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+{"text":"\\section{Introduction}\n\nThe basic structure of a pulsar wind nebula (PWN) is determined by the\nspin-down energy injected by the central pulsar and the interaction\nof the nebula with the interior regions of the supernova remnant\n(SNR) in which it evolves. Losses from synchrotron radiation in the\nnebular magnetic field, whose strength depends both on the nature\nof the injected wind and on the evolving size of the PWN, inverse-Compton\nscattering of ambient photons by the energetic electron population\nwithin the nebula, and adiabatic expansion as the nebula sweeps up\nthe surrounding supernova ejecta, all combine to determine emission\nstructure and long-term evolution of the nebula. (See Gaensler \\&\nSlane 2006 for a review.) Multiwavelength observations of PWNe\nprovide crucial information on the underlying particle spectrum\nwhich, in turn, strongly constrains both the magnetic field strength\nand the stage of evolution. Of particular importance is the spectrum\nof particles injected into the PWN. While this is typically assumed\nto be a power law, it has long been known that the spectrum of the\nCrab Nebula is not fully consistent with such a structure; changes\nin the spectral index in the optical band seem to indicate inherent\nfeatures in the injection spectrum, and the large population of\nradio-emitting particles may have a completely distinct origin from\nthat of the higher energy particles.\n\nComplex structure in the observed spectrum of a PWN can originate\nin a number of ways that are associated with the long-term evolution,\nincluding synchrotron losses and interaction with the SNR reverse\nshock (Reynolds \\& Chevalier 1984; Gelfand, Slane, \\& Zhang 2009).\nIn addition, recent studies of the spectrum immediately downstream of\nthe wind termination shock show that, at least in some cases, the\ninjection spectrum itself deviates significantly from a simple power\nlaw (Slane et al.  2008), and particle-in-cell simulations of the\nacceleration process produce spectra that are well-described by a\nMaxwellian population with a power law tail (Spitkovsky 2008).  Any\nsuch structure in the spectrum of the injected particles imprints\nitself on the broadband emission spectrum of the entire nebula. The\nresulting breaks or regions of curvature in the emission spectrum,\nand the frequencies at which they are observed, depend upon the\nenergy at which features appear in the electron spectrum as well\nas the means by which the photons are produced (e.g. synchrotron\nradiation or inverse-Compton emission).  To fully understand the\nnature of the particle injection, as well as the long-term evolution\nof PWNe, it is thus crucial to study the emission structure over\nthe entire electromagnetic spectrum.\n\nGamma-ray observations have provided an important window into the\nlate-phase structure of PWNe. While the X-ray emission from older\nPWNe is relatively weak due to the long-term decline in magnetic\nfield strength, inverse-Compton scattering of the cosmic microwave\nbackground (CMB), as well as other ambient photons, by the energetic\nparticles in the nebula produce very energetic photons, providing\nunique discovery space for these systems. Even in younger nebulae,\nthe $\\gamma$-ray emission provides a crucial probe of the particle\nspectrum in energy regions that are not accessible with other\nobservations. The inverse-Compton emission from photons with a\ncharacteristic temperature $T$ peaks at an energy $\\epsilon_{ic}\n\\approx 5 \\gamma^2 kT$ for scattering from electrons of energy\n$\\gamma m_e c^2$.  CMB photons scattered into the bandpass of the\n{\\em Fermi}\\ Large Area Telescope (LAT), for example, originate from\ninteractions with electrons in the approximate energy range 0.25\n-- 4 TeV. Such particles produce synchrotron radiation with photon\nenergies in the range $\\epsilon_{s} \\approx 0.03 - 7 B_{10}$~eV,\nwhere $B_{10}$ is the magnetic field strength in units of 10$\\mu$G.\nDepending on the magnetic field, which can range from $>100 \\mu$G\nor higher for young PWNe, down to $\\sim 5 \\mu$G for highly evolved\nsystems, the synchrotron emission from particles in this energy\nrange can be difficult to detect due to instrumentation limitations\n(at long radio wavelengths), absorption (in the optical\/UV range)\nor confusion with the bright sky Galactic background (in the\ninfrared).  {\\em Fermi}\\ measurements can thus provide a unique probe\nof the emission from a significant population of the particle\nspectrum in PWNe.\n\nHESS J1640$-$465\\ (see Figure 1) is an extended source of very high energy\n$\\gamma$-ray emission discovered with the High Energy Stereoscopic\nSystem (H.E.S.S.) during a survey of the Galactic plane (Aharonian\net al.  2006).  Centered within the radio SNR G338.3$-$0.0 (Shaver\n\\& Goss 1970), the deconvolved TeV image of the source has an RMS\nwidth of $2.7 \\pm 0.5$~arcmin (Funk et al. 2007). HI measurements\nshow absorption against G338.3$-$0.0 out to velocities corresponding\nto the tangent point, indicating a distance of at least 8 kpc\n(Lemiere et al. 2009), and thus implying a rather large size for\nthe PWN ($R_{PWN} > 6.4 d_{10} $~pc, where $d_{10}$ is the distance\nin units of 10~kpc).  X-ray observations with {\\em XMM}\\ (Funk et al.\n2007) and Chandra (Lemiere et al. 2009) establish the presence of\nan accompanying X-ray nebula and an X-ray point source that appears\nto be the associated neutron star. In addition, Aharonian et al.\n(2006) noted the presence of the unidentified {\\em EGRET} source\n3EG~J1639$-$4702 located at a nominal position 34$^\\prime$ from\nHESS J1640$-$465, but the very large error circle on its position made any\nassociation with HESS J1640$-$465\\ highly uncertain.  Here we report on\nobservations of HESS J1640$-$465\\ with the {\\em Fermi}-LAT.  The observations and\ndata analysis are summarized in Section 2, and a discussion of the\nobserved $\\gamma$-ray emission is discussed in the context of the\nevolutionary state of HESS J1640$-$465\\ in Section 3. Our conclusions are\nsummarized in Section 4.\n\n\\begin{figure}[t]\n\\epsscale{1.15}\n\\plotone{f1.ps}\n\\epsscale{1.0}\n\\caption{\n{\\em Fermi}\\ LAT image (2 - 200 GeV) of HESS J1640$-$465. The cyan circle indicates\nthe uncertainty in the centroid of the {\\em Fermi}\\ LAT source, the\nmagenta circle indicates the 95\\% encircled flux distribution of\nthe HESS image, and the white circle indicates the 95\\% probability\ncontour for the position of 3EG J1639$-$4702. The white contours\noutline radio emission from G338.3$-$0.0 while the black contours\nat the center outline extended X-ray emission observed with {\\em XMM}.\nA compact X-ray source detected with {\\em Chandra}\\ resides within the\nX-ray contours.\n}\n\\end{figure}\n\n\n\n\\section{Observations and Data Analysis}\n\nWe investigated $\\gamma$-ray events acquired from the region\nsurrounding HESS J1640$-$465\\ with the {\\em Fermi}-LAT during the period 2008 August\n5 to 2009 November 13. Standard event selection was applied, using\nthe ``diffuse'' class as defined in Atwood et al. (2009), using\nevents with zenith angles less than 105 degrees to minimize the\nportion of the Earth limb in the LAT field-of-view (Abdo et al.\n2009a). The ``Pass6 version 3'' instrument response functions were\nused.  Standard analysis tools available from the {\\em Fermi}\\ Science\nSupport Center (version v9r15p2) were used for the reduction and\nanalysis.  In this work, the mapcube file gll\\_iem\\_v02.fit is used\nto describe the Galactic $\\gamma$-ray emission, and the isotropic\ncomponent is modeled using the isotropic\\_iem\\_v02.txt table.  Data\nanalysis details follow those in Castro \\& Slane (2010).\n\nFor source detection and spatial analysis of the source, we used\nonly events with energies in the range 2 - 200 GeV converting in\nthe front section of the tracker, where the Point Spread Function\n(PSF) is narrower. At 2 GeV, the 68\\% containment radius of the PSF\nis approximately 18 arcmin, which is considerably larger than the\nsource of interest.  Based on an unbinned maximum likelihood analysis,\nusing the {\\tt gtlike} routine, a LAT source coincident with HESS J1640$-$465\\\nis detected with a significance of $\\sim 17 \\sigma$ based on the\nTS statistic (Mattox et al. 1996).  The counts map is shown in\nFigure 1, along with the uncertainty in the centroid position based\non our analysis (cyan circle). We have also included the HESS error\ncircle (magenta) and contours from the {\\em XMM}\\ observation of HESS J1640$-$465\\\n(black) and the MOST observation of G338.3$-$0.0 (white contours). The error\ncircle for 3EG~J1639$-$4702 is indicated by a dashed white\ncircle.\\footnote{EGRET position contours are not necessarily circular;\nthe radius of the contour shown, from Hartman et al. (1999), contains\nthe same solid angle as the formal 95\\% contour.} The centroid of\nthe LAT emission is located at $16^{\\rm h} 40^{\\rm m} 46^{\\rm s}$,\n$-46^\\circ 30^\\prime 44^{\\prime\\prime}$, in good agreement with the\nposition of HESS J1640$-$465, and the brightness distribution is consistent\nwith an unresolved source (Figure 2); models for a disk of extent\nbetween $0.05 - 0.2$~degrees significantly degrade the quality of\nthe fit, providing strong evidence for an unresolved source.\n\n\\begin{figure}[t]\n\\epsscale{1.20}\n\\plotone{f2.ps}\n\\epsscale{1.00}\n\\caption{\nProfile of the {\\em Fermi}\\ LAT emission from HESS J1640$-$465. The histogram\ncorresponds to the best-fit point source profile, including the\ndiffuse Galactic and extragalactic background which is shown separately\nin red.\n}\n\\end{figure}\n\n\nThe source spectrum was extracted using both front and back events,\nand covering the energy range $0.2 - 51.2$~GeV.  Standard background\nmodels were used to account for both Galactic and extragalactic\ndiffuse emission as well as instrumental background.  Contributions\nfrom field sources identified in the one-year {\\em Fermi}-LAT First\nSource Catalog (Abdo et al. 2010a) were included in the analysis.\nThe LAT spectrum for HESS J1640$-$465\\ is shown in Figure 3, and is well-described\nby a power law with $\\Gamma = 2.30 \\pm 0.09$ and a $F(>100{\\rm\\\nMeV}) = (2.8 \\pm 0.4) \\times 10^{-7}{\\rm\\ photons\\ cm^{-2}}{\\rm\\\ns}^{-1}$ [or $(1.7 \\pm 0.2)) \\times 10^{-10}{\\rm\\ erg\\ cm^{-2}\\\ns}^{-1}$ in the $0.1 - 300$~GeV band] based on spectral fits for\nwhich statistical error and systematic errors associated with\nuncertainties in the Galactic background level (estimated by\nartificially changing the Galactic background normalization by $\\pm\n3\\%$ -- see Abdo et al. 2009b) are added in quadrature.\n\nWe note that {\\em Fermi}\\ observations of pulsars reveal spectra with\ndistinct cutoffs above $\\sim 1 - 10$~GeV (e.g. Abdo et al. 2009).\nThe absence of such a cutoff in Figure 3 implies that that the bulk\nof the emission does not arise directly from an unseen pulsar in\nHESS J1640$-$465.  Our fits imply a lower limit of 40~GeV (at the 90\\% confidence\nlevel) for any exponential cutoff energy in the spectrum.  Joint\nfits with the HESS spectrum are also well-described by a power law.\nAddition of a second power law, with an exponential cutoff, does\nnot improve the fit. Such a fit can, however, accommodate $\\sim\n20\\%$ of the observed flux in the second power law for cutoff\nenergies between $1 - 8$~GeV (beyond which such a component is\nstatistically disfavored).\n\n\\begin{figure}[t]\n\\epsscale{1.2}\n\\plotone{f3.eps}\n\\epsscale{1.0}\n\\caption{\n{\\em Fermi}\\ LAT spectrum of HESS J1640$-$465. Statistical (systematic) uncertainties are\nindicated by solid (dashed) error bars. The dashed line corresponds to\nthe best-fit power law model described in the text.\n}\n\\end{figure}\n\n\nThe source 1FGL~J1640.8$-$4634, from the one-year catalog, is\ncoincident with the source we detect, and the quoted flux is in\nagreement with our measurements as well.  The flux of the source\nis consistent with that of 3EG~J1639$-$4702; the spectral index is\nsomewhat flatter, though also consistent within the uncertainties.\n\n\\section{Discussion}\n\nThe evolutionary state of a composite supernova remnant system is\nstrongly constrained by the observed size of the SNR and the inferred\nspin-down properties of the associated pulsar. Radio observations\nestablish a radius $R_{SNR} \\sim 11.6 d_{10}$~pc for G338.3$-$0.0.  The\nobserved extent of HESS J1640$-$465\\ constrains the radius of the PWN to $R_{PWN}\n> 6.4 d_{10}$~pc.\\footnote{Here, and throughout, $R_{SNR}$ refers to\nthe distance from the SNR center to the outer edge of its blast wave,\nwhile $R_{PSR}$ refers to the distance from the pulsar to the outer\nboundary of its wind nebula.}\nFor evolution in the Sedov phase, the SNR radius\nis \n\\begin{equation} R_{SNR} = 4.9 \\left(\\frac{E_{51}}{n_0}\\right)^{1\/5}\nt_3^{2\/5}{\\rm\\ pc} \\end{equation} \nwhere $E_{51}$ is the supernova explosion energy in units of\n$10^{51}$~erg, $n_0$ is the number density of the ambient medium,\nand $t_3$ is the SNR age in kyr. The evolution of the PWN radius\nthrough the SNR ejecta is given by (Chevalier 1977) \n\\begin{equation}\nR_{PWN} \\sim 1.1 \\dot{E}_{0,38}^{1\/5} E_{51}^{3\/10} M_{10}^{-1\/2}\nt_3^{6\/5} {\\rm\\ pc} \\end{equation} \nwhere $\\dot{E}_{0,38}$ is the initial spin-down power in units of\n$10^{38}{\\rm\\ erg\\ s}^{-1}$ and $M_{10}$ is the ejecta mass in units\nof $10 M_\\odot$. Figure 4 illustrates the evolution of the SNR and\nPWN radii under such assumptions for a range of values for the\nambient density and initial spin-down power, assuming an ejecta\nmass of $8 M_\\odot$ and $E_{51} = 1$. For the observed radius of\nG338.3$-$0.0, we see that the age must be around $5 - 8$~kyr for reasonable\nambient conditions.  