$, which had no change with the Pantheon Type Ia supernovae \cite{Alam:2021eb}.
Note that including the curvature parameter in a $\Lambda$CDM model does not help mitigating the Hubble tension, since a positive curvature density or open universe is required to reduce the distance to the CMB last scattering with the angular scale of the sound horizon scale being fixed.
It is important to realize that these results are based on a $\Lambda$CDM model and a standard thermal history to predict the sound horizon scale (see, e.g., \cite{Yang:2022ex,Semenaite:2022sa} and their references therein for studies in models extended beyond $\Lambda$CDM).
We aim to understand how these results and interpretations are altered when we consider a scenario which alters the sound horizon scale.\par
\begin{figure}[t]
\centering
\includegraphics [width=0.99\textwidth]{fig_OmKcomp.pdf}
\caption{A summary (but highly incomplete examples) of recent measurements (mean with upper and lower 68\% confidence level (C.L.)) of the spatial curvature parameter $\Omega_{K}$ after Planck 2018.
We show our main results in this work (red, see Sec.~\ref{sec:results}), the measurements from the observables of the cosmic expansion history (cyan), and the measurements combined with information from the large-scale structure (LSS, magenta).
The measurements include Planck \cite{Planck2020}, Planck after marginalized over $A_{L}$ \cite{DiValentino:2020na}, Planck + Sloan Digital Sky Survey (SDSS) Data Release 12 (DR12) \cite{Alam:2017mn}, Planck + SDSS DR16, Planck + SDSS DR16 + Pantheon,
Planck + cosmic chornometer (CC) compiled in \cite{Vagnozzi:2021ap} (V21),
DR16 with a Big Bang Nucleosynthesis prior \cite{eBOSS:2020yzd}, strong gravitational lensing (SGL) + cosmic chornometer (CC) \cite{Liu:2022cc}, Planck + Dark Energy Survey Year 3 (DES Y3) \cite{DESY3:2022ex}, Planck + the full shape analysis of the galaxy power spectra ($P(k)$) \cite{Simon:2022ar} (see also \cite{Vagnozzi:2021pd}), the full shape analysis of the galaxy $P(k)$ with a BBN prior \cite{Glanville:2022mn}, and Planck + the galaxy clustering ratio (CR) \cite{Bel:2022cr}.
}
\label{Figure:OmKcomp}
\end{figure}
As a working example, we consider an axion-like Early Dark Energy (EDE) model \cite{Poulin:2018aa}.
Among many possible solutions, the EDE model mitigates the Hubble tension by introducing an additional scalar field prior to the recombination epoch and consequently lowering the sound horizon scale \cite{Poulin:2019aa,Smith:2019ih}.
The EDE model has recently been given close attention because there is mild evidence of a non-zero EDE component from some observations of the CMB anisotropies.
Refs.~\cite{Hill:2022pr,Poulin:2021ac,Smith:2022aa} reported evidence at the $\sim 3\sigma$ level from Atacama Cosmology Telescope (ACT) and South Pole Telescope (SPT), while Ref.~\cite{Hill:2020pl,Smith:2022aa} showed little evidence from the Planck data.
It is worth noting that all of these previous work assumed spatial flatness for simplicity (see \cite{Fondi:2022tt} for an exception and \cite{Hayashi2022de} for including the isocurvature perturbation).\par
In this paper, we will focus mainly on the observables related to the background expansion, and thus we will not touch upon the large-scale structure (LSS) or the perturbation part (except CMB).
It is well recognized that allowing non-zero EDE is compensated with higher matter density from the fit to CMB data \cite{Hill:2020pl,Smith:2022aa}, which may conflict with the low-lensing (or $S_{8}$) tension (see e.g., \cite{Leauthaud:2017aa,Amon:2022aa}) as shown by Refs.~\cite{Ivanov:2020fs,Herold:2022pl,D'Amico:2021aa,Simon:2022ar}.
To simultaneously resolve the Hubble and $S_{8}$ tensions, Ref.~\cite{Reeves:2022ss} extended an EDE model by varying the neutrino masses.
Also, Ref.~\cite{Berghaus:2022aa} considered an EDE scenario where the scalar field decays into extra dark radiation, which could mitigate the $S_{8}$ tension.
Both of these studies showed that Planck combined with other measurements does \textit{not} prefer an EDE model over the flat $\Lambda$ CDM model.
Furthermore, Ref.~\cite{Philcox:2022dh} (see also \cite{Smith:2022ra}) measured the distance scale independently of the BAO from the broadband shape (through the matter-radiation equality) of the galaxy power spectrum, reporting a somewhat small Hubble constant, $H_{0}=64.8^{+2.2}_{-2.5}\,{\rm km\,s^{-1}\,Mpc^{-1}}$.
Again, we notice that these studies assume spatial flatness.
We show some examples of the $\Omega_{K}$ constraints with LSS probes in Fig.~\ref{Figure:OmKcomp}.\par
The structure of this paper is as follows.
In Sec.~\ref{sec:theory}, we outline a theoretical background of the observables and physics relevant to this paper.
It is followed by Sec.~\ref{sec:method} which describes the method and datasets.
In Sec.~\ref{sec:results}, we report our results, and give elaborated discussion and summary in Sec.~\ref{sec:summary}.
\section{Theoretical Background}
\label{sec:theory}
In this section, we briefly provide a theoretical background on the observables relevant to this paper.
The sound horizon is the comoving distance that a sound wave of the primordial plasma can travel from the beginning of the universe to the point of the last scattering surface of CMB photons, defined by
\begin{equation}
r_{s}(z_{*})=\int_{z_*}^{\infty} \frac{dz}{H(z)}c_{s}(z),
\label{eq:r_s}
\end{equation}
where $z=1/a-1$ is the cosmological redshift, $a$ is the scale factor of the universe, and $z_{*}\sim 1090$ is the redshift of the CMB last scattering.
The sound speed, $c_{s}(z)$, is given by $c_{s}(z)^{2}=c^{2}/3\{1+R(z)\}$ where $c$ is the speed of light and $R(z)=3\rho_{\rm b}(z)/\{4\rho_{\gamma}(z)\}$ is the baryon-to-photon ratio.
The wave propagation is affected by cosmic expansion through the factor of the Hubble expansion rate, $H(z)\equiv \dot{a}/a$.
The CMB temperature and polarization anisotropies allow us to infer the angular size of the sound horizon scale, that is,
\begin{equation}
\theta_{s} \equiv \frac{r_s(z_*)}{D_A(z_*)}.
\label{eq:2.3}
\end{equation}
Here $D_A(z_*)$ is the angular diameter distance at $z=z_{*}$, given by
\begin{equation}
D_A(z) = \frac{1}{1+z}S_{K}\left[\int^{z}_{0}\frac{cdz'}{H(z')}\right],
\end{equation}
where the comoving distance $S_{K}(x)$ is obtained as $\sinh(\sqrt{-K}x)/\sqrt{-K}$ for $K<0$, $x$ for $K=0$, or $\sin(\sqrt{K}x)/\sqrt{K}$ for $K>0$.
Notice that $D_{A}(z)\propto H_{0}^{-1}$, independently of the value of $K$.
A flat $\Lambda$CDM universe provides an excellent fit with the Planck CMB data with $100\theta_{s}=1.04110\pm 0.00031\,{\rm radian}$ and $r_{s}(z_{*})=144.43\pm 0.26\,{\rm Mpc}$ \cite{Planck2020}.
This leads to $H_{0}=67.36\pm 0.54\,{\rm km/s/Mpc}$ with $\Omega_{\rm m}=0.3153$ \cite{Planck2020} which is inconsistent with the local measurement of the Hubble constant, $H_{0}=73.0\pm 1.4\,{\rm km/s/Mpc}$ \cite{Alam:2021eb}.
Baryon Acoustic Oscillations (BAOs) can be precisely measured with a galaxy survey as another standard ruler to measure cosmic expansion at low redshift $z\lesssim 3$.
Since we measure the BAO scale from a three-dimensional galaxy map, a BAO survey allows us to simultaneously measure
\begin{equation}
\theta_{\rm BAO} = \frac{r_{s}(z_{d})}{D_A(z_{\rm BAO})}\,\,\,{\rm and}\,\,\,
c\Delta z_{\rm BAO} = r_{s}(z_{d})H(z_{\rm BAO}),
\label{theta_bao}
\end{equation}
where $z_{\rm BAO}$ is the typical redshift of a BAO survey, and $z_{d}\sim 1060$ is the redshift of the baryon-dragging epoch that occurs slightly after $z_{*}$.
Planck CMB data provide $r_{s}(z_{d})=147.09\pm 0.26\,{\rm Mpc}$ in a flat $\Lambda$CDM model.
Since $\theta_{s}$ or $\theta_{\rm BAO}$ is precisely measured with CMB or BAO and is proportional to $r_{s}(z_{*})H_{0}$ or $r_{s}(z_{d})H_{0}$, one way to alleviate the Hubble tension is to reduce the sound horizon scale by introducing new physics prior to the CMB last scattering surface.
A novel Early Dark Energy model exactly achieves this, motivated by an extremely light axion-like scalar field $\phi$ \cite{Poulin:2019aa}.
In general, such a scalar field is modeled with a potential
\begin{equation}
V(\phi)=m^{2}\phi^{2} \left\{1-\cos\left(\frac{\phi}{f}\right)\right\}^{n}
\label{2.5}
\end{equation}
where $f$ is the decay constant of the scalar field.
For the scalar field to be effective before the CMB epoch, the mass $m$ should be smaller than the mass scale corresponding to the Hubble horizon scale at CMB, $m\lesssim 10^{-27}\,{\rm eV}/c^{2}$, so that the field begins to oscillate around the potential minimum.
Since the potential minimum is locally $V\sim \phi^{2n}$ where the equation of state of the field is given by $w_{\phi}=(n-1)/(n+1)$, we consider $n=3$ such that the cosmic expansion is decelerated with $w_{\phi}<1/3$ (or $n>2$) and hence the sound horizon is reduced.
The phenomenology of this scalar field can be parameterized by the following effective parameters: $z_c$, critical redshift, which is the redshift at which the EDE contributes to its maximal fraction, $\theta_i \equiv \phi/f$, where $\theta_i$ shows the initial displacement of the scalar field.
The third parameter is $f_{\rm{EDE}} \equiv {\rm max}(\rho_{\rm{EDE}}(z)/\rho_{\rm tot}(z))$ which indicates the maximal fractional contribution to the total energy density of the universe (see, e.g., \cite{Hill:2020pl}).
Since $\theta_{s}$ is precisely measured by the frequency of the angular power spectra of the CMB temperature and polarization anisotropies, the EDE parameters are constrained by other physical effects on the CMB spectra.
The most prominent difference from the $\Lambda$CDM case is the reduction in the diffusion damping scale, which increases the relative power of the CMB spectra at high $\ell$ \cite{Poulin:2018aa}.
For our purpose, an EDE model serves as a well-motivated scenario that alters the sound horizon scales, $r_{s}(z_{*})$ and $r_{s}(z_{d})$. In a flat universe, the current dataset allowed $f_{\rm EDE}\lesssim 0.125$ (corresponding to the best-fit value in the ACT case, see Fig.~~\ref{Figure:checkflat}) which reduces both sound horizon scales by a factor of about 5\%.
Our main goal is to quantitatively study the impact of these reduced sound horizon scales on the constraint on the spatial curvature.
As a by-product, we provide the constraint on an EDE model, marginalizing over the spatial curvature.
\begin{table}[t]
\centering
\begin{tabular}{c c}
Parameters & Prior \\ [0.5ex]
\hline\hline
$100\theta_s$ & [0.5,10] \\
$\rm{ln}10^{10}A_s$ & [1.61,3.96] \\
$n_s$ & [0.8,1.2] \\
$\tau_{\rm{reio}}$ & [0.02,\rm{None}] \\
$\Omega_{K}$ & [-0.5,0.5] \\
$A_{L}$ & [0.1,2.1] \\
$\log_{10}z_{c}$ & [3.1,4.3] \\
$\theta_{i}$ & [0.1,3.1] \\
$f_{\rm EDE}$ & [0.001,0.5] \\
\end{tabular}
\caption{The assumed ranges of uniform priors.}
\label{table:prior}
\end{table}
\section{Method and Datasets}
\label{sec:method}
We fit the $\Lambda$CDM and EDE models to a series of cosmological data to find the best-fit model parameters and quantify the statistical uncertainties.
Here, we describe our method and the datasets.
Since direct sampling of likelihood functions in a high-dimensional parameter space is computationally infeasible, we adopt the Markov chain Monte Carlo (MCMC) method on the basis of Bayesian statistics.
Note that, instead of sampling posteriors, one could look at the profile likelihood and find confidence intervals using the frequentist approach which helps mitigate projection issues in high-dimensional parameter space \cite{Herold:2021ksg,Reeves:2022aoi, Herold:2022iib}.
We perform our series of MCMC sampling using the latest release of publicly available code, \texttt{MontePython-v3.5}\footnote{\href{https://github.com/brinckmann/montepython\_public}{https://github.com/brinckmann/montepython\_public}} \cite{Audren:2012wb,Brinckmann:2018cvx} interfaced with the \texttt{CLASS\_EDE}\footnote{ \href{https://github.com/mwt5345/class_ede}{https://github.com/mwt5345/class\_ede}} \cite{Hill:2020pl} Boltzmann solver as an extension to the \texttt{CLASS}\footnote{\href{https://github.com/lesgourg/class\_public}{https://github.com/lesgourg/class\_public}} \cite{class-code,CLASS2011} code that accounts for the EDE model.
We impose flat non-informative priors on both $\Lambda$CDM and EDE parameters, except the prior on $\tau_{\rm reio}$ in the case of SPT-3G and ACT DR4 (see the relevant texts below).
In the flat $\Lambda$CDM model, we consider the main cosmological parameters $\{ \Omega_{\rm b}h^{2}, \omega_{cdm}\equiv \Omega_{cdm}h^{2}, \theta_s, \ln[10^{10}A_s],n_{s}, \tau_{\rm{reio}} \}$ that account for the amount of baryon and CDM densities in the Universe, the ratio of the sound horizon to the angular diameter distance, the normalization amplitude and the spectral index of the primordial power spectrum, and the reionization optical depth, respectively.
When we consider a non-flat cosmology, we include two additional parameters, $\{\Omega_{\rm K},A_{L}\}$.
The reason we add the lensing parameter $A_{L}$ is that there exists a degeneracy between $\Omega_{\rm K}$ and $A_{L}$ in the Planck temperature and polarization data \cite{DiValentino:2020na} (see their Fig.~2), and this helps isolate the lensing information from the dataset we consider.
Regarding an EDE model, we add three more phenomenological parameters, $\{f_{\rm EDE}, \log_{10}z_{c}, \theta_{i}\}$ using the shooting method described in \cite{Smith:2019ih} to map them to the theoretical parameters $\{f,m\}$ mentioned in Eq.~(\ref{2.5}).
In modeling free-streaming neutrinos, we take into account the Planck collaboration convention as two massless species and one massive with $M_{\nu}=0.06\, \rm{eV}$.
In our analysis, we adopt the effective number of relativistic degrees of freedom to its standard model prediction as $N_{\rm eff}=3.046$.
To compute the non-linear matter power spectrum we use \texttt{Halofit} \cite{Smith:2002dz,Takahashi:2012hf} for CMB lensing, although again the information from CMB lensing is minimized by being marginalized over $A_{L}$.
The default prior choice is given in Table.~\ref{table:prior}.
Moreover, depending on the choice of the data set we use, we have additional nuisance parameters.
For example, for the case of the full Planck 2018 likelihood, we have $47$ nuisance parameters that would be added to our multidimensional parameter space.
For the case of SPT-3G 2018 likelihood, we have $20$ nuisance parameters.
In the case of the ACT DR4 likelihood, we only have one nuisance parameter which stands for the polarization efficiency.
To analyze the chains and produce plots, we use the \texttt{GetDist}\footnote{\href{https://getdist.readthedocs.io/}{https://getdist.readthedocs.io/}} Python package \cite{Lewis:2019xzd}.
We consider chains to be sufficiently converged, checking the Gelman-Rubin criterion $|R-1|\lesssim 0.1$.
In this work, we adopt the following datasets:
\begin{description}
\item[Planck 2018:] We consider the multifrequency TT, TE, and EE power spectra and covariances from Planck PR3 (2018)\footnote{\href{http://pla.esac.esa.int/pla/\#cosmology}{http://pla.esac.esa.int/pla/\#cosmology}} \cite{Planck:2018nkj,Planck2020,Planck:2019nip}, including the publicly available \texttt{Plik\_HM} high-$\ell$ likelihood consisting of $30 \leq \ell \leq 2508$ for TT and $30 \leq \ell \leq 1996$ for TE and EE spectra, and also from the \texttt{Commander} likelihood, we consider low-$\ell$ ($2 \leq \ell \leq 29$) TT data.
In addition, we include the Planck reconstructed CMB lensing potential power spectrum \cite{Planck:2018lbu}.
The gravitational lensing of the CMB has been detected by a high ($40 \sigma$) statistical significance in Planck 2018 \cite{Planck:2018lbu}. The multipole range of the Planck lensing power spectrum covers $8 \leq \ell \leq 400$.
In summary, we use Planck-high-$\ell$ \texttt{TTTEEE} $+$ Planck-low-$\ell$ \texttt{TT} $+$ Planck-low-$\ell$ \texttt{EE} + Planck-lensing, which we refer to this combination as ``Planck''.
\item[Planck $\ell\leq 650$:] When we use SPT-3G or ACT DR4 instead of Planck 2018, we combine the Planck TT data at $\ell \leq 650$.
For this purpose, we use the full \texttt{Plik} likelihood rather than the \texttt{Plik\_lite} version to change the range of the multipole.
This choice is motivated by the fact that Planck 2018 is fully consistent with WMAP \cite{Hinshaw_2013} up to this multipole \cite{Galli:2017pw} and that our results can be fairly compared with previous SPT and ACT results \cite{Hill:2022pr,Smith:2022aa}.
We also add a Gaussian prior on the optical depth as $\tau_{\rm reio} = 0.065 \pm 0.015$, following \cite{Hill:2022pr,ACT:2020gnv}.
\item[SPT-3G:] We use the most updated SPT-3G data from 2020 data release \cite{SPT-3G:2021eoc} which includes EE and TE power spectra spanning over the angular multipole range, $300 \leq \ell < 3000$, from observations of a $1500\, \rm{deg}^2$ survey field in three frequency bands centered at $95$, $150$, and $220\, \rm{GHz}$\footnote{Note that, while we prepare this work, the SPT-3G TT measurement has been recently released and we do include it in this work\cite{Balkenhol:2022rvc} }.
We adopt the publicly available full SPT-3G likelihood \cite{SPT-3G:2021eoc} for the \texttt{MontePython} environment\footnote{\href{https://github.com/ksardase/SPT3G-montepython}{https://github.com/ksardase/SPT3G-montepython}} \cite{Chudaykin:2022rnl}.
The likelihood function marginalizes posteriors over the super-smaple lensing parameter, the terms representing Galactic dust emission, polarized dust and the noise from the radio galaxies along with the all-sky calibration of temperature and polarization \cite{SPT-3G:2021eoc}.
We refer to this SPT-3G with Planck $\ell\leq 650$ (including the $\tau_{\rm reio}$ prior) as ``SPT''.
\item[ACT DR4:] We use the latest version of ACT data from the fourth data release, ACT DR4, \cite{ACT2020} from the 2013–2016 survey covering $>15000\,\, \rm{deg}^2$ including multi-frequency temperature and polarization measurements.
The TT, TE and EE power spectra have already been marginalized over various uncertainities such as foreground emission and systematic errors.
We use the \texttt{MontePython} support for the publicly available \texttt{actpollite dr4} likelihood implemented in \texttt{pyactlike}\footnote{\href{https://github.com/ACTCollaboration/pyactlike}{https://github.com/ACTCollaboration/pyactlike}}.
The range of multipoles for TE and EE power spectra spans over $326 < \ell < 4325$ while the TT power spectrum covers $ 576 < \ell < 4325 $ \cite{ACT:2020gnv}.
The only nuisance parameter in this likelihood is the overall polarization efficiency.
We call this ACT DR4 data with Planck $\ell\leq 650$ (including the $\tau_{reio}$ prior) as ``ACT''.
\item[Baryon Acoustic Oscillations (BAOs):] We consider the following datasets from BAO survey as a probe of the cosmic expansion history at low redshifts;
the 6df galaxy redshift survey at $z_{\rm BAO}=0.106$ \cite{Beutler_2011},
the Sloan Digital Sky Survey (SDSS) Data Release (DR) 7 main galaxy sample at $z_{\rm BAO}=0.15$ \cite{Ross:2014qpa},
and data compiled from the SDSS DR16 Baryon Baryon Oscillation Spectroscopic Survey (BOSS) \cite{BOSS:2017ba} and extended BOSS (eBOSS) measurements \cite{deSainteAgathe:2019voe,eBOSS:2020yzd,duMasdesBourboux:2020pck} that include
Luminous Red Galaxy sample at $z_{\rm BAO}=0.38$, $0.51$ and $0.698$\footnote{\href{https://github.com/CobayaSampler/bao_data/blob/master/sdss_DR16_BAOplus_LRG_FSBAO_DMDHfs8.dat}{https://github.com/CobayaSampler/bao\_data/blob/master/sdss\_DR16\_BAOplus\_LRG\_FSBAO\_DMDHfs8.dat}}, the QSO sample at $z=1.48$\footnote{\href{https://github.com/CobayaSampler/bao_data/blob/master/sdss_DR16_BAOplus_QSO_FSBAO_DMDHfs8.dat}{https://github.com/CobayaSampler/bao\_data/blob/master/sdss\_DR16\_BAOplus\_QSO\_FSBAO\_DMDHfs8.dat}}, (note that for these two aformaentioned cases the covariance matrices have been marginalized over the $f\sigma_8$ measurements), and also the $\rm{Ly}\alpha \times \rm{Ly}\alpha$\footnote{\href{https://github.com/CobayaSampler/bao_data/blob/master/sdss_DR16_LYAUTO_BAO_DMDHgrid.txt}{https://github.com/CobayaSampler/bao\_data/blob/master/sdss\_DR16\_LYAUTO\_BAO\_DMDHgrid.txt}} and $\rm{Ly}\alpha \times \rm{QSO}$ measurements at $z=2.334$\footnote{\href{https://github.com/CobayaSampler/bao_data/blob/master/sdss_DR16_LYxQSO_BAO_DMDHgrid.txt}{https://github.com/CobayaSampler/bao\_data/blob/master/sdss\_DR16\_LYxQSO\_BAO\_DMDHgrid.txt}}.
Note that we do not include the BAO measurement for the eBOSS Emission Line Galaxy as its contribution to the fit is minor.
In this work we do not consider the redshift-space distortion or full-shape galaxy power spectrum.
\end{description}
We do not include the distance ladder measurement of the Hubble constant, $H_{0}=73.2\pm 1.4\,{\rm km/s/Mpc}$ \cite{Riess:2021aa} in our fit, but compare it with the inferred values.
Moreover, since our primary interests are the observables relevant to the sound horizon scale, we do not include the data from Pantheon Type Ia supernovae.
Previous studies show that the impact of the Pantheon data is generally minor when combined with the BAO data (see e.g., Fig.~\ref{Figure:OmKcomp} and \cite{Yang:2022ex}).
\begin{figure}[t]
\centering
\includegraphics [width=1\textwidth]{fig_flatlcdm.pdf}
\caption{1D and 2D posterior distributions (68\% and 95\% C.L. for the 2D contours) of various cosmological parameters in a flat universe for both $\Lambda$CDM and EDE models.
Here we only use the CMB datasets, Planck (cyan/brown for EDE/$\Lambda$CDM), SPT (blue/orange), and ACT (gray/purple).
Hereafter, $H_{0}=73.2\,{\rm km/s/Mpc}$ in \cite{Riess:2021aa} and $r_{s}(z_{*})=144.43\,{\rm Mpc}$ in the fidicual flat $\Lambda$CDM fit \cite{Planck2020} are shown as references (vertical dashed lines).
}
\label{Figure:checkflat}
\end{figure}
\section{Results}
\label{sec:results}
We begin with the results of the fit with the CMB datasets only for the flat $\Lambda$CDM and EDE models in Fig.~\ref{Figure:checkflat}.
The purpose of showing Fig.~\ref{Figure:checkflat} is a sanity check that we have successfully reproduced the previous work;
Our fits in the flat $\Lambda$CDM are in excellent agreement with the previous work.
Consistently with \cite{SPT-3G:2021eoc}$, \omega_{\rm cdm}$ (and $r_{s}(z_{*})$) in the flat $\Lambda$CDM model for SPT is slightly smaller than that for Planck and ACT.
In the EDE model, Planck disfavors non-zero $f_{\rm EDE}$, while ACT and SPT somewhat prefer non-zero values, $f_{\rm EDE}< 0.231\,(0.214)$ at $95\%$ C.L. for ACT (SPT).
Non-zero $f_{\rm EDE}$ in ACT or SPT leads to reduction in the sound horizon scale, $r_{s}(z_{*})$ while $\theta_{s}$ is kept nearly fixed, yielding higher values of $H_{0}$ than flat $\Lambda$CDM cases \cite{Hill:2020pl,Smith:2022aa}.
\begin{figure}[t]
\centering
\includegraphics [width=1\textwidth]{fig_curvedcmb.pdf}
\caption{1D and 2D posterior distributions (68\% and 95\% C.L. for the 2D contours) of various cosmological parameters in a curved universe for both $\Lambda$CDM and EDE models.
Here we only use the CMB datasets, Planck (cyan/brown for EDE/$\Lambda$CDM), SPT (blue/orange), and ACT (gray/purple).
We show $\Omega_{K}=0$ as a reference (vertical dashed line).
}
\label{Figure:curvedcmb}
\end{figure}
Next, we present the CMB results including the spatial curvature in Fig.~\ref{Figure:curvedcmb}.
The constraints on $f_{\rm EDE}$ with these CMB datasets are not affected by $\Omega_{K}$ and $A_{L}$ and remain similar to the flat case in Fig.~\ref{Figure:checkflat}.
Meanwhile, we confirm the degeneracy among $\Omega_{K}$, $A_{L}$, and $H_{0}$ as expected, giving weak constraints on these parameters only with the CMB data.
However, the impact of the EDE model paramaters on the $\Omega_{K}$ constraints is minor \cite{Fondi:2022tt}.
We have curvature constraints with the CMB datasets even after marginalizing the EDE model parameters; $\Omega_K = -0.0116 ^{+0.0072}_{+0.0062}$ for Planck, $\Omega_K = -0.0526 ^{+0.0723}_{-0.0179}$ for ACT, and $\Omega_K = -0.0589 ^{+0.0776}_{-0.0336}$ for SPT (see also Table \ref{Table:main}).
The reason why Planck constrains $\Omega_{K}$ better than ACT and SPT is that we combine with the reconstructed CMB lensing in Planck.
Without the Planck CMB lensing, the Planck constraint on $\Omega_{K}$ is degraded from $\Delta \Omega_K \sim 0.0066$ to $\Delta \Omega_K \sim 0.0488$ in the $\Lambda$CDM case (see also \cite{Planck2020}).
To understand why the EDE model paramaters are constrained even with $\Omega_{K}$, we present the CMB angular power spectra and their residuals from the Planck $\Lambda$CDM best-fit model in the left panel of Fig.~\ref{Figure:spectra}.
Top, middle, and bottom panels show the residuals from TT, TE, and EE spectra, respectively.
If the data deviates from zero in each panel, a model may be required to include physics beyond the Planck $\Lambda$CDM model.
Solid and dashed curves show the ACT best-fit curves in EDE and EDE$+\Omega_{K}+A_{L}$ models.
We only show the ACT best-fit as it prefers the largest $f_{\rm EDE}$.
As demonstrated by the two curves, including $\Omega_K$ in the EDE model does not affect the acoustic feature of the CMB spectra and consequently has little impact on the EDE parameters.
This is not surprising, since $\Omega_{K}$ is constrained by the CMB mainly through $\theta_{s}$ with the sound horizon fixed.
We mitigate information from CMB lensing by marginalizing $A_{L}$.
\begin{figure}[t]
\centering
\includegraphics[width=7.2cm]{residuals_ttteee.pdf}
\qquad
\includegraphics[width=7cm]{BAO_ACT_datapoint_cutted.pdf}
\caption{Left Panel: ACT best-fit CMB angular spectra residuals (TT, TE and EE) for EDE (solid) and EDE+$\Omega_K + A_L$ (dashed). Cyan, orange and purple data points show the residuals of the Planck, SPT-3G and ACT measurements with respect to the flat $\Lambda$CDM best-fit in Planck.
Right Panel: The ratio of the sound horizon scale (Eqs.~\ref{eq:2.3}, \ref{theta_bao}) measurements in EDE+$\Omega_K + A_L$ model with respect to the Planck flat $\Lambda$CDM. Red data points in the left panel show the low-redshift BAO measurements from DR16 and the right panel shows the CMB measurements. The green curve represents the EDE+$\Omega_K + A_L$ best-fit model in the light of the ACT+BAO datasets. Black dashed/dotted-dashed curve shows the same model with slightly larger/smaller values for $\Omega_K$.}
\label{Figure:spectra}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics [width=1\textwidth]{fig_PlanckBAO.pdf}
\caption{1D and 2D posterior distributions (68\% and 95\% C.L. for the 2D contours) in a curved EDE model with Planck only (red) and Planck+BAO (blue).
We show $r_{s}(z_{d})=147.09\,{\rm Mpc}$ in the flat $\Lambda$CDM fit in Planck as a reference (vertical dashed line).
}
\label{Figure:PlanckBAO}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics [width=1\textwidth]{fig_ACTBAO.pdf}
\caption{The same as Fig.~\ref{Figure:PlanckBAO} but with ACT only (magenta) and ACT+BAO (green).
}
\label{Figure:ACTBAO}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics [width=1\textwidth]{fig_SPTBAO.pdf}
\caption{The same as Fig.~\ref{Figure:PlanckBAO} but with SPT only (purple) and SPT+BAO (orange).
}
\label{Figure:SPTBAO}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics [width=0.8\textwidth]{fhmw.pdf}
\caption{The same results from Figs.~\ref{Figure:ACTBAO} and \ref{Figure:SPTBAO} are combined for a clear comparison.
}
\label{Figure:combinedBAO}
\end{figure}
Thus, it is crucial to combine the CMB data with other low-redshift probes to break the degeneracy and improve the CMB curvature constraints.
In Figs.~\ref{Figure:PlanckBAO}, \ref{Figure:ACTBAO}, and \ref{Figure:SPTBAO}, we compare the CMB only result with the case combined with BAO for Planck, ACT, and SPT, respectively.
Not surprisingly, adding BAO to the CMB data is extremely powerful to precisely measure $\Omega_{K}$; $\Omega_K = 0.00011_{-0.0023}^{+0.0021}$ for Planck+BAO, $\Omega_K = -0.0056_{-0.0031}^{+0.0031}$ for ACT+BAO, and $\Omega_K =-0.0032_{-0.0038}^{+0.0036}$ for SPT+BAO.
It is interesting to confirm that the current CMB+BAO data prefer $\Omega_{K}=0$ within 95\% C.L. even after marginalizing over the EDE parameters and hence the sound horizon scale.
Since both the sound-horizon scales $r_{s}(z_{*})$ and $r_{s}(z_{d})$ are reduced by the existence of EDE by a similar amount, this result suggests that the paremeters important for the late-time universe such as $\Omega_{K}$ and $\Omega_{m}$ are mainly determined by the relative ratio of two standard rulers at two distinct epochs from the CMB and BAOs.
To demonstrate this, we show in Fig.~\ref{Figure:spectra} the ratio of the sound horizon scale for EDE$+\Omega_K+A_L$ model with respect to the best-fit Planck $\Lambda$CDM.
The middle panel shows the low-redshift BAO measurements, while the right panel shows the $\theta_s$ constraints of the same curved EDE model for Planck, ACT, and SPT measurements.
The green curve is our best-fit curved EDE model for ACT, equivalent to the best-fit for the green contours in Fig.~\ref{Figure:ACTBAO}.
The black dashed/dotted-dashed curves show the EDE$+\Omega_K+A_L$ model with slightly larger/smaller values for $\Omega_{K}$.
Note that, since the $y$ axis is divided by the sound horizon, the reduction of the sound horizon is taken into account.
The relative comparison between the CMB and BAO data points is crucial to break the degeneracies and therefore to precisely constrain $\Omega_K$.
Accurate determination of $\Omega_{K}$ and $\Omega_{m}$ with BAO data has a nonnegligible impact on other parameters by breaking degeneracies.
$\Omega_{K}$ and $\Omega_{m}$ are correlated with $H_{0}$ through geometric degeneracy positively and negatively, respectively.
Therefore, a higher $\Omega_{K}$ (or smaller $\Omega_{m}$) in CMB+BAO than CMB only generally leads to a higher $H_{0}$.
The $H_{0}$ value in Planck+BAO is not as high as the distance ladder measurement (the dashed vertical line), while those in ACT+BAO and SPT+BAO are more consistent with them.
This result is driven by the preferred value in $f_{\rm EDE}$ and hence how much the sound horizon scale is reduced.
Interestingly, the ACT+BAO posterior has a peak at non-zero $f_{\rm EDE}$ consistently with the CMB-only case, while the SPT case has a notable (but statistically insignificant) shift in $f_{\rm EDE}$.
To understand the reason behind this, we compare the ACT and SPT results in Fig.~\ref{Figure:combinedBAO}.
The peak of $\Omega_{m}$ for the SPT data only is slightly deviated from the value preferred by BAO, resulting in a lower value of $\omega_{cdm}$.
Since $\omega_{cdm}$ and $f_{\rm EDE}$ are positively correlated, the preferred value of $f_{\rm EDE}$ becomes smaller than the case without BAO.
On the contrary, this is not the case for ACT, as the peak of $\Omega_{m}$ for ACT remains unchaged with or without the addition of BAO.
\begin{table}[t]
\centering
\scalebox{0.8}{
\begin{tabular}{c||ccc|ccc}
& Planck+BAO& ACT+BAO& SPT+BAO & Planck & ACT & SPT \\
\hline
\hline
$f_{\rm EDE}$ & $0.034_{-0.024}^{+0.017}$ & $0.141_{-0.054}^{+0.031}$ & $0.070_{-0.065}^{+0.025}$ & $0.020_{-0.018}^{+0.005}$ & $0.134_{-0.050}^{+0.029}$ & $0.105_{-0.066}^{+0.052}$ \\
$\theta_{i}$ & $0.690_{-0.147}^{+0.169}$ & $1.569_{-1.037}^{+1.150}$ & $1.347_{-0.960}^{+0.865}$ & $0.731_{-0.299}^{+0.267}$ & $1.476_{-1.266}^{+1.353}$ & $1.645_{-1.311}^{+1.139}$\\
$\log_{10}(z_{c})$ & $3.507_{-0.095}^{+0.076}$ &$3.248_{-0.136}^{+0.052}$ & $3.487_{-0.269}^{+0.198}$ & $3.871_{-0.239}^{+0.319}$ & $3.245_{-0.126}^{+0.063}$ & $3.532_{-0.160}^{+0.084}$\\
\hline
$H_0\,[{\rm km/s/Mpc}]$ & $69.355_{-1.15}^{+0.974}$ & $72.451_{-2.504}^{+1.350}$ & $70.639_{-2.686}^{+1.322}$ & $64.081_{-2.550}^{+2.309}$ & $62.288_{-17.839}^{+12.784}$& $59.145_{-17.665}^{+11.059}$ \\
$\Omega_{K}$ & $0.0001_{-0.0023}^{+0.0021}$ & $-0.0056_{-0.0031}^{+0.0031}$ &$-0.0032_{-0.0038}^{+0.0036}$ & $-0.0116_{+0.0062}^{+0.0072}$ & $-0.0526_{-0.0179}^{+0.0723}$ & $-0.0589_{-0.0366}^{+0.0776}$ \\
$r_s(z_{*})\,[{\rm Mpc}]$ & $142.736_{-1.256}^{+1.646}$ & $138.233_{-2.083}^{+4.679}$ & $142.112_{-1.738}^{+4.81}$ & $144.275_{-0.287}^{+1.011}$ & $137.968_{-2.076}^{+4.513}$ & $139.291_{-3.832}^{+4.59}$ \\
\end{tabular}
}
\caption{The constraints on key cosmological parameters for the curved-EDE (EDE$+\Omega_{K}+A_{L}$) model. Each column corresponds to different dataset. Each value quoted as $\rm mean_{-(mean - lower \,\,68\% limit)}^{+(upper \,\,68\% limit - mean)}$ where ``mean'' refers to the mean value of the marginalized posterior distribution. }
\label{Table:main}
\end{table}
\section{Summary and discussion}
\label{sec:summary}
In this work, our main interest is to derive the constraint on the spatial curvature from the cosmic expansion history in a cosmological scenario with non-standard sound horizon scale.
More specifically, we use recent CMB datasets from Planck, ACT, and SPT as well as the most updated BAO measurement from SDSS DR16 in both of which the change in the sound horizon scale would impact the inference on the spatial curvature parameter.
As a working example, we adopt an axion-like EDE model which attracts attention in the community in light of the Hubble tension.
We extend our EDE cosmological parameter space by the spatial curvature and the lensing amplitude to use the information from the cosmic expansion history.
Our main findings are highlighted as follows:
\begin{itemize}
\item We find that, independent of the CMB datasets, the EDE model parameters are constrained only by the CMB power spectra as precisely as the flat case in previous work, even with $\Omega_{K}$ and $A_{L}$. Although Planck disfavors non-zero $f_{\rm EDE}$, ACT and SPT alone prefer non-zero value, $f_{\rm EDE}\lesssim 0.145$ (see Fig.~\ref{Figure:curvedcmb} and Table \ref{Table:main}).
Not surprisingly, the constraints on $\Omega_{K}$ and $H_{0}$ are weakly constrained by any CMB-only cases due to the geometric degeneracy.
\item We demonstrate that combining CMB with BAO is extremely powerful to constrain the spatial curvature even with the reduction of the sound horizon scales. In the case of ACT which prefers the largest amount of $f_{\rm EDE}\sim 0.14$ which reduces the sound horizon scales by about 5\%, we obtain $\Omega_{K}=-0.0056\pm 0.0031$ after marginalizing over the EDE parameters.
This constraint is as competitive as the Planck + DR16 BAO result in a $\Lambda$CDM model, $\Omega_{K}=-0.0056\pm 0.0018$ \cite{eBOSS:2020yzd} (see Figs.~\ref{Figure:OmKcomp}, \ref{Figure:spectra}, \ref{Figure:PlanckBAO}-\ref{Figure:SPTBAO} and Table \ref{Table:main}).
\end{itemize}
Let us clarify the difference between our work and similar other work.
Ref.~\cite{Fondi:2022tt} studied an EDE model that included $\Omega_{K}$ and $A_{L}$ using the Planck data and obtained $\Omega_{K}=0.0004\pm 0.0020$ when they combined with the SDSS DR12 BAO data.
Although this is in excellent agreement with our result for a similar case, $\Omega_{K}=0.0001^{+0.0021}_{-0.0023}$, there are a few minor differences between the two works.
First, they consider Planck only, while we also study ACT and SPT.
Second, their prediction of the EDE model is based on an approximated method implemented in \texttt{CAMB} to solve the linear perturbation equation for an EDE scalar field \cite{Smith:2019ih}.
Instead, we adopt the exact method implemented in \texttt{CLASS} for an EDE model, following \cite{Hill:2020pl,Smith:2022aa}.
In fact, we have tested the approximate method in \texttt{CAMB} and confirmed that we were unable to reproduce the previous work for ACT and SPT in \cite{Hill:2020pl,Smith:2022aa}.
However, the impact of the approximation on the Planck case is minor, since Planck does not prefer nonzero $f_{\rm EDE}$.
Finally, they vary the equation-of-state parameter for the late-time dark energy, $w$, while we fix it with the cosmological constant, $w=-1$.
As Ref.~\cite{Fondi:2022tt} showed in their Fig.~2, a degeneracy between $\Omega_{K}$ and $w$ is expected to some extent, but their Planck+BAO result is in perfect agreement with $w=-1$.
Along the similar line, there are similar studies which consider different dark energy scenarios which become prominent \textit{well after} the recombination (see e.g., \cite{Yang:2022ex,Khoraminezhad:2020cer,Cruz:2022oqk}).
We leave the impact of these other extended parameters on the spatial curvature for future work.
As motivated by Fig.~\ref{Figure:spectra}, it is desirable to obtain more accurate and precise data from CMB such as LiteBird, CMB-S4, and the Simons Array (see e.g., \cite{Chang:2022sn} for a recent review) as well as the BAO data filling at $z\gtrsim 1$ such as Hobby-Eberly Telescope Dark Energy Experiment \cite{Gebhardt:2021mm}, Dark Energy Spectroscopic Instrument \cite{DESI:2016pp}, Prime Focus Spectrograph \cite{Takada:2014aa}, and Roman Space Telescope \cite{Wang:2022tt}.
These new datasets will lead to a more precise and accurate constraint on the spatial curvature of the Universe (see e.g., \cite{Takada:2015ok,Leonard:2016ph,Sailer:2021aa}).
\acknowledgments
JS was supported by the OURE program at Missouri University of Science and Technology.
HK and SS acknowledge the support for this work from NSF-2219212.
SS is supported in part by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan.
|
2,869,038,154,805 | arxiv | \section{INTRODUCTION}
In the past 20 years, observations from space-based telescopes
revealed that the solar corona and transition region are much more
dynamic than had been thought. Surges, jets, macrospicules, bright
points and macroflares take place everywhere in the solar
atmosphere. Sometimes, surges, jets and macrospicules exhibit
rotating motions \citep{pik98, zha00, pat08, 2009ApJ...707L..37L,
2010A&A...510L...1K}, which is suggested to be an indicator of the
existence of Alfv\'{e}n waves in these structures (Sterling, 1998).
The mechanism providing such a highly dynamic corona and transition
region is considered to be related to the long-standing puzzle of
how the corona is heated to a million Kelvin \citep{asc04}.
Rotational motions in the photosphere (Brandt et al 1988) may create
rotating motions of surges (jets) and macrospicules which are driven
by localized downdrafts that collect the cold plasma returning to
the solar interior after releasing internal energy (e.g., Spruit et
al. 1990; Stein \& Nordlund 1998). Usually those vortices are small
(0.5 Mm) and last several minutes, but sometimes they could be large
and survive much longer (see, e.g. Attie et al. 2009). In the quiet
Sun, about 3.1$\times$10$^{-3}$ vortices Mm$^{-2}$ minute$^{-1}$
are found \citep{bon10}, and they do not have a preferred rotation
sense on the different hemispheres (Bonet et al. 2008). The vortices
on the photosphere can propagate upward along the field lines (e.g.,
Choudhuri et al. 1993; Zirker 1993; van Ballegooijen et al. 1998),
and govern the evolution of magnetic footpoints \citep{bal10}. In
the low chromosphere, vortical motions with possible propagation of
waves were also detected for the first time by Wedemeyer-B{\"o}hm \&
Rouppe van der Voort (2009). Moreover, they often wind up opposite
polarity field lines, facilitating magnetic reconnection and ensuing
energy release. For active regions, rotating motion is one of the
most important processes for sunspot evolution, and may be
responsible for the flaring \citep{ger03}. Thus those rotating
motions may provide a mechanism to build up the magnetic energy in
the corona that is eventually released during transients, which in
turn heats the corona.
In this Letter, we report the discovery of EUV cyclones and Rotating
Network magnetic Fields (RNFs) in the quiet Sun with the
observations from the Atmospheric Imaging Assembly (AIA; Lemen et
al. 2011) and the Helioseismic and Magnetic Imager (HMI; Wachter et
al. 2011) aboard the {\it Solar Dynamics Observatory} (SDO). The
paper is organized as follow. In Section 2 we describe the
observational data we use. An analysis of the data is presented in
Section 3. In Section 4, we conclude this study and discuss the
results.
\section{OBSERVATIONS}
SDO/AIA observes uninterruptedly the Sun's full disk at 10
wavelengths at a 12-second cadence and a 0.6$''$ pixel$^{-1}$
sampling. The measurements reflect various temperatures of the solar
atmosphere (from $\sim$5000 K to $\sim$2.5 MK) from the photosphere
to the corona. SDO/HMI, from polarization measurements, information
such as Doppler-velocity, line-of-sight (LOS) magnetic field
strength, and vectorial information of the magnetic field are
obtained. The data cover the full disk of the Sun with a spatial
sampling of 0.5 $''$ pixel$^{-1}$. The full disk LOS magnetograms,
used in this Letter, are taken at a cadence of 45 seconds.
\section{ANALYSIS}
The phenomenon of the EUV cyclones is seen everywhere in the quiet
Sun in the AIA data in all EUV channels of 171, 193, 304, 211, 131,
335, and 94 {\AA}. Those cyclones are found to be associated with
the magnetic fields that show conspicuous rotation when viewed with
the movies of the HMI time-series magnetograms. We use the term RNFs
to refer to those rotating magnetic fields. Here we present two
examples of the EUV cyclones to demonstrate their evolutionary
characteristics and properties. Shown in Figures. 1(A) and 1(B) is a
cyclone occurred on July 2 (see also movie1). The images were taken
in the 171 {\AA} channel. This cyclone was in the northern
hemisphere, close to the solar equator. It started at $\sim$2:00 UT,
rotated counter-clockwisely, and lasted 12 hours. The footpoint of
the cyclone on the photosphere, shown as a positive field patch in
the HMI LOS magnetograms (the white feature in Figures. 1(C) and
1(D)), also rotated counter-clockwisely. At 04:02:48 UT, the
magnetic flux in this patch was 2.9$\times$10$^{19}$ Mx, one order
greater than the flux in a typical network element($\sim$10$^{18}$
Mx, Zhang et al. 2006). The noise level of HMI 45 s magnetograms is
10.2 Mx cm$^{-2}$ (Liu et al. 2011), and the minimum flux obtained
for this level in HMI is 1.3 $\times$ 10$^{16}$ Mx. Its longer axis
was along the northeast-southwest direction. At 06:44:48 UT, this
axis rotated to southeast-northwest direction. It suggests that the
magnetic field rotated 64$^{\circ}$ within 162 minutes.
The other example was in the southern hemisphere on July 20. It
occurred at 07:00 UT, and lasted 9 hours. Figures. 2 (A) and (B)
show the cyclone at the 171 {\AA} images. The two circles denote the
place where a time-slice map is made. Different from the cyclone on
July 2, this cyclone rotated clockwisely (see movie2). From 11:23:48
UT to 13:20:48 UT, the positive magnetic fields (black contours in
Figures 2(C) and 2(D)), which the cyclone are rooted in, rotated
83$^{\circ}$ clockwisely (see (C) and (D) in Figure 2) with a speed
of 0.8$^{\circ}$ min$^{-1}$. Its flux is about 1.6$\times$10$^{19}$
Mx and then it cancelled with the nearby negative elements (white
contours in Figures 2(C) and 2(D)). We find that the negative
polarity with an absolute magnetic flux of 3.6$\times$10$^{18}$ Mx
disappeared and the positive polarity faded about
3.9$\times$10$^{18}$ Mx during the cancellation. The blue curve in
panel (D) is the contour of EUV brightening in panel (B). We can see
that the brightening corresponds to the neutral region, instead to
the positive polarity. Figure 2 (E) shows a time-slice map of the
AIA 171 {\AA} images. The X-axis refers to time, running from 2010
July 20 11:00 UT to 15:00 UT. The Y-axis refers to the angle
subtended by the dotted curves measured clockwise, whose origin is
the center of the circle and the reference direction is the West, as
shown in Panel (A). The dashed curve outlines the intersection of
the leading boundary of the cyclone. In the 3-hour time interval
(from 11:00 UT to 14:00 UT), the cyclone rotated about
360$^{\circ}$. In the later phase of the cyclone at 14:26:59 UT, an
EUV brightening appeared. This brightening continuously developed,
exhibiting subtle structures, and becoming a two-ribbon microflare
(the white patches in Figure 3). It lasted 2 hours, and disappeared
at 16:18:11 UT. Meanwhile a small-scale wave was triggered by the
microflare. It propagated with an average speed of 45 km s$^{-1}$,
and lasted almost 8 minutes. Its disturbance spread over an area of
6.3$\times$10$^{8}$ km$^{2}$, equivalent to a typical supergranular
cell.
Figure 4(A) shows the angular speed of the cyclone measured from the
time-slice map. It indicates that the cyclone underwent two
acceleration processes. At first the cyclone rotated with an angular
speed of 1$^{\circ}$ min$^{-1}$, 50 minutes later the speed reached
7$^{\circ}$ min$^{-1}$, and then slowed down to 1$^{\circ}$
min$^{-1}$ again. At 13:20 UT, the cyclone accelerated again. The
speed reached another peak (4.8$^{\circ}$ min$^{-1}$). Figure 4(B)
shows the propagating speed of the EUV wave. We determine the wave
front (denoted by stars in each panel in Figure 3) every 36-second,
and then compute the mean speed in the 36-second period. The
propagation speed of the wave was not constant. There were two
peaks, ($\sim$80 km s$^{-1}$ at 16:00:06 UT, and 85 km s$^{-1}$ at
16:05:30 UT).
Observations have shown that EUV cyclones are rooted in the RNFs. An
interesting question is then: how many RNFs are there on the solar
surface? To answer this question, we surveyed the 2010 July 8
AIA/HMI data when no active region was present on the solar disk. In
an area of 800$\times$980 square arcseconds near the disk center, we
found 388 RNFs (movie3 shows an example of the RNFs). The occurrence
of these events is 4.9 $\times$ 10$^{-4}$ arcsec$^{-2}$ day$^{-1}$.
The average lifetime of the RNFs is about 3.5 h. The distribution of
these RNFs is plotted in Figure 5. Their magnetic fluxes range from
1.24$\times$10$^{17}$ Mx to 5.85$\times$10$^{19}$ Mx. The average
flux is 1.03$\times$10$^{19}$ Mx, and the total flux of these RNFs
is about 4.0$\times$10$^{21}$ Mx, or 78\% of the unsigned magnetic
flux in that area. There are 179 and 209 RNFs in the northern and
southern hemispheres, respectively. The sense of the rotation shows
a weak hemisphere preference: 61\% of the RNFs in the northern
hemisphere (109 of the 179 RNFs) rotated counter-clockwisely, while
56\% of the RNFs in the northern hemisphere (116 of 209 RNFs)
rotated clockwisely. It leads to 5600 RNFs over the entire solar
surface each day. It infers that a total flux of
5.8$\times$10$^{22}$ Mx is rotating. Simultaneous AIA observations
show that among the cyclones relevant to these RNFs, 231 ones
($\sim$60\% of the total) are associated with microflares.
\section{DISCUSSION AND CONCLUSIONS}
In this Letter we report the discovery of EUV cyclones in the quiet
Sun. They are rooted in the RNFs, and last several to more than ten
hours, evidently different from surges (jets) and macrospicules,
which have lifetimes less than one hour \citep{zhan00}. About 60\%
EUV cyclones are associated with microflares and even small-scale
EUV waves at the later phase of cyclones. The microflares correspond
to the neutral region, instead to one polarity. HMI observations
show the ubiquitous presence of the RNFs. The sense of rotation
shows a weak hemisphere preference, which is also presented with
G-band observations in Vargas Dom{\'{\i}}nguez et al. (2011). The
average magnetic flux is 1.03$\times$10$^{19}$ Mx. Every day, we
infer that about 5600 RNFs are present over the entire solar
surface. The total unsigned flux is about 5.8$\times$10$^{22}$ Mx.
As revealed in this study, the brightenings (microflares) are not
corresponding to one polarity but to the neutral place where the
opposite polarities cancelled. This is consistent with the
morphological model shown in Figure 8 of Zhang et al. (2000b)., i.e,
when two network elements with opposite polarities converge together
and begin to cancel each other, a microflare appears at the
cancelling site.
Surface motion on the photosphere is one of the two mechanisms
proposed to build up the free energy in the corona (the other is the
emergence of magnetic field). It can produce current sheets parallel
to a separatrix that detach the interacting magnetic flux regions
(Sweet 1969; Lau 1993; see Somov 2006 for a review). In the
simulation of Gerrard et al. (2003), rotation of a sunspot with
inflow of a pore leads to a strong build-up of current which is
needed for magnetic reconnection. This current buildup is quite
similar to the conclusion of \cite{zha07, zha08}, i.e., the
rotational motions of sunspots relate to the transport of magnetic
energy and complexity from the low atmosphere to the corona and play
a key role in the onset of flares. In the quiet Sun, the random walk
of the footpoints of coronal loops in the solar granulation is
expected to cause the braiding of the field, which in turn leads to
a multitude of coronal reconnection events (Schrijver 2007).
Magnetic reconnection between sheared magnetic loops, indicating the
injection of magnetic helicity of mixed signs, works as a trigger
mechanism of solar flares \citep{kus04,kus05}. Thus, the phenomena
of RNFs, as well as the EUV cyclones, reported in this Letter,
likely contain the process of energy buildup and release in solar
corona.
What heats the solar corona remains one of the most important
questions in solar physics and astrophysics. \citet{kli06} has
proposed that three parts are involved to solve the coronal heating
puzzle, which are (1) identifying a source of energy and a mechanism
for converting the magnetic energy into heat, (2) determining how
the plasma responds to the heating, and (3) predicting the spectrum
of emitted radiation and its manifestation as observable quantities.
Our results may provide evidence to address the first two parts.
Continuous development of RNFs braids field lines of the magnetic
elements, manifested by the cyclones seen in the EUV observation.
Magnetic reconnection then takes place in the braiding fields,
releasing the stored energy that heats the corona. It is supported
by the observed brightenings (microflares) and EUV waves.
Up to now, one of the most common mechanisms to heat the corona is
the impulsive energy release (nanoflares, with times of minutes, and
dimensions of Mms) that occurs in the small-scale magnetic fields
(Parker 1988). AIA data indicate that microflares (or nanoflares)
are not impulsive events. Furthermore, they are accompanied with
other activities. For example, in the 2010 July 20 cyclone, the
microflare occurred 7 hours after the cyclone, followed by a
small-scale EUV wave. Small-scale waves in the quiet Sun may play a
role in coronal heating: they transport energy to other places.
The small-scale EUV waves reported here propogate at a speed of
35--85 km s$^{-1}$, much slower than the EUV waves originate from
active regions, which is in arange of 200-400 km s$^{-1}$ (see,
e.g., Thompson \& Myers 2009). The nature of the small-scale waves
is an interesting question that needs further investigation. It may
provide additional information that helps understand the EUV waves
that is still under intense debate (Wills-Davey \& Thompson 1999; Wu
et al. 2001; Ofman \& Thompson 2002; Schmidt \& Ofman 2010;
Delann\'{e}e 2000; Attrill et al. 2007; Attrill 2010; Liu et al.
2010).
\acknowledgments
The authors are indebted to the {\it SDO} teams for providing the
data. This work is supported by the National Natural Science
Foundations of China(G11025315, 40890161 and 10921303), the CAS
Project KJCX2-YW-T04, and the National Basic Research Program of
China under grant G2011CB811403.
|
2,869,038,154,806 | arxiv |
\section{Introduction}
\label{sec:Intro}
\subsection{The Fermi bubbles}
Non-thermal lobes emanate from the nuclei of many galaxies.
These lobes, thought to arise from starburst activity or an outflow from a super-massive black hole \citep[for reviews, see][]{VeilleuxEtAl05,KingPounds15}, play an important role in the theory of galaxy formation \citep[\emph{e.g.,} ][and references therein]{Benson10}.
The presence of a massive, bipolar outflow in our own Galaxy has long been suspected, largely based on X-ray and radio signatures on large \citep{Sofue00}, intermediate \citep{BlandHawthornAndCohen03} and small \citep{BaganoffEtAl03} scales, indicating an energetic, $\gtrsim 10^{55}\mbox{ erg}$ event \citep[][and references therein]{VeilleuxEtAl05}.
This scenario was revived by the discovery \citep[][henceforth \Su]{DoblerEtAl10, SuEtAl10} of two large $\gamma$-ray, so called Fermi, bubbles (FBs), symmetrically rising far from the Milky-Way plane yet morphologically connected, at least approximately and in projection, to the intermediate scale X-ray outflow features.
Due to their dynamical, nonthermal nature, and the vast energy implied by their presumed Galactic-scale distance, an accurate interpretation of the FBs is important for understanding the energy budget, structure, and history of our Galaxy.
The FBs, extending out to latitudes $|b|\simeq 55^\circ$,
are also seen in microwave synchrotron emission \citep{Dobler12, PlanckHaze13}, as the so-called microwave Haze \citep{Finkbeiner04}, a residual diffuse signal surrounding the Galactic centre (GC).
They may also morphologically coincide and with linearly polarized radio emission \citep{CarrettiEtAl13}, although the association of this signal with the FBs is unclear.
\subsection{Interpretation as a Galactic-scale phenomenon}
The tentative identification of the FBs as massive structures emanating from the GC, rather than small lobes of a nearby object seen in projection, was based mainly on the coincidence of the lobe's base, to within a few degrees, with the GC.
The FB edges were recently extracted robustly, without making any assumptions concerning the Galactic foreground, by applying gradient filters to the Fermi-LAT map; the resulting edges connect smoothly to the intermediate-low latitude X-ray features, strengthening the FB--GC coincidence to sub-degree scales \citep[][henceforth \citetalias{KeshetGurwich16_Diffusion}]{KeshetGurwich16_Diffusion}.
The implied, $\sim 10\mbox{ kpc}$ distance scale corresponds to a high, $\sim 4\times 10^{37}\mbox{ erg}\mbox{ s}^{-1}$ luminosity (\Su, \FT).
Additional, less direct indications for the FB--GC association include the radio emission being too faint for a nearby source confined to the already-magnetized thick Galactic disk, the orientation of the FB axis being nearly exactly perpendicular to the Galactic plane, in agreement with an extended structure bursting out of the Galactic disk \citepalias[][and references therein]{KeshetGurwich16_Diffusion}, and the low-latitude depolarization of the tentatively associated linearly polarized lobes \citep{CarrettiEtAl13}.
Another claim is the fairly high emission measure (EM) of possibly related high-latitude X-ray features \citep{KataokaEtAl13}; however, we argued in \citetalias{KeshetGurwich16_Diffusion} and conclusively show here that the X-ray signature was incorrectly interpreted.
Moreover, it is unclear if $\mbox{EM}\simeq 0.01\mbox{ cm}^{-6}\mbox{ pc}$ suffices to rule out a local structure.
\subsection{Underlying flow and edge shock}
In spite of their dramatic appearance in the $\gamma$-ray sky, the nature of the FBs is still debated.
Different models were proposed (\Su, \FT), interpreting
the FB edge as an outgoing shock \citep{FujitaEtAl13}, a termination shock of a wind \citep{Lacki14, MouEtal14}, or a discontinuity \citep{Crocker12, GuoMathews12, SarkarEtAl15};
the $\gamma$-ray emission mechanism as either hadronic \citep{CrockerAharonian11, FujitaEtAl13} or leptonic \citep{YangEtAl13};
the underlying engine as a starburst \citep{CarrettiEtAl13, Lacki14, SarkarEtAl15}, a jet from the the central massive black hole \citep{ChengEtAl11, GuoMathews12, ZubovasNayakshin12, MouEtal14}, or steady star-formation \citep{Crocker12};
and the cosmic-ray (CR) acceleration mechanism as first order Fermi acceleration, second order Fermi acceleration \citep{MertschSarkar11, ChernyshovEtAl14}, or injection at the GC \citep{GuoMathews12, Thoudam13}.
More clues regarding the nature of the FBs have gradually surfaced.
The microwave haze shows a hard spectrum, $\nu I_\nu\propto \nu^{-0.55\pm0.05}$ \citep[\emph{e.g.,} ][]{PlanckHaze13}, corresponding to CR electron (CRE) acceleration in a strong, $M\gtrsim5$ shock \citepalias{KeshetGurwich16_Diffusion}.
Metal absorption lines of $\{-235,+250\}\mbox{ km} \mbox{ s}^{-1}$ line of sight velocities in the spectrum of quasar PDS 456, located near the base of the northern FB, indicate an outflow velocity $\gtrsim 900\mbox{ km} \mbox{ s}^{-1}$ \citep{FoxEtAl15}.
Longitudinal variations in the
\MyMNRAS{\ion{O}{VII}}
\MyApJ{\ion{O}{7}}
and
\MyMNRAS{\ion{O}{VIII}}
\MyApJ{\ion{O}{8}}
emission line strengths, integrated over a wide latitude range covering the entire FBs, suggest a $\sim 0.4\mbox{ keV}$ FB multiphase plasma with a denser, slightly hotter edge, propagating through a $\sim 0.2\mbox{ keV}$ halo, thus suggesting a forward shock of Mach number $M=2.3_{-0.4}^{+1.1}$ \citep{MillerBregman16}.
By removing a FB template, \citet{SuFinkbeiner12} found a southeast--northwest, bipolar jet, with a cocoon on its southeast side; however, only the cocoon was so far confirmed to be significant (\FT).
The $\gamma$-ray spectrum of the FB shows very little variations with position along the edge; this alone indicates, when invoking CRE Fermi-acceleration, a strong shock with $M>5$ \citepalias{KeshetGurwich16_Diffusion}.
The spatially integrated \citepalias{SuEtAl10,FermiBubbles14} or locally measured \citepalias{KeshetGurwich16_Diffusion} $\gamma$-ray spectrum can be naturally explained for $\sim$few Myr old bubbles, without invoking ad-hoc energy cutoffs, only in a leptonic model featuring a $\sim 1\mbox{ GeV}$ cooling break (\PaperTwo); this spectrum is again consistent with CREs injected in a strong shock.
Finally, the edge spectrum is found to be slightly but uniformly and consistently softer than the FB-integrated spectrum \citepalias{KeshetGurwich16_Diffusion}; this is naturally explained by inward CRE diffusion in a Kraichnan-like magnetic turbulence if the FB edge is a forward shock.
\subsection{High latitude X-ray signature}
At the highest FB latitudes, an absorbed, $\sim0.3\mbox{ keV}$ X-ray component was reported \citep{KataokaEtAl13}, with a $\sim 60\%$ jump in the EM as one crosses outside the edge of the northern bubble.
Here and in what follows, we refer to the gas closer to (farther from) the GC as lying below, or equivalently inside (above, or equivalently outside) the edge.
The putative jump reported by \citet{KataokaEtAl13} would suggest that the FB edges are in fact a weak, $M\sim1.5$ reverse shock, terminating a wind.
However, as we pointed out in \citetalias{KeshetGurwich16_Diffusion}, these observations are complicated by the high level of dust and confusion with other structure in the northern hemisphere, and are equally --- if not more convincingly --- consistent with a drop, rather than a jump, in both south and north bubbles, which would suggest a forward shock.
Such a drop would furthermore be consistent with the evident X-ray drops at the intermediate-low latitude X-ray features \citep{BlandHawthornAndCohen03}, and with the other evidence outlined above.
Here we use the \emph{ROSAT} all sky survey \citep[RASS;][]{SnowdenEtAl97} to measure the high latitude X-ray signal associated with the FBs.
Modelling the FBs as an expanding shell, we derive the drop in flux and in temperature expected as one crosses outside the edge, in a transition spanning $\sim 2{^\circ}$ in projection, as well as the gradual brightening of the signal and cooling of the gas over $\sim10{^\circ}$ within the FBs towards the GC.
These signals are difficult to pick up directly due to various structure in the X-ray sky, the uncertainty in the precise location of the FB edge, the low surface brightness, and the \emph{ROSAT} X-ray background.
Nevertheless, the data can be stacked at fixed distances from the FB edge, which has already been traced directly with a few degree precision using the Fermi data in \citetalias{KeshetGurwich16_Diffusion}, to greatly enhance the signal well beyond the detection threshold.
Errors in edge location, variations in the radial profiles, and noise, are thus effectively averaged out, enabling a firm detection of the signal.
The paper is arranged as follows.
In \S\ref{sec:Model} we model the expected X-ray signal from the FBs.
The \emph{ROSAT} data and analysis procedure are presented in \S\ref{sec:Data}.
The X-ray structure perpendicular to the FB edges is extracted in \S\ref{sec:RadialProfiles}, and analyzed in \S\ref{sec:Analysis}.
The results are summarised and discussed in \S\ref{sec:Discussion}.
We use $1\sigma$ error bars, unless otherwise stated, and Galactic coordinates, throughout.
\section{FB model}
\label{sec:Model}
\subsection{Edge toy model}
To model the gas flow underlying the FBs, we begin with a toy model for the shape of the high latitude edge.
Consider a simple bipolar shock pattern, specified by the Galactocentric radius,
\begin{equation} \label{eq:FBEdgeFull}
R = R_0 \times \begin{cases}
1-\left(\theta/\theta_0\right)^2 & \mbox{for $0<\theta<\theta_1$ ;}\\
c_1/(\theta+\theta^8) & \mbox{for $\theta_1<\theta<\pi/2$}
\end{cases}
\end{equation}
for the north FB, and symmetrically $(\theta\to\pi-\theta$) about the Galactic plane for the south FB.
Here, $R_0$ is the peak height of the FB, and $\theta$ is the polar angle, measured in a frame with the GC at the origin.
For a FB seen in projection with a maximal latitude $b\simeq 53{^\circ}$, and for a Solar Galactocentric radius $r_\odot\simeq8.5\mbox{ kpc}$, we find that $R_0\simeq 10\mbox{ kpc}$.
A good fit for the top of the FBs requires $\theta_0\simeq \pi/5$; continuity of the edge and its first derivative then yield $\theta_1\simeq\pi/8.6$ and $c_1\simeq 0.24$.
The FB edge resulting from this two-parameter ($R_0$ and $\theta_0$) model is shown in projection in Figure \ref{fig:FBModel}.
For comparison, the figure also shows the FB edges, as extracted in \citetalias{KeshetGurwich16_Diffusion} (coarse grained edge number 1, therein and below), for both hemispheres.
At latitudes $|b|\gtrsim 15{^\circ}$, the model reasonably matches the observed edge on the better resolved, eastern side; lower latitudes are outside the scope of the present analysis.
The model does not, however, capture the east-west FB asymmetry, in particular the observed westward extension of the high $|b|$ bubbles.
Consequently, the high latitude ($|b>15{^\circ}$) solid angle $\sim 0.34$ of each observed FB is $\sim20\%$ larger than the corresponding, $\sim 0.28\mbox{ sr}$ solid angle of the FB in the toy model.
\subsection{Upstream, halo model}
Consider a scenario where the FBs arise from a rapid release of energy, leading to a supersonic outflow with a forward shock coincident with the FB edges.
The outflowing gas should form a massive shell, with density and pressure increasing outwards towards the shock.
This increase is expected to be gradual for the $\sim r^{-2}$ decline attributed to the density of the Galaxy's hot gas halo, into which the FBs are presumably expanding.
In particular, a $\beta$-model based on
\MyMNRAS{\ion{O}{VII}}
\MyApJ{\ion{O}{7}}
emission and absorption lines \citep{MillerBregman13} is consistent at $r\gg1\mbox{ kpc}$ radii with an isothermal sphere distribution,
\begin{equation}\label{eq:nu}
n_{e,u} \simeq n_{e,10} \left( \frac{r}{10\mbox{ kpc}} \right)^{-2} \mbox{ ,}
\end{equation}
where $n_{e,10}\equiv 4\times 10^{-4}n_{4}\mbox{ cm}^{-3}$ is the electron number density $n_e$ at $r=10\mbox{ kpc}$, and subscript $u$ (subscript $d$) denotes the upstream (downstream) plasma.
Specifying the flow underlying the FBs requires some assumption on the upstream temperature, $T_u$.
The halo temperature, based on
\MyMNRAS{\ion{O}{VII}}
\MyApJ{\ion{O}{7}}
emission and absorption, is \citep{MillerBregman13}
\begin{equation} \label{eq:Tu}
k_B T_u \equiv k_B T_h\simeq (0.1\mbox{--}0.2) \mbox{ keV} \mbox{ ,}
\end{equation}
where $k_B$ is Boltzmann's constant.
Assuming (henceforth) an adiabatic index $\Gamma=(5/3)$ and a cosmic element abundance with mean particle mass $\mu m_p$, where $\mu\simeq 0.6$, these temperatures correspond to an upstream sound velocity $c_{s,u}\simeq (170\mbox{--}190) \mbox{ km} \mbox{ s}^{-1}$.
Note that a somewhat higher, $k_B T_u\simeq 0.3\mbox{ keV}$ temperature was derived from X-rays using \emph{Suzaku} \citep{KataokaEtAl13}.
\MyApJ{\begin{figure}[h]}
\MyMNRAS{\begin{figure}}
\PlotFigsA{
\epsfxsize=8.5cm \epsfbox{\myfig{FBXROSATPrimakoffCGFull.eps}}
}
\caption{
Projected X-ray FB model: flux in the \emph{ROSAT} (0.1--2.4 keV) band (solid blue curves), $\log_{10}(F_X [n_{4}^2\mbox{ erg}\mbox{ s}^{-1}\mbox{ cm}^{-2}\mbox{ sr}^{-1}])$, and X-ray weighted temperature (dashed red contours), $T_X[M_{10}^2T_{0.15}\mbox{ keV}]$.
Also shown are edges 1 of the north (dot-dashed black) and (reflected about the Galactic plane) south (dotted green) FBs.
\label{fig:FBModel}
}
\end{figure}
\subsection{Flow model}
A spherical, strong shock propagating into a halo-like, $n\propto r^{-2}$ medium asymptotes to the Primakoff-like solution \citep{CourantFriedrichs48, Keller56}, in which the downstream distribution follows power-law profiles \citep{BernsteinBook80}
\begin{equation} \label{eq:PrimakoffScaling}
n\propto r \mbox{ ,}\quad P\propto r^3 \mbox{ ,}\quad\mbox{and}\quad v\propto r \mbox{ .}
\end{equation}
For simplicity, we assume that the linear (in $r$) velocity of the spherical Primakoff-like solution remains valid in our nonspherical model, when choosing the GC as the origin. Then \citep{BernsteinBook80}
\begin{equation} \label{eq:LinearVScaling}
\vect{v}(\vect{r},t)= (2t)^{-1}\vect{r} \mbox{ ,}
\end{equation}
where $t$ is the age of the FBs.
For simplicity, we assume here that electrons and ions are shock-heated to the same temperature, and revisit this issue in \S\ref{sec:Discussion}.
These assumptions now fix the structure of the flow throughout the modelled FBs.
The Rankine-Hugoniot jump conditions dictate that the shock velocity in the Galaxy frame is
\begin{equation}
v_s=v_u=\frac{\Gamma+1}{\Gamma-1+2M^{-2}} v_d \simeq \frac{4}{1+3M^{-2}} v_d \mbox{ ,}
\end{equation}
where $v_{u}$ and $v_{d}$ are the fluid velocities with respect to the shock.
Taking the velocities as radial, $v_d\simeq v_s-v(R)$, so
\begin{equation}
v_s = M c_{s,u} \simeq \frac{4v(R)}{3(1-M^{-2})} = \frac{2R/t}{3(1-M^{-2})} \mbox{ ,}
\end{equation}
giving
\begin{equation}
M=\frac{2}{3}\tilde{M}+\sqrt{1+\left(\frac{2}{3}\tilde{M}\right)^2} \mbox{ ,}
\end{equation}
where $\tilde{M}\equiv v(R)/c_{s,u}=R/(2c_{s,u}t)$ is the fluid velocity at $r=R$ normalized to upstream sound.
In the strong shock limit, this becomes $M\simeq(4/3)\tilde{M}$.
Equivalently, given a shock Mach number $M$, the flow velocity is fixed by $v(R)=3c_{s,u}(M-M^{-1})/4$, becoming $v(R)\simeq (3/4)Mc_{s,u}$ in the strong shock limit.
We may now write $t$ in terms of the Mach number $M=10 M_{10}$ at the top of the FB,
\begin{equation}
t = \frac{2R_0/(3c_{s,u})}{M-M^{-1}} \simeq \frac{2R_0}{3Mc_{s,u}} \simeq 3.3 M_{10}^{-1}T_{0.15}^{-1/2} \mbox{ Myr} \mbox{ ,}
\end{equation}
where $T_{0.15}\equiv k_B T_u/(0.15\mbox{ keV})$, and in the second equality we assumed the shock to be strong.
The thermal properties in the immediate downstream are now given by
\begin{equation} \label{eq:NScaling}
n_{e,d}=\frac{\Gamma+1}{\Gamma-1+2M^{-2}} n_{e,u} \simeq \frac{4}{1+3M^{-2}} n_{e,u} \mbox{ ,}
\end{equation}
which scales as $n_{e,u}\propto R^{-2}$ in the strong shock limit, and
\begin{equation}\label{eq:PScaling}
P_{e,d}=\left(\frac{2\Gamma M^2}{\Gamma+1}-\frac{\Gamma-1}{\Gamma+1}\right) P_{e,u} \simeq \frac{5M^2-1}{4} P_{e,u} \mbox{ ,}
\end{equation}
which for a strong shock becomes $\propto v(R)^2 P_{e,u}\propto R^0$, interestingly implying a constant downstream pressure along the entire shock surface at any given time.
\subsection{X-ray signature}
By combining the edge pattern in Eq.~(\ref{eq:FBEdgeFull}), the flow profiles in Eqs.~(\ref{eq:PrimakoffScaling}--\ref{eq:LinearVScaling}), the jump conditions in Eqs.~(\ref{eq:NScaling}--\ref{eq:PScaling}), and the upstream distribution in Eqs.~(\ref{eq:nu}--\ref{eq:Tu}), we may now compute the expected X-ray signature of the FBs.
The properties of the resulting signal, as seen in projection from the Solar system, are shown in Figures \ref{fig:FBModel} and \ref{fig:FBModelProj}.
They depend on the shock and upstream parameters, calibrated for simplicity at the top of the FB; in particular, we use the Mach number $M(r=10\mbox{ kpc})=10M_{10}$ and electron number density $n_e(r=10\mbox{ kpc})=4\times 10^{-4} n_4 \mbox{ cm}^{-3}$.
For comparison with the \emph{ROSAT} data analyzed in \S\ref{sec:Data}--\ref{sec:Analysis}, we model the full \emph{ROSAT}, $(0.1\mbox{--}2.4)\mbox{ keV}$ band.
The emission coefficient integrated over this enegry range, computed using the MEKAL model
\citep{MeweEtAl85, MeweEtAl86, Kaastra92, LiedahlEtAl95} in XSPEC v.12.5 \citep{Arnaud96}, does not strongly depend on $\sim \mbox{ keV}$ temperature,
\begin{equation} \label{eq:FxMEKAL}
j_{X}\simeq 9\times 10^{-25}\frac{n_{e}^2 Z_{0.3}^{0.6}}{T_{keV}^{0.1}}\mbox{ erg}\mbox{ s}^{-1}\mbox{ cm}^{-3}\mbox{ sr}^{-1} \mbox{ ,}
\end{equation}
where $T_{keV}\equiv(k_B T_e/1\mbox{ keV})$ is the electron temperature, $Z_{0.3}\equiv Z/(0.3Z_\odot)$ is the metallicity, and the fit pertains to the $T_{keV}\in[0.1,1.5]$, $Z_{0.3}\in[0.3,3]$ range.
Neglecting for simplicity the temperature and metallicity dependencies, we integrate the approximate $j_X\simeq C_X n_e^2$ along the line of sight $l$,
\begin{align}
F_{X} & \simeq \int C_X n^2 \,dl \simeq C_X \int \left(\frac{r}{R}n_{e,d}\right)^2 \, dl \nonumber \\
& \simeq \frac{16C_X n_{e,10}^2}{(1+3M^{-2})^2} \int \frac{(10\mbox{ kpc})^4 r^2}{R^6} \, dl \mbox{ .}
\end{align}
Here, $R=R(\vect{r})$ is the shock radius along the ray emanating from the GC and passing through $\vect{r}$.
The X-ray-weighted temperature can similarly be computed,
\begin{align}
T_X & \simeq
\frac{C_X}{F_X} \int \left(\frac{r^2}{R^2}T_{e,d}\right) \left(\frac{r}{R}n_{e,d}\right)^2 \, dl \nonumber \\
& \simeq \frac{500 M_{10}^2 C_X n_{e,10}^2 T_h}{F_X} \int \frac{(10\mbox{ kpc})^2 r^4}{R^6} \, dl \nonumber \\
& \simeq \frac{31M_{10}^2 T_h}{(10\mbox{ kpc})^2} \, \frac{\int (r^4/R^6) \, dl}{\int (r^2/R^6) \, dl} \mbox{ ,}
\end{align}
where in the last two lines we approximated the shock as strong.
The resulting, projected X-ray structure, as seen by a putative observer in the Solar system, assumed to be $r_\odot=8.5\mbox{ kpc}$ from the GC, is shown in Figure \ref{fig:FBModel}.
In \S\ref{sec:RadialProfiles} we measure the \emph{ROSAT} flux as a function of the angular distance $\psi$ from the FB edge.
For consistency with \citetalias{KeshetGurwich16_Diffusion}, we take $\psi<0$ to designate regions inside the bubble.
To boost the signal and allow such a measurement, we stack data along wide sectors, in particular sectors defined by intermediate ($15{^\circ}<|b|<30{^\circ}$) or high ($|b|>30{^\circ}$) latitudes.
In order to compare these results with the model, we apply the same procedure to the modelled X-ray signature in Figure \ref{fig:FBModel}; the resulting profiles are shown in Figure \ref{fig:FBModelProj}.
Figures \ref{fig:FBModel} and \ref{fig:FBModelProj} show both $F_X$ (solid) contours and $T_X$ (dashed) contours.
\MyApJ{\begin{figure}[h]}
\MyMNRAS{\begin{figure}}
\PlotFigsA{
\epsfxsize=8.5cm \epsfbox{\myfig{FBXROSATProfsPrimakoff.eps}}
}
\caption{
Stacked X-ray signature of the FB model as a function of angular distance $\psi$ from the edge (marked by a dotted black line): flux $F_X$ in the \emph{ROSAT} band (solid blue curves, left axis) and X-ray weighted temperature (dashed red, right axis), shown for both the high latitude ($|b|>30{^\circ}$, thick curves) and intermediate latitude ($15{^\circ}<|b|<30{^\circ}$, thin curves) sectors.
\label{fig:FBModelProj}
}
\end{figure}
\subsection{Other model properties}
The total energy in the modelled FBs is estimated by integrating the ion bulk kinetic energy and the thermal energy, which we temporarily assume to be equilibrated between ions and electrons (see \S\ref{sec:Discussion} for a discussion of this assumption).
In the above model, this yields
\begin{equation}
E_{FB} \simeq 2.4 \times 10^{56} M_{10}^2n_4 T_{0.15} \mbox{ erg}
\end{equation}
from the two bubbles combined; $\sim 45\%$ of this energy is in the form of bulk kinetic energy.
We confirm that in our Primakoff-like model, the $\gamma$-ray signature is broadly consistent with Fermi observations.
It rises across several degrees from the edge inward, remains quite spatially flat, and shows no limb brightening, in agreement with the data.
This is demonstrated in Figure \ref{fig:FBModelProjGamma}, showing the profiles of the $\gamma$-ray flux in the high and intermediate latitude sectors.
The figure depicts both $j_\gamma \propto n_e^0$ and $j_\gamma \propto n_e^1$ $\gamma$-ray emissivity models (each with its own arbitrary units); the former corresponds to the strong diffusion limit.
Interestingly, the $\propto n_e^1$ model provides a better fit to the results of \citetalias{KeshetGurwich16_Diffusion}, in which the stacked, low-energy signal appears to be stronger at lower latitudes than it is in high latitudes.
\MyApJ{\begin{figure}[h]}
\MyMNRAS{\begin{figure}}
\PlotFigsA{
\epsfxsize=8.5cm \epsfbox{\myfig{FBProfsGammaPrimakoff.eps}}
}
\caption{
Stacked $\gamma$-ray signature of the FB model for emissivities $j_\gamma\propto n_e^0$ (green, dot-dashed curves) and $j_\gamma\propto n_e^1$ (cyan, dotted), in the high (thick) and intermediate (thin) latitude sectors. An arbitrary normalization is applied for each emissivity model.
\label{fig:FBModelProjGamma}
}
\end{figure}
In our model, quasar PDS 456 would show $\sim\{-50,+200\}M_{10}T_{0.15}^{1/2}\mbox{ km}\mbox{ s}^{-1}$ absorption line velocities in the Galactic standard of rest (GSR), due to the FBs.
For our fiducial parameters, these offsets are somewhat smaller than, namely are only $\sim\{1/4,2/3\}M_{10}T_{0.15}^{1/2}$ times, the $\{-190,+295\}\mbox{ km}\mbox{ s}^{-1}$ GSR line velocities inferred from observations \citep{FoxEtAl15}.
The observed line velocities, and importantly, their ratio, are not well-reproduced here, but these values are sensitive to the assumed linear velocity and the adopted edge pattern at low latitudes, which are not well constrained.
Indeed, a $2.5{^\circ}$ eastward shift in the position of the $b=10{^\circ}$ modelled east FB edge would reproduce the observed values for $M_{10}^2T_{0.15}=1$.
Our analysis can be readily generalized for different choices of the 3D FB edge, flow, and upstream models.
As an example, consider the case where the upstream density is constant, giving rise to a Sedov-Taylor-Von Neumann-like profile behind the shock.
Here, the mass shell is more compact, compressed against the shock.
The resulting X-ray and $\gamma$-ray profiles are shown in Figure \ref{fig:ST}, for a fixed upstream density $n_e=4\times 10^{-4}n_4\mbox{ cm}^{-3}$.
The projected profiles of X-rays and of $j_\gamma\propto n_e^1$ $\gamma$-rays show clear limb brightening.
As we show, such profiles are inconsistent with both \emph{ROSAT} and Fermi-LAT observations; the X-ray data thus favor the standard, $n_e\propto r^{-2}$ upstream profile of the Primakoff-like model.
\MyApJ{\begin{figure}[h]}
\MyMNRAS{\begin{figure}}
\PlotFigsA{
\epsfxsize=8.5cm \epsfbox{\myfig{FBXROSATProfsGammaST.eps}}
}
\caption{
Stacked X-ray and $\gamma$-ray signatures of the FB for an underlying Sedov-Taylor-Von Neumann-like flow.
Notations and symbols combine those of Figures \ref{fig:FBModelProj} and \ref{fig:FBModelProjGamma}.
\label{fig:ST}
}
\end{figure}
\section{Data preparation and analysis}
\label{sec:Data}
We use the \emph{ROSAT} all sky survey \citep[RASS;][]{SnowdenEtAl97}, with the Position Sensitive Proportional Counter (PSPC) of the X-ray telescope (XRT).
The provided\footnote{http://hea-www.harvard.edu/rosat/rsdc.html} PSPC maps were binned onto $12'\times12'$ pixels, well above the native $1'.8$ radius for $50\%$ energy containment.
Point sources were removed to a uniform source flux threshold for which their catalog is complete over $90\%$ of the sky; the corresponding pixels are masked from our analysis.
We use the four high energy bands of the survey, denoted R4--R7, spanning the energy range $0.4\mbox{--}2.0\mbox{ keV}$ with a considerable overlap, as detailed in Table \ref{tab:ROSATBands}.
The low energy bands, R1--R3, are found to be too noisy for our analysis.
Figure \ref{fig:EdgesROSAT6} illustrates the analysis using the R6 band.
The figure, spanning $160{^\circ}$ in latitude and $70{^\circ}$ in longitude, was retrieved from SkyView \citep{McGlynnEtAl98} in a rectangular (CAR) projection with a $16'$ latitude resolution, chosen to slightly exceed the native, binned map resolution.
A biconal, heart shaped signature reminiscent of the model in Figure \ref{fig:FBModel} is evident at the base of the FBs, at low latitudes $|b|\lesssim 15{^\circ}$ \citep{BlandHawthornAndCohen03}.
However, this emission is seen to extend to higher latitude, at least in the southern bubble, as we demonstrate by smoothing the map on large scales (for illustrative purposes only; no smoothing is used in the subsequent analysis).
For example, using an $8{^\circ}$ Gaussian filter (panels $c$ and $d$) shows that the signal (highlighted as long-dashed yellow contours in panel $d$) extends to $|b|\simeq 40{^\circ}$ latitudes.
The signal is less clear in the northern bubble, which is known to be more contaminated \citepalias[\emph{e.g.,} ][]{FermiBubbles14} due to higher levels of dust and gas \citep[\emph{e.g.,} ][]{NarayananSlatyer16}, especially near the northeastern Loop I feature; the signal may nevertheless be discernible in its northwestern part.
\MyApJ{\begin{table}[h]}
\MyMNRAS{\begin{table}}
\begin{center}
\caption{\label{tab:ROSATBands} \emph{ROSAT} energy bands used in the analysis.
}
\begin{tabular}{cc}
\hline
Band name & Energy range [keV; $10\%$ of peak response] \\
\hline
R4 & 0.44--1.01 \\
R5 & 0.56--1.21 \\
R6 & 0.73--1.56 \\
R7 & 1.05--2.04 \\
\hline
\end{tabular}
\MyApJ{\tablenotetext{1}{$10\%$ of peak response; keV units.}}
\end{center}
\end{table}
\MyApJ{\begin{figure*}[t]}
\MyMNRAS{\begin{figure*}}
\PlotFigsA{
\centerline{
\begin{overpic}[width=4.86cm]{\myfig{ROSAT6Raw.eps}}
\put (6,91) {\normalsize \textcolor{white}{(a)}}
\end{overpic}
\begin{overpic}[width=4.3cm, trim=0 -0.29cm 0 0]{\myfig{ROSAT6Contours.eps}}
\put (1,92) {\normalsize \textcolor{white}{(b)}}
\end{overpic}
\begin{overpic}[width=4.3cm, trim=0 -0.29cm 0 0]{\myfig{ROSAT6GaussFilt.eps}}
\put (1,92) {\normalsize \textcolor{white}{(c)}}
\end{overpic}
\begin{overpic}[width=5.05cm, trim=0 0.15cm 0 0]{\myfig{ROSAT6GaussFiltContoursMarkup.eps}}
\put (1,91) {\normalsize \textcolor{white}{(d)}}
\end{overpic}
}
}
\caption{
\emph{ROSAT} band R6 ($0.73\mbox{ keV}<E<1.56\mbox{ keV}$) image in Galactic coordinates, with a rectangular (CAR) projection and a cube-helix \citep{Green11_Cubehelix} colormap. Shown are both the raw map (panels $a$ and $b$; color scale: $10^{-6}\mbox{ counts s}^{-1}\mbox{ arcmin}^{-2}$) and the map smoothed with an $8{^\circ}$ Gaussian filter (panels $c$ and $d$; arbitrary color scale). The four edge contours (see Table \ref{tab:EdgeSummary}) are overlaid in panels $b$ and $d$, as extracted in \citetalias{KeshetGurwich16_Diffusion} based on the Fermi-LAT map with a $6{^\circ}$ gradient filter (solid blue) and a $4{^\circ}$ filter (dashed red), or traced by eye in \citetalias{KeshetGurwich16_Diffusion} (dot-dashed cyan) and in \citetalias{SuEtAl10} (dotted purple). Notice that the bipolar, heart shape structure inside the bubbles extends to high latitudes in the south bubble (long-dashed yellow curves in panel $d$), and perhaps also in the west part of the north bubble.
\label{fig:EdgesROSAT6}
}
\end{figure*}
To highlight the association of the bipolar X-ray features with the Fermi bubbles, Figure \ref{fig:EdgesROSAT6} also shows (in panels $b$ and $d$) the FB edges extracted from the $\gamma$-ray data in \citetalias{KeshetGurwich16_Diffusion} and \citetalias{SuEtAl10} (see Table \ref{tab:EdgeSummary}), superimposed on the X-ray map.
The \citetalias{KeshetGurwich16_Diffusion} edges labeled 1 and 2, which we use to extract the stacked, radial profiles in \S\ref{sec:RadialProfiles}, are based on gradient filters of coarse-grained ($6{^\circ}$) and refined ($4{^\circ}$) angular scales, applied to the Fermi data.
Also shown are the edges extracted by eye in \citetalias{KeshetGurwich16_Diffusion} (edge 3) and in \citetalias{SuEtAl10} (edge 4).
\MyApJ{\begin{table}[h]}
\MyMNRAS{\begin{table}}
\begin{center}
\caption{\label{tab:EdgeSummary} Different FB edge contour tracing.
}
\begin{tabular}{cll}
\hline
Edge & Tracing method & Reference \\
\hline
1 & Gradient filter on a $6{^\circ}$ scale & \citetalias{KeshetGurwich16_Diffusion} \\
2 & Gradient filter on a $4{^\circ}$ scale & \citetalias{KeshetGurwich16_Diffusion} \\
3 & Traced by eye & \citetalias{KeshetGurwich16_Diffusion} \\
4 & Traced by eye & \citetalias{SuEtAl10} \\
\hline
\end{tabular}
\end{center}
\end{table}
For the subsequent analysis, we convert the \emph{ROSAT}/PSPC count rates into physical flux units using the R4--R7 filters in the PIMMS \citep[v4.8d;][]{Mukai93} tool.
For simplicity, the photon flux in each band is converted into the corresponding energy flux in one and the same, $[0.1,2.4]\mbox{ keV}$, wide energy \emph{ROSAT} band, henceforth denoted as $F_X$.
Hence, one may expect the exact same $F_X$ profile to be extracted from the different energy bands, provided that they are dominated by the same signal, with a comparable weighted temperature, and, importantly, that the correct temperature is used in the conversion.
Unabsorbed fluxes are reported, computed using weighted column densities based on the \citet{DickeyLockman90} HI analysis.
We also compute the emission measure, $\mbox{EM}\equiv\int n_e^2 \,dl$, corresponding to $F_X$, with \citep[\emph{e.g.,} ][]{RybickiBook79}
\begin{eqnarray} \label{eq:EM}
F_X & \simeq & \frac{\alpha c\sigma_T}{\sqrt{6\pi^3}} \int dl \, Z_i^2 n_e^2 \left(\frac{m_e c^2}{k_B T}\right)^{1/2} \int d\epsilon \, \bar{g}_{ff} e^{-\epsilon/k_B T} \nonumber \\
& \simeq & 2.9 \times 10^{-16} \frac{\bar{g}_b}{T_{keV}^{1/2}} \mbox{EM} \mbox{ erg} \mbox{ s}^{-1}\mbox{ cm}^{-2}\mbox{ sr}^{-1}\mbox{ ,}
\end{eqnarray}
where $\alpha$ is the fine-structure constant, $\sigma_T$ is the Thompson cross section, $c$ is the speed of light, and $m_e$ is the electron mass.
Here, we neglected temperature and metallicity variations along the line of sight, took the cosmic value of the mean squared atomic number, $Z_i^2\simeq 1.2$, and defined $\bar{g}_b$ as the integral (in the first line of Eq.~\ref{eq:EM}) of the weighted Gaunt factor $\bar{g}_{ff}e^{-\epsilon/k_B T}$ over photon energy $\epsilon$. We use the Gaunt factor approximations \citep[\emph{e.g.,} ][]{DeWittDeWitt73}
\begin{equation} \label{eq:Gaunt}
\bar{g}_{ff}\simeq \begin{cases}
\sqrt{3k_B T/(\pi\epsilon)} & \mbox{for $\epsilon\gg k_B T$} \, ; \\
(\sqrt{3}/\pi)[\ln(4k_B T/\epsilon)-\gamma] & \mbox{for $\epsilon \ll k_B T$} \mbox{ ,}
\end{cases}
\end{equation}
where $\gamma$ is Euler's constant.
\section{Stacked X-ray profiles}
\label{sec:RadialProfiles}
Next, we measure the profile of the X-ray brightness as a function of a varying angular distance $\psi$ from the edge.
The resulting $F_X(\psi)$ profile can then be compared to the model in Figure \ref{fig:FBModelProj}, testing the presence of a shell and providing an estimate of its parameters, in particular the plasma density and temperature.
In order to pick up the weak, diffuse signal, we analyze wide sectors along the FB, and map the pixels onto $\Delta\psi=2{^\circ}$ wide bins according to their distance $\psi$ from the edge.
The results do not appreciably change for other resolutions, but the statistical fluctuations become prohibitively large for $\Delta\psi\lesssim1{^\circ}$.
\subsection{South, high latitude profile}
\label{sec:FirstSector}
Consider first the wide, east+west, high latitude sector in the southern bubble, defined by $b<-30{^\circ}$.
Its X-ray profile measured with respect to the coarse-grained edge 1 is presented in Figure \ref{fig:ProfileBgt30}.
With the above choice of $\Delta\psi$, each angular bin corresponds to a large solid angle, ranging from $\sim63$ square degrees in the innermost bin, to $\sim121$ square degrees in the bin lying just below the edge, to even larger solid angles outside the edge.
The error bars represent the $1\sigma$ statistical confidence levels of each bin, assuming a Poisson distribution.
They do not include the dispersion in the signal among the pixels within the bin, as this is affected by the spatial non-uniformity of the gas distribution, FB asymmetry, gas clumping, and other effects which are beyond the present scope; the smoothness of the resulting signal indicates that our averaging process is meaningful.
(Even the R7 bump around $\psi=3{^\circ}$ is resolved at smaller $\Delta\psi$.)
\MyApJ{\begin{figure}[t]}
\MyMNRAS{\begin{figure}}
\PlotFigsA{
\epsfxsize=8.5cm \epsfbox{\myfig{XR1ST04XR1CGBgt30.eps}}
}
\caption{
The X-ray flux in the \emph{ROSAT} $[0.1,2.4]\mbox{ keV}$ energy band, as a function of the angular distance $\psi$ from the FB coarse-grained edge 1, for the southern, $b<-30{^\circ}$, east+west sectors.
Negative $\psi$ values correspond to the inner part of the bubble, \emph{i.e.,} closer to the GC.
Plotted is the flux difference, $\Delta F_X$ (left axis), with respect to the bin just inside the edge.
The flux is computed based on the four energy bands R4--R7 (thick, long-dashed, red curve, to thin, solid, blue curve; higher energy bands shown with increasingly thinner lines, shorter dashing, and bluer hue), using the best fit temperature, $k_B T_X=0.4\mbox{ keV}$; see text and Figure \ref{fig:ProfileBgt30T}.
The corresponding emission measure difference ($\Delta\mbox{EM}$; right axis) is computed using Eqs.~(\ref{eq:EM}--\ref{eq:Gaunt}).
\label{fig:ProfileBgt30}
}
\end{figure}
\MyApJ{\begin{figure*}[t]}
\MyMNRAS{\begin{figure*}}
\PlotFigsA{
\centerline{
\begin{overpic}[width=5.8cm]{\myfig{XR1ST0.8XR1CGBgt30.eps}}
\put (82,46) {\scriptsize \textcolor{black}{(a)}}
\end{overpic}
\begin{overpic}[width=5.8cm]{\myfig{XR1ST02XR1CGBgt30.eps}}
\put (80,47) {\scriptsize \textcolor{black}{(b)}}
\end{overpic}
\begin{overpic}[width=5.8cm]{\myfig{XR1ST015ZoomXR1CGBgt30.eps}}
\put (84,49) {\scriptsize \textcolor{black}{(c)}}
\end{overpic}
}
}
\caption{
Temperature dependence of the $b<-30{^\circ}$ sector shown in Figure \ref{fig:ProfileBgt30} (with the same notations and symbols), shown for $k_B T_X=0.8\mbox{ keV}$ (panel \emph{a}), $0.2\mbox{ keV}$ (panel \emph{b}), and $0.15\mbox{ keV}$ (panel \emph{c}).
The (downstream; $\psi<0$) mismatch between the R4--R6 bands at these temperatures (in both panels \emph{a} and \emph{b}) indicates that $T_X$ lies between $0.2\mbox{ keV}$ and $0.8\mbox{ keV}$;
such $k_BT_X<\mbox{ keV}$ temperature is also consistent with the unclear signal in the high energy band R7; see Figure \ref{fig:ProfileBgt30} and discussion in the text.
Outside ($\psi>0$) the FB,
energy bands R4 and R5 can be matched with $k_B T_X\simeq 0.15\mbox{ keV}$ (panel \emph{c}), but this cannot be confirmed by R6--R7.
\label{fig:ProfileBgt30T}
}
\end{figure*}
The signal in Figure \ref{fig:ProfileBgt30} shows a clear break at the location of the FB edge, with $F_X$ becoming noticeably stronger inward, in resemblance of the expected signature of the supersonic shell in Figure \ref{fig:FBModelProj}.
(Interestingly, in this sector the signal also strengthens outwards; see discussion below.)
Thus, stacking along the edge allows us to measure the weak, extended signal.
The emission measure is $\mbox{EM}\lesssim 0.02\mbox{ cm}^{-6}\mbox{ pc}$ at $(-4){^\circ}<\psi<0$.
As expected, this is somewhat lower than the \emph{Suzaku} \citep{KataokaEtAl13} signal and sensitivity in this region.
Indeed, the small field of view in the \emph{Suzaku} observation \citep[$\sim0.9$ square degrees per CCD;][]{KataokaEtAl13} renders its results sensitive to the substantial variations in foreground and signal along the edge, which are averaged out in our method.
In Figure \ref{fig:ProfileBgt30}, similar signatures are seen in each of the three low energy bands, R4--R6, but the signal is less clear in the high energy band, R7, suggesting that the electron temperature is somewhat lower than $\sim1\mbox{ keV}$.
Indeed, the R4--R6 signals agree with each other for the $k_B T_X\simeq 0.4\mbox{ keV}$ conversion temperature used to prepare this figure.
More precisely, this is the temperature we find far ($\psi\lesssim-7{^\circ}$) inside the edge, where the signal is strong.
Closer to, yet still inside, the edge, the mismatch between bands R4--R6 and the clearer R7 signal suggest a higher temperature; see also Figure \ref{fig:ProfileBgt30T}.
Notice that the temperature is indeed expected to decline with increasing distance inside the edge, by a factor of $\sim 2$ by $\psi=-10{^\circ}$; see Figure \ref{fig:FBModelProj}.
Figure \ref{fig:ProfileBgt30T} shows the same sector and edge, but with different temperatures assumed in the flux conversion: (from left to right) $0.8$, $0.2$, and $0.15$ keV.
The mismatch here between bands R4--R6 indicates that $T_X$ is indeed lower than $0.8\mbox{ keV}$ (shown in panel \emph{a}), yet higher than $0.2\mbox{ keV}$ (shown in panel \emph{b}).
We conclude that in this sector, far ($\psi\simeq -10{^\circ}$) inside the edge, $T_X\simeq 0.4\mbox{ keV}$ to within a factor of $\sim2$.
It is difficult to measure the temperature outside the FB edge, where the signal is weaker and the gas is colder than optimal for our energy bands.
Figure \ref{fig:ProfileBgt30T} shows (in panels \emph{b} and \emph{c}) that bands R4 and R5 are well matched for $k_BT_X\simeq 0.1$--$0.2\mbox{ keV}$; this would place these bands on the exponential decline of the signal.
The rising profile of $F_X$ with increasing $\psi>0$ outside the edge in bands R4--R6 suggests some high energy upstream contamination; see discussion in \S\ref{sec:Analysis}.
\subsection{High and intermediate latitude profiles}
In the above method, we measure the stacked X-ray profiles in ten smaller sectors, at both east and west longitudes, both high and intermediate latitudes, and in both hemispheres.
We use the sectors defined in \citetalias{KeshetGurwich16_Diffusion}, as summarized in Table \ref{tab:EdgeSectors}, labeled by lowercase letters \emph{a} through \emph{e}, with or without a hemispheric designation \emph{N} (north) or \emph{S} (south).
Defining the $\psi=0$ contour according to FB edge 1, which in turn is based on the coarse-grained gradient filter, yields the results shown in Figure \ref{fig:RadialProfilesCG}.
Results for the higher resolution gradient filter (more sensitive to sharp transitions), edge 2, are presented in Figure \ref{fig:RadialProfilesHR}.
\MyApJ{\begin{table}[h!]}
\MyMNRAS{\begin{table}}
\begin{center}
\caption{\label{tab:EdgeSectors} Different sectors along each bubble's edge.
}
\begin{tabular}{ccc}
\hline
Sector & Longitude range & Latitude range \\
\hline
a & $-5{^\circ}<l<5{^\circ}$ & $|b|>30{^\circ}$ \\
b & $l>0{^\circ}$ & $|b|>30{^\circ}$ \\
c & $l<0{^\circ}$ & $|b|>30{^\circ}$ \\
d & $l>0{^\circ}$ & $15{^\circ}<|b|<30{^\circ}$ \\
e & $l<0{^\circ}$ & $15{^\circ}<|b|<30{^\circ}$ \\
\hline
\end{tabular}
\\
\raggedright Sectors are also denoted by the above notation along with the letter \emph{N} (for northern hemisphere) or \emph{S} (southern hemisphere).
\end{center}
\end{table}
\MyApJ{\begin{figure*}[h]}
\MyMNRAS{\begin{figure*}}
\PlotFigsA{
\centerline{
\begin{overpic}[width=5.7cm]{\myfig{XR1NT04XR1CGBgt30Lm5to5.eps}}
\put (50,44) {\scriptsize \textcolor{black}{(a)}}
\end{overpic}
}
\vspace{-2.0cm}
\centerline{
\begin{overpic}[width=5.7cm]{\myfig{XR1NT04XR1CGBgt30Lgt0.eps}}
\put (50,36) {\scriptsize \textcolor{black}{(b)}}
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\hspace{5.7cm}
\begin{overpic}[width=5.7cm]{\myfig{XR1NT04XR1CGBgt30Llt0.eps}}
\put (50,44) {\scriptsize \textcolor{black}{(c)}}
\end{overpic}
}
\vspace{0.3cm}
\centerline{
\begin{overpic}[width=5.7cm]{\myfig{XR1NT04XR1CGB15to30Lgt0.eps}}
\put (50,46) {\scriptsize \textcolor{black}{(d)}}
\end{overpic}
\begin{overpic}[width=5.7cm]{\myfig{XR1NT04XR1CGB15to30Llt0.eps}}
\put (50,48) {\scriptsize \textcolor{black}{(e)}}
\end{overpic}
}
\centerline{
\begin{overpic}[width=5.7cm]{\myfig{XR1ST04XR1CGB15to30Lgt0.eps}}
\put (50,46) {\scriptsize \textcolor{black}{(d)}}
\end{overpic}
\begin{overpic}[width=5.7cm]{\myfig{XR1ST04XR1CGB15to30Llt0.eps}}
\put (50,44) {\scriptsize \textcolor{black}{(e)}}
\end{overpic}
}
\centerline{
\begin{overpic}[width=5.7cm]{\myfig{XR1ST04XR1CGBgt30Lgt0.eps}}
\put (50,44) {\scriptsize \textcolor{black}{(b)}}
\end{overpic}
\hspace{5.7cm}
\begin{overpic}[width=5.7cm]{\myfig{XR1ST04XR1CGBgt30Llt0.eps}}
\put (50,44) {\scriptsize \textcolor{black}{(c)}}
\end{overpic}
}
\vspace{-2.0cm}
\centerline{
\begin{overpic}[width=5.7cm]{\myfig{XR1ST04XR1CGBgt30Lm5to5.eps}}
\put (50,44) {\scriptsize \textcolor{black}{(a)}}
\end{overpic}
}
}
\caption{
X-ray flux (and emission measure) difference as a function of angular distance $\psi$ from edge 1
in various sectors (arranged roughly according to their position on the sky; see Table \ref{tab:EdgeSectors}).
We use the conversion temperature $k_B T_X=0.4\mbox{ keV}$, which gives the best fit far below the edge, for all sectors.
Notations and symbols are identical to those used in Figure \ref{fig:ProfileBgt30}.
\label{fig:RadialProfilesCG}
}
\end{figure*}
These figures show the difference of $F_X$ and EM with respect to the FB edge, which can be taken as the first bin either below or above the putative, edge 1 or edge 2 position.
In most cases, we define the edge value according to the bin just below the putative edge, as in Figures \ref{fig:ProfileBgt30}--\ref{fig:ProfileBgt30T}, but some sectors (\emph{aN}, \emph{cN}, \emph{dS}, and for edge 2 also \emph{cS}) yield better results with the first bin above the edge.
For simplicity, we assume a constant temperature within each sector when converting the \emph{ROAST}/PSPC counts to energy flux; the more realistic, $\psi$-dependent temperature is beyond the scope of the present work.
In all sectors that show a signal inside the FBs, the best fit is obtained with $k_BT_X\simeq 0.4\mbox{ keV}$ (up to a factor of $\sim2$) far ($\psi\simeq -10{^\circ}$) from the edge.
There is evidence for higher temperatures closer to the edge, as discussed in \S\ref{sec:FirstSector} above, but here the statistical errors become large.
\begin{comment}
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\end{overpic}
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\caption{
X-ray flux and emission measure differences as a function of angular distance $\psi$ from edge 1 (which is based on a coarse-grained gradient filter applied to the Fermi data in \citetalias{KeshetGurwich16_Diffusion}) in various sectors (arranged according to their position on the sky; see Table \ref{tab:EdgeSectors}).
We use the temperature $k_B T_X=0.5\mbox{ keV}$, which gives the best fit far below the edge, for all sectors except \emph{a}, for which $k_B T_X=1.0\mbox{ keV}$ yields a better fit.
Fluxes are measured with respect to the first bin below the edge, except sectors \emph{aN}, \emph{cN} and \emph{dS}, for which the first bin above the edge gives a better fit.
Notations and symbols are identical to those used in Figure \ref{fig:ProfileBgt30}.
\label{fig:RadialProfilesCG}
}
\end{figure*}
\end{comment}
\MyApJ{\begin{figure*}[h]}
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\put (50,47) {\scriptsize \textcolor{black}{(b)}}
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\caption{
Same as Figure \ref{fig:RadialProfilesCG}, but for edge 2.
\label{fig:RadialProfilesHR}
}
\end{figure*}
\begin{comment}
\MyApJ{\begin{figure*}[h]}
\MyMNRAS{\begin{figure*}}
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\put (50,47) {\normalsize \textcolor{black}{(b)}}
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\put (50,47) {\normalsize \textcolor{black}{(a)}}
\end{overpic}
}
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\caption{
Same as Figure \ref{fig:RadialProfilesCG}, but for edge 2 (which is based on a higher resolution gradient filter applied to the Fermi data in \citetalias{KeshetGurwich16_Diffusion}). Here we use the first bin above the edge, instead of below it, also for sector \emph{cS}.
\label{fig:RadialProfilesHR}
}
\end{figure*}
\end{comment}
\section{Signal Analysis and modelling}
\label{sec:Analysis}
As Figures \ref{fig:RadialProfilesCG} and \ref{fig:RadialProfilesHR} show, the high latitude ($|b|>30{^\circ}$) signal in the southern bubble is identified in both the southeast (sector \emph{bS}) and the southwest (sector \emph{cS}), independently. It is also seen if we consider only the bubbles' axis, restricting the analysis to the narrow longitudinal range $-5{^\circ}<l<5{^\circ}$ (sector \emph{aS}).
These southern signals are seen when using both edges 1 and 2, with small variations as expected from the differences in the precise edge locations.
We conclude that the signal is robustly confirmed in the southern bubble.
The northern bubble is known to be more prone to confusion, especially near Loop I in the northeast.
Nevertheless, the high-latitude signal can be seen in the north bubble as well, in sectors \emph{aN} and \emph{cN}, albeit not in the northeast sector \emph{bN} which is adjacent to Loop I.
This result, and the similarity between the north and south signatures, especially when using edge 2, support the presence of an underlying X-ray shell associated with both bubbles at high latitudes.
At intermediate ($15{^\circ}<|b|<30{^\circ}$) latitudes, only the southeast sector (\emph{dS}) shows clear evidence for the signal, using both edges 1 and 2;
the signal in the adjacent sector \emph{eS} is marginal.
These stacked profiles are considerably more noisy than at high latitudes; no signal is seen in the north. This is to be expected, due to confusion with the abundant X-ray structure near the Galactic plane, and the difficulty of tracing the $\gamma$-ray edges at low-latitude; both effects are more severe in the northern hemisphere.
As mentioned in \S\ref{sec:RadialProfiles}, all high-latitude sectors that show a signal are consistent with $k_BT_X\simeq 0.4\mbox{ keV}$ far ($\psi\simeq -10{^\circ}$) inside the edge.
We cannot confirm a latitude dependence of $T_X$, but this is not surprising considering the noisy signal at intermediate latitudes and the oversimplified, $\psi$-independent conversion temperature we use in each sector.
As one approaches the FB edge from below, a fairly sharp drop of $F_X$, spanning a few degrees, can be seen in sectors \emph{aS}, \emph{cS}, \emph{dS}, \emph{aN}, and with edge 2, also \emph{cN}.
But these drops are not as sharp and pronounced as in the model Figure \ref{fig:FBModelProj}, and no localized drops are seen in other sectors, in particular \emph{bS}, which is further discussed below.
The excessive smoothness of the measured profiles are likely a result of inaccuracies in tracing the edge position and orientation, along with variations in the actual gas profiles, projection effects, and noise.
The $b<-30{^\circ}$ signal in Figure \ref{fig:ProfileBgt30} monotonically rises with increasing $(-\psi)$ inward towards the GC.
This resembles the anticipated (\emph{cf.} Figure \ref{fig:FBModelProj}) signature at intermediate ($15{^\circ}<b<30{^\circ}$) latitudes, but is unlike the flattening of the modeled signal at high latitudes, which is seen in Figure \ref{fig:FBModelProj} to be nearly constant for $-10{^\circ}<\psi<-2{^\circ}$.
Such unexpected, non-flat behavior is seen in Figure \ref{fig:RadialProfilesCG} to be dominated by sector \emph{bS}; a similar trend is also seen in sector \emph{cN}.
In contrast, the expected flattening of the profile is seen in sectors \emph{aS}, \emph{cS}, and \emph{aN}.
Moreover, this flattening is more pronounced for these sectors when using edge 2 (see Figure \ref{fig:RadialProfilesHR}); here, sector \emph{cN} also shows a clear flattening.
We conclude that the detailed $\psi$-profile is broadly consistent with the model, but cannot be robustly inferred from the present analysis, as it is somewhat sensitive to the method of edge tracing, and may vary across the FBs.
This somewhat diminishes our ability to distinguish between different (presently over-simplified) models for the gas distribution.
Another difference between the measured (high-latitude, southern) profile in Figure \ref{fig:ProfileBgt30} and the model Figure \ref{fig:FBModelProj} pertains to the upstream: the measured signal strengthens away from the edge also with increasing positive $\psi$, outside the FB, instead of being flat or slightly decreasing due to the expected diminishing Galactic emission away from the plane.
This again is seen to be dominated by sector \emph{bS}, although \emph{cS} contributes here as well.
Again, a more consistent, flatter upstream profile is seen in sectors \emph{aS} and \emph{aN}, as well as \emph{cN} and \emph{dS}, and especially when using edge 2.
The unusual profiles both inside and outside the southeast edge \emph{bS} suggest that the upstream gas here differs from other sectors and from our model.
The detection of upstream structure in energy bands R4--R6 and even R7 suggests some high energy upstream contamination in this sector.
In spite of these caveats, we may carry out an approximate, quantitative comparison of the measurements in Figures \ref{fig:ProfileBgt30}, \ref{fig:RadialProfilesCG} and \ref{fig:RadialProfilesHR} with the model Figure \ref{fig:FBModelProj}.
The high latitude profiles reach a flux $F_X(\psi\simeq -10{^\circ})\simeq 5\times 10^{-8}\mbox{ erg}\mbox{ s}^{-1}\mbox{ cm}^{-2}$, to within a factor of $\sim2$, whereas the intermediate latitude sector \emph{dS} reaches a flux as high as three times this value.
The corresponding, normalized $n_4^2F_X$ values in the model are comparable to these values, so matching the observations with the model confirms the expected upstream densities.
We conservatively take the discrepancy factor in the flux to be $D_F\simeq 1$, with an uncertainty factor $\sim 3$, such that the upstream electron number density just outside the top of the bubbles is inferred to be roughly
\begin{equation}\label{eq:nDiscrepancy}
n_e(r=10\mbox{ kpc}) \simeq 4_{-2}^{+4}\times 10^{-4} \left(1D_F\right)^{1/2}\mbox{ cm}^{-3} \mbox{ .}
\end{equation}
Note that the error bars here and in Eq.~(\ref{eq:TDiscrepancy}) below reflect the variations in the measured and modeled signals, and are not statistical.
In the model, the normalized X-ray temperatures at $\psi=-10{^\circ}$ are approximately $k_BT_X\simeq 1.8T_{0.15}M_{10}^2\mbox{ keV}$ at high latitudes, and $\sim 0.7\mbox{ keV}$ at intermediate latitudes.
Only the high latitude signal temperature is adequately measured (up to a factor of 2), as $k_BT_X\simeq 0.4\mbox{ keV}$.
The implied $T_X$ discrepancy is a factor of $D_T\simeq1/4$, with an uncertainty factor $\sim 2$, so matching the model crudely yields
\begin{equation} \label{eq:TDiscrepancy}
M_{10}^2 k_B T_u \simeq 0.04_{-0.02}^{+0.04}\left(4D_T\right)\mbox{ keV} \mbox{ .}
\end{equation}
It is difficult to measure the upstream temperature with bands R4--R7, as illustrated by Figure \ref{fig:ProfileBgt30T}, but temperatures higher than $0.5\mbox{ keV}$ can be excluded.
For a high-latitude temperature $k_BT_u\simeq 0.3\mbox{ keV}$, as found by \emph{Suzaku} \citep{KataokaEtAl13}, the shock Mach number becomes $M\simeq 3.6$, up to an uncertainty factor of $\sim50\%$.
However, lower estimates of the upstream temperature \citep[$k_BT_u\simeq 0.2\mbox{ or }0.15\mbox{ keV}$ according to][and $k_BT_u\simeq0.15\mbox{ keV}$ suggested by Figure \ref{fig:ProfileBgt30T}]{MillerBregman13, MillerBregman16} would imply a stronger, $M\simeq 5$ shock. We conclude that $M\simeq 4$, to within a systematic uncertainty of $\sim2$.
\section{Summary and Discussion}
\label{sec:Discussion}
We analyze the \emph{ROSAT} all sky survey in search of the faint, high-latitude X-ray counterpart of the FB $\gamma$-ray signal.
First, we present a semi-analytic model that reproduces the $\gamma$-ray and low-latitude X-ray signatures of the FBs (see Figures \ref{fig:FBModel} and \ref{fig:FBModelProjGamma}), as well as other constraints, such as the strong shock inferred from microwave and $\gamma$-ray observations, and the absorption line velocities seen towards quasar PDS 456.
This model is then used to compute the signal expected from stacking the \emph{ROSAT} data along the FB edge (Figure \ref{fig:FBModelProj}).
Next, we use the FB edges identified previously \citepalias[by applying gradient filters to the Fermi-LAT map;][see Table \ref{tab:EdgeSummary} and Figure \ref{fig:EdgesROSAT6}]{KeshetGurwich16_Diffusion}, and stack the \emph{ROSAT} data at varying distance from the edge, in various sectors (see Table \ref{tab:EdgeSectors}) along the FBs (Figures \ref{fig:ProfileBgt30}--\ref{fig:RadialProfilesHR}).
The resulting high-latitude signal shows structure clearly associated with the FB edge, in all sectors in the southern hemisphere.
The signal can also be seen in the northern hemisphere, but only in the northwest sectors, far from Loop-I.
Owing to the stacking method, averaging the data over bins of several $10$ square degrees, the statistical errors are rendered manageably small.
Systematic errors due to precise edge localization, projection effects, and competing structure, are more important, but they too are largely washed out in the averaging process.
The similar stacked \emph{ROSAT} signature seen in the different sectors (both north and south, and in the south bubble at both east and west longitudes, and at high and even intermediate latitudes), its approximate agreement with the model predictions, and its robustness against small variations in the edge location and in the analysis parameters (resolution, emission model, absorption model; see below), support a high-significance detection.
The distinguishing characteristic of the signal is the high X-ray brightness found several degrees inside the FBs, declining towards, and dropping as one crosses outside, the FB edge.
This conclusively shows that the FBs are a forward, and not a reverse, shock.
The FBs must therefore arise from a rapid release of energy near the GC, ruling out competing wind or other slow energy release models.
Our results are consistent with the \emph{Suzaku} data \citep{KataokaEtAl13}, showing a similar effect at least in the cleaner, southern hemisphere.
Another important feature of the signal is the $\sim 0.4\mbox{ keV}$ temperature we infer for the emitting electrons far ($\psi\simeq-10{^\circ}$) inside the edge.
This is evident both from the weak signal in the high \emph{ROSAT} energy band 7 in most sectors, and from fitting the lower energy bands (see Table \ref{tab:ROSATBands}) to the stacked signal.
There is some evidence for a higher temperature closer to the edge and in the highest latitudes (see for example Figures \ref{fig:ProfileBgt30} and \ref{fig:ProfileBgt30T}), but here the data is more noisy.
A radially-increasing temperature inside the FBs, dropping as one crosses outside the edge, is indeed consistent with our forward shock model (see Figures \ref{fig:FBModel} and \ref{fig:FBModelProj}).
The inferred Mach number at the top of the FBs, assuming a thermal equilibrium between shocked electrons and ions, is $M\simeq 4$, with an uncertainty of $\sim2$.
Comparing the stacked results with the projected model in Figure \ref{fig:FBModelProj}, we find that the observed flux and temperature are fractions $D_F\simeq 1$ and $D_T\simeq 1/4$ of their expected values (for our putative model parameters), respectively, with uncertainty factors of $\sim 3$ and $\sim 2$.
This implies similar upstream densities (see Eq.~\ref{eq:nDiscrepancy}) but somewhat lower Mach numbers (Eq.~\ref{eq:TDiscrepancy}) than our fiducial values.
Accordingly calibrating our model, it corresponds to a total energy in (both) the FBs of
\begin{align}
E_{FB} & \simeq 2.4 \times 10^{56} M_{10}^2n_4 T_{0.15} \mbox{ erg} \\
& \simeq 6 \times 10^{55} (1D_F)^{1/2} (4D_T) \mbox{ erg} \mbox{ ,} \nonumber
\end{align}
released in a rapid event that took place
\begin{equation}
t \simeq 3.3 M_{10}^{-1}T_{0.15}^{-1/2} \mbox{ Myr} \simeq 6.6 (4D_T)^{-1/2} \mbox{ Myr}
\end{equation}
ago, near \citepalias[within $\sim1{^\circ}$;][]{KeshetGurwich16_Diffusion} the GC.
These model results should be corrected for the larger extent of the FBs to the west, indicating a higher, $E_{FB}\simeq 10^{56}\mbox{ erg}$ energy.
Importantly, the relatively low $T_X$ we infer is at some tension with the line of sight velocities towards PDS 456, observed to be 3--8 times larger than implied by the calibrated model.
To reproduce these velocities, the model would require a higher gas temperature, and thus would imply younger, more energetic FBs.
A possible resolution of this tension is shock-heating being stronger for ions than it is for electrons.
In particular, $T_i/T_e\simeq 10$ would reconcile the X-ray data with the observed line-of-sight velocities.
This would imply a very strong, Mach $\gtrsim 10$ shock at the top of young, $\lesssim 3\mbox{ Myr}$ FBs, containing a total energy $\sim10^{57}\mbox{ erg}$.
Note that the ion--electron equilibration time would then exceed the age of the bubbles.
The model calibration is based on the clear signals seen several degrees to $10{^\circ}$ inside the edge; the uncertainty in the radial dependence of the signal dominates our large systematic errors.
Other systematic uncertainties arise from the simplifying assumptions underlying the X-ray analysis (see \S\ref{sec:Data}); in particular approximating the temperature in each sector as fixed.
We tested our results by varying the analysis, for example by replacing the modelled \emph{ROSAT} filters by top-hat filters, and by replacing the absorption column densities by a fixed mean value; the results change within the systematic errors.
Additional systematic uncertainties, not included here, arise from our oversimplified model: we generalized the Primakoff-like spherical kinematics to a bipolar flow, and mostly neglected deviations from axisymmetry (except for a $\sim30\%$ correction to the overall energy budget).
The stacked \emph{ROSAT} signal, like the Fermi-LAT signal, shows no evidence for limb brightening. This indicates that the upstream density declines rapidly with radius; our upstream $n_e\propto r^{-2}$ model (Figures \ref{fig:FBModel}--\ref{fig:FBModelProjGamma}) fits the data much better than an upstream uniform, $n_e\propto r^0$ model (Figure \ref{fig:ST}).
In some sectors, the detailed agreement between model and data is quite good, including the monotonic inward strengthening of the signal at low latitudes vs. the flattening of the signal at high latitudes.
However, this is not observed in all sectors, and is seen (compare Figures \ref{fig:RadialProfilesCG} and \ref{fig:RadialProfilesHR}) to somewhat depend on the precise edge localization.
Accurately inferring the gas distribution underlying the FBs thus requires a more careful tracing of the edge, including deviations from axisymmetry and the consequent projection effects, and cleaning some of the noise, in particular a possible high-energy contaminant upstream of the southeast sector.
Our results constrain some fundamental aspects of the FB phenomenon.
First, the strong shock we deduce is consistent with the spectrum inferred from the microwave haze and from the absence of strong variations in the $\gamma$-ray spectrum along the edge \citepalias{KeshetGurwich16_Diffusion}.
This supports the interpretation of the haze and the $\gamma$-rays as arising from CREs, Fermi-accelerated by the shock.
The very low density we infer rules out hadronic models for the $\gamma$-ray signal, providing strong support for the competing, leptonic models (for a discussion, see \PaperTwo).
While the X-ray signal removes some of the degeneracies in the model, it does not by itself unequivocally prove that the FBs lie at a Galactic distance; the emission measure is quite low \citep[comparable and somewhat lower than reported by][]{KataokaEtAl13}, due to the high latitude.
\MyApJ{\acknowledgements}
\MyMNRAS{\section*{Acknowledgements}}
We thank Y. Lyubarski, R. Crocker, and Y. Naor for helpful discussions.
This research (grant No. 504/14) was supported by the ISF within the ISF-UGC joint research program, and by the GIF grant I-1362-303.7/2016, and received funding from the IAEC-UPBC joint research foundation grant 257.
We acknowledge the use of NASA's SkyView facility (http://skyview.gsfc.nasa.gov) located at NASA Goddard Space Flight Center.
|
2,869,038,154,807 | arxiv | \section{Introduction}
Anisotropic superhydrophobic (SH) surfaces have raised a considerable interest over the recent years. Such surfaces in the Cassie state, i.e., where the texture is filled with gas, can induce exceptional lubricating properties~\cite{bocquet2007,rothstein.jp:2010,vinogradova.oi:2012} and generate secondary flows transverse to the direction of the applied pressure gradient~\cite{feuillebois.f:2010b,schmieschek.s:2012,zhou.j:2012}. These are important for a variety of applications that involve a manipulation of liquids at the small scale and can be used to separate particles~\cite{LabChip} and enhance their mixing rate~\cite{ou.j:2007,nizkaya.tv:2017} in microfluidic devices.
During past decade the quantitative understanding of liquid flow past SH anisotropic surfaces was significantly
expanded. However, many fundamental issues still remain unresolved.
To quantify the drag reduction and transverse hydrodynamic phenomena
associated with SH surfaces with given area of gas and solid fractions it is convenient to construct the effective slip boundary condition applied at the imaginary homogeneous surface~\cite{vinogradova.oi:2011,Kamrin_etal:2010}, which mimics the real one and is generally a tensor~\cite{Bazant08}. Once eigenvalues of the slip-length tensor, which depend on the local slip lengths at the solid and gas areas, are determined, they can be used to solve various hydrodynamic problems. To calculate these eigenvalues, SH surface is usually modeled as a perfectly smooth with patterns of local boundary conditions at solid and gas sectors. It is widely accepted that
one can safely impose no-slip at the solid area, i.e., neglect slippage of liquid past smooth solid hydrophobic areas,
which is justified provided the nanometric slip is small
compared to parameters of the texture~\cite{joly.l:2006,vinogradova:03,charlaix.e:2005,vinogradova.oi:2009}. For gas sectors of SH surfaces the situation is much less clear. Prior work often applied shear-free boundary conditions at the flat gas areas~\cite{philip.jr:1972,priezjev.nv:2005,lauga.e:2003}. In this idealized description both a meniscus curvature and a viscous dissipation in the gas phase, which could affect the local slippage, are fully neglected.
Several groups have recently studied the effect of a meniscus curvature on the friction properties of SH surfaces~\cite{lauga2009,sbragaglia.m:2007,harting.j:2008,karatay.e:2013,crowdy.dg:2017}. Most of these studies neglected a viscous dissipation in the gas, by focussing on the connection of the meniscus protrusion angle and effective slip length (but note that no attempts have been made to connect the meniscus curvature and a local slip at the gas area). It has been generally concluded that if the protrusion angle is $\pm \pi/6$ or smaller the effective slip of the SH surface does not differ significantly from expected in the case of the flat interface, so that the model of a flat meniscus can always serve as a decent first-order approximation.
There is some literature describing attempts to provide a satisfactory theory of a local slip length, which take into account a viscous dissipation in the gas phase (and as far as we know, all these studies have modeled the liquid-gas interface as flat). We mention below what we believe are the most relevant contributions. To account for a dissipation in gas it is necessary to solve Stokes equations both for the liquid and for the gas phases (Fig.~\ref{fig_sketch}(a)), by imposing boundary conditions
\begin{equation}
z=0: \; \mathbf{u}= \mathbf{u}_g, \; \mu \dfrac{\partial \mathbf{u}_\tau}{\partial z}=\mu_g\dfrac{\partial \mathbf{u}_{g\tau}}{\partial z},
\label{BCcon}
\end{equation}
where $\mathbf{u}$ and $\mu $ are the velocity and the dynamic viscosity of the liquid, and $\mathbf{u}_g$ and $\mu _g$ are those of the gas, $\mathbf{u}_\tau=(u_x,u_y)$ is the tangential velocity. This problem has been resolved numerically for rectangular grooves~\cite{maynes2007,ng:2010}. It is however advantageous to replace the two-phase approach, by a single-phase problem with spatially dependent partial slip boundary conditions~\cite{cottin:2004,belyaev.av:2010a,bocquet2007}. For unidirectional (1D) surfaces they are normally imposed as
\begin{equation}
z=0: \; \mathbf{u}_\tau-b(y)\dfrac{\partial \mathbf{u}_\tau}{\partial z}=0,
\label{BCslip}
\end{equation}
where $b(y)$ is the local scalar slip length, which is varying in one direction only.
That the gas flow can be indeed excluded from the analysis being equivalently replaced by $b(y)$ is by no means obvious. Early work has suggested that the effect of gas-filled cavities is equivalent to the introduction of a slip length, proportional to their depth if shallow and to their spacing if deep~\cite{hocking.lm:1976}. To model the trapped gas effects theoretically a semi-analytical method, which predicted a non-uniform local slip length distribution across the liquid-gas interface, has been proposed~\cite{shoenecker.c:2014,shoenecker.c:2013}. During a last few years several authors have discussed that the local slip depends on the flow direction~\cite{shoenecker.c:2013,nizkaya.tv:2013,nizkaya2014gas}. Although they did not fully recognize it, their results are equivalent to a tensorial generalization of Eq.(\ref{BCslip})
\begin{equation}
z=0: \; \mathbf{u}_\tau-\mathbf{b}\dfrac{\partial \mathbf{u}_\tau}{\partial z}=0,
\label{BCslip_tensor}
\end{equation}
where $\mathbf{b}=b\{i,j\}$ is the second-rank local slip length tensor, which is represented by
symmetric, positive definite $2\times 2$ matrix diagonalized by a rotation with respect to the alignment of the SH grooves.
Recent work has elucidated a mechanism which transplants the flow in the gas to a local slip boundary condition at the liquid-gas interface~\cite{nizkaya2014gas}. This study has concluded that the non-uniform local slip length of a shallow texture is defined by the viscosity contrast and local thickness of a gas cavity, similarly to infinite systems~\cite{vinogradova.oi:1995a,miksis.mj:1994}. In contrast, the local slip length of a deep texture has been shown to be fully controlled by the dissipation at the edge of the groove, {i.e.} the point where three phases meet, but not by the texture depth as has sometimes been invoked for explaining the extreme local slip.
These results led to simple formulas describing liquid slippage at the trapped gas interface of 1D grooves of width $\delta$ and constant depth $e^*$~\cite{nizkaya2014gas}
\begin{equation}\label{bgas}
b_{\|,\perp} \simeq \dfrac{\mu }{\mu_g} \delta \beta_{\|,\perp},
\end{equation}
where $b_{\|,\perp}$ are eigenvalues of the slip length tensor, $\mathbf{b}$, and $\beta_{\|,\perp}$ are eigenvalues of the tensorial slip coefficient, $\boldsymbol{\beta }$. The latter become linear in $e^*/\delta$, when $e^*/\delta$ is small and saturate at large $e^*/\delta$.
The validity of this ansatz for rectangular grooves has been confirmed numerically~\cite{nizkaya2014gas} and, although indirectly, experimentally~\cite{nizkaya.tv:2016}.
Previous investigations have addressed the question of local slip at the gas areas of rectangular grooves with a constant depth only. We are unaware of any previous work that has quantified liquid slippage at gas areas of more general 1D SH surfaces. In this paper, we explore grooves with beveled edges and a non-uniform depth, which varies with $y$ and depends on the bevel angle, $\pi/2 - \alpha$ (see Fig.~\ref{fig_sketch}(b)). An obvious practical advantage of such a relief is that the manufactured grooves become mechanically more stable against bending compared to rectangular ones. This is especially important for dilute textures of large $\delta$, which induce larger slip. It is therefore very timely to understand important consequences of a bevel angle for a generation of liquid slippage at the gas areas of such grooves. Here we present theoretical arguments, which allow one to relate $b_{\|,\perp}$ to $e(y, \alpha)$. Our results show that
$b_{\|,\perp}$ are not really sensitive to a bevel angle when SH grooves are shallow and weakly slipping, but the large local slip at deep SH grooves is controlled by their width and bevel angle only. These two parameters could be used to tune the large slip at the gas areas of any grooved SH surface and constrain its attainable upper value.
Our paper is organized as follows. In Sec.~II we describe our model and justify the choice of model surfaces. Sec.~III gives a brief summary of our theoretical results for $b_{\|,\perp}$ obtained in some limiting situations. In Sec.~IV we describe a numerical method developed here to compute local slip length profiles. To solve numerically the two-phase hydrodynamic problem we consider separately flows in liquid and gas phases, which is in turn done by using different computational techniques. Our results are discussed in Sec.~V, and we conclude in Sec.~VI. Details of our asymptotic analysis can be found in Appendix~\ref{sec_app_edge}.
\begin{figure}[tb]
\includegraphics[width=0.75\columnwidth]{Nizkaya_Fig1.eps}
\caption{(a)~Sketch of the flow past a 1D superhydrophobic surface. (b)~Sketch of trapezoidal (solid curve) and arc-shaped (dashed curve) grooves of width $\delta$, maximal depth $e^{\ast}$, and bevel angle $\pi/2-\alpha$.}
\label{fig_sketch}
\end{figure}
\section{Model}
We consider creeping flow past 1D SH textures of period $L$ and gas area fraction $\phi$. The coordinate axis $x$ is
parallel to the grooves; the cross-plane coordinates are denoted by $y$ and $z$. The width of the groove is denoted as $\delta=L \phi$. The bevel angle of the grooves is $\pi/2 - \alpha$, where an angle $\alpha\leq \pi/2$ is defined relative to the vertical, and the depth of the grooves, $e(y, \alpha)$, is varying in only one direction (see Fig.~\ref{fig_sketch}(b)). Since we consider SH textures with air
trapped in the grooves in a contact with water, the ratio of liquid and gas viscosities is typically $\mu / \mu_g \simeq 50$, which is much larger than unity. Our results apply to a situation where the capillary and Reynolds numbers are sufficiently small, so that the liquid-gas interface does not deform, but not to the opposite case, where significant deformation of this interface is expected~\cite{seo.j:2015}.
\begin{figure}[tb]
\includegraphics[width=0.375\columnwidth]{Nizkaya_Fig2.eps}
\caption{Sketch of the contact angle, ``measured'' taking the horizontal as a reference, before, upon, and after the
corner.}
\label{fig_cassie}
\end{figure}
The texture material is characterized by the (unique) Young angle $\Theta$ (above $\pi/2$) measured taking the horizontal as a reference. Displacing a contact line to the sharp edge of a bevel angle $\pi/2 - \alpha$ we would observe an apparent variability of the contact angle from $\Theta$ to $\pi/2 - \alpha + \Theta$ since the line becomes pinned (see Fig.~\ref{fig_cassie}). Using simple geometry one can then conclude that in the Cassie state permitted protrusion angles of a meniscus are confined between $\pi/2 + \alpha - \Theta$ and $\pi - \Theta$, so that one of these (many) possible protrusion angles, which would be observed in practice, should be determined by a pressure drop between the liquid and the gas phases. Since positive protrusion angles (convex meniscus) are expected only when an external pressure is applied to the gas phase~\cite{karatay.e:2013}, we can exclude this artificial for SH surfaces case from our consideration. Now simple estimates suggest that for $\Theta = 2 \pi /3$, typical for SH texture materials, and when $\alpha = 0$, bounds, which constrain attainable protrusion angles are $0$ (flat meniscus) and $- \pi /6$ (concave meniscus). Tighter bounds for finite $\alpha$ further constrain the attainable protrusion angle. These angles are too small to significantly reduce the effective slip~\cite{lauga2009,karatay.e:2013}. Therefore, to highlight the effect of viscous dissipation on the slippage in simpler terms we here assume that liquid-gas interface to be flat, which implies that pressure in gas is equal to that in liquid. Such a situation occurs when trapped by texture gas is in contact with the atmosphere~\cite{nizkaya.tv:2016}.
Our aim is to investigate how the local depth $e(y,\alpha)/\delta$ modifies the slippage of liquid past gas areas. We will provide some general theoretical arguments and results valid for any shape of the SH grooves, and also some simulation results for representative SH grooves. As an initial illustration of our theoretical and computational approach we will consider a trapezoidal surface, where $\alpha$ and the maximal depth, $e^*$, can be varied independently and in the very large range
\begin{equation}
\label{eq:profile}
e(y, \alpha) =
\begin{cases}
y \cot \alpha, & y \le e^* \tan \alpha \\
e^*, & e^* \tan \alpha < y \le \delta - e^* \tan \alpha \\
(\delta - y) \cot \alpha, & \delta - e^* \tan \alpha < y
\end{cases}
\end{equation}
We should like to mention that motivated by a recent experiment~\cite{choi.ch:2006} there have been already attempts to determine an effective slip past trapezoidal SH grooves~\cite{Zhou_etal:2013}, but that work has simply assumed the trapezoidal shape of a local scalar slip and no attempt has been made to properly connect it with viscous dissipation in the confined gas.
Another type of SH surfaces we explore here are grooves bounded by arcs of circles of radii $\displaystyle \delta/(2 \cos \alpha)$. These arc-shaped grooves do not have areas of a constant depth, and the $e(y, \alpha)$ profile does not contain sectors where dependence on $\alpha$ and $\delta$ disappears
\begin{equation}
e(y, \alpha)=\frac{\delta}{2}\left(\sqrt{\frac{1}{\cos^2\alpha}-\left(\frac{2y}{\delta}-1\right)^2}-\tan\alpha\right).
\label{eq:arc_groove}
\end{equation}
Their maximal depth, $e^{\ast}$, is attained at the midplane, $y/\delta=0.5$, and is given by
\begin{equation}\label{earc}
e^{\ast} = \frac{\delta (1 - \sin \alpha)}{2 \cos \alpha}
\end{equation}
Eq.(\ref{earc}) shows that arc-shaped grooves can never be really deep since $ e^{\ast}/\delta$ cannot exceed $0.5$. It will be therefore instructive to compare their local slip with that of trapezoidal textures of the same $\delta$ and $e^{\ast}$.
\section{Theory}
Our aim is to calculate the eigenvalues of the local slip length tensor, $\mathbf{b}$, which can be found from the solution of the two-phase problem for the longitudinal (fastest) and
transverse (slowest) flow directions by imposing conditions, Eq.(\ref{BCcon})
\begin{equation}
b_{\parallel}=\left.\dfrac{u_{x}}{\partial u_{x}/ \partial z}\right|_{z=0} \equiv \dfrac{\mu }{\mu_g}\left.\dfrac{u_{g,x}}{\partial u_{g,x}/ \partial z}\right|_{z=0},
\label{bpadef}
\end{equation}
\begin{equation}
b_{\perp}=\left.\dfrac{u_{y}}{\partial u_{y}/ \partial z}\right|_{z=0} \equiv \dfrac{\mu }{\mu_g}\left.\dfrac{u_{g,y}}{\delta \partial u_{g,y}/ \partial z}\right|_{z=0}.
\label{bpedef}
\end{equation}
However, since all properties of $\boldsymbol{\beta }$ are inherited by the local slip-length tensor, $\mathbf{b}$, below we will focus more on calculations of scalar non-uniform eigenvalues, $\beta_{\parallel,\perp}$~\cite{note3}.
In the general case they can be calculated only numerically. However, for some limiting cases explicit expressions can be obtained.
The solution for local slip lengths can be found analytically close to the edge of the grooves, $y/\delta\ll 1$, and the details of our analysis are given in Appendix~\ref{sec_app_edge}. Here we highlight only the main results. Our theory predicts that
in the vicinity of groove edges, the eigenvalues of local slip length of any 1D texture augment from zero as
\begin{equation}
b_{\parallel} \simeq \frac{\mu }{\mu_g}\frac{2y}{\tan(\pi/4+\alpha/2)},\hspace{1cm} b_{\perp} \simeq \frac{b_{\parallel}}{4}.
\label{bxAsympt0}
\end{equation}%
In what follows that at small $y/\delta$ the slip coefficient grows as
\begin{equation}
\beta_{\parallel,\perp}\simeq \beta'_{\parallel,\perp} \dfrac{y}{\delta},
\label{angles}
\end{equation}
by
having slopes
\begin{equation}
\beta'_{\parallel} \simeq \dfrac{2}{\tan(\pi/4+\alpha/2)},\hspace{1cm}
\beta'_{\perp} \simeq \dfrac{1}{2\tan(\pi/4+\alpha/2)}.
\label{angles_prim}
\end{equation}
Note that when $\alpha = 0$, Eqs.(\ref{angles_prim}) predict $\beta'_{\parallel} \simeq 2$ and $\beta'_{\perp} \simeq 1/2$. If $\alpha = \pi/2$, i.e. there are no gas sectors, $\beta'_{\parallel,\perp} \simeq 0$.
We also remark that the linearity of the Stokes equations implies that near the second edge of the groove, $y/\delta = 1$, we have
\begin{equation}\label{angles2}
\beta_{\parallel,\perp}\simeq \beta'_{\parallel,\perp} \left(1 - \dfrac{y}{\delta}\right).
\end{equation}
It has been suggested before that local slip length of a shallow groove is defined only by the viscosity contrast and local thickness of a thin lubricating films~\cite{nizkaya.tv:2013}, similarly to infinite systems~\cite{vinogradova.oi:1995a,miksis.mj:1994}, but also depends strongly on the flow direction. Since in our case the local thickness depends on $\alpha$ the early results~\cite{nizkaya.tv:2013} can be generalized as
\begin{equation}
b_{\parallel}\simeq \dfrac{\mu }{\mu_g} e(y, \alpha),\hspace{1cm} b_{\perp}\simeq \frac{b_{\parallel}}{4},
\label{shallow0}
\end{equation}
and by using Eq.(\ref{bgas}) we can immediately formulate equations for eigenvalues of a slip coefficient in this limit
\begin{equation}
\beta_{\parallel}\simeq e(y, \alpha)/\delta,\hspace{1cm}\beta_{\perp}\simeq \beta_{\parallel}/4.
\label{shallow}
\end{equation}
Although Eqs.(\ref{angles}) and (\ref{angles2}) are valid for any groove depth, in the case of weakly slipping shallow groves the contribution of dissipation in the area $y/\delta\ll 1$ should be small compared to the central part of the gas areas, so that $\beta_{\parallel,\perp}$ are controlled by the groove depth as predicted by Eq.(\ref{shallow}). However, for strongly slipping deep grooves $\beta_{\parallel,\perp}$ should depend mainly on $\alpha$ since their maximal values are constrained by $\beta'_{\parallel,\perp}/2$. \label{refe1}This suggests that when grooves are deep, $\beta_{\parallel,\perp}$ are not sensitive to the shape of their bottom and should saturate at some $e(y, \alpha)/\delta$. Coming back to Eq.(\ref{bgas}) we conclude that local slip length at the gas areas of sufficiently deep grooves is defined by the viscosity contrast, their width, and bevel angle, but not by their depth. This implies that $\delta$ and $\alpha$ are the only two parameters that could be used to tune the large local slip at SH surfaces. An important remark would be that they also constrain its attainable upper value, and we can conclude that for rectangular grooves $b_{\parallel}$ should be inevitably below $\mu \delta/\mu_g$, and that $b_{\perp}$ is always smaller than $ \mu \delta/4\mu_g$. For grooves of the same $\delta$ with beveled edges these upper (and in fact unattainable) bounds will be even smaller.
\section{Computation of the two-phase flow near superhydrophobic grooves.}
A precise discussion of the flow in the vicinity of a SH surface requires a numerical solution of the self-consistent two-phase boundary problem, which is normally done by using finite element~\cite{girault2012} or boundary element~\cite{pozrikidis1992} methods. Here we suggest a simple alternative approach, which is easier to implement. We start by noting that Eq.(\ref{bgas}) does not contain any parameters associated with the flow of liquid. This implies that if Eq.(\ref{bgas}) is correct, then $\beta_{\parallel,\perp}$ are universal characteristics of the groove geometry only. However, applying the same $\beta_{\parallel,\perp}$ as boundary conditions yields different velocities at the gas-liquid interface, $u^{int}_{x,y}(y)=u_{x,y}|_{z=0}$, since the solution of the Stokes equations in liquid depends on $\mu/\mu_g$ and $\phi$. This suggests that one can construct a simple iterative scheme to solve two-phase problem. We apply a lattice-Boltzmann method (LBM)~\cite{benzi1992,Luo1998} to simulate gas flow. It is robust, easy to implement, and allows one to measure velocity $\mathbf{u}_{g,\tau}$ and shear rate $\partial \mathbf{u}_{g,\tau}/ \partial z$ at the interface independently.
We begin with a solution for $u^{int}_{x,y}$ obtained for an isolated perfect slip stripe, i.e. in the limit of small $\phi$~\cite{philip.jr:1972}
\begin{equation}
u_{x,y}^{int}/u^{\rm{max}}_{x,y} = \sqrt{1-(2y/\delta-1)^2},
\label{velprofile}
\end{equation}
and impose it as a boundary condition to calculate a flow in the gas phase. The eigenvalues of the local slip coefficient can then be calculated using Eqs.(\ref{bpadef}) and (\ref{bpedef}). Note that it is also convenient to use a more accurate formula $\displaystyle u_{x,y}^{int}/u_{x,y}^{\max }=\frac{\cosh ^{-1}\left[ \cos \left( \pi
y\right) /\cos \left( \pi \phi /2\right) \right] }{\cosh ^{-1}\left[ 1/\cos
\left( \pi \phi /2\right) \right] }$, which is valid for an arbitrary $\phi$~\cite{philip.jr:1972,lauga.e:2003}, and that this velocity profile is very close to given by Eq.(\ref{velprofile}) when $\phi \leq 0.5$.
These computed eigenvalues specify boundary conditions for liquid flow. We then solve Stokes equations in liquid numerically by using a method based on Fourier series~\cite{asmolov_etal:2013,nizkaya.tv:2013}. The solution satisfying these new partial slip conditions leads in turn to a better approximation for the velocity at the gas-liquid interface. At the next iteration we use this new velocity profile instead of Eq.(\ref{velprofile}) to compute more accurate values of $\partial \mathbf{u}_{g,\tau}/ \partial z$ and $\beta_{\parallel,\perp}$, and so on, until an exact solution is obtained.
We use D3Q19 implementation of LBM with the unit length set by the lattice step $a$, the time-step $\Delta t$, and mass density $\rho_0=1$. The kinematic viscosity of the fluid is defined through the relaxation time scale $\tau$ as $\nu=(2\tau-1)/6$ and kept constant in all the simulations $\nu=0.03$. The simulated system is built in such way that the upper boundary represents a liquid-gas interface ($z=0$ in Fig.~\ref{fig_sketch}(b)) and the groove walls are perpendicular to $yz$-plane. For each simulation run the height of the system $N_z$ is chosen equal to the maximal depth of the groove. To provide sufficient resolution of groove shapes we use a simulation domain of the size $N_y = \delta = 126 a$, $N_x = 8 a$, and $N_z = 12 a - 122 a$. To verify the results we have repeated separate runs with 2 times and 4 times larger space resolution and have shown that the maximal error due to discretization does not exceed~$1\%$. At the liquid-gas interface we set $\mathbf{u}_g|_{z=0}=(u^{int}_x,0,0)$ for longitudinal and $\mathbf{u}_g|_{z=0}=(0,u^{int}_y,0)$ for transverse grooves with $u_{x,y}^{int}/u^{\rm{max}}_{x,y}$ given by Eq.(\ref{velprofile}). The maximum velocity $u^{\rm{max}}_{x,y}$ at $y=\delta/2$ is taken equal to $10^{-3} a/\Delta t$. At the gas-solid interface, given by $z=-e(y)$, where $e(y)$ is defined by Eq.(\ref{eq:profile}) or Eq.(\ref{eq:arc_groove}), we impose no-slip boundary conditions.
For trapezoidal grooves we have varied $N_z/N_y=e/\delta$ from 0.1 to 0.97 and $\alpha$ from $0$ to $\pi/6$. For arc-shaped grooves the same values of $\alpha$ define $N_z=e^{\ast}$ (see Eq.(\ref{earc})). Finally, the periodic boundary conditions have been set along the $x$-axis. All simulations are made with an open-source package ESPResSo~\cite{ESPResSo}.
To assess the validity and convergence of the approach we compute the interface velocity, $u_{x,y}^{int}/u^{\rm{max}}_{x,y},$ and the local slip coefficients, $\beta_{\parallel,\perp}$, at fixed $\phi=0.5$, $\alpha=3 \pi/32$, and $e^*/\delta=0.97 $. Fig.~\ref{fig_iterations} shows the interface velocity profiles and $\beta_{\parallel,\perp}$ obtained after two iteration steps. We conclude that both longitudinal and transverse velocity at the gas sector nearly coincide with predictions of Eq.(\ref{velprofile}) taken as an initial approximation, but we remark and stress that at the second iteration step $u^{\rm{max}}_{x,y}$ becomes quite different from used as an initial guess. We also see that $\beta_{\parallel,\perp}$ computed at the first and the second iteration steps are practically the same. Therefore, in reality our iteration procedure converges extremely fast, so that below we use two iteration steps only.
\begin{figure}[tb]
\includegraphics[width=0.75\columnwidth]{Nizkaya_Fig3.eps}
\caption{(a) Dimensionless interface velocities $u_{x,y}^{int}/u^{\rm{max}}_{x,y}$ and (b)~eigenvalues of local slip coefficient $\beta_{\parallel,\perp}$ computed for a flow past trapezoidal grooves of $\alpha=3\pi/32$ and $\phi=0.5$. Symbols plot the results of the second iteration obtained for longitudinal (squares) and transverse (circles) grooves. Solid curves correspond to the first iteration step.}
\label{fig_iterations}
\end{figure}
Finally, to calculate the effective slip lengths for given computed local slip length profiles
\begin{equation}
b_{\mathrm{eff}}^{\|,\perp}=\left.\dfrac{\langle u_{x,y}\rangle}{\langle \partial_z u_{x,y}\rangle}\right|_{z=0}
\end{equation}
we solve the Stokes equation in liquid numerically by using the Fourier series method described before in~\cite{nizkaya.tv:2013}. In these calculations we vary $\phi$ in the range from 0.1 to 0.9.
\section{Results and discussion}
\begin{figure}[tb]
\includegraphics[width=0.75\columnwidth]{Nizkaya_Fig4.eps}
\caption{Streamlines of the gas flow in (a,b)~the trapezoidal and (c,d)~arc-shaped transverse grooves of (a,c)~$\alpha=0$ and (b,d)~$\alpha=3\pi/32$. Color indicates the magnitude of velocity. Black lines show gas-solid interface. }
\label{fig_strlines}
\end{figure}
We begin by studying the flow field in the gas phase of the grooves. For a longitudinal configuration the gas flows in $x$-direction only, and its velocity monotonously decays from gas-liquid interface to the bottom of the groove. The streamlines for a transverse case form a single eddy, and the locus of its center depends slightly on the shape of the groove and on $\alpha$. Fig.~\ref{fig_strlines} shows typical streamlines for a transverse flow in gas computed for trapezoidal grooves and arc-shaped grooves of $\alpha=0$ and $\alpha=3\pi/32$. The other parameters of a trapezoidal relief are taken the same as used in Fig.~\ref{fig_iterations}. Note that with these parameters the trapezoidal grooves are roughly twice deeper than arc-shaped. Thus, if $\alpha=0$, Eq.(\ref{earc}) gives $e^{\ast}/\delta = 1/2$ (cf. $e^*/\delta = 0.97$). Altogether the simulation results show that the velocity field in the gas phase is not a unique function of $\alpha$ and that it generally depends on the relief of grooves and their depth. We note, however, that when the grooves are sufficiently deep, the gas in them is almost stagnant near a bottom, so that increasing the depth further should not change the liquid flow past the SH surface.
\begin{figure}[tb]
\includegraphics[width=0.75\columnwidth]{Nizkaya_Fig5.eps}
\caption{(a) Longitudinal and (b)~transverse slip coefficients for trapezoidal grooves computed at $\alpha=3\pi/32$ (solid curves). From bottom to top $e^*/\delta=0.094$, $0.22$, $0.35$, $0.47$, $0.60$, and $0.97$. Dashed curves shows slip coefficient profiles for an arc-shaped groove with the same $\alpha$ and $e^{\ast}/\delta \simeq 0.37$. Dotted lines represent calculations with Eqs.(\ref{shallow}) made with $e^*/\delta=0.094$ and $0.22$. Dash-dotted lines are predictions of Eqs.(\ref{angles}) and (\ref{angles2}).}
\label{fig_bloc_depth}
\end{figure}
The computed flow field in the gas phase allows one to immediately deduce $\beta_{\parallel,\perp}$. Fig.~\ref{fig_bloc_depth} shows $\beta_{\parallel,\perp}$ for trapezoidal grooves of a fixed $\alpha=3\pi/32$ and several depths $e^*/\delta$. We see that for shallow grooves ($e^*/\delta \ll 1$) the $\beta_{\parallel,\perp}$
profiles can be approximated by trapezoids with the central region of a constant slip $\beta_{\parallel} \simeq e/\delta$ and $\beta_{\perp} \simeq e/4 \delta$ given by
Eqs.(\ref{shallow}), and linear regions near edges with local slip coefficients
described by Eqs.(\ref{angles}) and (\ref{angles2}). For relatively deep grooves, $e^*/\delta = O(1)$, $\beta_{\parallel,\perp}$ profiles practically converge into
single curves and are no longer dependent on $e^*/\delta$. We also see that the crossover between the regimes of shallow and deep grooves takes place at intermediate $e^*/\delta$, where $\beta_{\parallel,\perp}$ are controlled both by $e^*/\delta$ and $\alpha$. To illustrate this we have included in Fig.~\ref{fig_bloc_depth} the $\beta_{\parallel,\perp}$ curves computed for arc-shaped grooves of the same $\alpha=3\pi/32$, which implies their $e^{\ast}/\delta \simeq 0.37$. Remarkably, the $\beta_{\parallel,\perp}$ profiles are very close to computed for trapezoidal grooves of $e^*/\delta = 0.35$. Some small discrepancies in the shapes of $\beta_{\parallel,\perp}$ obtained for two types of textures indicate that in the crossover regime the shape of the grooves contributes to a slip length profile, but this should be seen as a second-order correction only.
\begin{figure}[tb]
\includegraphics[width=0.75\columnwidth]{Nizkaya_Fig6.eps}
\caption{(a)~Transverse slip coefficient profiles for deep trapezoidal grooves of $e^*/\delta=0.97$ (dash-dotted curves), shallow trapezoidal grooves of $e^*/\delta=0.094$ (solid curves). From top to bottom $\alpha=0,~\pi/16$, and $\pi/8$. Dotted line represents prediction of Eq.~(\ref{shallow}) at $e^*/\delta=0.094$. Dashed curves plot the results for arc-shaped grooves of the same values of $\alpha$, which implies $e^{\ast}/\delta$ is varying from 0.5 down to 0.32. (b)~The data sets for $\alpha=0$ reproduced in a larger scale in the vicinity of the edge, $y/\delta = 1$, together with $\beta_{\perp}$ curves computed at $\alpha=3\pi/32$. }
\label{fig_bloc}
\end{figure}
Now we focus on the role of $\alpha$ and study first only deep, $e^*/\delta=0.97$, and shallow, $e^*/\delta=0.094$, trapezoidal grooves The computed $\beta_{\perp}$ are displayed in Fig.~\ref{fig_bloc}(a). One can see that in the case of shallow textures $\beta_{\perp}$ generally does not depend on $\alpha$ and is consistent with Eq.(\ref{shallow}), but note that there are deviations from Eq.(\ref{shallow}) in the vicinity of the edges. For deep trapezoids it decreases with $\alpha$. These trends fairly agree with the predictions of Eqs.(\ref{angles}) and Eqs.(\ref{angles_prim}). We can now compare $\beta_{\perp}$ for deep trapezoidal grooves with that of arc-shaped grooves with the same bevel angle. The results of calculations are included to Fig.~\ref{fig_bloc}(a). The data show that at the central part of the gas sector the arc-shaped grooves induce smaller $\beta_{\perp}$ than trapezoidal ones, which simply reflects the fact that they are not deep enough. Indeed, with our values of $\alpha$, their maximal depth $e^{\ast}/\delta$ varies from $0.5$ down to $0.32$ indicating the crossover regime. However, in the vicinity of the groove edges $\beta_{\perp}$ for both textures appears to be the same. To examine this more closely, the data obtained with two values of $\alpha$ in the edge region is plotted in Fig.~\ref{fig_bloc}(b), and we see that in the vicinity of the point where three phases meet $\beta_{\perp}$ computed for different grooves indeed coincide. Altogether these results do confirm that $\beta'_{\perp}$ is determined by $\alpha$ only.
\begin{figure}[tb]
\includegraphics[width=0.75\columnwidth]{Nizkaya_Fig7.eps}
\caption{Computed (a)~$\beta^\prime_{\parallel}$ and (b)~$\beta^\prime_{\perp}$ plotted as a function of $\alpha$ (symbols). Squares and triangles indicate trapezoidal grooves of $e^*/\delta=0.97$ and $0.094$, circles plot the results for arc-shaped grooves. Calculations made with Eq.(\ref{angles_prim}) are shown by solid curves. }
\label{fig_angles}
\end{figure}
Fig.~\ref{fig_angles} includes $\beta'_{\parallel,\perp}$ curves computed for deep, $e^*/\delta=0.97$, and shallow, $e^*/\delta=0.094$, trapezoidal grooves and arc-shaped grooves. The calculations are made using several $\alpha$ in the range from $0$ to $\pi/6$, which implies that $e^{\ast}/\delta$ varies from 0.5 down to 0.29. The general conclusion from this plot is that $\beta'_{\parallel,\perp}$ does not depend on the groove shape and depth being a function of $\alpha$ only, so that our results fully verify Eq.(\ref{angles_prim}).
We now turn to the effective slip lengths, $b_{\mathrm{eff}}^{\|,\perp}$, and will try to understand if such a two-phase problem can indeed be reduced to a one-phase problem, and whether a SH surface can be modeled as a flat one with patterns of stripes with piecewise constant apparent local slip lengths $b^{c}_{\parallel,\perp}$. We first fit our numerical data for $b_{\mathrm{eff}}^{\|,\perp}$ to the known formulae~\cite{belyaev.av:2010a}:
\begin{equation}
\begin{array}{ll}
b_\mathrm{eff}^{\|}\simeq\dfrac{L}{\pi}\dfrac{\ln\left[\sec\left(\frac{\pi\phi}{2}\right)\right]}{1+\dfrac{L}{\pi b^c_{\parallel}}\ln\left[\sec\left(\frac{\pi\phi}{2}\right)+\tan\left(\frac{\pi\phi}{2}\right)\right]},
\\\label{belyaev}
b_\mathrm{eff}^{\perp}\simeq\dfrac{L}{2\pi}\dfrac{\ln\left[\sec\left(\frac{\pi\phi}{2}\right)\right]}{1+\dfrac{L}{2\pi b^c_{\perp}}\ln\left[\sec\left(\frac{\pi\phi}{2}\right)+\tan\left(\frac{\pi\phi}{2}\right)\right],}
\end{array}\end{equation}
taking $b^{c}_{\parallel,\perp}$ as fitting parameters, and then deduce $\beta^c_{\parallel,\perp}$, which can be defined as~\cite{nizkaya2014gas}
\begin{equation}
\beta^c_{\parallel,\perp} = \dfrac{\mu_g }{\mu} \frac{b^{c}_{\parallel,\perp}}{\delta}
\label{newb2}
\end{equation}%
We stress that with such a definition the eigenvalues of apparent slip coefficients, $\beta^c_{\parallel,\perp},$ of trapezoidal grooves depend only on $\alpha$ and
$e^*/\delta$, but not on $y/\delta$. The curves for $\beta ^c_{\parallel,\perp},$ computed for several $\alpha$ are plotted Fig.~\ref{fig_bapp}. For all $\alpha$ these functions saturate already at $e^*/\delta \geq 1,$ thereby imposing constraints on the attainable $b_{\mathrm{eff}}^{\|,\perp}$. Also included in Fig.~\ref{fig_bapp} are $\beta ^c_{\parallel,\perp},$ for arc-shaped texture of the same $\alpha$. In this case $e(y, \alpha)/\delta$ is nonuniform throughout the cross-section, and we make no attempt to calculate its average or effective value at a given $\alpha$. Instead, Fig.~\ref{fig_bapp} is intended to indicate the range of $\beta ^c_{\parallel,\perp},$ that is expected for arc-shaped grooves, so that in this case we simply plot data as a function of $e^{\ast}/\delta$. One can see that data for arc-shaped grooves nearly coincide with results for trapezoidal grooves at $\alpha = \pi/8$, but at smaller angles there is
some discrepancy, especially when $\alpha=0$ and for a transverse case. The discrepancy is always in the direction of smaller $\beta ^c_{\parallel,\perp},$ than found for trapezoidal grooves of $e=e^{\ast}$, indicating that effects of the groove shape become discernible at intermediate depths. We stress, however, that both for trapezoidal and arc-shaped grooves the values of $\beta ^c_{\parallel,\perp},$ are close and show the same trends. So our results for two types of grooves provide a good sense of the possible local slip of 1D texture of any shape.
These results can be used to predict upper attainable local and effective slip lengths of complex grooves if their bevel angle and maximal depth are known. Indeed, Fig.~\ref{fig_bapp} allows one to immediately evaluate $\beta ^c_{\parallel,\perp},$ and the apparent local slip lengths can be found by using Eq.(\ref{newb2}). Once they are known, the eigenvalues of the effective length tensor can be calculated analytically by using Eqs.(\ref{belyaev}).
\begin{figure}[tb]
\includegraphics[width=0.75\columnwidth]{Nizkaya_Fig8.eps}
\caption{Apparent slip coefficients, $\beta_{\parallel,\perp}^c$, as a function of $e^*/\delta$ computed for trapezoidal grooves (filled symbols). From top to bottom $\alpha=0$, $\pi/16$, and $\pi/8$. Error bars were determined from the deviation among repeated calculations made at different $\phi$. When error bars are absent, uncertainties are
smaller than the symbols. Solid curves are plotted to guide the eye. Open symbols correspond to $\beta_{\parallel,\perp}^c$ of arc-shaped grooves of the same $\alpha$.}
\label{fig_bapp}
\end{figure}
\section{Conclusion}
We have analyzed liquid slippage at gas areas of 1D SH surfaces and developed an asymptotic theory, which led to
explicit expressions for the longitudinal and transverse slip lengths near the edge of the groove, i.e. the point, where three phases meet. The theory predicts that the local slip lengths in the vicinity of this point always increase linearly with the slope determined solely by a bevel angle of grooves. We have also shown that at a given viscosity contrast the eigenvalues of local slip tensor of strongly slipping deep grooves are fully determined by their width and the value of their bevel angle, but not by their depth as sometimes invoked for explaining the extreme local slip. Thus, it is not necessary to deal with very deep grooves to get a largest possible local slip at the gas areas. However, the eigenvalues of a local slip tensor of weakly slipping shallow grooves are not really sensitive to the bevel angle and are determined mostly by their depth.
Altogether, our study shows that for a given width, SH grooves with beveled edges are less efficient than rectangular ones for drag reduction purposes. However, the use of grooves with beveled edges appears as a good compromise between the positive effect of their stability against bending and a moderate reduction of the slippage effect due to a bevel angle. A very large local slip length can be induced by using wide grooves with beveled edges, which would be often impossible for wide rectangular grooves due to their bending instability.
To check the validity of our theory, we have proposed an approach to solve numerically the two-phase hydrodynamic problem by considering separately flows in liquid and gas phases, which can in turn be done by using different techniques. Our method significantly facilitates and accelerates calculations compared to classical two-phase numerical schemes. Generally, the numerical results have fully confirmed the theory for limiting cases. They have also clarified that at intermediate depths a particular shape of 1D SH surface modifies the local slip profiles, but only slightly.
Our strategy and computational approach can be extended to a situation of a sufficiently curved meniscus. Another fruitful direction could be to consider more complex 2D textures, which include various pillars and holes. Thus, they may guide the design of textured surfaces with superlubricating potential in microfluidic devices, tribology, and more. It would be also interesting to revisit recent analysis of an effective slip of SH surfaces and its various implications since our results suggest that instead of a piecewise constant slip length at the gas areas, a local tensorial slip should be employed to obtain more accurate eigenvalues of the effective slip length tensor. Finally, we mention that our approach can be immediately applied to compute local slip lengths of grooves filled by immiscible liquid of low or high viscosity.
\begin{acknowledgments}
This research was partly supported by the the Russian
Foundation for Basic Research (grants No. 15-01-03069).
\end{acknowledgments}
|
2,869,038,154,808 | arxiv | \section{Introduction}\label{s1}
Graph-based representations are powerful tools to analyze real-world structured data that encapsulates pairwise relationships between its parts~\cite{DBLP:conf/nips/DefferrardBV16,DBLP:journals/tnn/ZambonAL18}. One fundamental challenge arising in the analysis of graph-based data is to represent discrete graph structures as numeric features that preserve the topological information. Due to the recent successes of deep learning networks in computer vision problems, many researchers have devoted their efforts to generalizing Convolutional Neural Networks (CNNs)~\cite{DBLP:conf/cvpr/VinyalsTBE15,DBLP:journals/cacm/KrizhevskySH17} to the graph domain. These neural networks on graphs are now widely known as Graph Convolutional Networks (GCNs)~\cite{DBLP:journals/corr/KipfW16}, and
have proven to be an effective way to extract highly meaningful statistical features for graph classification problems~\cite{DBLP:conf/nips/DefferrardBV16}.
Generally speaking, most existing state-of-the-art graph convolutional networks are developed based on two strategies, i.e., a) the spectral and b) the spatial strategies. Specifically, approaches based on the spectral strategy employ the property of the convolution operator from the graph Fourier domain that is related to spectral graph theory~\cite{DBLP:journals/corr/BrunaZSL13}. By transforming the graph into the spectral domain through the eigenvectors of the Laplacian matrix, these methods perform the filter operation by multiplying the graph by a series of filter coefficients~\cite{DBLP:journals/corr/BrunaZSL13,DBLP:conf/nips/RippelSA15,DBLP:journals/corr/HenaffBL15}. Unfortunately, most spectral-based approaches demand the size of the graph structures to be the same and cannot be performed on graphs with different sizes and Fourier bases. As a result, approaches based on the spectral strategy are usually applied to vertex classification tasks. Methods based on the spatial strategy, on the other hand, generalize the convolution operation to the spatial structure of a graph by propagating features between neighboring vertices~\cite{DBLP:journals/corr/VialatteGM16,DBLP:conf/nips/DuvenaudMABHAA15,DBLP:conf/nips/AtwoodT16}. Since spatial-based approaches are not restricted to the same graph structure, these methods can be directly applied to graph classification problems. Unfortunately, most existing spatial-based methods have relatively poor performance on graph classifications. The reason for this ineffectiveness is that these methods tend to directly sum up the extracted local-level vertex features from the convolution operation as global-level graph features through a SumPooling layer. As a result, the local topological information residing on the vertices of a graph may be discarded.
To address the shortcoming of the graph convolutional networks associated with SumPooling, a number of methods focusing on local-level vertex information have been proposed. For instance, \cite{DBLP:conf/icml/NiepertAK16} have developed a different graph convolutional network by re-ordering the vertices and converting each graph into fixed-sized vertex grid structures, where standard one-dimensional CNNs can be directly used. \cite{DBLP:conf/aaai/ZhangCNC18} have developed a novel Deep Graph Convolutional Neural Network model to preserve more vertex information through global graph topologies. Specifically, they propose a new SortPooling layer to transform the extracted vertex features of unordered vertices from the spatial graph convolution layers into a fixed-sized vertex grid structure. Then a traditional convolutional operation can be performed by sliding a fixed-sized filter over the vertex grid structures to further learn the topological information. The aforementioned methods focus more on local-level vertex features and outperform state-of-the-art graph convolutional network models on graph classification tasks. However, they tend to sort the vertex order based on the local structure descriptor of each individual graph. As a result, they cannot easily reflect the accurate topological correspondence information between graph structures. Furthermore, these approaches also lead to significant information loss. This usually occurs when they form a fixed-sized vertex grid structure and some vertices associated with lower ranking may be discarded. In summary, developing effective methods to preserve the structural information residing in graphs still remains a significant challenge.
To overcome the shortcoming of the aforementioned methods, we propose a new graph convolutional network model, namely the Aligned Vertex Convolutional Network, to learn multi-scale features from local-level vertices for graph classification. One key innovation of the proposed model is the identification of the transitively aligned vertices between graphs. That is, given three vertices $v$, $w$ and $x$ from three sample graphs, assume $v$ and $x$ are aligned, and $w$ and $x$ are aligned, the proposed model can guarantee that $v$ and $w$ are also aligned. More specifically, the new model utilizes the transitive alignment procedure to transform different graphs into fixed-sized aligned vertex grid structures with consistent vertex orders. Overall, the main contributions are threefold.
\textbf{First}, we propose a new vertex matching method to transitively align the vertices of graphs. We show that this matching procedure can establish reliable vertex correspondence information between graphs, by gradually minimizing the inner-vertex-cluster sum of squares over the vertices of all graphs through a $k$-means clustering method.
\textbf{Second}, with the transitive alignment information over a family of graphs to hand, we show how the graphs of arbitrary sizes can be mapped into fixed-sized aligned vertex grid structures. The resulting Aligned Vertex Convolutional Network model is defined by adopting fixed-sized one-dimensional convolution filters on the grid structure to slide over the entire ordered aligned vertices. We show that the proposed model can effectively learn the multi-scale characteristics residing on the local-level vertex features for graph classifications. Moreover, since all the original vertex information will be mapped into the aligned vertex grid structure through the transitive alignment, the grid structure not only precisely integrates the structural correspondence information but also minimises the loss of structural information residing on local-level vertices. As a result, the proposed model addresses the shortcomings of information loss and imprecise information representation arising in existing graph convolutional networks associated with SortPooling or SumPooling.
\textbf{Third}, we empirically evaluate the performance of the proposed model on graph classification problems. Experiments on benchmark graph datasets demonstrate the effectiveness.
\section{Transitive Vertex Alignment Method}\label{s2}
One main objective of this work is to convert graphs of arbitrary sizes into the fixed-sized aligned vertex grid structures, so that a fixed-sized convolution filter can directly slide over the grid structures to learn local-level structural features through vertices. To this end, we need to identify the correspondence information between graphs.
In this section, we introduce a new matching method to transitively align the vertices. We commence by designating a family of prototype representations that encapsulate the principle characteristics over all vectorial vertex representations in a set of graphs $\mathbf{G}$. Assume there are $n$ vertices from all graphs in $\mathbf{G}$, and the associated $K$-dimensional vectorial representations of these vertices are $\mathbf{{R}}^K =\{\mathrm{R}_1^K,\mathrm{R}_2^K,\ldots,\mathrm{R}_n^K\}$. We employ $k$-means~\cite{witten2011data} to locate $M$ centroids over all representations in $\mathbf{{R}}^K$. Specifically, given $M$ clusters $\Omega=(c_1,c_2,\ldots,c_M)$, the aim of $k$-means is to minimize the objective function
\begin{equation}
\arg\min_{\Omega} \sum_{j=1}^M \sum_{\mathrm{R}_i^K \in c_j} \|\mathrm{R}_i^K- \mu_j^K\|^2,\label{kmeans}
\end{equation}
where $\mu_j^K$ is the mean of the vertex representations belonging to the $j$-th cluster $c_j$. Since Eq.(\ref{kmeans}) minimizes the sum of the square Euclidean distances between the vertex points $\mathrm{R}_i^K$ and the centroid point of cluster $c_j$, the set of $M$ centroid points $\mathbf{PR}^K=\{\mu_1^K,\cdots,\mu_j^K,\cdots,\mu_M^K\}$ can be seen as a family of $K$-dimensional \textbf{prototype representations} that encapsulate representative characteristics over all graphs in $\mathbf{G}$.
To establish the correspondence information between the graph vertices over all graphs in $\mathbf{G}$, we align the vectorial vertex representations of each graph to the prototype representations in $\mathbf{PR}^K$. Our alignment is similar to that introduced in~\cite{DBLP:conf/icml/Bai0ZH15} for point matching in a pattern space. Specifically, for each sample graph $G_p(V_p,E_p)\in {\mathbf{G}}$ and the associated $K$-dimensional vectorial representation of each vertex $v_i\in V_p$, we compute a $K$-level affinity matrix in terms of the Euclidean distances between the two sets of points as
\begin{align}
A^K_p(i,j)=\|\mathrm{R}_i^K - \mu_j^K\|_2.\label{AffinityM}
\end{align}
where $A^K_p$ is a ${|V_p|}\times {M}$ matrix, and each element $R^K_p(i,j)$ represents the distance between the vectorial representation $\mathrm{{R}}_{p;i}^K$ of vertex $v_\in V_p$ and the $j$-prototype representation $\mu_j^K\in \mathbf{PR}^K$. If the value of $A^K_p(i,j)$ is the smallest in row $i$, the vertex $v_i$ is aligned to the $j$-th prototype representation. Note that for each graph there may be two or more vertices aligned to the same prototype representation. We record the correspondence information using the $K$-level correspondence matrix
$C^K_p\in \{0,1\}^{|V_p|\times M}$
\begin{equation}
C^K_p(i,j)=\left\{
\begin{array}{cl}
1 & \small{\mathrm{if} \ A^K_p(i,j) \ \mathrm{is \ the \ smallest \ in \ row } \ i} \\
0 & \small{\mathrm{otherwise}}.
\end{array} \right.
\label{CoMatrix}
\end{equation}
For a pair of graphs $G_p$ and $G_q$, if their vertices $v_p$ and $v_q$ are aligned to the same prototype representation, we say that $v_p$ and $v_q$ possess similar characteristics and are also aligned. Thus, we can identify the transitive alignment information between the vertices of all graphs in $\mathbf{G}$, by aligning their vertices to the same set of prototype representations. The alignment process is equivalent to assigning the vectorial representation $\mathrm{{R}}_{p;i}^K$ of each vertex $v_i\in V_p$ to the mean $\mu_i^K$ of the cluster $c_i^K$. Thus, the proposed alignment procedure can be seen as an optimization process that gradually minimizes the inner-vertex-cluster sum of squares over the vertices of all graphs through $k$-means, and can establish reliable vertex correspondence information over all graphs.
\section{Learning Vertex Convolutional Networks}\label{s3}
In this section, we develop a new vertex convolutional network model for graph classification. Our idea is to employ the transitive alignment information over a family of graphs and convert the arbitrary sized graphs into fixed-sized aligned vertex grid structures. We then define a vertex convolution operation by adopting a set of fixed-sized one-dimensional convolution filters on the grid structure. With the new vertex convolution operation to hand, the proposed model can extract the original aligned vertex grid structure as a new grid structure with a reduced number of packed aligned vertices, i.e., the extracted multi-scale vertex features learned through the convolutional operation is packed into the new grid structure. Finally, we employ the Softmax layer to read the extracted vertex features and predict the graph class.
\subsection{Aligned Vertex Grid Structures of Graphs}
In this subsection, we show how to convert graphs of different sizes into fixed-sized aligned vertex grid structures. For each sample graph $G_p(V_p,E_p)$ from the graph set $\mathbf{G}$ defined earlier, assume each of its vertices $v_p\in V_p$ is represented as a $c$-dimensional feature vector. Then the features of all the $n$ ($n=|V_p|$) vertices can be encoded using the $n\times c$ matrix $F_p$ (i.e., $F_p\in \mathbb{R}^{n\times c}$). If $G_p$ are vertex attributed graphs, $F_p$ can be the one-hot encoding matrix of the vertex labels. For un-attributed graphs, we propose to use the vertex degree as the vertex label. Based on the transitive alignment method defined in Section~\ref{s2}, we commence by identifying the family of the $K$-dimensional prototype representations in $\mathbf{PR}^K=\{\mu_1^K,\ldots,\mu_j^K,\ldots,\mu_M^K \}$ of $\mathbf{G}$. For each graph $G_p\in \mathbf{G}$, we compute the $K$-level vertex correspondence matrix $C^K_p$, where the row and column of $C^K_p$ are indexed by the vertices in $V_p$ and the prototype representations in $\mathbf{PR}^K$, respectively. With $C^K_p$ to hand, we compute the $K$-level aligned vertex feature matrix for $G_p$ as
\begin{equation}
{X}_{p}^{K}= (C^K_p)^T F_p,\label{alignDB}
\end{equation}
where ${X}_{p}^{K}$ is a $M\times c$ matrix and each row of ${X}_{p}^{K}$ represents the feature of a corresponding aligned vertex. Since ${X}_{p}^{K}$ is computed by mapping the original feature information of each vertex $v_p\in V_p$ to that of the new aligned vertices indexed by the corresponding prototypes in $\mathbf{PR}^K$, it encapsulates all the original vertex feature information of $G_p$.
For constructing the fixed-sized aligned vertex grid structure for each graph $G_p\in \mathbf{G}$, we need to establish a consistent vertex order for all graphs in $\mathbf{G}$. As the vertices are all aligned to the same prototype representations, the vertex orders can be determined by reordering the prototype representations. To this end, we construct a prototype graph that captures the pairwise similarity between the prototype representations, then we reorder the prototype representations based on their degree. This process is equivalent to sorting the prototypes in order of average similarity to the remaining ones. Specifically, for the $K$-dimensional prototype representations in $\mathbf{PR}^K$, we compute the prototype graph as $G_{\mathrm{R}}(V_{\mathrm{R}},E_{\mathrm{R}})$, where each vertex $v_j\in V_{\mathrm{R}}$ represents the prototype representation $\mu_j^K\in \mathbf{PR}^K$ and each edge $(v_j,v_k)\in E_{\mathrm{R}} $ represents the similarity between a pair of prototype representations $\mu_j^K\in \mathbf{PR}^K$ and $\mu_k^K\in \mathbf{PR}^K$. The similarity between two vertices of $G_{\mathrm{R}}$ is computed as \begin{equation}
s(\mu_j^K,\mu_k^K)=\exp (-\frac{\| \mu_j^K-\mu_k^K \|_2}{K}).
\end{equation}
The degree of each prototype representation $\mu_j^K$ is $D_R(\mu_j^K)=\sum_{k=1}^{M}s(\mu_j^K,\mu_k^K)$. We sort the $K$-dimensional prototype representations in $\mathbf{PR}^K$ according to their degree $D_R(\mu_j^K)$. Then, we rearrange ${X}_{p}^{K}$ accordingly.
Finally, note that, to construct reliable grid structures for graphs, we employ the depth-based (DB) representations as the vectorial vertex representations to compute the required $K$-level vertex correspondence matrix $C_p^K$. The DB representation of each vertex is defined by measuring the entropies on a family of $k$-layer expansion subgraphs rooted at the vertex~\cite{DBLP:journals/pr/BaiH14}, where the parameter $k$ varies from $1$ to $K$. It is shown that such a $K$-dimensional DB representation encapsulates rich entropy content flow from each local vertex to the global graph structure, as a function of depth. The process of computing the correspondence matrix $C_p^K$ associated with DB representations is shown in the appendix file. When we vary the largest layer $K$ of the expansion subgraphs from $1$ to $L$ (i.e., $K\leq L$), we compute the final \textbf{aligned vertex grid structure} for each graph $G_p$ as
\begin{equation}
{X}_{p}= \sum_{K=1}^L \frac{{X}_{p}^{K}}{L},\label{AlignV}
\end{equation}
where ${X}_{p}$ is also a $M\times c$ matrix as same as ${X}_{p}^{K}$. Clearly, Eq.(\ref{AlignV}) transforms the original graphs $G_p\in \mathbf{G}$ of arbitrary sizes into a new aligned vertex grid structure with the same vertex number. Moreover, note that, the aligned vertex grid structure ${X}_{p}$ also preserve the original vertex feature information through the $K$-level aligned vertex feature matrix ${X}_{p}^{K}$.
\subsection{The Aligned Vertex Convolutional Network}
In this subsection, we develop a new Aligned Vertex Convolutional Network model that learns local-level vertex features for graph classifications. This model is defined by adopting a set of fixed-sized one-dimensional convolution filters on the aligned vertex grid structures and sliding the filter over the ordered aligned vertices to learn features, in a manner analogous to the standard convolution operation. Specifically, for each graph $G(V,E)\in \mathbf{G}$ and its associated aligned vertex grid structure ${X}\in \mathbb{R}^{M\times c}$ (\textbf{i.e., $M$ aligned vertices each with $c$ feature channels}), we denote the element of $X$ in the $e$-th row and $s$-th column as ${X}_{e,s}$, i.e., the $s$-th feature channel of the $e$-th aligned vertex. We pass ${X}$ to the convolution layer. Assume the size of the receptive field is $m$, i.e., the size of the one-dimensional convolution filter is $m$, the vertex convolution operation associated with $1$-stride takes the form
\begin{equation}
Z_{e,h}=\sigma(\sum_{s=1}^{c}(\sum_{j=1}^{m}W_{j}^{h,s}{X}_{e+j-1,s})+b^{h}),\label{V_convolution}
\end{equation}
where $Z_{e,h}$ is the element in the $e$-th row and $h$-th column of the new grid structure $Z$ after the convolution operation, the parameter $e$ satisfies $e\leq M-m+1$, $W_{j}^{h,s}$ is the $j$-th element of the convolution filter that maps the $s$-th feature channel of $X$ to the $h$-th feature channel of $Z$, $b^h$ is the bias of the $h$-th convolution filter, and $\sigma$ is the activation function.
An example of the vertex convolution operation defined by Eq.(\ref{V_convolution}) are show in Figure~\ref{f:vconv}. The vertex convolution operation consists of two computational steps. In the first step, the convolution filter $\sum_{s=1}^{c}(\sum_{j=1}^{m}W_{j}^{h,s}{X}_{e+j-1,s})$ is applied to map the $e$-th aligned vertex $X_{e,:}$ as well as its neighbor vertices $X_{e+j-1,:}$ ($j=2,3$) into a new feature value, associated with all the $c$ feature channels of these vertices. Specifically, Figure~\ref{f:vconv}.(1) exhibits this process. Here, assume the vertex index $e=2$, the convolution filter size $m=3$, and we focus on the $2$-nd aligned vertex $X_{2,:}$ of ${X}\in \mathbb{R}^{M\times c}$. The convolution filter $\sum_{s=1}^{c}(\sum_{j=1}^{m}W_{j}^{h,s}{X}_{2+j-1,s})$ represented by the red lines first maps the $s$-th feature channels of the $2$-nd aligned vertex $X_{2,:}$ as well as its neighbor vertices $X_{3,:}$ and $X_{4,:}$ into a new single value by $\sum_{j=1}^{m}W_{j}^{h,s}{X}_{2+j-1,s}$, and then sums up the values computed through all the $c$ channels as the $h$-th feature channel of $Z_{2,:}$. Moreover, we need to slide the convolution filter over all the aligned vertices, and this requires three convolution filters represented by the green, red and blue lines respectively. The weights for the three filters are shared, i.e., they are in fact the same filter. Finally, the second step $\sigma(\mathcal{X}_h +b^h)$, where $\mathcal{X}_h:=\sum_{s=1}^{c}(\sum_{j=1}^{m}W_{j}^{h,s}{X}_{e+j-1,s})$, applies the
Relu function associated with the bias $b^h$ and outputs the final result as $Z_{e,h}$.
\begin{figure}
\vspace{-0pt}
\centering
\includegraphics[width=0.80\linewidth]{vertex_convolution.pdf}
\vspace{-20pt}
\caption{The procedure of the vertex convolution.}\label{f:vconv}
\vspace{-20pt}
\end{figure}
To further extract the multi-scale features of a graph associated with its aligned vertex grid structure ${X}\in \mathbb{R}^{M\times c}$, we stack multiple vertex convolution layers defined as follows
\begin{equation}
Z_{e,h}^t=\sigma(\sum_{s=1}^{c}(\sum_{j=1}^{m} W_{j}^{t,h,s}{X}_{e+j-1,s}^{t-1})+b^{t,h}),\label{vcon_operation}
\end{equation}
where $X^0$ is the input aligned vertex grid structure $X$, and the corresponding notations of the symbols are listed in Table~\ref{T:notation}. After a number of vertex convolution operations, we employ the Softmax layer to read the extracted features computed from the vertex convolution layers and predict the graph class for graph classifications.
\begin{table}
\vspace{-0pt}
\centering {
\footnotesize
\caption{Important Terms and Notations}\label{T:notation}
\vspace{0pt}
\begin{tabular}{|c|c|}
\hline
~Symbol ~ ~ &~Defitions~ \\ \hline \hline
~node $e$~ &~ $\textrm{the e-th vertex}$ ~ \\ \hline
~$Z_{e,h}^{t}$~ &~ $\textrm{the h-th feature channel of vertex (e) in layer t}$ ~ \\ \hline
~$W^{t,h,s}$~ &~ $\textrm{the filter that maps to the h-th feature channel in}$ ~ \\
~$ $~ &~ $\textrm{layer t from the s-th feature channel in layer t-1 }$ ~ \\ \hline
~$W_j^{t,h,s}$~ &~ $\textrm{the j-th element of the filter that maps to the h-th }$ ~ \\
~$ $~ &~ $\textrm{feature channel in layer t from the s-th feature }$ ~ \\
~$ $~ &~ $\textrm{channel in layer t-1}$ ~ \\ \hline
~$b^{t,h}$~ &~ $\textrm{the bias of the h-th filter in layer t}$ ~ \\ \hline
~$\sigma$~ &~ $\textrm{the activate function, e.g., Relu function}$ ~ \\ \hline
~$c_{t-1}$~ &~ $\textrm{the number of filters in layer t-1}$ ~ \\ \hline
\end{tabular}
}\vspace{-10pt}
\end{table}
\begin{table*}
\centering {
\tiny
\scriptsize
\vspace{-0pt}
\caption{Information of the Graph Datasets}\label{T:GraphInformation} \vspace{0pt}
\begin{tabular}{|c||c||c||c||c||c||c||c||c|}
\hline
~Datasets ~ & ~MUTAG ~ & ~PROTEINS~& ~D\&D~ & ~GatorBait~ & ~Reeb ~ & ~IMDB-B~ & ~IMDB-M~ & ~RED-B~\\ \hline \hline
~Max \# vertices~ & ~$28$~ & ~$620$~ & ~$5748$~ & ~$545$~ & ~$220$~ & ~$136$~ & ~$89$~ & ~$3783$~\\ \hline
~Mean \# vertices~ & ~$17.93$~ & ~$39.06$~ & ~$284.30$~ & ~$348.70$~ & ~$95.42$~ & ~$19.77$~ & ~$13.00$~ & ~$429.61$~\\ \hline
~\# graphs~ & ~$188$~ & ~$1113$~ & ~$1178$~ & ~$100$~ & ~$300$~ & ~$1000$~ & ~$1500$~ & ~$2000$~ \\ \hline
~\# vertex labels~ & ~$7$~ & ~$61$~ & ~$82$~ & ~$78$~ & ~$32$~ & ~$-$~ & ~$-$~ & ~$-$~ \\ \hline
~\# classes~ & ~$2$~ & ~$2$~ & ~$2$~ & ~$30$~ & ~$20$~ & ~$2$~ & ~$3$~ & ~$2$~ \\ \hline
~Description~ & ~Bioinformatics~ & ~Bioinformatics~& ~Bioinformatics~ & ~Vision~ & ~Vision~ & ~Social~ & ~Social~ & ~Social~ \\ \hline
\end{tabular}
} \vspace{-15pt}
\end{table*}
\textbf{Discussions:} Comparing to existing state-of-the-art graph convolution networks, the proposed Aligned Vertex Convolution Network (AVCN) model has a number of advantages.
\textbf{First}, unlike the Neural Graph Fingerprint Network (NGFN) model~\cite{DBLP:conf/nips/DuvenaudMABHAA15} and the Diffusion Convolution Neural Network (DCNN) model~\cite{DBLP:conf/nips/AtwoodT16} that both employ a SumPooling layer to directly sum up the extracted local-level vertex features from the convolution operation as global-level graph features. The proposed AVCN model focuses more on learning local structural features through the proposed aligned vertex grid structure. Specifically, Figure~\ref{f:vconv} indicates that the associated vertex convolution operation of the proposed AVCN model can convert the original aligned vertex grid structure into a new grid structure, by packing the aligned vertex features from the original grid structure into the new grid structure. Thus, \textbf{the new grid structure can be seen as a new extracted aligned vertex grid structure with a reduced number of aligned vertices}. As a result, the proposed AVCN model can gradually extract multi-scale local-level vertex features through a number of stacked vertex convolution layers, and encapsulate more significant local structural information than the NGFN and DCNN models associated with SumPooling.
\textbf{Second}, similar to the proposed AVCN model, both the PATCHY-SAN based Graph Convolution Neural Network (PSGCNN) model~\cite{DBLP:conf/icml/NiepertAK16} and the Deep Graph Convolution Neural Network model~\cite{DBLP:conf/aaai/ZhangCNC18} need to rearrange the vertex order of each graph structure and transform each graph into the fixed-sized vertex grid structure. Unfortunately, both the PSGCNN and the DGCNN models sort the vertices of each graph based on the local structural descriptor, ignoring consistent vertex correspondence information between different graphs. By contrast, the proposed AVCN model associates with a transitive vertex alignment procedure to transform each graph into an aligned fixed-sized vertex grid structure. As a result, only the proposed AVCN model can integrate the precise structural correspondence information over all graphs under investigations.
\textbf{Third}, when the PSGCNN model and the DGCNN model form fixed-sized vertex grid structures, some vertices with lower ranking will be discarded. This in turn leads to significant information loss. By contrast, the required aligned vertex grid structures for the proposed AVCN model can encapsulate all the original vertex features from the original graphs. As a result, the proposed AVCN overcomes the shortcoming of information loss arising in the PSGCNN and DGCNN models.
\section{Experiments}\label{s4}
In this section, we compare the performance of the proposed AVCN model to both state-of-the-art graph kernels and deep learning methods on graph classification problems on eight standard graph datasets. These datasets are abstracted from bioinformatics, computer vision and social networks. A selection of statistics of these datasets are shown in Table.\ref{T:GraphInformation}.
\textbf{Experimental Setup:} We evaluate the performance of the proposed AVCN model on graph classification problems against a) six alternative state-of-the-art graph kernels and b) six alternative state-of-the-art deep learning methods for graphs. Specifically, the graph kernels include 1) Jensen-Tsallis q-difference kernel (JTQK) with $q=2$~\cite{DBLP:conf/pkdd/Bai0BH14}, 2) the Weisfeiler-Lehman subtree kernel (WLSK)~\cite{shervashidze2010weisfeiler}, 3) the shortest path graph kernel (SPGK) \cite{DBLP:conf/icdm/BorgwardtK05}, 4) the shortest path kernel based on core variants (CORE SP)~\cite{DBLP:conf/ijcai/NikolentzosMLV18}, 5) the random walk graph kernel (RWGK)~\cite{DBLP:conf/icml/KashimaTI03}, and 6) the graphlet count kernel (GK)~\cite{DBLP:journals/jmlr/ShervashidzeVPMB09}. The deep learning methods include 1) the deep graph convolutional neural network (DGCNN)~\cite{DBLP:conf/aaai/ZhangCNC18}, 2) the PATCHY-SAN based convolutional neural network for graphs (PSGCNN)~\cite{DBLP:conf/icml/NiepertAK16}, 3) the diffusion convolutional neural network (DCNN)~\cite{DBLP:conf/nips/AtwoodT16}, 4) the deep graphlet kernel (DGK)~\cite{DBLP:conf/kdd/YanardagV15}, 5) the graph capsule convolutional neural network (GCCNN)~\cite{DBLP:journals/corr/abs-1805-08090}, and 6) the anonymous walk embeddings based on feature driven (AWE)~\cite{DBLP:conf/icml/IvanovB18}.
\begin{figure}
\vspace{-0pt}
\centering
\includegraphics[width=0.85\linewidth]{GCN_arc1.pdf}
\vspace{-10pt}
\caption{An example of the ACVN architecture.}\label{f:vcn_arc}
\vspace{-22pt}
\end{figure}
\begin{table*}
\centering {
\tiny
\scriptsize
\caption{Classification Accuracy (In $\%$ $\pm$ Standard Error) for Comparisons with Graph Kernels.}\label{T:ClassificationGK}
\vspace{0pt}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
~Datasets~& ~MUTAG ~ & ~PROTEINS~ & ~D\&D~ & ~GatorBait~ & ~Reeb~ & ~IBDM-B~ & ~IBDM-M~ & ~RED-B~\\ \hline \hline
~\textbf{AVCN}~ & ~$\textbf{89.06}\pm0.90$~ & ~$\textbf{75.71}\pm0.65$~ & ~$\textbf{80.90}\pm0.97$~ & ~$\textbf{19.00}\pm.75$~ & ~$\textbf{67.00}\pm0.91$ ~& ~$\textbf{73.20}\pm.90$ & ~$51.14\pm.87$ & ~$\textbf{91.00}\pm0.20$\\ \hline
~JTQK~ & ~$85.50\pm0.55$~ & ~$72.86\pm0.41$~ & ~$79.89\pm0.32$~ & ~$11.40\pm0.52$~ &~$60.56\pm0.35$~ &~$72.45\pm0.81$~ & ~$50.33\pm0.49$~ & ~$77.60\pm0.35$\\ \hline
~WLSK~ & ~$82.88\pm0.57$~ & ~$73.52\pm0.43$~ & ~$79.78\pm0.36$~ & ~$10.10\pm0.61$~ &~$58.53\pm0.53$~ &~$71.88\pm0.77$~ & ~$49.50\pm0.49$~ & ~$76.56\pm0.30$\\ \hline
~SPGK~ & ~$83.38\pm0.81$~ & ~$75.10\pm0.50$~ & ~$78.45\pm0.26$~ & ~$9.00\pm0.75$~ &~$55.73\pm0.44$~ &~$71.26\pm1.04$~ & ~$\textbf{51.33}\pm0.57$~ & ~$84.20\pm0.70$\\ \hline
~CORE SP~& ~$88.29\pm1.55$~ & ~$-$~ & ~$77.30\pm0.80$~ & ~$-$~ &~$-$~ &~$72.62\pm0.59$~ & ~$49.43\pm0.42$~ & ~$90.84\pm0.14$\\ \hline
~ GK~ & ~$81.66\pm2.11$~ & ~$71.67\pm0.55$~ & ~$78.45\pm0.26$~ & ~$8.40\pm.83$~ &~$22.96\pm0.65$~ &~$65.87\pm0.98$~ & ~$45.42\pm0.87$~ & ~$77.34\pm0.18$\\ \hline
~RWGK~ & ~$80.77\pm0.72$~ &~$74.20\pm0.40$~ & ~$71.70\pm0.47$~ & ~$7.00\pm0.77 $~&~$32.47\pm0.69$~ &~$67.94\pm0.77$~ & ~$46.72\pm0.30$~ & ~$72.73\pm0.39$ \\ \hline
\end{tabular}
} \vspace{-15pt}
\end{table*}
\begin{table*}
\centering {
\tiny
\scriptsize
\caption{Classification Accuracy (In $\%$ $\pm$ Standard Error) for Comparisons with Graph Convolutional Neural Networks.}\label{T:ClassificationGCNN}
\vspace{0pt}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
~Datasets~& ~MUTAG ~ & ~PROTEINS~ & ~D\&D~ & ~IBDM-B~ & ~IBDM-M~ & ~RED-B~ \\ \hline \hline
~\textbf{AVCN}~ & ~$\textbf{89.06}\pm0.90$~& ~$75.71\pm0.65$~& ~$\textbf{80.90}\pm0.97$~ & ~$\textbf{73.20}\pm.90$ & ~$51.14\pm.87$ & ~$\textbf{91.00}\pm0.20$\\ \hline
~DGCNN~ & ~$85.83\pm1.66$~& ~$75.54\pm0.94$~& ~$79.37\pm0.94$~ & ~$70.03\pm0.86$ & ~$47.83\pm0.85$ & ~$76.02\pm1.73$\\ \hline
~PSGCNN~ & ~$88.95\pm4.37$~& ~$75.00\pm2.51$~& ~$76.27\pm2.64$~ & ~$71.00\pm2.29$ & ~$45.23\pm2.84$ & ~$86.30\pm1.58$\\ \hline
~DCNN~ & ~$66.98$~ & ~$61.29\pm1.60$~& ~$58.09\pm0.53$~ & ~$49.06\pm1.37$ & ~$33.49\pm1.42$ & ~$-$\\ \hline
~GCCNN~& ~$-$~ & ~$\textbf{76.40}\pm4.71$~& ~$77.62\pm4.99$~ & ~$71.69\pm3.40$ & ~$48.50\pm4.10$ & ~$87.61\pm2.51$\\ \hline
~DGK~ & ~$82.66\pm1.45$~ & ~$71.68\pm0.50$~& ~$78.50\pm0.22$~ & ~$66.96\pm0.56$ & ~$44.55\pm0.52$ & ~$78.30\pm0.30$\\ \hline
~AWE~& ~$87.87\pm9.76$~ & ~$-$~ & ~$71.51\pm4.02$~ & ~$73.13\pm3.28$ & ~$\textbf{51.58}\pm4.66$ & ~$82.97\pm2.86$\\ \hline
\end{tabular}
} \vspace{-10pt}
\end{table*}
For the experiment, \textbf{the proposed AVCN model uses the same network structure on all graph datasets}. Specifically, we set the channel of each vertex convolution operation as $32$, and the number of the prototype representations as $M=64$, i.e., the vertex numbers of the aligned vertex grid structures for the graphs in any dataset are all $64$. To extract different hierarchical multi-scale local vertex features, we propose to input the aligned vertex grid structure of each graph to a family of paralleling stacked vertex convolution layers associated with different convolution filter sizes. Specifically, the architecture of the AVCN model is $C_{64}^{4:(3;5;7;9)}$-$C_{64}^{4:(3;5;7;9)}$-$C_{64}^{4:(3;5;7;9)}$-$F_{64}$. Here, $C_k^{f:(i,j,x,y)}$ denotes a vertex convolution layer consisting of $f$ paralleling vertex convolution filters each with $k$ channels, and the filter sizes are $i$, $j$, $x$ and $y$ respectively. $F_k$ denotes a fully-connected layer consisting of $k$ hidden units. An example of the architecture $C_5^{2:(3;5)}$-$C_5^{2:(3;5)}$-$F6$ for the proposed AVCN model are shown in Figure~\ref{f:vcn_arc}. We set the stride of each filter in $C_k^{f:(i,j,x,y)}$ layer as $1$. With extracted patterns learned from the paralleling stacked vertex convolution layers to hand, we concatenate them and add a new fully-connected layer followed by a Softmax layer to learn the graph class. We set the dropout rate for the fully connected layer as $0.5$. We employ the rectified linear units (ReLU) as the active function for the convolution layers. The only hyperparameter that we need to be optimized is the learning rate, the number of epochs, and the batch size for the mini-batch gradient decent algorithm. To optimize the our AVCN model, we utilize the Stochastic Gradient Descent with the Adam updating rules. Finally, note that, our AVCN model needs to construct the prototype representations to identify the transitive vertex alignment information over all graphs. In this evaluation we proposed to compute the prototype representations from both the training and testing graphs. Thus, our model is an instance of transductive learning~\cite{DBLP:conf/uai/GammermanAV98}, where all graphs are used to compute the prototype representations but the class labels of the testing graphs are not used during the training process. For our model, we perform $10$-fold cross-validation to compute the classification accuracies, with nine folds for training and one fold for testing. For each dataset, we repeat the experiment 10 times and report the average classification accuracies and standard errors in Table.\ref{T:ClassificationGK}.
For the alternative kernel methods, we set the parameters of the maximum subtree height for both the WLSK and JTQK kernels as $10$, based on the previous empirical studies in the original papers. For each alternative graph kernel, we perform $10$-fold cross-validation associated with the LIBSVM implementation of C-Support Vector Machines (C-SVM) to compute the classification accuracies. We repeat the experiment 10 times for each kernel and dataset and we report the average classification accuracies and standard errors in Table.\ref{T:ClassificationGK}. Note that for some kernels we directly report the best results from the original corresponding papers, since the evaluation of these kernels followed the same setting of ours. On the other hand, for the alternative deep learning methods, we report the best results for the PSGCNN and DGK models from their original papers. Note that, these methods were evaluated based on the same setting with the proposed AVCN model. For the DCNN model, we report the best results from the work of Zhang et al.,~\cite{DBLP:conf/aaai/ZhangCNC18}, following the same setting of ours. For the AWE model, we report the classification accuracies of the feature-driven AWE, since the author have stated that this kind of AWE model can achieve competitive performance on label dataset. Finally, note that the PSGCNN model can leverage additional edge features, most of the graph datasets and the alternative methods do not leverage edge features. Thus, we do not report the results associated with edge features in the evaluation. The classification accuracies and standard errors for each deep learning method are shown in Table.\ref{T:ClassificationGCNN}. Note that, the alternative deep learning methods have been evaluated on the Reeb and GatorBait datasets abstracted from computer vision by any author, we do not include the accuracies for these methods.
\textbf{Experimental Results and Discussions:} Table.\ref{T:ClassificationGK} and Table.\ref{T:ClassificationGCNN} indicate that the proposed AVCN model can outperform the alternative state-of-the-art methods including either the graph kernels or the deep learning methods for graphs. Specifically, for the alternative graph kernels, only the accuracy of the SPGK kernel on the IBDM-M dataset is a little higher than that of the proposed AVCN model. On the other hand, for the alternative deep learning methods, only the accuracies of the GCCNN model on the PROTEINS dataset and the AWE model on the IMDB-M dataset are a little higher than those of the proposed AVCN model. The reasons for the effectiveness are threefold. First, these alternative graph kernels are typical examples of R-convolution kernels and are based on measuring any pair of substructures, ignoring the correspondence information between the substructures. By contrast, the proposed model associated with aligned vertex grid structure incorporates the transitive alignment information between graphs, and thus better reflect graph characteristics. Furthermore, the C-SVM classifier associated with graph kernels can only be seen as a shallow learning framework~\cite{DBLP:conf/icassp/ZhangLYG15}. By contrast, the proposed model can provide an end-to-end deep learning architecture, and thus better learn graph characteristics. Second, similar to the alternative graph kernels, all the alternative deep learning methods also cannot integrate the correspondence information between graphs into the learning architecture. Especially, the PSGCNN and DGCNN models need to reorder the vertices and some vertices may be discarded, leading to information loss. By contrast, the associated aligned vertex grid structures can preserve all the information of original graphs. Third, unlike the proposed model, the DCNN model needs to sum up the extracted local-level vertex features as global-level graph features. By contrast, the proposed model can learn richer multi-scale local-level vertex features. The experiments demonstrate the effectiveness of the proposed model.
\section{Conclusion}\label{s6}
In this paper, we have developed a new aligned vertex convolutional network model for graph classification. The proposed model cannot only integrates the precise structural correspondence information between graphs but also minimises the loss of structural information residing on local-level vertices. Experiments demonstrate the effectiveness of the proposed vertex convolution network model.
\bibliographystyle{named}
|
2,869,038,154,809 | arxiv | \section{Introduction}
Breast cancer is one of the most common causes of mortality in the female population in the world~\cite{acs2017breast}. It accounts for around $25\%$ of all the cancers diagnosed in women~\cite{alshanbari2015breast}. For traditional diagnostic tools like mammography, even experienced radiologists can miss $10-30\%$ of breast cancers during routine screenings~\cite{cheng2003computer}. With the advent of digital imaging, whole-slide imaging has gained attention from the clinicians and pathologists because of its reliability. Whole-slide images (WSIs) have been permitted for diagnostic use in the USA~\cite{fda}. They are the high-resolution scans of conventional glass slides with Hematoxylin and Eosin (H\&E) stained tissue. There are four types of tissue in breast biopsy: \textit{normal}, \textit{benign}, \textit{in situ carcinoma}, and \textit{invasive carcinoma}. Fig.~\ref{fig:cancer} shows examples of the four types of breast tissue. In clinical testing, the pathologists diagnose breast cancer based on 1) the percentage of tubule formation, 2) the degree of nuclear pleomorphism, and 3) the mitotic cell count~\cite{elston1991pathological}.
\begin{figure}[t]
\begin{center}
\captionsetup[subfigure]{labelformat=empty}
\subfloat[(a) normal]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/normal.png}}\vspace{-0.05cm}
\subfloat[(b) benign]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/benign.png}}\vspace{-0.05cm}
\subfloat[(c) in situ]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/insitu.png}}\vspace{-0.05cm}
\subfloat[(d) invasive]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/invasive.png}}\vspace{-0.05cm}
\end{center}
\caption{Examples of different types of tissue. The microscopy images (patches of WSIs at $200\times$ magnification) are labeled according to the predominant tissue type in each image.}
\label{fig:cancer}
\end{figure}
Convolutional Neural Networks (CNNs) can be trained in an end-to-end manner to distinguish the different types of cancer, by extracting high-level information from images through stacking convolutional layers. Breast cancer classification has been fundamentally improved by the development of CNN models~\cite{wang2018classification}.
However, breast cancer segmentation in WSIs is still underexplored. WSIs are RGB images with high resolution (e.g. $80000 \times 60000$). Constrained by the memory, WSIs cannot be directly fed into the network. One solution is to crop the WSIs to small patches for patch-wise training~\cite{bandi2017comparison}. Given a fixed input size, however, there is a trade-off between accuracy and the inference speed. One can efficiently reduce the inference cost by cropping the WSIs to larger patches and rescaling the patches to a smaller input size, but this results in a loss of detail and sacrifices accuracy. In WSIs, the suspicious cancer areas our regions of interest (ROIs), are sparse, since most regions are normal tissue or the glass slide. The four classes are therefore highly imbalanced. Further, the pixel-wise annotation of breast cancer segmentation requires domain knowledge and extensive human labor and the ground truth labels are often noisy at the pixel-level. Training on patches with a small field of view can therefore easily lead to overfitting.
In this paper, we propose a semantic segmentation framework, Reinforced Auto-Zoom Net (RAZN). When a pathologist examines the WSIs with a digital microscope, the suspicious areas are zoomed in for details and the non-suspicious areas are browsed quickly (See Fig.~\ref{fig:zoom} for an intuition.). RAZN is motivated by this attentive zoom-in mechanism. We learn a policy network to decide the zoom-in action through the policy gradient method~\cite{sutton1998reinforcement}. By skipping the non-suspicious areas (normal tissue), noisy information (glass background) can be ignored and the WSIs can be processed more quickly. By zooming in the suspicious areas (abnormal tissue), the data imbalance is alleviated locally (in the zoomed-in regions) and more local information is considered. Combining these two can efficiently reduce overfitting for the normal tissue, which is caused by the imbalanced data, and lead to improved accuracy. However, since the zoom-in action is selective, the inference can at the same time be fast.
The previous studies on zoom-in mechanism focus on utilizing multi-scale training to improve prediction performance. The Hierarchical Auto-Zoom Net HAZN~\cite{xia2016zoom} uses sub-networks to detect human and object parts at different scales hierarchically and merges the prediction at different scales, which can be considered as a kind of ensemble learning. Zoom-in-Net~\cite{wang2017zoom} zooms in suspicious areas generated by attention maps to classify diabetic retinopathy. In both HAZN and Zoom-in-Net, the zoom-in actions are deterministic. So in the training phase, the patches will be upsampled and trained even if it may not decrease the loss. In RAZN, the zoom-in actions are stochastic, and a policy is learned to decide if the zoom-in action can improve the performance.
\begin{figure}[t]
\begin{center}
\captionsetup[subfigure]{labelformat=empty}
\subfloat[(a)]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/scale2.png}}\vspace{-0.05cm}
\subfloat[(b)]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/scale3.png}}\vspace{-0.05cm}
\subfloat[(c)]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/scale4.png}}\vspace{-0.05cm}
\subfloat[(d)]{\includegraphics[width=0.22\linewidth,height=0.22\linewidth]{figures/scale5.png}}\vspace{-0.05cm}
\end{center}
\caption{Zoom-in process. The regions bounded by the red boxes are zoomed in sequentially with zoom-in rate 2. All zoomed-in regions are resized to the same resolution for visualization. The white regions in (a), (b) and (c) are the background glass slide.}
\label{fig:zoom}
\end{figure}
This paper makes the following contributions: 1) we propose an innovative framework for semantic segmentation for images with high resolution by leveraging both accuracy and speed; 2) we are the first to apply reinforcement learning to breast cancer segmentation; 3) we compare our framework empirically with multi-scale techniques used in the domain of computer vision and discuss the influence of multi-scale models for breast cancer segmentation.
\section{Reinforced Auto-Zoom Net}
In clinical practice, it is impossible for a clinician to go through each region of a WSI at the original resolution, due to the huge image size. The clinician views regions with simple patterns or high confidence quickly at coarse resolution and zooms in for the suspicious or uncertain regions to study the cells at high resolution. The proposed RANZ simulates the examining process of a clinician diagnosing breast cancer on a WSI. Another motivation of RAZN is that the characteristics of the cancer cells have different representations at different field of view. For semantic segmentation tasks on common objects, the objects in the same category share discriminative features and attributes. For example, we can differentiate a cat from a dog based on the head, without viewing the whole body. However, in cancer segmentation, the basic unit is the cell, which consists of nucleus and cytoplasm. The difference between the cells is not obvious. Instead of checking only a single cell, the diagnosis is based on the features of a group of cells, such as the density, the clustering and the interaction with the environment. RANZ is designed to learn this high-level information.
RAZN consists of two types of sub-networks, policy networks $\{f_\theta\}$ and segmentation networks $\{g_\phi\}$. Assume the zoom-in actions can be performed at most $m$ times and the zoom-in rate is $r$. There is one base segmentation network $f_{\theta_0}$ at the coarsest resolution. At the $i$th zoom-in level, there is one policy network $g_{\phi_i}$ and one segmentation network, $f_{\theta_i}$. In the inference time, with fixed field of view and magnification level, we have a cropped patch $x_0$ with shape $[H,W,3]$, like Fig.~\ref{fig:zoom} (a). Then $g_{\phi_1}$ will take $x_0$ as an input and predict the action, \textit{zoom-in} or \textit{break}. If the predicted action is break, $f_{\theta_0}(x_0)$ will output the segmentation results and the diagnosis for $x_0$ is finished. If the predicted action is zoom-in, a high-magnification patch $\bar{x}_0$ with corresponding zoom-in rate will be retrieved from the original image. $\bar{x}_0$, with shape $[rH,rW,3]$, will be cropped into $x_1$, which is $r^2$ patches of shape $[H,W,3]$. Then each patch of $x_1$ will be treated as $x_0$ for the next level of zoom-in action. Fig.~\ref{fig:zoom} (b) is a central crop of $x_1$. The process is repeated recursively until a pre-defined maximum magnification level is reached. In this work, we propose this novel idea and focus on the situation of $m = 1$. $m > 1$ will be discussed in future work. An overview of the architecture is illustrated in Fig.~\ref{fig:arch}.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.9\linewidth]{figures/arch.png}
\end{center}
\caption{Illustration of the proposed framework when $m = 1$ and $r = 2$. In the inference phase, given a cropped image $x_0$, the policy network outputs the action, zoom-in (red arrows) or break (blue arrows). In the training phase, the policy network will be optimized to maximize the reward (purple arrows), which is determined by the segmentation prediction.}
\label{fig:arch}
\end{figure}
The segmentation networks are Fully Convolutional Networks (FCNs) \cite{long2015fully} and share the same architecture. However, unlike parameter sharing in the common multi-scale training in semantic segmentation~\cite{chen2018deeplab}, each network is parameterized by independent $f_\theta$, where $f_{\theta_i}: \mathbb{R}^{H \times W \times 3} \rightarrow \mathbb{R}^{H \times W \times C}$ and $C$ is the number of classes. The reason for choosing independent networks for each zoom-in level is that CNNs are not scale-invariant \cite{goodfellow2016deep}.
Each FCN can thus learn high-level information at a specific magnification level. Given input image $x$ and segmentation annotation $y$, the training objective for each FCN is to minimize
\begin{equation}
J_{\theta_i}(x, y) = - \frac{1}{HW}\sum_{j}\sum_{c} y_{j,c}\log f_{\theta_i}(x)_{j,c} \; ,
\label{eq:1}
\end{equation}
where $j$ ranges over all the $H\times W$ spatial positions and $c \in \{0,...,3\}$ represents the semantic classes (cancer type).
At $m = 1$, the framework is a single-step Markov Decision Process (MDP) and the problem can be formulated by the REINFORCE rule \cite{williams1992simple}. The policy network projects an image to a single scalar, $g_{\phi_1}: \mathbb{R}^{H \times W \times 3} \rightarrow \mathbb{R}$. Given the state $x_0$, the policy network defines a policy $\pi_{\phi_1}(x_0)$. The policy samples an action $a \in \{0, 1\}$, which represents break and zoom-in, respectively. We have
\begin{equation}
\textit{p} = \sigma(g_{\phi_1}(x_0)) \;,
\label{eq:2}
\end{equation}
\begin{equation}
\pi_{\phi_1}(x_0) = \textit{p}^a (1-\textit{p})^{1-a} \; ,
\label{eq:3}
\end{equation}
where $\sigma(\cdot)$ is the sigmoid function and $\pi_{\phi_1}(x_0)$ is essentially a Bernoulli distribution. The motivation of RAZN is to improve the segmentation performance and it is therefore natural to define the reward such that it minimizes the segmentation loss. Based on Equation~\ref{eq:1}, we have $J_{\theta_0}(x_0, y_0)$, $J_{\theta_1}(x_1, y_1)$, where $x_1$ is the transformed $x_0$ after zoom-in and cropping operations. It is practical in reinforcement learning training to utilize the advantage function to reduce variance \cite{rennie2017self} and we therefore define the reward as
\begin{equation}
\textit{R}(a) = a \frac{J_{\theta_1}(x_1, y_1) - J_{\theta_0}(x_0, y_0)}{J_{\theta_0}(x_0, y_0)}.
\label{eq:4}
\end{equation}
So when $a = 1$, the reward is positive if $J_{\theta_1}(x_1, y_1) > J_{\theta_0}(x_0, y_0)$, and the reward is negative if $J_{\theta_1}(x_1, y_1) < J_{\theta_0}(x_0, y_0)$ . The denominator in Equation~\ref{eq:4} functions as a normalizer to prevent reward explosion. To prevent $\textit{p}$ from saturating at the beginning, we adopt the bounded Bernoulli distribution
\begin{equation}
\tilde{\textit{p}} = \alpha \textit{p} + (1 - \alpha) (1 - \textit{p}).
\label{eq:5}
\end{equation}
We have $\tilde{\textit{p}} \in [1-\alpha, \alpha]$. The training objective is to maximize the expected reward or to minimize the negative expected reward
\begin{equation}
J_{\phi_1}(x_0) = - \mathbb{E}_{a \sim \pi_{\phi_1}(x_0)}[\textit{R}(a)].
\label{eq:6}
\end{equation}
The optimization of the policy network is implemented through policy gradient methods \cite{williams1992simple,sutton1998reinforcement,sutton2000policy}, where the expected gradients are
\begin{equation}
\frac{\partial}{\partial \phi_1} J_{\phi_1}(x_0) = - \mathbb{E}_{a \sim \pi_{\phi_1}(x_0)} [\textit{R}(a) \frac{\partial}{\partial \phi_1} \text{log}(a \tilde{\textit{p}} + (1 - a)(1 - \tilde{\textit{p}}))]
\label{eq:7}
\end{equation}
We adopt an alternating training strategy to update both networks. The training procedure of RAZN is illustrated in Algorithm~\ref{algo:1}.
\begin{algorithm}[t]
\caption{Training of RAZN when $m = 1$}
\label{algo:1}
\begin{algorithmic}[1]
\Require $x_0$
\State Get $J_{\theta_0}(x_0, y_0)$ and $J_{\theta_1}(x_1, y_1)$
\State Sample action $a$ through $\pi_{\phi_1}(x_0)$
\State Get $\textit{R}(a)(x_0)$
\State Update $\phi_1$ by minimizing $J_{\phi_1}(x_0)$
\If{$a = 1$}
\State Update $\theta_1$ by minimizing $J_{\theta_1}(x_1, y_1)$
\Else
\State Update $\theta_0$ by minimizing $J_{\theta_0}(x_0, y_0)$
\EndIf
\end{algorithmic}
\end{algorithm}
\section{Experiments}
\paragraph{\bf{Dataset}}
The dataset used in this study is provided by Grand Challenge on Breast Cancer Histology Images \footnote{https://iciar2018-challenge.grand-challenge.org/dataset}. The dataset contains 10 high-resolution WSIs with various image size. WSIs are scanned with Leica SCN400 at $\times 40$ magnification.
The annotation was performed by two medical experts. As annotation of WSIs requires a large amount of human labor and medical domain knowledge, only sparse region-level labels are provided and annotations contain pixel-level errors. In this dataset, the white background (glass slide) is labeled as \textit{normal} by the annotators. The dataset is unbalanced for the four cancer types.
\paragraph{\bf{Implementation}}
Experiments are conducted on a single NVIDIA GTX Titan X GPU. In this study, $m = 1$, $r = 2$ and $\alpha = 0.8$. The backbone of $f_{\theta_i}$ is ResNet18 \cite{he2016deep}, with no downsampling performed in conv3\_1 and conv4\_1. $g_{\phi_1}$ is also based on the ResNet18 architecture. However, each block (consisting of 2 residual blocks \cite{he2016deep}) is replaced by a $3 \times 3$ convolution followed by batch normalization and ReLU non-linearity.
The computational cost for the policy network is $7.1\%$ of the segmentation networks. The input size to the segmentation networks and the policy network is fixed to $256 \times 256$. We use the Adam optimizer \cite{kingma2015adam} for both the policy network and segmentation networks and use a step-wise learning rate policy with decay rate 0.1 every 50000 iterations. The initial learning rate is 0.01.
\paragraph{\bf{Multi-scale}}
Given a $256 \times 256$ patch, we consider two resolutions in order to simulate the zoom-in process. A coarse resolution (Scale 1), where the patch is downsampled to $64 \times 64$ and a fine resolution patch (Scale 2), where the patch is downsampled to $128 \times 128$. The patches are then resized back to $256 \times 256$ using bilinear interpolation. To evaluate the efficiency of the proposed framework, we compare our model with two multi-scale models. The first multi-scale model is the segmentation network $f_\theta$ with multi-scale training \cite{chen2018deeplab}, denoted as MS. We only consider two scales in this experiment (Scale 1 and Scale 2). Similarly, another multi-scale model is the multi-scale fusion with attention \cite{chen2016attention}, which is denoted as Attention. The training details of all models are the same. All models are trained with 200000 batches.
\begin{table}[t]
\centering
{\setlength{\tabcolsep}{1pt}
\begin{tabular}{ccccc|c}
\hline
& non-carcinoma & carcinoma & mIOU & Weighted IOU & Relative Inference Time\\ \Xhline{4\arrayrulewidth}
Scale 1 & 0.45 & 0.32 & 0.38 & 0.07 & 1.00 \\ \hline
Scale 2 & 0.46 & 0.31 & 0.39 & 0.07 & 4.01 \\ \hline
MS \cite{chen2018deeplab} & 0.32 & 0.04 & 0.18 & 0.01 &5.06\\ \hline
Attention \cite{chen2016attention} & 0.43 & 0.29 & 0.36 & 0.06 & 5.16\\ \hline
RAZN & 0.49 & 0.49 & 0.49 & 0.11 & 2.71 $\pm$ 0.57 \\ \hline
\end{tabular}
}
\caption{Comparison of the performance. Non-carcinoma includes \textit{normal} and \textit{beign}. Carcinoma includes \textit{in situ carcinoma} and \textit{invasive carcinoma}.}
\label{tab:1}
\end{table}
\paragraph{\bf{Performance}}
We compare two key indicators of the performance, which are the segmentation performance and the inference speed. We use intersection over union (IOU) as the metric for segmentation performance. We report mean IOU, which is just the average IOU among four classes. Due to the imbalanced data, we also report weighted IOU, where the weight is proportional to the inverse of the frequency of the labels of each class.
Further, we report relative inference time for the proposed RAZN and the baseline methods compared to the inference time for the model that only considers Scale 1. We report the average relative inference time over 100 patches. Lower values of relative inference time represent faster inference speed. The results are presented in Table~\ref{tab:1}. Note, we report the mean and the standard deviation for RAZN, as the inference time will vary depending on whether zooming is required for a given patch or not.
It can be shown that RAZN actually performs better than the single scale and the multi-scale baselines.
MS's performance is the worst of our benchmarks. MS exaggerates the imbalance problem by augmenting the data, which can confuse the network. We also hypothesize that the cell size is not the critical factor that influences the segmentation of cancer and that MS, therefore, aims to model unnecessary information on this task.
Similarly, attention models memorize the scale of the object by fusing the results from different scales.
However, when the object is not well-defined at certain scales, like in our task the cancer (group of dense cells), the network may learn to fit noise.
Our results illustrate that RAZN instead is more robust when data is noisy and imbalanced, providing an overall accuracy improvement at low inference time.
\section{Discussion and Conclusions}
We proposed RAZN, a novel deep learning framework for breast cancer segmentation in WSI, that uses reinforcement learning to selectively zoom in on regions of interest. The results show that the proposed model can achieve improved performance, while at the same time reduce inference speed compared to previous multi-scale approaches. We also discuss the use of multi-scale approaches for breast cancer segmentation. We conclude that cancer cells are different from general objects due to their relative small and fixed size. Multi-scale approaches may not work for a noisy and imbalanced data.
In future work, we aim to extend the model to study the multiple zoom-in actions situation ($m>1$) and will investigate the potential of more complex segmentation backbone models to improve overall performance.
\vspace{0.3cm}
\noindent{\bf{Acknowledgements.}} We thank ICIAR 2018 Grand Challenge on Breast Cancer Histology Images for providing the data for this study.
\bibliographystyle{splncs03}
|
2,869,038,154,810 | arxiv | \section{INTRODUCTION}
The developments of new radioactive ion beam facilities and new detection techniques have largely extended our knowledge of nuclear
physics from stable nuclei to unstable nuclei far from the $\beta$-stability line, the so-called exotic nuclei. Novel and striking features have
been found in the nuclear structure of exotic nuclei, such as the halo
phenomenon~\cite{tanihata1985measurements,meng1996relativistic,meng1998giant,zhou2010neutron,Meng2015Halos} and the disappearance of traditional
magic numbers and occurrence of new ones~\cite{ozawa2000new}.
In order to describe the exotic nuclei with large space distribution, theoretical approaches should be developed in coordinate space or coordinate-equivalent space.
The density functional theory (DFT) and its covariant version (CDFT) have been proved to be effective theories for the description of exotic
nuclei~\cite{meng1996relativistic, meng1998giant,meng2016relativistic, PPNP2006,mengNPA98, dobaczewski1996mean, Bender2003Self, Pei2008Deformed}.
In comparison with its nonrelativistic counterpart, the CDFT has many attractive advantages, such as the natural
inclusion of nucleon spin freedom, new saturation property of nuclear matter~\cite{Volum16, RING1996PPNP, meng2016relativistic}, large spin-orbit splittings in
single particle energies, reproducing the isotopic shifts of Pb isotopes~\cite{sharma1993anomaly}, natural inclusion of time-odd mean field, and
explaining the pseudospin of nucleons and spin symmetries of antinucleons in
nuclei~\cite{ginocchio1997pseudospin,meng1998pseudospin,zhou2003spin,liang2015hidden}.
In most CDFT applications, the harmonic oscillator basis expansion method has been widely used, which is an very efficient approach and has achieved a great success in not only the description of the single-particle motion in nuclei \cite{liang2015hidden} but also the self-consistent description of nuclear collective modes, such as rotations \cite{Afanasjev1999PhysicsReport,Peng2008maganetic_roration,Yao2014searching,Zhao2015cranking_RHB,Zhao2015Rod-shaped,Afanasjev2016superdeform}, vibrations \cite{Niksic2002DDME1, Paar2004QRPA_spin&isospin, Vretenar2005PhysicsReport, Niksic2006BRMF_config&mix, Litvinova2006CDFT&PVC, Yao2008CPL3DAngularMP, Niksic2009BRMF_5DCH, Yao2009&3DAMP, Yao2010Config&mix_amp, LiZP20105DCH, Yao2011Config&mix_lowlying, Litvinova2011Dynamics&PVC, LiZP20115DCH, Li2013Simutaneous, Yao2014Kr74, Zhou2016molecularNe20}, and isospin excitations, by restoring the symmetries and/or considering quantum fluctuations, see also \cite{meng2016relativistic} for details.
For exotic nuclei with large spatial distribution, a large basis space is needed to get a quick convergence.
Due to the incorrect asymptotic behavior of the harmonic oscillator wave functions, this method is not appropriate for halo or giant halo nuclei \cite{dobaczewski1996mean,zhou2003spherical,PPNP2006}.
In contrast, the solution of the Dirac equation for single nucleons in coordinate space or coordinate-equivalent space is preferred.
For the spherical system, the conventional shooting method works quite well \cite{mengNPA98}, which however is rather complicated for the deformed system~\cite{Price1987Self-consistent}.
Therefore the Dirac Woods-Saxon basis expansion method was developed~\cite{zhou2003spherical} and has been widely used to solve the deformed Dirac
equation~\cite{zhou2010neutron,li2012deformed}, which, however, is highly computationally time consuming for the heavy system.
The imaginary time method (ITM)~\cite{davies1980imaginary} is a powerful approach for the self-consistent mean-field calculations in a
three-dimensional (3D) coordinate space.
The ITM has been successfully employed in nonrelativistic self-consistent mean-field
calculations~\cite{bonche2005solution,Maruhn2014CPC}. For a long time, there exist doubts about the access of the ITM to the Dirac equation due to the Dirac sea, i.e., the relativistic ground state within the Fermi sea is a saddle point rather than a minimum. This is the so-called variational collapse
problem~\cite{zhang2009first,zhang2009solving,ZhangIJMPE2010,hagino2010iterative,tanimura20153d}. To avoid the variational collapse, Zhang \textit{et al }~\cite{zhang2009first,ZhangIJMPE2010} applied the ITM to the Schr\"{o}dinger-equivalent form of the Dirac equation in the spherical case. The same
method is used to solve the Dirac equation with a nonlocal potential in Refs.~\cite{zhang2009first,zhang2009solving}. Based on the idea of Hill and
Krauthauser~\cite{hill1994solution}, Hagino and Tanimura proposed the inverse Hamiltonian method (IHM) to avoid variational
collapse~\cite{hagino2010iterative}. This method solves the Dirac equation directly and the Dirac spinor is obtained
simultaneously.
Meanwhile when the IHM method is applied to lattice space in numerical calculations, another challenge appears, i.e., fermion doubling problem~\cite{salomonson1989relativistic,tanimura20153d} due to the replacement of the
derivative by the finite difference method~\cite{salomonson1989relativistic,tanimura20153d}.
This problem appears also in lattice quantum chromodynamics (QCD)~\cite{wilson1977new,kogut1983lattice}, which has been solved by Wilson's fermion method~\cite{wilson1977new,kogut1983lattice}. In Ref.~\cite{tanimura20153d}, Tanimura, Hagino, and Liang followed the same idea and realized the relativistic calculations on 3D lattice by introducing high-order Wilson term. However, the high-order Wilson term modified the original Dirac Hamiltonian and the single particle energies and wave functions need be corrected. Although the corrections can be done with the perturbation theory, numerically it is much more involved.
Another problem is that the high-order Wilson term introduces artificial symmetry breaking to the system~\cite{tanimura20153d}.
In this paper, we propose a new recipe for the imaginary time method to solve the Dirac equation in 3D lattice space, where the variational collapse
problem is avoided by the IHM, and the Fermion doubling problem is avoided by performing the spatial derivatives of the Dirac equation in momentum space
with the help of discrete Fourier transform, the so-called spectral method~\cite{Shen2011Spectral}.
This method is demonstrated by solving the Dirac equation for a given spherical potential in 3D lattice space and comparing with the results obtained by the shooting method.
By extending this method to solve the Dirac equations for an axial deformed, non-axial deformed, and octupole deformed potential, the corresponding single particle energy levels are obtained.
The corresponding quantum numbers of these energy levels are obtained respectively by projection.
The paper is organized as follows, the variational collapse and the Fermion doubling problems will be briefly
introduced in Sec. \ref{numerical_method} together with the inversion Hamiltonian method and the spectral method. In Sec. \ref{Numericaldetails} the parameters for Woods-Saxon type potentials and the numerical details are presented. Sec. \ref{results&discussion} is devoted to results and discussions. Summary and perspectives are given in Sec. \ref{summary}.
\section{THEORETICAL FRAMEWORK}\label{numerical_method}
\subsection{VARIATIONAL COLLAPSE AND INVERSE HAMILTONIAN METHOD}
\subsubsection{IMAGINARY TIME METHOD}
The ITM is an iterative method for mean-field problem. The idea of ITM is to replace time with an imaginary number, and the evolution of the wave function reads~\cite{davies1980imaginary},
\begin{equation}
\textrm{e}^{-{\rm i}\hat{h}t}|\psi_0\rangle\xrightarrow{t\rightarrow-{\rm i}\tau}\textrm{e}^{-\hat{h}\tau}|\psi_0\rangle,
\end{equation}
where $|\psi_0\rangle$ is an initial wave function and $\hat{h}$ is the Hamiltonian.
With the eigenstates $\{\phi_k\}$ of the Hamiltonian $\hat{h}$ corresponding to the eigenenergies $\{\varepsilon_k\}$, the evolution of the wave function
$|\psi(\tau)\rangle=\textrm{e}^{-\hat{h}\tau}|\psi_0\rangle$ can be written as,
\begin{equation}\label{Eq_imaginary}
|\psi(\tau)\rangle=\textrm{e}^{-\hat{h}\tau}|\psi_0\rangle=\sum_k\textrm{e}^{-\varepsilon_k\tau}|\phi_k\rangle\langle\phi_k|\psi_0\rangle,
\end{equation}
where $\varepsilon_1\leq\varepsilon_2\leq\cdots$.
For $\tau\rightarrow\infty$, $|\psi(\tau)\rangle$ will approach the ground state
wave function of $\hat{h}$ as long as $\langle\phi_1|\psi_0\rangle\neq0$.
In practice, the imaginary time $\tau$ is discrete with the interval $\Delta \tau$, i.e., $\tau = N \Delta \tau$.
The wave function at $\tau=(n+1)\Delta \tau$ is obtained from the wave function at $\tau=n\Delta \tau$ by expanding the exponential evolution operator $\textrm{e}^{-\Delta \tau\hat{h}}$ to the linear order of $\Delta \tau$,
\begin{equation}\label{Eq_iterative}
|\psi^{(n+1)}\rangle\propto\left(1-\Delta \tau\hat{h}\right)|\psi^{(n)}\rangle.
\end{equation}
Since this evolution is not unitary, the wave function should be normalized at every step.
In order to find excited states, one can start with a set of initial wave functions and orthonormalize them during the evolution by the Gram-Schmidt method. This method has been successfully
employed in the 3D coordinate-space calculations for nonrelativistic systems \cite{bonche2005solution,Maruhn2014CPC}.
\subsubsection{VARIATIONAL COLLAPSE}
For the static Dirac equation,
\begin{equation}\label{Dirac_equation}
\{-\textrm{i}\bm{\alpha\cdot\nabla}+V(\bm{r})+\beta[m+S(\bm{r})]-m\}\psi(\bm{r})=\varepsilon\psi(\bm{r}),
\end{equation}
with $\bm{\alpha}$ and $\beta$ the Dirac matrix, $V(\bm{r})$ the vector potential, $S(\bm{r})$ the scalar potential, and $\psi(\bm{r})$ the Dirac spinor, its eigenenergy spectrum extends from the continuum in the Dirac sea to the continuum in the Fermi sea.
Because of the existence of the Dirac sea, the evolution in Eq. \eqref{Eq_imaginary} inevitably dives into the Dirac sea (negative energy states) as $\tau\rightarrow\infty$, which is the so-called variational collapse problem~\cite{ZhangIJMPE2010}.
\subsubsection{INVERSE HAMILTONIAN METHOD}
To avoid the variational collapse, Hagino and Tanimura proposed the inverse Hamiltonian method~\cite{hagino2010iterative} to find the wave function of the
Dirac Hamiltonian $\hat{h}$ by,
\begin{equation}\label{Eq_inverseH}
\lim_{\tau\rightarrow\infty}\textrm{e}^{\tau/(\hat{h}-W)}|\psi_0\rangle,
\end{equation}
where $W$ is an auxiliary parameter introduced to locate the interested eigenstate.
With a given $W$, the spectrum of $\hat{h}$ can be labeled as
\begin{equation}
\cdots\leq\varepsilon_{-2}\leq\varepsilon_{-1}<W<\varepsilon_1\leq\varepsilon_2\leq\cdots,
\end{equation}
where $\cdots, \varepsilon_{-2},\varepsilon_{-1}$ and $\varepsilon_{1}, \varepsilon_{2}, \cdots$ are the eigenenergies of the Dirac Hamiltonian $\hat{h}$. Accordingly, the spectrum of $1/(\hat{h}-W)$ reads,
\begin{equation}
\frac{1}{\varepsilon_{-1}-W}\leq\frac{1}{\varepsilon_{-2}-W}\leq\cdots \leq\frac{1}{\varepsilon_{2}-W}\leq\frac{1}{\varepsilon_{1}-W}.
\end{equation}
The evolution of the wave function in Eq. \eqref{Eq_inverseH} will lead to the eigen wave function $|\phi_1\rangle$ corresponding to the eigenvalue $\varepsilon_{1}$,
\begin{align}\label{Eq_iterativeIHM}
&\lim_{\tau\rightarrow\infty}\textrm{e}^{\tau/(\hat{h}-W)}|\psi_0\rangle\nonumber\\
&~~=\lim_{\tau\rightarrow\infty}\sum_k\textrm{e}^{\tau/(\varepsilon_k-W)}
|\phi_k\rangle\langle\phi_k|\psi_0\rangle\propto|\phi_1\rangle,
\end{align}
as long as $\langle\phi_1|\psi_0\rangle\neq0$.
In practice, the imaginary time evolution in Eq. \eqref{Eq_inverseH} is performed iteratively,
\begin{equation}
|\psi^{(n+1)}\rangle \propto \left(1+\frac{\Delta \tau}{\hat{h}-W}\right)|\psi^{(n)}\rangle.
\label{Eq_inversWF}
\end{equation}
The wave function also should be normalized at every step.
The inverse of the Hamiltonian in Eq. \eqref{Eq_inversWF},
$\displaystyle \frac{\Delta \tau} {\hat{h}-W} | \psi^{(n)}\rangle$,
can be solved iteratively by the conjugate residual method~\cite{saad2003iterative}.
To find excited states, with a set of initial wave functions there are two options for choosing $W$.
One can take a fixed $W$, then evolve the set of wave functions and orthonormalize them during the evolution by the Gram-Schmidt method.
Alternatively, one can take the set of $W_i$ for each eigenstate $i$ to evolve the whole set of wave functions. The details can be found in Sec.~\ref{Numericaldetails}, where an efficient method for choosing $W_i$ is suggested to achieve a fast convergence.
\subsection{FERMION DOUBLING PROBLEM AND SPECTRA METHOD}
\subsubsection{FERMION DOUBLING PROBLEM}
For a Dirac equation on 3D lattice, there exists a so-called Fermion doubling problem due to
the replacement of the first derivatives in the Dirac equation \eqref{Dirac_equation} by the finite difference
method \cite{salomonson1989relativistic,tanimura20153d}. Taking the one-dimensional Dirac equation as an example,
\begin{equation}\label{Eq_1dDirac}
(-\textrm{i}\alpha\partial_x+\beta m)\psi(x)=\varepsilon\psi(x),
\end{equation}
its solution has the form
\begin{equation}
\psi(x)=\tilde{\psi}(k)\exp(\textrm{i}kx).
\end{equation}
If one approximates the derivative $\partial_x$ in Eq. \eqref{Eq_1dDirac} with a three-point differential formula with the mesh size $d$, the Dirac
equation \eqref{Eq_1dDirac} becomes,
\begin{equation}\label{Eq_3pointDirac}
\left[\frac{1}{d}\alpha\sin(kd)+\beta m\right]\tilde{\psi}(k)=\varepsilon\tilde{\psi}(k).
\end{equation}
The dispersion relation obtained from Eq.~\eqref{Eq_3pointDirac} reads,
\begin{equation}\label{Eq_3pointDispersion}
\varepsilon^2=\frac{1}{d^2}\sin^2(kd)+m^2,
\end{equation}
which differs from the exact one,
\begin{equation}\label{Eq_exactDispersion}
\varepsilon^2=k^2+m^2.
\end{equation}
For the dispersion relation \eqref{Eq_3pointDispersion} obtained with the three-point differential formula, there are two momenta corresponding to one energy in the momentum interval $[0,d/\pi]$. The lower momentum corresponds to the physical solution, while the higher momentum corresponds to a spurious solution. As illustrated in Ref.~\cite{tanimura20153d}, this problem persists even with the more accurate finite differential formula.
Similar spurious solution problem in radial Dirac equations are also demonstrated in Ref.~\cite{zhao2016spherical}.
\subsubsection{SPECTRAL METHOD}
To avoid the Fermion doubling problem, the derivative in Eq.~\eqref{Eq_1dDirac} can be performed in momentum space,
\begin{equation}
\left[\alpha k+\beta m\right]\tilde{\psi}(k)=\varepsilon\tilde{\psi}(k),
\end{equation}
which yields the exact dispersion relation; i.e., the fermion doubling problem is avoided naturally.
This is the so-called spectral method, i.e., to perform spatial derivatives in momentum space. In the following, this method is illustrated in a 1D case and it is straightforward to generalize this method to the 3D case.
We assume that there are even $n_x$ discrete grid points $x_\nu$ in coordinate space distributing symmetric with
the origin point,
\begin{equation}
x_\nu=\left(-\frac{n_x-1}{2}+\nu-1\right)dx,~~\nu=1,...,n_x,
\end{equation}
same number of grid points $k_\mu$ in momentum space,
\begin{equation}
k_\mu=
\begin{cases}
(\mu-1)dk,&\mu=1,...,n_x/2,\\
(\mu-n_x-1)dk,&\mu=n_x/2+1,...,n_x,
\end{cases}
\end{equation}
and the steps in coordinate space $dx$ and in momentum space $dk$ are related by,
\begin{equation}
dk=\frac{2\pi}{n_x\cdot dx}.