The PWN has ideally expanded to a larger radius\nby this time, which means that under real conditions the SNR reverse\nshock has already encountered and disrupted the PWN.  This is\nillustrated in Figure 4 where the solid blue and red curves show\nthe SNR and PWN radii (respectively) as a function of time for the\nmodel shown in Gelfand et al. (2009) with $\\dot{E_0} = 10^{40} {\\rm\\\nerg\\ s}^{-1}$, $M_{ej} = 8 M_\\odot$, $n_0 = 0.1 {\\rm\\ cm}^{-3}$,\nand $E_{51} = 1$.  Here the early SNR evolution is calculated using\nthe solution from Truelove \\& McKee (1999), and the SNR and PWN\nevolution is treated self-consistently.  The SNR curve initially\nrises more quickly than the dashed blue curves, which assume a Sedov\nsolution from the onset, but approaches the Sedov solution at later\ntimes. The solid red curve shows a distinct reduction in the radius\nof the PWN upon compression by the SNR reverse shock. Such a reverse\nshock interaction is consistent with both the inferred size of HESS J1640$-$465\\\nand with the observed offset between the putative pulsar and the\nsurrounding nebula (Funk et al.  2007, Lemiere et al. 2009).\n\nA full exploration of the parameter space constrained by both the\nspatial and spectral properties of HESS J1640$-$465\\ and G338.3$-$0.0\\ is beyond the scope of\nthis initial investigation. Here we have modeled the PWN emission\nfollowing the description in Lemiere et al. (2009). A simple power\nlaw injection of particles from the pulsar into a 1-zone nebula is\nassumed, and radiative losses are assumed to be dominated by\nsynchrotron radiation.\\footnote{This assumption is valid for magnetic\nfields larger than $\\sim 1 \\mu$G, which is typically the case in\nPWNe. We note, however, that at late times (ages of order 10 kyr,\nas suggested for HESS J1640$-$465), when the magnetic field\ndecreases due to expansion, IC losses can begin to be significant.}  The\ntime evolution of the pulsar spin-down is determined by \n\\begin{equation} \\dot{E}(t) = \\dot{E_0} \\left(1 +\nt\/\\tau_0\\right)^{-\\frac{b+1}{b-1}} \\end{equation} \nwhere $\\tau_0$ is a characteristic spin-down timescale for the\npulsar and $b$ is the pulsar braking index. Pulsations from the\nputative pulsar have not yet been detected. To estimate the current\nspindown power, we use the empirical relationship for $\\dot{E}\/L_x$\nobtained by Possenti et al.  (2002), which yields $\\dot{E} = 4\n\\times 10^{36} {\\rm\\ erg\\ s}^{-1}$.  Based on the observed scatter\nin the $\\dot{E}\/L_x$, the uncertainty on this estimated value may\nbe a factor of 10 or larger.  The maximum particle energy is\nassumed to be limited by the condition that the particle gyroradii\ndo not exceed the termination shock radius (de Jager \\& Harding\n1992).  The lower limit on the particle energy is a free parameter,\nand the overall normalization is set by the wind\nmagnetization parameter $\\sigma$, equal to the ratio of power in\nparticles to that in the Poynting flux. The magnetic field in the\nnebula is assumed to evolve as \n\\begin{equation} B(t) = B_0\/\\left[\n1 + (t\/\\tau_0)^\\alpha\\right] \\end{equation} \nwhere $\\alpha$ is a\nfree parameter.\n\n\\begin{figure}[t]\n\\epsscale{1.2}\n\\plotone{f4.ps}\n\\epsscale{1.0}\n\\caption{\nTime evolution of the SNR and PWN radii for a range of values for\nthe ambient density and initial spin-down power of the pulsar. The\nsolid curves correspond to models from Gelfand et al. (2009) using\n$\\dot{E_0} = 10^{40} {\\rm\\ erg\\ s}^{-1}$, $M_{ej} = 8 M_\\odot$, $n_0\n= 0.1 {\\rm\\ cm}^{-3}$, and $E_{51} = 1$.  See text description for\ndetails.\n}\n\\end{figure}\n\n\nThe broadband emission model results are shown in Figure 5 where\nwe plot the {\\em Fermi}\\ and H.E.S.S spectra along with the radio upper\nlimit from GMRT observations (Giacani et al. 2008) and spectral\nconfidence bands derived from Chandra (Lemiere et al. 2009).  The\nblack curves represent the model prediction for the synchrotron\n(left) and inverse-Compton (right) emission that best describes the\nX-ray and TeV $\\gamma$-ray spectra, similar to results from Lemiere\net al. (2009); the parameters for the model, which were adjusted ``by\nhand'' to provide good agreement with the radio, X-ray, and TeV\n$\\gamma$-ray data, as well as the inferred size of the PWN, \nare summarized in the\ncaption.  As seen in Figure 5, this model significantly underpredicts\nthe observed {\\em Fermi}-LAT emission.\n\n\\begin{figure}[t]\n\\epsscale{1.2}\n\\plotone{f5.ps}\n\\epsscale{1.0}\n\\caption{\nElectron spectrum (upper) and broadband emission model (lower) for\nHESS J1640$-$465\\ assuming the evolutionary history described in the text. The\nblack curves represent a PWN with an age $T = 10$~kyr, and $B(T) =\n5 \\mu$G, assuming $\\dot{E}(T) = 4 \\times 10^{36} {\\rm\\ erg\\ s}^{-1}$\nand an injection spectrum with $\\sigma = 10^{-3}$, $\\gamma = 2.5$,\n$\\tau_0 = 500$~yr, and $E_{\\rm min} = 115$~GeV. The magnetic field\nevolution is characterized by $\\alpha = 0.65$. The magenta curves\nrepresent the scenario with a low-energy Maxwellian electron component\nreplacing the low-energy portion of the electron power-law spectrum.\nThe mean temperature for the IR and optical photon fields are 15~K\nand 5000~K, respectively, and the energy densities relative to the\nCMB are 4 and 1.15. The dashed curve in the upper panel represents\nthe truncated portion of the power law that was replaced by a\nMaxwellian. The dashed blue curve in the lower panel represents a\nmodel for which all of the $\\gamma$-ray emission results from pion\ndecay.\n}\n\\end{figure}\n\n\nAs discussed above, our spectral fits can formally accommodate up\nto about $\\sim 20\\%$ of the observed flux in a pulsar-like component\ncharacterized by a power law with an exponential cutoff energy\nbetween 1 and 8 GeV. This corresponds to an energy flux (above\n100~MeV) of $\\sim 3.8 \\times 10^{-11}{\\rm\\ erg\\ cm}^{-2}{\\rm\\\ns}^{-1}$. For the spin-down power suggested by its X-ray luminosity,\nthe available energy flux from the pulsar that powers HESS J1640$-$465\\ is $\\sim\n3.3 \\times 10^{-10} d_{10}^{-2}{\\rm\\ erg\\ cm}^{-2}{\\rm\\ s}^{-1}$.\nThus, as much as 10\\% of its spin-down could conceivably be\ncontributing directly to $\\gamma$-rays in the LAT band. There are\nknown radio pulsars within the field of 1FGL~J1640.8$-$4634 as well.\nPSR~J1637$-$4642, for example, is located within $\\sim 37$~arcmin\nand has a total energy flux of $2 \\times 10^{-10} {\\rm\\ erg\\\ncm}^{-2}{\\rm\\ s}^{-1}$ given its estimated distance and measured\nspin-down. Thus, while our spectral fits do not require a pulsar-like\ncomponent, it it quite feasible that one or more of these sources\ncontributes as much as 20\\% of the observed flux. Inspection of\nFigure 5 makes it clear, however, that the remaining LAT emission\nstill vastly exceeds the predicted flux from HESS J1640$-$465.\n\nWe note that simple power-law injection models for Vela X, another\nevolved PWN, fail to reproduce the observed broadband spectrum\n(LaMassa, Slane, \\& de Jager 2009). The presence of an excess\npopulation of low-energy electrons has been suggested, and models\nfor the inverse-Compton scattering of photons from this population\npredict an excess of $\\gamma$-rays in the GeV range (de Jager,\nSlane, \\& LaMassa 2009). This excess has been confirmed with\nobservations by both AGILE (Pellizzoni et al. 2010) and {\\em Fermi}\\\n(Abdo et al. 2010b).  Motivated by this result, we modified the\nevolved power law spectrum from our model for HESS J1640$-$465\\ by truncating\nthe lower end of the power law and adding a distinct low-energy\ncomponent. Based on results from simulations of shock acceleration\n(Spitkovsky 2008), we chose a Maxwellian distribution for this\npopulation.  Our resulting (ad hoc) particle spectrum is shown in\nthe upper panel in Figure 5, and the resulting broadband emission\nis shown in the magenta curves in the lower panel. Here we have\nadjusted the normalization of the Maxwellian to reproduce the\nemission in the {\\em Fermi}-LAT band, which is produced primarily by\nupscattered infrared (IR) photons from local dust. The energy density\nand mean temperature of the IR photon field was adjusted slightly\nto improve the agreement between the data and the model, but the\nvalues (listed in the caption) are within reasonable expectations\n(see, e.g., Strong et al. 2000). We find a mean value of $\\gamma\n\\approx 2 \\times 10^5$ for the electrons in the Maxwellian component,\nand roughly 5\\% of the total electron energy in the power law tail,\nconsistent with results from particle-in-cell simulations.\\footnote{A.\nSpitkovsky, private communication.} The associated pair multiplicity\nrelative to the integrated Goldreich-Julian injection rate is of\norder $10^6$, similar to that inferred for the Crab Nebula as well\nas several other PWNe (see Bucciantini et al. 2010).  Recent work\nby Fang \\& Zhang (2010) uses a similar input distribution to\nsuccessfully model the emission for several PWNe including HESS J1640$-$465.\nHowever, their results for HESS J1640$-$465\\ underpredict the observed GeV\nemission from this source, apparently due to use of a slightly lower\nbulk Lorentz factor and a larger fraction of the total energy in\nthe power law tail than we have used in this analysis.\n\nWe note that the Maxwellian shape for the low-energy electron\npopulation is not unique. Indeed even a very narrow Gaussian\ndistribution can produce the GeV $\\gamma$-ray emission without\nexceeding the radio upper limit for the synchrotron emission. For\nany of these models, the total energy in electrons requires a larger\ninitial spin-down period than assumed for the simple power-law\ninjection models -- by a factor of roughly 3 for the adopted\nMaxwellian distribution. This is within the uncertainty in our\nassumed value based on scaling from the X-ray luminosity of the\nPWN. It is also important to note that while our proposed model is\nconsistent with the observed properties of this system, there are\npotential degeneracies in the effects of different model parameters,\nmeaning that the proposed scenario is not necessarily unique.\n\n\nAn alternative scenario for the $\\gamma$-ray emission is that it\narises from the SNR itself, and not the PWN.  Nonthermal bremsstrahlung\nhas been suggested as a mechanism for the production of $\\gamma$-rays\nin SNRs (e.g. Bykov et al. 2000). This process can be dominant for\nelectron-to-proton ratios of order 1.  However, the value typical\nof local cosmic rays is closer to $10^{-2}$, and even smaller values\nappear favored in models for $\\gamma$-ray emission from SNRs (e.g.\nMorlino et al. 2009, Zirakashvili \\& Aharonian 2010, Ellison et al.\n2010), so that this process is typically not dominant.  The dashed\nblue curve in Figure 5 represents a model for the emission from the\ncollision of protons accelerated in the SNR with ambient material,\nleading to $\\gamma$-rays from the production and subsequent decay\nof neutral pions. The $\\gamma$-ray spectrum is calculated based on\nKamae et al. (2006) using a scaling factor of 1.85 for helium and\nheavy nuclei (Mori 2009), and we have used a power law distribution\nof protons with $dN_p\/dE \\propto E^{-2.4}$ to best reproduce the\nobserved spectrum. Assuming a shock compression ratio of 4 and that\n25\\% of the total supernova energy appears in the form of relativistic\nprotons, an ambient density $n_0 \\approx 100 {\\rm\\ cm}^{-3}$ is\nrequired to produce the model shown in Figure 5. This is much higher\nthan can be accommodated for the observed size of the SNR and the\nlack of observed thermal X-ray emission from the SNR.  Such high\ndensities are found in dense molecular clouds, suggesting that the\n$\\gamma$-rays could be produced by particles that stream away to\ninteract with high-density material outside the SNR. However, only\nthe most energetic particles can escape the acceleration region,\nwhich is in conflict with the proton spectrum we require to match\nthe data. Moreover, the observed TeV emission appears to originate\nfrom within the SNR boundaries, making such an escaping-particle\nscenario appear problematic.  Based on this, along with the lack\nof a spectral cutoff that might suggest emission from a central\npulsar, we conclude that the GeV $\\gamma$-ray emission most likely\narises from the PWN.\n\nIt is well-known that a distinct low-energy electron population\nresides in the Crab Nebula, although the origin is not well-understood.\nAtoyan (1999) has suggested that the spin-down timescale $\\tau_0$\n(see Equation 1) may itself be time-dependent, resulting in a large\nenergy input in the earliest epoch of pulsar spin-down, when\nsignificant synchrotron and adiabatic losses would result in rapid\ncooling of the electrons. Studies of the broadband emission from\n3C~58 indicate an injection spectrum that differs from a pure power\nlaw as well (Slane et al. 2008). As noted above, observations of\nVela~X appear to require an excess of low-energy electrons which\nmay be a relic population or could have been produced through rapid\nsynchrotron losses associated with the increased magnetic field\nstrength during the reverse-shock crushing stage.  More complete\nmodeling of the broadband emission of HESS J1640$-$465, accounting for the full\ndynamical evolution of the system, including the effects of the\nreverse shock on the PWN, is required to assess the overall energetics\nand underlying particle spectrum more completely, but is beyond the\nscope of the results we report here.\n\n\\section{Conclusions}\n\nBroadband studies of HESS J1640$-$465\\ have identified this source as a likely\nPWN, with X-ray observations providing images of an extended nebula\nas well as the putative pulsar powering the system. Modeling of the\nPWN evolution based on inferred parameters of the pulsar imply\ndetectable emission in the GeV $\\gamma$-ray band, and our investigations\nof the {\\em Fermi}-LAT observations of this region reveal clear evidence\nof such emission, consistent with the source 1FGL~J1640.8$-$4634. \nThe flux and spectrum we derive are consistent with that\nof the previously-identified source 3EG~J1639$-$4702, and the much-improved\nposition makes it likely that the emission arises from HESS J1640$-$465. The\nlack of a spectral cutoff rules out an association with emission\ndirectly from the pulsar that powers the nebula, and the flux is\ninconsistent with $\\gamma$-rays from G338.3$-$0.0, in which HESS J1640$-$465\\ \nresides.\n\nWe have investigated the radio, X-ray, and $\\gamma$-ray emission\nfrom HESS J1640$-$465\\ in the context of a simple one-zone model in which power\nis injected into a nebula at a time-dependent rate consistent with\nthe current observed X-ray emission from the system. We find that\nmodels constrained by the observed size of the associated SNR, as\nwell as limits on the size of the PWN, require an approximate age\nof 10~kyr and a current magnetic field strength of only $\\sim 4\n\\mu$G, consistent with expectations for the late-phase evolution\nof a PWN. The conditions in such an evolved PWN yield a considerably\nhigher $\\gamma$-ray flux, relative to the X-ray flux, than in younger\nsystems where the higher magnetic field results in significant\nsynchrotron radiation.\n\nThe observed {\\em Fermi}-LAT emission from HESS J1640$-$465\\ significantly exceeds\nthat predicted by our broadband models. We propose that the excess\nemission is a signature of a distinct population of low-energy\nelectrons similar to that inferred from studies of the Crab Nebula\nand Vela~X, although the nature of this electron component is not\nwell constrained. Deeper radio observations are needed to place\nstronger constraints on this population. Sensitive searches for\npulsations from the central pulsar are of particular importance to\nconstrain the spin-down properties of the system, which can only\nbe very roughly constrained at present. There has been considerable\nsuccess in identifying such pulsars with the {\\em Fermi}-LAT, but the\nlack of an obvious pulsar-like spectrum in HESS J1640$-$465\\ may argue for more\nlikely success with deep radio timing searches.\n\n\\acknowledgments\n\nThe work presented here was supported in part by NASA Contract\nNAS8-39073 (POS) and {\\em Fermi}\\ Grant NNX09AT68G.  JDG is supported\nby an NSF Astronomy and Astrophysics Postdoctoral Fellowship under\naward AST-0702957.  POS, SF, and YU are grateful to the KITP in\nSanta Barbara, where elements of the work presented here were first\ndiscussed during a KITP program. The authors would like to thank\nDon Ellison, Luke Drury, Felix Aharonian, and David Smith for helpful\ndiscussions during the preparation of this manuscript.\n\nThe $Fermi$ LAT Collaboration acknowledges support from a number\nof agencies and institutes for both development and the operation\nof the LAT as well as scientific data analysis. These include NASA\nand DOE in the United States, CEA\/Irfu and IN2P3\/CNRS in France,\nASI and INFN in Italy, MEXT, KEK, and JAXA in Japan, and the\nK.~A.~Wallenberg Foundation, the Swedish Research Council and the\nNational Space Board in Sweden. Additional support from INAF in\nItaly and CNES in France for science analysis during the operations\nphase is also gratefully acknowledged.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\n\nStar formation is ultimately controlled by the processes that regulate the formation of density enhancements in molecular clouds. In our current picture, the density statistics of the interstellar medium are heavily affected by supersonic turbulence \\citep[for a review, see][]{hen12}. \nThe density statistics depend on characteristics such as the total turbulent and magnetic energy \\citep[e.g.,][FK13 hereafter]{pad97mnras, nor99, vaz01, kow07, mol12, fed13}, the driving mechanism of the turbulence \\citep[e.g.,][FK12, hereafter]{fed10, fed12}, the equation of state \\citep[e.g.,][]{pas98, gaz13}, and the driving scale \\citep[e.g.,][]{fis04, bru09}. Constraining these characteristics is fundamental for virtually all analytic star formation theories.\n\n\nWe have previously employed near-infrared dust extinction mapping in analyzing column density statistics of molecular clouds \\citep[][KT13 hereafter]{kai09, kai11a, kai11b, kai13}. This technique is sensitive and well-calibrated at low column densities, making it suitable to study the mass reservoirs of molecular clouds. Exploiting this advantage, we studied how the clouds gather gas to the regime where star formation occurs. We used an easily accessible characteristic to quantify this, namely the dense gas mass fraction\\footnote{We purposefully use here the term \"dense gas mass fraction\" instead of \"cumulative mass function\" (CMF) from our previous works. This is to avoid confusion with the \"core mass function\" that is commonly used in literature.} (DGMF, hereafter), defined as a function that gives the fraction of the cloud's mass above a column density value\n\\begin{equation}\n\\mathrm{d}M' (> N) = \\frac{M(> N)}{M_\\mathrm{tot}},\n\\label{eq:dgmf}\n\\end{equation}\nwhere $M(> N)$ is the mass above the column density $N$ and $M_\\mathrm{tot}$ is the total mass. The DGMF is linked to the probability density function (PDF), $p(N)$, of column densities, which gives the probability to have a column density between $[N, N+dN]$, via\n\\begin{equation}\ndM' = \\int_N^{N_\\mathrm{high}} p(N') dN' \/ \\int_{N_\\mathrm{low}}^{N_\\mathrm{high}} p(N') dN',\n\\label{eq:dgmf-pdf}\n\\end{equation}\nwhere $[N_\\mathrm{low}, N_\\mathrm{high}]$ is the probed column density range. The reason for analyzing DGMFs instead of PDFs is simply the intuitive connection to the total mass reservoir of the cloud. Previously, DGMFs have been analyzed by, e.g., \\citet{kai09} who showed that starless clouds contain much less dense gas than star-forming clouds and by \\citet{lad10} who used them to derive a star-formation threshold.\n\nFrom the theoretical point-of-view, the form of the DGMF can be controlled by any of the forces affecting the cloud's density structure. The key parameters describing these forces are\\footnote{However, see the discussion on the caveat related to the Reynolds numbers of simulations in Section \\ref{subsec:sim-DGMFs}} \\emph{i)} the sonic Mach number, ${\\cal M}_\\mathrm{s}$, \\emph{ii)} the turbulence \\emph{driving} \\citep{fed08, fed10}, which is commonly denoted by $b$, with $b=1\/3$ corresponding to purely solenoidal driving and $b=1$ to fully compressive driving, and \\emph{iii)} the magnetic field strength, $B$, reflected by the Alfv\\'en Mach number, ${\\cal M}_\\mathrm{A}$. These parameters relate to density fluctuations via \\citep[][]{nor99, pri11, mol12}\n\\begin{equation}\n\\sigma^2_{\\ln{\\rho \/ \\langle \\rho \\rangle}} = \\ln{(1+b^2 {\\cal M_\\mathrm{s}}^2   \\frac{\\beta}{\\beta+1})},\n\\label{eq:b}\n\\end{equation}\nwhere $\\sigma_{\\ln{\\rho \/ \\langle \\rho \\rangle}}$ is the standard deviation of logarithmic, mean-normalized densities and $\\beta = 2{\\cal M}^2_\\mathrm{A} \/ {\\cal M}^2_\\mathrm{s}$. This form of Eq. \\ref{eq:b} \\citep{mol12} is valid up to moderate magnetic field strengths, ${\\cal M}_\\mathrm{A} \\gtrsim 6$. The strength of the ${\\cal M}_\\mathrm{s}$ - density coupling is of great importance for analytic star formation theories, because it directly affects the star formation rates and -efficiencies (SFE) they predict \\citep[e.g.,][see FK12]{kru05, hen11, pad11}. \n\nIn this work, we will estimate how the different physical parameters affect the observed DGMFs of molecular clouds. To this goal, we will analyze numerical turbulence simulations and derive predictions for observable DGMFs. We will then compare the predictions to the results of \\citet{kai09, kai11b} and KT13 \\citep[see also][]{lad10}.\n\n\\section{Simulation data}          \n\\label{sec:data}\n\n\n\n\nWe analyze a set of magneto-hydrodynamic simulations of isothermal, driven turbulence in a periodic box, including self-gravity and sink particles to follow gas accretion onto protostars (see FK12). Each simulation is a time-series which starts ($t=0$) when the turbulence is fully developed and the gravity is switched on. Then, the evolution is followed as a function of SFE, defined as the fraction of mass accreted into sink particles. The formation of the first sink particle occurs at $\\mathrm{SFE=0\\%}$.\nThe sink particles affect their surroundings because of gas accretion, and we eliminated them from the simulations. The issue is described in Appendix \\ref{app:sinks}. Here we quote the main result: the DGMFs of ${\\cal M}_\\mathrm{s} = 10$ simulations (which we directly compare with observations) with $512^3$ cells are unaffected by sink particles below $N(\\mathrm{H}_2) < 11 \\times 10^{21}$ cm$^{-2}$. They are $70$\\% accurate up to $N(\\mathrm{H}_2) \\approx 25 \\times 10^{21}$ cm$^{-2}$. We also show in Appendix \\ref{app:resolution} that the resolution does not affect the DGMFs in this range.\n\n\n   \\begin{figure*}\n   \\centering\n\\includegraphics[bb = 0 4 960 225, clip=true, width=\\textwidth]{fig1.eps}\n      \\caption{DGMFs of four simulations (black lines) with ${\\cal M}_\\mathrm{s} = 10$, processed to mimic those observed with near-infrared dust extinction mapping technique. The solid lines show the DGMFs at $t=0$ and the dotted lines at time-steps $\\mathrm{SFE} = \\{1, 3, 10\\}\\%$. The panels also show with dashed lines the mean DGMF of nearby starless clouds (blue) and of Taurus \\citep[][red]{kai09}, and of a sample of IRDCs (KT13, green). \n                    }\n         \\label{fig:DGMFs-nir}\n   \\end{figure*}\n\n\nThe simulations were scaled so that their virial parameters, $\\alpha_\\mathrm{vir, 0} = 5 \\sigma^2_\\mathrm{v} L \/ (6 G M)$ where $\\sigma_\\mathrm{v}$ is the 3-D velocity dispersion and $L$ the size of the simulation, were close to unity. Observations have shown that molecular clouds, on average, show $\\alpha_\\mathrm{vir, 0} \\approx 1$ \\citep[e.g.,][]{hey09}. However, this definition is an idealized approximation. The actual virial parameters, $\\alpha_\\mathrm{vir} = 2|E_\\mathrm{kin}|\/|E_\\mathrm{pot}|$, vary by more than an order of magnitude in the simulations. However, the actual virial parameters do not affect density PDFs greatly (FK12). If the virial parameter is \"low-enough\" to allow some collapse, the density structure is determined by other parameters \\citep[FK12;][]{mol12}.\n\nTo make a realistic comparison with observations, we processed the simulations with ${\\cal M}_\\mathrm{s} = 5-10$ to mimic data derived using near-infrared dust extinction mapping \\citep{lom01}. First, column density data from simulations was re-gridded to $60 \\arcsec \/ \\mathrm{pixel}$ and smoothed to \nthe $FWHM = 120\\arcsec$ resolution (0.09 pc at $150$ pc distance). The native resolution of the simulations with ${\\cal M}_\\mathrm{s} > 10$ is coarser than this, and we could not smooth them (we do not compare them with the lower ${\\cal M}_\\mathrm{s}$ simulations). Then, the column densities outside $N(\\mathrm{H}_2) = [3, 25] \\times 10^{21}$ cm$^{-2}$ were discarded, approximating the dynamic range of extinction mapping. The lower limit of the range was chosen to be high enough that it is possible to define separate structures in simulations using (approximately) closed contours of constant column density. This is because observationally, \"clouds\" are commonly defined in this manner \\citep[e.g.,][]{lad10}. Finally, Gaussian noise with $\\sigma (N) = 0.018N(\\mathrm{H}_\\mathrm{2}) + 0.2 \\times 10^{21}$ cm$^{-2}$ was added, following typical uncertainties in the data of \\citet{kai09}. This procedure was repeated for three different projections of the simulation data, and the DGMFs from them were averaged to form the final DGMF.\n\nWe examined the effects of the resolution and noise to the DGMFs. We experimented with the resolution of 0.03 pc that studies employing \\emph{Herschel} data of nearby clouds will reach \\citep[e.g.,][]{sch13}. Similar resolution is reached by combined near- and mid-infrared extinction mapping when applied to infrared dark clouds (IRDCs, KT13). The effect of the resolution and noise to the DGMFs was practically negligible.\n\n\\section{Results and Discussion}          \n\\label{sec:results}\n\\label{sec:discussion}\n\n\\subsection{Dependence of the DGMF on physical parameters}          \n\\label{subsec:sim-DGMFs}\n\n\nWe derived the DGMFs for the simulations up to $\\mathrm{SFE} = 10$\\%. Figure \\ref{fig:DGMFs-nir} shows the DGMFs of four simulations with ${\\cal M}_\\mathrm{s}=10$ and $b = \\{1\/3, 0.4, 1\\}$. For the case $b=0.4$, a non-magnetized and magnetized simulation is shown. The DGMFs at early stages ($t = 0$ and $\\mathrm{SFE} = 0$\\%) are well-described by exponential functions, $dM' \\propto e^{\\alpha N}$. When star formation begins, the DGMFs flatten. Their shapes remain close to an exponential function, or curve upwards approaching a powerlaw shape. This behavior is similar in all models. Since the DGMFs are close to exponential functions in the range $N(\\mathrm{H_2}) = 3-11 \\times 10^{21}$ cm$^{-2}$, we quantified their shapes through fits of exponentials. This yielded the range $\\alpha = [-0.41, -0.023]$ in all models.