\end{equation}
The function in coordinate space $f(x_\nu)$ and the function in momentum space $\tilde{f}(k_\mu)$ are connected by
the discrete Fourier transform,
\begin{subequations}
\begin{align}
&\tilde{f}(k_\mu)=\sum_{\nu=1}^{n_x}\exp(-\textrm{i} k_\mu x_\nu)f(x_\nu),\label{Eq_DFT}\\
&f(x_\nu)=\frac{1}{n_x}\sum_{\mu=1}^{n_x}\exp(\textrm{i} k_\mu x_\nu)\tilde{f}(k_\mu).\label{Eq_inversDFT}
\end{align}
\end{subequations}
From Eq.\eqref{Eq_inversDFT}, the $m$-th order derivative of $f(x_\nu)$ can be found as,
\begin{equation}
\begin{split}
f^{(m)}(x_\nu)&=\frac{1}{n_x}\sum_{\mu=1}^{n_x}\exp(\textrm{i} k_\mu x_\nu)(\textrm{i} k_\mu)^m\tilde{f}(k_\mu)\\
&=\frac{1}{n_x}\sum_{\mu=1}^{n_x}\exp(\textrm{i} k_\mu x_\nu) \tilde{f}^{(m)}(k_\mu).
\end{split}
\end{equation}
Here $\tilde{f}^{(m)}(k_\mu)$ corresponds to the Fourier transform of the $m$-th order derivative of $f(x_\nu)$,
\begin{equation}\label{Eq_f&fm}
\tilde{f}^{(m)}(k_\mu)=(\textrm{i} k_\mu)^m\tilde{f}(k_\mu).
\end{equation}
In summary, the procedures to perform derivatives in coordinate space are as follows:
(1) calculate $\tilde{f}(k_\mu)$ from $f(x_\nu)$ by the discrete Fourier transform in Eq.~\eqref{Eq_DFT};
(2) calculate $\tilde{f}^{(m)}(k_\mu)$ by Eq. \eqref{Eq_f&fm};
(3) calculate the $m$-th order derivative $\tilde{f}^{(m)}(x_\nu)$ from $\tilde{f}^{(m)}(k_\mu)$ by
the inverse discrete Fourier transform as in Eq. \eqref{Eq_inversDFT}.
The spectral method has the advantage to perform the spatial derivatives with a good accuracy. The information of all grids is used in calculating the spatial derivative of any grid. Different from the finite differential method,
all grids are treated on the same footing and the grids near the boundaries do not need special numerical techniques.
\section{NUMERICAL DETAILS}\label{Numericaldetails}
In the following, we will solve the Dirac equation on 3D lattice in which the variational collapse problem is avoided by the inverse Hamiltonian method, and the fermion doubling problem is avoided by performing spatial derivatives in momentum space with the help of the discrete Fourier transform, i.e., spectral method.
The vector potential $V(\bm{r})$ and the scalar potential $S(\bm{r})$ in Eq.~\eqref{Dirac_equation} are Woods-Saxon type potentials satisfying,
\begin{equation}\label{Eq_potential}
\begin{split}
&V(\bm{r})+S(\bm{r})=\frac{V_0}{1+\exp[(r-R_0F(\Omega))/a]},\\
&V(\bm{r})-S(\bm{r})=\frac{-\lambda V_0}{1+\exp[(r-R_{ls}F(\Omega))/a_{ls}]},
\end{split}
\end{equation}
where $F(\Omega)$ is a function of $\Omega=(\theta, \varphi)$ with potential deformation parameters $\beta_{20}$, $\beta_{22}$ and $\beta_{30}$,
\begin{equation}\label{Eq_deformed}
F(\Omega)=1+\beta_{20}Y_{20}(\Omega)+\beta_{22}[Y_{22}(\Omega)+Y_{2(-2)}(\Omega)]+\beta_{30}Y_{30}(\Omega).
\end{equation}
The deformation parameters $\beta_{20}$ and $\beta_{22}$ in Eq.~\eqref{Eq_deformed} are related to Hill-Wheeler coordinates $\beta$ and $\gamma$~\cite{Hill1953PhysicalReview,ring2004nuclear} by
\begin{equation}\label{Eq_HWcoordinates}
\begin{cases}
\beta_{20}=\beta\cos\gamma,\\
\beta_{22}=\frac{1}{\sqrt{2}}\beta\sin\gamma.
\end{cases}
\end{equation}
The adopted Woods-Saxon potential parameters in Eq. \eqref{Eq_potential} are listed in Table \ref{Tab_parameter}, which correspond to the neutron potential in
$^{48}$Ca~\cite{koepf1991WoodsSaxon}.
\begin{table}[h
\caption{\label{Tab_parameter}The parameters in the Woods-Saxon type potential Eq. \eqref{Eq_potential} adopted in the present 3D lattice calculations.}
\begin{ruledtabular}
\begin{tabular}{cccccc}
$V_0$ [MeV] &$R_0$ [fm] &$a$ [fm] &$\lambda$ &$R_{ls}$ [fm] &$a_{ls}$ [fm]\\
\hline
-65.796 &4.482 &0.615 &11.118 &4.159 &0.648
\end{tabular}
\end{ruledtabular}
\end{table}
In the calculations, the box sizes $L=23$~fm and step sizes $d=1$~fm are respectively chosen along $x$, $y$ and $z$ axes if not otherwise specified. The imaginary time step size $\Delta T$ is taken 100~MeV.
For the $i$-th level, the upper component of the initial wave function is generated from a nonrelativistic harmonic oscillator state and the corresponding lower component is taken the same as the upper one.
The energy shift $W_i$ is taken as
\begin{equation}
W_i=\varepsilon_i - \Delta W_i,
\end{equation}
where $\varepsilon_i$ is the expectation value of the Dirac Hamiltonian for the $i$-th level.
The choice of $\Delta W_i$ is as follows: $\Delta W_1=6$ MeV and for $i > 1$,
\begin{equation}
\begin{split}
&~~\Delta W_i =
\begin{cases}
\varepsilon_i-\varepsilon_{i-1},&\varepsilon_i-\varepsilon_{i-1} > \Delta W_1 \\
\Delta W_{i-1},&\varepsilon_i-\varepsilon_{i-1}\leqslant \Delta W_1
\end{cases}
\end{split}
\end{equation}
The convergence in the evolution of the wave functions for our interested states is determined by $\sqrt{\langle \hat{h}^2 \rangle_i - \langle\hat{h}\rangle_i^2}$ smaller than the required accuracy
$\delta_i = 10^{-4}$~MeV if not otherwise specified.
To speed up the convergence, the Dirac Hamiltonian is diagonalized within the space of the evolution wave functions every 10 iterations, and the eigenfunctions thus obtained are taken as initial wave functions for future iteration. A similar technique is also used in Ref.~\cite{Maruhn2014CPC}.
\section{RESULTS AND DISCUSSION}\label{results&discussion}
\subsection{SPHERICAL POTENTIAL}
In this section, the Dirac equation with a given potential is solved in 3D lattice space by the new method (denoted as \textit{3D lattice}). First we examine the convergence feature of the present 3D lattice calculation for a spherical potential in Eq. \eqref{Eq_potential}.
The results will be compared with those obtained by the shooting method (denoted as \textit{shooting})~\cite{mengNPA98} with a box size $R=20$~fm and a step size $dr=0.01$~fm.
With the potential parameters in Table \ref{Tab_parameter}, the evolution of single particle energies as a function of iteration times is shown in Fig. \ref{Fig_iteration}. There are in total of 40 bound single particle states in the 3D lattice calculation and some of them are degenerate in energy due to the spherical symmetry.
For clarity, only one energy level of the degenerate ones is shown to illustrate the evolution of single particle energies.
The single particle energies obtained by the shooting method are also shown for comparison. It can be seen that the deeper levels converge more quickly. After the 39th iteration, the accuracy of energy for all bound levels is smaller than $10^{-4}$~MeV. A distinct feature is observed at the 10th iteration where the convergence of 1p$_{1/2}$, 1d$_{3/2}$, and 2s$_{1/2}$ states is speeded up due to the diagonalization of the Hamiltonian within the space of the evolution wave functions. In fact, it will cost tens of thousands of iteration steps to reach the convergence tolerance without this diagonalization procedure.
\begin{figure}
\centering
\includegraphics[width=0.45\textwidth]{Fig1_SP_EE}\\
\caption{Evolution of single particle energies in the spherical Woods-Saxon potential in Eq. \eqref{Eq_potential} as a function of
iteration times. Convergence is achieved after the 39th iteration where the energy dispersions of all bound single particle levels are
smaller than $10^{-4}$~MeV. As a comparison, the results obtained by the shooting method are also given.
}\label{Fig_iteration}
\end{figure}
In Fig. \ref{Fig_Energydifference}, the absolute deviations of single particle energies between the 3D lattice calculation and the shooting method are given as a function of single particle
energy for different step sizes $d$ and box sizes $L$.
In Fig. \ref{Fig_Energydifference} (a), for $d=1.0$ fm and $L=23.0$ fm, the absolute deviations of single particle energies are smaller than $10^{-3}$ MeV, except the weakly
bound states 1f$_{5/2}$, 2p$_{3/2}$, and 2p$_{1/2}$.
In Fig. \ref{Fig_Energydifference} (b), for $d=0.8$ fm and $L=23.2$ fm, the absolute deviations of single particle states are less than $10^{-4}$ MeV, except 2p$_{3/2}$ and 2p$_{1/2}$.
And in Fig. \ref{Fig_Energydifference} (c), for $d=0.8$ fm and $L=31.2$ fm, all absolute deviations including 2p$_{3/2}$ and 2p$_{1/2}$ are smaller than $10^{-4}$MeV.
These results indicate that smaller step size can definitely improve the accuracy but not for the weakly bound states with low orbital angular momentum. By choosing suitable step and box sizes, accurate descriptions for all the bound states including the weakly bound states 2p$_{3/2}$ and 2p$_{1/2}$ can be achieved in the 3D lattice calculations.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Fig2_SP_ED}\\
\caption{Absolute deviations of single particle energies between the 3D lattice calculation and the shooting method as a function of single particle energy for (a) step sizes $d=1.0$ fm and box sizes $L=23.0$ fm, (b) $d=0.8$ fm and $L=23.2$ fm, and (c) $d=0.8$ fm and $L=31.2$ fm.
The spherical quantum numbers are listed in (b).
}\label{Fig_Energydifference}
\end{figure}
It is interesting to investigate the spatial distributions of states and examine their agreements with the results obtained by the shooting method.
In Fig.~\ref{Fig_densityplane}, as examples, the distributions of the states corresponding to 1d$_{5/2}$ in $z=0$ plane are illustrated.
The states corresponding to 1d$_{5/2}$ are six degenerate single-particle states in the 3D lattice calculations.
Their spatial distributions are respectively shown in Figs.~\ref{Fig_densityplane} (a)-(f), and Fig.~\ref{Fig_densityplane} (g) exhibits their average in the $z=0$ plane.
As there is no symmetry restriction in the 3D lattice calculations, the six states are randomly oriented in space.
However, their average spatial distribution does show the spherical symmetry as shown in Fig.~\ref{Fig_densityplane} (g), which is consistent with the given spherical potential.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Fig3_SP_SD}\\
\caption{Spatial distributions of the states corresponding to 1d$_{5/2}$ in $z=0$ plane in the 3D lattice calculation.
Figures (a)-(f) are the density
distributions of the states in 1d$_{5/2}$, and (g) is their average spatial distributions.}
\label{Fig_densityplane}
\end{figure}
To compare with the radial density distribution obtained by the shooting method, one can average the density distributions in the 3D lattice calculation,
\begin{equation}\label{Eq_radialdensity}
\rho_{nlj}(r)=\frac{1}{2j+1} \sum_{i\in\{nlj\}}\psi_i^\dag(\bm{r})\psi_i(\bm{r}).
\end{equation}
In Fig.\ref{Fig_radialdensity}, the radial density distributions for 1s$_{1/3}$, 1d$_{5/2}$, and 2s$_{1/2}$ in the 3D lattice calculation (open circles) in comparison with the shooting method (solid line) are given,
in which a factor $4\pi r^2$ has been multiplied in order to amplify the radial density distribution at large distance. It can be clearly seen that the two distributions are in perfect agreement with each other.
The data points in the 3D lattice calculation are denser for large $r$ because the grid points used are uniform in the 3D lattice space.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Fig4_SP_RDD}\\
\caption{Radial density distributions for 1s$_{1/3}$, 1d$_{5/2}$, and 2s$_{1/2}$ in the 3D lattice calculation
(open circles) in comparison with the shooting method (solid line). The radial density distribution in the 3D lattice calculation is extracted by Eq.\eqref{Eq_radialdensity}.
}\label{Fig_radialdensity}
\end{figure}
\subsection{DEFORMED POTENTIALS}
For the Dirac equations with the deformed potentials in Eq. \eqref{Eq_potential}, the single particle energies as functions of deformation parameters $\beta$, $\gamma$, and $\beta_{30}$ are given in Fig. \ref{Fig_deformedlevel}, which respectively correspond to axial, non-axial, and reflection-asymmetric deformed potentials.
In Fig. \ref{Fig_deformedlevel}(a), the potentials have both the
space reflection symmetry and axial symmetry with $\gamma=0$, $\beta_{30}=0$, and $\beta$ from $0$ to $0.3$.
In Fig. \ref{Fig_deformedlevel}(b), the potentials break the axial symmetry while keeping the space reflection symmetry
with $\beta=0.3$, $\beta_{30}=0$, and $\gamma$ from $0^\circ$ to $30^\circ$.
In Fig. \ref{Fig_deformedlevel}(c), the potentials break both the
space reflection symmetry and axial symmetry with $\beta=0.3$, $\gamma=30^\circ$, and $\beta_{30}$ from $0$ to $1.0$.
Although there is no symmetry restriction in the 3D lattice calculations, we can search for good quantum numbers from the expectations of physical operators.
For spherical cases, total angular momentum $j$ and orbital angular momentum $l$ can be calculated by the expectation of $\hat{\bm{j}}^2$ and $\hat{\bm{l}}^2$ with the upper components of the wave functions.
For axial cases, the $z$ component of the total angular momentum $|m_z|$ can be calculated by the expectation of $\hat{j}_z^2$.
For the space reflection symmetry case, the parity can be calculated by the expectation of the parity operator $\hat{P}=\beta \hat{P}_{\bm{r}}$, where $\beta$ is the Dirac matrix and $\hat{P}_{\bm{r}} F(\bm{r})= F(\bm{-r})$.
From Fig. \ref{Fig_deformedlevel}, it can be seen that the levels in the spherical case are split into $(2j+1)/2$ levels with the potential changing from spherical to deformed. However, the Kramers degeneracy remains as there is no time odd potential.
For the axial case, the levels with lower (higher) $|m_z|$ values shift downwards (upwards) consistent with the
Nilsson model.
Comparing Fig. \ref{Fig_deformedlevel}(a) and Fig. \ref{Fig_deformedlevel}(b), it can be seen that the spectrum changes more modestly with $\gamma$ than with $\beta$.
In Fig. \ref{Fig_deformedlevel} (c), all levels trend to shift downwards with $\beta_{30}$, which shows its instability in fission.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Fig5_SP_DE}\\
\caption{Single-particle levels in the deformed Woods-Saxon potential as functions of the deformation parameters $\beta$, $\gamma$, and $\beta_{30}$.
The red and blue lines represent the levels with
positive and negative parity respectively.
The shapes shown in the top panel correspond to the deformed parameters
$(\beta,\gamma,\beta_{30})=(0,0^\circ,0)$, $(0.3,0^\circ,0)$, $(0.3,30^\circ,0)$, $(0.3,30^\circ,0.7)$, respectively.
}\label{Fig_deformedlevel}
\end{figure}
To examine the compositions and their evolution of the single-particle level with deformation parameters $\beta$, $\gamma$, and $\beta_{30}$, levels A , B and C in Fig. \ref{Fig_deformedlevel} are chosen as examples.
The results are illustrated in Fig. \ref{Fig_components}.
In the left panels, the compositions of each level are obtained by overlapping the wave functions with the wave functions obtained with $(\beta,\gamma,\beta_{30})=(0,0^\circ,0)$.
In the middle panels, the compositions of each level are obtained by overlapping the wave functions with the wave functions obtained with $(\beta,\gamma,\beta_{30})=(0.3,0^\circ,0)$.
In the right panels, the parity compositions of each level are obtained by the expectation of the parity operator.
In the left panels, there is only small mixing with other orbits for level A compared to levels B and C. It can be understood as follows. This is due to the special character of level A with $|m_z|=7/2$ and parity~$=-$.
The possible mixing is from the 1h$_{11/2}$ orbit which lies high in energy.
Similar conclusions can be drawn for the levels $|m_z|=3/2$ originating from 1p$_{3/2}$ and $|m_z|=5/2$ originating from 1d$_{5/2}$.
In the middle panels, for level A, there is a dramatic change for $|m_z|=7/2$ and $|m_z|=1/2$ components when $\gamma$ approaches $30^\circ$. This is due to the interaction between level A and the level originating from $1f_{5/2}$ and $|m_z|$=1/2 at $\gamma = 30^\circ$, as
shown in energy levels in Fig. \ref{Fig_deformedlevel}(b).
In the right panels, for the octupole deformed case, the parity composition of level B and C changes rigorously due to complicated interaction between levels.
For Level A, the main composition is negative-parity as it mainly interacts with negative-parity dominated levels.
All these can be understood from Fig. \ref{Fig_deformedlevel} (c).
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{Fig6_SP_C}\\
\caption{Compositions of levels A , B and C in Fig. \ref{Fig_deformedlevel} as functions of the deformation parameters $\beta$, $\gamma$, and $\beta_{30}$. The quantum numbers are given for each compositions. The total probabilities are shown as black dashed lines.}\label{Fig_components}
\end{figure}
\section{SUMMARY AND PERSPECTIVES}\label{summary}
In summary, a new method to solve Dirac equation in 3D lattice space is proposed with the inverse Hamiltonian method to avoid variational collapse and the spectral method to avoid the Fermion doubling problem.
This method is demonstrated in solving the Dirac equation for a given spherical potential in 3D lattice space.
In comparison with the results obtained by the shooting method, the differences in single particle energies are smaller than $10^{-4}$~MeV, and the densities are almost identical, which demonstrates the high accuracy of the present method.
Applying this method to Dirac equations with an axial deformed, non-axial deformed, and octupole-deformed potential without further modification, the single-particle levels as functions of the deformation parameters $\beta$, $\gamma$, and $\beta_{30}$ are shown together with their compositions.
Efforts in implanting this method on the CDFT to investigate nuclei without any geometric restriction are in progress.
Possible applications include solving the Dirac equation in an external electric potential (deformation constrained calculation) to investigate nuclei with an arbitrary shape, and in an external magnetic potential (Coriolis term) to investigate rotating nuclei with arbitrary shape and an arbitrary rotating axis.
Moreover, the 3D time-dependent CDFT is also envisioned to be developed to investigate the relativistic effects in heavy-ions collisions and other nuclear reactions.
\begin{acknowledgments}
We thank P. Ring for helpful discussions. This work was supported in part by the Major State 973 Program of China (Grant No. 2013CB834400), the National Natural Science Foundation of China (Grants No. 11335002, No. 11375015, No. 11461141002, No. 11621131001).
\end{acknowledgments}
|
2,869,038,154,811 | arxiv | \section{Introduction} \label{Intro}
Computer simulation is a powerful tool for modeling and evaluating scientific ideas and engineering solutions in many fields. Simulations are often used by engineers to design and test individual components of an imaging system, including lenses and sensors. Image system simulations model how properties of the scene, optical components, and sensor interact to produce the final sensor image. These end-to-end simulations can be used to design rendering methods for consumer photography or to develop machine learning algorithms \cite{jiang2017learning, liu2021isetauto}.
Together with our colleagues, we implemented image systems simulation software to model the complete image processing pipeline of a digital camera, including scene radiance, image formation by the optics, sensor capture, image processing and display rendering. This work, initiated in 2003, was developed to support the design of digital imaging sensors for consumer photography and mobile imaging \cite{Farrell2012-ma, Farrell2003-rb}. The original simulations were validated by comparing simulated and real digital camera images using calibrated 2D test charts \cite{Farrell2008-sc, Chen2009-bj}.
In recent years, we added the ability to model three-dimensional scenes into the image systems simulation \cite{Liu2019-eo}. Using physically based ray tracing \cite{Pharr2016-yb}, we calculate the spectral irradiance image of complex, high-dynamic range (HDR) 3D scenes. This extension opens the opportunity for accurately simulating natural scenes with complex lighting, high dynamic range, occluding objects, and surface inter-reflections.
We have used three-dimensional scenes to design and evaluate imaging systems for a range of applications, including AR/VR \cite{Lian2018-kc}, underwater imaging \cite{Blasinski2017-lr}, autonomous driving \cite{Blasinski2018-em, Liu2019-ds}, fluorescence imaging \cite{Farrell2020-tb, Lyu2021-ae} and models of the human image formation \cite{Lian2019-xr}. In each case, we carried out certain system validations - typically by comparing some aspect of the simulation with real data. In this work, we validate the software by comparing camera images measured using a constructed Cornell Box \cite{Goral1984-wd, Meyer1986-pb} with simulations of the scene and camera (Figure \ref{Fig-scenepreview}). The box contains one small and one large rectangular object placed near the side walls of the box. The box is illuminated through a hole at the top, so that some interior positions are illuminated directly and others are in shadow. The walls of the box are covered with red and green paper, providing colorful indirect illumination of the rectangular objects.
The Cornell Box was designed to include important features found in complex natural scenes, such as surface-to-surface inter-reflections and shadows. In its original use, the Cornell Box was used to judge the visual similarity between a real constructed box and computer graphics renderings of the box displayed on a CRT. Ray tracing algorithms capture the main effects of the lighting and inter-reflections \cite{Goral1984-wd, Cohen1986-ld, Meyer1986-pb,Sumanta-qr}.
Our study is designed to assess the accuracy of end-to-end simulations that include a physical description of a 3D spectral radiance, realistic optics, and an electronic image sensor. We validate the end-to-end simulations -- from the Cornell box to the sensor digital values -- by comparing simulated and real camera images.
\section{Related work} \label{relwork}
A number of groups have recognized the importance of image systems simulation using the principles of optics and sensor simulations \cite{Farrell2003-rb, Costantini2004-mn,Konnik2014-mu,Toadere2013-au, florin2009simulation, fiete2014modeling}. There are many commercial offerings that provide engineers with simulation tools for a range of applications (e.g., dSpace, ANSYS, Anyverse, Omniverse-Nvidia, Zemax, CodeV).
Among the first systems were those developed for remote sensing. Schott et al. \cite{schott1999advanced} developed image systems software (DIRSIG) that includes radiometric descriptions of scenes, ray tracing through atmospheric media, and sensor capture. The goal of the software is to create synthetic broad-band, multi-spectral and hyper-spectral imagery. Some aspects of the simulations are validated \cite{White1996-lv, Peterson2004-la, dirsig-validation-se}. The code is distributed in binary form to those who pay to attend a course and who hail from a specific set of countries. A large number of related papers describing in-house simulations have also been reported (e.g., \cite{GISGeography2014-ez,Borner2001-cn}).
Garnier et al \cite{garnier1998general, garnier1999infrared} describe an end-to-end image systems simulation of an infrared (IR) sensor model. Their simulation traces rays from a scene, through an optics model, and to a sensor array. The optics model combines both thin-lens equations and blurring kernels applied to the final image. The computational concepts are very similar to the methods we use, and their paper is a good overview of the physics of image formation and sensor capture. No experimental validation of this work is published and we were unable to locate the software.
Image systems simulations have been a powerful tool in our research and have enabled us to build soft-prototypes of imaging sensors before they were built. For example, we used simulations to develop the L\textsuperscript{3} (local, linear and learned) method for processing RGBW sensor data \cite{Lansel:2011, Tian:2014, Tian:2015, Jiang:2017}. We also used simulations to design a spectrally-adaptive imaging system for underwater imaging \cite{blasinski2017computational} and a camera that is capable of measuring the fluorescence of oral bacteria \cite{Farrell2020-tb, Lyu2021-ae}.
More recently, we used simulations to compare the performance of neural networks for detecting vehicles in complex daytime scenes. In one study, we explored the effect that sensor pixel size has on performance \cite{liu2020neural}. And, in another study, we compared the performance of imaging sensors that are capable of collecting radiance or depth information, and a hybrid imaging sensor that can capture both radiance and depth simultaneously on the same sensor array \cite{liu2021isetauto}.
The ISETCam and ISET3d software we describe relies on physically based ray tracing that is capable creating quite complex natural scenes \cite{Pharr2016-yb}. The software can describe radiance in any spectral band, and it can specify a wide variety material properties including fluorescence \cite{Lyu2021-ae}. We added the capability of modeling arbitrary imaging optics specified by a ray transfer function \cite{Goossens-RTF} and thus compute the sensor irradiance. The ISETCam software models many properties and types of imaging sensors, including sensors with arbitrary color filter arrays and sensors with microlens arrays to capture the light field.
As far as we are aware, at this time ISETCam and ISET3d are the only open-source, freely available image systems simulation software. Using ISETCam and ISET3d, scientists and engineers can check each others' work and extend the system for their specific applications.
\section{Contributions}
In this paper, we evaluate the accuracy of the simulations by comparing the pixel values in real camera images of a real Cornell Box with the pixel values in simulated camera images of a simulated Cornell Box. Our specific contributions include:
\begin{enumerate}
\item Quantifying the accuracy of end-to-end image system simulations using a real 3D scene (Cornell Box) and a real camera (Google Pixel 4a), including
\begin{enumerate}
\item Scene with shadows, lighting and surface inter-reflections
\item Optical blur, relative illumination, and depth of field
\item Sensor noise, channel cross-talk, and chromatic responses
\end{enumerate}
\item Providing the data as well as the open-source and free software to support transparency and reproducibility
\begin{enumerate}
\item {https://github.com/ISET/isetcam}
\item {https://github.com/ISET/iset3d}
\item {https://github.com/ISET/isetcornellbox}
\end{enumerate}
\end{enumerate}
\section{Simulation pipeline modeling} \label{parameters}
This Section reviews the image systems simulation pipeline modeling. Section \ref{val} describes the validation measurements. Section \ref{Discussion} reviews the results. Section and \ref{connfuture} describes ongoing and future work.
The image systems simulation validation includes three main parts: (1) a 3D scene radiance description that models light sources, asset geometry, and material properties, (2) an optical model that maps rays from the scene onto the sensor, and (3) a model of how the irradiance at the sensor is converted into unprocessed digital values.
The validation measures how closely the simulation predicts the minimally processed sensor data obtained from the Google Pixel 4a. We focus on matching the unprocessed sensor data because remaining components of the image processing are implemented in proprietary software.
\subsection{Scene construction}
\begin{figure}[!tb]
\includegraphics[width=1\columnwidth]{figures/figure_ScenePreview.png}
\caption{Real and simulated Cornell box. (a) An image of the Cornell box and its light source in the lab. (b) Geometric representation of the Cornell box model and camera position in Cinema 4D. (c) Measured spectral power distribution of light source. (d) Measured spectral reflectance of white, red and green surfaces.}
\label{Fig-scenepreview}
\end{figure}
We constructed a Cornell Box and created a computer graphics model of the same box in our simulation environment. The Cornell Box was constructed using wood and covered with white matte paper. We added red and green matte paper to the left and right walls, respectively. We placed a light behind a diffusing filter over an opening at the top of the box (Figure \ref{Fig-scenepreview} (a)). The spectral power distribution of the light source and the spectral reflectances of the surfaces in the Cornell Box were measured using a PR670 spectroradiometer \ref{Fig-scenepreview} (b-c). In different experiments we added a miniature Gretag color chart, a slanted bar target, and a 3D printed Stanford Bunny \cite{stanford-bunny}.
The Cornell Box is a good compromise between simplicity and complexity: it includes a light source and a set of objects within a closed environment that includes many inter-reflections and shadows. There are only a few materials, making it possible to accurately model the scene. The light and objects within the box create significant shadows and a relatively high dynamic range image. Thus, the scene is useful for testing complex rendering, dynamic range, and sensor noise. The surfaces with different reflectances can be used to evaluate sensor chromatic responses and surface inter-reflections. For example, we were able to assess the estimated spectral quantum efficiency of the sensor by adding the Gretag color chart to the scene. Similarly, we assessed the optics model by varying the position of the slanted bar target added to the scene.
A Cornell Box was constructed using geometric specifications of a model of the Cornell Box represented in Cinema 4D (Figure \ref{Fig-scenepreview} (b)), a commercial graphics modelling software. The meshes and their positions are exported as a set of text files that can be read by PBRT. We model the light source as an area light, and we model the surface materials as mainly Lambertian, but with a small specular term. The reflectance distribution function for each surface is documented in the software we share. The measured spectral data of the light source (Figure \ref{Fig-scenepreview} (c)) and surface reflectances (Figure~\ref{Fig-scenepreview} (d)) are stored in data files that are read into PBRT.
\subsection{Optics modelling}
\begin{figure}[!hb]
\includegraphics[width=1\columnwidth]{figures/figure_RTF_Zemax.png}
\caption{The ray transfer method of lens modeling. (a) We use a Zemax 'black-box model' to estimate the ray-transfer function. This function describes the mapping between rays from the sensor plane that are incident at the entrance pupil, through the unknown optics, to rays in a plane after exiting from the optics. The position of the incident and output rays are described by the four-dimensional vectors, $(x_i,y_i,u_i,v_i)$ and $(x_o,y_o,u_o,v_o)$. The first two entries define the position of the ray in the aperture, and the second two entries define the ray direction. We use Zemax to calculate 128,561 input-output ray pairs. We then fit a set of four polynomials relating the four input ray parameters to each of the four output ray parameters (Equation \ref{eq:polynomial}). (b-d) The Zemax model generates a line spread function (LSF) , modulation transfer function (MTF) and relative illumination function, all at 550 nm. The ray-transfer function implementation in PBRT matches the Zemax calculations on these functions \cite{Goossens-RTF}. We evaluate the model accuracy by comparing measurements with these simulations.}
\label{Fig-rtfzemax}
\end{figure}
Optics are a critical component of image systems and determine important properties, such as lens shading (vignetting), spatial resolution, and depth of field. It is possible to use standard ray tracing to quantify these properties when the shapes, positions, and indices of refraction of the optical components are known. In some cases, including the project analyzed in this paper, the lens design is proprietary.
To solve this problem, we used the approach described in \cite{Goossens-RTF}, who point out that for ray-tracing purposes we only need to know the "ray-transfer function" of the optics. This function maps rays entering the optics into outgoing rays. We can simulate a proprietary lens by finding an equivalent ray-transfer function using a "black box" model provided by the vendor. This model enables a customer to calculate how a ray in the incident light field (at a position and angle in the aperture) is mapped to a position and angle of a ray in the exit pupil, without revealing the lens design (Figure \ref{Fig-rtfzemax}).
We use the methods in \cite{Goossens-RTF} to calculate the polynomials that describe the Google Pixel 4a ray-transfer function. Conceptually, four input ray parameters are related to four output ray parameters by four polynomials. These polynomials can be calculated for each sampled wavelength:
\begin{equation}
\label{eq:polynomial}
\begin{cases}
x_o =& \text{poly}_1(x_i, y_i, u_i, v_i)\\
y_o =& \text{poly}_2(x_i, y_i, u_i, v_i)\\
u_o =& \text{poly}_3(x_i, y_i, u_i, v_i)\\
v_o =& \text{poly}_4(x_i, y_i, u_i, v_i)\\
\end{cases}.
\end{equation}
The $x,y$ values describe the position of the rays within the input or output aperture of the optics. The $u,v$ values are the first two elements of the unit vector that describes the direction of the ray with respect to the input or output aperture. Only two components are needed because the direction is a unit vector. As described in \cite{Goossens-RTF}, this formulation can be simplified for the common case when the optics are rotationally symmetric, and we assumed rotational symmetry here.
The ray-transfer function can be used to calculate many summary measures of the optics, including the linespread function, modulation transfer function, and the relative intensity map (Figure~\ref{Fig-rtfzemax}b-d). It is convenient to assess the accuracy of the model by comparing these predictions with measurements, and we do so in the Results section.
\subsection{Sensor modelling}
The optical irradiance at the sensor surface is converted to voltages and a digital output value using a sensor model. The Google Pixel 4a uses a Sony IMX363 sensor; we summarize the sensor specifications (e.g. pixel size, fill factor, sensor resolution) in \ref{appendix:sensor}. Some of these values were published by Sony and others were estimated in lab experiments described below. To perform these experiments, we used OpenCamera\cite{opencamera}, a free and open-source software application that controls camera gain and exposure duration, to obtain the nearly unprocessed digital values from the sensor.
\subsubsection{Sensor spectral quantum efficiency}
To calibrate the sensor spectral quantum efficiency (QE), we captured images of a Macbeth Color Checker (MCC) under three different illuminants. The spectral power distribution of the illuminants and the reflectance functions of the 24 MCC patches are plotted in Figure \ref{Fig-QEcalibration}(ab). As part of the estimation, it is necessary to reduce relative illumination (vignetting) effects. We did this by measuring and correcting for the relative illumination and by placing the MCC in the center of the camera image where the change in relative illumination is smaller.
We began the calibration by comparing the measured RGB values with those predicted by using the published sensor color channel quantum efficiencies (QE) and the transmissivity of a NIR filter as described in \cite{lyu2021validation}. There is a substantial mismatch for this prediction which we attribute to optical and electrical cross-talk and channel gains. To account for these factors, we found a positive $3 \times 3$ matrix, $M$, that transforms the spectral QE. The matrix was estimated by minimizing the least square error between the measurement ($r^\prime, g^\prime, b^\prime$) and the original prediction ($r, g, b$):
\begin{ceqn}
\begin{align}
\label{eq:transform}
\displaystyle{\min_{M}}~||\begin{bmatrix}r^\prime& g^\prime & b^\prime\end{bmatrix} - \begin{bmatrix} r& g& b\end{bmatrix} M||^2
\end{align}
\end{ceqn}
where $\begin{bmatrix} r^\prime, g^\prime, b^\prime \end{bmatrix}$ and $\begin{bmatrix} r,g,b \end{bmatrix}$ are $72 \times 3$, (24 patches under three illuminants). We solve for $M$ subject to a non-negativity constraint
\begin{ceqn}
\begin{align}
\label{eq:condition}
M(i, j) \geq 0, 1 \leq i, j \leq 3
\end{align}
\end{ceqn}
The fitted matrix is:
\begin{ceqn}
\[M=
\begin{bmatrix}
0.5636 & 0.0807 & 0.0069\\
0 & 0.5917 & 0\\
0 & 0.2470 & 0.7098
\end{bmatrix}\]
\end{ceqn}
The diagonal entries of $M$ are channel gains. There is one significant off-diagonal value that represents cross-talk between the blue and green channels. After incorporating the correction matrix, the simulated sensor data match the measurement (Figure \ref{Fig-QEcalibration}d) within a few percent (8.9\%). The three spectral quantum efficiency curves for the channels are shown before and after correction in the Figure \ref{Fig-QEcalibration} c-d, respectively.
\begin{figure}[!ht]
\includegraphics[width=1\columnwidth]{figures/figure_QECalibration.png}
\caption{Sensor spectral quantum efficiency calibration. (a) Scatter plot comparing measured and simulated RGB values of the MCC based on nominal color filter quantum efficiency. (b) Scatter plot comparing measured and simulated RGB values after correction for cross-talk and channel gain. (c) Nominal color filter quantum efficiency and (d) quantum efficiency after correction.}
\label{Fig-QEcalibration}
\end{figure}
\subsubsection{Sensor noise model}
We estimated the following noise sources \cite{emva-sensor-noise}:
\begin{enumerate}
\item \textbf{Photon noise (or shot noise)} is a natural consequence of the photoelectric effect. The number of electrons generated is Poisson distributed.
\item \textbf{Dark current noise} is the leakage current within each pixel in the absence of light. The electrons are generated by thermal effects and crystallographic defects in the depletion region.
\item \textbf{Dark signal non-uniformity} (DSNU) is random distribution of offset levels across pixels measured at short exposure durations in the absence of light.
\item \textbf{Photo response non-uniformity} (PRNU) describes the variation of the gain (slope) when measuring the electrical signal as a function of incident photons.
\item \textbf{Read noise} arises from electrical noise in the circuitry that reads out the pixel value.
\item \textbf{Reset noise} arises from small fluctuations in the voltage level achieved when resetting the pixel prior to an acquisition.
\end{enumerate}
Some types of noise differ with each acquisition (temporal noise). Other types of noise are variations across the sensor that remain consistent across acquisitions (fixed pattern). Certain potential sources of variation, such as column fixed pattern noise, are very small and not considered in IMX363 sensor modelling.
\subsubsection{Combined noise model estimates}
The combined effect of different noise sources is present in the sensor digital values. The expected noise is signal-dependent, largely because the photon noise is signal-dependent. To evaluate the accuracy of the simulated noise we compared the standard deviation of the simulation and measurement in multiple regions that span a range of signal levels (Section \ref{val}).
\section{Results} \label{val}
\subsection{Qualitative appearance comparison}
The simulated and measured sensor image data appear quite similar (Figure \ref{Fig-qualitative}). Both represent the high dynamic range of the scene. For example, the area light source is saturated while the shadows and corners are very dark. Light reflected from the colored walls can be seen on the sides of the cubes.
\begin{figure}[!t]
\includegraphics[width=1\columnwidth]{figures/figure_QualitativeComp.png}
\caption{Qualitative comparisons between the simulation and measurement from two camera positions. (a) and (b) are measurements; (c) and (d) are simulations.}
\label{Fig-qualitative}
\end{figure}
We do not expect to observe a pixel-by-pixel match between the simulated and measured camera images. This is partly due to small differences in the positions of the objects and camera in the simulated and real scene. Furthermore, we modeled the scene illumination as a uniform area light. This approximation introduces some small differences. Hence, to characterize the simulation accuracy we compare a series of quantitative summary measures. In the next sections we evaluate relative illumination, optical blur, depth of field, chromatic channel responses and sensor noise.
\subsection{Relative illumination}
In many cell phone cameras, the optics introduce a substantial change in the relative illumination with field height. Using the Zemax black-box model of the Google Pixel 4a lens, we expect a fall-off from the center to the 3mm field height to be approximately 60\% (Figure \ref{Fig-lensvignettcomp}). The relative illumination, calculated using the ray-transfer method described in \cite{Goossens-RTF}, matches the Zemax prediction very closely. The measured relative illumination differs, falling off by approximately 70\%. The most likely cause of the differences is that we did not model the microlens array placed on the surface of nearly all commercial sensors. This information is not included in the Zemax black box model, and we did not have access to the microlens description.This effect is widely described in the literature \cite{SVS-Vistek_GmbH_undated-sd}. In subsequent calculations, we use the measured relative illumination as part of the simulation.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.7\columnwidth]{figures/figure_LensVignettComp.png}
\caption{Comparison of simulated and measured relative illumination. The curves show the relative illumination from the center of the sensor to a 3.5 mm field height (the sensor diagonal is 7 mm). The Zemax black box model relative illumination and ray-transfer prediction in PBRT are nearly identical (black). Neither closely matches the measured relative illumination (red), which falls-off more rapidly.}
\label{Fig-lensvignettcomp}
\end{center}
\end{figure}
\subsection{MTF and depth of field}
\begin{figure}[!b]
\begin{center}
\includegraphics[width=0.9\columnwidth]{figures/figure_SlantedBarPrev.png}
\caption{The line spread function (LSF) was estimated from a simulated image the contained two slanted edges. These were positioned at two different depths (0.3 m and 0.5 m from the camera position) and field heights (0.7 mm and 0.8 mm) that matched the distances and field heights of the slanted edge target that was captured with the Google Pixel 4a camera. The LSF is estimated by selecting a small region of the slanted edge and using measurements from multiple rows to estimate a densely sampled step-edge response. The derivative of the step-edge response is the LSF. The modulation transfer function (MTF) is the magnitude of the Fourier Transform of the LSF.}
\label{Fig-slantedbarpreview}
\end{center}
\end{figure}
The line spread function (LSF) measures how light from a thin line is spread across the sensor surface, serving as a summary measure of spatial blur. The LSF is narrow for a line at the focal distance, broadening for lines placed closer or farther. We measured the LSF function of the camera using the methods defined by the ISO 12233 standard \cite{iso12233}. To validate the RTF, two slanted edge targets were placed in the Cornell box at 0.3 m and 0.5 m from camera position. Two images were acquired, one with the focal length set to each of these distances. We expect that when one slanted edge target is in focus, the LSF for the other slanted edge target will be wider.
The simulated images in Figure \ref{Fig-slantedbarpreview} illustrate the geometry. The LSFs computed from the measured and simulated images are shown in Figure \ref{Fig-lsfmtfdof}. Panel (a) compares the LSFs with the camera focused at 0.3 m and panel (b) compares the LSFs with the camera focused at 0.5 m. As expected, in both cases the LSF is broader when the line is away from the focal distance.
To further compare the data, we transformed the LSFs to modulation transfer functions (MTFs) in panels (c) and (d). The MTF describes the reduction in image contrast as a function of image spatial frequency. The narrow LSF becomes a broader MTF. Both the MTF and LSF measure the depth-dependence, which defines the depth of field. This will vary with the lens and its aperture opening. The Google Pixel 4a does not adjust the aperture, and for this reason we compared using focal distance.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=1\columnwidth]{figures/figure_LSF_MTF_DoF.png}
\caption{Comparison of the spatial blur and depth of field. (a,b). The measured (blue) and simulated (red) LSFs are compared. The curves show the LSFs for lines at two different distances. Panel a shows data with a focal distance of 0.3m and Panel b with the focal distance of 0.5m. In each case the narrow LSF is for the line at the focal distance, and the broader LSF is for the line that is too near or too far. (c,d) The LSF data are transformed into the corresponding MTF curves where the broader MTF is for the line that is in focus. The measured and simulated lines were at slightly different field heights (0.7 mm and 0.8 mm), but this difference did not have a substantial impact on the measurements.}
\label{Fig-lsfmtfdof}
\end{center}
\end{figure}
\subsection{Sensor noise}
In this section we describe our evaluation of the sensor noise model. The model includes photon noise and several classes of temporal (reset, dark current, and read) and fixed pattern (DSNU, PRNU) noise. These sources contribute differently to the total noise, and some of these sources depend on the sensor irradiance level.
\begin{figure}
\begin{center}
\includegraphics[width=1\columnwidth]{figures/figure_SensorNoiseVal.png}
\caption{Sensor noise estimated in measurement and simulation. (a) We randomly tested many small ($10 \times 10$) image regions. We accepted a region for a noise estimate if it contained no dead pixels and the values were consistent with a uniform spectral irradiance. The yellow boxes show the regions that were accepted. See text for details. (b) Within each uniform region, the standard deviation increases with mean signal level. The simulated data match well with the measurement.}
\label{Fig-sensornoiseval}
\end{center}
\end{figure}
To assess the accuracy of the sensor noise model, we compared simulated and measured data from image regions with a wide range of digital levels. Specifically, we selected 500 uniform patches using criteria described in \ref{appendix:uniform}. The regions are shown overlaid on a measured image in Figure \ref{Fig-sensornoiseval}a). The sensor noise, measured as the standard deviation of the green pixels, increases with the mean level (Figure \ref{Fig-sensornoiseval}b). The relationship between the standard deviation and the mean is similar whether estimated using the measured or simulated data. Hence, we conclude that the sensor noise model is a good approximation to the measured data.
\subsection{Inter-reflections}
\begin{figure*}[!ht]
\begin{center}
\includegraphics[width=2\columnwidth]{figures/figure_Interreflection.png}
\caption{Comparisons of simulated and measured digital values in the complex scene including inter-reflections. (a-c) Acquired images of an MCC at three different positions along the rear wall. The arrows show the likely paths for inter-reflections from the light source off the nearby walls onto the MCC. The white rectangle highlights the position of the achromatic patches on the MCC. (d-f) The digital values from the Google Pixel 4a images (dashed lines) are compared with the simulated values (solid lines). See text for details.}
\label{Fig-interreflection}
\end{center}
\end{figure*}
Next, we compare the simulated and measured digital values for an image that includes substantial inter-reflections. We acquired and simulated images of the Cornell box that contained an MCC at the rear wall. Figure \ref{Fig-interreflection})a-c show images that were measured when the MCC was positioned either closer to the left wall (red), right wall (green) or half-way in between. The graphs (Figure \ref{Fig-interreflection} d-f) below each image plot the digital values in a horizontal line passing through the achromatic series of the MCC in that image. The solid lines plot the digital values from the measured image and the points plot the digital values from the simulated image, with one free scalar adjustment. This adjustment was necessary because the angle of the simulated MCC is perfectly parallel to the rear wall, but the measured MCC was tilted slightly. Hence, the amount of light scattered from the achromatic series was larger in the measured image than in the simulated image. We therefore scaled all of the simulated values in the horizontal line passing through the achromatic series of the MCC by a single positive constant that brought the two sets of data into reasonable agreement.
There are several features worth noting when comparing the simulated and measured digital values. First, notice that the ratio between the green:red channel ratio is close to one (1.08) at the left (red) wall and becomes significantly higher near the right (green) wall (1.28). This change is due to the light reflected onto the achromatic patches from the nearby walls. Second, the variance in the simulated and acquired data is quite similar at all levels. Finally, there is a difference in the black line regions between the simulated and measured data. This arises because the black lines in the simulated MCC were set to zero reflectance, but the black lines in the real MCC are slightly more reflective. We left this imperfection in the simulation to show that it is possible to see such small mismatches. Note also that the lowest digital value contains the black level offset (64).
\subsection{Validation of chromatic channel and relative illumination}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=1\columnwidth]{figures/figure_ColorFilterHeight.png}
\caption{Validation of estimated color filter QE. The scatter plots compare the mean simulated and measured value for the 24 patches under three different illuminants. The color of each point is a measurement in the R, G or B channels. Panels a-c show these fits. (a) Without lens vignetting, channel gain, or cross-talk correction the fits are quite poor. (b) Correcting for lens vignetting alone improves the fit. (c) Correcting for lens vignetting, channel gain and cross-talk places all the points near the identity line. (d) The histogram shows the distribution of measured and simulated digital values; most of the comparisons are based on relatively small digital values.}
\label{Fig-colorfilterheight}
\end{center}
\end{figure}
Finally, we performed one more assessment of the accuracy of the sensor QE by comparing simulated and measured R, G and B digital values of the Macbeth color checker. This validation was performed with a miniature version of the MCC in the center of a Gretag light booth. The booth has 3 different lights nominally labeled "illuminant A", "CWF" and "day". For each light, we captured images using the Google Pixel 4a camera in 8 different positions, for a total of 576 color patch comparisons (3 illuminants x 8 positions x 24 color patches). Moving the camera while keeping the position of the MCC fixed places the MCC at different field heights and preserves the same illumination on the MCC. To correctly simulate these sensor responses, we must have an accurate model of both the color channels and the relative illumination.
We simulated RGB values in several steps. First, we multiplied the spectral reflectance of each of the 24 MCC color patches with the spectral power of the illumination and the spectral quantum efficiencies (QE) of the Google Pixel 4a camera. Second, we corrected for relative illumination by dividing the measured RGB values by the measured relative illumination (see Figure \ref{Fig-lensvignettcomp}). Third, we used the conversion gain (see Appendix \ref{appendix:conversion}) to calculate the sensor RGB digital values. Finally, we added the digital black level (64) to the simulated values.
Figure \ref{Fig-colorfilterheight} compares 1728 measured and simulated mean RGB values (576 patches with 3 values) at several steps in the calculation. We first compare values without accounting for relative illumination and using the nominal sensor QE functions (Figure \ref{Fig-QEcalibration}a). Next, we compare the measured and simulated RGB values after correcting for sensor gain and cross-talk (see Figure \ref{Fig-QEcalibration}b). Finally, we make the same comparison also accounting for the effect of relative illumination (Figure \ref{Fig-colorfilterheight}c). Each additional step improves the fit between simulated and measured data. The histogram panel \ref{Fig-colorfilterheight}d illustrates the distribution of digital values in the data set.
\section{Discussion} \label{Discussion}
We assessed simulation accuracy using a series of quantitative measurements: Our end-to-end image systems simulation matches the camera data in many ways. Simulations of the optical blur and depth of field agreed with the measurements to within a few percent (Figures \ref{Fig-lsfmtfdof} and \ref{Fig-colorfilterheight}). Simulations of the spatial pattern of sensor responses within a complex scene (Figure \ref{Fig-interreflection}) and the sensor noise (Figure \ref{Fig-sensornoiseval}) are also in close agreement with the measurements. The quantitative agreement between the simulated and measured camera image data are a strong validation of the end-to-end simulation methodology.
The validation is based on a three-dimensional scene that includes significant shadows and inter-reflections. The light scattered from the colored walls are a substantial factor in illuminating the sides of the cubes. The cubes cast significant shadows on the walls. All of these features are congruent in the simulated and measured camera images.
Modeling commercial camera optics is a long-standing challenge in image systems simulation. Goossens et al. \cite{Goossens-RTF} developed the ray-transfer method that enables us to model a Google Pixel 4a lens even though the specific design is unknown. We found that key properties determined by the optics - relative illumination, optical blur and depth of field - can be matched even though the specific lens components are not specified. These results show that the ray transfer function obtained from a Zemax black model for the Google Pixel 4a camera can be substituted for a lens model in PBRT.
Prior work had already established the accuracy of linear plus noise sensor models for planar scenes with simple lighting \cite{Farrell2008-sc,Chen2009-bj}. Image system simulation validation based on 3D scenes with fluorescent materials and sensors were described in \cite{Lyu2021-ae}. The tests here extend those validations by comparing the R, G and B digital values in simulated and measured camera images of the MCC under multiple light sources and different camera positions. In both of these cases, using different cameras, we found it necessary to estimate the channel gains and crosstalk.
After including these calibrations, the mean pixel values in the simulated and measured camera images of the MCC lie near the identity line (Figure \ref{Fig-colorfilterheight}c). The agreement between the mean pixel values in the simulated and measured camera images of the MCC test chart, illuminated with three different lights, confirms the general accuracy of the sensor model. The standard deviation of the points around the identity line, however, is larger than expected based on sensor and photon noise alone (Figure \ref{Fig-sensornoiseval}b). To explain this additional variance may require more precise accounting for factors such as the relative illumination.
Accurate sensor modeling is necessary for designing novel sensor architectures. Soft-prototyping using realistic scene data can shorten development time by enabling designers to select imaging components and avoid mistakes. Image systems simulation also has value in developing subsequent signal processing and machine learning algorithms. Consider, for example, that machine learning developers for driving applications are using synthetic camera images to augment data sets and create specific training scenarios (e.g., \cite{omniversevalidation-Kamel2021,anyverse-mainpage,waymo-simulationcity}. Each of those vendors emphasize different types of validations for their simulations. A contribution of this work is the focus on properties of the optics and sensor that are beyond what we have found in the literature.
Correctly modeling the optics and sensor properties, including pixel size and color filter arrays, is important for many applications. For example, neural networks trained on physically-based simulations of camera images can detect cars in real camera image data almost as well as neural networks trained on the real camera image data \cite{liu2020neural}. Furthermore, neural networks trained on camera images generated by physically-based simulations performed better than neural networks trained on camera images generated by raster-based graphics or by ray-traced graphics rendering methods that did not include the correct optics and sensor modeling \cite{Liu2019-ds, liu2020neural,Liu2021-mr,liu2021isetauto}.
The results of our study support the use of physically based end-to-end image systems simulations to create large data sets that are automatically and accurately labeled for training neural networks. The results also support using the simulations to assess how hardware changes will impact system performance. These analyses can be helpful in fields apart from consumer photography, such as medical diagnosis, driving, and robotics.
\section{Future work} \label{connfuture}
The results that we report in this paper give us confidence in our ability to accurately simulate a camera imaging complex, natural scenes. Some of the limitations in this work are a good target for future work.
For example, ISETCam and ISET3d include the ability to simulate microlens arrays. We did not have this information, and the missing information produced a difference between the simulated and measured relative illumination. We are investigating methods for accounting for the microlens array when this information is not provided.
Related, we are simulating sensors with multiple photosites beneath each microlens to measure the light field. A particularly simple form of these sensors, the dual pixel architecture, is already in wide use for setting focus. A useful extension of the analysis here is to simulate dual pixel and light field sensors. These simulations will help us understand how much information can reliably be extracted from these devices.
We are also exploring other types of imaging systems, such as time-of-flight and gated-CMOS cameras. Our simulation of such devices has been enabled by the ability to use ray tracing to calculate the path length of each ray from the sensor into the scene, including the effect of optical elements and filters that are in the light path. Further work that accounts for the participating media on the light path (fog, smoke), and the bidirectional scattering distribution functions of the materials at the relevant wavelengths, will be helpful in the design of such systems and in synthesizing realistic data sets. Such data sets can be used to extract information about the scene using different algorithms, or simply to create labeled training data for machine learning applications.
As we noted, an important goal for this project is to build trust in these simulations. We accomplish this by assessing the accuracy of our simulations and by making our image systems simulation software open-source and freely available. We continue to update our software repositories with new data and functions that are part of our ongoing research so that others can check our work and build on it. By making these soft-prototyping tools easily available, we hope to advance the design of imaging sensors for future imaging applications.
\appendices
\section{Render configuration} \label{appendix:render configurations}
The optical images of the Cornell Box scene were rendered using PBRT-V3 \cite{Pharr2016-yb} with the RTF modifications. The images were rendered at the same spatial sampling resolution as the Sony sensor (3024 $\times$ 4032). The rendering parameters were set to 6 bounces and 3072 rays per pixel. These parameters were chosen to reduce the rendering noise to a very small level. Rendered on a CPU with 40 cores, the rendering time was 15 hours. Experiments with PBRT-V4 and a 3080 Nvidia GPU suggest that the rendering time for the same scene will be approximately 1.5 hours.
\section{Sensor parameters} \label{appendix:sensor}
Sensor parameters used for simulation are in Appendix Table \ref{tab:sensor}. The pixel size, fill factor, well capacity, voltage swing, conversion gain, black level offset and quantization are provided by the manufacturer. We also estimated conversion gain (see Appendix \ref{appendix:conversion}).
We estimated the dark signal nonuniformity (DSNU), photoresponse nonuniformity (PRNU), dark voltage and read noise from lab measurements. Specifically we used an integrating sphere to produce images of uniform intensity. We then acquired a set of images for a range of integration times. We fit the measurements from a collection of different exposure times to estimate these quantities (see methods in \cite{Farrell2012-ma}).
\begin{table}[!h]
\caption{Sony IMX363 sensor specification.}
\label{tab:sensor}
\begin{center}
\begin{tabular}{|p{0.25\columnwidth}|p{0.30\columnwidth}|p{0.30\columnwidth}|}
\hline
\textbf{Properties} & \textbf{Parameters} & \textbf{Values (units)} \\ \hline\hline
\multirow{2}{*} {Geometric} & Pixel Size & [1.4, 1.4] ($\mu$m) \\
& Fill Factor & 100 ($\%$) \\ \hline
\multirow{6}{*} {Electronics} & Well Capacity & 6000 ($e^-$) \\
& Voltage Swing & 0.4591 (volts) \\
& Conversion Gain & $ 0.1707 (dv/e^-$) \\
& Analog Gain & 1 \\
& Black Level Offset & 64 (dv) \\
& Quantization Method & 10 (bit) \\ \hline
\multirow{4}{*} {\makecell[l]{Noise Sources \\ @Analog gain=1}} & DSNU & 0.038 (mV) \\
& PRNU & 0.54 (\%) \\
& Dark Voltage & 0.02 (mV/sec) \\
& Read Noise & 0.226 (mV) \\ \hline
\end{tabular}
\end{center}
\end{table}
\section{Estimating conversion gain} \label{appendix:conversion}
We estimate the conversion gain by analyzing the pixel values in an image of a bright scene, where we expect shot noise to be dominant. In the ideal case, the number of excited electrons, which is the signal $\Tilde{S}$, follows a Poisson distribution \cite{Janesick2007-sc}:
\begin{equation}
\Tilde{S} \sim \text{Poisson}(\mu) .
\end{equation}
The mean and variance are equal. In practice, the observed signal also depends on the photo-response nonuniformity (PRNU) and dark signal nouniformity (DSNU). The measured DSNU is very small, and so we do not include it in the calculations. The PRNU $\Tilde{P}$ follows a normal distribution:
\begin{equation}
\Tilde{P} \sim 1 + \mathcal{N}(0, \sigma_{\text{prnu}}).
\end{equation}
The observed signal $\Tilde{O}$ will be the product of these two independent random variables:
\begin{equation}
\Tilde{O} = \Tilde{S} \Tilde{P}
\end{equation}
The expected value of the number of observed electrons can be expressed in terms of the mean and variance of these two random variables
\begin{equation}
\label{eq:obsmean}
\begin{split}
\text{E}(\Tilde{O}) = \text{E}[\Tilde{S}\Tilde{P}]
= \text{E}(\Tilde{S})\text{E}(\Tilde{P})
= \mu
\end{split}
\end{equation}
The variance of the observed signal $\text{V}(\Tilde{O})$ is:
\begin{flalign}
\label{eq:obsvariance}
\begin{split}
\text{V}(\Tilde{O}) = &\text{V}[\Tilde{S}\Tilde{P})]\\
= & [\text{E}(\Tilde{S})]^2 \text{V}(\Tilde{P}) + [\text{E}(\Tilde{P})]^2 \text{V}(\Tilde{S}) + \text{V}(\Tilde{S})\text{V}(\Tilde{P}) \\
= &\mu^2\sigma_{\text{prnu}}^2+\mu+\mu\sigma_{\text{prnu}}^2
\end{split}
\end{flalign}
Since we can only measure digital values DV the conversion gain $\alpha$ is defined as converting electrons directly to digital value that has unit of $(dv/e^-)$:
\begin{equation}
\begin{split}
\text{DV}= \alpha \Tilde{O}
\end{split}
\end{equation}
The relationships of expectation and variance between digital value and observed signal are:
\begin{flalign}
\begin{split}
\text{E}(\text{DV})= & \alpha \text{E}(\Tilde{O}) \\
\text{V}(\text{DV})= & \alpha^2 \text{V}(\Tilde{O})
\end{split}
\end{flalign}
\begin{comment}
The conversion gain then can be expressed as the ratio of these equations.
\begin{flalign}
\begin{split}
\frac{\alpha^2 \text{V}(\Tilde{O})}{\alpha \text{E}(\Tilde{O})} & = \frac{\text{V}(DV)}{\text{E}(DV)} \\
\alpha & = \frac{\text{E}(\Tilde{O})\text{V}(DV)}{\text{V}(\Tilde{O})\text{E}(DV)} \\
\end{split}
\end{flalign}
\begin{equation}
E(\Tilde{O}) = \mu = \frac{E(DV)}{\alpha}
\end{equation}
\end{comment}
Substituting in Equations \ref{eq:obsmean} and \ref{eq:obsvariance} we have
\begin{flalign}
\begin{split}
\text{V}(DV) = & \alpha^2 \text{V}(\Tilde{O}) \\
= & \alpha^2 (\mu^2\sigma_{\text{prnu}}^2 + \mu + \mu\sigma_{\text{prnu}}^2)\\
= & \alpha^2 (\frac{[\text{E}(DV)]^2 \sigma_{\text{prnu}}^2}{\alpha^2} + \frac{\text{E}(DV)}{\alpha} + \frac{\text{E}(DV)}{\alpha}\sigma_{\text{prnu}}^2) \\
= & [\text{E}(DV)]^2 \sigma_{\text{prnu}}^2 + \alpha \text{E}(DV)(1 + \sigma_{\text{prnu}}^2)
\end{split}
\end{flalign}
Finally, rearranging terms we have a solution for $\alpha$ in terms of the measured digital values from a uniform scene and the measured PRNU.
\begin{equation}
\begin{split}
\alpha =
\frac{\textup{V}(\text{DV})-[\text{E}(\text{DV})]^2\sigma_{\text{prnu}}^2}{\text{E}(\text{DV})(1+\sigma_{\text{prnu}}^2)} .
\end{split}
\end{equation}
To estimate the mean and variance of the digital values of a bright uniform scene, we used 1500 pixel values taken from 15 images acquired using an integrating sphere (see Appendix \ref{appendix:sensor}). We took 10 $\times$ 10 crops near the center of each image; this minimized the influence of lens vignetting. The estimated conversion gain 0.1677 $dv/e^-$ is close to the value provided by the manufacturer (0.1707 $dv/e^-$), differing by only 1.8\%. Figure \ref{Fig-conversiongain} shows the histogram of the digital values (panel a) and estimated number of electrons (b). The distribution of digital values is very different from Poisson. After applying the estimated conversion gain to estimated the number of electrons, the distribution is close to Poisson, as expected.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=1\columnwidth]{figures/figure_ConversionGain.png}
\caption{Histogram of (a) digital values. The mean is equal to 320.7 and the variance is equal to 57.8. (b) After scaling by the inverse of the conversion gain we can obtain an estimate of the number of electrons. In this case, the distribution mean and variance are more nearly equal as expected in Equation (6). This is consistent with the expected Poisson character of the light.}
\label{Fig-conversiongain}
\end{center}
\end{figure}
\section{Uniform region identification} \label{appendix:uniform}
To estimate sensor noise, we chose regions in the image with uniform response levels. We randomly sampled a large number of $10 \times 10$ square regions in both the simulated and measured images. For each region we extracted the 50 green channel responses and selected regions based on an analysis of the digital values.
We rejected any region from measured images that contained an obviously 'dead' pixel or significant image non-uniformity. To accomplish this, we first converted the median value of the green pixel responses to electrons, using the conversion gain (See Appendix \ref{appendix:conversion}). In the absence of sensor noise, a uniform region would have a variance equal to the mean, as predicted by the Poisson distribution of photon noise. We rejected any region that had a pixel value more than 3 standard deviations from the mean. We performed the same analysis on the simulated data, though in that case we attribute the rejection to rendering noise.
Second, we defined a criteria for non-uniformity. We held out 20\% of green values and fit the remaining values with a 2\textsuperscript{nd} order spatial polynomial. We then compared the root mean squared error (RMSE) of the held out data to the polynomial fit and to a constant value set to the mean of the green pixel values. We rejected a region as non-uniform if the RMSE of the fit to the constant was larger (2\% greater) than the RMSE of the polynomial fit.
\section*{Acknowledgement} \label{acknowledge}
We thank Krithin Kripakaran for constructing the Cornell Box. We thank Max Furth and Eric Tang for building a model of the Cornell Box in Cinema 4D. We thank Zhenyi Liu, Henryk Blasinski and David Cardinal for their contributions in developing and supporting the ISET3d code and also for many helpful discussions. We thank Gordon Wan, Jamyuen Ko, Guangxun Liao, Bonnie Tseng and Ricardo Motta for their advice, support and encouragement for this project.
\bibliographystyle{ieeetr}
|
2,869,038,154,812 | arxiv | \section{Introduction}
A multi-gap superconductor is characterized by separate superconducting gaps opening on distinctly different parts of the Fermi surface \cite{PhysRevLett.3.552}. The interest in this phenomenon and the emergent new physics was invigorated after the experimental discovery of two-gap superconductivity in bulk MgB$_2$ in 2001 \cite{Nagamatsu2001}. MgB$_2$ consists of planes of boron in a honeycomb lattice alternated by planes of Mg-atoms sitting above the centers of the honeycomb tiles. It is therefore akin to intercalated graphite \cite{Weller2005}, with Mg in the role of the dopant. In MgB$_2$, in-plane $\sigma$-bonds coexist with out-of-plane $\pi$-bonds, and separately give rise to two superconducting gaps for bulk MgB$_2$: the stronger $\sigma$-gap $\Delta_{\sigma}(0)\sim 7$ meV and the weaker $\pi$-gap, $\Delta_{\pi}(0) \sim 2 - 3$ meV \cite{0953-2048-16-2-305,Choi2002,PhysRevB.91.214519,PhysRevB.87.024505,PhysRevB.92.054516}.
Competition and coupling between the multiple condensates in a multi-gap superconductor can lead to rich new physics \cite{0953-2048-28-6-060201}. In that sense, one expects superconductors with three or more gaps to be far more exciting than the two-gap ones, due to additional competing effects and possible quantum frustration between the condensates \cite{PhysRevB.81.134522}. To date discovered effects specific to multi-gap superconductors include novel vortical and skyrmionic states \cite{PhysRevLett.89.067001,PhysRevLett.107.197001}, giant-paramagnetic response \cite{rogerio}, hidden criticality \cite{PhysRevLett.108.207002}, and time-reversal symmetry breaking \cite{PhysRevB.81.134522,PhysRevB.87.134510}, to name a few. A major roadblock for the experimental confirmation of these predictions is the lack of distinctly multi-gap (beyond two-gap) superconductors. In recent years two such materials were proposed theoretically by Gross and coworkers, using density functional theory for superconductors \cite{PhysRevLett.115.097002}. One is molecular hydrogen, which under very high pressure develops three superconducting gaps on different Fermi sheets \cite{PhysRevLett.100.257001}. However, due to anisotropy two of the gaps strongly overlap. The other material is CaBeSi, a MgB$_2$-like compound in which splitting of the $\pi$-bands was predicted to give rise to three-gap superconductivity \cite{PhysRevB.79.104503}, but with impractically low $T_{\mathrm{c}} \cong 0.4$ K.
Here, we follow a different route, namely that of atomically-thin instead of bulk superconductors. Recently, owing to immense experimental progress \cite{0953-2048-30-1-013003,0953-2048-30-1-013002}, superconductivity was realized down to monolayer thickness in several materials -- ranging from electron-phonon-based superconductors, such as In and Pb \cite{Qin2009,Zhang2010}, NbSe$_2$ \cite{doi:10.1021/acs.nanolett.5b00648,Ugeda2016,Xi2016} and doped graphene \cite{Profeta2012,2053-1583-1-2-021005,Ludbrook22092015,Kanetani27112012,Chapman2016,2053-1583-3-4-045003}, to materials with non-conventional coupling mechanisms, such as La$_{2-x}$Sr$_x$CuO$_4$ \cite{Bollinger2011} and FeSe \cite{Ge2015}. The promise for extremely low power, ultra-lightweight and ultra-sensitive electronic devices warrants further progress in ultrathin superconductivity \cite{Golod2015,Najafi2015,Lowell2016}. Quantum confinement in the vertical direction generally separates subbands in ultrathin films, innating multi-band and thereby potentially multi-gap superconductivity \cite{PhysRevB.87.064510}. We here note an additional, natural connection between two-dimensional and multi-gap superconductors, much less explored to date: surface states can equally host new superconducting gaps without equivalent in the bulk material.
In this paper, we start from the known bulk two-gap superconductor MgB$_2$, and show how the gap spectrum changes at the thinnest limit. It was predicted that, albeit not being the thermodynamic ground state, such structures are mechanically stable and could be grown owing to kinetic barriers \cite{PhysRevB.80.134113}, such that few-monolayer MgB$_2$ has already been synthesized experimentally on a Mg-substrate \cite{Cepek2004}. Using a combination of first-principles calculations and anisotropic Eliashberg theory, we reveal a major influence of an emerging surface state on superconductivity in these ultrathin films. This contribution hybridizes with those of the $\sigma$- and $\pi$-bands in a highly nontrivial manner, changing the multi-gap physics with every additional monolayer. This finally leads to pure three-gap superconductivity in one-monolayer MgB$_2$, retained up to a high critical temperature of 20 K (highest among monolayer superconductors without coupling to a substrate). This superconductivity originating from the surface state could not be detected by a previous study of few-monolayer MgB$_2$ based on the tight-binding formalism, in which surface states (electronic as well as vibrational) were completely omitted \cite{PhysRevB.74.094501}. We further demonstrate that this three-gap superconductivity remains robust even under strain, where tensile strain of just $\sim$ 4\% boosts $T_{\mathrm{c}}$ to above 50 K. Such small strain was previously found to increase $T_{\mathrm{c}}$ in bulk MgB$_2$ by at most 10\% \cite{PhysRevLett.93.147006,PhysRevB.73.024509}, or nearly not at all in both electron- \cite{0295-5075-108-6-67005} and hole-doped \cite{PhysRevLett.111.196802,C5NR07755A} graphene (only strain beyond 5\% is predicted to have significant influence there). Considering that such straining can be conveniently realized by growing the monolayer MgB$_2$ on substrates with a somewhat larger lattice constant (e.g., Si$_{1+x}$C$_{1-x}$ or Al$_x$Ga$_{1-x}$N alloys, with a lattice constant tunable by $x$) \cite{PhysRevB.73.024509}, we expect our results to be of immediate experimental relevance.
\begin{figure*}[ht]
\centering
\includegraphics[width=1\linewidth]{fig2.pdf}
\caption{(Color online) (a) and (b) The distribution of the superconducting gap spectrum of 2-ML and 4-ML MgB$_2$, respectively, on the Fermi surface, calculated from anisotropic Eliashberg theory with \textit{ab initio} input. Both are anisotropic two-gap superconductors, with surface condensates \textit{S} and \textit{S'} hybridized with the $\pi$ condensate. (c) The density of states in the superconducting state for 2 and 4 MLs, calculated at $T=1$ K, showing the overall two gap-nature as well as the anisotropy of the gap spectrum. The critical temperatures found for 2 and 4 MLs MgB$_2$ are 23 K and 27 K respectively.}
\label{fig:fig2}
\end{figure*}
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\linewidth]{fig3.pdf}
\caption{(Color online) The overall \textit{e-ph} coupling $\lambda(\textbf{q})=\sum_{\nu} \lambda_{\nu}(\textbf{q})$ (i.e., summed over all phonon nodes) as a function of phonon wave vectors $\textbf{q}$ for (a) 1-ML, (b) 2-ML and (c) 4-ML MgB$_2$.}
\label{fig:fig3}
\end{figure}
\section{Monolayer MgB$_2$}
Our investigation starts from first-principles calculations (using ABINIT \cite{Gonze20092582,suppl_mat}) of one monolayer (ML) of MgB$_2$. It consists of one Mg- and one B-layer, the latter in a honeycomb lattice, and thus structurally similar to doped graphene. The resulting Fermi surface is shown in Fig.~\ref{fig:fig1}(a). It consists of two $\sigma$-bands (around $\Gamma$), a $\pi$-band (around K), and a surface band \textit{S}. While, as we mentioned above, the former two are also present in bulk MgB$_2$, the surface band originates from the Mg-plane facing vacuum. It is thus characteristic of two-dimensional forms of MgB$_2$ and has predominant Mg-$p$ character, as opposed to the B-$p$ character of the other bands. Next, we calculated the electron-phonon (\textit{e-ph}) coupling in 1-ML MgB$_2$ from first principles, employing density functional perturbation theory (DFPT) \cite{PhysRevB.54.16487,Gonze20092582}. With this input, the anisotropic Eliashberg equations (i.e., taking into account the full spatial dependence) were solved self-consistently \cite{PhysRevB.92.054516,PhysRevB.94.144506,suppl_mat}.
We describe the Coulomb repulsion with $\mu^{*}=0.13$, yielding correct $T_{\mathrm{c}}$ for bulk MgB$_2$. This value is also in line with previously established values \cite{Choi2002,PhysRevB.70.104522}. The Coulomb pseudopotential is not expected to change drastically in the 2D limit, owing to the layered structure of MgB$_2$. Namely, superconductivity of the dominant $\sigma$-bands is quasi-two-dimensional even in bulk MgB$_2$, so the same is expected for the screening.
In Fig.~\ref{fig:fig1}(a) we show the resulting superconducting gap spectrum on the Fermi surface, $\Delta(\textbf{k}_{\mathrm{F}},T)$, at $T=1$ K, as well as the distribution of the gap, $\rho(\Delta)$. This result shows that 1-ML MgB$_2$ is a distinctly three-gap superconductor, with separate gaps opening on the $\sigma$-, $\pi$- and \textit{S}-bands. The gap amplitudes are about half of those of bulk MgB$_2$, with Fermi surface averages at zero temperature of $\langle \Delta_{\sigma}(0) \rangle = 3.3$ meV, $\langle \Delta_{S}(0) \rangle = 2.7$ meV and $\langle \Delta_{\pi}(0) \rangle = 1.4$ meV. The critical temperature of $T_{\mathrm{c}}=20$ K, compared to the bulk $T_{\mathrm{c}} \cong 39$ K \cite{0953-2048-16-2-305,Choi2002,PhysRevB.91.214519,PhysRevB.87.024505,PhysRevB.92.054516}, follows the same trend.
To corroborate further the predicted three-gap superconductivity in 1-ML MgB$_2$, we calculated the density of states (DOS) in the superconducting state $N_{\mathrm{S}}$, using Eliashberg relations \cite{PhysRevB.92.054516,suppl_mat}. The result displayed in Fig.~\ref{fig:fig1}(b) shows that $N_{\mathrm{S}}$ for 1-ML MgB$_2$ consists of three distinct and narrow peaks, corresponding to the three superconducting gaps. As $N_{\mathrm{S}}$ determines the superconducting tunneling properties, the predicted three-gap superconductivity can be verified with low-temperature scanning tunneling spectroscopy \cite{0953-2048-16-2-305}.
Last but not least, we show that three-gap superconductivity in 1-ML MgB$_2$ is very robust with temperature. Fig.~\ref{fig:fig1}(c) displays the calculated temperature evolution of the superconducting gap spectrum, proving that the three superconducting gaps are well separated up to 18 K, very close to $T_{\mathrm{c}} = 20$ K.
\section{Evolution with added monolayers}
To provide a deeper understanding of the origin of three-gap superconductivity in 1-ML MgB$_2$, we studied what changes when adding monolayers to the system, considering in particular 2- and 4-ML thick MgB$_2$. The superconducting gap spectra, obtained using anisotropic Eliashberg theory, are displayed in Fig.~\ref{fig:fig2} (a) and (b). One observes in Fig.~\ref{fig:fig2}(a) that a hexagonal band lying between the \textit{S}-band and the $\sigma$-bands develops an additional gap in 2-ML MgB$_2$. This band is a split-off band of the $\sigma$-bands (with B-$p$ character), indicated with \textit{S'} as it originates from a surface state of the free B-surface. The superconducting gap opening on band \textit{S'} is weakly linked to the gaps opening on the $\pi$- and \textit{S}-bands, but (barely) separate from the gap on the $\sigma$-bands, making 2-ML MgB$_2$ an anisotropic two-gap (nearly single-gap) superconductor. In 4-ML MgB$_2$ we find a higher degree of hybridization between the $\pi$-, \textit{S}- and \textit{S'}-condensates, forming an anisotropic gap clearly separated from the $\sigma$-gap. In Fig.~\ref{fig:fig2}(c) we show the corresponding DOS in the superconducting state. For 2-ML MgB$_2$, $N_{\mathrm{S}}$ clearly reflects the anisotropy of the gap spectrum, while for 4-ML MgB$_2$ $N_{\mathrm{S}}$ consists of two broader peaks, resulting from the strong hybridization between the condensates. The critical temperatures we obtained from the solution of the anisotropic Eliashberg equations are larger than that of 1-ML MgB$_2$, namely 23 K and 27 K for 2-ML and 4-ML MgB$_2$ respectively (still well below the bulk value of 39 K \footnote{We note here that this result is different from that obtained in Ref.~\citenum{MORSHEDLOO20151} for 2-ML MgB$_2$, where $T_{\mathrm{c}}$ was found to exceed the bulk value. The difference can be traced back to the unreasonably low Coulomb pseudopotential used in this work, to compensate the lack of multi-band effects in their isotropic Eliashberg approach.}).
The transition from three-gap superconductivity in ML MgB$_2$ to anisotropic two-gap superconductivity and 2-ML and 4-ML MgB$_2$ can be explained by means of the \textit{e-ph} coupling field shown in Fig.~\ref{fig:fig3}. In all cases, the \textit{e-ph} coupling peaks for phonon wave vectors $\textbf{q} \simeq 0$ (i.e., $\Gamma$), which promotes intraband coupling, giving rise to separate condensates on different sheets. However, in Fig.~\ref{fig:fig3} one observes also a clear evolution towards stronger coupling at non-zero wave vectors going from a ML to thicker structures. These emerging coupling channels enable scattering between different sheets, notably between the close-lying \textit{S}, \textit{S'} and $\pi$-bands. This leads to the hybridization between the corresponding condensates shown in Fig.~\ref{fig:fig2}.
Our results show thus a drastic change from the distinctly three-gap superconductivity in single ML MgB$_2$ to very anisotropic two-gap superconductivity by addition of even a single monolayer. Bearing in mind that the superconducting gap opening on the surface band in very thick MgB$_2$ films was found experimentally to be nearly degenerate with the gap on the $\sigma$ band \cite{Souma03}, we expect further rich behavior of the gap spectrum as the MgB$_2$ film is made progressively thicker beyond 4 MLs. Besides accompanying fundamental physics, this strong variation of the gap structure with the number of MLs opens perspectives for nano-engineered superconducting junctions using one single material with spatially varied thickness on the atomic scale. Such local control of thickness is readily available for, e.g., Pb films \cite{0953-2048-30-1-013003,0953-2048-30-1-013002}.
\begin{figure}[ht]
\centering
\includegraphics[width=1\linewidth]{fig4.pdf}
\caption{(Color online) Phonons and electron-phonon coupling of biaxially strained 1-ML MgB$_2$ calculated using DFPT. (a) The phonon dispersion for strains of $-4.5\%$, $+0\%$ and $+4.5\%$. Increasing strain leads to lower phonon frequencies. (b) The E$_{2g}$ phonon mode of the B-atoms that gives the strongest contribution to the electron-phonon coupling. (c) The isotropic Eliashberg function under different strains, $\alpha^2F(\omega)=\langle\langle\alpha^2F({\bf k\, k'},\omega)\rangle_{{\bf k}'_{\mathrm{F}}}\rangle_{{\bf k}_{\mathrm{F}}}$ (i.e., the double Fermi surface average). The peaks originating from the E$_{2g}$ mode are indicated by arrows. The resulting electron-phonon coupling $\lambda$ is shown as inset.}
\label{fig:fig4}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.95\linewidth]{fig5.pdf}
\caption{(Color online) The superconducting spectrum of a biaxially strained 1-ML MgB$_2$. (a) The distribution of the superconducting gap for $+4.5\%$ tensile strain as a function of temperature, displaying the same three gaps ($\pi$, \textit{S} and $\sigma$) as in the unstrained case (Fig.~\ref{fig:fig1}). The calculation shows an enhancement of the critical temperature to $T_{\mathrm{c}}=53$ K. (b) The maximum value of the superconducting gap, $\Delta_{\mathrm{max}}$, as a function of temperature and strain. Superconductivity depletes upon compression and is strongly boosted with tensile strain. (c) $T_{\mathrm{c}}$ as a function of the film thickness, and as a function of strain for a 1-ML MgB$_2$. The bulk value, $T_{\mathrm{c}}=39$ K, is shown for comparison.}
\label{fig:fig5}
\end{figure}
\section{Strained monolayer MgB$_2$}
In experiments, the preferred growth method of atomically thin MgB$_2$ is epitaxial growth on a substrate \cite{Cepek2004}. Due to the ever-present lattice mismatch in that case, we consider the effect of strain on the three-gap superconductivity predicted here. We concentrate on biaxial strain applied with respect to the in-plane cell parameter, namely the Mg-Mg distance with equilibrium value $a=3.04$ \AA. In Fig.~\ref{fig:fig4}(a) we compare the equilibrium phonon band structure of 1-ML MgB$_2$ with the cases of $-4.5$\% compressive strain and $+4.5$\% tensile strain. In the tensile case, interatomic charge densities get depleted as the distances between atoms increase. Consequently, the interatomic bonds become less stiff, resulting in a decrease of phonon frequencies. In the compressive case, the exact opposite occurs. In Fig.~\ref{fig:fig4}(b) we show the E$_{2g}$ phonon mode of the B-atoms, which is the mode harbouring the strongest \textit{e-ph} coupling in 1-ML MgB$_2$. As such, this mode dominates the Eliashberg function $\alpha^2F$, shown in Fig.~\ref{fig:fig4}(c). The peaks in $\alpha^2F$ due to the E$_{2g}$ mode (indicated by arrows) are stronger and more pronounced in 1-ML MgB$_2$ compared with bulk MgB$_2$ \cite{PhysRevB.92.054516}, in particular in equilibrium and under tensile strain. The shift to lower energy (following the general trend for the phonons) and amplification of this peak due to tensile strain lead to a significant enhancement of the \textit{e-ph} coupling, as shown in the inset of Fig.~\ref{fig:fig4}(c). As follows from the above discussion, it is a general principle that tensile strain lowers the energy of the phonon modes, resulting in enhanced \textit{e-ph} coupling, since $\lambda=2\int_0^\infty \mathrm{d}\omega\omega^{-1}\alpha^2F(\omega)$ is weighted by $\omega^{-1}$ \cite{Grimvall}. However, the effect is particularly strong in 1-ML MgB$_2$ due to the occurrence of the E$_{2g}$ phonon mode, which not only goes down in energy but also develops stronger intrinsic coupling to electrons, as follows from the evolution of the Eliashberg function shown in Fig.~\ref{fig:fig4}(c). A similar trend in the \textit{e-ph} coupling under the influence of strain has been found in both electron- and hole-doped graphene \cite{0295-5075-108-6-67005,PhysRevLett.111.196802,C5NR07755A}, although much less pronounced.
With this first-principles input for strained 1-ML MgB$_2$, we solved again the anisotropic Eliashberg equations. We found that the Fermi surface is almost unaltered w.r.t.~that shown in Fig.~\ref{fig:fig1}(a), in the studied range of straining of $-4.5$\% to $+4.5$\%. This, in combination with the robust coupling to the E$_{2g}$ mode, leads to three-gap superconductivity in ML MgB$_2$ being conserved under all strains considered here \footnote{We note that for compressive strains exceeding $-1.5$\% $\sigma$- and \textit{S}-gaps become hybridized, albeit their contributions can still be distinguished. Their partial overlap is not due to new physics -- it is provoked by a general depletion of the superconducting gap values, forcing the gaps closer together.}. In Fig.~\ref{fig:fig5}(a) we show the temperature evolution of the gap spectrum of 1-ML MgB$_2$ subject to tensile strain of $+4.5$ \%, proving the robustness of the three-gap superconductivity even under a considerable amount of strain. Owing to the enhanced \textit{e-ph} coupling [cf.~Fig.~\ref{fig:fig4}(c)] the superconducting gaps are much larger than in the equilibrium case. For $+4.5$ \% strain, the average gaps amount to $\langle \Delta_{\sigma}(0) \rangle = 10.0$ meV, $\langle \Delta_{S}(0) \rangle = 8.4 $ meV and $\langle \Delta_{\pi}(0) \rangle =4.3$ meV, with a corresponding critical temperature as high as $T_{\mathrm{c}} = 53$ K. In Fig.~\ref{fig:fig5}(b) we show the temperature evolution of the maximum ($\sigma$) gap value, comparatively for different strains. It reveals that upon compression, superconductivity is greatly suppressed ($T_{\mathrm{c}}$ drops to 11 K for $-4.5$\% strain), while it is strongly boosted when the ML is subject to tensile strain. The changes are particularly drastic for such limited amounts of strain, in comparison to, e.g., superconducting doped graphene \cite{0295-5075-108-6-67005,PhysRevLett.111.196802,C5NR07755A}. In Fig.~\ref{fig:fig4}(c) we show the evolution of $T_{\mathrm{c}}$ with the number of monolayers and with strain. It is apparent that the effect of strain on superconductivity is stronger, with a ML strained at $+3$\% already surpassing bulk MgB$_2$ as to its $T_{\mathrm{c}}$. A major difference between both manipulations we considered is that strain preserves the three-gap superconductivity of monolayer MgB$_2$, while increasing thickness strongly changes the gap spectrum with every added monolayer, as shown in Fig.~\ref{fig:fig2}.
\section{Conclusion}
In summary, we presented the formation and evolution of three-gap superconductivity in few-monolayer MgB$_2$, by solving the anisotropic Eliashberg equations with full \textit{ab initio} input. We showed that the electronic surface band, originating from the free Mg-surface, plays a major role in ultrathin MgB$_2$, and hosts a third superconducting gap that coexists with the bulk-like $\pi$- and $\sigma$-gaps. These gaps are distinctly separate in 1-monolayer MgB$_2$, where the resulting three pronounced peaks in the superconducting tunneling spectrum provide a clear signature for experimental validation of our prediction. The shown three-gap superconductivity is moreover very robust with temperature, persisting even close to the critical temperature of 20 K. With only $\sim 4$\% tensile strain, \textit{e-ph} coupling is greatly enhanced and superconductivity is boosted to temperatures beyond $50$ K. As more monolayers are added to the film, different condensates hybridize, changing the multi-gap spectrum drastically with every added monolayer. Our investigation therefore establishes atomically thin MgB$_2$ as a unique system to explore tunability of high-$T_\mathrm{c}$, multi-gap superconductivity, and its possible applications in ultrathin cryogenic electronics engineered by strain and atomically controlled thickness.
\begin{acknowledgments}
\noindent This work was supported by TOPBOF-UAntwerp, Research Foundation-Flanders (FWO), the Swedish Research Council (VR) and the R{\"o}ntgen-{\AA}ngstr{\"o}m Cluster. The first-principles calculations have been carried out on the HPC infrastructure of the University of Antwerp (CalcUA), a division of the Flemish Supercomputer Centre (VSC), supported financially by the Hercules foundation and the Flemish Government (EWI Department). Eliashberg theory calculations were supported through the Swedish National Infrastructure for Computing (SNIC).
\end{acknowledgments}
\section*{Appendix}
\begin{appendix}
\section{Density functional (perturbation) theory calculations}
Our density functional theory (DFT) calculations make use of the Perdew-Burke-Ernzerhof (PBE) functional implemented within a planewave basis in the ABINIT code \cite{Gonze20092582}. Electron-ion interactions are treated using norm-conserving Vanderbilt pseudopotentials \cite{PhysRevB.88.085117}, taking into account Mg-2$s^2$2$p^6$3$s^2$ and B-2$s^2$2$p^1$ as valence electrons. An energy cutoff of 60 Ha for the planewave basis was used, to achieve convergence of the total energy below 1 meV per atom. In order to simulate the atomically thin films, we used unit cells that include 25 \AA~of vacuum. A dense $22 \times 22 \times 1$ $ \Gamma$-centered Monkhorst-Pack \textbf{k}-point grid is used for an accurate description of the Fermi surfaces. The lattice parameters were obtained using a conjugate-gradient algorithm so that forces on each atom were minimized below 1 meV/\AA. Strain was implemented by changing the in-plane lattice parameter w.r.t.~the equilibrium value thus obtained.\\
\indent To calculate phonon dispersions and electron-phonon coupling, density functional perturbation theory (DFPT) calculations were carried out, also within the framework of ABINIT. The total number of perturbations due to atomic displacements to be treated (in other words, the number of phonon branches) amounts to $3\cdot N_{\mathrm{atoms}}$, ranging from 9 for a ML to 36 for 4 MLs. Thus, the phonon spectrum and electron-phonon coupling coefficients, matrix elements of the perturbative part of the Hamiltonian \cite{PhysRevB.54.16487}, are obtained. We carried out the DFPT calculations on a $22 \times 22 \times 1$ electronic $\textbf{k}$-point grid and a $11 \times 11 \times 1~\textbf{q}$-point grid (a subgrid of the $\textbf{k}$-point grid) as phonon wave vectors.
\section{Fully anisotropic Eliashberg theory calculations}
In order to describe superconductivity of MgB$_2$ on an \textit{ab initio} level, we solve self-consistently the coupled anisotropic Eliashberg equations \cite{Choi2002},
\begin{align}
Z^{\phantom{\dagger}}_{{\bf k},n}&= 1+ \frac{\pi T}{\omega_n}\sum_{{\bf k'},n'}\frac{\delta(\xi^{\phantom{\dagger}}_{\bf k'})}{N_F}\lambda({\bf kk'},nn') \nonumber \\ \label{el1} & \times \frac{\omega_{n'}}{\sqrt{\omega_{n'}^2 + \Delta^2_{{\bf k'},n}}} \\
\Delta^{\phantom{\dagger}}_{{\bf k},n}Z^{\phantom{\dagger}}_{{\bf k},n}&=\pi T\sum_{{\bf k'},n'}\frac{\delta(\xi^{\phantom{\dagger}}_{\bf k'})}{N_F}\left[\lambda({\bf kk'},nn') - \mu^*(\omega_c)\right] \nonumber \\ \label{el2} & \times \frac{\Delta^{\phantom{\dagger}}_{{\bf k'},n'}}{\sqrt{\omega_{n'}^2 + \Delta^2_{{\bf k},n}}}
\end{align}
using the \textit{ab initio} calculated electron band structure contained in $\xi^{\phantom{\dagger}}_{\bf k}$ and phonon and electron-phonon coupling contained in $\lambda({\bf kk'},nn')$. In the above, T is temperature, $\omega_n=\pi T(2n+1)$ are fermion Matsubara frequencies, $Z^{\phantom{\dagger}}_{{\bf k},n}$ is the mass renormalization function, $\Delta^{\phantom{\dagger}}_{{\bf k},n}$ describes anisotropic even-frequency spin singlet superconductivity, $N_{\mathrm{F}}$ is the electronic density of states at the Fermi level and $\mu^*(\omega_c)$ is the Anderson-Morel Coulomb pseudopotential which comes with a cut-off $\omega_c$. The momentum dependent electron-phonon coupling is
\begin{align}
\lambda({\bf k-k'},n-n')&= \nonumber \\
\int_0^\infty d\omega \, \alpha^2F({\bf k\, k'},\omega) & \frac{2\omega}{\left(\omega_n-\omega_{n'}\right)^2+\omega^2}
\end{align}
with the momentum dependent Eliashberg function
\begin{eqnarray}
\alpha^2F({\bf k\, k'},\omega)=N_F\sum_{\nu} |g^\nu_{\bf q}|^2 \delta(\omega-\omega^{\phantom{\dagger}}_{{\bf q}\nu}),
\end{eqnarray}
where ${\bf q}={\bf k}-{\bf k'}$ and where $g^\nu_{\bf q}$ and $\omega^{\phantom{\dagger}}_{{\bf q}\nu}$ are the phonon branch-resolved electron-phonon scattering matrix elements and phonon frequencies, respectively. From the above, one can obtain the isotropic Eliashberg function as
\begin{eqnarray}
\alpha^2F(\omega)=\langle\langle\alpha^2F({\bf k\, k'},\omega)\rangle_{{\bf k}'_{\mathrm{F}}}\rangle_{{\bf k}_{\mathrm{F}}},
\end{eqnarray}
where $\langle\ldots\rangle_{{\bf k}_{\mathrm{F}}}=\frac{1}{N_{\mathrm{F}}}\sum_{\bf k}\delta(\xi^{\phantom{\dagger}}_{\bf k})\left(\ldots\right)$ is the Fermi surface average.
The quasiparticle density of states that is proportional to single-particle tunneling measurements, is given by
\begin{eqnarray}
N_{\mathrm{S}}(\Omega)\propto\sum_{\bf k}A({\bf k},\Omega)\approx N_{\mathrm{F}}\Big\langle \int_{-\infty}^{\infty}d\xi A_{\bf k}(\xi,\Omega)\Big\rangle_{{\bf k}_{\mathrm{F}}}
\end{eqnarray}
with the spectral function,
\begin{eqnarray}
A({\bf k},\Omega)=-\frac{1}{\pi}\textrm{Im}\left[\hat{G}_R^{\phantom{\dagger}}({\bf k},\Omega)\right]_{11}
\end{eqnarray}
where $\left[\hat{G}_R^{\phantom{\dagger}}({\bf k},\Omega)\right]_{11}$ is the (11) element of the retarded matrix Green's function, obtained after analytic continuation of the full matrix Green's function,
\begin{eqnarray}\label{Gfun}
\hat{G}^{\phantom{\dagger}}_{{\bf k},n}=\left[i\omega_n Z^{\phantom{\dagger}}_{{\bf k},n}\hat{\rho}_0-\xi^{\phantom{\dagger}}_{\bf k}\hat{\rho}_3-\Delta^{\phantom{\dagger}}_{{\bf k},n}\hat{\rho}_1\right]^{-1}.
\end{eqnarray}
The coupled equations (\ref{el1}--\ref{el2}), supplemented by the electron and phonon band structure and the electron-phonon coupling, calculated by first principles, were solved self-consistently in Matsubara space and the converged solutions were then analytically continued to real frequencies. In order to ensure a good accuracy, we imposed a strict convergence criterion of $\frac{x_n-x_{n-1}}{x_n}<10^{-6}$ and allowed up to 1000 iteration cycles. In all the calculations presented here we set $\mu^*(\omega_c)=0.13$ for the Coulomb pseudopotential with a cut-off frequency $\omega_c>0.5$ eV. We have also checked that $\omega_c$ is sufficiently large and that results are well converged with this cut-off. The analytic continuation was performed numerically by employing the high-accuracy Pad\'e scheme based on symbolic computation \cite{PhysRevB.61.5147,PhysRevB.92.054516} with a chosen precision of 250 decimal digits. After this procedure, we calculate the retarded momentum dependent Green's function, the tunneling spectra and the superconducting gap-edge.
\end{appendix}
|
2,869,038,154,813 | arxiv | \section*{Introduction}
Cosmic strings are topological defects that could have been
created during cosmological phase transitions in the early
universe \cite{vilenkin1}. Beyond all question, these defects
cannot be responsible for the primordial density perturbations in
the Cosmic Microwave Background radiation. However the interest in
cosmic strings has been renewed since they can be related to a
number of physical phenomena \cite{vilenkin2,brane,saharian}.
\noindent Cosmic strings cause gravitational phenomena due to the
extremely large mass per unit length these have. Therefore, even
if the string is a straight line, it affects drastically spacetime
around it. For a straight string, spacetime is flat except from a
small deficit angle where curvature has a conical singularity.
This angle can cause many astrophysical effects, such as doubling
images of distant objects (for example quasars), or even cause
gravitational lensing. Furthermore, wiggly cosmic strings could
explain structure formation. In addition, emission of
gravitational waves can be explained with the help of cosmic
strings.
\noindent Astrophysical phenomena could find their origin in
cosmic string theories. Examples of such astrophysical phenomena
are high energy cosmic rays in our galaxy and primordial galactic
magnetic fields. The later are related to superconducting cosmic
strings \cite{witten} from which ultra-high currents (that consist
of charged localized matter) are emitted.
\noindent In our study we shall use straight cosmic strings.
Although real strings are not straight, they can be thought to be
chains of small straight segments. Thus, calculations for straight
strings can be very useful. The background we shall use is that of
de Sitter spacetime. This spacetime is a maximal symmetric
solution to Einstein's equations, with $R^1\times S^3$ topology.
Due to the high symmetry that this gravitational background has,
many problems are exactly solvable. Therefore studies of such
spacetimes can shed light to more difficult problems.
\noindent We focus our study on massive fermions around cosmic
strings in de Sitter spacetime. We find a hidden $N=2$
supersymmetric quantum mechanics algebra underlying the system and
we relate the number of zero mode solutions of the equation of
motion to the Witten index of the supersymmetric algebra.
\noindent The connection of a supersymmetric algebra with a Dirac
fermionic system is not accidental. It is known that Dirac
operators $H$ can be split into even and odd parts, that is,
$H=H_{+}+H_{-}$ (with $H_+$ denoting the even part and $H_-$
denoting the odd part), and this fact is actually closely related
with the general notion of supersymmetry \cite{thaller}.
Particularly, when $H_{+}$ anti-commutes with $H_{-}$, then $H$ is
called a supersymmetric Dirac operator. Supersymmetric quantum
mechanics in Dirac and gauge theories related has been studied in
recent works. For the connection of extra dimensional gauge models
and $N=2$ supersymmetric quantum mechanics algebra ($N=2$ susy QM
thereafter) see \cite{japs}. Also localized fermions around
superconducting cosmic strings are connected with $N=2$ susy QM
algebra, see \cite{oikonomoustrings}.
\medskip
This article is organized as follows: First we briefly review the
$N=2$ supersymmetric quantum mechanics algebra, next we derive the
equations of motion of the fermions around the cosmic strings and
relate the fermionic system with the $N=2$ susy QM algebra.
Finally we present the conclusions along with a discussion.
\section*{Supersymmetric Quantum Mechanics}
We briefly review the $N=2$ supersymmetric quantum mechanics
\cite{susyqm,susyqm1} algebra, relevant to our analysis.
\noindent Consider a quantum system, described by the self-adjoint
Hamiltonian operator $H$ and characterized by the set of
self-adjoint operators $\{Q_1,...,Q_N\}$. The quantum system is
supersymmetric, if,
\begin{equation}\label{susy1}
\{Q_i,Q_j\}=H\delta_{i{\,}j}
\end{equation}
with $i=1,2,...N$. The $Q_i$ are the supercharges and the
Hamiltonian ``$H$" is called supersymmetric (from now on ``susy")
Hamiltonian. The algebra (\ref{susy1}) constitutes the N-extended
supersymmetry. Owing to the anti-commutativity one has,
\begin{equation}\label{susy3}
H=2Q_1^2=2Q_2^2=\ldots =2Q_N^2=\frac{2}{N}\sum_{i=1}^{N}Q_i^2.
\end{equation}
A supersymmetric quantum system is said to have unbroken
supersymmetry, if its ground state vanishes, that is $E_0=0$. When
$E_0>0$, susy is said to be broken.
\noindent In order susy is unbroken, the ground states that belong
in the Hilbert space of all the eigenstates, must be annihilated
by the supercharges,
\begin{equation}\label{s1}
Q_i |\psi_0\rangle=0.
\end{equation}
\subsection*{$N=2$ supersymmetric quantum mechanics algebra}
The $N=2$ algebra consists of two supercharges $Q_1$ and $Q_2$ and
a Hamiltonian $H$, which obey,
\begin{equation}\label{sxer2}
\{Q_1,Q_2\}=0,{\,}{\,}{\,}H=2Q_1^2=2Q_2^2=Q_1^2+Q_2^2
\end{equation}
We introduce the operator,
\begin{equation}\label{s2}
Q=\frac{1}{\sqrt{2}}(Q_{1}+iQ_{2})
\end{equation}
and the adjoint,
\begin{equation}\label{s255}
Q^{\dag}=\frac{1}{\sqrt{2}}(Q_{1}-iQ_{2})
\end{equation}
The above two operators satisfy,
\begin{equation}\label{s23}
Q^{2}={Q^{\dag}}^2=0
\end{equation}
and are related to the Hamiltonian as,
\begin{equation}\label{s4}
\{Q,Q^{\dag}\}=H
\end{equation}
The Witten parity, $W$, for a $N=2$ algebra is defined as,
\begin{equation}\label{s45}
[W,H]=0
\end{equation}
and
\begin{equation}\label{s5}
\{W,Q\}=\{W,Q^{\dag}\}=0
\end{equation}
Also $W$ satisfies,
\begin{equation}\label{s6}
W^{2}=1
\end{equation}
By using $W$, the Hilbert space $\mathcal{H}$ of the quantum
system is spanned to positive and negative Witten parity subspaces
which are defined as,
\begin{equation}\label{shoes}
\mathcal{H}^{\pm}=P^{\pm}\mathcal{H}=\{|\psi\rangle :
W|\psi\rangle=\pm |\psi\rangle \}
\end{equation}
Therefore, the Hilbert space $\mathcal{H}$ is decomposed into the
eigenspaces of $W$, hence $\mathcal{H}=\mathcal{H}^+\oplus
\mathcal{H}^-$. Each operator acting on the vectors of
$\mathcal{H}$ can be represented by $2N\times 2N$ matrices. We use
the representation:
\begin{equation}\label{s7345}
W=\bigg{(}\begin{array}{ccc}
I & 0 \\
0 & -I \\
\end{array}\bigg{)}
\end{equation}
with $I$ the $N\times N$ identity matrix. Bring to mind that
$Q^2=0$ and $\{Q,W\}=0$, hence the supercharges are of the form,
\begin{equation}\label{s7}
Q=\bigg{(}\begin{array}{ccc}
0 & A \\
0 & 0 \\
\end{array}\bigg{)}
\end{equation}
and
\begin{equation}\label{s8}
Q^{\dag}=\bigg{(}\begin{array}{ccc}
0 & 0 \\
A^{\dag} & 0 \\
\end{array}\bigg{)}
\end{equation}
which imply,
\begin{equation}\label{s89}
Q_1=\frac{1}{\sqrt{2}}\bigg{(}\begin{array}{ccc}
0 & A \\
A^{\dag} & 0 \\
\end{array}\bigg{)}
\end{equation}
and also,
\begin{equation}\label{s10}
Q_2=\frac{i}{\sqrt{2}}\bigg{(}\begin{array}{ccc}
0 & -A \\
A^{\dag} & 0 \\
\end{array}\bigg{)}
\end{equation}
The $N\times N$ matrices $A$ and $A^{\dag}$, are generalized
annihilation and creation operators. The action of $A$ is defined
as $A: \mathcal{H}^-\rightarrow \mathcal{H}^+$ and that of
$A^{\dag}$ as, $A^{\dag}: \mathcal{H}^+\rightarrow \mathcal{H}^-$.
In the representation (\ref{s7345}), (\ref{s7}), (\ref{s8}) the
Hamiltonian $H$, can be cast in a diagonal form \footnote{The
diagonal form of a Hamiltonian is most welcome, since the spectral
analysis of the Hamiltonian can be reduced to the analysis of
simpler operators. However if an off diagonal form is preferred,
one can perform a Foldy-Wouthuysen transformation
\cite{thaller}.},
\begin{equation}\label{s11}
H=\bigg{(}\begin{array}{ccc}
AA^{\dag} & 0 \\
0 & A^{\dag}A \\
\end{array}\bigg{)}
\end{equation}
Therefore the total supersymmetric Hamiltonian $H$, consists of
two superpartner Hamiltonians,
\begin{equation}\label{h1}
H_{+}=A{\,}A^{\dag},{\,}{\,}{\,}{\,}{\,}{\,}{\,}H_{-}=A^{\dag}{\,}A
\end{equation}
We define the operator $P^{\pm}$. The eigenstates of $P^{\pm}$,
denoted as $|\psi^{\pm}\rangle$ are called positive and negative
parity eigenstates which satisfy,
\begin{equation}\label{fd1}
P^{\pm}|\psi^{\pm}\rangle =\pm |\psi^{\pm}\rangle
\end{equation}
Using the representation (\ref{s7345}), the parity eigenstates are
represented in the form,
\begin{equation}\label{phi5}
|\psi^{+}\rangle =\left(%
\begin{array}{c}
|\phi^{+}\rangle \\
0 \\
\end{array}%
\right)
\end{equation}
and also,
\begin{equation}\label{phi6}
|\psi^{-}\rangle =\left(%
\begin{array}{c}
0 \\
|\phi^{-}\rangle \\
\end{array}%
\right)
\end{equation}
with $|\phi^{\pm}\rangle$ $\epsilon$ $H^{\pm}$.
\noindent In order to have unbroken supersymmetry, there must be
at least one state in the Hilbert space (we denote it as
$|\psi_{0}\rangle$ ) with vanishing energy eigenvalue, that is
$H|\psi_{0}\rangle =0$. This implies that $Q|\psi_{0}\rangle =0$
and $Q^{\dag}|\psi_{0}\rangle =0$. For a ground state with
negative parity,
\begin{equation}\label{phi5}
|\psi^{-}_0\rangle =\left(%
\begin{array}{c}
0 \\
|\phi^{-}_{0}\rangle \\
\end{array}%
\right)
\end{equation}
this would imply that $A|\phi^{-}_{0}\rangle =0$, while for a
positive parity ground state,
\begin{equation}\label{phi6s6}
|\psi^{+}_{0}\rangle =\left(%
\begin{array}{c}
|\phi^{+}_0\rangle \\
0 \\
\end{array}%
\right)
\end{equation}
it would imply that $A^{\dag}|\phi^{+}_{0}\rangle =0$. A ground
state can either have positive or negative Witten parity.
Nevertheless, when the ground state is degenerate, both cases can
occur. When $E\neq 0$, the number of positive parity eigenstates
is equal to the negative parity eigenstates. Yet, this does not
hold for the zero modes. Zero modes are fully described by the
Witten index. Let $n_{\pm}$ be the number of zero modes of
$H_{\pm}$ in the subspace $\mathcal{H}^{\pm}$. For a finite number
of zero modes (which implies the operator $A$ is Fredholm
\footnote{If an operator $A$ is Fredholm, this implies that it has
discrete spectrum. In addition the Fredholm property is ensured if
$\mathrm{dim{\,}ker}A<\infty$. Equivalently if an operator is
trace-class, then it is by definition Fredholm. For a extensive
analysis on these issues, see \cite{thaller}}), $n_{+}$ and
$n_{-}$, the quantity,
\begin{equation}\label{phil}
\Delta =n_{-}-n_{+}
\end{equation}
is called the Witten index. When the Witten index is non-zero
integer, supersymmetry is unbroken and if it is zero, it is not
clear whether supersymmetry is broken. If $n_{+}=n_{-}=0$
supersymmetry is obviously broken, but if $n_{+}= n_{-}\neq 0$
supersymmetry is not broken.
\noindent The Fredholm index of the operator $A$ is closely
related to the Witten index. The former is defined as,
\begin{equation}\label{ker}
\mathrm{ind} A = \mathrm{dim}{\,}\mathrm{ker}
A-\mathrm{dim}{\,}\mathrm{ker} A^{\dag}=
\mathrm{dim}{\,}\mathrm{ker}A^{\dag}A-\mathrm{dim}{\,}\mathrm{ker}AA^{\dag}
\end{equation}
Indeed we have,
\begin{equation}\label{ker1}
\Delta=\mathrm{ind} A=\mathrm{dim}{\,}\mathrm{ker}
H_{-}-\mathrm{dim}{\,}\mathrm{ker} H_{+}
\end{equation}
If the operator $A$ is not Fredholm, then the Witten index is not
defined as in (\ref{phil}) and (\ref{ker1}). However there exists
a heat-kernel regularized index, both for the operator $A$ (which
we denote $\mathrm{ind}_tA$) and for the Witten index, $\Delta_t$.
The regularized index for the operator $A$ is defined as:
\begin{equation}\label{heatkerw}
\mathrm{ind}_tA=\mathrm{tr}e^{-tA^{\dag}A}-\mathrm{tr}e^{-tAA^{\dag}}
\end{equation}
with $t>0$ and the trace is taken over the eigenfunctions
corresponding to the zero modes of $A$. The regularized Witten
index is equal to:
\begin{equation}\label{hetreg}
\Delta_t=\lim_{t\rightarrow \infty}\mathrm{ind}_tA
\end{equation}
In the following we shall extensively use the definitions we gave
and the notation we used in this section.
\section*{Fermions Around de Sitter Cosmic Strings and $N=2$ SUSY QM}
Consider and infinitely long straight cosmic string. Due to the
cylindrical symmetry, the line element in cylindrical coordinates
$(r,\phi,z)$ is (we use the notation of \cite{saharian}):
\begin{equation}\label{metric}
\mathrm{d}s^2=g_{ \mu
\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=\mathrm{d}t^2-e^{2t/a}(\mathrm{d}r^2+r^2\mathrm{d}\phi^2+\mathrm{d}z^2),
\end{equation}
with $r\geq 0$, $-\infty <z <\infty$. The points $\phi$ and
$\phi_{0}$ on the hypersurface $z=const$ and $r=const$ are
considered to be identical. The parameter $a$ is equal to
$a=\sqrt{3/\Lambda}$, with $\Lambda$ the cosmological constant.
\noindent The Dirac equation in the curved spacetime reads
\cite{Jost,Nakahara,eguchi,saharian},
\begin{equation}\label{spinor}
i\gamma^{\mu}\nabla_{\mu}\psi-m\psi=0,{\,}{\,}{\,}{\,}\nabla_{\mu}=\partial_{\mu}+\Gamma_{\mu}
\end{equation}
where $\gamma^{\mu}$ and $\Gamma^{\mu}$, stand for the curved
spacetime gamma matrices and spin connection respectively. Using
the vierbeins $e^{\mu}_{(a)}$, we can connect the curved spacetime
gamma matrices, to the flat spacetime ones:
\begin{equation}\label{gammadir}
\gamma^{\mu}=e^{\mu }_{(a)}\gamma^{(a)}, {\,}{\,}\Gamma_{\mu
}=\frac{1}{4}\gamma^{(a)}\gamma^{(b)}e^{\nu
}_{(a)}e_{{(b)}{\,}{\nu ;\mu }}
\end{equation}
with $g^{\mu \nu}=e^{\mu}_{(a)}e^{\nu}_{(b)}\eta^{ab}$ and
$\mu=0,1,2,3,4$. The flat space spacetime Dirac matrices are,
\begin{equation}\label{gammaflat}
\gamma^{(0)}=\bigg{(}\begin{array}{ccc}
1 & 0 \\
0 & -1 \\
\end{array}\bigg{)},{\,}{\,}{\,}{\,}{\,}\gamma^{(a)}=\bigg{(}\begin{array}{ccc}
0 & \sigma_{\alpha} \\
-\sigma_{\alpha} & 0 \\
\end{array}\bigg{)}{\,}{\,}{\,}\alpha=1,2,3
\end{equation}
with $\sigma_i$ the Pauli matrices. Using the vierbeins,
\begin{equation}\label{connections}
e^{\mu}_{(a)}=e^{-t/a} \left ( \begin{array}{cccc}
e^{t/a} & 0 & 0 & 0 \\
0 & \cos(q \phi) & -\sin(q\phi)/r & 0 \\
0 & \sin(q\phi) & \cos(q\phi) & 0 \\
0 & 0 & 0 & 1 \\
\end{array} \right)
\end{equation}
with $q=2\pi/\phi_0$, the curved spacetime gamma matrices can be
written as, $\gamma^{0}=\gamma^{(0)}$ and
\begin{equation}\label{gammalast}
\gamma^{i}=e^{t/a}\left (\begin{array}{ccc}
0 & \beta^i \\
-\beta^i & 0 \\
\end{array}\right )
\end{equation}
The matrices $\beta^i$ are equal to,
\begin{equation}\label{beta1}
\beta^{1}=\left (\begin{array}{ccc}
0 & e^{-iq\phi} \\
e^{iq\phi} & 0 \\
\end{array}\right ){\,}{\,}{\,}{\,}{\,} \beta^{2}=-\frac{i}{r}\left (\begin{array}{ccc}
0 & e^{-iq\phi} \\
-e^{iq\phi} & 0 \\
\end{array}\right ){\,}{\,}{\,}{\,}{\,}
\beta^{3}=\bigg{(}
\begin{array}{ccc}
1 & 0 \\
0 & -1 \\
\end{array} \bigg{)}
\end{equation}
Finally the spin connections are,
\begin{equation}\label{Gammabig}
\Gamma_0=0,{\,}{\,}\Gamma_i=-\frac{1}{2a}\gamma^0\gamma_i+\frac{1-q}{2}\gamma^{(1)}\gamma^{(2)}\delta_i^2,
{\,}{\,}{\,}i=1,2,3
\end{equation}
Decomposing the spinor $\psi$ in the following form,
\begin{equation}\label{bispinor}
\psi=\left(%
\begin{array}{c}
\phi \\
\chi \\
\end{array}%
\right)
\end{equation}
the fermionic equations of motion around a cosmic string, in de
Sitter background, can be cast as:
\begin{align}\label{eqmotion}
&D_{+}\phi+\left(\beta^l\partial_l+\frac{1-q}{2r}\beta^1\right
)\chi=0\\ \notag
&D_{-}\chi+\left(\beta^l\partial_l+\frac{1-q}{2r}\beta^1\right
)\phi=0
\end{align}
In the above equations, $D_{\pm}$, stand for:
\begin{equation}\label{operators}
D_{\pm}=\partial_r-\frac{1}{\tau}\left(\frac{3}{2}\pm ima \right )
\end{equation}
with $\tau=-ae^{t/a}$, $-\infty<\tau <0$. Note that
$D_{+}=D_{-}^*$, which means the complex conjugate of $D_{+}$ is
$D_{-}$. Also the operator
$\mathcal{B}=\beta^l\partial_l+\frac{1-q}{2r}\beta^1$ is
self-adjoint. Using the operators $D_{\pm}$ and also
$\mathcal{B}$, we can form an $N=2$ susy QM algebra. Indeed, we
define,
\begin{equation}\label{dmatrix1}
D=\left(%
\begin{array}{cc}
D_{+} & \beta^i\partial_i+\frac{1-q}{2r}\beta^1
\\
\beta^i\partial_i+\frac{1-q}{2r}\beta^1
& D_{-} \\
\end{array}%
\right)
\end{equation}
acting on,
\begin{equation}\label{wee33}
|\phi^{-}\rangle=\left(%
\begin{array}{c}
\phi \\
\chi \\
\end{array}%
\right).
\end{equation}
Upon taking the adjoint we obtain,
\begin{equation}\label{dmatrix23}
D^{\dag}=\left(%
\begin{array}{cc}
D_{-} & \beta^i\partial_i+\frac{1-q}{2r}\beta^1
\\
\beta^i\partial_i+\frac{1-q}{2r}\beta^1
& D_{+} \\
\end{array}%
\right)
\end{equation}
acting on,
\begin{equation}\label{cartman}
|\phi^{+}\rangle=\left(%
\begin{array}{c}
\chi \\
\phi \\
\end{array}%
\right)
\end{equation}
The equation $D|\phi^{-}\rangle =0$ (we used the notation of
(\ref{phi5}) for reasons that will become clear shortly) yields
the solutions of the equation of motion (\ref{eqmotion}). Hence,
it is easy to see that the zero modes of the operator $D$
correspond to the solutions of the equation of motion
(\ref{eqmotion}). Notice that, the zero modes of the operator
$D^{\dag}$ are $\chi$ and $\phi$, as can be easily checked
(actually the equation $D^{\dag}|\phi^{+}\rangle =0$ yields the
equations of motion (\ref{eqmotion})). Using the operators $D$ and
$D^{\dag}$, we can define the supercharges $Q$ and $Q^{\dag}$,
\begin{equation}\label{wit2}
Q=\bigg{(}\begin{array}{ccc}
0 & D \\
0 & 0 \\
\end{array}\bigg{)}, {\,}{\,}{\,}{\,}{\,}Q^{\dag}=\bigg{(}\begin{array}{ccc}
0 & 0 \\
D^{\dag} & 0 \\
\end{array}\bigg{)}
\end{equation}
Also the Hamiltonian of the system can be written in terms of $D$
and $D^{\dag}$, in the following diagonal form,
\begin{equation}\label{wit4}
H=\bigg{(}\begin{array}{ccc}
DD^{\dag} & 0 \\
0 & D^{\dag}D \\
\end{array}\bigg{)}
\end{equation}
It is obvious that the above matrices obey, $\{Q,Q^{\dag}\}=H$,
$Q^2=0$, ${Q^{\dag}}^2=0$, $\{Q,W\}=0$, $W^2=I$ and $[W,H]=0$.
Thus we can see that an $N=2$ susy QM algebra underlies the
fermionic system. This property is it self particularly useful,
especially for spectral problems of Dirac fields around defects
(straight on deformed). Let us now see what are the implications
of supersymmetry in the fermionic system and it's zero mode
solutions. We are particularly interested in the zero mode
solutions but, we shall also discuss at the end of this section
the implications of supersymmetry on the eigenfunctions of the
Hamiltonian with $E\neq 0$.
\noindent The operator $D$ is not Fredholm because it has not
discrete spectrum. This can be easily seen from equation
(\ref{eqmotion}). Indeed it can be written as:
\begin{equation}\label{bigequation}
\left
(\partial_r^2+\frac{1}{r}\partial_r+\frac{1}{r^2}\partial_{\phi}^2+\partial_r^2+\frac{q-1}{r}\beta^1\beta^2\partial_{\phi}-\frac{(q-1)^2}{4r^2}-D_{+}D_{-}\right
)\phi =0
\end{equation}
The solution reads,
\begin{equation}\label{onesolution}
\psi_{\sigma}(x)\sim \left ( \begin{array}{c}
C_1 H_{1/2-ima}^{(1)}(\gamma \eta )J_{\beta_1}(\lambda r) \\
C_2 H_{1/2-ima}^{(1)}(\gamma \eta )J_{\beta_1}(\lambda r)e^{iq\phi} \\
C_3 H_{-1/2-ima}^{(1)}(\gamma \eta )J_{\beta_1}(\lambda r)\\
C_4 H_{-1/2-ima}^{(1)}(\gamma \eta )J_{\beta_1}(\lambda r)e^{iq\phi}\\
\end{array} \right )
\end{equation}
with ``$\sigma$" characterizing the quantum numbers of the system,
two of which continuously vary from zero to infinity ($\lambda$
and $k$). For a complete analysis on the solutions see
\cite{saharian}. Accordingly, the definitions
(\ref{heatkerw}) and (\ref{hetreg}) for the regularized indices
hold.
\noindent Since the two equations $D|\phi^{-}\rangle =0$ and
$D^{\dag}|\phi^{+}\rangle=0$ have the same solutions, namely
$\chi$ and $\phi$, this implies that
$\mathrm{ker}D=\mathrm{ker}D^{\dag}$, which in turn implies
$\mathrm{ker}DD^{\dag}=\mathrm{ker}DD^{\dag}$. Hence the operators
$e^{-tD^{\dag}D}$ and $e^{-tDD^{\dag}}$ have the same trace, that
is $\mathrm{tr}e^{-tD^{\dag}D}=\mathrm{tr}e^{-tDD^{\dag}}$. This
means that the regularized index of the operator $D$ is zero and
hence the regularized Witten index is zero. Therefore,
supersymmetry is unbroken. In order to further clarify this point,
recall that supersymmetry is unbroken in two cases: when the
Witten index is non-zero integer and if it is zero, but the number
of zero modes satisfy $n_{+}= n_{-}\neq 0$. If on the contrary
$n_{+}=n_{-}=0$ supersymmetry is obviously broken. In the
non-Fredholm operator case, the numbers $n_{+}$ and $n_{-}$, that
is, the zero modes of the operator $D$ and $D^{\dag}$
respectively, are replaced by $\mathrm{ker}D$ and
$\mathrm{ker}D^{\dag}$. Since
$\mathrm{ker}D=\mathrm{ker}D^{\dag}\neq 0$, we conclude that just
as in the Fredholm operators case, supersymmetry is unbroken.
Clearly this corresponds to the case we described below equation
(\ref{onesolution}).