\n\n\nWe examined the dependence of the DGMF slopes on the driving of turbulence and magnetic field strength ($B$) in the simulations with ${\\cal M}_\\mathrm{s} = 10$. The results are shown in Fig. \\ref{fig:slopes} (left and center). Most importantly, \\emph{the DGMF slope responds most sensitively to the turbulence driving}, changing by a factor of $4.8-8.5$ when $b$ changes from $1\/3$ to 1. The slopes depend clearly less on $B$. The non-magnetic simulations show significantly shallower slopes than magnetized ones, but if $ B \\gtrsim 3$ $\\mu$G, the slopes are uncorrelated with it. \n\n\nThe DGMF slopes depend on the SFE. The dependency is stronger in magnetized than in non-magnetized simulations: the spreads of the slopes in the range $\\mathrm{SFE}=[1, 10]\\%$ for these cases are $0.09$ and 0.03, respectively. The mean difference in the slopes of non-magnetized and magnetized runs is 0.05. The early stages ($t=0$, $\\mathrm{SFE}=0$\\%) show clearly steeper slopes than the higher SFEs. \nWe also examined the relationship between the DGMF slopes and ${\\cal M}_\\mathrm{s}$. For this, we derived the DGMFs in the native resolution of the simulations (smoothing would greatly reduce the size of the low-${\\cal M}_\\mathrm{s}$ runs). Therefore, the results should be compared to observations with caution. Figure \\ref{fig:slopes} shows the DGMF slopes and ${\\cal M}_\\mathrm{s}$ in simulations with $b=1\/3$. The slopes are non-responsive to ${\\cal M}_\\mathrm{s}$, except when ${\\cal M}_\\mathrm{s} = 5$. \n\n\nThe DGMFs can vary also due to \\emph{i)} the random nature of turbulence (\"cloud-to-cloud\" variations) and \\emph{ii)} projection effects. The former can be examined by comparing simulations that have the same input parameters, but different random number seeds (e.g., \\#12, 14, and 17, see Table \\ref{tab:sinks}). Unfortunately, we only had three simulation pairs with varying random number seeds. The mean difference in the DGMF slopes among these was 0.08 at the early stages ($t = 0$, $\\mathrm{SFE} = 0\\%$). However, for time-steps $\\mathrm{SFE} \\ge 1$ the mean difference was only 0.02. The projection effects were studied by examining the standard deviation of the slopes derived for three different projections of all models. The mean standard deviation of the slopes in all models was 0.03.\n\n\n   \\begin{figure*}\n   \\centering\n\\includegraphics[bb= 20 5 480 355, clip=true, width=0.33\\textwidth]{fig2.eps}\\includegraphics[bb= 20 5 480 355, clip=true, width=0.33\\textwidth]{fig3.eps}\\includegraphics[bb= 20 5 480 355, clip=true, width=0.33\\textwidth]{fig4.eps}\n      \\caption{Exponential slopes of the DGMFs as a function of $b$ (\\emph{left}), $B$ (\\emph{center}), and ${\\cal M}_\\mathrm{s}$ (\\emph{right}). The solid black lines show the timestep $t = 0$, and the dotted lines $\\mathrm{SFE}=\\{0, 1, \\dots, 10\\}\\%$. The blue, red, and green shaded regions indicate the slopes observed in starless and star-forming nearby clouds \\citep{kai09} and in IRDCs (KT13, however, see the discussion on these data in Section \\ref{subsec:comp_with_obs}). The median masses of each set of the clouds, $M_\\mathrm{1\/2}$, are shown in the panels.\n                    }\n         \\label{fig:slopes}\n   \\end{figure*}\n\n\nWe note that the effective Reynolds numbers of our simulations ($\\lesssim$$10^4$) are lower than that of the interstellar medium ($\\sim$$10^7$). It is not clear how this affects the predicted statistical properties. \\citet{alu13} has rigorously shown that the direct influence of driving on the kinetic energy is restricted to scales larger than the smallest scale at which the turbulence is stirred. However, numerical \\citep{fed10} and analytic \\citep{gal11} works have found differences in flow statistics in the range that can be considered to be the \"inertial range\" of compressible turbulence simulations. Resolution studies of the simulations suggest that the driving-induced differences remain when the Reynolds number increases. As this issue cannot be addressed with the current computational methods, our results are also subject to it.\n\n\\subsection{Comparing the predictions with observations}          \n\\label{subsec:comp_with_obs}\n\nFigures \\ref{fig:DGMFs-nir} and \\ref{fig:slopes} show observed DGMFs to be compared with the simulated ones. Figure \\ref{fig:DGMFs-nir} shows the mean DGMF of quiescent clouds (LDN1719, Lupus V, Cha III, and Musca) and a DGMF of a typical star-forming cloud (Taurus) from \\citet{kai09}, and a mean DGMF of ten IRDCs from KT13. Figure \\ref{fig:slopes} shows the ranges of the observed slopes from \\citet{kai09}, which span $\\alpha = [-0.17, -0.45]$ for 13 nearby star-forming clouds and $\\alpha = [-0.35, -1.2]$ for four quiescent clouds. The range of IRDC slopes from KT13 is also shown. We note that the DGMFs of IRDCs in KT13 were derived from a slightly different column density range than those of nearby clouds (they begin from $N(\\mathrm{H}_2) \\approx 7 \\times 10^{21}$ cm$^{-2}$). Thus, the comparison of them with the other data should be considered only suggestive.\n\n\nThe dependence of the DGMF slopes on the turbulence driving allows us to constrain $b$ (see Fig. \\ref{fig:slopes}). None of the simulations shows as steep slopes as observed in starless clouds. From the non-magnetized simulations, only those with $b=1\/3$ are in agreement with the nearby star-forming clouds.  Magnetic fields can steepen the slopes by about $0.05$ (Fig. \\ref{fig:slopes}, center). Therefore, from the magnetized runs those with $b=1\/3$, or $b=0.4$ and $B \\ge 3 \\ \\mu$G agree with star-forming clouds. \\emph{The fully compressive simulations produce a greatly higher fraction of dense gas than observed in nearby clouds}. The comparison suggests a low $b$ for nearby molecular clouds on average, possibly lower than previously estimated by \\citet{pad97apj} and \\citet{bru10taurus} in Taurus, $b \\approx 0.5$.\n\nThe DGMF slopes correlate with the SFE, depending on whether the cloud is magnetized or not. Since in the current view clouds have magnetic fields \\citep[][]{cru12}, the spread of slopes is likely the most realistic in magnetized simulations (i.e., 0.1, see Fig. \\ref{fig:slopes}). Thus, it seems that part of the spread in the observed slopes originates from the SFEs of the clouds. We used a Monte Carlo simulation to estimate whether all the variation in the observed slopes can originate from changes in the SFE and statistical variations. We assumed that the changes due to SFE are uniformly distributed between $[0, 0.1]$ and the statistical variations are normally distributed with $\\sigma = 0.04$. The test showed that the probability that 13 clouds span a range $>0.28$ is $0.2\\%$. Note that the range of the observed slopes can be wider. KT13 showed that IRDCs possibly have flatter DGMFs than nearby clouds (Fig. \\ref{fig:slopes}). In conclusion, it seems likely that the spread of the observed DGMF slopes cannot be explained by statistical variations and changes in the SFE alone. Changes in the clouds' average compression provides one possible source to account for this variation.\n\nOne interesting question for the future is to examine the effect of cloud mass to the DGMFs. There are no very massive clouds in the nearby cloud sample (median mass $0.5 \\times 10^4$ M$_\\odot$). In contrast, the median mass of the IRDCs is $5 \\times 10^4$ M$_\\odot$, which is ten times higher. This could contribute to the differences seen in the slopes of the two cloud sets. However as discussed earlier, comparing DGMFs of IRDCs with nearby clouds is not without caveats. The question could be properly addressed by a study of a statistical sample of IRDCs, or a study of the nearest high-mass clouds (e.g., Orion, Cygnus, Rosette) employing \\emph{Herschel} data.\n\nThe weak dependence of the DGMF slopes on ${\\cal M}_\\mathrm{s}$ appears to be an effect of the narrow column density range we examine (note that the results were derived from simulations that have differing physical resolutions and are only suggestive). The density PDF is expected to respond to ${\\cal M}_\\mathrm{s}$ following Eq. \\ref{eq:b}, which should reflect to the DGMFs. However, it appears that in the range of $N(\\mathrm{H}_2) = 3-11 \\times 10^{21}$ cm$^{-2}$ the effect is insignificant. This result is in agreement with \\citet{goo09} who did not detect any dependence between column density PDF widths and CO linewidths in Perseus. However, we recently measured the column density PDF widths using a high-dynamic-range technique (KT13) and concluded that if a wider range is examined, the PDF widths correlate with ${\\cal M}_\\mathrm{s}$. \n\n\nWhen comparing observed DGMFs with simulations, it should be kept in mind that in simulations \"driving\" is well-defined and ideal: energy is injected at large scales, with certain characteristics such as the divergence and curl. In real clouds, energy is likely injected at multiple scales and the characteristics of the driving can depend on the scale. However, if some of these driving modes excite more compression than others, particular regions in a cloud, and hence, also clouds \\emph{on average}, can show characteristics of the flows produced with ideal driving with different mixtures of solenoidal and compressive modes.\n\n\nFinally, we comment on the relation between the DGMFs and column density PDFs. The column density PDFs of nearby clouds are log-normal below $N(\\mathrm{H}_2) \\lesssim 3 \\times 10^{21}$ cm$^{-2}$. In the range $N(\\mathrm{H}_2) = 3-25 \\times 10^{21}$ cm$^{-2}$, they are in agreement with either powerlaws or (wide) log-normals (KT13). It is not established if the PDFs above $N(\\mathrm{H}_2) \\gtrsim 3 \\times 10^{21}$ cm$^{-2}$ are log-normals (KT13) or powerlaws \\citep[][see Fig. \\ref{fig:pdfs}]{sch13}. Importantly, it follows from Eq. \\ref{eq:dgmf-pdf} that  \\emph{a log-normal PDF yields an exponential DGMF and a powerlaw PDF yields a powerlaw DGMF.} The simulated DGMFs in the range $N(\\mathrm{H}_2) \\gtrsim 3-25 \\times 10^{21}$ cm$^{-2}$ appear exponential at the early stages. Therefore, the column density PDFs at these stages are close to log-normals. When the simulations evolve, the DGMFs become closer to powerlaws (see FK13). This means that the underlying column density PDF transits from a log-normal to a powerlaw.\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have examined the relationship between the dense gas mass fraction (DGMF), star formation, and turbulence properties in molecular clouds by comparing DGMFs derived from isothermal, magneto-hydrodynamic, self-gravitating turbulence simulations to observed ones. Our conclusions are as follows. \t\n\\begin{enumerate}\n\n   \\item Simulations predict close-to exponential DGMFs for molecular clouds in the column density range of $N(\\mathrm{H}_2) = 3-11 \\times 10^{21}$ cm$^{-2}$. The DGMF slopes span the range $\\alpha = [-0.41, -0.023]$, being clearly steeper at the early stages of the simulations compared to the stages when stars are forming ($\\mathrm{SFE} \\geq 1$\\%). These predictions are accurate on a 70\\% level up to $N(\\mathrm{H}_2) \\approx 25 \\times 10^{21}$ cm$^{-2}$.\n   \n   \\item The DGMF slopes depend strongly on the turbulence driving ($b$). They depend less, but  significantly, on the exact SFE. The dependence on the SFE is stronger in magnetized than non-magnetized cases. Generally, the effect of the magnetic field to the DGMF is small. Also ${\\cal M}_\\mathrm{s}$ has a negligible effect on the slopes in the examined column density range. The statistical variations are comparable to those arising from varying SFE. However, how compressive the turbulence is (i.e., parameter $b$) is the largest single factor in determining the slope of the DGMF in the simulations. \n   \n   \\item The observed DGMFs can be used to constrain the turbulence driving parameter $b$. The DGMFs of nearby clouds are only reproduced by simulations that are driven by relatively non-compressive forcing, i.e., $b = 1\/3$ or 0.4. The fully compressive simulations ($b = 1$) over-estimate the DGMFs greatly. Massive IRDCs can show flatter DGMFs that are in agreement with more compressive driving. The spread of the observed DGMFs cannot be explained by different SFEs and statistical variations alone. Variations in the clouds' average compression level offer one explanation to account for the observed spread. \n   \n\\end{enumerate}\n\n\n\\begin{acknowledgements}\nThe work of JK was supported by the Deutsche Forschungsgemeinschaft priority program 1573 (\"Physics of the Interstellar Medium\"). C. F. acknowledges a Discovery Projects Fellowship from the Australian Research Council (grant DP110102191).\t\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe origin of the galactic halo has been an interesting question ever since halo stars and clusters became the defining objects of population II (\\citealt{Baade1944}, \\citealt{OConnell1958}). Halo subdwarfs were soon recognized to be metal-poor (\\citealt{Chamberlain1951}), as were red giant stars branch (RGB) stars in glubular clusters of the halo (\\citealt{Helfer1959}). Halo cepheids are now called Type II Cepheid (T2C) stars, and were first recognized by \\citet{Joy1937}, and found to be metal-poor by \\citet{Abt1954} and \\citet{Rodgers1963}. Short period cepheids were found in metal-poor globular clusters by \\citet{Arp1955}, and catalogued by \\citet{Clement2001}.\\footnote{The most recent update is to be found at http:\/\/www.astro.utoronto.ca\/~cclement\/cat\/listngc.html}\n\nTo the surprise of some, \\citet{Woolley1966} found that the halo cepheids showed thick disk kinematics and, hence, were likely to be only moderately metal-poor. The short period stars received the name BL~Her, a star which has a period of 1.3 days, and is known to show a metallicity of --0.16 (\\citealt{Maas2007}).\n\nIn an effort to further understand the T2C stars of the galactic halo, \\citet{Maas2007} derived metallicities of 19 stars, 7 of which have periods of 3 days or less. Except of one star, UY~Eri, they showed very modest depletions of heavy elements and, hence, fit the classification of BL~Her stars. To further expand the database for T2C stars in the halo, we have obtained high resolution spectra of field T2C stars, 10 of which have periods below 3 days. The stars were selected to be bright enough for the available telescopes, and to be conveniently placed when observing time was assigned.\n\nIn this paper, we report on the chemical composition of the 10 T2C field stars with periods from 0.9 to 3 days. The upper limit marks the edge of an almost empty gap from 4 to 9 days in the distribution shown in Figure~1 of \\citet{Soszynski2008} and Figure~6 of \\citet{Schmidt2011}. The separation of both cepheid strip candidates and Type II Cepheids into metal-normal and metal-poor stars can be seen in Figure~8 of \\citet{Schmidt2011}.\n\n\\begin{table*}[t]\n    \\scriptsize\n    \\caption{Details of Observations and Stellar Parameters}\n   \n    \\label{tab:atmparam}\n    \\begin{adjustbox}{width=\\textwidth,center=\\textwidth}\n    {\\begin{tabular}{rcccrccccrl}\n    \\hline \\hline\n    Star & P(days) & Telescope & JD (2400000+) & Exp (s) & Phase & ${T_{\\rm eff}}$ {} (K) & $\\log \\mathrm{g}$ {} & $\\mathrm{V_{t}}$ {} (km s$^{-1}$) & [Fe\/H] & Remarks \\\\\n    \\hline\n    UY~CrB & 0.929 & APO & 53162.6736 & 3600 & 0.491 & 6150 & 1.8 & 2.0 & --0.43 & \\\\\n     & & APO & 53456.9981 & 1800 & 0.245 & 6700 & 2.0 & 2.4 & --0.32& \\\\\n     & & APO & 53626.7528 & 1800 & 0.839 & 6300 & 2.5 & 2.0 & --0.47& \\\\\n     NSV 10788 & 1.081 & VLT & 56124.7937 & 1200 & 0.451 & 6250 & 2.3 & 2.8 & --2.41 & \\\\\n     V716~Oph & 1.116 & APO & 52336.9382 & 1200 & 0.154 & 7000 & 1.8 & 2.7 & --1.64 & \\\\\n     & & APO & 52417.7674 & 1800&0.595 & 6100 & 2.2 & 2.6 & --1.67 & \\\\\n     & & APO & 52447.6805 & 1800&0.380 & 6600 & 2.6 & 2.6 & --1.56 & \\\\\n     & & APO & 52448.6667 & 1800&0.284 & 6700 & 2.2 & 2.6 & --1.72 & \\\\\n     & & APO & 52449.7423 & 1800&0.248 & 6750 & 2.0 & 2.3 & --1.62 & \\\\\n     BF~Ser & 1.165 & APO & 52336.8431 & 2400 & 0.652 & 5800 & 1.0 & 2.3 & --2.15 & \\\\\n     & & APO & 52417.7090 & 1800 & 0.038 & 7300 & 2.2 & 3.0 & --2.04 & \\\\\n     & & APO & 52804.7590 & 900 & 0.145 & 7000 & 2.0 & 3.0 & --2.08 & H$\\alpha$ double or emission \\\\\n     & & APO & 53457.8495 & 1800 & 0.526 & 6300 & 2.1 & 2.2 & --2.04 & H$\\alpha$ double or emission \\\\\n     BL~Her & 1.307 & APO & 53163.8646 & 900 & 0.104 & 7000 & 2.2 & 2.2 & --0.12 &    \\\\\n     & & OHP & 49572.3784 & 900 & 0.147 & 6650 & 2.5 & 2.2 & --0.20 &   \\\\\n     XX~Vir & 1.348 & APO &52417.6479 & 1800 & 0.070 & 7500 & 2.2  & 2.3 & --1.62& \\\\\n     & & APO &52449.7188 & 1800 & 0.858 & 6100 & 2.5  & 2.8 & --1.51& \\\\\n     & & APO &53541.6750 & 1800 & 0.791 & ... & ... & ... & ... &   \\\\\n     V1287~Sco & 1.956 & VLT &56126.4862 & 1500 & 0.487 & 5950 & 2.2 & 3.5 & --1.94 & H$\\alpha$ emission \\\\\n     V553~Cen & 2.061 & ESO & 56748.6532 & 90 & 0.818 & 6060 & 2.2 & 2.7 & 0.01 & see \\citet{Wallerstein1996} \\\\\n     UY~Eri & 2.213 & APO & 52984.7985 & 1800 & 0.889 & 6400 & 2.0 & 2.0 & --1.83 & H$\\alpha$ emiss or double \\\\\n     & & APO & 53687.7856 & 1800 & 0.511 & 6200 & 1.8 & 2.6 & --1.66 & \\\\\n     & & APO & 54044.7828 & 1800 & 0.809 & 6000 & 1.9 & 2.6 & --1.70 & H$\\alpha$ emission \\\\\n     AU~Peg & 2.402 & APO & 53626.7292 & 900 & 0.848 & 6008 & 2.0 & 2.8 & 0.33 & orbit phase =  0.538 \\\\\n     & & APO & 53687.6390 & 900 & 0.109 & 5544 & 1.5 & 2.3 & 0.21 & orbit phase = 0.681 \\\\\n    \\hline\n    \\end{tabular}}\n    \\end{adjustbox}\n   \n    \\begin{itemize}\n    \\item[] {\\it Remarks: \\\\}\n    \n(a) -- {\\it NSV 10788} is a star with P = 1.08 days and a metallicity of [Fe\/H] = --2.4. The lightcurve shows an amplitude of 0.5 magnitudes. Thus, classification of this star as a cepheid is uncertain.\n\n(b) -- For {\\it BF~Ser}, all four spectra were taken at important phases. This star is very likely a member of the UY~Eri group. It is a metal poor-star with a pulsation period of 1.2 days.\n\n(c) -- {\\it XX~Vir} is a high latitude star of the UY~Eri group with a short pulsation period of 1.3 days. For one spectrum, with emission in iron lines, the chemical composition was not determined.\n\n(d) --  {\\it V1287~Sco} is a variable star of UY~Eri type. There is not much information in the literature about this star. H$\\alpha$ emission is seen in the spectrum. The lines are slightly asymmetric.\n\n(e) -- {\\it V553~Cen} is a star recognized by \\citet{Evans1983} as a rare C-rich Cephied. Its composition was investigated by \\citet{Wallerstein1998}.\n\n(f) -- {\\it AU~Peg} is a spectroscopic binary \\citep{Harris1984} with an orbital period of 53.3 days. The chemical composition of this star was studied by \\citet{Harris1984} and \\citet{Maas2007}. It appears to be slightly metal rich, which may be due in part to mass transfer from its unobserved companion.\n\n    \\end{itemize}\n\\end{table*}\n\n\\section{Observations and Data Reduction}\n\\label{sec:obs}\n\nSeven objects were observed with the 3.5-m telescope at the Apache Point Observatory with the ARC Echelle Spectrograph (ARCES). By using a prism as a cross-disperser, the APO echelle captures the entire spectrum from 3500 \\AA{} to 10\\,400 \\AA{} with a resolving power of 31\\,500. However, the red-sensitive 2048x2048 chip has decreasing sensitivity for cool stars at shorter wavelengths and beyond 9000 \\AA{}. The observations were obtained as a part of a program to derive the chemical composition of certain Type II Cepheids and RR Lyrae stars. The exposure times were approximately 10--30 minutes. The estimated S\/N ratio per pixel at the continuum level, depending upon the wavelength interval, is approximately 80--150. The uncertainty in the determination of velocities is a few tenths of a km s$^{-1}$.\n\nTwo objects were observed with the cross-dispersed echelle spectrograph, UVES, at the Very Large Telescope.\\footnote{Based on observations collected at the European Organization for Astronomical Research in the Southern Hemisphere under ESO programme 089.D-0489(A).} The red arm was used, which covers the wavelength region between 4200 \\AA{} and 11\\,000 \\AA{}. Standard instrumental settings were used to achieve wavelength coverage from 4790--5760 \\AA{} and 5830--6810 \\AA{} with a resolution of 0.16 \\AA{}. The observations were done in service mode. For some objects, several spectra were obtained. The exposure times were approximately 20--30 minutes. The primary data reduction, such as bias subtraction, flat-field correction, wavelength calibration, sky subtraction and spectra extraction was performed with the UVES pipeline (\\citealt{Ballester2000}).\n\nOne object, the carbon cepheid, V553~Cen, was taken from the ESO archive. The star was observed with the echelle spectrograph HARPS at the ESO La Silla 3.6-m telescope. The spectral range is 4000--6800 \\AA{} with a resolving power of 100\\,000. For BL~Her, an additional spectrum was taken from the archive of the Elodie spectrograph at the Observatoire de Haute-Provence 1.93-m telescope (R=40\\,000, $\\lambda$ 4000-6800 \\AA{}). The observed stars and their atmospheric parameters are given in Table 1.\n\n\\section{Spectroscopic Analysis}\n\\label{sec:spec}\n\nTo determine the effective temperature, T$_{eff}$, we used the line depth ratio method of \\citet{Kovtyukh2007}. The uncertainties in this method range from 30--100 K depending on the S\/N. To set the $\\log{g}$ value, we required that the iron abundance, as determined from lines of FeI and FeII, be equal. \n\nElemental abundances were determined using LTE and NLTE approximations combined with atmospheric models by \\citet{Castelli2004}, computed for the parameters of each star. The solar abundances were computed for lines from the solar spectrum \\citep{Kurucz1984} with $\\log{gf}$ from the Vienna Atomic Line Database (VALD) \\citep{Kupka1999}, and the solar model by \\citet{Castelli2004}. They are listed in \\citet{Lemasle2015}. \n\n\\begin{table}\n    \\centering\n    \\caption{Table of $\\log (gf)$ and $\\chi_{\\rm{exc}}$ for the Na and O lines used.}\n    \\label{tab:loggf}\n    \\begin{tabular}{crr}\n    \\hline \\hline Lambda & $\\chi_{\\rm{exc}}$ & $\\log (gf)$\\\\ \\hline\n    & Sodium & \\\\ \\hline\n    5682.63 & 2.09 & --0.71 \\\\\n    5688.19 & 2.10 & --0.41 \\\\\n    6154.23 & 2.09 & --1.56 \\\\\n    6160.75 & 2.10 & --1.26 \\\\\n    \\hline\n    & Oxygen & \\\\ \\hline\n    6300.30 & 0.00 & --9.717 \\\\\n    7771.94 & 9.11 & 0.369 \\\\\n    7774.17 & 9.11 & 0.223 \\\\\n    7775.39 & 9.11 & 0.001 \\\\\n    8446.25 & 9.52 & --0.462 \\\\\n    8446.36 & 9.52 & 0.236 \\\\\n    8446.76 & 9.52 & 0.014 \\\\ \n    \\hline\n    \\end{tabular}\n\\end{table}\n\n\\subsection{Oxygen}\n\nThe NLTE model of the oxygen atom was first described by \\citet{Mishenina2000}, and then updated by \\citet{Korotin2014}. The model consists of 51 OI levels of singlet, triplet, and quintet systems, and the ground level of the OII ion. Fine structure splitting was taken into account only for the ground level and the 3p5P level (the upper level of the 7772,4,5 triplet lines). A total of 248 bound-bound transitions were included. Oxygen line parameters are listed in Table \\ref{tab:loggf}. The high excitation OI triplet suffers from departure from LTE (\\citealt{Parsons1964}, \\citealt{Altrock1968}, \\citealt{Amarsi2016}). Its strength depends sensitively on the star's surface gravity. In stars of high luminosity, the triplet is greatly enhanced since radiative effects dominate recombination and ionization, as compared to collisional excitation, which is controlled by the local temperature.\n\n\\subsection{Sodium}\n\nSodium line parameters are given in Table \\ref{tab:loggf}. We derived the sodium abundances by line profile fitting. The NLTE atomic model of sodium was presented by \\citet{Korotin1999} and then updated by \\citet{Dobrovolskas2014}. The updated sodium model currently consists of twenty energy levels of NaI and the ground level of NaII. In total, 46 radiative transitions were taken into account for the calculation of the population of all levels. For four stars, the Na D lines were saturated, so we used the pair of lines at 5682 \\AA{} and 5688 \\AA{}. For stars with nearly solar metallicity, we used the weaker pair at 6154 \\AA{} and 6160 \\AA{}. We chose not to use the 8183 \\AA{} and 8194 \\AA{} lines due to blending with absorption by atmospheric lines, which depends on the stellar radial velocity and the humidity above the observatory.\n\nThe results of the abundance analysis on our program stars is given in Table \\ref{tab:MetalCMn}.\n\n\\section{Discussion}\n\\label{sec:disc}\n\nThe T2C stars have been divided into W Vir stars with periods from 9 to 30 Days (though stars with periods greater than 20 days often show mild RV Tau properties); and the BL Her stars with periods less than 3 days.\\footnote{The gap from 3 to 9 days is violated by variable 3 in the globular cluster M10.} We advocate for the  separation of the 1--3 day stars into the BL Her group with near-solar metallicity, and the distinctly metal-poor group that we will call the UY Eri class. The latter are to be found also in metal-poor globular clusters \\citep{Clement2001}. The relatively metal-rich globulars with [Fe\/H] $\\textgreater$ --1.0 do not have cepheids with periods less than 3 days. Of 10 stars in Table 1, 6 fall into the metal poor UY Eri group, all of which show [Fe\/H] $\\textless$ --1.5.\n\nOur database may be enhanced by including the abundances in \\citet{Maas2007}. Just by chance, all of their short period stars, except for UY~Eri, fall into the more metal-rich BL~Her class. Hence, we have combined the two sources to develop mean abundance ratios for the BL~Her subgroup, and used only our data for the metal-poor UY~Eri group. Our abundance data are summarized in Table~\\ref{tab:OurStats}.\n\nFor our small sample, formal probable errors are less indicative of the accuracy of a quantity than one might hope. Hence, we show the dependence of [O\/Fe] and the dependence of [$\\alpha$\/Fe] on [Fe\/H] in Figure~\\ref{XFe}, using the elements Mg, Si, Ca, and Ti. The dependence of [O\/Fe] on [Fe\/H] shows a dependence that is similar to what is seen in both giants and dwarfs, with a rise of [O\/Fe] from near 0.0 at [Fe\/H] = 0.0 to +0.7 between [Fe\/H] = --1.5 to --2.5. For [$\\alpha$\/Fe], the pattern is similar to many other investigations of the $\\alpha$-elements, rising from near 0.0 to +0.5 for the metal-poor UY~Eri Group.\n\nThe metal-rich BL Her group show abundances of 22 elements that differ little in their ratios relative to iron, with the possible exception of sodium. For the UY~Eri group, the abundances of oxygen, and the $\\alpha$-elements from Mg to Ti  show excesses that are comparable to the excesses in most metal-poor red giants and main sequence stars.\n\nWe await the discovery and observation of additional T2C stars by Gaia and eventually the Large Synoptic Survey Telescope (LSST). The Gaia spectra in the 8400--8800 \\AA{} region may be used to derive metallicities from the strenth of the Ca II IR triplet as has been calibrated for RR Lyrae stars by \\citet{Wallerstein2012}.\n\n\\subsection{Carbon, Nitrogen, and Oxygen}\n\nThere are a few CI lines of high excitation in these stars. Despite our NLTE analyses, the scatter of [C\/Fe] is too great to consider the mean values of [C\/Fe] to be definitive. A significant excess in [N\/Fe] for the BL~Her stars is approximately what is expected from the enhancement of nitrogen by the CNO cycle and internal mixing. For the metal-poor UY~Eri stars, the nitrogen lines are too weak for analysis. For oxygen, the analysis depends upon the OI triplet at 7772,4,5 \\AA{} and the 8446.63 \\AA{} lines. As we have shown in Figure~\\ref{XFe}, for the metal-poor UY~Eri stars, the mean excess in [O\/Fe] is $0.72 \\pm 0.09$, which is typical for metal-poor stars.