\noindent The relation $D|\phi^{-}\rangle=0$ (we use the notation
of relations (\ref{phi6}) and (\ref{phi5})) implies
$Q|\psi^{-}\rangle=0$. This means that the ground state $\Psi^{-}$
is actually a negative parity eigenstate. In such a way, the
negative Witten parity zero mode eigenstate $|\psi^{-}_0\rangle$,
of the Hamiltonian $H^-$, is:
\begin{equation}\label{neg}
|\psi^{-}_0\rangle=\left(%
\begin{array}{c}
0\\
0\\
\phi \\
\chi \\
\end{array}%
\right)
\end{equation}
In the same vain, the positive parity zero mode eigenstate is,
\begin{equation}\label{negkar}
|\psi^{+}_0\rangle =\left(
\begin{array}{c}
\chi \\
\phi \\
0 \\
0 \\
\end{array}
\right)
\end{equation}
Therefore, owing to supersymmetry, the $\chi$ and $\phi$
components of the Dirac spinor constitute the positive and
negative parity solutions of the $N=2$ supersymmetric system.
\noindent Let us see the implications
of supersymmetry on the eigenfunctions of the Hamiltonian with
$E\neq 0$. The Hamiltonians $H_+$ and $H_-$, are known to be
isospectral for eigenvalues different from zero
\cite{thaller,susyqm}, that is,
\begin{equation}\label{isosp}
\mathrm{spec}(H_{+})\setminus \{0\}=\mathrm{spec}(H_{-})\setminus
\{0\}
\end{equation}
In addition, the following relations hold,
\begin{equation}\label{positivee}
Q|\psi^{-}_0\rangle=\sqrt{E}|\psi^{+}_0\rangle{\,}{\,}{\,}\mathrm{and}{\,}{\,}{\,}Q^{\dag}|\psi^{+}_0\rangle=\sqrt{E}|\psi^{-}_0\rangle,
\end{equation}
with $E$ the common eigenvalues of the Hamiltonians $H_+$ and
$H_-$. In turn these imply,
\begin{equation}\label{pose}
D|\phi^{-}\rangle=\sqrt{E}|\phi^{+}\rangle
{\,}{\,}{\,}\mathrm{and}{\,}{\,}{\,}D^{\dag}|\phi^{+}\rangle=\sqrt{E}|\phi^{-}\rangle.
\end{equation}
Apart from these, there are many interesting mathematical
properties that the supersymmetric system possesses, but these are
beyond the scope of this article \footnote{For example the Krein's
spectral function $\xi(\lambda)$, which is equal to zero because
the Witten index is zero.}.
We have to note that supersymmetry of any kind around topological defects and in
various gravitational backgrounds, has been studied in many works,
for example \cite{schwarzchild,monopole,taub}. In reference
\cite{schwarzchild}, there was found that there is a close
connection between supersymmetric quantum mechanics and the
mechanics of a spinning Dirac particle moving in Schwarzschild
spacetime. In addition in reference \cite{monopole} it was found
that a fermion in a monopole field possesses a rich supersymmetry
structure. The connection of supersymmetry with the Taub-NUT
spacetime was studied in \cite{taub}.
\noindent Studies involving de Sitter space are of particular
importance. Indeed there has been a great deal of work on de
Sitter space solutions in supergravity and string theory (see
\cite{linde} and references therein). The connection of de Sitter
space and supercritical string theory was presented in
\cite{critstring}. In addition, de Sitter space is known to have
finite entropy. De Sitter entropy can be understood as the number
of degrees of freedom in a quantum mechanical dual \cite{horava}.
Moreover, de Sitter space is closely connected with $N=2$
supersymmetry \cite{renata} and also these two are connected with
hybrid inflation solutions. Furthermore, $N=4$ supersymmetric
quantum mechanics is very closely related to the description of
particle dynamics in de Sitter space.
\noindent Obviously spacetime supersymmetry and supersymmetric
quantum mechanics are not the same, nevertheless the connection is
profound, since extended (with $N=4,6...$) supersymmetric quantum
mechanics models describe the dimensional reduction to one
(temporal) dimension of $N=2$ and $N=1$ Super-Yang Mills models
\cite{pashev}.
\noindent The interconnection of supersymmetric theories is an
interesting mathematical property that is closely connected with
the construction of superconformal quantum mechanical models
\cite{pashev}. In addition, a number of physical applications,
renders studies of supersymmetric quantum mechanics models to be
of valuable importance (due to the simplicity these have), such as
low energy dynamics of black hole models, see \cite{pashev}.
\noindent Localized fermion solutions around vortices and other
defects frequently appear in superconductor studies. Thus in view
of a micro-description of superconductors, through holographic
superconductors \cite{hol}, the existence of a supersymmetric
quantum mechanic algebra could be of particular importance.
\noindent From the above, it is clear that supersymmetric quantum
mechanics is a very useful tool to make complex field theories
more easy to handle. In addition, studying fermionic solutions
around defects in de Sitter space is very interesting, since these
problems may be reductions of more evolved problems.
\noindent But there is another valuable property of the
supersymmetric quantum mechanics supercharges that we did not
mentioned. The set of the $N=2$ supersymmetric quantum mechanics
supercharges are invariant under an R-symmetry. Furthermore, the
Hamiltonian is invariant under this symmetry \cite{susyqm}.
Particularly, the real superalgebra (\ref{susy1}) and
(\ref{sxer2}) is invariant under the transformation,
\begin{equation}\label{rsymetry}
\left (\begin{array}{c}
Q'_1 \\
Q'_2 \\
\end{array}\right )=\left (\begin{array}{ccc}
\cos a & \sin a \\
-\sin a & \cos a \\
\end{array}\right ) \cdot \left ( \begin{array}{c}
Q_1 \\
Q_2 \\
\end{array} \right )
\end{equation}
Correspondingly the complex supercharges $Q$ and $Q^{\dag}$ are
transformed under a global $U(1)$ transformation:
\begin{equation}\label{supercharges}
Q^{'}=e^{-ia}Q, {\,}{\,}{\,}{\,}{\,}{\,}{\,}{\,}
{\,}{\,}Q^{'\dag}=e^{ia}Q^{\dag}
\end{equation}
This R-symmetry is a symmetry of the Hilbert states corresponding
to the spaces $\mathcal{H}^{+}$ and $\mathcal{H}^{-}$. Furthermore
the two spaces can have different transformation parameters. To
make this clear, let $\psi^{+}$ and $\psi^{-}$ denote the Hilbert
states corresponding to the spaces $\mathcal{H}^{+}$ and
$\mathcal{H}^{-}$. Then the $U(1)$ transformation of the states
is,
\begin{equation}
\psi^{'+}=e^{-i\beta_{+}}\psi^{+},
{\,}{\,}{\,}{\,}{\,}{\,}{\,}{\,}
{\,}{\,}\psi^{'-}=e^{-i\beta_{-}}\psi^{-}
\end{equation}
It is clear that the parameters $\beta_{+}$ and $\beta_{-}$ are
different global parameters. Consistency with relation
(\ref{supercharges}) requires that $a=\beta_{+}-\beta_{-}$. We
must note that a global $U(1)$ symmetry is expected around a
cosmic string \cite{meierovich}. Furthermore the breaking of a
local $U(1)$ leaves a global $U(1)$ in cosmic string models
\cite{vilenkin1}. Thus this global $U(1)$ symmetry is really
interesting. Nevertheless we must further study whether there is a
connection between the susy R-symmetry and the remnant global
$U(1)$ symmetry of phenomenological models around strings. It
would then be interesting to study more realistic models, but this
is beyond the scopes of this article.
\noindent The cylindrical line element (\ref{metric}) has played a
crucial role in our analysis. An important question that is raised
is whether the outcomes of the de-Sitter cosmic string framework
hold for more general spacetimes. The answer in this question is
not so trivial and we should be very cautious in generalizing our
results. We have applied our results to cosmic strings in flat
spacetime and also in anti-de Sitter spacetimes and no
supersymmetry algebra underlies the fermionic system in these two
cases. Therefore such an analysis must be carried away with great
detail and caution and certainly would be invaluable. Also
possible generalizations to include bosonic or vector
configurations could be interesting. But this analysis is beyond
the purposes of this article.
\noindent Before closing this section we must mention some
supersymmetric quantum mechanics developments, relevant to our
work. The
supersymmetry studied in this article is similar to what takes
place in some periodic systems where $n_+=n_-=1\neq 1$, that is,
susy is exact though the Witten index is equal to zero
\cite{pluskai1,pluskai2,pluskai3}.
\noindent The peculiarity of the periodic systems
\cite{pluskai1,pluskai2,pluskai3} is that besides an $N=2$
supersymmetry, these systems possess much more rich structure
(related to their special nature being finite gap systems) which
is related to the existence of a hidden supersymmetry. Since in
our case, our system is characterized by the same property
$n_+=n_-\neq 0$, maybe it also possess a hidden, bosonized
supersymmetry. Hidden, bosonized supersymmetry is related to the
existence of a grading operator, which has a nonlocal nature
(reflection operator) \cite{pluskai4,pluskai5,pluskai6}. We hope
we pursuit further these issues in a future publication.
\section*{Conclusions}
In this paper we found a hidden $N=2$ susy QM algebra, underlying
the fermionic solutions of the Dirac equation around a straight
cosmic string in de Sitter spacetime. The negative and positive
Witten parity states of the susy algebra can be written in terms
of the fermionic solutions of the equations of motion. The Witten
index is zero and additionally the number of zero modes for the
two supercharges are equal. We therefore concluded that the system
has unbroken supersymmetry. It was tempting to check whether this
susy algebra underlies the fermion system around a cosmic string
but with a flat background. It turns out that there is no
supersymmetry in the flat case. It stands to reason to argue that,
the presence of spacetime curvature is responsible for the susy
structure in the de Sitter fermion system. However this must not
be the case, since in the superconducting string case, the
background spacetime is also flat, still, an $N=2$ susy QM algebra
is present \cite{oikonomoustrings}. We hope to comment on this
issues in the future.
\noindent Before closing we must note that there is a mathematical
property stemming from $N=2$ susy QM, that can be very useful
perhaps when studying perturbations of the metric around the
string, or in the case the string is not straight (also perhaps in
the case of an electromagnetic field around the string. However we
must further study the last case to be sure. We hope to do so in a
future work). If these effects can be described by a matrix $C$,
which anti-commutes with the Witten operator and is also
symmetric \footnote{Also the operator $Ce^{-D^{\dag}D}$ must be
trace-class}, then,
\begin{equation}\label{indperturbation}
\mathrm{ind}_{t}(D+C)=\mathrm{ind}_{t}D
\end{equation}
This means that there is a correspondence between the solutions of
the equation \linebreak $(D+C)\psi=0$ and these of the equation
$D\psi=0$. This is very useful, regarding fermionic spectral
problems around one dimensional defects.
.
|
2,869,038,154,814 | arxiv | \section{Introduction}
Irreversibility is the telltale sign of nonequilibrium dissipation~\cite{maes2003time,parrondo2009entropy}. {Systems operating far-from-equilibrium utilize part of their free energy budget to perform work, while the rest is dissipated into the environment. Estimating the amount of free energy lost to dissipation is mandatory for a complete energetics characterization of such physical systems. For example, it is essential for understanding the underlying mechanism and efficiency of natural Brownian engines, such as RNA-polymerases or kinesin molecular motors, or for optimizing the performance of artificial devices \cite{ gnesotto_Chase2018broken,li2018quantifying,brown2017toward}. Often the manifestation of irreversibility} is quite dramatic, signalled by directed flow or movement, as in transport through mesoscopic devices \cite{datta1997electronic}, traveling waves in nonlinear chemical reactions~\cite{castets1990experimental}, directed motion of molecular motors along biopolymers~\cite{astumian1994fluctuation}, and the periodic beating of a cell's flagellum~\cite{brokaw1979calcium,battle2016broken} or cilia~\cite{vilfan2006hydrodynamic}.
This observation has led to a handful of experimentally-validated methods to identify irreversible behavior by confirming the existence of such flows or fluxes~\cite{gnesotto_Chase2018broken,gladrow2016broken,fodor_Wijland2016nonequilibrium,zia2007probability}.
However, in the absence of directed motion, it can be challenging to determine if an observed system is out of equilibrium, especially in small noisy systems where fluctuations could mask any obvious irreversibility~\cite{rupprecht2016fresh}.
One such possibility is to observe a violation of the fluctuation-dissipation theorem~\cite{martin2001comparison,mizuno2007nonequilibrium,bohec2013probing}; though this approach requires not just passive observations of a correlation function, but active perturbations in order to measure response properties, which can be challenging in practice.
Thus, the development of noninvasive methods to quantitatively measure irreversibility and dissipation are necessary to characterize nonequilibrium phenomena.
Our understanding of the connection between irreversibility and dissipation has deepened in recent years with the formulation of stochastic thermodynamics, which has been verified in numerous experiments on meso-scale systems \cite{liphardt2002equilibrium,collin2005verification,toyabe2010experimental,xiong2018experimental}. {Within this framework, it is possible to evaluate quantities as the entropy along single non-equilibrium trajectories \cite{seifert2005entropy}.}
A cornerstone of this approach is the establishment of a quantitative identification of dissipation, or more specifically entropy production rate ${\dot S}$, as the Kullback-Leibler divergence (KLD) between the probability ${\mathcal P}(\gamma_t)$ to observe a trajectory $\gamma_t$ of length $t$ and the probability ${\mathcal P}(\tilde\gamma_t)$ to observe the time-reversed trajectory $\tilde\gamma_t$ \cite{kawai2007dissipation,maes1999fluctuation,maes2003time,roldan2018arrow,horowitz2009illustrative,gaveau2014dissipation,gaveau2014relative}:
\begin{equation}
\dot S \geq \dot S_{\rm KLD} \equiv \lim_{t\to\infty} \frac{k_{\rm B}}{t} D [{\cal P}(\gamma_t)||{\cal P}( \tilde\gamma_t)],
\label{eq:KPB}
\end{equation}
where $k_{\rm B}$ is Boltzmann's constant.
The KLD between two probability distributions $p$ and $q$ is defined as $D[p||q]\equiv\sum_x p(x)\ln (p(x)/q(x))$ and is an information-theoretic measure of distinguishability \cite{cover2012elements}.
{For the rest of the paper we take $k_{\rm B} = 1$, so the entropy production rate has units of $time^{-1}$. The entropy production $\dot S$ in Eq. \eqref{eq:KPB} has a clear physical meaning. It is the usual entropy production defined in irreversible thermodynamics by assuming that the reservoirs surrounding the system are in equilibrium. For instance, in the case of isothermal molecular motors hydrolyzing ATP to ADP+P at temperature $T$, the entropy production in Eq. \eqref{eq:KPB} is $\dot S=r\Delta\mu/T-\dot W/T$, where $r$ is the ATP consumption rate, $\Delta\mu=\mu_{\rm ATP}-\mu_{\rm ADP}-\mu_{\rm P}$ is the difference between the ATP, and the ADP and P chemical potentials, and $\dot W$ is the power of the motor \cite{parrondo2002energetics}. In many experiments, all these quantities can be measured except the rate $r$. Therefore, the techniques that we develop in this paper can help to estimate the ATP consumption rate, even at stalling conditions.}
The equality in \eqref{eq:KPB} is reached if the trajectory $\gamma_t$ contains all the meso- and microscopic variables out of equilibrium.
Hence the relative entropy in \eqref{eq:KPB} links the statistical time-reversal-symmetry breaking in the mesocopic dynamics directly to dissipation.
Based on this connection, estimators of the relative entropy between stationary trajectories and their time reverses allow one to determine if a system is out of equilibrium or even bound the amount of energy dissipated to maintain a nonequilibrium state.
Such an approach, however, is challenging to implement accurately as it requires large amounts of data, especially when there is no observable current \cite{roldan2010estimating}.
Despite the absence of observable average currents, irreversibility can still leave a mark in fluctuations.
Consider, for example, a particle hoping on a 1D lattice, as in Fig.~\ref{fig:fig1}, where {up} and {down} jumps have equal probabilities, but the timing of the jumps have different likelihoods. Although there is no net drift on average, the process is irreversible, since any trajectory can be distinguished from its time-reverse due to the asymmetry in jump times.
Thus, beyond the sequence of events, the timing of events can reveal statistical irreversibility.
{Such a concept was used, for example, to determine that the \textit{E. Coli} flagellar motor operates out of equilibrium based on the motor dwell-time statistics \cite{tu2008nonequilibrium}.}
\begin{figure}
\centering
\includegraphics[width=.75\linewidth]{./Fig1}
\caption{ Brownian particle jumping on an one-dimensional lattice. Jumps up and down are equally likely, but with asymmetric jump rates. As a result, the irreversibility of the dynamics is contained solely in the timing fluctuations.}
\label{fig:fig1}
\end{figure}
In this work, we establish a technique that allows one to identify and quantify irreversibility in fluctuations in the timing of events, by applying Eq.~(\ref{eq:KPB}) to stochastic jump processes with arbitrary waiting time distributions, that is, semi-Markov processes, also known as continuous time random walks (CTRW) in the context of anomalous diffusion.
{Such models emerge in a plethora of contexts \cite{kindermann2017nonergodic, schulz2014aging,metzler2014anomalous} ranging from economy and finance \cite{scalas2006application} to biology, as in the case of kinesin dynamics \cite{fisher2001simple} or in the anomalous diffusion of the Kv2.1 potassium channel \cite{weigel2011ergodic}.} In fact, as we show below and in {the Methods section}, semi-Markov processes result in experimentally-relevant scenarios where one has access only to a limited set of observables of Markov kinetic networks with certain topologies.
We begin by reviewing the semi-Markov framework, where we present our main result of the entropy production rate estimator. Next, we apply our approach to general hidden networks, where an observer has access only to a subset of the states, comparing our estimator with previous proposals for partial entropy production that are zero in the absence of currents. Finally, we address a particularly important case of molecular motors, where their translational motion is easily observed, but the biochemical reactions that power their motion are hidden. Remarkably, our technique allows us to even reveal the existence of parasitic mechano-chemical cycles at stalling -- where the observed current vanishes or the motor is stationary -- simply from the distribution of step times. {In addition, our quantitative lower bound on the entropy production rate can be used to shed light on the efficiency of molecular motors operation and on the entropic cost of maintaining their far-from-equilibrium dynamics \cite{horowitz2017minimum,horowitz2017information,pietzonka2016universal,brown2017allocating,large2018optimal}.}
\section{Results}
\subsection{Irreversibility in semi-Markov processes}
\label{sec:ismc}
A semi-Markov process is a stochastic renewal process $\alpha(t)$ that takes values in a discrete set of states, $\alpha=1,2,\dots$. The renewal property implies that the waiting time intervals $t_\alpha$ in a given state $\alpha$ are positive, independent, and identically distributed random variables. If the system arrives to state $\alpha$ at $t=0$, the probability to jump to a different state $\beta$ at time $[t,t+{\rm d}t]$ is $\psi_{\beta\alpha}(t){\rm d}t$, with $\psi_{\beta\alpha}(t)$ being the probability density of transition times \cite{cinlar2013introduction}. These densities are not normalized, with $p_{\beta\alpha}\equiv \int_0^\infty \psi_{\beta\alpha}(t)dt$ being the probability for the next jump to be $\alpha\to \beta$ given that the walker arrived at $\alpha$. We assume that the particle eventually leaves any site $\alpha$, {\it i.e.}, $\psi_{\alpha\alpha}(t)=0$ and $\sum_{\beta} p_{\beta\alpha}=1$, so the matrix $p_{\beta\alpha}$ is a stochastic matrix.
Its normalized (right) eigenvector ${R_\alpha}$ with eigenvalue $1$, then represents the fraction of visits to each state $\alpha$.
The waiting time distribution at site $\alpha$, $\psi_\alpha(t)=\sum_\beta\psi_{\beta\alpha}(t)$, is normalized with average waiting time $\tau_{\alpha}$. We can also define the waiting time distribution conditioned on a given jump $\alpha\to \beta$ as $\psi(t|\alpha\to \beta)\equiv{\psi_{\beta\alpha}(t)}/{p_{\beta\alpha}}$, which is already normalized.
Consider now a generic semi-Markovian trajectory $\gamma_t$ of length $t$ with $n$ jumps, which is fully described by the sequence of jumps and jump times,
$\gamma_t=\{ \alpha_1\xrightarrow{t_1} \alpha_2 \xrightarrow{t_2} \dots \xrightarrow{t_{n-1}} \alpha_{n} \xrightarrow{t_{n}} \alpha_{n+1}\}$
with $\sum_n t_n=t$,
occurring with probability
${\cal P}(\gamma_t)=\psi_{\alpha_2,\alpha_1}(t_1)\psi_{\alpha_3,\alpha_2}(t_2)\dots \psi_{\alpha_{n+1},\alpha_{n}}(t_n)$.
In order to characterize the dissipation of this single trajectory, we must define its time reverse $\tilde\gamma_t=\{ \alpha_n\xrightarrow{t_n} \alpha_{n-1} \xrightarrow{t_{n-1}}\dots \xrightarrow{t_2}\alpha_1\xrightarrow{t_1} \alpha_{0} \}$
whose probability is given by
${\cal P}(\tilde\gamma_t)=\psi_{ \alpha_0, \alpha_1}(t_1)\dots \psi_{ \alpha_{n-1}, \alpha_{n}}(t_n)$, see Methods and Fig. \ref{figtrajalpha}.
Directly applying \eqref{eq:KPB} to this scenario shows that the KLD between the probability distributions of the forward and backward trajectories can be split into two contributions (see Methods):
\begin{equation}\label{Eq:def_KLD_EP_rate}
\dot S_{\rm KLD}=\dot S_{\rm aff}+\dot S_{\rm WTD}.
\end{equation}
The first term, $\dot S_{\rm aff}$, or {affinity entropy production}, results entirely from the divergence between the state trajectories, regardless of the jump times, $\sigma\equiv\{ \alpha_1,\alpha_2,\dots,\alpha_{n+1}\}$ and $\tilde\sigma\equiv\{ \alpha_n,\dots, \alpha_1,\alpha_0\}$, that is, it accounts for the affinity between states:
\begin{equation}
\dot S_{\rm aff} =\frac{1}{\cal T}\sum_{\alpha\beta} p_{\beta\alpha}R_\alpha\ln\frac{p_{\beta\alpha}}{p_{ \alpha\beta}}=\frac{1}{\cal T}\, \sum_{\alpha<\beta} J^{\rm ss}_{\beta\alpha} \ln\frac{p_{\beta\alpha}}{p_{\alpha\beta}} ,
\label{saff}
\end{equation}
where $J^{\rm ss}_{\beta\alpha}=p_{\beta\alpha}R_\alpha - p_{\alpha\beta}R_\beta $ is the net probability flow per step, or current, from $\alpha$ to $\beta$, and the factor ${\cal T}=\sum_\alpha \tau_\alpha R_\alpha$ is the mean duration of each step, which can be used to transform the units from per-step to per-time~\cite{bedeaux1971relation}. We see that the affinity entropy production vanishes in the absence of currents, as it occurs in arbitrary Markov systems \cite{roldan2010estimating,roldan2012entropy}.
The contribution due to the waiting times is expressed in terms of the KLD between the waiting time distributions
\begin{equation}
\dot S_{\rm WTD}=\frac{1}{\cal T}\sum_{\alpha\beta\mu} p_{\mu\beta}p_{\beta\alpha}R_\alpha D\left[\psi(t|\beta\to \mu)||\psi(t|\beta\to \alpha)\right],\label{d2}
\end{equation}
which is the main result of this paper and allows one to detect irreversibility in stationary trajectories with zero current.
Notice that $R_\alpha$ being the occupancy of state $\alpha$, $p_{\beta\alpha}R_\alpha$ is the probability to observe the sequence $\alpha\to \beta$ in a stationary forward trajectory, while $p_{\mu\beta}p_{\beta\alpha}R_\alpha $ is the probability to observe the sequence $\alpha\to \beta\to \mu$.
Equation (\ref{Eq:def_KLD_EP_rate}) is the chain rule of the relative entropy applied to the semi-Markov process and the core of our proposed estimator. In the special case of Poisson jumps, $D\left[\psi(t|\beta\to \mu)||\psi(t| \beta\to \alpha)\right]=0$ since all waiting time distributions for jumps starting at a given site $\beta$ are equal (see Methods), and we recover the standard expression for the relative entropy of Markov processes $\dot S =\dot S_{\rm aff}$. It is worth mentioning that previous attempts to establish the entropy production of semi-Markov processes failed to identify the term $S_{\rm WTD}$ because they assumed that the waiting time distributions were independent of the final state, as occurs in Markov processes \cite{esposito2008continuous,maes2009dynamical,wang2007detailed}.
However, such a strong assumption does not hold in many situations of interest, as in the ones discussed below.
\subsection{Decimation of Markov chains and second-order semi-Markov processes} Semi-Markov processes appear when sites are decimated from Markov chains of certain topologies. Fig.~\ref{fig:fig2} shows representative examples. In Fig.~\ref{fig:fig2}a-b, we show two models of a molecular motor that runs along a track with sites $\{\dots,i-1,i,i+1,\dots\}$ and has six internal states. If the spatial jumps (red lines) and the transitions between internal states (black lines) are Poissonian jumps, then the motor is described by a Markov process. On the other hand, when the internal states are not accessible to the experimenter, the waiting time distributions corresponding to the spatial jumps $i\to i\pm 1$ are no longer exponential and the motion of the motor must be described by a semi-Markov process. Fig.~\ref{fig:fig2}a show an example where the decimation of internal states directly yields a semi-Markov process ruling the spatial motion of the motor. The second example, sketched in Fig.~\ref{fig:fig2}b, is more involved since the upward and the downward jumps end in different sets of internal states. As a consequence, the waiting time distribution of, say, the jump $i\to i+1$, depends on the site that the motor visited before site $i$. Then, the resulting dynamics must be described by a second order semi-Markov process, that is, one has to consider the states $\alpha(t)=[i_{\rm prev}(t),i(t)]$, where $i(t)$ is the current position of the motor and $i_{\rm prev}(t)$ is the previous position, right before the jump.
The same applies to generic kinetic networks, as the one depicted in Fig.~\ref{fig:fig2}c. Suppose that the original network is Markovian with states $i=1,\dots,5$.
However, if the experimenter only has access to states $1$ and $2$, with the rest clumped together into a hidden state $H$, then the resulting dynamics is also a second-order semi-Markov process with the reduced set $i=1,2,H$.
For second-order semi-Markov processes the affinity entropy production reads
\begin{equation}
\dot S_{\rm aff} =\frac{1}{\cal T} \sum_{i,j,k}p(ijk)\ln\frac{p([ij]\to [jk])}{p([kj]\to [ji])} \label{d10main},
\end{equation}
where $p(ijk)\equiv R_{[ij]} p([ij]\to [jk])$ is the probability to observe the sequence $i\to j\to k$. This entropy is still proportional to the current for one dimensional processes and therefore vanishes in the absence of flows in the observed dynamics, see Methods.
The entropy production contribution due to the irreversibility of the waiting time distributions is:
\begin{equation}\label{Eq:EP_rate_WTD_2nd_semi_markov}
\dot S_{\rm WTD}=\frac{1}{\cal T} \sum_{i,j,k} p(ijk) D\left[\psi(t|[ij]\to [jk])||\psi(t|[kj]\to [ji])\right].
\end{equation}
{Let us emphasize that the calculation of $\dot S_{\rm WTD}$ requires collecting statistics on sequences of two consecutive jumps, \emph{i.e.}, $i\to j \to k$}.
We now proceed to apply these results to generic cases of simple kinetic networks and molecular motors.
\begin{figure
\centering
\includegraphics[width=.9\linewidth]{./Fig2}
\caption{Decimation of Markov processes. \textbf{a-b} Molecular motor model: An observer with access only to the position (vertical axis) cannot resolve the internal states (circles). \textbf{a} Decimation to position results in a first-order Markov process, since spatial jumps connect the same internal state. \textbf{b} Decimation results in a second-order semi-Markov process, where the waiting time distribution for spatial transitions depends on whether the motor previously jumped down or up. \textbf{c} Hidden kinetic network: An observer unable {to} resolve states 3, 4, and 5, treats them as a single hidden state $H$. The resulting decimated network is a second-order semi-Markov process on the three states $1$, $2$, and $H$, where the non-Poissonian waiting time distributions for transitions out of state $H$ depend on the past.}
\label{fig:fig2}
\end{figure}
\subsection{Hidden networks}\label{sec:hidden}
We first apply our formalism to estimate the dissipation in kinetic networks with hidden states, which have received increasing attention in recent years owing to their many practical and experimental implications \cite{andrieux2007entropy,kawai2007dissipation,roldan2010estimating,roldan2012entropy,shiraishi2015fluctuation,polettini2017marginal}.
Consider a network where $\omega_{ij}$ is the transition rate from state $j$ to $i$, with $\pi_i$ the steady-state distribution. The total entropy production rate at steady-state is \cite{van2015ensemble}
\begin{equation}\label{saff}
\dot S= \sum_{i<j} \left( \omega_{ji}\pi_i- \omega_{ij}\pi_j\right) \ln\frac{\omega_{ji}\pi_i}{\omega_{ij}\pi_j},
\end{equation}
where the positivity of $\dot S$ stems from the positivity of each individual term in the sum \cite{esposito2015stochastic,shiraishi2015fluctuation,horowitz2017minimum}.
{In order to calculate the total entropy production $\dot{S}$ according to Eq. \eqref{saff}, full knowledge of the steady state probability distribution $\{\pi_i\}$ and the transition rates between all the microstates $\{\omega_{ij}\}$ is required. }
We would like to assign a partial entropy production rate when one only has access to a limited set of states and transitions.
To be concrete, we focus on the scenario depicted in Fig.~\ref{fig:fig2}c, where only states $1$ and $2$ can be observed. Previously, two approaches for assigning partial entropy production rate in such a case have been defined in the literature, both of which provide a lower bound on the total entropy production rate \cite{gilijordan2017}: the passive partial entropy production rate due to Shiraishi and Sagawa \cite{shiraishi2015fluctuation}, and the informed partial entropy production rate due to Polettini and Esposito \cite{polettini2017marginal,polettini2018effective}
The passive partial entropy production rate $\dot{S}_{\rm PP}$ for the single observed link is simply given by the corresponding term in \eqref{saff}
\begin{equation}
\dot{S}_{\rm PP} = (\omega_{12}\pi_2 - \omega_{21}\pi_1 )\ln\frac{\omega_{12}\pi_2}{\omega_{21}\pi_1},
\end{equation}
where the observer is assumed to have access to the steady-state populations of the two states, $\pi_1$ and $\pi_2$, as well as the transition rates between them.
The informed partial entropy production $\dot{S}_{\rm IP}$ for the single link requires additional information: the observer is assumed to have control over the transition rates of the observed link, without affecting any of the hidden transitions, such that they can stall the corresponding current and record the ratio of populations in the two observed states, $\pi_1^{\rm stall}/\pi_2^{\rm stall}$. The stalling distribution $\pi^{\rm stall}_i$ produces an effective thermodynamic description of the observed subsystem \cite{polettini2017marginal} and an effective affinity with which the informed partial entropy production rate is calculated:
\begin{equation}
\dot{S}_{\rm IP} = (\omega_{12}\pi_2 - \omega_{21}\pi_1 )\ln\frac{\omega_{12}\pi_2^{\rm stall}}{\omega_{21}\pi_1^{\rm stall}}.
\end{equation}
Although the informed partial entropy production was proven to produce a better estimation of the total dissipation compared to the passive partial entropy production, {\it i.e.}, $\dot{S}_{\rm PP}\leq\dot{S}_{\rm IP}\leq\dot{S}$ \cite{gilijordan2017}, both vanish at stalling conditions. Hence, even if the system is in a nonequilibrium steady-state, when the current over the observed link is zero, these estimators cannot give a nontrivial lower bound on the total entropy production.
To be fair, we point out that each estimator uses different information.
For the KLD estimator, we assume that the observer can record whether the system is in states $1$ or $2$, or in the hidden part of the network, $H$, which is a coarse-grained state representing the unobserved subsystem. In this case, the resulting contracted network has three states, $\{1,2,H\}$. Jumps between states $1$ and $2$ follow Poissonian statistics, as in a general continuous-time Markov process, with the same rates as in the original network. On the other hand, jumps from $H$ to $1$ or $2$ are not Poissonian and depend on the state just prior to entering the hidden part.
To apply our results for semi-Markov processes, we thus have to consider the states $\alpha(t)=[i_{\rm prev}(t) \;i(t)]$, where $i(t)=1,2,H$ is the current state and $i_{\rm prev}(t)=1,2,H$ is the state right before the last jump.
To make the equations more compact, we will use the short-hand notation $\prescript{}{i}{j} \equiv [i\;j]$ for the remainder of this section.
\begin{figure*}
\centering
\includegraphics[width=\textwidth]{./Fig3.pdf}
\caption{ {Hidden network. \textbf{a} Four-state network, as seen by an observer, with access only to states $1,2,H$. \textbf{b-c} Illustration of a trajectory over the four possible states \textbf{b}, where the gray region corresponds to the hidden part. The resulting observed semi-Markov dynamics \textbf{c}. \textbf{d} Kernel density estimation of the wait time distributions at $F=F^{\rm stall}$. \textbf{e} Estimated total entropy production rate ${\dot S}$ (solid red line), entropy production for semi-Markov model ${\dot S}_{\rm KLD}$(dashed blue curve), informed-partial entropy production rate ${\dot S}_{\rm IP}$ (dashed-dotted black curve), the passive-partial entropy production rate ${\dot S}_{\rm PP}$ (dotted green curve), and the experimental entropy production rate estimated according to the semi-Markov model ${\dot S}^{\rm Exp}_{\rm KLD}$ (blue crosses). \textbf{f-g} Relative error (ratio of experimental entropy production rate to analytical value) for three random trajectories as a function of the number of steps at $F=F^{\rm stall}$ \textbf{f}, and $F=3\beta^{-1}L^{-1}$ \textbf{g}, showing faster convergence away from the stalling force. Inset: P-value for rejecting the null hypothesis that the experimental data was sampled from a zero mean distribution as a function of the number of steps for $F=F^{\rm stall}$ (blue curve), and $F=3\beta^{-1}L^{-1}$ (red curve), showing that the average is statistically significant different from zero. {The numerical simulations were done {using the Gillespie algorithm} with the following transition rates: $\omega_{12}=2{\rm s^{-1}},\ \omega_{13}=0{\rm s^{-1}},\ \omega_{14}=1{\rm s^{-1}},\ \omega_{21}=3{\rm s^{-1}},\ \omega_{23}=2{\rm s^{-1}},\ \omega_{24}=35{\rm s^{-1}},\ \omega_{31}=0{\rm s^{-1}},\ \omega_{32}=50{\rm s^{-1}},\ \omega_{34}=0.7{\rm s^{-1}},\ \omega_{41}=8{\rm s^{-1}},\ \omega_{42}=0.2{\rm s^{-1}},\ \omega_{43}=75{\rm s^{-1}}$, where the diagonal elements were chosen to have zero sum coloums.}}}
\label{fig:hidden}
\end{figure*}
Similarly to Eq.~(\ref{Eq:def_KLD_EP_rate}), the semi-Markov entropy production rate for hidden networks, $\dot{S}_{\rm KLD}$, consists of two contributions: the affinity estimator $\dot S_{\rm aff}$ and the WTD estimator $\dot S_{\rm WTD}$.
In this case, the affinity estimator, \eqref{d10main}, is given by
\begin{equation}
\dot S_{\rm aff}=\frac{J^{\rm ss}_{21}}{\cal T}\,
\ln\frac{p(\prescript{}{1}{2} \to \prescript{}{2}{H} )
p(\prescript{}{2}{H} \to \prescript{}{H}{1} )
p(\prescript{}{H}{1} \to \prescript{}{1}{2} )}
{p(\prescript{}{1}{H} \to \prescript{}{H}{2} )
p(\prescript{}{H}{2} \to \prescript{}{2}{1} )
p(\prescript{}{2}{1} \to \prescript{}{1}{H} )},
\end{equation}
where $J^{\rm ss}_{21}$ is the stationary current per step from $1$ to $2$, defined as $J^{\rm ss}_{21}=R_{[12]}-R_{[21]}$. As expected, this term vanishes when detailed balance holds and the current is zero (Methods).
Applying {Eq.~\eqref{Eq:EP_rate_WTD_2nd_semi_markov}} to the semi-Markov process results in the following expression for the contribution of the hidden estimator
\begin{eqnarray}
&&\dot S_{\rm WTD} = \\ &&
\frac{p(1 H 2)}{\cal T} D\left[\psi(t|{ \prescript{}{1}{H}\to \prescript{}{H}{2} })||\psi(t| \prescript{}{2}{H}\to \prescript{}{H}{1})\right] \nonumber +\\ &&
\frac{p(2 H 1)}{\cal T} D\left[\psi(t|\prescript{}{2}{H}\to \prescript{}{H}{1})||\psi(t|{ \prescript{}{1}{H}\to \prescript{}{H}{2} })\right] \nonumber ,
\end{eqnarray}
where $p(ijk)=R_{[ij]}p(\prescript{}{i}{j} \to \prescript{}{j}{k} )$. In Methods, we further show that for a network of a single cycle of states the informed partial entropy production $\dot S_{\rm IP}$ equals the affinity estimator $\dot S_{\rm aff}$ defined in \eqref{d10main}. Summarizing, we have the hierarchy
$\dot{S}_{\rm PP}\leq\dot{S}_{\rm IP}=\dot S_{\rm aff}\leq \dot S_{\rm KLD}\leq\dot{S}$.
Let us apply the hidden semi-Markov entropy production framework to a specific example of a network with {four states, two of which are hidden (Fig.~\ref{fig:hidden}a). We have chosen a random $4\times4$ matrix, with non-negative off diagonal entries and zero sum columns, as a generator of a continuous-time Markov jump process over the four states. The rates over the observed link were varied according to $\omega_{12}(F)=\omega_{12}e^{\beta FL}$ and $\omega_{21}(F)=\omega_{21}e^{-\beta FL}$ over a range of values of a force $F$ that included the stalling force $F^{\rm stall}$, {where $\beta=1/T$ is the inverse temperature and $L$ is a characteristic length scale}. For each value of $F$, we contracted the dynamics to the three states, $1$, $2$, and $H$, (Fig.~\ref{fig:hidden}b-c), and estimated the waiting time distributions $\psi(t|{_2 H}\to {_H 1})$ and $\psi(t|{_1 H}\to {_H 2})$ using a kernel density estimate with a positive support \cite{terrell1992variable,botev2010kernel} (Methods), depicted in Fig.~\ref{fig:hidden}d. From those distributions, we derived the hidden semi-Markov entropy production rate $\dot{S}_{\rm KLD}$ (Fig.~\ref{fig:hidden}e). We further calculated both the passive- and informed-partial entropy production rates to compare all the estimators to the total entropy production rate (Fig.~\ref{fig:hidden}e). Our results clearly demonstrate the advantage of using the waiting time distributions for bounding the total entropy production rate compared to the two other previous approaches. Our framework can reveal the irreversibility and the underlying dissipation, even when the observed current vanishes, without the need of manipulating the system.
The KLD entropy production rate was also estimated from simulated experimental data, obtained by sampling random trajectories of $10^7$ jumps using the Gillespie algorithm \cite{gillespie1977exact}. The simulated trajectories (Fig.~\ref{fig:hidden}b) were coarse-grained into the set of states of the hidden semi-Markov model (Fig.~\ref{fig:hidden}c), and the hidden semi-Markov entropy production rate for the simulated experimental data, $\dot{S}^{\rm Exp}_{\rm KLD}$, was estimated as above (Fig.~\ref{fig:hidden}e, blue crosses). In order to {assess} the rate of convergence with increasing number of simulated steps, we calculated the $\dot{S}^{\rm Exp}_{\rm KLD}$ for different fractions of the $10^7$ steps trajectories, showing less {than} $20\%$ error above $10^5$ steps at stalling, and less then $5\%$ error away from stalling for trajectories with as little as $10^4$ steps (Fig.~\ref{fig:hidden} f-g). Let us stress that the hidden semi-Markov entropy production rate averaged over three simulated experimental trajectories produced a lower bound on the total entropy production rate, which was strictly positive and statistically significant different from zero ($p<0.05$, Fig.~\ref{fig:hidden}g, inset) for all trajectory lengths tested.
\subsection{Molecular motors}
A slight modification of the case analyzed in the previous section allows us to study molecular motors with hidden internal states. We are interested in the schemes previously sketched in Fig.~\ref{fig:fig2}a-b, where a motor can physically move in space or switch between internal states. The observed motor position is labeled by $\{...,i-1,i,i+1,...\}$.
All jumps are Poissonian and obey local detailed balance, with an external source of chemical work, $\Delta\mu$, and an additional mechanical force $F$ that can act only on the spatial transitions.
Analogous to the previous example, the observed dynamics is a second-order semi-Markov process.
To make the following equations more intuitive, we use the graphical notation \upup for two consecutive upward jumps ($i-1\to i \to i+1$), \downup for a downward jump followed by and upward one, \updown for an upward followed by a downward jump, and \downdown for two consecutive downward jumps.
Notice that the probabilities are normalized as $p_{\uus}+p_{\dus}=p_{\dds}+p_{\uds}=1$.
Similar to \eqref{Eq:def_KLD_EP_rate}, we have the decomposition of the KLD estimator into a contribution from state affinities given by
\begin{equation}
\dot S_{\rm aff}=\frac{J^{\rm ss}}{\cal T}\,\ln\frac{p_{\uus}}{p_{\dds}},
\end{equation}
where the current per step is $J^{\rm ss}=R_{\rm up} -R_{\rm down}$ with $R_{\rm up}=R_{[i,\;i+1]}$ ($R_{\rm down}=R_{[i,\;i-1]}$) corresponding to the occupancy rate of states moving upward (downward).
The contribution due to the relative entropy between waiting time distributions is
\begin{eqnarray}
\dot S_{\rm WTD} &=& \frac{1}{\cal T}\, R_{\rm up} p_{\uus} D[\psi(t|\uus)||\psi(t|\dds)]+ \nonumber \\
&& \frac{1}{\cal T}\,R_{\rm down}p_{\dds}D[\psi(t|\dds)||\psi(t|\uus)].
\end{eqnarray}
As in the previous examples, the latter term can produce a lower bound on the total entropy production rate even in the absence of observable currents, in which case $\dot S_{\rm aff}=0$. Without chemical work ($\Delta\mu=0$), however, the waiting time distributions of the $\upup$ and $\downdown$ processes become identical and the contribution of $\dot S_{\rm WTD}$ vanishes as well.
{Let us apply the molecular motor semi-Markov entropy production framework to a specific example.}
We consider the following two-state molecular motor model of a power stroke engine that works by hydrolizing ATP against an external force $F$, see Fig.~\ref{fig:powerstroke}a.
\begin{figure*}[!ht]
\centering
\includegraphics[width=\linewidth]{./Fig4.pdf}
\caption{Molecular motor. \textbf{a} Illustration: Active states (red boxed squares) can use a source of chemical energy while passive states (circles) cannot. The chemical energy is used to power the motor against and external force $F$. \textbf{b} Illustration of a trajectory for four positions, where the hidden internal active state is denoted by the red shaded regions. \textbf{c-d} Waiting time distributions $\psi(t)$ for the up-up (red) and down-down (blue) transitions at stalling for $\Delta \mu = 0\beta^{-1} $ \textbf{c} and $\Delta \mu = 10\beta^{-1}$ \textbf{d}. Notice that the distributions are only different in the presence of a chemical drive. \textbf{e} Total entropy production rate $\dot{S}$ (red), affinity estimator $\dot{S}_{\rm aff}$ (green), and entropy production for semi-Markov model $\dot{S}_{\rm KLD}$ for $\Delta\mu=0\beta^{-1}$ (left) up to $\Delta \mu=10\beta^{-1}$ (right), as a function of force $F$, centered at the stall force. \textbf{f} Same as \textbf{e} at stalling as a function of chemical drive. The affinity estimator ${\dot S}_{\rm aff}$ offers a lower bound constrained by the statistical uncertainty due to the finite amount of data (green shaded region). Calculations were done using the parameters $k_s=1{\rm s^{-1}}$, $k_0=0.01{\rm s^{-1}}$, and the trajectories were sampled using the Gillespie algorithm \cite{gillespie1977exact}}
\label{fig:powerstroke}
\end{figure*}
The state of the motor is described by its physical position and its internal state, which can be either {active}, that is, capable of hydrolyzing ATP, or {passive}. We label the active and passive states as $i'$ and $i$, respectively with $i=0,\pm 1,\pm 2,\dots$. Owing to the translational symmetry in the system, all the spatial positions are essentially equivalent. The position of the motor is accessible to an external observer, whereas the two internal states $i$ and $i'$ are indistinguishable. An example of a trajectory is illustrated in Fig. \ref{fig:powerstroke}b.
The chemical affinity $\Delta \mu$, arising from ATP hydrolysis, determines the degree of nonequilibrium in our system and biases the transitions $i'\leftrightarrow i+1$, whereas the external force $F$ affects all the spatial transitions, regardless of the internal state. The transition rates between the two internal states are defined as $\omega_{i'i} =\omega_{ii'}=k_{\rm s}$. Transition rates between passive states obey local detailed balance: $\omega_{i,i+1}/\omega_{i+1,i}=e^{\beta FL}$, where $L$ is the length of a single spatial jump.
From the active state, the system can use the ATP to move upward with rates verifying local detailed balance $\omega_{i',i+1} /\omega_{i+1,i'} =e^{\beta\left(FL-\Delta\mu \right)}$.
The resulting waiting time distributions are shown in Fig.~\ref{fig:powerstroke}c-d, and the estimated entropy production rates as a function of external force are depicted in Fig. \ref{fig:powerstroke}e, with chemical potential ranging from $\Delta \mu=0\beta^{-1}$ to $10\beta^{-1}$. The total entropy production rate ${\dot S}$ is calculated using \eqref{saff}.
As expected, the dissipation increases with the nonequilibrium driving force, and vanishes when {$\Delta\mu=FL=0\beta^{-1}$}. Notice that the affinity estimator $\dot{S}_{\rm aff}$ does not provide a lower bound to the total entropy production rate ${\dot S}$ at stalling, as it is not statistically different from zero (Fig.~\ref{fig:powerstroke}f), and thus cannot distinguish between nonequilibrium and equilibrium processes. In contrast, the semi-Markov estimator $\dot{S}_{\rm KLD}$, which accounts for the asymmetry of the waiting time distributions provides a nontrivial positive bound, even in the absence of observable current.
\section{Discussion}
We have analytically derived an estimator of the total entropy production rate using the framework of semi-Markov processes. The novelty of our approach is the utilization of the waiting time distributions, which can be non-Poissonian, allowing us to unravel irreversibility in hidden degrees of freedom arising in any time-series measurement of an arbitrary experimental setup. Our estimator can thus provide a lower bound on the total entropy production rate even in the absence of observable currents. {Hence, it can be applied to reveal an underlying nonequilibrium process, even if no net current, flow, or drift, are present.}
We stress that our method fully quantifies irreversibility. Owing to the direct link between the entropy production rate and the relative entropy between a trajectory and its time-reversal, as manifested in Eq. \eqref{eq:KPB}, our estimator provides the best possible bound on the dissipation rate utilizing time-irreversibility. {One can consider utilizing other properties of the waiting time distribution to bound the entropy production, through the thermodynamics uncertainty relations \cite{Gingrich_Horowitz_Dissipation_Bounds,SeifertBaratoUncertainty,li2018quantifying}, for example.}
We have illustrated our method with two possible applications: a situation where only a subsystem is accessible to an external observer and a molecular motor whose internal degrees of freedom cannot be resolved. Using these examples, we have demonstrated the advantage of our semi-Markov estimator compared to other entropy production bounds, namely, the passive- and informed-partial entropy production rates, both of which vanish at stalling conditions.
In summary, we have developed an analytic tool that can expose irreversibility otherwise undetectable, and distinguish between equilibrium and nonequilibrium processes. This framework is completely generic and thus opens opportunities in numerous experimental scenarios by providing a new perspective for data analysis.
\section{Methods}
\subsection{Semi-Markov processes, waiting time distributions and steady states}
A semi-Markov stochastic process is a renewal process $\alpha(t)$ with a discrete set of states
$\alpha=1,2,\dots, N$. The dynamics is determined by the probability densities of transition times $
\psi_{\beta\alpha}(t)$, which are defined as $\psi_{\beta\alpha}(t)dt$ being equal
to the probability that the system jumps from state $\alpha$ to state $\beta$ in the time
interval $[t,t+dt]$ if it arrived at site $\alpha$ at time $t=0$. By definition $
\psi_{\alpha\alpha}(t)=0$. When the system is a particle jumping between the sites of a
lattice, the semi-Markov process is also called a continuous time random walk (CTRW). For
clarity, we will assume this CTRW picture, that is, the system in our discussion
will be a particle jumping between sites $\alpha$.
The probability densities $\psi_{\beta\alpha}(t)$ are not normalized:
\begin{equation}
p_{\beta\alpha} =\int_0^\infty \psi_{\beta\alpha}(t)dt
\end{equation}
is the probability that, given that the particle arrived at site $\alpha$, the next jump is $\alpha\to \beta$. We will assume that the particle eventually leaves any site $\alpha$, i.e., $\sum_\beta p_{\beta\alpha}=1$.
Then
\begin{equation}
\psi_\alpha(t)=\sum_\beta\psi_{\beta\alpha}(t)
\end{equation}
is normalized and it is the probability density of the residence time at site $\alpha$. It is also called the waiting time distribution. Its average
\begin{equation}
\tau_\alpha=\int_0^\infty dt\,t\,\psi_\alpha(t)
\end{equation}
is the mean residence time or mean waiting time. We can also define the waiting time distribution conditioned on a given jump $\alpha\to \beta$,
\begin{equation}
\psi(t|\alpha\to \beta)=\frac{\psi_{\beta\alpha}(t)}{p_{\beta\alpha}},
\end{equation}
which is normalized. The function $\psi_{\beta\alpha}(t)$ is in fact the joint probability distribution of the time $t$ and the jump $\alpha\to \beta$.
The transition probabilities $p_{\beta\alpha}$ determine a Markov chain given by the visited states $\alpha_1,\alpha_2,\alpha_3,...$, regardless of the times when the jumps occur.
The transition matrix of this Markov chain is $\{ p_{\beta\alpha}\}$ and the stationary probability distribution $R_\alpha$ verifies
\begin{equation}\label{eqvisits}
R_\beta=\sum_\alpha p_{\beta\alpha}R_\alpha ,
\end{equation}
i.e., the distribution $R_\alpha$ is the right eigenvector of the stochastic matrix $\{ p_{\beta\alpha}\}$ with eigenvalue 1.
Moreover, if the Markov chain is ergodic, then the distribution $R_\alpha$ is precisely the fraction of visits the system makes to site $\alpha$ in the stationary regime. Thus, we call $R_\alpha$ the {\em distribution of visits}.
From the distribution of visits one can easily obtain the stationary distribution of the process $\alpha(t)$,
\begin{equation}\label{pss}
\pi_\alpha=\frac{\tau_\alpha R_\alpha}{\cal T},
\end{equation}
since the particle visits the state $\alpha$ a fraction of steps $R_\alpha$ and spends an average time $\tau_\alpha$ in each step. The normalization constant ${\cal T}\equiv \sum_\alpha R_\alpha \tau_\alpha$ is the average time per step.
The stationary current in the Markov chain from state $\alpha$ to $\beta$ is
\begin{equation}
J^{\rm ss}_{\beta\alpha}=p_{\beta\alpha}R_\alpha -p_{\alpha\beta}R_\beta .
\end{equation}
This is in fact the current per step in the original semi-Markov system since, in an ensemble of very long trajectories, it is the net number of particles that jump from $\alpha$ to $\beta$ divided by the number of steps. Since the duration of a long stationary trajectory with $K$ steps ($K\gg 1$) is $K{\cal T}$, the current per unit of time is
$J^{\rm ss}_{\beta\alpha}/{\cal T}$.
Notice that the average time per step ${\cal T}$ acts as a conversion factor that allows one to express currents, entropy production, etc. either as per step or as per unit of time.
\subsection{The Markovian case}
If the process $\alpha(t)$ is Markovian, then the jumps are Poissonian and transition time densities are exponential. Let $\omega_{\beta\alpha}$ be the rate of jumps from $\alpha$ to $\beta$. The mean waiting time at site $\alpha$ is the inverse of the the total outgoing rate:
\begin{equation}
\tau_\alpha=\frac{1}{\sum_\beta \omega_{\beta\alpha}},
\end{equation}
and the waiting time distributions are
\begin{equation}
\psi_{\beta\alpha}(t)= \omega_{\beta\alpha} e^{- t/\tau_\alpha} \qquad
\psi_\alpha(t)=\psi(t|\alpha\to \beta)=\frac{e^{- t/\tau_\alpha}}{\tau_\alpha} .
\end{equation}
with jump probabilities $p_{\beta\alpha}=\tau_\alpha\omega_{\beta\alpha}$. Notice that the waiting time distribution $\psi(t|\alpha\to \beta)$ does not depends on $\beta$. The distribution of visits $R_\alpha$ verifies
\begin{equation}
R_\beta=\sum_\alpha \tau_\alpha{\omega_{\beta\alpha}}R_\alpha,
\end{equation}
and the stationary distribution $\pi_\alpha$ obeys
\begin{equation}
\frac{\pi_\beta}{\tau_\beta}=\sum_\alpha \omega_{\beta\alpha} \pi_\alpha,
\end{equation}
which is the equation for the stationary distribution that one obtains from the master equation
\begin{equation}
\dot P_\beta(t)=\sum_\alpha\left[ \omega_{\beta\alpha} P_\alpha(t)-\omega_{\alpha\beta}P_\beta(t)\right] =\sum_\alpha \omega_{\beta\alpha} P_\alpha(t)-\frac{P_\beta(t)}{\tau_\beta}.
\end{equation}
\subsection{Decimation of Markov chains}
Semi-Markov processes arise
in a natural way when states are removed or decimated from Markov processes with certain topologies. Consider a Markov process where two sites, $1$ and $2$, are connected through a closed network of states $i=3,4,\dots$ that we want to decimate, as sketched in Fig.~\ref{fig:fig2}c.
{If the observer cannot discern between states $i=3,4,\dots$, the resulting three-state process with $i(t)=1,2,H$ is a second-order semi-Markov chain.}
We want to calculate the effective transition time distribution $\psi^{\rm decim}_{21}(t)$ from state $1$ to state $2$ in terms of the distributions $\psi_{ij}(t)$ of the initial Markov chain. For this purpose, we have to sum over all possible paths from 1 to 2 through the decimated network.
\ignore{
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=3.5cm]{./hiddenSI0.pdf}
\caption{Decimation of a Markov chain. If the observer cannot discern between states $3,4,5$, the resulting three-state process with $i(t)=1,2,H$ is a second-order semi-Markov chain.}
\label{figdecim}
\end{center}
\end{figure}
}
Consider first the paths with exactly $n+1$ jumps, like $\gamma_{n+1}=\{1\to i_1\to i_2 \dots i_{n}\to 2 \}$, where $i_k=3,4,\dots$. The probability that such a path occurs with an exact duration $t$ is
\begin{widetext}
\begin{equation}\label{pgammat}
P(\gamma_{n+1},t)=\int_{\sum t_k=t} dt_1\, dt_2\dots\, dt_{n+1}\, \psi_{i_1,1}(t_1)\,\psi_{i_2,i_1}(t_2)\dots \psi_{2,i_{n}}(t_{n+1}).
\end{equation}
\end{widetext}
This is a convolution. If one performs the Laplace transform on all time-dependent functions, generically denoted by a tilde,
\begin{equation}
\tilde\psi(s)\equiv \int_0^\infty dt\, e^{-st} \psi(t)
\end{equation}
then \eqref{pgammat} simplifies to
\begin{equation}
\tilde P(\gamma_{n+1},s)=\tilde \psi_{i_1,1}(s)\,\tilde \psi_{i_2,i_1}(s)\dots \tilde \psi_{2,i_{n}}(s).
\end{equation}
The transition time distribution $\psi^{\rm decim}_{21}(t)$ in the decimated network is the sum of $P(\gamma_{n+1},t)$ over all possible paths with an arbitrary number of steps. For Laplace transformed distributions, this is written as
\begin{equation}
\tilde\psi^{\rm decim}_{21}(s)=\sum_{n=0}^\infty \sum_{\{i_1,\dots,i_n\}} \tilde \psi_{i_1,1}(s)\,\tilde \psi_{i_2,i_1}(s)\dots \tilde \psi_{2,i_{n}}(s).
\end{equation}
where the sum runs over all possible paths, that is, the indexes $i_k=3,4,\dots$ take on all possible values corresponding to decimated sites.
Then the sum can be expressed in terms of the matrix $\Psi(t)$ whose entries are the transition time densities
$[\Psi(t)]_{ji} = \psi_{ji}(t)$, $i,j=3,4,\dots$. If $\tilde\Psi(s)$ is the corresponding Laplace transform of that matrix, one has
\begin{eqnarray}
\tilde\psi^{\rm decim}_{21}(s)&=&\sum_{n=0}^\infty
\sum_{i,j}\tilde\psi_{i,1}(s)\left[ \tilde\Psi(s)^n\right]_{ji}\tilde\psi_{2,j}(s)\nonumber\\&=&\sum_{i,j}\tilde\psi_{i,1}(s)\left[ {\mathbb I}-\tilde\Psi(s)\right]^{-1}_{ji}\tilde\psi_{2,j}(s)
\end{eqnarray}
which is a sum only over all the decimated sites $i,j=3,4,\dots$ that are connected to sites 1 and 2, respectively.
The decimation procedure can be used to derive transition time distributions in a kinetic network when the observer cannot discern among a set of states, say $3,4,5,\dots$, that are generically labelled as $H$ for hidden, as in Fig.~\ref{fig:fig2}c.
For the specific case of the figure, the effective transition time distribution from site 1 to site $H$, for instance, can be written as
\begin{equation}
\psi^{\rm eff}_{H1}(t)= \psi_{31}(t) +\psi_{51}(t),
\end{equation}
whereas the distributions for jumps starting at $H$ depend on the previous state. For instance, if $H$ is reached from 1, the random walk within $H$ starts at site 3 with probability $p_{31}/(p_{31}+p_{51})$ and site 5 with probability $p_{51}/(p_{31}+p_{51})$. The transition time distribution corresponding to the jump $[1H]\to [H2]$ is
\begin{widetext}
\begin{equation}
\psi^{\rm eff}_{[H2]\leftarrow [1H]}(s)= \frac{p_{31}}{p_{31}+p_{51}}\left[ {\mathbb I}-\tilde\Psi(s)\right]^{-1}_{43}\tilde\psi_{24}(s)+
\frac{p_{51}}{p_{31}+p_{51}}\left[ {\mathbb I}-\tilde\Psi(s)\right]^{-1}_{45}\tilde\psi_{24}(s)
\end{equation}
\end{widetext}
where the matrix $\tilde\Psi(s)$ is a $3\times 3$ matrix corresponding to the Laplace transform of the transition time distributions among sites 3, 4, and 5.
\subsection{Irreversibility in semi-Markov processes}
Here we calculate the relative entropy between a stationary trajectory $\gamma$ and its time reversal $\tilde\gamma$ in a generic semi-Markov process. A trajectory $\gamma$ is fully described by the sequence of jumps (see Fig.~\ref{figtrajalpha}):
\begin{widetext}
\begin{equation}
\gamma=\{ (\alpha_1\to \alpha_2,t_1), (\alpha_2\to \alpha_3,t_2), \dots, (\alpha_{n-1}\to \alpha_n,t_{n-1}),(\alpha_n\to \alpha_{n+1},t_n)\}
\end{equation}
\end{widetext}
and occurs with a probability (conditioned on the initial jump $\alpha_0\to\alpha_1$ at $t=0$)
\begin{equation}
P(\gamma)=\psi_{\alpha_2,\alpha_1}(t_1)\psi_{\alpha_3,\alpha_2}(t_2)\dots \psi_{\alpha_{n+1},\alpha_n}(t_n).
\end{equation}
\begin{figure}[ht!]
\begin{center}
\includegraphics[height=3cm]{./figtraj_alpha.pdf}
\caption{A trajectory $\gamma$ of a semi-Markov process.}
\label{figtrajalpha}
\end{center}
\end{figure}
The reverse trajectory is
\begin{equation}
\tilde\gamma=\{ (\tilde \alpha_n\to \tilde \alpha_{n-1}, t_n),\dots, (\tilde \alpha_2\to \tilde \alpha_{1}, t_2),(\tilde \alpha_1\to \tilde \alpha_{0}, t_1) \},
\end{equation}
where we assume, for the sake of generality, that states can change under time reversal, $\tilde\alpha$ being the time-reversal of state $\alpha$.
The probability to observe $\tilde\gamma$, conditioned on the initial jump $\tilde\alpha_{n+1}\to\tilde\alpha_n$ at $t=0$, is
\begin{equation}
P(\tilde\gamma)=\psi_{\tilde \alpha_0,\tilde \alpha_1}(t_1)\psi_{\tilde \alpha_1,\tilde \alpha_2}(t_2)\dots \psi_{\tilde \alpha_{n-1},\tilde \alpha_n}(t_n).
\end{equation}
It is again convenient to consider the forward and backward trajectories without the waiting times, i.e.,
\begin{eqnarray}
\sigma &=&\{ \alpha_1, \alpha_2, \alpha_3, \dots, \alpha_n,\alpha_{n+1} \} \\
\tilde\sigma &=&\{ \tilde\alpha_n, \tilde\alpha_{n-1}, \dots, \tilde\alpha_{2}, \tilde\alpha_{1}, \tilde\alpha_0 \} ,
\end{eqnarray}
and the probability to observe those trajectories are
\begin{eqnarray}
P(\sigma) &=& p_{\alpha_2,\alpha_1} p_{\alpha_3,\alpha_2} \dots p_{\alpha_{n+1},\alpha_n} \\
P(\tilde\sigma) &=& p_{\tilde\alpha_0,\tilde\alpha_1} p_{\tilde\alpha_1,\tilde\alpha_2} \dots p_{\tilde\alpha_{n-1},\tilde\alpha_n}
\end{eqnarray}
The initial jumps of $\gamma$ and $\tilde\gamma$ do not contribute to the entropy production in the stationary regime. Then the relative entropy per jump reads
\begin{widetext}
\begin{eqnarray}
\delta S_{\rm KLD}&=&\lim_{n \to \infty} \frac{1}{n}\sum_\gamma P(\gamma)\ln\frac{P(\gamma)}{P(\tilde \gamma)}\nonumber\\
&=& \lim_{n \to \infty} \frac{1}{n}\sum_\sigma \int_0^\infty dt_1\dots \int_0^\infty dt_n\,
\psi_{\alpha_2,\alpha_1}(t_1)\dots \psi_{\alpha_{n+1},\alpha_n}(t_n)
\left[\ln \frac{\psi_{\alpha_2,\alpha_1}(t_1)}{\psi_{\tilde \alpha_0,\tilde \alpha_1}(t_1)}+\dots
\ln \frac{\psi_{\alpha_{n+1},\alpha_n}(t_n)}{\psi_{\tilde \alpha_{n-1},\tilde \alpha_n}(t_n)}\right]\label{d0}.
\end{eqnarray}
Each time integral can be written as
\begin{equation}
\int_0^\infty dt\, \psi_{\mu\beta}(t) \ln \frac{\psi_{\mu\beta}(t)}{\psi_{\tilde \alpha \tilde \beta}(t)}=p_{\mu\beta}\ln\frac{p_{\mu\beta}}{p_{\tilde \alpha \tilde \beta}}+p_{\mu\beta} D\left[\psi(t|\beta\to \mu)||\psi(t|\tilde \beta\to \tilde \alpha)\right],
\end{equation}
\end{widetext}
where $\alpha,\beta,\mu$ is a substring of the forward trajectory $\sigma$ ($\alpha=\alpha_k$, $\beta=\alpha_{k+1}$, $\mu=\alpha_{k+2}$). Inserting this expression in \eqref{d0},
\begin{widetext}
\begin{eqnarray}
\delta S_{\rm KLD} &=&\lim_{n \to \infty} \frac{1}{n}D[P(\sigma)||P(\tilde\sigma)] +\sum_{\alpha\beta\mu} p_{\mu\beta}p_{\beta\alpha}R_\alpha D\left[\psi(t|\beta\to \mu)||\psi(t|\tilde \beta\to \tilde \alpha)\right] \nonumber \\
&=& \sum_{\alpha\beta} p_{\beta\alpha}R_\alpha\ln\frac{p_{\beta\alpha}}{p_{\tilde \alpha\tilde \beta}} +\sum_{\alpha\beta\mu} p_{\mu\beta}p_{\beta\alpha}R_\alpha D\left[\psi(t|\beta\to \mu)||\psi(t|\tilde \beta\to \tilde \alpha)\right].\label{d1}
\end{eqnarray}
\end{widetext}
Notice that $ p_{\beta\alpha}R_\alpha$ is the probability to observe the sequence $\alpha,\beta$ in the stationary forward trajectory and $p_{\mu\beta}p_{\beta\alpha}R_\alpha $ is the probability to observe the sequence $\alpha,\beta,\mu$.
Finally, we can obtain the expression used in the main text for the entropy production per unit of time dividing by the conversion factor ${\cal T}$ (average time per step), that is $\dot S=\delta S/{\cal T}$. The result is
\begin{equation}
\dot S_{\rm KLD} =\dot S_{\rm aff} +\dot S_{\rm WTD}, \label{d1t}
\end{equation}
where the entropy production corresponding to the affinity of states reads
\begin{equation}
\dot S_{\rm aff} =\frac{1}{\cal T}\sum_{\alpha\beta}p_{\beta\alpha}R_\alpha \ln\frac{p_{\beta\alpha}}{p_{\tilde \alpha\tilde \beta}}\label{d1taff},
\end{equation}
and the one corresponding to the waiting time distributions is
\begin{equation}
\dot S_{\rm WTD}=\frac{1}{\cal T}\sum_{\alpha\beta\mu} p_{\mu\beta} p_{\beta\alpha}R_\alpha D\left[\psi(t|\beta\to \mu)||\psi(t|\tilde \beta\to \tilde \alpha)\right].\label{d1wtd}
\end{equation}
If $\alpha=\tilde\alpha$, then the affinity entropy production can be written as
\begin{equation}
\dot S_{\rm aff} =\frac{1}{\cal T}\, \sum_{\alpha<\beta} J^{\rm ss}_{\beta\alpha} \ln\frac{p_{\beta\alpha}}{p_{\alpha\beta}} ,
\end{equation}
which vanishes in the absence of currents.
\subsection{2nd-order semi-Markov processes}
A 2nd-order semi-Markov process $i(t)$ also describes the trajectory of a system that jumps among a discrete set of states $i=1,2,\dots$. However, $i(t)$ is not semi-Markov because the transition time distributions depend on the previous state $i_{\rm prev}(t)$ visited right before the last jump. Hence, the vector $\alpha(t)\equiv [ i_{\rm prev} (t)\ i(t)]$ is indeed a semi-Markov process.
\begin{figure}[htbp]
\begin{center}
\includegraphics[height=4.6cm]{./figSI_2otraj.pdf}
\caption{A trajectory $\gamma$ of a second-order semi-Markov process.}
\label{figtraj2o}
\end{center}
\end{figure}
To quantify the irreversibility of a second-order Markov chain, we introduce the time-reversal state of $\alpha=[ij]$, which is $\tilde\alpha =[ji]$. However, this is not enough to reconstruct the backward trajectory, since there is a shift compared to the simple semi-Markov case, as illustrated in Fig. \ref{figtraj2o}. In the forward trajectory, the system spends a time $t_k$ in state $\alpha_k= [i_{k-1}i_k]$, with $k=1,\dots,n$, whereas in the backward trajectory it spends the same time $t_k$ in state $\tilde\alpha_{k+1}=[i_{k+1}i_{k}]$. Consequently, the probabilities of the forward and backward trajectories are, respectively,
\begin{align}
P(\gamma) &=\psi_{\alpha_2,\alpha_1}(t_1)\psi_{\alpha_3,\alpha_2}(t_2)\dots \psi_{\alpha_{n+1},\alpha_n}(t_n) \\
P(\tilde\gamma) &=\psi_{\tilde \alpha_1,\tilde \alpha_2}(t_1)\psi_{\tilde \alpha_2,\tilde \alpha_3}(t_2)\dots \psi_{\tilde \alpha_{n},\tilde \alpha_{n+1}}(t_{n}).
\end{align}
Repeating the arguments of the previous section, one obtains
\begin{widetext}
\begin{equation}
\delta S_{\rm KLD} = \sum_{\alpha\beta} p_{\beta\alpha}R_\alpha\ln\frac{p_{\beta\alpha}}{p_{\tilde \alpha\tilde \beta}} +\sum_{\alpha\beta} p_{\beta\alpha}R_\alpha D\left[\psi(t|\alpha\to \beta)||\psi(t|\tilde \beta\to \tilde \alpha)\right].\label{d12o}
\end{equation}
\end{widetext}
The contribution to the entropy production (per step) due to the state affinities now reads
\begin{eqnarray}
\dot S_{\rm aff} &=& \frac{1}{\cal T} \sum_{i,j,k}R_{[ij]} p([ij]\to [jk])\ln\frac{p([ij]\to [jk])}{p([kj]\to [ji])} \nonumber \\
&=& \frac{1}{\cal T} \sum_{i,j,k}p(ijk)\ln\frac{p([ij]\to [jk])}{p([kj]\to [ji])},
\label{d10}
\end{eqnarray}
and the contribution due to the waiting time distributions is given by
\begin{widetext}
\begin{align}
\dot S_{\rm WTD}& = \frac{1}{\cal T} \sum_{i,j,k} R_{[ij]} p([ij]\to [jk]) D\left[\psi(t|[ij]\to [jk])||\psi(t|[kj]\to [ji])\right]\nonumber \\
&=\frac{1}{\cal T} \sum_{i,j,k} p(ijk) D\left[\psi(t|[ij]\to [jk])||\psi(t|[kj]\to [ji])\right]\label{d1b},
\end{align}
\end{widetext}
where $p(ijk)=R_{[ij]} p([ij]\to [jk])$ is the probability to observe the sequence $i\to j\to k$ in the trajectory and $p(ij)=R_{[ij]}$ is the probability to observe the sequence $i\to j$.
It is interesting to particularize \eqref{d10} to a ring with $N$ sites. This is the case of our examples -- the hidden network and the molecular motor. In this case, in the stationary regime,
\begin{eqnarray}
p(ijk)-p(kji)&=&p(ijk)+p(iji)-p(iji)-p(kji)\nonumber\\&=&p(i j)-p(ji)=J^{\rm ss}
\end{eqnarray}
since each site has only two neighbours and therefore $p(ijk)+p(iji)=p(ij)$ for any triplet of contiguous sites $ijk$. Here $J^{\rm ss}$ is the stationary current between any pair of contiguous sites.
Hence, we can write the affinity as
\begin{widetext}
\begin{eqnarray}
\dot S_{\rm aff} &=& \frac{1}{\cal T} \sum_{i<j<k}\left[p(ijk)-p(kji)\right]\ln\frac{p([ij]\to [jk])}{p([kj]\to [ji])} \nonumber \\
&=& \frac{J^{\rm ss}}{\cal T} \,\ln\frac{p([1,2]\to [2,3])\dots p([N-1,N]\to [N,1])p([N,1]\to [1,2])}{p([1,N]\to [N,N-1])\dots p([3,2]\to [2,1])p([2,1]\to [1,N])} \label{saffring}
\end{eqnarray}
\end{widetext}
which is proportional to the current. The argument of the logarithm also vanishes at zero current (see \eqref{GDB} below); consequently, the affinity entropy tends to zero quadratically when as the force is tuned to the stalling condition. This is the usual behavior in linear irreversible thermodynamics, but recall that for semi-Markov processes the affinity entropy production misses the non-equilibrium signature that is present in the waiting time distributions and is assessed by $\dot S_{\rm WTD}$.
\subsection{Affinity and informed partial entropy production}
Here we show that the informed partial entropy production equals the affinity entropy production for the case of a generic hidden kinetic network proposed in the main text where the observed network forms a single cycle.
First, let us generalize the detailed balance condition for a second-order Markov ring with three states, $i=1,2,H$, and zero stationary current. The stationary distribution $R_{[ij]}$ verifies the master equation \eqref{eqvisits}:
\begin{align}
R_{[12]}&= R_{[H1]}p([H1] \to [12])+R_{[21]}p([21]\to [12]) \nonumber \\
R_{[2H]}&= R_{[12]}p([12] \to [2H])+R_{[H2]}p([H2]\to [2H])\\
R_{[H1]}&= R_{[2H]}p([2H] \to [H1])+R_{[1H]}p([1H]\to [H1]). \nonumber
\end{align}
If the current vanishes, $R_{[ij]}=R_{[ji]}$ for all $i,j$, and these equations
reduce to
\begin{align}
R_{[12]}p([21]\to [1H])&=R_{[H1]}p([H1]\to [12]) \nonumber \\
R_{[2H]}p([H2]\to [21])&=R_{[12]}p([12]\to [2H])\\
R_{[H1]}p([1H]\to [H2])&=R_{[2H]}p([2H]\to [H1]). \nonumber
\end{align}
Multiplying the three equations we get the generalized detailed balance condition:
\begin{widetext}
\begin{equation}\label{GDB}
J^{\rm ss}=0\Rightarrow p([1H]\to [H2])p([H2]\to [21])p([21]\to [1H])=p([2H]\to [H1])p([H1]\to [12])p([12]\to [2H]).
\end{equation}
\end{widetext}
In the observable network, the transitions from states 1 and 2 are still Poissonian and independent of the previous state:
\begin{align}
p([H2]\to [21])=\tau_2\omega_{12}\qquad & p([21]\to [1H])=\tau_1\omega_{H1} \nonumber \\
p([H1]\to [12])=\tau_1\omega_{21}\qquad & p([12]\to [2H])=\tau_2\omega_{H2} .
\end{align}
At stall force, the generalized detailed balance condition \eqref{GDB} holds and can be written as
\begin{equation}
p([1H]\to [H2])\omega^{\rm stall}_{12}\omega_{H1}=p([2H]\to [H1])\omega_{21}^{\rm stall}\omega_{H2},
\end{equation}
where we have taken into account that only the rates $\omega_{12}$ and $\omega_{21}$ are tuned in the protocol proposed by Polettini and Esposito to obtain the informed partial entropy production.
The current in the direction $1\to 2\to H \to 1$ in the stationary regime can be written as $J^{\rm ss}/{\cal T}=\omega_{12}\pi_2-\omega_{21}\pi_1$. Then, at stall force $\omega^{\rm stall}_{12}\pi^{\rm stall}_2 =\omega^{\rm stall}_{21}\pi^{\rm stall}_1$.
With all these considerations, the argument of the logarithm in Eq.~(9) of the main text can be written as
\begin{align}
\frac{\omega_{12}\,\pi^{\rm stall}_2}{\omega_{21}\,\pi^{\rm stall}_1} &=
\frac{\omega_{12}\,\omega^{\rm stall}_{21}}{\omega_{21}\,\omega^{\rm stall}_{12}}
=\frac{\omega_{12}\,p([1H]\to [H2])\omega_{H1}}{\omega_{21}\,p([2H]\to [H1])\omega_{H2}
}\nonumber \\
&=\frac{p([1H]\to [H2])p([H2]\to [21])p([21]\to [1H])}{p([2H]\to [H1])p([H1]\to [12])p([12]\to [2H])}.
\end{align}
Comparing Eq.~(9) in the main text with \eqref{saffring}, one immediately gets $\dot S_{\rm IP}=\dot S_{\rm aff}$.
\section{Acknowledgements}
IAM and JMRP acknowledge funding from the Spanish Government through grants TerMic (FIS2014-52486-R) and Contract (FIS2017-83709-R). IAM acknowledges funding from Juan de la Cierva program. GB acknowledges the Zuckerman STEM Leadership Program. JMH is supported by the Gordon and Betty Moore Foundation
as a Physics of Living Systems Fellow through Grant No. GBMF4513.
\section{Author contributions}
JMRP conceived the project. IAM and GB performed the numerical simulations and analyzed the data.
All authors discussed the results and wrote the manuscript.
\section{Competing interests}
The authors declare no competing interests.
\section{Data availability}
The data that support the findings of this study are available from the corresponding author upon reasonable request.
\section{Code availability}
Source code is available from the corresponding authors upon reasonable request.
|
2,869,038,154,815 | arxiv | \section{Introduction. Statement of Results.}
A \emph{graph} is a connected one dimensional compact
polyhedron. Ghrist and Abrams (\cite{abramsthesis},
\cite{abramsgeom},\cite{ghrist}) have recently called attention to
the $n-$point unordered configuration space of a graph $X$,
denoted here as $\cal U \rm _n^{top}(X)$. This is the space of $n-$element subsets of
$X$ (see Definition \ref{unx}, (\ref{confspace})). It is an
aspherical space with the homotopy type of a finite polyhedron,
for each $n$ and $X$ (see \cite{ag}). Its fundamental group is the
\emph{$n-$string braid group of $X$,} denoted $B_n(X,c)$, if $c$ is
a base point of $\cal U \rm _n^{top}(X)$. The group $B_n(X,c)$ is therefore torsion free; it
can have arbitrarily high finite cohomological dimension. Abrams and
Ghrist (\cite{abramsgeom}, \cite{ghrist}) have put forward the
following striking conjecture:
\guess \label{conj} If $X$ is a planar graph, then the $n-$string
braid group $B_n(X,c)$ is a right angled Artin group for each $n$.
A \emph{right angled Artin group} is a group having a presentation
in which the only relations are commutators between generators. It
is known (\cite{abramsgeom}) that the planar condition cannot be
removed from the above conjecture.
The purpose of this paper is to prove:
\thm \label{raathm}For each $n$, Conjecture \ref{conj} is true if
$X$ is a linear tree.
A \emph{linear tree} is a contractible graph $X$ containing an
interval $I$ (a homeomorphic copy of $[0,1]$) such that each node
of $X$ is in $I$.
(Recall that the \emph{nodes} in a graph $X$ are the points of
degree $\geq 3$. The \emph{degree} of any point $x\in X$ is the
number of points in $Link_X(x)$).
Theorem \ref{raathm} is a direct consequence of our main theorem
(\ref{mainthm}) below. Before stating it we give a more careful
definition of the objects just mentioned.
\begin{definition} \label{unx} A subset $c$ with $n$ elements in a
space $X$ is called an $n-$point configuration in $X$. For any
$n\geq 0$ and any topological space $X$, the unordered $n-$point
configuration space of $X$ is:
\begin{equation}\label{confspace} \cal U \rm _n^{top}(X) = \{ c \subset X| \quad |c|=n\} .
\end{equation}
If $(X,d)$ is a metric space, then $\cal U \rm _n^{top}(X)$ is topologized by using
the Hausdorff metric on closed sets of $X$. Explicitly, then:
\begin{equation}
d(c,c') = max\{d(x,c'), d(y,c) |\quad x\in c, \;y\in c'\}\quad
\forall c,c'\in \cal U \rm _n^{top}(X).
\end{equation}
Equivalently, one can give $ \cal U \rm _n^{top}(X)$ the quotient topology of the map
\[(X^n-\Delta) \overset{\pi}{\longrightarrow} \cal U \rm _n^{top}(X) : \quad \pi(x_1,
x_2, \dots x_n)= \{x_1,x_2,\dots x_n\}\] where $\Delta=\{(x_1,
x_2, \dots x_n)| \quad x_i= x_j \text{ for some } i\neq j\}$.
Note \;$ \cal{U}\rm^{top}_0(X)= \ast, $\; and $\cal U \rm
_1^{top}(X) = X$.
The \emph{$n-$string braid group} of a space $X$ is:
\begin{equation}\label{braid}
B_n(X, c) = \pi_1( \cal U \rm _n^{top}(X), c),
\end{equation}
where $c\in \cal U \rm _n^{top}(X)$ is a base point.
\end{definition}
$B_n$ is a functor from the category of topological spaces $X$
with fixed base configurations $c\in \cal U \rm _n^{top}(X)$ and isotopy classes of
injective continuous maps that preserve base configurations.
But in order to finesse the many changes of base configuration
required, we employ the following artifice. Let $(X,A)$ be a
pair where $A$ is a nonempty {\em simply connected} subspace of
$X$. The \emph{fundamental group} of $(X,A)$ denoted $\pi_1(X,A)$
is the set of homotopy classes of maps $(I, {\partial} I)
\overset{\sigma}{\longrightarrow} (X,A)$
The multiplication in this group is:
\[
[\sigma][\tau] = [\sigma\cdot \rho\cdot \tau]\text{ where $\rho$
is any path in $A$ from $\sigma(1)$ to $\tau(0)$}
\]
This group is functorial in such $(X,A)$ and isomorphic to
$\pi_1(X,x_0)$ if $x_0\in A$. In particular, if $I $ denotes an
interval (a subspace homeomorphic to $[0,1]$) in $X$, then $\cal
U\rm_n^{top}(I)$ is a contractible subset of $\cal U \rm _n^{top}(X)$. We define the
\emph{n-point braid group of (X,I)} by:
\begin{equation}\label{braid XI}
B_n(X;I) = \pi_1( \cal U \rm _n^{top}(X), \cal U\rm_n^{top}(I)).
\end{equation}
\begin{definition} \label{i_p}(The Endpoint Inclusion Map). Let $X$ be
a graph. Let $p$ be an endpoint of $X$ (that is, $degree(p) =1$).
For each $n\geq 1$, we define a map
\[ B_{n-1}(X,c)\overset{\iota_p}{\longrightarrow} B_n(X,c') \]
as follows. Choose an isotopy of $1_X$, say $\{r_t:X\to X
\;\;|\;\; 0\leq t\leq 1\}$, which is stationary outside a small
neighborhood of $p$ and which satisfies $r_1(X)\subset X-\{p\}.$
Then $r_1$ induces a map
\[\cal U\rm_{n-1}^{top}(X)\overset{\i_p}{\longrightarrow}\cal U \rm _n^{top}(X):\quad i_p(d) = \{p\}\cup
r_1(d) \quad \forall \;d\in \cal U\rm_{n-1}^{top}(X).
\]
In turn, $i_p$ induces a map of fundamental groups, denoted:
\[B_{n-1}(X,c)\overset{\iota_p}{\longrightarrow} B_n(X,c'), \text{ if } c'=\iota_p (c)
\]
(or $\iota_{p\;X}$ when such explicitness is needed).
Now if $I$ is an interval in $X$ containing $p$ and if the isotopy
fixes $X-I$, then $i_p$ induces
\[B_{n-1}(X;I)\overset{\iota_p}{\longrightarrow} B_n(X;I)\]
which is independent of the isotopy chosen. Write
$\iota_p^k:B_{n-k}(X;I)\to B_n(X;I)$ for the $k$-fold iteration of
this map. Abrams (\cite{abramsgeom}, Lemma 3.4, p.24) shows that
$\iota_p$ is injective if $X$ is any graph.
\end{definition}
(Note: Abrams proves this for the {\em pure} braid groups. These
have finite index in our braid groups $B_n(X;I)$, which are
torsion free. This implies that $\iota_p$ is injective).
Our main theorem will say that the groups $B_n(X;I)$, for
$n=0,1,2,\dots$, admit right angled Artin presentations which are
all related by the maps $\iota_p$ above. A \emph{right angled
Artin presentation} of a group $G$, denoted $\langle \beta,\cal R
\rangle $, consists of a subset $\beta$ of $G$, and a subset ${\cal R}$
of $F(\beta)$, the free group on $\beta$, such that ${\cal R}$
consists of elements of the form $(xyx^{-1}y^{-1})$, where $x,y\in
\beta$, and the following sequence is exact
\[
1\to N\to F(\beta)\overset{j}{\longrightarrow}G\to 1,
\]
where $j$ denotes the natural homomorphism, and $N$ denotes the
smallest normal subgroup
containing ${\cal R}$.
Here is the main theorem.
\thm \label{mainthm}Let $X$ be a tree. Let $p$ be an endpoint of
$X$. Let $I\subset X$ be an interval containing $p$ and every node
of $X$. Then for each integer $n\geq0$ there is a right angled
Artin presentation, $\langle \beta(n), {\cal R}(n)\rangle$ for
$B_n(X;I)$ such that
\begin{equation}
\iota_{p}(\beta(n-1))\subset \beta (n) \quad \text{and }
\iota_{p\ast}({\cal R}(n-1))\subset {\cal R}(n)\quad \forall n\geq 1.
\label{nat}
\end{equation}
Here $\iota_{p\ast}:F(\beta(n-1))\to F(\beta(n))$ is the
homomorphism induced by the function $ \iota_p:\beta(n-1)\to \beta(n)$.
\
It is easy to see that Theorem \ref{raathm} follows from Theorem
\ref{mainthm} because every interval in a tree $X$ lies in a
bigger interval $I$ containing an endpoint of $X$.
\
Here is an outline of the rest of this paper. In
Section~\ref{sec2}, we study the case of a \emph{star} (a tree
with one node). If $X$ is a star and $I$ is an interval whose
endpoints are endpoints of $X$, we show that $B_n(X;I)$ is a free
group admitting a basis $\beta(n)$, for each $n$, such that
$\iota_p(\beta(n-1))\subset \beta(n)$ if $p$ is \emph{either}
endpoint. In Section \ref{sec3}, we prove a theorem computing the
braid group $B_n(X;I)$ if $X$ is the one point union of two graphs
along a common endpoint. In section \ref{sec4}, we use the
previous results to prove the main theorem, \ref{mainthm}.
We wish to thank Aaron Abrams here for suggesting this problem to us, in 2003. His comments
about it were encouraging and helpful.
\section{Braid Groups of Stars.}\label{sec2}
A \emph{star} $S$ is a tree with no more than one node. The
node is denoted $v$. If $S$ is a tree with no nodes (an interval), then
we assume further that a fixed interior point $v$ of $S$ is given.
It turns out that, for any star $S$ and any integer $n\geq 0$, the
$n-$point configuration space $\cal U\rm_n^{top}(S)$ contains a
compact one dimensional polyhedron $D_n(S)$ which is a deformation
retract of $S$. This is constructed in \cite{mdoig}. We review
the construction here. We then use $D_n(S)$ to prove the
following:
\prop \label{starprop} Let $S$ be a star. Let $p$ and $q$ be two
distinct endpoints of $S$, and let $I$ be the interval $[p, q]$.
Then for each $n\geq 0$, $B_n(S;I)$ is a free group. Moreover,
there is a basis $\beta(n)$ for $B_n(S;I)$ such that, if $n\geq
1$,
\[ \iota_p(\beta(n-1))\subset \beta (n) \text{ and}
\quad\iota_q(\beta(n-1))\subset \beta (n). \]
Note: The rank of this free group $B_n(S;I)$ is
\[
1+(k-1)\binom{n+k-2}{k-1} - \binom{n-k-1}{k-1}
\]
where $k$ is the number of endpoints of $S$. This is proved by
Doig in \cite{mdoig}, but an equivalent formula for the
corresponding \emph{pure}, braid group of $S$ appears earlier in
Ghrist \cite{ghrist}, Prop. 4.1.
\
Before beginning the proof of \ref{starprop}, we now choose a
fixed metric $d$ on the star $S$. To this end, fix the integer
$n\geq0$. The metric $d$ will be a constant multiple of the
standard simplicial metric $\rho$ on $S$. The star $S$ has a {\em
canonical} simplicial structure in which the vertices are the
points of degree different from $2$. For the corresponding
simplicial metric $\rho$, $[p, q]$ is isometric to $[0,2]$ and $v$
is the midpoint of $[p, q]$. Define the metric $d$ by:
\begin{equation}\label{metric}
d(x,y) = C\rho(x,y) \quad \forall \, x,y\in S
\end{equation}
where $C$ is a fixed constant such that $C\geq max(1, n-1) $.
\
\con\label{DnS}(of $D_n(S)$; compare \cite{mdoig}).
Let $S$ be a star. Let \;$ c\in\cal U\rm_n^{top}(S)$ be any
$n-$point configuration in the star $S$. For each endpoint $p$ of
$S$, we define
\begin{equation}\label{arm}
A_p(c)= c\cap [p,v].
\end{equation}
We call this an \emph{arm} of $c$ \rm if $A_p(c)\neq \emptyset$.
We say $c$ is \emph{regular} if it satisfies the following rules:
\begin{enumerate}\label{regular}
\item For all $x,y\in c $ with $x\neq y, \; d(x,y)\geq 1.$ Also
$d(x,y) =1$ if $x$ and $y $ \;lie in a single arm of $c$ and
$[x,y]\cap c=\{x, y\}$. \item If $v\notin c$, and if $A_p(c)$ is
an arm of $c$, then there is a point $x\in c$ such that $d(x,
A_p(c)) =1$.
\end{enumerate}
In English: successive points in a single arm of $c$ are one unit
apart; if $v\notin c$, the innermost point of each arm of $c$ has
distance one from some other arm, and has distance at least one
from every other arm.
Therefore $c$ has at least one arm $A_p(c)$ such that
$d(v,A_p(c))\leq \frac{1}{2} $. There is {\em at most} one arm of
$c$ that satisfies: $0<d(v,A_p(c))<\frac{1}{2}$ (the
\emph{governing arm} in the language of \cite{mdoig}). When
$A_p(c)$ is the unique governing arm of $c$, every other arm,
$A_q(c)$, satisfies: $d(v, A_q(c))= 1-d(v, A_p(c))$.
The subspace $D_n(S) $ of $\cal U\rm_n^{top}(S)$ can now be
defined:
\begin{equation}\label{dns}
D_n(S) := \{c\in \cal U\rm_n^{top}(S)| \; c \text{ is regular}\}.
\end{equation}
Note $D_0(S) = \ast =\cal U\rm_0^{top}(S)$, and $D_1(S) =\{v\}$.
$D_n(S)$ has the structure of a 1-dimensional compact polyhedron,
as we now explain. Its vertices come in two types. The Type I
vertices are those configurations $c\in D_n(S)$ such that $v\in c
$. The Type II vertices are those $c\in D_n(S)$ such that
$d(v,c)=\frac{1}{2}$. For Type II vertices, note that
$d(v,A_p(c))=\frac{1}{2}$ for each arm $A_p(c)$.
Each 1-simplex of $D_n(S)$ has a single Type I vertex $c$ and a
single Type II vertex $c'$. We denote this 1-simplex $[c,c']$ or
$[c',c]$. For each Type II vertex $c'$, there is a single
1-simplex $[c', c]$ for each endpoint $p$ of $S$ such that
$A_p(c')\neq \emptyset$. The other endpoint $c$ of this 1-simplex
$[c',c]$ is defined as the unique Type I vertex such that:
\begin{equation}\label{type one vertex} |A_p(c)| = |A_p(c')|,\textrm{ and
}|A_q(c)| = |A_q(c')|+1 \;\; \text{for any \emph{other} endpoint }
q. \end{equation} The points of the 1-simplex $[c', c]$ are $c'$,
$c$, and those $e\in D_n(S)$ such that
\begin{equation} |A_q(e)| = |A_q(c')| \;\text{ for all endpoints
$q$ of $S$, and } 0<d(v, A_p(e))<\frac{1}{2}. \end{equation} The
rule $e\mapsto d(v,A_p(e))$ gives a homeomorphism from $[c',c]$
onto the interval $[0,\frac{1}{2}]$.
Each point $e\in D_n(S)$, other than a vertex, belongs to a unique
1-simplex $[c', c]$. The Type II vertex $c'$ is specified by the
requirement that $|A_q(e)|=|A_q(c')| $ for all endpoints $q$. The
endpoint $p$, and also (by (\ref{type one vertex})) the Type~I
vertex $c$, \;is specified by the requirement that \;
$0<d(v,A_p(e))<\frac{1}{2}$.
This completes the construction of the 1-dimensional polyhedron
$D_n(S)$.
\
Doig proves in \cite{mdoig} that $D_n(S)$ is a strong deformation
retract of $\cal U\rm_n^{top}(S)$.
Incidentally, if we change the metric $d$ on $S$ to $d'$, by
changing the constant $C$ to $C'$, then $D_n(S, d)$ is isometric
to $D_n(S, d')$. But the two are not identical.
Before beginning the proof of Proposition \ref{starprop}, we need
to relate $\iota_p$ to the deformation retract $D_n(S)$.
If $p$ and $q$ are two endpoints of $S$ and $I=[p, q]$, then
$D_n(I)\subset D_n(S)$ and $D_n(I)$ is an interval.
Analogous to the map $i_p$ of Definition \ref{i_p} is a simplicial
inclusion map $ \tilde{i}_p:D_{n-1}(S)\to D_n(S)$, defined by:
\[\tilde{i}_p(c)=\{x\}\cup c \quad \quad\forall c\in D_{n-1}(S),
\] where $x$ is the unique point of $[p, v]$ such that $\{x\}\cup c \in D_n(S)$.
(We have chosen the constant $C$ above, so that $d(p,c)\geq 1.$ This
ensures that there {\em is} such a point $x$). It is elementary to see
that
$\tilde{i}_p(D_n(I))\subset D_n(S)$, and that the following diagram
commutes up to homotopy:
\[ \begin{CD}
(D_{n-1}(S), D_{n-1}(I))@>\tilde{i}_p>> (D_n(S), D_n(I))\\
@V\text{incl.}VV @V\text{incl.}VV\\
(\cal U\rm^{top}_{n-1}(S), \cal U\rm^{top}_{n-1}(I))@>{i}_p>> (\cal
U\rm^{top}_{n}(S), \cal U\rm^{top}_{n}(I))
\end{CD}
\]
Therefore we can identify the group $B_n(S;I)$ with
$\pi_1(D_n(S), D_n(I))$, and we can identify the homomorphism
$\iota_p: B_{n-1}(S; I)\to B_n(S;I)$ with the map
$(\tilde{i}_p)_*: \pi_1(D_{n-1}(S), D_{n-1}(I))\to \pi_1(D_n(S),
D_n(I))$.
\begin{proof} (of Proposition \ref{starprop}):
Suppose a maximal tree ${\cal T}$ of $D_n(S)$ is chosen, containing
$D_n(I)$. Because $dim(D_n(S))= 1$, $B_n(S;I)$ is a free group,
and a basis for $B_n(S;I)$ is given by those 1-simplices of
$D_n(S)$ which are not in ${\cal T}$. Therefore, it is enough to
exhibit, for each $n$, a maximal tree ${\cal T}(n)$ for $D_n(S)$ such
that
\begin{equation}\label{T} {\cal T}(n)\supset D_n(I) \quad \text{
and } \quad {\cal T}(n-1)= \tilde{i}_p^{-1}({\cal T}(n))=
\tilde{i}_q^{-1}({\cal T}(n)).
\end{equation}
We do this now.
Let $c^{(n)}$ be the unique point of $D_n(S)$ such that
$c^{(n)}\subset [p, v]$. The configuration $c^{(n)}$ is a Type I vertex.
For each vertex $c$ of $D_n(S)$ except $c^{(n)}$, we are going to
construct a \emph{successor vertex} $s(c)$ such that $[c, s(c)]$ is a
1-simplex and $s^k(c) = c^{(n)}$ for some $k>0$. Then $ {\cal T}(n)$
will consist of the union of these $[c, s(c)].$
Number the endpoints of $S: \; p_1, p_2,\dots p_m$. Ensure that
$p=p_1$ and $q=p_2$. Define $s(c)$ as follows. If $c$ is a Type I
vertex, then $s(c)$ is the unique Type II vertex satisfying:
\begin{equation}\label{scI}
|A_{p_1}(s(c))| = |A_{p_1}(c)|; \quad\qquad |A_{p_j}(s(c))|=
|A_{p_j}(c)|-1 \text{ if } j\neq 1
\end{equation}
If $c$ is a Type II vertex, let $r=r(c)$ be the biggest index for
which $A_{p_{r(c)}}\neq \emptyset$. Note $r\geq 2 $ (by
\ref{regular}.2.). Define $s(c)$ as the unique Type I vertex
satisfying:
\begin{equation}\label{scII}
|A_{p_r}(s(c))| = |A_{p_r}(c)|;\qquad |A_{p_j}(s(c))| =
|A_{p_j}(c)|+1 \text{ if } j\neq r.
\end{equation}
For any vertex $c$ of $D_{n-1}(S)$ except $c^{(n-1)}$, we have, by
(\ref{scI}) and
(\ref{scII})
\begin{equation}\label{scIII} \tilde{i}_{p_j}(s(c)) =
s(\tilde{i}_{p_j}(c)), \text { for }
j\leq 2.
\end{equation}
By (\ref{type one vertex}), $c$ and $s(c)$ span a 1-simplex for each
vertex $c\neq c^{(n)}$.
It is also clear from (\ref{scI}) and (\ref{scII}) that, for each
vertex $c$, there is an
integer
$k\geq 0$ such that $s^k(c) = c^{(n)}$.
Therefore we define:
\begin{equation}\label{Tn}
{\cal T}(n) = \cup\{[c, s(c)]\; \;|\; c \text{ is a vertex of } D_n(S)
\text{ other than }
c^{(n)}\}.
\end{equation}
${\cal T} (n)$ is a tree containing every vertex of $D_n(S);$ therefore it
is maximal.
By (\ref{scIII}), $\; \tilde{i}_{p_j}({\cal T}(n-1))\subset {\cal T}(n)$, for
$j=1$ and $2.$ But since ${\cal T}(n-1)$ is a \emph{maximal} tree in
$D_{n-1}(S)$, and ${\cal T}(n)$ contains no cycles, this
implies:
\[
{\cal T}(n-1) = \tilde{i}_p^{-1}({\cal T}(n))= \tilde{i}_q^{-1}({\cal T}(n)).
\]
Finally we must show that $D_n(I)\subset {\cal T}(n)$. The key is to
note that each Type II vertex $c$ of $D_n(I)$ belongs to exactly
two 1-simplices. One of these is $[c, s(c)]$. The other is $[c,
d]$, where $d$ is that Type I vertex of $D_n(I)$ such that
$c=s(d)$. It follows that $D_n(I)\subset {\cal T}(n)$. This completes
the proof of Proposition \ref{starprop}.
\end{proof}
\section{The Endpoint Union of Two Graphs.}\label{sec3}
This section is devoted to computing the $n$-point braid group of the
union of two graphs which intersect at a single endpoint of each.
Our goal is Proposition~\ref{exact} below.
We let $X$ be a graph of the form $X=S \cup T$ where $S$ and $T$
are graphs. Assume that $\{q\} = S \cap T$, a single point, and
that $q$ is an endpoint of $S$ and of $T$. Let
$j:S \longrightarrow X $ and $ j': T\longrightarrow
X,$ be inclusion maps. These induce maps of braid groups with the same
names. Let
$I$ and
$J$ be intervals in
$S$ and $T$ respectively so that $\{q\}=I \cap J$. Form the free
product $B_n(S;I) \ast B_n(T;J)$. Let $N$ be the smallest normal
subgroup of this product containing each of the commutator subgroups
$[\iota_q^k B_{n-k}(S;I),\iota_q^{n-k} B_k(T;J)],~1<k<n$. Then we have:
\begin{prop}\label{exact}
With the hypotheses above, the following sequence is exact: \[1
\rightarrow N \rightarrow B_n(S;I) \ast B_n(T;J) \stackrel{j \ast
j'}{\longrightarrow} B_n(X;I \cup J) \rightarrow 1.\]
\end{prop}
We will need:
\begin{lemma} \label{exactlem}
Let $A_0 \subseteq A_1 \subseteq \dots \subseteq A_n$ and $B_0
\subseteq B_1 \subseteq \dots \subseteq B_n$ be two increasing
sequences of groups. Suppose that the following diagram is a
pushout diagram:
\[ G
\]
\[
\nearrow \qquad \qquad \qquad \uparrow \qquad \qquad \qquad \nwarrow
\]
\[\qquad A_n\times B_0 \qquad \qquad \qquad A_{n-1}\times B_1\qquad
\qquad\qquad A_{n-2}\times B_2\quad \dots
\]
\[
\nwarrow i\times 1 \qquad \nearrow 1\times i'\qquad
\nwarrow i\times 1 \quad \nearrow 1\times i'
\]
\[
A_{n-1}\times B_0 \qquad \qquad \qquad A_{n-2}\times
B_1\quad \dots
\]
(i.e., $G$ is the direct limit of the diagram obtained by
deleting
$G$ and the maps to $G$). Here $i$ and $i'$ denote inclusions.
Then there is an exact sequence:
\[1
\rightarrow N \rightarrow A_n \ast B_n \stackrel{j \ast
j'}{\longrightarrow} G \rightarrow 1\] where $N$ is the smallest
normal subgroup of $A_n*B_n $ containing $[A_{n-k},B_{k}]$ for each
$k=0,1,2,
\dots, n$. Here $j$ and $j'$ are restrictions of the limit maps
$A_n\times B_0\to G$, and $ A_0 \times B_n\to G$ to $A_n$ and $B_n$
respectively.
\end{lemma}
\begin{proof}
If $n=1$ this is clear. Working inductively, we let $G'$ be the
limit of the diagram obtained by omitting $A_0$ and $ B_n$. This
gives us $G'=A_n \ast B_{n-1} / N'$ where $N'$ is the smallest
normal subgroup containing all $[A_{n-k},B_k]$ for $k=0,1,2,
\dots, n-1$. It also gives a new diagram whose limit is
obviously $A_n \ast B_n /N$ where $N$ is the smallest normal
subgroup containing $N'$ and $[A_0,B_n]$.
\end{proof}
\begin{proof} ({\em of Proposition~\ref{exact}}):
For each $k=0,1,\dots,n$, let
\[U(k)=\{c \in \cal U \rm _n^{top}(X)|\quad|c \cap S|
\geq k, |c \cap T| \geq n-k\},\]
and let $j_k:U(k)\to \cal U \rm _n^{top}(X)$ denote the inclusion map. Since $|S \cap T|=1$,
we see that
$ U(j) \cap U(k) = \emptyset$ if $|k-j|>1$ and that $\cal U \rm _n^{top}(X)=\bigcup_{k=0}^n
U(k)$. Each $c\in U(k)$ can be written uniquely in the form:
\[c=c_S(k) \cup c_T(n-k)\] where:\newline (i) $c_S(k) \subset
S$;\quad (ii) $c_T(n-k) \subset T$;\quad (iii) $|c_S(k)|=k;\quad
(iv) |c_T(n-k)|=n-k$. Necessarily, $c_S(k) \cap
c_T(n-k)=\emptyset$.
One sees easily that there is a homotopy equivalence
\[U(k)
\stackrel{h_k}{\longrightarrow} \cal U\rm_k^{top}(S) \times \cal
U\rm_{n-k}^{top}(T): \qquad h_k(c)= \left(c_S(k),c_T(n-k)
\right).\] It sends $U(k) \cap \cal U\rm_n^{top}(I \cup J)$ to
$\cal U\rm_k^{top}(I) \times \cal U\rm_{n-k}^{top}(J)$. Similarly
\[U(k)\cap U(k-1) = \{c \in \cal U \rm _n^{top}(X)~|~q \in c, |c \cap S|=k, |c
\cap T|=n-k+1\}.
\] Therefore there is a homotopy equivalence for each $k\geq 1$:
\[U(k) \cap U(k-1) \stackrel{j'_k}{\longrightarrow}\cal U\rm_{k-1}^{top}(S) \times
\cal U\rm_{n-k}^{top}(T):\quad j_k'(c)= \left(c_S(k-1),c_T(n-k)
\right).
\]
Passing to fundamental groups of these spaces, we obtain the
pushout diagram below by using the version of Van Kampen's Theorem
in Hatcher (\cite{hatcher} 1.20, p.43):\newpage
\[
\qquad B_n(X;I\cup J)\]
\[
j_0 \nearrow \qquad \qquad \qquad j_1\uparrow \qquad \qquad
\qquad \nwarrow j_2
\]
\[B_0(S;I)\times B_n(T;J) \quad \quad B_1(S;I)\times B_{n-1}(T;J)
\quad \quad B_2(S;I)\times B_{n-2}(T;J)\dots\]
\[ \nwarrow i\times 1 \qquad 1\times i' \nearrow \qquad \qquad
\nwarrow i\times 1 \qquad 1\times i' \nearrow \]
\[ B_0(S;I)\times B_{n-1}(T;J) \qquad \qquad B_1(S;I)\times
B_{n-2}(T;J)\dots \]
\
Now, by
Lemma~\ref{exactlem}, where $j=j_n$, $j'=j_0$, we get the exact
sequence \[1 \rightarrow N \rightarrow B_n(S;I) \ast B_n(T;J)
\overset{j*j'}\rightarrow B_n(X;I \cup J) \rightarrow 1.\]
\end{proof}
\section{Proof of the Main Theorem.}\label{sec4}
\begin{proof}
(of the Main Theorem \ref{mainthm}): If the tree $X$ has less than
two nodes, the proof is already clear from
Proposition~\ref{starprop}. Therefore we assume $X$ has at least
two nodes. We write $X$ as: $X= S\cup T$, where $S$ is a star with one
node, $v$, $T$ is a tree, $S\cap T =\{q\}$, where $q$ ia an endpoint
of both $S$ and $T$, and
$p$ is an endpoint of $X $ lying in $S$.
We can
moreover assume $I=J\cup K$, where $J= [p,q]$ and $K$ is an interval in
$T$, containing $q$ and each node of $T$.
From Proposition
\ref{starprop} we have a basis $\beta_S(n)$ for the free group
$B_n(S;J)$ for each $n$, such that
\begin{equation}\label{betan}
\iota_p(\beta_S(n-1))\subset \beta_S(n);\quad
\iota_q(\beta_S(n-1))\subset \beta_S(n).
\end{equation}
By induction on the number of nodes of $X$, we have a right angled
Artin presentation $\langle \beta_T(n); {\cal R}_T(n)\rangle $ of
$B_n(T;K)$ for each $n= 0, 1, 2, \dots$ so that
\begin{equation}
\label{artinT} \iota_q(\beta_T(n-1)) \subset \beta_T(n),\qquad
\iota_{q*}({\cal R}_T(n-1))\subset {\cal R}_T(n), \quad \forall n\geq 1.
\end{equation}
By Proposition \ref{exact} we have an exact sequence \[1\to
N\rightarrow B_n(S;J)*B_n(T;K) \overset{j_S*j_T} {\longrightarrow}
B_n(X;I)\to 1 \] where $j_S$ and $j_T$ are induced by the
inclusions $S\subset X$ and $ T\subset X$ and $N$ is the smallest
normal subgroup of the free product containing each of the sets:
\[ [\iota_q^k\beta_S(n-k), \,\iota_q^{n-k}\beta_T(k)] . \]
Set \begin{align} \beta_X(n) &= j_S(\beta_S(n))\cup j_T(\beta_T(n))\label{artpres1}\\
{\cal R}_X(n) &= j_{T*}({\cal R}_T(n))\cup [j_{S*}\iota^{k}_{q*}\beta_S(n-k),
\; j_{T*}\iota^{n-k}_{q*}\beta_T(k)]\label{artpres2}
\end{align} where $j_{S*}:F(\beta_S(n))\to F(\beta_X(n))$ and
$j_{T*}:F(\beta_T(n))\to F(\beta_X(n))$ are induced by $j_S$ and
$j_T$.
Clearly $\langle \beta_X(n); {\cal R}_X(n)\rangle$ is an Artin
presentation of $B_n(X;I)$ for each $n$. To complete the
argument, we must prove (\ref{nat}). First note that
\begin{align} \iota_{p X}\circ j_T &= j_T\circ \iota_{q T}: B_{n-1}(T;K)\to B_n(X;
I)\label{comm1}\\
\iota_{p X}\circ j_S &= j_S\circ \iota_{p S}: B_{n-1}(S;J)\to B_n(X;
I)\label{comm2}\\
\iota_{p S}\circ \iota_{q S} &= \iota_{q S}\circ \iota_{p S}: B_{n-2}(S;J)\to
B_n(S;J)\label{comm3}
\end{align} because the corresponding diagrams of spaces commute up to homotopy.
It follows that
\begin{equation}\label{ibx} \iota_p(\beta_X(n-1))\subset \beta_X(n)
\end{equation}
by (\ref{artpres1}). Also, by (\ref{artinT}), (\ref{artpres2}) and
(\ref{ibx}) we have
\begin{equation}
(\iota_{p \, X})_*j_{T*}
({\cal R}_T(n-1)) = j_{T*}(\iota_{q\, T})_*({\cal R}_T(n-1))\subset
j_{T*}({\cal R}_T(n))\subset {\cal R}_X(n).\label{last}
\end{equation}
Finally by ( \ref{comm1}), (\ref{comm2}) and (\ref{comm3}) we have:
\begin{multline*} \iota_{p\;X}[j_{S\;*}(\iota^k_{q\;S})_*\beta_S(n-1-k),\quad
j_{T\;*}(\iota_{q\;T}^{n-1-k})_*\beta_T(k)]\\
\subset [j_{S\;*}(\iota^k_{q\;S})_*\beta_S(n-k),
j_{T\;*}(\iota_{q\;T}^{n-k})_* \beta_T(k)]
\end{multline*}
which, with (\ref{ibx}) and (\ref{last}) implies \[(\iota_{p
X})_*{\cal R}_X(n-1) \subset {\cal R}_X(n). \] This completes the proof of
Theorem \ref{mainthm}.
\end{proof}
|
2,869,038,154,816 | arxiv | \section{Introduction}
Let $k$ be an algebraically closed field.
The Zariski Cancellation Problem for Affine Spaces
asks whether the affine space $\A^n_k$ is cancellative, i.e.,
if $\V$ is an affine $k$-variety such that
$\V \times \A^1_k \cong \A^{n+1}_k$,
does it follow that $\V \cong \A_k^n$? Equivalently,
if $A$ is an affine $k$-algebra
such that $A[X]$ is isomorphic to the polynomial ring $k[X_1, \dots, X_{n+1}]$,
does it follow that $A$ is isomorphic to $k[X_1, \dots, X_n]$?
The affine line $\A^1_k$ was shown to be cancellative by
S. S. Abhyankar, P. Eakin and W. J. Heinzer
(\cite{AEH}) and the affine plane $\A^2_k$ was shown to be cancellative by
T. Fujita, M. Miyanishi and T. Sugie
(\cite{F}, \cite{MS}) in characteristic zero and
by P. Russell (\cite{R}) in positive characteristic.
However, in \cite{G}, the author showed when ch. $k>0$,
the affine space $\A^3_k$ is not cancellative by proving that
a threefold constructed by Asanuma in \cite{A} is
not isomorphic to the polynomial ring $k[X_1, X_2, X_3]$.
In this paper, we shall show when ch. $k>0$,
the affine space $\A^n_k$ is not cancellative
for any $n\ge 3$ (Theorem \ref{zar}).
This completely settles the Zariski's Cancellation problem
in positive characteristic.
\section{Preliminaries}\label{pri}
We shall use the notation $R^{[n]}$
for a polynomial ring in $n$ variables over a ring $R$.
We shall also use the following term from affine algebraic geometry.
\smallskip
\noindent
{\bf Definition.}
An element $f \in k[Z, T]$ is called a {\it line} if $k[Z, T]/(f) = k^{[1]}$.
A line $f$ is called a {\it non-trivial line} if $k[Z, T]\neq k[f]^{[1]}$.
We recall the definition of an exponential map
(a formulation of the concept of $\Ga_a$-action)
and associated invariants.
\smallskip
\noindent
{\bf Definition.}
Let $A$ be a $k$-algebra and let $\phi: A \to A^{[1]}$ be a
$k$-algebra homomorphism. For an indeterminate $U$ over $A$, let the notation
$\phi_U$ denote the map $\phi: A \to A[U]$.
$\phi$ is said to be an {\it exponential map on $A$} if $\phi$ satisfies the following two properties:
\begin{enumerate}
\item [\rm (i)] $\varepsilon_0 \phi_U$ is identity on $A$, where
$\varepsilon_0: A[U] \to A$ is the evaluation at $U = 0$.
\item[\rm (ii)] $\phi_V \phi_U = \phi_{V+U}$, where
$\phi_V: A \to A[V]$ is extended to a homomorphism
$\phi_V: A[U] \to A[V,U]$ by setting $\phi_V(U)= U$.
\end{enumerate}
The ring of $\phi$-invariants of an exponential map $\phi$ on $A$
is a subring of $A$ given by
$$
A^{\phi} = \{a \in A\,| \,\phi (a) = a\}.
$$
An exponential map $\phi$ is said to be {\it non-trivial} if $A^{\phi} \neq A$.
For an affine domain $A$ over a field $k$, let
${\rm EXP} (A)$ denote the set of all exponential maps on $A$.
The {\it Derksen invariant} of $A$ is a subring of $A$ defined by
$$
{\rm DK} (A) = k[ f \,| \, f \in A^{\phi}, \phi ~\text{a non-trivial exponential map}].
$$
We recall below a crucial observation (cf. \cite[Lemma 2.4]{G}).
\begin{lem}\label{r1}
Let $k$ be a field and $A= k^{[n]}$, where $ n > 1$. Then ${\rm DK} (A) = A$.
\end{lem}
We shall also use the following result proved in \cite[Lemma 3.3]{G}.
\begin{lem}\label{lem1}
Let $B$ be an affine domain over an infinite field $k$.
Let $f \in B$ be such that $f - \lambda $ is a prime element of $B$
for infinitely many $\lambda \in k$. Let $\phi$ be
a non-trivial exponential map on $B$ such that
$f \in B^{\phi}$. Then there exist infinitely many $\beta \in k$ such that each $f - \beta$
is a prime element of $B$ and $\phi$
induces a non-trivial exponential map $\hat{\phi}$ on $B/(f - \beta)$.
Moreover, $B^{\phi}/(f - \beta)B^{\phi}$ is contained in $(B/(f - \beta))^{\hat{\phi}}$.
\end{lem}
We shall also use the following result proved in \cite[Theorem 3.7]{G2}.
\begin{thm}\label{dka}
Let $k$ be a field and $A$ be an integral domain defined by
$$
A= k[X,Y,Z,T]/(X^m Y - F(X, Z, T)), {\text{~~where~~}} m > 1.
$$
Set $f(Z, T):= F(0, Z, T)$. Let $x$, $y$, $z$ and $t$ denote,
respectively, the images of $X$, $Y$, $Z$ and $T$ in $A$.
Suppose that $\dk(A) \neq k[x,z,t]$.
Then the following statements hold.
\begin{enumerate}
\item [\rm (i)] There exist $Z_1, T_1 \in k[Z, T]$ and $a_0, a_1 \in k^{[1]}$ such that
$k[Z, T]=k[Z_1, T_1]$ and $f(Z, T) = a_0(Z_1) + a_1(Z_1)T_1$.
\item [\rm(ii)] If $k[Z, T]/(f)= k^{[1]}$, then $k[Z, T]=k[f]^{[1]}$.
\end{enumerate}
\end{thm}
An exponential map $\phi$ on a graded ring $A$ is said to be {\it homogeneous}
if $\phi: A \to A[U]$ becomes homogeneous
when $A[U]$ is given a grading induced from $A$ such that $U$ is a homogeneous element.
We state below a result on homogenization of exponential maps due to
H. Derksen, O. Hadas and L. Makar-Limanov (\cite{DHM}, cf. \cite[Theorem 2.3]{G}).
\begin{thm}\label{MDH}
Let $A$ be an affine domain over a field $k$ with an admissible
proper $\bZ$-filtration and $\gr(A)$ the induced $\bZ$-graded domain.
Let $\phi$ be a non-trivial exponential map on $A$.
Then $\phi$ induces a non-trivial homogeneous exponential map $\bar{\phi}$ on
$\gr (A)$ such that $\rho( {A^{\phi}}) \subseteq {\gr (A)}^{\bar{\phi}}$.
\end{thm}
\section{Main Theorems}
Throughout the paper, $k$ will denote a field
(of any characteristic unless otherwise specified) and
$A$ an integral domain defined by
$$
A= k[X_1, \dots, X_{m},Y,Z,T]/({X_1}^{r_1} \cdots {X_m}^{r_m} Y - F(X_1, \dots, X_m, Z, T)),
$$
where $r_i >1$ for each $i$, $1\le i\le m$.
The images of $X_1$, $\dots$, $X_m$, $Y$, $Z$ and $T$ in $A$ will be denoted by
$x_1$, $\dots$, $x_m$, $y$, $z$ and $t$ respectively.
Set $B: = k[x_1,\dots, x_m, z, t] (= k^{[m+2]})$.
We note that $B \hookrightarrow A \hookrightarrow B[({x_1\cdots x_m})^{-1}]$.
For each $m$-tuple $(q_1, \dots, q_m) \in \bZ^m$,
consider the $\bZ$-grading on $B[({x_1\cdots x_m})^{-1}]$
given by
$$
B[({x_1\cdots x_m})^{-1}]= \bigoplus_{i \in \bZ}B_i, \text{~where~}
B_i = \bigoplus_{(i_1, \dots, i_m) \in \bZ^m, q_1i_1+q_2i_2+ \dots+ q_mi_m = i}k[z, t]{x_1}^{i_1}{x_2}^{i_2}\dots{x_m}^{i_m}.
$$
Each element $a \in B[({x_1\cdots x_m})^{-1}]$ can be uniquely written as
$a= \sum_{{\ell_a} \le j\le {u_a}} a_j$, where $a_j \in B_j$.
(Note that if $a \in B$ then $a_j \in B$ for each $j$, ${\ell_a} \le j\le {u_a}$.)
We call $u_a$ the degree of $a$ and $a_{u_a}$ the leading homogeneous summand of $a$.
Using this grading on $B[({x_1\cdots x_m})^{-1}]$, we can define a proper
$\bZ$-filtration $\{A_n\}_{n \in \bZ}$ on $A$
by setting $A_n := A \cap \bigoplus_{i \le n}B_i$.
Then $x_j \in A_{q_j}\setminus A_{q_j-1}$, $1\le j\le m$
and $z, t \in A_{0}\setminus A_{-1}$.
Since $A$ is an integral domain, we have $F \neq 0$ and hence $F_{u_F} \neq 0$.
Thus, $y \in A_b\setminus A_{b-1}$, where
$b= u_F-(q_1r_1+ \cdots + q_mr_m)$.
Let $\gr{A}$ denote the induced graded ring
$\bigoplus_{n \in \bZ}A_{n}/ A_{n-1}$.
Note that each element $h \in A$ can be uniquely written as sum of monomials of the form
\begin{equation}\label{eqn}
{x_1}^{i_1} \cdots {x_m}^{i_m} z^{j_1}t^{j_2}y^{\ell},
\end{equation}
where $i_j \ge 0$, $1\le j \le m$ and $\ell \ge 0$ satisfying
that if $\ell > 0$ then $i_{s} < r_s$ for at least one $s$, $1\le s \le m$.
Therefore, it can be seen that, if
$h \in A_{n}\setminus A_{n-1}$ then $h$ can be uniquely written as
sum of monomials of the form as in equation (\ref{eqn})
and each of these monomials lies in $A_n$.
Thus, the filtration defined on $A$ is admissible with the
generating set $\Gamma := \{x_1, \dots, x_m, y, z, t\}$
and so $\gr{A}$ is generated by image of $\Gamma$ in $\gr{A}$
(cf. \cite[Remark 2.2 (2)]{G}).
We now exhibit a structure of $\gr{A}$ when $F_{u_F}$ is not divisible by $x_j$ for any $j$.
\begin{lem}\label{fil}
Suppose that $F_{u_F}$ is not divisible by $x_j$ for any $j$, $1\le j\le m$,
then $\gr{A}$ is isomorphic to
$$
D = \dfrac{k[X_1, \dots, X_m,Y,Z,T]}{({X_1}^{r_1} \cdots {X_m}^{r_m} Y - F(X_1, \dots, X_m, Z, T)_{u_F})}.
$$
\end{lem}
\begin{proof}
For $a \in A$, let $\gr(a)$ denote the image of $a$ in $\gr{A}$.
Then, as discussed above, $\gr{A}$ is generated by $\gr(x_1)$, $\dots$,
$\gr(x_m)$, $\gr(y)$, $\gr(z)$ and $\gr(t)$.
Note that if $a \in B (\subseteq A)$, then $\gr(a) = \gr(a_{u_a})$.
As ${x_1}^{r_1} \cdots {x_m}^{r_m}{y} (= F)\in A_{u_F} \setminus A_{u_F-1}$
and hence ${x_1}^{r_1} \cdots {x_m}^{r_m}{y}- F(x_1, \dots, x_m, z, t)_{u_F} \in A_{u_F-1}$. Therefore,
$$
\gr({x_1})^{r_1} \cdots \gr({x_m})^{r_m}\gr({y})- \gr(F_{u_F}) =0 {\text{~~in~~}} \gr{A}.$$
Since $F_{u_F}$ is not divisible by $x_j$ for any $j$,
$1\le j\le m$, $D$ is an integral domain.
As $\gr{A}$ can be identified with a subring of
$\gr (B[({x_1\cdots x_m})^{-1}]) \cong B[({x_1\cdots x_m})^{-1}]$,
we see that the elements $\gr(x_1),\dots, \gr(x_m),\gr(z),\gr(t)$ of $\gr{A}$
are algebraically independent over $k$. Hence $\gr{A} \cong D$.
\end{proof}
\begin{lem}\label{dkA}
We have $B (=k[x_1, \dots, x_m, z, t])\subseteq \dk(A)$.
\end{lem}
\begin{proof}
Define $\phi_1$ by $\phi_1(x_j) =x_j$ for each $j$, $1\le j\le m$,
$\phi_1(z) = z$, $\phi_1(t) =t+{x_1}^{r_1} \cdots {x_m}^{r_m}U$ and
\begin{equation*}
\phi_1(y) = \frac{F(x_1, \dots, x_m,z,t+{x_1}^{r_1} \cdots
{x_m}^{r_m}U)}{{x_1}^{r_1} \cdots {x_m}^{r_m}} = y + U\alpha(x_1, \dots, x_m, z, t, U)
\end{equation*}
and define $\phi_2$ by $\phi_2(x_j) =x_j$ for each $j$, $1\le j\le m$,
$\phi_2(t) = t$, $\phi_2(z) =z+{x_1}^{r_1} \cdots {x_m}^{r_m}U$,
\begin{equation*}
\phi_2(y) = \frac{F(x_1, \dots, x_m, z+{x_1}^{r_1} \cdots {x_m}^{r_m}U, t)}
{{x_1}^{r_1} \cdots {x_m}^{r_m}} = y+ U \beta(x_1, \dots, x_m, z, t, U).
\end{equation*}
Note that $\alpha(x_1, \dots, x_m, z, t, U), \beta(x_1, \dots, x_m, z, t, U) \in k[x_1, \dots, x_m, z, t, U]$
and that $k[x_1, \dots, x_m,z]$ and $k[x_1, \dots, x_m, t]$ are algebraically closed in $A$
of transcendence degree $m+1$ over $k$.
It then follows that $\phi_1$ and $\phi_2$ are nontrivial exponential maps on $A$
with $A^{\phi_1} = k[x_1, \dots, x_m,z]$ and $A^{\phi_2} = k[x_1, \dots, x_m,t]$.
Hence $k[x_1, \dots, x_m, z, t]\subseteq \dk(A)$.
\end{proof}
We now prove a generalisation of Theorem \ref{dka}.
\begin{prop}\label{main}
Suppose that $f(Z, T):= F(0,\dots, 0, Z, T) \neq 0$ and that $\dk(A) = A$.
Then the following statements hold.
\begin{enumerate}
\item [\rm (i)] There exist $Z_1, T_1 \in k[Z, T]$ and $a_0, a_1 \in k^{[1]}$ such that
$k[Z, T]=k[Z_1, T_1]$ and $f(Z, T) = a_0(Z_1) + a_1(Z_1)T_1$.
\item [\rm (ii)] Suppose that
$k[Z, T]/(f) = k^{[1]}$. Then $k[Z, T] = k[f]^{[1]}$.
\end{enumerate}
\end{prop}
\begin{proof}
(i) We prove the result by induction on $m$. The result is true for $m= 1$ by Theorem \ref{dka}.
Suppose that $m >1$.
Set $B:= k[x_1, x_2, \dots, x_m, z, t] (\subseteq A)$. Since $\dk(A) = A$,
there exists an exponential map $\phi$ on $A$ such that $A^{\phi} \nsubseteq B$.
Let $g \in A^{\phi} \setminus B$. Since $g \notin B$,
by equations (\ref{eqn}),
there exists a monomial in $g$ which is of the form
${x_1}^{i_1} \dots {x_m}^{i_m} z^{j_1}t^{j_2}y^{\ell}$
where $\ell >0$ and $i_{s}< r_s$ for some $s$, $1\le s \le m$.
Without loss of generality, we may assume that $s=1$.
Consider the proper $\bZ$-filtration on $A$
with respect to $(-1, 0, \dots, 0) \in \bZ^m$
and let $\overline{A}$ denote the induced graded ring.
For $h \in A$, let $\bar{h}$ denote the image of $h$ in $\overline{A}$.
Since $A$ is an integral domain, we have
$F(0, x_2, \dots, x_m, z, t) \neq 0$ and so,
$F(x_1, \dots, x_m, z, t)_{u_F}=
F(0, x_2, \dots, {x_m}, {z}, {t})= G$ (say).
Since $f(Z, T) \neq 0$, by Lemma \ref{fil}, we have
$$
\overline{A} \cong
k[X_1, \dots, X_{m},Y,Z,T]/({X_1}^{r_1} \cdots {X_m}^{r_m} Y - G).
$$
By Theorem \ref{MDH}, $\phi$ induces a non-trivial homogeneous
exponential map $\bar{\phi}$ on $\overline{A}$ such that $\bar{g} \in \overline{A}^{\bar{\phi}}$.
By the choice of $g$ and the filtration defined on $A$,
we have $\bar{y} \mid \bar{g}$.
Since $\overline{A}^{\bar{\phi}}$ is factorially closed in $\overline{A}$
(cf. \cite[Lemma 2.1 (i)]{G}),
it follows that $\bar{y} \in \overline{A}^{\bar{\phi}}$.
We now consider the proper $\bZ$-filtration on $\overline{A}$
with respect to $(-1, -1, \dots, -1) \in \bZ^m$
and let $\widetilde{A}$ denote the induced graded ring.
For $\bar{h} \in \overline{A}$, let $\widetilde{h}$ denote
the image of $\bar{h}$ in $\widetilde{A}$.
Since $f(Z, T) \neq 0$, we have
${G}_{u_G}= f(z, t)$
and hence by Lemma \ref{fil},
$$
\widetilde{A} \cong k[X_1, \dots, X_{m},Y,Z,T]/
({X_1}^{r_1} \cdots {X_m}^{r_m} Y - f(Z, T)).
$$
Again by Theorem \ref{MDH}, $\bar{\phi}$ induces a non-trivial homogeneous
exponential map $\widetilde{\phi}$ on $\widetilde{A}$ such that $\widetilde{y} \in \widetilde{A}^{\widetilde{\phi}}$.
Since $\td_k \widetilde{A}^{\widetilde{\phi}} = m+1$, there exist $m$ algebraically independent elements
in $\widetilde{A}^{\widetilde{\phi}}$ over $k[\widetilde{y}]$.
If $\widetilde{A}^{\widetilde{\phi}} \subseteq k[\widetilde{y},\widetilde{z},\widetilde{t}]$,
then since $m \ge 2$ and $\widetilde{A}^{\widetilde{\phi}}$ is algebraically closed in $\widetilde{A}$,
we have $\widetilde{A}^{\widetilde{\phi}} = k[\widetilde{y},\widetilde{z},\widetilde{t}]$.
Since $\widetilde{x_1}^{r_1} \cdots \widetilde{x_m}^{r_m} \widetilde{y} (= f(\widetilde{z}, \widetilde{t})) \in
k[\widetilde{y},\widetilde{z},\widetilde{t}]$,
we have $\widetilde{x_1}, \dots, \widetilde{x_m}, \widetilde{y} \in \widetilde{A}^{\widetilde{\phi}}$ as
$\widetilde{A}^{\widetilde{\phi}}$ is factorially closed in $\widetilde{A}$. This would contradict
the fact that $\widetilde{\phi}$ is non-trivial. Thus, there exists an homogeneous element
$h \in \widetilde{A}^{\widetilde{\phi}} \setminus k[\widetilde{y},\widetilde{z},\widetilde{t}]$.
Hence, $h$ contains a monomial which is divisible by $\widetilde{x_i}$ for some $i$, $1\le i\le m$. Without loss of generality
we may assume that $\widetilde{x_2 }$ divides a monomial of $h$.
Now again consider the proper $\bZ$-filtration on $\widetilde{A}$
with respect to the $m$-tuple $(0, 1, 0, \dots, 0) \in \bZ^{m}$
defined as in Lemma \ref{fil} and the induced graded ring $\gr{\widetilde{A}}$ of $\widetilde{A}$.
Then $\gr{\widetilde{A}} \cong \widetilde{A}$.
For $\widetilde{a} \in\widetilde{A}$,
let $\gr(a)$ denote the image of $\widetilde{a}$ in $\gr{\widetilde{A}}$.
Then $\gr(x_2) \mid \gr(h)$ in $\gr{A}$. Again by Theorem \ref{MDH},
$\widetilde{\phi}$ induces a non-trivial homogeneous
exponential map $\gr{\phi}$ on $\gr{\widetilde{A}}$ such that
$\gr(y)$ and $\gr(h) \in {\gr{A}}^{\gr{\phi}}$.
Since $\gr(x_2) \mid \gr(h)$ in $\gr{A}$, we have $\gr(x_2) \in {\gr{A}}^{\gr{\phi}}$.
Let $F$ be an algebraic closure of the field $k$.
Then $\gr{\phi}$ induces a non-trivial exponential map $\psi$ on
$$
E=\gr{\widetilde{A}} \otimes_k F\cong \dfrac{F[X_1,\dots, X_m, Y, Z, T]}
{({X_1}^{r_1} \cdots {X_m}^{r_m} Y - f(Z, T))} = F[\hat{x_1}, \dots, \hat{x_m}, \hat{z}, \hat{t}, \hat{y}],
$$
such that $F[\hat{y}, \hat{x_2}]\subseteq E^{{\psi}}$.
Since $\hat{x_2} -\lambda$ is a prime element in $E$
for every $\lambda \in F^*$, by Lemma \ref{lem1},
$\psi$ induces a non-trivial exponential map on
$$
E/ (\hat{x_2} -\lambda)E (\cong \dfrac{F[X_1, X_3, \dots, X_{m},Y,Z,T]}
{(\lambda^{r_2}{X_1}^{r_1}{X_{3}}^{r_3}\cdots{X_m}^{r_m} Y -f(Z, T))}
$$
for some $\lambda \in F^*$
such that the image of $\hat{y}$ lies in the ring of invariants.
Therefore, using Lemma \ref{dkA},
we have $\dk(E/ (\hat{x_2} -\lambda)E)= E/ (\hat{x_2} -\lambda)E$.
We are thus through by induction on $m$.
(ii) Suppose that
$f(Z, T)$ is a line in $k[Z, T]$.
Then $A/(x_1, \dots, x_m) = k[Y, Z, T]/(f(Z, T)) \cong k^{[2]}$
and hence $(A/(x_1, \dots, x_m))^*= k^*$. By (i) above,
there exist $Z_1, T_1 \in k[Z, T]$ and $a_0, a_1 \in k^{[1]}$ such that
$k[Z, T]=k[Z_1, T_1]$ and $f(Z, T) = a_0(Z_1) + a_1(Z_1)T_1$.
If $a_1(Z_1)=0$, then $f(Z, T)= a_0(Z_1)$ must be a linear polynomial in $Z_1$
and hence a variable in $k[Z, T]$.
Now suppose $a_1(Z_1) \neq 0$. As $f(Z, T)$ is irreducible in $k[Z, T]$,
$a_0(Z_1)$ and $a_1(Z_1)$ are coprime in $k[Z_1]$.
Hence $A/(x_1, \dots, x_m) \cong k[Z_1, \frac{1}{a_1(Z_1)}]^{[1]}$
and since $(A/(x_1, \dots, x_m))^*= k^*$, we have $a_1(Z_1) \in k^*$.
This again implies that $f(Z, T)$ is a variable in $k[Z, T]$.
\end{proof}
We now deduce a result for a field of positive characteristic.
\begin{cor}\label{ce}
Suppose that ch. $k >0$ and that
$f(Z,T) \in k[Z, T]$ is a non-trivial line in $k[Z, T]$.
Then, $A \ncong k^{[m+2]}$.
\end{cor}
\begin{proof}
Suppose, if possible that $A \cong k^{[m+2]}$. Then, by Lemma \ref{r1}, $\dk(A)= A$. Therefore,
by Proposition \ref{main}(ii), $k[Z, T]= k[f]^{[1]}$. This contradicts the given hypothesis.
\end{proof}
\begin{thm}\label{sta}
Let $k$ be a field of any characteristic and
$A$ an integral domain defined by
$$
A= k[X_1, \dots, X_{m},Y,Z,T]/({X_1}^{r_1} \cdots {X_m}^{r_m} Y - f(Z, T)),
$$
where $r_i >1$ for each $i$.
Suppose that $f(Z, T)$ is a line in $k[Z, T]$. Then
$$
A^{[1]}=k[X_1, \dots, X_m]^{[3]} \cong k^{[m+3]}.
$$
\end{thm}
\begin{proof}
Let $h \in k[Z, T]$ be such that $k[Z, T]= k[h] +(f)k[Z, T]$.
Let $P , Q \in k^{[1]}$ be such that
$Z= P(h) + fP_1(Z, T)$ and
$T = Q(h) + fQ_1(Z, T)$ for some $P_1, Q_1 \in k[Z, T]$.
Let $W$ be an indeterminate over $A$.
Set
\begin{eqnarray*}
W_1 : &=& {X_1}^{r_1} \cdots {X_m}^{r_m} W + h(Z, T) \\
Z_1:&=& (Z- P(W_1))/{X_1}^{r_1} \cdots {X_m}^{r_m}\\
T_1: &=& (T- Q(W_1))/{X_1}^{r_1} \cdots {X_m}^{r_m}.
\end{eqnarray*}
Let $y$ be the image of $Y$ in $A$.
We show that $A[W]= k[X_1, \dots, X_m, Z_1,T_1, W_1]$.
Set $B: = k[X_1,\dots, X_m, Z_1,T_1, W_1]$. We have
\begin{eqnarray*}
Z &=& P(W_1)+ {X_1}^{r_1} \cdots {X_m}^{r_m}Z_1, \\
T &=& Q(W_1)+ {X_1}^{r_1} \cdots {X_m}^{r_m}T_1,\\
y &=& \frac{f(Z,T)}{{X_1}^{r_1} \cdots {X_m}^{r_m}} =
\frac{f({X_1}^{r_1} \cdots {X_m}^{r_m}Z_1 + P(W_1),{X_1}^{r_1} \cdots {X_m}^{r_m} T_1 + Q(W_1))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} \\
&=& \frac{f(P(W_1), Q(W_1))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} + \alpha(X_1,\dots, X_m, Z_1, T_1, W_1),\\
W &=& \frac{W_1- h(Z, T)}{{X_1}^{r_1} \cdots {X_m}^{r_m}} = \frac{W_1- h({X_1}^{r_1} \cdots {X_m}^{r_m}Z_1 + P(W_1),
{X_1}^{r_1} \cdots {X_m}^{r_m}T_1 + Q(W_1))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} \\
&=& \frac{W_1- h(P(W_1), Q(W_1))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} + \beta(X_1, \dots, X_m, Z_1, T_1, W_1)
\end{eqnarray*}
for some $\alpha , \beta \in B$.
Since $f(P(W_1), Q(W_1)) = 0$ and $h(P(W_1), Q(W_1)) = W_1$, we see that $y, W \in B$.
Hence, $A[W] \subseteq B$.
We now show that $B \subseteq A[W]$. We have,
\begin{eqnarray*}
Z_1 &= & \frac{Z- P(W_1)}{{X_1}^{r_1} \cdots {X_m}^{r_m}}
= \frac{Z- P({X_1}^{r_1} \cdots {X_m}^{r_m} W + h(Z, T))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} \\
&=& \frac{Z-P(h(Z, T))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} + \gamma(X_1, \dots, X_m, Z, T, W)=
\frac{f(Z, T)P_1(Z, T)}{{X_1}^{r_1} \cdots {X_m}^{r_m}}+\gamma(X_1,\dots, X_m, Z, T, W)\\
&=& P_1(Z, T) y + \gamma(X_1,\dots, X_m, Z, T, W),
\end{eqnarray*}
and
\begin{eqnarray*}
T_1 &= & \frac{T- Q(W_1)}{{X_1}^{r_1} \cdots {X_m}^{r_m}} =
\frac{T- Q({X_1}^{r_1} \cdots {X_m}^{r_m}W + h(Z, T))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} \\
&=& \frac{T-Q(h(Z, T))}{{X_1}^{r_1} \cdots {X_m}^{r_m}} + \delta(X_1,\dots, X_m, Z, T, W)=
\frac{f(Z, T)Q_1(Z, T)}{{X_1}^{r_1} \cdots {X_m}^{r_m}}+\delta(X_1,\dots, X_m, Z, T, W)\\
&=& Q_1(Z, T) y + \delta(X_1,\dots, X_m, Z, T, W)
\end{eqnarray*}
for some $\gamma, \delta \in A[W]$.
Thus, $Z_1, T_1 \in A[W]$. Hence $B \subseteq A[W]$.
Since $B = k[X_1, \dots, X_m]^{[3]}$, the result follows.
\end{proof}
\begin{thm}\label{zar}
When $k$ is a field of positive characteristic,
Zariski's Cancellation Conjecture does not hold for the affine $n$-space
$\A^n_k$ for any $n \ge 3$.
\end{thm}
\begin{proof}
There exists non-trivial line in $k[Z, T]$ when ch $k$ $= p>0$. For instance, we may take the
Nagata's line $f(Z, T)= Z^{p^2} + T + T^{qp}$, where $q$ is a prime other than $p$.
The result now follows from Theorem \ref{sta} and Corollary \ref{ce}.
\end{proof}
\small{
|
2,869,038,154,817 | arxiv | \section{Introduction}
The time-dependent Schr{\"o}dinger equation (TDSE) governs the behavior of $N$ quantum particles,
\begin{equation}
\label{eqn:tdse}
i \partial_t \Psi(r_1, r_2, ..., r_N,t) = \hat{H}(r_1, r_2, ..., r_N,t) \Psi(r_1, r_2, ..., r_N,t),
\end{equation}
where $\hat{H}$ is the Hamiltonian and $\Psi$ is the many-body wave function. In $d$-dimensional space, the many-body Coulomb interaction in the potential term of $\hat{H}$ leads to a coupled system of partial differential equations (PDE) in $dN+1$ variables. Hence (\ref{eqn:tdse}) can only be solved for simple model problems, such as for one electron in three dimensions or two electrons in one dimension. To simulate electron dynamics in molecules and materials, a widely used approach is time-dependent density functional theory (TDDFT), in which the many-body wave function $\Psi$ is replaced with the Kohn-Sham wave function $\Phi(r)$ to give the time-dependent Kohn-Sham (TDKS) equation \citep{maitra2016perspective, ullrich2011time-dependent}:
\begin{equation}
\label{eqn:tdks3d}
i \partial_t \Phi (r,t) = \sum_{i=1}^N [ -(1/2)\nabla^2_i + v^{\text{ext}}(r_i,t) + v^H[n](r_i,t) + v^{XC}[n, \Psi_0, \Phi_0](r_i,t) ] \Phi(r,t).
\end{equation}
Because $\Phi(r)$ is constructed as a product of non-interacting single-particle orbitals $\phi(r_i)$, (\ref{eqn:tdks3d}) decouples into $N$ separate evolution equations in $3+1$ variables. Assuming all terms in (\ref{eqn:tdks3d}) are specified, one can use (\ref{eqn:tdks3d}) to simulate molecular systems for which numerical simulation of (\ref{eqn:tdse}) is intractable.
In (\ref{eqn:tdks3d}), the many-body Coulomb interaction between electrons is replaced by known Hartree $v^H$ and unknown exchange-correlation $v^{XC}$ single-particle potentials, with the latter incorporating many-body effects. TDDFT is formally an exact theory, as the Runge-Gross and Van Leeuwen theorems proved the existence of a time-dependent electronic potential and the unique mapping to the time-dependent electron density, which is generated from the KS orbitals of the TDKS equation \citep{runge1984density, vanLeeuwen1999mapping}.
The challenge in TDDFT is to construct $v^{XC}$ potentials that yield an electron density $n$ that is identical to the exact time-dependent many-body electron density generated from the TDSE. Previous work has shown that the unknown $v^{XC}$ formally depends on the initial many-body wave function $\Psi_0$, the initial KS state $\Phi_0$, and the electron density at all points in time $n(r, s<t)$ \citep{maitra2002memory}. Although the development of $v^{XC}$ for electrons is a very active area of research, almost all exchange-correlation potentials make use of the so-called ``adiabatic approximation'' that only takes into account the instantaneous electron density, leading to significant inaccuracies in electron dynamics due to the lack of memory in $v^{XC}$. This leads to a natural question: can we learn $v^{XC}$ from data? In the context of time-independent density functional theory (DFT), researchers have answered this question affirmatively, successfully learning static, ground state potentials from the exact ground state electron density \citep{nagai2018neural, Burke2021review}.
For machine learning of $v^{XC}$ to proceed in the TDDFT context, a first obstacle is generating training data. In recent work, \cite{suzuki2020machine} produces such data by working with an electron-hydrogen scattering model problem such that (\ref{eqn:tdse}) is numerically solvable. For this problem, $v^{XC} = v^{X} + v^{C}$ with $v^{X}$ known; the goal is to learn the correlation potential $v^{C}$ from data. The one-dimensionality enables one to solve for the values of $v^{C}$ on a spatial and temporal grid \citep{elliott2012universal}. With grid-based values of both $v^{C}$ and $n$, the electron density computed from the solution of (\ref{eqn:tdse}), \cite{suzuki2020machine} trains neural network models of the $v^{C}[n]$ functional. To our knowledge, this is the only prior work on learning $v^{XC}$ for TDDFT.
We revisit the electron-hydrogen scattering model problem and develop methods to learn $v^{C}$ using the TDKS equation as a constraint. This constraint leads naturally to adjoint methods. We view the $v^C$ functional as a control that guides TDKS propagation; the idea of using adjoints to compute gradients is standard in optimal control \citep{bryson1975applied, hasdorff1976gradient}. However, the derivations and applications of the adjoint method, to \emph{learn} $v^C$ models with memory for the TDKS equation, are to our knowledge considered here for the first time\footnote{See Section \ref{sect:relationship} in the Appendix for further details.}. We derive adjoint systems for two settings: (i) to learn pointwise values of $v^C$ on a grid, and (ii) to learn the functional dependence of $v^C$ on the electron density at two points in time. We apply our methods to train both types of models, and study their training and test performance. In particular, we train a neural network model of $v^C[n]$ with memory that, when used to solve the TDKS equations for initial conditions outside the training set, yields qualitatively accurate predictions of electron density.
\section{Methods}
\label{sect:methods}
To set up the problem of learning $v^C$ from data, we first consider the forward problem: that of solving the TDKS equation, assuming that all potentials have been specified. Our approach will be to discretize in space and time before optimizing, i.e., before formulating a Lagrangian and deriving an adjoint system.
\paragraph{Forward Problem.} Define the 1D electron density created from KS orbitals
\begin{equation}
\label{eqn:ndef}
n(x,t) = 2 | \phi(x,t) |^2,
\end{equation}
the soft-Coulomb external potential
\begin{equation}
\label{eqn:vextdef}
v^{\text{ext}}(x) = -((x+10)^2+1)^{-1/2},
\end{equation}
and the soft-Coulomb interaction potential
\begin{equation}
\label{eqn:Weedef}
W^{ee}(x',x) = ((x' - x)^2 + 1)^{-1/2}.
\end{equation}
In what follows, $v^C[\phi](x,t)$ stands for the correlation potential, which we allow to depend functionally on $\phi(x,s)$ for $s \leq t$, evaluated at position $x$ and time $t$. By (\ref{eqn:ndef}), we view $v^C[n]$ functionals as a subset of $v^C[\phi]$ functionals. We will specify models for $v^C$ in more detail below.
Let $\phi$ and $n$ stand for $\phi(x,t)$ and $n(x,t)$. Then the time-dependent Kohn-Sham system in one spatial dimension in atomic units (a.u.) is
\begin{subequations}
\label{eqn:tdks1d}
\begin{align}
i \partial_t \phi &= -\frac{1}{2} \partial_{xx} \phi + v^{\text{ext}}(x,t) \phi + v^H[n](x,t) \phi + v^X[n](x,t) \phi + v^C[\phi](x,t) \phi \\
v^H[n](x,t) &= \int_{x'} W^{ee}(x',x) n(x',t) \, dx' \\
v^X[n](x,t) &= -\frac{1}{2} v^H[n](x,t)
\end{align}
\end{subequations}
Note that $v^C[\phi]$ may depend on $\phi$ non-locally in time, while both $v^H$ and $v^X$ depend on $\phi$ nonlinearly and non-locally in space. To solve (\ref{eqn:tdks1d}), we use a finite-difference discretization of (\ref{eqn:tdks1d}) on the spatial domain $x \in [L_{\text{min}}, L_{\text{max}}]$ and temporal domain $t \in [0, T]$. Fix spatial and temporal grid spacings $\Delta x > 0$ and $\Delta t > 0$; let $x_j = L_{\text{min}} + j \Delta x$ and $t_k = k \Delta t$. Then our spatial grid is $\{x_j\}_{j=0}^{j=J}$ with $J \Delta x = L_{\text{max}} - L_{\text{min}}$, and our temporal grid is $\{t_k\}_{k=0}^K$ with $K \Delta t = T$.
Suppose the correlation functional has been specified in one of two ways: (i) for a particular trajectory, we have access to the \emph{values} of $v^C(x_j, t_k)$ at all spatial and temporal grid points $x_j$ and $t_k$, or (ii) we have a model $v^C[\phi]$ that takes as input $\phi(x,s)$ for $s \leq t_k$ and produces as output $v^C(x_j, t_k)$ for all $j$. Then, given an initial condition $\phi(x,0)$, the \emph{forward problem} is to solve (\ref{eqn:tdks1d}) numerically on the grids defined above.
Our first step is to discretize (\ref{eqn:tdks1d}) in space. Let $\ensuremath{\boldsymbol{\phi}}(t)$ be the $(J+1) \times 1$ column vector
\begin{equation}
\ensuremath{\boldsymbol{\phi}}(t) = [\phi(x_{0}, t), \ldots, \phi(x_J, t)]^T,
\end{equation}
where $^T$ denotes transpose. We use the integers $0, \ldots, J$ to index this vector, so that $[\ensuremath{\boldsymbol{\phi}}(t)]_{j} = \phi(x_j, t)$. We discretize $\partial_{xx}$ with a fourth-order Laplacian matrix $\Delta$, defined in (\ref{eqn:discreteLaplacian}), such that
\begin{equation}
\label{eqn:laplacian}
\partial_{xx} \phi(x,t) \bigr|_{x=x_j} = [\Delta \ensuremath{\boldsymbol{\phi}}(t)]_{j} + O(\Delta x^4).
\end{equation}
Besides the $\partial_{xx}$ term, the remaining terms on the right-hand side of (\ref{eqn:tdks1d}) result in a diagonal matrix multiplied by $\ensuremath{\boldsymbol{\phi}}$. The only term that requires further numerical approximation is the integral term. For this purpose, we define the symmetric matrix $W_{j,j'} = ((x_{j'} - x_j)^2 + 1)^{-1/2} \Delta x$.
Let $\circ$ denote the entry-wise product of vectors and let $|\ensuremath{\boldsymbol{\phi}}(t)|^2$ be the vector of entry-wise magnitudes $|\phi(x_j,t)|^2$. With quadrature weights $w_{j}$ given by Simpson's rule, the integral at $x=x_j$ becomes
\begin{align}
\int_{x'} ((x' - x_j)^2 + 1)^{-1/2} | \phi(x',t)|^2 dx' &= \sum_{j'} W_{j,j'} \phi(x_{j'},t) \phi^\ast(x_{j'},t) w_{j'} + O(\Delta x^{4}) \nonumber \\
\label{eqn:quadrature}
&= \left[ W ( |\ensuremath{\boldsymbol{\phi}}(t)|^2 \circ \ensuremath{\boldsymbol{w}} ) \right]_{j} + O(\Delta x^{4})
\end{align}
Let $K = -(1/2) \Delta$ be the spatially discretized kinetic operator, with $\Delta$ defined by (\ref{eqn:discreteLaplacian}) in the Appendix. Evaluating both sides of (\ref{eqn:tdks1d}) at $x=x_j$ for all $j$ at once, we arrive at the following nonlinear system of ordinary differential equations (ODE) for $\ensuremath{\boldsymbol{\phi}} = \ensuremath{\boldsymbol{\phi}}(t)$:
\begin{equation}
\label{eqn:tdksODE}
i \frac{d}{dt} \ensuremath{\boldsymbol{\phi}} = K \ensuremath{\boldsymbol{\phi}} + V(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^C) \ensuremath{\boldsymbol{\phi}},
\end{equation}
where $V(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^C)$ is a diagonal matrix whose diagonal is the vector
$\ensuremath{\boldsymbol{v}}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^{C}) = -((\ensuremath{\boldsymbol{x}} + 10)^2 + 1)^{-1/2} + W ( |\ensuremath{\boldsymbol{\phi}}|^2 \circ \ensuremath{\boldsymbol{w}} ) + \ensuremath{\boldsymbol{v}}^{C}$.
Here $\ensuremath{\boldsymbol{x}}$ is the vector whose $j$-th entry is $x_j$ and operations involving $\ensuremath{\boldsymbol{x}}$ should be interpreted entry-wise. We have deliberately kept $\ensuremath{\boldsymbol{v}}^{C}$ general to encompass both the cases where (i) $\ensuremath{\boldsymbol{v}}^{C}(t)$ is a vector of time-dependent parameters whose $j$-th entry is $v^C[\phi](x_j,t)$, and (ii) $\ensuremath{\boldsymbol{v}}^{C}$ is a function that takes as input, e.g., $\{ \ensuremath{\boldsymbol{\phi}}_k, \ensuremath{\boldsymbol{\phi}}_{k-1}, \ldots \ensuremath{\boldsymbol{\phi}}_0 \}$ and produces as output the values $v^C[\phi](x_j, t_k)$.
To solve (\ref{eqn:tdksODE}), we apply operator splitting \citep{castro2004propagators}, resulting in the fully discretized propagation equation
\begin{equation}
\label{eqn:tdksDISC}
\ensuremath{\boldsymbol{\phi}}(t_{k+1}) = \exp(-i K \Delta t/2) \exp(-i V(\ensuremath{\boldsymbol{\phi}}(t_k), \ensuremath{\boldsymbol{v}}^{C}_k) \Delta t) \exp(-i K \Delta t/2) \ensuremath{\boldsymbol{\phi}}(t_k).
\end{equation}
We choose this method for two reasons. First, the propagator is unitary and hence preserves the normalization of $\ensuremath{\boldsymbol{\phi}}$ over long times. Second, as $V$ is diagonal, both the matrix exponential of $-i V \Delta t$ and its Jacobian with respect to $\ensuremath{\boldsymbol{v}}^C$ are simple to calculate. The ease with which we can compute derivatives of the right-hand side of (\ref{eqn:tdksDISC}) balances its second-order accuracy in time.
Note that $K$ is time-independent and symmetric. For small systems, $\exp(-i K \Delta t/2)$ can be computed by diagonalizing $K$. If $K = S D S^{-1}$, then $\exp(-i K \Delta t/2) = S \exp(-i D \Delta t/2) S^{-1}$. During the initial part of our codes, we compute this kinetic propagator once and store it for future use. The adjoint derivations below can be extended straightforwardly to higher-order version of operator splitting, as long as the discrete propagation scheme involves alternating products of kinetic and potential propagation terms as in (\ref{eqn:tdksDISC}), with $V$ and $\ensuremath{\boldsymbol{v}}^C$ exponentiated diagonally.
\paragraph{Learning Problem.} Assume we have access to observed values of electron density on the grid---we denote these observed or reference values by $\tilde{n}(x,t)$. Using (\ref{eqn:tdksDISC}) as a constraint, the first learning (or inverse) problem we consider is to use optimization to solve for $\ensuremath{\boldsymbol{v}}^{C}_k$ at each $k$. We think of this as learning $v^C(x_j, t_k)$ on a grid.
Suppose we start from an initial condition $\ensuremath{\boldsymbol{\phi}}(0)$ and an estimate $\ensuremath{\boldsymbol{v}}^{C}$. We iterate (\ref{eqn:tdksDISC}) forward in time and obtain a trajectory $\ensuremath{\boldsymbol{\phi}}(t_k)$ for $0 \leq k \leq K$. We then form $n(x_j,t_k) = |\phi(x_j,t_k)|^2$. In this subsection, $\phi$ and $n$ are the predicted wave function and density when we use the estimated correlation potential $\ensuremath{\boldsymbol{v}}^{C}$. Let $\mathcal{P}_K = \exp(-i K \Delta t/2)$ and abbreviate $\ensuremath{\boldsymbol{\phi}}_k = \ensuremath{\boldsymbol{\phi}}(t_k)$, $\ensuremath{\boldsymbol{v}}^C_k = \ensuremath{\boldsymbol{v}}^C(t_k)$. Define the discrete-time propagator
\begin{equation}
\label{eqn:discprop}
\ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^{C}) = \mathcal{P}_K \exp(-i V(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^C) \Delta t) \mathcal{P}_K \ensuremath{\boldsymbol{\phi}},
\end{equation}
so that (\ref{eqn:tdksDISC}) can be written as the discrete-time system $\ensuremath{\boldsymbol{\phi}}_{k+1} = \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}_k, \ensuremath{\boldsymbol{v}}^{C}_k)$. Both sides of this system are complex-valued. In order to form a real-valued Lagrangian and take real variations, we split both $\ensuremath{\boldsymbol{\phi}}$ and $\ensuremath{\boldsymbol{F}}$ into real and imaginary parts: $\ensuremath{\boldsymbol{\phi}} = \ensuremath{\boldsymbol{\phi}}^R + i \ensuremath{\boldsymbol{\phi}}^I$ and $\ensuremath{\boldsymbol{F}}_{\Delta t} = \ensuremath{\boldsymbol{F}}_{\Delta t}^R + i \ensuremath{\boldsymbol{F}}_{\Delta t}^I$. Superscript $R$ and $I$ denote, respectively, the real and imaginary parts of a complex quantity. Let the uppercase $\ensuremath{\boldsymbol{\Phi}}$, $\ensuremath{\boldsymbol{\Lambda}}$, and $\ensuremath{\boldsymbol{V}}^{C}$ denote the collections of all corresponding lowercase $\ensuremath{\boldsymbol{\phi}}_k$, $\ensuremath{\boldsymbol{\lambda}}_k$, and $\ensuremath{\boldsymbol{v}}^{C}_k$ for all $k$. Let $\dagger$ denote conjugate transpose. Then we form a real-variable Lagrangian that consists of an objective function (the $L^2$ distance between predicted and training electron densities) together with a constraint that the evolution of $\ensuremath{\boldsymbol{\phi}}$ satisfy the TDKS system (\ref{eqn:tdks1d}).
\begin{multline}
\label{eqn:lag}
\mathscr{L}(\ensuremath{\boldsymbol{\Phi}}^R, \ensuremath{\boldsymbol{\Phi}}^I, \ensuremath{\boldsymbol{\Lambda}}^R, \ensuremath{\boldsymbol{\Lambda}}^I, \ensuremath{\boldsymbol{v}}^{C}) = \frac{1}{2} \sum_{k=0}^{K} \sum_{j=0}^{J} ( 2 \phi^R(x_j, t_k)^2 + 2 \phi^I(x_j, t_k)^2 - \tilde{n}(x_j, t_k) )^2 \\
- \sum_{k=0}^{K-1} [\ensuremath{\boldsymbol{\lambda}}_{k+1}^R]^T ( \ensuremath{\boldsymbol{\phi}}_{k+1}^R - \ensuremath{\boldsymbol{F}}^R_{\Delta t}(\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{v}}^{C}_k) ) + [\ensuremath{\boldsymbol{\lambda}}_{k+1}^I]^T ( \ensuremath{\boldsymbol{\phi}}_{k+1}^I - \ensuremath{\boldsymbol{F}}^I_{\Delta t}(\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{v}}^{C}_k) ).
\end{multline}
Setting $\delta \mathscr{L} = 0$ for all variations $\delta \ensuremath{\boldsymbol{\phi}}_k^R$ and $\delta \ensuremath{\boldsymbol{\phi}}_k^I$ for $k \geq 1$, we obtain\footnote{The variations are computed in more detail in the Appendix.}
\begin{subequations}
\label{eqn:adjsys}
\begin{align}
\label{eqn:adjfin}
\ensuremath{\boldsymbol{\lambda}}_K &= 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_K|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_K) \circ \ensuremath{\boldsymbol{\phi}}_K \right] \\
\label{eqn:adjeqn}
\begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k}^I \end{bmatrix}^T &= 4 (2 |\ensuremath{\boldsymbol{\phi}}_k|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_k) \circ \begin{bmatrix} \ensuremath{\boldsymbol{\phi}}_k^R \\ \ensuremath{\boldsymbol{\phi}}_k^I \end{bmatrix}^T +
\begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+1}^I \end{bmatrix}^T \ensuremath{\boldsymbol{J}}_{\ensuremath{\boldsymbol{\phi}}} \ensuremath{\boldsymbol{F}}_{\Delta t} (\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{v}}^{C}_k).
\end{align}
\end{subequations}
Here $\ensuremath{\boldsymbol{J}}_{\ensuremath{\boldsymbol{\phi}}} \ensuremath{\boldsymbol{F}}_{\Delta t}$ denotes the Jacobian of $\ensuremath{\boldsymbol{F}}$ with respect to $\ensuremath{\boldsymbol{\phi}}$---see (\ref{eqn:blockjacob}) in the Appendix for its block matrix form. We use (\ref{eqn:adjfin}) as a final condition and iterate (\ref{eqn:adjeqn}) backward in time for $k = K-1, \ldots, 1$. Having computed $\ensuremath{\boldsymbol{\Lambda}}$ from (\ref{eqn:adjsys}), we return to (\ref{eqn:lag}) and compute the gradient with respect to $\ensuremath{\boldsymbol{v}}^{C}_{\ell}$:
\begin{equation}
\label{eqn:nablavc}
\nabla_{\ensuremath{\boldsymbol{v}}^{C}_{\ell}} \mathscr{L} = \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{\ell+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{\ell+1}^I \end{bmatrix}^T \nabla_{\ensuremath{\boldsymbol{v}}^{C}_{\ell}} \begin{bmatrix} \ensuremath{\boldsymbol{F}}_{\Delta t}^R \\
\ensuremath{\boldsymbol{F}}_{\Delta t}^I \end{bmatrix} (\ensuremath{\boldsymbol{\phi}}_{\ell}^R, \ensuremath{\boldsymbol{\phi}}_{\ell}^I, \ensuremath{\boldsymbol{v}}^{C}_{\ell}).
\end{equation}
Given a candidate $\ensuremath{\boldsymbol{v}}^{C}$, we solve the forward problem to obtain $\ensuremath{\boldsymbol{\Phi}}$. We then solve the adjoint system to obtain $\ensuremath{\boldsymbol{\Lambda}}$. This provides everything required to evaluate (\ref{eqn:nablavc}) for each $\ell$. In order to complete these derivations, we need the gradients of the discrete-time propagator $\ensuremath{\boldsymbol{F}}$; these can be found in the Appendix.
\paragraph{Learning $v^C$ Functionals.} The previous derivation enables learning \emph{pointwise} values of $v^C$ on a grid in $(x,t)$ space. Here we rederive the adjoint method to enable learning the \emph{functional dependence} of $v^C[\phi](x,t)$ on $\phi(x,t)$ and $\phi(x,t - \Delta t)$. We take as our model
$v^C[\phi] = v^C(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}})$.
The parameters $\ensuremath{\boldsymbol{\theta}}$ determine a particular functional dependence of $v^C$ on the present and previous Kohn-Sham states $\ensuremath{\boldsymbol{\phi}}$ and $\ensuremath{\boldsymbol{\phi}}'$. At spatial grid location $x_j$ and time $t_k$, the model $v^C$ is
\begin{equation}
\label{eqn:vcmemorymodel2}
v^C[\phi](x_j, t_k) = [ \ensuremath{\boldsymbol{v}}^C(\ensuremath{\boldsymbol{\phi}}_{k}, \ensuremath{\boldsymbol{\phi}}_{k-1}; \ensuremath{\boldsymbol{\theta}}) ]_j.
\end{equation}
In short, we intend $\ensuremath{\boldsymbol{\phi}}'$ to be the Kohn-Sham state at the time step \emph{prior} to the time step that corresponds to $\ensuremath{\boldsymbol{\phi}}$. Our goal is to learn $\ensuremath{\boldsymbol{\theta}}$. This requires redefining the following quantities:
\begin{align*}
V(\ensuremath{\boldsymbol{\phi}}; \ensuremath{\boldsymbol{\theta}}) &= \operatorname{diag}( \mathbf{v}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}})) \\
\ensuremath{\boldsymbol{v}}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) &=-((\ensuremath{\boldsymbol{x}} + 10)^2 + 1)^{-1/2} + W ( |\ensuremath{\boldsymbol{\phi}}|^2 \circ \ensuremath{\boldsymbol{w}} ) + \ensuremath{\boldsymbol{v}}^{C}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \\
\ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) &= \mathcal{P}_K \exp(-iV(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) \mathcal{P}_K \ensuremath{\boldsymbol{\phi}}.
\end{align*}
The Lagrangian still has the form of an objective function together with a dynamical constraint:
\begin{multline}
\label{eqn:newlag}
\mathscr{L}(\ensuremath{\boldsymbol{\Phi}}^R, \ensuremath{\boldsymbol{\Phi}}^I, \ensuremath{\boldsymbol{\Lambda}}^R, \ensuremath{\boldsymbol{\Lambda}}^I, \ensuremath{\boldsymbol{\theta}}) = \frac{1}{2} \sum_{k=0}^{K} \sum_{j=0}^{J} ( 2 \phi^R(x_j, t_k)^2 + 2 \phi^I(x_j, t_k)^2 - \tilde{n}(x_j, t_k) )^2 \\
- \sum_{k=1}^{K-1} [\ensuremath{\boldsymbol{\lambda}}_{k+1}^R]^T ( \ensuremath{\boldsymbol{\phi}}_{k+1}^R - \ensuremath{\boldsymbol{F}}^R_{\Delta t}(\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_{k-1}^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{\phi}}_{k-1}^I; \ensuremath{\boldsymbol{\theta}}) ) \\
+ [\ensuremath{\boldsymbol{\lambda}}_{k+1}^I]^T ( \ensuremath{\boldsymbol{\phi}}_{k+1}^I - \ensuremath{\boldsymbol{F}}^I_{\Delta t}(\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_{k-1}^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{\phi}}_{k-1}^I; \ensuremath{\boldsymbol{\theta}}) )
\end{multline}
Setting $\delta \mathscr{L} = 0$ for all variations $\ensuremath{\boldsymbol{\phi}}^R_k$ and $\ensuremath{\boldsymbol{\phi}}^I_k$ for $k \geq 1$, we obtain the following adjoint system:
\begin{subequations}
\label{eqn:newadj}
\begin{align}
\label{eqn:lambK}
\ensuremath{\boldsymbol{\lambda}}_K &= 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_K|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_K) \circ \ensuremath{\boldsymbol{\phi}}_K \right] \\
\label{eqn:lambKm1}
\ensuremath{\boldsymbol{\lambda}}_{K-1} &= 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_{K-1}|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_{K-1}) \circ \ensuremath{\boldsymbol{\phi}}_{K-1} \right] \\
&\qquad + [\ensuremath{\boldsymbol{\lambda}}_{K}^R]^T \nabla_{\ensuremath{\boldsymbol{\phi}}} \ensuremath{\boldsymbol{F}}_{\Delta t}^R(\ensuremath{\boldsymbol{\phi}}_K, \ensuremath{\boldsymbol{\phi}}_{K-1}; \ensuremath{\boldsymbol{\theta}}) + [\ensuremath{\boldsymbol{\lambda}}_{K}^I]^T \nabla_{\ensuremath{\boldsymbol{\phi}}} \ensuremath{\boldsymbol{F}}_{\Delta t}^I(\ensuremath{\boldsymbol{\phi}}_K, \ensuremath{\boldsymbol{\phi}}_{K-1}; \ensuremath{\boldsymbol{\theta}}) \nonumber \\
\label{eqn:newadjiter}
\begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k}^I \end{bmatrix}^T &= 4 (2 |\ensuremath{\boldsymbol{\phi}}_k|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_k) \circ \begin{bmatrix} \ensuremath{\boldsymbol{\phi}}_k^R \\ \ensuremath{\boldsymbol{\phi}}_k^I \end{bmatrix}^T \\
&\qquad + \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+1}^I \end{bmatrix}^T \ensuremath{\boldsymbol{J}}_{\ensuremath{\boldsymbol{\phi}}} \ensuremath{\boldsymbol{F}}_{\Delta t} (\ensuremath{\boldsymbol{\phi}}_k, \ensuremath{\boldsymbol{\phi}}_{k-1}; \ensuremath{\boldsymbol{\theta}}) + \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+2}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+2}^I \end{bmatrix}^T \ensuremath{\boldsymbol{J}}_{\ensuremath{\boldsymbol{\phi}}'} \ensuremath{\boldsymbol{F}}_{\Delta t} (\ensuremath{\boldsymbol{\phi}}_{k+1}, \ensuremath{\boldsymbol{\phi}}_k; \ensuremath{\boldsymbol{\theta}}) \nonumber
\end{align}
\end{subequations}
The key difference between (\ref{eqn:newadjiter}) and (\ref{eqn:adjeqn}) is that the right-hand side of (\ref{eqn:newadjiter}) involves $\ensuremath{\boldsymbol{\lambda}}$ at two points in time. The adjoint system is now a linear delay difference equation with time-dependent coefficients. Additionally, the derivatives of $F_{\Delta t}$ needed to evaluate (\ref{eqn:newadj}-\ref{eqn:nablatheta}) are different---see the Appendix.
For a candidate value of $\ensuremath{\boldsymbol{\theta}}$, we solve the forward problem to obtain $\ensuremath{\boldsymbol{\phi}}$ on our spatial and temporal grid. Then, to compute gradients, we begin with the final conditions (\ref{eqn:lambK}-\ref{eqn:lambKm1}) and iterate (\ref{eqn:newadjiter}) backwards in time from $k = K-2$ to $k=1$. Having solved the adjoint system, we compute the gradient of $\mathscr{L}$ with respect to $\ensuremath{\boldsymbol{\theta}}$ via
\begin{equation}
\label{eqn:nablatheta}
\nabla_{\ensuremath{\boldsymbol{\theta}}} \mathscr{L} = \sum_{k=1}^{K-1} \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+1}^I \end{bmatrix}^T \nabla_{\ensuremath{\boldsymbol{\theta}}} \begin{bmatrix} \ensuremath{\boldsymbol{F}}_{\Delta t}^R \\
\ensuremath{\boldsymbol{F}}_{\Delta t}^I \end{bmatrix} (\ensuremath{\boldsymbol{\phi}}_k, \ensuremath{\boldsymbol{\phi}}_{k-1}; \ensuremath{\boldsymbol{\theta}}).
\end{equation}
\begin{figure}
\centering \includegraphics[width=5in]{vCgridresults.pdf}
\caption{Training results for the problem of learning pointwise values of $\ensuremath{\boldsymbol{V}}^C$ on a grid consisting of $K = 30000$ points in time and $J = 600$ points in space. The adjoint method succeeds in producing $\ensuremath{\boldsymbol{V}}^C$ values that yield TDKS solutions such that the corresponding electron densities (red) match those computed from the 2D Schr\"odinger equation (black).}
\label{fig:vCgridtrain}
\end{figure}
\section{Modeling and Implementation Details}
\label{sect:deets}
\paragraph{Modeling Correlation Functionals.}
In this work, all models of the form (\ref{eqn:vcmemorymodel2}) consist of dense, feedforward neural networks. For models of the form $\ensuremath{\boldsymbol{v}}^C(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}})$, we treat the real and imaginary parts of $\ensuremath{\boldsymbol{\phi}}$ and $\ensuremath{\boldsymbol{\phi}}'$ as real vectors each of length $J+1$. Hence for $J = 600$, we have an input layer of size $4(J+1)$. We follow this with three hidden layers each with $256$ units and a scaled exponential linear unit activation function \citep{selupaper}. The output layer has $J+1$ units to correspond to the vector-valued output $\ensuremath{\boldsymbol{v}}^C$. For models in which $\ensuremath{\boldsymbol{v}}^C$ depend on $\ensuremath{\boldsymbol{n}}$ and $\ensuremath{\boldsymbol{n}}'$, we take the real and imaginary parts of $\ensuremath{\boldsymbol{\phi}}$ and $\ensuremath{\boldsymbol{\phi}}'$ as inputs and use them to immediately compute $\ensuremath{\boldsymbol{n}}$ and $\ensuremath{\boldsymbol{n}}'$, which we then concatenate and feed into an input layer with $2(J+1)$ units. The remainder of the network is as above. We experimented with other activation functions, convolutional layers, and smaller numbers of units and layers---none of these models produced satisfactory results during training. We have not explored models larger than those described above.
\paragraph{Generation of Training Data.}
To generate training data, we solve (\ref{eqn:tdse}) for a model system consisting of $N=2$ electrons. The model consists of a one-dimensional electron scattering off a one-dimensional hydrogen atom. Hence (\ref{eqn:tdse}) becomes a partial differential equation (PDE) for a wave function $\Psi(x_1,x_2,t)$. We discretize this PDE using finite differences on an equispaced grid in $(x_1, x_2)$ space with $J=1201$ points along each axis. Here $-80 \leq x_1, x_2 \leq 40$, so that $\Delta x = 0.1$. After discretizing the kinetic and potential operators in space, we propagate forward in time until $T = 0.72$ fs, using second-order operator splitting with $\Delta t = 2.4 \times 10^{-5}$ fs (or, in a.u., $\Delta t \approx 9.99219 \times 10^{-4}$). Note that this is $1/100$-th the time step used by \cite{suzuki2020machine}. For further details of the spatial and temporal discretizations, consult the Appendix.
After discretization, the wave function $\Psi(x_1,x_2,t)$ at time step $k$ is a complex vector $\ensuremath{\boldsymbol{\psi}}_k$ of dimension $(J+1)^2$. For the initial vector $\ensuremath{\boldsymbol{\psi}}_0$, we follow \cite{suzuki2020machine} and use a Gaussian wave packet that represents an electron initially centered at $x=10$ a.u., approaching the H-atom localized at $x=-10$ a.u., with momentum $p$. We generate training/test data by repeating our numerical solution of the Schr\"odinger system for initial conditions with $p \in \{-1.0, -1.2, -1.4, -1.5, -1.6, -1.8\}$. From the resulting time series of wave functions, we compute the time-dependent one-electron density $n(x,t)$; \emph{we refer to this as the TDSE electron density below}.
\section{Results}
\label{sect:results}
\begin{figure}
\centering \includegraphics[width=5in]{vCp15TESTphi256results.pdf}
\caption{We use $300$ time steps (corresponding to $0.72$ fs) of the $p=-1.5$ data together with the adjoint method to train a neural network model of $v^C$ that depends on the current and previous $\phi$. Using the learned $v^C$, we propagate (\ref{eqn:tdks1d}) for $60$ additional time steps and plot the test set results (in red) against the reference electron density (in blue).}
\label{fig:vCp15TEST}
\end{figure}
\paragraph{Pointwise Results.} Our first result concerns learning the pointwise values of $\ensuremath{\boldsymbol{V}}^C$. Here we use the same fine time step $\Delta t = 2.4 \times 10^{-5}$ used to generate the training data. However, we increase $\Delta x$ by a factor of $2$, taking $J = 600$ and sampling the initial condition $\ensuremath{\boldsymbol{\phi}}_0$ at every other grid point. We retain this subsampling in space in all training sets/results that follow. Still, our unknown $\ensuremath{\boldsymbol{V}}^C$ consists of a total of $30000 \cdot 601$ values.
We learn $\ensuremath{\boldsymbol{V}}^C$ by optimizing an objective function that consists of the first line of (\ref{eqn:lag}) together with a regularization term. The regularization consists of a finite-difference approximation of $\mu \sum_{k} \sum_{j} ( \partial_x v^C(x_j, t_k) )^2$, with $\mu = 10^{-5}$. For training data, we use only the TDSE one-electron densities computed from the $p=-1.5$ initial condition. To optimize, we use the quasi-Newton L-BFGS-B method, with gradients $\nabla_{\ensuremath{\boldsymbol{V}}^C} \mathscr{L}$ computed via the procedure described just below (\ref{eqn:nablavc}). We initialize the optimizer with $\ensuremath{\boldsymbol{V}}^C \equiv 0$ and use default tolerances of $10^{-6}$.
In Figure \ref{fig:vCgridtrain}, we present the results of this approach. Each panel shows a snapshot of both the training electron density (in black, computed from TDSE data) and the electron density $n = 2 |\phi|^2$ (in red) obtained by solving TDKS (\ref{eqn:tdks1d}) using the learned $\ensuremath{\boldsymbol{V}}^C$ values. Note the close quantitative agreement between the black and red curves. The overall mean-squared error (MSE) across all points in space and time is $2.035 \times 10^{-6}$. Note that no exact $\ensuremath{\boldsymbol{V}}^C$ data was used; the learned $\ensuremath{\boldsymbol{V}}^C$ does not match the exact $\ensuremath{\boldsymbol{V}}^C$ quantitatively, but does have some of the same qualitative features.
This problem suits the adjoint method well: regardless of the dimensionality of $\ensuremath{\boldsymbol{V}}^C$, the dimensionality of the adjoint system is the same as that of the discretized TDKS system. Note that, for this one-dimensional TDKS problem (\ref{eqn:tdks1d}), it is possible to solve for $\ensuremath{\boldsymbol{V}}^C$ on a grid \citep{elliott2012universal}. If we encounter solutions of \emph{higher-dimensional, multi-electron} ($d \geq 2$ and $N \geq 2$) Schr\"odinger systems from which we seek to learn $\ensuremath{\boldsymbol{V}}^C$, we will not be able to employ an exact procedure. In this case, the adjoint-based method may yield numerical values $\ensuremath{\boldsymbol{V}}^C$, with which we can pursue supervised learning of a functional from electron densities $\ensuremath{\boldsymbol{n}}$ to correlation potentials $\ensuremath{\boldsymbol{V}}^C$.
\begin{figure}
\centering \includegraphics[width=5in]{vCphimemoryresults.pdf}
\caption{We plot training and test results at time $t = 0.432$ fs for the adjoint method, applied to estimating neural network models $v^C[\phi]$ that, at time $t$, depends on both $\phi(x,t)$ and $\phi(x, t-\Delta t)$. Propagating TDKS (\ref{eqn:tdks1d}) with the learned $v^C$ yields the red curves.}
\label{fig:vCphimemory}
\end{figure}
\paragraph{Functional Results.} Next we present results in which we learn $v^C$ functionals. In preliminary work, we sought to model $v^C[\phi](x,t)$ as purely a function of $\phi(x,t)$, a model without memory. These models did not yield satisfactory training set results, and hence were abandoned. We focus first on models $v^C[\phi](x,t)$ that allow for arbitrary dependence on the real and imaginary parts of both $\phi(x,t)$ and $\phi(x,t-\Delta t)$. The input layer is of dimension $4(J+1)$---see \ref{sect:deets}.
To train such a model, we again apply the L-BFGS-B optimizer with objective function given by the first line of (\ref{eqn:newlag}) and gradients computed with the adjoint system (\ref{eqn:newadj}-\ref{eqn:nablatheta}). We initialize neural network parameters $\ensuremath{\boldsymbol{\theta}}$ by sampling a mean-zero normal distribution with standard deviation $\sigma = 0.01$. For training data, we subsample the $p=-1.5$ TDSE electron density time series by a factor of $100$ in time, so that $\Delta t = 2.4 \times 10^{-3}$ fs and the entire training trajectory consists of $K=301$ time steps. We retain this time step in all training sets and results that follow.
We omit the training set results here (see Figure \ref{fig:vCp15TRAIN} in the Appendix) as they show excellent agreement between training and model-predicted electron densities. The overall training set mean-squared error (MSE) is $7.668 \times 10^{-6}$. In Figure \ref{fig:vCp15TEST}, we display test set results obtained by propagating for $60$ additional time steps beyond the end of the training data. On this test set, we see close quantitative agreement near $t=0.72$ fs, which slowly degrades. Still, the learned $v^C$ leads to TDKS electron densities that capture essential features of the reference trajectory. Note that no regularization was used during training of the $v^C$ functional, leading to a learned $v^C$ that is not particularly smooth in space. We hypothesize that, with careful and perhaps physically motivated regularization, the learned $v^C$ will yield improved test set results over longer time intervals.
In the next set of results, we retrain our model using TDSE electron densities with initial momenta equal to $p=-1.0$ and $p=-1.8$. We train two models: a $v^C[\phi](x,t)$ model that depends on $\phi$ at times $t$ and $t - \Delta t$, and a $v^C[n](x,t)$ model that depends on $n$ at times $t$ and $t - \Delta t$. We view the $v^C[n]$ model as more constrained because here the first hidden layer can depend on $\phi(x,t)$ and $\phi(x,t-\Delta t)$ \emph{only through} the electron densities $n(x,t)$ and $n(x,t-\Delta t)$. We keep all other details of training the same. The final training set MSE values are $4.645 \times 10^{-5}$ for the $v^C[\phi]$ model and $8.098 \times 10^{-5}$ for the $v^C[n]$ model.
In Figures \ref{fig:vCphimemory} and \ref{fig:vCnvecmemory}, we plot both training and test set results for these models. Here we have chosen a particular time ($t = 0.432$ fs) and plotted the electron density at this time for six different trajectories, each with a different initial momentum $p$. We have chosen this time to highlight the large, obvious differences between the $p=-1.0$ and $p=-1.8$ curves. The $p=-1.0$ and $p=-1.8$ panels contain training set results; here the TDKS electron densities (in red, produced using the learned $v^C$) lie closer to the ground truth TDSE electron densities (in black).
Note that, despite the greater freedom enjoyed by the $v^C[\phi]$ model, its generalization to trajectories \emph{outside the training set} ($-1.2 \leq p \leq -1.6$) is noticeably worse than that of the more constrained $v^C[n]$ model. In fact, the $v^C[n]$ model's results (Figure \ref{fig:vCnvecmemory}, in red) show broad qualitative agreement with the test set TDSE curves (in blue). The test set MSE values are $9.363 \times 10^{-4}$ for the $v^C[\phi]$ model and $2.482 \times 10^{-4}$ for the $v^C[n]$ model. Overall, these results support the view that $v^C$ should depend on $\phi$ through $n$. Again, we hypothesize that if we were to filter out short-wavelength oscillations in the electron density---perhaps by regularizing the $v^C[n]$ model or by training on a larger set of trajectories---the agreement could be improved.
\begin{figure}
\centering \includegraphics[width=5in]{vCnvecmemoryresults.pdf}
\caption{We plot training and test results at time $t = 0.432$ fs for the adjoint method, applied to estimating a neural network model $v^C[n]$ that, at time $t$, depends on both $n(x,t)$ and $n(x, t-\Delta t)$. Propagating TDKS (\ref{eqn:tdks1d}) with the learned $v^C[n]$ yields the red curves.}
\label{fig:vCnvecmemory}
\end{figure}
\paragraph{Conclusion.} For a low-dimensional model problem, we have developed adjoint-based methods to learn the correlation potential $v^C$ (either its pointwise values or its functional dependence) using data from TDSE simulations. There is intrinsic value in being able to train a model for $v^C$ without knowing exact values of $v^C$ in advance, as for higher-dimensional systems it will be impossible to obtain an exact $v^C$. Additionally, the ability of the trained $v^C[n]$ model to generalize outside its training set warrants further investigation. Overall, the results show the promise of learning $v^C$ via TDKS-constrained optimization.
\acks{This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award Number DE-SC0020203. This research used resources of the National Energy Research Scientific Computing Center (NERSC), a U.S. Department of Energy Office of Science User Facility located at Lawrence Berkeley National Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award BES-m2530 for 2021.
We acknowledge computational time on the Pinnacles cluster at UC Merced (supported by NSF OAC-2019144). We also acknowledge computational time on the Nautilus cluster, supported by the Pacific Research Platform (NSF ACI-1541349), CHASE-CI (NSF CNS-1730158), and Towards a National Research Platform (NSF OAC-1826967). Additional funding for Nautilus has been supplied by the University of California Office of the President.}
\section{Appendix}
\label{sect:Appendix}
\subsection{Details of Forward Solution Methods}
\label{sect:forwardmethods}
For forward time-evolution of the 1D TDKS system (\ref{eqn:tdks1d}), we use the following fourth-order Laplacian:
\begin{equation}
\label{eqn:discreteLaplacian}
\Delta = \frac{1}{12 \Delta x^2} \begin{bmatrix}
-30 & 16 & -1 & & \\
16 & \ddots & \ddots & \ddots & \\
-1 & \ddots & \ddots & \ddots & \ddots \\
& \ddots & \ddots & \ddots & \ddots & -1 \\
& & \ddots & \ddots & \ddots & 16 \\
& & & -1 & 16 & -30
\end{bmatrix}.
\end{equation}
For the 2D Schr\"odinger model system, the Hamiltonian is $\hat{H} = \hat{K} + \hat{V}$ with a kinetic operator $\hat{K} = -(1/2) \nabla^2$; here $\nabla^2$ is the two-dimensional Laplacian. The electronic potential $\hat{V}$ consists of a sum of electron-nuclear and electron-electron terms:
\begin{equation}
\label{eqn:schropot}
\hat{V}(x_1, x_2) = v^\text{ext}(x_1) + v^\text{ext}(x_2) + W^{ee}(x_1, x_2),
\end{equation}
with $v^\text{ext}$ and $W^{ee}$ defined via the soft-Coulomb potentials (\ref{eqn:vextdef}) and (\ref{eqn:Weedef}), respectively.
For forward time-evolution of the 2D Schr\"odinger model system, we use a discrete Laplacian $\Delta_2$ that consists of
\[
\Delta_2 = \Delta \otimes I + I \otimes \Delta,
\]
where $I$ is the $(J+1) \times (J+1)$ identity matrix and $\otimes$ denotes the Kronecker product. The resulting $\Delta_2$ is a fourth-order approximation to the two-dimensional Laplacian $\nabla^2 = \partial_{x_1 x_1} + \partial_{x_2 x_2}$. In the Schr\"odinger system, spatially discretizing the kinetic operator yields the matrix $K = -(1/2) \Delta_2$, which is of dimension $(J+1)^2 \times (J+1)^2$ with $J = 1200$. To compute the kinetic portion of the propagator,
\[
\mathcal{P}_K = \exp(-i K \Delta t/2),
\]
we used a straightforward fourth-order series expansion of the matrix exponential:
\[
\mathcal{P}_K \approx \sum_{j=0}^4 \frac{ (-i K \Delta t/2)^j }{j!}.
\]
With $\Delta t = 2.4 \times 10^{-5}$ fs (or, in a.u., $\Delta t \approx 9.99219 \times 10^{-4}$), this series approximation of the matrix exponential incurs negligible error.
The potential portion of the propagator, $\mathcal{P}_V = \exp(-i V \Delta t)$, is a purely diagonal matrix---the $(J+1)^2$ entries along its diagonal consist of a flattened version of the $(J+1) \times (J+1)$ matrix obtained by evaluating (\ref{eqn:schropot}) on our finite-difference spatial grid.
Equipped with $\mathcal{P}_K$ and $\mathcal{P}_V$, both of which are time-independent, we propagate forward using second-order operator splitting as in (\ref{eqn:discprop}):
\begin{equation}
\label{eqn:schroprop}
\ensuremath{\boldsymbol{\psi}}_{k+1} = \mathcal{P}_K \mathcal{P}_V \mathcal{P}_K \ensuremath{\boldsymbol{\psi}}_k.
\end{equation}
As a numerical method for the TDSE (\ref{eqn:tdse}), operator splitting goes back at least to the work of \cite{fleck1976time} and \cite{feit1982solution}. Starting from an initial condition $\ensuremath{\boldsymbol{\psi}}_0$ represented as a complex vector of dimension $(J+1)^2$, we iterate for $K=30000$ steps until we reach a final time of $0.72$ fs. We have implemented the above Schr\"odinger solver using sparse linear algebra and CuPy.
\paragraph{Initializing TDKS Simulations with Memory.} When we solve (\ref{eqn:tdks1d}) with a correlation potential with memory, \emph{e.g.}, $v^C[\phi]$ that depends on both $\phi(x,t)$ and $\phi(x,t - \Delta t)$, how do we initialize the simulation? Our solution is to start with the wave function data generated by solving the TDSE (as above). With this data, we apply the methods from \citet[Appendix E]{ullrich2011time-dependent} to compute exact Kohn-Sham states $\ensuremath{\boldsymbol{\phi}}_k$ corresponding to $k=0$ and $k=1$ with $\Delta t = 2.4 \times 10^{-3}$ fs. We use the exact $\ensuremath{\boldsymbol{\phi}}_0$ and $\ensuremath{\boldsymbol{\phi}}_1$ to initialize our TDKS simulations when we use a $v^C$ model with memory.
\subsection{Derivation of the Adjoint System}
\label{sect:adjderiv}
Variations of (\ref{eqn:lag}) with respect to $\ensuremath{\boldsymbol{\Lambda}}^R$ and $\ensuremath{\boldsymbol{\Lambda}}^I$ give the real and imaginary parts of the equality constraint (\ref{eqn:tdksDISC}). For the variation with respect to $\ensuremath{\boldsymbol{\Phi}}^R$, we obtain
\begin{align*}
&\delta \mathscr{L} = \frac{d}{d\epsilon} \biggr|_{\epsilon=0} \mathscr{L}(\ensuremath{\boldsymbol{\Phi}}^R + \epsilon \delta \ensuremath{\boldsymbol{\Phi}}^R, \ensuremath{\boldsymbol{\Phi}}^I, \ensuremath{\boldsymbol{\Lambda}}^R, \ensuremath{\boldsymbol{\Lambda}}^I, \ensuremath{\boldsymbol{v}}^{C}) \\
&= \sum_{k=0}^{K} \sum_{j=0}^{J} ( 2 \phi^R(x_j, t_k)^2 + 2 \phi^I(x_j, t_k)^2 - \tilde{n}(x_j, t_k) )( 4 \phi^R(x_j, t_k) \delta \phi^R(x_j,t_k) ) \\
&\quad - \sum_{k=0}^{K-1} [\ensuremath{\boldsymbol{\lambda}}_{k+1}^R]^T \delta \ensuremath{\boldsymbol{\phi}}_{k+1}^R \\
&\quad + \sum_{k=0}^{K-1} [\ensuremath{\boldsymbol{\lambda}}_{k+1}^R]^T \nabla_{\ensuremath{\boldsymbol{\phi}}^R} \ensuremath{\boldsymbol{F}}^R_{\Delta t}(\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{v}}^{C}_k) \delta \ensuremath{\boldsymbol{\phi}}^R_k + [\ensuremath{\boldsymbol{\lambda}}_{k+1}^I]^T \nabla_{\ensuremath{\boldsymbol{\phi}}^R} \ensuremath{\boldsymbol{F}}^I_{\Delta t}(\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{v}}^{C}_k) \delta \ensuremath{\boldsymbol{\phi}}^R_k \\
&= \sum_{k=0}^K 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_k|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_k) \circ \ensuremath{\boldsymbol{\phi}}_k^R \right]^T \delta \ensuremath{\boldsymbol{\phi}}^R_k - \sum_{k=1}^{K} [\ensuremath{\boldsymbol{\lambda}}_{k}^R]^T \delta \ensuremath{\boldsymbol{\phi}}_{k}^R + \sum_{k=0}^{K-1} \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+1}^I \end{bmatrix}^T \nabla_{\ensuremath{\boldsymbol{\phi}}^R} \begin{bmatrix} \ensuremath{\boldsymbol{F}}_{\Delta t}^R \\
\ensuremath{\boldsymbol{F}}_{\Delta t}^I \end{bmatrix}
\delta \ensuremath{\boldsymbol{\phi}}_{k}^R.
\end{align*}
Analogously, for the variation with respect to $\ensuremath{\boldsymbol{\Phi}}^I$, we obtain
\[
\delta \mathscr{L} = \sum_{k=0}^K 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_k|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_k) \circ \ensuremath{\boldsymbol{\phi}}_k^I \right]^T \delta \ensuremath{\boldsymbol{\phi}}^I_k - \sum_{k=1}^{K} [\ensuremath{\boldsymbol{\lambda}}_{k}^I]^T \delta \ensuremath{\boldsymbol{\phi}}_{k}^I + \sum_{k=0}^{K-1} \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+1}^I \end{bmatrix}^T \nabla_{\ensuremath{\boldsymbol{\phi}}^I} \begin{bmatrix} \ensuremath{\boldsymbol{F}}_{\Delta t}^R \\
\ensuremath{\boldsymbol{F}}_{\Delta t}^I \end{bmatrix}
\delta \ensuremath{\boldsymbol{\phi}}_{k}^I.
\]
Setting $\delta \mathscr{L} = 0$ for all variations $\delta \ensuremath{\boldsymbol{\phi}}_k^R$ and $\delta \ensuremath{\boldsymbol{\phi}}_k^I$ for $k \geq 1$, we obtain the following backward-in-time system for $\ensuremath{\boldsymbol{\Lambda}}$:
\begin{subequations}
\label{eqn:adjsysappendix}
\begin{align}
\label{eqn:adjfinappendix}
\ensuremath{\boldsymbol{\lambda}}_K &= 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_K|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_K) \circ \ensuremath{\boldsymbol{\phi}}_K \right] \\
\label{eqn:adjReqn}
[\ensuremath{\boldsymbol{\lambda}}_{k}^R]^T &= 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_k|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_k) \circ \ensuremath{\boldsymbol{\phi}}_k^R \right]^T + \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+1}^I \end{bmatrix}^T \nabla_{\ensuremath{\boldsymbol{\phi}}^R} \begin{bmatrix} \ensuremath{\boldsymbol{F}}_{\Delta t}^R \\
\ensuremath{\boldsymbol{F}}_{\Delta t}^I \end{bmatrix} (\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{v}}^{C}_k)\\
\label{eqn:adjIeqn}
[\ensuremath{\boldsymbol{\lambda}}_{k}^I]^T &= 4 \left[ (2 |\ensuremath{\boldsymbol{\phi}}_k|^2 - \tilde{\ensuremath{\boldsymbol{n}}}_k) \circ \ensuremath{\boldsymbol{\phi}}_k^I \right]^T + \begin{bmatrix} \ensuremath{\boldsymbol{\lambda}}_{k+1}^R \\ \ensuremath{\boldsymbol{\lambda}}_{k+1}^I \end{bmatrix}^T \nabla_{\ensuremath{\boldsymbol{\phi}}^I} \begin{bmatrix} \ensuremath{\boldsymbol{F}}_{\Delta t}^R \\
\ensuremath{\boldsymbol{F}}_{\Delta t}^I \end{bmatrix} (\ensuremath{\boldsymbol{\phi}}_k^R, \ensuremath{\boldsymbol{\phi}}_k^I, \ensuremath{\boldsymbol{v}}^{C}_k)
\end{align}
\end{subequations}
We can write the Jacobian as a block matrix:
\begin{equation}
\label{eqn:blockjacob}
\ensuremath{\boldsymbol{J}}_{\ensuremath{\boldsymbol{\phi}}} \ensuremath{\boldsymbol{F}}_{\Delta t} = \begin{bmatrix} \nabla_{\ensuremath{\boldsymbol{\phi}}^R} \ensuremath{\boldsymbol{F}}^R_{\Delta t} & \nabla_{\ensuremath{\boldsymbol{\phi}}^I} \ensuremath{\boldsymbol{F}}^R_{\Delta t} \\
\nabla_{\ensuremath{\boldsymbol{\phi}}^R} \ensuremath{\boldsymbol{F}}^I_{\Delta t} & \nabla_{\ensuremath{\boldsymbol{\phi}}^I} \ensuremath{\boldsymbol{F}}^I_{\Delta t}
\end{bmatrix}.
\end{equation}
\subsection{Gradients of the TDKS Propagator}
\label{sect:TDKSgrads}
Here we consider gradients of the propagator $\ensuremath{\boldsymbol{F}}_{\Delta t}$ defined in (\ref{eqn:discprop}). Note that $\ensuremath{\boldsymbol{F}}_{\Delta t}$ also satisfies
\begin{equation*}
\ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^{C}) = \ensuremath{\boldsymbol{F}}^R_{\Delta t}(\ensuremath{\boldsymbol{\phi}}^R, \ensuremath{\boldsymbol{\phi}}^I, \ensuremath{\boldsymbol{v}}^{C}) + i \ensuremath{\boldsymbol{F}}^I_{\Delta t}(\ensuremath{\boldsymbol{\phi}}^R, \ensuremath{\boldsymbol{\phi}}^I, \ensuremath{\boldsymbol{v}}^{C}).
\end{equation*}
\paragraph{Gradients of $\ensuremath{\boldsymbol{F}}$ when we seek pointwise values of $\ensuremath{\boldsymbol{v}}^C$.} Because $V$ is diagonal, the $\ell$-th element of $\ensuremath{\boldsymbol{F}}$ is
\[
\left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^C) \right]_{\ell} = \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{v}}^C) \Delta t) \mathcal{P}_{K;q, r} \phi_r.
\]
First let us compute the derivative of this $\ell$-th element with respect to $\phi_m^R$. We obtain
\begin{multline}
\label{eqn:dFdphiR}
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^C) \right]_{\ell} }{\partial \phi_m^R} = \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{v}}^C) \Delta t) (-i \Delta t) \frac{\partial V_{qq}}{\partial \phi_m^R} \mathcal{P}_{K;q, r} \phi_r \\
+ \sum_{q} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{v}}^C) \Delta t) \mathcal{P}_{K;q, m},
\end{multline}
with
\[
\frac{\partial V_{qq}}{\partial \phi_m^R} = \frac{\partial}{\partial \phi_m^R} \sum_{s} W_{q,s} w_s \phi_s \phi_s^\ast = 2 W_{q,m} w_m \phi_m^R.
\]
The derivative with respect to $\phi_m^I$ is similar:
\begin{multline}
\label{eqn:dFdphiI}
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^C) \right]_{\ell} }{\partial \phi_m^I} = \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{v}}^C) \Delta t) (-i \Delta t) \frac{\partial V_{qq}}{\partial \phi_m^I} \mathcal{P}_{K;q, r} \phi_r \\
+ \sum_{q} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{v}}^C) \Delta t) \mathcal{P}_{K;q, m} i,
\end{multline}
with
\[
\frac{\partial V_{qq}}{\partial \phi_m^I} = \frac{\partial}{\partial \phi_m^I} \sum_{s} W_{q,s} w_s \phi_s \phi_s^\ast = 2 W_{q,m} w_m \phi_m^I.
\]
Taking the real and imaginary parts of (\ref{eqn:dFdphiR}-\ref{eqn:dFdphiI}), we obtain all necessary elements of the block Jacobian (\ref{eqn:blockjacob}). Next we compute the derivative of the $\ell$-th element of $\ensuremath{\boldsymbol{F}}$ with respect to the $m$-th element of $\ensuremath{\boldsymbol{v}}^{C}$:
\begin{equation*}
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{v}}^C) \right]_{\ell} }{\partial v^C_m } = \sum_{r} \mathcal{P}_{K;\ell,m} \exp(-i V_{mm}(\ensuremath{\boldsymbol{\phi}},\ensuremath{\boldsymbol{v}}^C) \Delta t) (-i \Delta t) \mathcal{P}_{K;m, r} \phi_r,
\end{equation*}
which follows from
\[
\frac{\partial V_{qq}}{\partial v^C_m} = \delta_{qm} = \begin{cases} 1 & q = m \\ 0 & q \neq m. \end{cases}
\]
\paragraph{Gradients of $\ensuremath{\boldsymbol{F}}$ when we model the functional dependence of $\ensuremath{\boldsymbol{v}}^C$ on present and past states.} The $\ell$-th element of $\ensuremath{\boldsymbol{F}}$ is now
\[
\left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \right]_{\ell} = \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}; \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) \mathcal{P}_{K;q, r} \phi_r.
\]
Using the new expression for $V$, we derive
\begin{multline}
\label{eqn:newdFdphiR}
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \right]_{\ell} }{\partial \phi_m^R} = \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) (-i \Delta t) \frac{\partial V_{qq}}{\partial \phi_m^R} \mathcal{P}_{K;q, r} \phi_r \\
+ \sum_{q} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) \mathcal{P}_{K;q, m},
\end{multline}
with
\[
\frac{\partial V_{qq}}{\partial \phi_m^R} = 2 W_{q,m} w_m \phi_m^R + \frac{\partial v^C_q}{\partial \phi_m^R}.
\]
The derivative with respect to $\phi_m^I$ is similar:
\begin{multline}
\label{eqn:newdFdphiI}
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \right]_{\ell} }{\partial \phi_m^I} = \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) (-i \Delta t) \frac{\partial V_{qq}}{\partial \phi_m^I} \mathcal{P}_{K;q, r} \phi_r \\
+ \sum_{q} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) \mathcal{P}_{K;q, m} i,
\end{multline}
with
\[
\frac{\partial V_{qq}}{\partial \phi_m^I} = 2 W_{q,m} w_m \phi_m^I + \frac{\partial v^C_q}{\partial \phi_m^I}.
\]
The derivatives with respect to the past state $\ensuremath{\boldsymbol{\phi}}'$ are
\begin{subequations}
\label{eqn:newdFdphiprime}
\begin{align}
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \right]_{\ell} }{\partial \phi_m^{',R}} &= \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) (-i \Delta t) \frac{\partial v^C_{q}}{\partial \phi_m^{',R}} \mathcal{P}_{K;q, r} \phi_r \\
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \right]_{\ell} }{\partial \phi_m^{',I}} &= \sum_{q, r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) (-i \Delta t) \frac{\partial v^C_{q}}{\partial \phi_m^{',I}} \mathcal{P}_{K;q, r} \phi_r
\end{align}
\end{subequations}
Taking the real and imaginary parts of (\ref{eqn:newdFdphiR}-\ref{eqn:newdFdphiI}-\ref{eqn:newdFdphiprime}), we obtain all necessary elements of both block Jacobians in (\ref{eqn:newadjiter}).
Finally, we need
\begin{equation*}
\frac{\partial \left[ \ensuremath{\boldsymbol{F}}_{\Delta t}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \right]_{\ell} }{\partial \theta_m} = \sum_{q,r} \mathcal{P}_{K;\ell,q} \exp(-i V_{qq}(\ensuremath{\boldsymbol{\phi}}, \ensuremath{\boldsymbol{\phi}}'; \ensuremath{\boldsymbol{\theta}}) \Delta t) (-i \Delta t) \frac{\partial V_{qq}}{\partial \theta_m} \mathcal{P}_{K;q,r}\phi_r.
\end{equation*}
\subsection{Further Implementation Details}
We implemented the adjoint method in JAX. Derivatives of the $v^C$ model are computed via automatic differentiation. XLA compilation enables us to run the code on GPUs. To optimize via L-BFGS-B, we use scipy.optimize. All source code is available upon request.
\subsection{Training Set Results}
Here we consider training a model $v^C[\phi](x,t)$ that allows for arbitrary dependence on the real and imaginary parts of both $\phi(x,t)$ and $\phi(x,t-\Delta t)$. The input layer is of dimension $4(J+1)$---see \ref{sect:deets}.
To train such a model, we apply the L-BFGS-B optimizer with objective function given by the first line of (\ref{eqn:newlag}) and gradients computed with the adjoint system (\ref{eqn:newadj}-\ref{eqn:nablatheta}). We initialize neural network parameters $\ensuremath{\boldsymbol{\theta}}$ by sampling a mean-zero normal distribution with standard deviation $\sigma = 0.01$. For training data, we subsample the $p=-1.5$ TDSE electron density time series by a factor of $100$ in time, so that $\Delta t = 2.4 \times 10^{-3}$ fs and the entire training trajectory consists of $K=301$ time steps. We retain this time step in all training sets and results that follow.
In Figure \ref{fig:vCp15TRAIN}, we show the resulting model's results on the training set. The trained $v^C[\phi]$ functional, when used to solve the TDKS equation (\ref{eqn:tdks1d}), yields electron densities that agree closely with the reference TDSE electron densities.
\begin{figure}
\includegraphics[width=6in]{vCp15TRAINphi256results.pdf}
\caption{We use $300$ time steps (corresponding to $0.72$ fs) of the $p=-1.5$ data (black) together with the adjoint method to train a neural network model $v^C[\phi]$ that depends on the current and previous $\phi$. We plot in red the results of using the learned $v^C$ to propagate (\ref{eqn:tdks1d}) from $t=0$ to $t=0.72$ fs. Note the close agreement.}
\label{fig:vCp15TRAIN}
\end{figure}
\subsection{Relationship to Existing Literature on Optimal Control for TDKS Systems}
\label{sect:relationship}
Here we contrast our work with prior work on optimal control for TDKS systems, specifically work that involves the adjoint method.
First let us view our work through the lens of optimal control: we generate reference data by first solving the two-dimensional TDSE. Our cost function is then the mismatch between (i) electron densities computed from TDKS, and (ii) electron densities computed from the time-dependent wave functions obtained from TDSE, all on a discrete temporal grid. We view $v^C$ as a control that, properly chosen, guides TDKS to produce the same electron densities that would have been produced by solving TDSE.
A common feature of both present and prior work is the idea of incorporating the TDKS equation as a time-dependent constraint---see Eq. (8) in \cite{PhysRevLett.109.153603}, Eq. (43) in \cite{castro2013optimal}, and Eq. (3.4) in \cite{sprengel2018investigation}. Upon taking functional derivatives, this leads naturally to adjoint systems, which have been analyzed in detail for TDKS systems \citep{Borzi2012, sprengel2017theoretical}. In particular, \cite{sprengel2017theoretical} and \cite{sprengel2018investigation} develop and analyze optimal control problems for multidimensional TDKS systems. In all prior work we have seen, the correlation potential $v^C$ is taken as \emph{adiabatic} with fixed functional form throughout the solution of the optimal control problem.
In prior work, the control $u$ is distinct from $v^C$. In \cite{PhysRevLett.109.153603}, the control $u$ governs the Fourier spectrum of the amplitudes of an applied electric field. In \cite{sprengel2018investigation}, the control $u$ influences the system through potentials such as $V_u(x) = x^2$ and $V_u(x) = x \cdot p$, modeling the control of a quantum dot.
In \cite{PhysRevLett.109.153603}, the objective is to balance (i) maximization of charge transfer from one potential well to a neighboring potential well with (ii) minimization of the intensity of the applied field. The resulting cost function models both parts of this physical objective. In \cite{sprengel2018investigation}, the authors do include in their cost function the $L^2$ distance between the electron density computed from TDKS and a reference electron density, all in continuous time. They apply this to the problem of guiding TDKS towards a target trajectory that itself was computed by solving TDKS.
Viewed in this context, the distinguishing features of the present work are as follows: treating $v^C$ itself as the control, allowing $v^C$ to be \emph{non-adiabatic} and depend on both present and pass electron densities, and applying this method to match electron density trajectories computed from TDSE. By treating $v^C$ as the object of interest, and by guiding TDKS trajectories to match TDSE trajectories, the present work addresses the system identification problem of learning $v^C$ from data.
\newpage
|
2,869,038,154,818 | arxiv |
\section{Introduction}
Nucleus-nucleus collisions at ultrarelativistic energies probe the properties
of nuclear matter under extreme conditions \cite{QM99}. Of particular
interest is the question whether ordinary nuclear matter undergoes
a (phase) transition to quark-gluon matter, as predicted by lattice
calculations of quantum chromodynamics (QCD) \cite{lattice}.
A nuclear collision can be viewed as a sequence of
nucleon-nucleon collisions. At sufficiently high energies,
multi-particle production leads to the formation of a
region of high energy and particle number density.
With increasing beam energy, multi-particle production through
processes where the partons inside the nucleons directly
interact with each other,
becomes more and more important. The dominant partonic
particle-production mechanism is so-called
mini-jet production \cite{eskola}.
If the momentum transfer in these interactions is sufficiently large,
mini-jet production is reliably computable within
perturbative QCD (pQCD). It was estimated \cite{eskola2} that
about half of the total transverse energy created
in $Au+Au$ collisions at the Relativistic Heavy-Ion Collider
(RHIC) at BNL resides in mini-jets, while at CERN's
Large Hadron Collider (LHC) the transverse energy created is
almost exclusively due to mini-jets.
The transverse energy can be used to
estimate the energy densities created in the initial stage of the collision.
If matter is in thermodynamical equilibrium,
both for RHIC and LHC energies the corresponding
energy densities are found to be
large enough for nuclear matter to be in the
quark-gluon phase.
The remaining question is whether high-energy density matter formed
in ultrarelativistic nuclear collisions lives
sufficiently long enough to actually reach thermodynamical equilibrium, so
that it can be identified with the quark-gluon phase seen in
lattice QCD calculations.
Thermodynamical equilibrium means that matter is in thermal, mechanical,
and chemical equilibrium. In general, the question of thermodynamical
equilibration can only be decided with {\em microscopic\/} transport models
\cite{micromodels}. In this paper, we study a simpler problem which
allows us to use a {\em macroscopic\/} transport model.
We assume that matter reaches thermal and mechanical equilibrium
after a proper time $\tau_0 \sim 0.1 - 0.25$ fm/c.
This is justified given the fact that the rate for elastic
collisions between quarks and gluons, which establish thermal and
mechanical equilibrium, is much larger than for inelastic
collisions which establish chemical equilibrium.
Thermal and mechanical equilibrium will be referred to as
{\em kinetic equilibrium\/} in the following.
Under these assumptions, and given initial values for
the energy density and the quark, antiquark and gluon number densities,
we can then employ {\em ideal
fluid dynamics\/} to study the subsequent evolution of the kinetically
equilibrated quark-gluon phase, coupled to {\em rate equations\/}
which determine the chemical composition of the system away
from chemical equilibrium.
This problem has been previously studied by Bir\'{o} {\it et al.} \cite{Biro},
and by Srivastava {\it et al.} \cite{MMS,woundedmms}.
As in these previous studies, we assume that
chemical equilibration is driven mainly by the two-body reaction
$gg \leftrightarrow q{\bar q}$, and by gluon multiplication as well as
fusion, $gg \leftrightarrow ggg$. We also
terminate the evolution when the plasma reaches the
hadronization energy density,
$\epsilon_h \equiv 1.45$ GeV/fm$^{3}$. For a completely
equilibrated plasma, this corresponds to a hadronization temperature of
$T_h \sim 0.17$ GeV. We do not study the evolution in the mixed and
purely hadronic phases.
Our motivation to reinvestigate this subject is the following.
First of all, the question whether the quark-gluon
phase created in ultrarelativistic heavy-ion collisions is chemically
equilibrated is highly relevant for the experiments
commencing at the RHIC collider in the fall of this year.
Since \cite{Biro,MMS,woundedmms} were published, newer results on
mini-jet production in nuclear collisions at RHIC and LHC energies
have been obtained \cite{eskola2}. We therefore decided to
compare the evolution computed with initial conditions obtained from
the so-called ``self-screened parton cascade'' (SSPC) model
(used in \cite{MMS,woundedmms}) with that employing the more recent pQCD
estimates of \cite{eskola2}.
Second, the authors of \cite{Biro,MMS,woundedmms} simplified
the rate equations using an approximate, so-called ``factorized''
phase-space distribution function for the quarks and gluons (see below).
In our treatment we use the full distribution function in the rate equations,
and so are in a position to assess the validity of the factorization
assumption.
Third, baryon stopping is non-negligible in nuclear collisions at
SPS energies \cite{Stop0}, and there is mounting evidence
from theoretical studies that, even at RHIC energies,
the midrapidity region is not completely net-baryon free
\cite{RossiVeneziano}. Our analysis therefore also accounts for non-zero
net-baryon number.
Finally, the numerical algorithms used here are different from
the ones employed before \cite{MMS,woundedmms}. Thus, they
constitute an independent check on the validity of the conclusions
reached previously.
This paper is organized as follows. In section 2 we discuss the
macroscopic transport equations. Starting
from a single-particle phase-space distribution in kinetic
equilibrium, we show that these equations are given
by ideal fluid dynamics, coupled to rate equations which determine
the densities of the individual particle species. We also prove that
the entropy never decreases during the time evolution of the system.
In section 3 we discuss the initial conditions for the transport
equations. Section 4 is devoted to purely longitudinal boost-invariant
expansion. We show that the difference in the equilibration process
is small when using the full phase-space distribution function instead
of the factorized distribution function of \cite{Biro,MMS,woundedmms}.
We compare results obtained with the initial conditions of the SSPC model
\cite{MMS,woundedmms} with those when using pQCD estimates \cite{eskola2}.
We also study the sensitivity of the equilibration process on the
initial time, $\tau_0$, and the strong coupling constant, $\alpha_s$.
In section 5 we include (cylindrically symmetric) transverse expansion
as well, and present results for the SSPC model. We conclude in section
6 with a summary of our results and an outlook.
Our units are $\hbar = c = k_B =1$, and the metric tensor is
$g^{\mu \nu} = {\rm diag}\, (+,-,-,-)$.
\section{Macroscopic transport equations}
In this section, we discuss the macroscopic transport equations on which
our results are based.
We assume that elastic collisions between quarks (antiquarks)
and gluons are sufficiently frequent to establish
kinetic equilibrium. Ideal fluid dynamics can then be invoked to
follow the evolution of the energy and momentum densities in the
system, while rate equations for the inelastic reactions
$gg \leftrightarrow ggg$ and $gg \leftrightarrow q \bar{q}$
determine the number density of quarks, antiquarks, and gluons.
\subsection{Phase-space distribution}
Once kinetic equilibrium is achieved, all particles in an (infinitesimal)
volume element at space-time point $x\equiv x^\mu = (t, {\bf x})$
have the same temperature, $T(x)$,
and move with a common average 4-velocity, $u(x) \equiv u^\mu(x)$.
In that case, the single-particle phase-space
distribution for particles of species $i$
assumes the same form as in (local) thermodynamical
equilibrium, except that the fugacity (or the chemical potential)
is not equal to its equilibrium value:
\begin{equation} \label{1}
f_i(x, k \cdot u) = \lambda_i(x)\, \frac{d_i}{(2 \pi)^3} \,
\frac{1}{\exp \left[ k \cdot u(x)/T(x) \right] + \lambda_i(x)
\, \theta_i } \,\, ,
\end{equation}
where $k \equiv k^\mu = (E_i,{\bf k})$ is the 4-momentum; $E_i =
\sqrt{{\bf k}^2 + m_i^2}$ is the on-shell energy of particles of species
$i$ with 3-momentum ${\bf k}$ and rest mass $m_i$. The fugacity
$\lambda_i(x) = \exp[\mu_i(x)/T(x)]$ controls the number density of
particle species $i$, $\mu_i$ is the chemical potential.
$\theta_i = \pm 1$ for
fermions or bosons, respectively. $d_i$ denotes the number of
internal degrees of freedom for particles of species $i$ (like spin,
isospin, color, {\em etc.}).
In \cite{Biro,MMS,woundedmms}, instead of (\ref{1}) a so-called
{\em factorized\/} distribution function was used, in which the
fugacity $\lambda_i$ in the denominator is approximated by 1,
\begin{equation} \label{fac}
f_i^{\rm fac}(x,k \cdot u) \equiv \lambda_i(x) \, \frac{d_i}{(2 \pi)^3} \,
\frac{1}{\exp \left[ k \cdot u(x)/T(x) \right] + \theta_i } \,\, .
\end{equation}
We will comment on the validity of this approximation below.
\subsection{Energy-momentum conservation}
With a distribution function of the type (\ref{1}),
the energy-momentum tensor assumes the so-called {\em
ideal-fluid\/} form
\begin{equation} \label{tmunu}
T^{\mu \nu}(x) \equiv \sum_i \int \frac{{\rm d}^3{\bf k}}{E_i} \,
k^\mu \, k^\nu \, f_i(x,k \cdot u) = [\epsilon(x) + p(x)]\, u^\mu(x)\,
u^\nu(x) - p(x)\, g^{\mu \nu}\,\, ,
\end{equation}
where
\begin{equation}
\epsilon(x) \equiv \sum_i \int {\rm d}^3{\bf k}
\, E_i\, f_i(x,E_i) \,\,\, , \,\,\,\,
p(x) \equiv \sum_i \int {\rm d}^3{\bf k}
\, \frac{{\bf k}^2}{3 E_i}\, f_i(x,E_i)
\end{equation}
are the energy density and pressure in the local
rest frame of a fluid element moving with 4-velocity $u^\mu$.
While it is clear that thermal equilibrium requires all particle
species to have the same temperature, it is less obvious
why the phase-space distribution (\ref{1}) ensures mechanical
equilibrium as well. To see this, consider the tensor
decomposition $T^{\mu \nu}$ as given by (\ref{tmunu}). In the rest
frame of the fluid element, the pressure is completely isotropic,
$p\equiv T^{ii}/3 = T^{xx} = T^{yy} = T^{zz}$, which is synonymous
to mechanical equilibrium. If
(\ref{1}) depends on more than {\em one\/} 4-vector $u$,
additional tensors would appear on the right-hand side of (\ref{tmunu}),
such as $u^\mu_k u^\nu_l$, and the diagonal components of the
stress tensor $T^{ij}$ would no longer be identical.
Hence, the common 4-velocity $u$ in (\ref{1}) ensures
mechanical equilibrium.
In our case, matter consists of massless gluons, quarks, and
antiquarks, $i= g,\, q,\, \bar{q}$, with energy density
\begin{eqnarray} \label{e}
\epsilon & = & \epsilon_g + \epsilon_q + \epsilon_{\bar{q}} \,\, ,\\
\epsilon_g & = & \lambda_g\, T^4\, \frac{d_g }{2\pi^2}
\int_0^\infty {\rm d} z \, \frac{z^3}{e^z - \lambda_g} \,\, , \label{e1} \\
\epsilon_q & = & \lambda_q\, T^4\, \frac{d_q}{2\pi^2}
\int_0^\infty {\rm d} z \, \frac{z^3}{e^z + \lambda_q} \,\, , \\
\epsilon_{\bar{q}} & = & \lambda_{\bar q}\, T^4\, \frac{d_q}{2\pi^2}
\int_0^\infty {\rm d} z \, \frac{z^3}{e^z + \lambda_{\bar{q}}} \,\, ,
\label{e3}
\end{eqnarray}
where $d_g \equiv 2 (N_c^2-1)$ is the number of internal degrees
of freedom for gluons, $N_c = 3 $ is the number of colors, and
$d_q \equiv 2 N_c N_f$, is the number of internal degrees of freedom
for quarks and antiquarks, with $N_f$ being the number of
massless flavors.
Throughout our analysis, we use $N_f = 2.5$, mimicking the effect
of the nonzero mass of the strange quark by taking $d_s = 0.5\, d_q$.
This approximation interpolates between the region of high temperature,
$T \gg m_s \simeq 150$ MeV, where $N_f \simeq 3$, and the region of
low temperature, $m_{u,d} \ll T \ll m_s$, where $N_f \simeq 2$.
Since we follow the time evolution of the temperature from very
high $T$ down to $T \simeq T_c \sim m_s$, taking $N_f = 2.5$ is not
a particularly good approximation. We nevertheless make this choice
to be able to compare our results to earlier work \cite{Biro,MMS,woundedmms}.
Also, the chemical reaction rates, cf.\ eqs.\ (\ref{Rs}) below, which
appear on the right-hand side of the rate equations (\ref{r1}) -- (\ref{r3}),
are more complicated in the case of massive particles \cite{pkoch}.
The extension of our present study to nonzero strange quark mass is, however,
important and will be pursued in a subsequent publication \cite{dmedhr2}.
For massless bosons, the equilibrium value of the fugacity,
$\lambda_i^{\rm eq}$, is always equal to 1. The reason is the
following. For bosons,
the equilibrium value of the chemical potential, $\mu_i^{\rm eq}$,
has to be smaller than the rest mass,
$\mu_i^{\rm eq} \leq m_i$. On the other hand, bosons which carry a
conserved charge, $\mu_i^{\rm eq} \neq 0$, always come in pairs with
their own antiparticles, {\it e.g.}, $\pi^+$ and $\pi^-$, $K$ and
$\bar{K}$, {\it etc.}. Without loss of generality, we can therefore choose
$\mu_i^{\rm eq} \geq 0$ (with the chemical potential of the
associated antiparticle $\mu_{\bar{i}}^{\rm eq}= - \mu_i^{\rm eq} \leq 0$).
Then, for massless bosons, $\mu_i^{\rm eq} \equiv 0$, such that
$\lambda_i^{\rm eq} = 1$.
For massless fermions, there is no such constraint, although
$\lambda_q^{\rm eq} \equiv 1/ \lambda_{\bar{q}}^{\rm eq}$ in all cases
and $\lambda_q^{\rm eq} \equiv \lambda_{\bar{q}}^{\rm eq} = 1$
for vanishing net-baryon density.
The pressure is related to the energy density through
\begin{equation} \label{pe3}
p \equiv \frac{\epsilon}{3}\,\, ,
\end{equation}
the well-known equation of state for an ultrarelativistic ideal gas.
Note that, in order to derive this relation, we {\em only\/} required
the system to be in {\em kinetic\/} equilibrium, {\it i.e.},
eq.\ (\ref{pe3}) is valid even when the system is {\em not\/}
in chemical equilibrium!
With the factorized distribution function (\ref{fac}),
the integrals in (\ref{e1}) -- (\ref{e3})
can be performed analytically, as they
no longer depend on the fugacities.
The energy densities simplify to
\begin{equation} \label{efac}
\epsilon^{\rm fac}_g = a_2\, \lambda_g \, T^4 \;\;\;\; , \;\;\;\;\;
\epsilon^{\rm fac}_q = b_2\, \lambda_q\, T^4 \;\;\;\; , \;\;\;\;\;
\epsilon^{\rm fac}_{\bar{q}} = b_2\, \lambda_{\bar{q}}\, T^4
\,\, ,
\end{equation}
where $a_2 \equiv 8\pi^2/15$ and $b_2 \equiv 7\pi^2 N_f/40$.
The equation of state (\ref{pe3}) remains valid.
\begin{figure}
\vspace{7cm}
\caption{The ratio $\epsilon^{\rm fac}_i/\epsilon_i$ for
gluons (solid line) and quarks (dashed line) as a function of
the fugacity $\lambda_i$.}
\special{psfile=fig1.ps voffset=-20 hoffset=80 hscale=48 vscale=40}
\label{fig1}
\end{figure}
In Fig.\ \ref{1} we show the ratio $\epsilon^{\rm fac}_i/\epsilon_i$
for quarks and gluons as a function of the fugacity $\lambda_i$.
This ratio does not depend on the temperature.
For bosons, the factorized expression tends to
overestimate the correct result, for fermions, it underestimates it.
In the interval $0 \leq \lambda_i \leq 1$,
the error is maximized at $\lambda_i = 0$, and decreases monotonically
as $\lambda_i \rightarrow 1$. At $\lambda_i = 0$, it is
$8 \% $ for bosons, and $5 \% $ for fermions.
Energy and momentum is locally conserved,
\begin{equation} \label{2}
\partial_\mu T^{\mu \nu}(x) = 0 \,\, .
\end{equation}
With (\ref{tmunu}),
these are the equations of {\em ideal fluid dynamics}, which can
be used to compute the evolution of the energy density and the fluid
4-velocity of the system. The fluid-dynamical equations (\ref{2}) are
closed by specifying the equation of state of matter under
consideration, {\it i.e.}, the pressure as a function of energy density,
$p= p(\epsilon)$. In our case, this equation of state is simple,
see eq.\ (\ref{pe3}).
\subsection{Rate equations}
The 4-current of the number of particles of species $i$ is
\begin{equation} \label{nmu}
N_i^\mu(x) \equiv \int \frac{{\rm d}^3{\bf k}}{ E_i} \,
k^\mu \, f_i(x,k \cdot u) = n_i(x)\, u^\mu(x) \,\, ,
\end{equation}
where
\begin{equation}
n_i(x) \equiv \int {\rm d}^3{\bf k}\, f_i(x,E_i)
\end{equation}
is the number density of particle species $i$
in the rest frame of a fluid element.
This density is controlled by the value of the fugacity, $\lambda_i$.
In general, the fugacities for
the different particle species are independent
thermodynamic variables, and one
has to specify additional equations of motion which determine
the number densities. These are the {\em rate equations}.
To close the coupled set of fluid-dynamical equations and rate equations,
one has to be specify an
equation of state which, in general, also depends on the number densities of
the various particle species, $p= p(\epsilon,n_1,n_2, \ldots)$.
The reason why there is actually no such dependence for the system considered
here, eq.\ (\ref{pe3}), is that the quarks and gluons are
considered to be massless. This has the important consequence that,
in our case,
chemical non-equilibrium does {\em not\/} affect the evolution of the
energy and momentum densities of the fluid!
This certainly changes in the case of a nonzero strange quark mass,
although, for large temperatures $T \gg m_s$, the dependence of the
pressure $p$ on $n_s$ is relatively weak.
The number densities of gluons, quarks, and antiquarks
in the rest frame of a fluid element are
\begin{eqnarray} \label{ng}
n_g & = & \lambda_g \, T^3\, \frac{d_g}{2\pi^2} \int_0^{\infty} {\rm d} z\,
\frac{z^2}{e^z - \lambda_g} \,\, , \\
n_q & = & \lambda_q \, T^3 \, \frac{d_q}{2\pi^2} \int_0^{\infty}
{\rm d} z\, \frac{z^2}{e^z + \lambda_q} \,\, , \label{nq} \\
n_{\bar q} & = & \lambda_{\bar{q}}\, T^3\,
\frac{d_q}{2\pi^2} \int_0^{\infty}
{\rm d} z\, \frac{z^2}{e^z + \lambda_{\bar q}} \,\, . \label{nbarq}
\end{eqnarray}
With the factorized distribution function (\ref{fac}) these expressions
simplify to
\begin{equation} \label{nfac}
n_g^{\rm fac} = a_1 \, \lambda_g \, T^3 \,\,\, , \,\,\,\,
n_q^{\rm fac} = b_1\, \lambda_q \,T^3 \,\,\, , \,\,\,\,
n_{\bar{q}}^{\rm fac} = b_1 \, \lambda_{\bar{q}}\, T^3 \,\, ,
\end{equation}
where $a_1 \equiv 16\, \zeta(3)/\pi^2$ and $b_1 \equiv 9\, \zeta(3)\, N_f/
(2\pi^2)$.
\begin{figure}
\vspace{7cm}
\caption{The ratio $n^{\rm fac}_i/n_i$ for
gluons (solid line) and quarks (dashed line) as a function of
the fugacity $\lambda_i$.}
\special{psfile=fig2.ps voffset=-20 hoffset=80 hscale=48 vscale=40}
\label{fig2}
\end{figure}
In Fig.\ \ref{fig2}, we show the ratio $n_i^{\rm fac}/n_i$ as a function of
fugacity. As for the ratio of energy densities, Fig.\ \ref{fig1},
this ratio does not depend on the temperature, and
the factorized expression overestimates the correct result for bosons,
while it underestimates it for fermions. Again, the error decreases as
a function of $\lambda_i$, but it is about twice as large as for
the energy densities. The maximum error (at $\lambda_i =0$) is about
$20 \%$ for gluons and $10 \%$ for quarks.
In the absence of chemical equilibrium, the number of particles
of species $i$ is determined from the {\em rate equation}
\begin{equation} \label{5}
\partial_\mu N_i^\mu(x) = R_i(x) \,\, .
\end{equation}
In chemical equilibrium, $R_i$ vanishes.
In our case of a quark-gluon system, we
assume that the $R_i$ are determined
by the reactions $gg \leftrightarrow ggg$ and
$gg \leftrightarrow q{\bar q}$.
The rate equations (\ref{5}) can then be written in the form \cite{Biro}
\begin{eqnarray} \label{r1}
{\partial }_{\mu}( n_g u^{\mu} ) & = & {\rm R}_3 \, n_g
\left( 1 - \frac{n_g}{\tilde{n}_g} \right) -
2 \, {\rm R}_2\, n_g \left( 1 - \frac{n_q \,n_{\bar{q}} \, \tilde{n}_g^2}{
\tilde{n}_q \, \tilde{n}_{\bar{q}}\, n_g^2} \right) \,\,, \\
{\partial }_{\mu}( n_q u^{\mu} )& = &
{\rm R}_2 \, n_g
\left( 1 - \frac{n_q \, n_{\bar{q}} \, \tilde{n}_g^2 }{\tilde{n}_q\,
\tilde{n}_{\bar{q}} \, n_g^2} \right) \,\, , \\
{\partial }_{\mu}( n_{\bar q} u^{\mu} )& = &
{\rm R}_2\, n_g \left(1 - \frac{n_q\, n_{\bar{q}}\, \tilde{n}_g^2 }
{\tilde{n}_q\, \tilde{n}_{\bar{q}}\, n_g^2} \right) \,\,, \label{r3}
\end{eqnarray}
where the $\tilde{n}_i$ are the number densities computed at the
same temperature $T$, but for $\lambda_i = 1$.
The terms ${\rm R}_2$ and ${\rm R}_3$ are given by \cite{Biro}
\begin{equation} \label{Rs}
{\rm R}_2 \simeq 0.24\, N_f\,\alpha_s^2\, \lambda_g \,
T \, \ln \left( 1.65/\alpha_s\, \lambda_g \right) \,\,,
\,\,\,\, {\rm R}_3 \simeq 2.1 \,\alpha_s^2 \, T \, \sqrt{2\lambda_g -
\lambda_g^2} \ \ ,
\end{equation}
with $\alpha_s$ being the strong coupling constant.
We take $\alpha_s = 0.3$ throughout this analysis,
unless otherwise stated. Note that, while the densities on the
left-hand side of eqs.\ (\ref{r1}) -- (\ref{r3}) in general contain the
full phase-space distribution function (\ref{1}), the right-hand sides
of these equations and the terms
(\ref{Rs}) are derived assuming the factorized distribution (\ref{fac})
\cite{Biro}. This is in principle inconsistent. At least
close to equilibrium, however, the right-hand sides of the rate
equations (\ref{r1}) -- (\ref{r3}) are small, and we are allowed to
neglect this inconsistency.
On the other hand, far away from equilibrium we expect
other contributions to the chemical reaction rates to be more
important than the difference between factorized and full
distribution functions. Such contributions are, for
instance, multi-gluon processes $gg \rightarrow n\, g$ \cite{shuryakxiong}.
In this work, we only include the lowest-order rates (\ref{Rs}).
In chemical equilibrium, the right-hand sides of the rate equations
(\ref{r1}) -- (\ref{r3}) vanish by definition, and we obtain
\begin{equation}
\partial_\mu ( n_i^{\rm eq} u^\mu ) = 0 \,\,, \,\,\,\, i = g, q, \bar{q}
\,\, . \label{ra}
\end{equation}
We can now derive a relation for the time evolution
of the ratio of the parton number density $n_i$ to its corresponding
equilibrium value $n_i^{\rm eq}$.
By writing $n_i = n_i^{\rm eq} \, n_i/n_i^{\rm eq}$ on the left-hand
sides of the rate equations (\ref{r1}) -- (\ref{r3}), we
derive with eq.\ (\ref{ra}) the condition
\begin{equation} \label{comoving}
u^\mu \partial_\mu \left( \frac{n_i}{n_i^{\rm eq}} \right) =
\frac{R_i}{n_i^{\rm eq}} \,\, .
\end{equation}
This means that, for $R_i/n_i^{\rm eq} > 0$, the comoving time derivative
of the ratio of density to equilibrium density is positive,
and thus this ratio is bound to grow with increasing proper time in each fluid
element. Vice versa, this ratio decreases with time if
$R_i /n_i^{\rm eq} < 0$.
In other words, in the rest frame of a fluid element, $n_i/n_i^{\rm eq}$
cannot decrease (increase) if $R_i/n_i^{\rm eq} > 0 \, (< 0)$.
\subsection{Entropy}
Local thermodynamical equilibrium implies the conservation of entropy.
If the system is not chemically equilibrated, the entropy is bound
to grow, until chemical equilibrium is reached \cite{MMS,EntropyProd}.
To see this, we make the following observation:
the form of the distribution function (\ref{1}) or (\ref{fac})
is exactly that of a distribution function in
local thermodynamical equilibrium, except that
the fugacities, $\lambda_i$, assume
values which are {\em not\/} the same as
in chemical equilibrium, $\lambda_i^{\rm eq}$.
Note that in this case there are {\em no\/}
further dissipative terms \cite{thankstoJR} in the tensor
decomposition of (\ref{tmunu}) and (\ref{nmu}) \cite{DRproc}.
Nevertheless, the system is {\em not\/} in thermodynamical equilibrium, and
the actual values of $\epsilon, \, p$, and $n_i$ will differ
from the equilibrium values,
$\epsilon^{\rm eq},\, p^{\rm eq}$, and $n_i^{\rm eq}$.
Since the distribution function has the same form as
in thermodynamical equilibrium, we can
imagine each fluid element to actually {\em be\/} in chemical equilibrium
with a (local) particle bath which determines the value of the
fugacities instead of the rate equations (\ref{5}).
In other words, we can imagine local changes of particle number
to be induced by a change of the parameters of the particle bath
instead by chemical reactions within the fluid element itself.
As long as these changes happen faster than those of the
macroscopic fluid variables, one can view the fluid element to
remain in (local) thermodynamic equilibrium, and
apply thermodynamical relationships.
The assumption that microscopic reaction rates are much larger
than the rate of change of macroscopic fluid variables
is certainly an over-idealization in view of
the characteristic time scales in relativistic heavy-ion collisions.
So is, in fact, the ideal fluid limit which assumes
complete (local) thermodynamical equilibrium. Nevertheless, our
point of view is that ideal fluid dynamics still offers valuable information
about the collective behavior of the system. Our approach
is the simplest possible extension of ideal fluid dynamics.
A better approximation would be to solve the
equations for dissipative relativistic fluid dynamics in the
presence of chemical reactions \cite{raju}. In general, chemical
non-equilibrium gives rise to additional diffusion terms
which do not appear in our treatment. So far, the
complexity of these equations has discouraged attempts to apply them
to the description of nuclear collision dynamics.
Accepting these {\it caveats}, we now proceed by contracting (\ref{2})
with $u_\nu$,
\begin{equation}
0= u_\nu \partial_\mu T^{\mu \nu} = u \cdot \partial \epsilon +
(\epsilon + p) \partial \cdot u \,\, ,
\end{equation}
and use the first law of thermodynamics,
\begin{equation}
{\rm d} \epsilon = T\, {\rm d} s + \sum_i \mu_i\, {\rm d}n_i\,\, ,
\end{equation}
($s$ is the entropy density),
as well as the fundamental relation of thermodynamics,
\begin{equation}
\epsilon + p = T \, s + \sum_i \mu_i\, n_i\,\,,
\end{equation}
to derive
\begin{equation}
0 = T \, (u \cdot \partial s + s \, \partial \cdot u)
+ \sum_i \mu_i (u \cdot \partial n_i + n_i \, \partial \cdot u)\,\, .
\end{equation}
Equations (\ref{5}) and (\ref{nmu}), together with the definition of
$\lambda_i$, can be used to conclude that
\begin{equation} \label{noneqS}
\partial_\mu S^\mu = - \sum_i \ln \lambda_i \, R_i\,\, ,
\end{equation}
where $S^\mu = s\, u^\mu$.
In chemical equilibrium, $R_i$ vanishes, and entropy is conserved.
Consider now the case of gluons, or quarks when the net-baryon number is
zero. In this case, the equilibrium value of the
fugacity $\lambda_i^{\rm eq} = 1$, {\it i.e.},
$\mu_i^{\rm eq} = 0$.
Now assume that the {\em actual\/}
particle number density $n_i$ is {\em smaller\/} than
the equilibrium value, corresponding to $\lambda_i < \lambda_i^{\rm eq} =1$.
The rate equations drive $n_i$ towards equilibrium,
{\it i.e.}, the right-hand side has to be positive, $R_i >0$, in order
to produce particles of species $i$.
Since $\ln \lambda_i <0$, the change in entropy is positive.
On the other hand, if $n_i$ is larger than in equilibrium, $\lambda_i >
\lambda_i^{\rm eq} =1$,
the rate equations reduce the number of particles of species
$i$, {\it i.e.},
the right-hand side is negative, $R_i<0$. Again, entropy increases,
since $\ln \lambda_i > 0$.
Now suppose that $\lambda_i^{\rm eq} >1$, for instance for quarks
when the net-baryon number is positive. Naively, one would think
that the previous argument fails in this case, and one
might worry whether this could lead to a situation where
entropy actually {\em decreases\/} towards its equilibrium value,
which contradicts the second law of thermodyamics. This is, however,
not the case. In a relativistic system, one can create quarks
only in pairs with antiquarks. In order to conserve the net-baryon
number, $\partial_\mu (N_q^\mu - N_{\bar{q}}^\mu) \equiv 0$,
the right-hand side of the rate equation for the quark
number density, $R_q$, {\em must\/} be equal to the right-hand
side of the rate equation for the antiquark number density, $R_{\bar{q}}$,
see eqs.\ (\ref{r2a}) -- (\ref{r3a}) below.
In eq.\ (\ref{noneqS}), these two terms in the sum over $i$ can then
be combined to $- (\mu_q + \mu_{\bar{q}}) R_q/T$.
In equilibrium, $\mu_q^{\rm eq} = - \mu_{\bar{q}}^{\rm eq}$, such
that $\mu_q + \mu_{\bar{q}} = \delta \mu_q + \delta \mu_{\bar{q}}$,
where $\delta \mu_i \equiv \mu_i - \mu_i^{\rm eq}$. If the
quark (and consequently, the antiquark) number density is {\em smaller\/}
than its equilibrium value, $\delta \mu_q+ \delta \mu_{\bar{q}}< 0$, the
right-hand side of the rate equation is positive, $R_q > 0$. Consequently,
the sum of the quark and antiquark term on the right-hand side of
(\ref{noneqS}) is positive, leading again to an increase in entropy.
The argument is similar in the case when the quark number density is
larger than in equilibrium, or when $\lambda_i^{\rm eq} < 1$. In all cases, the
entropy increases.
\section{Initial conditions}
In order to solve the macroscopic transport equations discussed in
the last section, one has to specify the initial conditions.
In this paper, we use results from two different approaches,
both for RHIC and LHC energies.
The first approach is the SSPC model employed in \cite{MMS,woundedmms}.
We decided to use these initial conditions for two reasons. First,
the values for initial energy and parton densities obtained in this
approach constitute an upper bound
of what is expected to be created in ultrarelativistic nuclear collisions
at RHIC energies. Second, it allows us to directly compare the
results of \cite{MMS,woundedmms} with ours and point out
possible discrepancies.
The second approach is that of \cite{eskola2}. This approach is
purely based on pQCD, and thus does not contain the contribution from
the soft background, which constitutes about half of the total
transverse energy at RHIC energies.
In this sense, at RHIC this second approach
constitutes a lower bound for the initial energy and parton number densities.
Table \ref{table1} contains the initial values for energy and parton number
density for each approach and collision energy.
The SSPC values are taken from \cite{MMS},
while the values shown for the pQCD approach
\cite{eskola2} are computed
from the ${\rm d}N/{\rm d}\eta$ and ${\rm d}E_T/{\rm d}\eta$
values near midrapidity ($-0.5\leq \eta \leq 0.5$)
using a volume element $\Delta V = \pi R^2 \tau_0 \Delta \eta$ \cite{Bjorken},
such that
\begin{equation} \label{dV}
\frac{{\rm d}E_T}{{\rm d}\eta} \equiv \epsilon_0 \,\frac{{\rm d}V}{{\rm d}\eta}
= \epsilon_0 \tau_0 \pi R^2 \;\;\;\; , \;\;\;\;\;\;
\frac{{\rm d}N_i}{{\rm d}\eta} \equiv n_i^0 \, \frac{{\rm d}V}{{\rm d}\eta}
= n_i^0 \tau_0 \pi R^2 \;\;\;\; , \;\;\;\;\; i = g,\, q,\,\bar{q} \,\, ,
\end{equation}
where $R=1.12\, A^{1/3}$ fm.