\n\n\\subsection{Light Elements}\n\nFor sodium, we see small excesses that are approximately equal to their uncertanties for both groups of stars. In most, if not all, globular clusters, some red giants show a significant excess of sodium in second generation stars \\citep{Carretta2009}. Our abundances are based on the lines arising from the 2.1 eV excited level. We did not use the Na D lines because they are usually too strong and are sometimes blended with circumstellar or interstellar lines. For the alpha elements, Mg, Si, Ca, and Ti, the BL~Her stars do not show a difference in their abundances to that of iron, as is usually seen in stars with near-solar metallicity. S is omitted from the $\\alpha$-elements because its lines are too weak for analysis, except in UY~Eri. For the UY~Eri stars, we find a mean value of [$\\alpha$\/Fe] = $0.35\\pm0.19$ which is typical of metal-poor stars.\n\n\\begin{figure}\n\\epsscale{1.2}\n\\plotone{f1.pdf}\n\\caption{~The dependence of [O\/Fe] and [$\\alpha$\/Fe] on [Fe\/H] for the observed stars. \\label{XFe}}\n\\end{figure}\n\n\\begin{table*}\n   \n    \\tiny\n   \n   \n    \\caption{Relative to Fe Abundances in Type II Cephieds (C--Ti)}\n    \\label{tab:MetalCMn}\n    \\begin{adjustbox}{width=\\textwidth,center=\\textwidth}\n    {\\begin{tabular}{rrrrrrrrrrrrrrr}\n    \\hline \\hline\n    & P, days & [Fe\/H] & C & O & Na & Mg & Al & Si & S & Ca & Sc & Ti \\\\\n    \\hline\n  \n   \n    UY~CrB & 0.929 & --0.40 & 0.65 & 0.67 & 0.52 & 0.31 & 0.35 & 0.24 & 0.25 & 0.20 & 0.01 & 0.23 \\\\\n    NSV 10788 & 1.081 & --2.41 & ...  & ... & ... & 0.52 & ... & ... & ... & 0.49 & 0.27 & 0.37 \\\\\n    V716~Oph & 1.116 & --1.64 & 0.43 & 0.72 & 0.36 & 0.21 & ... & 0.14 & ... & 0.53 & 0.23 & 0.49 \\\\\n    BF~Ser & 1.165 & --2.08 & 0.13 & 0.73 & ... & 0.31& ... & ... & ... & 0.75 & 0.20 & 0.41 \\\\\n    BL~Her & 1.307 & --0.16 & 0.36 & 0.35 & 0.63 & 0.01 & 0.00 & --0.01 & 0.10 & 0.01 & 0.06 & 0.15 \\\\\n    XX~Vir & 1.348 & --1.55 & ... & 0.83 & ... & 0.19 & ... & 0.05 & ... & 0.46 & 0.26 & 0.39 \\\\\n    V1287~Sco & 1.956 & --1.94 & ... & ... & ... & 0.55 & ... & ... & ... & 0.52 & 0.25 & 0.47  \\\\\n    V553~Cen & 2.061 & 0.01 & 0.78 & --0.11 & 0.43 & 0.19 & 0.12 & 0.00 & 0.10 & 0.23 & 0.28 & 0.28 \\\\\n    UY~Eri & 2.213 & --1.73 & --0.19 & 0.61 & 0.18 & 0.37 & ... & 0.06 & 0.33 & 0.60 & 0.06 & 0.45 \\\\\n    AU~Peg & 2.402 & 0.26 & 0.17 & 0.15 & --0.04 & --0.07 & 0.18 & --0.11 & 0.15 & --0.14 & --0.27 & --0.04 \\\\\n    \\hline\n    \\end{tabular}}\n    \\end{adjustbox}\n    \\\\\n   \n\\end{table*}\n\n\\section{Conclusions}\n\\label{sec:conc}\n\nWe have found that the T2C stars with periods of 1--3 days may be divided into a nearly normal metal group, usually called BL~Her stars, and a newly recognized group with [Fe\/H] = --1.5 to --2.4. The BL~Her group probably belong to the thick disk, while latter are similar to short period cepheids in metal-poor globular clusters. With a few exceptions, it appears that globulars with [Fe\/H] $\\textgreater$ --1.0 do not have cepheid members of any metallicity. \n\nThe relationship of globular clusters and the general halo of our galaxy is not clear. \\citet{Martell2016} have shown that only a very small percentage of the general halo could have come from the evaporation of stars from globular clusters. Since all stars seem to form in groups and clusters, it is possible that the single halo stars are descendents of small, loose clusters no longer recognizable as such. In addition, the populations of variable stars in the halo and globulars are not the same. \\\\\n\nWe thank Giuseppe Bono, Charli Sakari, and Joanne Hughes for reading the manuscript and making some good suggestions. This research has been supported by the Kennilworth Fund of the New York Community Trust.\n\n\\begin{table}\n    \\centering\n    \\caption{Mean abundances of UY~Eri stars from this study, and BL~Her stars from this study and \\citet{Maas2007} for critical elements relative to Fe.}\n    \\label{tab:OurStats}\n    \\begin{tabular}{lrrrrc}\n    \\hline \\hline\n    Element & Mean$_{BL}$ & $\\sigma_{BL}$ & Mean$_{UY}$ & $\\sigma_{UY}$ & Mean$_{\\mid BL-UY \\mid}$\\\\\n    \\hline\n    O & 0.24 & 0.29 & 0.72 & 0.27 & 0.48 \\\\\n    Na & 0.37 & 0.26 & 0.27 & 0.20 & 0.10 \\\\\n    Mg & 0.08 & 0.16 & 0.35 & 0.20 & 0.27 \\\\\n    Si & 0.02 & 0.13 & 0.13 & 0.14 & 0.11 \\\\\n    S & 0.14 & 0.07 & 0.50 & 0.26 & 0.36 \\\\\n    Ca & 0.05 & 0.16 & 0.54 & 0.26 & 0.49 \\\\\n    Ti & 0.14 & 0.13 & 0.41 & 0.17 & 0.27 \\\\\n    $[$Fe\/H$]$ & --0.07 & 0.24 & --1.89 & 0.29 & 1.82 \\\\\n    \\hline\n    \\end{tabular}\n\\end{table}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe successes of quantum computing in the past decade have laid the foundations for the interdisciplinary field of quantum machine learning (QML), where parameterized quantum circuits (PQC) are used as machine learning models to extract features from the training data. It was argued~\\cite{amirapaper} that such quantum neural networks (QNN) could have higher trainability, capacity, and generalization bound than their classical counterparts. Practically, hybrid quantum neural networks (HQNN) have shown promise in small-scale benchmarking tasks~\\cite{boston-housing,schetakis2022review} and larger-scale industrial tasks~\\cite{vw-paper,pharma}. Nevertheless, the utility, practicality, and scalability of pure QNNs are still unclear. Furthermore,~\\cite{schuld-advantage} provided a thorough overview of this field, showing that while classical machine learning is solving large real-world problems, QNNs are mostly tried on synthetic, clean datasets and show no real-world immediate advantages in its current state\\footnote{This is true for pure quantum models trained on classical datasets. There are numerous successes in applying quantum models to quantum-native problems~\\cite{hhl,vqe,discocat}.}. It also suggested that the QNN research focus ought to be shifted from seeking quantum advantage to new research questions, such as finding a new, advantageous quantum neuron. This work explores a new quantum neuron that argues for moving beyond using the Pauli-, single-qubit gates for encoding data, and instead employing higher-dimensional unitaries through gate decomposition. \n\nSince quantum gates are represented by elements of compact groups, Fourier analysis is a natural tool to analyse QNNs. \\cite{schuld_fourier} showed that certain quantum encodings can create an asymptotically universal Fourier estimator. A universal Fourier estimator is a model that can fit a trigonometric series on any given function and as the number of terms in the series approaches infinity, the fit becomes an asymptotically perfect estimate. This estimator can initially infer the coarse correlations in the supplied data, and by increasing the number of Fourier terms, it can incrementally judge the more granular properties of the dataset. This provides an adjustable, highly-expressive machine-learning model. \n\nIn \\cite{schuld_fourier}, the authors showed that a QNN can operate as such in two ways: a \\emph{sequential} single-qubit architecture with $n$ repetitions of the same encoding could yield $n$ Fourier bases, which could also arise from an $n$-qubit architecture with the encoding gates applied to all qubits in \\emph{parallel}.\n\nThe sequential architecture is widely shown to be an efficient Fourier estimator, demonstrated both theoretically \\cite{schuld_fourier} and empirically~\\cite{data_reuploading,mo_paper}. However, sequential circuits are often deep \\cite{mo_paper}, and assuming that near-term quantum computers will have a non-negligible noise associated with each gate, these circuits can experience noise-induced barren plateaus~\\cite{noise-bp}. Barren plateaus are a phenomenon observed in optimization problems where the gradients of the model vanish, rendering them impossible to train. More importantly, a single qubit can be simulated efficiently in a classical setting, so this architecture brings no quantum advantage.\n\nIn contrast, the parallel setting offers the exponential space advantage of quantum computing but poses two challenges for large numbers of qubits:  \n\\begin{itemize}\n    \\item An exponentially growing parameter count with the number of qubits, required to span the entire group space $\\mathcal{SU}(2^n)$~\\cite{haar_measure_paper}, which could also lead to noise-induced barren plateaus. Spanning the full space is especially important for a priori problems, where the best model lies somewhere in the Hilbert space, but the machine learning scientist has no prior knowledge of how to parameterize a circuit to reach this point. In addition, gradient calculations in QNNs -- as of this publication -- use the parameter-shift rule discovered and presented in \\cite{parameter-shift}. This method requires two evaluations of the QNN to find the derivative of the circuit with respect to each of the trainable parameters. An exponentially growing number of trainable parameters translates to an exponentially increasing amount of resources required for gradient computation. In mitigation, \\cite{poly1,poly2} showed that a polynomially-growing number of parameters could generate a similar result based on the quantum $t$-design limits~\\cite{t-design}.\n  \n   \\item Strongly-parameterised QNNs have a vanishing variance of gradient which decreases exponentially with the number of qubits~\\cite{bp}. This means that for a large number of qubits if one were to initialize her QNN randomly, they will encounter a barren plateau. This happens because the expectation value of the derivative of the loss function with respect to each variable for any well-parameterized quantum circuit is zero, and its variance decreases exponentially with the number of qubits. Mitigation methods have been suggested in \\cite{bp-mit1,bp-mit2}, and most notably in \\cite{log-nobp} it was shown that by relaxing the well-parameterized constrain to only include a logarithmically growing circuit depth with the number of qubits in the system and use local measurements, the circuit is guaranteed to evade barren plateaus. \\cite{bp-zx} developed a platform based on ZX-calculus\\footnote{A graphical language, based on tensor networks, to analyze quantum circuits\\cite{zx1}.} to explore which QNN architectures are affected by the barren plateau phenomenon and found that strongly-entangling, hardware efficient circuits suffer from them. In contrast with the previously-mentioned cases of barren plateaus, the latter is not noise-induced and thus this is a problem that needs to be addressed even in the fault-tolerant future of quantum computing.\n\\end{itemize} \n\nTherefore, the practicing QML scientist is limited in choosing her QNN architectures for general data science problems: they need to be shallow or employ only a few qubits.\n\nThis contribution suggests a modification to the encoding strategies in \\cite{schuld_fourier} to increase the growth of the Fourier bases in a QNN from linear in the number of qubits\/number of repetitions, to exponential. The proposed encoding is constructed by decomposing large unitary generators into local Pauli-Z rotations. This improves the expressivity of the QNNs without requiring additional qubits or encoding repetitions. The increased expressivity is a product of eliminating the encoding degeneracies of the quantum kernel, making efficient use of the available Hilbert space by assigning a unique wave-vector to each of its dimensions. However, such encodings could introduce a greater risk of limiting the model's Fourier accessibility\\footnote{Each of these bases has a Fourier amplitude as well as a phase angle, both of which need to be altered to fit a model on the training data. However, depending on the architecture (of both the encoding layers and the trainable layers) these quantities may be limited in the values they can represent. This could significantly limit their ability to represent various functions. This will henceforth be referred to as the \\emph{Fourier accessibility} of the quantum architectures.}.\n\n\n\nSec.~\\ref{sec:lin_arch} provides a review of how angle-embedded QNNs approximate their input distributions by fitting to them a truncated Fourier. Specifically, Sec.~\\ref{sec:lin_arch} reviews the linear encoding architectures and how their number of Fourier bases grow linearly with the number of repetitions -- sequential linear in Sec.~\\ref{sec:seq_lin} -- as well as the number of qubits -- parallel linear in Sec.~\\ref{sec:par_lin}. Then, Sec.~\\ref{sec:exp_arch} introduces the same two architectures but with a slight modification to represent an exponentially-growing number of Fourier bases. To use these architectures in practice, Sec.~\\ref{sec:training} showcases a comparison in the training performance of these architectures and shows that the parallel exponential has a superior training performance to the other architectures on a synthetic, one-dimensional dataset. Finally, Sec.~\\ref{sec:critical} critically evaluates the work and suggests areas for future investigation. \n\n\n\n\n\\section{Background review -- linear architectures} \\label{sec:lin_arch}\n\nAs discussed in \\cite{schuld_fourier}, all quantum neural networks that use angle embedding\\footnote{Ref.~\\cite{schuld_kernel} explores other embedding strategies too, such as basis embedding and amplitude embedding. This work primarily focuses on angle embedding. Nevertheless, it is worth noting that in the circuit model all circuit parameters enter as angles at some level of the description.} as their encoding strategy produces a truncated Fourier series approximation to the dataset. \n\\cite{schuld_fourier} also specifically explored two families of architectures of quantum neurons: a single-qubit architecture with a series of sequential $\\mathcal{SU}(2)$ gates, and a multi-qubit architecture with parallel $\\mathcal{SU}(2)$ encoding gates. In this section, the results and the architectures introduced in \\cite{schuld_fourier} are explored in depth in Sec.~\\ref{sec:exp_arch} two alternative neuron architectures are presented with the capability of achieving an exponentially higher Fourier expressivity for the same number of gates.\n\nConsider a quantum neuron that maps a real feature $x \\in \\mathcal{X}$ onto the quantum circuit via a parametric gate $S(x)=e^{-i\\mathcal{G}x}$. In most common architectures the only parametric gates are single qubit rotations $\\{R_x,R_y,R_z\\}$. For this work, the Pauli-Z generated rotations are used without any loss of generality $\\mathcal{G}=\\frac{1}{2}\\sigma_z=\\frac{1}{2}\\big(\\begin{smallmatrix}\n  1 & 0\\\\\n  0 & -1\n\\end{smallmatrix}\\big)$\\footnote{extra gates to convert between different generators can be absorbed into the variational gates -- see \\cite{schuld_fourier}}, then the embedding gate takes a simple form $S(x)= \\big(\\begin{smallmatrix}\n e^{-i x \/2 } & 0\\\\\n  0 & e^{i x \/2}\n\\end{smallmatrix}\\big)$. In general, the dependence of the expected value of any observable on the parameter $x$ is then given by\n\\begin{equation*}\n    \\langle M\\rangle = \\bra{\\Psi}(S^\\dag(x)\\otimes \\mathbb{1}) M (S(x)\\otimes \\mathbb{1})\\ket{\\Psi} = (c_0+c_0^*) + c_{1} e^{i x}+ c_1^* e^{-i x}\n\\end{equation*}\nwith some complex parameters $c_0$ and $c_1$ which depend on the rest of the circuit and the measurements.\n\nThis expected value is a function of the feature $x$ with a very simple Fourier series. The \\textit{data re-uploading method}~\\cite{data_reuploading} is a natural way to construct neurons that give rise to richer Fourier series. These are architectures where several parametric gates depend on the same $x$. It is the most straightforward to consider gates that have a hardwired dependence on the feature\\footnote{In principle, one may also consider gates $S(x;\\theta)= e^{-i f_\\theta(x) \\mathcal{G}}$ with a \\textit{variational} dependence on the feature $x$, where the parameters $f_\\theta$ are to be learned. However, this could present additional challenges related to the computation of gradients and increased number of parameters}. In particular, such that the expected value of any observable takes the form of a discrete Fourier series \n\\begin{equation}\n\\label{eqn:fourier_general}\n    f(x,\\theta) =\\sum_{k} c_k(\\theta) e^{ikx},\n\\end{equation}\nwhere $\\theta$ the variational parameters and  $c_k \\in \\mathbb{C}$ with $c^{*}_{-k} = c_k$ for real observables. In Sec.~\\ref{sec:seq_lin} and \\ref{sec:par_lin}, two architectures exhibited in \\cite{schuld_fourier} are reviewed. The \\textit{Fourier expressivities} of these architectures are of particular interest, that is the list of wavenumbers $\\{k_1,k_2,\\dots\\}$ appearing in the exponents in Eq.~\\eqref{eqn:fourier_general}.\n\n\\subsection{Sequential linear}\\label{sec:seq_lin}\n\nThe single-qubit sequential linear method uses repetitions of the same single-qubit encoding gate $S(x)$ interlaced in-between trainable variational layers. Fig.~\\ref{fig:archs}a shows this implementation with generalized variational gates. Since the eigenvalues of each unitary are $e^{\\pm i \\frac{1}{2}x}$, it is\nstraightforward to observe (see e.g. App.~\\ref{appendix:seq_lin}) that after $n$ encoding layers the expected value of any observable takes the form\n\\begin{equation}\n\\label{eqn:fourier_first}\n    f(x,\\theta) =\\sum_{k=-n}^{n} c_k(\\theta) e^{ikx},\n\\end{equation}\nThus, the repetitions have an additive effect such that for $n$ repetitions the final list becomes $\\{-n,-n+1,\\cdots,0,\\cdots,n-1,n\\}$. Each of the wavenumbers in the list gives rise to a sinusoidal term with the same frequency. Therefore, for $n$ repetitions of the encoding $S(x)= e^{-i\\mathcal{G}x}$, $n$ distinct Fourier bases are generated. \n\n\n\\begin{figure*}\n     \\centering\n     \\includegraphics[width=1\\linewidth]{fig1.pdf}\n        \\caption{The four general circuits under analysis in this paper. In the exponential architectures, the first encoding is kept the same and the subsequent encoding gates are multiplied by the coefficients in Eq.~(\\ref{eqn:numbers}). The parallel circuits have a CNOT layer at the end to ensure that all qubits are cooperating in the training by propagating the $\\pi$-measurement through all quantum wires.}\n\\label{fig:archs}\n\\end{figure*}\n\n\n\\subsection{Parallel linear}\\label{sec:par_lin}\nIn the parallel setting, the single-qubit encoding gates are applied in parallel on separate qubits -- see Fig.~\\ref{fig:archs}c. Similarly to the sequential encoding, for $n$ parallel rotations $n$ Fourier bases are produced. This is due to the commutativity between the parallel rotations as they act on separate qubits. The generator $\\mathcal{G}$ becomes:\n\\begin{equation}\n    \\mathcal{G} =  \\frac{1}{2} \\sum_{q=1}^{r} \\sigma_z^{(q)},\n\\end{equation}\nwhere $q$ is the qubit index, $r$ indicates the total number of qubits, and $\\sigma_z^{(q)}$ is the Pauli-Z matrix applied to the $q^\\text{th}$ qubit. In App.~\\ref{appendix:par_lin_proof}, it is shown that $\\mathcal{G}$ -- being a square matrix of dimensions $2^r$ -- has $2r+1$ unique eigenvalues. This suggests a high degree of degeneracy in its eigenspectrum. As before, the subtraction of these values from themselves yields a list of wavenumbers ranging from $-r$ to $r$ generating $r$ Fourier bases.\n\n\\subsection{Redundancy}\n\nBoth in the sequential and parallel linear architectures there is a lot of redundancy in how the feature is encoded into the circuit. This is the easiest to see for the parallel architecture, where most of the eigenvalues of $\\exp(i x \\mathcal{G})$ are largely degenerate as the encoding commutes with qubit permutations.\n\n\n\\section{Results -- exponential architectures}\\label{sec:exp_arch}\n\nIn this section, two new families of architectures are suggested that can encode an exponential number of Fourier bases for a given number of repetitions\/parallel encodings. The basis of this generalization is to modify each \"subsequent\" appearance of the encoding gate in the circuit by a re-scaling of the generator $S(x)\\to S(m x)$ with an integer $m$. Keeping the factors $m$ integer guarantees that this procedure results in a discrete Fourier series in the form of Eq.~\\eqref{eqn:fourier_general}.\n\n\\subsection{Sequential exponential}\\label{sec:seq_exp}\nIt was shown in Sec.~\\ref{sec:seq_lin} that the wavenumbers created in the linear models are highly degenerate. By modifying the circuit encoding this degeneracy can be reduced, resulting in adding new wavenumbers to the list. This is accomplished by altering the generators in the individual encoding layers. In the linear case, the diagonal elements of the generator $\\lambda_i$ always belonged to the list $\\{-\\frac{1}{2},\\frac{1}{2}\\}$, but could be altered by scaling the generator $\\mathcal{G}$ in each layer. In practice, this is achieved by scaling the embedded data $x$, and then mathematically associating with the generator. The resultant function becomes\n\n\\begin{eqnarray*}\n    f(x,\\theta) =\\left( W_{1,j_1}^{(0)\\dagger}W_{j_1,j_2}^{(1)\\dagger}\\cdots M_{k',k} \\cdots W_{i_2,i_1}^{(1)} W_{i_1,1}^{(0)}\\right)\\\\\ne^{\\left((\\lambda^{(1)}_{j_1}+\\lambda^{(2)}_{j_2}+\\cdots)-(\\lambda^{(1)}_{i_1}+\\lambda^{(2)}_{i_2}+\\cdots)\\right)x},\n\\end{eqnarray*}\n\nwhere $\\lambda^{(l)}_i = a_l \\lambda_i = \\frac{1}{2}\\{-a_l,a_l\\}$ for $a_l \\in \\mathbb{N}$~\\footnote{In comparison, for linear architectures $a_l=1$ for all $l$.}. In this work, $a_l$ scales as follows $a_l =\\{2^0,2^1,2^2,\\cdots,2^{n-1}+1\\}$. The motivation behind this choice is the sum of powers of 2, $\\sum_{i=0}^{n-1} 2^i = 2^{n} - 1$, where the largest wavenumber possible, $2^n$, is obtained by taking all the positive contributions from the list of eigenvalues, i.e. $k_{max}= \\sum_{i=0}^{n-1} 2^i + 2^{n-1} + 1 = 2^n$. Next, one can switch the signs of the positive values to negative starting from the smallest term to produce all integers from $-2^n$ to $2^n$. This generates $2^n$ Fourier frequencies. Fig.~\\ref{fig:archs}b shows a quantum circuit encoded using the sequential exponential strategy with 2 layers. App.~\\ref{appendix:constrained} demonstrates that this network produces extreme constraints on the Fourier accessibility, and thus is an undesirable choice for general data modelling. However, this scheme motivates extending this idea to parallel architectures.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig2.pdf}\n    \\caption{Variational gates used in single -- (a) and 2-qubit (b) experiments.}\n    \\label{fig2:variational}\n\\end{figure*}\n\n\\subsection{Parallel exponential}\\label{sec:exp_par}\nTo perform this extension, it is appropriate to proceed with a 2-qubit example. The parallel linear method described in Sec.~\\ref{sec:par_lin} produces the generator:\n\\begin{equation}\n\\mathcal{G^{\\text{lin}}}=\\frac{1}{2} \\left(\\sigma_z \\otimes \\mathbb{I} + \\mathbb{I} \\otimes \\sigma_z\\right) = \\begin{pmatrix}\n  1 & 0 & 0 & 0\\\\\n  0 & 0 & 0 & 0\\\\\n  0 & 0 & 0 & 0\\\\\n  0 & 0 & 0 & -1\n\\end{pmatrix}\n\\end{equation}\nThis matrix has 3 unique eigenvalues, $\\lambda \\in \\{-1,0,1\\}$, and when subtracted from itself -- yielding wavenumbers $L^{\\text{(lin)}}_k = \\{-2,-1,0,1,2\\}$ -- it can produce 2 Fourier bases with frequencies $\\{1,2\\}$. One could generate a matrix with more unique values. For example,\n\\begin{equation}\n\\mathcal{G^{\\text{exp}}}=\\frac{1}{2} \\left(3\\sigma_z \\otimes \\mathbb{I} + \\mathbb{I} \\otimes  \\sigma_z\\right)=\\begin{pmatrix}\n  2 & 0 & 0 & 0\\\\\n  0 & -1 & 0 & 0\\\\\n  0 & 0 & 1 & 0\\\\\n  0 & 0 & 0 & -2\n\\end{pmatrix},\\label{eqn:exp_matrix}\n\\end{equation}\nis a generator with 4 unique eigenvalues that generate 9 wavenumbers $\\{-4, -3, -2, -1, 0, 1, 2, 3, 4\\}$. This generator can be constructed using the quantum circuit shown in Fig.~\\ref{fig:archs}d. In this case, a $\\mathcal{SU}(4)$ generator is employed. This is decomposed into 2 $\\mathcal{SU}(2)$ generators, one using the group parameter, $x$, and the other $3x$. This can be generalized to $n$ qubits as one can extend the matrix for larger numbers of qubits, i.e. for $n$ qubits $\\mathcal{G}$ would be a diagonal matrix starting from $-2^{(n-1)}$ up to $2^{(n-1)}$, producing $2^n$ Fourier bases. The quantum circuit associated with this generator is an application of Pauli-Z rotations of $x$ with frequencies increasing in the following way: \\begin{equation}\\label{eqn:numbers}\n   \\mathcal{L} = \\{2^0,2^1,2^2,...,2^{n-1}+1\\},\n\\end{equation}where $n$ is the number of qubits. Note the similarities between the sequential and parallel encodings and their symmetries in the way the circuits are constructed. One also recognizes similarities between the parallel encoding and Kitaev's quantum phase estimation algorithm~\\cite{kitaev1995quantum}, albeit in this case $x$ is a classical feature.\n\nThis can be significantly more expressive than the parallel linear method, but this advantage needs to be accompanied by Fourier accessibility since if the Fourier values of these newly-acquired bases cannot be altered, there would be no advantage in pursuing this setting. Sec.~\\ref{sec:training} shows there is a significant advantage in using parallel exponential encoding in a simple toy example.\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig3.pdf}\n    \\caption{Training losses indicate a training advantage for the parallel exponential, and that the sequential exponential architecture performs only marginally better than the linear architectures. The training was done on QMware hardware\\cite{qmware} using the PennyLane Python package~\\cite{pennylane}. The Adam optimizer was employed to minimize a mean squared loss function with a learning rate of $\\epsilon=0.1$ and with uniformly-distributed parameters $\\theta \\in [0,2\\pi]$.}\n    \\label{fig3:losses}\n\\end{figure*}\n\n\\begin{figure*}[t]\n    \\centering\n    \\includegraphics[width=\\linewidth]{fig4.pdf}\n    \\caption{The QNNs fit the best possible truncated Fourier series on the top-hat function. The parallel exponential architecture provides the best fit, and even though the sequential exponential architecture has access to the same 4 Fourier frequencies, it fails to access all of them efficiently and as a result, it performs sub-optimally. The linear architectures perform similarly to each other, potentially arising from their high Fourier accessibility to the 2 Fourier frequencies that they can represent.}\n    \\label{fig4:tophat}\n\\end{figure*}\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\textwidth]{fig5.pdf}\n    \\caption{The parallel exponential fit to the top-hat function on a simulator vs on a quantum processor. The noisy solid line is the evaluation of this network for 100 equally-spaced points using 100 shots.}\n    \\label{fig5:qpu}\n\\end{figure*}\n\n\n\n\\subsection{Training}\\label{sec:training}\nIn this section, the training performance of these 4 architectures on a simple dataset is compared. Each architecture is trained to reproduce a one-dimensional top-hat function. Fig.~\\ref{fig4:tophat} shows the ground truth as well as the fitting performances of these architectures, and Fig.~\\ref{fig3:losses} shows their training performance. It is clear that the parallel exponential architecture fits a closer function to the ground truth, and in contrast, the sequential exponential architecture has the worst performance of all models. Furthermore, the Fourier decompositions of the models in Fig.