From the four values $\epsilon_0,\, n_g^0,\, n_q^0$, and $n_{\bar{q}}^0$,
using eqs.\ (\ref{e}) and (\ref{ng}) -- (\ref{nbarq})
one can unambiguously extract $T_0, \, \lambda_g^0,\, \lambda_q^0$, and
$\lambda_{\bar{q}}^0$, and, from the latter, the values
$n_i^0/\tilde{n}_i^0$ given in the last three columns of Table \ref{table1}.
The initial time $\tau_0$ in the SSPC and pQCD approaches is chosen
as $\tau_0 \equiv 1/p_0$, where $p_0$ is the infrared momentum cutoff
required to regularize the mini-jet production cross sections.
The values for $\tau_0$ given in Table \ref{table1} arise from the cutoffs
$p_0 \sim 0.8$ GeV for the SSPC approach and $p_0 \sim 2$ GeV for
the pQCD approach. The transverse momentum $p_T \geq p_0$ of a
produced mini-jet
determines the time scale for the respective parton to come on mass-shell.
Only {\em after\/} the partons are on their respective mass-shell,
elastic scattering processes between them will drive the
system towards kinetic equilibrium. How the system approaches
kinetic equilibrium has to be studied in the framework of
kinetic theory; our approach does not make any statements about
this pre-equilibrium stage of a heavy-ion collision. In the following,
we take $\tau_0= 1/p_0$, which is certainly the
{\em earliest\/} possible time for the system to reach kinetic equilibrium.
In the next section, we shall also use larger (and thus more conservative)
values for $\tau_0$, in order to test the sensitivity of our results
on this important parameter.
For RHIC initial conditions, the energy density in pQCD
is much smaller than for the SSPC model, despite the smaller initial
kinetic equilibrium time $\tau_0$. The reason is the absence of the
soft background contribution in the former approach. For LHC, pQCD gives larger
values for the initial energy density than the SSPC model. If
scaled to the same initial proper time, however, the initial energy densities
are approximately equal. This reflects the fact that the soft
background contribution is rather small at LHC energies.
Nevertheless, despite the similarity of the initial energy densities, in pQCD
the gluons are close to chemical equilibrium at
LHC energies, while this is not the case for the SSPC approach.
Note that there is a small initial net-baryon number density
$(n_q^0 - n_{\bar{q}}^0)/3$ in the pQCD approach, as opposed to
the SSPC model where vanishing net-baryon number is assumed.
\begin{table}
\begin{center}
\caption {Initial conditions for the fluid-dynamical expansion phase in
central $Au+Au$ collisions at BNL RHIC and $Pb+Pb$ collisions at
CERN LHC energies from the SSPC model and pQCD.}
\bigskip\bigskip
\begin{tabular}{|c|c|c|c|c|c|c|c|} \hline
\multicolumn{1}{|c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } \\
\multicolumn{1}{|c|}{ Approach } &
\multicolumn{1}{c|}{ Energy } &
\multicolumn{1}{c|}{ ${\tau}_0$ } &
\multicolumn{1}{c|}{ ${\epsilon}_0$ } &
\multicolumn{1}{c|}{ $T_0$ } &
\multicolumn{1}{c|}{ $n_g^0/\tilde{n}_g^0$ } &
\multicolumn{1}{c|}{ $n_q^0/\tilde{n}_q^0$ } &
\multicolumn{1}{c|}{ $n_{\bar q}^0/\tilde{n}_{\bar q}^0$ } \\
\multicolumn{1}{|c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ (fm/c) } &
\multicolumn{1}{c|}{ (GeV/fm$^{3}$) } &
\multicolumn{1}{c|}{ (GeV) } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } \\
\multicolumn{1}{|c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } &
\multicolumn{1}{c|}{ } \\ \hline
& & & & & & & \\
SSPC & RHIC & $0.25$ & $61.4$ & $0.66$ & $0.34$ & $0.064$ & $0.064$ \\
SSPC & LHC & $0.25$ & $425$ & $1.01$ & $0.43$ & $0.082$ & $0.082$ \\
& & & & & & & \\ \hline
& & & & & & & \\
pQCD & RHIC & $0.10$ & $23.9$ & $0.889$ & $0.042$ & $0.0077$ & $0.0048$ \\
pQCD & LHC & $0.10$ & $1057$ & $1.09$ & $0.99$ & $0.064$ & $0.061$ \\
& & & & & & & \\ \hline
\end{tabular}
\label{table1}
\end{center}
\end{table}
\section{Boost-invariant longitudinal expansion}
In this section, we study purely longitudinal expansion
(in the $z$-direction) with boost-invariant initial conditions
\cite{Bjorken}. In this case,
physics is constant along the space-time hyperbolae
$\tau = \sqrt{t^2-z^2} =const.$.
Our aim is to establish how the parton equilibration process is affected
when the full phase-space distribution (\ref{1}) is used instead
of the factorized distribution (\ref{fac}). We will furthermore
compare the time evolution of the parton densities and the
entropy in the SSPC and pQCD scenarios, and
investigate the dependence of equilibration on the initial proper time
$\tau_0$ and the strong coupling constant $\alpha_s$.
\subsection{Factorized vs.\ full phase-space distribution}
For boost-invariant longitudinal expansion,
energy-momentum conservation (\ref{2}) reads
\begin{equation} \label{enmomcons}
\dot{\epsilon} + \frac{\epsilon+p}{\tau} = 0 \,\, ,
\end{equation}
with $\dot{\epsilon} \equiv {\rm d}\epsilon/{\rm d}\tau$,
$\tau$ being the proper time. The rate equations
(\ref{r1}) -- (\ref{r3}) assume the form
\begin{eqnarray} \label{r1a}
\dot{n_g} + \frac{n_g}{\tau} & = & {\rm R}_3 \, n_g
\left( 1 - \frac{n_g}{\tilde{n}_g} \right) -
2 \, {\rm R}_2\, n_g \left( 1 - \frac{n_q \,n_{\bar{q}} \, \tilde{n}_g^2}{
\tilde{n}_q \, \tilde{n}_{\bar{q}}\, n_g^2} \right) \,\,, \\
\dot{n_q} + \frac{n_q}{\tau} & = & {\rm R}_2 \, n_g
\left( 1 - \frac{n_q \, n_{\bar{q}} \, \tilde{n}_g^2 }{\tilde{n}_q\,
\tilde{n}_{\bar{q}} \, n_g^2} \right) \,\, , \label{r2a} \\
\dot{n_{\bar{q}}} + \frac{n_{\bar{q}}}{\tau} & = &
{\rm R}_2\, n_g \left(1 - \frac{n_q\, n_{\bar{q}}\, \tilde{n}_g^2 }
{\tilde{n}_q\, \tilde{n}_{\bar{q}}\, n_g^2} \right) \,\,. \label{r3a}
\end{eqnarray}
Equations (\ref{enmomcons}) -- (\ref{r3a}) are
four ordinary differential equations in the variable $\tau$,
containing four unknowns,
$T,\, \lambda_g,\, \lambda_q$, and $\lambda_{\bar{q}}$.
They are numerically solved with a standard Runge--Kutta
integration routine.
At this point note that, while the evolution of the fluid energy
and momentum densities are completely decoupled from the evolution of the
parton densities on account of the equation of state (\ref{pe3}), energy
used up in parton production will be reflected by the temperature
falling faster than in complete (local) thermodynamical
equilibrium. In the latter case, $T \sim {\tau}^{-1/3}$
\cite{Bjorken}.
For the factorized distribution, the evolution equations
(\ref{enmomcons}) -- (\ref{r3a}) can be further simplified \cite{Biro}:
\begin{eqnarray}
\frac{\dot{\lambda}_g + b(\dot{\lambda}_q + \dot{\lambda}_{\bar{q}})}{
\lambda_g + b(\lambda_q + \lambda_{\bar{q}} ) } +
4\, \frac{\dot{T}}{T} + \frac{4}{3 \tau} & = & 0\,\, ,\\
\frac{\dot{\lambda}_g}{\lambda_g} +
\frac{3\, \dot{T}}{T} + \frac{1}{\tau} & = &
{\rm R}_3 \, (1-\lambda_g) - 2\, {\rm R}_2 \left( 1 -
\frac{\lambda_q\, \lambda_{\bar{q}}}{\lambda_g^2}\right) \,\,, \\
\frac{\dot{\lambda}_q}{\lambda_q} +
\frac{3\, \dot{T}}{T} + \frac{1}{\tau} & = & R_2 \, \frac{a_1}{b_1}
\left( \frac{\lambda_g}{\lambda_q} -
\frac{\lambda_{\bar{q}}}{\lambda_g}\right) \,\, , \label{r2b} \\
\frac{\dot{\lambda}_{\bar{q}}}{\lambda_{\bar{q}}} +
\frac{3\, \dot{T}}{T} + \frac{1}{\tau} & = &R_2 \, \frac{a_1}{b_1}
\left( \frac{\lambda_g}{\lambda_{\bar{q}}} -
\frac{\lambda_q}{\lambda_g}\right) \,\, , \label{r3b}
\end{eqnarray}
with $b=b_2/a_2=21N_f/64$.
We are now in a position to check how the equilibration process
differs when using the
factorized phase-space distribution function (\ref{fac})
as compared with the full distribution function (\ref{1}).
Fig.\ \ref{fig3} shows the proper time evolution of
quark and gluon number densities, $n_q$ and $n_g$,
normalized to the corresponding
values for $\lambda_i = 1$, $\tilde{n}_q$ and $\tilde{n}_g$,
for SSPC initial conditions at RHIC and LHC
(cf.\ Table \ref{table1}).
The results for the factorized distribution function
tend to slightly overestimate the degree of equilibration, but the
deviation is of the order of a few percent only. This was to be expected
from Fig.\ \ref{fig2}.
\begin{figure}
\vspace*{10cm}
\caption{Parton densities for the SSPC RHIC (left) and
LHC scenarios (right), normalized to the corresponding
values for $\lambda_i = 1$ (denoted as $\tilde{n}_i$ in the text).
The solid line shows results for the full phase-space distribution function,
while the dashed line those for the factorized
distribution function $f^{\rm fac}$.}
\special{psfile=fig3a.ps voffset=0 hoffset=-50 hscale=48 vscale=52}
\special{psfile=fig3b.ps voffset=0 hoffset=204 hscale=48 vscale=52}
\label{fig3}
\end{figure}
\subsection{Comparison of SSPC and pQCD scenarios}
We now compare the proper time evolution of the parton densities and the
entropy in the SSPC and pQCD scenarios.
We exclusively use the full distribution function (\ref{1})
in the following. The entropy in a given rapidity unit is
${\rm d}S/{\rm d} \eta = s \, \tau \,\pi R^2$, where $\pi R^2$
is the transverse area of the expanding system. Since this area is constant
in a purely longitudinal expansion, we may consider the product of
entropy density and proper time, $s \, \tau$, as a measure for the
entropy per rapidity unit.
\begin{figure}
\vspace*{15.5cm}
\special{psfile=fig4c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig4d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig4a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig4b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{RHIC SSPC and pQCD scenarios (see Table \ref{table1}):
proper time evolution of the gluon (upper left panel),
quark (upper right panel), and antiquark densities
(lower left panel), and the entropy (lower right panel). The densities
are normalized to their corresponding values for
$\lambda_i=1$, the entropy ($\sim s \tau$) to its initial value. }
\label{fig4}
\end{figure}
In Fig.\ \ref{fig4} we show the parton equilibration process for
RHIC initial conditions (see Table \ref{table1}), for the
SSPC model (solid lines) and pQCD (dashed lines). For the SSPC case,
the hadronization energy density $\epsilon_h = 1.45$ GeVfm$^{-3}$ is
reached at a proper time $\tau_h=4.15$ fm/c, at which we stop
the time evolution. The relative gluon density, $n_g/\tilde{n}_g$, reaches
about $0.7$, and the relative quark density, $n_q/\tilde{n}_q$, about
$0.5$. The entropy increases $13$ \% before hadronization.
For pQCD initial conditions, the QGP phase has little
time to evolve, hadronizing only about $0.7$ fm/c
after thermalization. The partons have no time to equilibrate, and the
entropy increases only $\sim 5$ \%.
While the SSPC scenario was considered to be net-baryon free,
the pQCD scenario shows a slight difference in the quark and
antiquark initial densities, corresponding to an initial baryon number
density of about $0.12$ f${\rm m}^{-3}$. We checked that
our Runge-Kutta solver respects baryon number conservation, which in the
purely longitudinally expanding geometry implies that the product
$n_B\, \tau = (n_q - n_{\bar{q}})\tau /3$ is constant throughout the
expansion. As a consequence of nonzero net-baryon number, the
equilibrium ratio of $n_q /\tilde{n}_q$ will be larger than
1, while that of $n_{\bar{q}}/\tilde{n}_{\bar{q}}$ will be
smaller than 1, because $\tilde{n}_i$ is computed with $\lambda_i = 1$
instead of the correct equilibrium value $\lambda_q^{\rm eq}=
1/\lambda_{\bar{q}}^{\rm eq} > 1$. For a small initial baryon
number density of about $0.12\, {\rm fm}^{-3}$, however, the deviation
of $\lambda_i^{\rm eq}$ from 1 is negligible.
\begin{figure}
\vspace*{15.5 cm}
\special{psfile=fig5c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig5d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig5a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig5b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{As in Fig.\ \ref{fig4}, for LHC initial conditions.}
\label{fig5}
\end{figure}
The LHC case is shown in Fig.\ \ref{fig5}.
The parton species are seen to approach equilibrium after about
10 fm/c. We note that this time is well before a transverse rarefaction
front can penetrate to the center of the QGP (see discussion in
section \ref{transverse}). The reason why partons equilibrate is
that the system starts at much higher temperature and higher initial values
for the fugacities than at RHIC (see Table \ref{table1}).
Particularly noteworthy is that, in the pQCD case,
the gluons are almost completely equilibrated already at $\tau_0$,
while the quarks and antiquarks are not. Consequently, quark--antiquark
production processes drive the gluons temporarily {\em out\/} of equilibrium.
Entropy production is of the order of $13 \%$ (SSPC) to $17 \%$ (pQCD).
\subsection{Sensitivity to {\boldmath {\bf $\tau_0\ {\rm and} \ \alpha_s$}}}
We now investigate the sensitivity of the equilibration
process to variations of the initial proper time, $\tau_0$, and
the strong coupling constant, $\alpha_s$.
As discussed above, the values for the initial time $\tau_0$
in Table \ref{table1} are most certainly {\em lower\/} bounds.
We therefore consider values $\tau_0 \geq 0.25$ fm/c for the
SSPC scenario and $\tau_0 \geq 0.1$ fm/c for the pQCD approach.
In varying $\tau_0$, we keep the produced transverse energy,
${\rm d}E_T/{\rm d}\eta$, and the parton numbers, ${\rm d}N_i/{\rm d}\eta$,
constant. By eq.\ (\ref{dV}), the energy density and parton number
densities must then decrease like $\sim 1/\tau_0$. In
essence, this means that we allow the system to
evolve {\em without\/} doing longitudinal work. This is certainly
an idealization: only when the pressure vanishes,
the system does not perform any longitudinal work, but in our case
pressure builds up as the system approaches kinetic equilibrium.
To facilitate the presentation, we only give the {\em final\/}
values for the parton densities and the entropy at the end of the
expansion of the QGP phase. The evolution of the QGP is terminated either
at the {\em hadronization time}, $\tau_h$, when
the longitudinal expansion has cooled the system down to an energy
density $\epsilon_h = 1.45$ GeV/fm$^{3}$, or
at the {\em rarefaction time}, $\tau_{\rm rarefac}$, when
a transverse rarefaction wave reaches the center of the system
(see discussion in section \ref{transverse}).
The time evolution equation for the
energy density, (\ref{enmomcons}), with the equation of state (\ref{pe3}),
has the solution $\epsilon/\epsilon_0 = (\tau_0/\tau)^{4/3}$.
According to eq.\ (\ref{dV}), the product $\epsilon_0 \tau_0 =
({\rm d}E_T/{\rm d} \eta)/(\pi R^2)\,$ is constant for constant
transverse energy per unit rapidity and constant transverse area.
Consequently, the hadronization time $\tau_h$ grows with the initial proper
time $\tau_0$ according to
$\tau_h = \tau_0^{1/4}\, \left[ ({\rm d}E_T/{\rm d} \eta)/(\pi R^2 \epsilon_h)
\right]^{3/4}$. The time spent in the QGP phase is therefore
\begin{equation} \label{tauh}
\Delta \tau \equiv
\tau_h - \tau_0= \tau_0^{1/4}\, \left( \frac{{\rm d}E_T/{\rm d} \eta}{
\pi R^2 \epsilon_h} \right)^{3/4} - \tau_0\,\,.
\end{equation}
This time increases for small $\tau_0$, has a maximum at
$\tau_0^* = ({\rm d}E_T/{\rm d} \eta)/( 4^{4/3}\pi R^2 \epsilon_h)$, and
then decreases again.
For the SSPC model at RHIC, $\tau_0^* \simeq 1.67$ fm/c, while for
pQCD at RHIC, $\tau_0^* \simeq 0.27$ fm/c. At LHC energies, the values
for $\tau_0^*$ are quite similar, for the SSPC model,
$\tau_0^* \simeq 11.54$ fm/c, and for pQCD,
$\tau_0^* \simeq 11.48$ fm/c.
The hadronization time grows proportional to the initial
transverse energy. For LHC energies, the transverse energy is so
large that a transverse rarefaction
wave (see section \ref{transverse}),
travelling with sound velocity $c_s = 1/\sqrt{3}$ into matter at rest,
reaches the center of the system {\em before\/} the
longitudinal expansion has cooled matter down to $\epsilon_h$.
At $z=0$, this transverse rarefaction wave reaches the center at time
$\tau_{\rm rarefac} = \tau_0 + R/c_s$.
For a $Pb$ nucleus,
$R = 1.12\, A^{1/3} \simeq 6.6$ fm and
$ \tau_{\rm rarefac}\simeq \tau_0 + 11.5$ fm/c.
While we use $\tau_h$ to terminate the time evolution at RHIC
energy, for the LHC case, the time evolution is terminated
at $\tau_{\rm rarefac}$ instead of $\tau_h$.
\begin{figure}
\vspace*{15cm}
\special{psfile=fig6c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig6d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig6a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig6b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-1cm}
\caption{Hadronization values of the relative densities and the
entropy for ${\alpha}_s = 0.2,\, 0.3, \ldots,\, 0.6$, for the SSPC model
with RHIC initial conditions.
Upper left: gluons, upper right: quarks, lower left: antiquarks, lower right:
$s\tau /s_0{\tau}_0$. The dashed line is the time the system spends
in the QGP phase. }
\label{fig6}
\end{figure}
Fig.\ \ref{fig6} shows the case for the SSPC model with
RHIC initial conditions.
The hadronization values of the densities are closer to equilibrium
for larger values of $\alpha_s$. This is obvious,
since then the right-hand sides of the
rate equations (\ref{r1a}) -- (\ref{r3a}) are larger, cf.\ eq.\ (\ref{Rs}),
driving the system faster towards equilibrium. Consequently, also
entropy production increases with $\alpha_s$.
On the other hand, the hadronization values of the densities are
further away from equilibrium
for increasing values of the initial time $\tau_0$. The reason
is that, according to eq.\ (\ref{dV}), the values for the initial
energy density and the parton densities decrease for increasing $\tau_0$
(the parton number rapidity density ${\rm d}N_i/{\rm d}\eta$ and
transverse energy rapidity density ${\rm d}E_T/{\rm d}\eta$ are kept
constant). Surprisingly, this does not have an effect on the initial
temperature which is to all intents and purposes independent of $\tau_0$.
To understand this,
consider the factorized expressions (\ref{efac}) and (\ref{nfac});
reducing energy and parton densities by the same factor is
achieved by reducing the fugacities only, keeping the temperature constant.
However, the reduction of the fugacities
leads to smaller initial values for the parton densities.
This puts the initial
densities further away from their equilibrium values. This difference is,
with increasing $\tau_0$, increasingly harder
to overcome during the lifetime $\Delta \tau$, eq.\ (\ref{tauh}),
of the QGP phase.
The amount of entropy produced in the RHIC scenarios
is proportional to the lifetime of the QGP phase, {\it i.e.}, the
time during which chemical reactions drive the system towards
thermodynamical equilibrium and thus increase the entropy.
Note that equilibration is never complete at RHIC, unless one uses
rather large values of $\alpha_s$.
The pQCD RHIC scenario is depicted in Fig.\ \ref{fig7}.
The parton densities behave as in Fig.\ \ref{fig6}.
The main quantitative difference is that, due to the small initial values of
the energy and parton densities, even for large values
of $\alpha_s$ the system {\em never\/} reaches chemical equilibrium.
The behavior of the entropy follows again that of the
lifetime of the QGP.
\begin{figure}
\vspace*{15.5cm}
\special{psfile=fig7c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig7d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig7a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig7b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{As in Fig.\ \ref{fig6}, for pQCD initial conditions.}
\label{fig7}
\end{figure}
The LHC case is shown in Figs.\ \ref{fig8} and \ref{fig9}.
Despite different initial conditions, the relative
densities at $\tau_{\rm rarefac}$ are remarkably
similar in both the SSPC model and pQCD.
Again, this ratio increases with increasing $\alpha_s$ and decreasing
$\tau_0$. The entropy increases with $\alpha_s$ and $\tau_0$.
However, in this case
the time the system spends in the QGP phase is constant,
$\Delta \tau \equiv \tau_{\rm rarefac} - \tau_0 = R/c_s \simeq
11.5$ fm/c for $Pb$.
The increase in entropy can only be explained by the fact that
the time integral over the right-hand side of
eq.\ (\ref{noneqS}) is larger, if the system is further away
from equilibrium (which is the case for larger values of $\tau_0$).
As already seen in Fig.\ \ref{fig5}, entropy
production is stronger in the pQCD case.
Equilibration is nearly complete for large values of $\alpha_s$,
independent of the value of $\tau_0$.
The only scenario where the QGP will not reach full chemical
equilibration is when $\alpha_s$ is small, and the initial
time is large.
\begin{figure}
\vspace*{15.5cm}
\special{psfile=fig8c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig8d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig8a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig8b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{As in Fig.\ \ref{fig6}, for LHC initial conditions.
The largest value of $\alpha_s$ taken here is $0.4$; the curves
for larger values are indistinguishable from this case.}
\label{fig8}
\end{figure}
\begin{figure}
\vspace*{14.8 cm}
\caption{As in Fig.\ \ref{fig8}, for pQCD initial conditions.}
\special{psfile=fig9c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig9d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig9a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig9b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\label{fig9}
\end{figure}
\section{Longitudinal and transverse expansion}\label{transverse}
In this section, we consider (cylindrically symmetric)
transverse expansion in addition to boost-invariant longitudinal
expansion. In this case, the time evolution is no longer given
by ordinary differential equations, and one has to resort to numerical
algorithms which solve the equations of fluid dynamics, eqs.\
(\ref{2}), and the rate equations (\ref{r1}) -- (\ref{r3}).
The numerical scheme to solve the fluid-dynamical equations
used in this analysis is the relativistic
Harten--Lax--Van Leer--Einfeldt (RHLLE) algorithm \cite{RHLLE3},
with geometrical corrections performed using Sod's method
\cite{RHLLE4}. Since the rate equations are
conservation equations with source terms, they can also
be solved with the RHLLE algorithm, with the right-hand sides
being treated with Sod's method. Because of boost-invariance,
it is sufficient to solve the equations in the central plane, $z=0$,
where $\tau \equiv t$.
Our results are exclusively for the SSPC model and for the
initial values given in Table \ref{table1};
we did not vary $\tau_0= 0.25$ fm/c or $\alpha_s=0.3$.
We first show results for a box profile $\sim \Theta(R-r)$, and then
for a wounded-nucleon profile $\sim 3/2 \sqrt{1-r^2/R^2}$.
To facilitate comparison with \cite{MMS}, for the box profile
we use $R=7$ fm both at RHIC and LHC energies. For the wounded-nucleon
profile we use $R=1.12\, A^{1/3}$ to compute the actual values for
the radii of $Au$ nuclei (at RHIC) and $Pb$ nuclei (at LHC).
\subsection{Box profile}
In Fig.\ \ref{fig10} we show the evolution of the system in
the $t-r$ plane for RHIC initial conditions. In order to
highlight the consequences of transverse expansion and facilitate
comparison with \cite{MMS,woundedmms}, we plot
constant energy contours $\epsilon (r,t) = {\epsilon}_0/N^{4/3}$. If
there were no transverse expansion of the system, the energy
contours for $N\geq 2$ would be parallel to the $N=1$ contour. We see
that a rarefaction wave travels into the system from the surface, but
does not reach the center before the interior cools below the
hadronization energy density $\epsilon_h$ at $\tau_h=4.15$
fm/c. This agrees with the Runge-Kutta analysis of the purely
longitudinal expansion presented above.
It disagrees with Fig.\ 1 of \cite{MMS}, where the hadronization time
is of the order of 5 fm/c. Upon hadronization, the relative gluon
density, $n_g/\tilde{n}_g$, has reached roughly $0.7$, while the relative
quark (and antiquark) density is $n_q/\tilde{n}_q \simeq 0.5$.
Modest transverse velocities develop near the surface of the system.
We find them to be smaller than those in Fig.\ 1 of \cite{MMS}. This
difference is due to the different numerical scheme used in \cite{MMS}
(the SHASTA algorithm of \cite{Kataja}; we use the RHLLE algorithm
\cite{RHLLE3}).
In contrast to \cite{MMS}, Fig.\ 1, we do not see that these velocities drive
the fluid locally away from chemical equilibrium; in our case, the
respective fluid elements continue to equilibrate.
Let us give a simple argument as to why this must be the case.
Consider a fluid element on a given energy density contour in
the $t-r$ plane (upper left panel of Fig.\ \ref{fig10}).
Then, as the system expands, $\partial \cdot u >0$,
the energy density drops as a function of proper time,
$u \cdot \partial \epsilon = - (\epsilon + p) \partial \cdot u < 0$.
Since any other contour lying outside the original contour
(relative to the origin) corresponds to a smaller energy density,
it also corresponds to later proper times. Therefore, in going from the
first contour to the second, the proper time in the rest frame of a
fluid element increases. Since the actual parton number densities
are smaller than their equilibrium values, and consequently the
right-hand sides of the rate equations positive, $R_i >0$,
we conclude from eq.\ (\ref{comoving}) that
$n_i/n_i^{\rm eq}$ must increase in going from the first contour
to the second. In order to compare
with $n_i/\tilde{n}_i$, note that $\tilde{n}_i$ is computed
at the same temperature $T$ as $n_i$, with the fugacity $\lambda_i$ set
to 1. On the other hand, while $\lambda_i^{\rm eq}$ is also equal to 1
for the SSPC case considered here (on account of vanishing net-baryon density),
the temperature $T^{\rm eq}$ in $n_i^{\rm eq}$ is in general {\em different\/}
from $T$. Let us assume that the initial temperature for
an evolution in complete local equilibrium is the same as that for
the evolution including chemical equilibration. This has the
consequence that initially $\tilde{n}_i = n_i^{\rm eq}$, and thus
$n_i/\tilde{n}_i = n_i/n_i^{\rm eq}$ at $\tau_0$. As noted earlier in
section 4.1, in complete local equilibrium the temperature falls
less rapidly than in chemical non-equilibrium, $T^{\rm eq} > T$ at all
times $\tau > \tau_0$. Therefore,
$n_i/\tilde{n}_i = (n_i/n_i^{\rm eq})(T^{\rm eq}/T)^3 \geq n_i/n_i^{\rm eq}$,
with the equality holding at $\tau_0$.
Thus, not only does $n_i/\tilde{n}_i$ not decrease, it increases even
faster than $n_i/n_i^{\rm eq}$ during the evolution.
Consequently, in the $n_i/\tilde{n}_i-r$ diagram
(upper right and lower left panels of Fig.\ \ref{fig10}), the
second contour must lie outside the first contour (with respect to
the origin). This is exactly what we find numerically.
\begin{figure}
\vspace*{15.5cm}
\special{psfile=fig10c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig10d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig10a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig10b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{SSPC RHIC scenario: evolution in the $t-r$ plane.
Upper left panel: energy density
contours. The dashed line represents the advance of the rarefaction front.
Upper right panel: relative density of gluons. Lower left
panel: relative density of quarks (antiquarks identical). Lower right panel:
transverse velocity. }
\label{fig10}
\end{figure}
\begin{figure}
\vspace*{15.5cm}
\special{psfile=fig11c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig11d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig11a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig11b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{As in Fig. \ref{fig10}, for the SSPC LHC scenario.}
\label{fig11}
\end{figure}
Fig.\ \ref{fig11} shows the SSPC LHC scenario. In this case,
a rarefaction wave has enough time to penetrate into the center
of the system, albeit only shortly before the energy density
falls to $\epsilon_h=1.45$ GeV/fm$^{3}$.
Until the rarefaction wave reaches the center
at proper time $\tau = \tau_0 + R/c_s$,
the evolution of the system at $r=0$ is given by a purely
longitudinal expansion, as discussed in the previous section.
Once again, the results obtained with the RHLLE algorithm agree well with the
analysis using the Runge--Kutta method. As seen above,
equilibration of the parton species is nearly complete.
The hadronization time agrees well with that found in \cite{MMS},
Fig.\ 3. However, in contrast to that analysis we
again find that the transverse velocities
developing in the rarefaction front do not impede chemical equilibration.
In particular, we do not find the tendency to drive the
system out of equilibrium at larger transverse distances $r$, and
to overshoot the equilibrium values close to the origin, as in
Fig.\ 3 of \cite{MMS}. As explained above, in the $n_i/\tilde{n}_i -r$
diagram, contours corresponding to smaller energy density (larger proper
times) must lie outside those corresponding to larger energy density
(smaller proper times) relative to the origin.
\subsection{Wounded-nucleon profile}
Throughout the analysis presented above
a box-profile function has been assumed.
While this is instructive and facilitates
direct comparison between the results from a purely longitudinal
expansion and those including transverse expansion (as long as the
transverse rarefaction wave has not yet reached the center of the
system), a box profile is clearly an idealization.
In this section we repeat the analysis of the SSPC RHIC and
LHC scenarios with a more realistic
nuclear profile function, the so-called wounded-nucleon profile
\begin{equation} \label{ta}
T_A(r) = \frac{3}{2}\, \sqrt{1-\frac{r^2}{R^2}}\,\, ,
\end{equation}
where $r$ is the transverse coordinate. We generate an $r$--dependence
of our initial energy and parton density profiles by multiplying
the initial values of Table \ref{table1} with eq.\ (\ref{ta}).
Clearly, this implies higher
initial densities at the center of the plasma. However, the
fluid-dynamical expansion begins with a non-zero density gradient already at
$\tau_0$, which, without the factor $3/2$ in (\ref{ta}),
would lead to faster cooling than a constant
nuclear profile function.
\begin{figure}
\vspace*{15.5cm}
\special{psfile=fig12c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig12d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig12a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig12b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{As in Fig.\ \ref{fig10}, with an initial wounded-nucleon profile.}
\label{fig12}
\end{figure}
In Fig.\ \ref{fig12} we present the analogue of Fig.\ \ref{fig10}, now
computed with an initial wounded-nucleon density profile.
We observe that the lifetime of the plasma phase is
actually larger than in Fig.\ \ref{fig10}, $\tau_h=4.70$ fm/c,
instead of $\tau_h=4.15$ fm/c. Due to the initial density gradients,
transverse velocities develop now over the entire volume of the
system.
The cooling effects of the flow become apparent when considering
purely longitudinal expansion with an
initial energy density of $3/2 \times 61.4$ GeV/fm$^{3}$.
In this case, the QGP lives longer, $\tau_h=5.63$ fm/c.
The increase of the initial energy and parton densities by a factor
of $3/2$ in the center of the system has the effect that
the partons at small and intermediate $r$ equilibrate a little
further in the wounded-nucleon scenario than in the box-profile
scenario. The relative gluon density reaches values around
0.8 (0.7 for the box profile)
and the relative quark density values around 0.6 (compared with 0.5 in
the box profile case). In general, however,
the box-profile and the wounded-nucleon-profile relative parton
density calculations agree to a remarkable extent.
\begin{figure}
\vspace*{15.5cm}
\special{psfile=fig13c.ps voffset=0 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig13d.ps voffset=0 hoffset=204 hscale=48 vscale=40}
\special{psfile=fig13a.ps voffset=215 hoffset=-50 hscale=48 vscale=40}
\special{psfile=fig13b.ps voffset=215 hoffset=204 hscale=48 vscale=40}
\vspace*{-2cm}
\caption{As in Fig.\ \ref{fig11}, with an initial wounded-nucleon profile.
The dotted line in the upper left panel depicts the hadronization hypersurface
for an initially box-profiled plasma (of $R=7$ fm), shown
in Fig.\ \ref{fig11}.}
\label{fig13}
\end{figure}
In Fig.\ \ref{fig13} we show the LHC case, where the effects of the
wounded-nucleon profile become more apparent. The plasma cools
markedly more rapidly than in the box-profile case,
hadronizing at a proper time $\tau_h=9.37$ fm/c.
This is more than $3$ fm/c earlier than for
an initial box profile, in which case $\tau_h$ is essentially
moments after $\tau_{\rm rarefac}$, the arrival of the rarefaction
wave at $r=0$.
While the plasma's lifetime is significantly reduced, parton chemistry,
driven by high temperatures and high gluon densities, is
sufficiently rapid to equilibrate the plasma prior to hadronization.
Qualitatively, the transverse
velocities along contours of constant energy density in Figs.\ \ref{fig12}
and \ref{fig13} agree
with the analysis given in \cite{woundedmms}. No
parton evolution plots are shown in \cite{woundedmms}.
\section{Conclusions}
We have investigated chemical equilibration of quarks and
gluons at RHIC and LHC energies. Assuming that a kinetically
equilibrated quark-gluon plasma is created at these energies, we
have used ideal fluid dynamics coupled to rate equations for the parton
densities \cite{Biro,MMS,woundedmms}
to study chemical equilibration in the further evolution of the system.
Our work is motivated by the
upcoming experiments at RHIC in the fall of this year, and by recent
pQCD estimates on parton production in the initial stage of ultrarelativistic
nuclear collisions \cite{eskola2}. These estimates serve as initial
conditions to the fluid-dynamical as well as the rate equations.
Besides pQCD initial conditions, we also considered
those given by the so-called SSPC model \cite{MMS,woundedmms}, in order
to compare our results to previous studies.
At RHIC, the SSPC model constitutes an upper limit for
the initial energy and parton densities,
while pQCD sets a lower limit, as the effect of
soft background fields is neglected in this approach.
It was recently argued \cite{keijorecent} that
the transverse momentum cutoff $p_0$ should be chosen to match
the saturation scale, $p_{\rm sat} \sim 1$ GeV for RHIC,
rather than $\sim 2$ GeV chosen here. Then, the initial
conditions obtained from pQCD are rather similar to those in the
SSPC approach.
The produced transverse energy density can also be computed within
classical Yang--Mills theory \cite{rajualex}; the values obtained in this way
support the SSPC estimates.
Our results can be summarized as follows. Chemical equilibration of
partons is never complete at RHIC energies. The highest degree of
equilibration can be reached with SSPC initial conditions and an
(unphysically) large value of the strong coupling constant,
$\alpha_s \sim 0.6$. With initial conditions from pQCD, the system
never comes close to equilibration.
The situation changes completely at LHC energies. Here, the initial
energy and parton densities are already so large that chemical
equilibration is complete in almost all scenarios considered.
Only if $\alpha_s \sim 0.2$ or smaller and the initial
proper time $\tau_0$ is large, equilibration was seen to be incomplete.
Within the macroscopic transport approach employed here,
entropy production due to chemical non-equilibrium processes was found to
be large, in some cases $\sim 30 \%$. This has implications for estimating
the {\em initial conditions\/} in ultrarelativistic nuclear collisions.
Commonly, one assumes that the evolution of the system
is entropy conserving. Then, since $s \simeq const. \times n$,
the initial entropy $\sim s_0 \tau_0$ is
estimated from the total multiplicity of hadrons in the final state
by the formula ${\rm d}N_h/{\rm d}\eta \simeq const. \times
s_0 \tau_0 \pi R^2$. However, if entropy grows due to chemical reactions,
this formula would severely overestimate the entropy in the initial
state.
We furthermore established that the use of a factorized phase-space
distribution as in \cite{Biro,MMS,woundedmms} is permissible as
the error is of the order of a few percent only. We also gave an
analytical proof, as well as numerical evidence, that
transverse flow does not drive the system away from chemical equilibrium, in
contrast to the results of \cite{MMS}.
Future studies should focus on the further evolution of the system
into the mixed and hadron phases. To this end, previously developed
non-equilibrium models could be resurrected \cite{barz}.
Further applications of the methods presented here include the study
of strangeness and charm equilibration at RHIC and LHC energies
\cite{dmedhr2}.
Besides this, the reaction rates should be
improved using the full phase-space distribution function instead
of the factorized one. Other improvements of the expressions for
the reaction rates include the
running of the strong coupling constant. As shown in Ref.\ \cite{Wong},
equilibration is faster when the scale of the running coupling constant
is allowed to vary with the temperature $T$.
\\[1cm]
{\bf Acknowledgements}\\ ~~ \\
We thank S.\ Bass, J.\ Cleymans, A.\ Dumitru, K.\ Eskola, K.\ Kajantie,
B.\ M\"uller, M.\ Mustafa, J.\ Rafelski,
D.\ Srivastava, and R.\ Venugopalan for valuable discussions. D.H.R.\
thanks RIKEN, Brookhaven National Laboratory, and the U.S.\ Department of
Energy for providing the facilities essential for the completion of this work.
\input{bib}
\end{document}
|
2,869,038,154,819 | arxiv | \section{Motivation}
A remarkable property of quantum mechanics, is the so called quantum
Zeno effect \cite{Misra}. This effect, is that frequent observation
slow down the evolution of the state, with the limit of continuous
observation leading to ``freezing'' of the state\footnote{To be more
precise, restriction to a subspace.}. This has been experimentally
verified. The intuitive explanation, is that the interaction of the
observer with the system leads to this apparent paradox. It would
therefore be interesting to see whether this effect persists if we
consider a closed system. We would try to see what is the
probability of a closed system remaining in a particular subspace of
its Hilbert space with no external observer. This directly relates
to the arrival time problem as well (e.g. \cite{HaZa,Wallden2006}).
Having said that, we should emphasize that in closed systems, we
cannot in general assign probabilities to histories, unless they
decohere and it is this property that resolves the apparent paradox
that arises.
\section{This paper}
This contribution is largely based on Ref.\cite{Wallden2006}. In
Section \ref{Intro}
we revise the quantum Zeno effect and the decoherent histories, and
introduce a new formula for the restricted propagator that will be
of use further. In Section \ref{Prob} we see what probabilities we
would get if we had decoherence, that highlights the persistence of
the quantum Zeno effect. In Section \ref{Decoh Cond} we get the decoherence
condition that in Section \ref{Arrival} is stressed how restrictive is by considering the arrival time problem. We
conclude in Section \ref{Concl}.
\section{Introductory material}\label{Intro}
\subsection{Quantum Zeno effect}
In standard Copenhagen quantum mechanics,
the measurement is represented by projecting the state to a
subspace defined by the eigenstates that correspond to the range of
eigenvalues of the measured physical quantity. The latter is
represented by a self-adjoint operator. The state, otherwise evolves
unitarily: $\hat U(t)=\exp (-i\hat Ht)$, where $\hat H$ is the
Hamiltonian. It is then a mathematical fact, that frequent
measurement, of the same quantity (subspace) leads to slow down of
the evolution, i.e. decreases the probability that the state evolves
outside the subspace in question. This resembles the ancient Greek,
Zeno paradox ($ Z\acute{\eta}\nu\omega\nu$), and thus the name.
The continuum measurement limit, leads to zero probability of
leaving the observed subspace. The state continues to evolve
(unitarily), but restricted in the subspace of observation
\cite{Facchi2}. This implies that if we project to a one-dimensional
subspace, the state stops evolving. In most literature, the question
is of a particle decaying or not, so the last comment applies. In
particular, the above phenomenon is still present for infinite
dimensional Hilbert spaces, but provided that the restricted
Hamiltonian ($H_r=PHP$) is self-adjoint, as we will see later.
\subsection{Decoherent histories}
Decoherent histories approach to quantum theory is an alternative
formulation designed to deal with closed systems and it was
developed by Griffiths \cite{Gri84}, Omn\`es \cite{Omn88a}, and
Gell-Mann and Hartle \cite{GH90b}. There is no external observer, no
a-priori environment-system split. The main mathematical aim of this
approach, is to see when is it meaningful to assign probabilities to
a history of a closed quantum system and of course to determine this
probability.
Here we will revise the standard non-relativistic quantum mechanics
in decoherent histories formulation. To each history
($\underline\alpha$) corresponds a particular class operator
$C_{\underline\alpha}$,
\beq\label{1.6} C_{\underline{\alpha}}=P_{{\alpha}_n}e^{-iH(t_n-t_{n-1})}
P_{{\alpha}_{n-1}}\cdots e^{-iH(t_2-t_1)}P_{{\alpha}_1}\eeq Where $P_{{\alpha}_1}$ etc
are projection operators corresponding to some observable, $H$ is
the Hamiltonian, and $t_n$ is the total time interval we consider.
This class operator corresponds to the history, the system is at the
subspace spanned by $P_{\alpha_1}$ at time $t_1$ at $ P_{\alpha_2}$
at time $t_2$ and so on. The probability for this history, provided
we had some external observer making the measurement at each time
$t_k$ would be
\beq\label{3.2 candidate}
p(\underline{\alpha})=D(\underline{\alpha},\underline{\alpha})={\mathrm{Tr}}(C_{\underline{\alpha}}\rho
C^\dag_{\underline{\alpha}})\eeq where $\rho$ is the initial state. In the
case of a closed system, Eq.(\ref{3.2 candidate}) fails in general
to be probability due to interference\footnote{The additivity of
disjoint regions of the sample space is not satisfied by
Eq.(\ref{3.2 candidate})}.
There are, however, certain cases where we can assign probabilities.
This happens if for a complete set of histories, they pairwise obey
\beq\label{3.2 decoherence}
D(\underline{\alpha},\underline{\beta})={\mathrm{Tr}}(C_{\underline{\alpha}}\rho
C^\dag_{\underline{\beta}})=0\quad\forall\quad
\underline{\alpha}\neq\underline{\beta}\eeq In that case, the complete set of
histories is called \emph{decoherent} set of histories and we can
assign to each history of this set the probability of Eq.(\ref{3.2
candidate}). In order to achieve a set of histories that satisfy
Eq.(\ref{3.2 decoherence}) in general we need to consider coarse
grained histories, or/and very specific initial state
$\rho$\footnote{Note that the interaction of a system with an
environment that brings decoherence, in the histories vocabulary, is
just a particular type of coarse graining where we ignore the
environments degrees of freedom.}.
To sum up, in decoherent histories we need to first construct a
class operators that corresponds to the histories of
interest\footnote{Note that the same classical question can be
turned to quantum with several, possibly inequivalent ways. Due to
this property, the construction of the suitable class operator is
important for questions such as for example, the arrival time or
reparametrization invariant questions.}, and then confirm that these
histories satisfy Eq.(\ref{3.2 decoherence}). Only then we can give
an answer.
\subsection{The restricted propagator}
A mathematical object that will be needed for computing the suitable
class operators, is the restricted propagator. This is the
propagator restricted to some particular region $\Delta$ (of the
configuration space) that corresponds to a subspace of the total
Hilbert space denoted by ${\mathcal{H}}_\Delta$. The most common (but not the
most general) is the path integral definition:
\begin{equation}
g_r(x,t\mid x_0,t_0)=\int_\Delta \mathcal{D}x \exp(iS[x(t)])=\langle
x| g_r(t,t_0)| x_0\rangle
\end{equation}
The integration is done over paths that remain in the region
$\Delta$ during the time interval $[t,t_0]$. The $S[x(t)]$ is as
usual the action. The operator form of the above is given by
\cite{Halliwell:1995jh,Halliwell:2005nv}:
\begin{equation} \label{restricted operator}
g_r(t,t_0)=\lim_{\delta
t\rightarrow 0} P e^{-iH(t_n-t_{n-1})}P\cdots P e^{-iH(t_1-t_0)}P
\end{equation}
With $t_n=t$, $\delta t\rightarrow 0$ and $n\rightarrow \infty$
simultaneously keeping $\delta t \times n=(t-t_0)$. $H$ is the
Hamiltonian operator. $P$ is a projection operator on the restricted
region $\Delta$. We therefore have
\beq g_r(x,t\mid x_0,t_0)={\langle} x|g_r(t,t_0)|x_0{\rangle}\eeq Note here that
the expression Eq.(\ref{restricted operator}) is the defining one
for cases that the restricted region is not a region of the
configuration space, but some other subspace of the total Hilbert
space ${\mathcal{H}}$. The differential equation obeyed by the restricted
propagator is:
\begin{equation} \label{restricted differential}
(i\frac{\partial}{\partial t}-H) g_r(t,t_0)=[P,H] g_r(t,t_0)
\end{equation}
Which is almost the Schr\"{o}dinger equation, differing by the
commutator of the projection to the restricted region with the
Hamiltonian.
The most useful form, for our discussion was derived in Ref.
\cite{Wallden2006}
\begin{equation} \label{restricted zeno}
g_r(t,t_0)=P\exp\left(-i(t-t_0)P H P\right)P
\end{equation}
Note that $PH P$ is the Hamiltonian projected in the subspace
${\mathcal{H}}_\Delta$. To prove Eq.(\ref{restricted zeno}) we multiply Eq.
(\ref{restricted differential}) with $P$ we will then get
\beq (i\frac{\partial}{\partial t}-PHP) g_r(t,t_0)=0\eeq using the
fact that $P[H,P]P=0$ and that the propagator has a projection $P$
at the final time. This is Schr\"{o}dinger equation with Hamiltonian
$PHP$. It is evident that this leads to the full propagator in
${\mathcal{H}}_\Delta$ provided that the operator $PHP$ is self-adjoint in
this subspace \cite{Facchi2}\footnote{A detailed proof from
Eq.(\ref{restricted operator}) can be found in \cite{Wallden2006}.}.
\section{Quantum Zeno histories}
In this section we will examine the question \emph{what is the
probability for a system to remain in a particular subspace, during
a time interval $\Delta t=t-t_0$}. We will see the probabilities and
decoherence conditions for the general case, and then see what this
implies for the arrival time problem, which is just a particular
example.
\subsection{The class operator and probabilities}\label{Prob}
There are several ways of turning the above classical proposition to
a quantum mechanical one. The most straight forward is the
following. We consider a system being in one subspace by projecting
to that, and the history of always remaining in that subspace
corresponds to the limit of projecting to the region evolving
unitarily but for infinitesimal time and then projecting again, i.e.
taking the ${\delta} t$ between the propositions going to zero. The class
operator for remaining always in that subspace follows from
Eq.(\ref{1.6}) by taking each $P_{{\alpha} k}$ being the same ($P$) and
taking the limit of $(t_k-t_{k-1})$ going to zero for each $k$. We
then have
\beq C_{\alpha} (t,t_0)=g_r(t,t_0)\eeq and the class operator for not
remaining at this subspace during all the interval is naturally
\beq C_{\beta} (t,t_0)=g(t,t_0)-g_r(t,t_0)\eeq with
$g(t,t_0)=\exp(-iH(t-t_0))$ the full propagator.
Let us, for the moment, assume that the initial state $|\psi{\rangle}$ is
such, that we do have decoherence. We will return later to see when
this is the case. The (candidate) probability is
\beq\label{4.1 zeno prob}
p(\alpha)={\langle}\psi|g_r^\dagger(t,t_0)g_r(t,t_0)|\psi{\rangle}\eeq Following
Eq.(\ref{restricted zeno}) it is clear\footnote{Provided $PHP$ is
self-adjoint in the subspace. This is true for finite dimensional
Hilbert spaces and has been shown to be true for regions of the
configuration space in a Hamiltonian with at most quadratic momenta
\cite{Facchi2}.} that
\beq g_r^\dagger(t,t_0)g_r(t,t_0)=P\eeq which then implies
\beq p({\alpha})={\langle}\psi|P|\psi{\rangle}\eeq For an initial state that is in the
subspace defined by $P$, the probability to remain in this subspace
is one. This is the usual account of the quantum Zeno effect. As it
is stressed in other literature, to have the quantum Zeno is crucial
that the restricted Hamiltonian $H_r=PHP$ to be self-adjoint
operator in the subspace. Note, that this only states that the
system remain in the subspace, but it does not ``freeze'' completely
and in particular follows unitary evolution in the subspace with
Hamiltonian, the restricted one $H_r$. The form of
Eq.(\ref{restricted zeno}) of the restricted propagator makes the
latter comment more transparent.
\subsection{Decoherence condition}\label{Decoh Cond}
All this is well understood for open systems with external
observers. To assign the candidate probability of Eq.(\ref{4.1 zeno
prob}) as a proper probability of a closed system, we need the
system to obey the decoherence condition, i.e.
\beq D({\alpha},{\beta})={\langle}\psi|C^\dag_{\beta} C_{\alpha}|\psi{\rangle}=0\eeq and this implies
that \beq\label{4.2 decoh cond}
{\langle}\psi|g_r^\dagger(t,t_0)g(t,t_0)|\psi{\rangle}={\langle}\psi|P|\psi{\rangle}\eeq
which is a very restrictive condition and only very few states
satisfy this, as we will see in the arrival time example. The
condition, essentially states that the overlap of the time evolved
state ($g(t,t_0)|\psi{\rangle}$) with the state evolved in the subspace
($g_r(t,t_0)|\psi{\rangle}$) should be the same at the times $t_0$ and
$t$. Given that the restricted Hamiltonian leads, in general, to
different evolution, the condition refers only to very special
initial states with symmetries, or for particular time intervals
$\Delta t$.
\section{Arrival time problem}\label{Arrival}
The arrival time problem is the following: \emph{What is the
probability that the system crosses a particular region $\Delta$ of
the configuration space, at any time during the time interval
$\Delta t=(t-t_0)$.} One can attempt to answer this, by considering
what is the probability that the system remains always in the
complementary region $\bar\Delta$. So if $\mathcal{Q}$ is the total
configuration space, we have $\Delta\cup\bar\Delta=\mathcal{Q}$ and
$\Delta\cap\bar\Delta=\emptyset$. Taking this approach to the
arrival time problem, the relation with the quantum Zeno histories
is apparent, since it is just the special case, where the subspace
of projection is a region of the configuration space ($\bar\Delta$)
and the Hamiltonian is quadratic in momenta, i.e.
\bea \bar P&=&\int_{\bar\Delta}|x{\rangle}{\langle} x|dx\nonumber\\ \hat
H&=&\hat p^2/2m+V(\hat x)\eea
This particular case is infinite dimensional, but as shown in Ref.
\cite{Facchi2} the restricted Hamiltonian is indeed self-adjoint and
the arguments of the previous section apply.
Before proceeding further, we should point out that one could
construct different class operators that would also correspond to
the (classical) arrival time question. For example, one could
consider having POVM's\footnote{Positive Operator Valued Measure}
instead of projections at each moment of time, or could have a
finite (but frequent) number of projections (not taking the limit
where ${\delta} t\rightarrow 0$). These and other approaches are not
discussed here.
Let us see now, what the quantum Zeno effect implies about the
arrival time. It states that a system initially localized outside
$\Delta$ will always remain outside $\Delta$ (if it decoheres) and
therefore we can only get zero crossing probabilities. This is
definitely surprising, since for a wave packet that is initially
localized in $\bar\Delta$ and its classical trajectory crosses
region $\Delta$, we would expect to get crossing probability one.
The resolution comes due to the decoherence condition as will be
argued later.
Returning to the decoherence condition Eq.(\ref{4.2 decoh cond}) we
see that there is the overlap of the time evolved state with the
restricted time evolved state. In the arrival time case, the
restricted Hamiltonian corresponds to the Hamiltonian in the
restricted region ($\bar\Delta$) but with infinite potential walls
on the boundary (i.e. perfectly reflecting). We then get decoherence
in the following four cases.
\begin{itemize}
\item[(a)] The initial state $|\psi{\rangle}$ is in an energy eigenstate,
and it also vanishes on the boundary of the region.
\item[(b)] The restricted propagator can be expressed by the method
of images\footnote{Note that the restricted propagator can be
expressed using the method of images, if and only if there exist a
set of energy eigenstates, vanishing on the boundary, that when
projected on the region $\bar\Delta$ forms a dense subset of the
subspace ${\mathcal{H}}_{\bar\Delta}$, i.e. span ${\mathcal{H}}_{\bar\Delta}$. This is
equivalent with requiring that the restricted energy spectrum (i.e.
spectrum of the restricted Hamiltonian $H_r$) is a subset of the
(unrestricted) energy spectrum, which is not in general the case.}
and the initial state shares the same symmetry.
\item[(c)] The full unitary evolution in the time interval $\Delta
t$ remains in the region $\bar\Delta$.
\item[(d)]Recurrence: Due to the period of the Hamiltonian and
the restricted Hamiltonian their overlap happens to be the same
after some time $t$ as it was in time $t_0$. This depends
sensitively on the time interval and it is thus of less physical
significance.
\end{itemize}
It is now apparent that most initial states do not satisfy any of
those conditions. In particular, the wavepacket that classically
would cross the region $\Delta$, will not satisfy any of these
conditions, and we would not be able to assign the candidate
probability as a proper one, and thus we avoid the paradox. The
introduction of an interacting environment to our system, (that
usually produces decoherence by coarse-graining the environment)
does not change the probabilities and contrary to the intuitive
feeling, it does not provide decoherence for the particular type of
question we consider. This still leave us with no answer for any of
the cases that the system would classically cross the region. The
latter implies, that the straight forward coarse grainings we used,
were not general enough to answer fully the arrival time question
\footnote{For more details, examples and discussion see Ref.
\cite{Wallden2006}.}.
As a final note, we should point out that the quantum Zeno effect in
the decoherent histories, has implications for the decoherent
histories approach to the problem of time (e.g. Refs.
\cite{Halliwell:2005nv,Wallden2006}).
\section{Conclusions}\label{Concl}
We examined the quantum Zeno type of histories of a closed system,
using the decoherent histories approach. We show that the quantum
Zeno effect is still present, but only for the very few cases that
we have decoherence. The situation does not change with the
introduction of interacting environment. We see that while in the
open system quantum Zeno, the delay of the evolution arises as
interaction with the observer, in the closed system we have the
decoherence condition ``replacing'' the observer and resolving the
apparent paradox.
\paragraph{Acknowledgments:} The author is very grateful to Jonathan J. Halliwell for many
useful discussions and suggestions, and would like to thank the
organizers for giving the opportunity to give this talk and hosting
this very interesting and nice conference.
|
2,869,038,154,820 | arxiv | \section{Introduction}
Some standard model (SM) parameters have been measured with such a high
precision that has allowed to constrain the values of other SM parameters,
or even new physics, through the use of radiative corrections\cite{lepa},
as can be exemplified by the correct agreement between the predicted
top mass and the observed value \cite{topmass}. Finding the Higgs boson
remains as the final step to confirm the theoretical scheme of the SM. The
present lowest experimental bound on the Higgs mass is $m_H>90.4$ \rm{GeV}{}
\cite{mhexp}, this is a direct search limit. On contrary to the top quark
case, radiative correction are only logarithmically sensitive to the Higgs
mass, and thus it is more difficult to obtain an indirect bound. However,
fits with present data seems to favor a light SM Higgs \cite{altanew}.
Henceforth, it is interesting to ask how this conclusion will change if
one goes beyond the SM.
The framework of effective Lagrangians, as a mean to parametrize physics
beyond the SM in a model independent manner, has been used extensively
recently \cite{effecl}. Within this approach, the effective lagrangian is
constructed by assuming that the virtual effects of new physics modify the
SM interactions, and these effects are parametrized by a series of
higher-dimensional nonrenor\-ma\-lizable operators written in terms of the
SM fields. The effective linear Lagrangian can be expanded as follows:
\begin{equation}
{\mathcal L}_{\mathrm{eff}}= {\mathcal L}_{\mathrm{SM}}
+ \sum_{i,n} {\frac{\alpha_i}{\Lambda^n} } O^i_n
\end{equation}
where $\mathcal{L}_{\mathrm{SM}}$ denotes the SM renormalizable
lagrangian. The terms $O^i_n$ are $SU(3) \times SU(2)_L \times U(1)_Y$
invariant operators. $\Lambda$ is the onset scale where the appearance of
new physics will happen. The parameters $\alpha_i$ are unknown in this
framework, although "calculable" within a specific {\it full theory}
\cite{ital}. This fact was used in Ref. \cite{effecc} to show that, in a
weakly coupled full theory, a hierarchy between operators arises by
analyzing the order of perturbation theory at which each operator could be
generated {\it e.g.} by integrating the heavy degrees of freedom. Some
operators can be generated at tree-level, and it is natural to assume that
their coefficients will be suppressed only by products of coupling
constants; whereas the ones that can be generated at the 1-loop level, or
higher, will be also suppressed by the typical $1/16\pi^2$ loop factors.
This allows us to focus on the most important effects the high-energy
theories could induce, namely those coming from tree-level generated
dimension-six operators.
In this letter we address two related questions. First, we study how the
effective lagrangian affects the determination of the W boson mass,
and how this could affect the bounds on the Higgs mass. The second item
under consideration will be to re-examine the effects on Higgs-vector
boson production at hadron colliders by using the results obtained from
the first part.
\section{The SM $W$ mass}
We shall use the results of Ref. \cite{degrassi,coefi}, which parametrize
the bulk of the radiative corrections to $W$ mass through the
following expression:
\begin{equation}
m_W=m_W^o \ [1+ d_1 ln (\frac{M_h}{100})+d_2 C_{em}+
d_3 C_{top} +d_4 C_{as} + d_5 ln^2(\frac{M_h}{100}) ],
\end{equation}
where the coefficients $d_i$ are given in table 2 of Ref.
\cite{coefi}, they incorporate the full 1-loop effects, and
some dominant 2-loop corrections. The factors $C_i$ are
given by:
\begin{eqnarray}
C_{em}&=&\frac{\Delta \alpha_h}{0.0280} -1, \nonumber \\
C_{top}&=&(\frac{m_t}{175 GeV})^2 -1, \nonumber \\
C_{as}&=&\frac{\alpha_s(M_Z)}{0.118} -1,
\end{eqnarray}
and they measure the dependence on the fine structure
constant, top mass and strong coupling constant, respectively.
The reference $W$ mass, $m_W^o=80.383$ \rm{GeV}, is obtained with the
following values: $mt=175$ \rm{GeV}, $\alpha_s=.118$,
$\Delta \alpha=0.0280$, and $m_H=100$ \rm{GeV}.
By using eqs. (2-3), one finds a bound on the SM Higgs mass,
$m_t=176 \pm 2$ \rm{GeV}, $\Delta \alpha=.0280$
and $\alpha_s=.118$, of $170< m_H<330$ \rm{GeV}.
This result agree with several other studies \cite{lepa} which
anticipates the existence of a light Higgs boson.
Including the effects of new physics will modify the W mass value. That
effect should be combined with the previous SM corrections in order to
determine to what extent new physics could change
the bounds obtained for the Higgs mass.
\section{Modification to the $W$ mass}
The complete set of effective operators is large but
the analysis simplifies because loop-level dimension six operators and
tree-level dimension eight ones generate subdominant effects with respect
to tree-level generated dimension six effective operators \footnote{This
is not the case when tree-level dimension six operators do not contribute
\cite{yomi,mythes}.}. The effective contributions to the input parameters
in the formulas (2-3), besides $m_W$, can be show to dissappear after
suitable redefinitions \cite{mythes}. Just two operators contribute:
\begin{eqnarray}
O_{\phi}^{(1)}&=&(\phi ^{\dagger }\phi ) (D^\mu \phi ^{\dagger }
D_\mu \phi), \nonumber \\
O_{\phi}^{(3)}&=&(\phi ^{\dagger } D_{\mu} \phi) (D^{\mu}
\phi)^{\dagger } \phi.
\end{eqnarray}
The notation is as usual, $\phi$ denotes the Higgs doublet,
and $D_{\mu}$ is the usual covariant derivative.
An interesting characteristic of these operators is that
only $O_{\phi}^{(1)}$ gives contribution to $m_W$; although both
operators modify the coefficients of the vertices $HVV$ they leave
intact its Lorentz structure.
This approach is equivalent to the most usual one used in the literature
in which more operators are allowed but they are constrained in a
one-per-one basis, so that "unnatural" cancellations are not allowed.
We shall now include the contribution to $m_W$ arising from the effective
operators of eq. (4) into eqs. (2). The formulae for the $W$ mass
becomes:
\begin{equation}
m_W |_{\mathrm{eff}} \ = \ m_W |_{\mathrm{SM}} \
(1+\frac{1}{4} \alpha_{\phi}^{(1)} (\frac{v}{\Lambda})^2)
\end{equation}
where $m_W |_{\mathrm{SM}}$ corresponds to the $W$ mass as defined
in eq. (2), and the term in the parentheses arises from
the effective Lagrangian.
To study the inter-relations between the Higgs mass and
$\alpha_{\phi}^{(1)}$ and $\Lambda$, we must set the allowed $m_W-m_H$
region. We are going to use the future expected
minimum uncertainty for the W mass: $\Delta m_W=\pm.01$ \rm{GeV}{}
for a nominal central value of $m_W=80.33$ \rm{GeV},
and expand $m_H$ between the recent experimental bound,
$m_H \ge 90$ \rm{GeV}{} \cite{mhexp}, and the perturbative limit,
$m_H \le 700$ \rm{GeV} (Fig. 1). We take an optimum scenarios also for
the top quark mass ($ m_t=176 \pm 2$ \rm{GeV}).
$\alpha$ and $\Lambda$ should take values such that the resulting masses
must satisfy the above constraints. Since we assumed the
effective Lagrangian is derived from a weakly coupled
theory, it is reasonable to impose\footnote{We will choose
the $\alpha_i$ signs that give the maximum and minimum values
for the quantities of interest.} $|\alpha_{\phi}^{(1)}| \le 1$
, while $\Lambda$ is set
to values greater or equal than 1 \rm{TeV}. This scale is a conventional one,
it can be justified in some specific high-energy models.
It turns out that the case when
$|\alpha_{\phi}^{(1)}|=1$ and $\Lambda=1$ \rm{TeV}, simultaneously,
is already excluded by present data.
\begin{figure}
\begin{center}
\epsfysize=8cm \epsfbox{mwmh.ps}
\end{center}
\caption{$m_W$ versus $m_H$ for $m_t=176 \pm 2$ \rm{GeV}, both in the Standard
Model (SM) and in the maximum effective contributions (Eff).}
\label{uno}
\end{figure}
\begin{figure}
\begin{center}
\epsfysize=6cm \epsfbox{anpar.ps}
\end{center}
\caption{Constrained region for $\alpha_{\phi}^{(1)}$ and $\Lambda$.
$m_t=176$ \rm{GeV}.}
\label{dos}
\end{figure}
Figure \ref{dos} shows the level curves in the $\alpha-\Lambda$
plane according with the bounds discussed before for $m_W$ and $m_H$.
The allowed region of parameters corresponds to the area located to the
right of curves A, B, C, D. Allowing for an enlargement of the allowed
Higgs mass bound from $170 <m_H<330$ \rm{GeV}{} in the SM to $90 < m_H < 700$
\rm{GeV}{} into the effective Lagrangian case. The curve A is obtained
by taking $m_H=700$ \rm{GeV}{} and $m_W=80.32$ \rm{GeV}, while for the curve B we
use $m_H=90$ \rm{GeV}{} and $m_W=80.34$ \rm{GeV}. The shadowed area between the
curves (BD) marks the parameters region where no new physics effects can
be disentangled from the SM uncertainties. These results were obtained by
considering $m_t=176$ \rm{GeV}; it is found that there are not substantial
changes by adding the top mass uncertainty to the effective contributions.
It is also found that the allowed ranges for $\Lambda$ and
$\alpha_{\phi}^{(1)}$ are as follow: for $\Lambda=1$ \rm{TeV},
$.011< \alpha_{\phi}^{(1)} <.060$ and
$-.01010< \alpha_{\phi}^{(1)} < -.01019$. This correspond to Higgs masses
between the perturbative limit and upper SM bound in the first interval,
and between the lower SM bound (obtained from radiative corrections to
$m_W$) and the experimental limit for the second interval. In the
complementary case, i.e. taking $\alpha_{\phi}^{(1)}$ equal to its maximum
value, it is found that $4.1< \Lambda < 9.5$ \rm{TeV}, and $4.64 < \Lambda <
10$ \rm{TeV} with same considerations for the Higgs mass as in the first case.
Then, independiently of the value $\alpha_{\phi}^{(1)}$ effects arising
from a scale beyond of 10 \rm{TeV} can not be disentangled from SM top
uncertainties.
\section{Associated $W(Z)$ and $H$ production}
In this paper we also re-examined accordingly the modifications to the
mechanism of associated production of Higgs boson with a
vector particle ($W,Z$), due to the effective Lagrangian updating the
results obtained in Ref. \cite{lorent}. The corresponding Lagrangian
to be used is :
\begin{equation}
{\mathcal L}_{HVV} = \frac{m_Z}{2} (1 + f_1) H Z_{\mu} Z^{\mu}
+g m_W (1+f_2) H W^+_{\mu}W^{-\mu},
\end{equation}
where the parameter $f_i$ are functions of
$\epsilon_j=\alpha_j (\frac{v}{\Lambda})^2$
given as follows:
\begin{equation}
f_1=\dfrac{1}{2}(\epsilon_{\phi}^{(1)}
+\epsilon_{\phi}^{(3)} ), \hspace{2cm}
f_2=\dfrac{3}{4} (2\epsilon_{\phi}^{(1)}
-\epsilon_{\phi}^{(3)} ).
\end{equation}
The ratio of the effective cross-section
to the SM one for the processes $p\bar{p} \to H+V$ have been evaluated.
The parton convolution part is factored out, and only remains the ratio
of partonic cross-sections, thus the result is valid for both FNAL and
LHC. The expressions for the cross-sections ratios are:
\begin{eqnarray}
R_{HW}&=&\frac{\sigma_{\mathrm{eff}}(p\bar{p} \to H+W)}
{\sigma_{\mathrm{SM}}( p\bar{p} \to H +W)} \nonumber \\
&=& (1+f_2)^2
\end{eqnarray}
\begin{eqnarray}
R_{HZ}&=&\frac{\sigma_{\mathrm{eff}}( p\bar{p} \to H+Z)}
{\sigma_{\mathrm{SM}}( p\bar{p} \to H +Z)} \nonumber \\
&=& (1 + f_1)^2
\end{eqnarray}
For the operators under consideration, the cross-sections
ratio is independent of the Higgs mass, and as
the best values, it is found that the cross-section is only
slightly modified. The behavior of the cross-section ratios
are shown in fig. \ref{tres}, for typical values of $\alpha$
and $\Lambda$ as found in section 4. We consider the same
estimate for $\alpha_{\phi}^{(3)}$ as we got for $\alpha_{\phi}^{(1)}$.
\begin{figure}[!h]
\begin{center}
\epsfysize=6cm \epsfbox{ratiohv.ps}
\end{center}
\caption{Ratio of the effective cross section to the SM one for the
associated $HW,HZ$ production. It is assumed that $\alpha_{\phi}^{(3)}$
behaves in the same way as $\alpha_{\phi}^{(1)}$ does.}
\label{tres}
\end{figure}
As it can seen from figure \ref{tres},
these two processes result almost insensible to new physics
effects arising from the dimension-six operators $O_{\phi}^{(1,3)}$,
since their effects is of the order $10^{-3}$.
The corresponding contributions that arises from
the effective operators that were neglected are, at least, 2
orders of magnitude below that ones that are considered here. Of course,
a more detailed study is needed in order to include the modifications
to the expected number of events for discovery as a function of $m_H$,
and that case is under study.
\section{Conclusions}
We have studied the modifications that new physics imply for
the bound on the Higgs mass that is obtained from radiative corrections
to electroweak observables, within the context of effective
lagrangians. We found that the SM bound
$170 \le m_H \le 330$ \rm{GeV}{} that is obtained from a precise determination
of the $W$ mass, can be substantially modified by the presence of
dimension-6 operators that arise in the linear realization
of the effective Lagrangian approach. A Higgs mass as heavy as
700 \rm{GeV}{} is allowed for scales of new physics of the order of
1 \rm{TeV}, with a corresponding value for $|\alpha_{\phi}^{(1)}|$ of
the order of $10^{-2}$. Aswell we found that even for
$|\alpha_{\phi}^{(1)}|=1$, new physics effects arising from scales
$\Lambda > 10$ \rm{TeV}{} can not be separate from the uncertainties on
the top quark mass, in an optimum scenarios for the
observables considered here. Those results give us the landmark for the
decoupling limits for both $\alpha_i$ and $\Lambda$.
Accordingly, it is found that such operators do not
produce a significant modification for the present (FNAL) or future (LHC)
studies for the associated production mechanism $p\bar{p} \to H+V$.
\ack
We acknowledge financial support from CONACYT and SNI (MEXICO). We also
acknowledge to M.A. P\'erez for discussions.
|
2,869,038,154,821 | arxiv | \section{Introduction}\label{sec:intro}
\begin{figure}[t]
\centering
\subfloat[]{%
\label{fig:fig1_heatmap_a}
\includegraphics[clip,keepaspectratio,width=0.49\columnwidth]{images/fig1_heatmap_a.pdf}%
}
\subfloat[]{%
\label{fig:fig1_heatmap_b}
\includegraphics[clip,keepaspectratio,width=0.49\columnwidth]{images/fig1_heatmap_b.pdf}%
} \\
\vspace{-2.0ex}
\subfloat[]{%
\label{fig:fig1_heatmap_c}
\includegraphics[clip,keepaspectratio,width=0.49\columnwidth]{images/fig1_heatmap_c.pdf}%
}
\subfloat[]{%
\label{fig:fig1_heatmap_d}
\includegraphics[clip,keepaspectratio,width=0.49\columnwidth]{images/fig1_heatmap_d.pdf}%
}
\caption{Typical failure cases of conventional methods. The ground-truth human bounding boxes, object bounding boxes, object classes, and action classes are drawn with red boxes, blue boxes, blue characters, and yellow characters, respectively.}
\label{fig:fig1_heatmap}
\vspace{-2.0ex}
\end{figure}
Human-object interaction (HOI) detection has attracted enormous interest in recent years for its potential in deeper scene understanding~\cite{chao_wacv2018, gkioxari_cvpr2018, gao_bmvc2018, qi_eccv2018, gupta_iccv2019, li_cvpr2019, wan_iccv2019, wang_iccv2019, zhou_iccv2019, li_cvpr2020, liao_cvpr2020, ulutan_cvpr2020, wang_cvpr2020, zhou_cvpr2020, lin_ijcai2020, yang_ijcai2020, xu_tmm2020, gao_eccv2020, liu_eccv2020, kim_bumsoo_eccv2020, zhi_eccv2020, hai_eccv2020, zhong_eccv2020, kim_dong_eccv2020}.
Given an image, the task of HOI detection is to localize a human and object, and identify the interactions between them, typically represented as $\langle${\it human bounding box, object bounding box, object class, action class}$\rangle$.
Conventional HOI detection methods can be roughly divided into two types: two-stage methods~\cite{chao_wacv2018, gao_eccv2020, gao_bmvc2018, gkioxari_cvpr2018, gupta_iccv2019, zhi_eccv2020, kim_dong_eccv2020, li_cvpr2020, li_cvpr2019, lin_ijcai2020, liu_eccv2020, ulutan_cvpr2020, wan_iccv2019, hai_eccv2020, xu_tmm2020, yang_ijcai2020, zhong_eccv2020, zhou_iccv2019, qi_eccv2018, wang_iccv2019, zhou_cvpr2020} and single-stage methods~\cite{liao_cvpr2020, wang_cvpr2020, kim_bumsoo_eccv2020}.
In the two-stage methods, humans and objects are first individually localized by off-the-shelf object detectors, and then the region features from the localized area are used to predict action classes.
To incorporate contextual information, auxiliary features such as the features from the union region of a human and object bounding box, and locations of the bounding boxes in an image are often utilized.
The single-stage methods predict interactions using the features of a heuristically-defined position such as a midpoint between a human and object center~\cite{liao_cvpr2020}.
While both two- and single-stage methods have shown significant improvement,
they often suffer from errors attributed to the nature of convolutional neural networks (CNNs) and the heuristic way of using CNN features.
Figure~\ref{fig:fig1_heatmap} shows typical failure cases of conventional methods.
In Fig.~\ref{fig:fig1_heatmap_a}, we can easily recognize from an entire image that a boy is washing a car.
It is difficult, however, for two-stage methods to predict the action class ``wash" since they typically use only the cropped bounding-box regions.
The regions sometimes miss contextually important cues located outside the human and object bounding box such as the hose in Fig.~\ref{fig:fig1_heatmap_a}.
Even though the features of union regions may contain such cues, these regions are frequently dominated by disturbing contents such as background and irrelevant humans and objects.
Figure~\ref{fig:fig1_heatmap_b} shows an example where multiple HOI instances are overlapped.
In such a case, CNN-based feature extractors are forced to capture features of both instances in the overlapped region, ending up in obtaining contaminated features.
The detection based on the contaminated features easily results in failures.
The single-stage methods attempt to capture the contextual information by pairing a target human and object from an early stage in feature extraction and extracting integrated features rather than individually treating the targets.
To determine the regions from which integrated features are extracted, they rely on heuristically-designed location-of-interest such as a midpoint between a human and object center~\cite{liao_cvpr2020}.
However, such reliance sometimes causes a problem.
Fig.~\ref{fig:fig1_heatmap_c} shows an example where a target human and object are located distantly.
In this example, the midpoint is located close to the man in the middle, who is not relevant to the target HOI instance.
Therefore, it is difficult to detect the target on the basis of the features around the midpoint.
Fig.~\ref{fig:fig1_heatmap_d} is an example where the midpoints of multiple HOI instances are close to each other.
In this case, CNN-based methods tend to make mis-detection due to the same reason as the one for the failure in Fig.~\ref{fig:fig1_heatmap_b}, \ie contaminated features.
To overcome these drawbacks, we propose QPIC, a query-based HOI detector that detects a human and object in a pairwise manner with image-wide contextual information.
QPIC has a transformer~\cite{vaswani_nips2017} as a key component.
The attention mechanism used in QPIC scans through the entire area of an image and is expected to selectively aggregate contextually important information according to the contents of an image.
Moreover, we design QPIC's queries so that each query captures at most one human-object pair.
This enables to separately extract features of multiple HOI instances without contaminating them even when the instances are located closely.
These key designs of the attention mechanism and query-based pairwise detection make QPIC robust even under the difficult conditions such as the case where contextually important information appears outside the human and object bounding box (Fig.~\ref{fig:fig1_heatmap_a}), the target human and object are located distantly (Fig.~\ref{fig:fig1_heatmap_c}), and multiple instances are close to each other (Fig.~\ref{fig:fig1_heatmap_b} and~\ref{fig:fig1_heatmap_d}).
The key designs produce so effective embeddings that the subsequent detection heads may be fairly simple and intuitive.
To summarize, our contributions are three-fold:
(1) We propose a simple yet effective query-based HOI detector, QPIC, which incorporates contextually important information aggregated image-wide. To the best of our knowledge, this is the first work to introduce an attention- and query-based method to HOI detection.
(2) We achieve significantly better performance than state-of-the-art methods on two challenging HOI detection benchmarks.
(3) We conduct detailed analysis on the behavior of QPIC in relation to that of conventional methods, and reveal some of the important characteristics of HOI detection tasks that conventional methods could not capture but QPIC does relatively well.
\section{Related Work}
Two-stage HOI detection methods~\cite{chao_wacv2018, gao_eccv2020, gao_bmvc2018, gkioxari_cvpr2018, gupta_iccv2019, zhi_eccv2020, kim_dong_eccv2020, li_cvpr2020, li_cvpr2019, lin_ijcai2020, liu_eccv2020, ulutan_cvpr2020, wan_iccv2019, hai_eccv2020, xu_tmm2020, yang_ijcai2020, zhong_eccv2020, zhou_iccv2019, qi_eccv2018, wang_iccv2019, zhou_cvpr2020} utilize Faster R-CNN~\cite{ren_nips2015} or Mask R-CNN~\cite{he_iccv2017} to localize targets. Then, they crop features of backbone networks inside the localized regions. The cropped features are typically processed with multi-stream networks. Each stream processes features of target humans, those of objects, and some auxiliary features such as spatial configurations of the targets, and human poses either alone or in combination. Some of the two-stage methods~\cite{qi_eccv2018, ulutan_cvpr2020, hai_eccv2020, yang_ijcai2020, zhou_iccv2019} utilize graph neural networks to refine the features. These methods mainly focus on the second stage architecture, which uses cropped features to predict action classes. However, the cropped features sometimes lack contextual information outside the cropped regions or are contaminated by features of irrelevant targets, which results in the degradation of the performance.
Recently, single-stage methods~\cite{liao_cvpr2020, wang_cvpr2020, kim_bumsoo_eccv2020} that utilize integrated features from a pair of a human and object have been proposed to solve the problem in the individually cropped features.
Liao \etal~\cite{liao_cvpr2020} and Wang \etal~\cite{wang_cvpr2020} proposed a point-based interaction detection method that utilizes CenterNet~\cite{zhou_center_arxiv2019} as a base detector.
This method predicts action classes using integrated features collected at a midpoint between a human and object center.
In particular, Liao \etal's PPDM~\cite{liao_cvpr2020} achieves simultaneous object and interaction detection training, which is the most similar to our training approach. Kim \etal~\cite{kim_bumsoo_eccv2020} proposed UnionDet, which predicts the union bounding box of a human-object pair to extract integrated features. Although these methods attempt to capture contextual information by integrated features, they are still insufficient and sometimes contaminated due to the CNN's locality and heuristically-designed location-of-interests.
Our method differs from conventional methods in that we leverage a transformer to aggregate image-wide contextual features in a pairwise manner. We use DETR~\cite{carion_eccv2020} as a base detector and extend it for HOI detection.
\section{Proposed Method}
\begin{figure*}[t]
\centering
\includegraphics[keepaspectratio,width=0.9\linewidth]{images/fig2_arch.pdf}
\caption{Overall architecture of the proposed QPIC.}\label{fig:fig2_arch}
\vspace{-2.0ex}
\end{figure*}
To effectively extract important features for each HOI instance taking image-wide contexts into account, we propose to leverage transformer-based architecture as a base feature extractor.
We first explain the overall architecture in Sec.~\ref{subsec:architecture} and show that the detection heads following the base feature extractor can be simplified due to the rich features obtained in the base feature extractor.
In Sec.~\ref{subsec:int_train}, we show the concrete formulation of the loss function involved in the training.
Finally we explain how to use our method to detect HOI instances given a new image in Sec.~\ref{subsec:inference}.
\subsection{Overall Architecture}\label{subsec:architecture}
Figure~\ref{fig:fig2_arch} illustrates the overall architecture of QPIC.
Given an input image $\bm{x} \in \mathbb{R}^{3 \times H \times W}$, a feature map $\bm{z_b} \in \mathbb{R}^{D_{b} \times H' \times W'}$ is calculated by an arbitrary off-the-shelf backbone network, where $H$ and $W$ are the height and width of the input image, $H'$ and $W'$ are those of the output feature map, and $D_{b}$ is the number of channels.
Typically $H' < H$ and $W' < W$.
$\bm{z_b}$ is then input to a projection convolution layer with a kernel size of $1 \times 1$ to reduce the dimension from $D_b$ to $D_c$.
The transformer encoder takes this feature map with the reduced dimension $\bm{z_c} \in \mathbb{R}^{D_{c} \times H' \times W'}$ to produce another feature map with richer contextual information on the basis of the self-attention mechanism.
A fixed positional encoding $\bm{p} \in \mathbb{R}^{D_{c} \times H' \times W'}$~\cite{bello_iccv2019, parmar_icml2018, carion_eccv2020} is additionally input to the encoder to supplement the positional information, which the self-attention mechanism alone cannot inherently incorporate.
The encoded feature map $\bm{z_e} \in \mathbb{R}^{D_{c} \times H' \times W'}$ is then obtained as
$\bm{z_e} = f_{enc}\left(\bm{z_c}, \bm{p}\right)$, where $f_{enc}\left(\cdot, \cdot \right)$ is a set of stacked transformer encoder layers.
The transformer decoder transforms a set of learnable query vectors $\bm{Q} = \{\bm{q}_{i} | \bm{q}_{i} \in \mathbb{R}^{D_{c}}\}_{i=1}^{N_{q}}$ into a set of embeddings $\bm{D} = \{\bm{d}_{i} | \bm{d}_{i} \in \mathbb{R}^{D_{c}}\}_{i=1}^{N_{q}}$ that contain image-wide contextual information for HOI detection, referring to the encoded feature map $\bm{z_e}$ using the attention mechanism.
$N_q$ is the number of query vectors.
The queries are designed in such a way that one query captures at most one human-object pair and an interaction(s) between them.
$N_q$ is therefore set to be large enough so that it is always larger than the number of actual human-object pairs in an image.
The decoded embeddings are then obtained as
$\bm{D} = f_{dec}\left(\bm{z_e}, \bm{p}, \bm{Q}\right)$,
where $f_{dec}\left(\cdot, \cdot, \cdot \right)$ is a set of stacked transformer decoder layers.
We use a positional encoding $\bm{p}$ again to incorporate the spatial information.
The subsequent interaction detection heads further processes the decoded embeddings to produce $N_q$ prediction results.
Here, we note that one or more HOIs corresponding to a human-object pair are mathematically defined by the following four vectors:
a human-bounding-box vector normalized by the corresponding image size $\bm{b}^{(h)} \in [0, 1]^4$,
a normalized object-bounding-box vector $\bm{b}^{(o)} \in [0, 1]^4$,
an object-class one-hot vector $\bm{c} \in \{0, 1\}^{N_{obj}}$, where $N_{obj}$ is the number of object classes,
and an action-class vector $\bm{a} \in \{0, 1\}^{N_{act}}$, where $N_{act}$ is the number of action classes.
Note that $\bm{a}$ is not necessarily a one-hot vector because there may be multiple actions that correspond to a human-object pair.
Our interaction detection heads are composed of four small feed-forward networks (FFNs):
human-bounding-box FFN $f_{h}$,
object-bounding-box FFN $f_{o}$,
object-class FFN $f_{c}$,
and action-class FFN $f_{a}$, each of which is dedicated to predict one of the aforementioned 4 vectors, respectively.
This design of the interaction detection heads is fairly intuitive and simple compared with a number of state-of-the-art methods such as the point-detection and point-matching branch in PPDM~\cite{liao_cvpr2020} and the human, object, and spatial-semantic stream in DRG~\cite{gao_eccv2020}.
Thanks to the powerful embeddings that contain image-wide contextual information, QPIC does not have to rely on a rather complicated and heuristic design to produce the prediction.
One thing to note is that
unlike many existing methods~\cite{chao_wacv2018, gao_eccv2020, gao_bmvc2018, gkioxari_cvpr2018, gupta_iccv2019, zhi_eccv2020, kim_dong_eccv2020, li_cvpr2020, li_cvpr2019, lin_ijcai2020, liu_eccv2020, ulutan_cvpr2020, wan_iccv2019, hai_eccv2020, xu_tmm2020, yang_ijcai2020, zhong_eccv2020, zhou_iccv2019, qi_eccv2018, wang_iccv2019, zhou_cvpr2020}, which first attempt to detect humans and objects individually and later pair them to find interactions, it is crucial to design queries in such a way that one query directly captures a human and object as a pair to more effectively extract features for interactions.
We will experimentally verify this claim in Sec.~\ref{subsec:head}.
The prediction of normalized human bounding boxes
$\{\bm{\hat{b}}^{(h)}_i | \bm{\hat{b}}^{(h)}_i \in [0, 1]^4\}_{i=1}^{N_q}$,
that of object bounding boxes $\{\bm{\hat{b}}^{(o)}_i | \bm{\hat{b}}^{(o)}_i \in [0, 1]^4\}_{i=1}^{N_q}$,
the probability of object classes $\{\bm{\hat{c}}_i | \bm{\hat{c}}_i \in [0, 1]^{N_{obj} + 1}, \sum_{j=1}^{N_{obj} + 1}{\bm{\hat{c}}_i}(j)=1\}_{i=1}^{N_q}$, where $\bm{v}(j)$ denotes the $j$-th element of $\bm{v}$,
and the probability of action classes $\{\bm{\hat{a}}_i | \bm{\hat{a}}_i \in [0, 1]^{N_{act}}\}_{i=1}^{N_q}$,
are calculated as
$\bm{\hat{b}}^{(h)}_{i} = \sigma\left(f_{h}\left(\bm{d}_{i}\right)\right),
\bm{\hat{b}}^{(o)}_{i} = \sigma\left(f_{o}\left(\bm{d}_{i}\right)\right),
\bm{\hat{c}}_{i} = \varsigma\left(f_{c}\left(\bm{d}_{i}\right)\right),
\bm{\hat{a}}_{i} = \sigma\left(f_{a}\left(\bm{d}_{i}\right)\right)
$
, respectively.
$\sigma, \varsigma$ are the sigmoid and softmax functions, respectively.
Note that $\bm{\hat{c}}_{i}$ has the $(N_{obj}+1)$-th element to indicate that the $i$-th query has no corresponding human-object pair, while an additional element of $\bm{\hat{a}}_{i}$ to indicate ``no action'' is not necessary because we use the sigmoid function rather than the softmax function to calculate the action-class probabilities for co-occuring actions.
\subsection{Loss Calculation}\label{subsec:int_train}
The loss calculation is composed of two stages: the bipartite matching stage between predictions and ground truths, and the loss calculation stage for the matched pairs.
For the bipartite matching,
we follow the training procedure of DETR~\cite{carion_eccv2020} and use the Hungarian algorithm~\cite{kuhn_1955}.
Note that this design obviates the process of suppressing over-detection as described in~\cite{carion_eccv2020}.
We first pad the ground-truth set of human-object pairs with $\phi$ (no pairs) so that the ground-truth set size becomes $N_q$.
We then leverage the Hungarian algorithm to determine the optimal assignment $\hat{\omega}$ among the set of all possible permutations of $N_q$ elements $\bm{\Omega}_{N_q}$,
i.e. $\hat{\omega} = \argmin_{\omega \in \bm{\Omega}_{N_q}}{\sum_{i=1}^{N_q}{\mathcal{H}_{i,\omega(i)}}}$, where $\mathcal{H}_{i,j}$ is the matching cost for the pair of $i$-th ground truth and $j$-th prediction.
The matching cost $\mathcal{H}_{i, j}$ consists of four types of costs:
the box-regression cost $\mathcal{H}^{(b)}_{i, j}$, intersection-over-union (IoU) cost $\mathcal{H}^{(u)}_{i, j}$, object-class cost $\mathcal{H}^{(c)}_{i, j}$, and action-class cost $\mathcal{H}^{(a)}_{i, j}$.
Denoting $i$-th ground truth for
the normalized human bounding box by $\bm{b}_{i}^{(h)} \in [0, 1]^4$,
normalized object bounding box by $\bm{b}_{i}^{(o)} \in [0, 1]^4$,
object-class one-hot vector by $\bm{c}_{i} \in \{0, 1\}^{N_{obj}}$,
and action class by $\bm{a}_{i} \in \{0, 1\}^{N_{act}}$,
the aforementioned costs are formulated as follows.
\begin{align}
\mathcal{H}_{i, j} ={} & \mathbbm{1}_{\{i \not\in \bm{\Phi}\}}\left[\eta_{b} \mathcal{H}^{(b)}_{i, j} + \eta_{u} \mathcal{H}^{(u)}_{i, j} + \eta_{c} \mathcal{H}^{(c)}_{i, j} + \eta_{a} \mathcal{H}^{(a)}_{i, j}\right], \\
\mathcal{H}^{(b)}_{i, j} ={} & \max\left\{\left\|\bm{b}^{(h)}_{i} - \bm{\hat{b}}^{(h)}_{j}\right\|_{1}, \left\|\bm{b}^{(o)}_{i} - \bm{\hat{b}}^{(o)}_{j}\right\|_{1}\right\},\label{eq:cost_box} \\
\nonumber\mathcal{H}^{(u)}_{i, j} ={} & \max\left\{-GIoU\left(\bm{b}^{(h)}_{i}, \bm{\hat{b}}^{(h)}_{j}\right),\right.\\
&{} \qquad\qquad \left. -GIoU\left(\bm{b}^{(o)}_{i}, \bm{\hat{b}}^{(o)}_{j}\right)\right\},\label{eq:cost_iou} \\
\mathcal{H}^{(c)}_{i, j} ={} & -\bm{\hat{c}}_j(k)\quad s.t.\quad\bm{c}_i(k)=1, \\
\mathcal{H}^{(a)}_{i, j} ={} & -\frac{1}{2}\left(
\frac{\bm{a}^{\intercal}_{i}\bm{\hat{a}}_{j}}
{\left\|\bm{a}_{i}\right\|_{1} + \epsilon}
+ \frac{\left(\bm{1} - \bm{a}_{i}\right)^{\intercal}\left(\bm{1} - \bm{\hat{a}}_{j}\right)}
{\left\|\bm{1} - \bm{a}_{i}\right\|_{1} + \epsilon}
\right),\label{eq:cost_action}
\end{align}
where $\bm{\Phi}$ is a set of ground-truth indices that correspond to $\phi$,
$GIoU\left(\cdot, \cdot \right)$ is the generalized IoU~\cite{rezatofighi_cvpr2019},
$\epsilon$ is a small positive value introduced to avoid zero divide,
and $\eta_b$, $\eta_u$, $\eta_c$, and $\eta_a$ are the hyper-parameters.
We use two types of bounding-box cost $\mathcal{H}^{(b)}_{i, j}$ and $\mathcal{H}^{(u)}_{i, j}$ following~\cite{carion_eccv2020}.
In calculating $\mathcal{H}^{(b)}_{i, j}$ and $\mathcal{H}^{(u)}_{i, j}$, instead of minimizing the average of a human and object-bounding-box cost, we minimize the larger of the two to prevent the matching from being undesirably biased to either if one cost is significantly lower than the other.
We design $\mathcal{H}^{(a)}_{i, j}$ so that the costs of both positive and negative action classes are taken into account.
In addition, we formulate it using the weighted average of the two with the inverse number of nonzero elements as the weights rather than using the vanilla average.
This is necessary to balance the effect from the two costs because the number of positive action classes is typically much smaller than that of negative action classes.
The loss to be minimized in the training phase is calculated on the basis of the matched pairs
as follows.
\begin{align}
\mathcal{L} ={} & \lambda_{b} \mathcal{L}_{b} + \lambda_{u} \mathcal{L}_{u} + \lambda_{c} \mathcal{L}_{c} + \lambda_{a} \mathcal{L}_{a}, \\
\nonumber\mathcal{L}_{b} ={} & \frac{1}{|\bar{\bm{\Phi}}|} \sum_{i=1}^{N_{q}} \mathbbm{1}_{\{i \not\in \bm{\Phi}\}}\left[
\left\|\bm{b}^{(h)}_i - \bm{\hat{b}}^{(h)}_{\hat{\omega}\left(i\right)}\right\|_{1} \right.\\
&{} \qquad\qquad\qquad\qquad \left. + \left\|\bm{b}^{(o)}_i - \bm{\hat{b}}^{(o)}_{\hat{\omega}\left(i\right)}\right\|_{1}\right], \\
\nonumber\mathcal{L}_{u} ={} & \frac{1}{|\bar{\bm{\Phi}}|} \sum_{i=1}^{N_{q}} \mathbbm{1}_{\{i \not\in \bm{\Phi}\}}\left[2 -
GIoU\left(\bm{b}^{(h)}_i, \bm{\hat{b}}^{(h)}_{\hat{\omega}\left(i\right)}\right) \right.\\
&{} \qquad\qquad\qquad\qquad \left. - GIoU\left(\bm{b}^{(o)}_i, \bm{\hat{b}}^{(o)}_{\hat{\omega}\left(i\right)}\right)\right], \\
\nonumber\mathcal{L}_{c} ={} & \frac{1}{N_{q}}\sum_{i=1}^{N_{q}}\left\{
\mathbbm{1}_{\{i \not\in \bm{\Phi}\}}\left[-\log\bm{\hat{c}_{\hat{\omega}\left(i\right)}}(k)\right]\right.\\
\nonumber&{} \qquad\quad \left. + \mathbbm{1}_{\{i \in \bm{\Phi}\}}\left[-\log\bm{\hat{c}_{\hat{\omega}\left(i\right)}}(N_{obj}+1)\right]\right\}\\
&{} \qquad\qquad\quad s.t.\quad\bm{c}_i(k)=1, \\
\nonumber\mathcal{L}_{a} ={} & \frac{1}{\sum_{i=1}^{N_{q}}\mathbbm{1}_{\{i \not\in \bm{\Phi}\}}\left\|\bm{a}_i\right\|_{1}}\sum_{i=1}^{N_{q}}\left\{
\mathbbm{1}_{\{i \not\in \bm{\Phi}\}}\left[l_{f}\left(\bm{a}_i, \bm{\hat{a}_{\hat{\omega}\left(i\right)}}\right)\right]\right.\\
&{} \qquad\qquad \left. + \mathbbm{1}_{\{i \in \bm{\Phi}\}}\left[l_{f}\left(\bm{0}, \bm{\hat{a}_{\hat{\omega}\left(i\right)}}\right)\right]
\right\},
\end{align}
where $\lambda_{b}$, $\lambda_{u}$, $\lambda_{c}$ and $\lambda_{a}$ are the hyper-parameters for adjusting the weights of each loss, and $l_{f}\left(\cdot, \cdot\right)$ is the element-wise focal loss function~\cite{lin_iccv2017}.
For the hyper-parameters of the focal loss, we use the default settings described in~\cite{zhou_center_arxiv2019}.
\subsection{Inference for Interaction Detection}\label{subsec:inference}
As previously mentioned, the detection result of an HOI is represented by the following four components, $\langle${\it human bounding box, object bounding box, object class, action class}$\rangle$.
Our interaction detection heads are designed so intuitively that all we need to do is to pick up the corresponding information from each head.
Formally, we set the prediction results corresponding to the $i$-th query and $j$-th action as $\langle\bm{\hat{b}^{(h)}_{i}}, \bm{\hat{b}^{(o)}_{i}}, \argmax_{k}{\bm{\hat{c}_{i}}(k)}, j\rangle$.
We define a score of the HOI instance as
$\left\{\max_{k}{\bm{\hat{c}_{i}}(k)}\right\}\bm{\hat{a}_{i}}(j)$,
and regard this instance to be present if the score is higher than a threshold.
\section{Experiments}
\subsection{Datasets and Evaluation Metrics}
We conducted extensive experiments on two HOI detection datasets: HICO-DET~\cite{chao_wacv2018} and V-COCO~\cite{gupta_arxiv2015}.
We followed the standard evaluation scheme.
HICO-DET contains 38,118 and 9,658 images for training and testing, respectively.
The images are annotated with 80 object and 117 action classes.
V-COCO, which originates from the COCO dataset, contains 2,533, 2,867, and 4,946 images for training, validation, and testing, respectively.
The images are annotated with 80 object and 29 action classes.
For the evaluation metrics, we use the mean average precision (mAP).
A detection result is judged as a true positive if the predicted human and object bounding box have IoUs larger than 0.5 with the corresponding ground-truth bounding boxes, and the predicted action class is correct. In the HICO-DET evaluation, the object class is also taken into account for the judgment. The AP is calculated per object and action class pair in the HICO-DET evaluation, while that is calculated per action class in the V-COCO evaluation.
For HICO-DET, we evaluate the performance in two different settings following~\cite{chao_wacv2018}: {\it default setting} and {\it known-object setting}.
In the former setting, APs are calculated on the basis of all the test images, while in the latter setting, each AP is calculated only on the basis of images that contain the object class corresponding to each AP.
In each setting, we report the mAP over three set types: a set of 600 HOI classes ({\it full}), a set of 138 HOI classes that have less than 10 training instances ({\it rare}), and a set of 462 HOI classes that have 10 or more training instances ({\it non-rare}).
Unless otherwise stated, we use the default full setting in the analysis.
In V-COCO, a number of HOIs are defined with no object labels.
To deal with this situation, we evaluate the performance in two different scenarios following V-COCO's official evaluation scheme.
In scenario 1, detectors are required to report cases in which there is no object, while in scenario 2, we just ignore the prediction of an object bounding box in these cases.
\subsection{Implementation Details}
We use ResNet-50 and ResNet-101~\cite{he_cvpr2016} as a backbone feature extractor.
Both transformer encoder and decoder consist of 6 transformer layers with a multi-head attention of 8 heads. The reduced dimension size $D_c$ is set to 256, and the number of query vectors $N_q$ is set to 100.
The human- and object-bounding-box FFNs have 3 linear layers with ReLU activations, while the object- and action-class FFNs have 1 linear layer.
For training QPIC, we initialize the network with the parameters of DETR~\cite{carion_eccv2020} trained with the COCO dataset. Note that for the V-COCO training, we exclude the COCO's training images that are contained in the V-COCO test set when pre-training DETR\footnote{A few previous works inappropriately use COCO train2017 set for pre-training, whose images are contained in the V-COCO test set.}.
QPIC is trained for 150 epochs using the AdamW~\cite{loshchiloy_iclr2019} optimizer with the batch size 16, initial learning rate of the backbone network $10^{-5}$, that of the others $10^{-4}$, and the weight decay $10^{-4}$. Both learning rates are decayed after 100 epochs. The hyper-parameters for the Hungarian costs $\eta_b, \eta_u, \eta_c,$ and $\eta_a$, and those for the loss weights $\lambda_b, \lambda_u, \lambda_c,$ and $\lambda_a$ are set to 2.5, 1, 1, 1, 2.5, 1, 1, and 1, respectively.
Following~\cite{liao_cvpr2020}, we select 100 high scored detection results from all the predictions for fair comparison.
Please see the supplementary material for more details.
\subsection{Comparison to State-of-the-Art}\label{subsec:sota}
\begin{table}[t]
\caption{Comparison against state-of-the-art methods on HICO-DET. The top, middle, and bottom blocks show the mAPs of the two-stage, single-stage, and our methods, respectively.}
\label{table:comp_hico}
\centering
\small
\setlength{\tabcolsep}{2pt}
\begin{tabular}{@{}lcccccc@{}}
\toprule
& \multicolumn{3}{c}{Default} & \multicolumn{3}{c}{Known object} \\
\cmidrule(lr){2-4}\cmidrule(lr){5-7}
Method & full & rare & non-rare & full & rare & non-rare \\
\midrule
FCMNet~\cite{liu_eccv2020} & 20.41 & 17.34 & 21.56 & 22.04 & 18.97 & 23.13 \\
ACP~\cite{kim_dong_eccv2020} & 20.59 & 15.92 & 21.98 & -- & -- & -- \\
VCL~\cite{zhi_eccv2020} & 23.63 & 17.21 & 25.55 & 25.98 & 19.12 & 28.03 \\
DRG~\cite{gao_eccv2020} & 24.53 & 19.47 & 26.04 & 27.98 & 23.11 & 29.43 \\
\midrule
UnionDet~\cite{kim_bumsoo_eccv2020} & 17.58 & 11.72 & 19.33 & 19.76 & 14.68 & 21.27 \\
Wang \etal~\cite{wang_cvpr2020} & 19.56 & 12.79 & 21.58 & 22.05 & 15.77 & 23.92 \\
PPDM~\cite{liao_cvpr2020} & 21.73 & 13.78 & 24.10 & 24.58 & 16.65 & 26.84 \\
\midrule
Ours (ResNet-50) & 29.07 & 21.85 & 31.23 & 31.68 & 24.14 & 33.93 \\
Ours (ResNet-101) & \textbf{29.90} & \textbf{23.92} & \textbf{31.69} & \textbf{32.38} & \textbf{26.06} & \textbf{34.27} \\
\bottomrule
\end{tabular}
\vspace{-1.0ex}
\end{table}
\begin{table}[t]
\caption{Comparison against state-of-the-art methods on V-COCO. The split of the blocks are the same as Table~\ref{table:comp_hico}.}
\label{table:comp_vcoco}
\centering
\small
\begin{tabular}{@{}lcc@{}}
\toprule
Method & Scenario 1 & Scenario 2 \\
\midrule
VCL~\cite{zhi_eccv2020} & 48.3 & -- \\
DRG~\cite{gao_eccv2020} & 51.0 & -- \\
ACP~\cite{kim_dong_eccv2020} & 53.0 & -- \\
FCMNet~\cite{liu_eccv2020} & 53.1 & -- \\
\midrule
UnionDet~\cite{kim_bumsoo_eccv2020} & 47.5 & 56.2 \\
Wang \etal~\cite{wang_cvpr2020} & 51.0 & -- \\
\midrule
Ours (ResNet-50) & \textbf{58.8} & \textbf{61.0} \\
Ours (ResNet-101) & 58.3 & 60.7\\
\bottomrule
\end{tabular}
\vspace{-2.0ex}
\end{table}
\begin{figure*}[t]
\centering
\subfloat[AP depending on the distance between a human and object center.]{%
\label{fig:fig5_distance_analysis}
\includegraphics[clip,keepaspectratio,width=1.\columnwidth]{images/fig5_distance_analysis.pdf}%
}
\subfloat[AP depending on the larger area of a human and object bounding box.]{%
\label{fig:fig5_area_analysis}
\includegraphics[clip,keepaspectratio,width=1.\columnwidth]{images/fig5_area_analysis.pdf}%
}
\caption{Performance analysis on different spatial distribution of HOIs evaluated on HICO-DET.}\label{fig:space_analysis}
\vspace{-2.0ex}
\end{figure*}
We first show the comparison of our QPIC with the latest HOI detection methods including both two- and single-stage methods in Table~\ref{table:comp_hico}.
As seen from the table, QPIC outperforms both state-of-the-art two- and single-stage methods in all the settings.
QPIC with the ResNet-101 backbone yields an especially significant gain of 5.37 mAP (relatively 21.9\%) compared with DRG~\cite{gao_eccv2020} and 8.17 mAP (37.6\%) compared with PPDM~\cite{liao_cvpr2020} in the default full setting.
Table~\ref{table:comp_vcoco} shows the comparison results on V-COCO. QPIC achieves state-of-the-art performance among all the baseline methods. QPIC with the ResNet-50 backbone achieves a 5.7 mAP (10.7\%) gain over FCMNet~\cite{liu_eccv2020}, which is the strongest baseline. Unlike in the HICO-DET result, the ResNet-50 backbone shows better performance than the ResNet-101 backbone probably because the number of training samples in V-COCO is insufficient to train the large network.
Overall, these comparison results demonstrate the dataset-invariant effectiveness of QPIC.
We then investigate in which cases QPIC especially achieves superior performance compared with the strong baselines.
To do so, we compare the performance of QPIC in detail with DRG~\cite{gao_eccv2020} and PPDM~\cite{liao_cvpr2020}, which are the strongest baselines of the two- and single-stage methods, respectively.
We use the ResNet-50 backbone for QPIC in this comparison.
Note that hereinafter the distance and area are calculated in normalized image coordinates.
Figure~\ref{fig:fig5_distance_analysis} shows how the performances change as the distance between the center points of a paired human and object bounding box grows.
We split HOI instances into bins of size 0.1 according to the distances, and calculate the APs of each bin that has at least 1,000 HOI instances.
As shown in Fig.~\ref{fig:fig5_distance_analysis}, the relative gaps of the performance between QPIC and the other two methods become more evident as the distance grows.
The graph suggests three things;
HOI detection tends to become more difficult as the distance grows,
the distant case is especially difficult for CNN-based methods,
and QPIC relatively better deals with this difficulty.
The possible explanation for these results is that the features of the CNN-based methods, which rely on limited receptive fields for the feature aggregation,
cannot include contextually important information or are dominated by irrelevant information in the distant cases, while the features of QPIC are more effective thanks to the ability of selectively extracting image-wide contextual information.
Figure~\ref{fig:fig5_area_analysis} presents how the performances change as the areas of target human and object bounding boxes grow.
We pick up the larger area of a target human and object bounding box involved in each HOI instance.
We then split HOI instances into bins of size 0.1 according to the area, and calculate the APs of each bin that has at least 1,000 HOI instances.
As illustrated in Fig.~\ref{fig:fig5_area_analysis},
the gaps of the APs between the conventional methods and QPIC tend to grow as the area increases.
This is probably because of the combination of the following two reasons;
if the area becomes bigger, the area tends to more often include harmful regions such as another HOI instance,
and the conventional methods mix up the irrelevant features in such situation, whereas the attention mechanism and the query-based framework enable to selectively aggregate effective features in a separated manner for each HOI instance.
These results reveal that the QPIC's significant improvement shown in Table~\ref{table:comp_hico} and Table~\ref{table:comp_vcoco} is likely to be brought by its nature of robustness to diverse spatial distribution of HOIs, probably originating from its capability of aggregating image-wide contextual features for each HOI instance. This observation is further confirmed qualitatively in Sec.~\ref{sec:quality}.
\subsection{Ablation Study}\label{subsec:quantitative}
To understand the key ingredients of QPIC's superiority shown in Sec.~\ref{subsec:sota}, we analyze the key building blocks one by one in detail. We first analyze the interaction detection heads in Sec.~\ref{subsec:head} and subsequently analyze the transformer-based feature extractor in Sec.~\ref{subsec:feature_extractor}.
\vspace{-2.0ex}
\subsubsection{Analysis on Detection Heads}\label{subsec:head}
\begin{figure*}[t]
\centering
\subfloat[Interaction detection with point matching.]{%
\label{fig:fig3_variants_b}
\includegraphics[clip,keepaspectratio,width=0.92\columnwidth]{images/fig3_variants_b.pdf}%
}
\subfloat[Interaction detection with two-stage like approach.]{%
\label{fig:fig3_variants_a}
\includegraphics[clip,keepaspectratio,width=1.\columnwidth]{images/fig3_variants_a.pdf}%
}
\caption{Implemented variants for analyzing detection heads. These heads are on top of our transformer-based feature extractor.}\label{fig:variants}
\vspace{-2.0ex}
\end{figure*}
\begin{table}[t]
\caption{Evaluation results of the various detection heads.}
\label{table:variants}
\centering
\small
\setlength{\tabcolsep}{4pt}
\begin{tabular}{@{}ccc@{}}
\toprule
Base method & Detection heads & HICO-DET (mAP) \\
\midrule
\multirow{3}{*}{\shortstack{Ours\\ (ResNet-50)}} & Simple (original) & 29.07 \\
& Point matching (Fig.~\ref{fig:fig3_variants_b}) & 29.04 \\
& Two-stage like (Fig.~\ref{fig:fig3_variants_a}) & 26.18 \\
\midrule
\multirow{2}{*}{{\shortstack{PPDM~\cite{liao_cvpr2020}\\ (Hourglass-104)}}} & Simple & 17.45 \\
& Point matching (original) & 21.73 \\
\bottomrule
\end{tabular}
\vspace{-2.0ex}
\end{table}
\paragraph{Feasibility of simple heads.}
As previously mentioned, the inference process of QPIC is simplified thanks to the enriched features from the transformer-based feature extractor.
To confirm that this simple prediction is sufficient for QPIC, we investigated if the detection accuracy increases by leveraging a typical point-matching-based detection heads presented in~\cite{liao_cvpr2020}, which is one of the best performing heuristically-designed heads.
Figure~\ref{fig:fig3_variants_b} represents the implemented heads.
A notable difference from the original simple heads lies in that the interaction detection heads output center points of target humans and objects instead of bounding boxes. Consequently, the outputs from the interaction detection heads need to be fused with the outputs from the object detection heads with point matching.
Note that in this implementation, duplicate detection results that share an identical human-object pair needs to be suppressed by some means such as non-maximum suppression.
Table~\ref{table:variants} shows the evaluation results.
As seen from Table~\ref{table:variants}, the point-matching-based heads exhibit no performance improvement over the simple heads, which indicates that the simple detection heads are enough and we do not have to manually design complicated detection heads.
\vspace{-2.0ex}
\paragraph{Importance of pairwise detection.}
Although the detection heads can be as simple as we present, we claim that there is a crucial aspect that must be covered in the design of the heads. It is to treat a target human and object as a pair from early stages rather than to first detect them individually and later integrate the features from the cropped regions corresponding to the detection, as typically done in two-stage approaches.
We assume the features from the cropped regions do not contain enough contextual information because sometimes the regions in an image other than a human and object bounding boxes play a crucial role in HOI detection (see Fig.~\ref{fig:fig1_heatmap_a} for example).
We verify this claim by looking into the performance of the two-stage like detection-heads on top of our transformer-based feature extractor, which is exactly the same as original QPIC.
Figure~\ref{fig:fig3_variants_a} illustrates the implemented detection heads.
This model first derives object detection results from the object detection heads.
Then, the results are used to create all the possible human-object pairs.
The features of each pair is constructed by concatenating the features from the human and object bounding boxes.
The interaction detection head predicts action classes of all the pairs on the basis of the concatenated features.
As seen from Table~\ref{table:variants}, two-stage like method yields worse performance than the original.
This observation indicates that the two-stage methods, which rely on individual feature extraction, do not perform well even with our strong feature extractor, and suggests the importance of the pairwise feature extraction in heads for HOI detection.
\vspace{-2.0ex}
\subsubsection{Analysis on Feature Extractor}\label{subsec:feature_extractor}
\paragraph{Importance of a transformer.}
To confirm that a transformer-based feature extractor is key to make the simple heads sufficiently work for HOI detection as discussed in Sec.~\ref{subsec:head}, we replace QPIC's transformer-based feature extractor by a CNN-based counterpart and examine how the performance changes.
We utilize the Hourglass-104 backbone used in PPDM~\cite{liao_cvpr2020} in this experiment.
Table~\ref{table:variants} shows the performance of the original point-matching-based PPDM as well as its simple-heads variant.
The simple-heads variant directly predicts all the information corresponding to a human-object pair on the basis of the features extracted in the feature-extraction stage, just as QPIC's simple heads do.
More concretely, not only a human point, an object point, and action classes, but also a human-bounding-box size, an object-bounding-box size, and an object class are directly predicted on the basis of the features at the midpoint between the human and object centers.
As Table~\ref{table:variants} shows, the simple-heads variant exhibits far worse performance than QPIC.
This implicates that the CNN-based feature extractor is not as powerful as our transformer-based feature extractor, so the simple heads cannot be leveraged with it.
In addition, we find that the point-matching-based heads, which is the original version of PPDM, achieve higher performance than the simple ones, implying that there is a room for increasing accuracy by heuristically designing the heads if the feature extractor is not so powerful, which is not the case with our powerful transformer-based feature extractor.
\vspace{-2.0ex}
\paragraph{Importance of a decoder.}
\begin{figure*}[t]
\centering
\subfloat[]{%
\label{fig:fig4_qualitative_a}
\includegraphics[clip,keepaspectratio,width=0.23\textwidth]{images/fig4_qualitative_a.pdf}%
}
\subfloat[]{%
\label{fig:fig4_qualitative_b}
\includegraphics[clip,keepaspectratio,width=0.23\textwidth]{images/fig4_qualitative_b.pdf}%
}
\subfloat[]{%
\label{fig:fig4_qualitative_c}
\includegraphics[clip,keepaspectratio,width=0.23\textwidth]{images/fig4_qualitative_c.pdf}%
}
\subfloat[]{%
\label{fig:fig4_qualitative_d}
\includegraphics[clip,keepaspectratio,width=0.23\textwidth]{images/fig4_qualitative_d.pdf}%
}
\caption{Failure cases of conventional detectors (top row, same as Fig.~\ref{fig:fig1_heatmap}) and attentions of QPIC (bottom row). In (b) and (d), the attentions corresponding to different HOI instances are drawn with blue and orange, and the areas where two attentions overlap are drawn with white.}
\label{fig:fig4_qualitative}
\vspace{-2.0ex}
\end{figure*}
\begin{table}[t]
\caption{Effect of the transformer encoder and decoder.}
\label{table:ablation}
\centering
\small
\begin{tabular}{@{}cccc@{}}
\toprule
\multirow{2}{*}{\shortstack{Transformer\\ encoder}} & \multirow{2}{*}{\shortstack{Transformer\\ decoder}} &\multirow{2}{*}{\shortstack{HICO-DET\\ (mAP)}}&\multirow{2}{*}{\shortstack{COCO\\ (mAP)}} \\
&&&\\
\midrule
&& 18.89 & 34.6 \\
\ding{51} && 20.07 & 35.1 \\
&\ding{51}& 26.75 & 38.7 \\
\ding{51} &\ding{51}& 29.27 & 43.5 \\
\bottomrule
\end{tabular}
\vspace{-2.0ex}
\end{table}
To further dig into the transformer to find out the essential component for HOI detection, we compare four variants listed in Table~\ref{table:ablation}.
The model without the decoder leverages the point-matching-based method like PPDM~\cite{liao_cvpr2020} on top of the encoder's output (with encoder) or on top of the base features (without encoder).
The model with the decoder utilizes the point-matching-based heads (Fig.~\ref{fig:fig3_variants_b}) for fair comparison.
We use the ResNet-101 backbone for all the variants.
As seen from Table~\ref{table:ablation}, the transformer encoder yields merely slight improvement on HICO-DET
(2.52 and 1.18 mAP with and without the decoder, respectively),
while the decoder remarkably boosts the performance
(9.20 and 7.86 mAP with and without the encoder, respectively).
These results indicate that the decoder plays a vital role in HOI detection.
Additionally, we evaluate the performance on COCO to compare the degrees of improvement in object detection and HOI detection.
As seen in the table, the relative performance improvement brought by the decoder for object detection (on COCO) is 23.9\% and 11.8\% with and without the encoder, respectively, while that for HOI detection (on HICO-DET) is 45.8\% and 41.6\% with and without the encoder, respectively.
This means that the decoder is more effective in an HOI detection task than in an object detection task.
This is probably because the regions of interest (ROI) are mostly consolidated in a single area in object detection tasks, while in HOI detection tasks, the ROI can be diversely distributed image-wide.
CNNs, which rely on localized receptive fields, can deal with the former case relatively easily, whereas the image-wide feature aggregation of the decoder is crutial for the latter case.
\subsection{Qualitative Analysis}\label{sec:quality}
To qualitatively reveal the characteristics of QPIC and the main reasons behind its superior performance over existing methods, we analyze the failure cases of existing methods and QPIC's behavior in the cases.
The top row in Fig.~\ref{fig:fig4_qualitative} shows the failure cases shown in Fig.~\ref{fig:fig1_heatmap}, and the bottom row illustrates the attentions of QPIC on the images.
Figure~\ref{fig:fig4_qualitative_a} and~\ref{fig:fig4_qualitative_b} show the cases where DRG fails to detect the action classes, but QPIC does not.
As previously discussed, the regions in an image other than a human and object bounding box sometimes contain useful information.
Figure~\ref{fig:fig4_qualitative_a} is a typical example, where the hose held by the boy is likely to be the important contextual information.
Two-stage methods that utilize only the region features, namely the human and object bounding box (and sometimes the union region of the two), cannot fully leverage the contextual information, whereas QPIC successfully places the distinguishing focus on such information and leverages it as shown in the attention map.
Furthermore, the region features are sometimes contaminated by other region features when target bounding boxes are overlapped.
Figure~\ref{fig:fig4_qualitative_b} shows such an example, where the hand of the blocking man is contained in the bounding box of the catching man.
The typical two-stage methods, which rely on region features, cannot exclude this disturbing information, resulting in incorrect detection.
QPIC, however, can selectively aggregate only the helpful information for each HOI as shown in the attention map, resulting in the correct detection.
Figure~\ref{fig:fig4_qualitative_c} and~\ref{fig:fig4_qualitative_d} illustrate the failure cases of PPDM, whose detection points are drawn in yellow circles.
As discussed in Sec.~\ref{subsec:sota}, features of heuristic detection points are sometimes dominated by irrelevant information such as the non-target human in Fig.~\ref{fig:fig4_qualitative_c} and another HOI features in Fig.~\ref{fig:fig4_qualitative_d}.
Consequently, the detection based on those confusing features tends to result in failures.
QPIC alleviates this problem by incorporating the attention mechanism that selectively captures image-wide features as shown in the attention maps, and thus correctly detects these HOIs.
Overall, these qualitative analysis demonstrates the QPIC's capability of acquiring image-wide contextual features, which lead to its superior performance over the existing methods.
\section{Conclusion}
We have proposed QPIC, a novel detector that can selectively aggregate image-wide contextual information for HOI detection. QPIC leverages an attention mechanism to effectively aggregate features for detecting a wide variety of HOIs.
This aggregation enriches HOI features, and as a result, simple and intuitive detection heads are realized.
The evaluation on two benchmark datasets showed QPIC's significant superiority over existing methods.
The extensive analysis showed that the attention mechanism and query-based detection play a crucial role for HOI detection.
{\small
\bibliographystyle{ieee_fullname}
|
2,869,038,154,822 | arxiv | \section{Introduction}
Curved spacetimes describing highly compact astrophysical objects
may be characterized, according to the Einstein field equations, by
null circular geodesics (closed light rings) \cite{Bar,Chan,ShTe} on
which photons and gravitons can orbit the central self-gravitating
compact object. These null orbits are interesting from both a
theoretical and an astrophysical points of view and their physical
properties have been studied extensively by physicists and
mathematicians during the last five decades (see
\cite{Bar,Chan,ShTe,Pod,Ame,Ste,Goe,Mas,Dol,Dec,CarC,Hodf,Hodt1,Hodt2}
and references therein).
As demonstrated in \cite{Pod,Ame}, the optical appearance of a
highly compact collapsing star is determined by the physical
properties of its null circular geodesic \cite{Pod,Ame}. Likewise,
the intriguing phenomenon of strong gravitational lensing by highly
compact objects is related to the presence of light rings in the
corresponding curved spacetimes \cite{Ste}. In addition, as
explicitly shown in \cite{Goe,Mas,Dol,Dec,CarC,Hodf}, the discrete
quasinormal resonant spectra of compact astrophysical objects are
related, in the eikonal limit, to the physical properties (the
circulation time and the characteristic instability time scale) of
the null circular geodesics that characterize the corresponding
curved spacetimes \cite{Notekonon,Konon,Zde2n}.
Interestingly, it has recently been proved that spherically
symmetric black-hole spacetimes must posses at least one light ring
\cite{Hodlb}. In particular, the theorem presented in \cite{Hodlb}
has revealed the fact that the innermost null circular geodesic of
an asymptotically flat black hole must be located in a highly
compact spacetime region which is characterized by the dimensionless
lower bound \cite{Hodlb,Noteunits}
\begin{equation}\label{Eq1}
{{m(r^{\text{in}}_{\gamma})}\over{r^{\text{in}}_{\gamma}}}\geq
{1\over3}\ \ \ \ \text{for black holes}\ ,
\end{equation}
where $r^{\text{in}}_{\gamma}$ and $m(r^{\text{in}}_{\gamma})$ are
respectively the innermost (smallest) radius of the light ring which
characterizes the black-hole spacetime and the total gravitational
mass contained within this sphere.
Ultra-compact objects, spatially regular {\it horizonless} matter
configurations which, like black-hole spacetimes, possess light
rings, have attracted much attention in recent years as possible
exotic alternatives to the canonical black-hole spacetimes
\cite{Kei,Hodt0,Mag,CBH,Carm,Hodmz1,Hodmz2,Zde1}. In particular, in
a very interesting work, Novotn\'y, Hlad\'ik, and Stuchl\'ik
\cite{Zde1} (see also \cite{Zde2n}) have recently studied
numerically the physical properties of spherically symmetric
self-gravitating isotropic fluid spheres with a polytropic
pressure-density equation of state of the form \cite{Hdt}
\begin{equation}\label{Eq2}
p(\rho)=k_{\text{p}}\rho^{1+1/n}\ ,
\end{equation}
where the dimensionless physical parameter $n$ is the polytropic
index of the fluid system \cite{Hdt}. It is worth emphasizing that
the self-gravitating ultra-compact trapping polytropic spheres were
first mentioned in \cite{Stuch2n}.
Interestingly, it has been explicitly demonstrated numerically in
\cite{Zde1} that spatially regular polytropic spheres may possess
{\it two} light rings
$\{r^{\text{in}}_{\gamma},r^{\text{out}}_{\gamma}\}$ (see
\cite{CBH,Hodmz2} for related discussions) which are characterized
by the compactness inequality
\begin{equation}\label{Eq3}
{{m(r^{\text{in}}_{\gamma})}\over{r^{\text{in}}_{\gamma}}}\leq
{{m(r^{\text{out}}_{\gamma})}\over{r^{\text{out}}_{\gamma}}}\ .
\end{equation}
In particular, the intriguing fact has been revealed in \cite{Zde1}
that the spherically symmetric self-gravitating horizonless
polytropic spheres may be characterized by closed light rings (null
circular geodesics) with the remarkably small compactness relation
\begin{equation}\label{Eq4}
{{m(r^{\text{in}}_{\gamma})}\over{r^{\text{in}}_{\gamma}}}<
{1\over3}\ \ \ \ \text{for horizonless polytropic spheres}\ .
\end{equation}
It is worth emphasizing the fact that the dimensionless relation
(\ref{Eq4}), observed numerically in \cite{Zde1} for the {\it
horizonless} self-gravitating polytropic spheres, violates the lower
bound (\ref{Eq1}) which, as explicitly proved in \cite{Hodlb},
characterizes the innermost light rings of spherically symmetric
black-hole spacetimes.
The main goal of the present paper is to study analytically the
physical and mathematical properties of the horizonless
ultra-compact polytropic matter configurations. In particular, below
we shall provide compact {\it analytical} proofs for the
characteristic intriguing relations (\ref{Eq3}) and (\ref{Eq4}) that
have recently been observed {\it numerically} in \cite{Zde1} for the
spherically symmetric spatially regular isotropic fluid stars.
\section{Description of the system}
Following the interesting physical model studied numerically in
\cite{Zde1}, we shall consider asymptotically flat isotropic matter
configurations which are characterized by the spherically symmetric
static line element \cite{Chan,Notesc}
\begin{equation}\label{Eq5}
ds^2=-e^{-2\delta}\mu dt^2 +\mu^{-1}dr^2+r^2(d\theta^2 +\sin^2\theta
d\phi^2)\ ,
\end{equation}
where $\delta=\delta(r)$ and $\mu=\mu(r)$. Spatially regular matter
configurations are characterized by the functional behavior
\cite{Hodt1}
\begin{equation}\label{Eq6}
\mu(r\to 0)=1+O(r^2)\ \ \ \ {\text{and}}\ \ \ \ \delta(0)<\infty\
\end{equation}
in the near-origin $r\to0$ limit. In addition, asymptotically flat
regular spacetimes are characterized by the simple large-$r$
functional relations \cite{Hodt1,May}
\begin{equation}\label{Eq7}
\mu(r\to\infty) \to 1\ \ \ \ {\text{and}}\ \ \ \ \delta(r\to\infty)
\to 0\ .
\end{equation}
The non-linearly coupled Einstein-matter field equations,
$G^{\mu}_{\nu}=8\pi T^{\mu}_{\nu}$, can be expressed by the
differential relations \cite{Hodt1,Noteprm}
\begin{equation}\label{Eq8}
\mu'=-8\pi r\rho+{{1-\mu}\over{r}}\
\end{equation}
and
\begin{equation}\label{Eq9}
\delta'=-{{4\pi r(\rho +p)}\over{\mu}}\ ,
\end{equation}
where the radially-dependent density and pressure functions
\begin{equation}\label{Eq10}
\rho\equiv -T^{t}_{t}\ \ \ \ \text{and}\ \ \ \ p\equiv
T^{r}_{r}=T^{\theta}_{\theta}=T^{\phi}_{\phi}
\end{equation}
denote the components of the isotropic energy-momentum tensor
\cite{Bond1}. We shall assume that the spherically symmetric
asymptotically flat matter configurations respect the dominant
energy condition \cite{HawEl}
\begin{equation}\label{Eq11}
0\leq |p|\leq\rho\ .
\end{equation}
From the Einstein equations (\ref{Eq8}) and (\ref{Eq9}) and the
conservation relation
\begin{equation}\label{Eq12}
T^{\mu}_{r ;\mu}=0\ ,
\end{equation}
one can derive the characteristic compact differential equation
\begin{equation}\label{Eq13}
P'(r)= {{r}\over{2\mu}}\big[{\cal R}(\rho+p)+2\mu(-\rho+p)\big]\
\end{equation}
for the gradient of the radially-dependent isotropic pressure
function
\begin{equation}\label{Eq14}
P(r)\equiv r^2p(r)\ ,
\end{equation}
where
\begin{equation}\label{Eq15}
{\cal R}(r)\equiv 3\mu-1-8\pi r^2p\ .
\end{equation}
Below we shall analyze the spatial behavior of the characteristic
dimensionless compactness function
\begin{equation}\label{Eq16}
{\cal C}(r)\equiv {{m(r)}\over{r}}\ ,
\end{equation}
where the mass $m(r)$ of the matter fields contained within a sphere
of radius $r$ is given by the simple integral relation \cite{Hodt1}
\begin{equation}\label{Eq17}
m(r)=4\pi\int_{0}^{r} x^{2} \rho(x)dx\ .
\end{equation}
Taking cognizance of Eqs. (\ref{Eq8}) and (\ref{Eq17}), one deduces
the simple dimensionless functional relation
\begin{equation}\label{Eq18}
\mu(r)=1-{{2m(r)}\over{r}}\ .
\end{equation}
For later purposes we note that asymptotically flat regular matter
configurations are characterized by the asymptotic radial behavior
\cite{Hodt1}
\begin{equation}\label{Eq19}
r^3p(r)\to 0\ \ \ \ \text{for}\ \ \ \ r\to\infty\ .
\end{equation}
\section{Null circular geodesics of spherically symmetric curved spacetimes}
In the present section we shall follow the analysis presented in
\cite{Chan,CarC,Hodt1} in order to determine the radii of the null
circular geodesics (closed light rings) which characterize the
spherically symmetric self-gravitating ultra-compact objects. We
first note that the energy $E$ and the angular momentum $L$ provide
two conserved physical parameters along the null geodesics of the
static spacetime (\ref{Eq5}) \cite{Chan,CarC,Hodt1}.
In particular, the effective radial potential
\cite{Chan,CarC,Hodt1,Notedot}
\begin{equation}\label{Eq20}
E^2-V_r\equiv \dot
r^2=\mu\Big({{E^2}\over{e^{-2\delta}\mu}}-{{L^2}\over{r^2}}\Big)\
\end{equation}
determines, through the relations \cite{Chan,CarC,Hodt1,Notethr}
\begin{equation}\label{Eq21}
V_r=E^2\ \ \ \ \text{and}\ \ \ \ V'_r=0\ ,
\end{equation}
the null circular trajectories (light rings) of the static spacetime
(\ref{Eq5}). Substituting Eqs. (\ref{Eq8}), (\ref{Eq9}), and
(\ref{Eq20}) into (\ref{Eq21}), one obtains the characteristic
functional relation \cite{Chan,CarC,Hodt1}
\begin{equation}\label{Eq22}
{\cal R}(r=r_{\gamma})=0\
\end{equation}
for the null circular geodesics of the
spherically symmetric ultra-compact objects.
\section{An analytical proof of the characteristic relation
${\cal C}(r^{\text{in}}_{\gamma})\leq{\cal
C}(r^{\text{out}}_{\gamma})$ for horizonless isotropic ultra-compact
objects}
The physical properties of spherically symmetric self-gravitating
isotropic ultra-compact objects have recently been studied
numerically in the interesting work of Novotn\'y, Hlad\'ik, and
Stuchl\'ik \cite{Zde1} (see also \cite{Zde2n}). Intriguingly, it has
been explicitly shown in \cite{Zde1} that the horizonless curved
spacetimes of these spatially regular compact matter configurations
generally possess two light rings
$\{r^{\text{in}}_{\gamma},r^{\text{out}}_{\gamma}\}$ (see
\cite{CBH,Hodmz2} for related studies) which are characterized by
the dimensionless compactness relation (\ref{Eq3}).
In the present section we shall use {\it analytical} techniques in
order to provide a compact proof for the intriguing property ${\cal
C}(r^{\text{in}}_{\gamma})<{\cal C}(r^{\text{out}}_{\gamma})$ [see
Eqs. (\ref{Eq3}) and (\ref{Eq16})] which characterizes the
horizonless isotropic ultra-compact objects. We first point out
that, taking cognizance of Eqs. (\ref{Eq6}), (\ref{Eq7}),
(\ref{Eq15}), and (\ref{Eq19}), one finds the simple asymptotic
relations
\begin{equation}\label{Eq23}
{\cal R}(r=0)=2\ \ \ \ \text{and}\ \ \ \ {\cal R}(r\to\infty)\to 2\
\end{equation}
for the dimensionless radial function ${\cal R}(r)$. From Eqs.
(\ref{Eq22}) and (\ref{Eq23}) one deduces that, for horizonless
ultra-compact matter configurations with non-degenerate light rings
\cite{CBH,Hodmz2,Noteep}, the function ${\cal R}(r)$ is
characterized by the inequality
\begin{equation}\label{Eq24}
{\cal R}(r)<0\ \ \ \ \text{for}\ \ \ \ r\in
(r^{\text{in}}_{\gamma},r^{\text{out}}_{\gamma})\
\end{equation}
in the radial region between the two light rings of the
ultra-compact objects.
Substituting the characteristic inequality (\ref{Eq24}) into Eq.
(\ref{Eq13}) and taking cognizance of the relation (\ref{Eq11}), one
finds that $P(r)$ is a monotonically decreasing function between the
two light rings of the horizonless compact object:
\begin{equation}\label{Eq25}
P'(r)<0\ \ \ \ \text{for}\ \ \ \ r\in
(r^{\text{in}}_{\gamma},r^{\text{out}}_{\gamma})\ .
\end{equation}
In particular, from Eqs. (\ref{Eq14}), (\ref{Eq15}), (\ref{Eq22}),
and (\ref{Eq25}), one deduces that the dimensionless function
$\mu(r)$ is characterized by the inequality
\begin{equation}\label{Eq26}
\mu(r^{\text{in}}_{\gamma})\geq\mu(r^{\text{out}}_{\gamma})\ ,
\end{equation}
or equivalently [see Eqs. (\ref{Eq16}) and (\ref{Eq18})]
\begin{equation}\label{Eq27}
{\cal C}(r^{\text{in}}_{\gamma})\leq{\cal
C}(r^{\text{out}}_{\gamma})\ .
\end{equation}
We have therefore provided a simple analytical proof for the
numerically observed relation (\ref{Eq3}) \cite{Zde1} which
characterizes the horizonless isotropic ultra-compact objects.
\section{Upper bound on the compactness of the inner light ring of isotropic
ultra-compact objects}
The characteristic compactness parameter ${\cal C}(r)\equiv m(r)/r$
of the self-gravitating ultra-compact objects can be computed using
the numerical procedure described in \cite{Zde1}. Intriguingly, as
demonstrated explicitly in \cite{Zde1}, the spatially regular
horizonless ultra-compact objects may be characterized by inner
light rings whose dimensionless compactness parameter ${\cal
C}(r^{\text{in}}_{\gamma})$ is well below the lower bound
(\ref{Eq1}) which, as explicitly proved in \cite{Hodlb},
characterizes the innermost null circular geodesics (light rings) of
spherically symmetric asymptotically flat black-hole spacetimes.
In Table \ref{Table1} we present, for various values of the
polytropic index $n$, the numerically computed dimensionless
compactness parameter ${\cal
C}^{\text{numerical}}(r^{\text{in}}_{\gamma})$ of the isotropic
ultra-compact objects \cite{Zde1,Notecmx,Too}. One finds that ${\cal
C}(r^{\text{in}}_{\gamma};n)$ is a monotonically decreasing function
of the dimensionless polytropic index $n$. Interestingly, we find
that the numerical results presented in Table \ref{Table1} are
described extremely well by the simple asymptotic formula (see Table
\ref{Table1})
\begin{equation}\label{Eq28}
{\cal C}(r^{\text{in}}_{\gamma};n)=\alpha
+{{\beta}\over{n}}+O(n^{-2}) \ \ \ \ \text{with}\ \ \ \
\alpha=0.2149 \ \ \ \text{and}\ \ \ \beta=0.1602\ .
\end{equation}
\begin{table}[htbp]
\centering
\begin{tabular}{|c|c|c|}
\hline $\ \text{Polytropic}\ $\ \ & \ ${\cal
C}^{\text{numerical}}(r^{\text{in}}_{\gamma})$ \ &
\ ${\cal C}^{\text{analytical}}(r^{\text{in}}_{\gamma})$ \ \\
$\ \ \text{index}\ \ n\ \ $\ \ & \ \ $\text{Ref.}$\ \cite{Zde1} \ \
&
\ \ $\text{Eq.}$\ (\ref{Eq28}) \ \ \\
\hline
\ \ $2.2$\ \ \ &\ \ $0.2906$\ \ &\ \ $0.2877$\ \ \\
\ \ $2.4$\ \ \ &\ \ $0.2824$\ \ &\ \ $0.2817$\ \ \\
\ \ $2.6$\ \ \ &\ \ $0.2767$\ \ &\ \ $0.2765$\ \ \\
\ \ $2.8$\ \ \ &\ \ $0.2723$\ \ &\ \ $0.2721$\ \ \\
\ \ $3.0$\ \ \ &\ \ $0.2683$\ \ &\ \ $0.2683$\ \ \\
\ \ $3.2$\ \ \ &\ \ $0.2649$\ \ &\ \ $0.2650$\ \ \\
\ \ $3.4$\ \ \ &\ \ $0.2620$\ \ &\ \ $0.2620$\ \ \\
\ \ $3.6$\ \ \ &\ \ $0.2594$\ \ &\ \ $0.2594$\ \ \\
\ \ $3.8$\ \ \ &\ \ $0.2570$\ \ &\ \ $0.2571$\ \ \\
\ \ $4.0$\ \ \ &\ \ $0.2549$\ \ &\ \ $0.2550$\ \ \\
\hline
\end{tabular}
\caption{Ultra-compact polytropic fluid spheres with light rings. We
present, for various values of the polytropic index $n$, the {\it
numerically} computed \cite{Zde1,Notecmx,Too} dimensionless
compactness parameter ${\cal
C}^{\text{numerical}}(r^{\text{in}}_{\gamma};n)$ of the isotropic
matter configurations. We also present the corresponding values of
the dimensionless compactness parameter ${\cal
C}^{\text{analytical}}(r^{\text{in}}_{\gamma};n)$ as calculated
directly from the simple analytical fit (\ref{Eq28}). One finds a
remarkably good agreement between the numerical results \cite{Zde1}
and the analytical formula (\ref{Eq28}). In particular, one deduces
from (\ref{Eq28}) the characteristic asymptotic value ${\cal
C}(r^{\text{in}}_{\gamma})\to 0.2149<1/4$ for $n\gg1$.}
\label{Table1}
\end{table}
What we find most interesting is the fact that the horizonless
ultra-compact isotropic objects are characterized by the
dimensionless asymptotic compactness parameter
\begin{equation}\label{Eq29}
{\cal C}(r^{\text{in}}_{\gamma};n\gg1)\simeq0.2149<1/4\ .
\end{equation}
As emphasized above, one immediately realizes that the asymptotic
value (\ref{Eq29}), which characterizes the spatially regular
horizonless matter configurations, is well below the lower bound
${\cal C}(r_{\gamma})\geq 1/3$ [see Eqs. (\ref{Eq1}) and
(\ref{Eq16})] which characterizes the corresponding spherically
symmetric black-hole spacetimes \cite{Hodlb}.
In the present section we shall provide an {\it analytical}
explanation for the {\it numerically} inferred asymptotic behavior
(\ref{Eq29}) of the dimensionless compactness parameter. In
particular, we shall now derive an upper bound on the compactness
parameter ${\cal C}(r^{\text{in}}_{\gamma};n)$ of the isotropic
ultra-compact objects in the $n\gg1$ \cite{Notestuch2n,Stuch2n}
limit of the polytropic index, which corresponds to the limiting
pressure-density relation [see Eq. (\ref{Eq2})]
\begin{equation}\label{Eq30}
p=k_{\text{p}}\rho\ .
\end{equation}
From Eqs. (\ref{Eq8}), (\ref{Eq13}), (\ref{Eq14}), and (\ref{Eq15}),
one finds the gradient relation
\begin{equation}\label{Eq31}
{\cal R}'(r=r_{\gamma})={{2}\over {r_{\gamma}}}\big[1-8\pi
r^2_{\gamma}(\rho+p)\big]\
\end{equation}
for the isotropic ultra-compact objects. In addition, taking
cognizance of Eqs. (\ref{Eq22}) and (\ref{Eq23}), one deduces that
the inner light ring of a spatially regular ultra-compact
horizonless matter configuration is characterized by the relation
${\cal R}'(r=r^{\text{in}}_{\gamma})\leq 0$ \cite{Notegnh}, or
equivalently [see Eq. (\ref{Eq31})]
\begin{equation}\label{Eq32}
8\pi r^2_{\gamma}(\rho+p)\geq1\ \ \ \ \text{for}\ \ \ \
r=r^{\text{in}}_{\gamma}\ .
\end{equation}
Taking cognizance of Eqs. (\ref{Eq15}), (\ref{Eq30}), and
(\ref{Eq32}), one obtains the lower bound
\begin{equation}\label{Eq33}
\mu(r^{\text{in}}_{\gamma};n\gg1)\geq
{{2k_{\text{p}}+1}\over{3(k_{\text{p}}+1)}}\ ,
\end{equation}
which yields the characteristic upper bound [see Eqs. (\ref{Eq16})
and (\ref{Eq18})]
\begin{equation}\label{Eq34}
{\cal C}(r^{\text{in}}_{\gamma};n\gg1)\leq
{{k_{\text{p}}+2}\over{6(k_{\text{p}}+1)}}\
\end{equation}
on the dimensionless compactness parameter which characterizes the
inner null circular geodesic (inner light ring) of the isotropic
ultra-compact objects. In particular, taking cognizance of Eqs.
(\ref{Eq11}) and (\ref{Eq30}), one deduces that, in the $n\gg1$
limit of the polytropic index, the physical parameter $k_{\text{p}}$
is bounded from above by the simple relation $k_{\text{p}}\leq1$.
Substituting the limiting value $k_{\text{p}}\to1^-$ into
(\ref{Eq34}), one obtains the characteristic upper bound
\cite{Notekp0,Notegrc}
\begin{equation}\label{Eq35}
{\cal C}(r^{\text{in}}_{\gamma};n\gg1,k_{\text{p}}\to1^-)<
{1\over4}\ .
\end{equation}
It is worth emphasizing the fact that the {\it analytically} derived
upper bound (\ref{Eq35}) on the dimensionless compactness parameter
is consistent with the asymptotic behavior (\ref{Eq29}) which stems
from the {\it numerical} studies \cite{Zde1} of the self-gravitating
ultra-compact isotropic fluid configurations. In particular, in this
section we have provided an explicit {\it analytical} proof to the
{\it numerically} observed intriguing fact that horizonless
ultra-compact objects can violate the lower bound (\ref{Eq1}) which
characterizes spherically symmetric black-hole spacetimes
\cite{Notebhs}.
\section{Summary}
Horizonless spacetimes describing self-gravitating ultra-compact
matter configurations with closed light rings (null circular
geodesics) have recently attracted much attention as possible
spatially regular exotic alternatives to canonical black-hole
spacetimes (see \cite{Kei,Hodt0,Mag,CBH,Carm,Hodmz1,Hodmz2,Zde1} and
references therein).
In particular, the physical properties of horizonless ultra-compact
isotropic fluid spheres with a polytropic equation of state have
recently been studied numerically in the physically important work
of Novotn\'y, Hlad\'ik, and Stuchl\'ik \cite{Zde1}. Interestingly,
it has been explicitly shown numerically in \cite{Zde1} that these
spherically symmetric spatially regular ultra-compact polytropic
matter configurations generally posses {\it two} closed light rings
(see also \cite{CBH,Hodmz2} for related discussions).
In the present paper we have used {\it analytical} techniques in
order to explore the physical and mathematical properties of the
ultra-compact polytropic stars. In particular, it has been
explicitly proved that the two light rings of these horizonless
matter configurations are characterized by the relation [see Eqs.
(\ref{Eq16}) and (\ref{Eq27})]
\begin{equation}\label{Eq36}
{\cal C}(r^{\text{in}}_{\gamma})\leq{\cal
C}(r^{\text{out}}_{\gamma})\ .
\end{equation}
Interestingly, we have further proved that, while spherically
symmetric black-hole spacetimes are characterized by the lower bound
${\cal C}(r^{\text{in}}_{\gamma})\geq1/3$ [see Eqs. (\ref{Eq1}) and
(\ref{Eq16})] \cite{Hodlb}, the spatially regular horizonless
ultra-compact objects are characterized by the opposite
dimensionless relation
\begin{equation}\label{Eq37}
{\cal C}(r^{\text{in}}_{\gamma};n\gg1,k_{\text{p}}\to1^-)<
{1\over4}\ .
\end{equation}
It is worth noting that the analytically derived upper bound
(\ref{Eq37}) on the characteristic dimensionless compactness
parameter ${\cal C}(r^{\text{in}}_{\gamma})$ is consistent with the
numerically inferred asymptotic behavior (\ref{Eq29}).
Finally, it is interesting to emphasize the fact that the analytical
results derived in the present paper provide a simple {\it
analytical} explanation for the interesting {\it numerical} results
that have recently presented by Novotn\'y, Hlad\'ik, and Stuchl\'ik
\cite{Zde1} for the physical properties of the self-gravitating
ultra-compact polytropic spheres.
\bigskip
\noindent {\bf ACKNOWLEDGMENTS}
This research is supported by the Carmel Science Foundation. I would
like to thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B.
Tea for stimulating discussions.
|
2,869,038,154,823 | arxiv | \section{Introduction}
Hierarchical clustering is the favoured scenario to describe
the formation and evolution of matter structures in the universe
(White \& Rees, 1978), and semi-analytic models of galaxy formation
proved themselves to be a powerful tool of investigation since the first
formulation (White \& Frenk, 1991). Over the years, many
such models have been developed (see for instance Balland et al. 2003,
Baugh et al. 2005, Bower et al. 2006,
Cattaneo et al. 2008, Cole et al. 2000, De Lucia et al. 2004, Hatton et al. 2003,
Kauffmann et al. 1993, Menci et al. 2006, Monaco et al. 2007, Somerville et al. 2008).
The successes and failures of these models are strictly linked to those of
the hierarchical scenario itself, ultimately depending on the mechanisms of
mass accretion of objects as a function of time. The large-scale
structure and the integrated properties of the galaxy population (such as the total stellar
mass density for instance) are well reproduced.
The detailed evolution of galaxies however presents several puzzling aspects,
such as e.g. the size of the disks of spirals (Burkert 2009), or the $\alpha$-enhancement
and the ages of the stellar populations in massive ellipticals (Thomas 1999,
Thomas et al. 1999, Cimatti et al. 2004, Nagashima et al. 2005,
Thomas et al. 2005, Pipino et al. 2009, Kormendy et al. 2009).
Moreover, global properties of the galaxy population such as the
evolution of the stellar mass function (Cole et al. 2001, Bell et al. 2003, Bundy et al 2006-2009,
Pozzetti et al. 2009, Colless et al. 2007) are still not reproduced in the models
(Bundy et al. 2007, Marchesini et al. 2009, Kajisawa et al. 2009, Kodama \& Bower 2003),
although there is controversy on this point (Drory et al. 2004, Benson et al. 2007).
It is debated if any of these problems
can possibly be resolved with enhanced resolution
in the simulations and more sophisticated recipes in the models.
One of the most problematic issues for the models is to reproduce
the abundance of high redshift luminous galaxies
(e.g., Conselice et al. 2007, Cimatti et al 2004, van Dokkum et al. 2004, 2006).
This difficulty is partly due to a mis-interpretation
of the nature of these objects.
The high-luminosity end of the galaxy population up to redshift $z \sim 2.5$
consists in fact of objects that look like the early-type galaxies in the local
universe, \textit{i.e.} they are characterized by very red colours in the optical
and near-IR, and high near-IR luminosities (Mancini et al. 2009, Cimatti et al 2004,
McCarthy et al. 2004, Daddi et al. 2005, Saracco et al. 2005, Kriek et al. 2006).
Local ellipticals showing the same photometry are old
(with stellar populations older than $\sim1$ Gyr),
passively evolving, and with stellar masses $M_{star} > 10^{11} \ M_{\odot}$.
Moreover, newer observations
are building the case for the presence of extremely red and IR-luminous objects at even higher redshifts
(Rodighiero et al. 2007, Mancini et al. 2009, Fontana et al. 2009).
The problem posed by the presence of these high redshift ($z>2$) red and luminous galaxies
stems from the consensus that they are massive objects evolving
passively, the so-called 'red \& dead' galaxies.
With the stellar population models currently used in the
semi-analytic models in the literature (for the most part
Bruzual \& Charlot 2003, hereafter BC03),
the only way to explain the high near-IR luminosities of high-redshift
galaxies is to advocate very high galaxy masses and very old ages
of the stellar populations. But these are not achieved in the actual
model realizations (except at low redshifts), because the
hierarchical mass assembly has an intrinsic difficulty in putting
together massive and old objects at early epochs. In fact, the hierarchical scenario
predicts a steady decline of the abundance
of massive galaxies with increasing redshift (van Dokkum et al. 2004).
In Tonini et al. (2009) we showed that the predictions of colours and
luminosities of galaxies at high redshift in a semi-analytic model
are greatly affected by the recipes in use for the stellar populations,
expecially the inclusion of the Thermally-Pulsing Asymptotic Giant Branch (TP-AGB).
As shown in M05, in stellar populations of intermediate age ($\leq 0.2 - 2$ Gyr)
the TP-AGB phase dominates the near-IR luminosity,
with a contribution up to $80 \%$ in the rest-frame K band, and contributes to up to
$40 \%$ of the bolometric luminosity (M05).
High-redshift galaxies, in which the mean age of the stellar populations
is in that range, are expected to be dominated by the TP-AGB emission in
the near-IR. This has been recently confirmed by SED-fitting of observations
made with the Spitzer Space Telescope (Maraston et al. 2006, Cimatti et al. 2008).
In Tonini et al. (2009) we included a complete treatment of the TP-AGB phase in the semi-analytic
model GalICS (Hatton et al. 2003), by implementing the M05 stellar
population models into the code, and showed that the rest-frame
$V-K$ colours at high redshift get redder by more than 1 magnitude.
Relatedly, the K-band mass-to-light ratio is
shifted towards luminosities 1 magnitude higher for a given galaxy mass.
Notably, actively evolving, star-forming high-redshift galaxies are predicted to
have $V-K$ colours and near-IR luminosities similar to those of local,
passively evolving massive systems (Tonini et al. 2009).
Once the stellar emission is correctly modeled with an exhaustive treatment
of all the significant phases of stellar evolution, a more accurate comparison
between the semi-analytic model predictions and the data is possible.
In particular, the
performance of the semi-analytic model in reproducing the observed colours and luminosities
in the near-IR becomes meaningful to test the hierarchical mass assembly at different
redshifts.
In the literature the comparison between galaxy formation models and data is typically
done by obtaining physical properties for the real objects through application of stellar
population models to data. However, this approach carries several degeneracies, including
the adopted population synthesis model, the recipe for star formation history, the choice
of metallicity, etc. When a realistic errorbar including all these variables ia attached
to the observationally derived quantity, such as in Marchesini et al. 2009 (and see also
Conroy et al. 2009 for a discussion), the results of such comparisons may not be clear-cut.
In this paper we adopt a different philosophy for the
comparison between model and data. Instead of using processed data in the
rest-frame system, we consider raw, unprocessed, apparent magnitudes straight
out of the catalogues. We then produce mock catalogues out of the simulation,
so that the output spectra of the model galaxies are redshifted in the observer's frame.
The model apparent magnitudes and colours can then be directly compared with
the observational data. This comparison yields direct information about the
physical quantities in the model in use.
This procedure is straightforward and does not add substantial degeneracy that can
jeopardize the comparison.
A degeneracy that clearly remain is how dust reddening affects the intrinsic stellar
emission, as recently pointed out by Guo \& White (2008) and Conroy et al. (2009).
However, we shall show that considering the intrinsic star formation rates in the
model and using data mapping the rest-frame near-IR, such an uncertainty plays actually a minor role.
The structure of the paper is as follows.
In Section 2 we briefly introduce the new semi-analytic model GalICS with the TP-AGB implementation
through the M05 models (as from Tonini et al. 2009). In Section 3 we
describe the data samples used for our analysis. In Section 4 we compare
the colour-magnitude and colour-colour relations predicted by the model
against samples of $z \sim 2$ galaxies. In Section 4 we compare the model
rest-frame K-band luminosity function in the M05 and Pegase cases with the predictions
by other semi-analytic models.
In Section 5 we discuss our results.
\section{The semi-analytic model of galaxy formation}
We produce the model galaxies through the hybrid semi-analytic model
GalICS (Hatton et al. 2003), and we defer the reader to its original
paper for details on the dark matter N-body simulation and the
implementation of the baryonic physics. In brief, the model builds up the galaxies
hierarchically, and evolves the metallicity consistently with the
cooling and star formation history (with the new implementation by Pipino et
al. 2009). Feedback recipes for supernovae-driven
winds and AGN activity are implemented in the code (the lattest with the improved
version of Cattaneo et al. 2006). Merger-driven
morphology evolution and satellite stripping and disruption are taken into account.
The semi-analytic model was originally supplied with two
different sets of input stellar population models, namely the PEGASE (Fioc \& Rocca-Volmerange, 1997)
and the Stardust (Hatton et al. 2003, Cattaneo et al. 2008). The Stardust model is
too rudimentary for the scope of this work, so we discarded it.
In Tonini et al. (2009) for the first time the M05 models were implemented into GalICS,
and the predictions of the models were compared with those obtained with PEGASE.
The same sets of SSP models are utilized in the present paper.
For the purpose of this work, the most significative difference between PEGASE and M05
is that the M05 includes higher energetics for the TP-AGB phase. The PEGASE
models in this respect produce results comparable to the more commonly used
BC03 models (see M05), with a TP-AGB recipe with much lower energetics
than the M05 (see M05, Fig. 18).
For both sets of SSPs (Single Stellar Populations), the adopted IMF is Salpeter, and
the metallicity range is $0.001 < Z < 0.04$ (where $Z_{\odot}=0.02$).
In the current implementation, dust extinction is taken into account.
The model spectra are reddened according to the ongoning star formation.
We adopt a Calzetti extinction curve and a colour-excess $E(B-V)$
proportional to the star formation rate for each single galaxy,
parameterized as $E(B-V) = 0.33 \cdot (Log(SFR)-2) + 1/3$.
This choice is supported by data analysis in the literature, in general for
samples of star-forming galaxies at redshifts around $z \sim 2$ (Daddi et al. 2007,
Maraston et al., in prep.),
and in particular it agrees with the SED fitting of the galaxies in the data samples
used in this paper, which are introduced in the next Section.
We do not randomize the inclination of disk galaxies (which reduces the dust effect
in face-on objects) but we redden the spectrum
by the total amount of extinction calculated for each galaxy,
therefore considering the maximal reddening for each object.
As it will be clear in the next Sections,
this choice proves to be instructive
in the comparison between GalICS runs with M05 and PEGASE input SSPs.
This reddening recipe is more sophisticated than simple screen-models used so far,
and follows the spirit of important developments in the field
(see for example Ferrara et al. 1999, Guo $\&$ White 2008). The results presented here have
been tested against different extinction curves (Large Magellanic Cloud, Milky Way and
Small Magellanic Cloud types), with the
conclusion that the main factor affecting the dust contribution is $E(B-V)$.
The comparison with data performed in this work requires the distinction between
actively star-forming galaxies and nearly-passive galaxies (passively evolving or
with small residual star formation). For this purpose, we split our model
galaxies according to their istantaneous star formation rate, with the
criterium that objects with SFR $\le 3 \ M_{\odot}/yr$ are considered nearly-passive,
and galaxies with SFR $> 10 \ M_{\odot}/yr$ are actively star-forming.
This value was chosen according to the star-forming
objects selected in Daddi et al. 2007 and in Maraston et al.
(in preparation), where SFR $>10 \ M_{\odot}/yr$ are robustly
determined from far-IR and the UV-slope method.
We build mock catalogues from the simulation, by redshifting the rest-frame spectra
at each timestep, to produce observer-frame
luminosities and colours, to compare directly with the data. We set the
model magnitudes to mimic the data catalogues in use, by
filtering the spectra with the same broadband filters used in the observations.
The broadband magnitudes thus obtained are further scattered with gaussian errors comparable to the
observational errors of our data samples (on average $\sigma = 0.1 mag$ at $z \sim 2$).
\section{Data selection}
We want to compare the predicted colours and near-IR luminosity of our
semi-analytic model against data of high-redshift galaxies,
to test whether our improved stellar populations implementation
allows to better match the properties of the red galaxy population.
Given the nature and origin of the TP-AGB light, we focus our analysis
on datasets for which the IR photometry is available. In fact, the optical to near-IR rest-frame
emission will get redshifted into the IRAC bands for redshifts $z \geq 2$.
We explicitly looked for samples of excellent photometry quality,
in order to minimize any source of uncertainty.
The first sample is from Maraston et al. (2006). It consists of 7 galaxies
selected in the optical by Daddi et al. (2005), with photometry extended
to cover the rest-frame near-IR. These galaxies were selected
from the Hubble Ultra Deep Field through the $BzK$ technique
introduced by Daddi et al. (2004), and have spectroscopic redshifts
between $1.4 \leq z \leq 2.7$ (see Maraston et al. 2006 for details).
They show early-type morphology, and SED fitting (Maraston et al. 2006) indicates
an average age between $0.2 < \tau < 2$ Gyr and masses $\sim 10^{11} \ M_{\odot}$,
a modest dust attenuation and a negligible amount of OB stars, so that these
objects qualify as nearly passive. In Daddi et al. (2005) they were interpreted
as progenitors of present-time massive ellipticals.
The second sample is taken from the
GOODS-S (Great Observatories Origins Deep Survey -South) catalogue from Daddi et al. (2007),
and consists of 95 galaxies with high-quality photometry and spectroscopic
redshifts between $1.7 \leq z \leq 2.3$. The $BzK$ method identifies them as
star-forming, which has been confirmed by SED fitting (Maraston et al., in prep.).
\section{Colour-magnitude and colour-colour relations at $z \sim 2$}
In this Section we compare the observed broadband magnitudes and colours of the data
samples with the predictions of the model. We translate the model results in the
observer-frame at each redshift, so that all magnitudes presented here are apparent.
The magnitudes are calculated in the AB system.
\begin{figure*}
\includegraphics[scale=0.9]{fig1.eps}
\caption{The observed-frame colour-magnitude relation Irac3 $vs$ H-Irac3
corresponding to rest-frame K $vs$ V-K at redshifts $z=1.5, 2, 2.5$
(\textit{from left to right}), for nearly-passive galaxies in the M05 runs
(\textit{red filled dots}) and the PEGASE runs (\textit{yellow empty dots}), compared with
data of nearly passively-evolving galaxies from Maraston et al. (2006) (\textit{black triangles
with errorbars}). The dust reddening adopted in the models is a LMC-type law
with colour-excess $E(B-V)=0.2$ at $z=1.5$, and a Calzetti law with $E(B-V) \propto SFR$
at $z=2,\ 2.5$.}
\label{7gals}
\end{figure*}
\subsection{Nearly passive galaxies}
Interesting candidates for studying the effects of the TP-AGB in the semi-analytic model
are galaxies dominated by intermediate-age stellar populations.
In such objects most of the stellar component is active in the TP-AGB
phase, thus maximizing its effect,
while star formation is subdominant, which reduces the complication of dust.
In Fig.~(\ref{7gals}) we compare the model
observed-frame colour-magnitude relation Irac3 $vs$ H-Irac3,
corresponding to rest-frame K $vs$ V-K, for the nearly-passive galaxies, with data of the 7 galaxies
at redshifts $z=1.5, 2, 2.5$ singled out by Maraston et al. (2006).
The model galaxies are plotted as \textit{red filled dots} for the TP-AGB run (M05) and
\textit{yellow empty dots} for the PEGASE run, while the data are represented by
\textit{black triangles with errorbars}.
The comparison between model and observations show that the M05 run with the TP-AGB
perfectly matches the data, in the whole redshift range. With PEGASE on the
other hand, the model galaxies feature much bluer colours, with an offset of about
0.5 mags on average between the two runs. In the central panel ($z=2$) it is
also evident that the PEGASE run fails to reproduce the Irac3 (rest-frame K)
luminosity of these objects, while the M05 run produces luminosities up to 1 mag higher
and easily accomodates the observed ones.
\begin{figure*}
\includegraphics[scale=0.9]{fig2.eps}
\caption{The theoretical colour-magnitude relation Irac3 $vs$ H-Irac3 at $z=2$, corresponding
to the rest-frame K $vs$ V-K, for the star-forming galaxies in the M05 run (\textit{left panel}) and
the PEGASE run (\textit{right panel}), compared with data.
The \textit{black dots} with errorbars are data
points from the GOODS-S catalogue (Daddi et al. 2007).
In both panels, the results are shown for runs with reddening (a Calzetti-type
extinction with colour-excess proportional to the SFR; \textit{cyan and magenta dots})
and without reddening (\textit{blue and green dots}).}
\label{colmag}
\end{figure*}
These galaxies show early-type morphology and lack any significant
emission from young stars. However, as expected at these high redshifts,
their star formation is not strictly zero\footnote{This
actually suits the semi-analytic model, since the hierarchical nature of
the simulation causes satellites to continually infall into bigger objects,
triggering spurious star formation. Residual star formation is also caused by
the cooling of hot halo gas onto the central objects, but while at these high redshifts
this is acceptable, it becomes a problem of the model at lower redshifts
(the so-called cooling catastrophe).}, hence the label 'nearly passive'.
Therefore, although dust reddening in these objects is modest in general, it is not negligible.
The degree of residual star formation in our sample is variable, and in particular
the galaxies at $z=1.5$ show a little bit more activity. In fact, they are SED-fitted
with a colour-excess due to dust
reddening even slightly higher than predicted by our dust model, so
for the plot at $z=1.5$
we adopted a dust-screen LMC-type law with $E(B-V)=0.2$ (from Maraston et al. 2006).
Even with a higher reddening however, the PEGASE run is not able to match these data.
\begin{figure*}
\includegraphics[scale=0.9]{fig3.eps}
\caption{The colour-colour relation J-H $vs$ H-Irac3 at $z=2$, corresponding to the
rest frame B-V $vs$ V-K, for the M05 run (\textit{left panel}) and the PEGASE run
(\textit{right panel}). The \textit{black dots} with errorbars are data
points from the GOODS catalogue (Daddi et al. 2007). The \textit{green triangles}
with errorbars are data from Maraston et al. (2006) of nearly-passively evolving galaxies.
For the M05 run, the \textit{red dots} represent nearly-passive galaxies (SFR$<3 \ M_{\odot/yr}$) and
\textit{cyan dots} represent the star-forming galaxies (SFR$>10 \ M_{\odot/yr}$); in the PEGASE
run, the same holds for \textit{yellow/magenta dots} respectively.}
\label{colcolmiocalzetti}
\end{figure*}
The observed galaxies lie among the most luminous galaxies in the model, as expected.
The slight luminosity overshooting of the simulated galaxies probably stems from the fact that
the simulated volume is much bigger than the survey (at $z=2$ the ratio of the areas
is a factor $\sim 107$). Moreover, the observed galaxies were selected in observed-frame
K band, which maps the $\sim$R at $z=2$, while we are plotting the Irac3 luminosity (rest-frame near-IR).
The stellar masses at $z \sim 2$ are estimated to be between $10^{10}-10^{11} \ M_{\odot}$,
and are in the range of masses produced by the semi-analytic model at the same
redshift. This match insures a fair comparison between the model stellar luminosities
and the data, by nailing down the mass-to-light ratios and thus
leaving no room for degeneracies in this sense (see Conroy et al. 2009).
We notice that these galaxies lie close to the top-mass end of the model
distribution; for instance, at $z=2.5$ the observed galaxy lies among the $7 \%$
most massive galaxies in the simulation (this amounts to $\sim 500$ objects in the simulated
volume). This shows that the semi-analytic model is performing well in terms of galaxy masses at
these redshifts, but that there is no margin for reaching the observed colours
without the TP-AGB.
We conclude that the TP-AGB appears to be a necessary ingredient of the model
in order to reproduce the colours and near-IR luminosity of these nearly-passive galaxies.
\subsection{Star-forming galaxies}
Fig.~(\ref{colmag}) shows the same plot as Fig.~(\ref{7gals}) for the model star-forming galaxies,
compared with the sample of star-forming galaxies
from GOODS, selected in the range $1.7 \leq z \leq 2.3$. The M05 run
portrayed in the \textit{left panel}, where \textit{cyan dots} represent
the prediction of the semi-analytic model with dust reddening, and the \textit{blue dots}
represent the case without reddening. The \textit{right panel} shows the
PEGASE run, with \textit{magenta dots} for the case with dust reddening, and \textit{green dots}
for the case without reddening.
As expected, the galaxies in the M05 run are much redder than in the PEGASE run
(Tonini et al. 2009). They are in excellent agreement with the data.
The M05 run reproduces both the observed amplitude
of the $H-Irac3$ colour and the slope of the colour-magnitude relation,
indicating that the SFR across the observed mass range is well represented in the model.
In fact, the model star-formation rates easily cover the range of the ones
derived from this sample of observed galaxies.
The PEGASE run on the other hand is completely off the data, producing much bluer
colours. Moreover, while the Irac3 luminosity range is correctly reproduced
by the M05, in the PEGASE run the model galaxies are much fainter, and the run
misses half of the sample in luminosity.
This happens because the TP-AGB increases the emission in Irac3 (rest-frame K) by 1 mag on average.
About $\sim 53.5 \%$ of the galaxies in the GOODS sample have magnitudes
$Irac3 \leq 22$, and these objects can be reproduced only with the M05 run, while the
run with the PEGASE recipe is far off the mark. On the other hand, only about the most luminous $\sim 10 \%$
of objects is not reproduced by the M05.
It is also clear that a higher dust reddening cannot be advocated to make up for the
absence of the TP-AGB emission and match the observations.
First of all, dust reddening tilts the colour-magnitude relation upwards
at the high-mass end, but does not increase the Irac3 (rest-frame K) luminosity.
Secondly, the magnitude of its effect on the H-Irac3 colour is well below the
shift introduced by the TP-AGB. Our reddening recipe is
physically associated with the intrinsic SFR of the model galaxies (a choice sustained
by observations), which at $z \sim 2$ spans a range of values up to a few $10^2 M_{\odot}/yr$,
corresponding to $E(B-V) \sim 0.3-0.4$ (e.g. Daddi et al. 2007), and yielding values of
$E(B-V) \sim 0.1-0.2$ for typical SFR produced by GalICS.
Either due to the limited mass resolution of the N-body simulation and/or the recipes
currently employed in GalICS, the hybrid model is not able to produce very massive
starbursts, so that the maximum model SFR might be
on the low side. The investigation of this issue is beyond the scope of this paper,
and we shall pursue it further in future work.
Fig.~(\ref{colcolmiocalzetti}) shows the colour-colour relation J-H $vs$ H-Irac3 at $z=2$
(corresponding to rest-frame B-V $vs$ V-K),
and compares again the M05 run (\textit{left panel}) and the PEGASE run
(\textit{right panel}) with the same set of data. In the panels,
\textit{red/yellow dots} represent nearly-passive galaxies (SFR$<3 \ M_{\odot/yr}$) and
\textit{cyan/magenta dots} represent the star-forming galaxies (SFR$>10 \ M_{\odot/yr}$).
The striking feature highlighted by this plot is that, with the M05 stellar populations,
the semi-analytic model is now able to produce star-forming galaxies that
are very blue in the optical and very red in the near-IR, and reproduces the data
very well. The run with the PEGASE recipe on the other hand, while covering the correct
range in $J-H$, is clearly off the data in the near-IR. Given that the optical
colours are correctly reproduced, we conclude that a higher amount of reddening
would be unrealistic.
This plot highlights the ability of GalICS to reproduce the mix of stellar populations
of different ages in these galaxies, by matching the optical blue colours. At the same time, the
GalICS $+$ M05 run can also match the near-IR colours thanks to the contribution
of the TP-AGB. Finally, the dust extinction adopted in this implementation lies in a
sensible range of values. Our model reproduces the properties of the observed galaxies at $z \sim 2$
without invoking unrealistic star-formation rates or dust reddening.
Notice the 3 nearly-passive galaxies of Maraston et al. (2006) in the plot
(\textit{green triangles}). As
expected, they are near the reddest in optical colours. Most of the sample of star-forming
galaxies is bluer than these objects, but the near-IR colours are comparable.
Correspondingly in the model, the nearly-passive galaxies are among the reddest
both in the near-IR and optical colours, but they are not distinguishable from
active galaxies on the basis of colours alone.
As already pointed out in Tonini et al. (2009), star formation does not dilute the TP-AGB
emission in the near-IR, so that the near-IR colours alone do not discriminate
between star-forming and passive. In fact, star-forming galaxies can be redder than passive
ones in the near-IR.
This proves that the so-called 'red \& dead' galaxies may not, in fact, be dead at all.
The same near-IR colours can be achieved in star-forming galaxies, because the
TP-AGB emission is not offset by the light produced by young stars.
\subsubsection{Effects of removing the TP-AGB from M05 models}
\begin{figure}
\includegraphics[scale=0.4]{fig4.eps}
\caption{Same plot as in Fig.~(\ref{colmag}), but with the comparison between
the PEGASE run (\textit{right panel}) and a run with Maraston models
without the TP-AGB phase (\textit{left panel}),
and the rest of the recipe unchanged. In both cases we adopt
the reddening described previously. The (\textit{black dots})
represent the same set of data of Fig.~(\ref{colmag}).}
\label{colmag2}
\end{figure}
Obviously, the M05 and the PEGASE models differ also for other recipes than
the TP-AGB implementation. For instance, in M05 young stellar populations
are modelled with the Geneva stellar evolutionary tracks, while the Padova tracks
are used in PEGASE.
In order to rule out that the significant difference in the predictions of colour and
luminosity shown in Fig.~(\ref{colmag}) is due to any effect other than the TP-AGB,
we performed a test run of
GalICS with Maraston (2005) models \textit{without the TP-AGB}, and with the rest of the recipe
unchanged. This quantifies the actual contribution of the TP-AGB in the predictions of the M05 run.
The result is shown in Fig.~(\ref{colmag2}), where this new run (represented by
\textit{cyan empty dots, left panel})
is compared with the PEGASE run (\textit{magenta dots, right panel})
runs, and the same sets of data.
The predicted luminosity and colour in the run with Maraston models without TP-AGB is
strikingly similar to those of the PEGASE run.
This shows that the TP-AGB emission is the main driver
of the success of the model in correctly predicting luminosities and colours of the observed
galaxies.
\subsubsection{Effects of nebular emission}
It has been pointed out (Leitherer et al. 1999, Zackrisson et al. 2008, Molla' et al. 2009)
that, among the factors that can affect the colours of star-forming galaxies,
nebular emission can play a significant role. This kind of emission is produced by
ionizing photons emitted from massive young stars when scattering with the gas surrounding
star-forming regions, originating a series of emission lines and a continuum flux.
The nebular emission is important in the presence of strong starbursts,
when stellar populations of ages $\tau \le 5-10$ Myr contribute significantly to the
total emission.
For the first time, we implemented this contribution into a semi-analytic code of galaxy
formation, using the publicly-available models of Molla' et al. (2009). For each age and
metallicity in our grid of input M05 models, we included the nebular emission on top of
the stellar emission, following the indications of Molla' et al. (2009).
Notice that this is not stricly physically sensible, in that
the ionizing photons are not subtracted from the stellar UV spectrum, so that energy
is not conserved for a given stellar population. This however has the advantage of
setting the maximum limit
of the possible contribution of the nebular emission to the total galaxy spectra.
\begin{figure*}
\includegraphics[scale=0.9]{fig5.eps}
\caption{The Irac3 $vs$ H-Irac3 colour-magnitude relation (\textit{left panel})
and the J-H $vs$ H-Irac3 colour-colour relation (\textit{right panel}) at $z=2$ for the run with the
M05 models (\textit{cyan dots})
and for the run with M05 models $+$ nebular emission (as in Molla' et al. 2009;
\textit{yellow triangles}), compared with
the GOODS-S catalogue (Daddi et al. 2007). The \textit{highlighted squares} represent
galaxies with SFR$>70 \ M_{\odot}/yr$ in both runs. We are only plotting the model galaxies
with reddening, for clarity.}
\label{popstar}
\end{figure*}
The result is presented in Fig.~(\ref{popstar}). The (\textit{left panel}) shows the
colour-magnitude relation Irac3 $vs$ H-Irac3 and the \textit{right panel}) shows the
colour-colour relation J-H $vs$ H-Irac3 at $z=2$ (observed frame). The M05 run is
represented by \textit{cyan dots}, and the M05 run with nebular emission is
represented by \textit{yellow triangles}. As before, the model predictions are compared
with the GOODS-S catalogue. Notice that the nebular emission leaves the colours
and luminosities of galaxies virtually unchanged in these runs, except at the very
high-luminosity red end. The \textit{highlighted squares} in the plot represent
galaxies with an instantaneous star-formation rate of SFR$>70 \ M_{\odot}/yr$. For some
of these objects, the nebular emission increases the H-Irac3 (rest-frame V-K) colour
by about 0.2-0.4 mags, while it is not so clear for the J-H (rest-frame B-V).
The reason why the nebular emission contribution is not more impressive is
that the semi-analytic model cannot produce very violent starbursts. The high-SFR
tail of the galaxy population at $z=2$ is around 100-200 $M_{\odot}/yr$,
but these galaxies are relatively rare in the simulation. As stated in
Leitherer et al. 1999, Zackrisson et al. 2008 and Molla' et al. 2009,
the nebular emission is a main factor in determining the colours of starburst
galaxies, where very young ($\tau < 5-10$ Myr) stellar populations represent
a significant fraction of the stellar mass. Hence we conclude that the inclusion
of nebular emission does not affect our results.
\section{Rest-frame K-band luminosity functions}
In the previous Sections we presented a qualitative comparison between the
spectral energy distributions of model and real galaxies, and showed that
the introduction of the TP-AGB in the semi-analytic model substantially improves
the model performance in reproducing the observed galaxies at $z \sim 2$.
A more quantitative approach is to compute the luminosity function for
the model galaxy population. After the correct stellar population models are
implemented in the semi-analytic model and the TP-AGB is included,
the luminosity function in the near-IR is a good proxy
for the galaxy stellar mass function, therefore representing a meaningful test of
the mass assembly in the hierarchical model. With one caveat, the role played by AGN feedback.
AGN (Active Galactic Nuclei) feedback was introduced in the galaxy formation
models to turn down the efficiency
of star formation at the high-mass end of the galaxy population, a fundamental recipe
in order to obtain the correct shape of the luminosity function
(see Benson et al. 2003, Binney 2004, Granato et al. 2004, Silk 2005, Bower et al. 2006,
Croton et al. 2006). In fact, if stars were
to follow the dark matter halo mass function (which is essentially scale-invariant), the
models would produce a severe overabundance of massive systems.
A mechanism is needed in the models to
suppress star formation in more massive systems, at specific
points in each galaxy history.
The most popular solution adopted in the models, a mechanism that can act on a galactic scale
and heat up the gas, thus preventing star formation, is the energy emission
from the central black hole following gas accretion, the so-called AGN feedback.
It has the advantage that it is preferentially active in massive objects and at high redshifts
($z > 1-2 $, see for instance Madau et al. 1996, Shaver et al. 1996).
However, the coupling between the AGN energy release and the gas in galaxies is still
poorly understood, so that this source of feedback is implemented a-posteriori
to fine-tune the models, given some set of observational constraints.
The most widely used calibration data set for semi-analytic models is the luminosity
function at $z=0$, for various photometric bands. Different stellar
population models in the semi-analytic code affect the galaxy spectral energy
distribution and mass-to-light ratio (M05, Tonini et al. 2009),
so before investigating the
predictions at high redshift, a fundamental step is to verify
that the model is still well calibrated at $z=0$.
\begin{figure}
\includegraphics[scale=0.4]{fig6.eps}
\caption{The original GalICS rest-frame K-band luminosity function at $z=0$ (Hatton et al. 2003;
\textit{green line}) compared with data from Cole et al. 2001, Jones et al. 2006,
Bell et al. 2003 and a combination of these 3 samples (\textit{thick black circles}).
The \textit{thick red line} represents again the M05 run, and the \textit{thick, cyan line}
represents the PEGASE run.}
\label{LForiginal}
\end{figure}
Fig.~(\ref{LForiginal}) shows the $z=0$ K-band rest-frame luminosity function for
GalICS as in the original paper (Hatton et al. 2003, \textit{green line}),
compared to data from Cole et al. (2001; \textit{pentagons}),
Jones et al. (2006; \textit{triangles}),
Bell et al. (2003; \textit{squares}) and a combination of these 3 samples
(\textit{thick black circles}).
In this magnitude range, the errorbars on each luminosity function are at most as large
as the spread between the different functions.
The \textit{red line} is the luminosity function obtained with the M05 run,
and the \textit{thick, cyan line} line represents the PEGASE run.
The difference between the luminosity function in the GalICS $+$ PEGASE run and the
original version of Hatton et al. (2003) is partly due to the difference between the PEGASE and
Stardust models (used in the original version of the code), and the fact that the current
version of GalICS includes the improved recipe for
AGN feedback introduced by Cattaneo et al. (2006) and the new chemical evolution model
implemented by Pipino et al. (2008). In general, the original version and our two new runs
agree reasonably well with the data, and in particular the TP-AGB allows the model to
perform better at the high-mass end.
The LF in the current version of GalICS
overpredicts the number of galaxies at intermediate luminosities,
regardless of the stellar population models in use.
This tension with the data is not critically relevant for the work presented here,
in that it regards the implementation of physics beyond the SSP models, and it does not affect the
comparison between the M05 and PEGASE runs.
\begin{figure}
\includegraphics[scale=0.4]{fig7.eps}
\caption{The difference between the rest-frame K-band
broadband magnitudes predicted in the M05 and PEGASE runs, $\Delta M_K = M_K (M05) - M_K (PEGASE)$,
as a function of stellar mass, in 4 redshifts bins from $z=0$ to $z=3$. }
\label{Lmass}
\end{figure}
Notice that, at $z=0$, the K-band luminosity function is
mildly dependent on the
input model SSPs. In particular, there is a small difference
between the M05 run
and the run with the Stardust models. This
may originate from the different temperatures of the RGB phase in the two models and also
from residual TP-AGB dominated populations in the simulations at low redshift.
In fact, the amount of intermediate-age population at low redshift is
much lower than at high look-back times; hence
at z=0 the K band mostly traces the Red Giant Branch in galaxies.
The offset between the M05 and the PEGASE runs is $\sim$ 0.2 mag and is due to
galaxies that recently had - or are having - star formation.
since for the majority of the galaxies
the bulk of the stellar populations are old and the TP-AGB phase is subdominant (M05).
This however does not
remain true at all redshifts, as shown in the previous Sections and in Tonini et al. (2009). The
TP-AGB becomes the dominant contributor to the near-IR luminosity at $z>1$, so that the
mass-to-light ratio in the K-band and in the nearing bands is significantly offset between
the M05 and the PEGASE runs.
To illustrate this point, Fig.~(\ref{Lmass}) shows the difference between
the predicted K-band magnitudes in the M05 and PEGASE runs, defined as
$\Delta M_K = M_K (M05) - M_K (PEGASE)$, for the same galaxy masses,
in 4 redshifts bins from $z=0$ to $z=3$.
At $z=0$ the M05 run produces galaxies on average brighter by 0.3 mag
than the run with the PEGASE recipe,
but at $z>1$ the M05 run gets brighter by
more than 1 mag, and the offset between the two runs shows a mild dependence on galactic mass.
This difference is going to be mirrored by the luminosity function at high redshift.
\begin{figure*}
\includegraphics[scale=0.8]{fig8.eps}
\caption{The rest-frame K-band luminosity function of the M05 (\textit{thick red line}) and
PEGASE (\textit{thick cyan line}) runs, compared with published LFs in the literature:
Bower et al. 2006 (\textit{solid}), Cole et al. 2000 (\textit{short-dashed}),
Baugh et al. 2005 (\textit{dotted}), Menci et al. 2006 (\textit{long dot-dashed}),
Monaco et al. 2007 (\textit{short dot-dashed}) and De Lucia \& Blaizot (2007)
(\textit{long-dashed}).}
\label{bower}
\end{figure*}
Fig.~(\ref{bower}), shows the $z=2.5$ luminosity functions in the rest-frame K band
for the M05 and PEGASE runs (\textit{thick red} and \textit{thick cyan} lines respectively).
As expected, there is an offset of roughly 1 mag between the functions (with the PEGASE run
underpredicting the number of bright galaxies), more pronounced
at the high-mass end due to the steeper slope of the function and to the mild dependence
of the offset with galaxy mass. The difference between the two functions is exclusively due
to different stellar population models implemented into the GalICS code.
The two functions are also compared with the predictions of other semi-analytic models
in the literature (see the caption for the line-coding),
all of which make use of either the BC03 models or the GRASIL SSPs of Silva et al. (1998)
(which implement the TP-AGB, but produce near-IR spectra very similar to BC03;
P. Monaco, private communication), but differ for the
various implementations of the baryonic
physics. In particular, the main factor shaping the bright end of the near-IR luminosity
function is the recipe for AGN feedback.
Two considerations are important here. The first is that, regardless of the fact that
all these models are set to match the $z=0$ luminosity function, their predictions
at high redshift diverge dramatically. This is in part due to the different
recipes for the baryonic physics adopted in each model. However, semi-analytic models
cannot match the $z=0$ luminosity function without AGN feedback, which in each case
is implemented ad-hoc to fine-tune
the model at $z=0$, based on energy arguments at best. The lack of physics in the AGN
recipe makes it degenerate with other model parameters, expecially at high redshift.
In fact, AGN activity supposedly
peaks around $1 < z < 3$, at epochs when the stellar emission is dominated by the TP-AGB in
the near-IR, so that these two factors compete in shaping the high-redshift luminosity function.
The second consideration is even more striking. The shift in the luminosity function caused
by the introduction of the TP-AGB emission in the model is comparable in magnitude to the
difference introduced by different AGN-feedback recipes. In fact, the M05 and PEGASE runs
actually bracket most of the other semi-analytic models at the high-mass end.
Given that the stellar emission is much better understood and constrained, the importance
of producing realistic and complete stellar population models \textit{before}
fine-tuning the AGN-feedback recipe is evident.
\section{Summary and Discussion}
In a recent work (Tonini et al. 2009) we introduced the complete treatment
of the TP-AGB phase of stellar
evolution into a semi-analytic model of galaxy formation, by inserting the
Maraston (2005) SSP models into the code GalICS. In the work presented here we compared
the predictions on the near-IR luminosities and colours of high-redshift
galaxies with data samples of nearly-passive and star-forming galaxies
around $z \sim 2$. Our main results are:
$\cdot$ the TP-AGB is fundamental to allow the semi-analytic model to reproduce
the observed optical and near-IR colours of both nearly-passive and star-forming galaxies
at $z \sim 2$;
the inclusion of the TP-AGB increases the Irac3 luminosity (rest-frame K)
and shifts the H-Irac3 (rest-frame V-K) colours by more than 1 magnitude;
$\cdot$ without the
TP-AGB, it is not possible to match the observed galaxy colours and luminosities
by a modification of the dust reddening recipe alone;
$\cdot$ the TP-AGB emission does not alter the optical luminosity and colours
of star-forming galaxies. On the other hand, star formation does not dilute
the TP-AGB emission in the near-IR. Even star-forming galaxies, very blue in
the optical, can be very red in the near-IR. Therefore the labelling of
red galaxies as 'red and dead' is misleading;
$\cdot$ the nebular emission, produced by young stellar populations, does not
add a significant contribution to the colours of star-forming galaxies, in the
range of star-formation rates covered by the model; for SFR$>70 \ M_{\odot}/yr$
the rest-frame V-K colour is reddened by 0.2-0.4 mags;
$\cdot$ the predicted mass-luminosity relation is affected by the inclusion
of the TP-AGB; for a given galaxy mass, the rest-frame K-band luminosity
is higher by more than 1 mag at $z>1$. As a consequence, the K-band luminosity
function predicted by the model with the TP-AGB shifts
redwards, expecially at the high-mass end, for $z>1$
(by $\sim 0.7$ mag at $z \sim 2.5$). The spread
in the luminosity function between runs with and without the TP-AGB
is comparable to the scatter caused by different AGN-feedback recipes
in the literature.
Note that the high-mass end of the luminosity function in the near-IR is dominated
by spheroids, or the progenitors of today's spheroids. If the use of the TP-AGB in the
semi-analytic model shifts the luminosity function by $\sim$1 mag at the high-mass end,
it means that the mass-to-light ratio is lower by a factor of $\sim$2.5 for a given
luminosity. When galaxy masses are inferred from observations by the use of these models,
they are lower by the same factor (as shown in M05). This may rise the question of whether there
is enough mass in spheroids at high redshift to account for the $\sim 50\%$ of stellar mass
in ellipticals measured in the local universe. However, the model correctly predicts the
stellar mass density at all redshifts, meaning that only the \textit{distribution of galaxy masses}
is at tension with observations, \textit{if the TP-AGB is not taken into account}.
In fact, hierarchical models in general predict a faster evolution
of the high-mass end of the stellar mass function than currently inferred from
observations (see for instance Conselice et al. 2007).
A more accurare derivation of galaxy masses through complete stellar population models
with the TP-AGB, coupled with more accurate predictions from hierarchical models with the
right input SSP, surely contribute to alleviate the discrepancy.
The inclusion of the TP-AGB allows the semi-analytic model to reproduce the very red end of the
galaxy population at $z \sim 2$, both for nearly-passive and for star-forming objects.
It allows the model to do so with a comfortable range of galaxy masses and dust reddening.
Most importantly, it contributes to a realistic
and comprehensive treatment of the galaxy light emission in galaxy formation models,
making them a much more precise tool to test our understanding of galaxy assembly.
The implementation of the TP-AGB allows the model to produce, at a given stellar
mass, redder and more luminous galaxies in the near-IR, expecially at high redshift
where the ages of the stellar populations peak around the epoch of maximal emission
from this stellar phase.
In case of nearly-passively evolving galaxies, the model can
reproduce the red colours and high K-band magnitudes without invoking too large
stellar masses or too old ages, which would be problematic in the hierarchical context.
In the case of star-forming galaxies, the TP-AGB still increases
the near-IR luminosity and makes the galaxies redder, without offsetting the
blue optical colours. Thus, observed red colours in the near-IR do not necessarily imply old ages
and passive evolution, a fact that again would be problematic for the hierarchical
picture at high redshift. In general, the introduction of the TP-AGB in the models
is a step forward in reconciling the hierarchical assembly mechanism with the
observations of the high-redshift universe.
\section*{Acknowledgments}
This project is supported by the Marie Curie Excellence Team Grant MEXT-CT-2006-042754
of the Training and Mobility of Researchers programme financed by the European Community.
We wish to acknowledge Danilo Marchesini, Alvio Renzini, Emanuele Daddi
and Bruno Henriquez for the
interesting and useful discussions. We also want to thank Mark Dickinson and Emanuele Daddi
for letting us use the GOODS-S data sample of star-forming galaxies.
Finally, we wish to thank the Referee for his comments and suggestions, which
helped to improve this work.
|
2,869,038,154,824 | arxiv | \section{Introduction}
In this note we state a bit detailed account about
MacPherson's Chern class transformation $C_*$ for quotient stacks defined in \cite{O1},
although all the instructions have already been made in that paper.
Our approach is also applicable for
other additive characteristic classes, e.g.,
Baum-Fulton-MacPherson's Todd class transformation \cite{BFM}
(see \cite{EG2, BZ} for the equivariant version) and more generally
Brasselet-Sch\"urmann-Yokura's Hirzebruch class transformation \cite{BSY} (see section 4 below).
Throughout we work over the complex number field $\mathbf{C}$
or a base field $k$ of characteristic $0$.
We begin with recalling $C_*$ for schemes and algebraic spaces.
These are spaces having trivial stabilizer groups.
In following sections we will deal with quotient stacks having affine stabilizers,
in particular, `(quasi-)projective' Deligne-Mumford stacks
in the sense of Kresch \cite{Kresch}.
\subsection{Schemes}
For the category of quasi-projective schemes $U$ and proper morphisms,
there is a unique natural transformation from the constructible function functor to
the Chow group functor, $C_*: F(U) \to A_*(U)$,
so that it satisfies the normalization property:
$$C_*(\jeden_U)=c(TU) \frown [U] \in A_*(U) \quad \mbox{ if $U$ is smooth. }$$
This is called the {\it Chern-MacPherson transformation},
see MacPherson \cite{Mac} in complex case ($k=\mathbf{C}$) and Kennedy \cite{Ken}
in more general context of $ch(k)=0$.
Here the naturality means the commutativity $f_*C_*=C_*f_*$ of $C_*$
with pushforward of proper morphisms $f$.
In particular, for proper $pt: U \to pt (={\rm Spec}(k))$,
the ($0$-th) degree of $C_*(\jeden_U)$ is equal to
the Euler characteristic of $U$: $pt_*C_*(\jeden_U)=\chi(U)$
(as for the definition of $\chi(U)$ in algebraic context, see \cite{Ken, Joyce}).
As a historical comment,
Schwartz \cite{Schwartz} firstly studied
a generalization of the Poincar\'e-Hopf theorem for
complex analytic singular varieties by introducing
a topological obstruction class for certain stratified vector frames,
which in turn coincides with MacPherson's Chern class \cite{BS}.
Therefore, $C_*(U):=C_*(\jeden_U)$ is usually
called the {\it Chern-Schwartz-MacPherson class} (CSM class) of a possibly singular variety $U$.
To grasp quickly what the CSM class is, there is a convenient way
due to Aluffi \cite{Aluffi1, Aluffi2}.
Let $U$ be a singular variety and
$\iota: U_0 \hookrightarrow U$ a smooth open dense reduced subscheme.
By means of resolution of singularities,
we have a birational morphism $p: W \to U$ so that $W=\overline{U_0}$ is smooth
and $D=W-U_0$ is a divisor with smooth irreducible components $D_1, \cdots , D_r$
having normal crossings.
Then by induction on $r$ and properties of $C_*$ it is shown that
$$C_*(\jeden_{U_0}) =p_*\left(\frac{c(TW)}{\prod (1+D_i)} \frown [W]\right) \in A_*(U).$$
(Here $c(TW)/\prod (1+D_i)$ is equal to the total Chern class of dual to
$\Omega_W^1(\log D)$ of differential forms with logarithmic poles along $D$).
By taking a stratification $U=\coprod_{j} U_j$, we have
$C_*(U) = \sum_j C_*(\jeden_{U_j})$.
Conversely,
we may regard this formula as an alternative definition of CSM class,
see \cite{Aluffi1}.
\subsection{Algebraic spaces}
We extend $C_*$ to the category of arbitrary schemes or algebraic spaces
(separated and of finite type).
To do this, we may generalize Aluffi's approach, or
we may trace the same inductive proof by means of Chow envelopes
(cf. \cite{Kimura})
of the singular Riemann-Roch theorem for arbitrary schemes \cite{FG}.
Here is a short remark.
An {\it algebraic space} $X$ is
a stack over $Sch/k$, under \'etale topology, whose stabilizer groups are trivial:
Precisely, there exists a scheme $U$ (called an {\it atlas})
and a morphism of stacks $u: U \to X$
such that for any scheme $W$ and any morphism $W \to X$
the (sheaf) fiber product $U\times_X W$ exists as a scheme,
and the map $U\times_X W \to W$ is an \'etale surjective morphism of schemes.
In addition, $\delta: R:=U\times_X U \to U \times_k U$ is quasi-compact,
called the {\it \'etale equivalent relation}.
Denote by $g_i: R \to U$ (i=1,2) the projection to each factor of $\delta$.
The Chow group $A_*(X)$
is defined using an \'etale atlas $U$
(Section 6 in \cite{EG}).
In particular, letting $g_{12*}:=g_{1*}-g_{2*}$,
$$\xymatrix{
A_*(R) \ar[r]^{g_{12*}} & A_*(U) \ar[r]^{u_*} & A_*(X) \ar[r] & 0
}$$
is exact (Kimura \cite{Kimura}, Theorem 1.8).
Then the CSM class of $X$ is given by
$C_*(X) = u_*C_*(U)$:
In fact, if $U' \to X$ is another atlas for $X$ with the relation $R'$,
we take the third $U''=U\times_X U'$ with $R''=R\times_X R'$,
where $p: U'' \to U$ and $q: U''\to U'$ are \'etale and finite.
Chow groups of atlases modulo ${\rm Im}\, (g_{12*})$ are mutually identified
through the pullback $p^*$ and $q^*$, and particularly,
$p^*C_*(U) = C_*(U'')=q^*C_*(U')$, that is checked by using
resolution of singularities or
the Verdier-Riemann-Roch \cite{Yokura} for $p$ and $q$.
Finally we put $C_*: F(X) \to A_*(X)$ by sending $\jeden_W \mapsto \iota_*C_*(W)$
for integral algebraic subspaces $W \stackrel{\iota}{\hookrightarrow} X$ and extending it linearly, and
the naturality for proper morphisms
is proved again using atlases.
This is somewhat a prototype of $C_*$ for quotient stacks described below.
\section{Chern class for quotient stacks}
\subsection{Quotient stacks}
Let $G$ be a linear algebraic group acting on a scheme or algebraic space $X$.
If the $G$-action is set-theoretically free,
i.e., stabilizer groups are trivial,
then the quotient $X \to X/G$ always exists as a morphism of algebraic spaces
(Proposition 22, \cite{EG}).
Otherwise, in general we need the notion of quotient stack.
The {\it quotient stack} $\mathcal{X}=[X/G]$ is
a (possibly non-separated) Artin stack over $Sch/k$, under fppf topology
(see, e.g., Vistoli \cite{Vistoli}, G\'omez \cite{Gomez} for the detail):
An object of $\mathcal{X}$ is a family of $G$-orbits in $X$
parametrized by a scheme or algebraic space $B$,
that is, a diagram $B \stackrel{q}{\leftarrow} P \stackrel{p}{\rightarrow} X$
where $P$ is an algebraic space,
$q$ is a $G$-principal bundle and $p$ is a $G$-equivariant morphism.
A morphism of $\mathcal{X}$ is a $G$-bundle morphism $\phi: P \to P'$ so that $p'\circ \phi = p$,
where $B' \stackrel{q'}{\leftarrow} P' \stackrel{p'}{\rightarrow} X$ is another object.
Note that there are possibly many non-trivial automorphisms $P \to P$ over
the identity morphism $id: B \to B$,
which form the stabilizer group associated to the object
(e.g., the stabilizer group of a `point' ($B=pt$) is non-trivial in general).
A morphism of stacks $B \to \mathcal{X}$ naturally corresponds to an object $B \leftarrow P \to X$,
that follows from Yoneda lemma:
In particular there is a morphism (called {\it atlas}) $u: X \to \mathcal{X}$
corresponding to the diagram $X \stackrel{q}{\leftarrow} G\times X \stackrel{p}{\rightarrow} X$,
being $q$ the projection to the second factor and $p$ the group action.
The atlas $u$ recovers any object of $\mathcal{X}$
by taking fiber products: $B \leftarrow P=B \times_\mathcal{X} X \rightarrow X$.
Let $f: \mathcal{X} \to \mathcal{Y}$ be a {\it proper} and
{\it representable} morphism of quotient stacks, i.e.,
for any scheme or algebraic space $W$ and morphism $W \to \mathcal{Y}$, the base change
$\mathcal{X} \times_\mathcal{Y} W \to W$ is a proper morphism of algebraic spaces.
Take presentations $\mathcal{X}=[X/G]$, $\mathcal{Y}=[Y/H]$,
and the atlases $u: X \to \mathcal{X}$, $u': Y \to \mathcal{Y}$.
There are two aspects of $f$: \\
\noindent
(Equivariant morphism):
Put $B:=\mathcal{X}\times_\mathcal{Y} Y$, which naturally has a $H$-action
so that $[B/H]=[X/G]$, $v: B \to \mathcal{X}$ is a new atlas,
and $\bar{f}:B \to Y$ is $H$-equivariant:
\begin{equation}\label{d1}
\xymatrix{
B \ar[r]^{\bar{f}} \ar[d]_{v} & Y \ar[d]^{u'} \\
\mathcal{X} \ar[r]_f & \mathcal{Y}
}
\end{equation}
(Change of presentations):
Let $P:=X \times_\mathcal{X} B$, then the following diagram is considered as
a family of $G$-orbits in $X$ and simultaneously as a family of $H$-orbits in $B$, i.e.,
$p: P \to X$ is a $H$-principal bundle and $G$-equivariant,
$q: P \to B$ is a $G$-principal bundle and $H$-equivariant:
\begin{equation}\label{d2}
\xymatrix{
P \ar[r]^{q} \ar[d]_{p} & B \ar[d]^{v} \\
X \ar[r]_{u} & \; \mathcal{X}.
}
\end{equation}
A simple example of such $f$ is given by
proper $\varphi: X \to Y$ with an injective homomorphism
$G \to H$ so that $\varphi(g.x)=g.\varphi(x)$ and $H/G$ is proper. In this case,
$P=H \times_k X$ and $B=H\times_G X$ with
$p: P \to X$ the projection to the second factor, $q: P \to B$ the quotient morphism.
\subsection{Chow group and pushforward}
For schemes or algebraic spaces $X$ (separated, of finite type) with $G$-action,
the {\it $G$-equivariant Chow gourp} $A_*^G(X)$ has been introduced
in Edidin-Graham \cite{EG},
and the {\it $G$-equivariant constructible function} $F^G(X)$ in \cite{O1}.
They are based on Totaro's algebraic Borel construction:
Take a Zariski open subset $U$ in
an $\ell$-dimensional linear representation $V$ of $G$
so that $G$ acts on $U$ freely.
The quotient exists as an algebraic space, denoted by $U_G=U/G$.
Also $G$ acts $X \times U$ freely,
hence the mixed quotient $X \times G \to X_G:=X\times_G U$
exists as an algebraic space.
Note that $X_G \to U_G$ is a fiber bundle with fiber $X$ and group $G$.
Define
$A_n^G(X):=A_{n+\ell-g}(X_G)$ ($g=\dim G$) and $F^G(X):=F(X_G)$
for $\ell \gg 0$.
Precisely saying, we take the direct limit over all linear representations of $G$,
see \cite{EG, O1} for the detail.
$A_n^G(X)$ is trivial for $n > \dim X$ but
it may be non-trivial for negative $n$. Also note that
the group $F^G_{inv}(X)$ of {\it $G$-invariant} functions over $X$ is a subgroup of $F^G(X)$.
Let us recall the proof that these groups are actually invariants of quotient stacks $\mathcal{X}$.
Look at the diagram (\ref{d2}) above.
Let $g=\dim G$ and $h=\dim H$. Note that $G\times H$ acts on $P$.
Take open subsets $U_1$ and $U_2$ of
representations of $G$ and $H$, respectively ($\ell_i=\dim U_i\; i=1,2$)
so that $G$ and $H$ act on $U_1$ and $U_2$ freely respectively.
Put $U=U_1 \oplus U_2$, on which $G \times H$ acts freely.
We denote the mixed quotients for spaces arising in the diagram (\ref{d2}) by
$P_{G\times H}:=P\times_{G\times H}U$, $X_G:=X\times_G U_1$ and $B_H:=B\times_H U_2$.
Then the projection $p$ induces the fiber bundle
$\bar{p}: P_{G\times H} \to X_G$ with fiber $U_2$ and group $H$,
and $q$ induces $\bar{q}: P_{G\times H} \to B_{H}$ with fiber $U_1$ and group $G$.
Thus, the pullback $\bar{p}^*$ and $\bar{q}^*$ for Chow groups are isomorphic,
$A_{n+\ell_1}(X_G) \simeq A_{n+\ell_1+\ell_2}(P_{G\times H}) \simeq A_{n+\ell_2}(B_H)$.
Taking the limit, we have the {\it canonical identification}
$$\xymatrix{
A^G_{n+g}(X) \ar[r]^{p^*\;\; }_{\simeq\;\;} & A^{G\times H}_{n+g+h}(P) & \ar[l]^{\;\;\simeq}_{\;\;q^*} A^H_{n+h}(B)
}$$
(Proposition 16 in \cite{EG}).
Note that $(q^*)^{-1}\circ p^*$ shifts the dimension by $h-g$.
Also for constructible functions, put the pullback $p^*\alpha:=\alpha \circ p$,
then we have $F^G(X) \simeq F^{G\times H}(P) \simeq F^H(B)$
via pullback $p^*$ and $q^*$ (Lemma 3.3 in \cite{O1}).
We thus define $A_*(\mathcal{X}):=A_{*+g}^G(X)$ and $F(\mathcal{X}):=F^G(X)$,
also $F_{inv}(\mathcal{X}):=F^G_{inv}(X)$, through the canonical identification.
Given proper representable morphisms of quotient stacks $f: \mathcal{X} \to \mathcal{Y}$
and any presentations $\mathcal{X}=[X/G]$, $\mathcal{Y}=[Y/H]$, we define
the pushforward $f_*: A_*(\mathcal{X}) \to A_*(\mathcal{Y})$ by
$$f_*^H\circ (q^{*})^{-1}\circ p^*: A_{n+g}^G(X) \to A_{n+h}^H(Y)$$
and also
$f_*: F(\mathcal{X}) \to F(\mathcal{Y})$ in the same way.
By the identification $(q^{*})^{-1}\circ p^*$,
everything is reduced to the equivariant setting (the diagram (\ref{d1})).
\begin{lemma} The above $F$ and $A_*$ satisfy the following properties: \\
{\rm (i)} For proper representable morphisms of quotient stacks $f$,
the pushforward $f_*$ is well-defined; \\
{\rm (ii)} Let $f_1: \mathcal{X}_1 \to \mathcal{X}_2$, $f_2: \mathcal{X}_2 \to \mathcal{X}_3$ and
$f_3:\mathcal{X}_1 \to \mathcal{X}_3$ be proper representable morphisms of stacks
so that $f_2\circ f_1$ is isomorphic to $f_3$,
then $f_{2*}\circ f_{1*}$ is isomorphic to $f_{3*}$
{\rm (}$f_{3*}=f_{2*}\circ f_{1*}$
using a notational convention in Remark 5.3, \cite{Gomez}{\rm )}.
\end{lemma}
\noindent {\sl Proof} :\;
Look at the diagram below, where $\mathcal{X}_i=[X_i/G_i]$ ($i=1,2,3$).
We may regard $\mathcal{X}_1=[X_1/G_1]=[B_1/G_2]=[B_3/G_3]$, and so on.
{\rm (i)} Put $f=f_1$, then the well-definedness of the pushforward $f_{1*}$ (in both of $F$ and $A_*$)
is easily checked by taking fiber products and by the canonical identification.
{\rm (ii)} Assume that there exists
an isomorphism of functors $\alpha: f_2\circ f_1 \to f_3$
(i.e., a $2$-isomorphism of $1$-morphisms).
Then two $G_3$-equivariant morphisms
$\bar{f}_2\circ \bar{f}_1$
and $\bar{f}_3$ from $B_3$ to $X_3$
coincide up to isomorphisms of $B_3$ and of $X_3$
which are encoded in the definition of $\alpha$,
hence their $G_3$-pushforwards coincide up to the chosen isomorphisms.
\hfill $\Box$
\
\begin{center}
{\small
\begin{xy}
(0,0)*+{ };
(25,0) *+{P_1}="P1"; (45,0) *+{X_1}="X1";
(35,13) *+{B_1}="B1"; (55,13) *+{\mathcal{X}_1}="XX1";
(25,20) *+{P'}="Pd"; (45,20) *+{P_3}="P3";
(45,26) *+{X_2}="X2"; (65,26) *+{\mathcal{X}_2}="XX2"; (85,26) *+{\mathcal{X}_3}="XX3";
(35,33) *+{B'}="Bd"; (55,33) *+{B_3}="B3";
(45,46) *+{P_2}="P2"; (65,46) *+{B_2}="B2"; (85,46) *+{X_3}="X3";
(64,20) *+{{}_{f_1}};
{\ar@{->} "P1";"X1"}; {\ar@{->} "P1";"B1"};
{\ar@{->} "X1";"XX1"};
{\ar@{->} "Pd";"P1"};{\ar@{->} "Pd";"P3"};{\ar@{->} "Pd";"Bd"};
{\ar@{->} "P3";"X1"};{\ar@{->} "P3";"B3"};
{\ar@{->} "Pd";"P3"};{\ar@{->} "Pd";"Bd"};
{\ar@{->} "B1";"XX1"};{\ar@{->} "B1";"X2"};
{\ar@{->} "Bd";"B3"};{\ar@{->} "Bd";"P2"};{\ar@{->} "Bd";"B1"};
{\ar@{->} "XX1";"XX2"};{\ar@{->}_{f_3} "XX1";"XX3"};
{\ar@{->} "X2";"XX2"};
{\ar@{->}^{f_2} "XX2";"XX3"};
{\ar@{->} "B3";"X3"};{\ar@{->} "B3";"B2"};{\ar@{->} "B3";"XX1"};
{\ar@{->} "P2";"X2"};{\ar@{->} "P2";"B2"};
{\ar@{->} "B2";"X3"};{\ar@{->} "B2";"XX2"};
{\ar@{->} "X3";"XX3"};
\end{xy}
}
\end{center}
\subsection{Chern-MacPherson transformation}
We assume that $X$ is a quasi-projective scheme or algebraic space
with action of $G$.
Then $X_G$ exists as an algebraic space,
hence $C_*(X_G)$ makes sense.
Take the vector bundle $TU_G:=X\times_G (U\oplus V)$ over $X_G$,
i.e., the pullback
of the tautological vector bundle $(U\times V)/G$ over $U_G$
induced by the projection $X_G \to U_G$.
Our natural transformation
$$C^{G}_*: F^{G}(X) \to A_*^{G}(X)$$
is defined to be the inductive limit of
$$
T_{U, *}:=c(TU_G)^{-1} \frown C_*: F(X_G) \to A_*(X_G)
$$
over the direct system of representations of $G$,
see \cite{O1} for the detail.
Roughly speaking, the $G$-equivariant CSM class
$C_*^G(X)\, (:=C_*^G(\jeden_X))$ looks like
``$c(T_{BG})^{-1}\frown C_*(EG\times_G X)$",
where $EG\times_G X \to BG$ means the universal bundle (as ind-schemes)
with fiber $X$ and group $G$,
that has been justified using a different inductive limit of Chow groups,
see Remark 3.3 in \cite{O1}.
\begin{lemma}
{\rm (i)} In the same notation as in the diagram (\ref{d2}) in 2.1,
the following diagram commutes:
$$\xymatrix{
F^{G}(X) \ar[d]_{C^{G}_*} \ar[r]^{p^{*}\;}_{\simeq\;\;} & F^{G\times H}(P) \ar[d]^{C^{G\times H}_*} \\
A^{G}_{*+g}(X) \ar[r]_{p^*\;}^{\simeq\;\;} & A^{G\times H}_{*+g+h}(P)
}
$$
{\rm (ii)} In particular, $C_*: F(\mathcal{X}) \to A_*(\mathcal{X})$ is well-defined. \\
{\rm (iii)} $C_*f_*=f_*C_*$ for proper representable morphisms $f: \mathcal{X} \to \mathcal{Y}$.
\end{lemma}
\noindent {\sl Proof} :\;
{\rm (i)} This is essentially the same as Lemma 3.1 in \cite{O1} which
shows the well-definedness of $C_*^G$.
Apply
the Verdier-Riemann-Roch \cite{Yokura} to
the projection of the affine bundle $\bar{p}: P_{G\times H} \to X_G$ (with fiber $U_2$),
then we have the following commutative diagram
$$\xymatrix{
F(X_G) \ar[d]_{C_*} \ar[r]^{\bar{p}^{*}\;}_{\;\;} & F(P_{G\times H}) \ar[d]^{C_*} \\
A_{*+\ell_1}(X_G) \ar[r]_{\bar{p}^{**}\;}^{\;\;} & A_{*+\ell_1+\ell_2}(P_{G\times H})
}
$$
where $\bar{p}^{**}=c(T_{\bar{p}}) \frown \bar{p}^*$
and $T_{\bar{p}}$ is the relative tangent bundle of $\bar{p}$.
The twisting factor $c(T_{\bar{p}})$ in $\bar{p}^{**}$ is
cancelled by the factors in $T_{U_1,*}$
and $T_{U,*}$:
In fact, since $T_{\bar{p}} = \bar{q}^*TU_{2H}$, $T_{\bar{q}} = \bar{p}^*TU_{1G}$ and
$$TU_{G\times H}= P \times_{G\times H} (T(U_1\oplus U_2))
= T_{\bar{p}}\oplus T_{\bar{q}},$$
we have
\begin{eqnarray*}
T_{U,*} \circ \bar{p}^* (\alpha)&=& c(TU_{G\times H})^{-1} \frown C_*(\bar{p}^*\alpha)\\
&=&
c(T_{\bar{p}}\oplus T_{\bar{q}})^{-1} c(T_{\bar{p}}) \frown \bar{p}^*C_*(\alpha)\\
&=&
c(T_{\bar{q}})^{-1} \frown \bar{p}^*C_*(\alpha)\\
&=&
\bar{p}^*(c(T{U_1}_G)^{-1} \frown C_*(\alpha)) \\
&=& \bar{p}^* \circ T_{U_1, *} (\alpha).
\end{eqnarray*}
Taking the inductive limit, we conclude that $C_*^{G\times H} \circ p^*=p^* \circ C_*^{G}$.
Thus {\rm (i)} is proved.
The claim {\rm (ii)} follows from {\rm (i)} .
By {\rm (ii)} , we may consider $C_*$ as the $H$-equivariant
Chern-MacPherson transformation $C_*^H$ given in \cite{O1},
thus {\rm (iii)} immediately follows from the naturality of $C_*^H$.
\hfill $\Box$
\
The above lemmas show the following theorem
(cf. Theorem 3.5, \cite{O1}):
\begin{theorem}\label{theorem}
Let $\mathcal{C}$ be the category whose objects are
{\rm (}possibly non-separated{\rm )} Artin quotient stacks $\mathcal{X}$
having the form $[X/G]$ of
separated algebraic spaces $X$ of finite type with action
of smooth linear algebraic groups $G$;
morphisms in $\mathcal{C}$ are assumed to be proper and representable.
Then for the category $\mathcal{C}$,
we have a unique natural transformation $C_*: F(\mathcal{X}) \to A_*(\mathcal{X})$
with integer coefficients
so that it coincides with the ordinary MacPherson transformation
when restricted to the category of quasi-projective schemes.
\end{theorem}
\subsection{Degree}
Let $g=\dim G$. The $G$-classifying stack $BG=[pt/G]$ has
(non-positive) virtual dimension $-g$, hence
$$A_{-n}(BG)=A_{-n+g}^G(pt)=A^{n-g}_G(pt)=A^{n-g}(BG)$$
for any integer $n$ (trivial for $n<g$).
We often use this identification.
In particular, $A_{-g}(BG)=A^0(BG)=\mathbf{Z}$.
Let $\mathcal{X}=[X/G]$ in $\mathcal{C}$ with $X$ projective and equidimensional of dimension $n$.
Then we can take the representable morphism $pt: \mathcal{X} \to BG$:
$$
\xymatrix{
G\times X \ar[r]^{q} \ar[d]_{p} & X \ar[r]^{\bar{pt}} \ar[d]^{u} & pt \ar[d] \\
X \ar[r]_{u} & \mathcal{X} \ar[r]_{pt} & BG
}
$$
Here are some remarks:
\begin{enumerate}
\item[(i)]
For a $G$-invariant function $\alpha \in F_{inv}(\mathcal{X})=F^G_{inv}(X)$,
it is obvious that $(q^*)^{-1}\circ p^*(\alpha)=\alpha$, hence we have
$$pt_*(\alpha) = \bar{pt}_*(q^*)^{-1} p^*(\alpha)= \bar{pt}_*\alpha=\int_X\alpha =\chi(X; \alpha),$$
which is called the {\it integral}, or {\it weighted Euler characteristic} of the invariant function $\alpha$.
In particular, by the naturality, $pt_*C_*(\alpha)=C_*(pt_*\alpha)=\chi(X; \alpha)$.
More generally, in \cite{O1} we have defined the $G$-degree of
{\it equivariant constructible function} $\alpha \in F(\mathcal{X})$
by $pt_*(\alpha) \in F^G(pt)=F(BG)$, which is a
`constructible' function over $BG$. Then
$pt_*C_*(\alpha)=C_*(pt_*\alpha) \in A^*(BG)$,
being a polynomial or power series in universal $G$-characteristic classes.
\item[(ii)]
For invariant functions $\alpha \in F_{inv}(\mathcal{X})$ and for $i<-g$ and $i>n-g$,
the $i$-th component $C_i(\alpha)$ is trivial.
A possibly nontrivial highest degree term
$C_{n-g}(\alpha) \in A_{n-g}(\mathcal{X}) \, (=A_{n}^G(X))$ is a linear sum of
the $G$-fundamental classes $[X_i]_G$
of irreducible components $X_i$ (the virtual fundamental class of dimension $n-g$) .
As a notational convention,
let $\jeden_\mathcal{X}^{(0)}$ denote the constant function $\jeden_X \in F^G_{inv}(X) \, =F_{inv}(\mathcal{X})$ for a presentation $\mathcal{X}=[X/G]$. In particular,
if $X$ is smooth, then
$$C_*(\jeden_\mathcal{X}^{(0)})=C^G_*(\jeden_X)=c^G(TX) \frown [X]_G \in A_{*+g}^G(X)=A_*(\mathcal{X}).$$
\item[(iii)]
From the viewpoint of the enumerative theory in classical projective algebraic geometry
(e.g. see \cite{P}), a typical type of degrees often arises in the following form:
$$\int \, pt_*(c(E) \frown C_*(\alpha)) \in A^0(BG)$$
for some vector bundle $E$ over $\mathcal{X}$ and a constructible function $\alpha \in F _{inv}(\mathcal{X})$.
\end{enumerate}
\section{Deligne-Mumford stacks}
It would be meaningful
to restrict $C_*$ to a subcategory of certain quotient stacks having finite stabilizer groups,
which form a reasonable class of Deligne-Mumford stacks
(including smooth DM stacks).
\begin{theorem}
Let $\mathcal{C}_{\rm DM}$ be the category of
Deligne-Mumford stacks of finite type which admits a locally closed embedding into
some smooth proper DM stack with projective coarse moduli space: morphisms in $\mathcal{C}_{\rm DM}$ are assumed to be proper and representable.
Then for $\mathcal{C}_{\rm DM}$
there is a unique natural transformation $C_*: F(\mathcal{X}) \to A_*(\mathcal{X})$
satisfying the normalization property: $C_*(\jeden_X)=c(TX)\frown [X]$ for smooth schemes.
\end{theorem}
This is due to Theorem 5.3 in Kresch \cite{Kresch} which states that
a DM stack in $\mathcal{C}_{\rm DM}$ is in fact realized by a quotient stack in $\mathcal C$.
In \cite{Kresch}, such a DM stack is called to be ({\it quasi-}){\it projective}.
\begin{remark}
{\rm
(i) In the above theorem,
the embeddability into smooth stack
(or equivalently the resolution property in \cite{Kresch}) is required,
that seems natural, since original MacPherson's theorem
requires such a condition \cite{Mac, Ken}.
In order to extend $C_*$ for more general Artin stacks with values in Kresch's Chow groups,
we need to find some technical gluing property. \\
(ii)
We may admit proper {\it non-representable} morphisms of DM stacks
if we use rational coefficients. In fact for such morphisms
the pushforward of Chow groups with rational coefficients is defined \cite{Vistoli}.
}
\end{remark}
\subsection{Modified pushforwards} \label{mod}
The theory of constructible functions for Artin stacks has been established by Joyce \cite{Joyce}.
Below let us work with $\mathbf{Q}$-valued constructible functions and Chow groups with $\mathbf{Q}$-coefficients.
For stacks $\mathcal{X}$ in $\mathcal{C}_{\rm DM}$,
each geometric point $x: pt= {\rm Spec}\, k \to \mathcal{X}$ has a finite stabilizer group $Aut(x)(={\rm Iso}_x(x,x))$.
Then the group of constructible functions $\underline{\alpha}$ in the sense \cite{Joyce}
is canonically identified with
the subgroup $F_{inv}(\mathcal{X})_\mathbf{Q}=F^G_{inv}(X)_\mathbf{Q}$
of invariant constructible functions $\alpha$ over $X$
in the following way (the bar indicates constructible functions over the set of all geometric points $\mathcal{X}(k)$):
For each $k$-point $x: pt \to \mathcal{X}$, the value of $\alpha$ over the orbit $x\times_\mathcal{X} X$ is given by
$|Aut(x)|\cdot \underline{\alpha}(x)$, that is,
$$F(\mathcal{X}(k))_\mathbf{Q} \simeq F_{inv}(\mathcal{X})_\mathbf{Q} \;\; (\; \subset F(\mathcal{X})_\mathbf{Q} \;), \quad \underline{\alpha} \leftrightarrow
\alpha=\jeden_\mathcal{X} \cdot \pi^*\underline{\alpha}, $$
where $\pi$ is the projection to $\mathcal{X}(k)$,
$\alpha\cdot \beta$ is the canonical multiplication on $F(\mathcal{X})_\mathbf{Q}$,
$(\alpha\cdot \beta)(x):=\alpha(x)\beta(x)$, and
$$\jeden_\mathcal{X}:=|Aut(\pi(-))| \in F_{inv}(\mathcal{X})_\mathbf{Q}.$$
It is shown by Tseng \cite{Tseng} that
if $\mathcal{X}$ is a smooth DM stack,
$C_*(\jeden_{\mathcal{X}})$ coincides with (pushforward of the dual to) the total Chern class
of the tangent bundle of the corresponding smooth inertia stack.
From a viewpoint of classical group theory,
it would be natural to measure {\it how large of the stabilizer group is}
by comparing it with a fixed group $A$,
that leads us to define a $\mathbf{Q}$-valued constructible function over $\mathcal{X}(k)$.
Here the group $A$ is supposed to be, e.g., a finitely generated Abelian group
(we basically consider $A=\mathbf{Z}^m$, $\mathbf{Z}/r\mathbf{Z}$, etc).
Accordingly to \cite{O1, O2}, we define
{\it the canonical constructible function measured by group $A$}
which assigns to any geometric point $x$
the number of group homomorphisms of $A$ into $Aut(x)$:
$$\underline{\jeden_{\mathcal{X}}^{A}}(x):=\frac{|\, {\rm Hom}\, (A, Aut(x))\, |}{|\, Aut(x)\, |} \; \in \mathbf{Q}.$$
The corresponding invariant constructible function is denoted by
$\jeden_\mathcal{X}^A \in F_{inv}(\mathcal{X})_\mathbf{Q}$, or often by
$\jeden_{X;G}^{A} \in F^G_{inv}(X)_\mathbf{Q}$
when a presentation $\mathcal{X}=[X/G]$ is specified. Namely,
the value of $\jeden_{X;G}^{A}$ on the $G$-orbit expressed by
$x: pt \to \mathcal{X}$ is $|\, {\rm Hom}\, (A, Aut(x))\, |$.
The function for $A=\mathbf{Z}$ is nothing but $\jeden_\mathcal{X}$
in our convention, and for $A=\{0\}$ it is $\jeden_\mathcal{X}^{(0)}=1$.
If $A=\mathbf{Z}^2$, the function counts the number of mutually commuting pairs in $Aut(x)$,
hence its integral corresponds to
the orbiforld Euler number (in physicist's sense), see \cite{O2}.
Define
$T^A_\mathcal{X}: F(\mathcal{X})_\mathbf{Q} \to F(\mathcal{X})_\mathbf{Q}$ by
the multiplication $T^A_\mathcal{X}(\alpha):=\jeden_{X;G}^{A} \cdot \alpha$.
This is a $\mathbf{Q}$-algebra isomorphism,
for $\jeden_{X;G}^{A}$ is an unit in $F(\mathcal{X})_\mathbf{Q}$.
A new pushforward is introduced
for proper representable morphisms $f:\mathcal{X} \to \mathcal{Y}$ in $\mathcal{C}_{\rm DM}$ by
$$f^{A}_*: F(\mathcal{X})_\mathbf{Q} \to F(\mathcal{Y})_\mathbf{Q}, \quad
\alpha \mapsto (T^A_\mathcal{Y})^{-1} \circ f_*\circ T^A_\mathcal{X}(\alpha).$$
Obviously,
$g^{A}_* \circ f^{A}_*=(g\circ f)^{A}_*$.
The following theorem says that there are
several variations of theories of integration with values in Chow groups
for Deligne-Mumford stacks:
\begin{theorem} Given a finitely generated Abelian group $A$,
let $F^{A}$ denote the new covariant functor of constructible functions
for the category $\mathcal{C}_{\rm DM}$, given by
$F^{A}(\mathcal{X})_\mathbf{Q}:=F(\mathcal{X})_\mathbf{Q}$ and the pushforward by $f^{A}_*$.
Then,
$C^{A}_*:=C_*\circ T^A_\mathcal{X}: F^{A}(\mathcal{X})_\mathbf{Q} \to A_*(\mathcal{X})_\mathbf{Q}$ is a natural transformation.
\end{theorem}
\noindent {\sl Proof} :\;
It is straightforward that
$f_*\circ C^{A}_*=f_*\circ C_*\circ T^A_\mathcal{X} = C_* \circ f_*\circ T^A_\mathcal{X}
= C_* \circ T^A_\mathcal{Y} \circ (T^A_\mathcal{Y})^{-1} \circ f_*\circ T^A_\mathcal{X}
=C_*^{A} \circ f_*^{A}.$ \hfill $\Box$
\section{Other characteristic classes}
The method in the preceeding sections
is applicable to other characteristic classes
(over $\mathbf{C}$ or a field $k$ of characteristic $0$).
As the most general additive characteristic class for
singular varieties, {\it the Hirzebruch class transformation}
$$T_{y*}: K_0(Var/X) \to A_*(X) \otimes \mathbf{Q}[y]$$
was recently introduced by Brasselet-Sch\"urmann-Yokura \cite{BSY}:
For possibly singular varieties $X$ (and proper morphisms between them),
$T_{y*}$ is a unique natural transformation from
the Grothendieck group $K_0(Var/X)$ of the monoid of
isomorphism classes of morphisms $V \to X$ to
the rational Chow group of $X$ with a parameter $y$ such that it satisfies that
$$T_{y*}[X\stackrel{id}{\to}X]=\widetilde{td_y}(TX) \frown [X], \quad
\mbox{for smooth $X$,}$$
where $\widetilde{td_y}(E)$ denotes
the modified Todd class of vector bundles:
$$\widetilde{td_y}(E)=\prod_{i=1}^r \left(\frac{a_i(1+y)}{1-e^{-a_i(1+y)}}-a_iy \right),$$
when $c(E)=\prod_{i=1}^r (1+a_i)$, see \cite{BSY, SY}.
Note that the associated genus is well-known Hirzebruch's $\chi_y$-genus,
which specializes to:
the Euler characteristic if $y=-1$,
the arithmetic genus if $y=0$,
and the signature if $y=1$.
Hence, $T_{y*}$ gives
a generalization of the $\chi_y$-genus to
homology characteristic class of singular varieties,
which unifies the following
singular Riemann-Roch type formulas in canonical ways:
\begin{itemize}
\item
($y=-1$)
the Chern-MacPherson transformation $C_*$ \cite{Mac, Ken};
\item
($y=0$)
Baum-Fulton-MacPherson's Todd class transformation $\tau$ \cite{BFM};
\item
($y=1$)
Cappell-Shaneson's homology $L$-class transformation $L_*$ \cite{CS}.
\end{itemize}
For a quotient stack $\mathcal{X}=[X/G] \in \mathcal{C}$ in Theorem 1,
we denote by $K_0(\mathcal{C}/\mathcal{X})$
the Grothendieck group of the monoid
of isomorphism classes of representable morphisms
of quotient stacks to the stack $\mathcal{X}$.
To each element $[\mathcal{V} \to \mathcal{X}] \in K_0(\mathcal{C}/\mathcal{X})$,
we take
a $G$-equivariant morphism $V \to X$
where $V:=\mathcal{V}\times_{\mathcal{X}} X$ with natural $G$-action
so that $\mathcal{V}=[V/G]$, and associate
a class of morphisms of algebraic spaces
$[V_G \to X_G] \in K_0(Var/X_G)$.
We then define
$$T_{y*}: K_0(\mathcal{C}/\mathcal{X}) \to A_*(\mathcal{X}) \otimes \mathbf{Q}[y]$$
by assigning to $[\mathcal{V} \to \mathcal{X}]$ the inductive limit
(over all $G$-representations) of
$$\widetilde{td_y}^{-1}(TU_G) \frown T_{y*}[V_G \to X_G] \in
A_*(X_G)\otimes \mathbf{Q}[y].$$
This is well-defined, because the Verdier-Riemann-Roch for $T_{y*}$ holds
(Corollary 3.1 in \cite{BSY})
and the same proof of Lemma 2 can be used in this setting.
Note that in each degree of grading
the limit stabilizes, thus the coefficient is a polynomial in $y$.
So we obtain an extension of $T_{y*}$ to the category $\mathcal{C}$,
and hence also to $\mathcal{C}_{\rm DM}$.
It turns out that at special values $y=0, \pm 1$,
$T_{y*}$ corresponds to:
\begin{itemize}
\item
($y=-1$) the $G$-equivariant Chern-MacPherson transformation \cite{O1},
i.e., $C_*$ as described in section 2 above;
\item
($y=0$)
the $G$-equivariant Todd class transformation \cite{EG, BZ},
given by the limit of $td^{-1}(TU_G) \frown \tau$;
\item
($y=1$) the $G$-equivariant singular $L$-class transformation
given by the limit of
$(L^*)^{-1}(TU_G) \frown L_*$, where $L^*$ is
the (cohomology) Hirzebruch-Thom $L$-class.
\end{itemize}
Applications will be considered in another paper.
|
2,869,038,154,825 | arxiv | \section*{Introduction}
A Nakayama algebra is an algebra whose quiver consists of a single oriented cycle with a finite number of monomial relations or a single path, with or without relations. These are called \emph{cyclic} and \emph{linear} Nakayama algebras respectively. There are many results about these algebras. In \cite{ringel_gorenstein}, Ringel introduced the ``resolution quiver'' $R(\Lambda)$ of a Nakayama algebra $\Lambda$ and used it to characterize when $\Lambda$ is Gorenstein and to give a formula for its Gorenstein dimension. In \cite{shen_resolution}, Shen showed that every component of the resolution quiver of a Nakayama algebra has the same weight and, in \cite{shen_homological}, Shen used this to obtain the following result.
\begin{customthm}{A}[Shen \cite{shen_homological}]\label{thm A} A Nakayama algebra has finite global dimension if and only if its resolution quiver has exactly one component and that component has weight 1.
\end{customthm}
It is important for the purpose of induction that this includes linear Nakayama algebras.
In \cite{igusa_cyclic}, the second author and Zacharia showed that a monomial relation algebra has finite global dimension if and only if the relative cyclic homology of its radical is equal to zero. For the case of a cyclic Nakayama algebra, they constructed a finite simplicial complex $L(\Lambda)$, which we will call the ``relation complex'' of $\Lambda$, and show that the reduced homology of $L(\Lambda)$ gives the lowest degree term in the relative cyclic homology of the radical of $\Lambda$. They further simplified their results to obtain the following.
\begin{customthm}{B}[Igusa-Zacharia \cite{igusa_cyclic}]\label{thm B} A Nakayama algebra $\Lambda$ has finite global dimension if and only if the Euler characteristic of the relation complex $L(\Lambda)$ is equal to one.
\end{customthm}
In fact, only cyclic Nakayama algebras were considered in \cite{igusa_cyclic}. However, if $\Lambda$ is a linear Nakayama algebra, it is easy to see that $L(\Lambda)$ is still defined and is contractible. Thus it has Euler characteristic one.
In this paper we will review these definitions and statements and prove the following statement showing that the two results above are equivalent.
\begin{customthm}{C}\label{thm C} For any cyclic Nakayama algebra $\Lambda$ the Euler characteristic of the relation complex $L(\Lambda)$ is equal to the number of components of its resolution quiver $R(\Lambda)$ with weight 1.
\end{customthm}
We prove Theorem \ref{thm C} using an idea that comes from \cite{sen_syzygy}. In fact, our argument proves Theorems \ref{thm A}, \ref{thm B} and \ref{thm C} simultaneously. (See Remark \ref{rem: How the proposition implies Thms ABC}.) We also obtain a new version of Theorem \ref{thm B}:
\begin{customthm}{B'}[Corollary \ref{cor: theorem Bp}]\label{thm Bp} A Nakayama algebra $\Lambda$ has finite global dimension if and only if its relation complex $L(\Lambda)$ is contractible.
\end{customthm}
These results arose in our study, joint with Gordana Todorov, of the concept of ``amalgamation'' \cite{fock_cluster} and the reverse process which we call ``unamalgamation''. Amalgamation is used in \cite{arkani_scattering} to construct plabic diagrams which are used in \cite{alvarez_turaev} to construct new invariants in contact topology.
Since the result of \cite{igusa_cyclic} holds for any monomial relation algebra, we also attempted to generalize the definition of the resolution quiver to such algebras. This lead us to consider examples which eventually led to a counterexample to the $\phi$-dimension conjecture. See \cite{hanson_counterexample} for more details about this story.
\section{Resolution Quiver of a Nakayama Algebra}
By a \emph{Nakayama algebra} of \emph{order} $n$ we mean a finite dimensional algebra $\Lambda$ over a field $K$ given by a quiver with relations where the quiver, which we denote $Q_n$, consists of a single oriented $n$-cycle:
\begin{center}
\begin{tikzcd}
1 \arrow[r,"x_1"] & 2 \arrow[r, "x_2"] & \cdots \arrow[r, "x_{n-1}"]& n \ar[lll,bend left=30,"x_n" above]
\end{tikzcd}
\end{center}
with monomial relations given by paths $y_i$ in this quiver written left to right. We write $y_i=x_{k_i}x_{k_i+1}\cdots x_{\ell_i}$ for $i=1,\cdots,r$ where $1\le k_1<k_2<\cdots<k_r\le n$ and $r\ge1$. If the relations have length $\ge2$ they are called \emph{admissible} and the algebra is called a \emph{cyclic Nakayama algebra}.
We also allow relations of length one. If there is only one of these, say $x_n=0$, we have the linear quiver
\begin{center}
\begin{tikzcd}
1 \arrow[r,"x_1"] & 2 \arrow[r, "x_2"] & \cdots \arrow[r, "x_{n-1}"]& n
\end{tikzcd}
\end{center}
modulo admissible relations and the resulting algebra is called a \emph{linear Nakayama algebra}. We also allow more than one relation of length one. If there are $m$ such relations, the algebra is a product of $m$ linear Nakayama algebras arranged in a cyclic order. These are the algebras that we consider under one formalism: $\Lambda=KQ_n/I$ where $KQ_n$ is the path algebra of $Q_n$ and $I$ is the ideal generated by the relations $y_i$. Unless otherwise noted, the relations $y_i$ will be \emph{irredundant}, i.e., no relation $y_i$ is a subword of another relation $y_j$. We sometimes say that the arrows of $Q_n$ are \emph{cyclically composable}.
A finitely generated right $\Lambda$-module $M$ is equivalent to a representation of the quiver $Q_n$ satisfying the relations $y_i=0$. I.e., for each vertex $i$ we have a finite dimensional vector space $M_i=Me_i$ where $e_i$ is the path of length 0 at $i$ and for each arrow $x_i:i\to i+1$ we have a linear map $M_i\to M_{i+1}$ given by the right action of $x_i$.
Note that $\Lambda$ is a graded algebra since all relations are homogeneous. Also, it has been known for a long time that all modules are uniserial. In hindsight we might say this is because $\Lambda$ is a string algebra (see for example \cite{butler_auslander}).
\begin{rem}\label{rem: relations and projectives}
The projective cover $P_j$ of the simple $S_j$ at vertex $j$ is easy to determine given the relations $y_i$: The composition series of $P_j$ starts at $j$ and continues around the quiver $Q_n$ until it completes one of the relations. In particular, the composition series of each projective gives a (possibly redundant) relation for the algebra. The composition series of projectives $P_j$ of length $|P_j|\le |P_{j+1}|$ give the minimal relations.
\end{rem}
\begin{ex}\label{first example} Let $\Lambda_1$ be $KQ_5$ modulo the relations $y_1=x_2x_3$, $y_2=x_3x_4$ and $y_3=x_5x_1x_2$. Then $P_1,P_2$ have composition series ending in $S_2,S_3$; $P_3$ has composition series $S_3,S_4$; and $P_4,P_5$ have composition series ending in $S_5,S_1,S_2$. Composition series are usually written vertically as:
\begin{center}
\begin{tikzpicture}
\node (A) at (0,.8) {$P_1\!:$};
\node at (.5,.8){1};
\node at (.5,.4){2};
\node at (.5,0){3};
\node (B) at (2,.8) {$P_2\!:$};
\node at (2.5,.8){2};
\node at (2.5,.4){3};
\node (C) at (4,.8) {$P_3\!:$};
\node at (4.5,.8){3};
\node at (4.5,.4){4};
\node (D) at (6,.8) {$P_4\!:$};
\node at (6.5,.8){4};
\node at (6.5,.4){5};
\node at (6.5,0){1};
\node at (6.5,-.4){2};
\node (D) at (8,.8) {$P_5\!:$};
\node at (8.5,.8){5};
\node at (8.5,.4){1};
\node at (8.5,0){2};
\end{tikzpicture}
\end{center}
The relations can be read off from the composition series of those projectives $P_i$ so that $|P_i|\le |P_{i+1}|$ which, in this case, are $P_2,P_3,P_5$.
\end{ex}
\begin{define}[Ringel \cite{ringel_gorenstein}]
The \emph{resolution quiver} $R(\Lambda)$ of a Nakayama algebra $\Lambda$ of order $n$ is the quiver having the same vertex set as $Q_n$ and one arrow $i\to j$ if the $i$-th projective module $P_i$ has socle $S_{j-1}$ (so that $j-i$ is the length of $P_i$ modulo $n$).
\end{define}
\begin{rem}\label{rem: Gustafson's function}
The function $j=f(i)$ used above can be given by
\[
f(i)=i+|P_i|-n\dim\mathrm{Hom}_\Lambda(P_n,P_i).
\]
This function originates in \cite{gustafson_global} and is called \emph{Gustafson's function}.
\end{rem}
For $\Lambda_1$ from Example \ref{first example}, the lengths of the projectives are $3,2,2,4,3$. So the five arrows of the resolution quiver are $1\to 4, 2\to 4$, $3\to 5, 4\to 3,5\to 3$ giving:
\begin{center}
\begin{tikzcd}
&\circled{1} \ar[rd]\\
R(\Lambda_1):&\circled{2} \arrow[r] & \circled{4}\ar{r} &\circled 3\ar[ r]& \circled 5 \ar[l, bend right=40]
\end{tikzcd}
\end{center}
The \emph{leaves} of the resolution quiver are the vertices which are not targets of arrow. The other vertices, which we call \emph{nodes}, are in bijection with the relations of the algebra.
\begin{define}
The \emph{weight} of an oriented cycle in the resolution quiver is the sum of the lengths of the projective modules $P_i$ for all $i$ in the cycle divided by $n$, the size of the quiver.
\end{define}
For $\Lambda_1$ from Example \ref{first example}, the resolution quiver has one oriented cycle going through vertices $3,5$. The weight is:
\[
wt=\frac{|P_3|+|P_5|}n=\frac{2+3}5=1.
\]
Ringel \cite{ringel_gorenstein} showed that each component of the resolution quiver has exactly one oriented cycle and that its weight is a positive integer. The weight of a component of $R(\Lambda)$ is defined to be the weight of its unique oriented cycle. Shen \cite{shen_resolution} proved that all components of $R(\Lambda)$ have the same weight. Then he proved the following.
\begin{thm}[Shen, \cite{shen_homological}]\label{Shen thm}
A cyclic Nakayama algebra $\Lambda$ has finite global dimension if and only if $R(\Lambda)$ has exactly one component with weight one.
\end{thm}
For Example \ref{first example}, $R(\Lambda_1)$ has one component with weight 1. So, $gl\,dim\,\Lambda_1<\infty$.
\begin{ex}\label{second RA example}
Let $\Lambda_2$ be given by the cyclic quiver $Q_5$:
\begin{center}
\begin{tikzcd}
&1 \arrow[r,"x_1"] & 2 \arrow[d, "x_2"] \\
5\ar[ur, bend left, "x_5"] &4\arrow{l}[above]{x_4} & 3 \arrow{l}[above]{x_3}
\end{tikzcd}
\end{center}
modulo the relations: $x_1x_2x_3, x_2x_3x_4x_5, x_4x_5x_1,x_5x_1x_2$ with projectives:
\begin{center}
\begin{tikzpicture}
\node (A) at (0,.8) {$P_1\!:$};
\node at (.5,.8){1};
\node at (.5,.4){2};
\node at (.5,0){3};
\node (B) at (2,.8) {$P_2\!:$};
\node at (2.5,.8){2};
\node at (2.5,.4){3};
\node at (2.5,0){4};
\node at (2.5,-.4){5};
\node (C) at (4,.8) {$P_3\!:$};
\node at (4.5,.8){3};
\node at (4.5,.4){4};
\node at (4.5,0){5};
\node at (4.5,-.4){1};
\node (D) at (6,.8) {$P_4\!:$};
\node at (6.5,.8){4};
\node at (6.5,.4){5};
\node at (6.5,0){1};
\node (D) at (8,.8) {$P_5\!:$};
\node at (8.5,.8){5};
\node at (8.5,.4){1};
\node at (8.5,0){2};
\end{tikzpicture}
\end{center}
Then, in the resolution quiver, we have $1\to 4, 2\to 1, 3\to 2, 4\to 2, 5\to 3$ giving:
\begin{center}
\begin{tikzcd}
R(\Lambda_2): & \circled 5 \ar[r]&\circled{3} \arrow[r] & \circled{2}\ar{r} &\circled 1\ar r & \circled 4 \ar[ll,bend right=30]
\end{tikzcd}
\end{center}
This is connected with weight:
\[
wt(R(\Lambda_2))=\frac{|P_2|+|P_1|+|P_4|}{n}=\frac{ 4+3+3}5=2
\]
So, by Theorem \ref{Shen thm}, $\Lambda_2$ has infinite global dimension.
\end{ex}
\begin{ex}\label{third RA example}
Let $\Lambda_3$ be $KQ_4$ modulo $rad^2=0$, then all projectives have length 2. So $R(\Lambda_3)$ is:
\begin{center}
\begin{tikzcd}
R(\Lambda_3):&\circled{1} \arrow[r,bend left=25] & \circled{3}\ar[l,bend left=25] &\circled{2} \arrow[r,bend left=25] & \circled{4}\ar[l,bend left=25]
\end{tikzcd}
\end{center}
This has two components, each with weight 1. So again, the algebra has infinite global dimension. In fact, $\Lambda_3$ is self-injective.
\end{ex}
\section{Cyclic Homology of a Monomial Relation Algebra}
When quoting results from \cite{igusa_cyclic}, we note that a cyclic Nakayama algebra is called a ``cycle algebra'' in \cite{igusa_cyclic} and the relation complex is denoted $K$. We changed this to $L$ to avoid confusion with the ground field $K$. We reserve the symbol ``$\Lambda$'' for Nakayama algebras.
For a monomial relation algebra $A$ over any field $K$, the following result was obtained in \cite{igusa_cyclic}.
\begin{thm}\label{thm: characterizing monomial algebras of finite gl dim}
$A$ has finite global dimension if and only if the cyclic homology of the radical of $A$ is zero.