~\\ref{fig6:decomposition} show that exactly 2 Fourier terms are accessed by the linear architectures and 4 by the exponential ones. Additionally, Fig.~\\ref{fig5:qpu} demonstrates the performance of the parallel exponential architecture on a trapped ion quantum processor. The IonQ implements a high-fidelity gate-based quantum processing unit through a process known as laser pumping trapped-ions explained in~\\cite{ionq-paper}. The hardware was shown to be one of the most accurate in recent benchmarking tests~\\cite{benchmarkingQMW}. We specifically used the hardware introduced in~\\cite{11qubit-ionq} with a single-qubit fidelity of $0.997$ and a 2-qubit fidelity of $0.9725$. The code implementation was done through Amazon Web Services (AWS) Braket, and the process of the forward pass for 100 data points took 4 hours and 11 minutes, due to the delays and queuing times. It can be observed that the low number of shots is the dominant source of noise here and higher shot counts could yield a smoother curve that is closer to the simulator. \n\n\\begin{figure*}[ht]\n     \\centering\n     \\includegraphics[width=\\linewidth]{fig6.pdf}\n        \\caption{Fourier decomposition of the four architectures after training to fit the top-hat function. We see that the linear architectures can only access two Fourier frequencies, whereas the exponential ones can access four.}\n        \\label{fig6:decomposition}\n\\end{figure*}\n\n\n\\subsection{Critical Evaluation}\\label{sec:critical}\nWhile the results for the parallel exponentials are encouraging, it is equally important to understand the limitations of this approach. Firstly, while the exponential growth in the number of Fourier frequencies is evident, this is not the higher limit of Fourier frequency growth. \\cite{schuld_fourier} showed that for $L$ repetitions of an encoding gate with a Hilbert space of dimension $d$, there is an upper limit to this growth of the form\n\n\\begin{equation}\nK < \\frac{d^{2L}}{2} -1,\n\\end{equation}\nwhere $K$ is the number of Fourier frequencies. This suggests that there is a potential for square-exponential growth, whereas the method discussed in this work only grows exponentially. In App.~\\ref{appendix:maths_problem} a mathematical problem is proposed, whose solution could unlock the maximum possible Fourier accessibility.\n\nSecondly, it is important to emphasize that the two parallel architectures are the same with a minor multiplicative factor added in the exponential case. This means that training them for a fixed number of epochs requires the same computational resources. However, adding more Fourier bases by eliminating the degeneracy of the network could result in under-parameterized models. Therefore, it is often necessary to parameterize the exponential architectures more heavily than the linear ones, indirectly affecting the required resources. Every Fourier frequency requires 2 degrees of freedom (real-valued parameters) and an exponentially-growing Fourier space requires the resources to grow exponentially, too. These resources could include the classical memory required to store the parameters or the classical optimizer that needs to calculate the gradient for these parameters. And lastly, extending this to many qubits will still result in barren plateaus.\n\n\\section{Conclusion and future work}\\label{sec:conclusion}\nThis work suggested two new families of QNN architectures, dubbed sequential and parallel exponential circuits, that provided an exponentially growing Fourier space. It was demonstrated that the former struggled with accessing these frequencies, but also that the latter showed an advantage in approximating a top-hat function. \n\nFuture work could focus on a quantitative understanding of the Fourier accessibility of these networks, such that the optimal variational parameterization could be chosen for a specific problem. Another possible direction for future work is to depart from hardwired encoding gates. A natural elementary step in this direction is to consider single-qubit gates of the form $S_i(x,w_i)= \\exp(- i \\, x w_i \\frac{1}{2}\\sigma_z )$, where the scaling factor $w_i$ is an independent scalar trainable parameter for each occurrence of the encoding gate in the circuit. In this case, the final wavevectors $k$ are linear combinations of the parameters $w_i$ that can be potentially trained efficiently.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{S1. First-principles calculations of phonon transport across a silicon interface}\nThe system used for calculating phonon transport across an intrinsic silicon (Si) interface is shown in Fig.~S\\ref{Fig.S1}(a). The system is the same as Fig. 1(a) of the main text, except that there is no vacuum gap region.\n\nA 1~${\\times}$~1~${\\times}$~4 supercell shown in Fig.~S\\ref{Fig.S1}(b) is used for computing the interatomic force constants via first-principles calculations based on the density functional theory. The interatomic force constants of cells 1 and 2 are extracted for the left and right leads, and also for the device region. The gap distance in contact is defined as the inter-planar spacing in the (001) orientation \\cite{Chiloyan2015,Xiao2017} (i.e., 0.137 nm). Surface reconstruction is not considered when the Si crystals are in contact.\n\n\\begin{figure}[p!]\n\\centering\n\\includegraphics[width=1\\linewidth]{Fig_S1.png}\n\\caption{(a) Schematic of the system used for calculating phonon heat transfer across an interface. The semi-infinite left and right leads are respectively maintained at temperatures $T_{\\mathrm{L}}$ = 305 K and $T_{\\mathrm{R}}$ = 300 K. (b) A 1~${\\times}$~1~${\\times}$~4 supercell, which is the combination of four conventional unit cells (black numbered boxes), is used for calculating the interatomic force constants.}\n\\label{Fig.S1}\n\\end{figure} \\clearpage\n\\newpage\n\n\\section{S2. Near-field radiation modeling}\nThe heat transfer coefficient due to near-field radiation between the left (L) and right (R) semi-infinite leads separated by a vacuum gap $d$ is calculated using fluctuational electrodynamics \\cite{Rytov1978,Polder1971}: \n\\begin{equation}\n\\label{radiative flux}\nh_\\mathrm{rad}  \\ = \\ \\frac{1}{\\pi^{\\mathrm{2}}(T_\\mathrm{L}-T_\\mathrm{R})}\\int_{0}^{\\infty}d{\\omega}\\left[\t\\Theta\\left(\\omega,T_\\mathrm{L}\\right) - \\Theta\\left(\\omega,T_\\mathrm{R}\\right)\\right]\n\\\\ \\\\\n\\int_{0}^{\\infty}dk_\\mathrm{\\rho}k_\\mathrm{\\rho}\\sum_{\\gamma=\\mathrm{TE,TM}}{\\cal T_\\mathrm{rad}^{\\gamma}}\\left(\\omega,k_\\mathrm{\\rho}\\right)\n\\end{equation} \nwhere $T_{j}$ is the temperature of semi-infinite lead $j$ ($j = \\mathrm{L}, \\mathrm{R}$), $\\omega$ is the angular frequency, $k_\\mathrm{\\rho}$ is the component of the wave vector parallel to a surface, and $\\Theta(\\omega, T_{j})$ is the mean energy of an electromagnetic state calculated as $\\hbar\\omega\/[\\mathrm{exp}({\\hbar\\omega\/k_\\mathrm{B}T_{j}})-1]$. The transmission functions in polarization state $\\gamma$ for propagating ($k_\\mathrm{\\rho}$ $<$ $k_\\mathrm{0}$) and evanescent ($k_\\mathrm{\\rho}$ $>$ $k_\\mathrm{0}$) electromagnetic waves in vacuum are calculated as:\n\\begin{equation}\n\\label{propagating transmission}\n{\\cal T_\\mathrm{rad,prop}^\\gamma}\\left(\\omega,k_\\mathrm{\\rho}\\right) \\ = \\frac{\\left(1-\\left|r_\\mathrm{0L}^{\\gamma}\\right|^{2}\\right)\\left(1-\\left|r_\\mathrm{0R}^{\\gamma}\\right|^{2}\\right)}{4\\left|1-r_\\mathrm{0L}^{\\gamma}r_\\mathrm{0R}^{\\gamma}e^{2i\\mathrm{Re}\\left(k_{\\mathrm{z0}}\\right)d}\\right|^{2}}\n\\end{equation}\n\\begin{equation}\n\\label{evanescent transmission}\n{\\cal T_\\mathrm{rad,evan}^\\gamma}\\left(\\omega,k_\\mathrm{\\rho}\\right) \\ = e^{-2\\mathrm{Im}\\left(k_\\mathrm{z0}\\right)d}\\frac{\\mathrm{Im}\\left(r_\\mathrm{0L}^{\\gamma}\\right)\\mathrm{Im\\left(r_\\mathrm{0R}^{\\gamma}\\right)}}{\\left|1-r_\\mathrm{0L}^{\\gamma}r_\\mathrm{0R}^{\\gamma}e^{-2\\mathrm{Im}\\left(k_{\\mathrm{z0}}\\right)d}\\right|^{2}}\n\\end{equation} where $k_\\mathrm{0}={\\omega}\/c_{0}$ is the magnitude of the vacuum wave vector with $c_{0}$ as the speed of light in vacuum, $k_\\mathrm{z0}$ is the component of the vacuum wave vector perpendicular to an interface, and $r_{0j}^\\gamma$ is the Fresnel reflection coefficient at the vacuum-lead $j$ ($j = \\mathrm{L}, \\mathrm{R}$) interface in polarization state $\\gamma$ \\cite{Yeh1988}. The local, frequency-dependent dielectric function of intrinsic Si provided in Refs. \\cite{Adachi1988,Aoki1991} is used in the calculations.\n\\newpage\n\n\\section{S3. Number of electrons and interatomic force constants in the vacuum gap region}\nThe number of electrons, $n_\\mathrm{e}$, in the middle of the vacuum gap within the 1~${\\times}$~1~${\\times}$~8 supercell is calculated by integrating the electron density within the volume specified by the red shaded box in Fig.~S\\ref{Fig.S2}(a). Specifically, the ratio $l\/d$, where $l$ is defined in Fig.~S\\ref{Fig.S2}(a) and $d$ is the average vacuum gap thickness, was varied from 0.4 to $5{\\times}10^{-6}$. For $l\/d$ values smaller than $5{\\times}10^{-3}$, the number of electrons follows a $d^{-8.15 \\pm 0.85}$ power law, as shown in Fig.~S\\ref{Fig.S2}(b). This converging trend indicates that the number of electrons in the middle of the vacuum gap can be calculated using $l\/d$ $<$ $5{\\times}10^{-3}$. \n\nThe interatomic force constants reported in Fig. 3(d) of the main text are calculated by extracting the $zz$ components of the interatomic force constants acting on the atoms contained in the volume of the device region specified by the orange box in Fig.~S\\ref{Fig.S2}(c). Only the $zz$ components of the interatomic force constants are used since phonon transport across a single-digit nanometer vacuum gap is quasi-one-dimensional \\cite{Chiloyan2015,Tokunaga2021}.\n\n\\begin{figure}[p!]\n\\centering\n\\includegraphics[width=1\\linewidth]{Fig_S2.png}\n\\caption{(a) Vacuum gap region of a 1~${\\times}$~1~${\\times}$~8 supercell after structural optimization. The number of electrons is calculated within the volume specified by the red shaded box. The length $l$ surrounds the middle of the average vacuum gap. (b) Number of electrons as a function of the average vacuum gap thickness $d$ and the ratio $l\/d$.  (c) The $zz$ components of the interatomic force constants acting on the atoms within the volume identified by the orange box are extracted, and their summations are reported in Fig. 3(d) of the main text.}\n\\label{Fig.S2}\n\\end{figure} \\clearpage\n\\newpage\n\n\\section{S4. Phonon dispersion relations and phonon density of states of bulk silicon calculated via the density functional theory}\nFigure S\\ref{Fig.S3} shows the phonon dispersion relations and phonon density of states of bulk Si obtained via first-principles calculations based on the density functional theory. In this work, acoustic and optical phonon modes are characterized by frequencies respectively lower and higher than 12 THz. This threshold is defined based on the boundary between longitudinal acoustic (LA) and longitudinal optical (LO) phonon modes located around 12 THz along the $\\Gamma$-$\\mathrm{X}$ direction, as shown in Fig. S\\ref{Fig.S3}.\n\n\\begin{figure}[p!]\n\\centering\n\\vspace{100pt}\n\\includegraphics[width=1\\linewidth]{Fig_S3.png}\n\\caption{Phonon dispersion relations and phonon density of states (DoS) of bulk Si obtained via the density functional theory \\cite{Gonze2002,Gonze2020}. Transverse optical, longitudinal optical, longitudinal acoustic, and transverse acoustic phonon modes are respectively denoted as TO, LO, LA, and TA. The order of the symmetric points within the first Brillouin zone used for generating the phonon dispersion relations is selected based on Refs. \\cite{Setyawan2010,Davis2011}. The horizontal dashed line is drawn at a frequency of 12 THz along the $\\Gamma$-$\\mathrm{X}$ direction.}\n\\label{Fig.S3}\n\\end{figure} \\clearpage \n\\newpage\n\n\\section{S5. Phonon populations}\nPhonon populations of the atomic layers adjacent to the vacuum gap in the left and right leads are calculated for vacuum gaps of 0.47, 0.69, 0.89, and 1.09 nm and are reported in Fig.~S\\ref{Fig.S4}. These atomic layers are denoted as layer 16 in the left lead and layer 1 in the right lead in Fig. 1(b) of the main text. Note that surface reconstruction is considered in the calculations. The phonon populations are obtained by multiplying the Bose-Einstein distribution function by the local phonon density of states of the device region obtained via the atomistic Green's function method. The local phonon density of states, $D_{\\mathrm{d}}$, is calculated as follows \\cite{Zhang_NHTPBF_2007}:\n\\begin{equation}\n\\label{pldos}\n    D_{\\mathrm{d}} = i\\left(G_{\\mathrm{d}}-G_{\\mathrm{d}}^{\\dagger}\\right)\\omega\n\\end{equation} where $G_{\\mathrm{d}}$ is the Green's function of the device region, the superscript $\\dagger$ denotes conjugate transpose, and $\\omega$ is the phonon frequency. \n\nFigure~S\\ref{Fig.S4} clearly shows that the populations of acoustic phonons characterized by frequencies lower than 12 THz largely surpass that of optical phonons for all vacuum gaps considered. \n\n\\begin{figure}[p!]\n\\centering\n\\includegraphics[width=0.9\\linewidth]{Fig_S4.png}\n\\caption{Phonon populations of the atomic layers adjacent to the vacuum gap in the left and right leads for: (a) $d$ = 0.47 nm. (b) $d$ = 0.69 nm. (c) $d$ = 0.89 nm. (d) $d$ = 1.09 nm. The vertical dashed line at 12 THz delimits the frequencies associated with acoustic and optical phonons.}\n\\label{Fig.S4}\n\\end{figure} \\clearpage\n\\newpage\n\n\\normalem\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}