\end{thm}
The statement uses the fact, proved in \cite{igusa_cyclic}, that the relative cyclic homology of the (Jacobson) radical of $A$ (as an ideal in $A$) is equal to the cyclic homology of $J=rad\,A$ as a ring without unit.
We first give the definition of cyclic homology when the ground field $K$ has characteristic zero. (See \cite{igusa_cyclicMatrix} or \cite{loday_cyclic}.) In the proof of Proposition \ref{prop: cyclic complex is good for any char} we outline the more complicated general definition used in \cite{igusa_cyclic}. In both cases we use Proposition 1.2 in \cite{igusa_cyclic}, which allows us to assume that the $X_i$ in the definition below are indecomposable. We use here the term \emph{radical morphism} to mean a morphism $f:X\to Y$ so that $g\circ f$ is nilpotent for any $g:Y\to X$. When $X,Y$ are projective this is equivalent to saying that $f(X)\subseteq rad\,Y$. The radical morphisms $X\to Y$ form a vector space which we denote $r(X,Y)$.
\begin{define}
In the case when the characteristic of the ground field is equal to zero, the cyclic homology $HC_\ast(J)$ of $rad\,A$ is defined as the homology of the \emph{cyclic complex} of $J=rad\,A$ given by
\begin{equation}\label{eq: cyclic complex}
0\leftarrow C_0(J)\xleftarrow{\overline b} C_1(J)/\mathbb{Z}_2 \xleftarrow{\overline b} C_2(J)/\mathbb{Z}_3 \xleftarrow{\overline b}\cdots
\end{equation}
where $C_p(J)$ is the direct sum of all sequences of $p+1$ cyclically composable radical morphisms between indecomposable projective modules: $X_0\to X_1\to \cdots\to X_p\to X_0$:
\begin{equation}\label{eq: Cp(J)}
C_p(J)=\coprod_{X_0,\cdots,X_p} r(X_0,X_1)\otimes r(X_1,X_2)\otimes\cdots\otimes r(X_{p-1},X_p)\otimes r(X_p,X_0).
\end{equation}
$C_p(J)/\mathbb{Z}_{p+1}$ is $C_p(J)$ modulo the action of the cyclic group $\mathbb{Z}_{p+1}$ where the action of the generator $t$ of $\mathbb{Z}_{p+1}$ is given by $t(f_0,f_1,\cdots,f_p)=(-1)^p(f_1,f_2,\cdots,f_p,f_0)$ and $\overline b:C_p(J)/\mathbb{Z}_{p+1}\to C_{p-1}(J)/\mathbb{Z}_p$ is the map induced by $b:C_p(J)\to C_{p-1}(J)$ given by
\[
b(f_0,\cdots,f_p)=\sum_{i=0}^{p-1} (-1)^{i} (f_0,\cdots,f_if_{i+1},f_{i+2},\cdots,f_p)+(-1)^p(f_pf_0,f_1,\cdots,f_{p-1}).
\]
\end{define}
Since the action of $t\in \mathbb{Z}_{p+1}$ on $C_p(J)$ and the map $b:C_p(J)\to C_{p-1}(J)$ are homogeneous of degree zero, the cyclic homology complex $C_\ast(J)/\mathbb{Z}_\ast$ is graded when $A$ is graded. In the cyclic Nakayama case, we can say more. To form a cycle, the degrees of the morphisms $X_i\to X_{i+1}$ must add up to a multiple of $n$. So $C_\ast(J)/\mathbb{Z}_\ast$ and $HC_\ast(J)$ are graded and have nonzero terms only in degrees which are positive multiples of $n$. Let $HC_\ast^n(J)$ denote the degree $n$ part of $HC_\ast(J)$.
\begin{prop}\label{prop: cyclic complex is good for any char}
The cyclic complex \eqref{eq: cyclic complex} will compute the degree $n$ part of the cyclic homology of $J=rad\,\Lambda$ for $\Lambda$ a Nakayama algebra of order $n$.
\end{prop}
\begin{rem}\label{rem: HC of linear Nakayama is zero}
In the case when $\Lambda$ is a linear Nakayama algebra, the cyclic complex \eqref{eq: cyclic complex} is zero in every degree and the cyclic homology of $J$ is zero in every degree as predicted since linear Nakayama algebras have finite global dimension.
\end{rem}
\begin{proof}
The cyclic homology of $J$ is in general defined to be the homology of a double complex which is given by replacing $C_p(J)/\mathbb{Z}_{p+1}$ in \eqref{eq: cyclic complex} by a chain complex $C_{p}(J)\otimes_{K\mathbb{Z}_{p+1}} P_\ast(\mathbb{Z}_{p+1})$, where $P_\ast(\mathbb{Z}_{p+1})$ is a free $K\mathbb{Z}_{p+1}$-resolution of $K$. However, in $C_p^n(J)$, the degree $n$ part of $C_p(J)$, the projective modules $X_i$ in \eqref{eq: Cp(J)} must all be distinct. So, the cyclic group $\mathbb{Z}_{p+1}$ acts freely on the set of summands of $C_p^n(J)$. In other words, $C_p^n(J)$ is a free $K\mathbb{Z}_{p+1}$-module. So, the complex $C_{p}^n(J)\otimes_{K\mathbb{Z}_{p+1}} P_\ast(\mathbb{Z}_{p+1})$ has homology only in degree zero where it is $C_p^n(J)/\mathbb{Z}_{p+1}$. Thus the degree $n$ part of the double complex defining $HC_\ast(J)$ collapses to the degree $n$ part of the chain complex \eqref{eq: cyclic complex} as claimed.
\end{proof}
\begin{ex}\label{ex: first calculation of 3rd example}
In Example \ref{third RA example}, $n=4$ and the degree $4$ part of $C_3(J)/Z_4$ contains
\[
P_4\to P_3\to P_2\to P_1\to P_4
\]
This is a cycle since the composition of any two morphisms is zero. It cannot be a boundary since each map has degree 1. Thus $HC_3(J)\neq0$ showing (again) that $\Lambda_3$ has infinite global dimension.
\end{ex}
In \cite{igusa_cyclic}, a topological interpretation of $HC^n_\ast(J)$ was given. In this definition we use the term ``internal vertex'' of a relation to mean any vertex $i$ so that $x_{i-1}x_i$ is a subword of the relation. For example, the internal vertices of $x_1x_2x_3$ are $2,3$. A relation of length $\ell\le n$ has $\ell-1$ internal vertices. A relation is said to ``cover'' a vertex if that vertex is an interior vertex of the relation.
\begin{define}\label{def: K(A)}
For any (connected) Nakayama algebra $\Lambda$ of order $n$ with $r$ relations we define the \emph{relation complex} to be the simplicial complex $L(\Lambda)$ having one vertex for every relation of length $\le n$. The vertices $v_0,\cdots,v_p$ span a $p$ simplex in $L(\Lambda)$ if and only if the corresponding relations do not cover all $n$ vertices of the quiver.
\end{define}
Let $CL(\Lambda)$ denote the cone of $L(\Lambda)$.
\begin{lem}\cite{igusa_cyclic}\label{lem: HC in terms of L(A)}
For any Nakayama algebra $\Lambda$, the degree $n$ part of the cyclic homology of $rad\,\Lambda$ is equal to the relative homology of $(CL(\Lambda),L(\Lambda))$ with coefficients in $K$:
\[
HC_p^n(rad\,\Lambda)=H_p(CL(\Lambda),L(\Lambda);K).
\]
\end{lem}
\begin{rem}\label{rem: chi(HC)=1-chi(L)}
This implies in particular that the Euler characteristic of $HC_\ast^n(rad\,\Lambda)$ is $\chi(CL(\Lambda))-\chi(L(\Lambda))=1-\chi(L(\Lambda))$.
\end{rem}
\begin{ex}\label{ex: L(A) is a cone when A is linear}
Suppose $\Lambda$ is a linear Nakayama algebra. In that case one of the relations, say $y_0$, has length 1 and such a relation has no internal vertices. Therefore, $y_0$ can be added to any simplex in $L(\Lambda)$. This means that $L(\Lambda)$ is a cone with cone point $y_0$. So, $L(\Lambda)$ is a contractible space and $H_\ast(CL(\Lambda),L(\Lambda);K)=0$ in every degree. By Remark \ref{rem: HC of linear Nakayama is zero}, this proves Lemma \ref{lem: HC in terms of L(A)} in the linear case.
\end{ex}
\begin{ex}\label{ex: second calculation of 3rd example}
For Example \ref{third RA example}, there are four relations of length 2 giving four vertices $v_0,v_1,v_2,v_3$. Since each relation covers only one vertex, the only set of relations which covers all four vertices is all of them. So, $L(\Lambda_3)$ in this case is a tetrahedron missing its interior. This is homeomorphic to a 2-sphere. So, $HC_p(rad\,\Lambda_3)=H_p(CL(\Lambda_3),K(\Lambda_3))$ is nonzero only when $p=3$, in agreement with the calculation in Example \ref{ex: first calculation of 3rd example}.
\end{ex}
\begin{thm}\cite{igusa_cyclic}\label{thm: HC char of fin gl dim}
For any Nakayama algebra $\Lambda$ the following are equivalent.
\begin{enumerate}
\item $\Lambda$ has finite global dimension.
\item $HC_\ast^n(rad\,\Lambda)=0$.
\item The Euler characteristic of the relation complex $L(\Lambda)$ is 1.
\end{enumerate}
\end{thm}
\begin{ex}\
\begin{enumerate}
\item In Example \ref{first example}, there are two relations neither of which has vertex $4$ in its interior. So, $L(\Lambda_1)$ is two points connected with one edge:
\begin{center}
\begin{tikzpicture
\coordinate (A) at (0,0);
\coordinate (B) at (1.5,0);
\draw[fill] (A) circle[radius=2pt];
\draw[fill] (B) circle[radius=2pt];
\draw[very thick] (A)--(B);
\end{tikzpicture}
\end{center}
This has Euler characteristic equal to 1. So, $gl\,dim\,\Lambda_1<\infty$.
\item In Example \ref{second RA example}, there are four relations $y_1,y_2,y_3,y_4$ where $y_2=x_2x_3x_4x_5$ and $y_4=x_5x_1x_2$ cover all 5 vertices but the other five pairs do not. Thus $R(\Lambda_2)$ has 4 vertices and 5 edges. One of the two triangles, with vertices $y_1=x_1x_2x_3,y_2,y_3=x_4x_5x_1$ covers $Q_5$. The other ($y_1,y_3,y_4$) does not. So, only the second face is filled in.
\begin{center}
\begin{tikzpicture
\coordinate (A) at (0,0);
\coordinate (B) at (1.5,0);
\coordinate (C) at (.75,1.3);
\coordinate (D) at (2.25,1.3);
\draw[fill, color=gray!40!white] (B)--(C)--(A)--cycle;
\foreach \x in {A,B,C,D}
\draw[fill] (\x) circle[radius=2pt];
\draw[very thick] (B)--(A)--(C)--(D)--(B)--(C);
\draw (A) node[left]{$y_4$};
\draw (C) node[left]{$y_1$};
\draw (B) node[right]{$y_3$};
\draw (D) node[right]{$y_2$};
\end{tikzpicture}
\end{center}
So, $\chi L(\Lambda_2)=4-5+1=0$ showing again that $\Lambda_2$ has infinite global dimension.
\item In Example \ref{third RA example}, computed in Example \ref{ex: first calculation of 3rd example} above, $L(\Lambda_3)$ is a 2-sphere with $\chi L(\Lambda_3)=2$. So, again, $gl\,dim\,\Lambda_3=\infty$.
\end{enumerate}
\end{ex}
\section{Comparison of Characterizations}
In this section we prove Theorem \ref{thm C}: For any Nakayama algebra $\Lambda$, the Euler characteristic of its relation complex $L(\Lambda)$ is equal to the number of components of its resolution quiver $R(\Lambda)$ of weight one. In Examples \ref{first example}, \ref{second RA example}, \ref{third RA example} this number is 1,0,2, respectively. The proof will be by induction on the number of vertices of $R(\Lambda)$ which do not lie in any oriented cycle. If this number is zero then $R(\Lambda)$ has no leaves. So, the number of relations is equal to $n$ the size of the quiver $Q_n$. In that case, the relations must all be of the same length, say $\ell$, and we have the following.
\begin{lem}
Let $\Lambda$ be $KQ_n$ modulo $rad^\ell$. Then the number of components of $R(\Lambda)$ is equal to $(n,\ell)$, the greatest common divisor of $n,\ell$, and the weight of each component is $w=\ell/(n,\ell)$. In particular, the weight is 1 if and only if $\ell$ divides $n$ in which case the number of components is $\ell$.
\end{lem}
\begin{proof}
Since the arrows of $R(\Lambda)$ go from $i$ to $i+\ell$ modulo $n$, the size of each oriented cycle is the smallest number $m$ so that $m\ell$ is a multiple of $n$. This number is $m=n/(n,\ell)$. The weight of the cycle is
\[
wt=\frac{m\ell}n=\frac{\ell}{(n,\ell)}.
\]
Since the total length of all the projectives is $n\ell$, the number of components is $n\ell/m\ell=n/m=(n,\ell)$ as claimed.
\end{proof}
\begin{lem}
Let $\Lambda$ be $KQ_n$ modulo $rad^\ell$. Then the Euler characteristic of $L(\Lambda)$ is
\[
\chi(L(\Lambda))=\begin{cases} \ell & \text{if $\ell$ divides $n$} \\
0 & \text{otherwise}
\end{cases}
\]
\end{lem}
\begin{proof}
By Remark \ref{rem: chi(HC)=1-chi(L)}, this is equivalent to the equation proved in Theorem 4.4 of \cite{igusa_cyclic} that $\chi(HC_\ast^n(rad\,\Lambda))=1-\ell$ if $\ell$ divides $n$ and is 1 otherwise. This includes the degenerate case when $\ell=1$ since, in this case, $L(\Lambda)$ is contractible by Example \ref{ex: L(A) is a cone when A is linear} and thus $\chi(L(\Lambda))=1=\ell$.
\end{proof}
These two lemmas imply that Theorem \ref{thm C} holds in the case when $R(\Lambda)$ is a union of oriented cycles or, equivalently, when $R(\Lambda)$ has no leaves. These lemmas also imply Theorems \ref{thm A} and \ref{thm B} in this case since $KQ_n$ modulo $rad^\ell$ has finite global dimension if and only if $\ell=1$.
Suppose now that $\Lambda$ is a Nakayama algebra of order $n$ and $j$ is a leaf of $R(\Lambda)$. By reindexing we may assume $j=n$.
\begin{prop}\label{prop: properties of L'} Let $n$ be a leaf of $R(\Lambda)$ and let $\Lambda'=\mathrm{End}_\Lambda(P')$, where $P'=P_1\oplus \cdots \oplus P_{n-1}$. Then we have the following.
\begin{enumerate}
\item $R(\Lambda')$ is equal to $R(\Lambda)$ with the leaf $n$ removed. In particular:
\begin{enumerate}
\item $R(\Lambda')$ has the same number of components as $R(\Lambda)$ and
\item $R(\Lambda')$ has one fewer vertex not in an oriented cycle than $R(\Lambda)$.
\end{enumerate}
\item $R(\Lambda')$ has the same weight as $R(\Lambda)$.
\item $L(\Lambda')$ is homotopy equivalent to $L(\Lambda)$. So, they have the same Euler characteristic.
\item $gl\,dim\,\Lambda'\le gl\,dim\,\Lambda\le gl\,dim\,\Lambda'+2 $. Thus, $gl\,dim\,\Lambda<\infty$ if and only if $gl\,dim\,\Lambda'<\infty$.
\end{enumerate}
\end{prop}
\begin{rem}\label{rem: How the proposition implies Thms ABC}
Suppose for a moment that $\Lambda'$ has all of these properties. The proof of Theorem \ref{thm C} then proceeds as follows. By induction on the number of vertices of the resolution quiver which are not in any oriented cycle, Theorem \ref{thm C} holds for $\Lambda'$. So $\chi(L(\Lambda'))$ is equal to the number of components of $R(\Lambda')$ with weight 1. But these numbers are the same for $\Lambda'$ and $\Lambda$. So, Theorem \ref{thm C} also holds for $\Lambda$. By (4), Theorems \ref{thm A}, \ref{thm B} also hold for $\Lambda$ assuming they hold for $\Lambda'$. Therefore, the existence of $\Lambda'$ satisfying the properties above will prove all three theorems at the same time!
\end{rem}
\subsection{Algebraic properties of $\Lambda'$}
Since $P'$ is a projective $\Lambda$-module, we have an exact functor
\[
\pi: \mathsf{mod}\text-\Lambda\to \mathsf{mod}\text-\Lambda'
\]
given by $\pi(M_\Lambda)=\mathrm{Hom}_\Lambda(\prescript{}{\Lambda'}{P'_\Lambda},M_\Lambda)$.
\begin{lem}\label{lem: id Sn=1}
The injective dimension of $S_n$ is at most one.
\end{lem}
\begin{proof}
Since $n$ is a leaf in $R(\Lambda)$, there is no relation for $\Lambda$ which ends in $x_{n-1}$. So, the injective envelope of $S_n$ modulo $S_n$ is injective making $S_n\to I_n\to I_n/S_n\to 0$ the injective copresentation of $S_n$.
\end{proof}
The following result appears in \cite{ChenYe_Gorenstein} and is one step in the construction of $\varepsilon(\Lambda)$ in \cite{sen_syzygy}.
\begin{prop}\label{prop: embedding of Lambda-prime}
The functor
\[
\alpha:\mathsf{mod}\text-\Lambda'\to \mathsf{mod}\text-\Lambda
\]
given by $\alpha(X_{\Lambda'})=X\otimes_{\Lambda'}P'$ is a full and faithful exact embedding which takes projectives to projectives and whose image contains the second syzygy of any $\Lambda$-module.
\end{prop}
\begin{proof} Since $\alpha(\Lambda')=P'$ and $\alpha$ is additive, $\alpha$ takes projective modules to projective modules.
For any $X\in \mathsf{mod}\text-\Lambda'$, consider a free presentation of $X$ of the form
\begin{equation}\label{eq: presentation of X}
(\Lambda')^p\xrightarrow f (\Lambda')^q\to X\to 0.
\end{equation}
Since $\alpha$ is right exact, we get a projective presentation of $\alpha(X)$:
\begin{equation}\label{eq: presentation of aX}
(P')^p\xrightarrow{f_\ast} (P')^q\to \alpha(X)\to 0.
\end{equation}
Since $\mathrm{Hom}_\Lambda(P',P')=\mathrm{Hom}_{\Lambda'}(\Lambda',\Lambda')$, the correspondence $f\leftrightarrow f_\ast$ is a bijection with $\pi$ sending \eqref{eq: presentation of aX} back to \eqref{eq: presentation of X}. If we do the same for another $\Lambda'$-module $Y$, the morphisms between $P'$-presentations in $\mathsf{mod}\text-\Lambda$ are in bijection with the morphisms between free presentations in $\mathsf{mod}\text-\Lambda'$. So, we get \[
\mathrm{Hom}_\Lambda(\alpha(X),\alpha(Y))=\mathrm{Hom}_{\Lambda'}(X,Y)\]
showing that $\alpha$ is full and faithful.
The image of the functor $\alpha$ contains all cokernels of all morphisms $(P')^p\to (P')^q$ as in \eqref{eq: presentation of aX}. Since $id\,S_n\le 1$, this includes all second syzygies of all $\Lambda$-modules.
To show that $\alpha$ is exact, consider a short exact sequence $0\to Z\to Y\to X\to 0$ in $\mathsf{mod}\text-\Lambda'$. Take free presentations of $X,Z$, and combine them to get a presentation of $Y$:
\begin{center}
\begin{tikzcd}
0 \ar[r] & (\Lambda')^{r}\ar[r]\ar[d,"h"] & (\Lambda')^{r+p}\ar[r]\ar[d,"g"]& (\Lambda')^{p}\ar[r]\ar[d,"f"] &0\\
0 \ar[r] & (\Lambda')^{s}\ar[r]\ar[d] & (\Lambda')^{s+q}\ar[r]\ar[d]& (\Lambda')^{q}\ar[r]\ar[d] &0\\
0 \ar[r] & Z \ar[r]\ar[d] & Y \ar[r]\ar[d] & X\ar[r]\ar[d] & 0\\
&0&0&0
\end{tikzcd}
\end{center}
Applying the functor $\alpha$ we get a right exact sequence:
\[
\alpha(Z)\xrightarrow k \alpha(Y)\to \alpha(X)\to 0.
\]
Since the exact functor $\pi=\mathrm{Hom}_\Lambda(P',-)$ sends this back to the sequence $0\to Z\to Y\to X\to 0$, $\ker\,k$ must be a direct sum of copies of $S_n$. However, by the Snake Lemma, the kernel of $f_\ast:(P')^p\to (P')^q$ maps onto $\ker\,k$. But this is impossible unless $\ker\,k = 0$, since $\mathrm{Ext}^2_\Lambda(\alpha(X),S_n)=0$ by Lemma \ref{lem: id Sn=1}. Therefore, $\alpha$ is exact as claimed.
\end{proof}
\begin{cor}\label{cor: gl dim of Lambda and Lambda-prime}
The global dimensions of $\Lambda$ and $\Lambda'$ differ by at most two:
\[
gl\,dim\,\Lambda'\le gl\,dim\,\Lambda\le gl\,dim\,\Lambda'+2.\]
In particular, $gl\,dim\,\Lambda<\infty$ if and only if $gl\,dim\,\Lambda'<\infty$.
\end{cor}
\begin{proof} Since $\alpha$ is an exact functor taking projectives to projectives, it takes a projective resolution of $M$ to a projective resolution of $\alpha(M)$. Since $\alpha$ is full and faithful, the projective resolution of $M$ does not split. This means $M, \alpha(M)$ have the same projective dimension. So,
\[
gl\,dim\, \Lambda'\le gl\,dim\,\Lambda.
\]
Given any $\Lambda$-module $M$ of projective dimension at least 3, by Proposition \ref{prop: embedding of Lambda-prime} there is a $\Lambda'$-module $N$ so that $\alpha(N)\equiv \Omega^2M$. Thus, \[
pd\,M=2+pd\,N\le gl\,dim\,\Lambda'\]
which implies that $gl\,dim\,\Lambda\le 2+gl\,dim\,\Lambda'$.
\end{proof}
The following essentially follows from Lemma 2.1 in \cite{shen_homological}.
\begin{prop}
The resolution quiver of $\Lambda'$ is equal to the resolution quiver of $\Lambda$ with the leaf $n$ removed. In particular, $R(\Lambda),R(\Lambda')$ have the same number of components. Furthermore, the weight of $R(\Lambda')$ is equal to the weight of $R(\Lambda)$.
\end{prop}
\begin{proof}
The length of $\pi(M)$ for any $\Lambda$-module $M$ is $|M|-\dim\mathrm{Hom}_\Lambda(P_n,M)$. Gustafson's function for $\Lambda$ on any $i<n$ is
\[
f_{\Lambda}(i)=i+|P_i|-n\dim \mathrm{Hom}_\Lambda(P_n,P_i)=i+|\pi(P_i)|-(n-1)\mathrm{Hom}_\Lambda(P_n,P_i),
\]
which is congruent to $f_{\Lambda'}(i)$ modulo $n-1$. Thus, for $i<n$, $i\to j$ in $R(\Lambda)$ if and only if $i\to j$ in $R(\Lambda')$. In particular, the cycles are the same.
Suppose that $j_1\to j_2\to \cdots \to j_m$ is a cycle in $R(\Lambda)$ with weight
\[
w=\frac1n \sum |P_{j_i}|.
\]
This implies that $\bigoplus P_{j_i}$ has each simple $\Lambda$-module $w$ times in its composition series. In particular, $w=\sum \dim\mathrm{Hom}(P_n,P_{j_i})$. The same cycle is also a cycle in $R(\Lambda')$ with weight
\[
w=\frac1{n-1} \sum |\alpha(P_{j_i})|=\frac1{n-1} \sum \left(
|P_{j_i}| -\dim\mathrm{Hom}_\Lambda(P_n,P_{j_i})
\right)= \frac{nw-w}{n-1}=w.
\]
So, $R(\Lambda)$ and $R(\Lambda')$ have the same weight.
\end{proof}
\subsection{Topology of $L(\Lambda')$} It remains to show that the relation complex of $\Lambda'$ is homotopy equivalent to that of $\Lambda$. This will follow from the following three lemmas where we note that only Lemma \ref{lem: redundant relations give the same L(L)} requires $n$ to be a leaf of the resolution quiver.
For any word $y$ in the letters $x_1,\cdots,x_n$ let $\delta(y)$ be $y$ with all occurrences of the letter $x_n$ deleted.
\begin{lem}\label{lem: relations for L'} Given that $\Lambda=KQ_n$ modulo relations $y_1,\cdots,y_r$ with $y_i=x_{j_i}x_{j_i+1}\cdots x_{\ell_i}$, $\Lambda'$ is equal to $KQ_{n-1}$ modulo the possibly redundant relations $y_i'$ given by
\[
y_i'=\begin{cases} x_{n-1}\delta(y_i) & \text{if }j_i=n\\
\delta(y_i)& \text{otherwise}
\end{cases}
\]
\end{lem}
\begin{proof}
We use the fact that the composition series of any projective module is a relation for the algebra and that the minimal relations come from projective modules $P_i$ with length $|P_i|\le |P_{i+1}|$.
If $j_i\neq n$ then $y_i$ gives the composition series of $P_{j_i}$ and $y_i'=\delta(y_i)$ gives the composition series of $\pi(P_{j_i})$. If $j_i=n$ then $y_i$ gives the composition series for $P_n$. Since $P_n$ maps onto the radical of $P_{n-1}$, $x_{n-1}y_i$ is a relation for $\Lambda$ and $y_i'=x_{n-1}\delta(y_i)$ is a relation for $\Lambda'$.
We conclude that $y_1',\cdots,y_r'$ are among the relations that hold for $\Lambda'$. We need to show that this list contains all the minimal relations for $\Lambda'$. So, suppose that $|\pi(P_i)|\le |\pi(P_{i+1})|$ and $z$ is the relation for $\Lambda'$ given by $\pi(P_i)$. If $i\le n-2$ then $|P_i|\le |P_{i+1}|$; so, $z=\delta(y)=y'$, where $y$ is the relation for $\Lambda$ given by $P_i$. Thus, $z$ is in the list. If $i=n-1$ then $|\pi(P_{n-1})|\le |\pi(P_1)|$, which implies that $|P_{n-1}|\le |P_1|+1$. So, either
\begin{enumerate}
\item[(a)] $|P_{n-1}|\le |P_n|$ in which case $z=\delta(y)=y'$, where $y$ is the relation for $\Lambda$ given by $P_{n-1}$ or
\item[(b)] $|P_n|=|P_{n-1}|-1\le |P_1|$, so $P_n$ gives a relation $y$ for $\Lambda$ and $y'=x_{n-1}\delta(y)=z$.
\end{enumerate}
In both cases the relation $z$ is in the list. So, $\{y_i'\}$ is a list of relations for $\Lambda'$ which contains all the minimal relations.
\end{proof}
\begin{lem}\label{lem: redundant relations give the same L(L)}
Let $L'(\Lambda')$ be the simplicial complex with vertices given by the possibly redundant relations $y_i'$ of Lemma \ref{lem: relations for L'} with $p+1$ such relations spanning a $p$-simplex if their interiors cover all of $Q_{n-1}$. Then $L'(\Lambda')$ is isomorphic to $L(\Lambda)$ as a simplicial complex.
\end{lem}
\begin{proof}
We have a bijection $y_i\leftrightarrow y_i'$ between the vertices of $L(\Lambda)$ and those of $L(\Lambda')$. So, we need to show that a collection of relations, $\{y_i\}$ for $\Lambda$ covers all the vertices of $Q_n$ if and only if the corresponding relations $y'_i$ cover all the vertices of $Q_{n-1}$. We claim the following.
\begin{enumerate}
\item If $y'_i$ covers vertex $n-1$ then $y_i$ covers $n-1$ and $n$.
\item $y_i$ covers vertex $k\neq n$ if and only if $y_i'$ covers $k$.
\end{enumerate}
To prove (1), suppose that $y'_i$ covers $n-1$. Then this relation contains $x_{n-2}x_{n-1}$ as a subword. Since $n$ is a leaf of $R(\Lambda)$, the relation $y_i$ cannot end in $x_{n-1}$. Thus, it must contain the subword $x_{n-2}x_{n-1}x_n$. So, $y_i$ covers $n-1$ and $n$ as claimed.
To prove (2), first suppose $y_i$ covers $k\neq n$. Then $\delta(y_i)$ will cover $k$ except in the case when $k=1$ and $x_nx_1$ are the first two letters of $y_i$. In that case $y_i'=x_{n-1}x_1\cdots$ still covers $k=1$. The converse follows from this and (1). This proves the lemma.
\end{proof}
\begin{lem}\label{lem: removing redundant relations is collapse}
The relation complex $L(\Lambda')$ is homotopy equivalent to one constructed using any redundant set of relations such as $L'(\Lambda')$.
\end{lem}
\begin{proof}
Suppose that the relation set used to construct $L'(\Lambda')$ has relations $z_0$ and $z$ so that $z_0$ is a subword of $z$. Let $L''(\Lambda')$ be obtained from $L'(\Lambda')$ by deleting the vertex $z$ and all open simplices containing $z$ as an endpoint. The union of these is called the ``open star'' of $z$ and the closure of this union is called the ``closed star'', $St(z)$ (see \cite[Chapter 2.C]{hatcher_algebraic}). The union of all simplices in $St(z)$ which do not contain $z$ as an endpoint is called the ``link'' of $z$ and denoted $Lk(z)$. Then
\[
L'(\Lambda')=L''(\Lambda')\cup_{Lk(z)}St(z),
\]
which means that $L'(\Lambda')=L''(\Lambda')\cup St(z)$ and $L''(\Lambda')\cap St(z)=Lk(z)$. Since $St(z)$ is a cone on $Lk(z)$, it is contractible. So, $L'(\Lambda')\simeq L''(\Lambda')$ if and only if $Lk(z)$ is contractible.
But, $St(z)$ is the cone on $z_0$ in $L'(\Lambda')$ since, for any set of relations including $z$ which does not cover all vertices of the quiver, adding $z_0$ will cover the same set of vertices (because $z_0$ is a subword of $z$). This implies that the link of $z$ is the cone on $z_0$ in $L''(\Lambda')$. So, $L'(\Lambda')\simeq L''(\Lambda')$. Repeat this to remove redundant relations one at a time without changing the homotopy type of $L'(\Lambda')$.
\end{proof}
\begin{ex}
For $\Lambda_1$ in Example \ref{first example}, $n=5$ is the only leaf. The relations $y_i$, $y_i'$ are given by:
\[
\begin{array}{c|cc}
y_i & y_i'\\
\hline
x_1x_2x_3 & x_1x_2x_3\\
x_2x_3x_4x_5 & x_2x_3x_4\\
x_4x_5x_1 & x_4x_1 & \text{($z_0$: subword of $z$)}\\
x_5x_1x_2 & x_4x_1x_2 & \text{($z$: redundant)}
\end{array}
\]
The relation complex of $\Lambda_1'$ is obtained from $L'(\Lambda_1')=L(\Lambda_1)$ by deleting all open simplices containing $z=y_4'=x_4x_1x_2$ as a vertex. So, $L(\Lambda_1')$ is the hollow triangle with vertices $y_1',y_2',y_3'$. One can see that this is a deformation retract of $L(\Lambda_1)$. The proof of Lemma \ref{lem: removing redundant relations is collapse} uses the fact that $Lk(z)=Lk(y_4')$ is a cone on $z_0=y'_3$ and is thus contractible. So, $L(\Lambda_1')$ is homotopy equivalent to $L(\Lambda_1)$.
\begin{center}
\begin{tikzpicture
\begin{scope}
\coordinate (X) at (-2,.7);
\draw (X) node{$L'(\Lambda_1')=L(\Lambda_1):$};
\coordinate (A) at (0,0);
\coordinate (B) at (1.5,0);
\coordinate (C) at (.75,1.3);
\coordinate (D) at (2.25,1.3);
\coordinate (BC) at (1.2,.6);
\coordinate (BC1) at (2,1);
\coordinate (BC2) at (.9,0);
\coordinate (BC3) at (2.5,.5);
\draw[thick, color=blue,<-] (BC)..controls (BC1) and (BC2)..(BC3);
\draw[color=blue] (BC3) node[right]{$Lk(z)$};
\draw[fill, color=gray!40!white] (B)--(C)--(A)--cycle;
\foreach \x in {A,D}
\draw[fill] (\x) circle[radius=2pt];
\draw[very thick] (B)--(A)--(C)--(D)--(B);
\draw[very thick, color=blue] (B)--(C);
\foreach \x in {B,C}
\draw[fill,color=blue] (\x) circle[radius=2pt];
\draw (A) node[left]{$z=y_4'$};
\draw (C) node[left]{$y'_1$};
\draw (B) node[right]{$z_0=y'_3$};
\draw (D) node[right]{$y'_2$};
\end{scope}
\begin{scope}[xshift=6cm]
\coordinate (X) at (-.6,.7);
\draw (X) node{$L(\Lambda_1'):$};
\coordinate (A) at (0,0);
\coordinate (B) at (1.5,0);
\coordinate (C) at (.75,1.3);
\coordinate (D) at (2.25,1.3);
\foreach \x in {C,B,D}
\draw[fill] (\x) circle[radius=2pt];
\draw[very thick] (B)--(D)--(C)--cycle;
\draw (C) node[left]{$y'_1$};
\draw (B) node[right]{$z_0=y'_3$};
\draw (D) node[right]{$y'_2$};
\end{scope}
\end{tikzpicture}
\end{center}
\end{ex}
From Lemmas \ref{lem: redundant relations give the same L(L)} and \ref{lem: removing redundant relations is collapse} we immediately obtain the following.
\begin{prop}\label{prop: L simeq Lp}
The relation complexes $L(\Lambda)$ and $L(\Lambda')$ are homotopy equivalent.\qed
\end{prop}
This concludes the proof of Proposition \ref{prop: properties of L'} which proves Theorems \ref{thm A}, \ref{thm B} and \ref{thm C} simultaneously. (See Remark \ref{rem: How the proposition implies Thms ABC}.) Theorem \ref{thm Bp} also follows:
\begin{cor}\label{cor: theorem Bp}
A Nakayama algebra has finite global dimension if and only if its relation complex is contractible.
\end{cor}
\begin{proof}
If $L(\Lambda)$ is contractible, its Euler characteristic is 1 and it thus has finite global dimension by Theorem \ref{thm C}. Conversely, if $\Lambda$ has finite global dimension then, by repeatedly removing the leaves of the resolution quiver we obtain an algebra $\Lambda''$ with $rad=0$, i.e., it is semi-simple. Since the relations of $\Lambda''$ have empty interiors, the relation complex of $\Lambda''$ is a solid simplex, so it is contractible. By Proposition \ref{prop: L simeq Lp}, $L(\Lambda)\simeq L(\Lambda'')$ is also contractible.
\end{proof}
\section{Comments}
The construction of $\Lambda'$ from $\Lambda$ is an example of ``unamalgamation'', which is the construction described in Lemma \ref{lem: relations for L'}. This is a partial inverse to the ``amagamation'' construction of Fock-Goncharov \cite{fock_cluster} used in \cite{arkani_scattering} to construct the Jacobian algebras of plabic diagrams \cite{postnikov_total}. These operations are partial inverses since we claim that $UAU=U$ and $AUA=A$, where $A,U$ stand for amalgamation and unamalgamation, respectively.
\begin{center}
\begin{tikzcd}
& 1' \ar[d]&1 \arrow[r,"x_1"] & 2 \arrow[d, "x_2"] \\
5\ar[ur, bend left, "x_5"] &4'\arrow{l}[above]{x_4} &4\arrow{u} & 3 \arrow{l}[above]{x_3}
\end{tikzcd}
$\text{amalgamation }\Rightarrow\over
\Leftarrow\text{ unamalgamation}$
\begin{tikzcd}
&1 \arrow[r,"x_1"] & 2 \arrow[d, "x_2"] \\
5\ar[ur, bend left, "x_5"] &4\arrow{l}[above]{x_4} & 3 \arrow{l}[above]{x_3}
\end{tikzcd}
\end{center}
The idea to consider removing the leaves $R(\Lambda)$ comes from \cite{sen_syzygy} where, for $\Lambda$ of infinite projective dimension, Emre \c{S}en constructs a Nakayama algebra $\varepsilon(\Lambda)$ which seems to have the property that $R(\varepsilon(\Lambda))$ is equal to $R(\Lambda)$ with all its leaves removed. He then proves that the $\phi$-dimension of $\Lambda$ is equal to 2 plus the $\phi$-dimension of $\varepsilon(\Lambda)$.
\section*{Acknowledgements}
Both authors thank Gordana Todorov who is working with them on the larger project of amalgamation and unamalgamation of Nakayama algebras. The first author thanks Job Rock for meaningful conversations. The second author thanks Emre \c{S}en for many discussions about Nakayama algebras and especially the construction of $\varepsilon(\Lambda)$ from \cite{sen_syzygy}. The second author thanks his other coauthor Daniel \'Alvarez-Gavela for working with him, in \cite{alvarez_turaev}, to reduce the calculation of higher Reidemeister torsion in \cite{IK93} to a calculations on plabic diagrams (\cite{postnikov_total}). Also, the second author would like to thank An Huang for directing him to the paper \cite{arkani_scattering} which inspired this line of research, particularly the transition from \cite{alvarez_turaev} to \cite{hanson_counterexample} and this paper. Finally, the authors thank the referee for helpful comments.
\newcommand{\etalchar}[1]{$^{#1}$}
|
2,869,038,154,826 | arxiv | \section{Introduction}
Recent experimental works in the $1s0d$ shell with large $\gamma$ detector arrays and heavy-ion fusion reactions have substantially extended the knowledge of relatively high spin states. However, these do not form well-behaved rotational bands amenable to study by collective models because rotational energies are comparable to single-particle energies. On the other hand microscopic configuration-interaction model calculations are feasible in these lighter nuclei. The USD family of effective interactions \cite{usd, usdAB} have been very successful in describing most lower-lying positive-parity states of nuclei with $8 \leq (N, Z) \leq 20$. However, higher spin states involve excitations into the $fp$ shell where orbitals contributing larger values of angular momentum are occupied, which is beyond the scope of the USD interaction. Also, neutron-rich isotopes quickly move beyond the $sd$ shell boundaries \cite{MOTOBAYASHI, huber, detraz, GUILLEMAUD, YANAGISAWA, vandana32Mg}.
Over the years, several configuration interaction models have made significant contribution towards explaining cross-shell excitations \cite{sdpf-nr, sdpf-u, psdpf, sdpf-m, sdpfumix}. A case in point is the ``Island of Inversion" (IoI). Perhaps in an inverse way the first contribution came from the failure of the otherwise very successful pure $sd$ interactions \cite{usd, usdAB} to reproduce the stronger binding energy measured for $^{31}$Na \cite{thibault}, pointing to the importance of effects outside the $sd$ shell. The WBMB \cite{wbmb} interaction, which was designed for the nuclei near $^{40}$Ca was successful in reproducing the inversion of some nuclei within the IoI. More recent shell model calculations using interactions like SDPF-M \cite{sdpf-m}, SDPF-U-MIX \cite{sdpfumix} have shown that the IoI phenomenon can be accounted for by a reduction of the $N = 20$ shell gap. Recently, a significant theoretical result was reported, see Ref. \cite{tsunoda}, showing the emergence of IoI effect from nucleon-nucleon forces stemming from the fundamental principles of QCD. This highlights the importance of certain cross $sd$ - $fp$ interaction terms that we assess in this work using experimental systematics.
In search for a single cross-shell interaction which works well over a wide range of nuclei, we have developed a new interaction \cite{fsu-38Cl} with parallel treatment of protons and neutrons by fitting the energies of 270 states in nuclei from $^{13}$C through $^{51}$Ti and $^{49}$V originated from the WBP interaction \cite{wbp} using well-established techniques. The present report is organized as follows: First we will discuss the development of the new FSU shell model interaction. The trend of the effective single particle energies (ESPEs) of the $0f_{7/2}$ and $1p_{3/2}$ orbitals for the $sd$-shell nuclei will be examined along with the comparison to the experimental data. Then we will move to the IoI region and test some predictions of the FSU interaction in this region. Finally, the experimentally observed fully aligned states with the $f_{7/2}^2$ configuration will be interpreted with the new shell model interaction.
\section{Development of the FSU empirical shell model interaction}
A modified version of the WBP \cite{wbp} interaction has been used as the starting point of the data fitting procedure. The WBP interaction was developed in order to address the cross-shell structure around $A = 20$.
While the $sd$-$fp$ cross-shell matrix elements of the WBP were taken from the WBMB \cite{wbmb} interaction which was developed for the nuclei around $^{40}$Ca, the different single particle energies and different implementation make WBP not good for the upper $sd$-shell nuclei. Yet, the WBP is a perfect starting point for a more modern, much broader assessment of the nuclear matrix elements. Our data set included nuclei from the upper mass region of the $p$, the full $sd$, and lower mass region of the $fp$ shells; where we systematically looked at states that involve a particle promotion across the harmonic oscillator shell, referred to as 1 particle- 1 hole excited states (1p1h).
In the $sd$-shell region of most interest and most data, the combined 0p0h and 1p1h space considered in the fit is
equivalent to $0\hbar\omega$ and $1\hbar\omega$ often referred to as the $N_{\rm max}=1$ harmonic oscillator basis truncation.
The resulting fit seamlessly spans from the $A=20$ region, where the low-lying intruder states are predominantly those with holes in the lower $p$-shell, to the island of inversion around $A=40$ where particles are promoted to $fp$ shell.
The ability to separate the center of mass exactly within the $N_{\rm max}=1$ harmonic oscillator basis truncation is an additional benefit of this strategy.
Before the current effort of developing the FSU interaction, a number of attempts have been made to modify the WBP interaction, mainly by changing the single particle energies (SPEs) of the $fp$-shell orbitals for a particular $sd$-shell nucleus and applying that for the nearby isotopes. For example, in the WBP-A \cite{wbp-a} version, the SPEs of the $f_{7/2}$ and $p_{3/2}$ orbitals were lowered in order to better explain the negative parity intruder states of $^{34}$P. This adjustment was quite successful in explaining the energy levels of $^{32}$P and $^{36}$P, however, WBP-A failed to predict the intruder states of $^{31}$Si. Hence, another version of the WBP, called the WBP-B was introduced \cite{wbp-b} by changing the SPEs of the $f_{7/2}$, $p_{3/2} $ and $p_{1/2}$ orbitals. In a different version, named WBP-M \cite{wbp-m}, all the SPEs of the $fp$ shell orbitals were changed in order to reproduce the energies and the ordering of the $3/2^-$ and $7/2^-$ states of $^{27}$Ne which eventually fixed the ordering of the same levels in $^{25}$Ne and $^{29}$Mg.
However, none of these modified versions was able to reproduce the experimental data for a large range of the nearby nuclei, and hence we have taken a step forward towards building a more general effective shell model Hamiltonian.
The model space for the WBP interaction and for the newly developed one consists of four major oscillator shells; $0s$, $0p$, $1s0d$, and $0f1p$.
The following steps briefly describe the development of the FSU interaction.
\begin{itemize}
\item The newly developed interaction starts from the WBP framework, the model consists of four major oscillator shells: $0s$, $0p$, $1s0d$, and $0f1p$.
Isospin invariance is assumed and Coulomb corrections to the binding energies are implemented using the standard procedures as discussed in Refs. \cite{wbmb,usd,usdAB}.
\item The single particle energies (SPE) and the two body matrix elements (TBME) of the $0s$ and $0p$ shells and across
$0s$ - $0p$ are same as those of the original WBP interaction and are not a part of the fit.
\item The TBMEs within the $sd$ shell are taken from the USDB \cite{usdAB} interaction and also are not a part of the fit.
\item The 6 monopoles between the orbitals of the $0p$ shell and $sd$ shell are modified simultaneously with the $sd$ shell single particle energies, thus changing the shell gap but ensuring that excitation energies of all $0\hbar \omega$ states in the $sd$ shell are identical to those from the USDB calculations.
\item $sd$ - $fp$ cross-shell matrix elements:
\begin{enumerate}
\item $1p_{1/2}$ orbital in the $fp$ shell is relatively high and not very sensitive to our data set. We thus fitted only one monopole term between the $1p_{1/2}$ and the $sd$ orbitals. This amounts to two fit parameters because we have allowed different strengths for isospin T=0 and T=1.
\item Only the monopole terms between $0f_{7/2}$ - $0d_{5/2}$ and $1p_{3/2}$ - $0d_{5/2}$ were considered since $d_{5/2}$ is deeply bound for $sd$ - $fp$ cross-shell nuclei. A total of 4 parameters were varied for T=0 and T=1.
\item For the remaining $0f_{7/2}$ - $1s_{1/2}$, $0f_{7/2}$ - $0d_{3/2}$, $1p_{3/2}$ - $0s_{1/2}$, and $1p_{3/2}$ - $0d_{3/2}$ all multipole-multipole density terms were fitted. A total of 24 parameters were varied.
\end{enumerate}
\item For the $fp$ shell, GXPF1A \cite{gxpf1a} was used as a starting Hamiltonian and all the TBMEs associated with only $0f_{7/2}$ and $1p_{3/2}$ were fitted; a total 30 TBMEs and hence 30 parameters were adjusted within the $fp$ shell.
\item All the matrix elements within the $sd$ and $fp$ shells as well as the $sd$ - $fp$ cross shell were scaled with $A^{-0.3}$. However, no scaling was adopted for the cross shell interactions between the lower $p$ and the $sd$ shells.
\item A total of 70 parameters were fitted using 270 experimentally observed states compiled in Ref. \cite{lubnaThesis} and \cite{nndc}. The experimental data was compiled from four groups
\begin{enumerate}
\item Intruder states sensitive to $p$ - $sd$ shell gap. This group consists of pure $p$ shell C and N isotopes and
nuclei between O to Si with states that have strong spectroscopic factors if populated via $(p,\, d)$ reactions.
\item Negative parity states in $sd$-shell populated via $(d,\, p)$ reactions which are sensitive to particle promotion from $sd$ to $fp$. High spin states, that gain spin from the promotion of a particle to $0f_{7/2}$ are of particular importance.
\item Neutron rich cross shell nuclei with $Z<20$ and $N>20$ where both $0\hbar\omega$ and $1\hbar\omega$ types of states were included in the fit.
\item Nuclei in $fp$ shell with $Z\ge 20$ and $N\ge 21$; the $0\hbar\omega$ states in these nuclei are critical for tuning the $0f_{7/2}$ - $1p_{3/2}$ gap.
\end{enumerate}
\item The fitting procedure followed the method described in [2], with 40 linear combinations of parameters being selected at each iteration. We reached the convergence after 6 iterations with an overall rms deviation from experiment of 190 keV.
\item All calculations were carried out within $N_{\rm max}=1$ truncation thus including $0\hbar\omega$ and $1\hbar\omega$ types of excitations that due to different parities do not mix. This truncation allows for exact identification and separation of the spurious center-of-mass excitations.
\item Tables of the matrix elements can be found in the Thesis publication of Lubna, Rebeka Sultana \cite{lubnaThesis}. Users are encouraged to contact the authors for help with the calculations, further details and updates.
\end{itemize}
All the shell model calculations were performed with the shell model code CoSMo \cite{COSMO}. A histogram of the differences between the experimental states included in the fit and those predicted with the FSU interaction is shown in Figure \ref {fig:fsu9Differences}.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.3]{Fig1}
\end{center}
\caption{Histogram of the differences in excitation energy between experiment and the FSU interaction fit. The root-mean-square
deviation is 190 keV.}
\label{fig:fsu9Differences}
\end{figure}
\section{Effective Single Particle Energy (ESPE)}
The evolution of the mean field, which is described by the position of the single particle levels and how they change
with number of protons and neutrons, is a particularly interesting and non-trivial question in the strongly-interacting two-component many-body systems of atomic nuclei. In most nuclei the single-particle strength is distributed over many states.
Systematic studies have been performed before with other shell model interactions \cite{sdpf-m, sdpf-u, smirnova} to understand the evolution of the ESPE. An experimental approach of determining the ESPEs has been to measure and sum up the energies of appropriate states weighted by the reaction spectroscopic factors.
This process is limited by decreasing cross sections for higher lying states and by difficulties in making spin assignments and in determining what fraction of the cross sections come from direct reaction components.
Theoretical approaches do not suffer from most of these experimental limitations, but have their own uncertainties. Perhaps chief among them being the uncertainty in the interaction Hamiltonian. The bare single-particle energies tell only a part of the story of the effective shell positions. The TBME have a major influence on the positions of the orbitals. In fact, the TBMEs shift the orbitals based on the number of particles in shells, and are the major reason that one interaction could fit such a wide range of nuclei.
How the newly developed FSU interaction describes the shell evolution is among the most interesting immediate questions that can be addressed. While the FSU interaction was fitted to the negative-party states in $sd$ nuclei,
the study of the ESPE extrapolates to a much broad spectrum of configurations not limited by those experimentally reachable with single-nucleon transfer reactions.
The evaluation of the ESPE relies on $0\hbar\omega$ and $1\hbar\omega$ calculations.
In order to determine the ESPE of the $0f_{7/2}$ and $1p_{3/2}$ orbitals, we have followed a procedure similar to the experimental approach, but using the theoretically computed energies and spectroscopic factors
in the following formula
\begin{equation}
\label{eq:eqn1}
\rm{ESPE}=\frac{\sum_{i=1} {\rm SF}_i \times E^*_i}{\sum_{i=1}{\rm SF}_i}
\end{equation}
In the above formula, $\rm {SF}_i$ is the spectroscopic factor for
$A\rightarrow A+1$ where a particle is placed onto a single-particle orbit of interest
above an even-even $A$ core. The $\rm{E}^*_i$ is the excitation energy of the i-th state in $A+1$ with the matching quantum numbers measured relative to the ground state energy of the core $A.$
It has been observed from the calculations that it is enough to consider 30 lowest states in the sum (\ref{eq:eqn1}),
by then the SF reach to a saturation and the ESPE converges.
From the formal theoretical perspective, Eq. (\ref{eq:eqn1}) represents single particle energies of the mean field arising from the shell model Hamiltonian.
The ESPEs obtained from the above formula across the $sd$ shell are plotted in Figure \ref{fig:ESPE} as a function of proton number $Z$. The points represent the ESPEs of the $0f_{7/2}$ and $1p_{3/2}$. The systematic crossing of the ESPEs of the $0f_{7/2}$ and $1p_{3/2}$ orbitals with increasing neutron number is evident in the figure. The crossing occurs between $Z = 10$ and $12$, suggesting that the $N = 28$ shell gap shifts to $N = 24$ with lower $Z$, which points to the inversion of $0f_{7/2}$ and $1p_{3/2}$ neutron orbitals. The ground state of $^{31}$Ne is tentatively assigned $3/2^-$ as is the first excited state in $^{27}$Ne \cite{nndc}. In $^{27}$Mg the lowest $3/2^-$ and $7/2^-$ states are essentially degenerate \cite {nndc}.\\ \\
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.33]{Fig2}
\end{center}
\caption{Neutron Effective Single Particle Energies (ESPEs) of $0f_{7/2}$ and $1p_{3/2}$ orbitals calculated with the FSU interaction. They represent the theoretical centroids of the energies of the $0f_{7/2}$ and $1p_{3/2}$ orbitals. In the ``normal" ordering the red diamonds ($1p_{3/2}$) lie above the black circles ($0f_{7/2}$). }
\label{fig:ESPE}
\end{figure}
This inversion of the $1p_{3/2}$ and $0f_{7/2}$ ESPE is related to the 2-body interactions between nucleons in the $sd$ and $fp$ shells; the effect of this interaction is density dependent and varies as a function of the shell fillings. In the FSU interaction these TBME emerge as a consequence of fitting the energies of the states in a wide range of nuclei. Over half a century ago Talmi and Unna \cite{talmi} attributed the inversion of the $1s_{1/2}$ and $0p_{1/2}$ orbitals to the same principle. Alternate explanations, especially for the $1s_{1/2}$ and $0p_{1/2}$ case, have been given in terms of the effects of weak binding on the mean field of low $\ell$ orbitals. Hoffman ${\it et\, al.}$ \cite{hoff} have explored the weak binding effect for pure single-particle shells in a Woods-Saxon potential and have shown that it is large near the threshold for neutron $s$ states.
While much smaller for $p$ states, there is still a crossing between the $0p_{1/2}$ and $0d_{5/2}$ orbitals at the threshold.
A similar effect for the $1p_{3/2}$ and $0f_{7/2}$ appears to be a contributing factor to the inversion shown in Figure \ref{fig:ESPE}. Nearly all crossings occur around ESPE$=0$ indicating that the levels become unbound. Indeed,
the centrifugal barrier for $\ell=3$ $f$ orbital is high which would make a transition into the continuum smooth, while for the $\ell=1$ $p$-wave the interaction with the continuum is strong and is pushing the level down as discussed in Ref. \cite{volyapc}.
It appears that the continuum effect is incorporated through the fitting of the effective interaction, but this can be a challenge for theoretical methods that do not take continuum of reaction states into account. This inversion of the $1p_{3/2}$ and $0f_{7/2}$ ESPE at high neutron excess also has implications for the IoI phenomenon discussed in the next section.
Another way of examining the systematics of shell evolution, which is closer to experiment, is from the positions of the states carrying the largest part of the single-particle strength. Such a comparison is shown in Table \ref{tab:dp} which lists the experimental and theoretical excitation energies of the lowest $3/2^+$, $7/2^-$, and $3/2^-$ states, of the even $Z$ odd mass nuclei, along with the predicted and measured $(d, \, p)$ reaction spectroscopic factors (SF). As mentioned before, there is more uncertainty in measuring the values of SF than excitation energies and in some cases the SF cannot (lack for appropriate targets) or have not been measured. With this in mind, the agreement between experiment and predictions using the FSU interaction for both excitation energies and SF is generally good. Also, the relatively large values of the SF show that these states represent the dominant single-particle states.
\begin{table}[]
\centering
\setlength{\tabcolsep}{0.85em}
\renewcommand{\arraystretch}{1.28}
\caption{Comparison of the experimentally observed $7/2^-$, $3/2^-$ and $3/2^+$ states of even $Z$ odd mass $sd$-shell nuclei to the predictions by the FSU interaction. The measured spectroscopic factors were taken from the NNDC \cite{nndc}. All the experimental spectroscopic factors were compiled from the $(d, \, p)$ reactions.}
\label{tab:dp}
\begin{tabular}[c]{|c|c|c|c|c|c|}
\hline
\multirow{2}{*}{Nucleus} & \multirow{2}{*}{J$^\pi$} & \multicolumn{2}{c|}{Energy} & \multicolumn{2}{c|}{(2J+1)SF} \\ \cline{3-6}
& & EXP &Th & EXP & Th \\ \hline
\multirow{3}{*}{$^{25}$Ne} & $7/2^-$ & 4030 & 3957 & 5.8 & 4.5 \\ \cline{2-6}
& $3/2^-$ & 3330 & 3471 & 3.0 & 1.9 \\ \cline{2-6}
& $3/2^+$ & 2030 & 2044 & 1.6 & 1.8 \\ \hline
\multirow{3}{*}{$^{27}$Ne} & $7/2^-$ & 1740 & 1634 & 2.8 & 3.9 \\ \cline{2-6}
& $3/2^-$ & 765 & 858 & 2.6 & 2.4 \\ \cline{2-6}
& $3/2^+$ & 0 &0 & 1.7 & 2.8 \\ \hline
\multirow{3}{*}{$^{25}$Mg} & $7/2^-$ & 3971 & 3902 & 2.2-3.3 & 3.9 \\ \cline{2-6}
& $3/2^-$ & 3413 & 3525 & 0.9-1.2 & 1.5 \\ \cline{2-6}
& $3/2+$ & 974 & 1098 & 0.8 & 0.9 \\ \hline
\multirow{3}{*}{$^{27}$Mg} & $7/2^-$ & 3761 & 3827 & 4.6 & 3.5 \\ \cline{2-6}
& $3/2^-$ & 3559 & 3644 & 1.6 & 2.2 \\ \cline{2-6}
& $3/2+$ & 984 & 994 & 2.4 &1.56 \\ \hline
\multirow{3}{*}{$^{29}$Mg} & $7/2^-$ & 1430 & 1719 & 3.0 & 4.4 \\ \cline{2-6}
& $3/2^-$ & 1094 & 1396 & 0.4 & 2.0 \\ \cline{2-6}
& $3/2^+$ & 0 &0 & 1.2 & 1.8 \\ \hline
\multirow{3}{*}{$^{29}$Si} & $7/2^-$ & 3623 & 3684 & 7.0 & 4.5 \\ \cline{2-6}
& $3/2^-$ & 4934 & 4373 & 2.2 & 2.3 \\ \cline{2-6}
& $3/2^+$ & 1273 & 1285 & 3.0 & 2.7 \\ \hline
\multirow{3}{*}{$^{31}$Si} & $7/2^-$ & 3134 & 2855 & 4.8 & 5.6 \\ \cline{2-6}
& $3/2^-$ & 3533 & 3435 & 1.6 & 2.8 \\ \cline{2-6}
& $3/2^+$ &0 &0 & 2.8 & 2.4 \\ \hline
\multirow{3}{*}{$^{33}$Si} & $7/2^-$ & 1435 & 1452 & & 6.0 \\ \cline{2-6}
& $3/2^-$ & 1981 & 1944 & & 2.9 \\ \cline{2-6}
& $3/2^+$ & 0 &0 & & 1.4 \\ \hline
\multirow{3}{*}{$^{35}$Si} & $7/2^-$ & 0 & 0 & 4.5 & 7.4 \\ \cline{2-6}
& $3/2^-$ & 910 & 909 & 2.8 & 3.7\\ \cline{2-6}
& $3/2^+$ & 974 & 936 & & \\ \hline
\multirow{3}{*}{$^{33}$S} & $7/2^-$ & 2935 & 2942 & 4.2 & 5.8 \\ \cline{2-6}
& $3/2^-$ & 3221 & 3386 & 3.5 & 2.3 \\\cline{2-6}
& $3/2^+$ & 0 &0 & 3.5 & 2.6 \\ \hline
\multirow{3}{*}{$^{35}$S} & $7/2^-$ & 1991 & 2042 & 5.4 & 6.4 \\ \cline{2-6}
& $3/2^-$ & 2348 & 2409 & 2.1 & 2.7 \\ \cline{2-6}
& $3/2^+$ & 0 & 0 & 1.7 & 1.5 \\ \hline
\multirow{3}{*}{$^{37}$S} & $7/2^-$ & 0 & 0 & 5.5 & 7.3 \\ \cline{2-6}
& $3/2^-$ & 646 & 573 & 1.8 & 3.5 \\ \cline{2-6}
& $3/2^+$ & 1398 & 1303 & & \\ \hline
\multirow{3}{*}{$^{37}$Ar} & $7/2^-$ & 1611 & 1543 & 6.1 & 6.3 \\ \cline{2-6}
& $3/2^-$ & 2491 & 2679 & 1.8 & 2.6 \\ \cline{2-6}
& $3/2^+$ & 0 & 0 & 2.2 &1.5 \\ \hline
\multirow{3}{*}{$^{39}$Ar} & $7/2^-$ & 0 & 0 & 5.0 & 6.7 \\ \cline{2-6}
& $3/2^-$ & 1267 & 1186 & 2.0 & 2.8 \\ \cline{2-6}
& $3/2^+$ & 1517 &1457 & & \\ \hline
\end{tabular}
\end{table}
Figure \ref{fig:fpdiff}(a) provides a pictorial summary of the relative positions between the $7/2^-$ and $3/2^-$ states as a function of the proton number $Z$. The black circles and red lines show the average values from Table \ref{tab:dp} for experiment and theory, respectively, while the black error bars represent the variation of the experimental differences. The observed trends are reproduced by theory, see Figure \ref{fig:fpdiff}(a). This graph agrees qualitatively with those in Figure \ref{fig:ESPE}. It demonstrates that the evolution of the separation between the $7/2^-$ and $3/2^-$ states is largely a function of the proton number $Z$ and that the
$3/2^-$ energies drop below the $7/2^-$ ones between $Z=14$ and 12. In contrast to the ESPEs which approximate the positions of the $0f_{7/2}$ and $1p_{3/2}$ orbitals the crossing between $0f_{7/2}$ and $1p_{3/2}$ happens between $Z = 10$ and 12. Together these results show that the trend is robust, but the question of the relative position of the orbitals is more complex and nuanced than was expected earlier.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.6]{Fig3}
\end{center}
\caption{(a) Average energy differences between the lowest $7/2^-$ and $3/2^-$ experimental levels in Table \ref{tab:dp}. The error bars give an indication of the range of values for different neutron numbers. Positive (negative) values of the ordinate correspond to the $3/2^-$ state above (below) the $7/2^-$ one. (b) Occupancies of the neutron $0f_{7/2}$ and $1p_{3/2}$ orbitals in neutron number $N = 20$ nuclei as a function of proton number $Z$ for the lowest 2p2h states. The values are shown as filled circles for the cases where the lowest 2p2h state is the ground state (IoI) and as open circles where the lowest 2p2h state is excited above the ground state.}
\label{fig:fpdiff}
\end{figure}
\section{Evolution of the N=20 Shell gap and the Island of Inversion (IoI)}
One of the first indications that the pure $sd$ shell model could not represent low-lying states in all $sd$ nuclei came from the experimentally measured mass of $^{31}$Na \cite {thibault}. The experimental mass was about 1.6 MeV lower than that predicted from the USD interaction \cite {usd}. This was further clarified by the USDA, USDB showing that states in the highest $N$ - $Z$ nuclei can not be fitted. A consistent over-prediction of 1 to 2 MeV of the ground state energies of these nuclei can be seen in Figure 9 of Ref. \cite {usdAB}. This region of nuclei is now known as the ``Island of Inversion" (IoI) and its origin has been discussed a lot. Most explanations center around the filled or almost filled neutron $sd$ shell and $fp$ intruder configurations leading, counter-intuitively, to lowering the energy of
the 2 particle- 2 hole (2p2h) state, with two nucleons being promoted from $sd$ to $fp$ shell, below that of the ``normal" 0p0h. Such lowering is associated with increased correlation energy or higher deformation, lowering Nilsson orbitals. However the effect fades away with filling of the proton $sd$ shell.
While a number of shell model calculations in the past have reproduced many aspects of the IoI, as discussed in the Introduction, here we study what the FSU interaction predicts for the Iol nuclei.
Concentrating on the IoI region, we consider the states where two nucleons are promoted from $sd$ to $fp$
referring to them as 2p2h states. These states were not a part of the fit and for this extrapolation to be meaningful the additional 2p2h states cannot be allowed to directly mix and renormalize the previously fitted 0p0h states.
Due to the valence space limitation the full $2\hbar\omega$ excitations from the sd space cannot be considered.
Moreover, our tests have shown that excitations from the $0s$ and $0p$ are nearly irrelevant for the validity of this discussion,
thus we did not include those states into our definition of 2p2h excitations. It also has been verified that the inevitable center-of-mass contamination in this truncation scheme is very low. We estimate that the errors from truncation and center-of-mass contamination amount to less then 200 keV uncertainty in the energies, which is of the same order as the rms deviation in the fit.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.35]{Fig4}
\end{center}
\caption{The experimentally known levels of $^{31}$Na compared to the lowest ones predicted using the FSU interaction for 0p0h, 1p1h, and 2p2h configurations.
The experimental levels agree well with the 2p2h results while the 0p0h states start almost 2.5 MeV higher in excitation energy. Only the two lowest calculated 1p1h
states are labeled because of the high level density above this.}
\label{fig:31Na}
\end{figure}
We first discuss the case of $^{31}$Na ($N=20$) \cite{thibault}. As shown in Figure \ref {fig:31Na}, the total binding energies for the first four 2p2h states were found to be below that of the lowest 0p0h state. The first three 2p2h states agree well with what is so far known experimentally, whereas
the lowest 0p0h state ($5/2^+$) appears much higher in energy and has a different spin from the experimentally observed ground state of $^{31}$Na.
\begin{figure*}
\begin{center}
\includegraphics[scale=0.55]{Fig5}
\end{center}
\caption{The lowest experimental energy levels of $N=20$ $sd$-shell nuclei compared to those calculated using the FSU shell model interaction for 0p0h and 2p2h configurations.
The levels of the known IoI nuclei $^{30}$Ne and $^{32}$Mg agree well with the 2p2h results while the lowest states in the higher Z nuclei agree much better with
the 0p0h results.}
\label{fig:IoI}
\end{figure*}
While the experimental information is limited, it is clear that the FSU interaction has depicted the correct picture of $^{31}$Na as one with the inverted configuration. As mentioned above, only the low $Z$ and $N \approx 20$ nuclei exhibit the IoI or inverted 2p2h - 0p0h behavior. To explore the transition from IoI to ``normal" behavior, Figure \ref {fig:IoI} compares experimentally measured energies with our calculations for the lowest levels in a sequence of $N=20$ even-$A$ $sd$ nuclei. For $Z = 10$ and $12$, not only do the lowest states have 2p2h character, but the whole $0^+,\, 2^+,\, 4^+$ 2p2h sequence agrees well with experiment. In addition to starting much higher in energy, the spacing between 0p0h states differs significantly from experiment. The story changes for $Z = 14$ $^{34}$Si, where the 0p0h $0^+$ state is substantially lower than the 2p2h one. The experimental second $0^+$ and first $2^+$ states are much closer to the 2p2h ones, while the second experimental $2^+$ level corresponds well with the 0p0h one. This shows the shape coexistence, also discussed in Ref. \cite{rotaru}. For $Z = 16$ and $18$ both the first experimental $0^+$ and $2^+$ states correspond with the 0p0h calculations. The second $0^+$ states in both the nuclei were discussed to have 2p2h dominant configurations \cite{wood, olness, flynn} and are in very good agreement with the FSU predictions. The $4^+$ states of $^{36}$S and $^{38}$Ar lie much closer to the calculated 2p2h ones. Note, that the FSU cross-shell interaction describes the transition from inverted 0p0h-2p2h order to normal as a function of $Z$ despite not having been fitted to any of these states.
This emergence of the IoI does not involve any $fp$ orbitals dropping below the $sd$ shell, at least not for spherical shape. The lowering in energy of the 2p2h configurations does not extend so much to 1p1h ones, as shown for $^{31}$Na in Figure \ref {fig:31Na}. The lowest 1p1h state (3579 keV, $3/2^-$) lies over an MeV above the lowest 0p0h state. So it is the promotion of a neutron pair to the $fp$ shell which favors the 2p2h configuration so much. The promotion of a neutron pair to the $fp$ orbital appears to lower its energy because of correlation energy in the shell model. Clearly, collective behaviors such as pairing and deformation and intricate interplay between them are central for the IoI phenomenon. Representing a mesoscopic phase transition, the picture is highly sensitive to the matrix elements of the effective Hamiltonian and in particular to the components describing short and long range limits of nucleon-nucleon in-medium interaction.
In a geometrical picture IoI can be associated with increased prolate deformation due to the promotion of a pair into a down-sloping Nilsson orbital whose excitation energy decreases rapidly with increasing deformation. An indication of this difference in deformation is shown in the lower panel of Figure \ref {fig:BE2}. For $^{30}$Ne and $^{32}$Mg the calculated B(E2) transition strengths from the lowest $2^+$ to ground states (both of which have 2p2h configurations) are relatively large at over 400 e$^2$fm$^4$, consistent with relatively high deformation, and agree well with experiment. In contrast the B(E2) strengths for $^{36}$S and $^{38}$Ar are rather low, consistent with near spherical shape.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.35]{Fig6}
\end{center}
\caption{Experimental E($2_1^+$) and B(E2: $0^+_1 \rightarrow 2^+_1$) values for the $N=20$ isotones are compared to those calculated by using the FSU interaction. The B(E2: $0^+_1 \rightarrow 2^+_1$) value of $^{34}$Si has not been calculated because of the different configurations associated with the $0^+_1$ and $2^+_1$ states.}
\label{fig:BE2}
\end{figure}
Figure \ref{fig:BEIoI} portrays the differences between experiment and theory of the binding energies around the IoI which
are sensitive to pairing correlations.
The calculated total binding energies are compared with the measured ground state masses from the 2016 mass evaluation \cite{mass16}. The Coulomb corrections to the total binding energies are included following procedures in Refs. \cite{wbmb,usd,usdAB}. The $N = 21$ 0p0h (2p2h) configurations have 1(3) nucleons in $fp$, and, $N = 22$ 2p2h actually have 4 $fp$ nucleons so the $fp$ matrix elements are tested along with the cross-shell ones. Looking at the $N = 20$ isotonic chain, the agreement is quite good with an RMS deviation of 276 keV comparing the experimental binding energies with the 2p2h results below $Z = 13$ and with 0p0h for higher $Z$. For $10 \le Z \le 12$ and $19 \le N \le 21$ the 2p2h inverted configuration is lower in energy and agrees better with experiment. Outside this range the 0p0h configuration is lower, which again agrees with experiment. For $N = 22$ it appears that promoting a second neutron pair to $fp$ is not energetically favorable.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.32]{Fig7}
\end{center}
\caption{The number displayed inside a box corresponding to an isotope is the difference in binding energy between experiment and shell model predictions using the FSU interaction with 0 or 2 particle-hole configurations. We call the states over-bound where the calculated states are more tightly bound than that of the experimental ones and under-bound when it is otherwise.}
\label{fig:BEIoI}
\end{figure}
A similar approach of calculating the 2p2h states was taken in Ref. \cite{wbmb} using the WBMB interaction. As mentioned earlier, the WBMB interaction was developed for the mass region near $^{40}$Ca by fitting the 1p1h states within the $sdfp$ model space. We have compared the differences in the 0p0h and 2p2h ground states calculated by using the WBMB and the FSU interactions for $N =20$ isotones in Table \ref{tab:wbmbFSU}. The predictions with the WBMB interaction were taken from Ref. \cite{wbmb}. From Table \ref{tab:wbmbFSU}, we see that both the interactions predict $^{30}$Ne, $^{31}$Na and $^{32}$Mg having their 2p2h ground state more tightly bound than that calculated for the 0p0h configurations, meaning that these nuclei are the members of the IoI. The FSU interaction predicts $^{29}$F also as an IoI nucleus, which was suggested recently by the Ref. \cite{29F}. The difference between the first two $0^+$ states in $^{34}$Si is known experimentally as 2719 keV \cite{rotaru}. The FSU interaction predicts it better as 2432 keV. The experimentally observed $0^+_2$ states in $^{36}$S and $^{38}$Ar are at 3346 and 3376 keV respectively, which are presumably 2p2h in nature. The FSU interaction predicts them at 3373 and 3140 keV respectively. In $^{37}$Cl the first 2p2h state was identified at 3708 keV energy \cite{37Cl}, whereas the FSU prediction is 3538 keV. The better predictability of the FSU interaction comes from a more extensive fit for a wide range of cross shell data as well as the use of a better Hamiltonian for the $fp$ shell, we believe.
\begin{table}[]
\centering
\setlength{\tabcolsep}{0.85em}
\renewcommand{\arraystretch}{1.5}
\caption{The ground state energies with 2p2h configurations are calculated with respect to those with the 0p0h configurations using the WBMB \cite{wbmb} and the FSU interactions. The symbols W, F and T in the WBMB calculations stand for weak coupling, full WBMB space and the truncated space respectively.}
\label{tab:wbmbFSU}
\begin{tabular}{|c|c|c|}
\hline
Nucleus & WBMB \footnote{Ref. \cite{wbmb}} & FSU \\ \hline
\multirow{2}{*}{$^{28}$O} & 3038: W & -755 \\
& 2956: F & \\ \hline
\multirow{2}{*}{$^{29}$F} & 1286: W & -2201 \\
& 1338: F & \\ \hline
\multirow{2}{*}{$^{30}$Ne} & -698: W & -2823 \\
& -788: F & \\ \hline
\multirow{2}{*}{$^{31}$Na} & -502: W & -2452 \\
& -764: T & \\ \hline
\multirow{2}{*}{$^{32}$Mg} & -926: W & -1666 \\
& -966: T & \\ \hline
$^{33}$Al & 854: W & 922 \\ \hline
\multirow{2}{*}{$^{34}$Si} & 1816: W & 2432 \\
& 1554: T & \\ \hline
$^{35}$P & 2698: W & 3264 \\ \hline
\multirow{2}{*}{$^{36}$S} & 3146: W & 3373 \\
& 3009: T & \\ \hline
\multirow{2}{*}{$^{37}$Cl} & 3195: W & 3538 \\
& 3091: T & \\ \hline
$^{38}$Ar & 2701: F & 3140 \\ \hline
\end{tabular}
\end{table}
Since the IoI involves excitations into the $fp$ shell, the question arises how the inversion of the $0f_{7/2}$ and $1p_{3/2}$ single particle energies at low $Z$, discussed above, affects our understanding of the IoI. The answer, within the context of the FSU interaction is shown in Figure \ref{fig:OccupIoI}.
This figure shows some of the $fp$ shell occupancies calculated for the lowest 2p2h states in Figure \ref {fig:IoI}. Occupancy here is defined as the average number of nucleons in a given orbital.
There is almost no proton $fp$ occupancy calculated for these nuclei and there is a relatively constant $\nu 1p_{1/2}$ occupancy of about 0.1 neutron. For $Z=10$ $^{30}$Ne, which is the most strongly inverted, the $\nu1p_{3/2}$ occupancy is about twice that of $\nu0f_{7/2}$.
With increasing $Z$, the ratio of $\nu 1p_{3/2}$ to $\nu 0f_{7/2}$ decreases steadily from about 2 to about 0.2 across this region. Of course, the energies of the 2p2h configurations rise above that of the 0p0h ones around $Z = 14$.
\begin{figure}[h!]
\begin{center}
\includegraphics[scale=0.4]{Fig8}
\end{center}
\caption{2p2h occupancies of the $\nu 0f_{7/2}$ and $\nu 1p_{3/2}$ orbitals for the first $0^+$, $2^+$ and $4^+$ calculated states using the FSU interaction for nuclei with $N=20$ and $Z$ between 10 and 18.}
\label{fig:OccupIoI}
\end{figure}
We note that considering that the degeneracy of the $f_{7/2}$ is twice that of $p_{3/2}$, at the level crossing or in the limit of strong pairing the ratio of occupancies of $\nu 1p_{3/2}$ to $\nu 0f_{7/2}$ should be about 0.5.
This indeed happens at around $Z=14$; however significant deviation from 0.5 suggests that pairing, or at least pair transfer between $f_{7/2}$ and $p_{3/2}$ is weak. Pair transfer and pair vibration, collective pairing condensation, interplay of paring and deformation in the IoI region, as well as the connection of these collective effects with the underlying matrix elements of the FSU Hamiltonian, all require more study and remain a challenge for the future. The occupancy trend is perhaps illustrated more clearly in Figure \ref{fig:fpdiff}(b) which shows the $\nu 1p_{3/2}$ and $\nu 0f_{7/2}$ occupancies of the lowest 2p2h states in the $N = 20$ nuclei as a function of proton number $Z$. Note, that for $^{34}$Si, the 2p2h $0^+$ state lies 2432 keV above the 0p0h ground state but the 2p2h $2^+$ level lies close in energy with the lowest experimental $2^+$ state.
Together these calculations imply that the $\nu 1p_{3/2}$ orbital plays a larger role in the IoI phenomenon than does the $\nu 0f_{7/2}$ one.
\section {Fully aligned states}
In describing the states used in the fit of the FSU interaction, we included only 0p0h(1p1h) configurations for natural(unnatural) parity sectors. In particular, no 2p2h configurations were used to adjust the interaction parameters. After the fitting, two early tests were performed to explore the predictive properties of the FSU interaction for 2p2h configurations. One was the calculation of the lowest 2p2h $7^+$ states in $^{34}$Cl and $^{36}$Cl \cite {fsu-38Cl}. These agreed within 200 keV with the experimental states. The other test was performed on $^{38}$Ar \cite {abromeit}, since experimental states up to $8^+$ and $(10^+)$ are known. Calculations using the USD family of interactions agree within 200 keV with the excitation energy of the lowest $2^+$ state of $^{38}$Ar, but over-predict the lowest experimental $4^+$ level by over 3 MeV. With only two holes in the $sd$ shell, the maximum spin from coupling two $0d_{3/2}$ protons is $2 \hbar$. The very high $4^+$ energy represents the cost of promoting a $0d_{5/2}$ proton to $0d_{3/2}$, but nature finds another less energetic way of achieving $4^+$. This must be by promoting an $sd$ nucleon pair to the $fp$ shell. A 2p2h calculation with the FSU interaction predicts the lowest $4^+$ level only 300 keV above the experimental one, and it predicts the $6+$ state 200 keV below experiment, while the predicted $8^+$ state is 100 keV above experiment as shown in Ref. \cite {abromeit}.
With this success we have searched for other states with confirmed 2p2h structure to compare with theory. One such group of excited states across the $sd$ shell are often called the ``fully aligned" states. One subgroup of fully-aligned states is the lowest $J^\pi = 7^+$ states. These states have been suggested to have both odd nucleons in the highest spin orbital around - $0f_{7/2}$ - and with their spins fully aligned, which, from the Pauli principle, is only possible for non-identical nucleons. For these calculations it is critical that the FSU interaction treats protons and neutrons on an equivalent basis. These fully-aligned $\pi f_{7/2} \otimes \nu f_{7/2}$ are yrast and strongly populated in high-spin $\gamma$-decay sequences. Stronger evidence of their unique nature comes from $(\alpha,d)$ reactions \cite{rivet, del, alphaD1, alphaD2, alphaD3, alphaD1962, alphaD1960} where they are the most strongly populated states with an orbital angular momentum transfer of $\ell = 6$. In most cases such states involve two nucleons beyond those in the dominant ground state configuration outside the $sd$ shell. The energies of these $7^+$ states (including those in $^{34}$Cl and $^{36}$Cl mentioned above) are graphed in Figure \ref {fig:aligned} along with calculated results using the FSU interaction. The agreement is excellent both in value and in the trend which extends from 10 MeV for the lightest nuclei down to 2 MeV for the heaviest and from 2p2h to 1p1h excitations relative to the ground state. The calculations also indirectly confirm the spin alignment with approximately equal proton and neutron occupancies in the $0f_{7/2}$ orbitals, even though most 2p2h states in these neutron-rich nuclei as discussed in the IoI section involve predominantly two neutron configurations.
\begin{figure*}
\centerline{\includegraphics[scale=0.65,angle=0]{Fig9}}
\caption{ Comparisons of the energies of fully aligned states in $sd$-shell nuclei with those predicted employing the FSU interaction. Many of the experimental points are confirmed by both selective population in $(\alpha,d)$ reactions and in high-spin $\gamma$ decay sequences and are displayed with solid black circles, while dotted black circles are used to represent states observed by only one of the two signatures. The structure of many of these aligned states involve the promotion of two (extra) nucleons to the $0f_{7/2}$ orbital and are shown with solid red diamonds. Those with at least one nucleon in the $0f_{7/2}$ orbital may require only one more promotion (1p1h excitation) and are shown with dotted red diamond symbols.}
\label{fig:aligned}
\end{figure*}
Fully aligned states are also known for some odd-$A$ nuclei where an $sd$ nucleon is also aligned in spin with the aligned $0f_{7/2}$ nucleons. Five such cases in Figure \ref {fig:aligned} are known experimentally as the strongest states populated in $(\alpha,d)$ reactions. They have an unpaired nucleon in the $0d_{3/2}$ orbital which contributes an extra spin of $3/2 \hbar$. Again the 2p2h and 1p1h calculations with the FSU interaction agree well. In lighter odd-A nuclei the aligned $sd$ nucleon could be in the $1s_{1/2}$ or $0d_{5/2}$ orbitals, leading to total spins of 15/2 or 19/2 and higher excitation energies. Their calculated energies are also shown in Figure \ref {fig:aligned}, but none have been seen in $(\alpha,d)$ reactions. A $(11/2^+,15/2^+)$ state which decays only to the lowest $13/2^+$ state and is very likely the $15/2^+$ fully aligned state has been reported \cite {nndc} in $^{31}$P, as shown in the figure would agree well with the predictions.
The last category of aligned states in the $sd$ shell consists those in even-even nuclei. Their excitations involve the breaking of a proton and a neutron pair and promotion of one of each nucleon to the $0f_{7/2}$ orbital. For example, all 4 unpaired nucleons coupled to maximum spin of $10^+$ if both unpaired $sd$ nucleons are in the $0d_{3/2}$ orbital. No $(\alpha,d)$ reactions to the fully aligned state in even-even nuclei are known because of the absence of stable odd-$Z$ odd-$N$ targets in the $sd$ shell. However, the lowest experimentally known $10^+$ state in $^{38}$Ar observed by other reactions does compare well with a 2p2h calculation using the FSU interaction, as shown in Figure \ref{fig:aligned}. In the case of $^{42}$Ca the analogous state would involve breaking a $\pi d_{3/2}$ pair, promoting one proton to $0f_{7/2}$, breaking the $\nu f_{7/2}$ pair and coupling them to maximum spin for a total of $11^-$. This state has been seen in $\gamma$ decay following fusion-evaporation and its energy agrees well with the FSU calculation. We hope that future experiments in the FRIB age will be able to test these predictions. This study of aligned states targets cross shell matrix elements of high angular momentum channels that describe
long-range effective in-medium nucleon nucleon interactions and play key role in determining nuclear shape and deformation.
\section{summary}
In this work we present an effective nuclear interaction Hamiltonian for shell model calculations, named FSU interaction.
The interaction targets a broad range of nuclei from $p$ to $fp$ shells with a particular emphasis on exotic nuclei with extreme proton to neutron ratio and on states that involve cross shell excitations. The interaction was fitted using
binding energies and $1\hbar\omega$ states that probe cross-shell matrix elements in nuclei from $^{13}$C through $^{51}$Ti and $^{49}$V. Additional details of the fit can be found in Refs. \cite{fsu-38Cl,lubnaThesis}.
This report provides a comprehensive study of nuclei in the region of the Island of Inversion, namely those nuclei between
$sd$ and $fp$ shells whose low-lying structure is dominated by cross shell excitations.
We use the newly obtained FSU interaction to infer information about the mean field and evolution of the
effective single particle energies (ESPE). The ESPEs of the $0f_{7/2}$ and $1p_{3/2}$ show the expected normal ordering,
where $0f_{7/2}$ is below $1p_{3/2}$ for $Z > 12$ and a consistent trend of a decreasing separation with decreasing $Z$ until the energy order reverses around $Z = 10$ to $12$. It is remarkable that the inversion happens near zero energy associated
with the decay threshold. The interaction with the continuum is not explicitly included but maybe captured as a part of the fit.
While there have been many indications of inverted shell ordering in the past, these results present a more systematic picture from a model very firmly rooted in data. Perhaps somewhat surprisingly, over the range explored here, the inversion appears to depend more on the proton number than on the neutron excess.
The lowest $3/2^+$, $7/2^-$, and $3/2^-$ experimental states are surveyed for a complementary view of shell evolution. These energies are compared with predictions of the FSU interaction in excitation energies and spectroscopic factors.
They present a similar picture of the $0f_{7/2}$ - $1p_{3/2}$ shell evolution as a function of proton number.
The success of the FSU interaction in reproducing the negative parity states of the $sd$-shell nuclei with the $1\hbar\omega$ configuration suggests that improved, over those in Ref. \cite{rp}, calculations of the rp process rates
can be performed in the future.
In this report, the FSU interaction was taken a step forward and applied to configurations involving promotion of two nucleons
from $sd$ to $fp$ (2p2h) in the region of IoI. In this region the nuclei are more tightly bound than predicted within the pure $sd$ model space (0p0h). The 2p2h configurations have lower binding energies and agree well with the measured ground state masses in the range $10 \le Z \le 12$ and $19 \le N \le 21$, while the 0p0h configurations are lower in energy and agree better with the measured masses elsewhere. The lowest $2^+$ states agree well with the 2p2h calculations in the region $Z = 14$ and with 0p0h for $Z = 16 - 18$. The results of the FSU interaction which was not fitted to these states reproduce
well both the IoI and the transition to normal behavior. $^{34}$Si with $Z = 14$ emerges as transitional with a 0p0h ground state and a 2p2h lowest $2^+$ state. It would be interesting to locate the experimentally $4^+_1$ state which is predicted as 2p2h at 5523 keV. Another implication of the FSU shell model calculations is that $\nu1p_{3/2}$ pairs dominate over $\nu0f_{7/2}$ ones in the IoI, but $\nu0f_{7/2}$ pairs dominate the lowest 2p2h states beyond the IoI.
This is an indication of a relative weakness of pairing that would act to equilibrate occupancy.
Interestingly the IoI coincides relatively well with the region where the $\nu1p_{3/2}$ orbital falls below the $\nu0f_{7/2}$ one.
Another success of the FSU interaction has been the calculation of the energies and occupancies of the fully aligned states, first identified in the early 1960's in $(\alpha,d)$ reactions and frequently observed in high-spin $\gamma$-decay cascades (most involve 2p2h excitations relative to the ground state). Their energies are reproduced very well across the mass range, and their occupancies prove the excitation of both protons and neutrons, even though pure neutron excitations are more common in other states. This is an important result that establishes values for the specific cross shell high angular momentum matrix elements that are responsible for long range effective nucleon-nucleon interaction and are particularly challenging to obtain from fundamental principles.
This work brings forward an interesting comparison between traditional shell model interactions with those arising from first principles methods. While the former are obtained from simply fitting SPEs and TBMEs to experimental data, the latter require renormalizations, many-body forces and explicit inclusion of the reaction continuum to achieve agreement with experiment. This dichotomy, presents a modern challenge to nuclear theory and deserves a full investigation.
The capability of the FSU interaction to explain the exotic phenomena of the nuclei carries the prospect that the interaction will be successful for more exotic nuclei or states. It is hoped that the interaction will prove valuable in the coming FRIB age. \\
\begin{acknowledgements}
This work was supported by U.S. National Science Foundation under grant No. PHY-1712953 (FSU), U.S. Department of Energy, office of Science, under Award No. DE-SC-0009883 (FSU). Part of the manuscript was prepared at LLNL under Contract DE-AC52-07NA27344.
\end{acknowledgements}
|
2,869,038,154,827 | arxiv | \section{Introduction}
Recommender systems \ric{that use} historical customer-product interactions to provide customer\iadh{s} with useful suggestions \iadh{have} been of interest to \iadh{both} academia and industry for many years. Various matrix completion\iadh{-based} methods \cite{rendle2009bpr,he2017neural,mnih2008probabilistic} have been proposed to predict the rating scores of \zaiqm{products (or items)} for customers (or users). Recently, many grocery recommendation \ric{models}~\cite{wan2018representing,wan2017modeling,grbovic2015commerce} were proposed \ric{that} \iadh{target} grocery shopping \ric{use-cases}. \ric{In} real grocery shopping platforms, such as Amazon and Instacart, users' interactions \iadh{with} items are sequential, personalized and more complex than those represented by a single rating score matrix. Thus \ric{effective} recommendation models \ric{for this use-case} are designed to learn representations of users and items so that context\iadh{ual} information, such as basket context~\cite{wan2018representing} and time context \cite{manotumruksa2018contextual}, \ric{are} captured \ric{within the} learned representations, which \ric{results in increased} recommendation performance.
In the grocery shopping \ric{domain}, prod2vec~\cite{grbovic2015commerce} and triple2vec~\cite{wan2018representing} are two state-of-the-art models that learn latent representations capturing the basket context\ric{,} based on the Skip-gram model for grocery recommendation. In these models, both the user's general interest (which item\iadh{s} the user likes) and the personalized dependencies between \iadh{items} (what items the user \ricB{commonly includes} in the same basket) are encoded by the embeddings of users and items. \ric{Furthermore, when combined with} negative sampling \ric{approaches}~\cite{mikolov2013distributed}, these Skip-gram-based models are able to scale to \ric{very} large shopping datasets. \ric{Meanwhile, through the} \iadh{incorporation of} basket \ric{contextual information during representation learning}, \ricB{significant} \ric{improvements} in grocery recommendation \iadh{have been observed}~\cite{grbovic2015commerce,wan2018representing}.
However, these representation models still have several defects: (1) they represent each user and item by single deterministic points in a low-dimensional continuous space, which limits the expressive ability of their embeddings and recommendation performance\iadh{s};
(2) their model\iadh{s} are simply trained by maximizing the likelihood of recovering the purchase history, which is a point estimate solution that \iadh{is} more sensitive to \ricB{outliers when training}~\cite{barkan2017bayesian}.
To alleviate the aforementioned problems, we propose a \emph{Variational Bayesian Context-Aware Representation} model, abbreviated as \emph{VBCAR{}}, which extends the existing Skip-gram based representation models for grocery recommendation in two directions. \ric{First,} it jointly models \iadh{the} representation of users and items in a Bayesian manner, \iadh{which} represents users and items as \ric{(Gaussian)} distributions and \ric{ensures} \iadh{that} these probabilistic representations \iadh{are} \ric{similar} to their prior distributions \ric{(using} the variational auto-encoder framework~\cite{kingma2014auto}\ric{). Second,} the model is optimized according to the amortized inference network that learns an efficient mapping from samples to
\zaiq{variational} distributions~\cite{shu2018amortized}, which is a \ric{method} for efficiently approximating maximum likelihood training. Having inferred the representation vectors of users and items, we can calculate the preference scores of items for each \iadh{user} based on these two types of Gaussian embeddings to make recommendation\iadh{s}.
Our contributions can be summarized as follows~\footnote{The code of our VBCAR model is publicly available from: https://github.com/mengzaiqiao/VBCAR}:
\begin{enumerate}
\item We propose a variational Bayesian context-aware representation model for grocery recommendation that jointly learns probabilistic user and item representations while the item-user-item triples in \iadh{the} shopping baskets can be \zaiq{reconstructed}.
\item We use the amortized inference neural network to infer the embeddings of both users and items, which can learn more expressive latent representations by integrating both the non-linearity and Bayesian \iadh{behaviour}.
\item We \iadh{validate} the effectiveness of our proposed model \iadh{using} a real large grocery shopping dataset.
\end{enumerate}
\section{Related Work}
In this section, we briefly discuss two lines of related work, \iadh{namely} methods for grocery recommendation and deep neural network-based methods for recommendation.
\ric{A grocery recommender} is a \iadh{type} of recommender system employed in the domain of grocery shopping to support consumers during their shopping \ric{process}. The most significant difference between the grocery recommendation \iadh{task} and other recommendation \iadh{tasks}, such as video recommendation~\cite{rendle2009bpr} and movie rating prediction~\cite{mnih2008probabilistic}, is that the basket context\iadh{ual} information is more \ric{common and important} in grocery shopping \ric{scenarios}. \iadh{However,} most existing matrix completion-based methods~\cite{rendle2009bpr,mnih2008probabilistic,he2017neural} are unable to incorporate such basket information. Hence, many approaches have been proposed to learn latent representations that incorporate the basket information to enhance the performance of grocery recommendation~\cite{le2017basket,wan2018representing,grbovic2015commerce}, among which Triple2vec~\cite{wan2018representing} is one of the most \ric{effective.} Triple2vec~\cite{wan2018representing} is a recent proposed approach, \iadh{which uses} the Skip-gram model to capture the semantics in \iadh{the} users' grocery basket for product representation and purchase prediction. In this paper, we also apply the Skip-gram model to calculate the likelihood of the basket-based purchase history, but we further \iadh{extend} it to the Bayesian framework \iadh{that represents users and items as Gaussian
distributions and optimize them} with the Amortized Inference~\cite{kingma2014auto,shu2018amortized}.
Besides the Skip-gram-based models, other deep neural network-based recommendation methods have also achieved success due to the \ric{highly} expressive \ric{nature} of deep learning techniques~\cite{he2017neural,NGCF19,liang2018variational}. For instance, the Neural Collaborative Filtering~\cite{he2017neural} model is a general framework that integrates deep learning into matrix factorization approaches \ric{using} implicit feedback. \iadh{Meanwhile}, Li et al. proposed a collaborative variational auto-encoder ~\cite{li2017collaborative} that learns deep latent representations from content data in an unsupervised manner and also learns implicit relationships between items and users from both content and rating\iadh{s}. \ric{Additionally, to better capture} context\iadh{ual} information, Manotumruksa et al.~\cite{manotumruksa2018contextual} proposed two gating mechanisms, i.e. a Contextual Attention Gate (CAG) and Time- and Spatial-based \iadh{Gates} (TSG), incorporating both time and geographical information for \ric{(venue) recommendation.} In this work, to further enhance the expressive ability of the learned embeddings for grocery recommendation, we propose to use the variational auto-encoder-based deep neural network~\cite{kingma2014auto} to approximately optimize the variational lower bound.
\section{Methodology}
In this section, we first briefly introduce the basic notations and the problem that we plan to address (Section{} \ref{sec:notation}). \iadh{Next, we} briefly review the Skip-gram model as well as a state-of-the-art representation model called Triple2vec~\cite{wan2018representing} \iadh{tailored to} grocery recommendation (Section{} \ref{sec:triple2vec}). \iadh{Then, we} present \iadh{our} proposed representation learning model, i.e. Variational Bayesian Context-Aware Representation (VBCAR{}), as well as show how to use the learned embeddings for downstream recommendation tasks.
\subsection{Problem Definition and Notations}
\label{sec:notation}
We use $\set{U}=\{u_1,u_2,\cdots,u_N\}$ to denote the set of users and $\set{I}=\{i_1,i_2,\cdots,i_M\}$ to denote the set of items, where $N$ is the number of users and $M$ is the number of items. Then, in \iadh{a} grocery shopping scenario, the users' purchase history can be represented as $\set{S}=\{(u,i,o)\mid u\in \set{U},i\in \set{I},o\in \set{O}\}$ with $\set{O}=\{o_1,o_2,\cdots,o_L\}$ being the set of orders (i.e. baskets). We also use $\mat{Z}^{u}\in \mathbb{R}^{N\times{D}}$ and $\mat{Z}^{i}\in \mathbb{R}^{M\times{D}}$ to denote \iadh{the} latent representation matrices for users and items, respectively, where $D$ denotes the dimension of these latent variables.
Given $\set{U}$, $\set{I}$, $\set{O}$ and $\set{S}$, the task we aim to address in our paper is to infer the latent representation matrices of users and items, i.e. $\mat{Z}^{u}$ and $\mat{Z}^{i}$, so that the missing preference scores of items for each user \ric{that estimate} future \ric{user} purchase probabilities can be predicted \ric{(using} these latent representation matrices\ric{)}.
\subsection{Skip-gram and Triple2vec}
\label{sec:triple2vec}
The Skip-gram model was originally designed for estimating \iadh{word representations} that \iadh{capture} co-occurrence relations between a word and its surrounding words in a sentence~\cite{mikolov2013distributed}. It aims \iadh{to maximize} the \ric{log-likelihood} of a target entity (word) $v$ predict\ric{ing} contextual entities (words) $C_v$:
\begin{equation}
\log p(C_{v} \mid v)= \sum_{v^{\prime} \in C_{v}} \log P\left(v^{\prime} | v\right),
\end{equation}
where $P\left(v^{\prime}\mid v\right)$ is defined by the softmax formulation $P\left(v^{\prime} | v\right)=\frac{\exp \left(f_{v}^{T} g_{v^{\prime}}\right)}{\sum_{v^{\prime \prime}} \exp \left(f_{v}^{T} g_{v^{\prime \prime}}\right)}$ with $f_v$ and $g_{v^{'}}$ being the latent representations of the target entity and its contextual entities\iadh{,} respectively.
The Triple2vec~\cite{wan2018representing} model further extends the Skip-gram model for capturing \iadh{co-purchase product} relationships within users\iadh{'} baskets according to sampled triples from \iadh{the} grocery shopping data\ric{. Here} each triple reflects two items purchased by the same user in the same basket. Specifically, Triple2vec samples a set of triples $\mathcal{T}=\{(u,i,j)\mid (u,i,o)\in \set{S}, (u,j,o)\in\set{S} \}$ from the purchase history $\set{S}$ as the purchase context for training and assumes that a triple \zaiqm{$(u,i,j) \in \mathcal{T}$} is generated by \iadh{a} probability $\sigma$ calculated by the function of \zaiqm{$p((u, i, j)\mid \mat{z}^{u}_u,\mat{z}^{i}_i,\mat{z}^{i}_j)$}:
\begin{align}
\label{eq:likelihood}
\sigma = \zaiqm{p((u, i, j)\mid \mat{z}^{u}_u,\mat{z}^{i}_i,\mat{z}^{i}_j)}=P(i | j, u)P(j | i, u)P(u | i, j),
\end{align}
where $\mat{z}^{u}_u\in \mat{Z}^{u}$ and $\mat{z}^{i}_i, \mat{z}^{i}_j \in \mat{Z}^{i}$ are the latent representations of user $u$ and items $i$ and $j$, respectively, $P(i\mid j, u)=\frac{\exp \left(\mat{z}^{iT}_{i} (\mat{z}^{i}_{j}+\mat{z}^{u}_{u})\right)}{\sum_{i^{\prime}} \exp \left(\mat{z}^{iT}_{i^{\prime}} (\mat{z}^{i}_{j}+\mat{z}^{u}_{u})\right)}$ and $P(u \mid i, j)=\frac{\exp \left(\mat{z}^{uT}_{u} (\mat{z}^{i}_{i}+\mat{z}^{i}_{j})\right)}{\sum_{u^{\prime}} \exp \left(\mat{z}^{uT}_{u^{\prime}} (\mat{z}^{i}_{i}+\mat{z}^{i}_{j})\right)}$.
The Skip-gram based models ~\cite{wan2018representing,grbovic2015commerce} can learn representations for users and items at scale, and with the aid of basket information they have \ric{previously been shown to be effective} for grocery recommendation. However, these models represent each user and item by single deterministic points in a low-dimensional continuous space, which limits the expressive ability of their embeddings and recommendation performance. To address \iadh{this} problem, we propose a new Bayesian Skip-gram model that represents users and items by Gaussian distributions, as illustrated in Section{} \ref{sec:baysskpgram}. \iadh{Then,} we describe how to approximately optimize the Bayesian Skip-gram model with \iadh{a} Variational Auto-encoder and the amortized Inference (Section{} \ref{sec:inference}). \ric{We provide an overview of our overall proposed model in} Figure{} \ref{fig:overview}.
\subsection{The Variational Bayesian Context-aware Representation Model}
\setlength{\abovedisplayskip}{5pt}
\setlength{\belowdisplayskip}{5pt}
\begin{figure*}
\begin{centering}
\includegraphics[width=10cm]{overview}
\par\end{centering}
\caption{\label{fig:overview}The architecture of our proposed VBCAR{} model. The model takes the user and item one-hot identity representation, i.e. $\mat{F}^u$ and $\mat{F}^i$, as input and outputs Gaussian distributions with means and variances as latent embeddings for all users and items. The model then uses the deterministic variables $\mat{Z}^u$ and $\mat{Z}^i$, reparameterized from their Gaussian distributions, to predict the sampled triples.}
\vspace{-1em}
\end{figure*}
\subsubsection{Bayesian Skip-gram Model}
\label{sec:baysskpgram}
Here we present our proposed Variational Bayesian Context-aware Representation model, i.e. VBCAR{}, \iadh{which} represents the users and items as random variables, i.e. $\mat{Z}^{u}$ and $\mat{Z}^{i}$ \iadh{that} are independently generated according to their priors. Like other probabilistic methods for embedding~\cite{meng2019co} and recommender systems~\cite{he2017neural,liang2018variational}, these priors are assumed to be the standard Gaussian distributions:
\begin{align}
p\left(\mat{Z}^{u}\right)=&\mathcal{N}\left(0, \alpha^{2} \mat{I}\right), &
p\left(\mat{Z}^{i}\right)=&\mathcal{N}\left(0, \alpha^{2} \mat{I}\right)
\end{align}
where $\alpha^{2}$ is \iadh{the} same hyperparameter for all the priors - we used the default setting of $\alpha=1$ in our paper, following~\cite{kingma2014auto}.
\ric{Consider past purchase} triples $(u,i,j)\in\set{T}$ that are sampled \ric{from historical} grocery shopping data~\cite{wan2018representing}. These sampled triples are \textit{positive examples} that should be precisely predicted according to the latent variables of users and items. We use $n+$ to denote the number of times that a given \iadh{triple} is observed in the \textit{total sample} $\set{T}$
Then $n+$ is a \textit{sufficient statistic} of the Skip-gram model, and it \iadh{contributes} to the likelihood $p(n+\mid \mat{z}^{u}_u,\mat{z}^{i}_i,\mat{z}^{i}_j)=\sigma^{n+}$~\cite{barkan2017bayesian}.
Thus, one needs to \ric{also} construct \ric{an associated} rejected triples set (i.e. $n-$, \textit{negative examples}) that are not in the total sample so that we can conduct \iadh{an} efficient \textit{negative sampling} for approximate optimization~\cite{mikolov2013distributed}. We let $n{\pm}=\{n+, n-\}$ be the combination of both positive and negative examples, then the likelihood of this complete purchase context is obtained by:
\begin{equation}
\log p\left(n{\pm} | \mat{Z}^{u}, \mat{Z}^{i}\right)=\sum_{(v_i, v_j, u_u) \in \mathcal{T}}\log\sigma + \sum_{(v_i, v_j, u_u) \notin \mathcal{T}}\log(1-\sigma),
\end{equation}
where $\sigma$ is calculated by the same function as in Triple2vec~\cite{wan2018representing} (i.e.\ $p((u, i, j)\mid \mat{z}^{u}_u,\mat{z}^{i}_i,\mat{z}^{i}_j)$ in Equation{} (\ref{eq:likelihood})).
\subsubsection{The Variational Evidence Lower Bound and Amortized Inference}
\label{sec:inference}
Since we assume that both $\mat{Z}^{u}$ and $\mat{Z}^{i}$ are random variables, \iadh{the} exact inference of their posterior density is intractable due to \iadh{the non-differentiable marginal likelihood} $p\left(n^{ \pm}\right)$~\cite{kingma2014auto}. Variational Bayes resolves this issue by constructing a tractable lower bound of the logarithm marginal likelihood and maximizing the lower bound instead~\cite{blei2017variational}. Following the Variational Autoencoding framework~\cite{kingma2014auto}, we also solve \iadh{this} problem by \iadh{introducing} the two variational distributions to formulate a tractable lower bound and \iadh{optimize} the lower bound by the Amortized Inference~\cite{shu2018amortized}. To infer the users' and items' embedding, we start by formulating the logarithm marginal likelihood of $n^{ \pm}$:
\begin{align}
\label{eq:elbo}
\log p\left(n^{ \pm}\right)=&\log\mathbb{E}_{q_{\phi}\left(\mat{Z}^{u},\mat{Z}^{i}\right)}\left[\frac{p\left(n^{ \pm},\mat{Z}^{u}, \mat{Z}^{i}\right)}{q_{\phi}\left(\mat{Z}^{u}, \mat{Z}^{i}\right)}\right]
\end{align}
\begin{align}
\nonumber \ge&\mathbb{E}_{q_{\phi}\left(\mat{Z}^{u}, \mat{Z}^{i}\right)}\left[\log\frac{p\left(n^{ \pm},\mat{Z}^{u}, \mat{Z}^{i}\right)}{q_{\phi}\left(\mat{Z}^{u}, \mat{Z}^{i}\right)}\right]\\\nonumber
=&\mathbb{E}_{q_{\phi}\left(\mat{Z}^{u}, \mat{Z}^{i}\right)}\left[\log p\left(n^{ \pm}\mid \mat{Z}^{u}, \mat{Z}^{i}\right)\right]\\\nonumber
&-KL\left(q_{\phi}\left(\mat{Z}^{u}, \mat{Z}^{i}\right)\|p(\mat{Z}^{u}, \mat{Z}^{i})\right)\\\nonumber
\overset{\underset{\mathrm{def}}{}}{=}&\set{L},
\end{align}
where the inequation of the second line is derived from the Jensen's inequality\iadh{;} $\set{L}$ is called the Evidence Lower BOund (ELBO) of the observed triple context~\cite{kingma2014auto}\iadh{;}
$KL(\cdot\|\cdot)$ is the Kullback-Leibler (KL) divergence and $q_{\phi}(\mat{Z}^{u}, \mat{Z}^{i})$ is the variational distribution, which can be factorized in a mean-field form:
\begin{align}
q_{\phi}(\mat{Z}^{u},\mat{Z}^{i})=q_{\phi_{1}}(\mat{Z}^{u})q_{\phi_{1}}(\mat{Z}^{i}),
\end{align}
where $\phi_{1}$ and $\phi_{2}$ are the trainable parameters of the inference models (encoders).
In order to get more expressive latent factors of users and items, we consider that the variational distributions $q_{\phi_1}(\mat{Z}^{u})$ and $q_{\phi_2}(\mat{Z}^{i})$ are Gaussian distributions and are encoded from the identity codes of users and items such that we have:
\begin{align}
q_{\phi_{1}}\left(\mat{Z}^{u}\mid \mat{F}^{u}\right)&=\mathcal{N}\left({\boldsymbol{\mu}}^{u}, \boldsymbol{\sigma}^{u2} \mathbf{I}\right),\\
q_{\phi_{2}}\left(\mat{Z}^{i}\mid \mat{F}^{i}\right)&=\mathcal{N}\left({\boldsymbol{\mu}}^{i}, \boldsymbol{\sigma}^{i2} \mathbf{I}\right),
\end{align}
where $\mat{F}^{u}\in \mathbb{R}^{N\times{F_1}}$ and $\mat{F}^{i}\in \mathbb{R}^{M\times{F_2}}$ are \iadh{the} identity representation (can be one-hot or binary encoded) of users and items respectively, with $F_1$ and $F_2$ being the dimension of their identity representation respectively, and $\boldsymbol{\mu}^{u}$, $\boldsymbol{\sigma}^{u2}$, $\boldsymbol{\mu}^{i}$ and $\boldsymbol{\sigma}^{i2}$ are inferred by the encoder networks. Specifically, the parameters of these Gaussian embeddings are encoded from their identity codes, i.e.\ $\mat{F}^{u}$ and $\mat{F}^{i}$, according to two two-layer fully-connected neural networks:
\begin{align}
\label{eq:inference1}
[\boldsymbol{\mu}^{u}, \boldsymbol{\sigma}^{u2} \mathbf{I}]=&\mat{W}^{u}_2\textbf{tanh}(\mat{W}^{u}_1\mat{F}^{u}+\mat{b}^{u}_1)+\mat{b}^{u}_2,\\
\label{eq:inference2}
[\boldsymbol{\mu}^{i}, \boldsymbol{\sigma}^{i2} \mathbf{I}]=&\mat{W}^{i}_2\textbf{tanh}(\mat{W}^{i}_2\mat{F}^{i}+\mat{b}^{i}_2)+\mat{b}^{i}_2,
\end{align}
\looseness -1 where \textbf{tanh} is the non-linearity activation function, \iadh{and} $\mat{W}^{u}_1$, $\mat{W}^{u}_2$, $\mat{b}^{u}_1$, $\mat{b}^{u}_2$, $\mat{W}^{i}_1$, $\mat{W}^{i}_2$, $\mat{b}^{i}_1$ \& $\mat{b}^{i}_2$ are trainable parameters of the neural networks.
Since we assume the priors and the variational posteriors are Gaussian distributions, the KL-divergence terms in Equation{} \iadh{(\ref{eq:elbo})} have analytical forms. By using the Stochastic Gradient Variational Bayes (SGVB) estimator and the reparameterization trick~\cite{kingma2014auto}, we can directly optimize \iadh{ELBO} by sampling deterministic and differentiable embedding samples from the \iadh{inferred variational} distributions:
\begin{align}
\mat{Z}^{u}&=\boldsymbol{\mu}^{u}+\boldsymbol{\sigma}^{u2} \odot \boldsymbol{\epsilon}^{(l)}, \boldsymbol{\epsilon}^{(l)}\sim \mathcal{N}\left(0, \mat{I}\right),\\\nonumber
\mat{Z}^{i}&=\boldsymbol{\mu}^{i}+\boldsymbol{\sigma}^{i2} \odot \boldsymbol{\epsilon}^{(l)}, \boldsymbol{\epsilon}^{(l)}\sim \mathcal{N}\left(0, \mat{I}\right),
\end{align}
to approximate and regularize maximum likelihood training, which is also \iadh{referred to as} \textbf{amortized inference}~\cite{shu2018amortized}.
\subsection{Recommendation Tasks}
\label{sec:reco_task}
Our model infers the embeddings of both users and items according to the variational auto-encoder and \iadh{represents} them by means of their variational Gaussian distributions. Since we have taken advantage of the basket information, having obtain\iadh{ed} the embedding of users and items \ric{in} our VBCAR{} \ric{model}, we can follow \iadh{a} similar \iadh{approach to}~\cite{wan2018representing} \ric{tackling both} next-basket product recommendation and within-basket product recommendation\ric{:}
\begin{enumerate}
\item Next-basket product recommendation: Recommending a given user $u$ with products for the next basket, we can obtain a preference score $s_{ui}=\textbf{dot} \left(\mat{z}_u^{u},\mat{z}_i^{i}\right)$\footnote{\textbf{dot} is the dot product for two vectors.} for each item $i$, then return the top-$K$ items with the highest preference scores.
\item Within-basket product recommendation: If \iadh{the} products in the current basket $b$ are given, we can first compute a preference score of item $i$ for user $u$ by: $s_{ui}=\textbf{dot}(\mat{z}_u^{u}+\sum_{i^{\prime}\in b}\mat{z}_i^{i^{\prime}},\mat{z}_i^{i})$, then return the top-$K$ preference \iadh{score} items as recommendation\iadh{s}.
\end{enumerate}
In this paper, we only evaluate the performance of our \iadh{model} based on the next-basket product recommendation and leave the evaluation of within-basket product recommendation for future work.
\section{Experiments}
\iadh{In the following, we first introduce the research questions we aim to answer in this paper (Section~\ref{rqs}). Next, we describe our experimental setup (Section~\ref{setup}), followed by our results and analysis (Section~\ref{results})}.
\subsection{Research \iadh{Q}uestions} \label{rqs}
\iadh{In this paper, we aim to answer the following two research questions:}
\begin{enumerate}[(\textbf{RQ}1)]
\item Can our \iadh{proposed} model outperform \iadh{the} Triple2vec model for grocery recommendation?
\item Can our Bayesian model learn more expressive representations of users and items than Triple2vec?
\end{enumerate}
\subsection{Experimental \iadh{Setup}} \label{setup}
\noindent\textbf{Dataset.}
We evaluate our model \iadh{using} the \textbf{Instacart}~\cite{wan2018representing} dataset, which is a public \iadh{large} grocery shopping \iadh{dataset} from the Instacart Online Grocery Shopping Website\footnote{\url{https://www.instacart.com/datasets/grocery-shopping-2017}}. This dataset contains over 3 million grocery orders and 33.8 million interactions from 0.2 million users and 50 thousand items. We first clean the dataset by filtering users and items \iadh{using a number of thresholds}. \ric{In particular,} users that have less than 7 orders or less than 30 items\ric{, as well as} items that were purchased by less than 16 users in the purchase history were removed. \iadh{Next, we} uniformly sample different percentages of users and items to construct different \iadh{sizes} of evaluation dataset. For model evaluation, we split all the sampled datasets into training (80\%) and testing (20\%) sets according to the temporal order of baskets. Table{} \ref{tab:dataset} shows the statistics of \ric{these} datasets.
\begin{table}[H]
\caption{Statistics of the datasets in \iadh{used in our} experiments.}
\vspace{-1em}
\label{tab:dataset}
\begin{tabular}{@{}l*{4}c@{}}
\toprule
\textbf{Percentage} & \textbf{\#Orders} & \textbf{\#Users} & \textbf{\#Items} & \textbf{\#Interactions}
\tabularnewline
\cmidrule{2-5}
\textbf{$5\%$} & 47,207 & 5,679 & 1,441 & 354,946
\tabularnewline
\textbf{$10\%$} & 154,285 & 11,888 & 3,124 & 1,103,361
\tabularnewline
\textbf{$25\%$} & 527,431 & 46,850 & 9,174 & 4,010,904
\tabularnewline
\textbf{$50\%$} & 1,186,957 & 59,549 & 16,121 & 12,217,555
\tabularnewline
\textbf{$100\%$} & 2,741,332 & 119,098 & 32,243 & 29,598,689
\tabularnewline
\bottomrule
\end{tabular}
\vspace{-0.5em}
\end{table}
\noindent\textbf{Baseline and Evaluation Metrics.}
\ric{To provide a} fair comparison, \ric{we use} Triple2vec~\cite{wan2018representing} as \ric{a state-of-the-art} baseline\ric{, since} Triple2vec \ric{incorporates} basket information \ric{in a similar way to our proposed VBCAR model.}
We evaluate the effectiveness of our model \ric{for next-basket (grocery) product recommendation}, where we evaluate the top-K items recommended by each model. We \ric{report the standard recommendation evaluation metrics Recall@K and NDCG@K}~\cite{NGCF19,he2017neural,wan2018representing} to \iadh{evaluate} the preference ranking performance. \ric{We report} result\iadh{s} \iadh{for} $K=10$ in our subsequent experiments\ric{, however we observed similar results when testing other} values of $K$ (e.g. 5 and 20)\ric{.}
\subsection{Result\iadh{s} and Analysis} \label{results}
\begin{table}[htp]
\caption{\zaiq{Overall performance on item recommendation. The best performing result is highlighted in bold; and ${}^*$ denotes a significant difference compared to the baseline result, according to the paired t-test p < 0.01.}}
\label{tab:over_perform}
\vspace{-1em}
\begin{tabular}{lcccc}
\toprule
\multirow{2}{*}{\textbf{Dataset}} & \multicolumn{2}{c}{\textbf{Triple2vec}} & \multicolumn{2}{c}{\textbf{VBCAR}} \\
\cmidrule(l){2-3} \cmidrule(l){4-5}
& NDCG@10 & Recall@10 & NDCG@10 & Recall@10 \\
\cmidrule{1-5}
\textbf{5\%} & \zaiq{0.557} & \zaiq{0.708} & \textbf{\zaiq{0.731${}^*$}} & \textbf{\zaiq{0.748${}^*$}} \\
\textbf{10\%} & \zaiq{0.558} & \zaiq{0.664} & \textbf{\zaiq{0.723${}^*$}} & \textbf{\zaiq{0.720${}^*$}} \\
\textbf{25\%} & \zaiq{0.626} & \zaiq{0.608} & \textbf{\zaiq{0.686${}^*$}} & \textbf{\zaiq{0.628${}^*$}} \\
\textbf{50\%} & 0.708 & 0.525 & \zaiqm{\textbf{0.719${}^*$}} & \zaiqm{\textbf{0.643${}^*$} }\\
\textbf{100\%} & 0.726 & 0.660 & \zaiqm{\textbf{0.768${}^*$}} & \zaiqm{\textbf{0.742${}^*$}} \\
\bottomrule
\end{tabular}
\vspace{-1em}
\end{table}
To answer \textbf{RQ1}, we evaluate our model as well as the triple2vec \iadh{baseline} on the task of item recommendation with \iadh{the} same size of triple samples (i.e.\ 1 million). Table{} \ref{tab:over_perform} shows the overall performance of our proposed model as well as \iadh{that of} the baseline method. For both the Triple2vec and our proposed VBCAR approach, we empirically set \iadh{the} embedding size to be 64 and train \iadh{both} models, with a batch size of 512 and a RMSprop optimizer. From Table{} \ref{tab:over_perform}, we can clearly see that our VBCAR{} model performs better than Triple2vec on all the datasets. This result suggests that our model can learn more expressive latent representations by integrating both non-linearity and \iadh{a} Bayesian \iadh{behaviour}.
\begin{figure}[t!]
\centering
\subfigure[NDCG@10 performance by different triple size]{\includegraphics[width = 2.2in]{sample_vs_ndcg.pdf}}\\
\subfigure[Recall@10 performance by different triple size]{\includegraphics[width = 2.2in]{sample_vs_recall.pdf}}
\vspace{-1em}
\caption{\label{fig:size_comp}Performance comparison for the various sample sizes on 5\% of the Instacart data.}
\end{figure}
To further validate this argument (\textbf{RQ2}), we also compare the recommendation performance of our VBCAR{} \iadh{model} \iadh{with} Triple2vec \iadh{using a} different number of triple sample\iadh{s}. Figure{} \ref{fig:size_comp} shows the NDCG@10 and Recall@10 \iadh{performances} \iadh{for} triple sample size\iadh{s} ranging from 50k to 500k. In \iadh{these experiments}, we set the embedding dimension \iadh{to} 64, \iadh{while the} other parameters \ric{for both models} are tuned to be optimal except \iadh{the} \ric{fixed} triple sample size. \ric{Again, w}e can clear\iadh{ly} observe that our VBCAR{} \iadh{model} outperforms Triple2vec in \iadh{terms of both metrics} and \iadh{on all} triple sample size\iadh{s}. \iadh{Moreover,} the gap between the performance of VBCAR{} \iadh{model} and Triple\iadh{2vec} is larger \iadh{on small sample sizes}. This result validates \iadh{our hypothesis} that our VBCAR{} model can learn more expressive latent representations with limited input samples.
\section{Conclusions}
In this paper, we have proposed the VBCAR \iadh{model}, a variational Bayesian context-aware representation model for grocery recommendation. Our model was built based on the variational Bayesian Skip-gram framework coupled with the amortized inference. Experimental result\iadh{s} on \iadh{the} Instacart dataset \iadh{show} that our VBCAR \iadh{model} can learn more expressive representations of users and items than Triple2vec and \iadh{does} significantly outperform Triple2vec \ric{under} \iadh{both the} NDCG and Recall \iadh{metrics}. Indeed, we observe up to a $31\%$ increase in recommendation effectiveness over Triple2vec (under NDCG@10). For future work, we plan to extend our model to infer latent representations for new users and new items by taking the side information \iadh{about} users and items into account.
\vspace{-0.5em}
\subsection*{Acknowledgements}
\zaiqm{The research leading to these results has received funding from the European Community's Horizon 2020 research and innovation programme under grant agreement $\text{n}^{\circ}$ 779747.}
\vspace{-0.5em}
\bibliographystyle{ACM-Reference-Format}
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